ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK H. SCHULZ-BALDES∗ and J. BELLISSARD† Universit´ e Paul-Sabatier UMR 5626, CNRS and Laboratoire de Physique Quantique 118, Route de Narbonne 31062-Toulouse Cedex France ∗ E-mail:
[email protected] † E-mail:
[email protected] We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo’s formula for the conductivity and hence lead to anomalies in Drude’s formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner’s n-orbital model as well as the Anderson model in coherent potential approximation.
1. Introduction 1.1. Anomalous electronic transport Quantum effects and interactions in various materials cause a great variety of behaviors for electronic transport at low temperature. Understanding why some materials are conductors and others insulators is a challenging central problem of solid state physics. The first attempt to get a microscopic theory of electronic transport goes back to the work of Drude [27] who wrote the conductivity in the form: nq 2 τ , (1) σ= m∗ where n is the charge carrier density, q is the carrier electric charge, m∗ is the carrier effective mass and τ is the collision time. The derivation of this formula was initially given in terms of kinetic transport of classical particles, but Sommerfeld and Peierls rederived it in the context of quantum theory of crystals which led to consequences in better agreement with experiments in metallic samples [5]. The main weakness of the theory lies in the definition of τ . The collision time is often understood as a phenomenological parameter that can be easily measured, but is difficult to interpret. This is the so-called relaxation time approximation (RTA). However, using more sophisticated theories, one can calculate it if various contributions such as electron-impurity, electron-phonon or electron-electron scattering are taken into account. It gives a temperature dependence in the form of a power law, namely 1 Reviews in Mathematical Physics, Vol. 10, No. 1 (1998) 1–46 c World Scientific Publishing Company
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τ (T ) ∼ T γ , where γ depends upon the type of collision which dominates dissipation [5]. Among the various mechanisms that may lead to a metal-insulator transition, one is the anomalous quantum transport. This means that, within a one-electron effective theory, the electronic wave packet diffuses anomalously through the systems instead of moving ballistically between collisions as free electrons in a perfect crystal. Such a mechanism is probably at the basis of the strange transport properties of quasicrystals at low temperature [17, 50]. The best samples are nowadays alloys made of very good metals, such as the Al62.5 Cu25 Fe12.5 , Al70.5 Pd22 Mn7.5 or Al70.5 Pd21 Re8.5 , crystallizing in a quasiperiodic icosahedral phase. They have conductivities comparable to doped semiconductors, namely 10−6 to 10−9 times less than for pure aluminum at 4K (see for instance [56, 64] for the following information); moreover, the conductivity increases with temperature, which is just the opposite of the behavior of a normal metal; the temperature dependence of the conductivity is neither of the exponential type exp(−E/kB T ) characterizing thermally activated processes, nor of the form exp(−aT −α ) as in Mott’s hopping conductivity or related mechanisms [68], but rather exhibits a power law behavior σ(T ) ∼ T β in the range 4 − 800K with 1 < β < 1.5 [64]; at last, the behavior of the conductivity as a function of the magnetic field shows the typical weak localization signature observed in slightly disordered metals [64]. On the other hand, several numerical results concerning the time behavior of tight binding models on a quasiperiodic lattice have shown that the spreading of the wave packet satisfies a power law behavior in time of the type [38, 29, 73] 2 ~ ~ |φi ∼ t2σdiff hφ|(X(t) − X(0))
as t → +∞ ,
(2)
~ is the position operator where |φi is the initial localized wave function and X for the particle. Depending on the strength of the quasiperiodic potential, the diffusion exponent σdiff may vary from 0 to 0.8 in a nearest neighbor model on an octagonal lattice [73]. Similar behavior were observed in the Harper model [38, 29], the Fibonacci Hamiltonian [38] and the kicked Harper model [3]. For quasicrystals, analytical calculations in one dimension [55] or more concrete phenomenological models in three dimension [42] have led to similar results and to predictions for the diffusion exponent σdiff . Experimental results, their theoretical interpretation and numerical simulations have therefore led the experts toward the idea that anomalous diffusion as described in Eq. (2) is the main reason for the strange transport properties of quasicrystals, at least for temperatures above approximately 4K (below 4K, electron-electron interactions may become more important). One consequence of (2), as we will show in this paper, is that the conductivity in RTA no longer satisfies the Drude formula (1), but rather: (3) σ ∼ τ 2σdiff −1 as τ → +∞ . Such a formula can be guessed by means of the following non-rigorous argument [72, 50]. By the Einstein relation, σ = q 2 N (EF )D(τ ) where N (EF ) is the density of states at the Fermi level EF , while D(τ ) is the diffusion coefficient. One has
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
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D(τ ) = L(τ )2 /3τ where L(τ ) is the mean free path, namely the distance spread by a typical wave packet during the time τ . Then (2) gives (3). Consequently, provided σdiff < 1/2 and τ depends on temperature as indicated above, the conductivity increases with temperature. For electrons in a periodic crystal, σdiff = 1 (ballistic quantum motion) and Eq. (3) gives a weak form of the Drude formula. Our aim in this paper is to put this type of argument on a rigorous ground. As we shall see, most of the work consists in defining the relevant mathematical framework liable to lead to (3). Then using many recent results on the characterization of singular continuous spectral measures, we will explain how to justify these arguments. 1.2. Homogeneous media The first difficulty in dealing with materials such as quasicrystals is their lack of translation invariance at the atomic scale. In particular, there is no Bloch theorem allowing analytical calculations. However, these materials are homogeneous, namely they become translation invariant at a larger scale. In the limit where the sample size is large, it becomes simpler to represent these systems in the infinite volume limit. This is the first assumption made here. It means that we exclude the description of mesoscopic devices from our framework. Our next assumption is to work within the one electron approximation, that is electron-electron and electron-phonon interactions are neglected in the Hamiltonian description and treated only in RTA by means of the collision time. This approximation is usually very good in metals because electrons can be described as quasiparticles dressed by interactions. As we have indicated previously, this is not a restriction when considering quasicrystals as long as the temperature is not smaller than 4K. Moreover, this framework includes electron-impurity scattering in the limit where impurities are quenched, namely their dynamic is not considered. The third assumption concerns the energy range within which the electronic motion is studied. For indeed, only electrons with energies within O(kB T ) from the Fermi level contribute to the electronic transport. This allows us to describe the electronic motion by means of an effective one-particle Hamiltonian, namely the restriction of the Schr¨ odinger operator to this energy interval. In practice, it is possible to work within the so-called tight binding representation (see for instance [8]): we consider the nuclei as fixed on the vertices of the crystalline lattice L and we restrict ourselves to bands crossing the Fermi level; if only one band contributes to the Fermi level, wave functions are represented by a square summable sequence ψ = (ψ(x))x∈L , namely an element of H = `2 (L); the Hamiltonian is then a bounded selfadjoint operator H = H ∗ on H with off-diagonal matrix elements hx|H|yi decreasing exponentially fast to zero as |x− y| → ∞; in practice, one considers only the nearest neighbors contribution to this operator. If M bands must be taken into account, the wave function gets a band index ψ = (ψm (x)), 1 ≤ m ≤ M , and H = `2 (L) ⊗ CM . In this way we avoid technicalities due to the unboundedness of the Schr¨ odinger operator without restricting the physical domain of applicability. In order to avoid further irrelevant technical difficulties, we will consider in this paper systems in
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which L is a Bravais lattice of the type Zd . In the case of quasicrystals, this procedure must be slightly modified using groupoids (see for instance [8, 12, 45, 21]), but leads to a very similar description. How now can homogeneity be described? Let us repeat the main arguments as given by one of the authors [8] leading to its mathematical definition: since the medium is translation invariant at large scale, translating H should give the same physical properties. So there is no reason to prefer H to any other of its translation. Moreover, changing from sample to sample is equivalent to looking at one unique infinite system through windows centered at different locations. This means that if we need to consider limits, H is now by definition homogeneous if the strong closure of the set of all translated, denoted by Ω, is compact. The choice of the strong topology is crucial: the norm topology leads to the too restricted notion of almost periodic operators while the weak topology would lead to a too wide notion of homogeneity. For an unbounded Hamiltonians, a similar construction holds when the strong resolvent topology is used [12]. Ω inherits a structure of a topological dynamical system by considering the action of the translation group on it. This dynamical system will be called the hull of H. This framework will be used to describe robust, namely sample independent, physical properties. We will focus here on the measurement of space averages (or equivalently sample averages) of observables. By observable we mean any bounded operator obtained from the Hamiltonian through the elementary operations of translation, sums, product with a scalar, product, involution and limits in the strong topology. In this way the observable algebra contains no more than the energy and the homogeneity of the system. On the other hand, space averaging is not uniquely defined in general for its definition requires the choice of a translation invariant ergodic probability measure P on the hull. This choice will be arbitrary in what follows because all our results are independent of it. However, one could wonder whether there is a preferred choice in nature. This is probably related to the question of existence (and uniqueness) of a Gibbs measure with respect to the translation dynamic, but we will not address this question here. These considerations lead to the construction of a C ∗ -algebra A that will be given in Secs. 2.1 and 2.2 below. This C ∗ -algebra has been called the Non-Commutative Brillouin Zone (NCBZ) of the homogeneous medium under consideration (see [8, 12]). In fact, in the periodic case, it coincides with the space of (matrix valued) continuous functions on the Brillouin zone. The probability P then defines a unique trace T on A which is nothing but the trace per unit volume. For a periodic crystal, it reduces to integration over the Brillouin zone. One advantage of this formalism is that quantities computed through the trace T are insensitive to perturbations of the Hamiltonian by compact operators (see Theorem 1 below). This is actually a delicate point. A compact perturbation of the Hamiltonian can physically be interpreted as a localized impurity in the crystal. An experimentalist considers relevant only those properties that are insensitive to a given localized impurity, unless there is a definite procedure to faithfully reproduce this impurity from sample to sample. In homogeneous media, such a control is usually out of reach.
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However, there are mathematical properties depending upon the occurence of a specific impurity. There have been quite a lot of works, starting with the result of Donoghue [26], showing that a rank one perturbation of a Hamiltonian with pure-point spectrum may produce a continuous spectrum (see e.g. [41] for a short review of historical references). One of the most striking results in this respect has been put in a systematic form in [71, 30, 60, 61]. In particular, for the Anderson model in a regime of pure point spectrum with exponentially localized eigenstates, a topologically generic rank-one perturbation of the Hamiltonian produces a singular continuous spectrum [61]. Moreover, the authors of [61] show that the corresponding robust property is that the spectral and diffusion exponent do not change under such a perturbation and are equal to zero. This result is mathematically non-trivial and remarkable. Their relevance in practical situation is however questionable. In most experimental situations or numerical simulations, physicists are looking at sample independent quantities such as the average localization length, the conductance or the magnetoresistance. In more delicate questions, they may even look at universal fluctuations (a question not investigated here). In any case, the occurence of only a volume independent finite number of impurities is out of reach in practice. This is the main reason why we chose a mathematical framework in which delicate results, such as the topological instability of spectral properties by compact perturbations, can be discarded. 1.3. Spectral and transport exponents As we have seen in Sec. 1.1, scaling laws are the rule if anomalous transport occurs. This leads us to address the question of a proper definition of scaling exponents. Several inequivalent definitions are available in the literature dealing with fractal and multifractal analysis [65, 36, 35, 28, 52]. In this work, we review several of them and discuss their relevance for our purpose. More precisely, we shall study the exponents characterizing the local behavior of the following three Borel measures: (i) the Density of States (DOS) expressing global properties related to the thermodynamics of the electron gas; (ii) the Local Density of States (LDOS), namely the spectral measure “with P-probability one”; (iii) the current-current correlation measure involved in Kubo’s formula for the conductivity. The basic definition of local exponents chosen here follows a suggestion of G. Mantica (see the second reference in [32]). Given a non-negative Lebesgue measurR1 ∼ able function f on (0, 1] we say f () ↓0 α whenever 0 d −1−γ f () converges for γ < α and diverges for γ > α. If f is monotonous, this is equivalent to α =liminf→0 log f ()/log . A similar definition holds for the behavior at infinity. Note that this definition ignores all kinds of subdominant contributions and is likely to be robust. The local exponent αν (E) of a Borel measure ν on R is introduced by Z E+ ∼ dν(E 0 ) ↓0 αν (E) . E−
We show that E 7→ αν (E) defines a function in L∞ (R, dν) which depends only on the measure class of ν (see Theorem 2 below). Note that, although a consequence of
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standard measure-theoretic arguments, this result cannot be found in the literature [65, 81, 32, 28, 22, 40, 53, 20, 52, 48, 61, 6]. Note further that other exponents defined in multifractal analysis do depend upon the measure in its own equivalence class (see Remark 11 in Sec. 3.3). These properties are important in view of applications to homogeneous systems. A multifractal analysis of the DOS in the vicinity of the Fermi level may be useful for the thermodynamical properties of the electron gas in our system because it gives more precise information about the DOS than the local exponents. However, multifractal properties of the LDOS are of no use in a homogeneous system where only the measure class of the spectral measure has some robustness. For indeed, by looking at the system through local windows chosen at random in the lattice, the corresponding spectral measures are in the same measure class and the entire equivalence class can be described in this way (with probability one). Therefore, we cannot expect exponents that do depend upon the spectral measure in its measure class to be relevant in practice. The local exponents take values in the interval [0, 1] ν-almost surely. For an absolutely continuous measure ν, αν (E) = 1 ν-almost surely. For a pure point measure ν, αν (E) = 0 ν-almost surely. Hence these exponents allow to distinguish between different singular continuous measures. Examples of Hamiltonians with singular continuous spectra have been studied over the years (see [4, 69, 7, 31, 24, 19, 75, 76, 9, 10, 25, 11, 18, 39, 44]) and the question of computing their spectral exponents is certainly worth studying [43]. The local exponents can be computed both numerically and analytically by using the Green’s function [82] (see also [61]): Z dν(E 0 ) ∼ αν (E)−1 ↓0 . (4) =m 0 R E − (E − ı) Note, however, that numerical computations may concern exceptional values of E for which αν (E) is larger than 1. The definition of local exponents extend to the spectral analysis of a self-adjoint operator on a separable Hilbert space. We show that the exponents are invariants of the operator itself independent of the states in the Hilbert space. For covariant families of such operators arising in homogeneous media as discussed in Sec. 1.2, the exponents are moreover P-almost surely constant and define the exponents αLDOS (E) of the LDOS. The DOS is always regular with respect to the LDOS in the sense that for typical energies one has αLDOS (E) ≤ αDOS (E). The definition of the diffusion exponent σdiff given in Eq. (2) is generalized by restricting the dynamics to an energy interval ∆ and by extending it to the case of homogeneous systems. One talks of ballistic motion whenever σdiff (∆) = 1 and of regular diffusion whenever σdiff (∆) = 1/2. Localization, a behavior strictly stronger than σdiff (∆) = 0, has been studied in [13, 14] and will be discussed in more detail in Sec. 2.6. For any other value of σdiff , the quantum diffusion is called anomalous. Guarneri’s inequality [32] (see also [20, 33, 48, 6]) gives a lower bound of the diffusion exponent by the local exponents of the LDOS: αLDOS ≤ d · σdiff ,
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
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where d is the space dimension (see Sec. 2 for a precise formulation). Guarneri’s inequality has several direct physical implications [15]. For dimension one, an absolutely continuous spectrum implies ballistic quantum motion. However, for d ≥ 2, one may have both absolutely continuous spectrum and quantum diffusion with σdiff ≥ 1/d. This is expected, in particular, for the three-dimensional Anderson model at low disorder where, on the basis of the renormalization group calculation, the diffusion exponent is conjectured to be σdiff = 1/2 [77, 1]. The same situation is expected for the Anderson model in dimension two provided spin-orbit coupling is added [37]. For three-dimensional quasicrystals, Guarneri’s inequality allows a diffusion exponent as low as 1/3 without forbidding an absolutely continuous spectrum. In Sec. 5, we give an example of a model having σdiff = 1/2. Note that this model is an operator theoretic version of the so-called coherent potential approximation of the Anderson model [51]. 1.4. Overview of this article After this motivating introduction, we have organized this article as follows. Section 2 contains the main results and discriminates between the new results and the ones already obtained elsewhere. In Secs. 2.1 and 2.2 we give an account of the mathematical framework introduced in [8, 12] and used to describe homogeneous media. We also give a precise formulation to the stability under compact perturbations of the Hamiltonian. Both well-known and new results on local spectral exponents of Borel measures are presented in Sec. 2.3 before being extended to self-adjoint operators in Sec. 2.4. Sections 2.5 through 2.8 are devoted to the exponents of the LDOS, DOS and current-current correlation function as well as the anomalous Drude formula. Section 3 contains proofs of the results of Secs. 2.3 and 2.4, as well as some complementary results. In particular, in Secs. 3.3 and 3.4 we give a few known or less known facts about multifractal analysis which are of interest. The remaining results of Sec. 2, all linked to homogeneous systems, are proved in Sec. 4. The content of this section has not yet been treated in the literature. Section 5 is devoted to the calculation of the diffusion exponent in the Anderson model with free random variables [51]. An appendix completes the study of the hull whenever an impurity (or a compact perturbation) is added to the Hamiltonian. 2. Notations and Results 2.1. Construction and stability of the hull ˆ =H ˆ ∗ be a bounded Hamiltonian acting on the one-particle Hilbert space Let H 2 d H = ` (Z ). Let us consider its hull ΩHˆ given by s
ˆ (a)−1 |a ∈ Zd } , ΩHˆ = {U (a)HU
(5)
where (U (a))a∈Zd is a projective unitary representation of Zd on H and the closure ˆ is homogeis taken with respect to the strong operator topology. By definition H neous if ΩHˆ is a compact metrisable space [8, 12]. The projective representation U
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induces a Zd -action T on ΩHˆ by homeomorphisms. Each point ω ∈ ΩHˆ describes a disorder or aperiodicity configuration of the crystal. A T -invariant and ergodic probability measure P on ΩHˆ gives the probability with which specific configurations are realized. We now have the following stability theorem showing that any quantity defined almost surely with respect to P is stable with respect to compact ˆ perturbations of the Hamiltonian H. ˆ is homogeneous, so is Theorem 1. Let Vˆ = Vˆ ∗ be a compact operator. If H ˆ + Vˆ . Then the symmetric difference of the compact hulls Ω ˆ 4Ω ˆ ˆ is at most H H H+V countable. Moreover, any T -invariant measure on ΩHˆ or ΩH+ ˆ has its support in ˆ V ΩHˆ ∩ΩH+ ˆ Vˆ . Hence an invariant measure on ΩH ˆ completely determines an invariant measure on ΩH+ ˆ. ˆ V ˆ in The proof is given in the appendix. In the sequel, we will drop the index H ΩHˆ whenever there is no ambiguity. 2.2. The non-commutative Brillouin zone ˆ Let us In the previous section, we constructed the hull of the Hamiltonian H. ∗ briefly review the construction of the corresponding crossed product C -algebra A ˆ [8, 12]. For quasicrystals, called the non-commutative Brillouin zone (NCBZ) of H ∗ the algebra is in general given by a C -algebra associated to a groupoid [12, 45, 21]. In that case the formulæ below have direct analogs except for Birkhoff’s Theorem (Eq. (8)) which is not yet proved in that context as far as we know. All the analysis of this article should transpose directly to that case. Let us first consider the topological vector space Cκ (Ω × Zd ) of continuous functions with compact support on Ω × Zd . It is endowed with the following structure of a ∗ -algebra by AB(ω, n) =
X
ıq
A(ω, l)B(T −l ω, n − l)e 2~ B.n∧l ,
l∈Zd
(6)
∗
A (ω, n) = A(T −n ω, −n) , where A, B ∈ Cκ (Ω × Zd ), ω ∈ Ω, n ∈ Zd , finally the antisymmetric real tensor P B = (Bi,j ) is a uniform magnetic field and B.n ∧ l = i,j Bi,j ni lj . For ω ∈ Ω, this ∗ -algebra is represented on H = `2 (Zd ) by πω (A)ψ(n) =
X
ıq
A(T −n ω, l − n)e 2~ B.l∧n ψ(l) ,
ψ ∈ `2 (Zd ) ,
(7)
l∈Zd
namely, πω is linear, πω (AB) = πω (A)πω (B) and πω (A)∗ = πω (A∗ ). In addition, πω (A) is a bounded operator. Let a projective unitary representations (U (a))a∈Zd on H be given by the magnetic translations: ıq
U (a)ψ(n) = e 2~ B.a∧n ψ(n − a) .
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
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Then the representations are related by the covariance condition U (a)πω (A)U (a)−1 = πT a ω (A) ,
a ∈ Zd .
Now kAk = supω∈Ω kπω (A)k defines a C ∗ -norm. This allows to define A = C ∗ (Ω × Zd , B) as the completion of Cκ (Ω×Zd ) under this norm. Clearly, the representations πω can be continuously extended to this C ∗ -algebra. This family of representations is strongly continuous in ω for any fixed A ∈ A. Finally, there exists an element ˆ where ω0 is the point H ˆ of Ω [12]. H ∈ A such that πω0 (H) = H Given an invariant and ergodic probability measure P on Ω, a trace T on all A is defined by Z 1 X dP(ω) h0|πω (A)|0i = lim hn|πω0 (A)|ni , (8) T (A) = l→∞ |Λl | Ω n∈Λl
where |ni is the state completely localized at n ∈ Zd . The Λl ’s are an increasing sequence of rectangles centered at the origin. The equality holds for almost all ω 0 by Birkhoff’s ergodic theorem. This shows that T is the trace per unit volume. Note that a compact perturbation of the Hamiltonian changes the C ∗ -algebra A, but not the trace per unit volume of observables. T gives rise to the GNS Hilbert space L2 (A, T ) and GNS representation πGNS . We denote by L∞ (A, T ) the von Neumann algebra πGNS (A)00 where 00 is the bicommutant. By a theorem of Connes [14], L∞ (A, T ) is canonically isomorphic to the von Neumann algebra of P-essentially bounded, weakly measurable and covariant families Aω of operators on H = `2 (Z2 ) endowed with the norm kAkL∞ = P−essinf kAω kB(H) . ω∈Ω
Consequently, the family of representations πω extends to a family of weakly measurable representations of L∞ (A, T ). Moreover, the trace T extends to L∞ (A, T ). To define a differential structure on A, consider the family of ∗ -automorphisms ρkj of A given by (ρkj A)(ω, n) = eıkj nj A(ω, n) ,
A ∈ A.
Then the d generators of ρkj , denoted by ∂j , j = 1 . . . d, are ∗ -derivations. We ~ = (X1 , . . . , Xd ) is the position operator in ~ = (∂1 . . . ∂d ). If X use the notation ∇ 2 d H = ` (Z ), (Xj φ)(n) = nj φ(n) ,
φ ∈ `2 (Zd ) , n = (n1 , . . . , nd ) ∈ Zd .
One can check that πω (ρkj (A)) = eıkj Xj (πω (A))e−ıkj Xj , ~ ~ πω (A)]. The differential elements of A are = ı[X, and πω (∇A) C k (A) = {A ∈ A|∂j1 . . . ∂jl A ∈ A, j1 , . . . , jl ∈ 1, . . . , d, l ≤ k} .
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2.3. Local exponents of Borel measures Definition 1. Let f and g be Lebesgue measurable non-negative functions on ∼ the intervals (0, b] and [b, ∞) respectively, b > 0. The behaviors f (x) x↓0 xα and ∼ η g(x) x↑∞ x are defined by Z a dx f (x) < ∞ , (9) α = sup γ ∈ R|∃ a ∈ R : 0 < a ≤ b; x xγ 0 Z η = inf γ ∈ R|∃ a ∈ R : b ≤ a < ∞; a
∞
dx g(x) <∞ . x xγ
(10)
Let M be the space of Borel probability measures on R with the vague topology. Definition 2. Let ν be in M and E ∈ R. The local exponent αν (E) of ν at E is defined by Z E+ ∼ dν(E 0 ) ↓0 αν (E) . E−
Remark that the local exponents are always bigger than or equal to 0. Note further that Fubini’s theorem immediately implies that Z 1 0 <∞ . (11) αν (E) = sup γ ∈ R| dν(E ) |E − E 0 |γ Theorem 2. Let ν, µ ∈ M. (i) For ν-almost all E, 0 ≤ αν (E) ≤ 1. (ii) If µ dominates ν, then αµ (E) ≤ αν (E) µ-almost surely and αν (E) = αµ (E) ν-almost surely. (iii) If ν is pure-point, then ν-almost surely αν (E) = 0. (iv) If ν is absolutely continuous, then αν (E) = 1 ν-almost surely. (v) (ν, E) ∈ M × R 7→ αν (E) is a Borel function. This result shows that ν-almost surely in E, the local exponent αν (E) only depends upon the measure class of ν. Note that item (ii) does not follow directly from the Radon–Nykodim theorem. For example, let f (E)dE, ∈ L1 (R), be an absolutely continuous measure. The function f may have singularities where the exponent is smaller than 1. However, according to Theorem 2, these points only have Lebesgue measure 0. Note also that αν (E) may be bigger than 1, but only on a set of zero ν-measure. In practice, one can use the following characterization which is an extension of the Charles de la Vall´ee Poussin Theorem [57]. The proof can be found in [82], see also [61]. Theorem 3 [82]. Let ν ∈ M and Gν (z) = function. The exponent β(E) is introduced by =m(Gν (E − ı))
∼
↓0
R
dν(E 0 ) R z−E 0 ,
β(E) .
Then β(E) = αν (E) − 1 whenever αν (E) ∈ [0, 2].
=m(z) > 0, its Green’s
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The previous results allow to associate to a given measure class [ν] a function αν ∈ L∞ (R, dν) taking values in the interval [0, 1] ν-almost surely. Of particular interest are the biggest and smallest typical exponent in a given Borel set (see [81]). Definition 3. Let ν ∈ M and ∆ ⊂ R be a Borel set. The upper and lower essential exponents are defined by α+ ν (∆) = ν −esssup αν (E) , E∈∆
α− ν (∆) = ν −essinf αν (E) . E∈∆
Corollary 1. Let µ, ν be two Borel measures on R and ∆ ⊂ R a Borel set. If + − − µ dominates ν, then α+ ν (∆) ≤ αµ (∆) and αµ (∆) ≤ αν (∆). If µ and ν are in the + + − same measure class, then αµ (∆) = αν (∆) and αµ (∆) = α− ν (∆). Proposition 1. Let ∆ ⊂ R be a Borel set, then ν ∈ M 7→ α+ ν (∆) and ν ∈ − M 7→ αν (∆) are Borel functions. The essential exponents are linked to the Hausdorff dimensions dimH (see for example [28]) associated to the measure ν and its support by the following theorem of Rodgers and Taylor [65] (our formulation varies only slightly from theirs). Theorem [65]. Let ν ∈ M and ∆ ⊂ R Borel. If S0 is the Borel set defined by S0 = {E ∈ ∆|αν (E) ≤ α+ ν (∆)} , then ν(S0 ) = ν(∆) and dimH (S0 ) = α+ ν (∆). There is no Borel set S ⊂ ∆ with dimH (S) < α+ ν (∆) satisfying ν(S) = ν(∆). Moreover, if a Borel set S ⊂ ∆ satisfies dimH (S) < α− ν (∆), then ν(S) = 0. The Hausdorff dimension of a measure is defined by [81] dimH (ν|∆ ) = inf {dimH (S)|ν(S) = ν(∆)} , S⊂∆
where the infimum is taken over Borel sets S ⊂ ∆. Rodgers’ and Taylors theorem shows that dimH (ν|∆ ) = α+ ν (∆). 2.4. Local exponents of a self-adjoint operator Let H be a self-adjoint operator acting on the separable Hilbert space H. The spectral theory associates to H a H-projection-valued Borel measure Π on R [57]. Furthermore, for any φ ∈ H, kφk = 1, let ρφ be the spectral measure of H relative to φ, namely for f ∈ C0 (R), Z Z dρφ (E) f (E) = hφ|f (H)|φi = hφ|Π(dE)|φi f (E) .
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In physics literature, ρφ is called the local density of states (LDOS). Definition 4. Let E ∈ R and ∆ be a Borel subset of R. The spectral exponent and essential spectral exponents of Π (or H) are defined by αΠ (E) = inf αρφ (E) , φ∈H
+ α+ Π (∆) = sup αρφ (∆) , φ∈H
− α− Π (∆) = inf αρφ (∆) . φ∈H
Theorem 4. There exists ψ ∈ H with + α+ Π (∆) = αρψ (∆) ,
− α− Π (∆) = αρψ (∆) .
This result shows that there are typical states in H giving the generic properties of the spectrum. 2.5. Density of states and local density of states Let H ∈ A be the Hamiltonian. The spectral projection of πω (H) is denoted by Πω . Theorem 5. For E ∈ R and a Borel subset ∆ ⊂ R, the exponents αΠω (E), − α+ Πω (∆) and αΠω (∆) are P-almost surely independent of ω. The common values are − denoted by αLDOS (E), α+ LDOS (∆) and αLDOS (∆), respectively. A related result was proved by Last [48]. Theorem 1 implies that all these exponents are stable with respect to compact perturbation of the Hamiltonian. Corollary 2. If πω (H) has pure-point spectrum in ∆, P-almost surely, then α± LDOS (∆) = 0. If πω (H) has absolutely continuous spectrum in ∆ for P-almost all − + ω ∈ Ω, then α± LDOS (∆) = 1. If 0 < αLDOS (∆) ≤ αLDOS (∆) < 1, then the spectrum is singular continuous in ∆ for P-almost all ω ∈ Ω. Another important spectral measure associated to H is the density of states (DOS) defined by Z Z (12) dN (E) f (E) = dP(ω) h0|πω (f (H))|0i , f ∈ C0 (R) . We denote αDOS (E) = αN (E) and for any Borel subset ∆ of R, we set: ± α± DOS (∆) = αN (∆) .
Theorem 6. For E ∈ R and a Borel subset ∆ ⊂ R, αLDOS (E) ≤ αDOS (E) ,
± α± LDOS (∆) ≤ αDOS (∆) .
Note that this implies the Hausdorff dimension of the DOS is bigger than or equal to the Hausdorff dimension of the LDOS.
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
13
2.6. Diffusion exponents and localization This section is devoted to dynamical quantum diffusion and quantum localization. The diffusion exponent allows to measure the importance of quantum interference effects due to the frozen (disorder or quasiperiodic) potential in the one-particle Hamiltonian. Collisions with time-dependent disorder such as collisions with phonons and its effects on diffusion are not considered here; in RTA these effects are treated by the phenomenological constant τrel in Kubo’s formula. Diffusion is supposed to be isotropic here, but this is only done for sake of notational simplicity. For a given Borel set ∆ ⊂ R, the mean square displacement operator is Z
T
2 (T ) = δXω,∆ 0
dt ~ ω (t) − X) ~ 2 Πω (∆) , Πω (∆)(X T
(13)
~ −ıtπω (H) . ~ ω (t) = eıtπω (H) Xe where X Definition 5. The diffusion exponent σdiff (∆) is defined by Z ∼ 2 dP(ω) h0|δXω,∆ (T )|0i T ↑∞ T 2σdiff (∆) .
(14)
Ω
Proposition 2. Let Π(∆) = χ∆ (H) ∈ L∞ (A, T ), where χ∆ is the characteristic function on the Borel set ∆ and suppose that H ∈ C 1 (A). Then Z 0
T
dt 2σdiff (∆) ~ −ıHt )|2 Π(∆)) T∼ T (|∇(e ↑∞ T . T
(15)
Theorem 7. Let H ∈ C 1 (A). Then: (i) 0 ≤ σdiff (∆) ≤ 1. ~ Vˆ ] is bounded. (ii) Let Vˆ be a compact operator on H = `2 (Zd ) such that [X, Let the invariant ergodic measure on ΩHˆ determine that on ΩH+ ˆ as in ˆ V ˆ ˆ ˆ Theorem 1, then the diffusion exponents σdiff (∆) of H and H + V are equal. (iii) Guarneri’s bound: for any open interval ∆ ⊂ R: 00α+ LDOS (∆) ≤ d · σdiff (∆) ,
(16)
whenever H ∈ C k (A) for some k > d/2. An inequality between spectral and diffusion exponents was first proved by Guarneri [32]. A further contribution is due to Combes [20]. Last improved the proof in order to show that it is the most continuous part of the spectrum which gives the lower bound of the diffusion exponent [48], see also [6]. The bound (16) links exponents associated to the covariant family of Hamiltonians irrespective of the choice of a specific vector in Hilbert space.
14
H. SCHULZ-BALDES and J. BELLISSARD
Let us conclude this section with a discussion of localization. The following localization criterion for a Borel subset ∆ ⊂ R was introduced in [13] motivated by the study of the quantum Hall effect [14]: Z T dt ~ −ıHt |2 Π(∆)) < ∞ . T (|∇e (17) l2 (∆) = lim sup T T →∞ 0 Note that it is strictly stronger than σdiff (∆) = 0 because no logarithmic divergences are allowed. Actually, (17) coincides with the localization criterion used by physicists: in physics literature, averages of products of Green functions are used; this leads to the current-current correlation measure m below (Theorem 11). In the Anderson model and a wide class of other models, the condition (17) has been shown to hold for the spectral subsets generally considered to be localized [14]. Theorem 8. Suppose that the localization condition (17) is satisfied for a Borel set ∆ ⊂ R. Then the following holds: (i) [13, 14] σdiff (∆) = 0 and πω (H) has pure-point spectrum in ∆ for P-almost every ω ∈ Ω. ~ Vˆ ] is bounded. Let the (ii) Let Vˆ be a compact self-adjoint operator such that [X, invariant ergodic measure on ΩHˆ determine that on ΩH+ ˆ V ˆ as in Theorem 1, ˆ and then the localization condition (17) is simultaneously satisfied for H ˆ ˆ H +V. (iii) [13, 14] There is an N -measurable function l on ∆ such that for every Borel subset ∆0 of ∆: Z 2 0 dN (E) l(E)2 . (18) l (∆ ) = ∆0
Let us notice that the criterion (17) can be weakened in the following way: let g be any increasing function on R+ such that limx→∞ g(x) = ∞ and consider Z Z T dt ~ ω (t) − X|)Π ~ dP(ω) h0|Πω (∆)g(|X lg (∆) = lim sup ω (∆)|0i . T →∞ Ω 0 T Then lg (∆) < ∞ suffices to get pure-point spectrum in ∆, P-almost surely and to insure that this property is stable by compact perturbations of the Hamiltonian. 2.7. Current-current correlation function In this section we give some useful formulæ for the calculation of the diffusion exponent. As illustrative application, the diffusion exponent of Wegner’s n-orbital model is calculated in Sec. 5. The current operator is defined (if H ∈ C 1 (A)) by ~ J~ = ∇(H) . The current-current correlation functions are the Borel measures mi,j on R2 given by [47] Z dmi,j (E, E 0 ) f (E)g(E 0 ) = T (∂i (H)f (H)∂j (H)g(H)) ,
R2
(19)
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
15
where f, g ∈ C0 (R). The right-hand side defines a positive and continuous bilinear form on C0 (R) × C0 (R) × Md (C). The Riesz–Markov theorem [57] then assures the existence of the Radon measures mi,j on R2 with finite mass. The cyclicity of the trace induces the following symmetry of mj,j with respect to the diagonal E = E 0 : Z Z 0 0 dmj,j (E, E ) f (E, E ) = dmj,j (E, E 0 ) (f (E, E 0 ) + f (E 0 , E)) . (20) E≥E 0
R2
Pd The isotropic part m is the measure m = j=1 mj,j /d. It is called the currentcurrent correlation measure or also the conductivity measure. It allows to calculate the diffusion exponent. Theorem 9. Given a Borel set ∆ ⊂ R and > 0, let diag(∆, ) be the set of points in ∆ × R within distance from the diagonal in R2 , then Z ∼ dm(E, E 0 ) ↓0 2(1−σdiff (∆)) . diag(∆,)
The Stieltjes transform of m is given by Z 1 1 . dm(E, E 0 ) Sm (z1 , z2 ) = (2πı)2 R2 (E − z1 )(E 0 − z2 )
(21)
If H = H0 + V with a translation invariant kinetic part H0 and a potential V ~ ) = 0, then Sm can be calculated by means of the 2-point Green’s satisfying ∇(V function: Sm (z1 , z2 ) =
1 1 d (2πı)2
X
~ 0 )|ri · hs|∇(H ~ 0 )|ti G2 (z1 , z2 , r, s, t, 0) , (22) h0|∇(H
r,s,t∈Zd
where G2 (z1 , z2 , r, s, s0 , r0 ) =
Z dP(ω)hr| Ω
1 1 |sihs0 | |r0 i. z1 − πω (H) z2 − πω (H)
(23)
Theorem 10. The diffusion exponent is given by Z ∼ da Sm (a + ı, a − ı) ↓0 1−2σdiff (R) . <e R
The localization criterion can also be expressed by means of the conductivity measure [14]. Theorem 11 [14]. The localization condition (17) is equivalent to Z 1 dm(E, E 0 ) < ∞. |E − E 0 |2 ∆×R
16
H. SCHULZ-BALDES and J. BELLISSARD
Let us define the Liouville operator LH acting on A ∈ A by LH (A) = ı[H, A]/~. Then the spectral measure ρJ~ of ıLH associated to the current operator J~ is defined by (for f ∈ C0 (R)) Z 1 ~ . (24) dρJ~ () f () = T (f (ıLH )(J~) · J) d Theorem 12. The spectral exponent αρJ~ (0) is given by αρJ~ (0) = 2(1 − σdiff (R)) . 2.8. Anomalous Drude formula In [14], we showed that the zero frequency, isotropic direct conductivity at inverse temperature β, chemical potential µ and relaxation time τrel is given by Z 0 2q 2 1 0 fβ,µ (E ) − fβ,µ (E) dm(E, E ) . (25) σβ,µ = 0 2 2 0 1 τrel ~ E≥E 0 E−E + E−E ~ τ2 rel
β(E−µ) −1
) , q is the particle charge Here fβ,µ (E) is the Fermi–Dirac function (1 + e and ~ is Planck’s constant. More details on the derivation of (25) will be given in a forthcoming work. Theorem 13. If β < ∞, the direct conductivity given in (25) satisfies σβ,µ
∼
τrel ↑∞
−1+2σdiff (R)
τrel
.
(26)
If τrel ∼ β α with α ≈ 1 − 5 as indicated in the introduction, a more detailed analysis shows that only exponents at the Fermi level µ intervene in the anomalous Drude formula. 3. Exponents: Generalities 3.1. Local regularity behavior In this section we compare different exponents characterizing the H¨ older regularity behavior of positive functions. Although not all of these exponents will be used in this article, we present them for the sake of completeness and later reference. Definition 6. Let f be a Lebesgue-measurable non-negative functions on the interval (0, b], b > 0. The exponents βˆ and β are defined by log f (x) , βˆ = lim sup log x x→0
β = lim inf x→0
log f (x) . log x
(27)
Remark 1. By convention a function vanishing in a neighborhood of the origin will have exponents equal to infinity. Proposition 3. Let f and g be Lebesgue-measurable non-negative functions on the interval (0, b], b > 0. Let α, βˆ and β be as in Definitions 1 and 6.
17
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
∼ ∼ (i) If f (x) x↓0 xα , then f (x) Log(x) x↓0 xα . ∼ (ii) (Calculation with Laplace transform) If f (x) x↓0 xα and α > −1, then Z 1 ∼ dt e−δt f (t) δ↑∞ δ −α−1 . 0
(iii) [40] The following equalities hold : f (x) β = sup γ ∈ R| lim supx↓0 γ < ∞ , x xγ ˆ <∞ . β = inf γ ∈ R| lim inf x↓0 f (x)
(28)
ˆ (iv) β ≤ α ≤ β. (v) For α > 0 and f non-decreasing (respectively, for α ≤ 0 and f nonincreasing), β = α. (vi) Suppose that both f and g are non-increasing or non-decreasing with corresponding exponents αf and αg as defined in (9). Then for δ > 0, f (x)δ
∼
x↓0
xδαf ,
f (x)g(x)
∼
x↓0
xαf +αg ,
f (x) + g(x)
∼
x↓0
xmin{αf ,αg } . (29)
Remark 2. These results transpose directly to the study of the behavior of ∼ a function at infinity as Rgiven in Definition 1. Note that in particular, if g(x) x↑∞ ∞ α −δx x , α > −1, and I(δ) = 1 dx e g(x), then Proposition 3(ii) and (iii) show that ) ( α = inf
γ ∈ R| lim sup δ γ+1 I(δ) = 0
.
δ↓0
Remark 3. The following example will show that there exist functions with β < α < βˆ and for which the conclusions of Proposition 3(v) do not hold. Let t > s > 1, u ∈ R, and consider nu for x ∈ In = 1 , 1 + 1 , ns ns ns+t (30) f (x) = 0 otherwise . Because of (28) we have βˆ = ∞ and β = − us . By explicit calculation one gets α=
t−1 u − . s s
(31)
As an example, take u = 0, s = 2 and t = 5, then β = 0, α = 2 and βˆ = ∞. To consider the function f δ is equivalent to replacing u by uδ and this leads, according to (31), to a exponent different from δα. Proof of Proposition 3. R1 (i) This follows from the fact that 0 dx x−1+ log x < ∞ for any > 0. (ii) For 0 > γ > −α − 1, the identity
18
H. SCHULZ-BALDES and J. BELLISSARD
Z 1
∞
dδ δ 1+γ
Z
1
dt e−δt f (t) =
0
Z
Z
1
dt f (t)tγ 0
∞
t
ds s1+γ
e−s
allows to conclude. (iii) is proved in [40]. (iv) Let us only show β ≤ α. The other inequality can be proved in a similar way. For any given δ > 0 there is a (δ) ≤ 1 such that log f (x) = β −δ. x≤(δ) log x inf
Then for x ≤ (δ), f (x) ≤ xβ−δ because log x < 0. Let now γ < β and choose δ such that β − δ − γ > 0, then Z (δ) Z (δ) dx f (x) dx β−δ−γ x ≤ < ∞, γ x x x 0 0 which shows γ ≤ α. (v) We only treat the case where α > 0 and f is non-decreasing. Take 0 < γ < α, then if x ≤ a/2, Z 2x Z 2x Z a f (x) 1 1 dy f (y) dy f (y) dy ≥ ≥ f (x) ≥ γ 1− γ , C(γ) = 1+γ yγ y yγ x γ 2 0 y x x y and therefore equality (28) implies that γ ≤ β and hence α ≤ β. Thanks to (iv) this gives α = β. (vi) is a direct consequence of (iii) and (v). 3.2. Local exponents and essential exponents We begin this section with the proof of Theorem 2. Then follow some comments on Definitions 2 and 3 and Theorem 2. In the rest of the section we prove the other results of Secs. 2.3 and 2.4 as well as some complementary results. The following lemma is known as the Hardy–Littlewood maximal inequality. We will need it in a slightly generalized form, nevertheless, its proof can be directly transposed from [67], for example. Lemma 1. Let µ, ν be two probability measures on R and h ∈ L1 (R, dν). The maximal function Mµ,ν,h is defined by Z 1 dν(E 0 ) h(E 0 ) . Mµ,ν,h (E) = sup µ([E − 3, E + 3]) ∈(0,1] (E−,E+) It is lower semicontinuous and satisfies for any positive λ: µ({E ∈ R|Mµ,ν,h (E) > λ}) ≤
1 khkL1 (R,dν) . λ
Lemma 2. Let µ, ν be two probability measures on R. Then αµ (E) ≤ αν (E) µ-almost surely.
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
19
Proof. If Mµ,ν,1 (E) < ∞, then ν((E − , E + )) < Cµ([E − 3, E + 3]) for all ∈ (0, 1] and some constant C > 0. Therefore αµ (E) ≤ αν (E). Thus αν (E) < αµ (E) implies Mµ,ν,1 (E) = ∞. Hence by Lemma 1 ! \ {E ∈ R|Mµ,ν,1 (E) > N } µ({E ∈ R|αν (E) < αµ (E)}) ≤ µ N ∈N
≤ lim
N →∞
1 = 0. N
Proof of Theorem 2. (i) Clearly the local exponents are all bigger than or equal to 0. The exponents of the Lebesgue measure are all equal to 1. Applying Lemma 2 to the measure ν and the Lebesgue measure shows that αν (E) ≤ 1 for ν-almost all E ∈ R. (ii) Apply Lemma 2 twice and use that µ-almost surely implies ν-almost surely. P (iii) Since ν is pure-point, it is of the form n∈N cn δ(E − En ), cn > 0. For each En , ν([En − , En + ]) ≥ cn such that αν (En ) = 0. Consequently αν (E) is equal to 0 for ν-almost all E, notably the En ’s. (iv) If ν is absolutely continuous, it is dominated by the Lebesgue measure. (ii) allows to conclude. (v) We will prove a stronger result in Proposition 4(iii) below. Remark 4. Proposition 3 implies that the exponents αν (E) are the same as those often considered in literature [81, 22, 53, 52, 48, 61, 6] because ν([E −, E +]) is a non-decreasing function of . Remark 5. Theorem 2 does not exclude singular continuous spectrum with exponents equal to 0 or 1. Remark 6. An absolutely continuous measure can have exceptional points where the exponent is not equal to 1. For example, consider dν(E) = h(E)dE ∈ M with h(E) = |E − E 0 |γ , γ > −1, on an interval around E 0 . Then αν (E 0 ) = 1 + γ. Remark 7. By definition γ < α− ν (∆) if and only if there exists a set Ξ ⊂ ∆ of zero ν-measure such that γ < αν (E) for all E ∈ ∆\Ξ. Furthermore γ < α+ ν (∆) if and only if there exists a set Ξ ⊂ ∆ of stictly positive ν-measure such that γ < αν (E) R 1 d R E+ 0 dν(E ) is bounded on for all E ∈ Ξ. Because the Borel function E 7→ 0 1+γ E− Ξ, Lusin’s RtheoremR then implies that there exists a set Ξ0 ⊂ Ξ of positive ν-measure 1 d E+ 0 0 such that 0 1+γ E− dν(E ) has a uniform bound for all E ∈ Ξ . Definition 7 [74, 20, 48]. The uniform dimension αuni ν (∆) of a measure ν on a Borel set ∆ ⊂ R is defined by ( ) Z E+ uni 0 γ ν(dE ) ≤ C ∀ < δ, E ∈ ∆ . αν (∆) = sup γ ∈ R|∃ C < ∞, δ > 0 : E−
20
H. SCHULZ-BALDES and J. BELLISSARD
− Remark 8. One clearly has αuni ν (∆) ≤ αν (∆). However, one does not necessarily have equality. For if
f (E) =
q−1 ∞ X X q=2
1 1 , 2 (q − 1) |E − p/q|1/2 q p=1
then f ∈ L1 ([0, 1]) and defines an absolutely continuous probability measure ν = z −1 f dx (if z > 0 is a normalization factor). Thus, for any Borel subset ∆ of uni [0, 1], α− ν (∆) = 1 whereas if ∆ contains some rational point, αν (∆) ≤ 1/2. Now we present some further technical results as well as proofs of the other results of Secs. 2.3 and 2.4. Lemma 3. Let N ∈ N, γ > 0. If ∆ ⊂ R is a Borel set, then ( ) Z 1 Z E+ d − dν(E) ≤ N for ν-a.a. E ∈ ∆ M (∆, γ, N ) = ν ∈ M| 1+γ 0 E− and
( M (∆, γ, N ) = +
Z ν ∈ M|∃Ξ ⊂ ∆, ν(Ξ) > 0, 0
1
d
Z
1+γ
E+
dν(E 0 ) ≤ N
E−
) for
E∈Ξ
are Borel sets in M. Furthermore, M± (∆, γ, ∞) = {ν ∈ M|γ < α± ν (∆)} are Borel sets. Proof. Let gk (x) be a continuous non-decreasing real function, equal to 0 for x < 0, equal to 1 for x > 1/k and 0 ≤ gk (x) ≤ 1 elsewhere. For χ ∈ C0 (R), δ > 0, N ∈ N and γ > 0, the function Z 1 Z d dν(E) χ(E) gk ν([E − , E + ]) − N ν ∈ M → Gk,δ,χ,N,γ (ν) = 1+γ R δ is a continuous function. It is non-increasing in δ. Since ∆ is a Borel set, there exists a sequence χn1 ,m1 ,...,nr ,mr ∈ C0 (R), increasing in the mj and decreasing in the ni , such that the characteristic function χ∆ is given by inf n1 supm1 . . . inf nr supmr χn1 ,m1 ,...,nr ,mr . Now G∆,γ,N (ν) = inf sup . . . inf sup sup sup Gk,δ,χn1 ,m1 ,...,nr ,mr ,N,γ (ν) n1 m1
nr mr
k
δ
is a Borel function in ν. By the dominated convergence theorem Z 1 Z d dν(E) g∞ ν([E − , E + ]) − N . G∆,γ,N (ν) = 1+γ ∆ 0
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
21
R 1 d If G∆,γ,N (ν) = 0 then there exists a set Ξ of ν-measure zero such that 0 1+γ ν([E − , E + ]) ≤ N ∀ E ∈ ∆\Ξ. Hence, G∆,γ,N (ν) = 0 implies ν ∈ M− (∆, γ, N ). Clearly ν ∈ M− (∆, γ, N ) implies G∆,γ,N (ν) = 0. Consequently M− (∆, γ, N ) = + ) is treated in a similar way. Finally, the G−1 ∆,γ,N ({0}) is a Borel set. M (∆, γ, NS 1 ± , N ). last result follows from M (∆, γ, ∞) = N,M∈N M± (∆, γ + M Proof of Proposition 1. Let h denote the application ν ∈ M 7→ α+ ν (∆). If I = (a, b) is an open interval, it is sufficient to show that h−1 (I) is a Borel set in order to deduce that h is a Borel function. With the notations of Lemma 3, h
−1
(I) = M (∆, a, ∞) ∩ +
\ n∈N
!C 1 M , ∆, b − , ∞ n +
so that h−1 (I) is a Borel set by Lemma 3. The case of α− ν (∆) is treated in a similar way. ˆ of H-projectionLet H be a separable Hilbert space. We consider the space M valued Borel measures on R [57] endowed with the weak and vague topology, that is (because of the polarization identity) Z Z ˆ M hφ|Πn (dE)|φif (E) → hφ|Π(dE)|φif (E) , Πn → Π ⇔ R
R
for all φ ∈ H and f in C0 (R). To every self-adjoint operator H the spectral theorem ˆ Convergence in the strong resolvent sense corresponds to associates a Π ∈ M. ˆ convergence in M. ˆ be a H-projection valued Borel measure on a separable Lemma 4. Let Π ∈ M Hilbert space H. Then there exists ψ ∈ H so that the spectral measure ρψ is in the same measure class as Π. Proof. The lemma being well known, we only sketch an outline of the proof. A countable family of normalized vectors (φi )i∈I is called Π-free if and only if R hφi |Π(dE)|φj if (E) = 0 for all i 6= j and all f ∈ C0 (R). The set of Π-free families is ordered by inclusion and Zorn’s lemma assures the existence of a maximal family (φi )i∈I . Set X 1 1 cI = √ . ψ = cI n+1 φn , 2 1 − 2−#I n∈I 2 It is now possible to verify that the spectral measure ρψ of Π dominates the spectral measure ρη of any η ∈ H. Proof of Theorem 4. With Lemma 4 choose φ ∈ H such that ρψ is in the same measure class as Π. Then ρψ dominates the spectral measures ρφ for all + + − φ ∈ H. Hence, by Theorem 2, α+ ρφ (∆) ≤ αρψ (∆) ≤ supη∈H αρη (∆) and αρφ (∆) ≥ − − αρψ (∆) ≥ inf η∈H αρη (∆) for all φ ∈ H.
22
H. SCHULZ-BALDES and J. BELLISSARD
Proposition 4. (i)
Z
1 d Z E+
0 Π(dE ) αΠ (E) = sup γ ∈ R|
0 1+γ E−
(
(ii) Let GΠ (z) =
R
Π(dE 0 ) R z−E 0
Z
<∞ .
(32)
B(H)
be the resolvent of Π. Suppose αΠ (E) ∈ [0, 2] then
1
=m(GΠ (E + ı))
1+γ d
0
)
B(H)
< ∞,
if and only if γ < αΠ (E) − 1. ˆ × R → αΠ (E) is a Borel function. (iii) (Π, E) ∈ M Proposition 5. Let D be a dense subset of H(∆) = Π(∆)H. Then + α+ Π (∆) = sup αρφ (∆) , φ∈D
− α− Π (∆) = inf αρφ (∆) . φ∈D
The proof of the following lemma follows the lines of the proof of Lemma 3. Lemma 5. Let N ∈ N, γ > 0. The set ( Z ˆ N ) = (Π, E) ∈ M ˆ × R|∀ φ ∈ H : S(γ,
1 0
d 1+γ
Z
)
E+
0
hφ|Π(dE )|φi ≤ N
,
E−
(33) ˆ × R. Furthermore, S(γ, ˆ ∞) = {(Π, E) ∈ M ˆ × R|γ < αΠ (E)} is a is closed in M Borel set. Proof of Proposition 4. (i) Let βΠ (E) be the exponent on the right-hand side of (32). Clearly βΠ (E) ≤ αΠ (E). To show βΠ (E) ≥ αΠ (E), let γ < αΠ (E). By the Schwarz inequality, the expression Z hψ|
1 0
Π([E − , E + ])|φi 1+γ d
sZ
≤ 0
1
d 1+γ
Z
E+ E−
sZ dρψ (E 0 ) 0
1
d 1+γ
Z
E+
dρφ (E 0 ) E−
R 1 d is bounded for all ψ, φ ∈ H. Consequently, the positive operator 0 1+γ Π([E − , E + ]) is everywhere defined. By the Hellinger–Toeplitz theorem [57] it is therefore a bounded operator. Hence γ < βΠ (E).
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
23
(ii) This follows from Theorem 3 and an application of the Hellinger–Toeplitz theorem to
Z 1
d
=m(GΠ (E − ı))
1+(γ−1) B(H)
0
Z
=
1
0
d 1+γ
2
Π(dE ) , 0 2 2 (E − E ) + B(H) R
Z
0
similar to (i). (iii) Let h be the application (Π, E) 7→ αΠ (E). Let I = (a, b) be an open interval. Then !C \ 1 −1 ˆ ˆ S b − ,∞ , h (I) = S(a, ∞) ∩ n n∈N
and Lemma 5 assures that h−1 (I) is a Borel set. Hence h is a Borel function. + Proof of Proposition 5. Let us put β = supφ∈D α+ ρφ (∆). Clearly β ≤ αΠ (∆). Let now ψ ∈ H be as in Theorem 4 and introduce Ξ(β) = {E ∈ ∆|αρψ (E) ≤ β}. As E 7→ αρψ (E) is a Borel function by Theorem 2, Ξ(β) is a Borel set. Thus H(β) = Π(Ξ(β))H is a closed linear subspace of H(∆). Now for any φ ∈ D, ρψ dominates ρφ and therefore αρφ (E) = αρψ (E) ρφ -almost surely by Theorem 2. Therefore
ρφ (∆) ≥ ρφ (Ξ(β)) ≥ ρφ ({E ∈ ∆|αρψ (E) ≤ α+ ρφ (∆)}) = ρφ (∆) . Hence ρφ (Ξ(β)) = kΠ(∆)φk2 = 1 and φ ∈ H(β) for all φ ∈ D. Because D is dense in H(∆) by hypothesis, H(∆) = H(β). Consequently, ρψ (Ξ(β)) = ρψ (∆) and + α+ ρψ (∆) ≤ β. Theorem 4 implies αΠ (∆) ≤ β. This shows the first equality. In order to show the second equality, one proceeds in a similar way using the set of all E ∈ ∆ such that β ≤ αρψ (E). 3.3. Multifractal dimensions The formulæ on which the multifractal analysis developed below is based are already explicit in the article of Hentschel and Procaccia [36]. The dimensions introduced are often referred to as generalized R´enyi dimensions [53, 52]. The main reason why this multifractal analysis is relevant for the quantum-mechanical study of solids is the following: the behavior of the Fourier transform of a measure at infinity which is of interest for physicists [38, 29, 40] can be rigorously linked to the 2-spectral dimension of the measure, its correlation dimension. This will be done in the next section. Moreover, the multifractal dimensions give lower bounds on the lower essential dimension. Note that there are other possibilities to define multifractal dimensions [36, 35, 52, 53].
24
H. SCHULZ-BALDES and J. BELLISSARD
Definition 8. Let ν ∈ M and ∆ ⊂ R be a Borel set. If ν(∆) 6= 0, let for q ∈ R 1 !p−1 p−1 Z E+ Z dν(E) dν(E 0 ) . (34) Iνq, (∆) = lim p↓q ν(∆) ∆ E− The q-spectral dimension αqν (∆) is defined by Iνq, (∆)
∼
↓0
q
αν (∆) ,
unless Iνq, (∆) is infinite for a set of ’s of positive Lebesgue measure (possible if q ≤ 1). We denote αqν = αqν (R). The dimensions α1ν and α2ν are called information and correlation dimension, respectively. Remark 9. The notation is chosen such that in good cases the dimensions αqν are equal to the Dq appearing in physics literature [36, 35, 29]. The dimensions αqν are rigorously linked to box-counting dimensions in [53, 52]. Remark 10. The limit in (34) is only introduced in order to study the case q = 1. Using the monotone convergence theorem one gets ! Z E+ Z dν(E) 1, 0 log dν(E ) . (35) Iν (∆) = exp ∆ ν(∆) E− This explains why one talks of information dimension. Proposition 6. Let ν ∈ M and let ∆ ⊂ R be a Borel set. (i) For q > 1, 0 ≤ αqν (∆) ≤ α− ν (∆). p (ii) [22] For p ≤ q, αν (∆) ≥ αqν (∆). (iii) (q − 1)αqν (∆) is a convex function of q. 1 (iv) α− ν (∆) ≤ αν (∆). Proposition 7. Let ν ∈ M and ∆ ⊂ R a Borel set. (i) If ν ∗ ν is the convolution of ν with itself, then αν∗ν (0) = α2ν (R). (ii) ( ) Z Z E+1 2 0 0 −γ dν(E) dν(E )|E − E | < ∞ . αν (∆) = sup γ ∈ R| ∆
(36)
E−1
(iii) [61] The correlation dimension can be calculated as Z 2 ∼ da|=mGν (a + ı)|2 ↓0 αν −1 . R
Remark 11. The multifractal dimensions are not measure class invariants. Let us give an example of an absolutely continuous measure for which the correlation dimension is smaller than 1: 1 dν(E) = const β e−E χ(E > 0) , E
25
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
where χ is the indicator function. It is a matter of calculation to verify that α2ν = min{1, 2(1 − β)}. Proof of Proposition 6. (i) Since q > 1, we get Iνq, (∆) ≤ 1 for all > 0. Hence αqν (∆) ≥ 0. Because Iνq, (∆) is increasing in , Proposition 3(v) and Fubini’s theorem shows that !q−1 Z 1 Z E+ Z dν(E) d 0 dν(E ) < ∞ . (q − 1)αqν (∆) = sup γ ∈ R| 1+γ ∆ ν(∆) 0 E− R 1 d R E+ ( E− Thus, for γ < (q − 1)αqν (∆) and ν-almost all E ∈ ∆, 0 1+γ 0 q−1 )) < ∞. Using again the monotonicity in and Proposition 3(v), Rdν(E 1 d R E+ γ dν(E 0 ) < ∞ for all γ 0 < q−1 and for ν-almost all E ∈ ∆, 0 1+γ 0 E− 0 namely γ ≤ αν (E) for ν-almost all E ∈ ∆. Hence γ 0 ≤ α− ν (∆) and (∆). therefore αqν (∆) ≤ α− ν (ii) Let us show that for q 6= 0, Iνq, (∆) is non-increasing functions of and a non-decreasing function in the variable q. This implies directly the result. Jensen’s inequality for the convex function f (t) = tβ , β ≥ 1 or β ≤ 0, is used in the following way: let p < 1 < q or 1 < p < q, then Z ∆
dν(E) ν(∆) ≥
Z ∆
Z
E+
q−1 !p−1 p−1
dν(E 0 )
E−
dν(E) ν(∆)
Z
E+
q−1 !p−1 p−1
dν(E 0 )
,
E−
that is, Iνq, (∆) ≤ Iνp, (∆). The case p < q < 1 is treated in a similar way. (iii) Let q = σq0 + (1 − σ)q1 with σ ∈ [0, 1]. By H¨older’s inequality one gets (Iνq, (∆))q−1 ≤ (Iνq0 , (∆))σ(q0 −1) (Iνq1 , (∆))(1−σ)(q1 −1) . Because the right-hand side is increasing in , Proposition 3(v) implies that it behaves as σ(q0 − 1)αqν0 (∆) + (1 − σ)(q1 − 1)αqν1 (∆). The exponent of the left-hand side is (q − 1)αqν (∆). The above inequality now implies that (q − 1)αqν (∆) ≥ σ(q0 − 1)αqν0 (∆) + (1 − σ)(q1 − 1)αqν1 (∆). (iv) Because Iν1, (∆) is monotone in , Proposition 3(iv) and Eq. (35) imply that the exponent α1ν (∆) is given by Z α1ν (∆)
= lim inf →0
∆
dν(E) log ν(∆)
R E+ E−
dν(E 0 )
log
.
By Fatou’s lemma and again Proposition 3(iv), α1ν (∆) ≥ ν − essinf E∈∆ αν (E).
26
H. SCHULZ-BALDES and J. BELLISSARD
Proof of Proposition 7. (i) By definition of the convolution Z Z E+ 0 dν ∗ ν(E ) = E−
dν(E 0 )dν(E 00 ) ,
diag(E,)
where diag(E, ) = {(E 0 , E 00 )| |E − E 0 + E 00 | < }. But for E = 0, this last expression is just equal to Iν2, (R). Therefore αν∗ν (0) = α2ν . (ii) follows by direct calculation using Fubini’s theorem. (iii) Because of Theorem 3 and αν∗ν (0) = α2ν (R), it is sufficient to show that Z 1 da|=mGν (a + ı)|2 . (37) =mGν∗ν (−2ı) = π R For this purpose, write out the right-hand side explicitly and use Fubini’s theorem. The contour of the integral over a can be closed by half of a circle in the upper half plane because the integrand falls off as 1/a4 at infinity. There are two poles within the closed contour at E + ı and E 0 + ı. The residue theorem then allows to show (37). 3.4. Asymptotic behavior of Fourier transforms In this section we study the asymptotic behavior of the Fourier transform of measures on the real line. It is governed by the correlation dimension of the measure. The first rigorous results in this direction were obtained by Strichartz [74]. He gave an estimate of the decrease of the Fourier transform by the uniform dimension of the measure (compare Theorem 15). Physicists interest began with the numerical works of [38] as well as [29]. The latter work also contains a formal derivation of Theorem 14. Wavelet transform was used in a more mathematical approach in [40], further related results appear in [48, 6, 34]. Here, we present two versions of these results as well as an application to a quantitative version of the RAGE-theorem. These results are not new and we give them for sake R of completeness. The Fourier transform of ν is given by Fν (t) = dν(E)eıtE . Further let Z T dt |Fν (t)|2 . (38) Cν (T ) = 0 T Theorem 14. Let ν be a measure on the real line and α2ν its correlation dimension. Then 2 ∼ Cν (T ) T ↑∞ T −αν . The next corollary follows directly from Theorem 14, the Schwarz inequality and the equivalent of Proposition 3(v) for the behavior at infinity. Corollary 3. Let ν, ν˜ be two measures on the real line. If the functions Cν (T ) and Cν˜ (T ) are eventually non-increasing, then Z T 1 dt ∼ |Fν (t)Fν˜ (t)| T ↑∞ T −α with α ≥ (α2ν + α2ν˜ ) ≥ min{α2ν , α2ν˜ } . T 2 0 Moreover, if ν ∗ ν˜ denotes the additive convolution, then one has α2ν∗˜ν ≥ (α2ν + α2ν˜ ).
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
27
Remark 12. It is certainly necessary to take the time average in Eq. (38). Consider, for example, an absolutely continuous measure dν(E) = f (E)dE with a C ∞ function f of compact support. This implies that |Fν (t)|2 ∼ t−2N for any N ∈ N although the 2-spectral dimension of ν is α2ν = 1. However, taking the time-average, one obtains Cν (T ) ∼ T −1 in agreement with Theorem 14. Theorem 15 [74]. Let ν ∈ M. If ν has uniform dimension αuni ν (∆) on a Borel (∆) and some positive constant set ∆ ⊂ R and f ∈ L2 (∆, dν), then for any β < αuni ν C: Z 2 Z T dt 2 ıEt dν(E) f (E) e ≤ C kf kL2 (∆,dν) T −β . (39) T ∆
0
Before coming to the proofs, let us study the links between the diffusion of presence probability under quantum mechanical time evolution and spectral properties of the underlying Hamiltonian. This leads to a quantitative version of the RAGEtheorem [57]. Corollary 4. Let φ, kφk = 1, belong to a Hilbert space H. Let H a be self-adjoint operator on H. Let ρφ be the spectral measure of H associated to φ. Then Z
T
dt −α2 ∼ |hφ|e−ıHt |φi|2 T ↑∞ T ρφ . T
0
(40)
Remark 13. Let φ, ψ ∈ H, kφk = 1, kψk = 1. If β is defined by Z 0
then
T
dt ∼ |hφ|e−ıHt |ψi|2 T ↑∞ T −β , T
Z Z 0 0 −γ dρφ,ψ (E) dρψ,φ (E )|E − E | < ∞ , β = sup γ ∈ R| R
(41)
R
where ρψ,φ is the complex spectral measure of H associated to φ, ψ. If either ρφ or ρψ has uniform dimension αuni , then one has β ≥ αuni as shows directly the spectral theorem and Theorem 15. Proof of Theorem 14. First note that 0 ≤ α2ν ≤ 1. We have to show the equivalence between Z d ν(E)dν(E 0 ) <∞ (42) |E − E 0 |γ E≥E 0 and
Z 1
∞
dT T 1−γ
Z E≥E 0
d ν(E)dν(E 0 )
2 sin((E − E 0 )T ) < ∞, (E − E 0 )T
provided 0 < γ < 1. Suppose that (42) holds. Because the integral Z ∞ Z ∞ sin s dT sin(|E − E 0 |T ) = ds 2−γ 0 |T )1−γ T (|E − E s 0 1 |E−E |
(43)
(44)
28
H. SCHULZ-BALDES and J. BELLISSARD
is absolutely convergent and behaves like O(1) as |E − E 0 | → 0, we may apply Fubini’s theorem to Z Z d ν(E)dν(E 0 ) ∞ dT sin((E − E 0 )T ) <∞ |E − E 0 |γ T ((E − E 0 )T )1−γ E≥E 0 1 in order to deduce (43). For the converse, let N be any integer. By (43), there is a constant C independent of N such that Z Z N sin((E − E 0 )T ) dT d ν(E)dν(E 0 ) C> 1−γ T (E − E 0 )T E−E 0 ≥0 1 Z > π E−E 0 ≥ N
d ν(E)dν(E 0 ) |E − E 0 |γ
Z
N (E−E 0 )
ds E−E 0
sin s , s2−γ
because the integrand long as E − E 0 < π/N . We have used Fubini’s R ∞is positive as theorem. Then 0 < 0 ds sin(s)/s2−γ < ∞ implies (42). Proof of Theorem 15. Use the Cauchy–Schwarz inequality and | sin(θ)| ≤ |θ|1−β for all β ∈ [0, 1], to get: Z 2 Z β Z Z T 2 dt ıEt 2 0 dν(E) f (E) e ≤ dν(E) |f (E)| dν(E ) . |E − E 0 |T ∆ ∆ ∆ 0 T Now if β < αuni ν (∆), by definition there exists constants C < ∞ and δ > 0 such that ν((E − , E + )) ≤ Cβ for all ≤ δ. Hence for β < β 0 < αuni ν (∆) one has (with changing constants C): Z E+δ Z δ Z 1 d 0 0 dν(E ) ≤β dν(E ) +C 0 β 1+β |E − E | R E−δ |E−E 0 | Z
δ
≤β 0
0 d Cβ + C 1+β
≤C. As this bound is uniform in E, this finishes the proof.
4. Spectral Exponents and Anomalous Quantum Diffusion 4.1. Spectral exponents of covariant Hamiltonians Let H = H ∗ ∈ A be a given covariant Hamiltonian family. It gives rise to a covariant family (Πω )ω∈Ω of projection-valued measures on H = `2 (Zd ) by Z Πω (dE) f (E) = πω (f (H)) , f ∈ C0 (R) . (45) R
For φ ∈ H, the corresponding spectral measure is denoted by ρω,φ . Let us introduce the measure X cn ρω,|ni , ρω = n∈Zd
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
29
P where the sequence (cn )n∈Zd of positive numbers satisfies n∈Zd cn = 1. By the following lemma, ρω is in the same measure class as Πω and the application Πω ∈ ˆ 7→ ρω ∈ M is continuous for fixed (cn )n∈Zd . M ˆ and introduce Lemma 6. Let (φn )n∈N be an orthonormal basis of H. Let Π ∈ M P∞ 1 the measure ρΠ = n=0 2n+1 ρφn . Then ρΠ is in the same measure class as Π and ˆ 7→ ρΠ ∈ M is continuous. the application Π ∈ M Proof. If a Borel set ∆ ⊂ R satisfies ρΠ (∆) = 0, then ρφn (∆) = 0 for all P n ∈ N. Hence for ψ = n∈N an φn , kψk = 1, the Schwarz inequality gives ρψ (∆) = 0. Therefore ρΠ dominates Π. On the other hand, kΠ(∆)k = 0 clearly implies ˆ ρΠ (∆) = 0. Hence ρΠ and Π are in the same measure class. Let now Πl → Π in M as l → ∞. For f ∈ C0 (R), by the dominated convergence theorem Z X 1 Z lim dρΠl (E) f (E) = hφn |Π(dE)|φn i f (E) l→∞ 2n+1 n∈N
Z = Therefore, ρΠl → ρΠ in M as l → ∞.
dρΠ (E) f (E) .
ˆ and ω ∈ Ω 7→ ρω ∈ M are Lemma 7. The applications ω ∈ Ω 7→ Πω ∈ M continuous. Proof. For f ∈ C0 (R), f (H) ∈ A and the application ω 7→ πω (f (H)) is ˆ strongly continuous. Because of Eq. (45) and the definition of the topology on M, ˆ The second this implies the continuity of the application ω ∈ Ω 7→ Πω ∈ M. statement now follows from Lemma 6. R Proof of Theorem 5. We denote the spectral projection ∆ Πω (dE) by Πω (∆) and introduce the function
Z 1
d
Πω ([E − , E + ]) Fω,E (γ) =
. 1+γ 0 The covariance implies that Fω,E (γ) = FT −n ω,E (γ) for all n ∈ Zd . The sets Ωγ,N = {ω ∈ Ω|Fω,E (γ) ≤ N } (n ∈ N ∪ {∞}) are therefore T -invariant. Moreover, ˆ N )} is Borel (see Lemma 5). By the ergodicity Ωγ,N = {ω ∈ Ω|(Πω , E) ∈ S(γ, of P, P(Ωγ,N ) = 0 or 1. The monotonicity of Fω,E (γ) in γ implies that, for γ < γ 0 , Ωγ 0 ,∞ ⊂ Ωγ,∞ . Hence there exists a γc such that for γ < γc , P(Ωγ,∞ ) = 1, and for γ > γc , P(Ωγ,∞ ) = 0. Because Ωγ,∞ = {ω ∈ Ω|γ < αΠω (E)}, γc = αΠω (E) Palmost surely. ± ± Let us now consider α± Πω (∆). The set Ωγ,∆ = {ω ∈ Ω|γ < αρω (∆)} is T invariant by Corollary 1 because ρω and ρT a ω are in the same measure class for any a ∈ Zd . Moreover, if M± (γ, ∆, ∞) are the sets defined in Lemma 3, then ± ± Ω± γ,∆ = {ω ∈ Ω|ρω ∈ M (γ, ∆, ∞)}. Lemmas 3 and 7 imply that Ωγ,∆ are Borel
30
H. SCHULZ-BALDES and J. BELLISSARD
sets. By ergodicity of P, they have either full or zero P-measure. As γ < γ 0 implies ± ± ± 0 ± Ω± γ 0 ,∆ ⊂ Ωγ,∆ , there exist critical values γc such that P(Ωγ 0 ,∆ ) = 1 for γ < γc ± ± and P(Ωγ,∆ ) = 0 for γc < γ. Proof of Theorem 6. Let γ < αLDOS (E). Then it follows from the arguments in the previous proof of Theorem 5 that there exists an N < ∞ such that the T -invariant set {ω ∈ Ω|(Πω , E) ∈ S(γ, N )} has full measure. On this set of full measure the Fω,E (γ)’s have uniform bound N , P-almost surely. Therefore the spectral exponent satisfies Z dP(ω) Fω,E (γ) < ∞ αLDOS (E) = sup γ ∈ R| Ω
Z ) Z 1 Z E+
d
0 Πω (dE ) < ∞ , ≤ sup γ ∈ R| dP(ω) 1+γ
Ω 0 E− (
such that αLDOS (E) ≤ inf φ∈H αNφ (E). Thus, αLDOS (E) ≤ αDOS (E). ± Let now Ω1 = {ω ∈ Ω|α± ρω (∆) = αLDOS (∆)}. If g is the continuous application ω ∈ Ω 7→ ρω ∈ M (by Lemma 7) and h the Borel function ρω ∈ M 7→ α± ρω (∆) (by (∆)}) is a Borel set. Moreover Ω1 is Proposition 1), then Ω1 = g −1 (h−1 ({α± LDOS T -invariant and has full P measure. Now let Ia = {(ω, E) ∈ Ω × R|αρT a ω (E) = αρω (E)} for a ∈ Zd . If k is the Borel function (ω, E) ∈ Ω × R 7→ αρT a ω (E) − αρω (E) (by Lemma 7 and Theorem 2(iv)), T then Ia = k −1 ({0}) shows that Ia is a Borel set. Hence I = a∈Zd Ia is also a Borel set. Because ρω and ρT a ω are in the same measure class, Theorem 2 gives ρω ({E ∈ R|(ω, E) ∈ Ia }) = 1 and by σ-additivity, ρω ({E ∈ R|(ω, E) ∈ I}) = 1. + Finally, let ∆ = {(ω, E) ∈ Ω × ∆|α− LDOS (∆) ≤ αρω (E) ≤ αLDOS (∆)}. By Lemma 7 and Theorem 2(iv), ∆ is a Borel set. It also satisfies ρω ({E ∈ ∆|(ω, E) ∈ ∆}) = ρω (∆) for P-almost all ω ∈ Ω. ˆ = I ∩ (Ω1 × R) ∩ ∆. It is a Borel set and ρω ({E ∈ ∆|(ω, E) ∈ Now we set ∆ ˆ ˆ then (T a ω, E) ∈ ∆ ˆ for all ∆}) = ρω (∆) for P-almost all ω ∈ Ω. If (ω, E) ∈ ∆, d ˆ ˆ a ∈ Z . If χ∆ ˆ is the indicator function of ∆, then the definition of ∆ and Fubini’s theorem give Z Z dP(ω)
dρω (E) χ∆ ˆ (ω, E) = N (∆) .
On the other hand, the invariance of P, ρω,|ni = ρT a ω,|n−ai and Fubini’s theorem imply that Z Z dP(ω) dρω (E) χ∆ ˆ (ω, E) =
X n∈N
Z cn
dP(ω)
! Z 1 X dρω,|n−mi (E) χ∆ ˆ (ω, E) . |Λ| m∈Λ
for any Λ ⊂ Zd . By Birkhoff’s theorem, in the limit of increasing rectangles centered at the origin Λ → Zd , the term in the parenthesis converges to N , P-almost surely.
31
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
ˆ Therefore, if we introduce the T -invariant Borel set ΩE = {ω ∈ Ω|(ω, E) ∈ ∆}, Fubini’s theorem gives Z N (∆) =
dN (E) P(ΩE ) . ∆
Hence P(ΩE ) = 1 for N -almost all E ∈ ∆. Consequently, because + ΩE = {ω ∈ Ω|α− LDOS (∆) ≤ αρω (E) = αρT a ω (E) ≤ αLDOS (∆)
∀ a ∈ Zd }
+ ˆ α− by definition of ∆, LDOS (∆) ≤ αLDOS (E) ≤ αLDOS (∆) for N -almost all E ∈ ∆. − − Because αLDOS (E) ≤ αDOS (E), αLDOS (∆) ≤ αDOS (∆) follows. + In order to show α+ LDOS (∆) ≤ αDOS (∆), it is now sufficient to show that
N −esssup αLDOS (E) = α+ LDOS (∆) . E∈∆
For this purpose, fix δ > 0 and introduce + Ξ = I ∩ (Ω1 × R) ∩ {(ω, E) ∈ Ω × ∆|α+ LDOS (∆) − δ ≤ αρω (E) ≤ αLDOS (∆)} .
˜ E = {ω ∈ Then ρω ({E ∈ R|(ω, E) ∈ Ξ}) > 0, P-almost surely. Now introduce Ω Ω|(ω, E) ∈ Ξ}. Repeating the same arguments as above, there exists a set ΞN of ˜ E has full P-measure, that positive N -measure such that the T -invariant Borel set Ω is for all E ∈ ΞN , + α+ LDOS (∆) − δ ≤ αLDOS (E) ≤ αLDOS (∆) .
As δ > 0 is arbitrary, this finishes the proof.
Remark 14. Usually one expects averaged exponents to be smaller than exponents obtained by taking an essential infimum over disorder configurations. Actually, it is easy to check that αDOS (E) ≤ P−essinf ω∈Ω αρω,|0i (E). On the other hand, it is possible that the inequality − P−essinf α− ρω,|0i (∆) < αDOS (∆) ω∈Ω
be realized. This is at the basis of Theorem 5. To understand the difficulty, let us consider a Hamiltonian with dense pure-point spectrum for P-almost all ω ∈ Ω. Therefore P−essinf ω∈Ω α− ρω,|0i (∆) = 0. It is however well known that a given E is P-almost surely not in the spectrum so that αDOS (E) may be strictly bigger than 0. The same may then hold for α− DOS (∆). As the exponents of the LDOS and the diffusion exponent, the exponents of the DOS do not depend on a given vector in Hilbert space as suggests the definition (12). For φ ∈ H, Nφ is defined by Z Z dNφ (E) f (E) = dP(ω)hφ|πω (f (H))|φi , f ∈ C0 (R) . The DOS is then N = N|0i .
32
H. SCHULZ-BALDES and J. BELLISSARD
Proposition 8. For any φ ∈ H, the measure Nφ is dominated by N . If E ∈ R and ∆ ⊂ R a Borel set, then αDOS (E) = inf αNφ (E) , φ∈H
+ α+ DOS (∆) = sup αNφ (∆) , φ∈H
− α− DOS (∆) = inf αNφ (∆) . φ∈H
Proof. By Theorem 2 and Corollary 1, it is sufficient to show that N dominates P Nφ for any φ ∈ H. Let ∆ ⊂ R be such that N (∆) = 0. If φ = n∈Zd φn |ni ∈ H, P then A = n φn U (n) ∈ L2 (A, T ) and Z dNφ (E) f (E) = T (f (H)AA∗ ) = hA|f (H)|AiL2 (A,T ) . Let An ∈ A be a Cauchy sequence in L2 (A, T ) converging to A such that An be a trigonometric polynomial. Then hAn |f (H)|An i converges to hA|f (H)|Ai for any bounded f (H). Since An is a trigonometric polynomial, hAn |χ∆ (H)|An i = 0, where χ∆ is the characteristic function of the Borel set ∆. Hence hA|χ∆ (H)|Ai = 0, that is Nφ (∆) = 0. 4.2. Diffusion exponents of covariant Hamiltonians Let us first generalize Definition 5. 2 (T ) be the mean square displacement operator defined Definition 9. Let δXω,∆ in (13). For φ ∈ H, Z ∼ 2 dP(ω) hφ|δXω,∆ (T )|φi T ↑∞ T 2σφ (∆) Ω
and 2 (T )|φi hφ|δXω,∆
∼
T ↑∞
T 2σω,φ (∆)
define the diffusion exponents σφ (∆) and σω,φ (∆). We set σ ˆφ (∆) = P−esssupω∈Ω σω,φ (∆) and σdiff (∆) = σ|0i (∆), where |0i is the state localized at the origin. Remark 15. Clearly σ ˆφ (∆) ≤ σφ (∆). A strict inequality may be possible if 2 (T )|φi has large fluctuations in ω. hφ|δXω,∆ Proposition 9. (i) For any φ ∈ `1 (Zd ), σφ (∆) ≤ σdiff (∆). (ii) If φ ∈ H satisfies hφ|Πω (∆)X 2 Πω (∆)|φi < ∞, then its diffusion exponent ~ ω (t) − X) ~ 2 in σω,φ (∆) can also be calculated by replacing the operator (X 2 ~ (13) by Xω (t). The rest of this section is devoted to proofs.
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
33
Proof of Proposition 2. By DuHamel’s formula, Z t −ıHs ~ ~ −ıHt ) = −ı ds e−ıH(s−t) ∇(H)e , ∇(e 0
where the integral is defined as a norm-convergent Riemann sum. Because H ∈ ~ −ıHt ) is an element of A and hence T (Π(∆)|∇(e ~ −ıHt )|2 ) is well defined. C 1 (A), ∇(e Using the definition of the gradient, the cyclicity of the trace T and [Π(∆), e−ıHt ] = 0, one gets ~ e−ıHt ] · [eıHt , X]e ~ −ıHt ) ~ −ıHt )|2 ) = T (Π(∆)eıHt [X, T (Π(∆)|∇(e ~ ~ 2 Π(∆)) . = T (Π(∆)(X(t) − X) By definition of the trace T on L∞ (A, T ), Z Z T dt 2 ~ ~ 2 Π(∆)) = T (Π(∆)(X(t) − X) dP(ω) h0|δXω,∆ (T )|0i . T Ω 0
This finishes the proof.
Proof of Theorem 7(i) and (ii). (i) σdiff (∆) is clearly bigger than or equal to 0. Because H ∈ C 1(A), DuHamel’s formula implies Z t ıHs ~ ~ ıHt )k ≤ dskeıH(t−s) ∇He k k∇(e 0
~ ≤ tk∇(H)k , so that Z 0
T
dt ~ −ıHt )|2 ) ≤ T (Π(∆)|∇(e T ≤
Z 0
T
dt ~ ıHt 2 k∇(e )k T
1 ~ k∇(H)k2 T 2 . 3
Hence the exponent σdiff (∆) is less than or equal to 1. ˆ ˆ ˆ (ii) Let us use the algebra AH, ˆ V ˆ common to H and H + V introduced in the appendix. Then both Hamiltonians H and (H + V ) are elements of AH, ˆ V ˆ ~ ˆ and hence so is V = (H + V ) − H. Because [X, V ] is bounded, V ∈ ˆ + Vˆ is well defined by C 1 (A ˆ ˆ ). Therefore the diffusion exponent of H H,V
(i). Because it is defined by means of the invariant ergodic measure P, Theorem 1 implies the result. Proof of Proposition 9. P P (i) Let φ = n∈Zd φn |ni with n∈Zd |φn | < ∞. By the Schwarz inequality Z Z 2 2 dP(ω)hφ|δXω,∆ (T )|φi ≤ dP(ω)h0|δXω,∆ (T )|0ikφk2`1 (Zd ) , Ω
Ω
34
H. SCHULZ-BALDES and J. BELLISSARD
where we have used ~ = U (a)(X ~ T a ω (t) − X)U ~ (a)∗ ~ ω (t) − X X
(46)
and the invariance of the measure P. Hence σdiff (∆) ≥ σφ (∆). (ii) Let |ψi = Πω (∆)|φi and γ > 2σφ (∆) ≥ 0, then by Schwarz’ inequality 2 s sZ Z ∞ Z T ∞ dT dt dT ~ ω (t)2 |ψi − ~ 2 |ψi < ∞ . hψ|X hψ|X T 1+γ 0 T T 1+γ 1 1 ~ 2 (t) is necessarily smaller Therefore the exponent defined by using only X ω than or equal to σφ (∆). A similar argument shows the converse inequality, such that the exponents coincide. Proof of Theorem 9. Using DuHamel’s formula and basic properties of projections, one gets Z T Z Z T 2 − 2 cos((E − E 0 )t) dt dt ~ −ıHt )|2 Π(∆)) = T (|∇(e dm(E, E 0 ) . (E − E 0 )2 ∆×R 0 T 0 T (47) Let now γ ∈ R be such that 2 − γ > 2σdiff (∆). Fubini’s theorem then leads to Z Z T Z ∞ 0 dT dt 0 2 − 2 cos((E − E )t) dm(E, E ) (E − E 0 )2 T 1+(2−γ) 0 T ∆×R 1 Z Z ∞ ds s − sin s =2 dm(E, E 0 ) |E − E 0 |−γ . 1−γ s3 ∆×R |E−E 0 | s The integral over s is bounded for γ ∈ (0, 2). Therefore, for σdiff (∆) < 1, Z dm(E, E 0 ) |E − E 0 |−γ < ∞ . 2(1 − σdiff (∆)) = sup γ ∈ R|
(48)
∆×R
For σdiff (∆) = 1 this is immediately clear. The theorem now follows by direct calculation using Fubini’s theorem. Proof of Theorem 11. This follows from similar calculations as in the proof of Theorem 9 above. Proof of Theorem 12. Let us consider ıLH as a self-adjoint operator on ~ ∈ L2 (A, T ) is L2 (A, T ). Then its spectral measure ρJ~ associated to J~ = ∇H defined by (24) (the existence is guaranteed for by the Riesz–Markov theorem). It can be verified by direct calculation that for any polynomial f , Z Z dρJ~() f () = dm(E, E 0 ) f (E − E 0 ) . By density this extends to any f ∈ C0 (R) such that the measures coincide. Using this in (48) for the case ∆ = R now allows to conclude. The second statement follows from the symmetry (20).
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
35
Proof of Theorem 10. Fubini’s theorem and a contour integration in the upper half plane shows in the first place that Z Z 1 . da Sm (a + ı, a − ı) = dm(E, E 0 ) 2πı E − E 0 − 2ı R R2 The latter is the Green’s function GρJ~ (2ı) of the measure ρJ~. By Theorem 3, ∼ α (0)−1 . On the other hand, αρJ~ (0) = 2(1 − σdiff (R)) by Eq. 48) =m GρJ~ (2ı) ↓0 ρJ~ for ∆ = R. The result follows. Proof of Theorem 13. The integrand in (25) is clearly positive such that we may apply Fubini’s theorem. After a change of variables we obtain Z ∞ dτrel ˆ) 1+γ σβ,µ (τrel , ω τrel 1 γ−1 Z ∞ Z fβ,µ (E 0 ) − fβ,µ (E) E − E 0 sγ q2 . dm(E, E 0 ) ds 2 = 0 ~ ~ E≥E 0 E−E ~ s −1 0 E−E
The integral over s is bounded for −1 < γ < 1. For β < ∞, the only singularity in the integrand of m comes from the factor (E − E 0 )γ−1 . The result now follows from Theorem 9. 4.3. The Guarneri bound The proof of the following result goes back to Guarneri [32], with considerable refinements due to [48, 6]. We refer the reader to [48, 6] for a proof. ˆ be its spectral Theorem. Let H be a Hamiltonian on H = `2 (Zd ) and Π ∈ M family. Let ∆ be a Borel spectral subset and φ ∈ H satisfying hφ|Π(∆)X 2 Π(∆)|φi < ∞. If ρφ is the spectral measure of H associated to φ and σφ (∆) is defined by Definition 5 with Ω reduced to one point, then α+ ρφ (∆) ≤ d · σφ (∆) . Remark 16. The result can be generalized to the study of other moments of the position operator: Z T X η dt ∼ hφ|Π(∆) |X|η (t) Π(∆)|φi = |n|η CT (φ, n, ∆) T ↑∞ T η·σφ (∆) . 0 T d n∈Z
η = 2 corresponds to the situation above. The inequality then reads α+ ρφ (∆) ≤ η d · σφ (∆). Proof of Theorem 7(iii). Let k be the smallest integer bigger than d/2. Let ~ k . If k = 2, then the hypothesis ~ k ), the domain of the operator |X| φ ∈ D(|X| 2 ~ kπω (∇ H)k < ∞ and ~ 2 H)|φi + 2πω (∇H) ~ ~ ~ 2 |φi ~ 2 πω (H)|φi = πω (∇ · X|φi + πω (H)|X| |X|
36
H. SCHULZ-BALDES and J. BELLISSARD
~ 2 ) invariant. By functional calculus, imply that πω (H) leaves the domain D(|X| 2 ~ f (πω (H)) also leaves D(|X| ) invariant for any f ∈ C ∞ (R). Similar arguments treat the case of other k’s. We set D(ω, ∆) = span{f (πω (H))|ni|n ∈ Zd , f ∈ C ∞ (R), supp(f ) ⊂ ∆} , where supp(f ) denotes the support of f . As ∆ is an open interval, D(ω, ∆) is dense ~ k ) and in H(ω, ∆) = Πω (∆)H. Moreover, D(ω, ∆) ⊂ D(|X| kφk`1 (Zd ) ≤
X n
1 |n|2k
! 12
X
! 12 |n| |φn | 2k
2
n
shows that D(ω, ∆) ⊂ `1 (Zd ). Finally, let D0 be a countable subset of D(ω, ∆) still dense in H(ω, ∆). Now for any φ ∈ D0 , Guarneri’s inequality given above shows that α+ ρω,φ (∆) ≤ d · σω,φ (∆). Thus sup α+ ρω,φ (∆) ≤ d sup σω,φ (∆) . φ∈D 0
φ∈D 0
Because of Proposition 5, the left-hand side is equal to α+ Πω (∆) and by Theorem 5 to + αLDOS (∆). Recall that σω,φ (∆) is smaller than or equal to σφ (∆) P-almost surely. Because D0 is countable, there exists a set Ω1 ⊂ Ω of full P-measure, such that σω,φ (∆) ≤ σφ (∆) for all φ ∈ D0 and ω ∈ Ω1 . Therefore D0 ⊂ `1 (Zd ) gives α+ LDOS (∆) ≤ d sup σφ (∆) ≤ d φ∈D 0
sup
σφ (∆) .
φ∈`1 (Zd )
But by Proposition 9(ii) the right-hand side is equal to σdiff (∆).
Proof of Theorem 8. (i) and (iii) are already proved in [14]. (The definition of l2 (∆) chosen in [14] was slightly different from (17) if, however, (17) is satisfied, it can be easily shown to be equivalent to the condition in [14].) (ii) follows directly from the proof of of Theorem 7(ii). 5. Example: Anderson Model with Free Random Variables As can be seen in Eq. (22), one needs to know the 2-particle Green function G2 of a given model in order to calculate the corresponding conductivity measure. There are not many interesting models known in which G2 can be calculated exactly. For Bloch electrons one can write out an explicit formula for the conductivity measure and then determine the diffusion exponent to be equal to 1. On the other hand we already discussed the situation for models with localization in Sec. 4.3. A class of solvable and non-trivial models has been studied by Wegner [80] and Khorunzhy and Pastur [47]. More recently, Neu and Speicher [51] considered a generalization of these models which will be the starting point here. The Hamiltonian in these models is given by the usual Anderson Hamiltonian H = H0 + H1 , where X X t|r−s| |rihs| , H1 = vr |rihr| , (49) H0 = r6=s∈Zd
r∈Zd
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
37
acting on `2 (Zd ), but the on-site disorder potentials vr are now supposed to be identically distributed free random variables in the sense of Voiculescu [79] instead of independent random variables. Random variables X1 , X2 , . . . are free if E(P1 (Xr(1) )P2 (Xr(2) ) . . . Pm (Xr(m) )) = 0 for any set of polynomials Pk satisfying E(Pk (Xr(k) )) = 0, k = 1, . . . , m whenever r(k) 6= r(k + 1), k = 1, . . . , m − 1. We suppose that t|n| ≤ |n|−d−1− for some > 0, so that ∂j (H) is bounded for any j = 1, . . . , d. As explained in [79, 51], Wegner’s n-orbital model in the integrable limit n → ∞ [80], the Anderson model in coherent potential approximation (CPA) and formally also the Lloyd model [49] are special cases of the Anderson model with free random variables. Strictly speaking this model is not covariant in the sense of Secs. 2.1 and 2.2. The hull is here a non-commutative manifold. We have not studied the general formalism in detail hoping that the generalization to this case is indeed straightforward. As a preamble, let us introduce the main tool for the addition of free random variables, notably Voiculescu’s R-transform [79]. Given a Herglotz function G(z) (namely, G(z) satisfies =m(G(z))=m(z) < 0), its R-transform is defined implicitly by the formula 1 . G(z) = z − R(G(z)) R(G(z)) has an analytic continuation to the upper half plane [51]. Examples: (i) Wigner’s semicircle law is given by the probability measure η (θ ∈ R): dη(E) =
1 p 2 4θ − E 2 χ(E 2 ≤ 4θ2 ) dE . 2πθ2
Its Green’s function and R-transform transform can be calculated explicitly: √ z − z 2 − 4θ2 , R(z) = θ2 z . G(z) = 2θ2 Note that =m(R(z)) ≤ θ2 =m(z) for =m(z) < 0. (ii) Let η be the uniform distribution on [−1, 1]. Then its Green’s function and R-transform transform are G(z) =
z−1 1 log , 2 z+1
R(z) =
1
tanhz −
1 . z
If z = x − ıy, y > 0, then =m(R(z)) =
y sin(2y) − . cos(2y) − cosh(2x) x2 + y 2
Hence R is not a Herglotz function in this example. Let us first summarize the main results of [51]. The probability distribution of the vr ’s is denoted by η. The space of disorder configurations Ω is a noncommutative measure space furnished with an expectation E which can be
38
H. SCHULZ-BALDES and J. BELLISSARD
calculated by free convolution technics [79]. Transposing the notations of Secs. 2.1 and 2.2 to the non-commutative hull Ω, let πω (H) be the representation of the Hamiltonian corresponding to an element ω ∈ Ω. The following Green’s functions are needed (=m z > 0): Z 1 1 |0i, G1 (z) = dη(vr ) , G0 (z) = h0| z − H0 z − vr R and G is the diagonal Green’s function of the full Hamiltonian H, that is, of the DOS. Voiculescu’s R-transforms [79] of these functions are denoted by R0 and R1 . Finally, the non-diagonal Green’s function G(r − s, z) and its Fourier transform ˜ z) are G(q, 1 |si G(r − s, z) = Eω hr| z − πω (H) Z dd q ıq·(r−s) ˜ G(q, z) , (50) e = d B (2π) where B = [−π, π)d is the Brillouin zone. Let further E0 (q) be the energy dispersion relation of H0 , that is the Fourier transform as in (50) of the function t determining the kinetic Hamiltonian H0 . Theorem (Neu and Speicher [51]). Consider the Anderson model with free random variables described by the Hamiltonian (49) and suppose the support of the measure η of the vr ’s to be compact. Then Green’s function satisfies the following equations (51) G(z) = G0 (z − R1 (G(z)) = G1 (z − R0 (G(z)) , and ˜ z) = G(q,
1 . z − E0 (q) − R1 (G(z))
(52)
Moreover, the 2-particle Green’s function defined in (23) satisfies the identity G2 (r, s, s0 , r0 , z1 , z2 ) = G(r − s, z1 )G(s0 , r0 , z2 ) + ·
X
R1 (G(z1 )) − R1 (G(z2 )) G(z1 ) − G(z2 )
G(r, t, z1 ) G2 (t, s, s0 , t, z1 , z2 ) G(t, r0 , z2 ) .
(53)
t∈Zd
Finally, the solution of the usual Anderson model in CPA is also given by (51) and (53). These results allow to calculate the diffusion exponent. Theorem 16. The diffusion exponent σdiff = σdiff (R) of the Anderson model with free random variables with compactly supported, absolutely continuous distribution is bigger than or equal to 1/2. The DOS is absolutely continuous. If moreover, =m(R1 (z)) ≤ C=m(z) for =m(z) < 0 and C ∈ R+ , then the diffusion exponent is
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
39
equal to 1/2. In particular, the diffusion exponent of the Wegner n-orbital model in the limit n → ∞ is equal to 1/2. Proof of Theorem 16. According to a theorem of Voiculescu [78], a measure obtained by free convolution of an absolutely continuous measure with any other measure is absolutely continuous. The DOS is the free convolution of the spectral measures of H0 and H1 (see [51]), such that it is absolutely continuous because H0 is so. Let us now calculate the Stieltjes transform Sm (z1 , z2 ) of the conductivity measure of the free Anderson model. Because the kinetic part in (49) is symmetric, the ~ 0 )|si only depend on r − s and therefore comparing with matrix elements hr|∇(H Eq. (22) shows that one actually needs to calculate the function G 2 (r, s, z1 , z2 ) =
X
G2 (r, t − s, t, 0, z1 , z2 ) .
t∈Zd
With the notations G 1 (r, z1 , z2 ) =
X
G(r − t, z1 ) G(t, z2 ) ,
R(z1 , z2 ) =
t∈Zd
R1 (G(z1 )) − R1 (G(z2 )) , G(z1 ) − G(z2 )
and using the fact that G2 (r, s, s0 , r0 , z1 , z2 ) only depends on three of its first four entries because the vr are identically distributed, summation of Eq. (53) leads to G 2 (r, s, z1 , z2 ) = G 1 (r + s, z1 , z2 ) + R(z1 , z2 ) G 2 (0, s, z1 , z2 ) G 1 (r, z1 , z2 ) . Putting r = 0 and solving for G 2 (0, s, z1 , z2 ) one gets G 2 (r, s, z1 , z2 ) = G 1 (r + s, z1 , z2 ) +
R(z1 , z2 ) G 1 (r, z1 , z2 ) G 1 (−s, z1 , z2 ) . 1 − R(z1 , z2 ) G 1 (0, z1 , z2 )
(54)
Remark that the quantity G 2 only depends on the one-particle Green function ˜ z) (cf. Eq. (52)) and in this model. Because E0 (q) is an even function, so is G(q, then Z dd q ıq·r ˜ 1 ˜ z2 ) , e G(q, z1 ) G(q, G (r, z1 , z2 ) = d B (2π) implies that G 1 (r, z1 , z2 ) is also an even function in its first variable. Now the ~ 0 )|si change sign as the sign of s is changed, and therematrix elements h0|∇(H fore the second term in (54) gives no contribution to the Stieltjes transform of ~ 0) the conductivity measure in (22). Passing to Fourier space the operator ∇(H becomes multiplication by the gradient of E0 (q) with respect to quasi-moments and subsequent summation in (22) leads to the result Sm (z1 , z2 ) Z 1 dd q ~ 1 . (55) |∇q E0 (q)|2 = d z1 − E0 (q) − R1 (G(z1 )) z2 − E0 (q) − R1 (G(z2 )) B (2π)
40
H. SCHULZ-BALDES and J. BELLISSARD
For Wegner’s n-orbital model, this equation has already been derived by Khorunzhy and Pastur [47]. Owing to Theorem 10 and Eq. (55), it is now necessary to consider the integral Z Z dd q ~ 1 da |∇q E0 (q)|2 , d (2π) |a + ı − E (q) − R1 (G(a + ı))|2 0 R B and to study its behavior in the limit → 0. Clearly the integral is strictly bigger than zero because of contributions for big |a| and therefore the diffusion exponent is bigger than or equal to 1/2 according to Theorem 10. In order to show that the above integral is bounded let us use its upper bound Z Z dd q 1 da d |a + ı − E (q) − R (G(a + ı))|2 (2π) 0 1 R B Z Z 1 = da dN0 (E) 2 |a + ı − E − R 1 (G(a + ı))| R R Z =m G(a + ı) , (56) = da =m(R 1 (G(a + ı))) − R where the last equality follows by direct calculation using the identity (51). The integrand in (56) is bounded by the hypothesis made on R1 and it falls off as 1/a2 at infinity. Therefore the integral is bounded and the diffusion exponent is equal to 1/2. In Wegner’s n-orbital model the vr are given by n×n hermitian random matrices with independent Gaussian entries. The model then becomes exactly solvable in the limit n → ∞. In fact, Voiculescu has shown that Gaussian random matrices are asymptotically free such that the vr in (49) are free in the limit n → ∞. The distribution of the vr is then given by Wigner’s semicircle law and thus the hypothesis on R1 is satisfied (cf. the above example). Remark 17. Let us investigate the Lloyd model. The vr then follow the Cauchy distribution with parameter γ > 0. Because all moments diverge, the calculations in [51] for the 2-point Green’s function are only formal. Actually, the formal calculations lead to erroneous results as the following arguments show. One has G1 (z) = 1/(z + ıγ sign(=mz)) and this implies that R1 (z) = ıγ sign(=mz). Use this for the calculation of Sm in (55) which is supposed to hold. Because the Green’s function is a Herglotz function, a contours integration leads to Z Z −1 1 dd q ~ da Sm (a + ı, a − ı) = |∇q E0 (q)|2 . <e 4π + γ B (2π)d R According to Theorem 10 the diffusion exponent would therefore be equal to 1/2. This would be independent of the dimension of the physical space and the strength of the coupling. In particular, it would imply regular diffusion in the onedimensional Lloyd model. But this is in contradiction with the rigorous results of Aizenman and Molchanov [2] which imply that the diffusion exponent is equal to zero in this situation as is shown in [14].
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
41
It is possible, however, to calculate the exponent of the DOS of the Lloyd model using Eq. (51) and Theorem 3. These are then shown to be equal to 1 which is a well known result [23]. In fact, it is shown in [16] that the calculations in [51] are justified for the 1-point Green’s function. Remark 18. Formula (55) also holds for the case where vr ’s are all equal to 0. Then R1 (z) = 0. Therefore a contour integration leads to Z Z 1 dd q ~ 1 ∼ −1 da Sm (a + ı, a − ı) = |∇q E0 (q)|2 ↓0 . <e d 2πı B (2π) 2ı R Comparison with Theorem 10 shows that σdiff (R) = 1 as expected for free particles. Remark 19. The two other integrable models considered by Khorunzhy and Pastur [47] have a conductivity measure which is absolutely continuous with respect to the Lebesgue measure on R2 [47, Eq. (2.29)]. For such an absolutely continuous conductivity measure, Theorem 9 implies directly a diffusion exponent 1/2. 6. Appendix Let us consider a topological dynamical system (Ω, T, Zd ), where Ω is a compact metrisable space and T is an action of Zd by homeomorphisms. A point ω ∈ Ω is called wandering whenever there is an open neighborhood U such that T a (U)∩U = ∅ for any a ∈ Zd∗ = Zd \{0}. The set Ωw of all wandering points is open. Hence its complement Ωnw , the set of all non-wandering points, is compact. Proposition 10. Any T -invariant probability P is supported on Ωnw . Proof. Let ω ∈ Ωw and U be a neighborhood satisfying T a (U) ∩ U = ∅ for any a ∈ Zd∗ . Then T a (U) ∩ T b (U) = ∅ for any a 6= b ∈ Zd . The σ-additivity and T -invariance of P imply X [ T a (U) = P(U) , P(Ω) ≥ P a∈Zd
a∈Zd
so that P(U) = 0. Therefore the support of P does not contain Ωw .
Let us now consider the non-wandering points of the hull (ΩHˆ , T, Zd ) associated ˆ ∈ Ω by ω0 . ˆ on H = `2 (Zd ) by (5). We denote H to a homogeneous Hamiltonian H a d The orbit of ω0 is the set Orb(ω0 ) = {T ω0 |a ∈ Z }. Lemma 8. Let us introduce the set d an ω0 = ω } . Ω∞ ˆ = {ω ∈ Ω|∃ sequence (an )n∈N in Z , lim |an | = ∞, s.t. lim T H n→∞
n→∞
w ∞ w If ω0 is not an element of Ω∞ ˆ , then ΩH ˆ = Orb(ω0 ). If ω0 ∈ ΩH ˆ , then ΩH ˆ = ∅. In H nw ∞ both cases, ΩHˆ = ΩHˆ .
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H. SCHULZ-BALDES and J. BELLISSARD
Proof. Let ω belong to Ω. Then there exists a sequence (an )n∈N , an ∈ Zd , such that limn→∞ T an ω0 = ω. Using the one-point compactification Zd ∪ {∞} of Zd , one can extract a convergent subsequence (ank )k∈N with limk→∞ T ank ω0 = ω. If limk→∞ ank = ∞, then ω ∈ Ω∞ ˆ , otherwise (ank ) is a stationary sequence on some H d a a ∈ Z and ω = T ω0 ∈ Orb(ω0 ). Hence Ω = Ω∞ ˆ ∪ Orb(ω0 ), but this decompostion H is not necessarily disjoint. nw ∞ We next show that Ω∞ ˆ ⊂ ΩH ˆ . For ω ∈ ΩH ˆ there exists an unbounded sequence H d an (an )n∈N in Z such that limn→∞ T ω0 = ω. Hence for any neighborhood U of ω there is a N ∈ N such that {T an ω0 |n ≥ N } ⊂ U. Choose an 6= am , n, m ≥ N . 6 ∅. As this Then T an ω0 ∈ U and T am −an (T an ω0 ) ∈ U so that T am −an (U) ∩ U = . holds for any neighborhood of ω, ω ∈ Ωnw ˆ H nw or ω ∈ Ω . In the first case, Orb(ω0 ) ⊂ Ωw Now either ω0 ∈ Ωw 0 ˆ ˆ ˆ . According H H H w nw . To deal with the second case, let to the above, Orb(ω0 ) = ΩHˆ and ΩHˆ = Ω∞ ˆ H us choose a metric on Ω compatible with the topology of Ω. Open balls of radius r around ω ∈ Ω are denoted by B(ω, r). Because ω0 ∈ Ωnw ˆ , there exists for any H 1 1 d ak k ∈ N an ak ∈ Z∗ such that B(ω0 , k ) ∩ T B(ω0 , k ) 6= ∅. Therefore the sequence ∞ T ak ω0 converges to ω0 . As ak 6= 0 for all k, ω ∈ Ω∞ ˆ . Hence Orb(ω0 ) ⊂ ΩH ˆ and H nw ∞ ΩHˆ = ΩHˆ = Ω. Proof of Theorem 1. Let Vˆ = Vˆ ∗ be a compact operator of H = `2 (Zd ). The ˆ and the perturbed Hamiltonian H ˆ + Vˆ are denoted by hulls of the Hamiltonian H ∗ ˆ ˆ ΩHˆ and ΩH+ ˆ V ˆ . V compact implies that s-limn→∞ U (an )V U (an ) = 0 for an unnw nw bounded sequence |an | → ∞. Hence Lemma 8 implies ΩHˆ = ΩH+ ˆ Vˆ . Proposition 10 and Lemma 8 show the other claims. ˆ and For a compact perturbation Vˆ , let us introduce the common hull of H ˆ + Vˆ by H s
−1 , U (a)H ˆ ˆ + Vˆ U (a)−1 )|a ∈ Zd } , ΩH, ˆ V ˆ = {(U (a)HU (a)
where the closure is taken with respect to the strong topology on the Cartesian product B(H) × B(H). By the arguments of the proof above, Ωnw ˆ V ˆ = {(ω, ω)|ω ∈ H,
d nw d Hence (Ωnw Ωnw ˆ }. ˆ V ˆ , T, Z ) and (ΩH ˆ , T, Z ) are homeomorhpic as dynamical H H, systems. Denoting by pHˆ (pH+ ˆ V ˆ ) the projection from ΩH, ˆ V ˆ to its first (second) component, we have the following picture:
ΩHˆ ∪ Ωnw ˆ H
←−
pˆ H
←→ pˆ H
ΩH, ˆ ˆ V ∪ Ωnw ˆ V ˆ H,
−→
pˆ ˆ H+V
←→
pˆ ˆ H+V
ΩH+ ˆ , ˆ V ∪ Ωnw ˆ V ˆ . H+
To conclude, let us consider the crossed product C ∗ -algebras AHˆ , AH+ ˆ and ˆ V AH, ˆ constructed from the dynamical systems on ΩH ˆ and ΩH, ˆ by the ˆ V ˆ , ΩH+ ˆ V ˆ V procedure of Sec. 2.1. There is an injection iHˆ : AHˆ ,→ AH, ˆ V ˆ which sends f ∈
ANOMALOUS TRANSPORT: A MATHEMATICAL FRAMEWORK
43
d C0 (ΩHˆ × Zd ) to a function iHˆ (f ) ∈ C0 (ΩH, ˆ × Z ) independent of the second ˆ V argument in ΩH, ˆ . Similarly one has iH+ ˆ : AH+ ˆ . On the other hand, ˆ V ˆ V ˆ Vˆ ,→ AH, ˆ V d nw d , T, Z ) and (Ω , T, Z ) are homeomorphic, because the dynamical systems (Ωnw ˆ V ˆ ˆ H, H nw they only give rise to one crossed product AHˆ . Let us introduce the ideal nw nw JH ˆ |πω (A) = 0 ∀ ω ∈ ΩH ˆ = {A ∈ AH ˆ }, nw nw nw and similarly JH+ ˆ or AH, ˆ) ˆ (or AH+ ˆ V ˆ V ˆ Vˆ and JH, ˆ V ˆ . Then AH ˆ is the quotient of AH nw nw nw (or J or J ). Hence by using the surjective applications with respect to JH ˆ ˆ Vˆ ˆ V ˆ H+ H, on the quotient, one has
AHˆ
−→
←−
AH, ˆ iH+ ˆ V ˆ Vˆ ˆ V ˆ AH+ & ↓ . Anw ˆ H iˆ H
Let now a trace T be defined on AHˆ (or AH+ ˆ V ˆ or AH, ˆ V ˆ ) by (8). As T vanishes nw nw nw (or J or J ) by Proposition 10, it is well defined on Anw on the ideals JH ˆ ˆ V ˆ ˆ V ˆ ˆ . H+ H, H References [1] E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, “Scaling theory of localization: Absence of quantum diffusion in two dimensions”, Phys. Rev. Lett. 42 (1979) 673–676. [2] M. Aizenman and S. Molchanov, “Localization at large disorder and at extreme energies: An elementary derivation”, Commun. Math. Phys. 157 (1993) 245–278; and M. Aizenman, “Localization at weak disorder: Some elementary bounds”, Rev. Math. Phys. 6 (1994) 1163–1182. [3] R. Artuso, G. Casati and D. Shepelyansky, “Fractal spectrum and anomalous diffusion in the kicked Harper model”, Phys. Rev. Lett. 68 (1992) 3826–3829. [4] J. Avron and B. Simon, “Almost periodic Schr¨ odinger operators II. The integrated density of states”, Duke Math. J. 50 (1983) 369–391. [5] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders Co, Philadelphia, 1976. [6] J.-M. Barbaroux, J. M. Combes and R. Montcho, “Remarks on the relation between quantum dynamics and fractal spectra”, to appear J. Ana. Appl. [7] J. Bellissard, P. Moussa and D. Bessis, “Chaotic states of almost periodic Schr¨ odinger operators”, Phys. Rev. Lett. 49 (1982) 701–704. [8] J. Bellissard, “K-theory of C∗ -algebras in solid state physics, in Statistical Mechanics and Field Theory: Mathematical Aspects”, Lecture Notes in Phys. eds. T. Dorlas, M. Hugenholtz, M. Winnink, Springer-Verlag, Berlin, 257 (1986) 99–156. [9] J. Bellissard, B. Iochum, E. Scoppola and D. Testard, “Spectral properties of onedimensional quasicrystals”, Commun. Math. Phys. 125 (1989) 527–543. [10] J. Bellissard, “Spectral properties of Schr¨ odinger operator with a Thue–Morse potential”, in Number Theory and Physics, eds. J. M. Luck, P. Moussa and M. Waldschmidt, Springer Proc. in Physics, Springer-Verlag, Berlin, 47 (1990) 140–150. [11] J. Bellissard, A. Bovier and J. M. Ghez, “Spectral Properties of a tight binding Hamiltonian with period doubling potential”, Commun. Math. Phys. 135 (1991) 379–399. [12] J. Bellissard, “Gap labelling Theorems for Schr¨ odinger operators”, in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J. Luck, C. Itzykson, SpringerVerlag, Berlin, 1991, pp. 538–630.
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THEORY OF CONNECTIONS ON GRADED PRINCIPAL BUNDLES T. STAVRACOU Centre de Physique Th´ eorique-CNRS Luminy, Case 907-F-13288 Marseille, Cedex 9, France E-mail:
[email protected] Received 26 December 1996 Revised 12 May 1997 1991 Mathematical Subject Classification: 58A50, 53C15 The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant–Berezin–Leites. We first review the basic elements of this theory establishing at the same time supplementary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the graded context. Next, we investigate the geometry of graded principal bundles; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of vertical derivations on a graded principal bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection and we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection. Keywords: graded manifold theory, actions of graded Lie groups, graded principal bundles, graded connections
1. Introduction Graded manifolds are geometrical objects introduced by Kostant [1], and Berezin, Leites [2, 3], as the natural mathematical tool in order to study supersymmetric problems. In particular, the formalism of graded manifolds provides the possibility to deal with classical dynamics of particles with spin. Indeed, it has been proved [4], that the extension of ordinary phase spaces by anti-commuting variables yields the classical phase spaces of particles with spin. Thus, Kostant’s article [1], can be also considered as a treatment on super-Hamiltonian systems and on their geometrical prequantization. Our aim here is to investigate several aspects of the graded differential geometry, unexplored up to today, and to use them in order to establish a theory of connections on graded principal bundles. This work is mainly motivated by the need to make clear why graded connection theory is convenient for the description of supersymmetric gauge theories, and how one could use this theory in practice, in order to obtain, for example, the supersymmetric standard model in the same spirit as one obtains the standard model of fundamental interactions from ordinary gauge 47 Reviews in Mathematical Physics, Vol. 10, No. 1 (1998) 47–79 c World Scientific Publishing Company
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theory. The notion of graded principal bundle has first appeared in [5, 6]: there, the notion of connection on such bundles has been also presented, but in a rather academic way, i.e. using jet bundles, bundles of connections and Atiyah’s exact sequence, conveniently generalized in this context. This point of view turned out to be particularly useful for mathematics, as it inspired and affected more recent developments in connection theory on far more sophisticated types of supermanifolds, see [7]. Our analysis focuses on the investigation of the geometry of graded principal bundles, as well as on the reformulation of the notion of graded connection in terms of graded distributions and Lie superalgebra-valued graded differential forms. We believe that our approach makes the theory of graded connections more accessible and close to applications in supersymmetric Yang–Mills theory, providing thus the possibility to this elegant and economical theory to be applied on specific problems of supersymmetric physics. Throughout the paper, we follow the original terminology of Kostant [1], and we avoid the use of the term “supermanifold” or “Lie supergroup”. This strict distinction between the terms “supermanifold” and “graded manifold” is justified if one wishes to avoid confusion with the DeWitt’s or Roger’s supermanifold theory [8–12]. In this latter approach, one constructs the supermanifold following the pattern of ordinary differential geometry (using supercharts, superdifferentiability, etc.). In this way, the supermanifold is a topological space, and in particular, a Banach manifold [8]. In the Kostant–Leites approach, the supermanifold is a pair which consists of a usual differentiable manifold equipped with a sheaf satisfying certain properties. The article is organized as follows. In Sec. 2, we recall the fundamentals of graded manifold theory from [1], which will be necessary for the subsequent analysis, introducing at the same time some new concepts and tools, useful in computing pullbacks of graded differential forms. The next section deals with graded Lie theory. After a review of the basics of the theory from [1, 5], we show how one can obtain a right action of a graded Lie group from a left one (and vice versa); we also explain how a graded Lie group action gives rise to graded manifold morphisms and derivations. A graded analog of the adjoint action of a Lie group on itself is analyzed, and the parallelizability theorem for graded Lie groups is proved. In Sec. 4 we study the main properties of Lie superalgebra-valued graded differential forms; the graded generalization of the Maurer–Cartan form is also discussed. In the next section we investigate the relation between graded distributions and graded Lie group actions. The free action in graded geometry is the central notion of this section for which we provide two equivalent characterizations. The main result is that each free action of a graded Lie group induces a regular graded distribution. Quotient graded structures are also examined. The notion and the geometry of graded principal bundle are analyzed in Secs. 6 and 7. Here, we adopt a definition of this object slightly different from that in [6], completing in some sense, the definition of the last reference. We prove that several properties of ordinary principal bundles remain valid in the graded context,
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making of course the appropriate modifications and generalizations. For example, we prove that the orbits of the structure graded Lie group are identical to the fibres of the projection and that a graded principal bundle is globally trivial, if and only if it admits a global graded section. As an application of the tools developed until now, we provide an alternative definition of the graded principal bundle, often very useful as we explain in Sec. 7. We define graded connections in Sec. 8, guided by the geometrical structure of the graded principal bundle: the sheaf of vertical derivations coincides with the graded distribution induced by the action of the structure graded Lie group. We discuss two equivalent definitions of the graded connection and show how one can construct a graded connection locally. As a direct application, we establish the existence theorem for graded connections. The graded curvature is the subject of Sec. 9. We prove the structure equation and the Bianchi identity for the curvature of a graded connection. We show that this notion controls the involutivity of the horizontal graded distribution determined by a graded connection. Finally, in the last section we comment some fundamental properties of the graded connection and its curvature, showing clearly the relation of these objects with super Yang–Mills theory. Notational conventions. For an algebra A, A∗ denotes its full dual and A◦ its finite dual (in contrast to Kostant’s conventions where A0 denotes the full, and A∗ the finite dual). If A is a sheaf of algebras over a differentiable manifold M , then 1A denotes the unit of A(M ) and mA the algebra multiplication. Throughout this article the term “graded commutative” means “Z2 -graded commutative”, unless otherwise stated. If E is a Z2 -graded vector space, then E0 and E1 stand for its even and odd subspaces: E = E0 ⊕ E1 . For an element v ∈ Ei , i = 0, 1, |v| = i denotes the Z2 -degree of v. Elements belonging only to E0 or E1 are called homogeneous (even or odd, respectively). 2. Graded Manifold Theory In this section, we review the basic notions of graded manifold theory [1]. We also introduce some new concepts following the pattern of ordinary differential geometry as well as tools that will be useful in the sequel. More precisely, the notions of vertical and projectable derivations are introduced; push-forward (and pull-back) of derivations under isomorphisms of graded manifolds is defined and some useful formulas for the computation of pull-backs of forms are established. We begin with the notion of the graded manifold [1]. A ringed space (M, A) is called a graded manifold of dimension (m, n) if M is a differentiable manifold of dimension m, A is a sheaf of graded commutative algebras, there exist a sheaf epimorphism % : A → C ∞ and for each open U , an open covering {Vi } such that A(Vi ) is isomorphic as a graded commutative algebra to C ∞ (Vi ) ⊗ ΛRn , in a manner compatible with the restriction morphisms of the sheaves involved. Here, C ∞ stands for the sheaf of differentiable functions on M , equipped with its trivial grading: C ∞ (U ) 0 = C ∞ (U ) for each open subset U ⊂ M . We note dim(M, A) = (m, n).
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An open U ⊂ M , for which A(U ) ∼ = C ∞ (U ) ⊗ ΛRn , is called an A-splitting neighborhood. A graded coordinate system on an A-splitting neighborhood U is a collection (xi , sj ) = (x1 , . . . , xm , s1 , . . . , sn ) of homogeneous elements of A(U ) with x1 , . . . , x˜m ) is an ordinary coordinate system on the |xi | = 0, |sj | = 1, such that (˜ i i open U , where x ˜ = %(x ) ∈ C ∞ (U ) and (s1 , . . . , sn ) are algebraically independent Qn elements, that is, j=1 sj 6= 0. We will say that U is an A-coordinate neighborhood if U is an A-splitting neighborhood admitting a graded coordinate system. It can also be shown [1], that if A1 (U ) is the ideal of nilpotent elements of A(U ), then the sequence 0 −→ A1 (U ) −→ A(U ) −→ C ∞ (U ) −→ 0
(2.1)
is exact, for each open subset U ⊆ M . Given two graded manifolds (X, A) and (Y, B), one can form their product (Z, C) = (X, A)×(Y, B) which is also a graded manifold [1]; the underlying manifold Z is simply X × Y and A(X) ⊗ B(Y ) is a dense subspace of C(Z). We can thus regard C(Z) as a completion of A(X) ⊗ B(Y ) and for this reason we will note C(Z) = A(X) ⊗ B(Y ). Let us recall here that the tensor product A(X) ⊗ B(Y ) is in the graded sense: the Z2 -grading of A(X) ⊗ B(Y ) is given by [A(X) ⊗ B(Y )]n = L i+j=nmod2 A(X)i ⊗ B(Y )j , i, j ∈ {0, 1}, whereas the product is defined by (a ⊗ b)(c ⊗ d) = (−1)k` ac ⊗ bd for a ∈ A(X), b ∈ B(Y )k , c ∈ A(X)` , d ∈ B(Y ) (this definition implying here the graded commutativity of A(X) ⊗ B(Y )). One can define a morphism between two graded manifolds as being a morphism of ringed spaces compatible with the sheaf epimorphism % [13], but often it is more convenient to do this in a more concise way: a morphism σ : (M, A) → (N, B) between two graded manifolds is just a morphism σ ∗ : B(N ) → A(M ) of graded commutative algebras. A very useful object in graded manifold theory is the finite dual A(M )◦ of A(M ) defined as A(M )◦ = {a ∈ A(M )∗ |a vanishes on an ideal of A(M ) of finite codimension} . Using general algebraic techniques (see for example [14]), one readily verifies that A(M )◦ is a graded cocommutative coalgebra, the coproduct ∆◦A and counit ◦A on A(M )◦ being given by ∆◦A a(f ⊗ g) = a(f g), ◦A (a) = a(1A ), ∀a ∈ A(M )◦ , f, g ∈ A(M ). The set of group-like elements of this coalgebra contains only elements of the form δp for p ∈ M , where δp : A(M ) → R is defined by δp (f ) = %(f )(p) = f˜(p) .
(2.2)
Furthermore, if σ : (M, A) → (N, B) is a morphism of graded manifolds, then the element σ∗ a ∈ B(N )∗ defined for a ∈ A(M )◦ by σ∗ a(g) = a(σ ∗ g) ,
∀g ∈ B(N )
(2.3)
vanishes on an ideal of finite codimension. We thus obtain a morphism of graded coalgebras σ∗ : A(M )◦ → B(N )◦ which respects the group-like elements and induces
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a differentiable map σ∗ |M : M → N . Let us recall here that the pairing between A(M )◦ ⊗ A(M )◦ and A(M ) ⊗ A(M ) is given by (a ⊗ b)(f ⊗ g) = a(f )b(g)(−1)|bkf | for any homogeneous elements a, b ∈ A(M )◦ , f, g ∈ A(M ). Another very important property of the graded coalgebra A(M )◦ is that the set of its primitive elements with respect to the group-like element δp , i.e. elements v ∈ A(M )◦ for which ∆◦A (v) = v ⊗ δp + δp ⊗ v, is equal to the set of derivations at p on A(M ), that is, the set of elements v ∈ A(M )∗ for which v(f g) = (vf )(δp g) + (−1)|vkf | (δp f )(vg) ,
∀f, g ∈ A(M )
(2.4)
if v and f are homogeneous. This set is a subspace of A(M )◦ which we call tangent space of (M, A) at p ∈ M and we note it by Tp (M, A). One easily verifies that the morphisms of graded manifolds preserve the subspaces of primitive elements: if v ∈ Tp (M, A), then σ∗ v ∈ Tq (N, B) where q = σ∗ |M (p). Hence, we have a welldefined notion of the tangent (or the differential) of the morphism σ at any point p ∈ M and we adopt the notation Tp σ : Tp (M, A) → Tq (N, B) ,
Tp σ(v) = σ∗ v .
(2.5)
In this context, the set Der A(U ) of derivations of A(U ) plays the rˆ ole of graded vector fields on (U, A|U ), U ⊂ M open. The difference with the ordinary differential geometry is that we cannot evaluate directly a derivation ξ ∈ Der A(U ) at a point p ∈ U in order to obtain a tangent vector belonging to Tp (M, A). Instead, we may associate to each ξ ∈ Der A(U ) a tangent vector ξ˜p ∈ Tp (M, A), ∀p ∈ U in the following way: for each f ∈ A(U ), we define ξ˜p (f ) = δp (ξf ) .
(2.6)
If U is an A-coordinate neighborhood of M and (m, n) = dim(M, A), the set of derivations Der A(U ) is a free left A(U )-module of dimension (m, n). Indeed, if (xi , sj ) are the graded coordinates on U and ξ ∈ Der A(U ), then there exist elements ξ i , ξ j ∈ A(U ) such that m X
X ∂ ∂ ξ + ξj j , ξ= i ∂x ∂s i=1 j=1 n
i
where the derivations ∂/∂xi and ∂/∂sj are defined by ∂ (s` ) = δk` 1U , ∂sk
∂ (xi ) = 0 , ∂sk
∂ k (s ) = 0 , ∂xi
∂ (xj ) = δij 1U , ∂xi
(2.7)
1U being the unit of A(U ). Clearly, this decomposition is not valid in general for derivations belonging to Der A(M ). Therefore, we give the following definition: Definition 2.1. A graded manifold (M, A) is called parallelizable if the set of derivations Der A(M ) admits a global basis on A(M ) consisting of m even and n odd derivations.
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The difficulty one encounters in ordinary manifold theory to push-forward (or to pull-back) a vector field by means of a differentiable mapping is also present in the context of graded manifolds. As in the case of ordinary manifolds, this is possible only if we use isomorphisms of the graded manifold structure. More precisely: Definition 2.2. Let σ : (M, A) → (N, B) be an isomorphism between the graded manifolds (M, A) and (N, B) and let U ⊂ M be an open subset such that U = σ∗−1 (V ), V ⊂ N open. If ξ ∈ Der A(U ), then we define the push-forward σ∗ ξ ∈ Der B(V ) as σ∗ ξ = (σ ∗ )−1 ◦ ξ ◦ σ ∗ . For the pull-back of η ∈ Der B(V ), we define σ ∗ η = (σ −1 )∗ η ∈ Der A(U ). In many situations it happens that some derivations may be related through a morphism (which is not necessarily an isomorphism) in a manner similar to those of pull-back. We give the following definition: Definition 2.3. Let σ : (M, A) → (N, B) be a morphism of graded manifolds. We call two derivations ξ ∈ Der A(M ) et η ∈ Der B(N ) σ-related if for each f ∈ B(N ) we have σ ∗ (ηf ) = ξ(σ ∗ f ). Especially, if we fix ξ and σ is an epimorphism, the derivation η, if it exists, is unique; in such a case we note η = σ∗ ξ, we call ξ a σ∗ -projectable derivation and σ∗ ξ its projection by means of σ · ξ will be called vertical derivation if it is σ∗ -projectable and σ∗ ξ = 0. It is easy to verify that the set Pro(σ∗ , A)(M ) of σ∗ -projectable derivations is a left B(N )-module and that the set Ver(σ∗ , A)(M ) of vertical derivations is a left A(M )-module. Indeed, for g ∈ B(N ) and ξ ∈ Pro(σ∗ , A)(M ) let us define g · ξ ∈ Der A(M ) as g · ξ = (σ ∗ g)ξ. Then for f ∈ B(N ) we find: (g · ξ)(σ ∗ f ) = (σ ∗ g) ξ(σ ∗ f ) = σ ∗ g · σ ∗ [σ∗ ξ(f )] = σ ∗ [g(σ∗ ξ)f ], which proves that g · ξ ∈ Pro(σ∗ , A)(M ) as well. We proceed analogously for Ver(σ∗ , A)(M ). The corresponding sheaves are denoted by Pro(σ∗ , A), Ver(σ∗ , A). The final part of this section is devoted to a short review of the basic properties of graded differential forms from [1]. We also develop some useful tools for computing pull-backs of forms using the previously mentioned concepts of push-forward and σ-related derivations. Let then (M, A) be a graded manifold of dimension (m, n). For an open U ⊂ M we consider the tensor algebra T(U ) of Der A(U ) with respect to its A(U )-module structure and the ideal J(U ) of T(U ) generated by homogeneous elements of the form ξ ⊗ η + (−1)|ξkη| η ⊗ ξ, for ξ, η ∈ Der A(U ). Let also Tr (U ) ∩ J(U ) = Jr (U ). We call set of graded differential r-forms (r ≥ 1) on U ⊂ M the set Ωr (U, A) of elements belonging to HomA(U) (Tr (U ), A(U )) which vanish on Jr (U ). For r = 0 ∞ M Ωr (U, A) by Ω(U, A). we define Ω0 (U, A) = A(U ) and we note the direct sum r=0
If α ∈ Ωr (U, A) and ξ1 , . . . , ξr ∈ Der A(U ), then we denote the evaluation of α on the ξ’s by (ξ1 ⊗ · · · ⊗ ξr |α), or simply (ξ1 , . . . , ξr |α). Clearly, the elements de Ω(U, A) have a (Z ⊕ Z2 )-bidegree and further we may define an algebra structure on Ω(U, A) which thus becomes a bigraded commutative
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algebra over A(U ) [1]. Here, we mention only the bigraded commutativity relation for this structure: if α ∈ Ωiα (U, A)jα , β ∈ Ωiβ (U, A)jβ , then αβ ∈ Ωiα +iβ (U, A)jα +jβ and αβ = (−1)iα iβ +jα jβ βα. Next consider the linear map d : Ω0 (U, A) → Ω1 (U, A) defined by (ξ|dg) = ξ(g) ,
ξ ∈ Der A(U ) ,
g ∈ A(U ) .
(2.8)
In a graded coordinate system (xi , sj ), i = 1, . . . , m, j = 1, . . . , n on U , we take dg =
m X i=1
X ∂g ∂g + dsj j . i ∂x ∂s j=1 n
dxi
(2.9)
One can extend this linear map to a derivation d : Ω(U, A) → Ω(U, A) of bidegree (1,0) such that d2 = 0 and d|Ω0 (U,A) gives Eq. (2.9) We will call d exterior differential on graded differential forms. Interior products and Lie derivatives with respect to elements of Der A(U ) also make sense in the graded setting. Indeed, if α ∈ Ωr+1 (U, A), then for ξ, ξ1 , . . . , ξr homogeneous elements of Der A(U ) we define Pr (2.10) (ξ1 , . . . , ξr |i(ξ)α) = (−1)|ξ| i=1 |ξi | (ξ, ξ1 , . . . , ξr |α) and we thus obtain a linear map i(ξ) : Ω(U, A) → Ω(U, A) of bidegree (−1, |ξ|). This is the interior product with respect to ξ. Lie derivatives are defined as usual by means of Cartan’s algebraic formula: Lξ = d ◦ i(ξ) + i(ξ) ◦ d, thus Lξ has bidegree (0, |ξ|). Furthermore, it can be proved that the morphism of graded commutative algebras σ ∗ : B(W ) → A(U ), U = σ∗−1 (W ) ⊂ M coming from a morphism σ : (M, A) → (N, B) of graded manifolds, can be extended to a unique morphism of bigraded commutative algebras σ ∗ : Ω(W, B) → Ω(U, A), which commutes with the exterior differential. As for the case of derivations, see relation (2.6), one can define for each graded ˜ p (with real values) on the differential form α ∈ Ωr (U, A) a multilinear form α tangent space Tp (M, A) for each p ∈ U . It suffices to set αp = δp (ξ1 , . . . , ξr |α) . (2.11) (ξ˜1 )p , . . . , (ξ˜r )p |˜ The set of forms on U obtained in this way is denoted by ΩrA (U ). We establish now a general method for the calculation of pull-backs of graded differential forms. Lemma 2.4. Let σ : (M, A) → (N, B) be a morphism of graded manifolds, W ⊂ N and U = σ∗−1 (W ) ⊂ M. Then, if α ∈ Ωr (W, B) and ξi ∈ Der A(U ) and ηi ∈ Der B(W ) are σ-related for i = 1, . . . , r, we have: (ξ1 , . . . , ξr |σ ∗ α) = σ ∗ (η1 , . . . , ηr |α) .
(2.12)
In particular, when σ is an isomorphism, the previous relation holds for each ξi ∈ Der A(U ), setting ηi = σ∗ ξi .
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Proof. Using the fact that σ ∗ is an isomorphism of bigraded commutative algebras and that Ω(W, B) is the exterior algebra of Ω1 (W, B), it is sufficient to prove this formula for α = df, f ∈ B(W ). If ξ ∈ Der A(U ) and η ∈ Der B(W ) are σ-related, then (ξ|σ ∗ α) = (ξ|d(σ ∗ f )) = ξ(σ ∗ f ) = σ ∗ (ηf ) = σ ∗ (η|df ) = σ ∗ (η|α). In particular, when σ is an isomorphism, each ξ is σ-related to η = σ∗ ξ. One easily proves that pull-backs on forms belonging to ΩrB (W ) can be expressed using the familiar formulas of ordinary differential geometry but taking as tangent of the morphism σ, the linear mapping defined through relations (2.3), (2.4). 3. Elements of Graded Lie Theory As we will see later in detail, graded Lie theory plays a central role in the geometry of graded principal bundles just as ordinary theory of Lie groups does in differentiable principal bundles. This section deals with the notion and elementary properties of graded Lie groups; more information can be found in [1]. Furthermore, we prove some facts about the theory of actions of graded Lie groups, which, to the best of our knowledge, have not ever appeared in the literature. We first give the definition of a graded Lie group [1, 5]. Definition 3.1. A graded Lie group (G, A) is a graded manifold such that G is an ordinary Lie group, the algebra A(G) is equipped with the structure of a graded Hopf algebra with antipode and furthermore, the algebra epimorphism % : A(G) → C ∞ (G) is a morphism of graded Hopf algebras. We denote by ∆A , A , sA the coproduct, counit and antipode of A(G), respectively. It is possible to prove that the finite dual A(G)◦ inherits also a graded Hopf algebra structure from A(G). The algebra multiplication on A(G)◦ is given by the convolution product: (a ? b) = (a ⊗ b) ◦ ∆A ,
∀a, b ∈ A(G)◦ ,
(3.1)
and the unit of A(G)◦ with respect to ? is the counit of A(G). One proves easily that the set of primitive elements of A(G)◦ with respect to δe (e is the identity of G), that is, the tangent space Te (G, A), is a graded Lie algebra, the bracket being given by [u, v] = u ? v − (−1)|ukv| v ? u, for homogeneous elements u, v. We call Te (G, A) Lie superalgebra of (G, A) and denote it by g. Clearly, g = g0 ⊕ g1 , where g0 = Te G. Using also the fact that % : A(G) → C ∞ (G) is a morphism of graded Hopf algebras one readily verifies that the convolution product ? is compatible with the group structure of G in the sense that δg ? δh = δgh . A very important property of the finite dual A(G)◦ is given by the following [1]. Theorem 3.2. For each graded Lie group (G, A), the finite dual A(G)◦ has the structure of a Lie–Hopf algebra. In fact, A(G)◦ = R(G) > E(g), where R(G) is the group algebra of G, E(g) is the universal enveloping algebra of g and > is the
THEORY OF CONNECTIONS ON GRADED PRINCIPAL BUNDLES
55
smash product of R(G) and E(g) with respect to the adjoint representation of G on the superalgebra g. For the adjoint representation of G on g, see also below in this section. There exist, in this setting, analogs of left and right translations on a Lie group. Definition 3.3. Let (G, A) be a graded Lie group and a ∈ A(G)◦ . Set ra = (id ⊗ a) ◦ ∆A and `a = (a ⊗ id) ◦ ∆A . We call the endomorphisms ra and `a of A(G) right and left translations respectively on (G, A). In order to justify this terminology, we first discuss some important properties of these endomorphisms. Proposition 3.4. (1) (2) (3) (4)
rA = `A = id ra?b = ra ◦ rb , `a?b = (−1)|akb| `b ◦ `a rb ◦ `a = (−1)|akb| `a ◦ rb , ∀a, b ∈ A(G)◦ If a ∈ A(G)◦ is group-like, then ra and `a are graded algebra isomorphisms.
We postpone the proof until the study of actions of graded Lie groups (see below in this section). Then, it will be clear that the previous proposition is immediate using the general techniques of actions (note that Proposition 3.4 has already appeared in [5] without proof). Part (4) of this proposition tells us that if a is group-like a = δg , g ∈ G, then there exist morphisms of graded manifolds Rg : (G, A) → (G, A), Lg : (G, A) → (G, A) such that Rg∗ = ra and L∗g = `a . It is interesting to calculate the coalgebra morphisms Rg∗ : A(G)◦ → A(G)◦ and Lg∗ : A(G)◦ → A(G)◦ . Consider for example Rg∗ . If b ∈ A(G)◦ and f ∈ A(G), we find: Rg∗ b(f ) = b(ra f )
! X |akI i f | i i (−1) (I f )a(J f ) =b i
=
X
(b ⊗ a)(I i f ⊗ J i f ) ,
i
P where we have set ∆A f = i I i f ⊗ J i f . Hence, Rg∗ b = b ? a and similarly Lg∗ b = a ? b. This means that ra and `a correspond to right and left translations, as one can see at the coalgebra level. Next, we introduce the graded analog of actions (see also [5]). Let then (G, A) be a graded Lie group and (Y, B) a graded manifold. We give the following definition. Definition 3.5. We say that the graded Lie group (G, A) acts on the graded manifold (Y, B) to the right if there exists a morphism Φ : (Y, B) × (G, A) → (Y, B) of graded manifolds such that the corresponding morphism of graded commutative
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ˆ A(G) defines a structure of right A(G)-comodule on algebras Φ∗ : B(Y ) → B(Y ) ⊗ B(Y ). Using the notion of left comodule, we may define the left action of (G, A) on (Y, B). More explicitly, if Φ is a right and Ψ is a left action, then the morphisms Φ∗ , Ψ satisfy the following properties: ∗
(id ⊗ ∆A ) ◦ Φ∗ = (Φ∗ ⊗ id) ◦ Φ∗ , (id ⊗ A ) ◦ Φ∗ = id , (∆A ⊗ id) ◦ Ψ∗ = (id ⊗ Ψ∗ ) ◦ Ψ∗ , (A ⊗ id) ◦ Ψ∗ = id .
(3.2)
(3.3)
Let now Ψr : (Y, B) × (G, A) → (Y, B) be the morphism of graded manifolds defined by Ψr∗ = (id ⊗ sA ) ◦ T ◦ Ψ∗ ,
(3.4)
where T is the twist morphism, T (a ⊗ b) = (−1)|akb| b ⊗ a. Lemma 3.6. The morphism Ψr defined by (3.4) is a right action of (G, A) on (Y, B). Proof. It suffices to prove that relations (3.2) are valid for the morphism Ψr∗ . We take: (id ⊗ ∆A ) ◦ Ψr∗ = id ⊗ (sA ⊗ sA ) ◦ T ◦ ∆A ◦ T ◦ Ψ∗ = id ⊗ (sA ⊗ sA ) ◦ T ◦ T ◦ (∆A ⊗ id) ◦ Ψ∗ = id ⊗ (sA ⊗ sA ) ◦ T ◦ T ◦ (id ⊗ Ψ∗ ) ◦ Ψ∗ = (id ⊗ sA ⊗ sA ) ◦ (T ◦ Ψ∗ ⊗ id) ◦ T ◦ Ψ∗ = (Ψr∗ ⊗ id) ◦ Ψr∗ . On the other hand, (id ⊗ A ) ◦ Ψr∗ = (id ⊗ A ◦ sA ) ◦ T ◦ Ψ∗ = (id ⊗ A ) ◦ T ◦ Ψ∗ = id , which completes the proof.
Similarly, for a right action Φ : (Y, B) × (G, A) → (Y, B) one can define in a canonical way, a left action Φ` : (G, A) × (Y, B) → (Y, B) as Φ`∗ = (sA ⊗ id) ◦ T ◦ Φ∗ . Then, the restriction Φ∗ |Y ×G : Y ×G → Y defines a right action of G on the manifold Y and furthermore, for the canonically associated left action Φ` , the restriction Φ`∗ |G×Y : G × Y → Y is a left action given by Φ`∗ |G×Y (g, y) = Φ∗ |Y ×G (y, g −1 ), as one expects. We have analogous facts for the left action Ψ. Observe here that the possibility to define Φ`∗ and Ψr∗ as morphisms of graded commutative algebras depends crucially on the fact that the antipode sA : A(G) → A(G) is a morphism of graded commutative algebras. Remark 3.7. The right action Φ`r canonically associated to Φ` equals to Φ: Φ = (id ⊗ sA ) ◦ T ◦ (sA ⊗ id) ◦ T ◦ Φ∗ = (id ⊗ sA ) ◦ T 2 ◦ (id ⊗ sA ) ◦ Φ∗ = Φ∗ , since T 2 = id, s2A = id. `r∗
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For a right action Φ, one may introduce for each a ∈ A(G)◦ and b ∈ B(Y )◦ , two linear maps (Φ∗ )a : B(Y ) → B(Y ) and (Φ∗ )b : B(Y ) → A(G) as follows: (Φ∗ )a = (id ⊗ a) ◦ Φ∗
and
(Φ∗ )b = (b ⊗ id) ◦ Φ∗ .
(3.5)
(Ψ∗ )b = (id ⊗ b) ◦ Ψ∗ .
(3.6)
Similarly, for a left action Ψ one defines (Ψ∗ )a = (a ⊗ id) ◦ Ψ∗
and
The following theorem clarifies the role of these maps. Theorem 3.8. (Φ∗ )A = (Ψ∗ )A = id (Φ∗ )a1 ?a2 = (Φ∗ )a1 ◦ (Φ∗ )a2 , (Ψ∗ )a1 ?a2 = (−1)|a1 ka2 | (Ψ∗ )a2 ◦ (Ψ∗ )a1 (Φ∗ )b ◦ (Φ∗ )a = (−1)|akb| ra ◦ (Φ∗ )b , (Ψ∗ )b ◦ (Ψ∗ )a = (−1)|akb| `a ◦ (Ψ∗ )b If a, b are group-like elements, then (Φ∗ )a is an isomorphism and (Φ∗ )b is a morphism of graded commutative algebras. In particular, if a = δg , b = δy , then we write the corresponding morphisms of graded manifolds as Φg : (Y, B) → (Y, B) and Φy : (G, A) → (Y, B), so Φ∗g = (Φ∗ )δg and Φ∗y = (Φ∗ )δy . Similarly for (Ψ∗ )a , (Ψ∗ )b . (5) If a is a primitive element with respect to δe , then (Φ∗ )a , (Ψ∗ )a ∈ Der B(Y ). We call these derivations the induced (by the action and the element a) derivations on B(Y ).
(1) (2) (3) (4)
Proof. (1) Evident, by the defining properties of the left and right action. (2) We prove this property only for the right action Φ; one proceeds in a similar way for the left action Ψ. We have: (Φ∗ )a1 ?a2 = (id ⊗ (a1 ? a2 )) ◦ Φ∗ = (id ⊗ a1 ⊗ a2 ) ◦ (id ⊗ ∆A ) ◦ Φ∗ = (id ⊗ a1 ⊗ a2 ) ◦ (Φ∗ ⊗ id) ◦ Φ∗ = (id ⊗ a1 ) ◦ Φ∗ ◦ (id ⊗ a2 ) ◦ Φ∗ = (Φ∗ )a1 ◦ (Φ∗ )a2 . (3) Again, we give the proof only for the right action. (Φ∗ )b ◦ (Φ∗ )a = (b ⊗ id) ◦ Φ∗ ◦ (id ⊗ a) ◦ Φ∗ = (−1)|akb| (id ⊗ a)(b ⊗ id) ◦ (id ⊗ ∆A ) ◦ Φ∗ = (−1)|akb| (id ⊗ a) ◦ ∆A ◦ (b ⊗ id) ◦ Φ∗ = (−1)|akb| ra ◦ (Φ∗ )b .
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(4) The fact that these maps are morphisms is evident because they are compositions of morphisms when a, b are group-like. Furthermore, (Φ∗ )a and (Ψ∗ )a are isomorphisms because their inverses exist, as one can check from parts (1) and (2). (5) A derivation ξ on the graded commutative algebra B(Y ) has the property ξ(f g) = ξ(f )g + (−1)|ξkf | f ξ(g) for each homogeneous element f ∈ B(Y ). This can be restated as follows: ξ ◦ mB = mB ◦ (ξ ⊗ id + id ⊗ ξ). We shall prove this property for ξ = (Φ∗ )a . We have: ξ ◦ mB = (mB ⊗ a ◦ mA ) ◦ (id ⊗ T ⊗ id) ◦ (Φ∗ ⊗ Φ∗ ) = (mB ⊗ (a ⊗ δe + δe ⊗ a)) ◦ (id ⊗ T ⊗ id) ◦ (Φ∗ ⊗ Φ∗ ) = mB ◦ [(id ⊗ a) ◦ Φ∗ ⊗ id] + mB ◦ [id ⊗ (id ⊗ a) ◦ Φ∗ ] = mB ◦ (ξ ⊗ id + id ⊗ ξ) . One proceeds similarly for (Ψ∗ )a . We note finally that if a is homogeneous, then |(Φ∗ )a | = |(Ψ∗ )a | = |a|. Corollary 3.9. Proposition 3.4. Proof. Since the coproduct ∆A on the Hopf algebra A(G) has the properties (id ⊗ ∆A ) ◦ ∆A = (∆A ⊗ id) ◦ ∆A and (id ⊗ A ) ◦ ∆A = (A ⊗ id) ◦ ∆A = id, it defines left and right actions L and R respectively of (G, A) on itself. Choosing thus (Y, B) = (G, A) in the previous theorem, we may write (L∗ )a = `a , (R∗ )a = ra ; Proposition 3.4 is then immediate. ˆ A(G) the linear Next, let (G, A) be a graded Lie group and L : A(G) → A(G) ⊗ map defined by L = [mA ◦ (id ⊗ sA ) ⊗ id] ◦ (id ⊗ T ) ◦ (id ⊗ ∆A ) ◦ ∆A .
(3.7)
Proposition 3.10. The linear map L is a morphism of graded commutative algebras defining thus a morphism of graded manifolds which we denote by AD : (G, A) × (G, A) → (G, A). Furthermore, AD is a left action of (G, A) on itself. Proof. L is a morphism of graded algebras as composition of morphisms; so we can write L = AD∗ for a morphism of graded manifolds AD : (G, A) × (G, A) → (G, A). We now check relations (3.3) for AD∗ . For the first one, the following identity is the key of the proof: (id ⊗ id ⊗ id ⊗ ∆A ) ◦ (id ⊗ id ⊗ ∆A ) ◦ (∆A ⊗ id) ◦ ∆A = (id ⊗ id ⊗ ∆A ⊗ id) ◦ (id ⊗ ∆A ⊗ id) ◦ (id ⊗ ∆A ) ◦ ∆A . Indeed, applying the two members of the previous identity on the same f ∈ A(G), we find after a long and cumbersome calculation the first of (3.3). For the second of (3.3), we proceed as follows:
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(A ⊗ id) ◦ L = (A ◦ mA ⊗ id) ◦ (id ⊗ sA ⊗ id) ◦ (id ⊗ T ) ◦ (id ⊗ ∆A ) ◦ ∆A = (A ⊗ A ◦ sA ⊗ id) ◦ (id ⊗ T ) ◦ (id ⊗ ∆A ) ◦ ∆A = (A ⊗ id ⊗ A ) ◦ (id ⊗ ∆A ) ◦ ∆A = (A ⊗ id) ◦ ∆A = id , which completes the proof.
We call the action AD adjoint action of (G, A) on itself. As in ordinary Lie theory, the adjoint action respects the primitive elements with respect to δe in the sense of the following proposition. Proposition 3.11. Let AD∗a : A(G)◦ → A(G)◦ , a ∈ A(G)◦ be defined as AD∗a (b) = AD∗ (a ⊗ b). Then, for an element a ∈ A(G)◦ group-like or primitive with respect to δe , AD∗a is a linear map on the Lie superalgebra g. Proof. Consider first the case where a is a group-like element, a = δg , g ∈ G. If v ∈ g, we have: AD∗a (v) = AD∗ (a⊗v) = a ? v ? a−1 , because for group-like elements the antipode s◦A of A(G)◦ is given by s◦A a = a−1 = δg−1 . It is then immediate to verify that if ∆◦A is the coproduct of A(G)◦ , we have: ∆◦A (AD∗a (v)) = AD∗a (v) ⊗ δe + δe ⊗ AD∗a (v) which means that AD∗a (v) belongs also to g. Proceeding in the same way for the case where a is primitive with respect to δe , we find AD∗a (v) = a ? v − (−1)|akv| v ? a = [a, v] ∈ g. Thus, for an element a ∈ A(G)◦ group-like or primitive with respect to δe , we take AD∗a ∈ End g. The previous proof makes clear that if a = δg , g ∈ G, then AD∗a is an iso−1 ◦ Lg∗ . For morphism of the Lie superalgebra g. Indeed, in this case AD∗a = Rg∗ the case where a is primitive with respect to δe , AD∗a coincides with the adjoint representation of g on itself, AD∗a = ad(a), where ad(a)(b) = [a, b], ∀b ∈ g. Remark 3.12. One can define a linear map Ψ∗a : B(Y )◦ → B(Y )◦ for each left action Ψ : (G, A) × (Y, B) → (Y, B) and a ∈ A(G)◦ by Ψ∗a (b) = Ψ∗ (a ⊗ b), ∀b ∈ B(Y )◦ . It is then easily verified that for a group-like, Ψ∗a is an isomorphism of graded coalgebras; furthermore, Ψ∗a = Ψg∗ if a = δg (see Theorem 3.8). We have analogous facts for a right action. Graded Lie groups provide an important and wide class of parallelizable graded manifolds as the following theorem asserts. Theorem 3.13. Each graded Lie group (G, A) is a parallelizable graded manifold. Proof. Let Φ : (G, A) × (G, A) → (G, A) be the right action such that Φ∗ = ∆A (see the proof of Corollary 3.9). Then, (Φ∗ )a = (id ⊗ a) ◦ Φ∗ = ra . If a ∈ g, then
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we know by Theorem 3.8 that (Φ∗ )a is a derivation on A(G). For g ∈ G, the ^ ∗ ) (g) ∈ T (G, A) is calculated by means of (2.6): (Φ ∗ ) (g) = ^ tangent vector (Φ a g a ∗ δg ◦ (Φ )a = δg ? a = Lg∗ (a); but Lg∗ is an isomorphism by Proposition 3.4. This means that if {ai , bj }o is a basis of the Lie superalgebra g, ai ∈ g0 , bj ∈ g1 , then n ^ ∗ ) i (g), (Φ ∗ ) j (g) is a basis of T (G, A) for each g ∈ G. By Proposition 2.12.1 ^ (Φ g a b of [1], we conclude that {(Φ∗ )ai , (Φ∗ )bj } is a global basis of Der A(G) for its left A(G)-module structure. 4. Lie Superalgebra-Valued Graded Differential Forms Let (Y, B) be a graded manifold and g a Lie superalgebra. We call g-valued graded differential form on (Y, B), an element of Ω(Y, B) ⊗ g. It is clear that the set Ω(Y, B, g) = Ω(Y, B) ⊗ g of these forms constitutes a (Z ⊕ Z2 ⊕ Z2 )-graded vector space; however, it is more convenient to introduce a (Z ⊕ Z2 )-grading as follows: if α ∈ Ω(Y, B, g) and deg(α) = (iα , jα , kα ) is its (Z ⊕ Z2 ⊕ Z2 )-degree, then we set |α| = (iα , jα + kα ) ∈ Z ⊕ Z2 . If {ei } is a basis of the Lie superalgebra g and P P α, β ∈ Ω(Y, B, g), then we may write α = i αi ⊗ ei and β = i β i ⊗ ei , for αi , β i ∈ Ω(Y, B). In the case where α and β are homogeneous with deg(α) = (iα , jα , kα ), deg(β) = (iβ , jβ , kβ ), we define the g-valued graded differential form [α, β] of degree deg ([α, β]) = (iα + iβ , jα + jβ , kα + kβ ) as: X β (−1)jβ kα αi β j ⊗ [eα (4.1) [α, β] = i , ej ] , i,j
P
i α if α = i αi ⊗ eα i with |α | = (iα , jα ) and ei = kα ; similarly for β. We extend this definition to non-homogeneous elements by linearity. Clearly, Eq. (4.1) gives the same result for every basis of the Lie superalgebra g. Thus, we have a bilinear 0 0 map [−, −] : Ωi (Y, B)j ⊗ gk × Ωi (Y, B)j 0 ⊗ gk0 → Ωi+i (Y, B)j+j 0 ⊗ gk+k0 with the following properties:
Lemma 4.1. If α, β, γ ∈ Ω(Y, B, g) are homogeneous, then we have: |αkβ| [β, (1) [α, β] = −(−1) α] |αkγ| [α, β], γ = 0. (2) S(−1)
In the previous relations, we have set |αkβ| = iα iβ + (jα + kα )(jβ + kβ ) and S means the cyclic sum on the argument which follows. Proof. Routine calculations using relation (4.1).
We realize that the space Ω(Y, B, g) possesses the structure of a (Z ⊕ Z2 )-graded Lie algebra, inherited from the the Lie superalgebra structure of g. The action now of elements of Ω(Y, B, g) on derivations can be seen as follows: P i ei ∈ Ω(Y, B, g) is an r-form and ξ1 , . . . , ξr ∈ Der B(Y ), we set if α = iα ⊗P (ξ1 , . . . , ξr |α) = i (ξ1 , . . . , ξr |αi ) ⊗ ei . Accordingly, one can extend the exterior differential d to a differential on g-valued graded differential forms, also noted by d in P P the following manner: if α = i αi ⊗ ei , then we set dα = i dαi ⊗ ei . The exterior
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differential d : Ω(Y, B, g) → Ω(Y, B, g) defined previously, is a derivation of degree |d| = (1, 0). By straightforward verification, we find that, if α, β are g-valued graded differential forms and α is homogeneous, then d[α, β] = [dα, β] + (−1)|αkd| [α, dβ]. In a similar way, one can extend the pull-back of graded differential forms under a morphism of graded manifolds σ : (Y, B) → (Z, Y) to a linear map σ ∗ : Ω(Z, Y, g) → Ω(Y, B, g) which commutes with the exterior differential, that is, d ◦ σ ∗ = σ ∗ ◦ d, and preserves the bracket [−, −] : σ ∗ [α, β] = [σ ∗ α, σ ∗ β]. We have analogous generalizations for the Lie derivative. The following properties of the bracket and the Lie derivative on g-valued graded differential forms will be useful; the proof proceeds by a straightforward calculation with graded differential forms and Lie superalgebra elements, and is left as an exercise for the reader. Lemma 4.2. If α, β ∈ Ω(Y, B, g) and ξ ∈ Der B(Y ) are homogeneous, then Lξkα| [α, Lξ β] (1) Lξ [α, β] = [Lξ α, β] + (−1)|L (2) id ⊗ ad(v) [α, β] = (id ⊗ ad(v))α, β + (−1)|vkα| α, (id ⊗ ad(v))β , ∀v ∈ g. One can restate Lemma 2.4 for the case of g-valued graded differential forms. For example, if σ is an isomorphism of graded manifolds, relation (2.12) becomes: (ξ1 , . . . , ξr |σ ∗ α) = (σ ∗ ⊗ id)(σ∗ ξ1 , . . . , σ∗ ξr |α) .
(4.2)
One can also define multilinear forms on the tangent spaces of a graded manifold taking its values in the Lie superalgebra g, using a simple modification of (2.11). We close this section by studying an example of Lie superalgebra-valued graded differential form, very useful for the theory of graded connections. If (G, A) is a graded Lie group, let g be its Lie superalgebra and R the right action of (G, A) on itself defined by the coproduct ∆A (see proof of Corollary 3.9). Then, if a ∈ g and (R∗ )a is the induced derivation, we define a 1-form θ ∈ Ω1 (G, A, g) by the relation ((R∗ )a |θ) = 1A ⊗ a. This defines θ completely because of the global parallelism of (G, A), see Theorem 3.13. Clearly, θ has two homogeneous parts of (Z ⊕ Z2 ⊕ Z2 )degrees (1,0,0) and (1,1,1), so θ has total degree |θ| = (1, 0). One can easily calculate the differential dθ, the pullback Rg∗ θ, g ∈ G, and the Lie derivative L(R∗ )a θ, a ∈ g, using Theorem 3.8, Proposition and formulas (2.10) and (2.12). We find: 1 dθ + [θ, θ] = 0 , 2 Rg∗ θ = (id ⊗ ADg−1 ∗ ) ◦ θ ,
(4.3)
L(R∗ )a θ = −(id ⊗ AD∗a ) ◦ θ . Let us calculate explicitly for example the pullback Rg∗ θ. Using relation (2.12) for the isomorphism Rg and Proposition 3.11, we obtain: ((R∗ )a |Rg∗ θ) = Rg∗ (Rg∗ (R∗ )a |θ) = Rg∗ ((R∗ )δg−1 ?a?δg |θ) = Rg∗ ((R∗ )ADg−1 ∗ (a) |θ) = 1A ⊗ ADg−1 ∗ (a) ,
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which means that Rg∗ θ = (id ⊗ ADg−1 ∗ ) ◦ θ. We call θ graded Maurer–Cartan form on (G, A). 5. Actions, Graded Distributions and Quotient Structures Two important notions in the study of graded Lie group actions on graded manifolds are those of graded distributions and quotient graded manifolds. In this section, we investigate the relation between graded distributions and free actions properly defined in the graded setting. In addition, we find a necessary and sufficient condition in order that the quotient defined by the action of a graded Lie group be a graded manifold. We first introduce the notion of the graded distribution (see also [15]). Definition 5.1. Let (M, A) be a graded manifold of dimension (m, n). We call graded distribution of dimension (p, q) on (M, A) a locally free sheaf E of A-modules with dimension (p, q), such that for each open U ⊂ M, E(U ) is a graded submodule of Der A(U ). The distribution will be called involutive if for each ξ, η ∈ E(U ), we have [ξ, η] ∈ E(U ), ∀U ⊂ M open. Thus, for each point x ∈ M there exists an open neighborhood U of x and elements ξi ∈ Der A(U ) 0 , i = 1, . . . , p, ηj ∈ Der A(U ) 1 , j = 1, . . . , q such that E(U ) = A(U ) · ξ1 ⊕ · · · ⊕ A(U ) · ξp ⊕ A(U ) · η1 ⊕ · · · ⊕ A(U ) · ηq . Given the graded distribution E on (M, A), one obtains, for each x ∈ M , a graded subspace Ex of Tx (M, A), calculating the tangent vectors ξ˜x ∈ Tx (M, A) via relation (2.6), for each ξ ∈ E(U ), x ∈ U . Clearly, Ex = (Ex )0 ⊕ (Ex )1 with dim(Ex )0 = 0 (x) ≤ p and dim(Ex )1 = 1 (x) ≤ q. Therefore, we make the following distinction: Definition 5.2. A graded distribution E of dimension (p, q) on the graded manifold (M, A) is called 0-regular if 0 (x) = p, and 1-regular if 1 (x) = q, for each x ∈ M . We say that E is regular if 0 (x) = p and 1 (x) = q, for each x ∈ M . For the subsequent analysis, a graded generalization of free actions will be necessary. Definition 5.3. We call the right action Φ : (Y, B) × (G, A) → (Y, B) free, if for each y ∈ Y the morphism Φy : (G, A) → (Y, B) is such that Φy∗ : A(G)◦ → B(Y )◦ is injective. Similarly, one defines the left free action. It is clear that if the graded Lie group (G, A) acts freely on (Y, B), then we obtain a free action of G on Y , but if only the restriction Φ∗ |Y ×G is a free action then, in general, the action Φ is not free. An equivalent characterization of the free action in graded Lie theory is provided by the following proposition for the case of a right action.
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Proposition 5.4. The action Φ : (Y, B) × (G, A) → (Y, B) is free if and ˜ = (Φ × π1 ) ◦ ∆ : (Y, B) × (G, A) → only if the morphism of graded manifolds Φ ˜ ∗ is injective. Here, ∆ denotes the diagonal morphism (Y, B) × (Y, B) is such that Φ on (Y, B) × (G, A) and π1 is the projection on the first factor. Proof. Consider elements a = δg ∈ A(G)◦ , b = δy ∈ B(Y )◦ group-like and u ∈ Ty (Y, B), w ∈ Tg (G, A) primitive. Then, a simple calculation gives ˜ ∗ (b ⊗ a) = Φy∗ (a) ⊗ b Φ ˜ ∗ (u ⊗ a + b ⊗ w) = [Φg∗ (u) + Φy∗ (w)] ⊗ b + Φg∗ (b) ⊗ u . Φ
(5.1) (5.2)
Suppose now that Φ is a free action; then the morphism Φy∗ : A(G)◦ → B(Y )◦ is ˜ ∗ is injective injective which implies immediately, thanks to (5.1) and (5.2), that Φ on all group-like and primitive elements. By Proposition 2.17.1 of [1], this is a ˜ ∗ to be injective on the whole necessary and sufficient condition for the morphism Φ ◦ graded coalgebra A(G) . The converse is immediate again by (5.1) and (5.2). Consider now a right action Φ : (Y, B) × (G, A) → (Y, B); by Theorem , we have a linear map I Φ : g → Der B(Y ) defined as I Φ (a) = (Φ∗ )a . We thus obtain a subspace DerΦ B(Y ) = im I Φ of the Lie superalgebra of derivations on B(Y ). As a matter of fact, DerΦ B(Y ) is a graded Lie subalgebra of Der B(Y ). Indeed, one readily verifies that for all a, b ∈ g we have [(Φ∗ )a , (Φ∗ )b ] = (Φ∗ )[a,b] , which means that I Φ ([a, b]) = [I Φ (a), I Φ (b)]. The following theorem provides an important property of free actions on graded manifolds. Theorem 5.5. Let Φ be a free right action of the graded Lie group (G, A) on the graded manifold (Y, B), dim(G, A) = (m, n). Then Φ induces a regular and involutive graded distribution E on (Y, B) of dimension (m, n). Proof. • Step 1. Let us first calculate the kernel of the Lie superalgebra morphism I Φ : g → Der B(Y ) when Φ is a free action. To this end, the following general property of actions is useful: ∗ ) (y) = Φ (a) , ^ (Φ a y∗
∀y ∈ Y , ∀a ∈ g .
(5.3)
∗ ) (y) = δ ◦ (Φ∗ ) = ^ For the proof of (5.3), we note only that, by relation (2.6), (Φ a y a Φ∗ (δy ⊗ a). Suppose now that I Φ (a) = 0 ⇔ (Φ∗ )a = 0; by (5.3), this implies that Φy∗ (a) = 0, ∀y ∈ Y . Since Φ is a free action, we know by Definition 5.3, that Φy∗ is injective for all y ∈ Y , which implies that a = 0. As a result, ker I Φ = 0, or I Φ is injective; hence, DerΦ B(Y ) is a graded Lie subalgebra of Der B(Y ) whose even and odd dimensions are m and n respectively: (DerΦ B(Y ))0 ∼ = g0 , (DerΦ B(Y ))1 ∼ = g1 . • Step 2. Let now DerΦ B be the correspondence U → DerΦ B(U ), where DerΦ B(U ) = PY U DerΦ B(Y ) , U ⊂ Y and PUV : Der B(U ) → Der B(V ) are the restriction maps for the sheaf Der B. Clearly, DerΦ B is a subpresheaf of Der B.
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Consider now the subpresheaf E = B·DerΦ B of Der B, E(U ) = B(U )·DerΦ B(U ) = PY U B(Y ) · DerΦ B(Y ) . E(U ) is the set of finite linear combinations of elements of DerΦ B(U ) with coefficients in B(U ). In order to prove that E is a sheaf, let us consider an open U ⊂ Y , an open covering {Uα }α∈Λ of U and elements Dα ∈ E(Uα ) such that PUα Uαβ (Dα ) = PUβ Uαβ (Dβ ), ∀α, β ∈ Λ when Uαβ = Uα ∩ Uβ 6= ∅. Then by the sheaf properties of Der B, there exists an element D ∈ Der B(U ) such that P PUUα (D) = Dα . But if we write Dα = i fαi PY Uα (Φ∗ )ei , fαi ∈ B(Uα ) and {ei } is a basis of g, then by step 1, we find easily that fαi = f i |Uα , f i ∈ B(U ), because Dα ’s coincide on the intersections Uαβ . This means that Dα = PY Uα (E), where P E = i F i (Φ∗ )ei with F i ∈ B(Y ) such that F i |U = f i , ∀i. It is then immediate that D = PY U (E) ∈ E(U ). • Step 3. It is evident that the sheaf E previously constructed, has the properties of a graded distribution. In fact, E(U ) is a graded submodule of Der B(U ) of dimension (m, n) for each open U ⊂ Y . This distribution is clearly regular thanks to relation (5.3) and to the fact that the action is free. It remains to show that P it is involutive. To this end, consider two elements ξ = i f i PY U (Φ∗ )ai and η = P j ∗ i j i j j g PY U (Φ )bj of E(U ), with f , g ∈ B(U ), a , b ∈ g. Then, direct calculation shows that X f i PY U (Φ∗ )ai g j PY U (Φ∗ )bj [ξ, η] = i,j
−(−1)|ξkη|
X
g j PY U (Φ∗ )bj f i PY U (Φ∗ )ai
i,j
+
X
(−1)|a
i
kgj | i j
f g PY U (Φ∗ )[ai ,bj ] ,
i,j
from which the involutivity is evident.
We focus now our attention on graded quotient structures defined by equivalence relations on graded manifolds (see [5] for a general treatment on this subject). A special case of equivalence relation is provided by the action of a graded Lie group on a graded manifold and this will be the interesting one for us. Definition 5.6. We call a right action Φ : (Y, B) × (G, A) → (Y, B) regular ˜ = (Φ × π1 ) ◦ ∆ : (Y, B) × (G, A) → (Y, B) × (Y, B) defines if the morphism Φ (Y, B) × (G, A) as a closed graded submanifold of (Y, B) × (Y, B). Recall here [1], that (R, D) is a graded submanifold of (Y, B) if D(R)◦ ⊂ B(Y )◦ and there exists a morphism of graded manifolds i : (R, D) → (Y, B) such that i∗ : D(R)◦ → B(Y )◦ is simply the inclusion; (R, D) will be called closed if, furthermore, ˜ ∗ B(Y )◦ ⊗ dim(R,D) < dim(Y, B). Then, the action is regular if the subset Φ A(G)◦ ⊂ B(Y )◦ ⊗ B(Y )◦ defines a graded submanifold of (Y, B) × (Y, B) in the sense of Kostant [1]. The following proposition generalizes in a natural way to the graded case, a fundamental result about quotients defined by actions in ordinary manifold theory.
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Proposition 5.7. The action Φ : (G, A) × (Y, B) → (Y, B) is regular if and only if the quotient (Y /G, B/A) is a graded manifold. Proof. Thanks to Theorem 2.6 of [5], it suffices to prove that the projections pi : (Y, B) × (Y, B) → (Y, B), i = 1, 2 on the first and second factors restricted to the ˜ are submersions. In other words, we must show image of (Y, B) × (G, A) under Φ ˜ ∗ : B(Y )◦ ⊗ A(G)◦ → B(Y )◦ , i = 1, 2 that the morphisms of graded coalgebras pi∗ ◦ Φ restricted to primitive elements are surjective. Consider an arbitrary primitive element V = u ⊗ a + b ⊗ w, for a = δg ∈ A(G)◦ , b = δy ∈ B(Y )◦ group-like and u ∈ Ty (Y, B), w ∈ Tg (G, A) primitive. Using ˜ ∗ (V ) = Φg∗ (u) + Φy∗ (w) and p2∗ Φ ˜ ∗ (V ) = u, relation (5.2), we find easily: p1∗ Φ ˜ are submersions, i = 1, 2. which proves that pi ◦ Φ Note here that if U ⊂ Y /G is an open subset, then the sheaf B/A is given by (B/A)(U ) = {f ∈ B(ˇ π −1 (U ))|Φ∗ f = f ⊗ 1A } ,
(5.4)
where π ˇ : Y → Y /G is the projection, [5]. Furthermore, the dimension of the quotient graded manifold (Y /G, B/A) is equal to dim(Y /G, B/A) = 2 dim(Y, B) − ˜ where im Φ ˜ denotes the closed graded submanifold defined by Φ. ˜ When dim(im Φ), Φ is a free action, then by Proposition 5.4, we take that dim(Y /G, B/A) = dim(Y, B) − dim(G, A). We make finally some comments about graded isotropy subgroups recalling their construction from [1], but in a more concise way. Consider a right action Φ : (Y, B) × (G, A) → (Y, B) and b ∈ B(Y )◦ a group-like element, b = δy , y ∈ Y . Let Hy (G, g) be the set of elements a ∈ A(G)◦ with the property Φ∗ (δy ⊗ a) = ◦A (a)δy .
(5.5)
Let Hy (G, g) ∩ G = Gy and Hy (G, g) ∩ g = gy , then (gy )0 is the Lie algebra of Gy . It is then clear that we can form the Lie–Hopf algebra R(Gy ) > E(gy ) because gy is stable under the adjoint action of Gy , see Proposition 3.11 and relation (5.5). By Proposition 3.8.3 of [1], R(Gy ) > E(gy ) corresponds to a graded Lie subgroup of (G, A). We denote this subgroup by (Gy , Ay ) and call it graded isotropy subgroup of (G, A) at the point y. 6. Graded Principal Bundles Graded principal bundles were first introduced in [5, 6]. Here, we discuss this notion with slight modifications suggested by the requirement that the definition of graded principal bundles reproduces well the ordinary principal bundles. Definition 6.1. A graded principal bundle over a graded manifold (X, C) consists of a graded manifold (Y, B) and an action Φ of a graded Lie group (G, A) on (Y, B) with the following properties: (1) Φ is a free right action (2) the quotient (Y /G, B/A) is a graded manifold, isomorphic to (X, C), such that the natural projection π : (Y, B) → (X, C) is a submersion.
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(3) (Y, B) is locally trivial that is, for each x ∈ X there exists an open neighborhood U of x and an isomorphism of graded manifolds φ : (V, B|V ) → (U × ˆ V = π∗−1 (U ) ⊂ Y , such that the isomorphism φ∗ of graded algeG, C|U ⊗A), bras is a morphism of A(G)-comodules, where the A(G)-comodule structures ˆ on C(U )⊗A(G) and B(V ) are given by id ⊗ ∆A and Φ∗ respectively. Furthermore, we require that φ∗ = mB ◦ (π ∗ ⊗ ψ ∗ ), where ψ : (V, B|V ) → (G, A) is a morphism of graded manifolds. The fact that φ∗ is a morphism of A(G)-comodules, that is, (φ∗ ⊗ id) ◦ (id ⊗ ∆A ) = Φ∗ ◦ φ∗
(6.1)
implies that ψ ∗ is also a morphism of A(G)-comodules: (ψ ∗ ⊗ id) ◦ ∆A = Φ∗ ◦ ψ ∗ .
(6.2)
One easily verifies that the underlying differentiable manifolds of Definition 6.1 form an ordinary principal bundle, and further, if the graded manifolds become trivial, ∞ ∞ , B = CY∞ , C = CX , then we obtain the definition of an in the sense that A = CG ordinary principal bundle. We refer the reader to [16] for a general and systematic treatment on the subject of principal bundles in differential geometry. Let us now compute the graded isotropy subgroups in the case where the action Φ is free (for example, this is the case of the graded principal bundle). The Lie group Gy which is defined as Gy = {g ∈ G|Φ∗ (b ⊗ δg ) = b} is equal to e because the action of G on Y is free. On the other hand, gy = {a ∈ g|Φ∗ (b ⊗ a) = 0} = 0, again because the action is free (the morphism Φy∗ is injective). Hence, in this case the graded isotropy subgroup (Gy , A y ) is simply (e, R), where R is the trivial sheaf over the identity e ∈ G, R(e) = R. In order to calculate the quotient graded manifold (G/Gy , A/A y ) which represents the orbit of the point y under the action of (G, A), we need the expression of the canonical right action Φ of the subgroup (Gy , A y ) on (G, A). If i : (Gy , A y ) → (G, A) is the inclusion, we have: Φ∗ = (id ⊗ i∗ ) ◦ ∆A and for the case of the free action, where (Gy , A y ) = (e, R), one finds that i∗ = A and finally Φ∗ = id. In view of relation (5.4), it is straightforward that A/A y = A. Therefore, Property 6.2. The orbits of a free action of (G, A) are always isomorphic as graded manifolds to (G, A). For the case now of the graded principal bundle, the orbit (Oy , By ) of (G, A) through y ∈ Y will be called fibre of (Y, B) over x = π∗ |Y (y) ∈ X. Using the graded version of the submersion Theorem [13], one can justify this terminology as follows. The pre-image π −1 (x, R) of the closed graded submanifold (x, R) ,→ (X, C) is a closed graded submanifold of (Y, B) whose underlying differentiable manifold is π∗−1 (x). So, if we write π −1 (x, R) = (π∗−1 (x), D), then for each z ∈ π∗−1 (x) we have: Tz (π∗−1 (x), D) = ker Tz π. We know already that π∗−1 (x) = Oy , the orbit under G of a point y ∈ Y such that π∗ |Y (y) = x. Furthermore, if δz = Φ∗ (δy ⊗
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δg ), g ∈ G and v ∈ Tg (G, A), then V = Φ∗ (δy ⊗ v) ∈ Tz (Oy , By ) and π∗ (V ) = 0. Consequently, Tz (Oy , By ) ⊂ ker Tz π. By a simple argument on dimensions, we obtain that Tz (Oy , By ) = Tz (π∗−1 (x), D). We conclude that the tangent bundles of π −1 (x, R) and (Oy , By ) are identical; but then, Theorem 2.16 of [1] tells us that these graded manifolds coincide. Next, we discuss an elementary example of graded principal bundle, the product bundle. In this case, one can directly verify the axioms of Definition 6.1. Nevertheless, there exist also graded principal bundles for which Definition 6.1 cannot be directly applied, even though this is possible for the corresponding ordinary principal bundles. For such cases, one may use an equivalent definition of the graded principal bundle, see next section. Example 6.3. Consider a graded manifold (X, C), a graded Lie group (G, A) and their product (Y, B) = (X, C) × (G, A). One has a canonical right action Φ : (Y, B) × (G, A) → (Y, B) defined as Φ∗ = id ⊗ ∆A . This action is free: if δy = δx ⊗δg ∈ C(X)◦ ⊗A(G)◦ is group-like and a ∈ A(G)◦ , then Φy∗ (a) = Φ∗ (δx ⊗δg ⊗a) = δx ⊗(δg ? a) = δx ⊗Lg∗ (a), which implies that Φy∗ is an injective morphism of graded coalgebras, because Lg∗ is an isomorphism. Evidently, the quotient Y /G is equal to ˆ A(G) for which X and the sheaf B/A over X is given by the elements f ∈ C(U ) ⊗ P Φ∗ f = f ⊗ 1A , U ⊂ X open. If we decompose f as f = i fi ⊗ hi , fi ∈ C(U ), hi ∈ A(G), we take easily ∆A hi = hi ⊗ 1A , hence hi ’s are such that A (hi )1A = hi . P We conclude that f is of the form f = i A (hi )fi ⊗ 1A = fC ⊗ 1A and finally (B/A)(U ) ∼ = C(U ), which proves that the quotient (Y /G, B/A) is isomorphic to ˆ ˆ → C(U )⊗A(G) admits the (X, C). Further, the identity map φ∗ = id : C(U )⊗A(G) decomposition of Definition (it suffices to choose π = π1 , ψ = π2 , the projections on the first and second factors respectively) and satisfies trivially the relation (6.1). 7. The Geometry of Graded Principal Bundles In this section we analyze three aspects of the geometry of graded principal bundles: the relation between the sheaf of vertical derivations and the graded distribution induced by the action of the structure group, a criterion of global triviality of the graded principal bundle, and, finally, a way to reformulate Definition 6.1 avoiding the use of local trivializations. For this and the subsequent sections, we will adopt the following notation in order to simplify the discussion: if a ∈ g and V ⊂ Y is an open, then the restriction PY V (Φ∗ )a (Theorem 5.5) will be simply denoted by (Φ∗ )a . It is well known that if Y (X, G) is an ordinary principal bundle, then the set of vertical vectors at y ∈ Y is equal to the set of induced vectors at the same point. In the previous section, we saw that the same is true for graded principal bundles; however, it is not evident that this property remains valid for the sheaves of vertical and induced derivations. Nonetheless, as the following proposition confirms, this is indeed the case. Proposition 7.1. Let (Y, B) be a graded principal bundle over (X, C) with structure group (G, A). If E is the natural graded distribution induced by the free
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action of (G, A) on (Y, B), then E is equal to the sheaf of vertical derivations, E = Ver(π∗ , B). Proof. We show first that E ⊂ Ver(π∗ , B); to this end, it is sufficient to prove that for each a ∈ g, we have π∗ (Φ∗ )a = 0. Indeed, if f is a homogeneous element of C(U ), U ⊂ X open, we take: π ∗ π∗ (Φ∗ )a (f ) = (Φ∗ )a (π ∗ f ) = (id ⊗ a)(π ∗ f ⊗ 1A ) = 0, since a is primitive with respect to δe . Now the following argument on dimensions completes the proof. A derivation D ∈ Ver(π∗ , B)(V ), V = π∗−1 (U ), is characterized by the property: π ∗ [(π∗ D)f ] = D(π ∗ f ) = 0, ∀f ∈ C(U ). This means in terms of coordinates that D does not depend on the graded coordinates on V obtained by pulling-back the graded coordinates of U via π ∗ . Using the fact that π ∗ is an injection, we find that the dimension of Ver(π∗ , B)(V ) equals to dim(Y, B) − dim(X, C) = dim(G, A) = dim E(V ). The fact that a local trivialization φ is an isomorphism of A(G)-comodules is expressed by relation (6.1) but it is also reflected in the induced derivations. The following lemma makes this precise, providing a relation between them. Lemma 7.2. If φ : (V, B|V ) → (U, C|U ) × (G, A), V = π∗−1 (U ), is a local trivialization of (Y, B), then the following relation is true for each a ∈ g: φ∗ (Φ∗ )a = id ⊗ (R∗ )a . Proof. We show first that φ∗ (Φ∗ )a ∈ Der A(G). Indeed, if fC ∈ C(U ), we take: φ∗ (Φ∗ )a (fC ⊗ 1A ) = (φ−1 )∗ ◦ (Φ∗ )a ◦ φ∗ (fC ⊗ 1A ) = (φ∗ )−1 (Φ∗ )a (π ∗ fC ) = 0 . ˆ A(G) of the form It is then sufficient to calculate φ∗ (Φ∗ )a on elements of C(U )⊗ P 1C ⊗ fA for fA ∈ A(G). Taking into account relation (6.2) and if ∆A fA = i I i fA ⊗ J i fA , one finds: φ∗ (Φ∗ )a (1C ⊗ fA ) = (φ−1 )∗ (id ⊗ a)(ψ ∗ ⊗ id)∆A fA X i = (−1)|akI fA | (φ∗ )−1 ψ ∗ (I i fA )a(J i fA ) i
=
X
(−1)|akI
i
fA |
(1C ⊗ I i fA )a(J i fA )
i
= 1C ⊗ (id ⊗ a)∆A fA = (id ⊗ (R∗ )a )(1C ⊗ fA ) ,
THEORY OF CONNECTIONS ON GRADED PRINCIPAL BUNDLES
69
where we have used that φ∗ (1C ⊗ I i fA ) = ψ ∗ (I i fA ) implies (φ−1 )∗ ψ ∗ (I i fA ) = 1 C ⊗ I i fA . Next we discuss the notion of section of a graded principal bundle and we show that graded and ordinary sections exhibit several analogous properties. Definition 7.3. Let U ⊂ X be an open on the base manifold (X, C) of a graded principal bundle (Y, B). We call graded section of (Y, B) on U a morphism of graded manifolds s : (U, C|U ) → (Y, B) having the property s∗ ◦ π ∗ = id. We write also π ◦ s = id : (U, C|U ) → (U, C|U ). A first property of graded sections is that to each local trivialization, one can associate in a canonical way a graded section. More precisely: Lemma 7.4. Let φ : (V, B|V ) → (U, C|U ) × (G, A), V = π∗−1 (U ), be a local ˆ A(G) → C(U ) is defined as E(fC ⊗ fA ) = trivialization. Then, if E : C(U ) ⊗ −1 ∗ δe (fA )fC , the map E ◦ (φ ) : B(V ) → C(U ) defines a morphism of graded manifolds with the properties of a graded section. Proof. The fact that E is a morphism of graded manifolds is evident because we may write E = id ⊗ δe . Therefore, there exists a morphism of graded manifolds s∗ = E ◦ (φ−1 )∗ . Now if fC ∈ C(U ), we have: s : (U, C|U ) → (Y, B) such that (s∗ ◦ π ∗ )(fC ) = E (φ−1 )∗ π ∗ fC and since φ∗ (fC ⊗ 1A ) = π ∗ fC , we finally obtain s∗ ◦ π ∗ = id. Conversely now, consider a graded section s : (U, C|U ) → (Y, B). We wish to show that there exists a local trivialization φ : (V, B|V ) → (U, C|U ) × (G, A) canonically associated to s, V = π∗−1 (U ) ⊂ Y . To this end, we first define a ˆ A(G) by φ˜∗ = (s∗ ⊗ id) ◦ Φ∗ . morphism of graded algebras φ˜∗ : B(V ) → C(U ) ⊗ Let φ˜∗ = Φ∗ ◦ (s∗ ⊗ id) : C(U )◦ ⊗ A(G)◦ → B(V )◦ be the corresponding morphism of graded coalgebras. Clearly, the differentiable mapping φ˜∗ |U×G : U × G → V is bijective. Consider now the tangent of φ at the arbitrary point (x, g) ∈ U × G. ˆ A) is a tangent vector at (x, g), u ∈ If z = u ⊗ δg + δx ⊗ w ∈ T(x,g) (U × G, C|U ⊗ Tx (U, C|U ), w = Tg (G, A), then: φ˜∗ (z) = Φg∗ (s∗ u) + (Φs∗ x )∗ (w) . This implies that T(x,g) φ˜ is injective, because Φg∗ is an isomorphism and Φy∗ is injective for each y ∈ Y (the action is free). Thus, T(x,g) φ˜ is an injection between two vector spaces of the same dimension and hence an isomorphism. Using now Theorem 2.16 of [1], we conclude that φ˜ is an isomorphism of graded manifolds. Let now π1 : (U, C|U ) × (G, A) → (U, C|U ) and π2 : (U, C|U ) × (G, A) → (G, A) be the projections. Then: Proposition 7.5. Let φ : (V, B|V ) → (U, C|U ) × (G, A), V = π∗−1 (U ), be the morphism of graded manifolds defined as φ∗ = mB ◦ (π ∗ ⊗ ψ ∗ ), where ψ ∗ = (φ˜∗ )−1 ◦ π2∗ . Then, φ is a local trivialization of the graded principal bundle (Y, B).
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Proof. We show first that φ∗ is an isomorphism. To this end, consider the composition φ˜∗ ◦ φ∗ : φ˜∗ ◦ φ∗ = (s∗ ⊗ id) ◦ Φ∗ ◦ mB ◦ (π ∗ ⊗ ψ ∗ ) = mCA ◦ [(s∗ ⊗ id) ⊗ (s∗ ⊗ id)] ◦ (Φ∗ ⊗ Φ∗ ) ◦ (π ∗ ⊗ ψ ∗ ) = mCA ◦ [(s∗ ⊗ id) ◦ Φ∗ ◦ π ∗ ⊗ (s∗ ⊗ id) ◦ Φ∗ ◦ ψ ∗ ] = mCA ◦ [(s∗ ⊗ id) ◦ Φ∗ ◦ π ∗ ⊗ π2∗ ] . Using now the fact that (s∗ ⊗ id) ◦ Φ∗ ◦ π ∗ = π1∗ , we find that φ˜∗ ◦ φ∗ = id, which proves that φ∗ is also an isomorphism. It remains to show relation (6.2), or equivalently, ψ∗ ◦ Φ∗ = m◦A ◦ (ψ∗ ⊗ id). Since φ˜∗ is an isomorphism between B(V )◦ and P i i C(U )◦ ⊗ A(G)◦ , we may write each element b ∈ B(V )◦ as b = Φ∗ i s∗ c ⊗ a 0 , for ci ∈ C(U )◦ and ai0 ∈ A(G)◦ . If a ∈ A(G)◦ , we take: ! ! X i i s∗ c ⊗ a 0 ⊗ a ψ∗ Φ∗ (b ⊗ a) = ψ∗ Φ∗ Φ∗ i
= ψ∗ Φ∗
X
!
s∗ c ⊗ i
(ai0
? a)
i
= ψ∗ Φ∗ (s∗ ⊗ id)
X i
= π2∗
X
! c ⊗ i
(ai0
? a)
!
ci ⊗ (ai0 ? a)
i
=
X
ci (1C )ai0 ? a
i
= ψ∗ b ? a , P P since ψ∗ b = π2∗ ( i ci ⊗ ai0 ) = i ci (1C )ai0 .
By Lemma 7.4 and Proposition 7.5 if we set U = X, it is straightforward that the condition of global triviality of a principal bundle remains valid in the graded setting. Corollary 7.6. A graded principal bundle (Y, B) is globally isomorphic to the product (X, C) × (G, A) if and only if it admits a global section s : (X, C) → (Y, B). We observe here that Lemma 7.4 and Proposition 7.5 remain valid if we replace the graded principal bundle (Y, B) by a graded manifold (Y, B) on which the graded Lie group (G, A) acts freely to the right in such a way that the quotient (X, C) = (Y /G, B/A) is a graded manifold, and the projection π : (Y, B) → (X, C) is a
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submersion. In that case, one can construct via Lemma 7.4 and Proposition 7.5 the local trivializations of Definition 6.1. In other words: Theorem 7.7. A graded principal bundle is a graded manifold (Y, B) together with a free right action Φ : (Y, B) × (G, A) → (Y, B) of a graded Lie group (G, A) such that : (1) the quotient (X, C) = (Y /G, B/A) is a graded manifold, (2) the projection π : (Y, B) → (X, C) is a submersion. As an immediate application, we examine if the principal bundles formed by Lie groups and closed Lie subgroups possess graded analogs. Example 7.8. Consider a graded Lie group (G, A) and a closed graded Lie subgroup (H, D) of (G, A). The natural right action Φ : (G, A) × (H, D) → (G, A) is given by Φ∗ = (id ⊗ i∗ ) ◦ ∆A , where i : (H, D) → (G, A) is the inclusion. Furthermore, we know that the quotient (G/H, A/D) is a graded manifold and the projection (G, A) → (G/H, A/D) is a submersion [1]. Now if g ∈ G, the morphism Φg∗ : D(H)◦ → A(G)◦ is given by Φg∗ (d) = δg ? d = Lg∗ (d), ∀d ∈ D(H)◦ . As a result, Φg∗ is injective, so the action Φ is free and Theorem 7.7 holds: (G, A) is a graded principal bundle over (G/H, A/D) with typical fibre (H, D). 8. Graded Connections We have seen that on each graded principal bundle there always exists a natural distribution induced by the action of the structure group which is equal to the sheaf of vertical derivations. The choice of a connection is essentially the choice of a complementary distribution. More precisely: Definition 8.1. Let (Y, B) be a graded principal bundle with structure group (G, A), over the graded manifold (X, C). A graded connection on (Y, B) is a regular distribution H ⊂ Der B of dimension dim H = dim(X, C) such that: (1) H ⊕ Ver(π∗ , B) = Der B, π : (Y, B) → (X, C) is the projection, (2) H is (G, A)-invariant. Let us explain the second statement in the previous definition: H will be called (G, A)-invariant if, for each open U ⊂ X and D ∈ H(π∗−1 (U )), the derivations Φ∗g D and [(Φ∗ )a , D] = L(Φ∗ )a D belong also to H(π∗−1 (U )), ∀g ∈ G, ∀a ∈ g. In order to put these conditions in a more compact form, we introduce the following notation Φ∗g D , if a = δg , g ∈ G (Φ∗ )a D = L ∗ D, if a ∈ g . (Φ )a Then, H will be (G, A)-invariant if (Φ∗ )a D ∈ H(π∗−1 (U )), for each element a grouplike or primitive with respect to δe . We can now reformulate this notion in terms of g-valued graded differential forms.
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Given the graded connection H sur (Y, B), we have: H(Y ) ⊕ E(Y ) = Der B(Y ), E = Ver(π∗ , B) (see Proposition 7.1), and each derivation ξ ∈ Der B(Y ) decomposes P as ξ = ξ H + i f i (Φ∗ )ei , where ξ H ∈ H(Y ), {ei } is a basis of g and f i ∈ B(Y ). P Then, we define a 1-form ω ∈ Ω1 (Y, B, g) setting (ξ|ω) = i f i ⊗ ei . Let us now calculate (Φ∗ )a ω, where by definition Φ∗g ω , if a = δg , g ∈ G (Φ∗ )a ω = L ∗ ω, if a ∈ g . (Φ )a Consider first the case a = δg . Then, by Eq. (4.2) we take ! ! X ∗ ∗ i ∗ f (Φ )ei |ω (ξ|Φg ω) = Φg ⊗ id Φg∗ i
and using the fact that Φg∗ (Φ∗ )ei = (Φ∗ )ADg−1 ∗ (ei ) , we obtain: (ξ|Φ∗g ω)
=
(Φ∗g
⊗ id)
X
! Φ∗g−1 f i
⊗ ADg−1 ∗ (ei )
= (id ⊗ ADg−1 ∗ )(ξ|ω) .
i
or Φ∗g ω = (id ⊗ ADg−1 ∗ ) ◦ ω .
(8.1)
(ξ|L(Φ∗ )a ω) = (−1)|akξ| [((Φ∗ )a ⊗ id) (ξ|ω) − ([(Φ∗ )a , ξ]|ω)]
(8.2)
If now a ∈ g, we have:
and it is sufficient to examine two cases: (1) ξ = ξ H (horizontal derivation): it is then immediate that (ξ|L(Φ∗ )a ω) = 0 (2) ξ = (Φ∗ )b , b ∈ g (vertical derivation): (ξ|L(Φ∗ )a ω) = (−1)|akb|+1 (Φ∗ )[a,b] |ω = − ((Φ∗ )b |(id ⊗ AD∗a )ω) where we have defined (ξ|(id ⊗ AD∗a )) ◦ ω) = (−1)|akξ| (id ⊗ AD∗a )(ξ|ω). We may thus write: L(Φ∗ )a ω = −(id ⊗ AD∗a ) ◦ ω .
(8.3)
Summarizing the previous results on the g-valued graded differential form ω, we have: P i P i ∗ (1) i f (Φ ei |ω) = i f ⊗ ei and |ω| = (1, 0), (2) (Φ∗ )a ω = (id ⊗ AD∗s◦A (a) ) ◦ ω, for each a group-like or primitive with respect to δe . Conversely now, if ω is a g-valued graded differential form with the previous two properties, then ker ω = H is a regular distribution on (Y, B) such that H ⊕ Ver(π∗ , B) = DerB. Consider D ∈ H(V ), V = π∗−1 (U ); if a = δg and ωV means the
THEORY OF CONNECTIONS ON GRADED PRINCIPAL BUNDLES
73
pull-back of ω under the inclusion (V, B|V ) ,→ (Y, B), we take: (D|Φ∗g ωV ) = (Φ∗g ⊗ id)(Φg∗ D|ωV ) = (id ⊗ ADg−1 ∗ )(D|ωV ) = 0; thus, Φg∗ D ∈ H(V ). Replacing g by g −1 , this gives Φ∗g D ∈ H(V ). On the other hand, if a ∈ g, we find: (D|L(Φ∗ )a ωV ) = −(−1)|akD| ([(Φ∗ )a , D]|ωV ) = 0. This means that L(Φ∗ )a D belongs also to H(V ). In summary: (Φ∗ )a D ∈ H(V ). We have thus proved the proposition. Proposition 8.2. The graded connection H of Definition 8.1 is described equivalently by a g-valued graded differential 1-form ω ∈ Ω1 (Y, B, g) of total Z2 degree zero such that : P i P i ∗ (1) i f (Φ )ei |ω = i f ⊗ ei , (2) (Φ∗ )a ω = (id ⊗ AD∗s◦A (a) ) ◦ ω, for each element a group-like or primitive with respect to δe . We call ω graded connection form. Graded principal bundles are by definition locally isomorphic to products of open graded submanifolds of the base space by the structure graded Lie group. So, it is quite natural to ask how one can construct graded connections on the trivial graded principal bundle of Example 6.3. Example 8.3. Let (Y, B) = (X, C) × (G, A) be as in Example 6.3. The right action Φ of (G, A) on (Y, B) is such that Φ∗ = id ⊗ ∆A . Consider now a g-valued P 1-form β ∈ Ω1 (X, C, g) and a basis {ek } of g. One can write β = i β k ⊗ ek , β k ∈ Ω1 (X, C) with |β| = (1, 0). If ξ ∈ Der C(X) and η ∈ Der A(G), we define a g-valued 1-form ω ∈ Ω1 (Y, B, g) as: X (ξ|β k ) ⊗ ((L∗ )ek |θ) + 1C ⊗ (η|θ) . (8.4) (ξ ⊗ 1A + 1C ⊗ η|ω) = k
In the previous relation L : (G, A) × (G, A) → (G, A) is the left action of (G, A) on itself, L∗ = ∆A and θ ∈ Ω1 (G, A, g) the graded Maurer-Cartan form on (G, A). Recall that the derivations (L∗ )ek and (R∗ )a are given by Theorem 3.8. We shall check now if the form ω in (8.4) has the properties of a connection form. (1) If a ∈ g, then (Φ∗ )a = id ⊗ (R∗ )a and ((Φ∗ )a |ω) = 1C ⊗ ((R∗ )a |θ) = 1B ⊗ a. Further, |ω| = (1, 0) because |β| = (1, 0) and |θ| = (1, 0). (2) Consider now an element g ∈ G; we shall calculate Φ∗g ω. To this end, we use formula (4.2), as well as the fact that each graded Lie group is parallelizable (Theorem 3.13), so it is sufficient to take η = (R∗ )a , a ∈ g. Thus, if we set D = ξ ⊗ 1A + 1C ⊗ (R∗ )a and a0 = ADg−1 ∗ (a), we find (D|Φ∗g ω) = (Φ∗g ⊗ id)(ξ ⊗ 1A + 1C ⊗ (R∗ )a0 |ω) X (ξ|β k ) ⊗ (Rg∗ ⊗ id)((L∗ )ek |θ) + 1B ⊗ ADg−1 ∗ (a) = k
=
X
(ξ|β k ) ⊗ (id ⊗ ADg−1 ∗ )((L∗ )ek |θ) + 1B ⊗ ADg−1 ∗ (a)
k
= (ξ ⊗ 1A + 1C ⊗ (R∗ )a |(id ⊗ ADg−1 ∗ ) ◦ ω) .
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T. STAVRACOU
Note that in the previous calculation we used that Rg∗ θ = (id ⊗ ADg−1 ∗ ) ◦ θ and Rg∗ (L∗ )ek = (L∗ )ek , which can be verified straightforwardly. (3) Let finally a ∈ g; we calculate the Lie derivative L(Φ∗ )a ω = (Φ∗ )a ω. One can use formula (8.2), which is valid for ω because ((Φ∗ )a |ω) = 1B ⊗ a. If we set now D = ξ ⊗ 1A + 1C ⊗ (R∗ )b and x = |akβ k |, we have: (D|(Φ∗ )a ω) = (−1)|akξ| ((Φ∗ )a ⊗ id)(ξ ⊗ 1A |ω) − (−1)|akb| (1C ⊗ (R∗ )[a,b] |ω) = (−1)x (ξ|β k ) ⊗ ((R∗ )a ⊗ id)((L∗ )ek |θ) − (−1)|akb| 1B ⊗ [a, b] = −(−1)x (ξ|β k ) ⊗ (id ⊗ AD∗a )((L∗ )ek |θ) − (−1)|akb| 1B ⊗ [a, b] = −(ξ ⊗ 1A |(id ⊗ AD∗a ) ◦ ω) − (1C ⊗ (R∗ )b |(id ⊗ AD∗a ) ◦ ω) which implies that (Φ∗ )a ω = −(id ⊗ AD∗a ) ◦ ω. In the previous calculation, summation over the repeated index k is understood. Note also that that we used the property (R∗ )a θ = −(id ⊗ AD∗a ) ◦ θ of the graded Maurer-Cartan form. Remark 8.4. Let s : (X, C) → (Y, B) be a graded section for the previous ˆ A(G) → C(X) of graded commuexample. Then we have a morphism s∗ : C(X) ⊗ ∗ tative algebras and therefore, a morphism σ : A(G) → C(X) given by σ ∗ (fA ) = s∗ (1C ⊗ fA ). This defines a morphism of graded manifolds σ : (X, C) → (G, A). Let now ξ ∈ Der C(X) and η ∈ Der A(G) be two σ-related derivations; then the derivations ξ 0 = ξ ⊗ 1A + 1C ⊗ η and ξ are s-related and one can use Lemma 2.4 in order to calculate the pull-back s∗ ω ∈ Ω1 (X, C, g). The result is: X β k (σ ∗ ⊗ id)((L∗ )ek |θ) + σ ∗ θ , s∗ ω = k
as one easily finds. One can use Example in order to construct graded connections on general graded principal bundles. Indeed, let (Y, B) be such a bundle with base space (X, C) and structure group (G, A), {Ui }i∈Λ a locally finite open covering of X by trivializing open sets (in the sense of Definition 6.1), and {fi } a graded partition of P unity subordinate to {Ui } : fi ∈ C(X)0 , suppfi ⊂ Ui and i fi = 1C [1]. Let also Vi = π∗−1 (Ui ) and ωi be the graded connection 1-forms that one can construct on (Vi , B|Vi ) ∼ = (Ui , C|Ui ) × (G, A) as in the previous example (see also Definition 6.1). If now D ∈ Der B(Y ), then we define (D|ωi ) ∈ B(Vi ) ⊗ g as (D|ωi ) = (Di |ωi ), Di = D|Vi . Then we have also (D|ωi ) · (π ∗ fi |Vi ) ∈ B(Vi ) ⊗ g and so, there exists an element of B(Y ) ⊗ g, which we denote by (D|ωi )π ∗ fi , such that [(D|ωi )π ∗ fi ]|Vi = (D|ωi )(π ∗ fi |Vi ) and whose support is a subset of Vi . ˜k ) we ˜k } is another open covering of X, then setting Wk = π −1 (U If now {U ∗ have an open covering of Y and for k fixed, Wk ∩ Vi is non-empty only for finitely many of the Vi . Taking the restrictions [(D|ωi )π ∗ fi ]|Wk ∈ B(Wk ) ⊗ g, only finite P many terms will be non-zero as i runs over Λ. Thus, the sum i [(D|ωi )π ∗ fi ]|Wk is
THEORY OF CONNECTIONS ON GRADED PRINCIPAL BUNDLES
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finite and well-defined as an element of B(Wk ) ⊗ g. Furthermore, the restrictions of such elements to the intersections Wk ∩ W` coincide and this means, by the sheaf properties of B, that there exists a unique element of B(Y ) ⊗ g, whose restriction to Wk gives the previous element of B(Wk ) ⊗ g. We denote this unique element by (D|ω) and by its linearity on D it determines a g-valued graded 1-form ω. The form ω is a graded connection form; this results without difficulty from the properties Φ∗g π ∗ fi = π ∗ fi and (Φ∗ )a π ∗ fi = 0, g ∈ G, a ∈ g, as well as from the fact that for each i ∈ Λ, ωi is a graded connection. We have thus proved the following: 8.5. Existence Theorem for graded connections. On each graded principal bundle there exists an infinity of graded connections. 9. Graded Curvature In ordinary differential geometry, one can define in a canonical way, for each connection, a Lie superalgebra-valued 2-form, the curvature form. In this section, we will define the curvature in the graded setting, using the notion of graded connection, previously developed. Let then ω ∈ Ω1 (Y, B, g) be a graded connexion form on the graded principal bundle (Y, B). Fix a basis {ek } of the Lie superalgebra g; then for each derivation P ξ ∈ Der B(Y ) one can find ξ H ∈ H(Y ) and f k ∈ B(Y ) such that ξ = ξ H + i f k (Φ∗ )ek . We have thus a canonical projection Der B(Y ) → H(Y ), ξ 7→ ξ H ; we call ξ H horizontal part of ξ. Clearly, this mechanism of taking the horizontal part of a derivation can be applied in the same way if instead of Y we put an open V ⊂ Y . Consider now a g-valued graded differential form φ ∈ Ωr (Y, B, g) and let φH be defined as (ξ1 , . . . , ξr |φH ) = (ξ1H , . . . , ξrH |φ), ξi ∈ Der B(Y ), i = 1, . . . , r. Definition 9.1. The covariant exterior derivative of φ ∈ Ωr (Y, B, g) is the g-valued graded differential form Dω φ ∈ Ωr+1 (Y, B, g) defined as Dω φ = (dφ)H . The curvature of the graded connection ω ∈ Ω1 (Y, B, g) is the covariant exterior derivative Dω ω; we use the notation F ω = Dω ω. We will next study in more detail the properties of F ω . As a general observation, we may say that F ω has, formally, properties analogous to those of the ordinary curvature. Proposition 9.2. The graded curvature F ω is given by the relation 1 F ω = dω + [ω, ω] . 2
(9.1)
This is the graded structure equation. Proof. It is sufficient to prove that for each ξ1 , ξ2 ∈ Der B(Y ), the following is true: 1 (ξ1H , ξ2H |dω) = (ξ1 , ξ2 |dω) + (ξ1 , ξ2 |[ω, ω]) . 2
(9.2)
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T. STAVRACOU
As we have seen |ω| = (1, 0); however, ω is not a homogeneous element with respect to the (Z ⊕ Z2 ⊕ Z2 )-grading of Ω(Y, B, g). More precisely, ω = ω0 + ω1 , where deg(ω0 ) = (1, 0, 0), deg(ω1 ) = (1, 1, 1) and if {ei , ej } is a basis of g with P i P j i |ei | = 0, |ej | = 1, we may write: ω = i ω ⊗ ei + j ω ⊗ ej , with ω ∈ 1 j 1 i j Ω (Y, B)0 , ω ∈ Ω (Y, B)1 . In particular, ω , ω vanish on horizontal derivaP j P i tions and if we decompose a ∈ g as a = i a ei + j a ej , we find immediately ((Φ∗ )a |ω i ) = ai 1B , ((Φ∗ )a |ω j ) = aj 1B . Let us now calculate the term 1 2 (ξ1 , ξ2 |[ω, ω]). P P P P (ξ1 , ξ2 |[ω, ω]) = ξ1 , ξ2 i ω i ⊗ ei + j ω j ⊗ ej , p ω p ⊗ ep + q ω q ⊗ eq =
i p 1+|ξ1 kξ2 | i p |ω )(ξ |ω ) + (−1) (ξ |ω )(ξ |ω ) ⊗ [ei , ep ] (ξ 1 2 2 1 i,p
P
+ + −
P i,q
P
(ξ1 |ω i )(ξ2 |ω q ) + (−1)1+|ξ1 kξ2 | (ξ2 |ω i )(ξ1 |ω q ) ⊗ [ei , eq ]
|ξ2 | j p x j p (ξ |ω )(ξ |ω ) + (−1) (ξ |ω )(ξ |ω ) ⊗ [ej , ep ] (−1) 1 2 2 1 j,p
P j,q
(−1)|ξ2 | (ξ1 |ω j )(ξ2 |ω q ) + (−1)x (ξ2 |ω j )(ξ1 |ω q ) ⊗ [ej , eq ] . (9.3)
In the previous calculation, the indices i, p label the even elements while j, q the odd ones. We also used the fact that if β1 , β2 ∈ Ω1 (Y, B), then (ξ1 , ξ2 |β1 β2 ) = (−1)|ξ2 kβ1 | (ξ1 |β1 )(ξ2 |β2 ) + (−1)1+|ξ1 kξ2 |+|ξ1 kβ1 | (ξ2 |β1 )(ξ1 |β2 ), see relation 4.1.9 of [1], and we have set x = 1 + |ξ1 kξ2 | + |ξ1 |. On the other hand, the term (ξ1 , ξ2 |dω) is calculated via (ξ1 , ξ2 |dω) = (ξ1 ⊗ id)(ξ2 |ω) − (−1)|ξ1 kξ2 | (ξ2 ⊗ id)(ξ1 |ω) − ([ξ1 , ξ2 ]|ω), which is an immediate generalization of 4.3.10 of [1]. We distinguish now the following cases: (1) ξ1 , ξ2 : horizontal ⇒ ξ1 = ξ1H , ξ2 = ξ2H . The graded structure equation holds, since 12 (ξ1 , ξ2 |[ω, ω]) = 0 and (ξi |ω) = 0. (2) ξ1 : horizontal, ξ2 = (Φ∗ )a , a ∈ g. The left-hand side of (9.2) is zero because ξ2H = 0. The right-hand side of the same equation reads: (ξ1 , (Φ∗ )a |dω) = (ξ1 ⊗ id)((Φ∗ )a |ω) − ([ξ1 , (Φ∗ )a ]|ω) = 0, because by the definition of the graded connexion, if ξ is horizontal, [ξ, (Φ∗ )a ] is horizontal too. Furthermore, it is clear that in this case, (9.3) gives (ξ1 , ξ2 |[ω, ω]) = 0. (3) ξ1 = (Φ∗ )a , ξ2 = (Φ∗ )b , a, b ∈ g. Clearly, the left-hand side of (9.2) is zero. Examining now the cases |a| = |b| = 0, |a| = |b| = 1 and |a| = 1, |b| = 0, we find always that the right-hand side is also zero. By the graded structure equation, it is clear that ω and F ω have the same Z2 total degree: |F ω | = (2, 0); therefore, F ω = F0ω + F1ω with F0ω ∈ Ω2 (Y, B)0 ⊗ g0 , F1ω ∈ Ω2 (Y, B)1 ⊗ g1 . Bianchi’s identity is a well-known property satisfied by the curvature in differential geometry. Using the generalization of the covariant derivative in the graded
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setting and the previous theorem, we may establish an analogous property in the context of graded manifolds. Proposition 9.3 (Bianchi’s Identity). Dω F ω = 0. Proof. Let us first calculate the differential dF ω . Using the graded structure equation and the fact that [ω, ω], ω = 0 (Jacobi identity), we find easily: dF ω = =
1 d[ω, ω] 2 1 ([dω, ω] − [ω, dω]) 2
= [dω, ω] = [F ω , ω] . Thus, dF ω = [F ω , ω] and (ξ1 , ξ2 , ξ3 |Dω F ω ) = (ξ1H , ξ2H , ξ3H |[F ω , ω]) = 0, because ω vanishes on horizontal derivations. We will show now that the graded curvature F ω satisfies the second property of the connexion ω described in Proposition 8.2. To this end, consider first a = δg , g ∈ G. Then: 1 Φ∗g F ω = Φ∗g dω + [ω, ω] 2 1 = dΦ∗g ω + Φ∗g [ω, ω] 2 1 = (id ⊗ ADg−1 ∗ )dω + (id ⊗ ADg−1 ∗ )[ω, ω] 2 = (id ⊗ ADg−1 ∗ )F ω . Suppose now that a ∈ g homogeneous; thanks to Lemma 4.2, direct calculation gives: (Φ∗ )a F ω = L(Φ∗ )a F ω 1 = L(Φ∗ )a dω + L(Φ∗ )a [ω, ω] 2 = dL(Φ∗ )a ω +
1 [L(Φ∗ )a ω, ω] + [ω, L(Φ∗ )a ω] 2
1 ([−(id ⊗ AD∗a )ω, ω] + [ω, −(id ⊗ AD∗a )ω]) 2 1 = − (id ⊗ AD∗a )dω + (id ⊗ AD∗a )[ω, ω] 2 = −d(id ⊗ AD∗a )ω +
= −(id ⊗ AD∗a )F ω , because |ω| = (1, 0), |L(Φ∗ )a | = (0, |a|). We have thus proved the following:
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Property 9.4. If a ∈ A(G)◦ is a group-like or primitive element with respect to δe , we have (Φ∗ )a F ω = (id ⊗ AD∗s◦A (a) ) ◦ F ω . We finally prove that the graded curvature provides a criterion for checking whether the horizontal distribution is involutive or not. Proposition 9.5. The graded horizontal distribution is involutive if and only if the graded curvature F ω of the connexion ω is zero: [H, H] ⊂ H ⇔ F ω = 0. Proof. It is sufficient to check the involutivity of H(Y ). If ξ, η ∈ H(Y ) and H is involutive, then: ([ξ, η]|ω) = 0 ⇒ (ξ, η|dω) + 12 (ξ, η|[ω, ω]) = 0 ⇒ (ξ, η|F ω ) = 0. But F ω vanishes identically on vertical derivations by its definition (relation (9.2)); thus F ω = 0. Conversely, suppose that F ω = 0. Then for each ξ, η ∈ H(Y ), we find: (ξ, η|F ω ) = 0 ⇒ (ξ, η|dω) + 12 (ξ, η|[ω, ω]) = 0 ⇒ (ξ, η|dω) = 0 ⇒ −([ξ, η]|ω) = 0, which implies that the derivation [ξ, η] is also horizontal, [ξ, η] ∈ H(Y ). 10. Concluding Remarks Consider a graded principal bundle (Y, B) equipped with a connection form ω ∈ Ω1 (Y, B, g). We know from the general theory of graded differential forms [1], that there always exists an algebra morphism κ : Ω(Y, B) → Ω(Y ) defined as follows: if i : (Y, C ∞ ) → (Y, B) is the morphism of graded manifolds determined by B(Y ) 3 f 7→ f˜ ∈ C ∞ (Y ) (see exact sequence (2.1)), then i∗ is just κ. On the other hand, the decomposition g = g0 ⊕ g1 induces a canonical projection π0 : g → g0 . So we have a linear map κ0 = κ ⊗ π0 : Ω(Y, B, g) → Ω(Y ) ⊗ g0 . P P i i α ⊗ e ∈ Ω(Y, B, g), then κ (α) = κ (α ) ⊗ (ei )0 , Explicitly, if α = i 0 0 i i where (αi )0 ∈ Ω(Y, B)0 and (ei )0 ∈ g0 are the even elements in the development of α; moreover, we easily realize that κ0 is not a (Z ⊕ Z2 )-graded Lie algebra morphism. As we have seen in the proof of Proposition 9.2, ω can be decomposed as ω = ω0 + ω1 , where ω0 ∈ Ω1 (Y, B)0 ⊗ g0 and ω1 ∈ Ω1 (Y, B)1 ⊗ g1 . Then P clearly, κ0 (ω) = κ0 (ω0 ) = i κ(ω i ) ⊗ ei , where i labels the even elements. Using the fact that the derivations induced on (Y, B) and (Y, C ∞ ) by the right actions of (G, A) and (G, C ∞ ) respectively (according to Theorem 3.8) are i-related, as well as the defining properties of a graded connection form, one can prove that κ0 (ω) is a connection form on the ordinary principal bundle (Y, C ∞ ). Furthermore, the curvatures of ω and κ0 (ω) are related through κ0 (F ω ) = F κ0 (ω) . The previous observations suggest that the connection theory on graded principal bundles is the suitable framework for the mathematical formulation of super-gauge field theories. The fact that the graded connection ω splits always as ω = ω0 + ω1 with κ0 (ω0 ) being a usual connection form, incorporates automatically the idea of supersymmetric partners: ω0 corresponds to the gauge potential of an ordinary Yang–Mills theory, while ω1 corresponds to its supersymmetric partner. A very interesting feature of this approach is that there is no graded connection form ω for which one of the terms ω0 or ω1 is zero, when the even and odd dimensions
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of the structure group are not zero. In physics terminology, all gauge potentials have super-partners. Similarly for the curvature F ω , the super-gauge field, we have F ω = F0ω +F1ω , F0ω ∈ Ω2 (Y, B)0 ⊗ g0 , F1ω ∈ Ω2 (Y, B)1 ⊗g1 . Thus, our approach sets the gauge potentials (resp. fields) and its supersymmetric partners on the same footing: they both “live” in a Lie superalgebra g as components of the same connection form (resp. curvature form). This is an essential difference between our approach and the standard treatment of this problem by means of DeWitt’s or Roger’s supermanifolds, (see [17] and references therein), where the connections take values in the even part of the Z2 -graded Lie module corresponding to a Lie supergroup. Acknowledgements I would like to thank Professor R. Coquereaux for his critical reading of the manuscript and for many stimulating discussions. References [1] B. Kostant, “Graded manifolds, graded Lie theory and prequantization”, Differential Geometric Methods in Mathematical Physics, Lecture Notes in Math. 570 pp. 177–306, Berlin, Heidelberg, New York: Springer-Verlag, 1977. [2] F. Berezin and D. Leites, “Supermanifolds”, Soviet Math. Dokl. 16 (1975) 1218–1222. [3] D. Leites, “Introduction to the theory of supermanifolds”, Russ. Math. Surv. 35 (1980) 1–64. [4] F. Berezin and M. Marinov, “Particle spin dynamics as the Grassmann variant of classical mechanics”, Ann. Phys. 104 (1977) 336–362. [5] A. L. Almorox, “Supergauge theories in graded manifolds”, Differential Geometrical Methods in Mathematical Physics Proc., Salamanca 1985, eds. P. L. Garc´ıa, A. P´erezRend´ on, Springer LNM 1251. [6] A. L. Almorox, “The bundle of graded frames”, Rend. Sem. Mat. Univ. Politech. Torino 43 (1985) 405–426. [7] C. Bartocci, U. Bruzzo and D. H. Ruip´ erez, The Geometry of Supermanifolds, Kluwer Academic Publ., 1991. [8] U. Bruzzo and R. Cianci, “Mathematical theory of super fibre bundles”, Class. Quantum Grav. 1 (1984) 213–226. [9] A. Jadczyk and K. Pilch, “Superspaces and supersymmetries”, Commun. Math. Phys. 78 (1981) 373–390. [10] A. Rogers, “A global theory of supermanifolds”, J. Math. Phys. 21 (1980) 1352–1365. [11] A. Rogers, “Graded manifolds, supermanifolds and infinite-dimensional Grassmann algebras”, Commun. Math. Phys. 105 (1986) 375–384. [12] B. DeWitt, Supermanifolds, Cambridge Univ. Press, 1992. [13] D. H. Ruip´erez and J. Mu˜ noz Masqu´e, “Global variational calculus on graded manifolds I: Graded jet bundles, structure 1-forms and graded infinitesimal contact transformations”, J. Math. Pures et Appl. 63 (1984) 283–309 and “Global variational calculus on graded manifolds II”, J. Math. Pures et Appl. 63 (1985) 87–104. [14] S. Montgomery, Hopf Algebras and their Actions on Rings, CBMS, Regional Conf. Series in Mathematics, Number 82, Amer. Math. Soc., 1993. [15] R. Giachetti and R. Ricci, “R-actions, derivations, and Frobenius theorem on graded manifolds”, Adv. Math. 62 (1986) 84–100. [16] S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, Interscience tracts in Pure and Appl. Math., John Wiley & Sons, 1963. [17] M. Grasso and P. Teofilatto, “Gauge theories, flat superforms and reduction of super fibre bundles”, Rep. Math. Phys. 25 (1987) 53–71.
EXAMPLES OF NONLINEAR CONTINUOUS TENSOR PRODUCTS OF MEASURE SPACES AND NON-FOCK FACTORIZATIONS∗ To Professor Jack Feldman on his 70th anniversary B. S. TSIRELSON School of Mathematics, Tel Aviv University Tel Aviv 69978, Israel E-mail :
[email protected]
A. M. VERSHIK∗ St. Petersburg branch of Mathematical Institute of Russian Academy of Science Fontanka 27, St. Petersburg 191011 Russia and Laboratoire de Probabilites, Universite Paris-6, France E-mail :
[email protected] Received 19 November 1996 Revised 15 April 1997
Contents 0. Introduction 81 1. Continuous Tensor Products of Hilbert Spaces and Measure Spaces 85 2. Measures on Flabby Sheaves and Nets of Borel Spaces 95 3. Inverse Limit Constructions and Criteria of Nonlinearity and Continuity of Factorizations 101 4. Nonlinearizable Factorizations Over Zero-Dimensional Space 114 5. Nonlinearizable Factorizations Over One-Dimensional Space 122 Appendix A. Logarithm 128 Appendix B. Topological Base and Boolean Base 134 Appendix C. Around the Asymptotically Fixed Point 140
0. Introduction The aim of this work is to construct a number of examples for random objects that are nonlinear generalizations of random processes with independent values such as white noise or Levy–Khinchin processes. The corresponding nonlinearizable (non-Fock) bosonic factorizations of Hilbert spaces are also considered. Our problem is related to: — continuous products of probability spaces, beyond those of Levy–Khinchin processes; — continuous tensor products of Hilbert spaces, beyond those of Fock type, and operator factorizations of type I. ∗ Partially
supported by the grant Russian Fund FI 96-01-00921 and MAE (France) through CNRS during visit Univ. Paris-6. 81
Reviews in Mathematical Physics, Vol. 10, No. 1 (1998) 81–145 c World Scientific Publishing Company
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We hope that our examples will be applied to: — representations of the gauge groups and C ∗ -algebras; — invariant measures on the space of solutions for some nonlinear hyperbolic equations, and nonlinear functional limit theorems; — families of sub-σ-fields on a measure space. Our starting point was the fundamental work of von Neumann on operator factorizations and the Fock exponent — a basic example of a continuous tensor product of Hilbert spaces (in other words, a type I operator factorization over the Boolean algebra of all measurable sets). Essential progress was then made by Araki and Woods [2]. All type I factorizations over an atomic Boolean algebra were described, and a necessary and sufficient criterion was given for a continuous factorization to be of the Fock type: this is the case when the set of so-called factorizable vectors is total in the Hilbert space considered. For the continuous case a factorization in the sense of [2] is defined over the complete Boolean algebra of all classes (mod 0) of measurable sets on a standard measure space. Probabilistic and measure-theoretical counterparts of those notions and constructions are discrete (for atomic cases) and continuous tensor products of measure spaces. Canonical examples are classical processes of Wiener or, more generally, Levy–Khinchin type. It is well known that, in the sense of Gel’fand–Ito, generalized processes with independent values (like white noise and Levy–Khinchin processes, see [14]) naturally generate continuous tensor products of measure spaces, and consequently factorizations of L2 spaces. Those factorizations of measure spaces need not be isomorphic in the measure-theoretical sense (for example, Poisson and Wiener processes generate non-isomorphic factorizations of measure spaces) but corresponding factorizations of L2 spaces (up to isomorphisms of Hilbert spaces) are always isomorphic to Fock factorizations of appropriate dimensions. It is not so difficult to find a factorization which is not isomorphic to a Fock factorization because of the absence of factorizable vectors (see eg. [23]) but it is much more complicated to construct such a factorization that is at the same time a factorization of measure spaces. This is the goal of our paper. In this paper examples of continuous tensor products of measure spaces over zero- and one-dimensional bases are constructed, which are not isomorphic in the measure-theoretical sense to classical continuous tensor products and generating factorizations of L2 - spaces, non-isomorphic to Fock factorizations. These main results are contained in Secs. 4 and 5. The factorization over R, constructed in Sec. 5, is of special interest. It can be obtained from a nonlinear hyperbolic system of partial differential equations in the same way as white noise from D’Alembert equation. The construction is symmetric under shifts of R. The factorization of Sec. 4 constructed over a Cantor set is invariant under measure preserving homeomorphisms of the set. In more invariant terms, we construct factorizations of the algebra of all bounded operators in the Hilbert space which are also factorizations of some maximal Abelian subalgebras, non-isomorphic to the Fock factorization because of the absence of
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non-trivial factorizable vectors. It is still unknown whether or not a non-Fock factorization — a factorization containing insufficient factorizable vectors (or even none) — exists over the complete Boolean algebra of all measurable sets (mod 0). We deal with smaller Boolean algebras that, unlike the complete one, reflect the topology of the underlying space; we have succeeded for dimensions 0 and 1 only. The notion of a measure space factorization, or a continuous tensor product of measure spaces, is so important that its definition (given in Sec. 1) will be outlined immediately. A measure (space) factorization over a Boolean algebra A (the base) consists of a measure space (Ω, F , P ), P (Ω) = 1, and a family {F(A)}A∈A of sub-σ-fields F(A) ⊂ F , satisfying the following conditions. F(A1 ∧ A2 ) = F(A1 ) ∩ F(A2 ) ; F(A1 ∨ A2 ) is the σ-field generated by F(A1 ) and F(A2 ) ; if A1 ∧ A2 = 0, then F(A1 ) and F(A2 ) are independent ; if An ↑ 1, then F(An ) ↑ F . Each measure factorization determines in a canonical way a type I factorization for the corresponding Hilbert space L2 . Well-known examples result from processes with independent values (Levy– Khinchin processes). For such a process, its measure space is factorized into a continuous direct product, and the corresponding L2 space into a continuous tensor product. The latter factorization is always of Fock type (cf. [20]). The Gaussian case (white noise) results in the well known isomorphism of the Wiener–Ito space and the Fock space, making explicit the correspondence between the measure factorization and the Hilbert factorization (see Sec. 1). Measure space factorizations generated by Levy–Khinchin processes were studied by Feldman [13] and the question of existence of other factorizations was posed, which he attributed to Kakutani. A measure-theoretic counterpart of the notion of a factorizable vector is as follows. A measurable function f on (Ω, F , P ) is called an additive (resp. multiplicative) integral, if for any finite partition of unity A1 , . . . , An (that is, A1 ∨ · · · ∨ An = 1 and Ak ∧ Al = 0 for k 6= l) there is a decomposition f = f1 + · · · + fn (resp. f = f1 · · · fn ), each fk being F(Ak )-measurable. We will construct measure factorizations admitting no integrals (neither additive, nor multiplicative), except for constants. This leads automatically to non-Fock factorizations for the corresponding L2 spaces, since factorizable vectors are not total: they are exhausted by the one dimensional space of constants. The above Boolean algebra A will be formed by appropriate regions of a topological space X. The natural context for constructing our measures is given by the sheaves theory: each measure will be constructed on the space of sections of a flabby sheaf of Borel spaces or, more generally, a net of Borel spaces. As far as we know, these notions have not been studied before. A net of Borel spaces is a commutative counterpart of the well-know notion of a net of local algebras (of quantum observables), see Sec. 2. The very construction of the required sheaves is not trivial, it is a
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kind of inverse spectra. The classical space of Schwarz distributions can be treated in this framework as a linear example. Factorizations invariant under a transformation group are especially interesting. Note that the Fock factorization is invariant under the group of all measure preserving (or even preserving the type of the measure) transformations of the base. In this connection, the following conjecture may be proposed: invariance under the whole group of measure preserving transformations is possible only for Fock factorization (Ito factorization for the commutative case). Our zero dimensional examples (see Sec. 4) give us nonlinear factorizations over the field Qp of p-adic numbers, invariant under the action of the p-adic affine group, while one dimensional examples (see Sec. 5) give nonlinear factorizations over reals. Our objects are constructed as inverse limits of rather simple random processes. This method is well known for processes of Levy–Ito type with linear maps, but these also admit many other descriptions (by characteristic functionals, by independent increment processes, and others). This is not the case for our examples. We use inverse limits with very nonlinear maps, so we cannot use characteristic functionals, and nonlinearity leads us to some (flabby) sheaves with no linear structure and nets of Borel spaces as the natural context. Roughly speaking, our measure is concentrated on sections of some sheaves though, no sheaves theory is really used in this work. For details see Sec. 2. Our construction can be compared to the classical one in another way: generalized processes of Levy type could be considered as random measures on some space. The value of such a measure on a set is an integral over this set of the initial generalized process. We also consider random functions on a family of sets on some space, but the family is not a σ-field, and the functions are far from being additive or multiplicative. In the zero dimensional case the family of sets consists of all elementary cylinders of the Cantor set, so our random functions are defined on vertices of a tree. However, the value of such a function on the union of two elementary cylinders is some nonlinear combination of its values on the components. The reader will find our main construction similar to random combinatorial models such as the hierarchical voting model. The one-dimensional case is technically more complicated but is considered in the same framework. Two conditions are important in our considerations. — Absence of additive integrals: roughly speaking, this means that no function defined on a hierarchical level can be represented as a sum of functions, each depending on a single variable belonging to the next level. — A simple continuity condition: roughly speaking, this means that any function on one level must be “almost measurable” with respect to the σ-field generated by most of the variables of a sufficiently higher level. Each condition can be readily satisfied separately (the latter, unlike the former, holds for the linear case). The latter condition allows us to deduce the absence of multiplicative integrals from that of additive integrals (see Appendix A “Logarithm”), which is much easier to check. The continuity of the base is also implied by that condition.
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For each of the two conditions, some criteria are given in Sec. 3. Hidden linearity is detected in a seemingly nonlinear construction (Example 4.16) by our criterion (Lemma 3.11). Continuity criteria (Theorem 4.4 and some lemmas of Sec. 3) use ideas related to coding ideas known in ergodic theory. It is difficult to describe all the connections of our topics with problems in probability theory, Hilbert spaces, algebras of operators, and representation theory. The classical constructions of continuous products for measure spaces using infinitely divisible distributions on groups and algebras do not work in our case, even if the group of reals is replaced by a non-Abelian group: Feinsilver [12] and Baxendale [4] have proved that infinitely divisible distributions on Lie groups and, respectively, on groups of diffeomorphisms, lead to the same factorizations as Abelian groups. Many authors have described a parallelism between the Wiener–Ito decomposition of L2 over a Gaussian process (say, white noise) and a Hilbert space equipped with a Fock factorization. Now it is well known from various viewpoints, and we will not repeat it (see [15, 22, 20]). Factorizations of Hilbert spaces, as well as operator factorizations, were studied in [21], [2] and others, where general definitions have been given. We have mentioned the main result of [2] about the characterization of Fock factorizations among type I factorizations. Later it became clear that in the framework of statistical and quantum physics, factorizations of types II and III (see [16] and references therein) occur much more often. Our intention to start from new non-trivial examples of type I factorizations may be considered as a step toward a theory of general factorizations and, especially, measure factorizations. We also hope that our examples are interesting from a purely probabilistic point of view. Factorizations usually appeared as a natural object in the theory of representations of CCR and CAR (see [5]), as well as a representation theory of current (gauge) groups and algebras. It is well known that many representations of infinite dimensional groups can be implemented in fermionic or bosonic Fock spaces. See the Araki construction in [22] or in [26] and [27] for current groups with semisimple values, see also [17] about the representation of Kac–Moody Lie algebras and so on. In the paper [26] the idea of “noncommutative distributions” was claimed. The use of Fock factorizations in all those cases is very natural, but it is also natural to use the new kind of factorizations, proposed here, for some new representations of those groups and algebras. Finally we want to emphasize connections with the theory of families of σ-fields (filtrations) which appeared in measure theory (see [25]), as well as in the theory of stochastic differential equations and Levy processes (see [10]): in our examples a new kind of filtration has appeared. 1. Continuous Tensor Products of Hilbert Spaces and Measure Spaces a. Hilbert factorizations Consider the complete lattice R(H) of all von Neumann algebras on a separable Hilbert space H. Lattice operations and constants are
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R1 ∧ R2 = R1 ∩ R2 , R1 ∨ R2 = (R1 ∪ R2 )00 , 0R(H) = {α1 : α ∈ C} , 1R(H) = B(H) , B(H) being the algebra of all bounded linear operators on H, and R0 = {r1 ∈ B(H) : rr1 = r1 r for all r ∈ R} being the commutant. Suprema and infima for arbitrary subsets of R(H) are defined similarly. Definition 1.1. A factorization of an operator algebra B(H) over a Boolean algebra A is a map Φ : A → R(H) such that Φ(A1 ∧ A2 ) = Φ(A1 ) ∧ Φ(A2 ) , Φ(A1 ∨ A2 ) = Φ(A1 ) ∨ Φ(A2 ) , 0
Φ(A0 ) = (Φ(A)) , Φ(0A ) = 0R(H) , Φ(1A ) = 1R(H) 0
for all A, A1 , A2 ∈ A. (Each Φ(A) is necessarily a factor, since Φ(A) ∩ (Φ(A)) = Φ(A) ∧ Φ(A0 ) = Φ(0A ).) The factorization Φ is called a type I factorization, if each factor Φ(A) is of type I. A factorized Hilbert space is a Hilbert space equipped with a type I factorization of its operator algebra. No kind of completeness of A or continuity of Φ is stipulated from the beginning; we will return to this matter at the end of this section. If the Boolean algebra A is finite, which means A = 2A for a finite set A, then a factorized Hilbert space over A is nothing but a tensor product ⊗α∈A Hα ; see [21] or [2, pp. 163–164]. This is the primary justification for the term “factorized Hilbert space”. A further justification: if A = 2A for countable A, and Φ respects infinite suprema, then H is a tensor product (ITPS); see [2, Sec. 4]. Type I factorizations will also be called “Hilbert space factorizations” or “Hilbert factorizations”. Note that factorizations built from non-type-I factors are of value for local quantum field theory; see [2, p. 161]. The principal example of a Hilbert factorization over a non-atomic A results from a direct integral of Hilbert spaces by taking exponential (Fock) space: H = Exp K ; Z K=
⊕
K(x) µ(dx) ;
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and for any µ-measurable A Z K(A) =
⊕
K(x) µ(dx) ,
A
H(A) = Exp (K(A)) , K = K(A) ⊕ K(A0 ) , H = H(A) ⊗ H(A0 ) , Φ(A) = B (H(A)) ⊗ 1H(A0 ) , A0 being the complement of A. See [2, Sec. 5]. Such factorizations will be called Fock, or linearizable. The last term reminds us that the multiplicative relation H = H(A) ⊗ H(A0 ) results (by some exponentiation process) from a linear relation, K = K(A) ⊕ K(A0 ); see also Appendix A. A complete unitary invariant of a Fock factorization is the dimension function x 7→ dim K(x) , and its spectrum (that is, a set of values), lying in {0, 1, 2, . . . ; ∞}, is a complete isomorphic invariant, see [2, p. 201]. We emphasize that the Fock factorization is defined over the whole Boolean algebra of classes mod 0 of measurable sets. Evidently, if dim K(x) = const, then the Fock factorization is invariant under the whole group of measure preserving transformations. Factorizable vectors (for a given Hilbert factorization Φ) were described by Araki and Woods in several ways. One definition [2, p. 207] is based on the decomposition H = H1 ⊗ · · · ⊗ Hn , Φ(Ak ) = 1H1 ⊗ · · · ⊗ 1Hk−1 ⊗ B(Hk ) ⊗ 1Hk+1 ⊗ · · · ⊗ 1Hn , associated with a partition (A1 , . . . , An ) of X. A vector Ψ ∈ H is called factorizable, if it is of the form Ψ = Ψ1 ⊗ · · · ⊗ Ψn , Ψk ∈ Hk , for any partition. An equivalent definition: Ψ is factorizable, if the corresponding one-dimensional projection operator PΨ is of the form PΨ = P1 ⊗ · · · ⊗ Pn ,
Pk ∈ Φ(Ak ) ,
for any partition. For one more definition see [2, Def. 5.2]. For a Fock factorization, a vector Ψ is factorizable if and only if it is of the form Ψ = c Exp f , Exp f =
c ∈ C, f ∈ K, ∞ X 1 √ f ⊗k , k! k=0
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see [2, Th. 6.2, and (5.2)], see also [15, 22]. These vectors are a total set, that is, the least closed subspace containing them is the whole H: span {Exp f : f ∈ K} = Exp K = H , see [2, Lemma 5.1]. The following remarkable theorem states that this property is characteristic for Fock factorizations; see [2, Th. 6.1]. The formulation is adapted to our terminology. Theorem. (Araki–Woods) Let A be a complete nonatomic Boolean algebra, and Φ a type I factorization satisfying Φ
_
! A
A∈S
=
_
Φ(A)
A∈S
for any S ⊂ A. (Compare it to the minimal up continuity condition introduced in Sec. 1d.) Then Φ is unitarily equivalent to a Fock factorization if and only if factorizable vectors are a total set. The hope is expressed in [2, p. 237] that these two equivalent conditions are satisfied for any such Φ. b. Measure factorizations We turn to the commutative case. Following [13], but adapting the presentation, consider the complete lattice Σ(P ) of all sub-σ-fields on a probability space (Ω, F , P ). The probability space is supposed to be non-atomic Lebesgue space, that is, isomorphic mod 0 to [0, 1] with the Lebesgue measure, and each σ-field of Σ(P ) is supposed to be contained in F and contain all negligible sets (that is, sets of zero probability). Lattice operations and constants are F 1 ∧ F2 = F1 ∩ F2 , F1 ∨ F2 is the σ-field generated by F1 ∪ F2 , 0Σ(P ) = {A ∈ F : P (A) = 0 or 1} , 1Σ(P ) = F . No orthocomplementation F 7→ F 0 is given on Σ(P ), in contrast to R(H). However, some F1 admit a (non-unique) independent complement F2 ; this means that F1 ∧ F2 = 0, F1 ∨ F2 = 1, and in addition F1 and F2 are independent: P (E1 ∩ E2 ) = P (E1 )P (E2 )
for all E1 ∈ F1 , E2 ∈ F2 ;
this independence relation will be denoted by F1 t F2 .
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Definition 1.2. A measure factorization over a Boolean algebra A is a map F : A → Σ(P ) such that F(A1 ∧ A2 ) = F(A1 ) ∧ F(A2 ) , F(A1 ∨ A2 ) = F(A1 ) ∨ F(A2 ) , F(A0 ) t F(A) , F(0A ) = 0Σ(P ) , F(1A ) = F for all A, A1 , A2 ∈ A. A factorized measure space (Ω, F, P ) (or a factorized probability space) is a probability space equipped with a measure factorization. A finite A leads to a product of several measure spaces, which is similar to the noncommutative (Hilbert) situation. The case of A = 2A with a countable A is also similar. The principal example of a measure factorization over a nonatomic A results from a process with independent values, as defined in [14, Vol. 4]. The simplest examples are derivations of Poisson and Wiener processes. A more general example is the derivative of an arbitrary process with independent increments, that is always an integral combination of Poisson and Wiener processes, according to Levy–Khintchin theory. These examples may be formulated on the more restrictive language of stochastic measures. Roughly speaking, a stochastic measure may be defined as a random process X = (X(A, ω))A∈A,ω∈Ω such that X(A ∪ B) = X(A) + X(B), the two summands being independent, whenever A∧B = 0. A more elaborate definition of a “decomposable process”, used in [13], stipulates a countable additivity in A, and a restriction of the domain of X to an ideal; we will not go into these details here. The problem is posed [13, Problem 1.9] whether any metric factorization F, defined on the Boolean algebra A of all Borel sets on [0, 1] and satisfying F(A1 ∪ A2 ∪ · · · ) = F(A1 ) ∨ F(A2 ) ∨ · · · for any A1 , A2 , . . . ∈ A, is “linearizable”, that is, generated by some collection of decomposable processes. Definition 1.3. Let (Ω, F, P ) be a factorized measure space over a Boolean algebra A. A measurable function f : Ω → R will be called an integral, if for any finite partition A1 , . . . , An of the unity element 1A there exist functions f1 , . . . , fn : Ω → R such that each fk is F(Ak )-measurable, and f = f1 + · · · + fn . A factorized measure space (Ω, F, P ) will be called linearizable (or integrable), if there exists a finite or countable collection of integrals fk : Ω → R generating the whole σ-field F = F(1A ). For any commutative topological group G, an integral valued in G can be defined similarly. In particular, the multiplicative group of all non-zero real or complex
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numbers may be used, giving multiplicative integrals (in contrast to additive integrals defined by 1.3). A semigroup is also acceptable; so, zero may be allowed as a value of a multiplicative integral. Lemma 1.4. Let two measure factorizations F1 : A1 → Σ(P ), F2 : A2 → Σ(P ) be given, using two Boolean algebras A1 , A2 , but a single probability space (Ω, F , P ). Suppose that for any A ∈ A1 _ {F2 (B) : B ∈ A2 , F2 (B) ⊂ F1 (A)} = F1 (A) . Then any integral (additive or multiplicative) on (Ω, F2 , P ) is also an integral on (Ω, F1 , P ). The proof is given in Appendix B. c. Measure type factorizations; the connection between the three kinds of factorizations Not only the conceptual similarity connects Hilbert and metric factorizations, but also the following construction. Any product of probability spaces determines a tensor product of Hilbert spaces: L2 ((Ω1 , F1 , P 1 ) × (Ω2 , F2 , P 2 )) = L2 (Ω1 , F1 , P 1 ) ⊗ L2 (Ω2 , F2 , P 2 ) . Accordingly, each A ∈ A determines a type I factor Φ(A) = B L2 (Ω, F (A), P |F (A) ) ⊗ 1 L2 (Ω, F (A0 ), P |F (A0 ) ) on the space H = L2 (Ω, F , P ). Hence, each metric factorization F determines its Hilbert factorization ΦF . From the noncommutative viewpoint, a metric factorization is nothing but a triple (Φ, Z, Ψ), consisting of a Hilbert factorization Φ : A → R(H), a maximal commutative subalgebra Z ⊂ B(H), and a factorizable vector Ψ ∈ H such that Z ∩ Φ(A) is a maximal commutative subalgebra of Φ(A) for any A ∈ A, and Ψ is Z-cyclic , that is, {RΨ : R ∈ Z} is dense in H; or, equivalently, RΨ 6= 0 for all R ∈ Z, R 6= 0. The correspondence between (Φ, Z, Ψ) and F : A → Σ(P ) is given by L∞ (F(A)) = Z ∩ Φ(A) , Z R dP = (RΨ, Ψ) for R ∈ Z . The notion of a factorizable vector for such a Hilbert factorization matches the notion of a square-integrable complex-valued multiplicative integral for the corresponding measure factorization.
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Waiving the factorizable vector Ψ but retaining the maximal commutative subalgebra Z we arrive at a measure type factorization. Remember that a measure type P is the set of all measures equivalent (that is, mutually absolutely continuous) to a given measure. Probability measures, finite measures, and σ-finite measures are equally acceptable in this context. The space L2 (P) is obtained by glueing all the spaces L2 (µ) for µ ∈ P, their connections identified by unitary operators Uν,µ : L2 (µ) → L2 (ν), Uν,µ f = (dµ/dν)1/2 f . The product of two measure type spaces (Ω1 , F1 , P1 ) and (Ω2 , F2 , P2 ) is defined evidently, and determines a tensor product of Hilbert spaces: L2 ((Ω1 , F1 , P1 ) × (Ω2 , F2 , P2 )) = L2 (Ω1 , F1 , P1 ) ⊗ L2 (Ω2 , F2 , P2 ) . A measure type factorization may be defined similarly to Def. 1.2. Example. Consider a stationary Gaussian generalized random process over R with a covariation function B such that 1 for t ∈ (0, ε) B(t) = t| log t|θ with some ε > 0, θ > 1. (This means a spectral density ∼ const · (log λ)−(θ−1) for λ → ∞.) Restrictions of this process to adjacent intervals (a, b) and (b, c) are interdependent. Applying the well-known Feldman’s criterion of equivalence of two Gaussian measures, it may be shown that the distribution of the restriction to (a, c) is equivalent to the product of distributions for (a, b) and (b, c). The spectral density decreases so slowly that the process is close to the white noise: its dependence is weak. From the other side, it is not so close to the white noise: no factorizable vectors exist for the corresponding Hilbert factorization (thus, non-Fock factorization). This example may be considered as a commutative (bosonic) counterpart of Power’s noncommutative (fermionic) example of a non-Fock factorization over R. d. Continuity Given a decreasing sequence A1 ≥ A2 ≥ · · · in A, we are interested to know, whether or not the intersection of all Φ(Ak ) or F(Ak ) is trivial. More generally, instead of a sequence we take an arbitrary set S ⊂ A (though, only a special kind of set S will be used in future). Consider the following eight conditions: ^ Φ(A) = Φ(0) ; (Hd) A∈S
_
Φ(A0 ) = Φ(1) ;
(Hu)
A∈S
^
Φ(B ∨ A) = Φ(B)
for all B ∈ A ;
(Hd0 )
Φ(B ∧ A0 ) = Φ(B)
for all B ∈ A ;
(Hu0 )
A∈S
_ A∈S
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^
F(A) = F(0) ;
(md)
F(A0 ) = F(1) ;
(mu)
A∈S
_ A∈S
^
F(B ∨ A) = F(B)
for all B ∈ A ;
(md0 )
F(B ∧ A0 ) = F(B)
for all B ∈ A .
(mu0 )
A∈S
_ A∈S
Clearly, “H” and “m” mean “Hilbert” and “metric”, while “d” and “u” mean “down” and “up”. Each property is unaffected, if S is augmented by adding all A1 ∧ · · · ∧ An for A1 , . . . , An ∈ S. Also, B may be added to S whenever B ⊃ A for some A ∈ S. That is, we may suppose that S is a filter in A (setting aside the trivial case when 0 ∈ S) or, identically, {A0 : A ∈ S} is an ideal in A, not containing 1A . Lemma 1.5. (1) For any Hilbert factorization Φ and any S ⊂ A, (Hd) ⇐⇒ ( Hd0 ) ⇐⇒ (Hu) ⇐⇒ (Hu0 ) . (2) For any metric factorization F and any S ⊂ A, (md) ⇐⇒ (md0 ) , (mu) ⇐⇒ (mu0 ) , (mu) =⇒ (md) ; the implication (md) =⇒ (mu) is generally false. (3) If Φ and F are connected as described above, then (Hu) ⇐⇒ (mu) , (Hd) =⇒ (md) ; the implication (md) =⇒ (Hd) is generally false. Proof of the positive statements is left to the reader. Examples verifying the negative statements will be considered below. The orthocomplementation R 7→ R0 in the lattice of von Neumann algebras is responsible for the equivalence (Hd) ⇐⇒ (Hu). The obstacle responsible for the absence of the implication (md) =⇒ (mu) is hidden in the metric framework, but becomes explicit in the Hilbert framework. A reasonable continuity condition for a factorization should stipulate one of the above conditions for a relevant class of sets S ⊂ A. A set with a nonzero lower
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bound is, of course, irrelevant. The more S, the weaker the condition. This is why the following conditions are called minimal. Remember that a maximal filter is called an ultrafilter, and a “maximal ideal” means an ideal maximal among ideals not containing 1A . An ultrafilter may have a nonzero lower bound; such a bound A is necessarily an atom, that is, B ≤ A implies B = 0A or B = A. We restrict ourselves to nonatomic Boolean algebras (that is, containing no atoms). Definition 1.6. A Hilbert factorization Φ : A → R(H) satisfies the minimal continuity condition, if \ Φ(A) = Φ(0) A∈S
for any ultrafilter S ⊂ A. Or, equivalently, if _ Φ(A) = Φ(1) A∈S
for any maximal ideal S ⊂ A. A measure factorization F : A → Σ(P ) satisfies the minimal down continuity condition, if \ F(A) = F(0) A∈S
for any ultrafilter S ⊂ A. The minimal up continuity condition for F is _ F(A) = F(1) A∈S
for any maximal ideal S ⊂ A. The minimal up continuity condition (stronger than the minimal down continuity) will be especially important for measure factorizations. This condition may be formulated in terms of a subadditive function on the Boolean algebra. Given a factorized measure space (Ω, F, P ) over A, choose a function f ∈ L2 (P ) generating the whole σ-field F = F(1A ), and define δf (A) = kf k2 − kE (f |F(A0 )) k2 for A ∈ A; that is, δf (A) is the averaged conditional variance of f given F(A0 ). Note that A ≤ B =⇒ σ(A) ≤ σ(B), and δf (A ∨ B) ≤ δf (A) + δf (B) for any A, B ∈ A. Indeed, projection operators PA : g 7→ E (g|F(A)) commute: PA PB = PA∧B = PB PA ; so, δf (A ∨ B) = kf − P(A∨B)0 f k2 = kf − PA0 PB 0 f k2 = kf − PA0 f k2 + kPA0 f − PA0 PB 0 f k2 ≤ kf − PA0 f k2 + kf − PB 0 f k2 = δf (A) + δf (B) .
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For an increasing sequence An ∈ A _ F(An ) = F if and only if lim δf (A0n ) = 0 . n
n
It follows that the minimal up continuity condition is equivalent to the following: inf δf (A0 ) = 0
A∈S
for any maximal ideal S ⊂ A. This, in its turn, is equivalent to the following: for any ε > 0 there is a finite partition A1 , . . . , An of 1A such that δf (Ak ) ≤ ε for each k. The function δf depends on f , but the condition does not depend on f . Theorem 1.7. Let a factorized measure space (Ω, F, P ) over a nonatomic Boolean algebra satisfy the minimal up continuity condition. Then the following items define one and the same σ-field Fint . (a) Fint is generated by all real-valued additive integrals. (b) The same as (a), but with square integrable integrals only. (c) Fint is generated by all complex-valued multiplicative integrals (the value zero being allowed). (d) The same as (c), but only the unit circle on the complex plane is allowed. The proof is given in Appendix A. The σ-field Fint defined above will be called the integral σ-field. Note 1.8. Theorem 1.7 remains valid when the Boolean algebra contains atoms, provided the equality ∨{F(A) : A ∈ S} = F(1A ) holds for all maximal ideals S. This means that F(A) is supposed to be trivial whenever A is an atom. e. Products Given two factorized Hilbert spaces (H1 , Φ1 ) and (H2 , Φ2 ) over the same Boolean algebra A, their tensor product (H, Φ) is defined naturally: H = H 1 ⊗ H2 , Φ(A) = Φ1 (A) ⊗ Φ2 (A) . The same applies for factorized measure spaces (Ωk , Fk , P k ): Ω = Ω1 × Ω2 ,
P = P1 ⊗P2,
F(A) = F1 (A) ⊗ F2 (A) , as well as for measure type factorizations. A generalization for any finite or countable number of factors is straightforward. Lemma 1.9. Let a factorized measure space (Ωk , Fk , P k ) over a nonatomic Boolean algebra A be given for each k running over a finite or countable set, and (Ω, F, P ) be their product. Suppose that each (Ωk , Fk , P k ) satisfies the minimal up
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continuity condition. Then (Ω, F, P ) also satisfies this condition, and its integral σ-field Fint is the tensor product of the integral σ-fields Fint,k corresponding to (Ωk , Fk , P k ). Proof. Let S be a maximal ideal of A. In order to prove that the whole σ-field F(1) is generated by all F(A) for A ∈ S, it suffices to check that the σ-field generated by them contains E1 × · · · × En for any n and any E1 ∈ F1 (1), . . . , En ∈ Fn (1). This follows from the fact that each Ek is a limit of sets from Fk (A), A ∈ S, which proves the minimal up continuity. Any integral fk on Ωk defines an integral on Ω by means of the canonical projection Ω → Ωk ; this is why Fint,k ⊂ Fint for each k. Consider an integral f on Ω; we have to prove that f is measurable with respect to ⊗Fint,k . To this end it suffices to check that f (ω1 , ω2 , . . .) considered as a function of ω1 ∈ Ω1 is an integral on Ω1 for almost all ω2 ∈ Ω2 , . . . This fact follows from the definitions, when the Boolean algebra A is countable. Otherwise a problem arises of an uncountable union of exceptional subsets of Ω2 × · · · However, we may choose a countable Boolean subalgebra A0 ⊂ A so that conditional expectation operators E ( · |F1 (A)) for A ∈ A0 are dense among E ( · |F1 (A)) for A ∈ A in the strong operator topology over L2 (Ω1 ). Then square integrable additive integrals over (Ω1 , F1 , P 1 ) and these over (Ω1 , F1 |A0 , P 1 ) are in a natural one-one correspondence, which completes the proof. 2. Measures on Flabby Sheaves and Nets of Borel Spaces a. Flabby sheaves of Borel spaces The notion of a sheaf is well known; see [18]. Applied most often in the context of Abelian groups, it can be adapted to various kinds of objects. The definition of a sheaf is reproduced below in the context of standard Borel spaces. Remember that a standard Borel space is a set, equipped with a σ-field of subsets, isomorphic either to the real line with its Borel σ-field, or to a finite or countable set with the σ-field of all subsets. For general theory of Borel spaces see [19] and [8, Appendix B]. Definition 2.1. A sheaf of Borel spaces Ω over a topological space X consists of: (a) standard Borel spaces Ω(U ) given for all open sets U ⊂ X, (b) Borel maps ρV,U : Ω(U ) → Ω(V ) given for all pairs of open sets V ⊂ U , satisfying the following conditions. (c) Ω(∅) consists of a single point; (d) ρU,U = idΩ(U) ; (e) ρW,U = ρW,V ◦ ρV,U for any triplet W ⊂ V ⊂ U of open sets; S (f) for any open set U ⊂ X, any open covering U = i∈I Ui , any sections ω1 , ω2 ∈ Ω(U ), the equality ρUi ,U ω1 = ρUi ,U ω2 for all i ∈ I implies ω1 = ω2 ; S (g) for any open set U ⊂ X, any open covering U = i∈I Ui , any family of sections ωi ∈ Ω(Ui ) satisfying ρUi ∩Uj ,Ui (ωi ) = ρUi ∩Uj ,Uj (ωj )
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for all pairs (i, j), there exists ω ∈ Ω(U ) such that ρUi ,U (ω) = ωi for all i. The sheaf Ω is called flabby, if (h) ρU,X is surjective for any open subset U of X (that is, any section on U has a global extension). A factorized Borel space is a Borel space identified with Ω(X) for some flabby sheaf of Borel spaces Ω over a topological space X. As usual, an element ω ∈ Ω(U ) is called a section of the sheaf Ω on U . If V is an open subset of U , we will write ω|V instead of ρV,U (ω), and call it the restriction of ω to V . Given a flabby sheaf of Borel spaces, Ω, each Ω(U ) may be identified with the quotient space of the “global” Borel space Ω = Ω(X) with respect to the equivalence relation, corresponding to the restriction map ρU,X (surjective due to 2.1(h)); see [8, B22]. This equivalence relation may be identified with the corresponding measurable partition ξU of Ω or sub-σ-field F(U ) of the Borel σ-field F ascribed to Ω. That is, a flabby sheaf of Borel spaces may be defined equivalently as a Borel space (Ω, F ) equipped with a family of countably generated sub-σ-fields F(U ), given for all open U , satisfying conditions that are evident reformulations of 2.1(c–g). An example of a sheaf of Borel spaces over Rn can evidently be obtained from any well-known function space, such as Lp (Rn ), C k (Rn ), Schwartz distributions, and many others. However, the condition 2.1(g) forces us to weaken restrictions near the boundary: ω ∈ Ω(U ), when ω ∈ Lp (V ) (resp. C k (V ), . . .) for any open V whose closure is a compact subset of U . This is why these sheaves are not flabby. The sheaves that will be constructed later will be flabby due to the following argument of compactness. Lemma 2.2. Suppose in addition to 2.1(a–g) that each Ω(U ) is equipped with a topology so that all restriction maps ρU,X are continuous, all single-point sets are closed, and Ω(X) is compact. Let U1 ⊂ U2 ⊂ · · · be open sets, U = U1 ∪ U2 ∪ · · · If ρUi ,X is surjective for each i, then ρU,X is surjective. The proof is left to the reader. b. From a topological space to a Boolean algebra Definition 2.3. Let Ω be a sheaf of Borel spaces over a topological space X. A compact set K ⊂ X is called Ω-thin if, for any open set U ⊂ X, the restriction map ρU\K,U is injective. (That is, a section on U can be restored from its restriction to U \ K.) If K1 , K2 are Ω-thin, then K1 ∪ K2 is Ω-thin. Indeed, ρU\(K1 ∪K2 ),U = ρ(U\K1 )\K2 ,U\K1 ◦ ρU\K1 ,U is a composition of two injective maps. A closed subset of a thin set is thin. Let U1 , U2 be open sets with Ω-thin boundaries. Then the same holds for U1 ∪U2 and U1 ∩ U2 . Indeed, ∂(U1 ∪ U2 ) ⊂ ∂U1 ∪ ∂U2 and ∂(U1 ∩ U2 ) ⊂ ∂U1 ∪ ∂U2 .
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Arbitrary open sets generally do not form a Boolean algebra. Indeed, the greatest open set disjoint with a given open set U is the interior of its complement: U 0 = Int (X \ U ) = X \ U \ ∂U = X \ U . Repeating this complementation, we have U 00 = Int U ⊃ U . Open sets U ⊂ X, satisfying the equivalent conditions U = Int U , ∂U = ∂U 0 , U 00 = U , form a Boolean algebra A(X) with the following operations: U1 ∨ U2 = Int U1 ∪ U2 ; U1 ∧ U2 = U1 ∩ U2 ; U 0 = Int (X \ U ) . Open sets with Ω-thin boundaries, satisfying the above conditions, form a Boolean subalgebra AΩ ⊂ A(X). We are mostly interested in sheaves satisfying the condition AΩ is a base of the topology , that is, any open set in X is a union of open sets belonging to AΩ . A zero-dimensional example: X is the totally disconnected compact Cantor set, Ω is arbitrary. The empty set is always Ω-thin, hence, clopen sets belong to AΩ . They form a basis of the topology. A one-dimensional example: X = R, Ω is such that finite sets are thin (the examples of Lp and C k satisfy this condition). Open intervals belong to AΩ and form a basis of the topology. A flabby sheaf of Borel spaces Ω over a topological space X determines a Borel space factorization over the Boolean algebra AΩ ; that is, a Borel space Ω(A) corresponds to each A ∈ AΩ , and Ω(A ∨ B) = Ω(A) × Ω(B)
whenever A ∧ B = 0 .
We may say that a Borel space factorized over a topological space X is thus factorized also over the Boolean algebra AΩ . (Though, AΩ may be trivial, since Ω-thin sets may be unable to cut X). Definition 2.4. Let Ω be a factorized Borel space over a topological space X. A product measure on Ω is a probability measure µ on Ω such that σ-fields F(U ), F(V ) are µ-independent for any open sets U, V ⊂ X with U ∩ V = ∅. If µ is a product measure on a factorized Borel space Ω, then (Ω, µ) may be treated as a factorized measure space (as defined in 1.2) over the Boolean algebra AΩ . Non-linearizable factorized measure spaces will be constructed in this way.
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c. Nets of Borel spaces Flabby sheaves of Borel spaces are so rare: no examples were shown to the reader till now! Non-flabby sheaves are widespread, but inconvenient for carrying measures. We are interested in measures on global sections, and have no need in inextensible local sections, produced by 2.1(g). The additional condition 2.1(h) is more important for us, than 2.1(g). What we really need, is always flabby, but not always a sheaf! This is why “nets of Borel spaces” will be introduced. They may be considered as a commutative counterpart for “nets of local algebras” well known in local quantum field theory. Remember that a Borel space is called countably separated, if it has a countable (or finite) family of Borel sets that separates points; see [19]. Definition 2.5. A net of Borel spaces Ω over a topological space X consists of (a) a standard Borel space Ω(X), and countably separated Borel spaces Ω(U ) given for all open sets U ⊂ X, (b) Borel maps ρV,U : Ω(U ) → Ω(V ) given for all pairs of open sets V ⊂ U , satisfying conditions (c)–(f) and (h) of Definition 2.1. Each Ω(U ) is an analytic Borel space and may be identified with a quotient space of Ω(X); see [19, Sec. 4]. We will mostly deal with standard Borel spaces, but in general a countably separated quotient space of a standard Borel space is analytic rather than standard; see, for instance, [11, Sec. 13.2]. A flabby sheaf of Borel spaces is both a sheaf and a net. An arbitrary sheaf of Borel spaces Ω produces a net of Borel spaces Ω0 , consisting of extensible sections: Ω0 (U ) = ρU,X (Ω(X)) . In particular, the spaces Lp (Rn ), C k (Rn ), Schwartz distributions, and many others may be considered as nets of Borel spaces over Rn . A net of Borel spaces may be defined equivalently as a Borel space (Ω, F ) equipped with a family of countably generated sub-σ-fields F(U ), given for all open U , satisfying conditions that are evident reformulations of 2.1(c–f,h). Both Ω-thin sets and product measures can be defined for a net of Borel spaces similarly to 2.3, 2.4. If µ is a product measure on a net of Borel spaces Ω, then, once again, (Ω, µ) may be treated as a factorized measure space over the Boolean algebra AΩ . However, the class of thin sets (and hence the Boolean algebra) may be increased as follows. Definition 2.6. Let µ be a product measure on a net Ω of Borel spaces over X. A compact set K ⊂ X is called µ-thin, if for any open set U ⊂ X the two σ-fields F(U ) and F(U \ K) coincide mod 0 with respect to µ. Each Ω-thin set is µ-thin. For the white noise the class of all µ-thin sets coincides with the class of all compact sets of zero Lebesgue measure. However, only the empty set is Ω-thin for the net of Borel spaces, formed by Schwartz distributions.
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Open sets U with µ-thin boundaries, satisfying the condition U 00 = U , form a Boolean subalgebra Aµ ⊂ A(X), and (Ω(X), µ) may be treated as a factorized measure space over Aµ ; it will be denoted by (Ω(X), F, µ), though F|Aµ is meant rather than F. Definition 2.7. Let Ω be a net of Borel spaces over X. An integral on Ω is a Borel function f : Ω(X) → R such that for any finite open covering U1 , . . . , Un of X there exist Borel functions fk : Ω(Uk ) → R for k = 1, . . . , n such that f (ω) = f1 (ω|U1 ) + · · · + fn (ω|Un ) for each ω ∈ Ω(X). The net Ω is called linearizable (or integrable), if there is a finite or countable set of integrals fk : Ω → R, generating the whole σ-field F = F(X). Given a product measure µ on Ω, a µ-integral on Ω is defined similarly; the distinction is that all functions are defined µ − mod 0, and the equality holds µ − mod 0. The net Ω is called µ-linearizable (or µ-integrable), if there is a finite or countable set of µ-integrals, generating the whole σ-field µ − mod 0. So, an integral is µ-integral for any product measure µ, and a linearizable net is µ-linearizable for any µ. The above are additive integrals; multiplicative integrals are introduced in the same way as was done after Definition 1.3. Theorem 2.8. Let µ be a product measure on a net Ω of Borel spaces over a separable metrizable space X, and Aµ be a base of the topology of X. Then a function on Ω(X) is an additive (resp. multiplicative) µ-integral on Ω if and only if it is an additive (resp. multiplicative) integral on the Aµ -factorized measure space (Ω(X), F, µ). The proof is given in Appendix B. Corollary 2.9. Let µ be a product measure on a net Ω of Borel spaces over X, and Aµ be a base of the topology of X. If a net Ω is µ-linearizable (as defined in 2.7), then the corresponding Aµ -factorized measure space (Ω(X), F, µ) is linearizable (as defined by 1.3). The spaces Lp (Rn ), C k (Rn ), Schwartz distributions, and many others form linearizable nets of Borel spaces (due to the existence of corresponding partitions of unity). Hence, these spaces cannot be used when constructing a non-linearizable factorized measure space. d. Simple continuity Definition 2.10. A product measure µ on a net Ω of Borel spaces over X is called simple continuous, if any point of X (more exactly, any single-point subset of X) is µ-thin.
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Lemma 2.11. Let µ be a product measure on a net Ω of Borel spaces over a separable metrizable space X without isolated points, and Aµ be a base of the topology of X. If µ is simple continuous, then the corresponding Aµ -factorized measure space (Ω(X), F, µ) satisfies the minimal up continuity condition (defined by 1.6). Proof. If S ⊂ Aµ is a maximal ideal, then the union US of all sets of S is either the whole X, or X without a single point. (Indeed, if two points x, y are missed, then S can be enlarged by adding a set of Aµ containing x but not y.) Due to simple continuity, F(US ) = F(X) µ − mod 0. Choose U1 ⊂ U2 ⊂ · · · of S so that their union is US , then F(Uk ) ↑ F(X) µ − mod 0. So, simple continuity allows us to use Theorem 1.7, thus, the integral σ-field Fint . The requirement not to contain isolated points may be dropped. Indeed, an isolated point x of X gives an atom {x} of Aµ , but the atom is harmless according to 1.8: {x} is thin, hence F({x}) is trivial. Given two nets Ω1 , Ω2 of Borel spaces over a topological space X, their product Ω = Ω1 × Ω2 is defined naturally: Ω(X) = Ω1 (X) × Ω2 (X) , F(U ) = F1 (U ) ⊗ F2 (U ) . Given two product measures µ1 , µ2 on Ω1 , Ω2 respectively, their product µ = µ1 ⊗µ2 is a product measure on Ω. The same for any finite or countable number of factors. Lemma 2.12. Let a net Ωk of Borel spaces over X and a product measure µk on Ωk be given for each k running over a finite or countable set. Let Ω be the product of all Ωk , and µ the product of all µk . If each µk is simple continuous, then µ is also simple continuous. Proof. Similar to the proof of the minimal up continuity in Lemma 1.9.
Corollary 2.13. Let the conditions of Lemma 2.12 be satisfied (including simple continuity), X be a separable metrizable space, and Aµ be a base of the topology of X. If each Ωk admits no nonconstant integrals (whether additive or multiplicative), then this holds for Ω, too. Proof. Lemmas 2.12 and 2.11 ensure the minimal up continuity of the corresponding factorized measure space. Theorems 2.8 and 1.7 show that the absence of nonconstant integrals (whether additive or multiplicative) means triviality of the corresponding integral σ-fields. Lemma 1.9 shows that Fint = ⊗Fint,k ; so, triviality of all Fint,k implies triviality of Fint . A connection to Hausdorff dimension is outlined below. Choose a function f ∈ L2 (µ) generating the whole σ-field F(X), and let δf (F ) = kf k2 − kE (f |F(X \ F )) k2
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for any closed F ⊂ X. Then δf (F ) = 0 for thin F , and δf (Fn ) ↓ δf (F ) for any decreasing sequence Fn ↓ F . Under the hypothesis that Aµ is a base of the topology of X, it can be proved that a compact set K ⊂ X is thin if and only if δf (K) = 0. Under the same hypothesis, the subadditivity of δf on Aµ implies its subadditivity for all compact sets K1 , K2 : δf (K1 ∪ K2 ) ≤ δf (K1 ) + δf (K2 ) . Let X be a compact metric space and µ a simple continuous product measure. Consider δf (ε) = sup{δf (F ) : diam (F ) ≤ ε} , then lim δf (ε) = 0
ε→0
due to compactness. For a compact set K ⊂ X let N (K, ε) be the smallest n such that K can be covered by n sets of diameter ≤ ε. Then, under the hypothesis, the subadditivity of δf implies δf (K) ≤ N (K, ε)δf (ε) for each ε. Hence, if lim N (K, ε)δf (ε) = 0 , ε→0
then K is µ-thin .
So, demanding that single-point sets are thin, we conclude that some uncountable compact sets are also thin. Although a base of open sets with thin boundaries is needed, this holds automatically in dimensions 0 and 1. 3. Inverse Limit Constructions and Criteria of Nonlinearity and Continuity of Factorizations a. Dimension zero All our constructions are inverse limits. Dealing with the zero-dimensional case, it is convenient to construct simultaneously a topological space X and a net of Borel spaces Ω over X. Take a sequence of finite sets X0 , X1 , X2 , . . . connected by maps Lk−1,k : Xk → Xk−1 . The simplest interesting case arises when |Xk | = 2k and |L−1 k−1,k (x)| = 2 for each x ∈ Xk−1 . This is nothing but a binary tree, shown in Fig. 1. The inverse limit X = lim Xk is the set of all branches x = (x0 , x1 , . . .) ←− of the tree; xk ∈ Xk , Lk−1,k xk = xk−1 . Being a closed subset of the product X0 × X1 × · · · , this X is a compact metrizable topological space, well known as the totally disconnected compact Cantor set. Maps Lk,∞ : X → Xk evidently arise, satisfying Lk−1,k Lk,∞ = Lk−1,∞ . Any m = 2, 3, . . . may be used instead of 2, giving an m-adic tree. The simplest interesting factorized Borel space Ωk over the finite topological space Xk is the finite set of all functions ωk : Xk → {0, 1}, with the natural restriction maps ωk 7→ ωk |U for U ⊂ Xk . In order to build an inverse limit, we have to connect these Ωk by maps Rk−1,k : Ωk (Xk ) → Ωk−1 (Xk−1 )
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B. S. TSIRELSON and A. M. VERSHIK ...... ....... ......... ..... . ...................... .... . .. .... .... . . .... . .. ... .... .... . . . . ... . . . .... . . . ... . ... .... . . . . ... . . . .... . . . ................. ................ . . . .. . . ..... 1 .... ..... 0 .... ......... ......... ......... ......... . . . . . . . . . . . .... . . ... . . .... .... .... .... ... .... .... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ...... .... ...... .... ...... ................... . . . . . . ... 00 ... ... 01 ... ... 10 ... ..... 11 .... ......... ......... ......... ......... ......... ......... ......... ......... . .. .... . .. .... . .. .... . .. .... .... .... .... .... .... .... .... .... ... ... ... ...
X0
X1
... ......... ... .. .. ... .
L0,1
... ....... .. .
L1,2
X2
...
Fig. 1. Inverse limit of finite topological spaces. (a)=≤0 ∈∨0 (X0 )
a=φ(a0 ,a1 )
.. .... ... .... .... . . . .. .... .... ... ... . . . ....
.... ... .... ... .... ... .... ... .... ... .... ... .
a0 =φ(a00 ,a01 ) ... .... .... ....
a00 =... .... ....
.... ....
.... .... .... ... a01 =... .... ... . ... .... .
≤0 =R0,1 ≤1
a1 =φ(a10 ,a11 ) . ... .... ... ....
a10 =... .... ....
.... ... .
.... ... .... ... . a11 =... .... . . . . ... .... .
(a0 ,a1 )=≤1 ∈∨1 (X1 ) ≤1 =R1,2 ≤2 (a00 ,a01 ,a10 ,a11 )=≤2 ∈∨2 (X2 )
...
Fig. 2. Inverse limit of factorized Borel spaces.
that are Lk−1,k -local in the following sense: if ω 0 = ω 00 on L−1 k−1,k (U ) ,
then Rk−1,k ω 0 = Rk−1,k ω 00 on U
for any U ⊂ Xk−1 and ω 0 , ω 00 ∈ Ωk (Xk ). Here and henceforth the phrase “ω 0 = ω 00 on U ” means that ω 0 |U = ω 00 |U , that is, ρU,X (ω 0 ) = ρU,X (ω 00 ). The above general formulation of locality becomes very easy for the elementary case considered; it means that (Rk−1,k ω)(x) is a function of ω(x0 ) and ω(x1 ), whenever L−1 k−1,k (x) = {x0 , x1 }. The simplest interesting case occurs when a single Boolean function ϕ : {0, 1} × {0, 1} → {0, 1} is used for all x on all levels: (Rk−1,k ω)(x) = ϕ (ω(x0 ), ω(x1 )); this ϕ will be called “the small map”. Figure 2 shows a sequence (ω0 , ω1 , . . .) of sections ωk ∈ Ωk (Xk ) satisfying the relation Rk−1,k ωk = ωk−1 for all k. All these sequences form the inverse limit Ω(X) = lim Ωk (Xk ) ; ←−
maps Rk,∞ : Ω(X) → Ωk (Xk ) arise evidently, satisfying Rk−1,k Rk,∞ = Rk−1,∞ . We have to equip Ω(X) with the structure of a factorized Borel space over X. Being a closed subset of the product Ω0 (X0 ) × Ω1 (X1 ) × · · · of finite sets, Ω(X) is a compact metrizable topological space, hence, a standard Borel space. Each vertex v of the binary tree determines a subtree and a clopen subset Av ⊂ X. The subtree carries its inverse limit Ω(Av ) in the same way as the whole tree carries Ω(X), and the corresponding restriction map Ω(X) → Ω(Av ) is the restriction to the subtree. Being continuous, this map is Borel. An arbitrary open set U ⊂ X is the union of a finite or countable number of disjoint Avk , and we define Ω(U ) as the product of Ω(Avk ). It is easy to see that
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these Ω(U ) form a sheaf of Borel spaces. Lemma 2.2 shows that the sheaf is flabby, since Ω(X) → Ω(A) is evidently surjective for any clopen A. What has been said above remains true for any m-adic tree, m = 2, 3, . . ., any finite number r = 2, 3, . . . of possible values for each function ωk , and any “small map” ϕ : {0, . . . , r − 1}m → {0, . . . , r − 1}. Also, all these data may vary from one vertex of the tree to another. A product measure µ on Ω(X) determines product measures µk = Rk,∞ µ on Ωk (Xk ), connected by relations Rk−1,k µk = µk−1 . For the elementary case of finite Xk , the term “product measure” keeps its traditional meaning, but for X it is defined by 2.4. Conversely, any sequence of product measures µk on Ωk (Xk ), satisfying Rk−1,k µk = µk−1 for all k, defines a product measure µ on Ω(X), µ = lim µk . ←−
The simplest product measure µk on Ωk (Xk ) is the uniform probability distribution on this finite set. Then the relation Rk−1,k µk = µk−1 means a simple condition for a “small map” ϕ : |ϕ−1 (a)| does not depend on a (here |ϕ−1 (a)| means the number of points in the inverse image). Among Boolean functions ϕ : {0, 1} × {0, 1} → {0, 1}, the only possibility is ϕ(a, b) = a + b + const (mod 2), except for degenerate cases ϕ(a, b) = a + const, ϕ(a, b) = b + const. More possibilities are raised by larger r and m. This is the subject of Sec. 4. Replace the finite range {0, . . . , r − 1} of each function ωk with a continuum, say, R, or (0, 1), or R/Z; all these are isomorphic as Borel spaces. The above inverse limit construction remains valid, except for one point: compactness used in the proof that the sheaf is flabby. Waiving this argument, we can obtain either a non-flabby sheaf, or a net of Borel spaces (consisting of extensible sections, as explained in Sec. 2(c)). However, the compactness argument holds, provided that the continuum S used is a compact metrizable topological space (say, [0, 1], or R/Z) and the “small map” ϕ : S × S → S is continuous. In this case Ω(X) is again a flabby sheaf. Examples will be given in Sec. 4. Thus, a section ω over X is a limit of “coarse grained sections” ωk over “coarse grained spaces” Xk . b. A nonzero dimension No “coarse grained spaces” will be used for the case X = R, since a connected X cannot be an inverse limit of finite sets. All “coarse grained sections” ωk will be defined over the same X = R. Accordingly, a small nonlocality will be admitted for the maps Rk,∞ transforming ω into ωk , and also for Rk−1,k . As a consequence, measures µk will not be product measures. However, the nonlocality will disappear in the limit k → ∞. Let X be a metric space, and U ⊂ X an open set; for any r > 0 define open sets U+r = {x ∈ X : ρ(x, U ) < r} , U−r = {x ∈ X : ρ(x, X \ U ) > r} , where ρ(x, S) = inf{ρ(x, y) : y ∈ S}.
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Definition 3.1. Let Ω1 , Ω2 be two nets of Borel spaces over a topological space X. A Borel map R : Ω1 (X) → Ω2 (X), satisfying the following condition, will be called local : if ω 0 = ω 00 on U , then Rω 0 = Rω 00 on U for any open U ⊂ X and any ω 0 , ω 00 ∈ Ω1 (X). If R is local, then maps RU : Ω1 (U ) → Ω2 (U ) are defined naturally; they are Borel maps. Definition 3.2. Let Ω be a net of Borel spaces over a metric space X, and r > 0. A new net Ω+r of Borel spaces over the same X, called a prolongation of Ω, is defined as follows: Ω+r (U ) = Ω(U+r ) , (ρ+r )V,U = ρV+r ,U+r : Ωr (U ) → Ωr (V ) . It is easy to see that conditions 2.1(c–f,h) are satisfied. However, 2.1(g) is usually violated, that is, Ω+r is not a sheaf, even if Ω is a flabby sheaf. Definition 3.3. Let Ω1 , Ω2 be two nets of Borel spaces over a metric space X, and r > 0. A Borel map R : Ω1 (X) → Ω2 (X) is called r-local, if it is local as a map from the prolongation (Ω1 )+r to Ω2 . So, r-locality means the following: if ω 0 = ω 00 on U+r ,
then Rω 0 = Rω 00 on U .
Borel maps RU : Ω1 (U+r ) → Ω2 (U ) are defined. Some examples (for X = R) follow. The differentiation operator Rω = ω 0 is local (for any reasonable pair of nets of Borel 2 spaces where it acts), as well as the nonlinear operator (Rω)(x) = (ω(x)) . The r-averaging operator Z x+r 1 ω(y) dy (Rω)(x) = 2r x−r is r-local. If R1 is r1 -local and R2 is r2 -local, then R2 R1 is (r1 + r2 )-local, whenever R1 : Ω1 (X) → Ω2 (X) and R2 : Ω2 (X) → Ω3 (X). We return to inverse limits. Let Ω0 , Ω1 , . . . be nets of Borel spaces over a metric space X, connected by maps Rk−1,k : Ωk (X) → Ωk−1 (X) , Rk−1,k being rk -local , X rk < ∞ . k
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Define Ω(X) as the set of all sequences ω = (ω0 , ω1 , . . .), ωk ∈ Ωk (X), such that Rk−1,k ωk = ωk−1 for all k. Taking into account that the graph of a Borel function is a Borel set (see [11, Lemma 13.2.2]), we see that Ω(X) is a Borel subset of the product Ω0 (X) × Ω1 (X) × · · · , hence, a standard Borel space. The relation “ω 0 = ω 00 on U ” for an open U ⊂ X and ω 0 , ω 00 ∈ Ω(X) means, by definition, that ωk0 = ωk00 on U−sk for all k; here and henceforth sk = rk+1 + rk+2 + · · · (see Fig. 3). The set U−sk may be empty for some k; then “ωk0 = ωk00 on U−sk ” holds trivially.
↑| | | | s0
↑ | | | | s1 | ↓| ↓
≤0
↑ |
≤0 =R0,1 ≤1
r1
| ↓...... ... .
r2 ... .......
.. ...
.. ....
. ....
. ....
. ....
. ....
U−s1 ....
. ....
|
.
... ..
≤1 .... ..
{z U
.... .
≤1 =R1,2 ≤2 .... .
≤2 ... ..
.... .
}
≤
Fig. 3. Inverse limit for a nonzero dimension.
The above equivalence relation defines the corresponding quotient space Ω(U ); it is countably separated, since each Ωk (U−sk ) is so. In order to prove that Ω is a net of Borel spaces, it remains to verify Condition 2.1(f): if U = ∪ Ui and ω 0 = ω 00 on each Ui , then ω 0 = ω 00 on U . We have ωk0 = ωk00 on each (Ui )−sk , hence on ∪(Ui )−sk ; this union, however, is in general less than U−sk . Lemma 3.4. Let X be a compact metric space. Then for any family of open sets Ui ⊂ X, i ∈ I, and any s ∈ (0, ∞) ! ! [ [ [ (Ui )−ε = Ui . ε∈(0,∞)
i
−s
i
−s
The proof is left to the reader. This lemma remains in force for a non-compact X such that any bounded closed subset of X is compact. For this property, it is necessary but not sufficient, that X be separable, complete, and locally compact. Imposing this condition on X, we can prove that ωk0 = ωk00 on U−sk . Indeed, the map Rk,k+l = Rk,k+1 · · · Rk+l−1,k+l : Ωk+l (X) → Ωk (X) 0 00 = ωk+l is (rk+1 +· · ·+rk+l )-local, that is, (sk −sk+l )-local. Hence, the equality ωk+l on ∪(Ui )−sk+l implies the equality
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ωk0 = ωk00
[ on (Ui )−sk+l
! −(sk −sk+l )
i
for any l. Taking the union in l and using the lemma, we obtain U−sk . So, the following definition is correct. Definition 3.5. Let X be a metric space such that any bounded closed subset of X is compact. Let nets of Borel spaces Ωk over X be given, and rk -local maps Rk−1,k : Ωk (X) → Ωk−1 (X) with Σrk < ∞. Their inverse limit Ω = lim Ωk is the ←− following net of Borel spaces over X: Ω(X) = {ω = (ω0 , ω1 , . . .) : ωk ∈ Ωk (X) , Rk−1,k ωk = ωk−1 } , ω 0 = ω 00 on U
iff ∀k ωk0 = ωk00 on U−sk ,
sk = rk+1 + rk+2 + · · · The map Rk,∞ : Ω(X) → Ωk (X), Rk,∞ ω = ωk , is sk -local; a product measure µ on Ω(X) corresponds to measures µk = Rk,∞ µ on Ωk (X) that are usually not product measures. Instead, µk is 2sk -dependent in the following sense. Definition 3.6. Let Ω be a net of Borel spaces over a metric space X, and r ∈ (0, ∞). A probability measure µ on Ω(X) is called r-dependent, if F(U ) and F(V ) are µ-independent for any open sets U, V ⊂ X such that ρ(U, V ) ≥ r, that is, ρ(x, y) > r whenever x ∈ U, y ∈ V . It is clear that a product measure on Ω = lim Ωk is nothing but a sequence of ←−
2sk -dependent measures µk on Ωk (X) connected by the relation Rk−1,k µk = µk−1 . Examples will be given in Sec. 5. c. Additive integrals for dimension zero A necessary condition will be given for the existence of nonconstant square integrable additive integrals. Their nonexistence for relevant cases will be drawn in Sec. 4. Combined with simple continuity, this fact will exclude all kinds of integrals due to Theorem 1.7. We return to the tree introduced in Subsec. (a). Let a product measure µ on the inverse limit Ω(X) be given. Several notions of integral may be associated to (Ω(X), µ). Definition 2.7 and Theorem 2.8 are applicable, but they are intended mainly for more general spaces. Dealing with a totally disconnected space X, it is natural to pay special attention to the Boolean algebra AX of clopen subsets of X. Lemma 3.7. For any µ-measurable function f : Ω(X) → R the following three conditions are equivalent: (a) f is a µ-integral (as defined by 2.7) on the net Ω(X); (b) f is an integral (as defined by 1.3) on (Ω(X), µ) considered as a factorized measure space over the Boolean algebra AX of clopen sets;
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(c) the same as (b) but for the Boolean algebra Aµ of open sets U ⊂ X with µ-thin boundaries, satisfying the condition U 00 = U . Proof. The transition (c) =⇒ (b) is trivial, since AX ⊂ Aµ . The transition (b) =⇒ (a) follows from the fact that for any finite open covering U1 , . . . , Un of X there is a clopen partition A1 , . . . , An of X such that A1 ⊂ U1 , . . . , An ⊂ Un . The transition (a) =⇒ (c) is the point of Theorem 2.8. One of these three equivalent conditions can be expressed easily in terms of random variables a, a0 , a1 , a00 , a01 , . . . shown on Fig. 2. Namely, the condition (b) means that f = f0 + f1 = f00 + f01 + f10 + f11 = · · · , f = f (a, a0 , a1 , a00 , a01 , a10 , a11 , a000 , . . .) , f0 = f0 (a0 , a00 , a01 , a000 , . . .) , f1 = f1 (a1 , a10 , a11 , a100 , . . .) , f00 = f00 (a00 , a001 , . . .) , . . . Remember that the minimal angle ∠(E, F ) between two subspaces E, F ⊂ H of a Hilbert space H is defined as the maximal number of [0, π/2] satisfying the inequality hx, yi ≤ kxk · kyk · cos ∠(E, F ) for all x ∈ E, y ∈ F . Hilbert spaces over R are meant; for the complex case, hx, yi should be replaced by Rehx, yi or, equivalently, |hx, yi|. Clearly, ∠(E, F ) =
π 2
iff E ⊥ F .
Lemma 3.8. ∠(E ⊗ H2 , F ⊗ H2 ) = ∠(E, F ) for any two Hilbert spaces H1 , H2 and any two subspaces E, F ⊂ H1 ; the angle between E⊗H2 and F ⊗H2 is calculated in the tensor product Hilbert space H1 ⊗ H2 . Lemma 3.9. Let subspaces E1 , E2 , F1 , F2 ⊂ H be such that E1 ⊥ E2 , F1 ⊥ F2 , E1 ⊥ F2 , E2 ⊥ F1 . Then ∠(E1 + E2 , F1 + F2 ) = min (∠(E1 , F1 ), ∠(E2 , F2 )) . Lemma 3.10. ∠(E, P (F )) ≤ ∠(E, F ) for any subspaces E, F ⊂ H and any Hermitian projection operator P on H such that E ⊂ P (H). The proofs of Lemmas 3.8–3.10 are left to the reader. Lemma 3.11. Let a product of two probability spaces be given: (Ω, F , µ) = (Ω1 , F1 , µ1 ) × (Ω2 , F2 , µ2 ). Let E1 ⊂ F1 and E2 ⊂ F2 be some sub-σ-fieds, and
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E1 ⊂ L2 (µ1 ), E2 ⊂ L2 (µ2 ) some subspaces, orthogonal to constants. Define a subσ-field E ⊂ F as E1 ⊗ E2 , and a subspace E ⊂ L2 (µ) as consisting of all sums f (ω1 , ω2 ) = f1 (ω1 ) + f2 (ω2 ) with f1 ∈ E1 , f2 ∈ E2 . Then ∠(E, L2 (E)) = min (∠(E1 , L2 (E1 )) , ∠(E2 , L2 (E2 ))) ; here L2 (E1 ) = L2 (Ω1 , E1 , µ1 ), and the same applies for L2 (E2 ), L2 (E). Proof. L2 (Ek ) = Fk ⊕ 1, where 1 is the one-dimensional space of constants, and Fk is its orthogonal complement in L2 (Ek ). So, L2 (E) = (F1 ⊕ 1) ⊗ (F2 ⊕ 1) = F1 ⊗ F2 ⊕ F1 ⊕ F2 ⊕ 1 (we identify F1 ⊗ 1 with F1 , and so on). Therefore ∠ (E, L2 (E)) = ∠ (E1 ⊕ E2 , F1 ⊗ F2 ⊕ F1 ⊕ F2 ) = ∠ (E1 ⊕ E2 , F1 ⊗ (F2 ⊕ 1) ⊕ 1 ⊗ F2 ) = min (∠(E1 ⊗ 1, F1 ⊗ (F2 ⊕ 1)), ∠(1 ⊗ E2 , 1 ⊗ F2 )) due to Lemma 3.9. Further, ∠(E1 ⊗ 1, F1 ⊗ (F2 ⊕ 1)) ≥ ∠(E1 , F1 ) due to Lemma 3.8. The converse inequality is trivial.
A generalization for n spaces is straightforward. The maps Rk,∞ : Ω(X) → Ωk (X) determine embeddings L2 (µk ) ⊂ L2 (µ). Being treated as subspaces of L2 (µ), these L2 (µk ) form an increasing sequence of subspaces, whose union is dense in L2 (µ). It follows that 0, when dim E > 0 , ∠(E, L2 (µk )) ↓ ∠(E, L2 (µ)) = π/2 , when dim E = 0 for any subspace E ⊂ L2 (µ). Define a subspace I2 (µ) ⊂ L2 (µ) as consisting of all f ∈ L2 (µ), satisfying R equivalent conditions (a)–(c) of Lemma 3.7, such that f dµ = 0. If dim I2 (µ) > 0, then ∠(I2 (µ), L2 (µk )) → 0 when k → ∞. Lemma 3.10 shows that ∠(I2 (µ), L2 (µk )) ≥ ∠(Pk+1 I2 (µ), L2 (µk )) , where Pk+1 is the orthogonal projection of L2 (µ) onto L2 (µk+1 ). An element of Pk+1 I2 (µ) is a sum of 2k+1 functions, each depending on a single variable av (v being a vertex of (k + 1)th level, identified with a word of length k + 1). For example, a function of P2 I2 (µ) is of the form f00 (a00 ) + f01 (a01 ) + f10 (a10 ) + f11 (a11 ) ,
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each term having zero mean. Denote the space of such functions by I2 (µ2 ); I2 (µk ) is defined similarly. If dim I2 (µ) > 0, then ∠(I2 (µk+1 ), L2 (µk )) → 0 when k → ∞. Lemma 3.11 (or rather its generalization for n spaces) shows that the angle ∠ (I2 (µk+1 ), L2 (µk )) is equal to the minimum of corresponding angles, calculated for each vertex of kth level. Restricting ourselves to the homogeneous case, where a single “small map” ϕ is used for all vertices of the tree (and, of course, a single probability distribution for all av ), we conclude that ∠ (I2 (µk+1 ), L2 (µk )) does not depend on k, hence it cannot tend to zero, unless it is zero for k = 0! The following proposition is thus proved. Proposition 3.12. For the homogeneous zero-dimension scheme shown on Fig. 2, the condition ∠ (I2 (µ1 ), L2 (µ0 )) = 0 is necessary for dim I2 (µ) > 0. The case of finite Ω0 (X0 ), that is, of a finite range {0, . . . , r − 1} for a0 (and each av ), is especially interesting. For this case, the angle ∠(I2 (µ1 ), L2 (µ0 )) is taken in a finite-dimensional Hilbert space; so, the zero angle condition means that these two spaces have a common nonzero vector. d. Additive integrals for a nonzero dimension Suppose that Ω = lim Ωk as defined by 3.5, µ is a product measure on Ω(X), and ←−
µk are the corresponding 2sk -dependent measures on Ωk (X) (see 3.6). Suppose also that Aµ is a base of the topology on X (see 2c). Define a subspace I2 (µ) ⊂ L2 (µ) as consisting of all square integrable zero mean µ-integrals (defined by 2.7) or, equivalently, all square integrable zero mean integrals on the corresponding Aµ factorized measure space (see 2.8). The condition ∠ (I2 (µ), L2 (µk )) → 0 for k → ∞ is necessary for dim I2 (µ) > 0 for the same reason as in the previous subsection, but µk is no longer a product measure; this is why I2 (µk ) is no longer defined, and the use of Lemma 3.11 becomes complicated. Given some U1 , . . . , Un ∈ Aµ such that ρ(Uk , Ul ) > 4s0 for k 6= l, we can construct an Aµ -partition V1 , . . . , Vn so that (Vk )−2s0 ⊃ U k for all k. Consider the σ-field Ek ⊂ F(Vk ) generated by the pair ; ω ω0 (Uk )+s0
Vk \U k
(see Fig. 4) and the space I2 (Uk ) of all F(Uk )-measurable elements of I2 (µ). Lemma 3.11 is applicable, giving ∠ (I2 (W ), L2 (E)) = min ∠ (I2 (Uk ), L2 (Ek )) , k
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....... s...0 ......
. ... ... ... . . ...
(Uk )+s0 }| {.
z
... ... ... .. ... ..
... ... ... .. ...
| {z } U {zk Vk
|
... ... ... . . .. ...
|
... ... ... ... ... .
... ... ... ... ... .
≤0
. ... .. ... . . ...
≤
} . ... ... ... . . ...
{z
... . ... ... .... ... .. .... ... ... . . . ... . ... ... .... .......... .. ......... .......... . ......... ........ . . . . . . ... . . .
... ... ... ... ... .
Ek
... . .. ... ... ... ... ..... ... ... ..
}|
Vk
... ... ... ... ... .
{z
. ... ... ... . . . ...
... ... ... ... ... .
Vk+1
}
≤0 ≤X\W
E = E 1 ∨ · · · ∨ En Fig. 4. How to use Lemma 3.11 for a nonzero dimension.
where W = U1 ∪ · · · ∪ Un and the σ-field E ⊂ F(X) is generated by the pair ; ω0 ; ω X\W
it is easy to see that E = E1 ∨ · · · ∨ En . Lemma 3.13. Let E1 , . . . , En and F be subspaces of a Hilbert space; then ! X X 1 Ek , F ≥ min sin ∠ Ek , F + El . sin ∠ n k k
l6=k
Proof. First, note that the inequality hx, yi ≤ kxk · kyk · cos ∠(E, F )
for all x ∈ E, y ∈ F ,
used in the definition of ∠(E, F ), is equivalent to the inequality kx − yk ≥ kxk · sin ∠(E, F ) for all x ∈ E, y ∈ F . This being said, denote ε = min sin ∠ Ek , F + k
then
X
El ,
l6=k
X
xk − y ± xl
≥ ε · kxk k
l6=k
for any x1 ∈ E1 , . . . , xn ∈ En , y ∈ F . So,
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111
X 1
kxk k ≤ y − xl ;
ε l
X n X
xk ; xk ≤ y −
ε
X
ε
X
xk ; xk − y ≥
n the last inequality means that sin ∠ (ΣEk , F ) ≥ ε/n.
This lemma is useful when combining sets W1 , . . . , Wn , satisfying a condition of the form ∠ (I2 (Wk ), L2 (Ek )) ≥ α for k = 1, . . . , n , where the σ-field Ek ⊂ F(X) is generated by ω on X \ W k and ω0 on the whole X. (A way of constructing such Wk was considered before the lemma.) Suppose that Wk are pairwise disjoint. Denote Ek = I2 (Wk ) and F = L2 (µ0 ). Then F+
X
El ⊂ L2 (µ0 ) + I2 (X \ W k ) ⊂ L2 (Ek ) ,
l6=k
and Lemma 3.13 gives sin ∠ (I2 (W ), L2 (µ0 )) ≥
1 sin α , n
where W = W1 ∪ · · · ∪ Wn . For simplicity, ω0 and s0 were used before, but all that has been said holds for ωm and sm with any m. Typically, each Wk is fine-grained, when m is large, but n of such sets can cover (up to a thin set) the whole of X or a large region of it, the number n being kept constant when m → ∞. This method will be applied in Sec. 5, giving ∠ (I2 (µ), L2 (µk )) ≥ ε for all k, which implies dim I2 (µ) = 0. e. Simple continuity for dimension zero The simple continuity condition (defined by 2.10) for a product measure µ within the tree framework (described in 3a) will be formulated in terms of the “small map” ϕ : S 2 → S. Here S may be a finite set {0, . . . , r − 1} with the uniform probability distribution, but an arbitrary probability space (S, µ0 ) is also acceptable; ϕ transforms µ0 ⊗ µ0 into µ0 . We restrict ourselves to the homogeneous case (ϕ, S, µ0 do not depend on a vertex, and ϕ(a, b) = ϕ(b, a)) on the binary tree; the generalization is straightforward. Simple continuity means that any single-point subset of X is µ-thin. Due to homogeneity we may consider only one point x0 ∈ X, that is, one branch of the tree; let it be the leftmost branch (root, 0, 00, 000, . . .). The whole σ-field F(X) is generated by random variables a, a0 , a1 , a00 , a01 , a10 , a11 , a000 , . . . (see Fig. 2); its sub-σ-field F(X \ {x0 }) is generated by all of them except for a, a0 , a00 , a000 , . . . Simple continuity means that these σ-fields coincide mod 0. A localization by an arbitrary open
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set U ⊂ X is stipulated by Definition 2.6; but this does not matter when Aµ is a base of the topology. The set V of all vertices of the tree, identified with the set of all words of 0’s and 1’s, may be decomposed into three subsets: the leftmost branch V0 = {root, 0, 00, 000, . . .}; its “boundary” V1 = {1, 01, 001, 0001, . . .}; and the rest V2 = V \V0 \V1 . The set V1 separates V0 from V2 , which leads to the following Markov-type property for the corresponding sub-σ-fields E0 , E1 , E2 ⊂ F(X) (Ek being generated by all av for v ∈ Vk ). Lemma 3.14. E0 and E2 are conditionally independent given E1 . Proof. We will prove that (a, a0 , a00 ) and (a10 , a11 ) are conditionally independent, given (a1 , a01 ); a generalization for any finite number of levels is straightforward, as well as a limiting procedure for an infinite tree. Consider random variables ξ = a00 , X = (a, a0 , a00 ) ,
η = (a01 , a10 , a11 ) , Y = (a1 , a01 ) ,
Z = (a10 , a11 ) .
Then ξ and η are independent, and X = f (ξ, Y ), Y = g(η), Z = h(η) for some measurable functions f, g, h. The sequence (η, g(η), ξ) = (η, Y, ξ) is a Markov chain; therefore, the sequence (Z, Y, X) = (h(η), Y, f (ξ, Y )) is also a Markov chain. The simple continuity means that E0 ⊂ E1 ∨E2 (mod 0); due to the above lemma, this is equivalent to E0 ⊂ E1 (mod 0), which means that a = f (a1 , a01 , a001 , a0001 , . . .) almost sure for some measurable function f . A similar expression for a0 , a00 , . . . follows due to homogeneity: a0 = f (a01 , a001 , a0001 , . . .) and so on. Only the following twocomponent random sequence is relevant: b0 b1 b2 · · · a a0 a00 · · · = ; a1 a01 a001 · · · c 0 c1 c2 · · · bk , ck are introduced in order to simplify the notation. The joint distribution of all bk , ck is uniquely determined by the following conditions: ck are independent; each ck is distributed according to µ0 ; the same for bk ; bk = ϕ(bk+1 , ck ) almost sure for all k; c0 , . . . , ck , bk+1 are independent (for any k) (the last condition implies the first one). The question is, whether ck determine bk uniquely, or not. The space (S k × S, µk0 ⊗ µ0 ) of all (c0 , . . . , ck−1 ; bk ) is a natural domain for the finite random sequence (b0 , c0 ; . . . ; bk−1 , ck−1 ; bk ). For k → ∞, a natural domain for all bk , ck arises as the inverse limit
EXAMPLES OF NONLINEAR CONTINUOUS TENSOR PRODUCTS
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113
S˜ = lim(S k × S, ϕ(k) ) , ←−
ϕ(k) : S k × S → S k−1 × S , ϕ(k) (c0 , . . . , ck−1 ; bk ) = (c0 , . . . , ck−2 ; ϕ(ck−1 , bk )) (remember that ϕ(ck−1 , bk ) = ϕ(bk , ck−1 )). This inverse limit is a measure space ˜ Σ, ˜ µ (S, ˜), whereupon bk , ck are naturally defined. The σ-field generated by all bk , ck ˜ The sub-σ-field Σ ˜ 0 ⊂ Σ, ˜ generated by all ck , is the limit of subis the whole Σ. (k) k on S × S corresponding to the first factor, S k . Simple continuity σ-fileds Σ ˜ An equivalent formulation: the union of all Σ(k) is dense in ˜ means that Σ0 = Σ. ˜ Σ. Another equivalent formulation: the conditional probability of any event, given Σ(k) , tends (when k → ∞) to the indicator function of the event. The following criterion is thus obtained. Theorem 3.15. Let (S, µ0 ) be a probability space and ϕ : S × S → S a measure preserving map, symmetric in the sense that ϕ(a, b) = ϕ(b, a). Then the following three conditions are equivalent. (1) The product measure µ on the inverse limit Ω(X), resulting from a binary tree construction with the local map ϕ, is simple continuous. (2) For any measurable subset A of the other inverse limit S˜ = lim(S k ×S, ϕ(k) ), ←− and any ε > 0, there are n and a measurable subset C ⊂ S n such that ˜ µ ˜ (A 4 C) < ε, where C is naturally transferred to S. (3) For any measurable A ⊂ S and any ε > 0 there are n and a measurable subset C ⊂ S n such that µn0 (C) > 1 − ε, and for any (c0 , . . . , cn−1 ) ∈ C µ0 {bn ∈ S : ϕ(1) · · · ϕ(n) (c0 , . . . , cn−1 ; bn ) ∈ A} ∈ [0, ε] ∪ [1 − ε, 1] . f. Simple continuity for a nonzero dimension Once again, Ω = lim Ωk , µ is a product measure on Ω(X), and µk are the ←−
corresponding 2sk -dependent measures on Ωk (X). The fact that µk are no longer product measures forces us to search for another approach to simple continuity. As we know (see the end of Sec. 2), simple continuity is naturally related to the Hausdorff dimension, which suggests the following approach. Let a metric dk be given on each Ωk (X) so that Rk−1,k are nonexpanding: dk−1 (Rk−1,k ω 0 , Rk−1,k ω 00 ) ≤ dk (ω 0 , ω 00 ) for any ω 0 , ω 00 ∈ Ωk (X). For a closed set F ⊂ X define F = sup{dk (ω 0 , ω 00 ) : ω 0 , ω 00 ∈ Ωk (X), ω 0 = ω 00 d k
on X \ F } .
For any r > 0 define F+r = {x ∈ X : ρ(x, F ) ≤ r}. If F satisfy the condition F+s → 0 for k → ∞ , k d k
then the restriction map Ω(X) → Ω(X \ F ) is injective (hence, bijective). Indeed, let ω 0 , ω 00 ∈ Ω(X), ω 0 = ω 00 on X \ F . Then ω 0 = (ωk0 ), ω 00 = (ωk00 ), and ωk0 = ωk00 on (X \ F )−sk = X \ F+sk due to the sk -locality of Rk,∞ . It follows that dk (ωk0 , ωk00 ) ≤
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|F+sk |dk . So, the increasing sequence dk (ωk0 , ωk00 ) tends to zero! This means that ωk0 = ωk00 for all k, that is, ω 0 = ω 00 . However, a localization by an arbitrary open set U ⊂ X is stipulated by Definition 2.6. This is why the above argument has to be used over U rather than X, and Ωk (U−sk ) has to be used rather than Ωk (X). This will be done in Sec. 5; all sets of small Hausdorff dimension will appear to be thin. Note also that this approach, being measure-free, leads to Ω-thin sets (that are µ-thin for any µ), while the probabilistic approach applied for dimension zero leads to µ-thin sets. 4. Nonlinearizable Factorizations Over Zero-Dimensional Space a. Combinatorial models Let m, r be natural numbers, and ϕ : {0, 1, . . . , r − 1}m → {0, 1, . . . , r − 1} a function, symmetric (under rearrangements of its m variables) and preserving the uniform measure; that is, each of the inverse images ϕ−1 (0), . . . , ϕ−1 (r − 1) contains exactly rm−1 points. Then, as was explained in Sec. 3(a), a flabby sheaf Ω is defined over the space X of all branches of an m-adic tree. Each section ω ∈ Ω(X) of this sheaf is a function on the tree, valued in {0, . . . , r − 1} and satisfying the local equation, determined by ϕ, at each vertex (see Fig. 2). Taking the uniform probability distribution on {0, . . . , r − 1} for each vertex, we obtain a product measure µ on Ω(X), and the corresponding measure factorization. This construction will be called the (m, r, ϕ)-model, the corresponding Ω and µ being denoted by Ωm,r,ϕ and µm,r,ϕ . Additive integrals for an (m, r, ϕ)-model may be defined in several equivalent ways, see Lemma 3.7. A necessary condition for their existence was given by Proposition 3.12. Due to finiteness of r, this condition means existence of functions f, g1 , . . . , gm (f being nonconstant) such that f (ϕ(a1 , . . . , am )) = g1 (a1 ) + · · · + gm (am ) for all a1 , . . . , am ∈ {0, . . . , r − 1}. The left-hand side is symmetric under rearrangements of a1 , . . . , am ; symmetrizing the right-hand side, we obtain g1 = · · · = gm = g. The first example of this situation appears for a (2, 4, ϕ)-model. Note that such a situation is possible only for r > m, since g(a1 ) + · · · + g(am ) takes on at least m + 1 (different) values, while f (ϕ(a1 , . . . , am )) takes on at most r values. However, a factorization generated by some (m, r, ϕ)-model, is also generated by some (m2 , r, ψ)-model. Indeed, we may skip levels 1, 3, 5, . . . of the tree, connecting the root immediately to all the m2 vertices of level 2, each of them — to m2 vertices of level 4, and so on. For m = 2 ψ(b0 , b1 , b2 , b3 ) = ϕ (ϕ(b0 , b1 ), ϕ(b2 , b3 )) (see Fig. 5), the general case being similar. It follows that a factorization, generated by some (m, r, ϕ)-model with r ≥ m, can be generated also by some (m, r, ϕ)-model with r < m, which excludes additive integrals! The following theorem is thus proved.
EXAMPLES OF NONLINEAR CONTINUOUS TENSOR PRODUCTS
a=φ(a0 ,a1 )= .... ..• ... ...... =φ(φ(a00 ,a01 ),φ(a10 ,a11 )) .... .... . . .... .. . . . ... .... .... ... ... . . . .... ... . .... . ... ... . . .... ... . . ... ... .... . . . a0 ..... .... a1 ...... ...... .• . ...• . .. ..... ... ....... . . . .... .... .... .. . . . . . . . .... .... .. . ... . . . . .... . . . .... . . .... .... .... .... ....... ..... ....... . ..... . . • • • . . . ...•.... .. ..... .. ..... .. ..... . . . . . . . . . . ... .... ...... . ... ... . . . .. . . . . . . . . . . ... . . . . . .. .. .. . . . . (2,r,φ)-model
...
115
b=ψ(b0 ,b1 ,b2 ,b3 ) .... .• ............ ........ .......... . . . .. ... .... ..... . . . . . ... ..... .... ... .... ... .... .... ... .. ... .... .. ... .... ... .. ... ... . . .... . . ... .. ... .... . . . ... . .... ... . . . . ... .... . ... . . . . ... . ... . ... .... . . . . ... .... . ... . . . . . . .... ... . .. . . . ... . . . ... . .. .... . . . . . ... .... b3 b0 ....... .. . . .... ........ ......... ........ . . . • • • ...... . . ..• . . ..... ....... ..... ....... ..... ....... . . . . ...... ..... . . . . . .. ... ... ..... ... .... .... ...... .. ... ... ...... .. ... ... ...... . . . . . . . . . . . . .. ... .... ..... .. ... .... ..... .. ... ... ..... .. ... .... ..... . . . . . . . . . . . . . .. ... .. ... .. .... .. ... ... .. ... .. ... .. ... .. (4,r,ψ)-model
Fig. 5. Skipping some levels.
Theorem 4.1. For any (m, r, ϕ)-model, all square integrable additive integrals are constant. Combining it with Lemma 2.11 and Theorem 1.7, we conclude: Corollary 4.2. For any simple continuous (m, r, ϕ)-model, all integrals (both additive and multiplicative) are constant. We start discussing simple continuity with an example of its violation. Example 4.3. Consider the (2, 2, ϕ)-model with ϕ(a, b) = a + b (mod 2). The equation f (ϕ(a, b)) = f (a)f (b) has a solution: f (0) = 1, f (1) = −1; it results in a nonconstant multiplicative integral. The violation of simple continuity in this model is spectacular: the above multiplicative integral is independent of the σ-field F(U ) for any open set U ⊂ X, U 6= X. Note, however, that the minimal down continuity (see 1.6) is satisfied. The simple continuity condition was reformulated in Sec. 3 in terms of a twocomponent random sequence b0 b1 b2 · · · c0
c1
c2
···
such that c0 , . . . , ck , bk+1 are independent, and bk = ϕ(bk+1 , ck ) for any k. The case m = 2 was considered in Sec. 3(e), in which case each bk is uniformly distributed on {0, . . . , r − 1}, as well as each ck . A generalization for any m is straightforward: bk is still uniformly distributed on {0, . . . , r − 1}, but ck is uniformly distributed on {0, . . . , r − 1}m−1 . Simple continuity means that b0 is equal almost everywhere to a function of c0 , c1 , . . . . We have b0 = ϕ(b1 , c0 ) = ϕ(ϕ(b2 , c1 ), c0 ) = . . . . A more convenient notation: b0 = ϕc0 b1 = ϕc0 ϕc1 b2 = · · · = ϕc0 · · · ϕck bk+1 , ϕc : {0, . . . , r − 1} → {0, . . . , r − 1} for c ∈ {0, . . . , r − 1}m−1 , ϕc b = ϕ(b, c) .
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Simple continuity demands that for large k, for most of c0 , . . . , ck , the function ϕc0 . . . ϕck : {0, . . . , r − 1} → {0, . . . , r − 1} is close to a constant, which means, due to discreteness, that it is absolutely constant! The maps ϕc belong to the semigroup of all maps {0, . . . , r − 1} → {0, . . . , r − 1}; they generate a subsemigroup. It is necessary for simple continuity (as was shown above) that this subsemigroup contains at least one constant map. Note that this necessary condition is violated by Example 4.3, since all ϕc are bijective. Interestingly, the above necessary condition is also sufficient. Indeed, consider the Markov chain ϕc0 , ϕc0 ϕc1 , ϕc0 ϕc1 ϕc2 , . . . ; this is a random walk on the finite semigroup of all maps {0, . . . , r−1} → {0, . . . , r− 1}. Constant maps form an absorbing set: if ϕc0 . . . ϕck−1 = const, then evidently ϕc0 . . . ϕck−1 ϕck = const for any ck . If at least one product ϕc0 . . . ϕck−1 is a constant map, then there is a positive probability of entering the absorbing set after k steps from any point of the semigroup. It follows that ultimately the Markov chain enters the absorbing set with probability 1. The following criterion is thus proved. Theorem 4.4. An (m, r, ϕ)-model is simple continuous if and only if at least one constant map is contained in the semigroup generated by the maps a1 7→ ϕa2 ,...,am (a1 ) ≡ ϕ(a1 , a2 , . . . , am ) . Example 4.5. Hierarchical voting model. Take m = 3, r = 2 (two-valued functions on a triadic tree) with the voting (majority) function ϕ that may be defined as the only symmetric function such that ϕ(a, b, b) = b for any a, b ∈ {0, 1}. The semigroup generated by maps ϕ(·, b, c) contains constant maps for a trivial reason: some of these generators are constant maps. This situation may be called a depth 1 fulfillment of the criterion (given by Theorem 4.4). So, the model is simple continuous. Due to Corollary 4.2, all integrals are constant. Example 4.6. Let r = 2, m being arbitrary. A symmetric function ϕ of m twovalued variables is clearly a function of their sum: ϕ(a1 , . . . , am ) = ψ(a1 + · · ·+ am ). The condition |ϕ−1 (0)| = |ϕ−1 (1)| = 2m−1 takes the form X m X m = = 2m−1 . k k k:ψ(k)=0
k:ψ(k)=1
The only such case known to us is the case symmetric in the following sense: ϕ(1 − a1 , . . . , 1 − am ) = 1 − ϕ(a1 , . . . , am ); in terms of ψ this means that ψ(k) + ψ(m − k) = 1 for k = 0, . . . , m. Clearly, there are 2m/2 such functions for even m (and none for odd m). The criterion 4.4 is fulfilled on depth 1 for all of these functions, except for two of them, satisfying ψ(k) + ψ(k + 1) = 1 for k = 0, . . . , m − 1, which means that either ϕ(a1 , . . . , am ) = a1 + · · · + am (mod 2), or 1 − ϕ is as above. These exceptional cases generalize Example 4.3; corresponding models have evident multiplicative integrals and are not simple continuous. All other models of the considered class are simple continuous, thus admitting no nonconstant integrals.
EXAMPLES OF NONLINEAR CONTINUOUS TENSOR PRODUCTS
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117
Example 4.7. Take m = 2, r = 3 (three-valued functions on a dyadic tree), and the following function ϕ: 2 2 0 ϕ = 2 0 1. 0 1 1 None of the maps ϕ(·, b) is constant, but there is a constant product of two of them: ϕ(ϕ(a, 2), 0) = 2 for all a = 0, 1, 2. Thus, the criterion of Theorem 4.4 is fulfilled (on the depth 2). Once again, all integrals are constant. Example 4.8. Take m = 2 and any odd number r ≥ 3; define ϕ(a, b) = a + b (mod r) when a 6= b, but ϕ(a, a) = a. The criterion 4.4 is fulfilled on depth r − 1 (at most), since the map ϕ(·, 1), applied r − 1 times, sends any number to 1. So, all integrals are constant. Note. Each example in this section is formulated as a sheaf (or net) Ω over the compact Cantor set X of all branches of an m-adic tree. The group of all automorphisms of the tree acts on Ω as a group of symmetries. This group is naturally embedded into the group of all measure preserving homeomorphisms of the Cantor set (its natural measure is meant). A larger group of symmetries may be introduced; the simplest example of a “new” symmetry is shown on Fig. 6. This new group is naturally embedded into the group of all homeomorphisms of the Cantor set, but these homeomorphisms do not preserve the measure. In contrast to the white noise, no kind of “Jacobian correction” is needed. Another turn: X may be identified with the space Zm of all m-adic integers. Each shift x 7→ x + x0 is an automorphism of the tree; thus, the additive group of Zm acts on Ω. An enlargement for the (σ-compact) space X of all m-adic rationals Qm is straightforward. The m-adic affine group, consisting of all transformations x 7→ ax + b, a, b ∈ Qm , a 6= 0, acts on such Ω. ... ... ... .... .... . . .... . .
... . ....... ..• . . . . .... .... .... .... .
X1
...
... . ...... X2 .. ... .. ... X3 . . ...... ...•... ...• ... .... ... ....
...
... .. ... ..... . ... ... . ..
∼
...
X1 X2 X3 ..... ..... ..... ..• ..• ..• .... .... .... .... .... .... ... ... ...
∼
.... . ..... X1 .... .. ... ... X2 .. .. ...•... ...•... ... .... ... ....
...
...
... X 3 ..... ..... ..• .... ...... . . .... ....
...
Fig. 6. An automorphism of Ω that is not induced by an automorphism of the tree.
b. Continuous models on trees Now we restrict ourselves to m = 2, while r becomes infinite in the sense that a continuous measure space (S, µ0 ) is used instead of {0, . . . , r − 1}. This construction will be called the (2, S, ϕ)-model. It is in general a net of Borel spaces; however, if S is a compact topological space and ϕ : S × S → S is continuous, then a flabby sheaf results. Remember that the notions of additive and multiplicative integrals, and simple continuity, are applicable to nets of Borel spaces, not only to flabby sheaves.
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A necessary condition for the existence of non-constant additive integrals was given by Proposition 3.12 in terms of the minimal angle between two subspaces in a Hilbert space. In terms of the embedding ϕ∗ : L2 (µ0 ) → L2 (µ0 ⊗ µ0 ) = L2 (µ0 ) ⊗ L2 (µ0 ) conjugate to ϕ, the two subspaces are ϕ∗ L02 (µ0 ) and L02 (µ0 ) ⊗ 1 ⊕ 1 ⊗ L02 (µ0 ); here L02 (µ0 ) = L2 (µ0 ) 1 is the subspace of zero mean functions. So, the condition ∠ ϕ∗ (L02 (µ0 )), L02 (µ0 ) ⊗ 1 ⊕ 1 ⊗ L02 (µ0 ) > 0 ensures that all square integrable additive integrals are constant. Lemma 4.9. Let E, F1 , F2 be subspaces of a Hilbert space such that F1 ⊥ F2 ; then cos2 ∠(E, F1 + F2 ) ≤ cos2 ∠(E, F1 ) + cos2 ∠(E, F2 ) . Proof. Let αk = ∠(E, Fk ), then for any x ∈ E, y1 ∈ F1 , y2 ∈ F2 hx, y1 + y2 i = hx, y1 i + hx, y2 i ≤ kxk · ky1 k · cos α1 + kxk · ky2 k · cos α2 p p ≤ kxk · ky1 k2 + ky2 k2 · cos2 α1 + cos2 α2 p = kxk · ky1 + y2 k · cos2 α1 + cos2 α2 .
Corollary 4.10. Let ϕ : S × S → S be a measure preserving (µ0 ⊗ µ0 → µ0 ) and symmetric (ϕ(a, b) = ϕ(b, a)) map, and there is ε >R0 such that Rthe following inequality holds for any function f ∈ L2 (S, µ0 ) satisfying f dµ0 = 0, f 2 dµ0 = 1: 2 Z Z f (ϕ(a, b)) µ0 (db) µ0 (da) ≤ 1 − ε . 2 S S Then the corresponding (2, S, ϕ)-model has no nonconstant square integrable additive integrals. Proof. The given inequality means that 1−ε ; cos2 ∠ ϕ∗ (L02 (µ0 )), L02 (µ0 ) ⊗ 1 ≤ 2 taking the symmetry into account, Lemma 4.9 gives cos2 ∠ ϕ∗ (L02 (µ0 )), L02 (µ0 ) ⊗ 1 ⊕ 1 ⊗ L02 (µ0 ) ≤ 1 − ε ,
which ensures the absence of integrals.
A more practical sufficient condition may be given, when ϕ(a, ·) sends µ0 to a measure having a density p(a, ·), that is, for any bounded measurable function f on S Z Z f (ϕ(a, b)) µ0 (db) = f (b)p(a, b) µ0 (db) S
for µ0 -almost all a ∈ S.
S
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Theorem 4.11. Let ϕ and p be related as above, and Z Z 1 |p(a, b) − 1|2 µ0 (da)µ0 (db) < . 2 S S Then the (2, S, ϕ)-model has no nonconstant square integrable additive integrals. R R Proof. Let f ∈ L2 (S, µ0 ), f dµ0 = 0, f 2 dµ0 = 1; then 2 2 Z Z Z Z f (ϕ(a, b)) µ0 (db) µ0 (da) = f (b)p(a, b) µ0 (db) µ0 (da) S
S
S
S
2 Z Z = f (b)(p(a, b) − 1) µ0 (db) µ0 (da) S S Z Z |p(a, b) − 1|2 µ0 (db)µ0 (da) ; ≤ S
S
it remains to use Corollary 4.10. Corollary 4.12. Suppose that for each a ∈ S there is a partition of S into measurable subsets Aa0 , Aa1 , . . . such that the restriction of ϕ(a, ·) to each Aar is a measure preserving map of Aar onto its image. If µ0 (Aa0 ) ≥ µ0 (Aa1 ) ≥ . . . and ess sup a∈S
∞ X
rµ0 (Aar ) <
r=1
1 , 4
then the (2, S, ϕ)-model has no nonconstant square integrable additive integrals. Proof. The corresponding density is p(a, b) = I0a (b) + I1a (b) + · · · , where Ira is the indicator function of the ϕ(a, ·)-image Bra of Aar . Hence Z Z 2 |p(a, b) − 1| µ0 (db) = −1 + p2 (a, b) µ0 (db) = −1 +
X
µ0 (Bra ∩ Bsa )
r,s
=2
X
µ0 (Bra ∩ Bsa )
r>s
≤2
X
µ0 (Aar )
r>s
=2
X r
rµ0 (Aar ) <
1 ; 2
it remains to use Theorem 4.11. Two examples of nonlinearizable factorizations with continuous fiber over Cantor sets are given below, followed by two examples of linearizable factorizations, one (4.15) being standard, the other (4.16) extravagant.
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Example 4.13. The product of a countable number of copies of the hierarchical voting model (described by Example 4.5) is a model with a continuous measure space (S, µ0 ) (the product of countably many finite sets), which inherits from Example 4.5 the properties of simple continuity and absence of integrals due to Lemma 2.12 and Corollary 2.13. Of course, the hierarchical voting model may be replaced with models of Examples 4.6–4.8 and others. The previous example is a product of finite models; the following one is, in some sense, their projective limit. Example 4.14. Choose an integer p ≥ 6 and consider the ring (∞ ) X k sk p , sk ∈ Z/pZ S = Zp = k=0
of p-adic integers, with its natural topology (the p-topology, coinciding with the product topology) and measure (Haar measure, coinciding with Bernoulli product measure). Following the idea of Example 4.8, we define ϕ : Zp × Zp → Zp by ! ! ! ∞ ∞ ∞ ∞ X X X X k k k k sk p , tk p sk p tk p , = + ϕ k=0
k=0
k=r
k=0
where r = min{k : sk 6= tk }; the formula looks asymmetric, but the function is symmetric: ϕ(s, t) = ϕ(t, s). Note that ϕ(s, t) = s + t whenever s0 6= t0 . Note also that 0th digits behave exactly as in Example 4.8. This fact gives us a key to simple continuity. Remember Theorem 3.15 and the two sequences bk , ck introduced before that theorem. The 0th digit c00 of c0 ∈ Zp is a function of 0th digits b00 , b01 , . . . of b1 , b2 , . . . ∈ Zp . The reason was pointed out in 4.8: c00 is uniquely determined by b00 , . . . , b0n if n satisfies the condition b0n = b0n−1 = · · · = b0n−p+2 = 1; such an n almost surely exists. Further, it is easy to see that the first digit c10 of c0 is uniquely determined by b00 , . . . , b0n and b10 , . . . , b1n , if n is chosen so that the sequence b00 , . . . , b0n ends with p−1 digits 1, and the sequence b10 , . . . , b1n contains a group of p−1 digits 1 in succession, preceding the similar group in b00 , . . . , b0n . You see, p-adic arithmetics stipulates carrying from 0th digit to first, but not the converse. Continuing this way, we see that the whole p-adic number c0 is a function of b0 , b1 , . . . almost everywhere. Thus, the model is simple continuous. The absence of integrals follows from Corollary 4.12. Indeed, given a = Σsk pk , the map ϕ(a, ·) coincides with a shift within each of the sets Aar = {b = Σtk pk : t0 = s0 , . . . , tr−1 = sr−1 , tr 6= sr }, and ∞ X
rµ0 (Aar ) =
r=1
∞ X r=1
r·
1 1 p−1 = , · pr p p−1
which is less than 1/4 for p ≥ 6. Examples 4.13 and 4.14 can be modified to achieve an affine p-adic invariance in the same way as it can be done for the combinatorial Examples 4.5–4.8 (see the note at the end of 4a). For comparison, two linearizable examples follow.
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Example 4.15. Take S = R, µ0 = N (0, 1) the standard √ normal distribution (onedimensional Gaussian measure), and ϕ(a, b) = (a + b)/ 2. In order to check simple continuity, we consider N (0, 1) random variables b√ k , ck such that c0 , . . . , ck , bk+1 are independent, and bk = ϕ(bk+1 , ck ) = (bk+1 + ck )/ 2. Then clearly k+1 k+1 1 1 b0 = √ c0 + c1 + · · · + 2− 2 ck + 2− 2 bk+1 , 2 2
the last term being small in L2 . It follows that b0 =
∞ X
2−
k+1 2
ck ;
k=0
the series converges in L2 , hence, almost surely. So, b0 is a function of c0 , c1 , . . ., which means that this (2, R, ϕ)-model is simple continuous. In fact, this is the well known white noise. It is defined this time not on (0, 1) with Lebesgue measure, but on the Cantor set X with its natural measure µ0 , but this distinction is inessential: the factorization may be extended by continuity from clopen sets to all measurable sets mod 0. A generalization to a stable distribution is straightforward. Of course, the model is linearizable, and all additive integrals are well-known stochastic integrals. Example 4.16. Take S = (0, ∞), µ0 (ds) = e−s ds, and ϕ(a, b) = min(2a, 2b). Clearly, ϕ sends µ0 ⊗ µ0 into µ0 . Check simple continuity: bk = ϕ(bk+1 , ck ) = min(2bk+1 , 2ck ); b0 = min 2c0 , 4c1 , . . . , 2k+1 ck , 2k+1 bk+1 ; the last term tends to ∞ almost surely due to the Borel–Cantelli lemma, since 2k bk > C with probability exp(−2−k C); therefore b0 = min 2k+1 ck k
almost surely, which means that the model is simple continuous. It is instructive that this model is linearizable in spite of the fact that the equation f (min(2a, 2b)) = g(a) + g(b) has only trivial solutions (indeed, the case of a ∈ (ε, ∞), b ∈ (0, ε) shows that g is constant on (ε, ∞)). No function of a single variable a = min(2a0 , 2a1 ) = min(4a00 , 4a01 , 4a10 , 4a11 ) = · · · can be an additive integral, but a function of all these variables can! Fix C ∈ (0, ∞) and define f2 as the number of violated inequalities among the four inequalities 4a00 ≥ C, 4a01 ≥ C, 4a10 ≥ C, 4a11 ≥ C. For each level n of the binary tree, define fn similarly; these fn form an increasing sequence of integer-valued random variables, and E fn = 2n (1 − exp(−2−n C)) → C when n → ∞. So, the limit f = lim fn exists; this f is an additive integral. Being a sum of indicators of independent events of arbitrarily small probabilities, this f has a Poisson distribution, thus suggesting the idea that the model is isomorphic to a combination of independent Poisson random point fields. Indeed, such an isomorphism exists, and is described below.
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Take a Poisson random point field on the product X ×(0, ∞) of the Cantor set X, equipped with its natural probability measure ν, by (0, ∞) equipped with Lebesgue measure. The mean number of random points within A × (s, t) is ν(A) · (t − s) for A ⊂ X and (s, t) ⊂ (0, ∞). Define a random variable Z(A) as the maximal number z ∈ (0, ∞) such that A × (0, z) contains no points of the random set. Then Z(A) has an exponential distribution with the mean 1/ν(A), and Z(A ∪ B) = min(Z(A), Z(B)) for disjoint Borel sets A, B ⊂ X. Denote ξ(A) = ν(A)Z(A), then ξ(A) is distributed according to µ0 , and ν(C) ν(C) ξ(A), ξ(B) ξ(C) = min ν(A) ν(B) whenever C = A ∪ B and A ∩ B = ∅. Restricting ourselves to dyadic intervals A, B, C, we obtain ξ(C) = min(2ξ(A), 2ξ(B)), thus returning to the (2, S, ϕ)-model with ϕ(a, b) = min(2a, 2b). 5. Nonlinearizable Factorizations Over One-Dimensional Space The framework introduced in Sec. 3(b) will be used for X = R: Rk,k−1 : Ωk (R) → Ωk−1 (R) ; Rk−1,k is rk -local; sk = rk+1 + rk+2 + · · · < ∞. The inverse limit was defined by 3.5: Ω(R) = {ω = (ω0 , ω1 , . . .) : ωk ∈ Ωk (R), Rk−1,k ωk = ωk−1 } , ω 0 = ω 00 on U
iff
∀k
ωk0 = ωk00 on U−sk
(see Fig. 3). A product measure µ on Ω(R) will be constructed via a sequence of 2sk -dependent measures µk on Ωk (R) connected by the relation Rk−1,k µk = µk−1 . In contrast to zero-dimensional case, µk is not a product measure; this is why its description will be much more complicated than before. We restrict ourselves to the shift-invariant case: each µk will be the distribution of a stationary sample-continuous random process, and rk -local maps Rk−1,k will commute with shifts. One linearizable example — the famous white noise — will be considered in short, and then the main, nonlinearizable example will be constructed. White Noise. Choose Rk,k−1 as convolution operators: ωk−1 = Rk−1,k ωk when Z x+rk
ωk−1 (x) = x−rk
Vk (y − x)ωk (y) dy
R with some positive function Vk , concentrated on (−rk , rk ) such that Vk (x) dx = 1. These functions Vk may be chosen C ∞ -smooth, then Ωk (R) = C ∞ (R), for example. The infinite convolution Uk = VRk+1 ∗ Vk+2 ∗ · · · is a positive C ∞ -smooth function, concentrated on (−sk , sk ), and Uk (x) dx = 1. Define µk as the Gaussian measure on C ∞ (R) with zero mean and the following covariance: Z Z ωk (x)ωk (y) µk (dωk ) = Uk (z − x)Uk (z − y) dz .
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Then the product measure µ may be identified with the distribution of the white noise. This construction comes from studying Schwartz distributions from the viewpoint of linear inductive (and projective) limits (see e.g. [1]). Our main construction is also a projective limit with very nonlinear operators. Nonlinearizable Example. Choose Rk,k−1 in the following way: ωk−1 = Rk−1,k ωk when √ Z M − 1 x+rk ωk (y) dy ωk−1 (x) = ϕ L 2rk x−rk for all x ∈ R. Large positive constants L, M and a function ϕ : R → R will be chosen later. The space C(R) of all continuous functions R → R is chosen for Ωk (R), though a smaller set may be used equally well. The net of Borel spaces Ωk arises naturally. So, Ω(R) = lim (C(R), Rk,k−1 ), the whole construction evidently being stationary ←−
(that is, shift invariant). A nonlinear function ϕ, stipulated in the above formula, will be responsible for the nonlinearizability, while the integration for the coarse graining. The constant M controls rk , sk that form geometric progressions: rk+1 =
1 rk ; M
sk =
1 rk . M −1
Three conditions are imposed on the function ϕ: antisymmetry ϕ(u) = −ϕ(−u) for all u ∈ R , Lipschitz condition |ϕ(u) − ϕ(v)| ≤ C|u − v|
for all u, v ∈ R
with some constant C, and ϕ(u) = 1
for all u ∈ [1, +∞) .
It follows immediately that ϕ is bounded: sup |ϕ(u)| = kϕk < ∞ . u∈R
Lemma 5.1. If
√
M > LC, then finite sets are Ω-thin.
Proof. Let U ⊂ R be an open set and K ⊂ U a finite set; we have to prove that the restriction map from Ω(U ) to Ω(U \ K) is injective. Let ω 0 = ω 00 on U \ K. Following the approach outlined in Sec. 3(f), we introduce √ k Z |ωk0 (x) − ωk00 (x)| dx , dk = LC M − 1 U−sk
then dk−1 ≤ dk , since
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Z U−sk−1
√ √ Z x+rk Z M − 1 M − 1 x+rk 00 0 dx ϕ L ω (y) dy − ϕ L ω (y) dy k k 2rk 2rk x−rk x−rk
√ Z x+rk Z M −1 |ωk0 (y) − ωk00 (y)| dy dx 2rk U−(sk +rk ) x−rk Z √ |ωk0 (y) − ωk00 (y)| dy . = CL M − 1
≤ CL
U−sk
However, ωk0 = ωk00 on (U \ K)−sk = U−sk \ K+sk and |ωk0 − ωk00 | ≤ 2kϕk everywhere, hence √ k dk ≤ LC M − 1 · 2kϕk · mes (K+sk ) . A finite set K satisfies mes (K+sk ) = O(sk ) = O(M −k ), hence dk → 0, which means dk = 0. In fact, all compact sets of Hausdorff dimension d small enough (satisfying √ LC M1 < M 1−d ) are proved to be Ω-thin. √ It follows that any product measure on Ω is simple continuous, when M > LC. It is doubtful that the required sequence of measures µk , describing stationary random processes ωk , connected by nonlinear maps Rk−1,k , can be written out explicitly. Only an existence theorem will be proved. To this end, the following rescaling and renaming is useful: x , ωk (x) = ϕ Lξk sk √ Z x M − 1 x+rk+1 ωk+1 (y) dy . = ξk sk 2rk+1 x−rk+1 The equation ωk−1 = Rk−1,k ωk takes the form ξk−1 = Rξk with an operator R not depending on k: 1 Rξ(x) = √ 2 M −1
Z
Mx+M−1
ϕ (Lξ(y)) dy . Mx−M+1
The 2sk -dependence for ωk (that is, for its distribution µk ) means 2-dependence for ξk . Equip C(R) with the topology of uniform convergence on compact sets, then R : C(R) → C(R) is a continuous map. Introduce the set M of all probability measures µ on C(R), satisfying the following three conditions. First, µ is invariant under shifts Tt : C(R) → C(R), Tt ξ(x) = ξ(x − t). Second, µ is invariant under the transformation ξ 7→ −ξ. Third, µ is 2-dependent; in other words, ξ|(−∞,−1] and ξ|[1,∞) are µ-independent. Equip M with the narrow topology (called also weak topology by probabilists, and weak∗ topology by functional analysts). It is easy to see that the map R : C(R) → C(R) sends any measure of M into a measure of M. This continuous map M → M will be also denoted by R, or by RL,M
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to emphasize its dependence on the parameters L, M . (One more parameter, the function ϕ, remains suppressed.) Unfortunately, M is not convex (because of the 2-dependency condition); this is why we do not know, whether or not RL,M has a nontrivial fixed point. However, we know of a measure that is asymptotically fixed for L, M → ∞; an exact formulation is given by Lemma 5.2 below. Denote by γ the Gaussian measure on C(R) with zero mean and the following covariation: Z 1 π−2 max 0, 1 − |x − y| ξ(x)ξ(y) γ(dξ) = 2 π 2 for all x, y ∈ R. It is clear that γ ∈ M. Lemma 5.2. For any neighbourhood U of γ in M there are L0 , M0 ∈ (0, ∞) and a nonempty compact set K ⊂ U such that RL,M (K) ⊂ K for all L ∈ [L0 , ∞) and M ∈ [M0 , ∞). The proof is given in Appendix C. Corollary 5.3. For any neighbourhood U of γ in M, for any L and M large enough (depending on U ), there is a product measure µ on Ω(R) such that the ˜k for ξk such that µ ˜k ∈ U corresponding measures µk on Ωk (R) give distributions µ for all k. Proof. Lemma 5.2 gives L0 , M0 , and K ⊂ U . Let L ≥ L0 , M ≥ M0 . The set K∞ =
∞ \
RL,M
k
(K)
k=1
˜1 ∈ is nonempty due to compactness, and RL,M (K∞ ) = K∞ . Choose any point µ L,M ˜2 ∈ K∞ so that R µ ˜2 = µ ˜1 ; and so on. Transformed measures µk K∞ . Choose µ determine the required product measure µ. √ The numbers L ∈ [L0 , ∞) and M ∈ [M0 , ∞) can be chosen so that M > LC; then Lemma 5.1 ensures that µ is simple continuous. Thus, in order to prove the absence of any integrals, it suffices to prove the absence of square integrable additive integrals. This will be done according to Sec. 3(d). Consider a set W ⊂ R of the form W =
N [
(6nsk − sk+1 , 6nsk + sk+1 ) .
n=−N
It will be shown that ∠ (I2 (W ), L2 (E)) ≥ α with some α > 0 not dependent on N and k; here the σ-field E ⊂ F(R) is generated by ω on R \ W and ωk on the whole R. Remember that sk = M sk+1 . Choose
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M to be an integer. A union of 3M sets congruent to W covers the large interval (−6N sk , 6N sk ) up to a finite (hence, thin) set. Using Lemma 3.13 as explained in Sec. 3(d), we obtain sin ∠ (I2 (−6N sk , 6N sk ), L2 (µk )) ≥
1 sin α 3M
for any N , therefore sin ∠ (I2 (R), L2 (µk )) ≥
1 sin α > 0 3M
for any k; so, the subspace I2 (R) of all zero mean square integrable additive integrals contains only zero. The angle ∠ (I2 (W ), L2 (E)) is estimated by using Lemma 3.11, as explained in Sec. 3(d): ∠ (I2 (W ), L2 (E)) = ∠ (I2 (−sk+1 , sk+1 ), L2 (A)) , where the σ-field A ⊂ F(R) is generated by ω on (−3sk , −sk+1 ) ∪ (sk+1 , 3sk ) and ωk on (−sk − sk+1 , sk + sk+1 ). The following lemma will be applied to this A, and B = F(−sk+1 , sk+1 ). Lemma 5.4. Let (Ω, F , P ) be a probability space, and two sub-σ-fields A ⊂ F , B ⊂ F be given. Suppose that a set A0 ∈ A satisfies the following condition: P (A ∩ B) = P (A)P (B) whenever A ∈ A, B ∈ B, and A ⊂ A0 (this may be expressed as “A and B are independent on A0 ”). Then cos ∠ L2 (A), L02 (B) ≤ 1 − P (A0 ) . Proof. Let f ∈ L02 (B), then E (f |A) vanishes on A0 , and R 2 P (A0 ) Ω |f | dP , hence Z Z |E (f |A)|2 dP = |E (f |A)|2 dP Ω
R A0
|f |2 dP =
Ω\A0
Z
|f |2 dP
≤ Ω\A0
Z |f |2 dP .
= P (Ω \ A0 )
Ω
We define the event A0 as the set of all ω satisfying the inequality √ Z s y k M ωk+1 (z) dz ≥ b(y − x) − c sk sk x 1 2 2 1 < x < y < −M or M < x < y < 3− M ; absofor all x, y such that −3 + M lute constants b, c ∈ (0, ∞) will be chosen later. If Corollary 5.3 is applied to U small enough and L, M large enough, then, as will be proved in Appendix C, the
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probability of A0 becomes arbitrarily close to the square of the probability that the following inequality holds for the Wiener process W : r π−2 (W (y) − W (x)) ≥ b(y − x) − c 2 π whenever 0 < x < y < 3. Clearly, this latter probability is positive. It remains to be proved that A and B are independent on A0 . Remember that B = F(−sk+1 , sk+1 ), while A is generated by two (non-scalar) random variables: ω on (−3sk , −sk+1 ) ∪ (sk+1 , 3sk ), and ωk on (−sk − sk+1 , sk + sk+1 ). It would be enough to prove that the latter is a function of the former within A0 , but we will see that it is a constant 1 identically! Remember the expression for ωk as ϕ of an integral of ωk+1 , and the fact that ϕ(u) = 1 for u ∈ [1, ∞); we see that what we need is the inequality √ Z M − 1 x+rk+1 ωk+1 (y) dy ≥ 1 L 2rk+1 x−rk+1 for x ∈ (−sk − sk+1 , sk + sk+1 ) = (−(M + 1)sk+1 , (M + 1)sk+1 ) and ω ∈ A0 . The interval (x − rk+1 , x + rk+1 ) of length 2rk+1 = 2(M − 1)sk+1 lies within the domain (−(3M − 1)sk+1 , −2sk+1 ) ∪ (2sk+1 , (3M − 1)sk+1 ), except for the interval (−2sk+1 , 2sk+1 ) or part of it. Taking into account that |ωk+1 (y)| ≤ kϕk everywhere, we have √ Z x+r k+1 M ωk+1 (y) dy sk x−rk+1 √ M 2(M − 1)sk+1 − 4sk+1 − 2c − · 4sk+1 · kϕk ≥b sk sk 4kϕk 3 b − 2c − √ , =2 1− M M which is clearly greater than r √ 2 M −1 2rk+1 M = · √ sk L M L M −1 provided that b > c, and L, M are large enough. This completes the proof of the absence of integrals. The nonlinearizable example is thus constructed. The constructed net Ω of Borel spaces is in fact a flabby sheaf of Borel spaces. This fact follows from Lemma 2.2 applied to the inverse limit topology on Ω(R) (and each Ω(U )), arising from the chosen topology on Ωk (R) (and each Ωk (U )). Remember that the latter topology is 0that of uniform convergence on any bounded piece of R. Compactness of Ω(X), needed for Lemma 2.2, follows from the fact that Rk,k−1 maps the whole C(R) into a compact set. Due to Lemma 2.2, the surjectivity of the restriction map Ω(X) → Ω(U ) may be verified only when U is a finite union of open intervals. This fact follows from our next lemma.
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Lemma 5.5. For any ω 0 , ω 00 ∈ Ω(R) there is one and only one ω ∈ Ω(R) such that ω = ω 0 on (−∞, 0) and ω = ω 00 on (0, +∞). Proof. The single-point set {0} is thin due to Lemma 5.1, which means that ω is unique. In order to prove the existence of ω, its approximation ω (n) is constructed (n) as follows. Given some n, we glue ωn0 |(−∞,0) with ωn00 |(0,+∞) , obtaining ωn ; this function is discontinuous at 0, but the formulas for Rk,k−1 may still be used, giving (n) (n) (n) ωn−1 , ωn−2 , . . . , ω0 . The same estimate as in the proof of Lemma 5.1 shows that (n)
ωk
converges, when n → ∞, for each k.
It is easy to see that the glueing operation (ω 0 , ω 00 ) 7→ ω is continuous. So, a flabby sheaf Ω of Borel spaces is constructed, Ω(R) being in addition a metrizable compact topological space. A product measure µ is constructed on Ω(R); this measure is invariant under translations (shifts), and the corresponding measure factorization is nonlinearizable. Our only description of a section ω ∈ Ω(R) is a sequence of ωk ∈ Ωk (R). Such a description looks rather scanty when compared to that of Schwartz distributions. A sample function of the white noise can be integrated in time, giving a Brownian sample path. A section ω ∈ Ω(R) cannot be integrated. Nevertheless some traditional questions may be applied to our case. A Markov process (Xt ) may be defined, each Xt taking its values in Ω ((−∞, 0)), as follows: Xt (ω) = (T−t ω)|(−∞,0) = T−t (ω|(−∞,t) ) ; here T−t is the shift. The process (Xt ) is sample continuous (which is easy to check). It is a Feller process (due to the continuous glueing mentioned after Lemma 5.5). The predictability theorem of Meyer (see for Example [7]) leads to an important conclusion about the filtration (that is, increasing one-parameter family of σ-fields) F(−∞, t): Corollary 5.6. Each martingale, adapted to this filtration, is continuous. Note also that the action of R on (Ω(R), µ) (by shifts) is a Bernoulli flow (as well as on white noise). Appendix A: Logarithm Theorem 1.7 is proved here by means of a logarithmic construction. A related construction was used by Araki and Woods [2] in the framework of linearizable Hilbert factorizations, restoring a Hilbert space K = Log H from its Fock exponential space H = Exp K. Another related construction is given by the Dol´eans stochastic differential equation (see [9, 12, 20, VI.3]) in the framework of linearly ordered time, which is different from the framework of Boolean algebras. Throughout this appendix (Ω, F, P ) is a factorized measure space over a nonatomic Boolean algebra A, the minimal up continuity condition is assumed, and integrals (additive and multiplicative) are complex-valued, unless otherwise stated; see 1.2, 1.3, and 1.6. All statements remain valid for the case of 1.8, when A may contain atoms, but F vanishes at each atom.
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Any additive integral g produces a multiplicative integral f = exp g. Any multiplicative integral f such that Re f > 0 produces an additive integral g = log f . Can we represent any multiplicative integral f as exp g with some additive integral g ? Is this done by the Araki–Woods logarithm? Additive integrals, satisfying a reasonable continuity condition (A is the Borel σ-field, and g(A) = g(A1 ) + g(A2 ) + · · · for any countable partition A1 , A2 , . . . of A) were studied by Feldman [13] under the name “decomposable processes”. Gaussian and Poissonian components were extracted [13, Th. 4.1]. A Gaussian g with zero mean satisfy the relation k exp gk2 = exp kgk2 , while a Poissonian g does not. However, Fock exponent always satisfy this relation: kExp gk2 = exp kgk2 ,
Exp g =
∞ X 1 √ g ⊗k . k! k=0
This shows that the Fock exponent, as well as the Araki–Woods logarithm, cannot agree with the usual (pointwise) exponent and logarithm in the general (nonGaussian) case. Note also that a multiplicative integral f such that 0 < E |f |2 < ∞ may vanish with a positive probability; such f surely cannot be written as exp g, while the Araki–Woods logarithm is still applicable, giving f = Exp g. The following is our counterpart of Lemma 6.4 from [2]. Lemma A1. Let f be a multiplicative integral, 0 < E |f |2 < ∞. Then E f 6= 0. Proof. An F(A)-measurable function on Ω corresponds to each A ∈ A; this function, defined uniquely up to a scalar factor, will be denoted by fA or f (A); so, f = f (A)f (A0 ) for each A. Suppose that E f = 0. The ideal of all A ∈ A such that E f (A) 6= 0 (maybe, only 0A ) is contained in a maximal ideal S. Due to the minimal up continuity, we can choose A1 ≤ A2 ≤ · · · , Ak ∈ S, so that F(A1 ) ∨ F(A2 ) ∨ · · · = F(1A ). Then f = lim E (f |F(Ak )). Taking into account that E (f |F(Ak )) = f (Ak ) · E f (A0k ), we see that E f (A0k ) 6= 0 for k large enough. Hence, A0k ∈ S, which is impossible, since Ak ∈ S. A multiplicative integral f may be normalized so that E f = 1. Each f (A) also may be normalized, E f (A) = 1, thus f (A ∨ B) = f (A)f (B) for any disjoint A, B. Introduce m(A) = log E |f (A)|2 , then m(A) ≥ 0 , m(A ∨ B) = m(A) + m(B)
whenever A, B are disjoint .
A function satisfying these conditions will be called a measure on A; note that no continuity is implied here. A measure m on A will be called nonatomic, if for any
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ε there exists a partition A1 , . . . , An such that m(A1 ) < ε, . . . , m(An ) < ε. An equivalent condition: m is nonatomic if and only if for any maximal ideal S of A, sup m(A) = m(1A ) .
A∈S
Lemma A2. (a) Any additive integral g ∈ L2 (P ) determines a nonatomic measure mg : mg (A) = E |g(A)|2 − |E g(A)|2 . 6 0 determines a nonatomic (b) any multiplicative integral f ∈ L2 (P ) with kf k = measure mf : mf (A) = log E |f (A)|2 − log |E f (A)|2 . The proof is left to the reader. The following is our counterpart of Lemma 6.7 from [2]. Lemma A3. Let f ∈ L2 (P ) be a multiplicative integral, E f = 1. Then the following limit exists in L2 (P ): X (f (Ak ) − 1) , g= lim max mf (Ak )→0
k
implying that A1 , . . . , An form a partition, and each f (Ak ) is normalized: E f (Ak ) = 1. In other words: for any ε > 0 there exists δ > 0 such that for any partition A1 , . . . , An satisfying mf (A1 ) ≤ δ, . . . , mf (An ) ≤ δ the following inequality holds: 2 n X (f (Ak ) − 1) ≤ ε . E g − k=1
Before the proof, note a useful general inequality: 2
E |(1 + Y1 ) · · · (1 + Yn ) − (1 + Y1 + · · · + Yn )| ≤ E |Y1 |2 + · · · + E |Yn |2 · 1 + E |Y1 |2 · · · 1 + E |Yn |2 − 1 ≤
2 1 + E |Y1 |2 · · · 1 + E |Yn |2 − 1
(A4)
for any independent random variables Y1 , . . . , Yn with E Yk = 0, E |Yk |2 < ∞. Indeed, (1 + Y1 ) · · · (1 + Yn ) − (1 + Y1 + · · · + Yn ) = ((1 + Y1 ) − 1) Y2 + ((1 + Y1 )(1 + Y2 ) − 1) Y3 + · · · + ((1 + Y1 ) · · · (1 + Yn−1 ) − 1) Yn , the terms being orthogonal.
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Proof of Lemma A3. Two partitions have to be compared. Without loss of generality we suppose that one partition is a subpartition of the other. After evident renaming we have Y Y Xk ; Xk = Xkl ; X= k
l
Xk are independent ;
all Xkl are independent ;
E Xk = 1 ;
E Xkl = 1 .
log E |Xk | = mk ≤ δ ; X mk = m ; 2
k
It is enough to prove that 2 X X X (Xkl − 1) ≤ 4 m2k , E (Xk − 1) − k k,l k since Σm2k ≤ mδ, and we may choose δ = ε/m. The summands corresponding to different k are orthogonal, and we will see that for each k 2 X (Xkl − 1) ≤ 4m2k . E Xk − 1 − l
Apply (A4) to Yl = Xkl − 1: 2 X (Xkl − 1) E Xk − 1 − l
≤
−1 +
Y
1 + E |Xkl − 1|
2
!2
l
=
−1 +
Y
!2 E |Xkl |2
l
2 2 Y = −1 + E Xkl l
= −1 + E |Xk |2
2
= (emk − 1)2 . It remains to note that exp(mk ) ≤ 1 + 2mk if δ is small enough, since mk ≤ δ. It is clear that the g constructed is an additive integral, and mg = mf . The transition from f to g is essentially the Araki–Woods logarithm for the commutative setup. The converse operation is described in the following lemma.
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Lemma A5. Let g ∈ L2 (P ) be an additive integral, E g = 0. Then the following limit exists in L2 (P ): Y (1 + g(Ak )) , f= lim max mg (Ak )→0
k
implying that A1 , . . . , An form a partition, and each g(Ak ) is centered: E g(Ak ) = 0. Proof. Starting as in the proof of Lemma A3, we have X X Yk , Yk = Ykl , Y = k
l
Yk are independent ,
all Ykl are independent ,
E Yk = 0 ,
E Ykl = 0 ,
E |Yk | = mk ≤ δ , X mk = m ;
E |Ykl |2 = mkl , X mkl = mk ,
2
k
l
and it is enough to prove that 2 Y X Y m2k . E (1 + Yk ) − (1 + Ykl ) ≤ em k k,l k Denote Xk =
Y
(1 + Yi ) ,
i≤k
Zk =
YY (1 + Yij ) , i≤k j
Zkl
YY Y = (1 + Yij ) · (1 + Ykj ) , i
Mk =
X
j≤l
mi .
i≤k
It is enough to prove that E |Xk − Zk |2 ≤ eMk
X
m2i
i≤k
for all k, which follows inductively from the inequality e−Mk E |Xk − Zk |2 ≤ m2k + e−Mk−1 E |Xk−1 − Zk−1 |2 or the slightly stronger inequality E |Xk − Zk |2 − E |Xk−1 − Zk−1 |2 ≤ mk mk eMk + E |Xk−1 − Zk−1 |2 .
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The latter will result from the following inequality by summation in l: 2 X Ykj − Zkl − E E Xk−1 1 + j≤l
2 X Xk−1 1 + Ykj − Zk,l−1 j≤l−1
≤ mkl mk eMk + E |Xk−1 − Zk−1 |2 . Finally,
Xk−1 1 +
X
Ykj − Zkl − Xk−1 1 +
j≤l
X
Ykj − Zk,l−1
j≤l−1
= Xk−1 Ykl − Zk,l−1 Ykl = Ykl (Xk−1 − Zk,l−1 ) , the former difference being orthogonal to its second term; hence, it remains to prove that 2 E |Ykl (Xk−1 − Zk,l−1 )| ≤ mkl mk eMk + E |Xk−1 − Zk−1 |2 . Now, Ykl is independent of Xk−1 − Zk,l−1 , and E |Ykl |2 = mkl , and E |Xk−1 − Zk,l−1 |2 = E |Xk−1 − Zk−1 + Zk−1 − Zk,l−1 |2 2 Y = E (Xk−1 − Zk−1 ) + Zk−1 1 − (1 + Ykj ) j≤l−1 2 Y 2 2 = E |Xk−1 − Zk−1 | + E |Zk−1 | · E 1 − (1 + Ykj ) , j≤l−1 but E |Zk−1 |2 =
Y Y
(1 + mij ) ≤ eMk−1 ,
i≤k−1 j
2 Y Y (1 + Ykj ) = −1 + (1 + mkj ) ≤ emk − 1 ≤ mk emk , E 1 − j≤l−1 j≤l−1 which completes the proof. The following theorem is thus obtained. Theorem A6. There is a natural one-one correspondence between all square integrable additive integrals g with E g = 0 and all square integrable multiplicative
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integrals f with E f = 1. If f ↔ g, then mf = mg , hence log E |f |2 = E |g|2 . (For the formulas connecting f and g see Lemmas A3 and A5.) Note. The minimal up continuity condition may be omitted, provided that only f, g with nonatomic mf , mg are considered. Then, however, some multiplicative integrals may have zero mean, thus escaping the theorem. Proof of Theorem 1.7. We have to prove the coincidence of the four σ-fields Fa , Fb , Fc , Fd , defined by the four items (a), (b), (c), (d) of the theorem. Trivially, Fb ⊂ Fa and Fd ⊂ Fc . The simple formula f = exp(iλg), λ ∈ R, shows that Fa ⊂ Fd . It remains to prove that Fc ⊂ Fb , that is, any multiplicative integral f is Fb -measurable. First, consider a square integrable multiplicative integral f . Theorem A5 gives the corresponding square integrable additive integral g such that f can be expressed through g(A), A ∈ A (but, in general, not through g itself!). For each A, both Re g(A) and Im g(A) are real-valued square integrable additive integrals, hence, Fb -measurable. Thus, f is Fb -measurable. Second, consider a multiplicative integral f valued on [0, ∞). For any λ ∈ R define exp(iλ log f ) , when f 6= 0 , fλ = 0, when f = 0 . Each fλ is also a multiplicative integral. Being bounded, it is square integrable, hence Fb -measurable. Taking two incommensurable values of λ, we conclude that f is Fb -measurable. Finally, consider an arbitrary multiplicative integral f . Its absolute value |f | is a multiplicative integral valued on [0, ∞), hence, Fb -measurable. Another multiplicative integral f /|f | , when f 6= 0 , 0,
when f = 0 ,
being bounded, is Fb -measurable. Their product f is thus Fb -measurable.
Appendix B: Topological Base and Boolean Base Theorem 2.8 and Lemma 1.4 are proved here. A product measure µ is given on a net Ω of Borel spaces over a separable metrizable space X, and a µ-integral f , additive or multiplicative. We have to prove that f is an integral on the Aµ -factorized measure space (Ω(X), F, µ), which is the main, “only if” part of Theorem 2.8. (Its “if” part and Lemma 1.4 are postponed to the end of this appendix.) Setting aside for a while the possible zero value of a multiplicative integral, we may join both cases (additive and multiplicative) by treating f as an integral valued in a commutative topological group G. A decomposition f = f1 + · · · + fn with F(Uk )-measurable
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fk exists for any finite open covering U1 , . . . , Un of X. We need to construct a decomposition f = f1 + · · · + fn with F(Ak )-measurable fk for any finite partition A1 , . . . , An of the unity within the Boolean algebra Aµ . The point is that Uk overlap, while Ak do not. It suffices to consider the case n = 2. Indeed, having a decomposition f = f1 + f23 with F(A1 )-measurable f1 and F(A2 ∨ A3 )-measurable f23 , and a similar decomposition f = f12 +f3 , we conclude that f = f1 +f2 +f3 , where f2 = f12 −f1 = f23 − f3 is measurable with respect to F(A1 ∨ A2 ) ∧ F(A2 ∨ A3 ) = F(A2 ). The same holds for higher n. Consider a two-element partition (A, B), B = A0 , A ∈ Aµ . Using the fact that Aµ is a base of the separable metrizable topology, we can cover the open set A by an increasing sequence of open sets Ak ∈ Aµ so that Ak ⊂ A. Similarly, choose Bk ∈ Aµ , Bk ↑ B, B k ⊂ B. The open set Ck = (Ak ∨ Bk )0 contains the common boundary of A and B. Three sets A, B, and Ck form an open covering of X for each k. There is a corresponding decomposition f = ϕk + ψk + χk , inspiring a naive hope for a limiting procedure k → ∞ giving f = ϕ + ψ. However, this transition is far from being straightforward. Theorem B1. Let G be a topological group, commutative, metrizable, and separable. Let (Ω, F, P ) be a factorized measure space over a Boolean algebra A. Let Ak and Bk be two increasing sequences in A such that Ak ∧ Bk = 0A for each k, and all the σ-fields F(Ak ), F(Bk ) together generate the whole σ-field F = F(1A ). Let a function f : Ω → G admit (for any k) a decomposition f = ϕk + ψk + χk such that ϕk : Ω → G is measurable with respect to F A = F(A1 ) ∨ F(A2 ) ∨ · · · , ψk w.r.t. F B = F(B1 ) ∨ F(B2 ) ∨ · · · , and χk w.r.t. F(Ck ), where Ck = (Ak ∨ Bk )0 . Then f admits a decomposition f = ϕ + ψ such that ϕ is F A -measurable and ψ is F B -measurable. The proof will be given after a definition and several lemmas. The reader can / L2 easily find a simple proof for the case of G = R and f ∈ L2 . The case G = R, f ∈ is not much more complicated. However, the case of a circle G = R/Z needs a quite different approach, presented below. Definition B2. Let (Ω1 , F1 , P 1 ) and (Ω2 , F2 , P 2 ) be two probability spaces, and µ, µ1 , µ2 , . . . be probability measures on (Ω1 × Ω2 , F1 ⊗ F2 ) whose projections to Ω1 coincide with P 1 and to Ω2 — with P 2 . The phrase “µk weakly converges to µ” will mean that µk (A × B) → µ(A × B) for any A ∈ F1 , B ∈ F2 . The same for any finite number of probability spaces. Lemma B3. In the above definition, it suffices to check the relation µk (A×B) → µ(A×B) for any A ∈ A1 , B ∈ A2 , where A1 ⊂ F1 , A2 ⊂ F2 are algebras, generating σ-fields F1 , F2 , resp. Proof. |µk (An ×Bn )−µk (A×B)| ≤ P 1 (An 4A)+P 2 (Bn 4B) for any k, n, and the same for µ. Choose An ∈ A1 , Bn ∈ A2 so that P 1 (An 4A) → 0, P 2 (Bn 4B) →
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0 and use the inequality |µk (A × B) − µ(A × B)| ≤ |µk (An × Bn ) − µ(An × Bn )| + |µk (An × Bn ) − µk (A × B)| + |µ(An × Bn ) − µ(A × B)|. Lemma B4. Let (Ω, F , P ) be the product of three probability spaces (Ωn , Fn , P n ), n = 1, 2, 3. Let µ, µ1 , µ2 , . . . be probability measures on (Ω, F ) such that µk → µ weakly. (Their projections to Ωn are thus supposed to coincide with P n for n = 1, 2, 3.) Suppose in addition that their projections to Ω2 × Ω3 coincide with P 2 ⊗ P3 . Then µk → µ weakly on the product of two spaces Ω1 × (Ω2 × Ω3 ). Proof. Apply Lemma B3 to the subalgebra of F2 ⊗ F3 generated by B × C, B ∈ F 2 , C ∈ F3 . Lemma B5. Let four probability spaces be given (Ωn , Fn , P n ), n = 1, 2, 3, 4, and two measure preserving maps f : Ω2 → Ω3 , g : Ω2 → Ω4 . Let µk → µ weakly on (Ω1 , F1 ) × (Ω2 , F2 ). (Their projections to Ωn are thus supposed to coincide with P n for n = 1, 2.) Consider the images ν, νk of µ, µk under the map h : Ω1 × Ω2 → Ω1 × Ω3 × Ω4 , h(ω1 , ω2 ) = (ω1 , f (ω2 ), g(ω2 )) . Then νk → ν weakly on Ω1 × Ω3 × Ω4 . Proof. Follows immediately from the definition. Proof of Theorem B1. Define FkA = F A ∩ F(Ck ), FkB = F B ∩ F(Ck ). Note that FkA ⊃ F(Al ∧ A0k ) for any l > k; hence, F(Ak ) ∨ FkA ⊃ F A ; in fact, F(Ak ) ∨ FkA = F A . The same for B. The four independent σ-fields F(Ak ), FkA , FkB , F(Bk ) generate F . Accordingly, (Ω, F , P ) may be identified with the product of four spaces. The following notation is convenient: Ω = Ω(Ak ) × Ω(A \ Ak ) × Ω(B \ Bk ) × Ω(Bk ) , despite the fact that A, B in general cannot be treated as elements of A. (Intuitively they may be thought of as belonging to a completion of A.) Similar notation will be used for σ-fields and measures; that is, (Ω, F , P ) = (Ω(Ak ), F(Ak ), P (Ak )) × (Ω(A \ Ak ), F(A \ Ak ), P (A \ Ak )) × (Ω(B \ Bk ), F(B \ Bk ), P (B \ Bk )) × Ω(Bk ), F(Bk ), P (Bk ) for any k. (Hopefully the reader will not be confused by the notation P (Ak ): this is not a measure of Ak .) Introduce measures µ, µk on the space Ω2 = Ω × Ω = Ω(A) × Ω(B) × Ω(A) × Ω(B) as follows. The measure µ is simply P 2 = P ⊗ P , which may be written as P (A) ⊗ P (B) ⊗ P (A) ⊗ P (B). We find it more convenient (though unconventional) to arrange the factors in two dimensions, P (A) P (B) µ=⊗ , P (A) P (B) rather than line them up. A finer decomposition: P (Ak ) P (A \ Ak ) P (B \ Bk ) P (Bk ) . µ=⊗ P (Ak ) P (A \ Ak ) P (B \ Bk ) P (Bk )
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In other words, µ may be defined as the distribution of the random assembly ω1 (Ak ) ω1 (A \ Ak ) ω1 (B \ Bk ) ω1 (Bk ) ω2 (Ak ) ω2 (A \ Ak ) ω2 (B \ Bk ) ω2 (Bk ) where all the eight elements are treated as independent random variables valued in corresponding spaces Ω(Ak ), . . . and having corresponding distributions P (Ak ), . . . The measure µk is, by definition, the distribution of the random assembly ω1 (Ak ) ω(A \ Ak ) ω(B \ Bk ) ω1 (Bk ) ω2 (Ak ) ω(A \ Ak ) ω(B \ Bk ) ω2 (Bk ) formed from six independent random variables, whose required distributions are evident from their notation. Both µ and µk may be considered as distributions of the four-element random assembly Ω(A) Ω(B) ω1 (A) ω1 (B) , ∈× Ω(A) Ω(B) ω2 (A) ω2 (B) these four elements being independent for µ, but dependent in a special way for µk . Note that the distribution of ω 0 (A) is just P (A) both for µ and for µk ; the same for the other three elements. In fact, µk → µ weakly on the product of the four probability spaces. This fact follows from Lemma B3 (generalized for four factors), since µ and µk coincide on sets of the form E1 F1 × E2 F2 with E1 , E2 ∈ F(Ak ) ⊂ F(A) and F1 , F2 ∈ F(Bk ) ⊂ F(B). You see, F(A1 ) ∪ F(A2 ) ∪ · · · is an algebra, generating the σ-field F(A); the same applies to B. We use Lemma B5 for duplicating each of the four elements of the random assembly; then we use Lemma B4 for pairing new elements according to the following scheme: Ω(A) Ω(B) ω1 (A) ω1 (B) × 3 ω2 (A) ω2 (B) Ω(A) Ω(B) Ω Ω (ω1 (A), ω1 (B)) (ω1 (A), ω2 (B)) h . ∈× 7−→ Ω Ω (ω2 (A), ω2 (B)) (ω2 (A), ω1 (B)) This map h sends µ, µk into some measures ν, νk on Ω4 such that νk → ν weakly. An additional requirement of Lemma B4 is fulfilled here: the distribution of each pair coincides with P both for νk and for ν. Now the function f : Ω → G comes into play via the set F ⊂ Ω4 consisting of all assemblies Ω Ω ω11 ω12 ∈× ω22 ω21 Ω Ω satisfying the equality f (ω11 ) + f (ω22 ) = f (ω12 ) + f (ω21 ) .
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Note that νk (F ) = 1 for each k. Indeed: νk (F ) = µk (h−1 (F )). The set h−1 (F ) consists of all assemblies ω1 (A) ω1 (B) ω2 (A) ω2 (B) satisfying the condition f (ω1 (A), ω1 (B)) + f (ω2 (A), ω2 (B)) = f (ω1 (A), ω2 (B)) + f (ω2 (A), ω1 (B)) . According to the definition of µk , we substitute ω1 (A) = (ω1 (Ak ), ω(A \ Ak )) ,
ω1 (B) = (ω(B \ Bk ), ω1 (Bk )) ,
ω2 (A) = (ω2 (Ak ), ω(A \ Ak )) ,
ω2 (B) = (ω(B \ Bk ), ω2 (Bk )) .
According to the given property of the function f , we substitute f = ϕk + ψk + χk , that is, f (ω1 (A), ω1 (B)) = f (ω1 (Ak ), ω(A \ Ak ), ω(B \ Bk ), ω1 (Bk )) = ϕk (ω1 (Ak ), ω(A \ Ak )) + χk (ω(A \ Ak ), ω(B \ Bk )) + ψk (ω(B \ Bk ), ω1 (Bk )) , and the same for the three other terms; the required equality becomes evident. The limiting procedure k → ∞ gives ν(F ) = 1. Indeed, the complement of F is the union of a sequence of product sets: Ω4 \ F =
[
×
n
E1n
F1n
E2n
F2n
for some E1n , E2n ∈ F(A) and F1n , F2n ∈ F(B). The point is that the complement of the closed set {(g1 , g2 , g3 , g4 ) : f (g1 ) + f (g2 ) = f (g3 ) + f (g4 )} ⊂ G4 is the union of a sequence of open product sets, since the topology of G is metrizable and separable. For each n the corresponding product subset of Ω4 \ F is of zero measure for each νk , hence for ν, as well. Thus, ν(F ) = 1, which means that f (ω1 (A), ω1 (B)) + f (ω2 (A), ω2 (B)) = f (ω1 (A), ω2 (B)) + f (ω2 (A), ω1 (B)) for P (A)-almost all ω1 (A), ω2 (A) ∈ Ω(A) and P (B)-almost all ω1 (B), ω2 (B) ∈ Ω(B). Choose ω2 (A), ω2 (B) so that the above equality holds for almost all ω1 (A), ω1 (B). Define ϕ0 : Ω(A) → G and ψ0 : Ω(B) → G by ϕ0 (ω1 (A)) = f (ω1 (A), ω2 (B)) ,
ψ0 (ω1 (B)) = f (ω2 (A), ω1 (B)) ;
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then f (ω1 (A), ω1 (B)) + f (ω2 (A), ω2 (B)) = ϕ0 (ω1 (A)) + ψ0 (ω1 (B)). It remains to put ϕ (ω(A), ω(B)) = ϕ0 (ω(A)) , ψ (ω(A), ω(B)) = ψ0 (ω(B)) − f (ω2 (A), ω2 (B)) , thus obtaining f = ϕ + ψ, which completes the proof of Theorem B1. The “only if” part of Theorem 2.8 is thus proved for all additive integrals, and for multiplicative integrals that do not vanish. Consider the case when G is the multiplicative semigroup of all complex numbers (including zero). Lemma B6. Defining an integral (additive or multiplicative) on a factorized measure space (see Definition 1.3), we may restrict ourselves to the case n = 2 (that is, two-element partitions of unity). Proof. The additive case (in fact, the group case) is already verified, see the second paragraph of this appendix: f = f1 +f23 = f12 +f3 , f2 = f12 −f1 = f23 −f3 . Consider the multiplicative (semigroup) case. We have f = f1 f23 = f12 f3 . The only problem is not to divide by zero when defining f2 = f12 /f1 = f23 /f3 . The partition A1 , A2 , A3 given in the Boolean algebra leads to the decomposition Ω = Ω1 ×Ω2 ×Ω3 such that f (ω1 , ω2 , ω3 ) = f1 (ω1 )f23 (ω2 , ω3 ) = f12 (ω1 , ω2 )f3 (ω3 ). If f is identically ω1 ) 6= zero, then there is nothing to prove. Otherwise choose ω ˜1 ∈ Ω1 so that f1 (˜ ω1 )f23 (ω2 , ω3 ) = 0 and the following equalities hold for almost all ω2 , ω3 : f1 (˜ ω1 , ω2 )f3 (ω3 ). Choose ω ˜ 3 ∈ Ω3 so that f3 (˜ ω3 ) 6= 0 and the f (˜ ω1 , ω2 , ω3 ) = f12 (˜ ω1 )f23 (ω2 , ω ˜ 3 ) = f (˜ ω1 , ω2 , ω ˜ 3) = following equalities hold for almost all ω2 : f1 (˜ ω1 , ω2 )f3 (˜ ω3 ). Define f2 (ω2 ) = f12 (˜ ω1 , ω2 )/f1 (˜ ω1 ) = f23 (ω2 , ω ˜ 3 )/f3 (˜ ω3 ), then f12 (˜ f (ω1 , ω2 , ω3 ) = f1 (ω1 )f2 (ω2 )f3 (ω3 ). Now we are in a position to finish the “only if” part of Theorem 2.8 for the multiplicative case. The proof proceeds in parallel to the additive case, giving f (ω1 (A), ω1 (B)) · f (ω2 (A), ω2 (B)) = f (ω1 (A), ω2 (B)) · f (ω2 (A), ω1 (B)) almost everywhere. In order to finish the proof, we need only ensure that f (ω2 (A), ω2 (B)) 6= 0. Such pairs (ω2 (A), ω2 (B)) form a set of positive measure (unless f is identically zero), and almost all of them ensure the above equality for almost all ω1 (A), ω1 (B). The “if” part of Theorem 2.8 for a compact space X follows immediately from the following simple fact: for any finite open covering U1 , . . . , Un of X there is a partition A1 , . . . , An of the unity within the Boolean algebra Aµ such that A1 ⊂ U1 , . . . , An ⊂ Un . However, the general case needs Lemma 1.4 (proved below). Proof of Theorem 2.8 (“If ” part). Let a finite open covering U1 , . . . , Un of X be given. Any base of X contains a countable subbase. Choose a sequence
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A1 , A2 , . . . ∈ Aµ such that A1 ∪ A2 ∪ · · · = X and for each k there is m(k) such that Ak ⊂ Um(k) . Put Bk = Ak \ (A1 ∪ · · · ∪ Ak−1 ) and define Clm = ∨{Bk : k = 1, . . . , l; m(k) = m}, then Cl1 ∨ · · · ∨ Cln = A1 ∨ · · · ∨ Al in Aµ . It follows that F(Cl1 ) ∨ · · · ∨ F(Cln ) increases to the whole σ-field F for l → ∞. Open sets Vm = C1m ∪ C2m ∪ · · · are disjoint, and Vm ⊂ Um for m = 1, . . . , n, and F(V1 ) ∨ · · · ∨ F(Vn ) = F . Lemma 1.4, applied to the finite Boolean algebra A1 built from n atoms V1 , . . . , Vn , and A2 = Aµ , shows that any integral on the Aµ -factorized measure space (Ω(X), F, µ) may be decomposed into n components, the mth component being measurable with respect to F(Vm ) ⊂ F(Um ). Proof of Lemma 1.4. We may suppose that F1 (A) is nontrivial for any nonzero A ∈ A1 (otherwise replace A1 with its quotient algebra). Due to Lemma B6, it suffices to consider a two-element partition (A, B), B = A0 , A ∈ A1 . Choose increasing sequences A1 , A2 , . . . ∈ A2 and B1 , B2 , . . . ∈ A2 such that F2 (Ak ) ↑ F1 (A) and F2 (Bk ) ↑ F1 (B). Note that Ak ∧ Bk = 0, since F2 (Ak ∧ Bk ) = F2 (Ak ) ∧ F2 (Bk ) = F2 (0). Take Ck = (Ak ∨ Bk )0 . An integral f on (Ω, F2 , P ) may be decomposed into ϕk , ψk , and χk so that ϕk is measurable w.r.t. F2 (Ak ) ⊂ F1 (A), ψk w.r.t. F2 (Bk ) ⊂ F1 (B), and χk w.r.t. F2 (Ck ). It remains to use Theorem B1 (and its multiplicative counterpart). Appendix C: Around the Asymptotically Fixed Point Lemma 5.2 is proved here, and some additional facts about the nonlinear transformation RL,M in a neighbourhood of the Gaussian measure γ within the set M of relevant probability measures on C(R) (these RL,M , γ, and M were intoduced before Lemma 5.2). Remember that RL,M was first defined as a map C(R) → C(R), but the corresponding transformation for measures is also denoted by RL,M . Introduce another transformation S L,M : C(R) → C(R), related to RL,M as follows: Z Mx 1 ξ(x) = √ ϕ (Lξ(y)) dy ; S M 0 r 1 1 1 M L,M ξ(x) = S L,M ξ x + 1 − − S L,M ξ x − 1 + . R 2 M −1 M M L,M
For any measure µ ∈ M the transformed measure S L,M µ is defined, but does not belong to M. The limit M → ∞ is treated by the following lemma. This is an exercise in a well-known technique (see [6]), with special attention to uniformity in L and µ ∈ M. Lemma C1. For any ε > 0 the set of all measures L,M µ : M ∈ [ε, ∞), L ∈ (0, ∞), µ ∈ M S is relatively compact in the narrow topology.
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Proof. First, a moment inequality: for any x > 0 4 E µ S L,M ξ(x) ZZZZ 4! = 2 M 0
× E µ ϕ (Lξ(y1 )) ϕ (Lξ(y2 )) ϕ (Lξ(y3 )) ϕ (Lξ(y4 )) dy1 dy2 dy3 dy4 Z Z y1 +2 Z Mx Z y3 +2 4!kϕk4 Mx ≤ dy1 dy2 dy3 dy4 M2 0 y1 y1 y3 ZZ 4!kϕk4 = · 4 dy1 dy3 M2 0
=
4!kϕk4 (M x)2 ·4 2 M 2
= 48kϕk4 x2 ; the point is that the expectation vanishes when y2 − y1 > 2Ror y4 − y3 > 2 due to independence (and symmetry). Of course, E µ (· · · ) denotes (· · · ) µ(dξ); similarly, P µ (· · · ) will denote µ{ξ : (· · · )}. Second, a maximal inequality: for any x > 0, u > 0 2kϕk L,M √ ξ(y) ≥ u + ≤ 2P µ S L,M ξ(x) ≥ u . P µ max S y∈[0,x] M The proof uses the Markov time 2kϕk . τ = min y : S L,M ξ(y) = u + √ M 2 2 , x], consider the case τ ∈ [0, x − M ]. We Leaving to the reader the case τ ∈ [x − M have Z Mτ Z Mτ +2 Z Mx ! 1 L,M ξ(x) = √ + + ϕ (Lξ(y)) dy . S M 0 Mτ Mτ +2 √ The middle integral cannot contribute more than 2kϕk/ M . The third integral is conditionally independent of the first one, the condition being a value of τ . This third integral is nonnegative with probability ≥ 1/2, thus ensuring √ S L,M ξ(x) ≥ u at least in half of the events when the maximum reaches u + 2kϕk/ M . Though, the conditioning by a single value of τ needs a justification, since such a condition is usually of zero probability. The discrete counterpart of τ gives the required result in the mesh refinement limit. The rest of the proof is standard; see [6], Theorem 8.4.
Include the measure γ in a one-parameter family of Gaussian measures γσ , or γ(σ), σ ∈ (0, ∞), with zero mean and the following covariance: 1 2 E γ(σ) ξ(x)ξ(y) = σ max 0, 1 − |x − y| ; 2
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p then γ = γ 2(π − 2)/π . We have E γ(σ) ϕ (Lξ(x)) = 0, since γ(σ) is symmetric and ϕ is antisymmetric under the sign change ξ 7→ −ξ. It follows that E γ(σ) ϕ (Lξ(x)) ϕ (Lξ(y)) = 0 for |x − y| ≥ 2 due to independence. Second moments can be calculated explicitly in the limit L → ∞ due to the fact that ϕ (Lξ(x)) → sign ξ(x), the sign correlation of two Gaussian random variables being a known function of their usual correlation: 1 2 lim E γ(σ) ϕ (Lξ(x)) ϕ (Lξ(y)) = arcsin max 0, 1 − |x − y| , L→∞ π 2 the dependence on σ disappearing in the limit. Second moments for RL,M ξ become simpler in the limit M → ∞: lim E γ(σ) Rξ(x)Rξ(y)
M→∞
Z Mx+M−1 Z My+M−1 1 dz1 dz2 E γ(σ) ϕ (Lξ(0)) ϕ (Lξ(z1 − z2 )) = lim M→∞ 4M Mx−M+1 My−M+1 Z 2 1 (2(M − 1) − |M (x − y) − z|) dz E γ(σ) ϕ (Lξ(0)) ϕ (Lξ(z)) · lim = M→∞ 4M −2 for |x − y| ≤ 2; so, lim E γ(σ) Rξ(x)Rξ(y) = BL (γσ ) · E γ(1) ξ(x)ξ(y) ,
M→∞
Z
2
E γ(σ) ϕ (Lξ(0)) ϕ (Lξ(x)) dx ,
BL (γσ ) = 0
and
Z
2
lim BL (γσ ) =
L→∞
0
1 π−2 2 arcsin 1 − x dx = 2 . π 2 π
(This is the reason why this value is especially important for us.) Define similarly Z 2 E µ ϕ (Lξ(0)) ϕ (Lξ(x)) dx BL (µ) = 0
for any µ ∈ M, then BL is a continuous function on M, and lim E µ Rξ(x)Rξ(y) = BL (µ) · E γ(1) ξ(x)ξ(y) ,
M→∞
lim E µ S L,M ξ(x)S L,M ξ(y) = 2BL (µ) · E W ξ(x)ξ(y) ,
M→∞
where W is the Wiener measure, defined as a Gaussian measure having E W ξ(x) = 0 and min (|x|, |y|) , when xy ≥ 0 , E W ξ(x)ξ(y) = 0, when xy ≤ 0 . A homothetic measure Wσ (or W (σ)) is defined naturally, E W (σ) ξ(x)ξ(y) = σ 2 E W ξ(x)ξ(y).
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The following lemma is an exercise in the functional central limit theorem (see [6]). Once again, we have to verify a uniformity in L and µ. Lemma C2. For any ε > 0 S L,M (µ) → W
p 2BL (µ)
when M → ∞
uniformly in L ∈ (0, ∞) and µ ∈ M such that BL (µ) ≥ ε. Note. It may seem that the uniformity statement is ambiguous, since the limit depends on the parameters. However, this is not the case, since the limit remains within a compact set {W (σ) : σ ∈ [0, 2kϕk]}. Proof. Let S˜L,M (µ) be the image of µ under the transformation 1 S L,M (ξ) . ξ 7→ p 2BL (µ) We have to prove that S˜L,M (µ) → W when M → ∞ uniformly in L and µ such that BL (µ) ≥ ε. It suffices to prove that the sequence νk = S˜Lk ,Mk (µk ) converges to W whenever Mk → ∞ and BLk (µk ) ≥ ε. Theorem 19.2 of [6] will be used. The sequence νk evidently has asymptotically independent increments as defined in [6] before the theorem. The sequence νk is tight (relatively compact) due to Lemma 2. The uniform integrability of ξ 2 (x) with respect to νk follows from the moment inequality given in the proof of Lemma 2. It follows that νk → W . Corollary C3. For any ε > 0 RL,M (µ) → γ
p BL (µ)
when M → ∞
uniformly in L ∈ (0, ∞) and µ ∈ M such that BL (µ) ≥ ε. Define a compact set Γ(σ1 , σ2 ) ⊂ M for 0 < σ1 < σ2 < ∞ as follows: Γ(σ1 , σ2 ) = {γσ : σ ∈ [σ1 , σ2 ]} . Lemma C4. For any σ1 , σ2 such that 0 < σ1 < σ2 < ∞ and any ε > 0 there exist L0 ∈ (0, ∞) and a neighbourhood U of Γ(σ1 , σ2 ) in M such that BL (µ) − 2 π − 2 ≤ ε for all µ ∈ U , L ∈ [L0 , ∞) . π Proof. Due to compactness of Γ(σ1 , σ2 ) it suffices to prove that BLk (µk ) → whenever Lk → ∞ and µk → γσ ∈ Γ(σ1 , σ2 ). We know that
2 π−2 π
Z
2
E γ(σ) sign ξ(0) sign ξ(x) dx = 2 0
π−2 . π
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Note that E µk sign ξ(0) sign ξ(x) → E γ(σ) sign ξ(0) sign ξ(x)
when k → ∞ ,
since the function ξ 7→ sign ξ(0) sign ξ(x) is continuous γσ -almost everywhere. It suffices to prove that E µk |ϕ (Lk ξ(0)) ϕ (Lk ξ(x)) − sign ξ(0) sign ξ(x)| → 0 when k → ∞ for each x. This follows from the fact that P µk (|Lk ξ(0)| ≤ 1) → 0
when k → ∞ ;
indeed, lim sup P µk (|Lξ(0)| ≤ 1) ≤ P γ(σ) (|Lξ(0)| ≤ 1) k→∞
for each L, and the right hand side tends to zero when L → ∞.
p Proof ofpLemma 5.2. Denote σ0 = 2(π − 2)/π. Take ε ∈ (0, σ02 ) so that p σ02 + ε) ⊂ Γ( σ02 − ε, p pU . Using Lemma C4, take L0 ∈ (0, ∞) and an open V containing Γ( σ02 − ε, σ02 + ε) so that V ⊂ U and BL (µ) ∈ [σ02 − ε, σ02 + ε] for all µ ∈ V and L ∈ [L0 , ∞). Using Corollary C3, take M0 ∈ (0, ∞) so that RL,M (µ) ∈ V for all M ∈ [M0 , ∞), L ∈ [L0 , ∞), µ ∈ V . The closure K of the set L,M (µ) : L ∈ [L0 , ∞), M ∈ [M0 , ∞), µ ∈ V R is compact due to Lemma 2. We have K ⊂ V ⊂ U and RL,M (K) ⊂ RL,M (V ) ⊂ K. One more fact, used in the proof of Lemma 5.4, will be proved here: if Corollary 5.3 is applied to U small enough, and L, M are large enough, then the distribution for the random function √ Z s x k M ωk+1 (z) dz V (x) = sk 0 is arbitrarily close (in the narrow topology) to the Wiener measure, uniformly in k. Only a finite interval (0 ≤ x ≤ 3) was used. Note that V (x) = S L,M ξk+1 (x), and the distribution of ξk+1 belongs to U . Lemma C4 shows that BL (µ) is bounded away from zero by an absolute constant, if L is large enough and U is small enough. It remains to apply Lemma C2. References [1] G. P. Akilov and A. M. Vershik, “On the reciprocal continuous extension of linear operations”, Vestn. Leningr. Univ. 13(7) (1958) 27–33. [2] H. Araki and E. J. Woods, “Complete Boolean algebras of type I factors”, Publications of the Research Institute for Mathematical Sciences, Kyoto Univ., Series A, 2(2) (1966) 157–242.
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[3] W. Arveson, “Continuous analogues of Fock space”, Memoirs AMS 80(409) (1989). “Continuous analogues of Fock space. II. The spectral C ∗ -algebra”, J. Funct. Anal. 90(1) (1990) 138–205. “Continuous analogues of Fock space. III: Singular states”, J. Operator Theory 22 (1989) 165–205. “Continuous analogues of Fock space IV: Essential states”, Acta Math. 164 (1990) 265–300. “Path spaces, continuous tensor products, and E0 -semigroups, Report PAM-619”, Center for Pure and Applied Math., Univ. of California, Berkeley, 1994. [4] P. Baxendale, “Brownian motions in the diffeomorphism group I”, Comp. Math. 53 (1984) 19–50. [5] F. A. Berezin, The Method of Second Quantization, Academic Press, New York, 1966. [6] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. [7] K. L. Chung and J. B. Walsh, “Meyer’s theorem on predictability”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 29 (1974) 253–256. [8] J. Dixmier, Les C ∗ -alg`ebres et leurs Repr´esentations, Gauthier-Villars, Paris, 1969. [9] C. Doleans-Dade, “Quelques applications de la formule de changement de variable pour les semimartingales”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 16 (1970) 181–194. [10] L. Dubins, J. Feldman, M. Smorodinsky and B. Tsirelson, “Decreasing sequences of σ-fields and a measure change for Brownian motion”, Ann. Probability 24 (2) (1996) 882–904. [11] R. M. Dudley, Real Analysis and Probability, Wadsworth & Brooks, Pacific Grove, CA, 1989. [12] P. Feinsilver, “Processes with independent increments on a Lie group”, Trans. AMS 242 (1978) 73–121. [13] J. Feldman, “Decomposable processes and continuous products of probability spaces”, J. Funct. Anal. 8 (1971) 1–51. [14] I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions, Academic Press, New York, 1964 (translated from Russian). [15] A. Guichardet, Symmetric Hilbert Spaces and Related Topics, Springer-Verlag, Berlin, 1972, Lect. Notes Math. 261. [16] S. S. Horuzhy, Introduction to Algebraic Quantum Field Theory, Kluwer Acad. Publ., Dordrecht, 1990 (translated from Russian). [17] V. G. Kac, Infinite Dimensional Lie Algebras, Third Edition, Cambridge, New York, 1990. [18] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer-Verlag, Berlin, 1990. [19] G. W. Mackey, “Borel structure in groups and their duals”, Trans. AMS 85 (1957) 134–165. [20] P.-A. Meyer, Quantum Probability for Probabilists, Springer-Verlag, Berlin, 1993, Lect. Notes Math. 1538. [21] F. J. Murray and J. von Neumann, “On rings of operators”, Ann. Math. 37 (1936) 116–229. [22] K. R. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory, Springer-Verlag, Berlin, 1972, Lect. Notes Math. 272. [23] R. T. Powers, “A non-spatial continuous semigroup of ∗ -endomorphisms of B(H)”, Publ. RIMS, Kyoto Univ., 23 (1987) 1053–1069. [24] A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, AMS, Providence, R.I., 1989 (translated from Russian). [25] A. M. Vershik, “Theory of decreasing sequences of measurable partitions”, St. Petersburg Math. J. 6 (4) (1994) 1–68. [26] A. M. Vershik, I. M. Gel’fand and M. I. Graev, “Representations of the group SL(2, R), where R is a ring of functions”, Uspehi Mat. Nauk 28(5) (1973) 83–128. [27] A. M. Vershik, I. M. Gel’fand and M. I. Graev, “Irreducible representations of the group GX and cohomology”, Funct. Anal. i Pril. 8(2) (1974) 67–69.
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM SERGIO ALBEVERIO Fakult¨ at f¨ ur Mathematik Ruhr-Universit¨ at Bochum D-44780 Bochum Germany
LEONID V. BOGACHEV∗ Faculty of Mechanics and Mathematics Moscow State University 119899 Moscow Russia E-mail:
[email protected] AMS 1991 subject classifications: 60J65, 60G55, 82A42 Received 18 December 1996 The survival problem for a Brownian particle moving among random traps is considered in the case where the traps are correlated in a particular fashion being gathered in clusters. It is assumed that the clusters are statistically identical and independent of each other and are distributed in space according to a Poisson law. Mathematically, such a trapping medium is described via a Poisson cluster point process. We prove that the particle survival probability is increased at all times as compared to the case of noncorrelated (Poissonian) traps, which implies the slowdown of the trapping process. It is shown that this effect may be interpreted as the manifestation of the trap “attraction”, thus supporting assertions on the qualitative influence of the trap “interaction” on the trapping rate claimed earlier in the physical literature. The long-time survival asymptotics (of Donsker–Varadhan type) is also derived. By way of appendix, FKG inequalities for certain functionals are proven and the limiting distribution for a Poisson cluster process, under clusters’ scaling, is determined. Key words and phrases: Trapping problem, Brownian motion, survival probability, Wiener sausage, trap correlations, clusterized traps, Poisson cluster point random process, long-time survival, Donsker–Varadhan type tail, FKG inequality.
Contents 1. Introduction 2. General Trapping Problem 2.1. Description of the trapping model 2.2. Survival probability 2.3. Poisson cluster point process 3. Clusterized Medium: Some Basic Quantities 3.1. Auxiliary lemmas 3.2. Laplace functional ∗ Research
148 151 151 152 153 154 154 159
supported in part by the Russian Foundation for Fundamental Researches, Grant 9501-00081, and by Volkswagen-Stiftung through Mathematisches Forschungsinstitut Oberwolfach (GUS-Projekt). 147
Reviews in Mathematical Physics, Vol. 10, No. 2 (1998) 147–189 c World Scientific Publishing Company
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3.3. Trap concentration 3.4. Emptiness probability 4. Probabilistic Consequences of Clusterization 4.1. Positivity of correlations 4.2. Trap attraction 4.3. Increase of the emptiness probability 5. Kinetical Consequences of Clusterization 5.1. Basic formula for the survival probability 5.2. Slowdown of trapping 5.3. Long-time survival Appendix A. FKG Inequality for Integral Functionals Appendix B. Limiting Distribution Under Clusters Scaling
162 164 166 166 169 170 172 173 173 176 181 183
1. Introduction The Brownian trapping problem refers to the motion of a Brownian particle in d-dimensional space Rd , d ≥ 1, among randomly distributed absorbing sinks (traps); it is well known in the theory of random media (see, e.g. review papers by Haus and Kehr [20], Havlin and Ben-Avraham [21], den Hollander and Weiss [22] and the bibliography therein). First proposed by Smoluchowski [39] for the theoretical study of the colloid coagulation kinetics, this model has by now become a conventional framework for the description of a wide range of processes in physics and physical chemistry — for instance, chemical reactions of the type A + B → A (see, e.g. Havlin and Ben-Avraham [21] and Rice [35]). A quantity of primary interest in the trapping problem is the so-called survival probability P (t), that is the probability for the Brownian particle to be untrapped for, at least, time t. The function P (t) decreases in the course of time, but its decay rate may, generally, be significantly affected by the statistical characteristics of the trapping environment. In the simplest case of a noncorrelated medium, where the traps are assumed to be distributed in space according to a Poisson law, the behavior of the survival probabilitya Pnc (t) has been studied fairly well (e.g. see Havlin and Ben-Avraham [21], Haus and Kehr [20] and den Hollander and Weiss [22] for an extensive survey of analytic, as well as approximation and simulation methods and results). One of the most important facts known in the noncorrelated case is a fractional exponential form of the long-time survival asymptotics first rigorously proven by Donsker and Varadhan [15] (see also Sznitman [41]): lim t−d/(d+2) log Pnc (t) = − γd c2/(d+2) .
t→∞
(1.1)
Here c > 0 denotes the intensity of the underlying Poisson law (which has the meaning of the trap concentration, i.e. the mean number of traps per volume) and γd > 0 is a certain dimension-dependent constant (see [15, 41]; an explicit form of γd is also cited in Sec. 5.3). In the past decade, a few attempts to allow for trap correlations in the trapping problem have also been made [47, 25, 26, 32, 46, 36, 33, 5, 6, 7, 45]. In particular, it was argued (mainly on physical grounds, see Torquato [46], Richards [36], a The subscript “nc” here and below refers to the case of noncorrelated traps.
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Ohtsuki [32], and Berezhkovskii et al. [5, 6, 7]) that trap “attraction” (“repulsion”) must induce the slowdown (respectively, acceleration) of the trapping as compared to the case of noncorrelated traps; that is to say, P (t) > Pnc (t) or P (t) < Pnc (t), depending on the qualitative type of trap “interaction” (in such a comparison, it is understood that the trap concentration c is fixed ). This conclusion has also been somewhat supported by the calculation of the long-time survival tail in some models (see Weiss and Havlin [47], Kayser and Hubbard [25], and Kerstein [26]). The most general rigorous result of that kind is due to Sznitman [45] who has shown that in the case of a Gibbsian medium, the Donsker–Varadhan type asymptotics remains valid with the concentration c replaced by the so-called pressure b p (see Ruelle [37]): lim t−d/(d+2) log P (t) = −γd p2/(d+2) .
t→∞
(1.2)
However, the role of trap correlations in the trapping process is far from being studied well enough. In the present paper, we consider the model introduced in Berezhkovskii et al. [3] of a microscopically inhomogeneous trapping medium with correlations of a special, “selective” type manifested through the formation of clusters of traps. In real systems, such correlations may occur, for example, with traps attached to certain supporting objects (e.g. to polymer chains, see Oshanin and Burlatsky [33]) or produced by some “localized” process on an initially homogeneous background (e.g. via radiation damage). We focus here on the simplest model of such kind, in which the trap clusters are assumed to be statistically identical and independent of each other, being distributed over space in uniform, uncorrelated fashion. Mathematically, such a model is set in terms of a Poisson cluster point random process (see Daley and Vere-Jones [14]). The goal of this paper is to provide a detailed exposition and rigorous proofs of the results announced earlier in Bogachev et al. [8] and Bogachev and Makhnovskii [9]. In particular, we establish here the effect of the trapping slowdown at all times; more specifically, we prove that P (t) > Pnc (t) for each t > 0. This is in agreement with the general conclusions of Refs. [25, 46, 36, 32, 5, 6, 7], since the Poisson cluster ensemble proves to be attractive (in some natural sense specified below). We also show that the long-time asymptotics for the P (t) is given by formula (1.1) with the concentration of clusters, c0 , in place of the total trap concentration, c: lim t−d/(d+2) log P (t) = −γd c0
2/(d+2)
t→∞
.
(1.3)
In our consideration, we proceed from a general representation of Brownian particle’s survival probability P (t) through the average of a certain functional of a Brownian path (incorporating the so-called Wiener sausage) in that the medium enters via its specific characteristic — the emptiness probability g(S), that is the b To the best of our knowledge, the behavior of the pressure as a function of the interaction (at fixed concentration) has not been rigorously analyzed so far. However, it seems clear, at least physically, that the attraction (repulsion) must decrease (respectively, increase) the pressure, which, on the comparison of (1.2) with (1.1), is consistent with the above mentioned qualitative results.
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probability for a given region S ⊂ Rd to be free from traps. Note that in the pure Poisson (noncorrelated) case, the emptiness probability is explicitly given by the well-known formula gnc (S) = exp(−c |S| ) ,
(1.4)
|S| standing for the volume of region S. Obviously, in the general case the problem of finding the emptiness probability g(S) is much more complicated and hardly yields to explicit analysis. However, for the Poisson cluster model, one can obtain a convenient expression for the g(S), which allows one to study the influence of cluster correlations in some detail. It is worth mentioning that the corresponding representation of the survival probability P (t) appears to be similar to that of Pnc (t), with the difference that the role of the Wiener sausage is played by its natural extension, the bunch of Wiener sausages, determined by the inner structure of a cluster. The paper is organized as follows. We start by presenting the general setting of the trapping problem (Secs. 2.1–2.2); a key point here is the representation of the survival probability P (t) in terms of the emptiness probability g(S) (Sec. 2.2). The Poisson cluster model of the trapping medium, being the main topic of the paper, is specified in Sec. 2.3. Some technical lemmas about indicator functions used throughout the paper are gathered in Sec. 3.1. Next, we derive the Laplace functional for the Poisson cluster point process (Sec. 3.2), which is a tool for finding the total trap concentration (Sec. 3.3) and the emptiness probability (Sec. 3.4). In Sec. 4, we establish some probabilistic consequences of the medium clusterization. First, we show here that the clustering correlations are positive (Sec. 4.1) by proving the corresponding inequality for the emptiness probability, which can be recognized as a particular form of FKG inequality (another form of FKG inequality is proved in Appendix A). In Sec. 4.2 the positivity of correlations is being related to the property of trap attraction. Finally, in Sec. 4.3 it is shown that in the clusterized case, the emptiness probability appears to be increased as compared to the noncorrelated case. Turning back to the trapping problem, in Sec. 5.1 we analyze the corresponding formula for the survival probability P (t); in particular, this formula is interpreted in geometrical terms via the notion of the bunch of Wiener sausages. The effect of the trapping slowdown, being a kinetical manifestation of the medium clusterization, is discussed in Sec. 5.2, where the survival probability P (t) is related to its noncorrelated counterpart, Pnc (t). Finally, in Sec. 5.3 we obtain the main result of this paper, that is, the long-time survival tail (1.3). It is worth pointing out that the derivation appears comparatively easy by virtue of the fact that an appropriate upper estimate for P (t), which is usually the most difficult part of the job, in our case is readily provided by a simple bound on the volume of the Wiener bunch (see Proposition 5.3 below). Appended to the main text (see Appendices A and B) are some results about the general properties of the Poisson cluster point process, which are not directly related to the trapping problem but may be of independent interest.
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
151
In particular, in Appendix B the limiting distribution of the Poisson cluster point process, under clusters’ scaling, is studied. 2. General Trapping Problem 2.1. Description of the trapping model Let us describe the general setting for the trapping problem. Denote by B = (Bt , t ≥ 0) a standard Brownian motion on Rd , d ≥ 1, starting from the origin, and let P0 and E0 stand for the corresponding Wiener measure and expectation, respectively. The Brownian particle evolves in a random environment arising through the presence of randomly located, static absorbing sinks (called traps). The traps are supposed to be constructed via translations of a given set K ⊂ Rd at the points {Zi } of some random point process Z, so that the resulting trapping set is of the form ∪i (K + Zi ) (generally speaking, traps are allowed to overlap). The probability law of the process Z will be denoted by P; correspondingly, E will stand for the expectation with respect to P. The model set K is assumed to be compact and nonpolar (i.e. accessible relative to the Brownian motion, see, e.g. Port and Stone [34]); the simplest choice is to take K to be a Euclidean closed ball: K = Vb (0) := x ∈ Rd : kxk ≤ b ,
(2.1)
the parameter b > 0 being sometimes called the reaction radius. In the simplest case, the traps are supposed to be perfect, which means that the diffusion is terminated instantly upon contacting a trap. Moreover, we will assume that, unless trapped, the diffusing particle does not interact with the random environment (in physical terminology, the process is diffusion-limited, see Havlin and Ben Avraham [21] and Haus and Kehr [20]). This amounts to saying that the Brownian motion B is independent of the random process Z; hence, the overall probability law governing the trapping process is the product measure P0 ⊗ P. In order to be more specific, we will think of the random point process Z = {Zi } as defined on a sample probability space (Ω, A, P), where Ω = {ω} is the set of all countable (finite or denumerably infinite) subsets of Rd without finite limiting points, and A stands for the σ-field on Ω generated by finite-dimensional Borel cylinders. The latter means that A is the smallest σ-field containing all subsets {ω ∈ Ω : #(ω ∩ S) = n}, where n is a non-negative integer, S ⊂ Rd is a bounded Borel set, and #(A) denotes the number of elements in A. Note that adopting such a probability space implies that the corresponding point process Z is simple, that is to say, the possibility of the occurrence of multiple points, Zi = Zi0 , is ruled out. It is physically natural to assume that the process Z is spatially homogeneous (stationary), so that its law P is translation invariant. This implies that the distribution of points {Zi } falling into a given (Borel) set S ⊂ Rd does not depend on translations of S. In particular, it follows that the mean number of points contained in S is proportional to the volume |S|:
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S. ALBEVERIO and L. V. BOGACHEV
E [ # {i : Zi ∈ S}] = c |S| ,
(2.2)
the constant c having the meaning of a trap concentration.c 2.2. Survival probability Let τ stand for the Brownian particle life-time (i.e. the time until first hitting any trap): (2.3) τ := min {t ≥ 0 : Bt ∈ ∪i (K + Zi )} , where K + z = {x ∈ Rd : x − z ∈ K}. In this work, we will be concerned with the survival probability P (t), defined as the probability for the Brownian particle to be untrapped until time t: P (t) := P0 ⊗ P {τ > t} . Note that, generally, P (0) ≤ 1, because the starting point of the Brownian motion may happen to be covered by a trap. Hence, somewhat more natural quantity is the normalized survival probability P (t | 0), conditioned on particle’s survival at the initial time moment: P (t | 0) := P {τ > t | τ > 0} = P (t)/P (0) .
(2.4)
By means of the formula of total probability and using the independence of the laws P0 and P, we obtain / ∪i (Zi + K))} P (t) = E0 P { ∩s≤t (Bs ∈ / Zi + K)} = E0 P { ∩s≤t ∩i (Bs ∈ / K)} = E0 P { ∩s≤t ∩i (Bs − Zi ∈ / −K)} = E0 P { ∩s≤t ∩i (Zi − Bs ∈ / Bs − K)} = E0 P { ∩i ∩s≤t (Zi ∈ / ∪s≤t (Bs − K))} . = E0 P { ∩i ( Zi ∈
(2.5)
Consider the region SK (t) in Rd “covered” by the set −K in the course of its translation along a Brownian path (Bs , 0 ≤ s ≤ t): SK (t) :=
S
s≤t (Bs
− K) = x ∈ Rd : ∃ s ∈ [0, t] such that x ∈ Bs − K , (2.6)
known as the Wiener sausage (Kac [23]; see also, e.g. [15, 41]). For example, in the particular case where K is a ball Vb (0), the Wiener sausage Sb (t) is the b-tubular neighborhood of the Brownian path: Sb (t) =
S s≤t
Vb (Bs ) = x ∈ Rd : mins≤t kBs − xk ≤ b .
(2.7)
c Throughout the paper, unless otherwise explicitly indicated in the notations, the value of the concentration c is assumed to be fixed.
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
153
As is easily seen, the last condition in (2.5) means that the Wiener sausage SK (t) does not contain the points {Zi }. Let g(S) stand for the emptiness probability regarded as a function of Borel sets S ⊂ Rd : g(S) := P {S ∩ {Zi } = ∅} = P {#{i : Zi ∈ S} = 0} .
(2.8)
Then from (2.5) we obtain the following representation for the survival probability which plays the key role in our analysis: P (t) = E0 [ g(SK (t)) ] .
(2.9)
Note that the “initial” sausage SK (0) coincides with the set K, so in view of Eqs. (2.4), (2.9), the normalized survival probability P (t | 0) is expressed as P (t | 0) = [g(K)]−1 E0 [ g(SK (t)) ] .
(2.10)
In the noncorrelated case where P is a Poisson law of intensity c, the emptiness probability is explicitly given by gnc (S) = exp (−c |S| ) .
(2.11)
Pnc (t) = E0 [ exp(−c |SK (t) |) ] .
(2.12)
Hence, (2.9) amounts to
Correspondingly, (2.10) then reads as Pnc (t | 0) = e c |K| E0 [ exp(−c |SK (t) |) ] .
(2.13)
The comparison of representations (2.9), (2.12) shows that the influence of trap correlations on the survival probability P (t) is primarily related to the difference between the respective emptiness probabilities, g(S) and gnc (S). We will study this question for the case of clusterized medium in Sec. 4.3 below. 2.3. Poisson cluster point process The clusterized trapping medium we are concerned with can be described by means of a cluster point random process Z (see Daley and Vere-Jones [14]). More specifically, suppose that each trap is ascribed to a certain group (called a cluster ) (i) so that the ensemble {Zi } can be represented in the form {Xi + Yj , j = 1, . . . , νi }. Here the points {Xi } play the role of cluster “centers” and the aggregates of vectors (i) (i) {Y1 , . . . , Yνi } determine the inner structure of clusters, that is the configuration of νi traps contained in the ith cluster (without restriction we assume that νi ≥ 1). (i) (i) Suppose that the random aggregates of random vectors {Y1 , . . . , Yνi } are independent of each other and of {Xi } and are identically distributed. The latter means that (independent) positive integer-valued random variables {νi } are identically distributed and, moreover, the distribution of an aggregate (i) (i) {Y1 , . . . , Yνi }, conditioned on the number of incorporated vectors: {νi = n},
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S. ALBEVERIO and L. V. BOGACHEV
n ∈ N, is determined by a (symmetric) distribution function Fn (y1 , . . . , yn ), yj ∈ Rd , not depending on i. So, each cluster is constructed — in a statistically like fashion — in two steps: first, the number of traps in a cluster is sampled, ν = n, and then n traps are distributed around the cluster center according to the distribution function Fn .d In the sequel, we will write P∗ for the model probability law that governs the random configuration of a cluster and E∗ for the corresponding expectation. Remarks. 1. With no loss in generality, we may and will assume that almost surely (P∗ -a.s.) the positions of any two traps in the cluster do not coincide: P∗
nS j6=j 0
o {Yj = Yj 0 } | ν = n = 0
(2.14)
(indeed, otherwise such trap configurations could be described via a lower order distribution function Fn0 , n0 < n). (i) 2. Note that a cluster point process Z = {Xi + Yj } is stationary provided so is the underlying process of cluster centers, X = {Xi }. From now on, we consider the simplest case of a Poisson cluster process (see Daley and Vere-Jones [14]) in that the trap centers {Xi } are assumed to be the points of a Poisson process (of intensity c0 ). For most of our results, we will also require that the mean number of traps in each cluster is finite: ν¯ := E∗ [ν] < ∞ .
(2.15)
It is worth noting that in the particular case ν ≡ 1, our model obviously amounts to the noncorrelated one since the Poisson ensemble is well known to be invariant under random (independent and identically distributed) translations (see Goldman [18], Daley and Vere-Jones [14], and also Remark in Sec. 3.2 below). So, henceforth we will imply that Z is a proper cluster process in the sense that the clusters are not degenerate; that is to say, ν ≥ 1 (P∗ -a.s.) and, moreover, ν ≥ 2 with positive probability: P∗ {ν ≥ 2} > 0 .
(2.16)
3. Clusterized Medium: Some Basic Quantities 3.1. Auxiliary lemmas In order not to be distracted later, in this section we gather some technical assertions stating certain properties and estimates, in particular for indicator functions, which will be instrumental in the sequel. First, introduce some general notations. We write S = {S} for the σ-field of Borel subsets of Rd . By F = {f } we denote the set of all real-valued, non-negative Borel functions on Rd and by M = {µ} the collection of all simple integer-valued d For a more formal construction of the cluster point process, see Daley and Vere-Jones [14].
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
155
measures on S, finite on bounded sets S ∈ S (the simplicity means that µ{x} ≤ 1 for each x ∈ Rd ). The bilinear form hµ, f i, µ ∈ M, f ∈ F, is defined as Z hµ, f i := f (x) µ(dx) (3.1) (here and below, unless otherwise indicated, the integration is over the whole Rd ). Given x ∈ Rd , let θx stand for the shift in Rd : θx (y) := x + y ,
y ∈ Rd .
Define the action of the shift operator θx on functions f ∈ F and measures µ ∈ M as follows: (θx f )(y) = f (θx y) = f (x + y) , (µθx )(S) = µ(θx−1 S) = µ(S − x) ,
y ∈ Rd , S ∈S.
(3.2) (3.3)
It is easy to check that the operator θx is symmetric: hµ, θx f i = hµθx , f i . For a set S ∈ S, the indicator function χ(x; S), x ∈ Rd , is defined as ( 1 if x ∈ S , χ(x; S) := 0 otherwise .
(3.4)
(3.5)
Two of its elementary properties will be of use: 1 − χ(x; S) = χ(x; S c ) , χ(x; S1 ∩ S2 ) = χ(x; S1 ) · χ(x; S2 ) ,
(3.6) (3.7)
where S c = Rd \ S is the complement of S. Lemma 3.1. For any bounded set S ∈ S with negligible boundary and any random vector Y, we have Z (3.8) E∗ [χ(x + Y ; S)] dx = |S| , where |S| is the volume of S. More generally, for any integrable function f ∈ F : Z Z E∗ [f (x + Y )] dx = f (z) dz . (3.9) Proof. The first identity is a particular case of (3.9) with f (x) = χ(x; S). In turn, (3.9) follows by Fubini’s theorem and the change of variables x + Y = z:
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S. ALBEVERIO and L. V. BOGACHEV
Z
∗
Z
∗
E [f (x + Y )] dx = E
f (x + Y ) dx
Z
= E∗
f (z) dz
Z =
f (z) dz .
The lemma is proved.
We will also need the n-point indicator function χ(x1 , . . . , xn ; S), x1 , . . . , xn ∈ R , n ≥ 1, defined by ( 1 if xj ∈ S for some j ∈ {1, . . . , n} , χ(x1 , . . . , xn ; S) := (3.10) 0 otherwise . d
Note that χ(x1 , . . . , xn ; S) can be expressed through ordinary (one-point) indicator functions as n Y χ(x1 , . . . , xn ; S) = 1 − (1 − χ(xj ; S)) . (3.11) j=1
For the aggregate of random vectors {Yj , j = 1, . . . , ν} determining the inner structure of a cluster (see Sec. 2.3), consider the corresponding (random) ν-point indicator ∞ X χ(Y1 , . . . , Yν ; S) = χ(Y1 , . . . , Yn ; S) · I{ν = n} , (3.12) n=1
where I(A) = I(ω; A), ω ∈ Ω, is the indicator of event A ∈ A. From the definition (3.10) we have S n (3.13) E∗ [χ(Y1 , . . . , Yn ; S)] = P∗ j=1 {Yj ∈ S} . Moreover, by taking the expectation of (3.12) and using the independence of ν and {Y1 , . . . , Yn }, n ≥ 1, we obtain E∗ [χ(Y1 , . . . , Yν ; S)] = P∗
S ν
{Y ∈ S} . j j=1
(3.14)
The next lemma comes as an extension of the previous one-point variant to the n-point case. First, introduce one more notation. Given a set S ∈ S and a collection of (fixed) vectors {y1 , . . . , yn }, define the region S ∗ = S ∗ (y1 , . . . , yn ) as the union of n copies of the set S shifted by the vectors −y1 , . . . , −yn : S ∗ = S ∗ (y1 , . . . , yn ) :=
Sn
j=1 (S
− yj ) .
(3.15)
Lemma 3.2. For each bounded set S ∈ S with negligible boundary and any random vectors Y1 , . . . , Yn one has
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
Z
E∗ [χ(x + Y1 , . . . , x + Yn ; S)] dx = E∗ |S ∗ (Y1 , . . . , Yn )| .
157
(3.16)
Moreover, this holds true with the number of vectors being random: Z
E∗ [χ(x + Y1 , . . . , x + Yν ; S)] dx = E∗ |S ∗ (Y1 , . . . , Yν )| ,
(3.17)
where the volume of the random region S ∗ (Y1 , . . . , Yν ) is defined as |S ∗ (Y1 , . . . , Yν )| =
∞ X
|S ∗ (Y1 , . . . , Yn )| · I{ν = n} .
(3.18)
n=1
Proof. Using the representation (3.11) and formulas (3.6), (3.7), we can write χ(x + Y1 , . . . , x + Yn ; S) = 1 −
n Y
(1 − χ(x; S − Yj ))
j=1
= 1−
n Y
χ (x; (S − Yj )c )
j=1
T n = 1 − χ x; j=1 (S − Yj )c S n = χ x; j=1 (S − Yj ) = χ(x; S ∗ ) .
(3.19)
By integrating Eq. (3.19) with respect to x, taking the expectation and applying Fubini’s theorem, we get relation (3.16). Its “random” version (3.17) then easily follows by the formula of total probability: Z
E∗ [χ(x + Y1 , . . . , x + Yν ; S)] dx =
∞ X
P∗ {ν = n} ·
Z
E∗ [χ(x + Y1 , . . . , x + Yn ; S)] dx
n=1
=
∞ X
P∗ {ν = n} · E∗ |S ∗ (Y1 , . . . , Yn )|
n=1
= E∗ |S ∗ (Y1 , . . . , Yν )| . The proof is complete.
Lemma 3.3. Let a1 , . . . , an , n ≥ 1, be arbitrary real numbers satisfying 0 ≤ ai ≤ 1, i = 1, . . . , n. Then the following identity holds:
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S. ALBEVERIO and L. V. BOGACHEV
n Y
(1 − aj ) = 1 −
j=1
n X
aj +
j=1
n X
aj
j=1
n X
n Y
ak
k=j+1
(1 − a` )
(3.20)
`=k+1
(in case an index set appears empty, the inner sum and product on the right-hand side are put equal to 0 and 1, respectively). Remark. In fact, this lemma expresses the result of partial multiplying out the product on the left-hand side of (3.20). Proof. We proceed by induction on the number n. The basis of induction is obvious since for n = 1 the inner sum on the right-hand side of (3.20) vanishes, according to the above convention. Suppose now that the identity (3.20) holds for some n ≥ 1. For the induction step, we have to prove that it stays valid on supplementing the aggregate {a1 , . . . , an } with a number an+1 , 0 ≤ an+1 ≤ 1. Multiplying (3.20) by (1 − an+1 ), we get n n n+1 n n Y X X X Y (1 − aj ) = 1 − aj + aj ak (1 − a` ) · (1 − an+1 ) j=1
j=1
j=1
= 1 − an+1 −
n X
k=j+1
aj +
j=1
+
n X
aj
j=1
= 1−
n+1 X
n X
j=1
aj an+1
j=1 n Y
ak
k=j+1
aj +
n X
`=k+1
(1 − a` ) (1 − an+1 )
`=k+1
n X
aj an+1 +
j=1
n X
n X
aj
j=1
n+1 Y
ak
k=j+1
(1 − a` ) .
(3.21)
`=k+1
Taking into account the index convention made in the statement of the lemma, we can rewrite the right-hand side of (3.21) to obtain n+1 Y
n+1 X
j=1
j=1
j=1
k=n+1
n+1 X
n+1 X
n+1 X
(1 − aj ) = 1 −
= 1−
j=1
aj +
aj +
n X
aj
aj
j=1
n+1 X
ak +
n+1 X
aj
j=1
ak
k=j+1
n+1 Y
n+1 X k=j+1
ak
n+1 Y
(1 − a` )
`=k+1
(1 − a` ) ,
`=k+1
which is just (3.20) with n + 1 in place of n. Thus the induction step is completed and our claim is proved. Corollary 3.4. In the conditions of Lemma 3.3, the following inequality holds: 1−
n Y j=1
(1 − aj ) ≤
n X j=1
aj .
(3.22)
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
Proof. The required inequality easily follows from identity (3.20).
159
Corollary 3.5. For any S ∈ S and each n ≥ 1, the following identity for the indicator functions holds: n n n X X Y χ(xj ; S) 1 − χ(xk ; S) (1 − χ(x` ; S)) . (3.23) χ(x1 , . . . , xn ; S) = j=1
k=j+1
`=k+1
Proof. In view of representation (3.11), our claim immediately follows from Lemma 3.3 by choosing aj = χ(xj ; S). Corollary 3.6. The n-point indicator function satisfies the following estimates: n X χ(xj ; S) . (3.24) χ(x1 ; S) ≤ χ(x1 , . . . , xn ; S) ≤ j=1
Proof. The lower bound easily follows from (3.11) by replacing χ(x2 ; S), . . . , χ(xn ; S) with zeroes. The upper bound follows from identity (3.23) (cf. also (3.22)). 3.2. Laplace functional Given a point random process Z = {Zi }, let µ be the associated random “counting” measure defined on Borel subsets S ∈ S of Rd by X χ(Zi ; S) = #{i : Zi ∈ S} , (3.25) µ(S) := i
being P∗ -a.s. finite for bounded S ∈ S. It is easily seen that µ ∈ M whenever the process Z is simple (see Sec. 2.1). (i) For a Poisson cluster process Z = {Xi + Yj }, denote by µ0 the “background” random measure associated with the underlying Poisson process X = {Xi }: X χ(Xi ; S) = #{i : Xi ∈ S} , (3.26) µ0 (S) := i
and let µ∗ be the random measure independent of µ0 , associated with trap configurations {Yj } in a single, “model” cluster centered at the origin: µ∗ (S) :=
ν X
χ(Yj ; S) .
(3.27)
j=1
Since the inner configurations of different clusters are assumed to be independent and identically distributed, the overall measure µ can be decomposed as X µ∗i θXi , (3.28) µ= i
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S. ALBEVERIO and L. V. BOGACHEV
where {µ∗i } is an aggregate of independent copies of the µ∗ and θx is the shift operator (see (3.2), (3.3)). We are now going to find the form of the Laplace functional for the Poisson (i) cluster process Z = {Xi + Yj } (or, equivalently, for the corresponding random measure µ), which is a standard tool in studying point processes. First, we recall the definition. Given a point random measure µ, its Laplace functional Lµ (·) is defined on the set of functions F by Lµ (f ) := E [exp(−hµ, f i)] , where (see (3.1))
Z hµ, f i =
f (x) µ(dx) =
X
f (Zi ) .
(3.29)
(3.30)
i
In the Poissonian case, the form of the Laplace functional is well known (e.g. see Daley and Vere-Jones [14]): Proposition 3.7. For a Poisson point measure µ0 (with intensity c0 ), its Laplace functional is given by Z Lµ0 (f ) = exp −c0 [1 − exp(−f (x))] dx .
(3.31)
This formula can be derived, for instance, via a standard technique through approximating a function f (x) by simple (i.e. piecewise constant) functions and verifying (3.31) for indicators and their linear combinations. For completeness, we adduce here the proof, however choosing another, more direct derivation. Proof. Introduce an auxiliary box ΛN = [−N, N ]d ⊂ Rd and consider a “cutoff” measure X χ(Xi ; S ∩ ΛN ) (3.32) µN 0 (S) := µ0 (S ∩ ΛN ) = i
(cf. (3.26)), thus only taking into account the points Xi contained in ΛN . Obviously, for each f ∈ F one has (f ) Lµ0 (f ) = lim LµN 0 N →∞
(3.33)
(e.g. by monotone convergence). Further, to calculate LµN (f ), we recall that 0 µ0 (ΛN ) = #{i : Xi ∈ ΛN } is a Poissonian random variable with the corresponding parameter c0 |ΛN |; moreover, it is well known (e.g. see Daley and Vere-Jones [14]) that, conditional on the event {µ0 (ΛN ) = n}, the points X1 , . . . , Xn are distributed over ΛN uniformly and independently of each other. So, by conditioning with respect to µ0 (ΛN ), we readily obtain:
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BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
LµN (f ) = E [exp(−hµN 0 , f i)] 0 !# " X χ(Xi ; ΛN )f (Xi ) = E exp − i
=
∞ X
"
P {µ0 (ΛN ) = n} · E exp
−
X
n=0
=e
i ∞ X
−c0 |ΛN |
(c0 |ΛN |)n
=e
−c0 |ΛN |
Z exp c0
Z
= exp −c0
1 |ΛN |
n!
n=0
! # χ(Xi ; ΛN )f (Xi ) µ0 (ΛN ) = n
n exp(−f (x)) dx
Z ΛN
exp(−f (x)) dx
ΛN
[1 − exp(−f (x))] .
ΛN
Finally, using (3.33) we pass to the limit as N → ∞ to get (3.31).
Proposition 3.8. (cf. Daley and Vere-Jones [14]). For the random measure µ associated with the Poisson cluster point random process Z described above, its Laplace functional is of the form Z ∗ Lµ (f ) = exp −c0 [1 − Lµ (θx f )] dx , (3.34) wheree
Lµ∗ (θx f ) = E∗ [exp(−hµ∗ , θx f i)] = E∗ exp −
ν X
f (x + Yj )
(3.35)
j=1
is the Laplace functional of the model measure µ∗ shifted by the vector x. The representation (3.34) can be proved by the “cut-off” method used above (see Proposition 3.7). Here we shall derive (3.34) as a direct corollary of (3.31). Proof. Substituting decomposition (3.28) into (3.29) and using Eq. (3.4), we have " !# " # X Y ∗ ∗ Lµ (f ) = E exp − hµi θXi , f i =E exp (−hµi , θXi f i) . (3.36) i
i
Now we apply the formula of total probability by conditioning with respect to the background process {Xi }. Since the measures µ∗i are identically distributed and independent of each other and of {Xi }, we can rewrite (3.36) in terms of the model measure µ∗ as e For f ∈ F fixed, L ∗ (θ f ) is a function of x. x µ
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S. ALBEVERIO and L. V. BOGACHEV
" Lµ (f ) = E E " =E
Y i
Y
" ∗
=E
"
i
# exp (−hµ∗i , θXi f i) {Xi }
# E [exp(−hµ , θXi f i)] {Xi }
Y
∗
∗
# Lµ∗ (θXi f )
i
= E∗ exp −
X
!# log Lµ∗ (θXi f )
i
= Lµ0 (f˜) ,
(3.37)
where f˜(x) := − log Lµ∗ (θx f ) .
(3.38)
Obviously, f˜ ∈ F . To complete the proof, it remains to substitute the expression (3.31) for Lµ0 (·) into the right-hand side of (3.37). Remark. In the particular case ν ≡ 1, by means of Lemma 3.1 (see (3.9)) formula (3.34) is readily reduced to (3.31): Z Lµ (f ) = exp −c0 (1 − E∗ [exp (−f (x + Y ))]) dx Z = exp −c0 (1 − exp (−f (z))) dz = Lµ0 (f ) .
(3.39)
That is to say, the case of “singleton” clusters, which amounts to random independent shifts of a Poisson ensemble, is actually the case with noncorrelated traps — for an arbitrary distribution of the model shift Y . 3.3. Trap concentration In this section we find the total trap concentration c for the Poisson cluster ensemble (the mean number of points {Zi } per volume, see (2.2)) by proceeding from the Laplace functional representation obtained in Proposition 3.8. Proposition 3.9. In the Poisson cluster case, the trap concentration is given by c = c0 ν¯ ,
(3.40)
where c0 is the concentration of clusters and ν¯ = E∗ [ν] stands for the mean number of traps in each cluster. In particular, c is finite whenever so is ν¯. Remark. In fact, the result (3.40) is “physically” quite clear. Indeed, the average number of traps contained in a large volume Λ is about c |Λ|; on the other hand, the same quantity is approximately given by the respective number of clusters,
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163
c0 |Λ|, times the mean number of traps in each cluster, ν¯. Then Eq. (3.40) “follows” by equating these two expressions. Proof. Pick f (x) in (3.34) in the form f (x) = zχ(x; S) with z > 0 a parameter, then (see (3.30)) Lµ (zχ(· ; S)) = E [exp (−zµ(S))] =
∞ X
exp(−nz) .
(3.41)
n=0
In particular, it follows that ∞ X ∂ = n · P {µ(S) = n} = E [µ(S)] . − Lµ (zχ(· ; S)) ∂z z=0 n=0
(3.42)
Applying this to (3.34) gives Z ∂ [1 − Lµ∗ θx (zχ(· ; S)] dx ∂z z=0 Z ∂ Lµ∗ θx (zχ(· ; S)) dx − = c0 ∂z z=0 Z = c0 E∗ [(µ∗ θx )(S)] dx .
E [µ(S)] = c0
(3.43)
(3.44)
Here, interchanging the differentiation and integration is justified by monotone convergence. Indeed, the derivative in (3.43) amounts to the limit Z Z lim z −1 [1 − Lµ∗ θx (zχ(· ; S))] dx = lim z −1 [1 − exp(−zhµ∗ , θx χ(· ; S)i)] dx , z↓0
z↓0
(3.45) and the monotone convergence theorem applies since, for any a ≥ 0, z −1 (1 − e−za ) ↑ a
as z ↓ 0 ,
as can be easily verified via differentiating. Further, by using Fubini’s theorem and Lemma 3.1 we can rewrite (3.44) to obtain Z ∗ ∗ E[µ(S)] = c0 E (µ θx (S)) dx = c0 E∗ = c0 E∗ ∗
= c0 E
Z
µ∗ (S − x) dx
Z Z
χ(y; S − x) µ∗ (dy) dx
Z Z
∗ χ(x + y; S) dx µ (dy)
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S. ALBEVERIO and L. V. BOGACHEV
∗
Z
|S| µ (dy) ∗
= c0 E
= c0 |S| · E∗ [ν] = c0 |S| ν¯ ,
(3.46)
where we used the fact that, P∗ -a.s., Z
µ∗ (dy) =
ν X
χ(Yj ; Rd ) =
j=1
ν X
1=ν.
j=1
So, recalling the definition of the concentration (see (2.2)), from (3.46) we obtain c = |S|−1 E [µ(S)] = c0 ν¯ ,
as required. 3.4. Emptiness probability
We are now in a position to derive from (3.34) a basic expression for the emptiness probability g(·) defined in (2.8). Proposition 3.10. For the Poisson cluster point random process, the emptiness probability g(S) is of the form Z S ν {x + Y ∈ S} dx (3.47) g(S) = exp −c0 P∗ j j=1 Z ∗ = exp −c0 E [χ(x + Y1 , . . . , x + Yν ; S)] dx .
(3.48)
Proof. As in the proof of Proposition 3.9, we take the function f in (3.34) to be f (x) = zχ(x; S) with z > 0. From (3.41) it is seen that Lµ (zχ(· ; S)) is a monotone decreasing function of z and also that lim Lµ (zχ(· ; S)) = P{µ(S) = n} = g(S) .
z→∞
Applying this to (3.34) gives Z g(S) = exp −c0 lim [1 − Lµ∗ θx (zχ(·; S))] dx z→∞
Z h i 1 − lim Lµ∗ θx (zχ(·; S)) dx = exp −c0
(3.49)
Z = exp −c0 [1 − P∗ {(µ∗ θx ) (S) = 0}] dx
(3.50)
z→∞
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BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
(passing to the limit under the integral sign in (3.49) is justified by monotone convergence). Further, using (3.2), the integrand in (3.50) can be rewritten as 1 − P∗ {µ∗ (S − x) = 0} = P∗ {µ∗ (S − x) > 0} S ν {Y ∈ S − x} = P∗ j j=1 = P∗
S
ν j=1
{x + Yj ∈ S} .
(3.51)
Combining (3.50) with (3.51) yields (3.47). Finally, (3.48) follows in view of (3.14). Corollary 3.11. The emptiness probability g(S) can be represented in the form g(S) = exp(−c0 E∗ |S ∗ (Y1 , . . . , Yν ) |) ,
(3.52)
where S ∗ (Y1 , . . . , Yν ) is the random region defined in Sec. 3.1 (see Lemma 3.2). Proof. The statement readily follows by combining expressions (3.48) and (3.17). Remarks. 1. It is worth mentioning that the form (3.52) of the emptiness probability is similar to that of (2.11) corresponding to the Poisson (noncorrelated) case. The presence of the concentration c0 in place of the total concentration c = c0 ν¯ is obviously due to the fact that, instead of the “ideal gas” of traps, in the Poisson cluster model we have the “ideal gas” of clusters. On the other hand, the effective “composite” volume S ν E∗ |S ∗ (Y1 , . . . , Yν )| = E∗ j=1 (S − Yj ) replacing the sum of “individual” volumes, ν X |S − Yj | , E∗ j=1
accumulates the information of a certain inner structure of clusters. 2. Conversely, the expression (2.11) for the Poissonian emptiness probability gnc (·) can be represented in a form similar to (3.48), namely gnc (S) = exp −c0
Z
E∗
ν X
χ(x + Yj ; S) dx
j=1
(cf. (3.48)). To check (3.53), it suffices to show that
(3.53)
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S. ALBEVERIO and L. V. BOGACHEV
Z
E∗
ν X
χ(x + Yj ; S) dx = |S| · E∗ [ν] ,
j=1
which follows by applying the formula of total probability (by conditioning on ν) and using the identity (3.8) of Lemma 3.1 in order to convert the integrals of indicator functions to volumes. 4. Probabilistic Consequences of Clusterization 4.1. Positivity of correlations In this section, we establish for the Poisson cluster point process the property of trap “attraction” in the sense of correlation positivity, which is expressed by means of an appropriate inequality for the emptiness probability. In what follows, the abbreviation P∗ -p.p. is to indicate that some (µ∗ -measurable) event is of positive probability with respect to P∗ . Proposition 4.1. For a Poisson cluster point process Z, the corresponding emptiness probability g(·) satisfies the inequality g(S1 ∪ S2 ) ≥ g(S1 ) · g(S2 ) ,
S1 , S2 ∈ S .
(4.1)
Moreover, if the process Z is proper (i.e. ν ≥ 2 P∗ -p.p., see Sec. 2.3) then there exist such disjoint sets S1 , S2 ∈ S (i.e. S1 ∩ S2 = ∅) that the inequality (4.1) is strict. Remark. It is clear that the inequality (4.1) is most informative just for disjoint sets S1 , S2 . Note that in the noncorrelated case (i.e. for a Poisson point process Z) we have identically gnc (S1 ∪ S2 ) ≡ gnc (S1 ) · gnc (S2 ) (4.2) for any disjointf S1 , S2 ∈ S (see (2.11)). Proof. In view of the basic formula (3.47), it suffices to verify that, for any x ∈ Rd , S ν ∗ Sν P∗ {x + Y ∈ S ∪ S } ≤ P {x + Y ∈ S } j 1 2 j 1 j=1 j=1 + P∗
S
ν j=1 {x
+ Yj ∈ S2 } ,
(4.3)
which is obvious once we rewrite this as P∗ (A1 (x) ∪ A2 (x)) ≤ P∗ (A1 (x)) + P∗ (A2 (x))
(4.4)
by setting Ai (x) := {µ∗ (Si − x) > 0} ,
i = 1, 2,
f More generally, for such S , S ∈ S that S ∩ S is Lebesgue negligible, i.e. |S ∩ S | = 0. 1 2 1 2 1 2
(4.5)
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167
Pν where µ∗ (·) = j=1 χ(Yj ; ·) is the random measure associated with model cluster configurations {Yj } (see (3.27)). As for the strict variant of inequality (4.1), note that by assuming the contrary (i.e. equality in (4.1)) we would have, instead of (4.4), that P∗ (A1 (x) ∪ A2 (x)) = P∗ (A1 (x)) + P∗ (A2 (x)) ,
(4.6)
almost everywhere in x (with respect to Lebesgue measure). In turn, from (4.6) it follows that for almost each x the events A1 (x), A2 (x) are P∗ -a.s. disjoint. Hence, in order to obtain a contradiction, we only have to show that there exist such S1 , S2 ∈ S that, on a set of points x of positive Lebesgue measure, the event A1 (x) ∩ A2 (x) occurs P∗ -p.p. To verify this, we first prove a technical lemma. Lemma 4.2. Suppose that the random measure µ∗ satisfies the condition (2.16). Then there exist such disjoint closed cubes T1 , T2 ⊂ Rd that P∗ -p.p., T1 , T2 are both charged by µ∗ , that is: P∗ {µ∗ (T1 ) > 0, µ∗ (T2 ) > 0} > 0 .
(4.7)
Once the lemma is proved, our claim follows easily. Indeed, let ρ :=
min
x1 ∈T1 ,x2 ∈T2
kx1 − x2 k > 0
denote the distance between the cubes T1 , T2 (ρ is strictly positive since T1 , T2 are constructed to be disjoint and closed). Let us take Si to be the (ρ/3)-neighborhood of Ti : Si := x ∈ Rd : ∃ y ∈ Ti such that kx − yk ≤ ρ/3 ,
i = 1, 2 .
Obviously, S1 , S2 are still disjoint. Since by condition (4.7) both S1 and S2 P∗ -p.p. contain charged subsets T1 , T2 , they are both P∗ -p.p. charged as well. This means (see (4.5)) that the probability of the event A1 (x) ∩ A2 (x) is positive for x = 0. Moreover, by continuity arguments it follows that under small shifts (of length kxk ≤ ρ/3), the sets S1 − x, S2 − x still contain T1 , T2 , respectively. Therefore, the probability of A1 (x) ∩ A2 (x) stays positive for all x ∈ Vρ/3 (0), and Proposition 4.1 is proved. So, it remains to prove the lemma. Proof of Lemma 4.2. 2 , N = 0, 1, . . . :
Let us partition the space Rd into cubes of side
−N
TkN := x = (x1 , . . . , xd ) ∈ Rd : ki 2−N ≤ xi ≤ (ki + 1)2−N ,
i = 1, . . . , d ,
k = (k1 , . . . , kd ) ∈ Zd . We claim that there exist N and two distinct cubes T1 = TkN1 , T2 = TkN2 (with k1 6= k2 ) such that condition (4.7) is satisfied. Indeed, otherwise we would have that for each N all the points Y1 , . . . , Yν P∗ -a.s. fall into a single cube, that is,
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S. ALBEVERIO and L. V. BOGACHEV
√ −N ∗ = 1. P max kYj1 − Yj2 k ≤ d 2 j1 ,j2
It then follows (by letting N → ∞) that P max kYj1 − Yj2 k = 0 = 1 , ∗
j1 ,j2
that is to say, all the points {Yj } P∗ -a.s. coincide with each other so that actually ν = 1 P∗ -a.s. This is a contradiction with the assumption that the cluster process is proper (see (2.16)). Moreover, similar arguments show that the cubes T1 , T2 can be chosen to be disjoint. The proof of the lemma and hence of Proposition 4.1 is complete. Proposition 4.1 may be recognized as a particular version of the so-called FKG inequality (e.g. see Liggett [31]) which, in our context, reads as follows. Let us be given two functionals, ϕ1 , ϕ2 , of realizations of the random point process Z, that is, random variables ϕi = ϕi (ω), i = 1, 2, defined on the basic probability space (Ω, A, P) (which is the sample space for the measure µ, see Sec. 2.1). Suppose that ϕi ’s are L2 -integrable: E|ϕi |2 < ∞, and increasing, that is, subject to the condition ϕi (ω) ≤ ϕi (ω 0 ) whenever ω ≤ ω 0 , with respect to some partial ordering ω ≤ ω 0 of Ω. Then the FKG inequality amounts to saying that E [ϕ1 · ϕ2 ] ≥ E [ϕ1 ] · E [ϕ2 ] .
(4.8)
In our case, let us set ϕi := I{µ(Si ) > 0} ,
i = 1, 2,
(4.9)
I{A} standing for the indicator of the event A. Obviously, the functionals ϕi are increasing relative to the natural partial ordering defined by ω ≤ ω0
whenever ω ⊆ ω 0
(4.10)
(recall that ω, ω 0 ∈ Ω are subsets of Rd , see Sec. 2.1). Then inequality (4.1) is rewritten as E [(1 − ϕ1 ) · (1 − ϕ2 )] ≥ (1 − Eϕ1 ) · (1 − Eϕ2 ) , whence (4.8) follows. Of course, there is a temptation to suggest that the inequality (4.8) is valid as well in the general case of arbitrary (increasing) ϕi ’s. Since this general question goes beyond the scope of the trapping problem addressed here, we will consider it elsewhere. However, by way of complement to the above, we include intoRthis paper the derivation of (4.8) in the case of integral functionals of the form ϕ = f dµ, f ∈ F (see Appendix A).
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4.2. Trap attraction Inequality (4.1) may probably seem as not referring directly to the trap attraction. We show here that actually Proposition 4.1 can be reformulated so as to state this property in intuitively more clear terms. To see this, we introduce, in addition to the emptiness probability g(S), the notion of the conditional emptiness probability g(S|•x0 ), x0 ∈ S, defined as g(S|•x0 ) := P {µ(S \ {x0 }) = 0 | µ{x0 } = 1} .
(4.11)
That is to say, we insert a “probe” trap at a point x0 of the region S and ask what the probability is that there will be no other traps in S. It is easy to realize that the comparison of this new emptiness probability to its unconditional counterpart, g(S), may serve as a natural tool of trying the “interaction” of traps. For instance, if the inequality g(S|•x0 ) ≤ g(S) appeared holding true for all S ⊂ Rd and each x0 ∈ S, it could be interpreted as a manifestation of trap “attraction”. Similarly, the opposite inequality: g(S|•x0 ) ≥ g(S), corresponds to our intuitive notion of “repulsion”. Of course, in general the qualitative type of trap interaction may not be universal, that is, purely attractive or repulsive (e.g. as in the case of Gibbsian interaction with a potential taking both positive and negative values). However, for the Poisson-cluster correlations, this is the case. Proposition 4.3. Let S ∈ S be an arbitrary set being the closure of its interior: S = S 0 , and let x0 be its (fixed) point. Then g(S|•x0 ) ≤ g(S) .
(4.12)
Proof. Let us write down (4.1) explicitly as P{µ(S1 ) = 0, µ(S2 ) = 0} ≥ P{µ(S1 ) = 0} · P{µ(S2 ) = 0} and replace the event {µ(S2 ) = 0} by the complementary event {µ(S2 ) ≥ 1} to obtain P{µ(S1 ) = 0, µ(S2 ) ≥ 1} ≤ P{µ(S1 ) = 0} · P{µ(S2 ) ≥ 1} .
(4.13)
Surround the point x0 with a (sufficiently small) ball Vε = Vε (x0 ) and put S1 = S \ Vε , S2 = S ∩Vε . Applying (4.13) to these sets and dividing by P{µ(S ∩Vε ) ≥ 1} > 0, we obtain P {µ(S \ Vε ) = 0 | µ(S ∩ Vε ) ≥ 1} ≤ P{µ(S \ Vε ) = 0} .
(4.14)
Passing here to the limit as ε → 0 and taking into account that the measure µ is simple (see Sec. 3.1), we finally get P{µ(S \ {x0 }) = 0 | µ{x0 } = 1} ≤ P{µ(S) = 0} , and (4.12) is proved.
(4.15)
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Remarks. 1. Again note that in the noncorrelated case, the inequality (4.12) is actually an identity: gnc (S|•x0 ) ≡ gnc (S) (4.16) (cf. (4.2)). 2. The point worth noting is that the qualitative character of correlations induced by the medium clusterization appears to be universally attractive — independently of the inner structure of clusters (i.e. of the particular distribution of {Yj }). This may seem somewhat surprising in view of possibly strong repulsion between traps belonging to a single cluster. The inequalities (4.1), (4.15) however indicate that the presence of trap separation into clusters, which can be interpreted as a kind of trap attraction, proves to be a decisive factor responsible for the overall effect. 4.3. Increase of the emptiness probability In this section, we give an answer to the important question (see remark at the end of Sec. 2.2) on how the emptiness probability g(·) is related to its noncorrelated counterpart, gnc (·).g Proposition 4.4. Suppose that the mean number of traps in each cluster is finite: ν¯ < ∞. Then, for any set S ∈ S g(S) ≥ gnc(S; c) .
(4.17)
Moreover, provided ν¯ > 1, there exists such set S that the inequality (4.17) is strict. Proof. By means of the basic representations (3.48), (3.53) for g(·) and gnc (·), respectively, our claim amounts to the following inequality for the indicator functions: ν X χ(x + Yj ; S) , (4.18) χ(x + Y1 , . . . , x + Yν ; S) ≤ j=1
which holds true by the upper estimate of Corollary 3.6. In order to prove the second part of the proposition, it suffices to show that one can find such bounded S that the inequality (4.18) is strict P∗ -p.p. on a set of x of positive Lebesgue measure. In turn, on account of identity (3.23), the equality in (4.18) would imply that the set S contains at most one point Yj . So, it remains first to take a (sufficiently large) set S which P∗ -p.p. contains at least two points, Yj 6= Yj 0 (so that the inequality (4.18) appears to be strict for x = 0) and then to employ the continuity arguments as in the proof of Proposition 4.1. Remark. Similarly to Sec. 4.1, inequality (4.17) can be incorporated into a more general framework. Indeed, let us again set ϕ = I{µ(S) > 0} and notice that (4.17) can be rewritten as g Let us remind that in both cases the trap concentration c is assumed to be the same.
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E [ϕ] ≤ Enc [ϕ]
(4.19)
(Enc refers to the expectation with respect to the Poisson probability law Pnc ). Then, inequality (4.19) expresses the property that the measures P and Pnc are comparable (see Liggett [31]): P ≤ Pnc . (4.20) To be more precise, (4.20) implies that the inequality (4.19) is valid for all increasing (A-measurable and integrable) random variables ϕ = ϕ(ω), ω ∈ Ω (cf. Sec. 4.1). Here we have to skip these questions (however, see Remark 2 in Appendix A). Proposition (4.4) can be extended to a more subtle, “conditional” version. Let us set (cf. (4.11)) g(S|S0 ) := P{µ(S) = 0 | µ(S0 ) = 0} = g(S)/g(S0 ) .
(4.21)
Proposition 4.40 . Let S0 , S ∈ S be arbitrary sets such that S0 is bounded and S0 ⊆ S. Then g(S|S0 ) ≥ gnc (S|S0 ) . (4.22) Proof. By means of representation (3.48), we rewrite (4.21) in the form g(S|S0 ) = exp
Z − c0
E∗ [χ(x + Y1 , . . . , x + Yν ; S)
−χ(x + Y1 , . . . , x + Yν ; S0 )] dx .
(4.23)
Analogously, by using (3.53), Z ν ν X X χ(x + Yj ; S) − χ(x + Yj ; S0 ) dx gnc (S|S0 ) = exp −c0 E∗ j=1
j=1
Z ν X χ(x + Yj ; S \ S0 ) dx . = exp −c0 E∗
(4.24)
j=1
Then, the proof of inequality (4.22) amounts to verifying that for each x, P∗ -a.s., χ(x + Y1 , . . . , x + Yν ; S) − χ(x + Y1 , . . . , x + Yν ; S0 ) ≤
ν X
χ(x + Yj ; S \ S0 ) . (4.25)
j=1
Observe that, thanks to the inclusion S0 ⊆ S, for each n ≥ 1 and any x1 , . . . , xn one has χ(x1 , . . . , xn ; S) ≤ χ(x1 , . . . , xn ; S0 ) + χ(x1 , . . . , xn ; S \ S0 ) .
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Using this inequality and also the upper estimate (3.24) of Corollary 3.6, the lefthand side of (4.25) is dominated by χ(x + Y1 , . . . , x + Yν ; S \ S0 ) ≤
ν X
χ(x + Yj ; S \ S0 ) ,
(4.26)
j=1
and we are done.
The following lower bound on the probability g(S), based on the simple idea of removing all but one trap in each cluster, will prove essential in the sequel. Proposition 4.5. For all S ∈ S g(S) ≤ gnc (S; c0 ) . Proof. Note that gnc (S; c0 ) can be written as (cf. (3.53)) Z ∗ gnc (S; c0 ) = exp −c0 E [χ(x + Y1 ; S)] dx .
(4.27)
(4.28)
The inequality (4.27) then follows by comparing (4.28) with the representation (3.48) and using the lower estimate of Corollary 3.6. As a result, by combining Propositions 4.4 and 4.5 we have the following twosided bounds on the emptiness probability g(·): gnc (S; c0 ) ≥ g(S) ≥ gnc (S; c) ,
(4.29)
which are valid independently of the specific inner structure of clusters. In fact, these estimates are quite clear in view of the “geometrical” representations (2.11) and (3.52) and the following inequalities on the volumes: |S| ≤ |S ∗ (Y1 , . . . , Yν )| ≤ ν |S| ,
(4.30)
which are evident by the definition of S ∗ (see (3.15)). Remark. It is worth pointing out that the bounds (4.29) may be interpreted as corresponding to the two opposite limiting cases. Namely, the upper bound, gnc (S; c0 ), corresponds to the case of extremely strong clustering correlations so that in each cluster the traps are superimposed on each other. In contrast, the lower bound is related to extremely weak interaction resulting in rather inflated and hence strongly overlapped clusters. These remarks are made precise by appropriate limit theorems for the Poisson cluster process included into Appendix B. 5. Kinetical Consequences of Clusterization Eventually, we come back to the trapping problem and apply the results of the preceding sections to studying in some detail the time behavior of the survival probability P (t).
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
173
5.1. Basic formula for the survival probability First, we write out the formula for the survival probability P (t) by substituting expressions (3.47) and (3.48) for the emptiness probability into the general representation (2.9). Proposition 5.1. The survival probability in the Poisson cluster model is of the form Z S ν {x + Y ∈ S (t)} dx (5.1) P (t) = E0 exp −c0 P∗ j K j=1 Z = E0 exp −c0 E∗ [χ(x + Y1 , . . . , x + Yν ; SK (t))] dx ,
(5.2)
where SK (t) is the Wiener sausage defined in (2.6). Note that the integrand in (5.1) has the meaning of the (conditional) death probability for a particle moving along a fixed Brownian path (Bs , s ≤ t) up to time t in the presence of a single trap cluster centered at point x: S ν ∗ P∗ j=1 {x + Yj ∈ SK (t)} = P0 ⊗ P {τx ≤ t | σ(Bs , s ≤ t)} (here τx = min{t ≥ 0 : Bt ∈ ∪νj=1 (K + x + Yj )} is the life-time of such a particle, cf. (2.3)). Using Eqs. (2.9) and (3.52), the survival probability P (t) can be represented in the “geometrical” form: ∗ (Y1 , . . . , Yν ; t)|)] . P (t) = E0 [exp (−c0 E∗ |SK
(5.3)
Here the (random) region SK∗ (Y1 , . . . , Yν ; t) is, according to the definition (3.15) of S ∗ , the union of ν identical copies of the Wiener sausage SK (t) modeled via the shifts by the vectors −Y1 , . . . , −Yν determining the arrangement of traps in a ∗ (t), which is a natural generalization of the Wiener cluster. We will refer to SK sausage SK (t) to the cluster case, as the bunch of Wiener sausages, or simply the Wiener bunch. Introducing this new object appears to be useful in the trapping context (see Berezhkovskii et al. [3]) and is of independent mathematical interest. Remark. The notion of the Wiener bunch arises quite naturally if the trapping process is observed from “particle’s point of view”. In such a (movable) coordinate system, Brownian motion is performed by the cluster (that single one which is implied in (5.1)) and the Wiener bunch is meanwhile “generated” by its jointly moving traps. 5.2. Slowdown of trapping We are now able to compare the survival probability P (t) with its noncorrelated counterpart, Pnc (t; c). Indeed, substituting the inequalities (4.29) for the emptiness probabilities into the basic representation (2.9), we immediately obtain:
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S. ALBEVERIO and L. V. BOGACHEV
Pnc (t; c0 ) ≥ P (t) ≥ Pnc (t; c) .
(5.4)
Another, somewhat less formal way to understand these inequalities is the following. Recalling the form (3.40) of the concentration c, we can rewrite Eq. (2.12) in a form similar to (5.3): Pnc (t; c) = E0 [exp(−c0 ν¯ |SK (t)|)] ,
(5.5)
where ν¯ = E∗ [ν]. Now, inequalities (5.4) easily follow in view of the estimates ∗ (t). It is also seen that the deviation (4.30) on the volume of the Wiener bunch SK of expression (5.3) from (5.5) is due to the difference between the volume of the Wiener bunch and the sum of the volumes of ν individual Wiener sausages forming the bunch. Moreover, a little thinking suggests that the second inequality in (5.4) must be strict since, with positive probability, the Wiener sausages contained in the bunch may intersect each other. So, we arrive at the following assertion. Proposition 5.2. For each t ≥ 0 Pnc (t; c0 ) ≥ P (t) ≥ Pnc (t; c) .
(5.6)
Moreover, if 1 < ν¯ < ∞ then for all t > 0 the right-hand side inequality in (5.6) is strict. Proof. We only have to prove the “strict” part. Assume to the contrary that P (t0 ) = Pnc (t0 ; c) for some t0 > 0, that is, ∗ (t0 )|) P (t0 ) − Pnc (t0 ; c) = E0 [exp(−c0 E∗ |SK ∗ (t0 )| ]))] = 0 . × (1 − exp(−c0 E∗ [ν|SK (t0 )| − |SK
(5.7)
By (4.30) the difference of volumes in (5.7) is non-negative: ∗ (t0 )| ≥ 0 , ν|SK (t0 )| − |SK
(5.8)
and hence equality (5.7) implies that, P0 -a.s., ∗ (t0 )| ] = 0 . E∗ [ν|SK (t0 )| − |SK
(5.9)
Now, we take the expectation of (5.9) with respect to P0 and apply Fubini’s theorem; one may do this because ∗ (t0 )| ] ≤ E0 E∗ [ν|SK (t0 )| ] E0 E∗ [ν|SK (t0 )| − |SK
= ν¯ · E0 |SK (t0 )| < ∞,
(5.10)
since ν¯ < ∞ by assumption and E0 |SK (t)| is known to be finite for all t (see, e.g. Kac [23] and Spitzer [40]). So, we have
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
∗ E∗ E0 [ν|SK (t0 )| − |SK (t0 )| ] = 0
175
(5.11)
whence it follows, again thanks to (5.8), that P∗ -a.s. ∗ E0 [ν|SK (t0 )| − |SK (t0 )| ] = 0 .
(5.12)
Geometrically, this means that, for P∗ -almost each configuration {Y1 , . . . , Yν }, the ∗ (t0 ) P0 -a.s. do not sausages (SK (t0 ) − Yj ) incorporated into the Wiener bunch SK intersect each other (up to Lebesgue-null sets). But this is a contradiction as can be seen as follows. Let us fix a configuration {Y1 , . . . , Yν } with ν ≥ 2 (the event {ν ≥ 2} is P∗ -p.p.) and prove that, for any j 6= j 0 , P0 {|(SK (t0 ) − Yj ) ∩ (SK (t0 ) − Yj 0 )| > 0} > 0 .
(5.13)
It suffices to check that the mean (with respect to P0 ) of the volume in (5.13) is positive. Expressing the volume through the indicator function, we have Z E0
χ(x; (SK (t0 ) − Yj ) ∩ (SK (t0 ) − Yj 0 )) dx Z
= Z =
(5.14)
P0 {∃ s, s0 ∈ [0, t0 ] : (−K + Bs − Yj ) ∩ (−K + Bs0 − Yj 0 ) 3 x} dx P0 {∃ s, s0 ∈ [0, t0 ] : Bs ∈ (x + Yj + K) , Bs0 ∈ (x + Yj 0 + K)} dx . (5.15)
The application of Fubini’s theorem in (5.14) is justified since the left-hand side of (5.14) is dominated by Z E0
χ(x; SK (t0 ) − Yj ) dx = E0 |SK (t0 ) − Yj | = E0 |SK (t0 )| < ∞.
(5.16)
Finally, it remains to note that the probability in (5.15) is positive for all x, thanks to the assumption (see Sec. 2.1) that the model set K is accessible [34]. Remark. From the kinetical point of view, the second inequality in (5.6) means that in the Poisson-cluster medium of traps, the trapping rate is decreased as compared to that in the noncorrelated (Poisson) case, which is to say that the trapping process slows down due to the medium clusterization. We would like to emphasize that this effect appears fairly general being valid independently of the inner structure of clusters. Moreover, since this result follows from the inequalities (4.17) which refer to the statistical properties of the medium only, in fact it takes place with an arbitrary law of the particle’s motion; it is also clear that the assumption of traps perfectness (see Sec. 2.1) is not critical as well.
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The slowdown result can also be stated for the normalized survival probabilities (see the definition in Sec. 2.1): Proposition 5.20 . For all t ≥ 0 P (t | 0) ≥ Pnc (t | 0) .
(5.17)
Moreover, if 1 < ν¯ < ∞, then for all t > 0 the inequality (5.17) is strict. Proof. The inequality (5.17) readily follows from formula (2.4) and Proposition 4.4. The strict version can be proved by a minor modification of the arguments used in Proposition 5.2, now exploiting the equality in the estimate (4.26) in place of (4.23). We conclude this section with a conjecture that the inequality similar to (5.17) is likely to be valid for more general conditional survival probabilities: P (t | s) ≥ Pnc (t | s) ,
(5.18)
where P (t | s) is defined as the probability of survival up to time t conditioned on the survival up to time s, s ≤ t. Actually, this is equivalent to proving that the quotient P (t)/Pnc (t; c) is increasing with time t. That this must be true is also evidenced from the comparison of the functions P (t) and Pnc (t; c) in the long-time limit (see Sec. 5.3 below) where one has log P (t)/Pnc (t; c) ∼ const · td/(d+2) 1 (provided ν¯ > 1). Note, however, that (5.18) does not immediately follow from the respective inequality for the conditional emptiness probabilities guaranteed by Proposition 4.4: gnc (SK (t)) g(SK (t)) ≥ , g(SK (s)) gnc (SK (s))
(5.19)
because, in view of the required inequality (5.18) and basic formula (2.9), the operation E0 of the Wiener expectation has to be applied in (5.19) both to the numerators and denominators, rather than to the whole fractions. The additional difficulty is related to possible self-intersections of the Wiener sausage, so that one cannot simply proceed by conditioning on the Brownian path (Bu , u ≤ s). 5.3. Long-time survival In this section, we obtain the rough (logarithmic) asymptotics for the survival probability P (t) as t → ∞. We start by reminding the reader the formulation of the classical result of Donsker and Varadhan [15] for the noncorrelated case and also outline the main idea of getting a proper lower bound, which will play a crucial role in our own derivation. The Donsker–Varadhan asymptotic law reads as follows:
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
lim t−d/(d+2) log Pnc (t; c) = −γd c2/(d+2) .
t→∞
177
(5.20)
Here d ≥ 1 is the space dimension, c is the intensity of the Poisson probability law which governs the distribution of the point trap process Z = {Zi }, and γd is an explicit numerical constant specified as [15] γd =
2/(d+2) vd
d/(d+2) d 2λd 1+ , 2 d
(5.21)
where vd = π d/2 /Γ(1 + d/2) is the volume of the Euclidean unit ball V1 ⊂ Rd and λd > 0 stands for the principal Dirichlet eigenvalue of the Laplacian − 12 ∆ in the ball V1 : Z 1 |∇φ(x)|2 dx . (5.22) λd = inf φ 2 The infimum is taken over smooth functionsRwith compact support in the open ball V10 subject to the normalization condition |φ(x)|2 dx = 1. Note that λd can be 2 , where ξp is the smallest zero of the Bessel expressed explicitly as λd = 12 ξd/2−1 function of the first kind of order p [1]. Remark. Historically, the asymptotics (5.20) was first discovered in the (exactly solvable) one-dimensional case by Balagurov and Vaks [2]. M. Kac conjectured, proceeding from the ideas of Lifschitz [30] with respect to some closely related problems in solid state physics, that the law (5.20)–(5.21) must be valid in all dimensions [23, 24].h Soon thereafter, Donsker and Varadhan [15] gave a complete proof of the tail (5.20); in so doing, they proceeded from the functional E0 [exp(−c|Sb (t)|)] (cf. (2.12)), with no trapping motivation. At this point, it is instructive to notice that the form of the asymptotics (5.20) does not depend on the size b. Thanks to this observation, the result (5.20) immediately extends to the case of an arbitrary (compact) trap set K with non-empty interior. Indeed, then there exists a ball Va0 contained in K and, on the other hand, the set K being compact is itself contained in some ball, Va . It is then evident that, on the replacement of the model trap set K by the smaller ball, Va0 , or by the bigger one, Va , the survival probability will be increased or decreased, respectively: P (t; a0 ) ≥ P (t) ≥ P (t; a)
(5.23)
(here the additional parameters a0 and a in the notation refer to the size of new spherical traps). Now, from the form of the asymptotics (5.20) notwithstanding the trap radius (assumed to be proven in the spherical case), it readily follows that h Incidentally, in the papers [23, 24] Kac suggested the term “Wiener sausage” for the tubular neighborhood of a Brownian path (Sb (t) in our notations, see Sec. 2.2). In fact, however, this object has been brought into attention in Probability Theory much earlier — the pioneering work dates back to Leontovitch and Kolmogorov [29] who studied the mean volume of the twodimensional Wiener sausage, thereafter followed by a series of papers on the long-time asymptotics of the Wiener sausage mean volume (see Spitzer [40], Getoor [17], Le Gall [28], Berezhkovskii et al. [4]) and also on the limit results of the type of Central Limit Theorem (see Le Gall [27]).
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(5.20) holds true for such more general K as well. Moreover, Sznitman [41] has proved that (5.20) is actually valid in the fairly general case of a compact nonpolar set K (i.e. a set being accessible, with positive probability, relative to Brownian motion starting from an arbitrary point [34]).i The idea of producing a lower bound on P (t) is very simple and by now well known (see, e.g. [26, 15, 41, 19, 22]). Namely, observe that the Brownian particle is obviously insured to survive until time t if: (a) a given ball VR (0) happens to be empty from traps (more specifically, the a-neighborhood of this ball, VR+a (0), is meant not to contain the points {Zi } — recall that K ⊆ Va ), and (b) the Brownian particle is forced to spend all the time up to t in the ball VR (0). The radius of the ball, R, will be adjusted later; in particular, it will appear to be growing with time. Since the trapping medium is assumed to be independent of the Brownian motion, the probability of such a conjunct event is given by the product of the marginal probabilities, and hence we obtain P (t) ≥ P{#{Zi ∈ VR+a } = 0} · P0 {Bs ∈ VR for all s ≤ t} .
(5.24)
Here, the first factor on the right-hand side of (5.24) is the emptiness probability g(VR+a ), in the noncorrelated case given by the formula gnc (VR+a ; c) = exp(−c|VR+a |) = exp(−c(R + a)d vd ) .
(5.25)
The second factor, being the “large deviation” probability for the Wiener process, is known to decrease exponentially in t [26, 15, 41]: log P0 max kBs k ≤ R ∼ −λd t/R2 s≤t
as
t→∞
(5.26)
(the symbol ∼ means that the ratio of the left- and right-hand sides tends to 1 as t → ∞). By scaling arguments, it is seen that the asymptotics (5.26) holds true also in the case where the radius R grows with time provided that t/R2 1. Combining equations (5.24)–(5.26) yields asymptotically as t → ∞: log P (t) ≥ −(1 + o(1)) c vd Rd + λd t/R2 .
(5.27)
Picking an “optimal” radius R = R∗ by minimizing the right-hand side of (5.27), we get 1/(d+2) 2λd t , (5.28) R∗ = R∗ (t) ∼ c d vd
i For ramifications and extensions along this line, see, e.g. Schmock [38], Bolthausen [11] and Sznitman [44] for the confinement property of surviving Brownian motion in one and two dimensions, Bolthausen [10] and Sznitman [42] for the case of shrinking traps, Bolthausen and den Hollander [12] for the case of decaying traps, and also Eisele and Lang [16] and Sznitman [43] for the case of Brownian motion with drift.
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
179
which, on the substitution into (5.27), gives lim inf t−d/(d+2) log Pnc (t; c) ≥ −γd c2/(d+2) t→∞
(5.29)
with γd defined in (5.21). Note that, in view of (5.20), the lower estimate (5.29) appears to be (logarithmically) sharp. (For the derivation of a consistent upper bound, which is much more complicated to obtain, see the seminal work by Donsker and Varadhan [15] and also the papers by Sznitman, e.g. [41, 45], where a more general technique of “enlargement of traps” has been developed.) Now we are in a position to prove the main result of this work. Proposition 5.3. Consider the trapping problem in a Poisson-clusterized medium and assume that ν¯ = E∗ [ν] < ∞. Then, the long-time asymptotics of the survival probability P (t) is given by (5.20) with c0 in place of c = c0 ν¯ : lim t−d/(d+2) log P (t) = −γd c0
2/(d+2)
t→∞
,
(5.30)
where γd is specified in (5.21). Proof. In view of the explicit result (5.20), our claim will be proved once we have shown that log P (t) ∼ log Pnc (t; c0 ) as
t → ∞,
(5.31)
where Pnc (t; c0 ) is meant to refer to the trapping problem with noncorrelated traps with the same form of traps, K, as is implied in P (t). The key observation which makes the proof quite easy is that an upper estimate on P (t) consistent with (5.30) is readily provided by the following general bound (see the first inequality in (5.6)): Pnc (t; c0 ) ≥ P (t) .
(5.32)
Hence, it suffices to obtain an appropriate lower estimate on P (t). To do this, we will employ the conventional method described above (see inequality (5.24)). It is not difficult to understand that, in order to be able to reproduce the derivation of (5.29) in our case, we only need to convince ourselves that the emptiness probability g(VR ) is logarithmically equivalent to gnc (VR ; c0 ) as R → ∞ (together with t → ∞): − log g(VR ) ∼ c0 vd Rd .
(5.33)
By the change of variables y = ρx, where ρ = R−1 → 0, the basic formula (3.48) is rewritten as Z (5.34) − log g(VR ) = c0 Rd E∗ [χ(y + ρY1 , . . . , y + ρYν ; V1 )] dy or, in the geometrical form (see (3.52)):
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S. ALBEVERIO and L. V. BOGACHEV
− log g(VR ) = c0 Rd E∗ |V1∗ (ρY1 , . . . , ρYν )| ,
(5.35)
where V1∗ (ρY1 , . . . , ρYν ) is the compound region formed of the shifted balls V1 − ρYj , j = 1, . . . , ν (see (3.15)). It is clear that, in the limit ρ → 0, these balls merge into a single ball of unit radius, and so one can expect that Eq. (5.35) passes over into − log g(VR ) ∼ c0 Rd |V1 | ,
(5.36)
which is just what is required. This is also seen by letting formally ρ → 0 in the integrand in (5.34): Z
Z
∗
E [χ(y + ρY1 , . . . , y + ρYν ; V1 )] dy =
lim
ρ→0
E∗ [χ(y; V1 )] dy = |V1 | .
(5.37)
Actually, it is not difficult to justify (5.37) by means of Corollary 3.5. To do this, we first apply Fubini’s theorem to change the order of the expectation and integration in (5.37) using that, according to (3.24), E∗
Z
χ(y + ρY1 , . . . , y + ρYν ; V1 ) dy ≤ E∗
ν Z X
χ(y + ρYj ; V1 ) dy (5.38)
j=1
ν Z X χ(z; V1 ) dz = E∗ [ν] · |V1 | = ν¯ vd < ∞ = E∗
(5.39)
j=1
(in each integral in the sum (5.38) we changed the variables: y + ρYj = z). Furthermore, by using the exact representation (3.23) and again changing the variables as above, the left-hand side of (5.38) is rewritten in the form Z X ν χ(y + ρYj ; V1 ) E∗ j=1
× 1 −
ν X
χ(y + ρYk ; V1 )
k=j+1
=E ∗
Z X ν
ν Y
(1 − χ(y + ρY` ; V1 )) dy
`=k+1
χ(z; V1 )
j=1
× 1 −
ν X k=j+1
χ(z + ρYkj ; V1 )
ν Y
(1 − χ(z + ρY`j ; V1 )) dz , (5.40)
`=k+1
where we set Ykj = Yk − Yj . Note that the integrand in (5.40) is dominated by the quantity
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
ν X
χ(z; V1 ) = ν χ(z; V1 )
181
(5.41)
j=1
which is integrable: ∗
E
Z
νχ(z; V1 ) dz = E∗ [ν] · |V1 | < ∞ .
(5.42)
Hence, the dominated convergence theorem applies to (5.40), and we can pass to the limit under the integral sign as ρ → 0. Note that for each z not belonging to the boundary of the ball V1 , in the limit ρ → 0 all the indicator functions in (5.40) are replaced by χ(z; V1 ) to yield ν ν ν X X Y χ(z; V1 ) · 1 − χ(z; V1 ) (1 − χ(z; V1 )) , (5.43) j=1
k=j+1
`=k+1
because whenever z is either an interior or exterior point of V1 , for all sufficiently small ρ so is the point z + ρYkj . For the boundary points z, such that kzk = 1, the limit heavily depends on the directions of the vectors Ykj ; fortunately, such points are of null Lebesgue measure and thus can be discarded. As a result, P∗ -a.s. for almost each z the limit of the integrand in (5.40) is given by expression (5.43) which, by applying again (3.22), can be converted back to χ(z, . . . , z; V1 ) = χ(z; V1 ). The integration of this indicator with respect to z gives the volume |V1 | = vd as claimed in (5.37), and we are finished. Remark. In the simplest case of bounded (e.g. nonrandom) clusters, the asymptotic result (5.30) easily follows from (5.20). Indeed, if max{Y1 , . . . , Yν } ≤ C < ∞, then we may replace each cluster by the trapping ball of radius C to obtain an appropriate lower bound on P (t) (cf. (5.23)). However, the general case of possibly unbounded clusters is not subject to an immediate application of the results obtained in [15, 41]. Appendices A. FKG Inequality for Integral Functionals Here we prove a Rversion of FKG inequality (see Sec. 4.1) for functionals ϕ of integral form: ϕ = f (x) µ(dx) with f ∈ F . Since f ≥ 0, such functionals are increasing (with respect to the natural partial ordering of the space Ω, see (4.10)). Proposition A.1. Let f1 ,Rf2 be arbitrary R integrable functions from F. Then for the random variables ϕ1 = f1 dµ, ϕ2 = f2 dµ the following inequality holds: E [ϕ1 · ϕ2 ] ≥ E [ϕ1 ] · E [ϕ2 ] .
(A.1)
Proof. Let us set f (x) = z1 f1 (x) + z2 f2 (x), z1 , z2 > 0, and note that, similarly to the proof of Proposition 3.9, the required averages can be obtained
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S. ALBEVERIO and L. V. BOGACHEV
by meansR of appropriate partial differentiating of the Laplace functional Lµ (f ) = E [exp(− (z1 f1 + z2 f2 ) dµ] with respect to the parameters z1 , z2 : ∂Lµ (f ) , (A.2) E [ϕi ] = − ∂zi z1 =z2 =0 and also
∂ 2 Lµ (f ) . E [ϕ1 · ϕ2 ] = ∂z1 ∂z2 z1 =z2 =0
(A.3)
Thanks to the explicit expression (3.34) for the Laplace functional, the calculation of derivatives (A.2), (A.3) is straightforward: Z Z ∂Lµ (f ) ∗ f E = c (x + y) µ(dy) dx − 0 i ∂zi z1 =z2 =0 Z ν X fi (x + Yj ) dx = c0 E∗ j=1
= c0 E∗
ν Z X
fi (x + Yj ) dx
j=1
= c0 E∗
ν Z X
fi (x) dx
j=1
Z = c0 ν¯
fi (x) dx ,
so that the right-hand side of (A.1) is of the form Z Z c20 ν¯2 f1 (x) dx f2 (x) dx .
(A.4)
(A.5)
Similarly, Z Z Z ∂Lµ (f ) 2 2 f1 (x) dx = c0 ν¯ f2 (x) dx + c0 ν¯ f1 (x)f2 (x) dx ∂z1 ∂z2 z1 =0,z2 =0 XZ +c0 E∗ f1 (x + Yj )f2 (x + Yk ) dx . (A.6) j6=k
Now, by comparing (A.6) with (A.5), we easily get inequality (A.1).
Remarks. 1. By applying inequality (A.1) to the purely Poissonian case, we see that it is actually reasonable to require that the supports of the functions f1 , f2 be disjoint. Under such an assumption, in the noncorrelated case the above inequality passes over to the identity, which is what one would naturally like to have.
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
183
2. With respect to the inequality P ≤ Pnc mentioned at the end of Sec. 4.3, let us note that for the functionals of integral form R considered above, inequality (4.19) degenerates to an equality since, given ϕ = f (x) µ(dx), both sides of (4.19) are R equal to c0 ν¯ f (x) dx (see (A.3), (A.5)). It is not difficult to understand that this equality is in fact a general consequence of the assumption that P and Pnc are equal “in mean” (i.e. have the same concentration c), which is seen by approximating the integrand f by linear combinations of indicator functions. B. Limiting Distribution Under Clusters Scaling In this section, we are concerned with the question on the limiting distribution of the Poisson cluster point process — in the limit of extremely contracted or, in contrast, extremely expanded clusters (see the end of Sec. 4.3). We will consider the simplest case where such “blowing down” and “blowing up” of the clusters is modeled via linear scaling. More specifically, consider the Poisson cluster process of (i) the form Zλ = {Xi + λYj } where λ > 0 is a scaling parameter. The corresponding “scaled” random measure associated to the process Zλ (see Sec. 3.2) will be referred to as µλ : X µλ (S) = µ∗λi θXi (S) , S ∈ S , (B.1) i
where µ∗λ (S) := µ∗ (λ−1 S) =
ν X
χ(λ Yj ; S)
(B.2)
j=1
(cf. (3.28), (3.27)). By Proposition 3.8, the Laplace functional Lµλ (·) of the measure µλ is of the form: Z ν X f (x + λYj ) dx . (B.3) Lµλ (f ) = exp −c0 1 − E∗ exp − j=1
The following proposition establishes the intuitively clear convergence (in distribution) of the process Zλ , as λ → 0, to the so-called compound Poisson process, that is the point process with random masses νi located at the Poisson points Xi (see Daley and Vere-Jones [14]).j Note that the Laplace functional for the compound Poisson process, which appears below in (B.4), is formally obtained from Eq. (3.34) by putting all Yj ’s equal to zero; this can be justified by repeating the arguments of the proof of Proposition 3.2. Proposition B.1. Let ν¯ = E∗ [ν] < ∞. Then Z lim Lµλ (f ) = exp −c0 (1 − E∗ [exp(−νf (x))]) dx λ→0
(B.4)
j To be more accurate, note that by definition the compound Poisson process is not simple, so the probability space (Ω, A, P) defined in Sec. 2.1 should be properly extended (see [14]).
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S. ALBEVERIO and L. V. BOGACHEV
for each function f ∈ F which is almost everywhere continuous and satisfies the condition Z [1 − exp(−f (x))] dx < ∞ . (B.5) Proof. Thanks to the basic formula (2.9) (see also (B.3)), we have to check that Z Z ν X f (x + λYj ) dx = E∗ [1 − exp(−νf (x))] dx . (B.6) lim E∗ 1 − exp − λ→0
j=1
Let us set f˜(x) := 1 − exp(−f (x)) .
(B.7)
Obviously, 0 ≤ f˜ ≤ 1 and, by assumption (B.5), f˜ is integrable. Then Eq. (B.6) takes the form Z Z ν Y (1 − f˜(x + λYj )) dx = E∗ [1 − (1 − f˜(x))ν ] dx . (B.8) lim E∗ 1 − λ→0
j=1
Using the inequality of Corollary 3.4 with aj = f˜(x + λYj ), we have 1−
ν Y
(1 − f˜(x + λYj )) ≤
j=1
ν X
f˜(x + λYj ) .
(B.9)
j=1
This allows us to apply Fubini’s theorem and interchange the integration and expectation in (B.8) since Z ν Y [1 − f˜(x + λYj )] dx E∗ 1 − j=1
ν Z X f˜(x + λYj ) dx ≤ E∗ j=1
ν Z X f˜(z) dz = E∗ j=1
Z = ν¯ · <∞
f˜(z) dz (B.10)
(here we used Eq. (3.9) of Lemma 3.1). Moreover, estimate (B.9) also shows that, by dominated convergence, it suffices to prove (P-a.s.) the limiting equation
185
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
Z lim
λ→0
1 −
ν Y
(1 − f˜(x + λYj )) dx =
Z [1 − (1 − f˜(x))ν ] dx .
(B.11)
j=1
Let us apply Lemma 3.3 to represent the integral in (B.11) as ν Z X Y X f˜(x + λYj ) · 1 − (1 − f˜(x + λY` )) dx f˜(x + λYk ) j=1
k>j
=
ν Z X
f˜(z) 1 −
j=1
X k>j
f˜(z + λYkj )
(B.12)
`>k
Y
(1 − f˜(z + λY`j )) dz ,
(B.13)
`>k
where in each integral we made the change of variables x + Yj = z and set Ykj = Yk − Yj . It follows that the difference between the integrals in (B.11) is of the form " # ν Z X Y Y X f˜(z + λYkj ) f˜(z) (1 − f˜(z + λY`j )) − f˜(z) (1 − f˜(z)) dz . (B.14) j=1
k>j
`>k
`>k
Using that 0 ≤ f˜ ≤ 1, it is easily seen that for each j (and fixed ν and Y1 , . . . , Yν ) the integrand in (B.14) is dominated by ν f˜(z) which is integrable, so that the dominated convergence theorem applies. It remains to note that thanks to a.e. continuity, for each fixed z the expression in the square brackets in (B.14) vanishes as λ → 0, which completes the proof. Now we consider the opposite limiting case λ → ∞ and show that the process Zλ converges to a Poisson process of intensity c = c0 ν¯. Proposition B.2. Assume that 1 < ν¯ < ∞. Then, for each function f ∈ F satisfying condition (B.3), we have Z (B.15) lim Lµλ (f ) = exp −c0 ν¯ [1 − exp(−f (x))] dx . λ→∞
Proof. By repeating the arguments used in the proof of Proposition B.1 and using the same notation (B.7), we are led from (B.15) to proving that, P∗ -a.s., Z Z ν Y ∗ ˜ 1 − f (x + Yj ) dx = E [ν] · f˜(x) dx . (B.16) lim 1− λ→∞
j=1
In turn, by means of the representation of the integrals in the form (B.12)–(B.13), the limit (B.16) amounts to Z Y ˜ f˜(z + λYkj ) f(z) (1 − f˜(z + λY`j )) dz = 0 , P∗ -a.s. , (B.17) lim λ→∞
`>k
where 1 ≤ j < k ≤ ν are fixed and Ykj = Yk − Yj (cf. (B.13)).
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S. ALBEVERIO and L. V. BOGACHEV
Using that 0 ≤ f˜ ≤ 1, we can estimate the integral (B.17) as Z 0≤
f˜(z)f˜(z + λYkj )
Y
Z (1 − f˜(z + λY`j )) dz ≤
f˜(z)f˜(z + λYkj ) dz . (B.18)
`>k
Since f˜ is integrable, for a given ε > 0 we can find such r > 0 that Z f˜(x) dx < ε ,
(B.19)
Vrc
where Vrc = Rd \ Vr and Vr = Vr (0) is the ball of radius r centered at the origin. Set (B.20) f˜r (x) = f˜(x) · χ(x; Vr ) . Then we can represent the integrand on the right-hand side of (B.18) as f˜(z)f˜(z + λYkj ) =
f˜(z) − f˜r (z) f˜(z + λYkj )
+ f˜r (z) f˜(z + λYkj ) − f˜r (z + λYkj ) + f˜r (z)f˜r (z + λYkj ) =: F1 (z, λ) + F2 (z, λ) + F3 (z, λ) .
(B.21)
Let us estimate the contribution of each summand in (B.20) into the last integral in (B.18). Uniformly in λ, we have Z F1 (z, λ) dz ≤
Z
Z f˜(z) − f˜r (z) dz =
f˜(z) dz < ε .
(B.22)
Vrc
Analogously, Z F2 (z, λ) dz ≤ =
Z f˜(z + λYkj ) − f˜r (z + λYkj ) dz Z f˜(u) − f˜r (u) du Z f˜(u) du
= Vrc
< ε.
(B.23)
As for the term F3 (z, λ), it is uniformly estimated from above as follows: F3 (z, λ) ≤ f˜r (z) ≤ f˜(z) . Hence, the dominated convergence theorem applies to show that
(B.24)
BROWNIAN SURVIVAL IN A CLUSTERIZED TRAPPING MEDIUM
Z lim
λ→∞
187
Z f˜(z)f˜r (z + λYkj ) dz
F3 (z, λ) dz = lim
λ→∞
Z =
Vr
lim
Vr λ→∞
f˜(z)f˜r (z + λYkj ) dz
= 0.
(B.25)
Here we used the fact that the limit within the integral in (B.25) equals zero. Indeed, because Ykj = Yk − Yj 6= 0 for k 6= j (see Sec. 2.3), it follows that, for each fixed z ∈ Vr , we have z + λYkj ∈ Vrc for all sufficiently large λ. Since the function f˜r is supported inside the ball Vr , this implies that the second factor in the integrand (B.25) vanishes as λ → ∞. As a result, combining (B.22), (B.23) and (B.25), we obtain that Z (B.26) lim sup f˜(z)f˜(z + λYkj ) dz ≤ 2ε , λ→∞
and we are done, since ε can be chosen to be arbitrarily small.
Remark. The result stated in Proposition B.2 should be compared to the known Goldman theorem on the Poisson limiting distribution for point processes under independent, identically distributed shifts of its points of “increasing amplitude” (see [18, 14]). Note that in our case, we consider a somewhat more general shifting mechanism since each point of the initial configuration {Xi } is being shifted by an aggregate of ν random vectors thus producing a cluster of points; in so doing, the i.i.d. assumption is only required of the aggregates of shifts. However, as mentioned above, we are restricted here by consideration of the one-parameter family of shifts. On the other hand, Goldman’s result applies to fairly general initial configurations of points which are only assumed to be so-called well-distributed (which, loosely speaking, amounts to the condition that the limiting fraction of points contained in a large volume is finite and translation invariant, see [18]). We conclude this section with the conjecture that such an extension must also hold true for the clusterized shifts considered above. Acknowledgments The authors are grateful to S. A. Molchanov and Ya. G. Sinai for their stimulating interest in this work and valuable remarks. Actually, the idea of using the Poisson cluster model in the trapping problem context is due to S. A. Molchanov (see the paper [3]). The second author, L. V. B., would also like to thank A. M. Berezhkovskii and Yu. A. Makhnovskii for numerous fruitful discussions of the physical aspects of the trapping problem. References [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions, Government Printing Office, Washington, D.C., 1964.
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[2] B. Ya. Balagurov and V. G. Vaks, “Random walks of a particle on lattices with traps”, Zh. Eksp. Teor. Fiz. 65 (1973) 1939–1945 (in Russian). English translation in Sov. Phys. JETP 38 (1974) 968–970. [3] A. M. Berezhkovskii, Yu. A. Makhnovskii, L. V. Bogachev and S. A. Molchanov, “Brownian-particle trapping by clusters of traps”, Phys. Rev. E47 (1993) 4564–4567. [4] A. M. Berezhkovskii, Yu. A. Makhnovskii and R. A. Suris, “Wiener sausage volume moments”, J. Stat. Phys. 5 (1989) 333–346. [5] A. M. Berezhkovskii, Yu. A. Makhnovskii, R. A. Suris, L. V. Bogachev and S. A. Molchanov, “Trap correlation influence on Brownian particle death. One-dimensional case”, Phys. Lett. A161 (1991) 114–117. [6] A. M. Berezhkovskii, Yu. A. Makhnovskii, R. A. Suris, L. V. Bogachev and S. A. Molchanov, “Diffusion-limited reactions with correlated traps”, Chem. Phys. Lett. 193 (1992) 211–214. [7] A. M. Berezhkovskii, Yu. A. Makhnovskii, R. A. Suris, L. V. Bogachev and S. A. Molchanov, “Trap correlation influence on diffusion-limited process rate”, Phys. Rev. A45 (1992) 6119–6122. [8] L. V. Bogachev, A. M. Berezhkovskii and Yu. A. Makhnovskii, “Brownian trapping with grouped traps”, in On Three Levels: Micro, Meso and Macroscopic Approaches in Physics, Proc. NATO ARW, Leuven, 1993, eds. M. Fannes, C. Maes and A. Verbeure, Plenum Press, New York, 1994, pp. 441–444. [9] L. V. Bogachev and Yu. A. Makhnovskii, “Brownian motion with absorption in a clusterized random medium”, Doklady Akad. Nauk 340 (1995) 300–302 (in Russian). [10] E. Bolthausen, “On the volume of the Wiener sausage”, Ann. Probab. 18 (1990) 1576–1582. [11] E. Bolthausen, “Localization of a two-dimensional random walk with an attractive path interaction”, Ann. Probab. 22 (1994) 875–918. [12] E. Bolthausen and F. den Hollander, “Survival asymptotics for Brownian motion in a Poisson field of decaying traps”, Ann. Probab. 22 (1994) 160–176. [13] J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications”, Phys. Rep. 195 (1990) 127–193. [14] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer, New York, 1988. [15] M. D. Donsker and S. R. S. Varadhan, “Asymptotics for the Wiener sausage”, Commun. Pure Appl. Math. 28 (1975) 525–565. [16] T. Eisele and R. Lang, “Asymptotics for the Wiener sausage with drift”, Probab. Theory Rel. Fields 74 (1987) 125–140. [17] R. K. Getoor, “Some asymptotic formulas involving capacity”, Z. Wahrsch. verw. Geb. 4 (1965) 248–252. [18] J. R. Goldman, “Stochastic point processes: limit theorems”, Ann. Math. Statist. 38 (1967) 771–779. [19] P. Grassberger and I. Procaccia, “The long time properties of diffusion in a medium with static traps”, J. Chem. Phys. 77 (1982) 6281–6284. [20] J. W. Haus and K. W. Kehr, “Diffusion in regular and disordered lattices”, Phys. Rep. 150 (1987) 263–406. [21] S. Havlin and D. Ben-Avraham, “Diffusion in disordered media”, Adv. Phys. 36 (1987) 695–798. [22] F. den Hollander and G. H. Weiss, “Aspects of trapping in transport processes”, in Contemporary Problems in Statistical Physics, ed. G. H. Weiss, SIAM, Philadelphia, 1994, pp. 147–203. [23] M. Kac, “Probabilistic methods in some problems of scattering theory”, Rocky Mount. J. Math. 4 (1974) 511–537.
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[24] M. Kac and J. M. Luttinger, “Bose–Einstein condensation in the presence of impurities II”, J. Math. Phys. 15 (1974) 183–186. [25] R. F. Kayser and J. B. Hubbard, “Reaction diffusion in a medium containing a random distribution of nonoverlapping traps”, J. Chem. Phys. 80 (1984) 1127–1130. [26] A. R. Kerstein, “Diffusion in a medium with connected traps”, Phys. Rev. B32 (1985) 3361–3363. [27] J.-F. Le Gall, “Fluctuation results for the Wiener sausage”, Ann. Probab. 16 (1988) 991–1018. [28] J.-F. Le Gall, “Sur une conjecture de M. Kac”, Probab. Theory Relat. Fields 78 (1988) 389–402. [29] M. A. Leontovitch and A. N. Kolmogorov, “Zur Berechnung der mittleren Brownschen Fl¨ ache”, Phys. Z. Sowjet. 4 (1933) 1–13. [30] I. M. Lifschitz, “About the structure of the energy spectrum and the quantum states of disordered systems”, Usp. Fiz. Nauk 83 (1964) 617–663 (in Russian). English translation in Sov. Phys. Uspekhi 7 (1965) 549–592. [31] T. M. Liggett, Interacting Particle Systems, Springer, New York, 1985. [32] T. Ohtsuki, “Diffusion-controlled trapping by extended traps”, Phys. Rev. A32 (1985) 699–701. [33] G. S. Oshanin and S. F. Burlatsky, “Reaction kinetics in polymer systems”, J. Stat. Phys. 65 (1991) 1109–1122. [34] S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York, 1978. [35] S. A. Rice, Diffusion-Limited Reactions, Elsevier, Amsterdam, 1985. [36] P. M. Richards, “Diffusion to nonoverlapping or spatially correlated traps”, Phys. Rev. B35 (1987) 248–256. [37] D. Ruelle, Statistical Mechanics, Benjamin, New York, 1969. [38] U. Schmock, “Convergence of the normalized one dimensional Wiener sausage path measures to a mixture of Brownian taboo processes”, Stochastics 29 (1990) 171–183. [39] M. von Smoluchowski, “Versuch einer mathematischen Theorie der Koagulationskinetik kolloider L¨ osungen”, Zeit. Phys. Chemie 29 (1917) 129–168. [40] F. Spitzer, “Electrostatic capacity, heat flow, and Brownian motion”, Z. Wahrsch. verw. Geb. 3 (1964) 110–121. [41] A. S. Sznitman, “Lifschitz tail and Wiener sausage I”, J. Funct. Anal. 94 (1990) 223–246. [42] A. S. Sznitman, “Long time asymptotics for the shrinking Wiener sausage”, Commun. Pure Appl. Math. 43 (1990) 809–820. [43] A. S. Sznitman, “On long excursions of Brownian motion among Poissonian obstacles”, in Stochastic Analysis, Proc. Durham Conf., 1990, eds. M. T. Barlow and N. H. Bingham, Cambridge Univ. Press, Cambridge, 1991, pp. 353–375. [44] A. S. Sznitman, “On the confinement property of two-dimensional Brownian motion among Poissonian obstacles”, Commun. Pure Appl. Math. 44 (1991) 1137–1170. [45] A. S. Sznitman, “Brownian survival among Gibbsian traps”, Ann. Probab. 21 (1993) 490–508. [46] S. Torquato, “Concentration dependence of diffusion-controlled reactions among static reactive sinks”, J. Chem. Phys. 85 (1986) 7178–7179. [47] G. H. Weiss and S. Havlin, “Trapping of random walks on the line”, J. Stat. Phys. 37 (1984) 17–25.
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY. A CLASS OF GALILEI-COVARIANT MODELS ∗ ´ J. DEREZINSKI
Department of Mathematical Methods in Physics Warsaw University Hoz˙ a 74, 00-682 Warszawa Poland E-mail :
[email protected] Received 4 April 1997 We describe N -body scattering in the case of identical particles (bosons and fermions).
Contents 0. Introduction 1. General Definitions 1.1. Hilbert spaces 1.2. Homomorphisms of commutative C ∗ - and W ∗ -algebras 1.3. Permutations 1.4. Clusters 1.5. Cartesian and tensor products associated with a cluster 1.6. Clusterings 1.7. Cartesian and tensor products associated with a clustering 2. Distinguishable Particles 2.1. Elementary particles 2.2. Clusters of elementary particles 2.3. Asymptotic particles 2.4. Clusters of asymptotic particles 2.5. Identification operator 2.6. Existence of the asymptotic velocity 2.7. Short-range case 2.8. Long-range case – free region 2.9. Long-range case – asymptotic interacting Hamiltonian 2.10. Long-range case – modified wave operators 3. Second Quantization in the Category of Sets 3.1. Second quantization of a set 3.2. “Third quantization” of a set 3.3. Permutations that preserve species 3.4. Clusters with composition n ∈ Γ(E) 3.5. Permutations that preserve species and a clustering 3.6. Clusterings associated with k ∈ Γtq (E) 3.7. Identification operators 4. Second Quantization in the Category of Z2 -Hilbert Spaces 4.1. Fock spaces
∗ Most
192 195 195 196 197 197 198 198 199 199 199 199 201 201 202 202 202 203 204 205 206 206 207 207 208 208 210 210 211 211
of this work was done while the author visited Centre de Math´ematiques, Ecole Polytechnique, 91128 Palaiseau Cedex, France. 191 Reviews in Mathematical Physics, Vol. 10, No. 2 (1998) 191–233 c World Scientific Publishing Company
´ J. DEREZINSKI
192
4.2. Operators on Fock spaces – functor Γ 4.3. Operators on Fock spaces – dΓ 4.4. Fock space as an associative algebra 4.5. Creation and annihilation operators 4.6. Operators on Fock spaces – dΓ(q) 4.7. “Third quantization” of a Z2 -Hilbert space 4.8. Fock space over a direct sum I 4.9. Fock space over a direct sum II 4.10. “Third quantization” of a Fock space 4.11. “Third quantization” of a direct sum 4.12. Algebras C∞ (Γ(χ)) 5. Identical Particles 5.1. Elementary particles 5.2. Asymptotic particles 5.3. Identification operators 5.4. Asymptotic velocity 5.5. Short-range case 5.6. Long-range case – free region 5.7. Long-range case – asymptotic interacting Hamiltonian 5.8. Long-range case – modified wave operators 5.9. Distinguishable particles revisited 5.10. Embedding identical particles into distinguishable particles 5.11. Existence and completeness of wave operators
213 213 214 214 215 215 216 216 217 218 218 219 219 222 224 224 225 226 227 228 229 230 231
0. Introduction Two basic formalisms are used in quantum mechanics to describe many-body systems. The first assumes that particles are distinguishable and the full Hilbert space is the tensor product of one-particle spaces. The second, which is physically more correct and is one of the basic ingredients of quantum field theory, assumes that particles are divided into species and within each species they are identical. The space of N -particle states is the symmetric (for bosons) or antisymmetric (for fermions) N th power of the one-particle space. One of the areas of non-relativistic quantum mechanics where there exists a very satisfactory mathematical understanding of physical phenomena is N -body scattering theory. Using the formalism of distinguishable particles it has been shown that wave operators exist and are complete for an arbitrary number of particles and a very large class of potentials [5, 13, 7, 16, 3, 4]. This article describes the construction of wave operators and their asymptotic completeness in the case of identical particles. We will be interested in the following class of Hamiltonians: XZ 1 a∗e (xe ) D2 ae (xe )dxe H= 2me e e∈E
+
X Z Z
vee0 (xe − xe0 )a∗e (xe )a∗e0 (xe0 )ae0 (xe0 )ae (xe )dxe dxe0 .
(0.1)
e,e0 ∈E
Here the index e ∈ E parametrizes various species of “elementary particles”, xe denotes the position of the eth particle, De its momentum, me its mass, a∗e (xe ), ae (xe ) are its creation and annihilation operators. The operators a∗e (xe ), ae (xe ) satisfy the
193
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
canonical commutation/anticommutation relations depending on whether the eth particle is bosonic or fermionic. The operator H acts on the physical Fock space which is the tensor product of (bosonic or fermionic) Fock spaces describing all the species of particles. We will also introduce the asymptotic free Hamiltonian X Z 1 as ∗ 2 an (xn ) D + µn an (xn )dxn . K = 2mn n as n∈E
Here the index n = (ne )e∈E parametrizes all possible composite particles — clusters P ne me is the mass of the cluster, formed of ne particles of the sth species, mn := e∈E
xn is its center of mass position, Dn is its momentum and µn describes its internal energies (which are the appropriate eigenvalues of H red – the full Hamiltonian with a removed center of mass motion). The operator K as acts on the asymptotic Fock space Γ(Has ) space, which is the tensor product of Fock spaces describing various possible composite particles. Note that the Hamiltonian K as has a much simpler structure than H – it is a second quantized one-body Hamiltonian. On the other hand, in general, Γ(Has ) is a much bigger Hilbert space than Γ(H) – composite particles are usually much more numerous than the elementary particles. Moreover, we will need to construct a certain natural identification operator J as that maps the asymptotic Fock space into the physical Fock space. The main result of this paper in the short-range case says that the wave operator Ω+ := s− lim eitH J as e−itK
as
t→∞
(0.2)
exists and is a unitary operator from Γ(Has ) to Γ(H). (The unitarity of Ω+ is usually called asymptotic completeness.) The analogous results in the long-range case (which includes the Coulomb potentials) are a little more complicated. One has two basic choices of describing the asymptotics of e−itH . The first possibility is to introduce the so-called interacting asymptotic Hamiltonian, which has the form X 1 as ∗ 2 an (xn ) D + µn an (xn )dxn H = 2mn n as n∈E
+
X
Z Z
as ∗ ∗ vnn 0 (xn − xn0 )an (xn )an0 (xn0 )an0 (xn0 )an (xn )dxn dxn0 .
n,n0 ∈E as
Then one can show the existence and completeness of the wave operator Ω+ := s− lim eitH J as e−itH 1as,free,+ , as
t→∞
(0.3)
where 1as,free,+ is a certain naturally defined projection onto the states that are asymptotically free for the Hamiltonian H as as t → ∞.
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´ J. DEREZINSKI
Alternatively, one can show the existence and completeness of modified wave operators Ωmod,+ := s− lim eitH J as e−itK t→∞
as
−iGas (t)
,
(0.4)
where Gas (t) is a certain time-dependent modifier (the N -body version of the Dollard modifier). Scattering in quantum field theory (including asymptotic completeness) is described in many physics books. It is a conceptual basis for constructing measurable quantities such as scattering cross-sections. It is usually presented in a somewhat heuristic manner, which is partly justified by the scarcity of rigorous results on this subject. We think that it is good to know that Hamiltonians of the form (0.1) possess a very satisfactory, complete and nontrivial mathematical scattering theory. Hamiltonians of the form (0.1) appear quite often in physics texts. They are used to describe systems of interacting non-relativistic particles. We think that it would be justified to call (0.1) “the standard Hamiltonian of non-relativistic matter”. Hamiltonians of the form (0.1) have some special features that distinguish them among other quantum-field-theoretical Hamiltonians. One of these features is the covariance with respect to Galilean transformations. Another special feature is the fact that they conserve the number of particles of each species. The proof of the existence and completeness of (0.2), (0.3) and (0.4) consists of two parts. In the first part, which is much more difficult, one proves the existence and completeness of wave operators for distinguishable particles. This was done in [5, 13, 7, 16, 3]. Our standard reference for these results is [4]. This paper is devoted to the second part of the proof, in which one shows how the results on distinguishable particles imply an analogous result on identical particles. All the functional-analytic difficulties are solved in the first part of the proof. The difficulties of the second part reside in the combinatorics and in the notation. The results of this paper have a rather intuitive physical content. It seems that in some form they belong to the folklore of quantum physics [8]. They are related to the Haag–Ruelle theory, which in the case of relativistic quantum field theory gives a framework for scattering theory (see e.g. [6] and references therein). A related result describing the location of the essential spectrum for identical particles was considered in [12, 2]. Similar models where considered in [10, 9]. Nevertheless, we are not aware of any rigorous description of the results of our paper in the literature. The formalism of quantum field theory (Fock spaces, field operators) is in principle adapted to treat problems with a variable number of particles. One might think that this formalism is not necessary to study particle-conserving Hamiltonians, such as (0.1). Nevertheless, this formalism turns out to be very convenient to formulate results on scattering for identical particles. Usually one defines separately a bosonic Fock space and a fermionic Fock space. In order to present our results in a concise way we found it convenient to define from the very beginning a mixed Fock space, whose one-particle subspace is the direct sum of bosonic and fermionic one-particle spaces.
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
195
We found it also convenient to endow a Fock space with a product, which we denote by 2. (A similar product was considered e.g. in [14]). This product is associative and is closely related to the usual creation operators. The Fock space over a one-particle space H, denoted Γ(H), is sometimes called the second quantization of H (although, admittedly, this name is not very logical, see [15]). In order to construct the asymptotic Fock space we will have to perform the second quantization twice, obtaining “the Fock space over a Fock space over H”, denoted Γtq (H) := Γ(Γ(H)) – “the third quantization of H”. (The name “third quantization” is probably even less appropriate than “second quantization”, but we did not find a better one). We will also need a certain natural map J tq from Γtq (H) to Γ(H). In its definition it is convenient to use the product 2. The map J tq is used to define the identification operator J as that appears in (0.2), (0.3) and (0.4). If one wants to describe results on N -body systems, one is tempted to introduce lengthy and cumbersome notation involving long sequences with lots of indices. We did our best to avoid it. In fact, we made an effort to describe our results in a possibly simple and short way. This desire forced us to reconsider some basic notions related to quantum field theory. In our paper we give an essentially selfcontained introduction to basic definitions of quantum field theory that we are using. Although most of this material is well known (see e.g. [11, vol II, 6, 1]), we introduce some objects that, it seems, are not commonly known and might be useful outside of our work. This is especially the case of the above-mentioned product 2 and the map J tq . Let us describe how our paper is organized. In Sec. 1 we fix some notation concerning tensor products of Hilbert spaces and Cartesian products of sets. In Sec. 2 we describe theorems on the existence and the completeness of wave operators for distinguishable particles. For a proof of these theorems we refer the reader to [4] and references therein. This section does not present new results as compared with e.g. [4]. Nevertheless, the formulation of the results is different. In [3] and [4] the language of generalized N -body Schr¨ odinger operators is used, which stresses the geometric aspects of the problem. In Sec. 2 we use a more physical (although more narrow) formalism of (distinguishable) particles. Various general constructions related to second quantization are described in Secs. 3 and 4. In Sec. 5 we describe and prove the main result of the paper, that is the existence and completeness of wave operators for identical particles. 1. General Definitions 1.1. Hilbert spaces Let H be a Hilbert space. The scalar product of φ, ψ ∈ H will be denoted by (φ|ψ). The set of bounded operators from H1 to H2 is denoted B(H1 , H2 ). We set B(H) := B(H, H). If A = (A1 , . . . , Ap ) is a vector of commuting self-adjoint operators and K is a Borel subset of Rp , then 1K (A) denotes the spectral projection of A onto K.
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If {Hi }∞ i=1 is a family of Hilbert spaces, then we define the algebraic direct sum ∞
⊕ Hi ,
(1.5)
fin,i=1
consisting of finite linear combinations and the usual direct sum of Hilbert spaces ∞
⊕ Hi ,
i=1
which is the closure of (1.5). 1.2. Homomorphisms of commutative C ∗ - and W ∗ -algebras In quantum mechanics we often need to use certain families of commuting observables. Usually, this is described using a certain number of commuting selfadjoint operators. It is however more flexible and appropriate to use to this end the language of commutative C ∗ - and W ∗ -algebras. In this section we recall some basic facts about this subject and describe the notation that we will use. If X is a locally compact space, then C∞ (X ) denotes the commutative C ∗ algebra of functions on X vanishing at ∞. Likewise, B ∞ (X ) denotes the W ∗ -algebra of bounded Baire functions on X (see Vol. I of [11]). Let H be a Hilbert space. Let C∞ (X ) 3 f → γ(f ) ∈ B(H)
(1.6)
be a ∗-homomorphism. Clearly, (1.6) has a unique extension to a normal ∗-homomorphism on B ∞ (X ), which we will denote by the same symbol B ∞ (X ) 3 f → γ(f ) ∈ B(H) . One says that γ is nondegenerate if γ(1) equals the identity in B(H). If X = Rp , then the homomorphism γ uniquely determines a vector of commuting self-adjoint operators A = (A1 , . . . , Ap ) such that γ(f ) = f (A) .
(1.7)
γ is nondegenerate iff A is densely defined. Note that we will sometimes use the notation of the rhs of (1.7), that is f (A), even if X is not Rp for any p ∈ N, and we cannot use the interpretation in terms of a vector of commuting self-adjoint operators. Suppose that Xi , i = 1, 2, are locally compact spaces and we are given ∗homomorphisms with commuting images, which are denoted by two notations C∞ (Xi ) 3 f 7→ γi (f ) = f (Ai ) ∈ B(H) ,
i = 1, 2 .
Then we have a natural ∗-homomorphism from C∞ (X1 × X2 ), which can be denoted in two fashions: C∞ (X1 × X2 ) 3 F 7→ γ1 ⊗ γ2 (F ) = F (A1 , A2 ) ∈ B(H) .
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
197
Suppose that E is a set and we are given a family of Hilbert spaces {He }e∈E . Let H = ⊕ He . e∈E
Equip E with the discrete topology. Then we have a natural ∗-homomorphism e) ∈ B(H) C∞ (E) 3 f 7→ f (ˆ defined as f (ˆ e) := ⊕ f (e) . e∈E
(For a space E with a discrete topology C∞ (E) denotes the set of complex-valued functions on E such that f ∈ C∞ (E) if for any > 0 there exists a finite subset E ⊂ E and |f | < outside E .) 1.3. Permutations If p ∈ N, then Sp denotes the group of permutations of {1, . . . , p}. If X1 , . . . , Xp are sets, then we define the map p
p
q=1
q=1
θ(σ) : × Xq → × Xσq by setting θ(σ)(x1 , . . . , xp ) := (xσ−1 1 , . . . , xσ−1 p ) . In particular, if X1 = . . . = Xp , then we obtain an action of Sp in X p . Let H1 , . . . , Hp be Hilbert spaces and σ ∈ Sp . Then we define the map p
p
q=1
q=1
Θ(σ) : ⊗ Hq → ⊗ Hσq by setting Θ(σ)(φ1 ⊗ · · · ⊗ φp ) := φσ−1 1 ⊗ · · · ⊗ φσ−1 p . In particular, if H = H1 = . . . = Hp , then we obtain an action of the group Sp on H⊗p . 1.4. Clusters Let J be a (finite or infinite) set. Anticipating the next section we will call its elements “particles”. Let C denote the family of finite subsets of J (the family of clusters). Every j ∈ J we identify with {j} ∈ C. If p ∈ N, then we set Cp := {c ∈ C : |c| = p} . (|c| denotes the number of elements of c). Clearly, G Cp , C= p∈N
where
F
denotes the disjoint union.
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Suppose that we fix a total order in c ∈ Cp (denoted <). Then we have a unique order-preserving bijection π : {1, . . . , p} → c (1.8) called the numbering of c. From now on we assume that a total order in J , denoted <, has been fixed. This fixes a total order in every c ∈ C, which will be denoted by the same symbol <. The corresponding map (1.8) will be denoted πc . 1.5. Cartesian and tensor products associated with a cluster Suppose that with every j ∈ J we associate a certain set Xj and a certain Hilbert space Hj . We define × Xj := Xπ(1) × · · · × Xπ(p) ,
⊗ Hj := Hπ(1) ⊗ · · · ⊗ Hπ(p) .
j∈c
(1.9)
j∈c
We will often use the shorthand notation Xc := × Xj ,
Hc := ⊗ Hj .
j∈c
j∈c
1.6. Clusterings Let A denote the family of finite subsets of C such that a ∈ A and b, b0 ∈ a implies b = b0 or b ∩ b0 = ∅. Elements of A are called clusterings. We identify c ∈ C with {c} ∈ A. We define also cfree := ∪ {{j}} ∈ C , j∈c
(all clusters in cfree are one-element). For any c ∈ C we set Ac := a ∈ A : c = ∪ b . b∈a
In other words, a ∈ Ac if a is a partition of c into disjoint subsets. The elements of Ac are called clusterings of c. Clearly, G Ac . A= c∈C
In every a ∈ A we fix a total order, denoted <. Let p ∈ N, c ∈ Cp and a ∈ Ac . The orders in c and in a determine a certain new total order
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The corresponding bijection (1.8) will be denoted πa . Define σa ∈ Sp σa := πa−1 πc . 1.7. Cartesian and tensor products associated with a clustering We have
b∈a
× Xj
× Xb = ×
b∈a
⊗ Hb = ⊗
b∈a
= Xπa (1) × · · · × Xπa (p) .
j∈b
⊗ Hj
b∈a
j∈b
= Hπa (1) ⊗ · · · ⊗ Hπa (p) ,
We will often use the shorthand notation Xa := × Xb ,
Ha := ⊗ Hb .
b∈a
b∈a
Suppose that c ∈ C and a ∈ Ac . Then we have natural identifications θ(σa ) : Xa → Xc ,
Θ(σa ) : Ha → Hc .
(1.10)
2. Distinguishable Particles 2.1. Elementary particles Suppose that the set J labels all the particles. If j ∈ J , then mj ∈ R+ denotes the mass of the jth particle, Xj ' Rp denotes its configuration space, xj ∈ Xj its position, Dj = 1i ∇xj is its momentum, hj is the finite dimensional Hilbert space of its internal degrees of freedom. In the case of Coulomb systems, zj is its charge. In Xj , we have two Euclidean structures, the basic metric x2j , and the “kinetic metric” mj x2j . We set Hj := L2 (Xj ) ⊗ hj . 2.2. Clusters of elementary particles Recall that in Subsec. 1.4 for any c ∈ C we defined the Euclidean space Xc and the Hilbert space Hc . Xc is equipped with the kinetic metric X mj x2j . (2.11) j∈c
Clearly, we have the natural identification Hc = L2 (Xc ) ⊗ ⊗ hj .
(2.12)
j∈c
xc := (xj )j∈c and Dc := (Dj )j∈c denote the position and the momentum operators of cluster c.
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The center-of-mass subspace of Xc is defined as Xccm := {xc ∈ Xc : xj1 = xj2 , j1 , j2 ∈ c} . The space of internal coordinates of cluster c ∈ C is X Xcred := (Xccm )⊥ = xc ∈ Xc : mj xj = 0 , j∈c
where ⊥ denotes the orthogonal complement wrt the kinetic metric. Clearly, Xc = Xccm ⊕ Xcred . We set mc :=
X
mj ,
zc :=
j∈c
X
zj .
j∈c
We define the following operators on Hc describing the center-of-mass position, the center-of-mass momentum and the kinetic energy: xcm c :=
1 X mj xj , mc j∈c Kc :=
Dccm :=
X
Dj ,
j∈c
X 1 Dj2 . 2m j j∈c
Note that Xcred = {xc ∈ Xc : xcm c = 0} ,
Dccm :=
1 ∇xcm . i c
We set Hcred := L2 (Xcred ) ⊗ ⊗ hj . j∈c
We have Hc ' L2 (Xccm ) ⊗ Hcred .
(2.13)
With respect to the decomposition (2.13) we can write: Kc =
1 (Dcm )2 ⊗ 1 + 1 ⊗ Kcred . 2mc c
If j, j 0 ∈ J and x ∈ X, then vjj0 (x) ∈ B(hj ⊗hj 0 ) is a self-adjoint operator called the potential of interaction of the jth and j 0 th particle. We assume that vjj (x) = 0 and (2.14) vjj0 (x)(1 + D2 )−1 is compact on L2 (X) ⊗ hj 0 ⊗ hj 0 . On Hc we define the potential Vc :=
1 X vjj0 (xj − xj 0 ) , 2 0 j,j ∈c
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Clearly, with respect to the decomposition (2.13) we can write: Vc = 1 ⊗ Vcred . Note that Vc is relatively bounded wrt Kc with an infinitesimal bound. Likewise, Vcred is relatively bounded wrt Kcred with an infinitesimal bound. We define the full Hamiltonian and the reduced Hamiltonian as the self-adjoint Hamiltonians on Hc and Hcred respectively: Hc := K + V ,
Hcred := K red + V red .
Clearly, Hc =
1 (Dcm )2 ⊗ 1 + 1 ⊗ Hcred . 2mc c
We also set Xcfree := {xc ∈ Xc : xj 6= xj 0 , j 6= j 0 , j, j 0 ∈ c} . 2.3. Asymptotic particles For c ∈ C we denote Hcas := Ran1pp (1 ⊗ Hcred ) ⊂ Hc ,
pp red red has c := Ran1 (Hc ) ⊂ Hc .
as cm cm acting on Hcas . We will also write Xcas instead We will write xas c , Dc for xc , Dc of Xccm . Clearly, Hcas = L2 (Xcas ) ⊗ has c .
Kcas := Hc
We set
Has c
red µas := H c c
,
has c
.
Clearly, Kcas =
1 (Das )2 ⊗ 1 + 1 ⊗ µas c . 2mc c
2.4. Clusters of asymptotic particles Let a ∈ A. We define Xaas := × Xbas ,
Haas := ⊗ Hbas ,
b∈a
b∈a
and a self-adjoint operator acting on Haas : X Kbas . Kaas := b∈a
xas a
(xas b )b∈a
Daas
:= and operators acting on Haas . We set
:=
(Dbas )b∈a
denote the position and the momentum
as as as 0 0 Xaas,free := {xas a ∈ Xa : xb 6= xb0 , b 6= b , b, b ∈ a} .
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2.5. Identification operator Let c ∈ C and a ∈ Ac . Recall that in Subsec. 1.7 we defined the Hilbert space Ha and the vector space Xa . We have Haas ⊂ Ha and Xaas ⊂ Xa . Now we define the identification operators jaas : Xcas → Xc , by setting
jaas := θ(σa )
, as
Xa
Jaas : Haas → Hc , Jaas := Θ(σa )
Has a
.
2.6. Existence of the asymptotic velocity The asymptotic velocity is a natural asymptotic observable that exists under very weak assumptions on the decay of potentials. It has a natural physical meaning, and it is useful in the proof of the asymptotic completeness of wave operators (for a much more restrictive class of potentials). Theorem 2.1. For every j, j 0 ∈ J assume |x| 2 −1 (1 + D2 )−1 ∈ L1 (dR) . (1 + D ) ∇x vjj0 (x)1[1,∞[ R Let c ∈ C. Then for every F ∈ C∞ (Xc ) there exists the limit x c e−itHc . s− lim eitHc F t→∞ t
(2.15)
(2.16)
There exists a unique vector of densely defined self-adjoint operators Pc+ such that the limit (2.16) equals F (Pc+ ). Moreover, we have [Pc+ , Hc ] = 0 , 1Xccm (Pc+ ) = Hcas . If Vc = 0, then Pc+ = Dc . 2.7. Short-range case Let us formulate the main result of N -body scattering theory for distinguishable particles in the short-range case. Theorem 2.2. For every j, j 0 ∈ J , assume |x| 2 −1 (1 + D2 )−1/2 ∈ L1 (dR) . (1 + D ) vjj0 (x)1[1,∞[ R
(2.17)
Let c ∈ C. Then for any a ∈ Ac , there exists itHc as −itKa Ω+ Ja e . a := s− lim e as
t→∞
(2.18)
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as Ω+ a is an isometry from Ha to Hc and as + Ω+ a K a = Hc Ω a , as as + + Ω+ a ja (Da ) = Pc Ωa .
Moreover, Hc = ⊕ RanΩ+ a . a∈Ac
2.8. Long-range case
free region
The remaining part of this chapter is devoted to wave operators in the long range case. From now on will assume that for any j, j 0 ∈ J s l vjj0 (x) = vjj 0 (x) + vjj 0 (x) , l s where vjj 0 (x) ∈ R and vjj 0 (x) ∈ B(hj ⊗ hj 0 ) (the long-range part of the potential is scalar in the internal variables). l In the case of vjj 0 (x) we will consider two types of assumptions. Either we will assume that it is a Coulomb potential l vjj 0 (x) :=
zj zj 0 , |x|
or we will give more general decay conditions. The following theorem gives a description of the scattering in the asymptotically free region. Note that the decay assumptions on the potentials are quite weak. Theorem 2.3. (1) For any j, j 0 ∈ J assume that |x| 2 −1 s (1 + D2 )−1/2 ∈ L1 (dR) , (1 + D ) vjj0 (x)1[1,∞[ R l lim vjj 0 (x) = 0 ,
|x|→∞
l ∇nx vjj 0 (x)1[1,∞[
(2.19)
|x| R
1−n
∈ hRi
L1 ([1, ∞[, dR) ,
n = 1, 2 .
Then we can find a function [0, ∞[×Xc 3 (t, ξc ) → Sc (t, ξc ) ∈ R such that there exists := s− lim eitHc e−itSc (t,Dc ) . Ωfree,+ c t→∞
The operator
Ωfree,+ c
is an isometry from Haas to Hc and it satisfies Kc = Hc Ωfree,+ , Ωfree,+ c c Dc = Pc+ Ωfree,+ , Ωfree,+ c c = Ran1free,+ , RanΩfree,+ c c
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where := 1Xcfree (Pc+ ) . 1free,+ c (2) If we replace the third line of the assumption (2.19) by |x| 1/2 l ∇x vjj hRi (x)1 ∈ L1 ([1, ∞[, dR) , 0 [1,∞[ R then we can use the so-called Dollard modifier: X S(t, ξc ) := (2mj )−1 ξj2 +
j,j 0 ∈c,
j∈c
where
X
Z
t l vjj 0
gjj0 (t) := 0
gjj0 (t)
j6=j 0
0 ξj 0 ξj 0 t dt0 . −t mj mj 0
(3) In the Coulomb case we can set zj zj 0 log t . gjj0 (t) := D 0 D mjj − mjj0 l (4) In the short-range case, that is if vjj 0 (x) = 0, we can set
gjj0 (t) = 0 . Then we simply obtain the usual free region wave operator: Ωfree,+ = Ω+ , c cfree where Ω+ is the wave operator for the free channel defined in Theorem 2.2. cfree 2.9. Long-range case
asymptotic interacting Hamiltonian
The asymptotics of e−itH on the whole physical space can be described in two basic ways. In this subsection we compare it with the evolution generated by a certain interacting asymptotic Hamiltonian H as on the subspace of its asymptotically free states. In the next subsection we will compare it with an evolution generated by the free asymtotic Hamiltonian K as with a time-dependent modifier. For disjoint b, b0 ∈ C we set X l l vbb vjj 0 (x) := 0 (x) . j∈b, j 0 ∈b0
In the Coulomb case we obtain l vbb 0 (x) =
Then we set as operators on Haas : Vaas :=
X
zb zb0 . |x|
l as as vbb 0 (xb − xb0 ) .
b,b0 ∈a b6=b0
Haas := Kaas + Vaas .
205
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
√ 3 − 1 and for any j, j 0 ∈ J |x| 2 −1 s ∈ L1 (dR) , (1 + D ) vjj0 (x)1[1,∞[ R
Theorem 2.4. Let µ =
l lim vjj 0 (x) = 0 ,
|x|→∞
l ∇x vjj 0 (x)1[1,∞[
(2.20)
|x| R
µ
hRi ∈ L1 ([1, ∞[, dR) .
Let c ∈ C and a ∈ Ac . Let Paas,+ be the asymptotic velocity defined as in Theorem 2.2 for the Hamiltonian Haas . Set 1aas,free,+ := 1Xaas,free (Paas,+ ) . Then there exists itHc as −itHa as,free,+ Ja e 1a . Ω+ a := s− lim e as
t→∞
as,free,+ ⊂ Haas to Hc satisfying Moreover, Ω+ a is an isometry from Ran1a as + Ω+ a Ha = Hc Ωa , as as,+ ) = Pc+ Ω+ Ω+ a ja (Pa a
and Hc = ⊕ RanΩ+ a a∈Ac
2.10. Long-range case
modified wave operators
In this subsection we present the second, more conventional approach to the long-range N -body scattering. Let us define Dollard-type modifiers. For disjoint b, b0 ∈ C we set: Z t Das Das0 l t0 b − t0 b dt0 , vbb (2.21) gbb0 (t) := 0 mb mb0 0 l where vbb 0 (x) was defined in (2.21). In the case of Coulomb potentials we set
zb zb0 log t . gbb0 (t) := Das Das0 mbb − mbb0 The Dollard modifier equals Gas a (t) :=
X
gbb0 (t) .
b,b0 ∈a b6=b0
Theorem 2.5. Suppose that the assumptions of Theorem 2.4 hold. Let c ∈ C and a ∈ Ac . Then there exists
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s− lim eitHc Jaas e−itKa
as
−iGas a (t)
t→∞
=: Ωmod,+ . a
(2.22)
is an isometry from Haas to Hc , Ωmod,+ a Kaas = Hc Ωmod,+ , Ωmod,+ a a jaas (Daas ) = Pc+ Ωmod,+ Ωmod,+ a a and Hc = ⊕ RanΩmod,+ . a a∈Ac
Note that Theorem 2.3 applied to Haas and Theorem 2.4 imply immediately Theorem 2.5. Note also that the wave operator Ωfree,+ of Theorem 2.3 constructed with the c of Theorem 2.5. Dollard modifier coincides with the wave operator Ωmod,+ cfree 3. Second Quantization in the Category of Sets 3.1. Second quantization of a set In Sec. 4 we will consider the well-known functor Γ, which to every Hilbert space associates the Fock space Γ(H). Γ is sometimes called the “second quantization”. An analogous (simpler but apparently less known) functor can be defined in the category of sets. We will denote it by the same letter Γ. Let E be a set. Anticipating Sec. 5 we will call the elements of E “species”. We define Γ(E) to be the set of sequences n = (ne )e∈E with values in N with all but a finite number of elements vanishing. If p ∈ N, then we set ( Γp (E) :=
n ∈ Γ(E) :
X
) ne = p
.
e∈E
Clearly, Γ(E) =
G
Γp (E) .
(3.23)
p∈N
Clearly, for any p ∈ N the group Sp acts on the Cartesian power E p . We have the natural isomorphism E p /Sp ' Γp (E) ,
(3.24)
where E p /Sp denotes the set of orbits. (3.23) and (3.24) should be compared with the definition of the Fock space (4.30). If e is a generic notation for an element of E, then a generic element of Γ(E) will be sometimes denoted e Note that if |E| = 1, then Γ(E) ' N. If f : E1 → E2 is a map, then we have a natural map Γ(f ) : Γ(E1 ) → Γ(E2 ).
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3.2. “Third quantization” of a set Γ(Γ(E)) will be denoted for shortnesss Γtq (E), where tq stands for “third quantizaton”. It is the family of sequences k = (km )m∈Γ(E) with values in N with all but a finite number of elements vanishing. If n ∈ Γ(E), then we set X k ∈ Γtq (E) : n = km m . Γtq n (E) := m∈Γ(E)
Clearly, Γtq (E) =
G
Γtq n (E) .
n∈Γ(E)
We define the map j tq : Γtq (E) → Γ(E) by setting j tq (k) := n if k ∈ Γn (E). 3.3. Permutations that preserve species Suppose that the set E is equipped with a total order <. Let p ∈ N and n ∈ Γp (E). We define δn : {1, . . . , p} → E by the following conditions: −1 δn {e} = ne ,
(1)
e∈E;
(2) 1 ≤ q < q 0 ≤ p ⇒ δn q ≤ δn q 0 . Let σ ∈ Sp . We say that σ ∈ Sn if σ δn−1 {e} = δn−1 {e} ,
e∈E.
Clearly, Sn is a subgroup of S p . Note that Sn ' × Sne , e∈E
|Sn | = Π ne ! =: n! .
(3.25)
e∈E
Treating δn as a “labelling” of numbers in {1, . . . , p} with “species” we can say that Sn is the subgroup of permutations that preserve species. Assume that {Xe }, e ∈ E, is a family of sets and {He }, e ∈ E, is a family of Hilbert spaces. Then for p ∈ N and n ∈ Γp (E), we define p
Xn := × Xene = × Xδn q , e∈E
q=1
p
Hn := ⊗ He⊗ne = ⊗ Hδn q . e∈E
q=1
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3.4. Clusters with composition n ∈ Γ (E) Now suppose that J is a set and we fix a map J → E. In other words, we choose a partition of J labelled by elements of E: G Je . (3.26) J = e∈E
Recall from Subsec. 1.4 that C denotes the family of clusters. Let n ∈ Γ(E). Then we set Cn := {c ∈ C : |c ∩ Je | = ne , e ∈ E} . (Cn is the family of clusters with composition n.) Clearly, G Cn . C= n∈Γ(E)
We fix a total order < in J . We can assume that, for e, e0 ∈ E, j ∈ Je , j 0 ∈ Je0 , e < e0 ⇒ j < j 0 . Recall from Subsec. 1.4, that we have the map πc : {1, . . . , p} → c . Note that if c ∈ Cn , then πc q ∈ Je ⇔ δn q = e . Suppose now that {Xj }, j ∈ J , is a family of sets and {Hj }, j ∈ J , is a family of Hilbert spaces. Recall that for any c ∈ C we defined in Subsec. 1.5 the set Xc and the Hilbert space Hc . Assume now that we have Xj = Xe ,
Hj = He ,
j ∈ Je ,
(3.27)
Let n ∈ Γ(E) and c ∈ Cn . Then Xc = Xn ,
Hc = Hn .
3.5. Permutations that preserve species and a clustering Let p ∈ N, and let A˜p denote the family of clusterings (partitions into disjoint subsets) of {1, . . . , p}. The group Sp acts on A˜p . We denote by σα ∈ A˜p the action of σ ∈ Sp on α ∈ A˜p . First note that given an order < in E, we can endow Γ(E) and Γ(E) × N with certain natural total orders. In fact, for n, n0 ∈ Γ(E), we will write n < n0 if for all e ∈ E ne0 = n0e0 ,
e0 < e ⇒ ne ≤ n0e .
For (m, i), (m0 , i0 ) ∈ Γ(E) × N, we will write (m, i) < (m0 , i0 ) if m < m0
or
(m = m0
and i < i0 ) .
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
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Let p ∈ N, n ∈ Γp (E), k ∈ Γtq n (E). Next we would like to define a certain natural clustering αk ∈ A˜p and a labelling δk : {1, . . . , p} → E. First we define a map δ˜k : {1, . . . , p} → Γ(E) × N by the following three conditions: (1) 1 ≤ q < q 0 ≤ p ⇒ δ˜k q ≤ δ˜k q 0 , S m × {1, . . . , km } , (2) δ˜k {1, . . . , p} = m∈Γ(E)
(3) if q ∈ N , Let
m ∈ Γq (E) ,
then
˜−1 δk (m × {i}) = q .
n αk := δ˜k−1 (m × {i}) : m ∈ Γ(E) ,
1 ≤ i ≤ km
o
be the clustering of {1, . . . , p} associated with k. Next we define δk : {1, . . . , p} → E as follows. Let q ∈ N, m ∈ Γq (E) and 0 ≤ i ≤ km Then for some 0 ≤ q0 ≤ p − q we have δ˜−1 (m × {i}) = {q0 + 1, . . . , q0 + q} ∈ αk . k
We set δk (q0 + s) := δm s ,
s = 1, . . . , q ,
where δm was defined in Subsec. 3.3. This ends the definition of δk . Let σ ∈ Sp . We say that σ ∈ Sk if the following two conditions hold: (1) σ δk−1 {e} = δk−1 {e} ,
e∈E,
(2) σαk = αk . Clearly, Sk is a subgroup of Sp . Note that Sk '
× S me
×
e∈E
m∈Γ(E)
|Sk | =
Π
m∈Γ(E)
km × Skm ,
km km ! =: k!tq . Π me !
(3.28)
e∈E
Sk has the interpretation of the subgroup of permutations that do not change species given by the labelling δk and do not change the clustering αk . Note that in general Sk is not a subgroup of Sn , because the labellings δn and δk are in general different.
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Assume again that {Xe }, e ∈ E, is a family of sets and {He }, e ∈ E, is a family of Hilbert spaces. We set Xk :=
×
m∈Γ(E)
km ×
Xeme
⊗
He⊗me
e∈E
Hk :=
⊗
m∈Γ(E)
e∈E
p
= × Xδk q , q=1
⊗km
p
= ⊗ Hδk q . q=1
Note that if σ ∈ Sk , then θ(σ) acts on Xk and Θ(σ) acts on Hk . 3.6. Clusterings associated with k ∈ Γtq (E) Suppose that J is a set satisfying (3.26). Let A denote the family of clusterings (see Subsec. 1.6). If k ∈ Γtq (E), then we define Ak := {a ∈ A : |{b ∈ a : b ∈ Cm }| = km , m ∈ Γ(E)} (clusterings containing exactly km clusters of composition m for each m ∈ Γ(E)). Clearly, G Ak . A= k∈Γtq (E)
Recall that E and Γ(E) are endowed with total orders. We assume that every a ∈ A is endowed with a total order < satisfying b, b0 ∈ a, n, n0 ∈ Γ(E), b ∈ Cn , b0 ∈ Cn0 n < n0 ⇒ b < b0 . Recall that in Subsec. 1.6 we defined a map πa : {1, . . . , p} → c . Note that if a ∈ Ak , then πa q ∈ Je ⇔ δk q = e . Suppose now that {Xj }, j ∈ J , is a family of sets and {Hj }, j ∈ J , is a family of Hilbert spaces. Recall that for any a ∈ A we defined in Subsec. 1.7 the set Xa and the Hilbert space Ha . If we assume (3.27) and a ∈ Ak , then we have Xa = Xk ,
Ha = Hk .
3.7. Identification operators Let p ∈ N. In this subsection we concentrate our attention on the “cluster” {1, . . . , p} and its clusterings. We will introduce some notation, which will essentially mimick notation of Subsecs. 1.6 and 3.6 adapted to the case of {1, . . . , p}.
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Let C˜p denote the family of subsets of {1, . . . , p}. Let n ∈ Γp (E), m ∈ Γ(E) and n if γ ∈ C˜p . We write γ ∈ C˜m γ ∩ δ −1 {e} = me , e ∈ E . n Recall that A˜p denotes the family of clusterings of {1, . . . , p}. If k ∈ Γtq n (E) and ˜ ˜ α ∈ Ap , then we write α ∈ Ak if n α ∩ C˜m = km . Clearly,
G
A˜p =
A˜k .
k∈Γtq n (E)
˜ Let p ∈ N, n ∈ Γp (E) and k ∈ Γtq n (E). In any α ∈ Ap we fix a total order < such that γ, γ 0 ∈ α ,
m, m0 ∈ Γ(E) ,
n γ ∈ Cm ,
n γ 0 ∈ Cm 0 ,
m < m0 ⇒ γ < γ 0 .
As in Subsec. 1.6 we then define πα ∈ Sp . We define also σα ∈ Sp by setting σα := πα−1 . Note the following facts: ˜ Proposition 3.1. Let p ∈ N, n ∈ Γp (E), k ∈ Γtq n (E) and α ∈ Ak . Then (0) σα (δk−1 {e}) = δn−1 {e}, e ∈ E. (1) σα αk = α. (2) The map −1 σσα , σα) ∈ Sk × A˜k Sn 3 σ 7→ (σσα is bijective. is a subgroup of Sn fixing the clustering α0 . (3) Let α0 ∈ A˜k . Then σα0 Sk σα−1 0 ∈ Sn and the map For any α ∈ A˜k we have σα σα−1 0 ∈ Sn /σα0 Sk σα−1 A˜k 3 α 7→ σα σα−1 0 0 is bijective. 4. Second Quantization in the Category of Z2 -Hilbert Spaces 4.1. Fock spaces A pair (H, ) where H is a Hilbert space and is a unitary involution in H is called a Z2 -Hilbert space. We define the bosonic and fermionic part of H: Hb = ker( − 1) , Clearly, H = Hb ⊕ Hf .
Hf := ker( + 1) .
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Let (H1 , 1 ), (H2 , 2 ) be Z2 -Hilbert spaces. Then a map A : H1 → H2 is said to be Z2 -linear if A1 = 2 A . Let p ∈ N. We introduce a twisted representation of the group of permutations Sp on H⊗p . Let σ ∈ Sp and I ⊂ {1, . . . , p} with |I| = q. Let π1 be the unique order preserving bijection of {1, . . . , q} onto I and π2 be the unique order preserving bijection of {1, . . . , q} onto σ(I). We define a permutation σI in Sq by setting σI := π2−1 σπ1 . If φi ∈ Hf for i ∈ I and φi ∈ Hb for i ∈ {1, . . . , p}\I, then Θ (σ)φ1 ⊗ · · · ⊗ φp := sgn(σI )Θ(σ)φ1 ⊗ · · · ⊗ φp . Note that if = 1, then Θ (σ) = Θ(σ) and if = −1, then Θ (σ) = sgn(σ)Θ(σ). We also set (4.29) sgn (σ) := Θ (σ)Θ(σ −1 ) . We define the projections Pp :=
1 X Θ (σ) , p! σ∈Sp
and the Hilbert spaces Γp (H) = Pp H⊗p ,
∞
Γ(H) = ⊕ Γp (H) .
(4.30)
p=0
Γ(H) is called the Fock space over (H, ). We set Γfin :=
∞
⊕
fin,p=0
Γp (H) .
Obviously, C = Γ0 (H). The vector 1 ∈ C will be denoted by ω and will be called the vacuum. Clearly, we have a natural embedding H = Γ1 (H) ⊂ Γ(H) . Let (H1 , 1 ), (H2 , 2 ) be Z2 -Hilbert spaces. Then so is (H1 ⊕ H2 , 1 ⊕ 2 ). We have a natural isomorhism Γ(H1 ⊕ H2 ) ' Γ(H1 ) ⊗ Γ(H2 ) . (See Subsec. 4.8.) In particular, Γ(H) ' Γb (Hb ) ⊗ Γf (Hf ) , where Γb (Hb ) is the bosonic Fock space over Hb and Γf (Hf ) is the fermionic Fock space over Hf .
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
4.2. Operators on Fock spaces
213
functor Γ
If A : H1 → H2 is a Z2 -linear map, then we define the operator Γp (A) : Γp (H1 ) → Γp (H2 ) by setting
Γp (A) := A⊗p
.
Γp (H1 )
We also define Γ(A) : Γ(H1 ) → Γ(H2 ) as Γ(A) := ⊕ Γp (A) . p∈N
Γ(A)
Note that
H1
= A.
Note that if A1 : H1 → H2 , A2 : H2 → H3 , then Γ(A2 )Γ(A1 ) = Γ(A2 A1 ) . One of important examples of the use of the functor Γ is the statistics operator Γ(). 4.3. Operators on Fock spaces
dΓ
If B is a Z2 -linear operator on H, then we define dΓp (B) : Γp (H) → Γp (H) as dΓp (B) :=
p X
1⊗q−1 ⊗ B ⊗ 1⊗p−q
Γp (H)
q=1
We also define dΓ(B) : Γ(H) → Γ(H) as
∞
dΓ(B) := ⊕ dΓp (B) . p=0
Note that dΓ(A)
H
= A,
[dΓ(A), dΓ(B)] = dΓ([A, B]) , edΓ(A) = Γ eA .
.
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4.4. Fock space as an associative algebra For φi ∈ Γpi (H), i = 1, 2, we define the product s (p1 + p2 )! Pp1 +p2 φ1 ⊗ φ2 . φ1 2φ2 := p1 !p2 ! We extend Using
2 to the whole Γfin(H) by linearity. Pp1 +p2 Pp2 ⊗ 1⊗p1 = Pp2 +p1 ,
we easily see that the operation
2 is associative.
Remark 4.1. Note that, in the case of a fermionic Fock space, the product is different from the usual exterior product ∧. In fact, φ1 ∧ φ2 :=
2
(p1 + p2 )! Pp1 +p2 φ1 ⊗ φ2 . p1 !p2 !
Let us note the following formula for a multiple 2-product. Let φi ∈ Γri (H), Pq i = 1, . . . , q, p = i=1 ri . Then s q q p! 2 Pp ⊗ φi . φi = q i=1 Πi=1 ri ! i=1 4.5. Creation and annihilation operators For φ ∈ H = Γ1 (H) we define the creation and annihilation operators a∗ (φ)ψ := φ2ψ ,
ψ ∈ Γfin (H) ,
a(φ) := (a∗ (φ))∗ . They satisfy the well-known commutation/anticommutation relations a(φ1 )a∗ (φ2 ) − a∗ (φ2 )a(φ1 ) = (φ1 |φ2 ) , a(φ1 )a(φ2 ) − a(φ2 )a(φ1 ) = 0 , a(φ1 )a(φ2 ) − a(φ2 )a(φ1 ) = 0 ,
φ1 , φ2 ∈ H .
One can also introduce the so-called multiple creation/annihilation operators, by setting for φ ∈ Γfin (H) a∗ (φ)ψ := φ2ψ ,
ψ ∈ Γfin (H) ,
Remark 4.2. In the literature one usually assumes that the one-particle space is H = L2 (X, dx), where X is a space with a measure dx. One introduces “operator valued measures” a∗ (x), a(x) and for φ ∈ H one writes Z Z a∗ (x)φ(x)dx = a∗ (φ) , a(x)φ(x)dx = a(φ) .
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
215
Then if φ ∈ Γp (H), ψ ∈ Γ(H), we have Z φ(x1 , . . . , xp )a∗ (x1 ) · · · a∗ (xp )ψdx1 . . . dxp . φ2ψ = (p!)−1/2 Xp
4.6. Operators on Fock spaces
dΓ(q)
Let q ∈ N and B ∈ B(Γq (H)). We define dΓ(q)p (B) : Γp (H) → Γp (H) by setting
dΓ(q)p (B) :=
0,
p! Pp B ⊗ 1⊗p−q (p−q)!
Γp (H)
p < q, ,
p ≥ q.
We also define dΓ(q) (B) : Γ(H) → Γ(H) as
∞
dΓ(q) (B) := ⊕ dΓ(q)p (B) . p=0
Clearly, if B is a Z2 -linear operator on H, then dΓ(1) (B) = dΓ(B). Remark 4.3. If B ∈ B(Γq (L2 (X, dx))) and the integral kernel of B equals b(x1 , . . . , xq , x01 , . . . , x0q ) , then
Z dΓ(q) (B) := X 2q ∗
b(x1 , . . . , xq , x01 , . . . , x0q )
× a (x1 ) . . . a∗ (xq2 )a(x0q1 ) . . . a(x01 )dx01 . . . dx0q1 dx1 . . . dxq2 . 4.7. “Third quantization” of a Z2 -Hilbert space Let (H, ) be a Z2 -Hilbert space. Let Γ(H) be the corresponding Fock space with the vacuum ω and the product 2. Clearly, (Γ(H), Γ()) is also a Z2 -Hilbert space. So we may define “the third quantization” of H, that is Γtq (H) := Γ(Γ(H)). Its vacuum will be denoted ω tq and its product 2tq . We define a map J tq : Γtq (H) → Γ(H) by setting J tq φ1 2tq . . . 2tq φq := φ1 2 . . . 2φq ,
φ1 , . . . , φq ∈ Γ(H) .
The above formula defines J tq on a total set in Γq (Γ(H)). It can be extended by linearity to a bounded operator on Γq (Γ(H)). Then it can be extended to an unbounded operator on a dense domain in Γtq (H).
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An alternative definition of J tq will be given in Subsec. 4.10. 4.8. Fock space over a direct sum I Let (H, ) be a Z2 -Hilbert space. Let Γ(H) be the corresponding Fock space with the vacuum ω and the product 2. Let E be a totally ordered set. Suppose that (He , e ), e ∈ E, are Z2 -Hilbert spaces. Let Γ(He ) be the corresponding Fock spaces with the vacua ωe and the product 2e . We assume that H = ⊕ He ,
= ⊕ e .
e∈E
(4.31)
e∈E
We would like to discuss a certain natural unitary operator U : ⊗ Γ(He ) → Γ(H) .
(4.32)
e∈E
If E is infinite, then the tensor product on the left of (4.32) is understood as the so-called restricted tensor product associated with the sequence of vectors ωe ∈ He . For a sequence φe ∈ Γ(He ), e ∈ E, with all but a finite number of φe ’s different from ωe , the operator U is defined as U ⊗ φe := e∈E
2
e∈E
φe .
4.9. Fock space over a direct sum II Next we would like to describe an equivalent characterization of the map U , which does not use the 2-product. Let p ∈ N and n ∈ Γp (E). We define Γn (H) := ⊗ Γne (He ) . e∈E
We define an operator acting on H⊗p : 1 X Θ (σ) . Pn := n! σ∈Sn
where n! was defined in 3.25. Clearly, Pn is a projection and Pn Hn = Γn (H) , where Hn was defined in Subsec. 3.3. Clearly, ⊗ Γ(He ) =
e∈E
⊕
n∈Γ(E)
For any p ∈ N and n ∈ Γ(E), we have s r p! n! P = = U n! p Γn (H) p! Γn (H)
Γn (H) .
X [σ]∈Sp /Sn
Θ (σ)
, Γn (H)
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
217
where we take a single representant in each class in Sp /Sn . We also have ⊕
U
n∈Γp (E)
Γn (H) = Γp (H) .
We will often use the identifications Γ(H) '
⊕
Γn (H) ,
n∈Γ(E)
Γp (H) '
⊕
Γn (H) .
n∈Γp (E)
Using these identifications we can write the following formula for the 2-product. ˜ Let n ∈ Γ(E), k ∈ Γtq n (E). Choose an arbitrary clustering α ∈ Ak and vectors φm,i ∈ Γ(H), i = 1, . . . , km , m ∈ Γ(E). Then
2 2 φm,i = m∈Γ(E) i=1 km
s Π
e∈E
km ne ! Pn Θ (σα ) ⊗ ⊗ φm,i . k m Πm∈Γ(E) (me !) m∈Γ(E) i=1
For σ ∈ Sp we define sgnn : Hn → Hn by setting
sgnn (σ) := sgn (σ)
Hn
.
Clearly, if we assume that e = ±1, e ∈ E, then sgnn (σ) = ±1. 4.10. “Third quantization” of a Fock space Next we would like to give an equivalent alternative characterization of the map J tq that does not use the 2-product. For r ∈ Γ(N) we set Γtq r (H) := ⊗ Γrp (Γp (H)) . p∈N
We have Γ(H) = ⊕ Γp (H) . p∈N
Hence by Subsec. 4.9 with E = N, we have a natural identification Γtq (H) '
⊕
r∈Γ(N)
Γtq r (H) .
Let p ∈ N and r ∈ Γp (N). Set Jrtq := J tq
Γtq r (H)
.
Then Jrtq : Γtq r (H) → Γp (H)
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and we have the identity r Jrtq
:=
p! P = r!tq p Γtq r (H)
s
r!tq X Θ (σα ) tq . p! Γr (H) ˜ α∈Ar
where p! is the usual factorial, r tq Π (q!) q rq ! =: r! ,
q∈N
and A˜r is the family of clusterings of {1, . . . , p} having rq q-element clusters. 4.11. “Third quantization” of a direct sum We return to the notation and assumptions of Subsec. 4.9. Let k ∈ Γtq (E). We define ⊗ (H) := ⊗ Γ (Γ (H)) = ⊗ Γ Γ (H ) . Γtq km m km me e k m∈Γ(E)
e∈E
m∈Γ(E)
Set Pk :=
1 X Θ (σ) . k!tq σ∈Sk
Clearly, Pk Hk = Γtq k (H) . We have the identifications: Γ(H) '
⊕
Γn (H) ,
n∈Γ(E)
Γtq (H) '
⊕
k∈Γtq (E)
Γtq k (H) .
Fix n ∈ Γ(E) and k ∈ Γtq n (E). Set
Jktq := J tq
Γtq (H) k
,
where J tq was defined in Subsec. 4.7. Then Jktq : Γtq k (H) → Γn (H) r
and Jktq
=
r n! k!tq X P Θ (σ ) = Θ (σ ) , tq α α 0 n k!tq n! Γtq (H) Γk (H) k ˜
(4.33)
α∈Ak
where n! was defined in 3.25 and k!tq was defined in 3.28 and we have chosen an ˜ arbitrary clustering α0 ∈ A. 4.12. Algebras C∞ (Γ(X )) Let X be a locally compact set. Then so is X p , Γp (X ) and Γ(X ). Let H be a Hilbert space. Let C∞ (X ) 3 f → γ(f ) ∈ B(H)
(4.34)
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
219
be a ∗-representation. For any p ∈ N this representation generates a natural representation from C∞ (X p ) into B(H⊗p ). It is covariant wrt the action of Sp on X p . Hence we obtain a representation Γp (γ) : C∞ (Γp (X )) → B(Γp (H)) . This leads to a representation Γ(γ) : C∞ (Γ(X )) → B(Γ(H))
(4.35)
defined as Γ(γ) := ⊕ Γp (γ) . p∈N
If instead of the notation involving γ we use the notation f (A) = γ(f ) ,
f ∈ C∞ (X ) ,
then Γ(γ) will be denoted F (A) = Γ(γ)(F ) ,
F ∈ C∞ (Γ(X )) .
5. Identical Particles 5.1. Elementary particles Let X be a finite dimensional Euclidean space — the configuration space of a single particle. x denotes the position operator in L2 (X) and D := 1i ∇x the momentum. Let (h, ) be a Z2 -Hilbert space, describing various kinds of particles. Let m be a Z2 -linear strictly positive operator on h, called the one-particle mass operator. In the case of Coulomb systems, let z be a Z2 -linear self-adjoint operator on h, called the charge operator. We assume that [m, z] = 0 . The full one-particle space is defined as H := L2 (X) ⊗ h . For any x ∈ X, let v(x) be a Z2 -linear operator on h ⊗ h. If τ ∈ S2 is a transposition, then we assume that τ v(x)τ = v(−x) , [v(x), m ⊗ 1] = [v(x), z ⊗ 1] = 0 . The operator v(x ⊗ 1 − 1 ⊗ x) acts on H ⊗ H and Γ2 (H) is its invariant subspace. It can be used to define the two-particle potential, denoted also v, as the linear operator . v : Γ2 (H) → Γ2 (H) , v := v(x ⊗ 1 − 1 ⊗ x) Γ2 (H)
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Suppose that e ∈ E labels the species of particles. For convenience we fix a total order in E. Let he be the Hilbert space describing the internal degrees of freedom of the particles of the eth species. Let me ∈ R+ be its mass, e = ±1 its statistics and He := L2 (X) ⊗ he its one-particle Hilbert space. In the case of Coulomb systems, let ze ∈ R be its charge. We asume that h = ⊕ he ,
and hence ,
e∈E
H = ⊕ He . e∈E
With respect to these decompositions we can write m = ⊕ me ,
= ⊕ e , e∈E
e∈E
v(x) =
z = ⊕ ze . e∈E
⊕ vee0 (x)
e,e0 ∈E
where for every e, e0 ∈ E we assume that vee0 (x)(1 + D2 )−1 is compact on L2 (X) ⊗ he ⊗ he0 .
(5.36)
Γ(H) is the physical Fock space. The full kinetic energy is defined as the operator on Γ(H): 2 D . K := dΓ 2m The full potential is defined as V := dΓ(2) (v) . V is relatively bounded wrt K with an infinitesimal bound. The Hamiltonian is defined as the self-adjoint operator on Γ(H) H := K + V . On Γ(H) Γ0 (H) (the orthogonal complement of the vacuum) we define the operators of the center-of-mass momentum and the center-of-mass position: Dcm := dΓ(D) ,
xcm := (dΓ(m))
−1
dΓ(xm) .
We have [xcm , Dcm ] = i1 . Hence, for X cm = X, Γ(H) Γ0 (H) ' L2 (X cm ) ⊗ Γ(H)red and the operators xcm , Dcm act only on L2 (X cm). We have [xcm , K] = (dΓ(m))−1 Dcm , [Dcm , K] = 0 . Hence K − (2dΓ(m))−1 (Dcm )2 commutes with xcm and Dcm . Therefore, on Γ(H) Γ0 (H)
221
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
K = (2dΓ(m))−1 (Dcm )2 + 1 ⊗ K red . V commutes with xcm and Dcm . Hence on Γ(H) Γ0 (H) V = 1 ⊗ V red . We define the reduced Hamiltonian as the self-adjoint operator on Γ(H)red H red := K red + V red . Obviously, on Γ(H) Γ0 (H) H = (2dΓ(m))−1 (Dcm )2 + 1 ⊗ H red . Note that [H, 1 ⊗ H red ] = 0 . Let us define the free region in Γ(X): Γ(X)free := {(nx )x∈X ∈ Γ(X) : nx ∈ {0, 1}, x ∈ X} , (all the particles are at different points) and the center-of-mass region in Γ(X): Γ(X)cm := {(nx )x∈X ∈ Γ(X) : |{x : nx > 0}| = 1} , (all the particles are at the same point). We set X := X × E. We have the obvious map X 3 (x, e) 7→ x ∈ X . It induces the map Γ(X ) → Γ(X) . free
cm
(5.37) free
cm
We define Γ(X ) and Γ(X ) as the preimages of Γ(X) and Γ(X) respect to (5.37). Note that we have the natural identifications Γ(X)cm ' X × N ,
with
Γ(X )cm ' X × Γ(E) .
The following ∗-homomorphisms have an obvious definition (see Subsec. 1.2): C∞ (E) 3 f 7→ f (ˆ e) := ⊕ f (e) ∈ B(h) , e∈E
C∞ (X) 3 f 7→ f (x) ∈ B(L2 (X)) , C∞ (X) 3 f 7→ f (D) ∈ B(L2 (X)) . We also have the tensor products of these homomorphisms: C∞ (X ) 3 f → f (x, eˆ) ∈ B(H) , C∞ (X ) 3 f → f (D, eˆ) ∈ B(H) .
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Sometimes we will use their second quantized extensions (see Subsec. 4.12): C∞ (Γ(X )) 3 F → F (x, eˆ) ∈ B(Γ(H)) , C∞ (Γ(X )) 3 F → F (D, eˆ) ∈ B(Γ(H)) . Recall that in Subsec. 4.8 we defined Γn (H) := ⊗ Γne (He ) . e∈E
Clearly, we can introduce Γn (H)red , by writing Γn (H) = L2 (X cm) ⊗ Γn (H)red . We have the identifications Γ(H) '
⊕
n∈Γ(E)
Γ(H)red '
Γn (H) ,
⊕
n∈Γ(E)
Γn (H)red .
Using these identifications we can decompose the mass, charge and statistics operators, as well as the Hamiltonians H and H red : dΓ(m) =
⊕
n∈Γ(E)
mn ,
H=
⊕
dΓ(z) =
⊕
n∈Γ(E)
H red =
Hn ,
n∈Γ(E)
zn , ⊕
Γ() =
⊕
n∈Γ(E)
n∈Γ(E)
n ,
Hnred .
5.2. Asymptotic particles Define the asymptotic one-particle space and the reduced asymptotic oneparticle space: Has := Ran1pp (1 ⊗ H red ) ⊂ Γ(H) ,
has := Ran1pp (H red ) ⊂ Γ(H)red . instead of xcm as , Dcm as , dΓ(m)
We will write xas , Das , mas , z as , as H dΓ(z) as , Γ() as . We will write X as instead of X cm. Clearly, H
H
Has ' L2 (X as ) ⊗ has . Define
k as := H
has
,
µas := H red
has
.
Clearly, k as = (2mas )−1 (Das )2 + 1 ⊗ µas . The family of asymptotic particles is defined as E as := {n ∈ Γ(E) : Has ∩ Γn (H) 6= {0}} .
H
Has
,
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
223
For n ∈ E as we set as Γn (E as ) := Γtq n (E) ∩ Γ(E ) .
We set Hnas := Has ∩ Γn (H) ,
as red has . n := h ∩ Γn (H)
Clearly, we can write the decompositions Has ' ⊕ Hnas ,
has ' ⊕ has n .
n∈E as
n∈E as
We can decompose the asymptotic one-particle operators as follows: mas = ⊕ mn , n∈E as
z as = ⊕ zn , n∈E as
k as = ⊕ kn ,
as = ⊕ n . n∈E as
µas = ⊕ µn .
n∈E as
n∈E as
The asymptotic Fock space is Γ(Has ). The 2-product in Γ(Has ) is denoted 2as ; creation and annihilation operators are denoted aas∗ (φ), aas (φ) for φ ∈ Has . The asymptotic Hamiltonian is K as := dΓ(k as ) . We define X as := X as × E as . We define Γ(X as )free similarly, as Γ(X as )free with X as , X as , E as replacing X, X , E. The following ∗-homomorphisms have an obvious definition eas ) := ⊕ f (n) ∈ B(has ) , C∞ (E as ) 3 f 7→ f (ˆ n∈E as
C∞ (X as ) 3 f 7→ f (xas ) ∈ B(L2 (X as )) , C∞ (X as ) 3 f 7→ f (Das ) ∈ B(L2 (X as )) . We also have the tensor products of these homomorphisms: C∞ (X as ) 3 f → f (xas , eˆas ) ∈ B(Has ) , C∞ (X as ) 3 f → f (Das , eˆas ) ∈ B(Has ) . Sometimes we will use their second quantized extensions C∞ (Γ(X as )) 3 F → F (xas , eˆas ) ∈ B(Γ(Has )) , C∞ (Γ(X as )) 3 F → F (Das , eˆas ) ∈ B(Γ(Has )) . For k ∈ Γ(E as ) we set Γk (Has ) :=
as ⊗ Γkm (Hm ).
m∈E as
Clearly, we have the decomposition Γ(Has ) '
⊕
k∈Γ(E as )
Γk (Has ) .
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We can decompose various operators, for example K as '
⊕
k∈Γ(E as )
Kkas .
5.3. Identification operators Recall that in Subsec. 4.7 we defined the maps J tq : Γtq (H) → Γ(H) ,
j tq : Γtq (X ) → Γ(X ) .
Recall that Has ⊂ Γ(H) and X as ⊂ Γ(X ). Hence Γ(Has ) ⊂ Γtq (H), Γ(X as ) ⊂ Γtq (X ) and we can define the identification operators J as : Γ(Has ) → Γ(H) , J as := J tq
as
Γ(Has )
,
j as : Γ(X as ) → Γ(X ) j as := j tq
Γ(X as )
.
tq as Let n ∈ E as and k ∈ Γn (E as ). Recall that Γk (Has ) ⊂ Γtq k (H)Γk (X ) ⊂ Γk (X ). We set tq tq as as Jkas := J as = J , j := j = j . k k k as as as as Γk (H )
Γk (H )
Γk (X
)
Γk (X
)
Clearly, Jkas : Γk (Has ) → Γn (H) ,
jkas : Γk (X as ) → Γn (X ) .
5.4. Asymptotic velocity Theorem 5.1. For every e, e0 ∈ E assume |x| (1 + D2 )−1 ∈ L1 (dR) . (1 + D2 )−1 ∇x vee0 (x)1[1,∞[ R Then for every F ∈ C∞ (Γ(X )) there exists the limit x itH , eˆ e−itH . s− lim e F t→∞ t The limit (5.39) defines a nondegenerate homomorphism denoted by C∞ (Γ(X )) 3 F 7→ F (P + ) ∈ B(Γ(H)) . We have [F (P + ), H] = 0 , 1Γ(X )cm (P + ) = Has . If V = 0, then F (P + ) = F (D, eˆ).
(5.38)
(5.39)
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5.5. Short-range case Let us state the main result of our paper in the short-range case. Theorem 5.2. For every e, e0 ∈ E assume |x| 2 −1 0 (1 + D2 )−1/2 ∈ L1 (dR) . (1 + D ) vee (x)1[1,∞[ R
(5.40)
Then Ω+ := s− lim eitH J as e−itK
as
t→∞
exists and is unitary from Γ(Has ) to Γ(H). It satisfies Ω+ K as = HΩ+ ,
(5.41)
F (P + )Ω+ = Ω+ F ◦ j as (Das , eˆas ) ,
(5.42)
for F ∈ C∞ (Γ(X )). The proof of this theorem will be given in Subsecs. 5.9, 5.10 and 5.11. Remark 5.3. It is possible and instructive to describe the result of the above theorem using creation/annihilation operators, in the spirit of the Haag–Ruelle theory. Let φ ∈ Has ∩ Γfin (H). Then for any ψ ∈ Γfin (H) there exist lim eitH a∗ (e−itH φ)e−itH ψ =: a+∗ (φ)ψ ,
t→∞
lim eitH a(e−itH φ)e−itH ψ =: a+ (φ)ψ ,
t→∞
(where a∗ (φ), a(φ) denote multiple creation/annihillation operators). We have a+ (φ) = Ω+ aas (φ)Ω+∗ a+∗ (φ) = Ω+ aas∗ (φ)Ω+∗ . Hence [a+ (φ1 ), a+∗ (φ2 )] = [a+ (φ1 ), a+∗ (φ2 )] = 0 , [a+ (φ1 ), a+∗ (φ2 )] = (φ2 |φ1 ) , a(φ)ω = 0 ,
φ1 , φ2 ∈ Has ,
φ1 , φ2 ∈ Has ,
φ ∈ Has .
Moreover, the set a+∗ (φ1 ) · · · a+∗ (φq )ω , spans the whole Γ(H).
φ1 , . . . , φq ∈ Has ,
q = 0, 1, . . .
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226
5.6. Long-range case
free region
From now on in the long-range case we will always assume that, for any e, e0 ∈ E, s l vee0 (x) = vee 0 (x) + vee0 (x) , l s vee 0 (x) ∈ R and vee0 (x) ∈ B(he ⊗ he0 ). We will set as operators on h ⊗ h
v s (x) :=
s ⊕ vee 0 (x) ,
e,e0 ∈E
v l (x) :=
l ⊕ vee 0 (x)
e,e0 ∈E
and as operators on Γ2 (H): v s := v s (x ⊗ 1 − 1 ⊗ x)
, Γ2 (H)
v l := v l (x ⊗ 1 − 1 ⊗ x)
. Γ2 (H)
l In the case of vee 0 (x) we will consider two types of assumptions. Either we will assume that it is the Coulomb potential l vee 0 (x) :=
in other words vl =
ze ze0 , |x|
z⊗z , |x ⊗ 1 − 1 ⊗ x| Γ2 (H)
or we will give more general decay conditions. The following theorem gives a description of scattering in the asymptotically free region. Note that the decay conditions on potentials are quite general. Theorem 5.4. (1) For any e, e0 ∈ E, assume that |x| 2 −1 s ∈ L1 (dR) , (1 + D ) vee0 (x)1[1,∞[ R l lim vee 0 (x) = 0 ,
|x|→∞
l ∇nx vee 0 (x)1[1,∞[
(5.43)
|x| R
∈ hRi1−p L1 ([1, ∞[, dR) ,
p = 1, 2 .
Then we can find a real function [0, ∞[×Γ(X ) 3 (t, ξ, e) → S(t, ξ, e) such that there exists Ωfree,+ := s− lim eitH e−itS(t,D,ˆe) . t→∞
The operator Ω
free,+
is an isometry from Γ(Has ) to Γ(H) and it satisfies Ωfree,+ K = HΩfree,+ , Ωfree,+ F (D, eˆ) = F (P + )Ωfree,+ , RanΩfree,+ = Ran1free,+ ,
where 1free,+ := 1Γ(X )free (P + ) .
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
227
(2) If we replace the third line of assumption (5.43) by |x| 1/2 l hRi ∈ L1 ([1, ∞[, dR) , ∇x vee0 (x)1[1,∞[ R then we can set S(t, D) := dΓ((2m)−1 D2 ) + dΓ(2) (g(t)) , D D v l t0 ⊗ 1 − t0 1 ⊗ dt0 . m m Γ2 (H) 0 (3) In the Coulomb case we can set z ⊗ z log t . g(t) := D D ⊗ 1 − 1 ⊗ Γ2 (H) m m
where
Z
t
g(t) =
(4) In the short-range case, that is under the condition (5.40) we can set g(t) = 0 . 5.7. Long-range case
asymptotic interacting Hamiltonian
Before we describe scattering for identical particles in the long-range case we need to introduce some additional definitions. For any n, n0 ∈ Γ(E) we set X l l ne n0e0 vee vnn 0 (x) = 0 (x) . e,e0 ∈E
We have has ⊗ has =
⊕ has n n,n0 ∈E as as as
⊗ has n0 .
We define the asymptotic potential on h ⊗ h v as (x) :=
⊕
n,n0 ∈E as
l vnn 0 (x) .
In the case of Coulomb potentials we set zn zn0 l , vnn 0 (x) = |x| and hence v as (x) :=
z⊗z . |x|
Now on Γ2 (Has ) we can set
v as := v as (x ⊗ 1 − 1 ⊗ x)
Γ2 (Has )
.
We define the second-quantized asymptotic potential and asymptotic Hamiltonian as operators on Γ(Has ): V as := dΓ(2) (v as ) , H as := K as + V as .
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228
√ 3 − 1 and for any e, e0 ∈ E, |x| s ∈ L1 (dR) , (1 + D2 )−1 vee 0 (x)1[1,∞[ R
Theorem 5.5. Let µ =
l lim vee 0 (x) = 0 ,
|x|→∞
l ∇x vee 0 (x)1[1,∞[
|x| R
(5.44) µ
hRi ∈ L1 ([1, ∞[, dR) .
Let P as,+ be the asymptotic velocity and 1as,free,+ be the projection on the asymptotically free states defined using the Hamiltonian H as . Then there exists Ω+ := s− lim eitH J as e−itH 1as,free,+ . as
t→∞
Moreover, Ω+ is unitary from Ran1as,free,+ ⊂ Γ(Has ) to Γ(H) and Ω+ H as = HΩ+ , Ω+ F ◦ j as (P as,+ ) = F (P + )Ω+ . Remark 5.6. Note that in the short-range case H as = K as ,
P as,+ = Das , eˆas ,
1as,free,+ = 1 ,
and Theorem 5.5 reduces to Theorem 5.2. 5.8. Long-range case
modified wave operators
Define the operator on Has ⊗ Has : Z t as Das as as 0D 0 t as ⊗ 1 − t 1 ⊗ as g (t) := v dt0 . m m Γ2 (H) 0 In the Coulomb case set
as as z ⊗ z log t g as (t) := Das as as ⊗ 1 − 1 ⊗ D as m m
. Γ2 (H)
The second quantized Dollard modifier equals Gas (t) := dΓ(2) (g as (t)) . Theorem 5.7. Suppose that the assumptions of Theorem 5.5 hold. Then there exists s− lim eitH J as e−itK t→∞
as
−iGas (t)
=: Ωmod,+ ,
Ωmod,+ is a unitary operator from Γ(Has ) to Γ(H) and Ωmod,+ K as = HΩmod,+ .
(5.45)
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
229
Ωmod,+ F ◦ j as (Das , eˆas ) = F (P + )Ωmod,+ . Clearly, Theorem 5.4 applied to H as and Theorem 5.5 imply Theorem 5.7. 5.9. Distinguishable particles revisited This subsection serves as a preparation for the proof of Theorems 5.5 and 5.7. In this subsection we recall and rephrase constructions and the results of Sec. 2 on distinguishable particles in a convenient manner. Our starting point is a system of bosonic and fermionic particles described in Subsecs. 5.1 and 5.2 labelled by E. We set J := E × N, we define the set of clusters C and of clusterings A as in Sec. 1. For (e, i) ∈ J we set h(e,i) := he , X(e,i) := Xe , m(e,i) := me , etc. For (e, i), (e0 , i0 ) ∈ J we set v(e,i)(e0 ,i0 ) (x) := vee0 (x). Using the formalism of Sec. 2 we introduce a system of distinguishable particles labelled by the set J = E × N with the above defined internal Hilbert spaces, configuration spaces, masses, potentials, etc. Let n ∈ Γ(E). Recall that in Subsec. 3.3 we defined the Hilbert space Hn := ⊗ He⊗ne . e∈E
Recall also that Hn equals Hc for all c ∈ Cn . The Hamiltonian Hc , defined in Subsec. 2.2 for c ∈ C, acts on Hc . In particular, if c ∈ Cn , it acts on Hn . The Hamiltonian Hc does not depend on c as long as ˜ n . It is a self-adjoint operator acting on Hn . Clearly, c ∈ Cn . It will be denoted H it satisfies ˜ n ] = 0 , σ ∈ Sn . ˜ n ] = [Θ (σ), H [Θ(σ), H ˜ n to avoid confusion with Hn defined in Subsec. 5.1.) (We use the tilde in H Let n ∈ Γ(E) and k ∈ Γtq (E). In Subsec. 3.5 we defined Hk :=
⊗
(Hm )⊗km .
m∈Γ(E)
It equals Ha for a ∈ Ak . Define also the Hilbert space Hkas :=
as ⊗km ⊗ (Hm ) .
m∈E as
It equals Haas for a ∈ Ak . The Hamiltonian Kaas , defined in Subsec. 2.3 for a ∈ A, acts on Haas . It does ˜ as . K ˜ as acts on the Hilbert not depend on a as long as a ∈ Ak . We denote it by K k k as space Hk . We have ˜ kas ] = 0 , ˜ kas ] = [Θ (σ), K [Θ(σ), K
σ ∈ Sk .
˜ as to avoid confusion with the Hamiltonian Hk defined in (We use the tilde in K k Subsec. 5.2.)
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230
˜ Let n ∈ Γ(E). k ∈ Γtq n (E) and α ∈ Ak . We have the maps
Jαas
Θ(σα ) : Hk → Hn := Θ(σα ) as : Hkas → Hn . Hk
The following theorem follows immediately from Theorem 2.2. Theorem 5.8. Let n ∈ E as , k ∈ Γ(E as ). Then for any α ∈ A˜k there exists ˜
˜ as
itHn as −itKk Jα e . Ω+ α := s− lim e t→∞
Moreover, Hn =
⊕
⊕ RanΩ+ α .
˜k k∈Γn (E as ) α∈A
Let us note also the following property of wave operators: Lemma 5.9. Let n ∈ E as , k ∈ Γ(E as ), α ∈ A˜k and σ ∈ Sn . Then + −1 Θ(σ)Ω+ α = Ωσα Θ(σσα σσα ) .
Proof. Note the following facts: ˜n. (1) σ ∈ Sn , hence Θ(σ) commutes with H −1 −1 ˜ as . (2) σσα σσα ∈ Sk , hence Θ(σσα σσα ) commutes with K k as as −1 (3) Θ(σ)Jα = Jσα Θ(σσα σσα ). Therefore, ˜
˜ as
itHn as −itKk Θ(σ)Ω+ Jα e α = s− lim Θ(σ)e t→∞
˜
˜ as
as −itKk −1 = s− lim eitHn Jσα e Θ(σσα σσα ) t→∞
=
−1 Ω+ σα Θ(σσα σσα ) .
5.10. Embedding identical particles into distinguishable particles Recall that, for any n ∈ Γ(E), we can embed the space Γn (H) in Hn : Γn (H) = Pn Hn . ˜ n and Moreover, Γn (H) is an invariant subspace for H ˜ n Hn = H . Γn (H)
Likewise, for k ∈ Γ(E as ), we can embed the space Γk (Has ) in Hkas : Γk (Has ) = Pk Hkas .
(5.46)
ASYMPTOTIC COMPLETENESS IN QUANTUM FIELD THEORY
231
˜ as and Moreover, Γk (Has ) is an invariant subspace for K k ˜ as Kkas = K . k as Γk (H )
Let n ∈ E
as
and k ∈ Γ(E ). Recall that the identification operator as
Jkas : Γk (Has ) → Γ(H) equals r Jkas
= r =
k!tq X Θ (σα ) n! Γk (Has ) ˜ α∈Ak
k!tq X sgnn (σα )Jαas . n! ˜ a∈Ak
We will show the following statement, which will imply Theorem 5.5: Theorem 5.10. Assume the hypotheses of Theorem 5.5. Let n ∈ E as and k ∈ Γ(E as ). Then as s− lim eitHn Jkas e−itKk =: Ω+ k t→∞
exists, is isometric and ⊕
k∈Γn (E as )
RanΩ+ k = Γn (H) .
We will also see that the wave operator for identical particles can be expressed in terms of wave operators for distinguishable particles as follows: r k!tq X + sgnn (σα )Ω+ Ωk = (5.47) α n! Γk (Has ) ˜ α∈Ak
5.11. Existence and completeness of wave operators In this subsection we are going to prove Theorem 5.10. Let us prove the existence of wave operators. Let φk ∈ Γk (Has ). We check that the limit r k!tq X ˜ ˜ as itHn as −itKkas Jk e φk = sgnn (σα )eitHn Jαas e−itK φk lim e t→∞ n! ˜ r =
α∈Ak
k!tq X sgnn (σα )Ω+ α φk n! ˜
(5.48)
α∈Ak
exists. The vectors in the above sum are orthogonal and have the same norm q + k!tq n! ˜ n! . Moreover, |Ak | = k!tq . Hence the norm of (5.48) is kφk k. Hence Ωk exists, is isometric and (5.47) is true.
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Let us prove now the asymptotic completeness. Let ψn ∈ Γn (H). By the asymptotic completeness (5.46), we know that there exists a unique family of vectors φ˜α , α ∈ A˜p , such that if k ∈ Γn (E as ) and α ∈ A˜k , then φ˜α ∈ Has and k
X
ψn =
˜ Ω+ α φα .
(5.49)
˜p α∈A
By setting φ˜α = sgnn (σα )φα , we can rewrite (5.49) as X
ψn =
sgnn (σα )Ω+ α φa .
˜p α∈A
Next we note that Pn ψn = ψn . Hence ψn =
X 1 X Θ (σ) sgnn (σα )Ω+ α φα n! ˜ α∈Ap
σ∈Sn
=
1 X X −1 sgnn (σσα )Ω+ σα Θ (σσα σσα )φα n! ˜ σ∈Sn α∈Ap
=
1 n!
k∈Γn (E as )
X
=
X
0 sgnn (σβ )Ω+ β Θ (σ )φα
˜k σ0 ∈Sk k∈Γn (E as ) β,α∈A
X
=
X
X
k!tq n!
X
sgnn (σβ )Ω+ β Pk ψα
˜k β,α∈A
Ω+ k φk ,
k∈Γn (E as )
r
where φk =
k!tq X Pk φα . n! ˜
(5.50)
α∈Ak
One may also remark that, using the uniqueness of φ˜α , we have Pk φα = φα ∈ Γk (Has ) and φα = φα0 for α, α0 ∈ A˜k . Therefore, we can rewrite (5.50) in a simpler fashion: for any α ∈ A˜k , we have r n! φα . φk = k!tq Acknowledgements I am very grateful to Prof. Laudenbach for his hospitality at Centre de Math´ematiques of Ecole Polytechnique. I also appreciate helpful discussions with my colleagues, especially with C. G´erard.
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This work is a part of the project Nr 2 P03A 029 08 financed in the years 1995–97 by Komitet Badan Naukowych. References [1] J. C. Baez, I. E. Segal and Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Univ. Press, Princeton, 1992. [2] P. Deift, W. Hunziker, B. Simon and E. Vock, “Pointwise bounds on eigenfunctions and wave packets in N -body quantum systems IV”, Commun. Math. Phys. 64 (1978) 1–34. [3] J. Derezi´ nski, “Asymptotic completeness for N −particle long-range quantum systems”, Ann. Math. 138 (1993) 427–476. [4] J. Derezi´ nski and C. G´erard, Asymptotic Completeness of N -Body Systems, Springer, 1996. [5] V. Enss, “Long-range scattering of two- and three-body quantum systems, Journ´ees Equations aux d´eriv´ees partielles”, Saint Jean de Monts, Juin 1989, Publications Ecole Polytechnique, Palaiseau, 1989. [6] J. Glimm and A. Jaffe, Quantum Physics, A Functional Integral Point of View, 2nd edition, 1987, Springer, New York, Berlin, Heidelberg. [7] G. M. Graf, “Asymptotic completeness for N −body short range quantum systems: A new proof”, Commun. Math. Phys. 132 (1990) 73–101. [8] G. M. Graf and W. Hunziker, oral communication. [9] C. J¨ akel, “Asymptotic triviality of Moeller operators in Galilei invariant quantum field theories”, Lett. Math. Phys. 21 (1991) 343–350. [10] H. Narnhofer and W. Thirring, “Galilei-invariant quantum field theories with pair interactions”, Int. J. Modern Phys. A6/17 (1991) 2937–2970. [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol I and II, 1976, vol. III, 1979, vol. IV, 1978, Academic Press, London. [12] A. G. Sigalov and I. M. Sigal, “Description of the spectrum of the energy operator of quantum mechanical systems that is invariant with respect to permutations of identical particles”, Theor. Math. Phys. 5 (1970) 990–1005. [13] I. M. Sigal and A. Soffer, “The N −particle scattering problem: asymptotic completeness for short-range quantum systems”, Ann. Math. 125 (1987) 35–108. [14] W. Slowikowski, “Commutative Wick algebras I”, Proc. Conf. on Vector Spaces, Measures and Applications, Dublin 1977, Lect. Notes in Math. 644, Springer, 1978. [15] S. Weinberg, “The Quantum Theory of Fields”, I, Cambridge Univ. Press, 1995. [16] D. R. Yafaev, “Radiation conditions and scattering theory for N -particle Hamiltonians”, Commun. Math. Phys. 154 (1993) 523–554
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II. VOLTERRA SYSTEMS AND SP(N) THEORIES CARLO MOROSI Dipartimento di Matematica Politecnico di Milano Piazza L. da Vinci 32 I-20133 Milano Italy
LIVIO PIZZOCCHERO Dipartimento di Matematica Universit` a di Milano Via C. Saldini 50 I-20133 Milano Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Milano Italy E-mail: [email protected] Received 26 October 1996 Revised 16 June 1997 A connection is suggested between the zero-spacing limit of a generalized N -fields Volterra (VN ) lattice and the KdV-type theory which is associated, in the Drinfeld–Sokolov classification, to the simple Lie algebra sp(N ). As a preliminary step, the results of the previous paper [1] are suitably reformulated and identified as the realization for N = 1 of the general scheme proposed here. Subsequently, the case N = 2 is analyzed in full detail; the infinitely many commuting vector fields of the V2 system (with their Hamiltonian structure and Lax formulation) are shown to give in the continuous limit the homologous sp(2) KdV objects, through conveniently specified operations of field rescaling and recombination. Finally, the case of arbitrary N is attacked, showing how to obtain the sp(N ) Lax operator from the continuous limit of the VN system.
1. Introduction In a previous paper [1] we have reconstructed some essential features of ordinary KdV theory (the Lax pairs, the biHamiltonian formulation, the hierarchy of conserved functionals and the associated vector fields, the one-soliton solutions) from the zero-spacing limit of the so-called Kac–Moerbeke (KM) system (in fact, we were unaware that the results presented therein on the vector fields and the Poisson structures had also been obtained in [2]). The mainstream in which these studies are inserted is the analysis of the procedures allowing to obtain the KdV-type theories from the zero-spacing limit of integrable Toda-type lattices. An introduction to the topic of integrable lattices
235 Reviews in Mathematical Physics, Vol. 10, No. 2 (1998) 235–270 c World Scientific Publishing Company
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and their continuous limits, together with some historical references, has been given in [1]; on this point, we take the occasion to cite also [3, 4]. The KM system considered in [1, 2] is a reduction of the Toda lattice: using the Flaschka coordinates (a, b) (where a = (a(i)) and b = (b(i)) are sequences, the index i running over the lattice sites) we can describe the KM flows as the restriction of (half of) the infinitely many commuting Toda flows to the submanifold b = 0. Of course, it would be desirable to extend as much as possible the above results to appropriate classes of lattices, recovering from the continuous limit of each class a family of KdV-type theories. In this paper, we pursue this program in connection with a series VN (N = 1, 2, . . . ) of integrable lattices, hereby denominated Volterra systems. We suggest a connection between the zero-spacing limit of these lattices and a class of KdV-type theories associated, in the Drinfeld–Sokolov classification [5], to the simple Lie algebras sp(N ) of the symplectic series.a The KM system is just the Volterra system V1 , whereas the ordinary KdV appears to be connected with the rank 1 Lie algebra sp(1) (in fact isomorphic to both algebras sl(2) and so(3), more frequently associated to this theory). In the present paper, we work out in full detail the V2 system, yielding the sp(2) KdV in the continuous limit. We also show how to obtain, for an arbitrary integer N , the Lax operator of the sp(N ) KdV theory from the zero-spacing limit of the VN Lax operator; this is a basic step for reconstructing the full scheme exploited in the cases N = 1, 2. Before going into technical details about the VN and sp(N ) theories, let us introduce the general ideas on which the limiting process is based. The first step in this framework is the interpolation of the lattice, i.e. the replacement of the sequences of phase space coordinates with smooth functions. For example, in the discrete version of KM theory a point of the phase space is a sequence (a(i)), indexed by the lattice sites; in the interpolated version, this sequence is replaced with a smooth function a of a continuous variable x. To help intuition, one can think the lattice sites to be arranged at the points xi = i, where is a fundamental spacing which will be sent ultimately to zero. We can interpolate in this way both infinite and periodic lattices, letting the x variable run over the real line R or the torus R/Z; in this paper, we will consider for definiteness the periodic case, but the results presented here could be adapted to the infinite case x ∈ R (with obvious technical adjustements, of the type described in [1]). In this framework, we are naturally led to consider the differential-difference operators acting on (conveniently defined spaces of) smooth functions of x. For each displacement η and each function p of x, let us denote by p[η] the shifted function given by p[η] (x) := p(x + η); furthermore, let ∆η be the shift operator sending p into p[η] . If f is a given function of x, f ∆η stands for the composition of the shift and the operator of multiplication by f . The Lax operators and the other relevant each N , sp(N ) is the Lie algebra of the symplectic group Sp(N ), formed by the 2N × 2N matrices preserving the standard symplectic form, see for instance [6].
a For
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
237
linear operators for the interpolated systems of interest can be represented within this scheme. For example the KM Lax operator, depending on the fundamental spacing , is L (a) = 12 (a∆ + a[−] ∆− ); this is obtained with the constraint b = 0 from the Toda Lax operator L (a, b) = 12 (a∆ + a[−] ∆− ) + b. The subsequent step in this scheme is the introduction of a transformation (“field rescaling”) relating the field variables of the interpolated lattice to another system of functions, to be ultimately identified with the field variables of the KdV-type theory associated to the lattice. The simplest example is the transformation a(x) = 2 + 2 u(x), giving the KM field variable a in terms of a new variable u. The final step is the 7→ 0 limit. In the particular case of KM theory, it is found that the differential-difference Lax operator goes into the KdV Lax operator Lsp(1) (u) = ∂ 2 +u (in the precise sense given by Eq. (2.10) of the next section). As a matter of fact, the KdV biHamiltonian structure, the vector fields and Hamiltonian functions of the hierarchy, as well as the fractional powers of the Lax operator can also be obtained from the 7→ 0 limits of conveniently defined linear combinations of homologous KM objects; giving the rules to form these recombinations, in [1] we have pointed out some formal analogy with the Wick reordering, well known to field theorists. In the present paper, the extension of the previous results to the V2 system and the sp(2) KdV is obtained through a new formulation of the recombination scheme, based on a system of real polynomials which are naturally associated to the problem under consideration. The need for this new formulation (also possible, but not indispensable for V1 ) arises from a peculiar feature of the VN systems for each N ≥ 2, i.e. the fact that these hierarchies, even though Hamiltonian and equipped with Lax pairs, do not seem to possess a second Hamiltonian structure (at least of a local type; this fact was already noticed for other integrable lattices, see [7]). Concerning the case of general N , we work out only the field rescaling and the continuous limit of the Lax operator. After these considerations, let us give the necessary informations about the systems considered in this paper. Let N be any positive integer; the phase space of the (interpolated) VN system is a set A of N -tuples a = (a1 , a2 , . . . , aN ) where, for each p, ap is a real smooth function of the variable x. The Lax operator for this system at any point a is the symmetric operatorb 1X ap ∆(2p−1) + ap[(−2p+1)] ∆(−2p+1) . L (a) := 2 p=1 N
(1.1)
The VN system entails infinitely many commuting vector fields Xs (s = 0, 1, 2, . . . ) on A, admitting the Lax formulation dL = [As , L ] dts b Here
(1.2)
and in the sequel, the notions of symmetry or skew-symmetry R for differential-difference R or differential operators are always referred to the pairing hψ, ψ0 i := dx ψψ0 , where denotes integration over the torus R/Z.
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(where d/dts stands for the Lie derivative along Xs ). The “companion operators” As in this equation are defined by As (a) := const.(L )2s skew (a) ,
(1.3)
where the normalization constant (to be fixed according to the specific needs) may depend on N and s; in this equation, skew denotes the “skew-symmetric part”: for Pjmax gj ∆j , each differential-difference operator G = j=−∞
Gskew :=
jX max
(gj ∆j − gj
[−j]
∆−j ) .
(1.4)
j=1
The vector fields Xs are Hamiltonian, the Hamiltonian functions being the traces of the even powers of L . For N = 1, 2, the explicit expressions of the Poisson tensor and all the details about the Hamiltonians can be found in Tables Ia and IIa; for N arbitrary, the VN Poisson tensor can be described in terms of R-matrix theory, as we will briefly explain hereafter. The VN theory can be viewed as a reduction of a Toda-like theory in 2N fields a = (a1 , a2 , . . . , aN ) and b = (b1 , b2 , . . . , bN ), based on the Lax operator 1X ap ∆(2p−1) + ap[(−2p+1)] ∆(−2p+1) 2 p=1 N
L (a, b) :=
1X + bp ∆(2p−2) + bp[(−2p+2)] ∆(−2p+2) , 2 p=1 N
(1.5)
which becomes the VN operator for b = 0. For N = 1, one has the standard Toda lattice; the description of KM system as a reduction of Toda can be found for example in [1]. It is well known that the Toda lattice can be described in terms of R-matrix theory; we will refer for definiteness to the R-matrix employed in [8, Sec. 5]. Here, the algebra of differential-difference operators is splitted as a direct sum of two Lie subalgebras (the skew-symmetric operators and the “nonpositive order” operators, formed with the shifts ∆j of orders j ≤ 0); the corresponding R-matrix gives in a standard way a linear and a quadratic Poisson structure [9, 10]. Both Poisson tensors can be employed for N arbitrary to construct the biHamiltonian formulation of the theory with the Lax operator (1.5). It turns out that the linear tensor cannot be restricted to the submanifold b = 0; on the contrary, the restriction is possible for the quadratic tensor, yielding the previously mentioned Hamiltonian formulation of the VN system. We now pass to the sp(N ) KdV-type theories. Let the positive integer N be arbitrary; the phase space of the corresponding theory is a set U of N -tuples u = (u1 , . . . , uN ) , where ul is a smooth function of x for each l. The Lax operator at a point u is the symmetric differential operator
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
239
1X (ul ∂ 2N −2l + ∂ 2N −2l ul ) . 2
(1.6)
N
Lsp(N ) (u) := ∂ 2N +
l=1
A family of infinitely many commuting vector fields Zssp(N ) (s = 0, 1, 2, . . . ) can be defined on U in terms of this operator; if d/dτs is the Lie derivative along Zssp(N ) , we have dLsp(N ) = [Bssp(N ) , Lsp(N ) ] , dτs
2s+1−2N 2N
Bssp(N ) (u) := const.(Lsp(N ) )+
(u) ,
(1.7)
where + takes the non-negative powers of ∂ in the algebra of pseudo-differential operators. The standard splitting of this algebra into the differential and the purely integral operators (formed, respectively, with the nonnegative and the negative powers of ∂) gives a well-known R-matrix. In this paper we will consider the Hamiltonian formulation of the vector fields Zssp(N ) , coming from the R-matrix linear Poisson structure (the second Hamiltonian formulation, coming from the quadratic Poisson structure, is less interesting for our purposes). Further details about this theory, including its Lie-theoretical interpretation, can be found in the fundamental Drinfeld–Sokolov paper [5] (according to the classification of these authors, what we are calling here the sp(N ) KdV is the theory induced removing the vertex c0 from the Dynkin diagram of the affine Lie algebra (1) CN = sp(N )[ζ, ζ −1 ]: see Table 4, p. 2022 to recover the Lax operator (1.6)). Let us summarize the organization of the paper. In Sec. 2 we briefly review the continuous limit of the KM = V1 system (here, some facts about KM and KdV theories are reported for conveniency in Tables Ia–b); we also give the new presentation of the recombination scheme already mentioned in this Introduction. In Sec. 3 we describe the V2 and sp(2) theories; the explicit expressions of the Poisson tensors and of the first vector fields and Hamiltonians are collected in Tables IIa–b. In Sec. 4 we construct the field rescaling for the V2 theory, introduce the recombination scheme and state the main theorem (Proposition 4.4) on the continuous limit of the Lax pairs and of the Hamiltonian scheme, recovering in this way the sp(2) KdV. In Sec. 5, we prove Proposition 4.4. In Sec. 6, we consider the VN theory for arbitrary N , and obtain the sp(N ) Lax operator from its continuous limit, after an appropriate field rescaling. Most of the explicit computations for Secs. 2–4 have been performed using the MATHEMATICA symbolic manipulator. The rapid increase with N of the complexity of the expressions for the VN vector fields is evident even from the comparison between the cases N = 1 and 2; the strength of the arguments employed in Sec. 5 to prove the main theorem is the fact that they are essentially independent from these explicit expressions, resting on general compatibility arguments for the Lax scheme. 2. The One-Component Case: A Review of the KM System and Its Continuous Limit Let us start directly from the interpolated KM = V1 system, in the sense explained in the Introduction. The phase space of this system is a set A of real
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C. MOROSI and L. PIZZOCCHERO
functions a : R/Z → R. At any point a ∈ A, the tangent and cotangent spaces will be denoted by Ta A and Ta∗ A; their elements can be represented as real R functions of x, written a˙ and δa respectively, and we have the pairing hδa, ai ˙ = dx δa a. ˙ The KM system envolves infinitely many vector fields Xs (s = 0, 1, 2, . . . ) on A, admitting the Lax formulation (1.2–3). The Lax operator is as in Eq. (1.1) (with N = 1 and a1 = a); it is written in Table Ia, together with the compatible Poisson tensors Q , P giving the biHamiltonian formulation of this system. At each point a, both of them are linear maps between the cotangent and the tangent space, sending any covector δa into the vector a˙ = Qa δa or a˙ = Pa δa. The zeroth Hamiltonian function is a Casimir of Q : Q df0 = 0; the Hamiltonians fs with s ≥ 1 are traces of the even powers of the Lax operator. RThe traceform tr on differential-difference operators is defined setting tr(G) := dx rem(G) for each Pjmax gj ∆j is, by definition, the G; the “reminder” rem(G) of an operator G = j=−∞ function g0 . After the KM system, let us consider the ordinary KdV theory. The phase space is a set U of real smooth functions u, defined on the torus R/Z; writing Eq. (1.6) with N = 1 and u1 = u, we obtain the familiar Lax operator. The main facts about KdV theory are summarized in Table Ib. Of course, the traceform Tr appearing in the definition R of the Hamiltonians is the Adler trace for pseudo-differential operators (Tr(F ) := dx Res(F ), where the residue Res(F ) is the coefficient of ∂ −1 in the operator F ). The zeroth Hamiltonian is a Casimir of the first Poisson tensor Qsp(1) . Table Ia: KM = V1 system Lax formulation for the sth vector field Xs : dL /dts = [As , L ] (s = 0, 1, 2, . . . ) L (a) :=
1 a∆ + a[−] ∆− , 2
(a) . As (a) := (L )2s skew
BiHamiltonian structure at a point a. Qa , Pa : Ta∗ A → Ta A, 1 aa[] ∆ − aa[−] ∆− , 8 1 2 aa[] a[2] ∆2 + a3 a[] + aa3[] ∆ Pa = 32 − aa3[−] + a3 a[−] ∆− − aa2[−] a[−2] ∆−2 .
Qa =
BiHamiltonian formulation: X0 = Q df0 = 0 , Hamiltonian functions Z f0 (a) := dx log a ,
Xs = Q dfs = P dfs−1
fs (a) :=
(s = 1, 2, 3, . . . ) .
1 tr(L )2s (a) (s = 1, 2, 3, . . . ) . 2s
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
First companions of the Lax operator: A0 (a) = 0 , A2 (a) =
1 (aa[] ∆2 − a[−] a[−2] ∆−2 ) 4
A1 (a) =
1 aa[2] a[3] a[] ∆4 + a3 a[] + aa2[2] a[] + aa2[−] a[] + aa3[] ∆2 16 − a2 a[−2] a[−] + a3[−2] a[−] + a[−2] a2[−3] a[−] + a[−2] a3[−] ∆−2 − a[−2] a[−4] a[−3] a[−] ∆−4 .
First Hamiltonians after f0 : f1 (a) = f2 (a)
1 4
Z
1 = 32
f3 (a) =
dx a2 , Z
1 192
dx a4 + 2a2 a2[] ,
Z
dx a6 + 3a4 a2[] + 3a2 a2[2] a2[] + 3a2 a4[] .
First vector fields: X0 (a) = 0 ,
X1 (a) =
1 a − a2[−] + a2[] , 16
X2 (a) =
1 a − a2 a2[−] − a2[−2] a2[−] − a4[−] + a2 a2[] + a2[2] a2[] + a4[] , 64
X3 (a) =
1 a − a4 a2[−] − a2 a2[−2] a2[−] − a4[−2] a2[−] − a2[−2] a2[−3] a2[−] 256 − 2a2 a4[−] − 2a2[−2] a4[−] − a6[−] + a4 a2[] + a2 a2[2] a2[] + a4[2] a2[] + a2[2] a2[3] a2[] + 2a2 a4[] + 2a2[2] a4[] + a6[] .
Table Ib: sp(1) KdV theory Lax formulation for the sth vector field Zssp(1) : dLsp(1) /dτs = [Bssp(1) , Lsp(1) ], (s = 0, 1, 2, . . . ) Lsp(1) (u) := ∂ 2 + u ,
s− 12
Bssp(1) (u) := 4s−1 (Lsp(1) )+
(u) .
(1) , Susp(1) : Tu∗ U → Tu U, BiHamiltonian structure at a point u. Qsp u
(1) = ∂, Qsp u
Susp(1) = ∂ 3 + 4u∂ + 2ux .
241
242
C. MOROSI and L. PIZZOCCHERO
BiHamiltonian formulation: (1) (1) (1) = 0 , Zssp(1) = Qsp(1) dhsp = S sp(1) dhsp Z0sp(1) = Qsp(1) dhsp s 0 s−1 (s = 1, 2, 3, . . . ) .
Hamiltonian functions (1) (u) := hsp s
4s 1 Tr(Lsp(1) )s+ 2 (u) (s = 0, 1, 2, . . . ) . 2s + 1
First companions of the Lax operator: B0sp(1) (u) = 0 ,
B0sp(1) (u) = ∂ ,
B2sp(1) (u) = 4∂ 3 + 6u∂ + 3ux .
First Hamiltonians: Z Z 1 1 (1) sp(1) dx u , h dx u2 , (u) := (u) = hsp 0 1 2 2 Z Z 1 1 sp(1) sp(1) 3 2 dx(2u − ux ) , h3 (u) = dx(5u4 − 10uu2x + u2xx ) . h2 (u) = 2 2 First vector fields: Z0sp(1) (u) = 0 ,
Z1sp(1) (u) = ux ,
Z2sp(1) (u) = uxxx + 6uux ,
Z3sp(1) (u) = uxxxxx + 10uuxxx + 20uxuxx + 30u2ux .
The starting point to construct the continuous limit of the KM system is the introduction of a transformation Ψ : U → A, u 7→ a = Ψ (u), with a := 2 + 2 u .
(2.1)
With a suitable specification of the function spaces A and U, this map is bijective with inverse (Ψ )−1 (a) = (a − 2)/2 , and allows to transport all the KM geometric structures of A onto U. For example, each KM vector field Xs on A induces a vector field on U, denoted for simplicity with the same symbol and given explicitly by Xs (u) := (T Ψ )−1 (u)Xs (a)|a=Ψ (u) (T standing for the tangent map). The key point in the limit process illustrated in [1] is a recombination method for the KM structures (including the Hamiltonians and the Lax pairs, besides the vector fields and Poisson tensors already considered in [2]). Let us define recursively a system of coefficients (cs,j ) (s = 0, 1, 2, . . . ; j = 0, 1, . . . , s), where c0,0 := 1 ,
(2.2)
243
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
cs+1,j :=
X s 2s + 2 2i − − (cs,i−1 − 4cs,i ) s+1 i i=1 cs,j−1 − 4cs,j 1
for j = 0 for j = 1, 2, . . . , s (2.3) for j = s + 1
r (here k denotes as usually the binomial coefficient, for each pair r, k). Furthermore, for s = 0, 1, 2, . . . , let us put
cs,−1
s X 1 2j cs,j . := −cs,0 log 2 − 2j j j=1
(2.4)
The first coefficients constructed in this way are: c0,−1 = − log 2 ,
c0,0 = 1 ,
c1,−1 = −1 + 2 log 2 ,
c1,0 = −2 ,
c1,1 = 1 ,
c2,0 = 6 ,
c2,1 = −6 , c2,2 = 1 ,
c2,−1 =
9 − 6 log 2 , 2
c3,−1 = −
55 + 20 log 2 , c3,0 = −20 , c3,1 = 30 , 3
(2.5)
c3,2 = −10 , c3,3 = 1 .
For s arbitrary, let us introduce the recombinations hs := cs,−1 +
s X
cs,j fj ,
Zs :=
j=0
s X
cs,j Xj ,
Bs :=
j=1
s X
cs,j Aj ,
(2.6)
j=1
S := P − 4Q .
(2.7)
We summarize the main results on the continuous limit in the following statements (see [1] for the proofsc ): Proposition 2.1. The 7→ 0 limit of the KM structures considered above are the homologous KdV structures. More precisely, lim
1
7→0 2s+2
(1) hs (u) = hsp (u) , s
(1) lim 3 Qu = Qsp , u
7→0
lim
1
7→0 2s−1
Zs (u) = Zssp(1) (u) .
lim Su = Susp(1) .
7→0
(2.8) (2.9)
c For uniformity with the rest of the present paper, the normalizations for the KM structures considered here are slightly different from the ones adopted in the previous paper. Denoting with 1 Xs , As = barred symbols the objects considered in [1], we have L = 12 L , fs = 41s fs , Xs = 4s+1 1 1 1 1 j−s A , Q = 4 Q , P = 16 P , cs,j = 4 cs,j , cs,−1 = − 4s cs (s = 0, 1, 2, . . . ; j = 0, 1, . . . , s). 4s+1 s The normalizations adopted for the KdV theory have not been changed.
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C. MOROSI and L. PIZZOCCHERO
lim
1
7→0 2
(L (u) − 2) = Lsp(1) (u) ,
lim
1
7→0 2s−1
Bs (u) = Bssp(1) (u) .
(2.10)
Let us illustrate, for example, how to obtain the second KdV Hamiltonian function via expansion and recombination of the KM Hamiltonians f0 , f1 , f2 . To this purpose, we take from Table Ia the explicit expressions of these functions, make the field rescaling (2.1) and expand up to O(8 ) in ; in this way, we find Z f0 (u)
=
dx log(2 + 2 u)
Z Z Z 4 6 2 2 dx u − dx u + dx u3 + O(8 ) , = log 2 + 2 8 24 Z 1 dx(2 + 2 u)2 f1 (u) = 4 (2.11) Z Z 1 4 2 2 dx u + dx u , =1+ 4 Z 1 dx (2 + 2 u)4 + 2(2 + 2 u() )2 (2 + 2 u)2 f2 (u) = 32 Z Z Z 9 1 3 = + 32 dx u + 4 dx u2 + 6 dx (3u3 − 2u2x ) + O(8 ) . 2 4 4 Now, we recall Eqs. (2.5–2.6) and put h2
:=
9 − 6 log 2 + 6f0 − 6f1 + f2 ; 2
(2.12)
the conclusion is h2 (u)
1 = 6 2
Z (1) dx (2u3 − u2x ) + O(8 ) = 6 hsp (u) + O(8 ) . 2
(2.13)
After this example, let us return to the general algorithm for finding the coefficients cs,j . For uniformity with the method applied in the next section to the V2 system, we give an alternative prescription for the determination of these coefficients. To this purpose, let us consider the function p0 (ξ) and the polynomials p1 (ξ), p2 (ξ), . . . , in one indeterminate ξ, defined setting log(2 + 2ξ) ps (ξ) := 1 2s (1 + ξ)2s 2s s
for s = 0 (2.14) for s = 1, 2, . . . .
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
245
Proposition 2.2. For s = 0, 1, 2, . . . , the coefficients cs,j (j = −1, 0, . . . , s) defined by Eqs. (2.2 − 2.4) are the unique (s + 2)-tuple of real numbers such that cs,−1 +
s X
cs,j pj (ξ) =
j=0
23s+1 2s + 1
s + 12 s+1
ξ s+1 + O(ξ s+2 ) .
(2.15)
Proof. As it was proved [1], the coefficients defined by (2.2–2.4) are such that cs,−1 +
s X
(1) cs,j fj (u) = 2s+2 hsp (u) + O(2s+4 ) ; s
(2.16)
j=0
from here, we will infer Eq. (2.15). To this purpose, let us apply Eq. (2.16) with u(x) = const. = 2. We note that f0 (2) = log(2 + 22 ) = p0 (2 ); to compute fs (2) for s > 0, we observe that (L )2s (2) = (1 + 2 )2s (∆ + ∆− )2s 2 2s
= (1 + )
2s X 2s k=0
k
∆(2k−2s) ;
(2.17)
(s = 0, 1, 2, . . . ) .
(2.18)
so, taking the trace, we find that fs (2) =
1 2s
2s s
(1 + 2 )2s = ps (2 )
(1) (2) = On the other hand, hsp s
4s 2s+1
R
1
dx Res(Lsp(1) )s+ 2 (2); but
1
1
(Lsp(1) )s+ 2 (2) = (∂ 2 + 2)s+ 2 =
+∞ X s+ k=0
k
1 2
2k ∂ 2s+1−2k .
(2.19)
Reading from this sum the coefficient of ∂ −1 , we conclude (1) hsp (2) = s
23s+1 2s + 1
s + 12 s+1
(s = 0, 1, 2, . . . ) .
(2.20)
Inserting the results (2.18) and (2.20) into Eq. (2.16) (with u = 2), and setting 2 = ξ, we obtain that the cs,j actually satisfy Eq. (2.15). Let us now prove the uniqueness of the system of coefficients fulfilling this equation. To this purpose, we employ again the cs,j defined by Eqs. (2.2–2.4) (recalling that cs,s = 1 for each s). Let us define qs (ξ) := cs,−1 +
s X j=0
cs,j pj (ξ)
(s = 0, 1, 2, . . . ) .
(2.21)
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C. MOROSI and L. PIZZOCCHERO
From now on s is fixed; we introduce the arrays 1 1 q0 (ξ) p0 (ξ) c := cs,−1 , cs,0 , . . . , cs,s−1 , 1 , p(ξ) := p1 (ξ) , q(ξ) := q1 (ξ) ... ... ps (ξ)
(2.22)
qs (ξ)
and the (s + 2) × (s + 2) lower triangular matrix
1
c 0,−1 c1,−1 C := ... ... cs,−1
0
0
0
...
...
1 c1,0
0 1
0 0
... ...
... ...
...
...
...
...
... cs,0
... cs,1
... ...
... ...
. ... ... 1
(2.23)
Then we can write Eqs. (2.21) and (2.15) as q(ξ) = C p(ξ) , c p(ξ) =
23s+1 2s + 1
s + 12 s+1
(2.24)
ξ s+1 + O(ξ s+2 ) .
(2.25)
Let c0 = c0s,−1 , c0s,0 , . . . c0s,s−1 , c0s,s be another system of coefficients satisfying Eq. (2.15); then (2.26) (c0 − c) p(ξ) = O(ξ s+2 ) . From Eqs. (2.24), (2.26), we infer that the array b = b−1 , b0 , b1 . . . bs−1 , bs defined by (c0 − c) = b C satisfies the condition b q(ξ) = O(ξ s+2 ) . Let us recall that the elements of the array q(ξ) are 1 and qj (ξ) =
(2.27) 23j+1 2j+1
j+ 12 j+1
ξ j+1 + O(ξ j+2 ) (j = 0, 1, . . . , s). Developing to order zero in ξ, we see that b q(ξ) = b−1 + O(ξ), so Eq. (2.27) implies b−1 = 0. Taking this fact into account, and developing to order 1, we find b q(ξ) = b0 ξ + O(ξ 2 ); so, using again Eq. (2.27), we infer b0 = 0. Iterating this argument, we conclude that each component of b is zero, which implies c0 = c. Summing up, c is the unique system of coefficients satisfying Eq. (2.15), and the proof is concluded. 3. The Two-Components Volterra (V2 ) System and the sp(2) Theory Each point in the phase space A of the V2 system is a pair a of real smooth functions on R/Z; for simplicity of notation, we denote these functions with a and r instead of using the symbols a1 and a2 prescribed by the Introduction. A
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
247
tangent vector a˙ ∈ Ta A and a covector δa ∈ Ta∗ A are both identified with pairs of functions, ˙ r) ˙ and (δa, δr), and we have the pairing R denoted respectively with (a, ˙ := dx (δa a˙ + δr r). hδa, ai ˙ A sequence of vector fields Xs (s = 0, 1, 2, . . . ) is defined on the phase space A; the Lax formulations are based on the operator (1.1) (with N = 2, a1 = a, a2 = r) and appear in Table IIa (where we use the notations d/dts for the Lie derivative along Xs , and skew for the “skew-symmetric part” of a differential-difference operator, defined by Eq. (1.4)). All these vector fields are Hamiltonian w.r.t. the Poisson tensor Q and the Hamiltonians fs : A → R introduced in Table IIa. It should be noted that X0 = Q df0 = 0, which means that f0 is a Casimir of the Poisson structure. To the best of our knowledge, a second Hamiltonian structure for the V2 system is not presently known; the methods employed successfully for the second KM Poisson structure (based on the Toda recursion operator [1] or the master symmetry approach [11]) could be applied in principle to the V2 case; unfortunately, the “second Poisson tensor” obtained in this way is a highly formal object, involving inverses of differential-difference operators (and the same seems to happen for all the VN theories with N ≥ 2). For this reason, we will leave aside the problem and confine our attention to the “first” Poisson tensor Q . In the next section, we will show that the 7→ 0 limit of the V2 system is the sp(2) theory; here, we review some of the basic facts about the latter. A point in the phase space U of this theory is a pair u, formed by two smooth functions on the torus; again we stray slightly from the notation of the Introduction and denote these functions with u, v rather than u1 , u2 . Tangent vectors and covectors at u are represented as pairs of functions u˙ = (u, ˙ v) ˙ and δu = (δu, δv), with the obvious pairing. A sequence of vector fields Zssp(2) (s = 0, 1, 2, . . . ) is defined on U through their Lax formulations, based on the differential operator of Eq. (1.6) (with N = 2, u1 = u, u2 = v) and its fractional powers of exponents 1/4, 3/4, 5/4, etc. The details are given in Table IIb (where we write d/dτs for the Lie derivative along the vector field Zssp(2) ). The first nontrivial vector field of the hierarchy is Z2sp(2) . All vector fields are Hamiltonian w.r.t. the Poisson tensor Qsp(2) reported (2) are (Adler) traces of the fractional powers of in the Table; the Hamiltonians hsp s sp(2) sp(2) . The vanishing of Z0 and Z1sp(2) means that the first two Hamiltonians L (2) (2) and hsp are Casimirs of Qsp(2) . The vector fields of the sp(2) theory also hsp 0 1 (2) admit a second Hamiltonian formulation Zssp(2) = S sp(2) dhsp s−2 ; however, this fact is not considered in the rest of the paper, so we do not even report the explicit expression of the Poisson tensor S sp(2) . As a final remark, we mention some facts related to the known isomorphism between sp(2) and the Lie algebra so(5) of the skew symmetric 5 × 5 matrices. Due to this isomorphism, the KdV theory discussed here could be as well interpreted in terms of so(5); this alternative description can be found, for example, in [12], where we also give the second Poisson tensor.d d It
is known that the flows and the biHamiltonian structure of a KdV-type theory are intrinsically associated to the underlying Lie algebra, while the Lax operator depends on the choice of a matrix representation of that algebra. As a matter of fact, sp(2) and so(5) give two inequivalent matrix representations of the same abstract Lie algebra, and this is the reason for the difference between the Lax operator of [12] and the one reported in Table IIb.
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C. MOROSI and L. PIZZOCCHERO
Table IIa: V2 system Lax formulation for the sth vector field Xs : dL /dts = [As , L ] L (a) :=
1 (r∆3 + a∆ + a[−] ∆− + r[−3] ∆−3 ) , 2
(s = 0, 1, 2, 3, . . . )
3 As (a) := − (L )2s (a) . skew 8
Poisson tensor at a point a = (a, r). Qa : Ta∗ A → Ta A Qar δa 3 Qaa a˙ δa := 7→ r˙ δr 16 Qra Qrr δr Qaa = − 2rr[] ∆3 − 2a[] r∆2 − aa[] ∆ + aa[−] ∆− + 2a[−]r[−2] ∆−2 + 2r[−2] r[−3] ∆−3 , Qar = − ar[] ∆ − ar + ar[−2] ∆−2 + ar[−3] ∆−3 , Qra = − a[3] r∆3 − a[2] r∆2 + ar + a[−] r∆− , Qrr = − rr[3] ∆3 + rr[−3] ∆−3 . Hamiltonian formulation: Xs = Q dfs (s = 0, 1, 2, 3, . . . ) Z 1 tr(L )2s (a) (s = 1, 2, 3, . . . ) . f0 (a) := dx log r , fs (a) := 2s First companions of the Lax operator: A0 (a) = 0, A1 (a) = −
3 {rr[3] ∆6 + (a[3] r + ar[] )∆4 + (aa[] + a[2] r + a[−] r[−] )∆2 32
− (a[−2] a[−] + ar[−2] + a[−3] r[−3] )∆−2 − (a[−] r[−4] + a[−4] r[−3] )∆−4 − r[−6] r[−3] ∆−6 } . First Hamiltonians after f0 : Z 1 dx (a2 + r2 ) , f1 (a) = 4 Z 1 dx (a4 + 2a2 a2[] + 4aa[2] a[] r + 2a2 r2 + 2a2[2] r2 + 2a2[3] r2 + r4 f2 (a) = 32 2 2 + 2r2 r[3] + 4aa[3]rr[] + 2a2 r[] ),
f3 (a) =
1 192
Z dx (a6 + 3a4 a2[] + 3a2 a2[2] a2[] + 3a2 a4[] + 6a3 a[2] a[] r + 6aa3[2]a[] r
+ 6aa[2]a2[3] a[] r + 6aa[2] a3[] r + 3a4 r2 + 3a2 a2[2] r2 + 3a4[2] r2 + 3a2 a2[3] r2
249
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
+ 6a2[2] a2[3] r2 + 3a2[4] a2[3] r2 + 3a4[3]r2 + 3a2 a2[] r2 + 3a2[2] a2[] r2 + 6aa[2] a[] r3 + 3a2 r4 + 3a2[2] r4 + 3a2[3] r4 + r6 + 6aa[4] a[3] a[] rr[2] 2 2 2 + 6a[2] a[4] a[3] r2 r[2] + 3a2 a2[] r[2] + 6aa[2] a[] rr[2] + 3a2[2] r2 r[2] 2 + 3a2[3] r2 r[4] + 6a[5] a[4] a[3] r2 r[3] + 6aa[5] a[] rr[2] r[3] 2 2 + 6a[5] a[2] r2 r[2] r[3] + 6a[6] a[3] r2 r[4] r[3] + 6aa[2] a[] rr[3] + 3a2 r2 r[3] 2 2 2 2 2 + 3a2[5] r2 r[3] + 6a2[2] r2 r[3] + 3a2[6] r2 r[3] + 6a2[3] r2 r[3] + 3r4 r[3] 2 2 4 + 3r2 r[6] r[3] + 3r2 r[3] + 6a2 a[2] a[3] a[] r[] + 6a3 a[3] rr[]
+ 6aa2[2] a[3] rr[] + 6aa2[4] a[3] rr[] + 6aa3[3]rr[] + 6aa[3] a2[] rr[] + 6a[2] a[3] a[] r2 r[] + 6aa[3] r3 r[] + 6a2 a[4] a[] r[2] r[] 2 + 6aa[2] a[4] rr[2] r[] + 6aa[3] rr[4] r[] + 6aa[5] a[4] rr[3] r[] 2 2 2 + 6aa[6] rr[4] r[3] r[] + 6aa[3] rr[3] r[] + 3a4 r[] + 3a2 a2[4] r[] 2 2 2 2 + 3a2 a2[3] r[] + 6a2 a2[] r[] + 6aa[2]a[] rr[] + 3a2 r2 r[] 2 2 2 3 4 + 3a2[3] r2 r[] + 3a2 r[4] r[] + 6aa[3] rr[] + 3a2 r[] ).
First vector fields: X0 (a)
= 0,
X1 (a)
3 = 32
X1,a (a) X1,r (a)
,
X2 (a)
3 = 128
X2,a (a) X2,r (a)
,
2 X1,a (a) := aa2[−] − aa2[] − 2a[2] a[] r − ar2 + 2a[−2] a[−] r[−2] + ar[−2] 2 2 + 2a[−3]r[−2] r[−3] + ar[−3] − 2a[3] rr[] − ar[] , 2 2 X1,r (a) := a2 r − a2[2] r + a2[−] r − a2[3] r + rr[−3] − rr[3] , X2,a (a) := a3 a2[−] + aa2[−2] a2[−] + aa4[−] − a3 a2[] − aa2[2] a2[] − aa4[] − 2a2 a[2] a[] r
− 2a3[2] a[] r − 2a[2] a2[3] a[] r − 2a[2] a3[] r − a3 r2 − aa2[2] r2 − aa2[3] r2 2 − 2aa2[]r2 − 2a[2] a[] r3 − ar4 − 2a[4] a[3] a[] rr[2] − aa2[] r[2] 2 − 2a[2] a[] rr[2] + 2a2 a[−2] a[−] r[−2] + 2a3[−2] a[−] r[−2] 2 2 + 2a[−2]a2[−3] a[−] r[−2] + 2a[−2] a3[−] r[−2] + a3 r[−2] + aa2[−2] r[−2] 2 2 3 4 + aa2[−3] r[−2] + 2aa2[−] r[−2] + 2a[−2] a[−] r[−2] + ar[−2] 2 2 + 2a[−4]a[−3] a[−] r[−2] r[−4] + aa2[−] r[−4] + 2a[−2] a[−] r[−2] r[−4]
250
C. MOROSI and L. PIZZOCCHERO
+ 2aa[−2]a[−3] a[−] r[−3] + 2a2 a[−3] r[−2] r[−3] + 2a2[−2]a[−3] r[−2] r[−3] + 2a2[−4] a[−3] r[−2] r[−3] + 2a3[−3] r[−2] r[−3] + 2a[−3]a2[−] r[−2] r[−3] + 2aa[−4]a[−] r[−4] r[−3] 2 + 2a[−3]r[−2] r[−3] r[−6] + 2r[−2] r[−3] a[−6] r[−6] r[−5] 2 2 2 + 2a[−2]a[−4] r[−2] r[−4] r[−3] + a3 r[−3] + aa2[−4] r[−3] + aa2[−3] r[−3] 2 2 2 2 + 2aa2[−]r[−3] + 2a[−2]a[−] r[−2] r[−3] + 2ar[−2] r[−3] 3 4 2 + 2a[−3]r[−2] r[−3] + ar[−3] − 2a[−] a[] r2 r[−] + 2a[−] a[] r[−2] r[−] 2 2 2 2 + aa2[−] r[−] − aa2[] r[−] − 2a[2] a[] rr[−] + 2a[−2]a[−] r[−2] r[−] 2 2 − 2a[2] a[] rr[3] − ar2 r[3] − 2aa[2] a[3] a[] r[] − 2a2 a[3] rr[]
− 2a2[2] a[3] rr[] − 2a2[4] a[3] rr[] − 2a3[3] rr[] − 2a[3] a2[] rr[] 2 − 2a[3] r3 r[] − 2aa[4] a[] r[2] r[] − 2a[2] a[4] rr[2] r[] − 2a[3] rr[4] r[] 2 2 2 2 2 − 2a[3] rr[3] r[] − a3 r[] − aa2[4] r[] − aa2[3] r[] − 2aa2[] r[] 2 2 2 2 3 4 − 2a[2] a[] rr[] − 2ar2 r[] − ar[4] r[] − 2a[3] rr[] − ar[]
− 2a[]rr[2] r[3] aa[5] − 2a[4] rr[3] r[] aa[5] − 2rr[4] r[3] r[] aa[6] 2 2 + ar[−3] r[−6] + 2a[−]r[−2] r[−4] a[−5] r[−5] 2 + 2a[−4]r[−2] r[−3] a[−5] r[−5] + 2a[−2] a[−] r[−2] r[−5] 2 2 2 3 + ar[−2] r[−5] + 2a[−3] r[−2] r[−3] r[−5] + 2a[−3]r[−2] r[−3] , X2,r (a) := a4 r − a4[2] r + 2a2 a2[−] r + a2[−2] a2[−] r + a4[−] r − 2a2[2] a2[3] r − a2[4] a2[3] r
− a4[3] r + a2 a2[] r − a2[2] a2[] r + a2 r3 − a2[2] r3 + 2a[−4] a[−] rr[−4] r[−3] 2 + a2[−] r3 − a2[3] r3 − 2a[2] a[4] a[3] rr[2] − a2[2] rr[2] 2 2 + 2aa[−2]a[−] rr[−2] + a2 rr[−2] + a2[−] rr[−4] 2 + 2a[−2]a[−3] a[−] rr[−3] + 2aa[−3] rr[−2] r[−3] + 2a2 rr[−3] 2 2 2 2 4 + a2[−4] rr[−3] + a2[−3] rr[−3] + 2a2[−] rr[−3] + r3 r[−3] + rr[−3] 2 2 2 + 2aa[−]a[] rr[−] − a2[2] rr[−] + a2[−] rr[−] − a2[3] rr[4] 2 2 2 4 − 2a2[2] rr[3] − 2a2[3] rr[3] − r3 r[3] − rr[3] − 2a[2] a[3] a[] rr[] 2 2 + a2 rr[] − a2[3] rr[] − 2a[4] a[3] rr[3] aa[5] − 2a[2] rr[2] r[3] aa[5]
251
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
2 2 − rr[3] aa2[5] − 2a[3] rr[4] r[3] aa[6] − rr[3] aa2[6] 2 2 2 2 + rr[−3] r[−6] − rr[3] rr[6] .
Table IIb: sp(2) theory Lax formulation for the sth vector field Zssp(2) : dLsp(2) /dτs = [Bssp(2) , Lsp(2) ] (s = 0, 1, 2, . . . ) L
sp(2)
4
2
(u) = ∂ + u∂ + ux ∂ +
1 uxx + v 2
2s−3
Bssp(2) (u) := 4s−2 (Lsp(2) )+ 4 (u) .
,
(2) : Tu∗ U → Tu U Poisson tensor at a point u = (u, v). Qsp u
δu δv
7→
0 u˙ := v˙ 4∂
4∂ 3 −∂ + 2u∂ + ux
(2) Hamiltonian formulation: Zssp(2) = Qsp(2) dhsp s
(2) (u) := hsp s
δu δv
.
(s = 0, 1, 2, . . . )
2s+1 22s−2 Tr(Lsp(2) ) 4 (u) . 2s + 1
First companions of the Lax operator: B0sp(2) (u) = B1sp(2) (u) = 0 ,
B2sp(2) (u) = ∂ ,
3 B3sp(2) (u) := 4∂ 3 + 3u∂ + ux . 2
First Hamiltonians: (2) hsp (u) 0
1 = 16
Z dx u ,
(2) (u) hsp 2
(2) (u) = hsp 3
1 128
(2) hsp (u) 1
Z
1 = 32
Z
1 = 32
Z dx (−u2 + 8v) ,
dx (−u3 + 8uv − u2x ) ,
dx (5u4 − 48u2 v + 192v 2 − 4uu2x + 32uxxv + 4u2xx ) .
First vector fields: Z0sp(2) (u) = Z1sp(2) (u) = 0 ,
Z2sp(2) (u) =
ux vx
,
uxxx − 3uux + 12vx . Z3sp(2) (u) = 3 −2vxxx + 3ux uxx + uuxxx + 3uvx 2
252
C. MOROSI and L. PIZZOCCHERO
4. Field Rescaling and Continuous Limit of the V2 System Let us keep all the notations of the previous section concerning the V2 system. We introduce a space U of pairs u = (u, v) and apply it into the phase space A of the Volterra system through the map Ψ : U → A, u 7→ a = Ψ (u), with 1 2 a := −3 − 4 u , (4.1) 1 1 2 4 r := + (u − u[] + u[2] ) + v . 3 4 If A and U are conveniently chosen, the map Ψ is one-to-one, with inverse (Ψ )−1 described by the relations 4 u = − 2 (3 + a) , (4.2) 1 1 8 + r + 4 (a − a[] + a[2] ) . v= 4 3 For a ∈ A and u = (Ψ )−1 (a), the tangent map Ta (Ψ )−1 : Ta A → Tu U sends a˙ ˙ with into u, 4 u˙ = − 2 a˙ , (4.3) 1 1 v˙ = 4 r˙ + 4 (a˙ − a˙ [] + a˙ [2] ) . The adjoint map Ta∗ (Ψ )−1 : Tu∗ U → Ta∗ A sends a covector δu into δa, with δa := − 4 δu + 1 (δv − δv [−] + δv[−2] ) , 2 4 1 δr := 4 δv .
(4.4)
Using this transformation, we can pull-back onto U all the V2 structures carried by the phase space A; we will keep the same notations for any object on A and its pull-back to U. By so doing, we induce at each point u the following objects: (i) the operators L (u) := L (a)|a=Ψ (u) ,
As (u) := As (a)|a=Ψ (u)
(s = 0, 1, 2, . . . ) ; (4.5)
the explicit expression of L (u) is obtained from Table IIa and Eq. (4.1). (ii) The vector fields Xs (u) := Ta (Ψ )−1 Xs (a)|a=Ψ (u)
(s = 0, 1, 2, . . . ) .
(4.6)
(iii) The Poisson tensor Qu : Tu∗ U → Tu U ,
Qu := Ta (Ψ )−1 ◦Qa ◦Ta∗ (Ψ )−1 a=Ψ (u) .
(4.7)
253
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
Explicitly, we have Qu
δu δv
3 Q 4 uu = 3 Q 46 vu −
3 Q uv 46 δu , δv 3 − Q 168 vv
(4.8)
where the matrix elements have the following expressions (with a and r given by Eq. (4.1)): Quu := 2rr[] ∆3 + 2a[] r∆2 + aa[] ∆ − aa[−] ∆− − 2a[−] r[−2] ∆−2 − 2r[−2] r[−3] ∆−3 , Quv := 2rr[] ∆3 + 2(a[] r − 2rr[] )∆2 + (aa[] − 2a[] r + ar[] + 2rr[] )∆ + (ar − aa[] + 2a[] r) + (aa[] − aa[−] )∆− + (aa[−] − ar[−2] − 2a[−]r[−2] )∆−2 + (2a[−] r[−2] − aa[−] − ar[−3] − 2r[−2] r[−3] )∆−3 + (2r[−2] r[−3] − 2a[−]r[−2] )∆−4 − 2r[−2] r[−3] ∆−5 , Qvu := 2r[2] r[3] ∆5 + 2(a[3] r[2] − r[2] r[] )∆4 + (a[2] a[3] + a[3] r − 2a[2] r[] + 2rr[] )∆3 + (2a[] r − a[2] a[] + a[2] r)∆2 + (aa[] − a[2] a[] )∆ + (aa[] − ar − 2a[] r) + (2ar[−] − aa[−] − a[−] r − 2rr[−] )∆− + 2(r[−2] r[−] − a[−] r[−2] )∆−2 − 2r[−2] r[−3] ∆−3 , Qvv := 2r[2] r[3] ∆5 + (2a[3] r[2] − 2r[2] r[3] − 2r[2] r[] )∆4 + (a[2] a[3] + a[3] r − 2a[3] r[2] + a[2] r[3] + rr[3] + 2r[2] r[3] − 2a[2] r[] + 2rr[] + 2r[2] r[] )∆3 + (2a[3] r[2] − a[] r[2] + 2a[2] r[] − 2rr[] − 2r[2] r[] − a[2] a[3] − a[2] a[] + a[2] r − a[3] r + 2a[] r + a[2] r[2] )∆2 + (a[2] a[3] + aa[] + a[3] r − a[2] r − 2a[]r + ar[] − 2a[2] r[] − a[] r[] + 2rr[] )∆ + (2a[] r + 2ar[−] − a[2] r[−] + a[] r[−] − 2rr[−] − aa[−] − a[2] a[] + ar − a[−] r)∆− + (aa[−] + aa[] − ar − 2a[] r + a[−] r + 2rr[−] + 2r[−2] r[−] + a[] r[−2] − ar[−2] − 2a[−] r[−2] − 2ar[−] )∆−2 + (2ar[−] − 2rr[−] − 2r[−2] r[−] − aa[−] − a[−] r − ar[−3]
254
C. MOROSI and L. PIZZOCCHERO
− rr[−3] − 2r[−2] r[−3] + 2a[−]r[−2] )∆−3 + (2r[−2] r[−3] + 2r[−2] r[−] − 2a[−] r[−2] )∆−4 − 2r[−2] r[−3] ∆−5 .
(4.9)
(iv) The Hamiltonians fs (u) := fs (a)|a=Ψ (u)
(s = 0, 1, 2, . . . ) .
(4.10)
We will expand the above objects as (formal) power series in ; any such P+∞ series will be of the form k=m k yk , where the nature of the coefficients yk depends on the object under consideration (they can be real numbers, real functions, differential operators, and so on); it may happen that the integer m be negative. From now on, for each integer l, O(l ) denotes a series where appears only with exponents ≥ l. Let us start from the expansion of the Lax operator and of the Poisson tensor. To this purpose, we write the jth power of the shift operator as ∆j = ej∂ =
+∞ k X j k=0
k!
k ∂ k .
(4.11)
Similarly, we express any shifted function u[j] or v[j] through its Taylor series. Both the Lax operator L (u) and the tensor Qu can be expanded as power series of differential operators; let us compute the lowest order terms. Proposition 4.1. One has 8 L (u) = − + 4 Lsp(2) (u) + O(5 ) , 3 1 sp(2) 1 Q u = 5 Qu + O 4 ,
(4.12) (4.13)
(2) as in Table IIb. with Lsp(2) (u) and Qsp u
Proof. By definition, it is L (u) =
1 3∂ (re + ae∂ + a[−] e−∂ + r[−3] e−3∂ ) , 2
(4.14)
with a and r as in Eq. (4.1). The determination of the lower order terms in this expansion amounts to a lenghty, but not difficult computation. For example, we have 1 1 r[−3] = + 2 (u[−3] − u[−2] + u[−] ) + 4 v[−3] 3 4 1 1 2 1 3 3 4 uxx + v + O(5 ) , = + u − ux + (4.15) 3 4 2 4 9 9 27 e−3∂ = 1 − 3∂ + 2 ∂ 2 − 3 ∂ 3 + 4 ∂ 4 + O(5 ) , 2 2 8
255
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
so that r[−3] e−3∂ =
1 1 1 − ∂ + 2 (6∂ 2 + u) − 3 (6∂ 3 + 3u∂ + 2ux ) 3 4 4 +
1 4 (18∂ 4 + 18u∂ 2 + 24ux∂ + 12uxx + 16v) + O(5 ) . 16
(4.16)
The other summands in Eq. (4.13) are expanded similarly; as a final result, one finds that all terms in , 2 and 3 mutually cancel, while the coefficient of 4 is the sp(2) Lax operator. Let us come to the Poisson tensor; from the representation (4.8), we can expect the expansion of Qu to contain negative powers of , of exponents ≥ −8. Explicit computation shows that the lowest order nonzero terms are in fact of order −5, and correspond to the sp(2) Poisson tensor. Indeed, we have Quu Qvu
4 = 5∂ + O
=O
1 4
1 4
,
,
Quv
Qvv
4 = 5∂ + O
1 4
,
1 = 5 (−∂ 3 + 2u∂ + ux ) + O
1 4
(4.17)
,
thus proving the thesis.
Now we consider the Hamiltonians, the vector fields and the companion operators As (u); to obtain from their expansions the homologous objects of the sp(2) theory, we need a recombination scheme of the type already applied to the KM system. To illustrate this scheme, we will first give the constructive rule for the coefficients of the recombination, and subsequently state the general recombination theorem. Definition 4.2. (i) p0 (ξ) is the function of a real parameter ξ defined by p0 (ξ) := log
1 1 + ξ + ξ2 3 2
;
(4.18)
(ii) ps (ξ) is the polynomial in ξ defined by ps (ξ) :=
1 rem(`(ξ, z)2s ) , 2s
(4.19)
where the “reminder” rem is the coefficient of the zero order term in z, and one considers the (2s)th power of the rational function 1 `(ξ, z) := 2
1 1 + ξ + ξ2 3 2
1 1 1 1 3 3+ ξ z+ . z + 3 − z 2 2 z
(4.20)
256
C. MOROSI and L. PIZZOCCHERO
The relation between the above objects and the V2 Hamiltonians is easily illustrated. First of all, let us note that, for u(x) = const. = 2, v(x) = const. = 1, Eq. (4.1) f0 becomes a = −3 − 12 2 , r = 13 + 12 2 + 4 . Let us evaluate the Hamiltonian at this point; comparing with Eq. (4.18), we see that f0 (2, 1) = p0 (ξ) ξ=2 . Also, if we consider the V2 Lax operator and compare with Eq. (4.20), we see that the expression of L (2, 1) is obtained from `(ξ, z) with the substitutions ξ 7→ 2 , z 7→ ∆ . From Table IIa, Eq. (4.19) and the above result on f0 we conclude that fs (2, 1) = ps (2 )
(s = 0, 1, 2, . . . ) .
(4.21)
We remark that, for each s ≥ 1, the polynomial ps has degree 4s in ξ. In particular, we have p1 (ξ) =
7 41 5 1 1 + ξ + ξ2 + ξ3 + ξ4 , 18 6 24 4 4
p2 (ξ) =
133 2 271 3 1143 4 23 5 2957 293 + ξ+ ξ + ξ + ξ + ξ 432 72 288 96 256 16 +
23 6 3 3 ξ + ξ7 + ξ8 , 64 16 32
(4.22)
54515 2 3065 3 1945195 4 2048495 176315 + ξ− ξ + ξ + ξ p3 (ξ) = 69984 7776 31104 144 41472 +
315455 5 1118665 6 77545 7 8095 8 255 9 ξ + ξ + ξ + ξ + ξ 13824 165888 9216 1152 128
+
5 10 5 5 ξ + ξ 11 + ξ 12 . 12 32 96
Futhermore, if we truncate the series p0 (ξ), say to order 4, we get 15 3 27 63 p0 (ξ) = − log 3 + ξ + ξ 2 − ξ 3 + ξ 4 + O(ξ 5 ) . 2 8 8 64
(4.23)
Definition 4.3. Let s be any nonnegative integer and consider an (s + 2)tuple of real numbers (cs,−1 , cs,0 , cs,1 , . . . , cs,s ). We say that the coefficients (cs,j ) (j = −1, . . . , s) satisfy the (Rs ) condition if cs,−1 +
s X j=0
cs,j pj (ξ) =
22s−2 2s + 1
s + 12 s+1
ξ s+1 + O(ξ s+2 ) .
(4.24)
We note that the determination of a system of coefficients satisfying the above condition amounts to solve a system of (s + 2) linear equations, in the same number of unknowns; for s = 0, 1, . . . , 6 we have checked that the system admits a unique solution. For s = 0, 1, 2, 3 the explicit expressions of the coefficients cs,j are:
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
c0,−1 =
1 log 3 , 12
c0,0 =
1 , 12
c1,−1 =
5 41 + log 3 , 108 54
c1,0 =
5 , 54
c2,−1
1 c1,1 = − , 6
257
(4.25)
77 1037 77 37 2 − log 3 , c2,0 = − , c2,1 = , c2,2 = − , =− 648 486 486 27 9
c3,−1 = −
584669 221 221 2675 332 16 − log 3 ,c3,0 = − , c3,1 = ,c3,2 = − ,c3,3 = . 96228 729 729 297 99 55
We conjecture that a unique solution exists for each nonnegative s. This would be granted if one were able to prove that, for each s, the matrix of elements pkj := coefficient of ξ j in pk (ξ) (k = 0, 1, . . . , s; j = 1, 2, . . . , s + 1) has a nonzero determinant, which is a nontrivial combinatorial problem. Let s be any nonnegative integer and (cs,−1 , cs,0 , . . . cs,s ) an (s + 2)-tuple of real numbers; define the recombinations hs := cs,−1 +
s X j=0
cs,j fj ,
Zs :=
s X
cs,j Xj ,
Bs :=
j=1
s X
cs,j Aj
(4.26)
j=1
(intending Z0 := 0 and B0 := 0). We note that, by construction, it is dL = [Bs , L ] , dθs
Zs = Q dhs
(4.27)
(denoting with d/dθs the derivative along Zs ). Proposition 4.4. If the sequence (cs,−1 , cs,0 , . . . cs,s ) satisfies the (Rs ) condition, at each point u ∈ U it is (2) (u) + O(2s+3 ) , hs (u) = 2s+2 hsp s
(4.28)
Zs (u) = 2s−3 Zssp(2) (u) + O(2s−2 ) ,
(4.29)
Bs (u) = 2s−3 Bssp(2) (u) + O(2s−2 ) .
(4.30)
This proposition tells us that the lowest order terms in the -expansion of the recombined V2 Hamiltonians and vector fields belong to the sp(2) theory; the next section will be fully devoted to the proof of this fact. As we will see, the (Rs ) condition is, essentially, the statement of Eq. (4.28) at u = (2, 1); so, proving Proposition 4.4 amounts to showing that Eqs. (4.28–4.30) hold everywhere on the phase space U if (4.28) is satisfied at this particular point . As an exercise, let us check directly the statements of Proposition 4.4 for the first Hamiltonian functions. If we consider the Hamiltonians of Table IIa, express a and r in terms of u and v via Eq. (4.1) and expand in , we find:
258
C. MOROSI and L. PIZZOCCHERO
Z Z 3 3 f0 (u) := − log 3 + 2 dx u − 4 dx (3u2 − 32v) 4 32 Z Z Z 9 3 8 9 dx (27u4 + 6 dx (u3 + 4u2x − 16uv) − 7 dx ux v − 64 4 1024 + 288uu2x + 112u2xx − 576u2v + 1152uxxv + 1536v 2) + O(9 ) , Z Z Z 5 1 41 1 + 2 dx u + 4 dx (3u2 + 16v) − 6 dx (u2x − 4uv) 18 12 96 32 Z Z 1 8 1 dx (7u2xx + 72uxxv + 96v 2 ) + O(9 ) , + 7 dx ux v + 8 384 Z Z 1 4 2957 293 2 + dx u + dx (453u2 − 1280v) f2 (u) := (4.31) 432 144 1152 Z Z 5 1 6 dx (31u3 − 348u2x + 960uv) + 7 dx ux v + 768 4 Z 1 8 dx (21u4 − 2784uu2x + 5072u2xx + 3072u2v + 12288 f1 (u) :=
+ 35328uxxv + 42240v 2) + O(9 ) , f3 (u) :=
Z Z 5 2048495 176315 2 + 4 dx (81429u2 − 369328v) dx u + 69984 15552 124416 Z Z 84965 7 5 6 dx (5075u3 − 30439u2x + 67972uv) + dx ux v + 41472 10368 Z 5 8 dx (28485u4 − 1432656uu2x + 1576324u2xx + 1334304u2v + 1990656 + 9825696uxxv + 12880896v 2) + O(9 ) .
The coefficients satisfying the (Rs ) condition for s = 0, 1, 2, 3 have been already computed; using their explicit expressions (4.25), together with the previous expansions, one sees that the recombinations hs (s = 0, 1, 2, 3) with these coefficients give at the lowest orders the corresponding sp(2) objects, in agreement with Proposition 4.4. For example, 221 584669 221 2675 332 16 − log 3 − f0 (u) + f1 (u) − f2 (u) + f (u) 96228 729 729 297 99 55 3 Z 1 8 = dx (5u4 − 48u2 v + 192v 2 − 4uu2x + 32uxxv + 4u2xx) + O(9 ) 128
h3 (u) = −
(2) = 8 hsp (u) + O(9 ) . 3
(4.32)
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
259
5. Proof of Proposition 4.4 Proposition 4.4 is a straightforward outcome of the subsequent three lemmas; most of the effort is required for proving the second one. Let us start evaluating the sp(2) Hamiltonians at the particular point u = (2, 1). Lemma 5.1. For s = 0, 1, 2, . . . , it is (2) (2, 1) = hsp s
22s−2 2s + 1
s + 12 s+1
.
(5.1)
Proof. The sp(2) Lax operator at u = (2, 1) is Lsp(2) (2, 1) = ∂ 4 + 2∂ 2 + 1 = (∂ + 1)2 , and so 2
(Lsp(2) )
2s+1 4
(2, 1) = (∂ 2 + 1)
2s+1 2
=
+∞ X s+ k=0
k
1 2
∂ 2s+1−2k .
(5.2)
The term of order −1 in ∂ in the above sum corresponds to k = s + 1, and the relation 2s+1 22s−2 (2) Res(Lsp(2) ) 4 (2, 1) hsp (2, 1) = (5.3) s 2s + 1
yields Eq. (5.1).
After these preliminaries, we fix a nonnegative integer, and consider an arbitrary (s + 2)-tuple (cs,−1 , cs,0 , cs,1 , . . . , cs,s ) of real coefficients (excluding that all of them are zero). The statements we will make in the forthcoming Lemma 5.2 do not depend on the (Rs ) condition, so for the moment we do not put it on the coefficients cs,j . Let us form on the phase space U the linear combinations hs , Zs and Bs as in Eq. (4.26). From the definitions of the Hamiltonians and the operators and from the explicit expression (4.1) of the field rescaling it is easily inferred that, at each point P u, there is an expansion hs (u) = γs + m m hs,m (u) where γs is a constant, m goes from 2 to +∞, and the hs,m are (integrals of) differential polynomial densities in u, v, without constant terms. We note that it cannot be hs,m (u) = 0 identically in u for each m, because the relation hs (u) = γs would imply the V2 Hamiltonians f0 , . . . , fs and the constant function 1 to be linearly dependent, which is clearly false. Definition 5.2. Throughout the rest of the section, q will denote the lowest integer such that the function hs,q is not identically zero on U. Summing up, we have hs (u) = γs +
+∞ X
m hs,m (u) ,
q ≥ 2,
hs,q 6= 0 ;
(5.4)
m=q
from here and from Eq. (4.13) we infer that the vector field Zs = Q dhs has an expansion
260
C. MOROSI and L. PIZZOCCHERO
Zs (u) =
+∞ X
m Zs,m (u) ,
Zs,q−5 = Qsp(2) dhs,q−5 .
(5.5)
m=q−5
As for the operator Bs , independently of any consideration envolving q we can state that there is an expansion of the form Bs (u) =
+∞ X
m Bs,m (u) ,
(5.6)
m=1
where the Bs,m (u) are differential operators of order ≤ m in ∂. The skew-symmetry of Bs (u) is inherited by all these operators (this also explains why the expansion (5.6) does not have a term of order zero in : the zero order part is necessarily a constant, and the skew-symmetry forces the constant to be zero). Finally, for the Lax operator we can write, recalling Eq. (4.12), +∞ X 8 m Lm (u) , L (u) = − + 4 Lsp(2) (u) + 3 m=5
(5.7)
where the Lm (u) are differential operators of order ≤ m. The coefficients of the powers of ∂ in all the operators Bs,m (u), Lm (u) are differential polyomials in the u variables. Lemma 5.2. (i) The integer q in Eq. (5.4) is even, so that we can write q = 2p + 2 with p = 0, 1, 2, . . . ; (ii) there is a nonzero constant κs such that, at any point u ∈ U, (2) (u) ; hs,q (u) = κs hsp p
(5.8)
Zs,q−5 (u) = κs Zpsp(2) (u) ;
(5.9)
Bs,q−5 (u) = κs Bpsp(2) (u) ;
(5.10)
(iii) it is also (iv) if q ≥ 6, it is furthermore, for q > 6 the lower order coefficients in the expansion (5.6) are zero: Bs,m (u) = 0
for 1 ≤ m ≤ q − 6 .
(5.11)
Proof. We will give ad hoc proofs for each value of q lower than 6, and a general argument for the higher values. Our analysis will also show that q cannot be odd. We start expanding the transformation (4.1) up to order 3 in . It is easily found that, for a = Ψ (u) and each integer k, 1 2 1 3 4 a[k] = −3 − 4 u − 4 k ux + O( ) , (5.12) 1 1 2 1 3 4 r[k] = + u + (1 + k) ux + O( ) ; 3 4 4
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
261
R R this implies hs (u) = const. + const. 2 dx u + const. 3 dx ux + O(4 ). But R R (2) (u) and dx ux = 0, so we can write dx u = 16hsp 0 (2) (u) + O(4 ) . hs (u) = const. + const. 2 hsp 0
(5.13)
Let us compare this result with Eq. (5.4). If q = 2, the constant multiplying 2 in Eq. (5.13) is nonzero; denoting this constant with κs , we can write (2) (u) , hs,2 (u) = κs hsp 0
(5.14)
which is just the statement (ii); statement (iii) follows immediately from Eq. (5.14), (2) = Z0sp(2) (incidentally, from the second relation (5.5) and the fact that Qsp(2) dhsp 0 we recall that this vector field is zero). So, the lemma is proved for q = 2; since Eq. (5.13) has no term of order 3 in , we also see that the case q = 3 cannot occur. All cases with q ≥ 4 can be attacked from the Lax formulation of the vector field Zs : at any point u, we write LZs L (u) = [Bs (u), L (u)] ,
(5.15)
where L is the Lie derivative (in view of the subsequent computations, the notation d/dθs of Eq. (4.27) is no more convenient). The strategy we employ is to insert the expansions (5.5) and (5.7) in the above Lax equation, and draw the appropriate conclusions. Let q = 4; substituting the expansions in Eq. (5.15), we see that the coefficient of 3 is LZs,−1 Lsp(2) (u) in the l.h.s, and zero in the r.h.s.. The equality LZs,−1 Lsp(2) (u) = 0 impliese Zs,−1 (u) = 0 .
(5.16)
But, as we know, Z1sp(2) (u) = 0, so statement (iii) is trivially satisfied in the present case. In order to derive (ii), we note that Eq. (5.16) and the second relation (5.5) imply Qsp(2) dhs,4 = 0. On the other hand, any Casimir of the Poisson tensor Qsp(2) (coming from a differential polynomial density in u, v, without constant terms) is a linear combination with constant coefficients of the fundamental R R (2) (2) 1 1 (u) = 16 (u) = 32 dx u and hsp dx (−u2 + 8v). Therefore, there Casimirs hsp 0 1 are two constants χs , κs such that (2) (2) (u) + κs hsp (u) ; hs,4 (u) = χs hsp 0 1
(5.17)
any vector field Y on the phase space U , it is (LY Lsp(2) )(u) = u∂ ˙ 2 + (v˙ + (1/2)u˙ xx )|u=Y ˙ (u) . From here, it is evident that LY Lsp(2) = 0 iff Y = 0.
e For
262
C. MOROSI and L. PIZZOCCHERO
but, from the expansion of Eq. (4.1) up to order 4 in , it is seen that no summand R of the form const. dx u can occur in the expression of hs,4 , and so χs = 0. On the contrary, κs cannot be zero, for this would contradict the assumption hs,4 6= 0. Summing up, statement (ii) is proved and the analysis of the case q = 4 is concluded. Let us show that the case q = 5 cannot occur. We assume by absurd the first nonzero term in Eq. (5.4) to be hs,5 . Expanding Eq. (5.15) and equating the coefficients of 4 in the two sides, we obtain LZs,0 Lsp(2) (u) = 0. So, we have 0 = Zs,0 = Qsp(2) dhs,5 , and the characterization just recalled for the sp(2) Casimirs (2) (2) (u) + ηs hsp (u) for some pair of constants χs , ηs . On the implies hs,5 (u) = χs hsp 0 1 other hand, analysing Eq. (4.1) up to order 5 in , we see that no terms proportional R R to dx u or dx (−u2 + 8v) can occur in hs,5 , yielding the conclusion χs = ηs = 0. This clearly contradicts the initial assumption that hs,5 does not vanish identically. The discussion of the case q = 6 starts again from Eq. (5.15). Equating the coefficients of 6 in the two sides of its expansion, we find LZs,1 Lsp(2) (u) = [Bs,1 (u), Lsp(2) (u)] ;
(5.18)
the above relation expresses the compatibility of the operator Bs,1 (u) with Lsp(2) (u), at each point u: the commutator of these operators is tangent to the “Lax submanifold” of the sp(2) theory. On the other hand, it is known [5] that any differential operator compatible with Lsp(2) (u) is a linear combination of the skew-symmetric 1
3
operators B2sp(2) (u) = (Lsp(2) )+4 (u), B3sp(2) (u) = 4(Lsp(2) )+4 (u), B4sp(2) (u) = 5
16(Lsp(2) )+4 (u), . . . (of orders 1, 3, 5, . . . in ∂) plus a linear combination of the symmetric operators 1, Lsp(2) (u), (Lsp(2) )2 (u), . . . (of orders 0, 4, 8, . . . ). But Bs,1 (u) is skew-symmetric and of order ≤ 1, so the above considerations imply Bs,1 (u) = κs B2sp(2) (u)
(5.19)
for some constant κs .f This fact, together with Eq. (5.18), gives Zs,1 (u) = κs Z2sp(2) (u) ;
(5.20)
(2) up since Zs,1 = Qsp(2) dhs,6 (see Eq. (5.15)), we also find that hs,6 is κs times hsp 2 to a Casimir, i.e.,
(2) (2) (2) (u) + χs hsp (u) + ηs hsp (u) . hs,6 (u) = κs hsp 2 0 1
(5.21)
Finally, the analysis of Eq. (4.1) up to order 6 in ensures that the constants χs , ηs are zero. Then κs is necessarily nonzero, and Eqs. (5.19–5.21) give the statements (ii), (iii) and (iv) for the present case q = 6. f In principle, κ could be a real functional of the variables u (say, the integral over x of some s u-dependent density). In fact, this is not the case: such a kind of u-dependence is excluded for all the differential operators Bs,m (u). The same considerations apply to the other constants (denoted with κs , χs , ηs , ξs,i ) appearing in the rest of the proof.
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
263
Let us pass to the general case q > 6, q even. Expanding Eq. (5.15), and equating the coefficients of 5 , 6 , 7 , . . . , q−2 in the two sides, we obtain q − 6 equations 0 = [Bs,1 (u), Lsp(2) (u)] , 0 = [Bs,2 (u), Lsp(2) (u)] + [Bs,1 (u), L5 (u)] , (5.22) 0 = [Bs,3 (u), Lsp(2) (u)] + [Bs,2 (u), L5 (u)] + [Bs,1 (u), L6 (u)] , ... ... ... 0 = [B sp(2) (u)] + [Bs,q−7 (u), L5 (u)] + · · · + [Bs,1 (u), Lq−3 (u)] s,q−6 (u), L whereas, equating the coefficients of q−1 , we get LZq−5 Lsp(2) (u) = [Bs,q−5 (u), Lsp(2) (u)] + [Bs,q−6 (u), L5 (u)] + · · · + [Bs,1 (u), Lq−2 (u)] .
(5.23)
The system (5.22) can be treated iteratively. The first equation tells us that Bs,1 (u) is in the centralizer of Lsp(2) (u); on the other hand, the centralizer of the Lax k operator consists of the linear combinations of the fractional powers (Lsp(2) ) 4 (u) (k ∈ Z), and the differential operators commuting with Lsp(2) (u) are the linear combinations of nonnegative integer powers, which are all symmetric. Since Bs,1 (u) is skew-symmetric, it is necessarily zero. Inserting this result into the second Eq. (5.22), we obtain [Bs,2 (u), Lsp(2) (u)] = 0, and the same argument employed above gives Bs,2 (u) = 0. Iterating this procedure, we obtain Bs,1 (u) = Bs,2 (u) = · · · = Bs,q−6 (u) = 0 .
(5.24)
From here and Eq. (5.23), we infer LZs,q−5 Lsp(2) (u) = [Bs,q−5 (u), Lsp(2) (u)] .
(5.25)
Now the characterization (recalled after Eq. (5.18)) for the operators compatible with Lsp(2) (u) and the skew-symmetry of Bs,q−5 (u) imply that the latter is a finite linear combination with constant coefficients of the operators Bisp(2) (u) = 2i−3
4i−2 (Lsp(2) )+ 4 (u) (i = 2, 3, 4, . . . ): Bs,q−5 (u) =
X
ξs,i Bisp(2) (u) .
(5.26)
i
In order to characterize the constants ξs,i , we apply Eq. (5.26) at the particular point u = (0, 0). Let us consider the transformation (4.1) at this point, and recall that Bs,q−5 (0, 0) is the coefficient of q−5 in the expansion of Bs (0, 0); also, let us explicitate Bisp(2) (0, 0). In this way we find Bs,q−5 (0, 0) = const. ∂ q−5 ,
Bisp(2) (0, 0) = 4i−2 ∂ 2i−3 .
(5.27)
264
C. MOROSI and L. PIZZOCCHERO
Inserting these results in Eq. (5.26), we obtain that ξs,i = 0 for 2i − 3 6= q − 5; if p is defined by q = 2p + 2, this implies ξs,i = 0 for i 6= p, and setting κs := ξs,p we can write Bs,q−5 (u) = κs Bpsp(2) (u) .
(5.28)
This result, together with Eq. (5.25), gives Zs,q−5 (u) = κs Zpsp(2) (u)
(5.29)
(2) and from the second relation (5.5) we also infer hs,q (u) = κs hsp (u) up to a Casimir p sp(2) , i.e., of Q
(2) (2) (2) hs,q (u) = κs hsp (u) + χs hsp (u) + ηs hsp (u) . p 0 1
(5.30)
R Again, we can exclude the appearing of any term proportional to either dx u or R dx (−u2 + 8v) in hs,q (u), so χs = ηs = 0, while κs 6= 0 to avoid a contradiction with the hypothesis hs,q 6= 0. On the grounds of Eqs. (5.24), (5.28–5.30) statements (ii), (iii) and (iv) are proved. To complete the proof of the lemma, we only have to show that the case q > 6, q odd cannot occur. Assume by absurd that the first nonzero term hs,q in the expansion (5.4) appears for q odd; then Eqs. (5.22–5.23) still hold, and their analysis leads again to the conclusion (5.26). The coefficients ξs,i are characterized by Eq. (5.27) where the exponent q − 5 is now even, while 2i − 3 is obviously odd. In conclusion, ξs,i = 0 for each i and Bs,q−5 (u) = 0, which also implies Zs,q−5 = Qsp(2) dhs,q = 0. The Casimir hs,q is necessarily a linear combination (2) (2) , hsp ; the usual analysis gives hs,q = 0, which contradicts the initial of hsp 0 1 assumption. Writing q = 2p + 2, we can express the statements of the previous Lemma 5.2 in the following way: (2) (u) + O(2p+3 ) (p ∈ {0, 1, 2, . . . }, κs 6= 0) , (5.31) hs (u) = γs + κs 2p+2 hsp p
Zs (u) = κs 2p−3 Zpsp(2) (u) + O(2p−2 ) ,
(5.32)
Bs (u) = κs 2p−3 Bpsp(2) (u) + O(2p−2 ) .
(5.33)
(At a first sight, Eq. (5.33) is granted by Lemma 5.1 only for q ≥ 6, i.e., p ≥ 2; however, if p = 0 or 1, (5.33) is again satisfied in a trivial way, for in these cases it is Bpsp(2) (u) = 0 and the exponent 2p − 2 is ≤ 0, while the operator Bs (u) has an expansion in positive powers of .) Lemma 5.3. Let the (s + 2)-tuple (cs,−1 , cs,0 , cs,1 , . . . , cs,s ) satisfy the (Rs ) condition (4.24). Then the integer p and the constants γs , κs appearing above are determined as follows:
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
p = s,
γs = 0 ,
κs = 1 ,
265
(5.34)
and the relations (4.28–4.30), appearing in Proposition 4.4, are satisified. Proof. Let us specialize Eq. (5.31) to the point u = (2, 1); on account of Lemma 5.1, we obtain hs (2, 1) = γs +
22s−2 2s + 1
s + 12 s+1
κs 2p+2 + O(2p+3 ) .
(5.35)
Ps On the other hand, Eqs. (4.26) and (4.21) give hs (2, 1) = cs,−1 + j=0 cs,j fj (2, 1) = Ps cs,−1 + j=0 cs,j pj (2 ). From here and from the (Rs ) condition (4.24) we infer hs (2, 1) =
22s−2 2s + 1
s + 12 s+1
2s+2 + O(2s+4 ) .
Comparing Eqs. (5.35–5.36) we obtain the thesis.
(5.36)
On the grounds of the previous results, the proof of Proposition 4.4 is now trivial; in fact, the statements (4.26–4.28) of this proposition are obtained inserting Eq. (5.34) into Eqs. (5.31–5.33). 6. On the Continuous Limit of the VN Theories Let N be an arbitrary integer, and A be the phase space of the VN system in N fields a = (a1 , a2 , . . . , aN ). The Lax operator for this theory is given by Eq. (1.3). Also, let us consider the sp(N ) KdV theory in N fields u = (u1 , . . . , uN ), and denote with U its phase space. Using the identity ∂mf =
m X m h=0
h
f(m−h) ∂ h
(6.1)
(where f(m−h) denotes the (m − h)th derivative of the function f ) we obtain for the Lax operator (1.6) the more explicit representation
L
sp(N )
(u) = ∂
2N
2N −1 [N −j/2] 1 X X 2N − 2l ul(2N −2l−j) ∂ j . (6.2) + (1 + δ2l,2N −j ) j 2 j=0 l=1
In order to connect the continuous limit of the VN system with the sp(N ) KdV, we search for a transformation Ψ := U → A u = (u1 , . . . , uN ) 7→ a = (a1 , . . . , aN ) = Ψ (u1 , . . . , uN )
(6.3)
such that, setting L (u) := L (a) a=Ψ (u) , one gets L (u) = const. + 2N Lsp(N ) (u) + O(2N +1 ) .
(6.4)
266
C. MOROSI and L. PIZZOCCHERO
The results obtained for N = 1, 2 suggest the following ansatz : ! p 2p−2l X X 2l (p = 1, 2, . . . , N ) . Cp,l,q ul[q] Ψ (u) = a , ap := Cp +
(6.5)
q=0
l=1
Here Cp and Cp,l,q (1 ≤ p ≤ N , 1 ≤ l ≤ p, 0 ≤ q ≤ 2p − 2l) are unknown constants, that we will often denote collectively with the symbol C. (We take the occasion to repeat that the subscript [. . . ] denotes the shift, while (. . . ) refers to x-derivatives. So, formulas (6.2) and (6.5) contain, respectively, the derivative of order 2N − 2l − j of the function ul and the shifted function ul[q] (x) = ul (x + q).) The map Ψ defined by Eq. (6.5) is invertible if the coefficients Cp,p,0 are nonzero for each p; the inverse map sends a into u, where ! l 2l−2p X X 1 (6.6) Dl,p,q ap[q] , ul = 2l Dl + p=1 q=0 the constants Dl and Dl,p,q being uniquely determined by the constants C. Let us show that the ansatz (6.5) reduces the problem of the continuous limit to the problem of solving a linear system with numerical coefficients. In fact, we have: Lemma 6.1. The condition (6.4) on the continuous limit of the VN Lax operator is fulfilled by the map Ψ if and only if (i) the coefficients Cp (1 ≤ p ≤ N ) satisfy a linear system of N equations; (ii) for each l between 1 and N, the (N + 1 − l)2 coefficients Cp,l,q (l ≤ p ≤ N, 0 ≤ q ≤ 2p − 2l) satisfy a linear system with the same number of equations. Proof. We start by computing the -expansion of the VN Lax operator, after having expressed the fields ap by means of Eq. (6.5). For each integer q, the shift of ap by q is ap[q] = Cp +
p +∞ X X
2l+r
r=0 l=1
2p−2l X h=0
(q + h)r Cp,l,h ul(r) r!
! .
(6.7)
From here and Eq. (4.11) we obtain, after some manipulations with multiple sums, L (u) =
N X
+∞ X
Cp +
p=1
m Lm (u)
(6.8)
m=2
where, for each m, Lm (u) is a differential operator with explicit expression
Lm (u) = bm (C) +
m−1 X j=0
[ m−j 2 ]
X l=1
bm,j,l (C)ul(m+j−2l) ∂ j .
(6.9)
267
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
In the above formula, we have put 0 N X bm (C) := 2 (2k − 1)m Ck
for m odd (6.10)
for m even
k=1
bm,j,l (C) :=
N 2k−2l X X 1 (−1)j q m−2l−j + (1 − 2k + q)m−2l−j Ck,l,q . 2(m − 2l − j)!j! q=0 k=l
(6.11) Our purpose is to pick up the coefficients C so as to fulfill Eq. (6.4), i.e. the conditions Lm (u) = 0 for
2 ≤ m ≤ 2N − 1 ,
L2N (u) = Lsp(N ) (u) .
(6.12)
Let us recall the symmetry of the operators Lsp(N ) (u) and L (u); the latter also implies all the operators Lm (u) to be automatically symmetric. On the other hand, P P k k for two symmetric differential operators F = k fk ∂ and G = k gk ∂ to be equal, it is sufficient that fk = gk for each even k. Due to this remark, Eq. (6.4) holds if and only if the coefficients of all even powers of ∂: (a) are zero in Lm (u), for 2 ≤ m ≤ 2N − 1; (b) are equal in L2N (u) and Lsp(N ) (u). This leads to the following set of equations: bm (C) = δm,2N
(2 ≤ m ≤ 2N ,
bm,j,l (C) = δm,2N (1 + δ2l,2N −j )
m even) ; 2N − 2l j
(6.14)
2 ≤ m ≤ 2N ,
0 ≤ j ≤ m −1,
j even ,
(6.13)
m−j 1≤l≤ 2
.
On account of the definition (6.10), the Eqs. (6.13) form a linear system of N equations in N unknowns Ck (k = 1, . . . , N ); so, statement (i) of this lemma is proved. Recalling the definition (6.11) we also see that, for each fixed l between 1 and N , the Eqs. (6.14) (with variable m and j) form a linear system in the unknowns Ck,l,q (k = l, . . . , N , q = 0, . . . , 2k − 2l). It is easily checked that the number of unknowns and the number of equations are both equal to (N + 1 − l)2 for each l, and this proves statement (ii) of this Lemma. On account of the previous lemma, the sp(N ) Lax operator can be attained as a continuous limit of the VN operator through a field rescaling of the form (6.5), provided that the linear systems mentioned in items (i) and (ii) have a solution; clearly, the solution exists and is unique if the square matrices associated to these systems have nonzero determinant.
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The N ×N matrix associated to the system (6.13) is essentially of a Vandermonde type, its determinant can be proved to be nonzero. As for the matrices of the systems (6.14) (one for each l between 1 and N ), showing that their determinants are nonzero is a nontrivial task, of which we do not have a proof for arbitrary N . Of course, from the treatment given in the previous sections for the cases of sp(1) and sp(2), we already know that the envolved determinants are nonzero for N = 1 and 2; we have checked directly that the determinants are nonzero also for N = 3, 4. In particular, by solving the systems (6.13–6.14) for N = 3 we have found the transformation Ψ sending (u1 , u2 , u3 ) into 15 1 2 a1 := + u1 , 4 8 3 2 5 1 4 a2 := − 8 − 16 (u1 − u1[] + u1[2] ) − 8 u2 , 1 3 + 2 (u1 − u1[] + u1[2] − u1[3] + u1[4] ) a3 := 40 16 1 − 4 (2u2 − 5u2[] + 2u2[2] ) + 6 u3 . 8
(6.15)
Composing with this tranformation the V3 Lax operator (given by Eq. (1.1) with 6 sp(3) (u) + O(7 ); the sp(3) Lax operator (see N = 3), we find L (u) = 16 5 + L Eq. (6.2)) is given explicitly by Lsp(3) (u) = ∂ 6 + u1 ∂ 4 + 2u1x ∂ 3 + (u2 + 3u1xx)∂ 2 1 + (2u1xxx + u2x )∂ + u3 + u1xxxx . 2
(6.16)
With convenient choices of the functional spaces A and U to which these triplets of fields belong, the map Ψ is one-to-one, with inverse sending (a1 , a2 , a3 ) into 1 u1 = 2 (−30 + 8a1 ) , 1 u = 4 (40 − 12a1 + 12a1[] − 12a1[2] − 8a2 ) , 2 16 7 1 7 u3 = 6 − 5 − 2 a1 + 11a1[] − 14a1[2] + 11a1[3] − 2 a1[4] . − 2a + 5a − 2a + a 2 3 2[] 2[2]
(6.17)
In a similar way, we have found a unique transformation of the form (6.5), with inverse as in Eq. (6.6), relating the V4 and sp(4) Lax operators.
ON THE CONTINUOUS LIMIT OF INTEGRABLE LATTICES II . . .
269
7. Comments and Acknowledgments The discussion of the 7→ 0 limit for the lattices considered in this paper and our previous one is centered on the algebraic/geometrical structures, and not on the behaviour of the solutions. In [1] the examination of this problem was confined to the continuous limit of the one-soliton solutions for the KM system and to some comments on the rigorous convergence theorem of Schwarz [13] for the 7→ 0 limit of the general initial data problem on the lattice. Here, we repeat some of the considerations of [1] and take the occasion for quoting, apart from references [13, 14] already mentioned therein, the paper of Lax, Levermore and Venakides [15]; we acknowledge the anonymous referee for this bibliographical indication. In [15] the authors analyze a limit process for the KM system, based on the rescaling a(x) 7→ u(x) := 14 a(x)2 which is different from the transformation (2.1) and more similar to the one considered by Brockett and Bloch [14] for the Toda lattice. Under the rescaling of [15], the zero-spacing limit of the KM vector field X1 is the vector field Z H (u) := uux , giving rise to the so-called Hopf (inviscid Burgers) evolution equation. The numerical experiment performed by the authors for the Cauchy problem with a particular initial datum shows the existence of a critical time tcrit , after which the solution u (t) of the lattice evolution equation develops irregular space oscillations, with wavelength proportional to . So, for t > tcrit the zero spacing limit of u (t) does not exist in a strong sense; this is the discrete counterpart of a shock phenomenon related to the limiting equation ut = uux , namely the fact that the x derivative of the solution u(t) develops a divergency in a finite time, for certain choices of the initial datum; such a kind of shock also occurs in the limiting continuous system of reference [14]. Let us point out some facts which can explain the difference between this singular behaviour and the regular behaviour outlined in [13] for the zero-spacing limit of the lattice solutions, under a limit process for the KM system similar to the one considered in [1] and yielding the KdV equation. It is known that the ordinary KdV and all the equations in the same hierarchy possess a global flow in each Sobolev space (see for instance [16] and references therein); from a technical viewpoint, global existence and uniqueness of the KdV flows can be traced back to the existence of conserved Hamiltonian densities, depending quadratically on the higher order x-derivative of the field u. These conservation laws can be employed to obtain a priori bounds on the Sobolev norms; discrete counterparts for such estimates are possible for the KM hierarchy, yielding the -independent a priori bounds which control in [13] the zero-spacing limit of the solutions. The situation is quite different for the Hopf hierarchy considered in [15]: the R conserved functionals have the form dx un (n = 1, 2, 3, . . . ) and do not control the derivatives of the field. With the rescaling considered therein, the conserved quantities of the KM system give the same functionals in the zero spacing limit, so they do not allow any bound on the derivatives of the interpolated solution; a similar remark can be made for the system discussed in [14].
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The KdV-type hierarchies obtained from the limit process in this paper do possess Hamiltonian densities which are quadratic in the higher derivatives of the fields (2) (2) and hsp in Table IIb contain the squares of (for instance, the Hamiltonians hsp 2 3 ux and uxx respectively; quadratic terms in uxxx, uxxxx , . . . and vx , vxx , . . . appear if one writes down the higher order Hamiltonians). So, in the situation considered here a regular behaviour ` a la Schwarz (globality of the flow for the limiting KdV’s and convergence of the solutions for all times, in the zero-spacing limit of the Volterra systems) seems much more likely than the singular behaviour of the dispersionless equations considered in [14, 15]. This paper has been partially supported by Consiglio Nazionale delle Ricerche, G. N. F. M, and by Ministero dell’Universit` a e della Ricerca Scientifica, Project “Metodi Geometrici e Probabilistici in Fisica Matematica”. References [1] C. Morosi and L. Pizzocchero, “On the continuous limit of integrable lattices I. The Kac-Moerbeke system and KdV theory”, Commun. Math. Phys. 180 (1996) 505–528. [2] Y. B. Zeng and S. Rauch-Woyciekowski, “Continuous limit for the Kac-van Moerbeke hierarchy and for their restricted flows”, J. Phys. A 28 (1995) 3825–3840. [3] M. Bruschi, S. V. Manakov, O. Ragnisco and D. Levi, “The nonabelian Toda latticediscrete analogue of the matrix Schr¨ odinger spectral problem”, J. Math. Phys. 21 (1980) 2749–2753. [4] I. Merola, O. Ragnisco and T. Gui-Zhang, “A novel hierarchy of integrable lattices”, Inverse Problems 10 (1994) 1315–1334. [5] V. G. Drinfeld and V. V. Sokolov, “Lie algebras and equations of Korteweg-de Vries type”, J. Sov. Math. 30 (1985) 1975–2036. [6] A. W. Knapp, “Lie Groups beyond an Introduction”, Birkh¨ auser, Basel, 1996. [7] B. A. Kupershmidt, “Discrete Lax equations and differential-difference calculus”, Ast´erisque 123 (1985) Soc. Math. de France. [8] C. Morosi and L. Pizzocchero, “R-matrix theory, formal Casimirs and the periodic Toda lattice”, J. Math. Phys. 37 (1996) 4484–4513. [9] W. Oevel and O. Ragnisco, “R-matrices and higher Poisson brackets for integrable systems”, Physica A161 (1989) 181–220. [10] L. C. Li and S. Parmentier, “Nonlinear Poisson structures and r-matrices”, Commun. Math. Phys. 125 (1989) 545-563. [11] P. A. Damianou, “Multiple Hamiltonian structures for Toda-type systems”, J. Math. Phys. 35 (1994) 5511–5541. [12] C. Morosi and L. Pizzocchero, “On the biHamiltonian interpretation of the Lax formalism”, Rev. Math. Phys. 7 (1995) 389–430. [13] M. Schwarz, “Korteweg-de Vries and nonlinear equations related to the Toda lattice”, Adv. in Math. 44 (1982) 132–154. [14] R. W. Brockett and A. Bloch, “Sorting with the dispersionless limit of the Toda lattice”, Proc. of the CRM Workshop on Hamiltonian systems, transformation groups and spectral transform methods (eds. J. Harnad and J. E. Marsden) (1990) 103–112. [15] P. D. Lax, C. D. Levermore and S. Venakides, “The generation and propagation of oscillations in dispersive initial value problems and their limiting behavior”, Important Developments in Soliton Theory (eds. A. S. Fokas and V. E. Zakharov) (1993), Springer, Berlin, 205–241. [16] M. Schwarz, “The initial value problem for the sequence of generalized Korteweg-de Vries equations”, Adv. in Math. 54 (1984) 22–56.
MASSLESSNESS IN n-DIMENSIONS EUGENIOS ANGELOPOULOS and MOURAD LAOUES Laboratoire Gevrey de Math´ ematique Physique Universite de Bourgogne 9 rue Alain Savary B.P. 400, F-21011 Dijon Cedex, France E-mail : [email protected] Received 27 June 1997 We determine the representations of the “conformal” group SO0 (2, n), the restriction of which on the “Poincar´ e” subgroup SO0 (1, n − 1).Tn are unitary irreducible. We study their restrictions to the “De Sitter” subgroups SO 0 (1, n) and SO0 (2, n − 1) (they remain irreducible or decompose into a sum of two) and the contraction of the latter to “Poincar´e”. Then we discuss the notion of masslessness in n dimensions and compare the situation for general n with the well-known case of 4-dimensional space-time, showing the specificity of the latter.
Introduction The formulation of a unifying theory which would include all fundamental interactions of physics is still an open problem, though the need for it was realized early in this century, when modern physics (relativistic and quantum) was introduced and in spite of unsuccessful efforts of many of its founders. A major difficulty consists in unifying the so-called gauge interactions and gravitation. A number of approaches to this question, appearing in various models such as the Kaluza–Klein theories or supergravity [7, 12], use an imbedding of the four-dimensional Minkowski spacetime into a higher dimensional one (that is, Rn endowed with a (1, n − 1)-Lorentz metric), then getting rid of the redundant spatial dimensions by various techniques, such as spontaneous compactification. On the other hand, an essential feature of relativity is the boundary character of the speed of light, which implies qualitatively distinct behaviours for massless and for massive particles. Mathematically, this is expressed by the (1,3) signature of Minkowski space and the distinction between the massless and massive case is kinematically expressed by distinct types of unitary irreducible representations (UIR) [14] of the kinematic group, the Poincar´e group P. Masslessness in four dimensions has been quite well studied from the group theoretical point of view. We shall start by recalling the relations between P and the De Sitter groups. Let Mρ be a four-dimensional manifold with constant curvature ρ. Its isometry group is the De Sitter group Gρ , which is isomorphic to SO0 (2, 3) (resp. SO0 (1, 4)) if ρ > 0 (resp. ρ < 0); a physical reason for the introduction of the curvature is that it provides efficient invariant infrared regularization in the limit of 271 Reviews in Mathematical Physics, Vol. 10, No. 3 (1998) 271–299 c World Scientific Publishing Company
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zero curvature [4]. Mρ is isomorphic to the homogeneous space Gρ /L where L is the Lorentz group SO0 (1, 3). In the limit ρ = 0, Mρ becomes the (flat) Minkowski space and Gρ contracts to the Poincar´e group P. Concerning representations, it may however happen that two nonequivalent UIRs of Gρ contract to the same massless representation of P. Moreover, if ρ < 0, the representations of P one gets by contraction have an unbounded energy spectrum. Now the conformal group G = SO0 (2, 4) acts on compactified Minkowski space. Massless UIR’s of P with discrete helicity extend uniquely to UIR of G acting on the same Hilbert space, and are the only ones with this property, besides the trivial [1]. It turns out that if such a UIR is extended to G, then restricted to the De Sitter subgroup SO0 (2, 3), and finally contracted to P, the initial representation of P is recovered. Therefore, from a kinematical point of view, the representations of the De Sitter group SO0 (2, 3) thus obtained provide a satisfactory tool for the extension of masslessness on Mρ . Furthermore one can define a gauge theory in the sense of Gupta–Bleuler and show that massless particles propagate on the light cone, and so on [2, 13]. Since the geometry of the n-dimensional Minkowski space-time is determined by its Lorentzian metric (1, n − 1), its kinematic group is Pn = SO0 (1, n − 1).Tn , and all groups related to it (Lorentz, conformal, De Sitter) are similarly defined. It is then natural to ask which properties of massless UIR’s of the Poincar´e group extend to n dimensions, independently of other considerations: this is the purpose of this paper. More precisely, we shall study here the following topics: (i) Which UIR’s U of Pn extend to irreducible representations d of the corresponding conformal group Gn = SO0 (2, n)? To be precise we are dealing with projective representations but it turns out that all the interesting ones are representations of the twofold covering for all groups concerned. (ii) Is the extended representation d unitary, and is it unique? (iii) Is the restriction d0 of d on the De Sitter subgroups SO0 (1, n) and SO0 (2, n − 1) irreducible? (iv) Can d0 be contracted to the initial one, U ? The answers to these questions for n = 4 were given in [2] (except for those concerning SO0 (1, 4)), some of the results being anterior to that paper: only (and all) UIR’s with zero mass and discrete helicity do extend to Gn with uniqueness and unitarity; the restriction to SO0 (2, 3) is irreducible unless the inducing representation of the little group is trivial (zero helicity) and it can be contracted back to the initial UIR. For structural reasons (all groups concerned have similar structure and real rank at most 2), a straightforward generalization was expected, at least for n even. It turned out, however, that though most features do indeed generalize, the constraints on the existence of the extension increase significantly with n. To be more precise, U must again be massless, that is induced by a UIR S of the little group SO0 (n − 2).Tn−2 (the Euclidean group in n − 2 dimensions). Not only S has to be
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trivial on the translations (the analogue of discrete helicity), but it must also be a very degenerate representation of SO0 (n − 2). Acceptable S are characterized by a discrete parameter 2s ∈ Z when n is even (2s is the helicity for n = 4), while for n odd S must be either trivial or spinorial. Also, the results for the De Sitter subgroup generalize, with the sole exception of the irreducibility of d0 for odd n: it reduces into the direct sum of two simple factors for spinorial S too. As far as SO0 (1, n) is concerned, d0 is always irreducible. We therefore see once more, in this simple (kinematical) group theoretical study, that the 4-dimensional space-time of special relativity and the related universes with constant curvature are really special. In higher dimensions the notion of masslessness becomes more involved and requires, in addition to zero mass, a degeneracy far greater than the requirement of discrete helicity in 4 dimensions. The paper is organized as follows. In Sec. 1 we fix notations for Gn , Pn and the normalizer Wn of Pn in Gn . Since we are interested in projective UIR’s we also present their universal coverings; in fact, as we shall see later, only twofold coverings are needed, corresponding to the covering of SO(n) by Spin(n). We also identify the compactified n-Minkowski space with the quotient Gn /Wn and describe the action of Gn on it. We then establish the unitary dual of Pn , using the orbit-stabilizer method, and discuss the possibility of extending a UIR U to Wn . Propositions 1.1 and 1.2 give the (expected) result: U must be massless, and induced by a representation S with trivial restriction on the Euclidean translation subgroup. Section 2 is devoted to determine which among the representations d of Gn can be viewed as extensions of massless UIR’s of Pn , using Lie algebraic methods. We begin by expressing the weight representations of the complexified so(N )C , in a way which can be used both for so(2, n) and for the compact real form so(n − 2) of the little group. We next translate into enveloping algebra properties the fact that the squared n-mass operator P µ Pµ is mapped to 0 by d, calling such a d a massless representation. After the study of low N , we determine the finite-dimensional ones, parameterizing them by a discrete parameter (Theorem 2.3). We next study infinite-dimensional ones, showing that on every so(n)-type the character of the so(2) which commutes with so(n) is fixed and increases in absolute value with the Casimir of so(n), keeping a fixed sign. Moreover, the lowest k-type must be a massless representation of so(n) itself: this constraint has no effect when n = 4, but does cut off a huge part in general (Theorem 2.4). Massless representations are unitary and possess an extremal weight. In the following paragraph we identify the UIR’s U S of Pn which extend to massless UIR’s of the conformal group Gn . The inducing S must be a finite-dimensional massless UIR of SO(n − 2). The expression of the generators of Gn as differential operators is uniquely determined by those of Pn . Section 3 discusses De Sitter subgroups. Irreducibility of the restriction is examined on k-types, which all have multiplicity one. As for the contraction to Poincar´e, the proof of [2] extends easily to the general case. The paper ends with a few remarks, where in particular we briefly recall and present in the light of the present study the known results for the lower dimensional cases n = 3 and n = 2.
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1. Poincar´ e and Conformal Group in n-dimensions (a) The n-Poincar´ e group Pn Let n ≥ 3 be a fixed integer. Let {eµ }µ∈J , with J = {0, 1, . . . , n − 1} be a basis of Rn ; J 0 = {1, . . . , n − 1} and define a quadratic form q, such that q(e0 ) = 1 = −q(eµ0 ) ∀ µ0 ∈ J 0 .
(1.1)
The associated symmetric bilinear form is denoted by g, with gµν = g(eµ , eν ), which is equal to q(eµ ) if µ = ν and equal to zero otherwise. The quadratic space (Rn , q) will be denoted R1,n−1 , and called the n-Minkowski space. It can be identified with its dual, the dual basis being {eµ }µ∈J , with e0 = e0
and ,
0
eµ = −eµ0
for µ0 ∈ J 0
(1.2)
and we shall write g µν = g(eµ , eν ), with the same properties as gµν . Any element x ∈ R1,n−1 has the form x = xµ eµ = xµ eµ ,
with
xµ = gµν xν ,
gµν g µλ = δνλ ,
(1.3)
where δ is the Kronecker symbol. Remark. The Einstein summation convention over the set J is used in (1.3). It will be used throughout this paper. The range of summation will not always be the same, and we shall use distinct index variables for distinct ranges of summation. 0 For instance we shall write xµ0 xµ instead of −Σ1≤µ0 ≤n−1 (xµ0 )2 , using the primed letter µ0 instead of µ to avoid confusion. If the range of summation is J, greek letters λ, µ, ν, . . . will always be used. The connected component of the Lie group of linear transformations of Rn which leave g invariant, SO0 (1, n − 1), will be called the n-Lorentz group and denoted by Ln . Its maximal compact subgroup is SO(n − 1). The twofold covering of the latter (universal if n > 3) will be denoted by Spin(n − 1) and the corresponding covering of Ln by Ln . The abelian group of translations of Rk , homeomorphic to Rk , will be denoted by Tk , for k ∈ N. The semidirect product Ln · Tn , where Ln acts canonically on Tn , will be called the n-Poincar´e group and denoted by Pn . Its twofold covering (universal if n > 3) Ln · Tn will be denoted by Pn . The Lie algebra pn of Pn is spanned by generators Xµν = −Xνµ ∈ ln = Lie (Ln ) and Pµ ∈ tn = Lie (Tn ), with µ, ν ∈ J, satisfying the commutation relations [Xλµ , Xνρ ] = gµν Xλρ − gλν Xµρ − gµρ Xλν + gλρ Xµν [Xλµ , Pν ] = gµν Pλ − gλν Pµ [Pλ , Pµ ] = 0
(1.4a) (1.4b) (1.4c)
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The element Pµ P µ = g µν Pµ Pν of the enveloping algebra U(pn ) commutes with all generators. In the classical case n = 4, when used in theoretical physics, it gives the squared mass of a particle. (b) The n-conformal group Gn Let I = {−1, 0, 1, . . . , n} = J ∪ Jˆ with Jˆ = {−1, n}. Extend the basis {eµ }µ∈J of Rn to the basis {eA }A∈I of Rn+2 , and extend the quadratic form q to Rn+2 by putting q(e−1 ) = 1 = −q(en ). The quadratic space thus obtained will be denoted R2,n ; the associated symmetric bilinear form will be again denoted by g, with gAB = C . g(eA , eB ), g AB = g(eA , eB ) for the dual basis, and gAB g BC = δA The connected group SO0 (2, n) which conserves the bilinear form g will be called the n-conformal group and denoted by Gn . Its maximal compact subgroup is SO(2)×SO(n), with universal covering (for n ≥ 3) R×Spin(n), that is, infinite-fold times twofold. The universal covering of Gn will be denoted Gn . The Lie algebra gn of Gn is spanned by generators XAB = −XBA (A, B ∈ I), with commutation relations: [XAB , XCD ] = gBC XAD − gAC XBD − gBD XAC + gAD XBC .
(1.5)
We shall denote by C the Casimir element of the enveloping algebra U(gn ), defined by 1 1 (1.6) C = XAB X BA = XAB g BC XCD g DA . 2 2 The stabilizer of the basis elements e−1 , en is obviously Ln . Moreover, the set of generators Xµ,−1 ± Xµ,n (µ ∈ J) spans an n-dimensional abelian subalgebra isomorphic to tn , on which ln acts like (1.4b), for either choice of the ± sign. The corresponding group elements have the (n + 2) × (n + 2) matrix form: 1 − q(t)/2 t ±q(t)/2 # , (1.7) 1In ±t# exp tµ (Xµ,−1 ± Xµ,n ) = −t ∓q(t)/2 ±t 1 + q(t)/2 where t# is the column vector (tµ ) and t the line vector (tµ ). The two Poincar´e subgroups thus obtained are conjugated in Gn , through the involutionary mapping: Θ = Ad exp(πXn−1,n ) . We shall write hereafter Pµ = Xµ,−1 + Xµ,n ;
Pˆµ = Xµ,−1 − Xµ,n
(1.8)
and we shall identify Tn as the subgroup spanned by exp(tµ Pµ ); the “other” translation subgroup will be denoted Tˆn , and the corresponding n-Poincar´e subgroups Pn and Pˆn . The twofold covering Pn is a subgroup of Gn (for n = 3 the universal covering of Pn is not contained in Gn ). The remaining generator D = Xn,−1 will be called the dilatation, it commutes with the Lorentz generators and its nonzero commutation relations are
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[D, Pµ ] = Pµ ;
[D, Pˆµ ] = −Pˆµ
(1.9)
One also has [Pµ , Pˆµ ] = −2(Xµν + gµν D) .
(1.10)
The normalizer Wn of Pn in Gn is the semidirect product Yn · Tn , with Yn = A × (W · Ln ), where A = {exp tD}t∈R and W = {1, w} is a group of order two, with w = exp(π(X0,−1 + Xn−1,n )). The action of w on Pµ is given by Adw (Pµ ) = εµ Pµ ;
ε0 = εn−1 = 1 ;
εj = −1 if 1 ≤ j ≤ n − 2 .
(1.11)
W · Ln is the non-connected group SO(1, n − 1). The group Wn is a maximal ˆ n = Yn · Tˆn of Pˆn , parabolic subgroup of Gn . The same holds for the normalizer W and one has the Bruhat-type decomposition: ˆ n Tn = Tˆn Wn , Gn ' Tˆn Yn Tn = W
(1.12)
that is, the set of elements which can be written in this form is a Zarisky open in Gn . To be more precise, Pn (resp. Pˆn ) stabilizes the point e = e−1 + en (resp. eˆ = e−1 −en ) of Rn+2 . The orbit of e under Gn is the isotropic cone minus the origin, that is Gn /Pn = Q = {y, y ∈ Rn+2 /y 6= 0 and yA y A = 0}. The group A × W sends e to λe (and eˆ to λˆ e), λ ∈ R−{0}, so that Gn /Wn = C0 = Q/(R−{0}) ∼ = (S1 ×Sn−1 )/Z2 is the set of directions of Q. The translation group Tn stabilizes the direction λe and acts transitively on the complementary subset of C0 . Thus the complementary ˆ n Tn in Gn is {g; ge ∈ e⊥ }. subset of W C0 is thus diffeomorphic to the compactified Tnc of Tn , that is Tnc = R1,n−1 ∪ C∞ where C∞ is the (n − 1)-dimensional compactified “light cone at infinity” [3, 8, 13]. Writing R2,n and R1,n−1 as line vectors we have the imbedding ϕ from R1,n−1 to Q: (1.13) ϕ(t) = e0 exp(tµ Pµ ) , with e0 = (1, 0, . . . , 0, 1); using (1.7) one has ϕ(t) = (1 − q(t), 2t, 1 + q(t)) .
(1.14)
One can thus define almost everywhere an action of Gn on R1,n−1 by means of the decomposition (1.12), writing, for t ∈ Tn and g ∈ Gn tg = γ(t, g)t0 ;
ˆ n , e0 g = ϕ(t0 ) γ(t, g) ∈ W
(1.15)
Clearly, if g = (∧, x) ∈ Wn with ∧ ∈ Yn , x ∈ Tn , one has t0 = t ∧ +x; if g=x ˆ = exp(ˆ xµ Pˆµ ) ∈ Tˆn one gets ˆµ )ˆ x)(1 − 2tµ x ˆµ + q(t)q(ˆ x))−1 t0 = (t − (tµ x
(1.16)
and t0 is defined when the denominator does not vanish. Elements of Tˆn acting on Tn are called special conformal transformations.
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(c) Representations of Pn and Gn We are interested in determining which unitary irreducible representations of Pn can be extended to Gn , or, conversely, which ones of Gn remain irreducible when restricted to Pn . It will appear that they can all be realized as functional spaces over the n-Minkowski space. We shall here begin by studying the UIR of Pn and operate a first selection among them; in the next chapter we shall study the representations of Gn which satisfy the necessary constraints, and give a complete description of the possible cases. Since Pn = Ln · Tn is a semidirect product with abelian normal subgroup Tn , its UIR’s are determined by the theory of Mackey [10]: let O be an orbit of the dual of Tn under the action of Ln , and Γ the stabilizer of a point in O; every UIR of Pn is equivalent to a representation U S induced by a UIR S of Γ. Different orbits, or non-equivalent representations of Γ for the same orbit, induce non-equivalent UIR of Pn . To construct U S one may proceed as follows: let V be the representation space of S; denote by x 7−→ xh the action of Ln on O, with x ∈ O and h ∈ Ln ; let ξ be the point of O stabilized by Γ, and let x 7−→ τx be a smooth injective mapping from O to Ln , so that x = ξτx ; denote by γ(x, h) the unique element of Γ satisfying γ(x, h)τ(xh) = τx h; let dµ be a quasi-invariant measure on O and let α be the positive function of O × Ln such that dµ(xh) = α(x, h)dµ(x). Let H = L2 (O, V, dµ) be the Hilbert space of V -valued functions f such that: Z kf (x)k2V dµ(x) < ∞ . O
For h ∈ Ln , t ∈ Tn and f ∈ H define U S , acting on H, by S f ](x) = [α(x, h)]1/2 · exp[ig(x, t)]Sγ(x,h) f (xh) , [U(h,t)
(1.17)
where i2 = −1. One thus has to determine the partition of the n-Minkowski space into orbits under Ln , and the corresponding stabilizers and their UIR. For n = 4 this was done by E. P. Wigner, who also established all the theoretical background neaded for this purpose, in a famous paper [4], anterior to the formulation of the general theory by G. Mackey. For other n ≥ 3, the resolution into orbits is a straightforward generalization of Wigner’s results; it is given in Table 1. Table 1. Orbits and little groups for the n-Poincar´ e group Type
Orbit
Stabilized point
Stabilizer
0
x=0
0
Ln
I±
q(x) = 0, ±x0 > 0
±(e0 + en−1 )
En−2
II± |m|
q(x) = m2 > 0, ±x0 > 0
± | m | e0
Spin(n − 1)
| m | en−1
Spin(1, n − 2)
III|m|
q(x) =
−m2
<0
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E. ANGELOPOULOS and M. LAOUES
In Table 1 the parameter |m| runs over positive real numbers; Spin(1, n − 2) denotes the twofold covering of SO0 (1, n − 2) (for n = 3 this covering is merely SO0 (1, 1) × Z2 ); En−2 is the twofold covering of the Euclidean group in n − 2 dimensions, Spin(n − 2) · Tn−2 (for n = 3 this reduces to Z2 × T1 ). One can immediately establish: Proposition 1.1. The UIR of Pn corresponding to orbits of types II, III and 0 (with the exception of the trivial one) cannot be extended to UIR of Gn . Proof. UIR of type 0 have trivial restriction on Tn . Since gn = tn ⊕ [ˆtn , tn ] ⊕ [ˆtn , [ˆt, tn ]], the trivial representation of Pn is the only possibility. e be the UIR of Gn obtained by extending Concerning types II and III, let U e on U . Since the parabolic subgroup Wn = Yn .Tn contains Pn , the restriction of U Wn must also be irreducible. Since Wn is a semidirect product, its UIR are again obtained by the orbit-stabilizer method. It turns out that the orbits of Tn under Wn are x = 0, xµ xµ = 0, xµ xµ < 0, and xµ xµ > 0, with sign(x0 ) fixed if xµ xµ ≥ 0; e to Pn is a direct integral, of representations thus, if xµ xµ 6= 0, the restriction of U over the parameter |m|, which is not irreducible. So let us focus to UIR of type I, called massless hereafter, by reference to the n = 4 case. Since Ln is an invariant subgroup of Yn , both groups acting on the same homogeneous space O, En−2 is an invariant subgroup of Γ0 , the stabilizer of ξ in Yn , and Yn /Ln = W × A ≈ Z2 × R+∗ is isomorphic to Γ0 /En−2 . More precisely one has (1.18) Γ0 = (W × A0 × Spin(n − 2)).Tn−2 such that Lie (Tn−2 ) is generated by elements Lj = Xj0 + Xj,n−1 , 1 ≤ j ≤ n − 2; Lie (Spin(n − 2)) = so(n − 2) is generated by Xj,k , 1 ≤ j, k ≤ n − 2; A0 = {exp t(X0,n−1 +Xn,−1 )}t∈R ≈ R+∗ and W = {1, exp π(X0,−1 +Xn−1,n )}; Γ0 consists of the elements of Gn which commute with P0 + Pn−1 = X0,−1 + X0,n + Xn−1,−1 + Xn−1,n . Let S 0 be the inducing representation of Γ0 and S its restriction to En−2 , so 0 that U S is the restriction to Pn of the representation U S of Wn . Since U S must be irreducible, S must be irreducible too. To determine the UIR of Γ0 , one can again apply Mackey’s theory of resolution into orbits. Without entering into many details, one can see that W × Spin(n − 2) 2 j stabilizes the “length” x2 = Σn−2 j=1 (xj ) = −xj x of an element x of Tn−2 , acting transitively on the corresponding sphere. On the other hand, λ ∈ A0 acts as a dilatation on Tn−2 , sending x to λx. If S 0 corresponds to a nonzero orbit, its restriction S is a direct integral of representations and U S is reducible. This leaves us with: Proposition 1.2. A necessary condition for a massless representation U S of Pn to extend to Gn is that the inducing representation S is a (finite-dimensional) UIR of Spin(n − 2).Tn−2 with trivial restriction to the normal subgroup Tn−2 .
MASSLESSNESS IN
n-DIMENSIONS
279
For every such choice of S and for either choice of sign(x0 ), U S extends to Wn , since S always extends to S 0 : one can always do this by choosing a one-dimensional UIR of A0 × W the choice being of course not unique. To see if the extension to Gn is possible, we shall use Lie algebraic methods. Before proceeding further, we shall give the expression of the infinitesimal operators of Pn , acting on a dense subspace of analytic vectors of H, the representation space of U S . To be more precise about H, the orbit O can be parametrized by Rn−1 − {0} : 2 1/2 . Since the orbit is massless, one has if (x0 , ~x ) ∈ Tn is in O, let k~xk = (Σn−1 µ0 =1 xµ0 ) 2 2 n−1 − {0} is given, x0 is fixed, its sign being determined x0 = k~xk , so that if ~x ∈ R by the choice of O. The quasi-invariant measure dµ is defined by dµ(x) = dn−1 ~x/kxk .
(1.19)
In fact dµ turns out to be invariant under the action of Ln (but not under the action of dilatations), so that the factor α in (1.17) equals 1. Putting Sjk = dS(Xjk ) acting on V , one obtains the following expressions: √ Pµ = −1 xµ X = Ljk + Sjk , 1 ≤ j, k ≤ n − 2 jk (1.20) Xj,n−1 = Lj,n−1 + Bj , 1 ≤ j ≤ n − 2 1≤j ≤n−2 X0j = x0 ∂j + Bj , X0,n−1 = x0 ∂n−1 where Lµ0 ν 0 = xµ0 ∂ν 0 − xν 0 ∂µ0 ,
Bj = (x0 + xn−1 )−1
n−2 X
xk Sjk .
(1.21)
k=1
We recall that we use the standard notation 0
∂µ0 = ∂/∂xµ = −∂/∂xµ0
(1 ≤ µ0 ≤ n − 1) .
(1.22)
This implies in particular, [∂µ0 , x0 ] = −xµ0 /x0 .
(1.23)
It is clear that U sends to zero the central element P Pµ of U(pn ). This feature will be the startpoint for the study of representations of gn , candidates to solve the problem. S
µ
2. Representations of so(2, n) Sending Pµ P µ to 0 (a) Weight representations of so(N )C and the Casimir element Let g be a symmetric nondegenerate bilinear form on RN , I a set of cardinality N, {eA }A∈I a basis of RN and gAB = g(eA , eB ). The orthogonal Lie algebra g = so(N, g) is spanned by generators XAB = −XBA such that [XAB , XCD ] = gBC XAD − gBD XAC − gAC XBD + gAD XBC
(2.1)
their action on RN being (with bracket notations) [XAB , eC ] = eA gBC − eB gAC
(2.2)
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E. ANGELOPOULOS and M. LAOUES
A If {eA } is the dual basis, with heA , eB i = δB , denoting by g again the associated A B AB bilinear form on the dual, with g(e , e ) = g , nondegeneracy implies g AB gBC = A . δC We shall use the tensor g for raising and lowering indices, writing for instance B XA for XAC g CB . The complexified Lie algebra gC is independent of the choice of g (up to isomorphism), the various real forms being obtained by a suitable choice of the basis {eA }, fixing RN in CN . We shall now introduce a Cartan subalgebra and a Borel-type decomposition in gC as follows:
Proposition 2.1. Let the indexing set I be {1, . . . , N } and assume (gAA )2 = (g AA )2 = 1, for every A ∈ I. Fix the constant γ by γ = N/2−Rank(g), that is γ = 0 if N is even and γ = 12 if N is odd. Let Iˆ = {γ + 1, γ + 2, . . . , N/2 be an indexing ˆ fix the constant ηa set of cardinality Rank(g); let qA = g(eA , eA ); for every a ∈ I, such that (2.3) ηa2 = −q2a−1 q2a (Hence ηa4 = 1 and ηa∗ = ηa−1 = ηa3 ) and define Ha ∈ gC by Ha = ηa X2a−1,2a .
(2.4)
The eigenvalues of adHa are 0, +1, −1; for every index A0 ∈ I − {2a − 1, 2a}, the linear combinations + = X2a,A0 + ηa q2a X2a−1,A0 X a A0 (2.5) − 0 0 = X2a−1,A + ηa q2a−1 X2a,A X a A0 are eigenvectors of adHa , satisfying ± H , X = a a A0 − + ηa X ,X = a B0 a A0 + + ,X = X a a B0 A0 Similarly the linear combinations X ε X a ε X a
+ b − b
= =
±X
A0
2 (XA0 B 0 + gA0 B 0 Ha )
(2.6)
− − ,X X = 0. a B0 a A0 ε a
ε0 b
ε a 2b ε X a 2b−1
X
± a
defined by + +
ηb q2b X ηb q2b−1
ε a 2b−1 ε X a 2b
(2.7)
MASSLESSNESS IN
n-DIMENSIONS
281
are simultaneous eigenvectors for every adHc , belonging to the eigenvalue ε1 if c = a, to ε0 1 if c = b and to 0 otherwise. Then (1) The elements Ha span a Cartan subalgebra h of gC . ± ˆ A0 < 2a − 1} span a nilpotent subalgebra n± of (2) The set {X a 0 , a ∈ I, A
gC , for either choice of the ± sign, such that n− ⊕ h ⊕ n+ is a Borel-type decomposition of gC . (3) When N is an even integer, all elements X
+ − a b
together with h span a
subalgebra l isomorphic to gl(N/2), while elements X ±±
−−
± ± a b
span abelian
⊕ n ) is a Cartan decomposition of subalgebras n , such that l ⊕ (n C ∗ g corresponding to the real form so (N ). (4) A Cartan–Weyl basis of gC is B0 =
++
i√ ηa ηb X 2
if N is even and
ε a
r B1/2 = B0 ∪
ε0 b
ηa X q1
, ε = ±, ε0 = ±
a
ε ,ε = ± a 1 a
if N is odd. Indeed one has, if {ea }a is the dual basis of {Ha }a :
i√ ηa ηb X 2
+ ± a b i√ ± ηa ηb X H, a 2 i√ ± ηa ηb X H, a 2
i√ − ∓ , ηa ηb X = Ha ± Hb a b 2 i√ ± ± = ±(ea + eb )(H) ηa ηb X b a 2 i√ ∓ ± = ±(ea − eb )(H) ηa ηb X b a 2
and, if N is odd: r ηa ηa + − X , X = 2Ha a 1 a 1 q1 q1 r r ηa ηa ± ± X X . H, = ±ea (H) a a 1 q1 q1 1
r
The root system is thus given by ∆0 = {εea + ε0 eb , ε = ±, ε0 = ±}a
± b ∓ b
282
E. ANGELOPOULOS and M. LAOUES
Remark. If q2a = −q2a−1 then the elements Ha and X
± a A0
belong to the
real form g from which we started: with suitable modification of the indexing set I, one sees that the dilatation operator or the Poincar´e translations Pµ , imbedded in the conformal Lie algebra, are of this form. On the contrary, such elements do not belong to the real form when q2a = q2a−1 since ηa is imaginary. We are now interested in relating the eigenvalue of the Casimir element C of g, defined by 1 1 (2.8) C = XAB g BC XCD g DA = XAB X BA 2 2 to the extremal weight which defines a finite-dimensional irreducible representation D of g. Though this result is classical, we shall give some details in view of further developments. So let I = I 0 ∪ I 00 , with I 0 = {1, . . . , N − 2} and I 00 = {N − 1, N }. Splitting the summations one has (2.9) C = C 0 + C 00 + B , where C 0 is the Casimir element of so(N − 2) and C 00 that of the complementary so(2), that is C 00 =
1 (XN −1,N X N,N −1 + XN,N −1 X N −1,N ) 2
= −q N q N −1 (XN −1,N )2 = (ηN/2 XN −1,N )2 = (HN/2 )2 while B = −XA00 A0 X A
(2.10) 00
A0
(2.11)
where primed and double-primed indices are summed over the sets I 0 and I 00 respectively. −+ 0 0 0 Develop now the expression BA 0 B 0 , symmetric in the indices A , B ∈ I , defined as follows: 1 + − + − −+ η = X + X X X BA 0B0 N/2 N/2 B 0 N/2 B 0 N/2 A0 N/2 A0 2 =
1 ηN/2 (XN −1,A0 + ηN/2 qN −1 XN,A0 )(XN,B 0 + ηN/2 qN XN −1,B 0 ) 2 1 + ηN/2 (XN −1,B 0 + ηN/2 qN −1 XN,B 0 )(NN,A0 + ηN/2 qN XN −1,A0 ) 2
=
1 (ηN/2 (XN −1,A0 XN,B 0 − XN,A XN −1,B 0 ) 2 −(q N −1 XN −1,A0 XN −1,B 0 + q N XN,A0 XN,B 0 ))
MASSLESSNESS IN
n-DIMENSIONS
283
1 + (ηN 2 (XN −1,B 0 XN,A0 − X,N B 0 XN −1,A0 ) 2 −(q N −1 XN −1,B 0 XN −1,A0 + q N XN,B 0 XN,A0 )) 1 00 00 = −HN/2 gA0 B 0 − g A B (XA00 A0 XB 00 B 0 + XA00 B 0 XB 00 A0 ) 2 0
(2.12)
0
Summing with g A B over the set I 0 of cardinality N − 2 yields B = (N − 2)HN/2 + B −+ , 0
(2.13)
0
−+ AB (notice that the permutation N ←→ N − 1 exchanges the where B −+ = BA 0B0 g + and − signs and transforms HN/2 to −HN/2 , while B is left unchanged). Thus one has (2.14) C = HN/2 (HN/2 + N − 2) + C 0 + B −+ .
Let now V be a finite-dimensional irreducible g-module, corresponding to the representation D. Let sN/2 be the eigenvalue of D(HN/2 ) with maximal real part, and let V 0 be the subspace V 0 = {ϕ ∈ V ; D(HN/2 )ϕ = sN/2 ϕ} .
(2.15)
If N = 2, then dim V = dim V 0 = 1 since the Lie algebra is abelian, and
D(C) = s21 = D(H12 ). If N > 2, then V 0 ⊆ ker D X
+ N/2 A0
for every A0 ∈ I 0 .
It follows that V 0 is an irreducible g0 -submodule, where g0 = so(N − 2) is the −+ 0 subalgebra generated by XA0 B 0 with A0 , B 0 ∈ I 0 and also D(BA 0 B 0 ) vanishes on V . Moreover, D is integrable to the compact real form since V is finite dimensional, so that D(Ha ) and ±D(Hb ) are conjugate for any choice of + or − and of a, b in ˆ it can be proved that this implies that every eigenvalue of D(Ha ± Hb ) (hence I: 2D(Ha )) is an integer; in particular, 2sN/2 ∈ N. If N = 3, then g0 = {0}, C 0 = 0, dim V 0 = 1 and one gets the well-known formula D(C) = s(s + 1) ,
with s = s3/2 .
(2.16)
If N > 3, one may apply the same procedure to the g0 -module V 0 , introducing the maximal eigenvalue sN/2−1 of D(HN/2−1 ) restricted on V 0 , and so on. Taking in account that |sa | ≤ |sa+1 | because D(Ha ) and D(Hb ) are conjugate for every a and b, one easily gets by induction: Theorem 2.1. The extremal weight of an irreducible finite-dimensional representation D of so(N ), N > 2, is determined by a sequence of positive numbers ˆ satisfying sa+1 − sa ∈ N, 2sa ∈ N, and such that sa , a ∈ I, X
N/2
D(C) =
a=γ+1
sa (sa + 2a − 2) .
(2.17)
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E. ANGELOPOULOS and M. LAOUES
There is an extremal weight vector ϕ 6= 0, spanning a one-dimensional subspace invariant by the Borel subalgebra h ⊕ n, such that D(n+ )ϕ = {0}, D(Ha )ϕ = sa ϕ if a > 1 and, when N is an even integer, D(H1 ) = ±s1 ϕ (representations with different choice of sign being inequivalent). One can also show that D is determined by the extremal weight up to equivalence, and that the representation space is D(U(n− ))ϕ. We shall denote such a representation here after by D(sN/2 , sN/2−1 , . . . , s1+γ ). The corresponding Coxeter–Dynkin diagrams are: 2S
S
3/2
5/2
-S
S
3/2
-S
a+1
a
S
N/2
-S
N/2-1
(Nodd)
S
2
S
1
S -S 3
2
S -S 4
3
S
a+1
-S
a
S
N/2
-S
N/2-1
(Neven)
S
2
S
1
Fig. 1.
Remark. Extremal weight representations of g with arbitrary range of the sa ’s can be defined, so that (2.17) still holds: I being the left ideal of U(g) corresponding to a one-dimensional representation of h ⊕ n+ , the left regular representation on U(g)/I has the desired form. Integrability over some real form implies restrictions on the range of sa . In particular, for the real form so(2, N − 2), we shall denote −2,ε by dN (α,~ s ) such a representation, where ε ∈ {−1, +1} and 2α 6∈ N; the spectrum −2,ε of dN (α,~ s ) (−εHN/2 ) is {α − k, k ∈ N} and the eigenspace corresponding to the maximal eigenvalue α is an irreducible so(N − 2)-module corresponding to the weight ~s = (s(N −2)/2 , . . . , s1+γ ).
(b) Massless representations Let us define the elements F¯AB of the enveloping algebra U of g = so(N )C by 1 1 CD CD C ¯ XAC g XDB +XBC g XDA = XA XCB − N − 1 XAB , (2.18) FAB = 2 2
MASSLESSNESS IN
n-DIMENSIONS
285
and the elements FAB as 1 2 FAB = F¯AB − gAB F¯CD g CD = F¯AB − g AB C . N N
(2.19)
The elements FAB are symmetric in the indices A, B (as well as the F¯AB ) and they span an irreducible g-submodule F of g ⊗ g under ad ⊗ ad. For N > 2 the dimension of F is N (N + 1)/2 − 1 = (N − 1)(N + 2)/2 (for N = 2, F is {0}); F is isomorphic, as a g-module, to the Cartan subspace p in the Cartan decomposition sl(N ) = so(N ) ⊕ p of sl(N ). Since F is irreducible, for every Y ∈ F the two-sided ideal U Y U of U contains F ; it follows: Lemma 2.1. Given a representation U of g, if there is Y ∈ F such that U (Y ) = 0, then U (Y 0 ) = 0 for every Y 0 in U Y U, and, in particular, for every Y 0 ∈ F. Split now the indexing set I into two disjoint sets I 0 and I 00 = {S, T }. Let, as in the preceeding section, η be such that η 2 = −g SS g T T and let H = ηXST . The eigenvectors of adH are given by + XA 0 = XSA0 − ηqS XT A0 ;
− XA 0 = XT A0 + ηqT XSA0
(2.20)
for every A0 in I 0 . Summing over A0 ∈ I 0 these expressions one gets 0
0
± ∓ AB = −η 2 (qT FS + qS FT ) − 2H 2 ± (N − 2)H + ηXA 0 XB 0 g
4 C N
(2.21)
and 0
0
0
0
+ + AB = FT T − 2ηqT FST − η 2 FSS XA 0 XB 0 g − − AB = FSS − 2ηqS FST − η 2 FT T XA 0 XB 0 g
(2.22)
One thus gets: Lemma 2.2. For every generator XST with qS2 = qT2 = 1, the expressions ± in which the summation runs over I − {S, T } and the XA 0 are the eigenvectors defined in (2.20), belong to F . In particular, if N = n + 2, I = {−1, 0, 1, . . . , n}, {ST } = {−1, n}, the element Pµ P µ of the Poincar´e enveloping algebra, canonically imbedded in U(so(2, n)), belongs to F . ± ∓ A0 B 0 , XA 0 XB 0 g
From these two lemmas it follows: Proposition 2.2. If a representation U of U(so(2, n)) satisfies U (Pµ P µ ) = 0, then U vanishes on F . Such a representation will be called massless hereafter.
286
E. ANGELOPOULOS and M. LAOUES
We shall begin the study of massless representations by establishing: Proposition 2.3. Let U be a representation of g which vanishes on F . Let N = N 0 + N 00 be any splitting of N into two positive integers, I = I 0 ∪ I 00 the corresponding splitting of the indexing set, g0 = so(N 0 ) and g00 = so(N 00 ) the corresponding subalgebras. Their Casimir elements C 0 and C 00 are related to the Casimir element C of g by N 0 − N 00 U (C) (2.23) U (C 0 ) − U (C 0 ) = N In particular, if N 00 = 1 and I 00 = {1} one has U (C 0 ) =
N −2 U (C) N
U (g AB X1A XB1 ) =
(2.24)
2q1 U (C) N
(2.25)
Proof. Using distinct summations over I 0 , I 00 and using the definition of FA0 B 0 one has 2gA0 B 0 A0 B 0 A0 B 0 C 0 D0 A00 B 00 C FA0 B 0 = g XD0 B 0 + XA0 A00 g XB 00 B 0 − X A0 C 0 g g N 0
0
0
00
= 2C 0 + XA0 A00 XB 00 B 0 g A B g A B − gA
00
B 00
0
0
FA00 B 00 = 2C 00 + XA00 A0 XB 0 B 00 g A B g A
00
B 00
2N 0 C N
−
2N 00 C N
and by substraction one gets the desired result, since U vanished on F .
(2.26a) (2.26b)
Let us now determine the irreducible massless representations. Starting from low values of N , one first establishes: Theorem 2.2. For N = 2, every representation is massless, F being {0}. For N = 3 the only irreducible massless representations are the trivial and the spinorial (two-dimensional) one. For N = 4, if g = g1 ⊕ g2 is the decomposition of so(4) into two ideals, each isomorphic to so(3), an irreducible representation is massless if and only if it vanishes on either g1 or g2 . Sketch of the Proof. For N = 3, g ⊗ g = F ⊕ g ⊕ C · C, and one can show (we leave this to the reader) that (C − 34 ) · g belongs to the ideal U F U, so that the quotient is a five-dimensional complex algebra, which turns out to be EndC (C2 )⊕C. For N = 4 one first sees that F is the span of all elements X1 X2 with Xi ∈ gi sot that UF = FU is the intersection of the two maximal ideals g1 U and g2 U, hence the result. So, from now on we shall suppose N ≥ 5.
MASSLESSNESS IN
n-DIMENSIONS
287
Examining first the finite-dimensional case one gets: Theorem 2.3. A representation D(sN/2 , . . . , s1+γ ) is massless if and only if |sa | = s for every a ∈ Iˆ where if N is even (and γ = 0) then 2s ∈ N while if N is odd (γ = 12 ) then s ∈ {0, 12 }. The corresponding value of the Casimir element is C=
1 1 Ns s + N − 1 . 2 2
(2.27)
Moreover, if N is even, an extremal weight subspace carries a one dimensional representation of the parabolic subgroup gl(N/2) ⊕ n++ , with trivial action of sl(N/2) and n++ . Proof. We shall calculate F¯AB on an extremal vector ϕ. Using the notations of the preceeding section and taking in account that n+ vanished on ϕ, let A, B < 2a−1 ˆ a calculation similar to (2.12) yields for some a ∈ I; X i,j∈{2a−1,2a}
1 (XAi XjB + XBi XjA )g ij ϕ = Ha gAB ϕ . 2
(2.28)
On the other hand, let I 0 = {1, . . . , 2b} and I 00 (b) = {2b − 1, 2b} ⊂ I 0 . Using distinct summations on primed and double-primed indices, with A0 , B 0 ∈ I 0 and A00 , B 00 ∈ I 00 (b), one has, using inductively (2.12): gA
00
B 00
0
0
XA00 A0 XBB 00 GA B ϕ = Hb (2Hb + 2b − 2)ϕ
(2.29)
hence X
" g
A00 B 00
F¯A00 B 00 ϕ = 2 Hb (Hb + b − 1) +
A00 ,B 00 ∈{2b−1,2b}
X
# Ha ϕ
(2.30)
a>b
Since FA00 B 00 vanishes, one obtains " # X 2 Cϕ = Hb (Hb + b − 1 + Ha ϕ N
(2.31)
a>b
Equalling the expressions obtained for b and b + 1, one gets for consecutive eigenvalues sb and sb+1 : 0 = sb (sb + b − 1) − sb+1 (sb+1 + b − 1) = (sb − sb+1 )(sb + sb+1 + b − 1)
(2.32)
For b ≥ 1 and N odd or b > 1 and N even one has 0 ≤ sb ≤ sb+1 so that one must have sb = sb+1 ; and for b = 1, N even, (2.32) becomes |s1 | = |s2 |: thus s = |sa | is constant. For b = N/2, (2.31) gives the values of the Casimir.
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E. ANGELOPOULOS and M. LAOUES
For N odd one also has, by taking A = B = 1 in (2.28) and summing all over ˆ a ∈ I: X 2 1 1 (2.33) sa = (N − 1)s = C = s s + N − 1 2 N 2 a∈Iˆ
hence s(s − 12 ) = 0. Notice also that X
ε ε0 ab
ϕ = 0 unless ε = ε0 = −, because otherwise an
eigenvalue equal to s + 1 would appear for some Ha , which is impossible. Since also Ha − Hb vanishes on ϕ, ϕ spans a one-dimensional representation of gl(N/2) ⊕ n++ for even N , as stated. It remains to show every representation of this form is a massless one. If s = 0 we have the trivial one which is massless, and if s = 12 we have a spinorial representation D and Ker D is a bilateral ideal of U containing F ; this ends the odd N case. For even N and s ≥ 1 one has D(g · g)ϕ = (D(n−− · n−− ) + D(n−− ) + C)ϕ. Diagonalizing the space F with respect to the Cartan subalgebra h one gets, among others, elements Fa++ and Fa−− such that [Ha , Fb±± ] = ±2δab Fb±± , and all these elements are in Ker D, since no elements of n−− · n−− or n−− have this P property. Writing h = h0 ⊕ CH with H = a Ha and h0 = h ∩ sl(N/2), one can substitute h0 with any conjugated subalgebra, and this does not affect ϕ. The new elements Fb±± thus obtained are distinct from the original ones, and as h varies the whole of F is spanned by such elements. It follows that D(F )ϕ = {0}, and since F U = UF, D(F ) vanishes on D(U)ϕ, so the representation D is massless. Consider now infinite-dimensional massless representations integrable to the universal covering of the conformal group. Putting n = N − 2, the maximal compact subalgebra is k = so(2) ⊕ so(n), and the complexified Cartan subspace pC is isomorphic to the k-module C2 ⊗ Cn . We shall again use the usual notations for the n-conformal algebra, that is the indexing set will be I = I 0 ∪ I 00 with ˆ for the Cartan subalgebra, I 0 = {1, . . . , n}, I 00 = {−1, 0} and the indexing set I, {0, n2 , n2 − 1, . . .}; we shall denote by H0 the central element ηX−1,0 (with η 2 = −1 and g−1,−1 = g00 = 1) of kC . The space H of the representation U is a direct sum of k submodules W (s0 , ~s ), where s0 is the eigenvalue of H0 and ~s the extremal weight of so(n) · pC acts on W (s0 , ~s ) like (C2 ⊗ Cn ) ⊗ W (s0 , ~s ): this tensor product splits in general into 2n components W (s0 + ε, ~s + ∆~s ) with ε = ±1 and ∆sa = (∆~s )a = ±1 for one a ∈ Iˆ−{0} (at most if n is odd, exactly if n is even), all remaining coordinates of ∆~s being 0 (if n is odd ∆~s = ~0 also exists in general). When ∆sa = ±1 and sa+1 = sa the corresponding component vanishes, since the resulting weight would not respect the ordering sa+1 ≥ sa . In particular, Cn ⊗ W always contains a component W ↑ for which ∆sn/2 = 1 (the maximal eigenvalue increases) and a component W ↓ for which ∆sn/2 = −1; this latter is nonzero only if sn/2 − 1 ≥ sn/2−1 . Assume now U irreducible and massless and take ϕ in W (s0 , ~s ). Because of (2.23) s0 is related to the Casimir C 0 of so(n) by
MASSLESSNESS IN
n-DIMENSIONS
(C 0 − s20 )ϕ =
289
n−2 Cϕ . n+2
(2.34)
Let |s0 | = εs0 . For ϕ in W (s0 , ~s ) one has ±ε ±ε ϕ. ϕ = (±2εs0 + 1)X H02 , X 0 A0 0 A0 On the other hand, if ϕ is an extremal vector then X ↑
00
00
+ ϕ n/2 A00
(2.35)
belongs to
W (s0 , ~s ) (with A ∈ {−1, 0} = I ) since the maximal eigenvalue increases, so that, by (2.17): + + ϕ. (2.36) ϕ = (2sn/2 + n − 1)X C 0, X n/2 A00 n/2 A00 Since the difference C 0 − H02 is constant, these two equations imply ±ε + ϕ = 0, (sn/2 + n/2 − 1 ∓ |s0 |)X 0 n/2 hence X
−ε
+ n/2
0
vanishes on ϕ; X
+ε 0
+ ϕ n/2
(2.37)
is an extremal vector of W (s0 , ~s )↑ 0
and the only non-vanishing component of ΣA0 ∈I−{n−1,n} λA X
+ ϕ, n/2 A0
so it is
nonzero (otherwise sn/2 , hence C 0 , would be bounded and U would be finite dimensional), and we get (2.38) |s0 | = sn/2 + n/2 − 1 , for every W (s0 , ~s ). It also follows that −ε W (s0 , ~s ) ⊂ W (s0 , ~s )↓ X 0 A0 and one can transform (2.21) − −ε + X X n/2 0 n/2 X
to 2 ε C − sn/2 (sn/2 + n/2) ϕ , ϕ=4 0 n+2
+ ε − −ε X ϕ n/2 0 n/2 0 2 C − (sn/2 − 1)(sn/2 − 1 + n/2) ϕ . =4 n+2
One also checks that X 0
ΣA0 ∈I 0 λA X
−ε A0
0
− n/2
−ε 0
(2.39)
(2.40a)
(2.40b)
ϕ is the only nonvanishing component in
ε (otherwise an eigenvalue of Hn/2 superior to sn/2 − 1 would
appear), and it is again an extremal vector. When sn/2 reaches its minimal value, s, every X
−ε 0
A0
ϕ is zero and (2.40b) gives
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E. ANGELOPOULOS and M. LAOUES
2C = (n + 2)(s − 1)(s − 1 + n/2) n . = (n + 2) · Inf |s0 | · Inf |s0 | − 2
(2.41)
It follows that −εH0 has a negative maximal value equal to −(s − 1 + n/2); an extremal vector ϕ for sn/2 = s is an extremal vector for the whole representation space and the nilpotent subalgebra n+ vanishes on ϕ. As for the remaining coordinates of ~s, one easily sees that they are all equal to s (or −s for the last one for even n), and that s = 0 or 1/2 when n is odd, the proof being exactly the same as in the finite-dimensional case. Using the notations of Theorem 2.1 and the Remark following it, one can summarize: Theorem 2.4. Every infinite-dimensional irreducible massless representation of so(2, n), for n ≥ 3, integrable to Gn , is a weight representation dn,ε (−(s+n/2−1),~ s ), D(~s ) being itself a massless representation of Spin(n), that is |sa | = s for every a. The eigenspace of εH0 corresponding to the eigenvalue (s + n/2 − 1 + k), k ∈ N, is an irreducible so(n)-module corresponding to the representation D(~s + (k, 0, . . . , 0)). The values of the Casimir element C is given by (2.41). In addition one has: are integrable Proposition 2.4. The massless representations dn,ε s) (−(s+ 1 n−1),~ 2
to unitary representations of Gn . Proof. From what precedes, every so(n)-submodule Wk has multiplicity one, it carries the representation D(~s + (k, 0, . . . , 0)) of so(n), and the unique eigenvalue of εH0 on it is s + k + 12 n − 1(k ∈ N). Since there is a natural k-invariant scalar product on each Wk and since p · Wk ⊂ Wk−1 ⊕ Wk+1 , it is sufficient to show that ||Xϕ||2 = q(X)||ϕ||2 for every X ∈ p such that [εH0 , X] = ±X, with q(X) ≥ 0; it is clear that q(X) belongs to the spectrum of X ∗ X. There is no loss of generality in assuming that ϕ is an extremal vector of Wk ; but then X must be proportional to either X+ = X
+ n/2
ε 0
or X− = X
− n/2
−ε , 0
with (X± )∗ = −X∓ and where X± ϕ is an extremal vector of Wk±1 ; from (2.40) one sees that X ∗ X is scalar, with n n ∗ − (s − 1) s − 1 + X+ = (s + k) s + k + X+ 2 2 n = (k + 1) 2s + k − 1 + 2 n n ∗ − (s − 1) s − 1 + X− = (s + k − 1) s + k − 1 + X− 2 2 n = k 2s + k − 2 + 2 and these expressions are positive for k ∈ N, s ≥ 0 and n ≥ 3.
MASSLESSNESS IN
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291
(c) Conformal imbedding of Poincar´ e massless representations Having determined all possible candidates, up to equivalence, we shall now examine whether a massless representation U of Poincar´e extends to one of them, and how. We shall proceed by combining the expressions of the generators of Pn given in (1.20) to obtain elements of the ideal UF. One first establishes: Proposition 2.5. Given the expressions (1.20) of the Poincar´e generators of U, if Pµ is identified with Xµ,−1 + Xµ,n (with µ ∈ J = {0, . . . , n − 1}), then the dilatation operator D = Xn,−1 , satisfying [D, Pµ ] = Pµ , is given by 0
D = xµ0 ∂ µ + (n − 2)/2 ,
µ0 ∈ {1, . . . , n − 1} .
(2.42)
Proof. Using a summation index λ ∈ J, and since g−1,−1 = −gn,n = 1, one has F−1µ + Fnµ = (X−1λ + Xnλ )Xµ λ + (Xn,−1 X−1,µ − X−1,n Xn,µ ) 1 − n(X−1,µ + Xnµ ) 2 n λ δ = Pλ Xµ λ + D + 1 − 2 µ
(2.43)
substituting the expressions of the generators, and putting FAB = 0, one gets, for every µ in J, 0 n (2.44) 0 = xµ D + 1 − − xµ ∂µ0 2 hence the result announced. Now, one can rewrite (2.23) as n−2 1 Xλµ X µλ = C + D2 2 n+2
(mod UF)
(2.45)
and one also has
1 2 1 λ ˆ ˆ (Pµ Pν + Pν Pµ ) = Xµ Xλν − (n − 2)Xµν − gµν D + C − Fµν , (2.46) 2 2 n+2
where Pˆν = Xν,−1 − Xνn , satisfying [D, Pˆν ] = −Pˆν . Substituting the expressions of the generators in (2.45) and (2.46), one obtains, after some calculations which we do not reproduce C=
n + 2 00 1 C − (n + 2)(n − 2) n−2 4
(2.47)
where C 00 = 12 Sij S ij (i, j ∈ {1, . . . , n − 2}) is the Casimir element of the inducing representation S; from (2.46) one gets expressions of the form: Pµ Pˆν + Pν Pˆµ = xν Gµ + xµ Gν + Eµν
(2.48)
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E. ANGELOPOULOS and M. LAOUES
with Eik = (Sij Skj + Skj Sij ) −
4 gik C 00 ; n−2
Eαk = σ(α)xi Eik (x0 + xn−1 )−1 ;
i, j, k ∈ {1, . . . , n − 2}
(2.49a)
α ∈ {0, n − 1}, σ(0)
= −σ(n − 1) = −1
(2.49b)
(2.49c) Eαβ = σ(α)σ(β)xi xk Eik (x0 + xn−1 )−2 ; α, β ∈ {0, n − 1} √ expressions (2.48) implies Since Pµ = −1 xµ , the consistency of the n(n+1) 2 that Eµν = 0; in particular Eik = 0, that is S is a massless representation of the little group Spin(n − 2). Carrying out the calculations, one finally obtains: Theorem 2.5. A massless representation of Pn (n ≥ 3) induced by the representation S of Spin(n − 2) · Tn−2 (trivial on Tn−2 ) extends to a massless UIR of Gn iff S itself is massless, that is of the form D(s, . . . , s, ±s), 2s ∈ N, if n is even and of the form D(s, . . . , s), s = 0 or 12 , if n is odd. The extension is unique, the form of the remaining generators (of gn ) being completely determined by those of pn in (1.20) : Xn,−1 = D is given by (2.42) and Pˆµ by √ −1Pˆx = xµ ∆ + 2(x0 + xn−1 )−1 Dµ + 2D∂µ (2.50) 2 with ∂0 = 0, ∆ = Σn−1 j=1 ∂j and
Dj = (L0k − Lk,n−1 )S k j Dn−1
(j, k ∈ {1, . . . , n − 2})
1 jk 1 = −D0 = L Skj + s s + n − 2 2 2
and the values of the Casimir element for tations are 1 1 00 C = (n − 2) s s + n − 2 ; C = 2 2
(2.50a) (2.50b)
the inducing and the extended represen 1 1 (n + 2)(s − 1) s + n − 1 . 2 2
(2.51)
Remark. The constraints upon S are relevant for n > 4. Indeed, for the classical case, n = 4, the little group is SO(2).T2 , and the elements Eij in (2.49) are identically zero: every such representation extends to the conformal group, as shown in [2]. For n = 3, so(n − 2) = {0} and all elements Sij vanish; notice that C 00 vanishes in (2.51) for n = 3 and for either s = 0 or s = 12 . However, the choice of S is relevant: it corresponds to the inducing representation of Spin(1) = {1, −1} and determines whether the center of Spin(3) = SU (2) is trivially represented (s = 0) or not (s = 12 ), the lowest so(3)-module occuring in the representation space having dimension 2s + 1. , Now, for given s, there are two possible choices for the extension, dn,ε (−(s+ 12 n−1),~ s) √ such that the spectrum of ε −1 X−1,0 is positive, so that it remains to identify which one is obtained. We shall show:
MASSLESSNESS IN
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293
Proposition 2.6. For a given sign ε of x0 = ε · |x0 |, the representation U S of Pn extends to dn,ε . (−(s+ 1 n−1),~ s) 2
Proof. On every so(3)-submodule Wk the absolute value of s0 is s + k + 12 n − 1, while the eigenvalues of Hn/2 run from −(s + k) to s + k, so that the spectrum of √ E = 2 −1(X−1,0 + Xn−1,n ) has the same sign as H0 and a lowest element equal,in absolute value, to n − 2. Substituting with the differential operators obtained one gets: E=
√
−1(−Pˆ0 − Pˆn−1 − P0 + Pn−1 )
= −(x0 + xn−1 )∆ − 2D∂n−1 + (x0 − xn−1 ) ,
0
D = xµ0 ∂ u +
n−2 2
0
Take f ∈ H so that f depends only on x0 = ε(−xµ xµ0 )1/2 , and denote by d0 the differential operator dxd 0 . For such an f one has Ef = ((x0 − xn−1 )(1 − d20 ) − (n − 2)d0 )f . If d0 2 f = f , that is, for example, if f (x0 ) = e±εx0 v, with v ∈ V , one gets Ef = ∓ε(n − 2)f . Since only e−εx0 v is a square-integrable function from Rn−1 to V, εE has a positive spectrum, and so does εH0 , hence the desired result. Remark. When S is trivial, the Fourrier transform on H sends it on the ˆ of L2 (R1,n−1 , dµ) which is the closure of all analytic functions satisfying subspace H µ ˆ is obtained from the action of dilatations ∂µ ∂ f = 0. The action of Gn /Pn on H and special conformal transformations on the n-Minkowski space. What we have ˆ ⊗ V can be shown is that, for S acting on V , the representation U S acting on H extended iff S is massless. 3. Massless Representations and the De Sitter Groups (a) Subgroups of Gn Let x ∈ Rn+2 . If its quadratic form q(x) is positive (resp. negative), its stabilizer Sn (x) is isomorphic to SO0 (1, n) (resp. SO0 (2, n − 1)). For distinct choices of x, Sn (x) and Sn (x0 ) are conjugated subgroups iff q(x) · q(x0 ) > 0, so we shall denote them by Sn± (for q(x) = ±|q(x)|) and call them the n-De Sitter subgroups of real rank 1 or 2 respectively, in analogy with the classical case n = 4. Clearly, Sn− = Gn−1 . When q(x) = 0 the stabilizer is isomorphic to Pn . We shall examine here the restriction to the twofold covering S¯n± of a massless representation d of Gn , establishing that it is either irreducible, or the direct sum of u contraction of Sn± , we shall establish two factors. Also, since Pn is a Wigner–Inon¨ ± that the restriction of d on S¯n can be contracted to its restriction on Pn .
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E. ANGELOPOULOS and M. LAOUES
¯± (b) Restriction of dn,ε s ) to Sn (−(s+ n −1),~ 2
We have already established that the Casimir element C 0 of Sn± is scalar and equal to C · (N − 1)/N , in (2.27). We shall next continue with: Lemma 3.1. Let g0 be the Lie algebra of Sn± , U 0 its enveloping algebra, and let ew , w ∈ I, be a basis vector stabilized by g0 . Let d be a massless representation of g acting on a Hilbert space H and W 0 be a g0 ∩ k invariant subspace of a k-type W. Let V0 , V1 be the prehilbert spaces V0 = d(U 0 )W 0 ;
V1 =
X
d(U 0 XAw )W 0
(3.1)
A
and H0 , H1 their closures. Then either H = H0 = H1 or H = H0 ⊕ H1 . Proof. Let x the g0 -invariant subspace of g spanned by the generators XAw , such that g = x ⊕ g0 . Since U 0 = ⊕k∈N U 0 S k (x), where S k (x) contains the fully symmetrized polynomials of degree k in the generators of x, it is sufficient to show that d sends S 2 (x) to S 0 (x) = C. But one has 0 0 XA0 w XB 0 w + XB 0 w XA0 w = gww XA0 D XD0 B + XB 0 D XD0 A0 ) − 2F¯A0 B 0 )
and since d(F¯A0 B 0 ) = gA0 B 0 2C/N ∈ C, d send S 2 (x) to U 0 .
(3.2)
Now, if G0 = Sn+ , x = {λA X−1,A }, g0 ∩ k = so(n), and the k-type W (k) is irreducible under the action of g0 ∩ k. The generator XA00 0 ∈ g (for A00 ∈ {1, . . . , n}) sends W (k) to W (k) ⊕ W (k ± 1) and so does [C 0 , XA00 0 ], C 0 being the Casimir of ± 0 so(n), so that, for every k ∈ N, there is a shift operator XA 00 ∈ d(U ), linear com0 bination of XA00 0 and [C , XA00 0 ], sending W (k) to W (k ± 1). Every W (k) being of multiplicity one, d(U 0 )ϕ contains every k-type of d, so that the closure of V0 is H and the restriction to G0 is irreducible. If G0 = Sn− , the situation is somewhat more complicated. Let e1 be the stabilized vector, so that x is spanned by {X1A }, k ∩ g0 being isomorphic to so(2) ⊕ so(n − 1). Assume first s = 0, so that H contains a trivial so(n)-submodule W (0). Let ϕ ∈ W (0): clearly X1A0 ϕ = 0 if A0 ∈ {2, . . . , n} and X is spanned by X
+ε 0
1
−ε 0
1
ϕ = 0 too, so that xW (0)
ϕ = ϕ+ . Since so(n − 1) commutes with X
+ε 0
1
, it stills
act trivially on xW (0), while the eigenvalue of H0 increases by 1 in absolute value, so that k ∩ g0 stabilizes xW (0). Moreover, for A0 ∈ {2, . . . , n}, X
−ε 0
ϕ+ = A0
X
ε −ε ε −ε X − X , X ϕ 0 1 0 A0 0 1 0 A0
√ = 2ε −1X1A0 ϕ = 0
so ϕ+ is an extremal weight vector of g0 , as well as ϕ, so that V0 ∩ V1 = {0}.
(3.3)
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295
Assume next s 6= 0 and n even (n ≥ 4). Since d(H1 ) = ±d(Ha ) on an extremal √ vector for every W (k), one has d( −1 X12 )ϕ = ±sϕ 6= 0 on an extremal vector of W (0), so that U(so(n − 1)) · (x ∩ k)W (0) = U(so(n − 1))W (0) = U(so(n))W (0) = W (0)
(3.4)
and V0 = V1 . Assume finally n odd and s = 12 . The lowest so(n)-type is a spinorial representation, and it is well known that such a representation of so(2r + 1)(r ∈ N) splits into two inequivalent spinorial representations of so(2r) of equal dimensions; they are labelled D( 12 , . . . , 12 , ± 12 ) with the two different choices of sign. Summarizing one has: Proposition 3.1. The representation dn,ε s ) remains irreducible when (−(s+ n −1),~ 2
restricted to SO0 (1, n). Its restriction on SO 0 (2, n − 1) when s = 0 is the direct sum ⊕ dn−1,ε ; dn−1,ε (−( n −1),~0 ) (− n ,~0 ) 2
for s =
1 2
2
and n = 2r + 1 odd, its restriction is ⊕ d2r,ε ; d2r,ε (−r, 1 ,..., 1 ,+ 1 ) (−r, 1 ,..., 1 ,− 1 ) 2
2
2
2
2
2
for s 6= 0 and n = 2r even, the restriction is irreducible and equal to d2r−1,ε (−s+r−1),∂~ s) , where ∂~s comes from ~s = (s, . . . , s, ±s) by dropping the last coordinate ±s. (c) Contraction of representations The Wigner–Inon¨ u contraction of Lie algebras [15] can be defined as follows: given a Lie algebra g and a continuous family Φα ∈ GL(g) of linear transformations of the underlying vector space, with 0 < α ≤ 1 and Φ1 = 1, a Lie algebra gα isomorphic to g is defined on the same underlying space by the Lie bracket: [X, Y ]α = Φ−1 α [Φα X, Φα Y ] .
(3.5)
If limα→0 (Φα ) is a non-invertible mapping and [X, Y ]0 = lim([X, Y ]α ) exists when α → 0, the Lie algebra g0 defined on the same underlying space is the contracted of g by the family {Φα }. Contraction of representations Uα of gα on Hα are defined in analogy. Here we shall limit ourselves to a fixed representation space H. Given a continuous family {Zα } of closed invertible linear transformations of H for 0 < α ≤ 1 with Z1 = 1, and a representation U1 = U of g1 = g, defined on a dense domain E of analytic vectors, the map
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E. ANGELOPOULOS and M. LAOUES
X 7−→ Uα (X) = Zα−1 U (Φα X)Zα
(3.6)
is a representation of gα ; indeed, one has [Uα (X), Uα (Y )] = Zα−1 [U (Φα Y ), U (Φα Y )]Zα = Zα−1 U ([Φα X, Φα Y ])Zα = Zα−1 U ([Φα [X, Y ]α )Zα = Uα ([X, Y ]α )
(3.7)
If the limit of Uα (X) exists for every X ∈ g when α → 0 (regardless of whether Zα has a limit), then U0 = limα→0 Uα is a representation of the contracted Lie algebra g0 : we shall say that it is the contracted of U1 through the family (Zα ). Let us apply this to g1 = Lie (Sn± ) = ln ⊕ y where y is spanned by the generators Yµ = Xµw , µ ∈ J, with w = n for Sn+ and w = −1 for Sn− ; one has [y, y] = ln . We shall define the familly {Φα } by Φα (Xµν ) = Xµν ;
Φα (Yµ ) = α(2 − α)Yµ .
(3.8)
Clearly, one has [Yµ , Yν ]α = α2 (2 − α)2 [Yµ , Yν ] ;
[Xµν , Yλ ]α = [Xµν , Yλ ]
(3.9)
so that the contacted algebra g0 is isomorphic to pn . Let now d be a massless representation of Gn on H, with analytic domain E, on which all operators of the Lie algebra are defined, their expressions being given by √ (1.20) (in particular d(Pµ ) = −1 xµ ) and (2.50). Let U = U1 be the restriction of d to g1 , so that one has 1 (d(Pµ ) ∓ d(Pˆµ )) . 2
(3.10)
(Zα ϕ)(x) = α(n−2)/2 ϕ(αx)
(3.11)
U (Yµ ) = Define the family {Zα } by
Zα is a unitary operator, equal to exp(d (Logα Xn,−1 )), which has no limit for α → 0. It satisfies Zα−1 d(Pµ )Zα = α−1 d(Pµ ) ; so that
Zα−1 d(Pˆµ )Zα = α d (Pˆµ )
α (d(Pµ ) ∓ α2 d(Pˆ )) Uα (Yµ ) = 1 − 2
(3.12)
(3.13)
while Uα (Xµν ) = U (Xµν ). It is clear that the limit of Uα (Yµ ) exists for α → 0, and it is equal to d(Pµ ). We have thus proved: Proposition 3.2. The restriction U on S¯n± of the massless representation d of Gn contracts to its restriction on Pn through the family of unitary operators {Zα }.
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297
4. Conclusion Comparing the results obtained here with the classical case n = 4, we first observe that the main features are conserved: only massless representations U S of Pn can be extended to ones of Gn , and when this is possible the extension d is unique: it is a unitary irreducible representation with extremal weight, vanishing on the two-sided ideal of the enveloping algebra generated by Pµ P µ . The form of the remaining Lie algebra generators is completely determined when those of pn are given (that is, d is not only fixed up to equivalence, but when U S is fixed inside its equivalence class so is d). Moreover, d is a representation of either Gn itself (when U S is one of Pn , that is s integer) or of a twofold covering (when s is half-integer). All representations d are realizable on a functional space over the corresponding Minkowski space or over a half-cone of its Fourier dual (in fact, when S is trivial d is equivalent to the representation induced by the trivial representation of the parabolic subgroup Wn ). The only feature which does not generalize concerns the restrictions imposed on the inducing representation S. For n = 4, the only restriction is that S is trivial on the translation subgroup T2 of the twofold covering of the Euclidean group E2 . This discards the so-called continuous spin representations and allows all helicities ±s ∈ 12 Z. For n > 4, S must still vanish on the translations, but there are additional constraints on S, depending on the parity of n: if n = 2r is even every coordinate of the extremal weight must equal in absolute value to the last one (the minimal one), which is equal to ±s. This constraint is automatically satisfied for n = 4, since so(n − 2) has rank 1 and the last coordinate of the weight is also the only one: the study of the case n = 4 alone gives no hint about this new constraint. For odd n the constraints are quite drastic: S may be either trivial or spinorial. This appears as a straightforward generalization of the case n = 3 [5], if one defines Spin(1) as Z2 . If, instead of increasing n, one decreases it to n = 2, one finds again that the only UIR of the simply connected P2 = SO0 (1, 1) · T2 (besides the trivial one) ¯ 2 = SO 0 (2, 2) are the massless ones: the massless orbits are the which extend to G connected components of the isotropic cone, that is, 4 half lines (instead of two half ones) and the stabilizer is just {1}, so that massless UIR vanish on the subgroup spanned by P0 + P1 or P0 − P1 , the factor group on which they are faithful being here isomorphic to the connected affine group (x 7−→ ax + b) of the real line. By Theorem 2.2, the extension to so(2, 2) = u+ ⊕ u− must vanish on one of the two factors u± (both isomorphic to so(2, 1)). The Casimir operator may take any value C and by (2.46) one gets Pˆ0 = (P0 )−1 (D2 − D + 12 C). There is no uniqueness of the extension, not even unitarity (C may be any complex number): lowering n to 2 removes all constraints. One should however mention that in this case the full conformal group is infinite, as is well known. We shall not discuss this case further here. Concerning the Poincar´e–De Sitter relations, the sequence “extension to Gn , then restriction to S¯n± , then contraction to Pn ” is cyclic for every n ≥ 3; the
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demonstration is practically identical with the one for n = 4 [2]. As for the irreduciblility of the restriction to the real rank two De Sitter subgroup S¯n− , the result for n even is a straightforward generalization of the case n = 4: the restriction splits into two simple factors if the inducing representation is trivial, otherwise it is irreducible. When n is odd, it splits into two simple factors for both s = 0 and s = 12 . The restriction on S¯n+ = SO0 (1, n) is always irreducible. Finally we should mention that there are some interesting open problems involving massless representations, such as their tensor products with other representations (in particular their tensor squares) or their appearance as factors in indecomposable representations. Acknowledgements The authors wish to thank Mosh´e Flato for suggesting the problem and his constant interest in this work and Daniel Sternheimer for helpful criticism of the manuscript. This work was partially supported by E.U. Program ERBCHRXCT 940701. References [1] E. Angelopoulos and M. Flato, “On unitary implementability of conformal transformations”, Lett. Math. Phys. 2 (1977/78) 405–412. [2] E. Angelopoulos, M. Flato, C. Fronsdai and D. Sternheimer, “Massless particles, conformal group and De Sitter universe”, Phys. Rev. D23 (1981) 1278–1289. [3] M. Cahen, S. Gutt and A. Trautman, “Spin structures on real projective quadrics”, J. Geom. Phys. 10 (1993) 127–154. [4] M. Flato and C. Fronsdai, “Spontaneously generated field theories, zero-center modules, colored singletons and the virtues of N = 6 supergravity”, in Essays on Supersymmetry, ed. C. Fronsdal, pp. 123–162, Math. Phys. Studies Vol. 8, D. Reidel Publ. Co., 1986. [5] M. Flato, C. Fronsdal and J. P. Gazeau, “Masslessness and light-cone propagation in 3 + 2 De Sitter and 2 + 1 Minkowski spaces”, Phys. Rev. D33 (1986) 415–420. [6] M. Flato and D. Sternheimer, “Remarques sur les automorphismes causals de l’espacetemps”, C. R. Acad. Sci. Paris 263 (1966) 935–938. [7] C. Fronsdal, Essays on Supersymmetry, ed. C. Fronsdal, Math. Phys. Studies Vol. 8, D. Reidel Publ. Co., 1986. [8] W. Kopczynski and S. L. Woronowiz, “A geometrical approach to the twistor formalism”, Rep. Math. Phys. 2 (1971) 35–51. [9] G. Mack and I. Todorov, “Irreducibility of the ladder representations of U (2, 2) when restricted to the Poincar´e subgroup”, J. Math. Phys. 10 (1969) 2078–2085. [10] G. Mackey, “Induced representations of locally compact groups I; II”, Ann. Math. 55 (1952) 101–139; Ann. Math. 58 (1953) 193–221. See also “Unitary representations of group extensions I”, Acta. Math. 99 (1958) 265–311. [11] J. Mickelsson and J. Nierderle, “Conformally covariant field equations”, Ann. Inst. Henri Poincar´e Sect. A (N.S) 23 (1975) 277–295. [12] A. Salam and J. Strathdee, “On Kaluza–Klein theory”, Ann. Phys. 141 (1982) 316–352. [13] I. Todorov, “Local Field Representations of the Conformal Group and their Applications”, in Mathematics and Physics ed. L. Streit, Vol. 1, 195–338, World Scientific, 1985.
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[14] E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group”, Ann. Math. 40 (1939) 149. [15] E. P. Wigner and E. Inon¨ u, “On the contraction of groups and their representations”, Proc. Nat. Acad. Sci. USA 39 (1953) 510–524.
SQUARE-INTEGRABILITY OF INDUCED REPRESENTATIONS OF SEMIDIRECT PRODUCTS P. ANIELLO Dipartimento di Fisica, Universit` a di Genova I.N.F.N., Sezione di Genova via Dodecaneso 33, 16146 Genova, Italy E-mail : [email protected]
GIANNI CASSINELLI Dipartimento di Fisica, Universit` a di Genova I.N.F.N., Sezione di Genova via Dodecaneso 33, 16146 Genova, Italy E-mail : [email protected]
ERNESTO DE VITO Dipartimento di Matematica, Universit` a di Modena via Campi 213/B, 41100 Modena, Italy and I.N.F.N., Sezione di Genova via Dodecaneso 33, 16146 Genova, Italy E-mail : [email protected]
ALBERTO LEVRERO Dipratimento di Fisica, Universit` a di Genova I.N.F.N., Sezione di Genova via Dodecaneso 33, 16146 Genova, Italy E-mail : [email protected] Received 30 June 1997 Revised 24 July 1997 We consider a semidirect product G = A ×0 H, with A abelian, and its unitary representations of the form IndG G0 (x0 m) where x0 is in the dual group of A, G0 is the stability group of x0 and m is an irreducible unitary representation of G0 ∩ H. We give a new selfcontained proof of the following result: the induced representation IndG G0 (x0 m) is square-integrable if and only if the orbit G[x0 ] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of IndG G0 (x0 m).
1. Introduction In the paper [6], Kleppner and Lipsman describe the Plancherel measure for a locally compact group with a closed normal Type I subgroup. As a consequence, they characterise the square-integrability of unitary representations of such groups. 301 Reviews in Mathematical Physics, Vol. 10, No. 3 (1998) 301–313 c World Scientific Publishing Company
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The study of Plancherel measure for nonunimodular groups has been continued by Duflo and Moore, [3]. They use as a fundamental tool for their investigation the notion of operator of formal degree for square-integrable representations. In our paper we use this notion and some of its most elementary properties to obtain a new selfcontained and more accessible proof of the characterisation of square-integrable representations of semidirect products with an abelian normal factor. More precisely, we consider a semidirect product G of a locally compact second countable (lcsc) topological abelian group A and a lcsc topological group H. We study the representations of the form IndG G0 (x0 m) where Ind is the Mackey unitary ˆ induction, x0 is an element in A, the dual group of A, G0 is the stability group of x0 with respect to the dual action of G on Aˆ and m is an irreducible unitary representation of G0 ∩ H. We prove that IndG G0 (x0 m) is square-integrable if and only if the orbit G[x0 ] has nonzero measure with respect to the Haar measure on Aˆ and m is square-integrable. Moreover, we give an explicit description of the formal degree of IndG G0 (x0 m). The need of an elementary proof of this result comes from the role that squareintegrable representations have in the construction of generalised coherent states, [1], and in wavelet analysis, [2, 8] (see the final section for a brief review of these applications). Indeed, quite recently, some authors, [2, 4], have considered the case of subrepresentations of the quasiregular representation on L2 (Rn ) for semidirect products of the form Rn ×0 H where H is a closed subgroup of GLn (R), obtaining, with an elementary proof, some of the results in [6] (nevertheless, it seems that these authors are not aware of this paper). 2. Preliminaries Let G be a locally compact topological group. We denote by K(G) the vector space of continuous functions of compact support on G and by B(G) the Borel σ-algebra of G. By Borel measure on G we mean a positive measure defined on B(G) and finite on the compact sets. Let µG be a left Haar measure on G and ∆G its modular function. In this paper we will use the word representation to mean a continuous irreducible unitary representation of G acting in a complex separable Hilbert space. A representation U of G in H is square-integrable if there exist two nonzero vectors φ, ψ ∈ H such that the map cφ,ψ from G to C cφ,ψ (g) = hφ, Ug ψi is in L2 (G, µG ). In the following we will use this result of Duflo and Moore (Theorem 3 of [3]). Theorem 1 (Duflo and Moore). Let U be a square-integrable representation of G acting in H. There exists a unique positive selfadjoint operator KU in H such that
303
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1. if φ, ψ ∈ H, φ 6= 0 the map cφ,ψ is in L2 (G, µG ) if and only if ψ ∈ −1/2 Dom KU ; −1/2 2. if φ, φ0 ∈ H and if ψ, ψ 0 ∈ Dom KU , then hcφ,ψ , cφ0 ,ψ0 iL2 (G,µG ) = hφ, φ0 iH
D
−1/2
KU
−1/2
ψ 0 , KU
E ψ
H
.
The operator KU is called the formal degree of the representation. From now on we let G be a lcsc topological group which is semidirect product of a closed abelian normal subgroup A and a closed subgroup H G = A ×0 H . The inner action of G on A is g[a] = gag −1 ,
a ∈ A, g ∈ G .
We denote by Aˆ the dual group of A, which is in a natural way an abelian lcsc topological group; G acts continuously on it by g[x](a) = x(g −1 ag),
ˆ g ∈ G. a ∈ A, x ∈ A,
We fix an element x0 of Aˆ and we denote by X the corresponding orbit X = {x ∈ Aˆ : x = g[x0 ], g ∈ G} and by G0 the stability subgroup of x0 , G0 = {g ∈ G : g[x0 ] = x0 } . We recall some properties of X as a measurable space. 1. X is a Borel set of Aˆ and we endow it with the σ-algebra ˆ : E ⊂ X} . A := {E ∈ B(A) X is a standard measurable space with respect to this σ-algebra and it is a transitive H-space with respect to the restriction of the action of G. 2. there exists a σ-finite H-quasi-invariant measure ν on (X, A), that is, Z λν (h, x)dν(x) h ∈ H, E ∈ A ν(h[E]) = E
where λν is a measurable function from H × X to (0, ∞) (see, for example, Theorem 5.19 of [9]); 3. there exists a measurable map q : X → H such that q(x0 ) = e q(x)[x0 ] = x,
where e is the identity, x∈X,
q(X) ∈ B(H)
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and, for any compact set K in H, the set {q(h[x0 ]) : h ∈ K} has compact closure in H (see, for example, Theorem 5.11, of [9]). Let H0 = G0 ∩ H, so that G0 = A ×0 H0 . We choose a representation m of H0 acting in a complex separable Hilbert space K and we consider the representation x0 m of G0 in K (x0 m)(ah) := x0 (a)m(h) a ∈ A, h ∈ H0 . We denote by IndG G0 (x0 m) the representation unitarily induced by x0 m from G0 to G, [7]. If the semidirect product is regular, any irreducible unitary representation of G is equivalent to a representation of the previous form. 3. Main Results The main result of the paper is the following theorem (compare with Corollary 11.1 of [6] and Proposition 9 of [3]). Theorem 2. Let G be a lcsc topological group, semidirect product of an abelian ˆ X be the orbit normal closed subgroup A and a closed subgroup H. Let x0 ∈ A, G[x0 ], G0 be the stability subgroup of x0 and m a representation of G0 ∩ H. Then the representation IndG ˆ (X) 6= 0 and m G0 (x0 m) is square-integrable if and only if µA is square-integrable. The results obtained in [2] and [4] follow easily from: Corollary 1. The representation IndG G0 (x0 I), where I is the trivial representation of G0 ∩ H on C, is square-integrable if and only if µAˆ (X) 6= 0 and G0 ∩ H is compact. We notice that, under the hypotheses of the previous corollary, the representation IndG G0 (x0 I) is equivalent to a subrepresentation of the quasi-regular representation of G in L2 (A, µA ), regarding A as the quotient space G/H. The representation IndG G0 (x0 m) can be realised in various unitarily equivalent ways. Since the square-integrability depends only on the equivalence class of the representation, we can freely choose the explicit form of IndG G0 (x0 m). Fixed a H-quasi-invariant measure ν and a measurable map q : X → H as in points 2 and 3 of Sec. 2, a suitable choice for IndG G0 (x0 m) is the representation U acting in L2 (X, ν, K) as (Uah φ)(x) = (λν (h−1 , x))1/2 x(a) m(q(x)−1 hq(h−1 [x]))φ(h−1 [x])
(1)
where x ∈ X, ah ∈ G and φ ∈ L2 (X, ν, K). Taking into account that the Hilbert space L2 (X, ν, K) can be identified with L2 (X, ν) ⊗ K, we have:
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Corollary 2. If U is square-integrable, its formal degree is KU = K0 ⊗ Km , where Km is the formal degree of m and K0 is the selfadjoint operator in L2 (X, ν) (K0 φ)(x) = ∆G (q(x))φ(x) ,
φ ∈ L2 (X, ν)
on its natural domain. 4. Proof of the Theorem In this section, we collect the proofs of the theorem and of its corollaries. We begin with two technical lemmas on the Haar measures on A, Aˆ and G. Lemma 1. For any h ∈ H and E ∈ B(A) we have µA (h[E]) = ρ(h)µA (E) where ρ : H → (0, ∞) is a continuous group homomorphism. Moreover the Haar measure µG is 1 µG = µA ⊗ µH ρ and the modular function of G satisfies ∆G (ah) =
∆H (h) , ρ(h)
a ∈ A, h ∈ H .
Proof. Let h ∈ H. Since a 7→ h[a] is a topological isomorphism, then B(A) 3 E 7→ µA (h[E]) ∈ [0, ∞] is a Haar measure on A so that there exists a strictly positive constant ρ(h) such that µA (h[E]) = ρ(h)µA (E) and the R map h 7→ ρ(h) is obviously a group homomorphism. Let f ∈ K(A) be such that A f (a)dµA (a) = 1, by the previous formula it follows that Z
f (h−1 [a]) dµA (a) = ρ(h)
Z f (a) dµA (a) = ρ(h) ,
A
so that ρ is continuous since the map h 7→ dominated convergence theorem.
A
R A
f (h−1 [a])dµA (a) is continuous by the
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Since G is a Hausdorff lcsc topological space, µA ⊗ 1ρ µH is a Borel measure. We show that it is left invariant. In fact, if f ∈ K(G) and g = a0 h0 ∈ G, then Z Z 1 1 dµH (h) = f (a0 h0 [a]h0 h) dµA (a) ⊗ dµH (h) f (gah)dµA (a) ⊗ ρ(h) ρ(h) Z ρ(h0 ) dµA (a) ⊗ dµH (h) = f (a0 h0 [a]h) ρ(h) Z 1 dµA (a) ⊗ dµH (h) = f (a0 ah) ρ(h) Z 1 dµA (a) ⊗ dµH (h) . = f (ah) ρ(h) With a similar computation we show that µA ⊗ ∆1H µH is the right Haar measure on G, hence ∆H (h) , a ∈ A, h ∈ H . ∆G (ah) = ρ(h)
This ends the proof.
ˆ be the Banach space of complex measures on B(A) ˆ and F the Fourier Let M (A) R ˆ defined as (F ν)(a) = ˆ x(a)dν(x). The same symbol F denotes transform on M (A) A ˆ µ ˆ ). We choose the Haar measure on Aˆ the Fourier–Plancherel operator on L2 (A, A ˆ µ ˆ ) onto L2 (A, µA ). is such a way that F is unitary from L2 (A, A ˆ we have Lemma 2. For any h ∈ H and E ∈ B(A) µAˆ (h[E]) = ρ(h−1 )µAˆ (E) . Proof. Arguing as in Lemma 1, we conclude that there exists a continuous homomorphism α : H → (0, ∞) such that µAˆ (h[E]) = α(h)µAˆ (E) . ˆ µ ˆ ) be such that F (f ) ∈ L1 (A, µA ). For all h ∈ H, using the Let now f ∈ L1 (A, A ˆ µ ˆ ), we have Fourier inversion theorem on L1 (A, A Z f h (x) := f (h[x]) =
h[x](a)(F f )(a)dµA (a) A
Z =
x(h−1 [a])(F f )(a)dµA (a)
A
Z
x(a)(F f )(h[a])ρ(h)dµA (a)
= A
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Z = ρ(h)
x(a) Z
307
Z ˆ A
A
Z x(a)
= ρ(h)
x(h[a])f (x)dµAˆ (x)dµA (a)
A
ˆ A
Z
h−1 [x](a)f (x)dµAˆ (x)dµA (a) Z
x(a)
= α(h)ρ(h) A
ˆ A
x(a)f (h[x])dµAˆ (x)dµA (a)
= α(h)ρ(h)f h (x) , ˆ µ ˆ ) and F (f h ) ∈ where in the last equality we have used the fact that f h ∈ L1 (A, A 1 −1 L (A, µA ). Hence α(h) = ρ(h ) for all h ∈ H. We now turn to the proof of Theorem 2. From now on let the notations be as in Theorem 2 and U be the realisation of IndG G0 (x0 m) given by formula (1). We split the proof in various lemmas. Lemma 3. If U is square-integrable, then µAˆ (X) 6= 0. Proof. If U is square-integrable there exist two nonzero vectors φ, ψ ∈ H such that the map cφ,ψ is in L2 (G, µG ). By the theorem of Fubini this implies that for µH − almost all h
(a 7→ hφ, Ua Uh ψiH ) ∈ L2 (A, µA )
Z
and we have hφ, Ua Uh ψiH =
x(a)hφ(x), (Uh ψ)(x)iK dν(x) . X
Since the function X 3 x 7→ hφ(x), (Uh ψ)(x)iK is ν-integrable, we can define the complex measure νh on X which has density hφ(x), (Uh ψ)(x)iK with respect to ν. Suppose that |νh |(X) = 0 (where |νh | is the total variation of νh ) for µH -almost any h, then the map X 3 x 7→ hφ(x), (Uh ψ)(x)iK is ν-negligible, hence cφ,ψ (ah) = 0 for all a ∈ A and for µH -almost all h, that is, cφ,ψ = 0 in L2 (G, µG ) and this is a contradiction, since U is irreducible and φ, ψ 6= 0. Hence there exists at least one h ∈ H such that |νh |(X) 6= 0 and (a 7→ hφ, Ua Uh ψiH ) ∈ L2 (A, µA ) .
(2)
Fix this h. The measure νh has a canonical extension to a complex measure on Aˆ which we still denote by νh . For all a ∈ A, Z x(a)dνh (x) = (F νh )(a) . cφ,ψ (ah) = ˆ A
Due to condition (2), F νh ∈ L2 (A, µA ). Using a standard result on Fourier transform (see, for example, Theorem 31.33 of [5]) νh is absolutely continuous with respect to µAˆ . Since |νh |(X) 6= 0, it follows that µAˆ (X) 6= 0 as claimed.
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If µAˆ (X) 6= 0, the restriction µX of the measure µAˆ to X is a H-quasi-invariant measure, due to Lemma 2. In this case we can choose µX as the quasi-invariant measure ν in the definition (1) of U . With this choice, λµX (h, x) = ρ(h)−1 and U acts on L2 (X, µX , K) as (Uah φ)(x) = ρ(h)1/2 x(a) m(q(x)−1 hq(h−1 [x]))φ(h−1 [x]) . Moreover, since for all s ∈ H0 the map ρ satisfies ρ(s−1 ) = Theorem 5.30 of [9]), we have ∆H0 (s) = ∆G (s)
∆H0 (s) ∆H (s)
(see for example
∀s ∈ H0 .
(3)
Lemma 4. Let G be as in Theorem 2. Suppose µAˆ (X) 6= 0 and U be the representation described above. Let φ, ψ be two nonzero vectors in L2 (X, µX , K), then the condition cφ,ψ ∈ L2 (G, µG )
(4)
holds if and only if both conditions for µX -almost all x h 7→ ∆G (h−1 ) |hφ(x), (Uh−1 ψ)(x0 )iK |2 ∈ L1 (H, µH ) and
Z 2 −1 ∆G (h ) |hφ(x), (Uh−1 ψ)(x0 )iK | dµH (h) ∈ L1 (X, µX ) x 7→
(5)
(6)
H
are satisfied. In this case we have Z Z 2 ∆G (h−1 ) |hφ(x), (Uh−1 ψ)(x0 )iK | dµH (h)dµX (x) . kcφ,ψ k2L2 (G,µG ) = X
(7)
H
Proof. According to Lemma 1 we have that µG = µA ⊗ ρ1 µH . Using the ˆ µ ˆ ) and again theorem of Fubini, the unitarity of the Fourier transform on L2 (A, A the theorem of Fubini, we conclude that the condition (4) is equivalent to the following two conditions: — For µX -almost all x ∈ X the function H 3 h 7→ ρ(h)−1 |hφ(x), (Uh ψ)(x)iK |
2
is in L1 (H, µH ); — the function Z X 3 x 7→ H
is in L1 (X, µX ).
ρ(h)−1 |hφ(x), (Uh ψ)(x)iK |2 dµH (h)
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First perform the change of variables h 7→ (q(x)h) in the integrals on H and observe that (Uq(x) Uh ψ)(x) = ρ(q(x))1/2 (Uh ψ)(x0 ). Next send h to h−1 and use Lemma 1. In doing so we obtain that the two previous conditions are equivalent to (5) and (6). Equation (7) is now consequence of the theorem of Fubini. We need a technical lemma that allows us to compute the integrals on H as integrals on X × H0 . Let β : X × H0 → H be the map β(x, s) = q(x)s
x ∈ X, s ∈ H0 .
Lemma 5. The map β is an isomorphism of measurable spaces. By suitably choosing the normalisation of µH0 , the image measure of (ρ ◦ β)µX ⊗ µH0 , under the map β, is the Haar measure µH . Proof. Since q is measurable, β is an isomorphism of measurable spaces. Hence, with the use of β we can regard X × H0 as an H-space, explicitly h[(x, s)] = (h[x], q(h[x])−1 hq(x)s) . Fix a Haar measure µH0 on H0 . We claim that (ρ ◦ β)µX ⊗ µH0 is a σ-finite Hinvariant measure on X × H0 . Since ρ is a continuous group homomorphism, ρµH0 is a σ-finite measure on H0 . Moreover, let (Kn )n≥1 be a sequence of compacts sets S of H such that n Kn = H and define En = {h[x0 ] : h ∈ Kn } . Then, as recalled in Sec. 2, q(En ) has compact closure, so that ρ ◦ q is bounded on En . Since µX is σ-finite, the same holds for (ρ ◦ q)µX . Moreover if h ∈ H and E ∈ A ⊗ B(H0 ), we have (ρ ◦ β) µX ⊗ µH0 (h[E]) Z Z ρ(q(x)s)χE (h−1 [x], q(h−1 [x])−1 h−1 q(x)s)dµH0 (s)dµX (x) = X
H0
Z Z = X
H0
Z Z = X
ρ(hq(h−1 [x])s)χE (h−1 [x], s)dµH0 (s)dµX (x) ρ(h−1 )ρ(hq(x)s)χE (x, s)dµH0 (s)dµX (x)
H0
= (ρ ◦ β) µX ⊗ µH0 (E) . The image measure of (ρ ◦ β)µX ⊗ µH0 with respect to β is a σ-finite H-invariant measure on H, hence it is the Haar measure µH , up to a constant factor that can be put equal to 1 by redefining µH0 . Lemma 6. With the assumptions of Lemma 4, the conditions (5) and (6) hold if and only if the following ones are satisfied:
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(a) m is a square-integrable representation of H0 (we denote by Km its formal degree); −1/2 for µX -almost all x ∈ X; (b) ψ(x) ∈ Dom Km (c) the function
2
−1/2 ψ(x) X 3 x 7→ ∆G (q(x))−1 Km K
1
is in L (X, µX ). In this case, we have that Z kcφ,ψ k2L2 (G,µG ) = kφk2H
2
−1/2 ∆G (q(x))−1 Km ψ(x) dµX (x) . K
X
(8)
Proof. By Lemma 5 and the theorem of Fubini, taking into account that (U(q(x0 )s)−1 ψ)(x0 ) = (ρ(q(x0 )s))−1/2 m(s−1 )ψ(x0 ) , the condition (5) is equivalent to the two conditions (5’) for µX -almost all x0 ∈ X the function 2 H0 3 s 7→ ∆G (q(x0 )s)−1 hφ(x), m(s−1 )ψ(x0 )iK is in L1 (H0 , µH0 ); (5”) the function X 3 x0 c 7→
Z
2 ∆G (q(x0 )s)−1 hφ(x), m(s−1 )ψ(x0 )iK dµH0 (s)
H0
is in L1 (X, µX ). Performing the change of variables s 7→ s−1 and using Eq. (3), condition (5’) becomes 2 H0 3 s 7→ ∆G (q(x0 ))−1 |hφ(x), m(s)ψ(x0 )iK | is in L1 (H0 , µH0 ). Since ψ 6= 0, this last relation is precisely the condition that m is square-integrable and that −1/2 ψ(x0 ) ∈ Dom Km
for µX -almost all x0 ∈ X. According to Theorem 1, Z 2 ∆G (q(x0 ))−1 |hφ(x), m(s)ψ(x0 )iK | dµH0 (s) H0
2
−1/2 ψ(x0 ) . = ∆G (q(x0 ))−1 kφ(x)k2K Km K
Using this equation, the conditions (5”) and (6) turn out to be equivalent to (c) and to the fact that φ is in L2 (X, µX , K). Equation (8) is now evident.
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We are now in the position to prove the results stated in Sec. 3. Proof of Theorem 2. By Lemma 3 we have µAˆ (X) 6= 0. Since the squareintegrability depends only on the equivalence class of U , we choose ν = µX and we explicitly realise U acting on L2 (X, µX , K). “if”: By definition of square-integrability of U there exist two nonzero vectors φ, ψ ∈ L2 (X, µX , K) such that cφ,ψ ∈ L2 (G, µG ). Applying Lemmas 4 and 6, we obtain that m is square-integrable. −1/2 is dense in K. Moreover the “only if”: Due to Theorem 1, the domain of Km domain of the multiplicative operator by the function x 7→ ∆G (q(x))1/2 is dense in L2 (X, µX , K), hence the conditions (b) and (c) of Lemma 6 are satisfied in a dense subspace of L2 (X, µX , K). Fix a nonzero vector ψ in this dense subspace and a nonzero φ in L2 (X, µX , K). Applying Lemmas 4 and 6, we have that cφ,ψ ∈ L2 (G, µG ). Proof of Corollary 1. It is an immediate consequence of the fact that the Haar measure of a topological group is finite if and only if the group is compact. Proof of Corollary 2. If U is the representation (1), with the choice ν = µX , apply Lemmas 4 and 6, taking into account the equation
−1/2 kcφ,ψ kL2 (G,µG ) = kφkH KU ψ . H
If ν is any σ-finite quasi-invariant measure on X, then it is equivalent to µX (see for example Theorem 5.19 of [9]). The thesis follows. 5. A Glance to Some Applications In this final section we summarise the role of square-integrable representations in the construction of wavelet transforms and coherent states. Let G be any lcsc topological group and U be a square-integrable representation −1 of G acting in a Hilbert space H. Fixed a vector ψ in the domain of KU 2 , normalised
−1 such that KU 2 ψ = 1, the isometry H
H 3 φ 7→ Wψ φ ∈ L2 (G, µG ) defined by Wψ φ(g) = cφ,ψ (g) = hφ, Ug ψiH is called, in the framework of signal analysis, the wavelet transform defined by the −1 mother wavelet ψ. The elements in the domain of KU 2 are called admissible vectors. In many concrete cases considered in the literature, G is a semidirect product and U is an irreducible subrepresentation of the representation V acting in L2 (A, µA ) as 1 (V(a,h) φ)(b) = ρ(h)− 2 φ(h−1 [ba−1 ]) .
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ˆ µ ˆ ) intertwines U The Fourier–Plancherel operator F from L2 (A, µA ) onto L2 (A, A with a representation of the form considered in Corollary 1, so that the squareintegrability condition turns out to be equivalent to the fact that the orbit X has positive measure and the stabilizer is compact. In this case the wavelet transform is an isometry from the closed subspace of L2 (A, µA ) {φ ∈ L2 (A, µA ) : supp(F φ) ⊂ X} to L2 (G, µG ) explicitly given by Z
ρ(h)− 2 φ(b)ψ(h−1 [ba−1 ]) dµA (b) 1
(Wψ φ)(a, h) = A
Z
1
= ˆ A
ρ(h) 2 x(a)(F φ)(x)(F ψ)(h−1 [x]) dµAˆ (x) .
The condition that ψ is admissible turns out to be
supp(F ψ) ⊂ X 1 ˆ µ ˆ) . x 7→ ∆G (q(x))− 2 (F ψ)(x) ∈ L2 (A, A
We notice that, since H0 is compact, then the restriction of ∆G to G0 is 1, so that ∆G (c(x)) is in fact independent on the choice of q. The above explicit formula for Wψ is the starting point for any further investigation, including numerical treatment of signals, and there is a huge amount of papers on these topics, dealing mainly with the case of the group “ax + b” or its n-dimensional generalisation, see for example [8] and references therein. Theorem 2 suggests the possibility of considering also wavelet transforms on spaces of vector valued functions. In the framework of quantum mechanics the square-integrable representations are relevant in order to describe the so-called coherent states. Roughly speaking, given a representation U of a group G in H and a vector ψ ∈ H, if the following integral decomposition (or resolution of the identity) holds Z hφ, Ug ψiUg ψdµG (g) φ ∈ H ,
φ= G
then the set {Ug ψ : g ∈ G} is called a set of coherent states for H. This intuitive idea can be made precise assuming that U is a square-integrable representation and ψ is admissible (and suitably normalised). Indeed, with these assumptions, the previous formula has to be interpreted in weak sense and, in doing so, it is just the relation stated in Theorem 1. Also in this case, the notion of square-integrability is at the root of an extremely useful construction, see for example [1] and references therein.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9]
S. T. Ali, J. P. Antoine and J. P. Gazeau, Rev. Math. Phys. 7 (1995) 1013. D. Bernier and K. F. Taylor, SIAM J. Math. Anal. 27 (1996) 594. M. Duflo and C. C. Moore, J. Funct. Anal. 21 (1976) 209. H. F¨ uhr, J. Math. Phys. 37 (1996) 6353. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer-Verlag, Berlin, 1970. ´ A. Kleppner and R. L. Lipsman, Ann. Sci. Ecole Norm. Sup. 5 (1972) 459. G. W. Mackey, Ann. Math. 55 (1952) 101. C. E. Heil and D. F. Walnut, SIAM Review 31 (1989) 628. V. S. Varadarajan, Geometry of Quantum Theory, Second Ed. Springer-Verlag, New York, 1985.
SOLITON LIKE SOLUTIONS OF A LORENTZ INVARIANT EQUATION IN DIMENSION 3∗
Dipart.
VIERI BENCI Matematica Appl. “U. Dini” Universit` a degli Studi di Pisa Via Bonanno, 25/b 50126 Pisa, Italy
DONATO FORTUNATO and LORENZO PISANI Dipartimento di Matematica Universit` a degli Studi di Bari Via Orabona, 4 70125 Bari, Italy Received 5 June 1997
Contents 1. Introduction 2. Statement of the Problem 2.1. Nonlinear wave equations and solitons 2.2. Statement of the main result 3. Functional Setting 3.1. The space H and the open set Λ 3.2. Topological charge and connected components of Λ 3.3. Properties of the energy functional 4. Existence Theorems 4.1. The splitting lemma 4.2. Existence of minima in the connected components of Λ 5. Mass and Energy 5.1. The mass matrix 5.2. Equivalence between mass and energy 6. Conclusions
315 316 316 319 321 321 323 326 329 329 334 338 338 340 343
1. Introduction A soliton is a solution of a field equation whose energy travels as a localized packet and which preserves its form under perturbations. In this respect solitons have a particle-like behavior and they occur in many questions of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics, plasma physics (see [7, 12, 16, 19]). Probably, the simplest equation which has soliton-like solutions is the sineGordon equation which is a semilinear hyperbolic equation defined in a two dimensional space-time. ∗ Supported
by M.U.R.S.T. (40% and 60% funds); the second and the third author were supported also by E.E.C., Program Human Capital Mobility (Conctract ERBCHRXCT 940494). 315
Reviews in Mathematical Physics, Vol. 10, No. 3 (1998) 315–344 c World Scientific Publishing Company
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The goal of this paper is to determine an equation defined in a four dimensional space-time which preserves the main peculiarities of the sine-Gordon equation, namely: • it has soliton solutions; • it is Lorentz invariant; • the linearized equation at u = 0 is the Klein–Gordon equation utt − c2 ∆u + ω02 u = 0 • the mass of a soliton which is the matrix mij defined by the formula pi = mij vj is in fact a scalar, namely mij = mδij (here pi is the momentum of the field and vi is the velocity of the soliton). • the solitons satisfy the celebrated Einstein equation E = mc2
(1.1)
The paper is organized as follows. In Sec. 2 we introduce our model equation and we state the existence result. In Sec. 3 we give the topological classification of static solutions by means of the topological charge. In Sec. 4 we prove the existence of static solutions with non-trivial charge. To prove this result we need to study the behaviour of sequences with bounded energy, in the spirit of the concentration-compactness principle. In the last section we define the mass of soliton and prove that it is a scalar which equals the energy. 2. Statement of the Problem 2.1. Nonlinear wave equations and solitons We introduce some notation. If n, k are positive integer, Rn+1 and Rk will denote respectively the physical space-time (typically n = 3) and the internal parameters space. A point in Rn+1 will be denoted by (x, t), with x ∈ Rn and t ∈ R. The fields we are interested in are maps u : Rn+1 → Rk ,
u = (u1 , . . . , uk )
Since we require the Lorentz invariance, we shall consider Lagrangian densities of the form L = L(u, ρ) , where ρ = c2 |∇u|2 − |ut |2 ,
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c being the light velocity, ∇u and ut denote respectively the Jacobian with respect to x and the derivative with respect to t. We shall consider 1 (2.1) L(u, ρ) = − α(ρ) − V (u) , 2 where α: R → R and the function V is defined in an open subset Ω ⊂ Rk . The action functional related to 2.1 is Z Z 1 L(u, ρ)dxdt = − α(ρ) − V (u) dxdt . S(u) = 2 R3+1 R3+1 So the Euler–Lagrange equation is ∂ 0 (α (ρ)ut ) − c2 ∇(α0 (ρ)∇u) + V 0 (u) = 0 , ∂t
(2.2)
where ∇(α0 (ρ)∇u) denotes the vector whose jth component is given by div (α0 (ρ)∇uj ) and V 0 denotes the gradient of V . When α(ρ) = ρ, Eq. (2.2) reduces to the semilinear wave equation u + V 0 (u) = 0 , where u = utt − c2 ∆u , ∆u =
∂2u ∂ 2u + · · · + . ∂x21 ∂x2n
Under suitable assumptions on α and V every static solution of (2.2) u(x, t) = ϕ(x) , i.e. a solution of −c2 ∇(α0 (ρ)∇u) + V 0 (u) = 0
(2.3)
has energy localized in a compact region. Let ϕ = ϕ(x1 , x2 , x3 ) be a solution of (2.3) and consider a vector v with |v| < c. For simplicity we take v = (v, 0, 0). Then it is easy to verify that x − vt 1 (2.4) uv (x1 , x2 , x3 , t) = ϕ q , x2 , x3 v 2 1− c is a solution of Eq. (2.2). Notice that the function u experiences a contraction of a factor 1 γ= q 1−
v 2 c
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in the direction of the motion; this is a consequence of the fact that (2.2) is Lorentz invariant. Moreover, it is clear that (2.4) has some kind of stability if the static solution of (2.2) is a local minimum of the energy functional Z 1 α(c2 |∇ϕ|2 ) + V (ϕ) dx . E(ϕ) = Rn 2 These observations lead to the following definition. Definition 2.1. We call soliton a solution of Eq. (2.2) having the form of Eq. (2.4), where ϕ is a local minimum of the energy functional. Before making our choice of the functions α and V , we make some remarks about the difference between the one dimensional case (n = 1) and the three dimensional case (n = 3). If n = 1, a classical nonlinear equation having soliton solutions is the sineGordon equation u + sin u = 0 . (2.5) Observe that (2.5) can be obtained from (2.2) choosing k = 1,
α(ρ) = ρ ,
V (ξ) = 1 − cos ξ .
The static solutions of (2.5) are critical point of the energy funtional Z 2 c 2 (ϕ(x)) ˙ + (1 − cos ϕ(x)) dx E(ϕ) = 2 R over the space H of the smooth real functions ϕ satisfying the asymptotic conditions ϕ(−∞) = 2k1 π ,
ϕ(+∞) = 2k2 π ,
(2.6)
with k1 , k2 ∈ Z; observe that the asymptotic values of ϕ are minima of the potential V , so that the energy E(ϕ) can be finite. Explicit calculations shows that these solutions possess a solitonic behaviour (see [7, 19]). Furthermore, since the space H can be divided in infinitely many connected components according to the asymptotics conditions (2.6), the solitons can be topologically classified (more precisely, since E(ϕ) = E(ϕ + 2k1 π), it is sufficient to fix k1 and consider the difference k2 − k1 ). Now consider the more realistic (from a physical point of view) case n = 3. Consider, as before, purely scalar fields (i.e. k = 1) and take α(ξ) = ξ. Then the energy functional for the static solutions is Z 2 c |∇ϕ|2 + V (ϕ) dx . (2.7) E(ϕ) = 2 R3 If the potential V is positive, then a simple scaling argument shows that any function ϕ minimizing (2.7) is necessarily trivial, i.e. it takes a constant value
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which is a minimum point of V ; this fact can be easily seen by imposing that d dλ E(ϕ(λx))|λ=1 = 0 (see [7]). So if we want to consider purely scalar fields and α(ξ) = ξ, we cannot take V (ξ) > 0. On the other hand, if we consider non-positive potential, we are forced to seek saddle points, instead of minima, and for these static solutions we have lack of stability. As an example we recall that, if we take V (ξ) =
1 2 1 4 ξ − ξ , 2 4
critical points of the energy functional Z 2 c 1 2 1 4 2 |∇ϕ| + ϕ − ϕ dx . E(ϕ) = 2 2 4 R3 have been found in [5] and [14] and for more general potentials in [4, 18]; but in [1] and [3] it has been proved that these static solutions are not stable. 2.2. Statement of the main result In this paper we consider the case n = 3. We propose a model equation like (2.2) whose static solutions can be topologically classified, as well as the static solutions of the sin-Gordon equation. Then we shall prove the existence of such a solution ϕ, obtained as minimum of the energy functional. We have seen that α linear implies that (2.2) does not have static soliton; thus we are forced to take α nonlinear: α(ρ) = ρ + a2 ρ2 + a3 ρ3 + · · · ; now we look for the simplest choice of α. A priori the simplest choice is α(ρ) = ρ + a2 ρ2 however this is a bad choice for the evolution Eq. (2.2), in fact α0 (ρ) might be negative for some values of ρ and this is a “disaster” for Eq. (2.2). Then it is natural to consider the next simple choice namely α(ρ) = ρ + a3 ρ3 ; this choice of α will be considered in this paper. However, from the proof of our theorem, it is not difficult to realize that more general choices are possible. We have decided to restrict ourselves to the simplest case since our aim is just to show that static solitons in dimension 3 do exist. Other choices are possible if we do not assume that the Lagrangian splits as in (2.1) as it is in the Skyrme model [17]. So we consider maps u : R3+1 → R4 (i.e. n = 3, k = 4) and we assume ε α(ρ) = ρ + ρ3 , 3
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being ε > 0. Then (2.2) can be written ∂ [(1 + ερ2 )ut ] − c2 ∇[(1 + ερ2 )∇u] + V 0 (u) = 0 , ∂t
(2.8)
u + ε6 u + V 0 (u) = 0
(2.9)
or where
∂ 2 [(c |∇u|2 − |ut |2 )2 ut ] − c2 ∇[(c2 |∇u|2 − |ut |2 )2 ∇u] . ∂t So the static solutions ϕ solve the equation 6 u =
−c2 ∆ϕ − εc6 ∇(|∇ϕ|4 ∇ϕ) + V 0 (ϕ) = 0 , which can be written −c2 ∆ϕ − εc6 ∆6 ϕ + V 0 (ϕ) = 0 . Then the energy functional becomes Z 2 c c6 |∇ϕ|2 + ε |∇ϕ|6 + V (ϕ) dx E(ϕ) = 2 6 R3
(2.10)
(2.11)
As the function V , we assume that it is defined on ¯ Ω = R4 \ {ξ} ¯ More precisely we assume and that it is positive and diverges for ξ → ξ. • V ∈ C 2 (Ω, R); • V (x) > V (0) = 0 for every x ∈ Ω; • there exist c, r > 0 such that, then V (ξ¯ + ξ) > c|ξ|−6
if |ξ| < r ,
(2.12)
¯ = 1. • for sake of simplicity |ξ| We observe that, for j = 1, . . . , 4, α0 (0) = 1 and, since 0 is a minimum for V , we can choose a base in R4 which diagonalizes V 00 (0) so that 2 m1 0 .. V 00 (0) = . 2 0 m4 Then, linearizing (2.9) at 0, we get a system of Klein–Gordon equations uj + m2j uj = 0 , (1 6 j 6 4) .
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The presence of ∆6 in (2.10) implies that the functions on which E is finite are continuous and decay to 0 at infinity; the presence of the singualr term V 0 (u) implies that such maps take value in Ω. So the non-trivial topological properties of Ω (namely π3 (Ω) = Z) permit, as in the sin-Gordon equation, to give a topological classification of static configurations. This classification is carried out by means of a topological invariant, the charge (see Subsec. 3.2), which depends only on the region where the function is concentrated, namely the support (see Definition 3.2). We point out that in other models (see [17, 8, 9]), the topological classification follows from the fact that the field U takes value in suitable manifolds. Now we state our existence result, whose precise statement will be given in Sec. 4, Theorem 4.2. Theorem 2.2. There exists ϕ weak solution of (2.10) (i.e. a static solution of Eq. (2.9)), which is obtained as minimum of the energy functional in the class of maps which are not homotopic to the null map, that is with charge different from 0. Remark 1. The functional E exibits an invariance for the symmetry group G = O(3) × R3 of rotations and traslations; indeed, for every function ϕ and g ∈ G, if we set ϕg (x) = ϕ(gx) , we have immediately E(ϕg ) = E(ϕ) . Then our theorem gives the existence of an orbit of minimum solutions. This orbit consists of two connected components, which are identified respectively by ϕ and ϕop (x) = ϕ(−x) . Since n = 3 is odd, ϕ and ϕop have opposite topological charge. 3. Functional Setting 3.1. The space H and the open set Λ In order to choose a suitable Banach space where we are going to minimize the functional (2.11), first we consider the parts of E which do not contain V . We call internal energy of ϕ: R3 → R4 , the functional Z Z c2 c6 |∇ϕ|2 dx + ε |∇ϕ|6 dx . (3.1) Ei (ϕ) = 2 R3 6 R3 Let H denote the closure of C0∞ (R3 , R4 ) with respect to the norm kϕk = k∇ϕkL2 + k∇ϕkL6 ;
(3.2)
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where we mean k∇ϕkL2
1/2 4
2 X
j = ;
∇ϕ L2
j=1
k∇ϕkL6
1/6 4
6 X
j = .
∇ϕ 6 L
j=1
Now let us recall some properties of the space H. Proposition 3.1. There exists a continuous imbedding of H in W 1,6 (R3 , R4 ). The proof easily follows from the Sobolev inequality, ∃C > 0
s.t. ∀ ϕ ∈ C0∞ (R3 , R4 )
kϕkL6 6 Ck∇ϕkL2 .
(3.3)
By Proposition 3.1 and well-known Sobolev embeddings, we get easily some useful properties of the Banach space H. 1. There exist two costants C0 , C1 > 0 such that, for every ϕ ∈ H kϕk∞ 6 C0 kϕk ,
(3.4)
|ϕ(x) − ϕ(y)| 6 C1 |x − y|1/2 k∇ϕkL6 .
(3.5)
lim ϕ(x) = 0 .
(3.6)
and 2. For every ϕ ∈ H, |x|→∞
3. If {ϕn } ⊂ H converges weakly in H to ϕ, then it converges uniformly on every compact set contained in R3 . Since the functions in H are continuous, we can consider the set Λ = {ϕ ∈ H|∀ x ∈ R3
ϕ(x) 6= ξ} ,
which is open in H; in fact, if ϕ ∈ Λ, by (3.6), we have ¯; 0 < d = inf 3 |ϕ(x) − ξ| x∈R
then, by using (3.4), we deduce that there exists a small neighborhood of ϕ in H contained in Λ. The boundary of Λ is given by ∂Λ = {ϕ ∈ H|∃¯ x ∈ R3
¯ . s.t. ϕ(¯ x) = ξ}
We can show that Λ has a rich topological structure, more precisely it consists of infinitely many connected components. These components are identified by the topological charge we are going to introduce.
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3.2. Topological charge and connected components of Λ ¯ we consider the 3-sphere centered at ξ¯ In the open set Ω = R4 \ {ξ} X = {ξ ∈ R4 | |ξ − ξ| = 1} . On Σ we consider the north and the south pole, denoted by ξN and ξS , with respect ¯ Since |ξ| = 1, we have to the axis joining the origin with ξ. ξN = 2ξ¯ ,
ξS = 0 .
We consider also the projection P : Ω → Σ ∀ξ ∈ Ω
P (ξ) = ξ¯ +
ξ − ξ¯ ¯ |ξ − ξ|
Remark 2. For every ξ ∈ Ω ¯ ξ¯ , P (ξ) = 2ξ¯ ⇔ ξ = (1 + |ξ − ξ|) therefore P (ξ) = 2ξ¯ ⇒ |ξ| > 1 . Using (3.6) and the previous remark we can give the following definition. Definition 3.2. For ϕ ∈ Λ, we call support of ϕ the compact set K(ϕ) = {x ∈ R3 |1 < |ϕ(x)|} , ¯ Then we define (topological) (the value 1 depends on the norm of the singularity ξ). charge of ϕ the Brower degree of P ◦ ϕ in the support of ϕ with respect to the north pole of Σ, namely ¯ . car(ϕ) = deg(P ◦ ϕ, int(K(ϕ)), 2ξ) Clearly the support of ϕ is the minimal intrinsic subset where the degree of P ◦ϕ with respect to 2ξ¯ stabilizes. More precisely the following proposition holds and it will be useful in order to study the properties of the charge. Proposition 3.3. For every ϕ ∈ Λ and for every R > 0 such that K(ϕ) ⊂ BR (0), ¯ . (3.7) car(ϕ) = deg(P ◦ ϕ, BR (0), 2ξ) Proof. Using the excision property of the degree, we have ¯ = car(ϕ) + deg(P ◦ ϕ, BR (0) \ int(K(ϕ)), 2ξ) ¯ . deg(P ◦ ϕ, BR (0), 2ξ)
(3.8)
where the second term on the right-hand side of (3.8) is 0. Indeed, by definition of K(ϕ), for every x ∈ R3 \ K(ϕ), |ϕ(x)| 6 1
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and, by Remark 2, P (ϕ(x)) 6= 2ξ¯ .
Then (3.7) easily follows.
On the other hand we notice that, if K(ϕ) consists of m connected components, K1 , . . . , Km , we can define also ¯ , car(ϕ, Kj ) = deg(P ◦ ϕ, Kj , 2ξ) then, by the additivity of the degree, car(ϕ) =
m X
car(ϕ, Kj ) .
j=1
In order to use some properties of the topological degree, we need to show the following result. Proposition 3.4. If a sequence {ϕn } ⊂ Λ converges to ϕ ∈ Λ uniformly on A ⊂ R3 , then also P ◦ ϕn converges to P ◦ ϕ uniformly on A. Proof. A simple calculation shows that, for every ξ, ξ1 ∈ Ω and δ > 0, ¯ |ξ1 − ξ| 6 δ < |ξ − ξ| v s u √ u ⇒ |P (ξ1 ) − P (ξ)| 6 2 t1 − 1 −
δ2 ¯2 . |ξ − ξ|
(3.9)
Let ϕ ∈ Λ and consider δ > 0 such that ¯. 0 < δ < d = inf 3 |ϕ(x) − ξ| x∈R
Let {ϕn } be a sequence in Λ converging to ϕ, uniformly on A. For n sufficiently large, we have ¯. ∀ x ∈ A |ϕn (x) − ϕ(x)| 6 δ 6 |ϕ(x) − ξ| Then, by (3.9), we get, for such n v s u √ u t sup |P (ϕn (x)) − P (ϕ(x))| 6 2 1 − 1 − x∈A
√ 6 2
s 1−
r 1−
δ2 ¯2 |ϕ(x) − ξ| δ2 . d2
Then, for every η > 0, we can choose δ sufficiently small so that sup |P (ϕn (x)) − P (ϕ(x))| < η , x∈A
for n large enough.
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Proposition 3.4 permits to prove the continuity of the charge with respect to the uniform convergence. For every A ⊂ R3 we set CA = R3 \ A . Theorem 3.5. For every ϕ ∈ Λ there exists r = r(ϕ) > 0 such that, for every ψ∈Λ kψ − ϕk∞ 6 r ⇒ car(ψ) = car(ϕ) . Proof. {ϕn } ⊂ Λ be uniformly convergent to ϕ ∈ Λ; we shall show that, for n sufficiently large, car(ϕn ) = car(ϕ). By (3.6), there exists R > 0 such that ∀ x ∈ CBR (0) |ϕ(x)| 6 1/4 ; then ¯ . car(ϕ) = deg(P ◦ ϕ, BR (0), 2ξ)
(3.10)
Since ϕn uniformly converges to ϕ, for n sufficiently large, ∀ x ∈ CBR (0) |ϕn (x)| 6 1/2 , which implies ¯ . car(ϕn ) = deg(P ◦ ϕn , BR (0), 2ξ)
(3.11)
Now, using the previous Proposition and the continuity of topological degree with respect to the uniform convergence, we get, for n large, ¯ = deg(P ◦ ϕ, BR (0), 2ξ) ¯ . deg(P ◦ ϕn , BR (0), 2ξ) So the conclusion follows by (3.10) and (3.11).
Now, for every q ∈ Z, we set Aq = {ϕ ∈ Λ|car(ϕ) = q} . By Theorem 3.5, it follows that Aq is open in H; since we have also S Aq ; • Λ= q∈Z
• Ap ∩ Aq = ∅ if
p 6= q;
we conclude that every Aq is a connected component of Λ. We observe that for every q ∈ Z, the component Aq is isomorphic to the component A−q . Indeed, if we set ϕop (x) = ϕ(−x) ,
(3.12)
ϕ ∈ Aq ⇒ ϕop ∈ A−q .
(3.13)
it is easy to see that
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We point out that (3.13) is due the fact that the space dimension is odd. Now we can define the set on which we are going to show the existence of a nontrivial minimum of the energy functional; we set [ Aq Λ∗ = q6=0
Finally we observe that, from Remark 2, it follows that ∀ ϕ ∈ Λ∗ :
kϕk∞ > 1 .
(3.14)
which implies that K(ϕ) is not empty. 3.3. Properties of the energy functional Clearly the energy functional E defined in (2.11) in coercive in the H norm, namely lim E(ϕ) = +∞ . kϕk→∞
Moreover, as immediate consequence of the continuous embedding of H in L∞ (see (3.4)), we have the following property. Proposition 3.6. There exists ∆∗ > 0 such that, for every ϕ ∈ Λ, kϕk∞ > 1 ⇒ E(ϕ) > ∆∗ .
(3.15)
Now we are going to study the properties of the second piece of the energy functional Z V (ϕ) dx . R3
Since R is unbounded, it is obvious that this integral can diverge, even if we choose ϕ ∈ Λ. 3
Remark 3. It is not difficult to see that E takes real values and it is continuous on M = Λ ∩ L2 (R3 , R4 ) . Nevertheless we cannot choose H ∩ L2 as “basic” function space since E is not coercive, in general, with respect to the norm k · k + k · kL2 . Such coercivity property holds if 0 is a non degenerate minimum of the potential V . First we are going to study the behaviour of E when ϕ approaches the boundary of Λ, in the spirit of a well-known result of Gordon (see [11]), concerning strongly attractive potentials. Lemma 3.7. Let {ϕn } ⊂ Λ be bounded in the H norm and weakly converging to ϕ ∈ ∂Λ, then Z V (ϕn ) dx → +∞ . R3
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¯ since V is Proof. Since ϕ ∈ ∂Λ, there exists x ¯ ∈ R3 such that ϕ(¯ x) = ξ; nonnegative, it is sufficient to show that there exists a small ball centered at x¯ such that Z V (ϕn ) dx → +∞ . (3.16) Bρ (¯ x)
By the uniform convergence on compact sets we have x) = ξ¯ . lim ϕn (¯
n→∞
(3.17)
x) and for n Now we show that there exists ρ > 0 such that, for every x ∈ Bρ (¯ sufficiently large, ¯ < r, (3.18) |ϕn (x) − ξ| where r has been introduced in (2.12). Since {ϕn } is bounded in H, then, in particular {∇ϕn } is bounded in L6 . Using (3.5) and the boundedness of {∇ϕn } in L6 , we have x)| 6 const |x − x ¯|1/2 , |ϕn (x) − ϕn (¯
(3.19)
for every x ∈ R3 . Then (3.18) easily follows from (3.17) and (3.19). We have also ¯ 6 const |x − x ¯|1/2 + o(1) . |ϕn (x) − ξ|
(3.20)
x), we have Now, using (3.18) and (2.12), for every x ∈ Bρ (¯ V (ϕn (x)) >
c ¯6 ; |ϕn (x) − ξ|
then, using (3.20), we obtain V (ϕn (x)) >
c . const |x − x ¯|3 + o(1)
x), we get (3.16). Integrating on Bρ (¯
From this lemma we immediately deduce that the sublevels of E are complete and that the following proposition holds. Proposition 3.8. Let {ϕn } ⊂ Λ be weakly converging to ϕ and such that E(ϕn ) is bounded, then ϕ ∈ Λ. From Proposition 3.8 we get also the following result. Proposition 3.9. For every a > 0, there exists d > 0 such that, for every ϕ ∈ Λ ¯ > d. E(ϕ) 6 a ⇒ min3 |ϕ(x) − ξ| x∈R
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Proof. Arguing by contradiction, assume that there exist a > 0 and a sequence {ϕn } ⊂ Λ such that E(ϕn ) 6 a ¯ < 1/n . min |ϕn (x) − ξ|
x∈R3
(3.21)
For every n ∈ N, by (3.6), there exists xn ∈ R3 such that ¯ = min |ϕn (x) − ξ| ¯ |ϕn (xn ) − ξ| 3 x∈R
Then we can consider the sequence ψn = ϕn (· + xn ) . Since E(ψn ) = E(ϕn ) 6 a ,
(3.22)
we have that {ψn } is bounded in H, then, up to subsequence, it weakly converges to ψ. Now, from the definition of ψn and (3.21), we obtain ψ(0) = lim ψn (0) = ξ¯ ; n→∞
therefore ψ ∈ ∂Λ. Taking into account (3.22) we have got a contradiction with Proposition 3.8. Now we prove the weakly lower semicontinuity of the energy functional E. Proposition 3.10. For every ϕ ∈ Λ and for every sequence {ϕn } ⊂ Λ, if {ϕn } weakly converges to ϕ, then lim inf E(ϕn ) > E(ϕ) . n→∞
Proof. The result is obvious when lim inf E(ϕn ) = +∞ , n→∞
then we have to study the case lim inf E(ϕn ) < +∞ . n→∞
Clearly lim inf Ei (ϕn ) > Ei (ϕ) . n→∞
where Ei is defined in (3.1). Now we have to study Z V (ϕn ) dx . R3
(3.23)
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Since {ϕn } converges to ϕ uniformly on every compact set, we fix a sphere BR (0) and we have Z Z V (ϕn ) dx = V (ϕ) dx . lim n→∞
BR (0)
BR (0)
On the other hand, since V is nonnegative, we have Z Z Z V (ϕn ) dx > lim inf V (ϕn ) dx = lim inf n→∞
n→∞
R3
BR (0)
and taking the limit for R → ∞, we obtain Z Z V (ϕn ) dx > lim inf n→∞
R3
V (ϕ) dx ,
BR (0)
V (ϕ) dx .
(3.24)
R3
By (3.23) and (3.24) the Proposition is completely proved.
Theorem 3.11. The minimum points ϕ ∈ Λ for the functional E are weak solutions of the system (2.10). Proof. Let ϕ ∈ Λ be a minimum point of E and ψ ∈ C0∞ (R3 , R). Let ej denote the jth vector of the canonical base in R4 . If s > 0 is sufficiently small, then ϕ + sψej ∈ Λ and E(ϕ + sψej ) < +∞. Since ϕ is a minimum point of E, differentiating E(ϕ + sψej ) with respect to s, we have d E(ϕ + sψej ) ds s=0 Z ∂V = (ϕ)ψ dx . c2 ∇ϕj · ∇ψ + εc6 |∇ϕ|4 ∇ϕj · ∇ψ + ∂ξj R3
0 =
4. Existence Theorems 4.1. The splitting lemma The proof of our main result is based on the following proposition, in the spirit of Concentration-Compactness principle for unbounded domains (see [2, 13]) Lemma 4.1 (Splitting lemma). Let {ϕn } ⊂ Λ∗ such that E(ϕn ) 6 a .
(4.1)
1 6 l 6 a/∆∗ ,
(4.2)
There exists l ∈ N, (∆∗ has been introduced in Proposition 3.6) and there exist ϕ1 , . . . , ϕl ∈ Λ, {x1n }, . . . , {xln } ⊂ R3 , R1 , . . . , Rl > 0 such that, up to subsequence,
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ϕn (· + xin ) * ϕi ;
(4.3)
kϕi k∞ > 1 ;
(4.4)
|xin − xjn | → ∞ l X
E(ϕi ) 6 lim inf E(ϕn ) ;
l [
(4.5) (4.6)
n→∞
i=1
∀x ∈ C
i 6= j ;
f or
! BRi (xin )
:
|ϕn (x)| 6 1 .
(4.7)
i=1
Then we have also car(ϕn ) =
l X
car(ϕi ) ;
(4.8)
i=1
l
X
i ϕi (· − xn ) lim sup ϕn −
n→∞ i=1
6 1.
(4.9)
∞
Remark 4. We notice that, from (4.4), it follows E(ϕi ) > ∆∗ .
(4.10)
Proof. The proof is divided in two parts. In the first part, with an iterative procedure, we prove the existence of l ∈ N, ϕ1 , . . . , ϕl ∈ Λ, {x1n }, . . . , {xln } ⊂ R3 , R1 , . . . , Rl > 0 such that (4.2)–(4.7) are satisfied; in the second part from these properties we shall easily deduce (4.8) and (4.9). For the sake of simplicity, whenever it is necessary, we shall tacitly consider a subsequence of {ϕn }. First of all we arbitrarily choose γ ∈]0, 1[. Let x1n ∈ R3 be a maximum point for |ϕn |; by (3.14) we have |ϕn (x1n )| > 1. We set ϕ1n = ϕn (· + x1n ) and we obtain
1
ϕn = ϕ1n (0) > 1 . ∞
(4.11)
Since E(ϕ1n ) = E(ϕn ) and the functional E is coercive, then the sequence {ϕ1n } is bounded in H and we have ϕ1n * ϕ1 ∈ H ,
(4.12)
From (4.11) it follows kϕ1 k∞ > 1 . Since {ϕ1n } ⊂ Λ and E(ϕ1n ) is bounded, by (4.12) and Proposition 3.8, we get ϕ1 ∈ Λ.
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Since E is weakly lower semicontinuous, we have E(ϕ1 ) 6 lim inf E(ϕ1n ) = lim inf E(ϕn ) n→∞
n→∞
(4.13)
Now, using (3.6), we consider R1 > 0 such that ∀ x ∈ CBR1 (0) |ϕ1 (x)| 6 γ ;
(4.14)
for simplicity we set Bn1 = BR1 (x1n ) . Now we distinguish two cases: either (A1) for n sufficiently large ∀ x ∈ CBn1 :
|ϕn (x)| 6 1 ;
or (B1) eventually passing to a subsequence, ∃ x ∈ CBn1
s.t. |ϕn (x)| > 1 .
In the case (A1) the first part of Proposition is proved with l = 1; let us consider the case (B1). Let x2n be a maximum point for |ϕn | in R3 \ Bn1 ; we have that |ϕn (x2n )| > 1. We set ϕ2n = ϕn (· + x2n ) and we obtain As for {ϕ1n }, we have that
2
ϕn = ϕ2n (0) > 1 . ∞ ϕ2n * ϕ2 ∈ Λ ,
(4.15)
kϕ2 k∞ > 1 .
(4.16)
2 xn − x1n → ∞ .
(4.17)
with Now we have to show that
We set yn = x2n − x1n and, arguing by conctradiction, we assume that the sequence {yn } is bounded in R3 ; then, up to subsequence, we have that yn → y˜ . y | > R1 ; then, using (4.14), Since |yn | = |x2n − x1n | > R1 , we have |˜ y )| 6 γ < 1 . |ϕ1 (˜
(4.18)
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On the other hand we have 1 6 ϕn (x2n ) = ϕn (yn + x1n ) = ϕ1n (yn ) , then, by (4.18), y)| 6 ϕ1n (yn ) − |ϕ1 (˜ y )| 6 ϕ1n (yn ) − ϕ1 (˜ y ) 0 < 1 − |ϕ1 (˜ y )| 6 ϕ1n (yn ) − ϕ1 (yn ) + |ϕ1 (yn ) − ϕ1 (˜ ! y )| . 6 sup ϕ1 (y) − ϕ (y) + |ϕ (yn ) − ϕ (˜ 1
n
|y−˜ y|61
1
1
Taking the limit for n → ∞ we get a contradiction. Now we show that E(ϕ1 ) + E(ϕ2 ) 6 E(ϕn ) .
(4.19)
Hereafter, for sake of simplicity, we set, for every ϕ ∈ Λ and A ⊂ R3 Z 2 c c6 |∇ϕ|2 + ε |∇ϕ|6 + V (ϕ) dx . E|A (ϕ) = 2 6 A For a fixed η > 0, there exists ρ > 0 such that E|CBρ (0) (ϕ1 ) < η/2 and E|CBρ (0) (ϕ2 ) < η/2 . From (4.17) it follows that the spheres Bρ (x1n ) and Bρ (x2n ) are disjoint for n sufficiently large, then we get: lim inf E(ϕn ) > lim inf E|Bρ (x1n ) (ϕn ) + E|Bρ (x2n ) (ϕn ) n→∞
n→∞
> lim inf E|Bρ (x1n ) (ϕn ) + lim inf E|Bρ (x2n ) (ϕn ) n→∞
=
n→∞
lim inf E|Bρ (0) (ϕ1n ) n→∞
+ lim inf E|Bρ (0) (ϕ2n ) n→∞
> E|Bρ (0) (ϕ1 ) + E|Bρ (0) (ϕ2 ) > E(ϕ1 ) + E(ϕ2 ) − η . From the arbitrariness of η, we get (4.19). Finally, as well as for ϕ1 , from (3.6) we get R2 > 0, such that ∀ x ∈ CBR2 (0) |ϕ2 (x)| 6 γ and we set Bn2 = BR2 (x2n ) Also in this second step we have an alternative: either (A2) for n sufficiently large, ∀ x ∈ C(Bn1 ∪ Bn2 ) :
|ϕn (x)| 6 1 ;
SOLITON LIKE SOLUTIONS OF A LORENTZ INVARIANT EQUATION IN DIMENSION 3
333
or (B2) up to a subsequence, ∃ x ∈ C(Bn1 ∪ Bn2 )
s.t. |ϕn (x)| > 1 .
If case (A2) holds true, the first part of Proposition is proved with l = 2; in the case (B2) we consider a maximum point of |ϕn | in C(Bn1 ∪ Bn2 ) and we repeat the same argument used in the case (B1). This alternative process terminates in a finite number of steps. Indeed, using (4.10), (4.6) and (4.1), we get (4.2); we notice that this estimate is independent on the sequence {ϕn }. Now we prove (4.8). We consider n sufficiently large so that (4.7) holds and Bni ∩ Bnj = ∅
for i 6= j .
(4.20)
Then we have, by the additivity property of the topological degree, ! l [ i B , 2ξ¯ car(ϕn ) = deg P ◦ ϕn , n
i=1
=
l X
¯ = deg(P ◦ ϕn , Bni , 2ξ)
i=1
l X
¯ . deg(P ◦ ϕin , BRi (0), 2ξ)
(4.21)
i=1
On the other hand, for every i ∈ {1, . . . , l}, since {ϕin } converges uniformly to ϕi on BRi (0) and ∀ x ∈ CBRi (0) |ϕi (x)| 6 γ < 1 , we obtain, for n large enough, ¯ = deg(P ◦ ϕ , BR (0), 2ξ) ¯ = car(ϕ ) . deg(P ◦ ϕin , BRi (0), 2ξ) i i i Then, substituting in (4.21), we obtain (4.8). Finally, in order to prove (4.9), we assume that, for every i ∈ {1, . . . , l}, (4.22) ∀ x ∈ Bni : ϕn (x) − ϕi (x − xin ) < γ . We shall prove that, for n large enough, l X 3 i ϕi (x − xn ) < 1 + lγ . ∀ x ∈ R : ϕn (x) −
(4.23)
i=1
Sl Indeed, if x ∈ i=1 Bni , then, by (4.20), there exists a unique index j ∈ {1, . . . , l} such that x ∈ Bnj , then l X X ϕi (x − xin ) ϕi (x − xin ) 6 ϕn (x) − ϕj (x − xjn ) + ϕn (x) − i=1
i6=j
< γ + (l − 1)γ = lγ < 1 + lγ .
(4.24)
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Sl On the other hand, if x ∈ / i=1 Bni , then, by (4.7), l l X X i ϕi (x − xin ) ϕi (x − xn ) 6 |ϕn (x)| + ϕn (x) − i=1
i=1
6 1 + lγ . Now fix η > 1; choosing γ sufficiently small we have 1 + lγ < η
(4.25)
(taking into account (4.2), this kind of choice can be made a priori in the proof). Substituting (4.25) in (4.23), we get l X ϕi (x − xin ) < η , ∀ x ∈ R3 : ϕn (x) − i=1
and, by the arbitrariness of η > 1, we obtain (4.9).
Pl Remark 5. Consider the function i=1 ϕi (· − xin ) which has been introduced in (4.9); using (3.6) and (4.5), it is not difficult to show that, for n large enough, l X
ϕi (· − xin ) ∈ Λ ,
i=1
car
l X
! ϕi (· − xin )
=
i=1
l X
car(ϕi ) .
i=1
4.2. Existence of minima in the connected components of Λ Finally we can give the proof of Theorem 2.2. We set E ∗ = inf E(Λ∗ ) . By (3.14) it follows ∆∗ 6 E ∗ . Theorem 4.2. There exists ϕ ∈ Λ∗ such that E(ϕ) = E ∗ . Proof. We consider a minimizing sequence {ϕn } ⊂ Λ∗ . It has obviously bounded energy; then we can apply Proposition 4.1. There exist l ∈ N and ϕ1 , . . . , ϕl ∈ Λ such that, up to a subsequence, l X i=1
E(ϕi ) 6 lim inf E(ϕn ) = E ∗ ; n→∞
car(ϕn ) =
l X i=1
car(ϕi ) .
(4.26)
(4.27)
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335
Since car(ϕn ) 6= 0, from (4.27) we deduce that there exists i ∈ {1, . . . , l}, for sake of simplicity i = 1, such that car(ϕ1 ) 6= 0. Then, by (4.26), we obtain E∗ >
l X
E(ϕi ) > E(ϕ1 ) > E ∗ ;
i=1
so we get E(ϕ1 ) = E ∗ .
Remark 6. We recall that, for every ϕ ∈ Λ, car(ϕop ) = −car(ϕ) , E(ϕop ) = E(ϕ) , where ϕop (x) = ϕ(−x). Then Theorem 4.2 can be restated in two different ways: • there exists ϕ ∈ Λ∗+ , such that E(ϕ) = inf E(Λ∗+ ) , where Λ∗+ =
[
Aq ;
q>0
• there exists at least two solutions of (2.10) having opposite topological charge (more precisely, taking in account Remark 1, in Subsec. 2.2, we have the existence of two connected manifolds of solutions). For every q ∈ N, we set Eq = inf E(Aq ) = inf E(A−q ) Now we want to study when the minima Eq are attained. It is obvious that the functional E takes its absolute minimum 0 in the class A0 . On the other hand, by Theorem 4.2, there exists at least q¯ = car(ϕ) 6= 0 such that Eq¯ is attained. We can show the following properties. Proposition 4.3. For every q 6= 0, a sufficient condition to guarantee that Eq is attained is that there exists a minimizing sequence in Aq which satisfies the properties of Proposition 4.1 with l = 1. So, if Eq is not attained, then every minimizing sequence in Aq satisfies the properties of Proposition 4.1 with l > 2. Proof. Let {ϕn } be a minimizing sequence in Aq which satisfies the properties of Proposition 4.1 with l = 1. From (4.8), we have car(ϕ1 ) = car(ϕn ) = q , which implies E(ϕ1 ) > Eq .
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On the other hand, by (4.6), Eq = lim inf E(ϕn ) > E(ϕ1 ) . n→∞
So we conclude E(ϕ1 ) = Eq .
Proposition 4.4. For every q 6= 0, if Eq < 2 E ∗ , then the value Eq is attained in Aq . Proof. Let {ϕn } be a minimizing sequence in Aq . For n sufficiently large we have (4.28) E(ϕn ) < 2E ∗ . Using Proposition 4.1, there exist l ∈ N and ϕ1 , . . . , ϕl ∈ Λ such that, up to a subsequence, l X
E(ϕi ) 6 lim inf E(ϕn ) = Eq ; n→∞
i=1
q = car(ϕn ) =
l X
car(ϕi ) .
(4.29)
(4.30)
i=1
By (4.30) there exists i ∈ {1, . . . , l} such that car(ϕi ) 6= 0; by (4.28) and (4.29) such index is unique; for sake of simplicity we mean i = 1. Then we have car(ϕ1 ) = car(ϕn ) = q and, substituting in (4.29), Eq >
l X
E(ϕi ) > Eq +
i=1
So we obtain
l X
E(ϕi ) .
i=2
l X
E(ϕi ) = 0
i=2
and we have necessarily l = 1. Then we can apply Proposition 4.3. Let S = {q ∈ N∗ |Eq is attained}
Theorem 4.2 states that S is not empty; the following Theorem provides a more precise information. Theorem 4.5. The ideal spanned by S coincides with Z. Remark 7. From Theorem 4.5 we deduce the following alternative. Either the set S contains 1, that is E attains its minimum in the classes A±1 ; or the set S
SOLITON LIKE SOLUTIONS OF A LORENTZ INVARIANT EQUATION IN DIMENSION 3
337
contains at least two elements, that is E has two (pairs of) minima, with different charge. Proof. Arguing by contradiction, assume that A = Z \ span{S} = 6 ∅. Then we set
[
A=
Aq
q∈A
EA = inf E(A) Let {ϕn } ⊂ A be such that E(ϕn ) → EA . Since A ⊂ Λ∗ , we can apply Proposition 4.1: there exist l ∈ N, ϕ1 , . . . , ϕl ∈ Λ such that, up to a subsequence,
l X
E(ϕi ) > ∆∗ > 0 ;
(4.31)
E(ϕi ) 6 lim inf E(ϕn ) = EA ;
(4.32)
n→∞
i=1
car(ϕn ) =
l X
car(ϕi ) .
(4.33)
car(ϕi ) = q¯ ;
(4.34)
i=1
For simplicity we set l X i=1
substituting in (4.33), we get car(ϕn ) = q¯ ,
(4.35)
q¯ ∈ A .
(4.36)
then, since {ϕn } ⊂ A, it follows Then, since Aq¯ ⊂ A and using (4.35), we get EA 6 Eq¯ 6 E(ϕn ) . So we conclude that {ϕn } is a minimizing sequence in Aq¯. From (4.34) and (4.36), we deduce that there exists i ∈ {1, . . . , l}, for simplicity / we mean i = 1, such that car(ϕ1 ) ∈ A; indeed, arguing by contradiction, if car(ϕi ) ∈ A for all i ∈ {1, . . . , l}, i.e. car(ϕi ) ∈ span{S}, then by (4.34), q¯ ∈ span{S}, which contradicts (4.36). On the other hand Eq¯ is not attained, since q¯ ∈ A ⊂ Z \ S, then, by Proposition 4.3, we get l > 2. Therefore, using again (4.32) and (4.31), we get the contradiction EA > E(ϕ1 ) +
l X i=2
E(ϕi ) > EA + (l − 1)∆∗ > EA .
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Remark 8. We notice that our static problem (2.10) have some similarities with the problem of finding homoclinic solutions for dynamical systems with a singular potential; for this latter problem the results we know are confined to planar dynamical systems, where the solutions are classified by the winding number (see [15, 6] and references therein). 5. Mass and Energy 5.1. The mass matrix In order to show the relativistic behaviour of solitons defined in Sec. 2, we need to define their inertial mass. The most natural thing to do is to define it as the ratio between the momentum and the velocity of the soliton. However this definition is possible only if the momentum and the velocity are parallel; otherwise the mass is a tensor. So the first thing to do is to compute the momentum and then the mass tensor of our solitons. Let u(x, t) = u(x1 , x2 , x3 , t) be a solution of (2.2) (where for simplicity we set c = 1). It is well known (see e.g. [10]) that the momentum of u is given by Z (5.1) pk = T0,k dx , k = 1, 2, 3 where Ti,k is the energy-momentum tensor defined by Tki =
n X j=1
∂u ∂L j − δik L ; ∂uj ∂xi ∂ ∂xk
with
x0 := t
(5.2)
In our case, for k 6= 0, we have Tk,0 = −α0 (ρ) uxk · ut
(5.3)
so we can easily compute the momentum of the soliton (2.4). Lemma 5.1. Let uv be the solution of (2.2) (with c = 1) given by the (2.4), then the momentum is given by Z p1 (uv ) = vγ Z p2 (uv ) = v Z p3 (uv ) = v where
0
2
α (|∇ϕ| )
∂ϕ ∂x1
2 dx
α0 (|∇ϕ|2 )
∂ϕ ∂ϕ dx ∂x1 ∂x2
α0 (|∇ϕ|2 )
∂ϕ ∂ϕ dx ∂x1 ∂x3
1 . γ= √ 1 − v2
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339
Proof. By (5.3) we have Z
0
p1 (uv ) = −
|∇uv | − 2
α
∂uv ∂t
2 !
∂uv ∂uv dx ; ∂x1 ∂t
v 2 exploiting the Lorentz invariance of |∇uv |2 − ( ∂u ∂t ) , we get
"
|∇x uv | − 2
∂uv ∂t
where
2 # (x1 , x2 , x3 ) = |∇ξ ϕ|2 (ξ1 , ξ2 , ξ3 )
ξ1 = γ(x1 − vt) ξ2 = x2 ξ3 = x3
Thus Z
α0 (|∇ξ ϕ|2 )
p1 (uv ) = − Z
∂uv ∂uv dx1 dx2 dx3 ∂x1 ∂t
∂ϕ ∂ξ1 ∂ϕ ∂ξ1 1 dξ1 dξ2 dξ3 ∂ξ1 ∂x1 ∂ξ1 ∂t γ 2 Z ∂ϕ = γv α0 (|∇ξ ϕ|2 ) dξ1 dξ2 dξ3 . ∂ξ1 α0 (|∇ξ ϕ|2 )
=−
Now let us compute p2 : Z p2 (uv ) = −
α Z
=− Z =
|∇uv | − 2
α0 (|∇ξ ϕ|2 )
α0 (|∇ξ ϕ|2 ) Z
=v
0
∂uv ∂t
2 !
∂uv ∂uv dx ∂x2 ∂t
∂uv ∂uv dx1 dx2 dx3 ∂x2 ∂t
∂ϕ ∂ϕ 1 γv dξ1 dξ2 dξ3 ∂ξ2 ∂ξ1 γ
α0 (|∇ξ ϕ|2 )
∂ϕ ∂ϕ dξ1 dξ2 dξ3 . ∂ξ2 ∂ξ1
The mass matrix {mij } of uv is defined by the following relation: pi (uv ) =
4 X j=1
mij (uv )vj ,
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then, by the previous lemma we get: Z m11 (uv ) = γ Z m12 (uv ) = Z m13 (uv ) =
α0 (|∇ϕ|2 )
∂ϕ ∂x1
2 dx
α0 (|∇ϕ|2 )
∂ϕ ∂ϕ dx ∂x1 ∂x2
α0 (|∇ϕ|2 )
∂ϕ ∂ϕ dx . ∂x1 ∂x3
and by symmetry arguments we get: Z mii (uv ) = γ Z
α0 (|∇ϕ|2 )
α0 (|∇ϕ|2 )
∂ϕ ∂xi
2 dx
∂ϕ ∂ϕ dx , ∂xi ∂xj
(5.4) i 6= j
(5.5)
Now let us compute the energy of a soliton. By (5.2) we have Z E(uv ) = T0,0 dx
(5.6)
mij (uv ) =
for
So the rest mass matrix is defined as follows: Z ∂ϕ ∂ϕ dx . mij (uv ) = α0 (|∇ϕ|2 ) ∂xi ∂xj 5.2. Equivalence between mass and energy
Since the direct computation of T0,0 is quite involved, we start to compute the energy of a static solution; in this case it is easy to see that Z Z 1 2 α(|∇ϕ| ) + V (ϕ) dx (5.7) E(ϕ) = T0,0 dx = 2 At this point we need the following lemma Lemma 5.2. Let ϕ be a critical point of the energy (2.7) (and hence a weak solution of Eq. (2.3)); then 1 2
Z
Z α(|∇ϕ|2 ) dx +
Z V (ϕ) dx =
α0 (|∇ϕ|2 )
∂ϕ ∂xi
2 dx, i = 1, 2, 3
and, consequently Z Z Z 1 1 2 α(|∇ϕ| ) dx + V (ϕ) dx = α0 (|∇ϕ|2 )|∇ϕ|2 dx . 2 3 whenever the integrals converge (as before, we have assumed c = 1).
(5.8)
(5.9)
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341
Proof. Let ϕλ (x) = ϕ(λx1 , x2 , x3 ) ; then, setting y = (λx1 , x2 , x3 ) Z Z 1 α(|∇ϕλ |2 ) dx + V (ϕλ ) dx E(ϕλ ) = 2 2 2 2 ! Z Z ∂ϕ ∂ϕ ∂ϕ λ−1 2 α λ + + dy + λ−1 V (ϕ) dy = 2 ∂x1 ∂x2 ∂x3 Since ϕ is a critical point of E, d E(ϕλ ) =0 dλ λ=1 and since λ−2 d E(ϕλ ) = − dλ 2 Z
2
α λ
0
+
α −2
−λ
Z
2
λ
∂ϕ ∂x1
∂ϕ ∂x1
2 +
2 +
∂ϕ ∂x2
∂ϕ ∂x2
2 +
2 +
∂ϕ ∂x3
∂ϕ ∂x3
2 !
2 !
dy ∂ϕ ∂x1
2 dy
Z V (ϕ) dy
we get 1 − 2
Z
Z 2
α(|∇ϕ| ) dy +
0
2
α (|∇ϕ| )
∂ϕ ∂x1
2
Z dy −
V (ϕ) dy = 0
which gives (5.8). Equation (5.9) is obtained adding (5.8) with i = 1, 2, 3.
By the above lemma we get the following theorem. Theorem 5.3. If ϕ is a solution of (2.3) (with c = 1); then mii (uv ) = γE(ϕ) mij (uv ) = 0
if i 6= j
Proof. Since the matrix mij (uv ) is symmetric, we can put ourselves in a new reference frame x0 = T x such that m0ij is diagonal. In this reference frame (where we omit the 0 ), by (5.4) Z mij (uv ) = γδij
0
2
α (|∇ϕ| )
∂ϕ ∂xi
2 dx ,
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where δij is the Kroneker symbol. Then, by Lemma 5.2 and (5.7) mij (uv ) = γE(ϕ)δij Thus mij (uv ) is the identity matrix times γE(ϕ). This implies that mij (uv ) = γE(ϕ)δij in any reference frame. By the above lemma we get the following facts: (i) since the mass matrix is a scalar times the identity, the mass is in fact a scalar; (ii) the mass is equal to the rest energy times γ. In particular, by Lemma 5.2, the mass gets the following form: Z γ α0 (|∇ϕ|2 )|∇ϕ|2 dx m(uv ) = 3
(5.10)
and we have the equations p(uv ) = m(uv )v
(5.11)
m(uv ) = γE(ϕ)
(5.12)
and
Notice that Z (p(uv ), E(uv )) =
Z T0,1 dx,
Z T0,2 dx,
Z T0,3 dx,
T0,0 dx
is a 4-vector since Eq. (2.2) is Lorentz invariant; so, its Lorentzian norm |p(uv )|2 − E(uv )2
(5.13)
is independent of v. Using this fact we get: Theorem 5.4. If ϕ is a solution of (2.3) (with c = 1) and uv is the function (2.4), then (5.14) E(uv ) = γE(ϕ) Proof. Since (5.13) is independent of v, |p(uv )|2 − E(uv )2 = −E(ϕ)2 ; then by (5.11) and Theorem 5.3 E(uv )2 = E(ϕ)2 + |p(uv )|2 = E(ϕ)2 + m(ϕ)2 γ 2 |v|2 = E(ϕ)2 + E(ϕ)2 γ 2 |v|2 |v|2 = 1+ E(ϕ)2 = γ 2 E(ϕ)2 . 1 − |v|2
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SOLITON LIKE SOLUTIONS OF A LORENTZ INVARIANT EQUATION IN DIMENSION 3
Comparing (5.14) with (5.12) we get the equality between mass and energy: E(uv ) = m(uv ) .
(5.15)
If c 6= 1, the above formula gives the celebrated Einstein equation E(uv ) = m(uv )c2 . It is interesting to note that this formula is a consequence not only of the Lorentz invariance of the Langrangian L but also of the fact that uv is a soliton (and hence Lemma 5.2 holds). 6. Conclusions Equation (2.8) probably is the simplest Lorentz invariant equation which has static solitons. Nevertheless, these solitons have some interesting properties since they behave as relativistic bodies, namely: • they experience a relativisic contraction in the direction of the motion; • the rest mass is a scalar and not a tensor; • the mass equals the energy; • the mass increases with the velocity by the factor γ. Our Lagrangian is Lorentz invariant, thus it is reasonable to expect at least some of these features. However, it is somewhat surprising that they can be deduced from a single equation without extra assumptions. Moreover, this equation might be interpreted as the equation of an “elastic medium” in a Newtonian space-time. Thus, this model shows, from a purely formal point of view, that the main features of the special relativity can be deduced from a partial differential equation in a Newtonian space-time. References [1] D. Anderson and G. Derrick, “Stability of time dependent particle like solutions in nonlinear field theories”, J. Math Phys. 11 (1970) 1336–1346. [2] V. Benci and G. Cerami, “Positive solutions of some nonlinear elliptic problems in exterior domains”, Arc. Rational Mech. Anal. 99 (1987) 283–300. [3] H. Berestycki and T. Cazenave, “Instabilit`e del etats stationnaires dans des equations de Schr¨ odinger et de Klein–Gordon non lin´eairs”, C.R.A.S. Paris, s´erie I 293 (1981) 488–492. [4] H. Berestycki and P. L. Lions, “Existence d’ondes solitaires dans des problems non lin´eairs du type Klein–Gordon”, C.R.A.S. Paris, s´erie A 287 (1978) 503–506. [5] M. S. Berger, “On the existence and structure of stationary states for a nonlinear Klein–Gordon equation”, J. Funct. Analysis 9 (1972) 249–261. [6] P. Caldiroli and L. Jeanjean, “Homoclinics and heteroclinics for a class of conservative singual Hamiltonian systems”, J. Differential Equations, 136 (1997) 76–114. [7] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, New York, 1982. [8] M. J. Esteban, “A direct variational approach to Skyrme’s model for Meson fields”, Commun. Math. Phys. 105 (1986) 571–591. [9] M. J. Esteban and P. L. Lions, “Skyrmions and symmetry”, Asymptotic Anal. 1 (988) 187–192.
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[10] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, N.J., 1963. [11] W. Gordon, “Conservative dynamical systems involving strong forces”, Trans. A.M.S. 204 (1975) 113–135. [12] S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker Inc., New York, Basel, Hong Kong, 1996. [13] P. L. Lions, “The Concentration-Compactness Principle in the calculus of variations. The limit case. Parts I and II”, Part I — Rev. Mat. Iber. 1.1 (1985) 145–200; Part II — Rev. Mat. Iber. 1.2 (1985) 45–121. [14] S. I. Pohozaev, “Eigenfunctions of the equation ∆u + λf (u) = 0”, Sov. Math. Dolk. 5 (1965) 1408–1411. [15] P. H. Rabinowitz, “Homoclinic for a singular Hamiltonian system”, to appear in Geometric Analysis and the Calculus of Variations, ed. J. Jost, International Press. [16] R. Rajaraman, Solitons and Instantons, North Holland, Amsterdam, Oxford, New York, Tokyo, 1998. [17] T. H. R. Skyrme, “A nonlinear field theory”, Proc. Roy. Soc. A260 (1961) 127–138. [18] W. A. Strauss, “Existence of solitary waves in higher dimensions”, Commun. Math. Phys. 55 (1977) 149–162. [19] G. B. Witham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS OF THE AKNS HIERARCHY F. GESZTESY and R. RATNASEELAN Department Of Mathematics University Of Missouri Columbia, MO 65211 USA E-mail : [email protected] E-mail : [email protected] Received 26 April 1997 We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall–Chaundy curves, Baker–Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.
1. Introduction The principal aim of this paper is an alternative elementary algebraic approach to the entire AKNS hierarchy in the spirit of previous treatments of the Korteweg– de Vries (KdV), Boussinesq, and Toda hierarchies. More precisely, we advocate a fundamental polynomial formalism to recursively construct Lax pairs for the AKNS hierarchy, that is, pairs (D, En+1 ) of matrix-valued differential expressions of order one (i.e., D) and n + 1 (i.e., En+1 ) with D of the Dirac-type. In addition, we establish the basic algebro-geometric setup for special classes of solutions of the AKNS hierarchy including solitons, rational solutions, algebro-geometric quasiperiodic solutions, and limiting cases thereof. Our treatment includes a systematic approach to Burchnall–Chaundy curves, Baker–Akhiezer functions, trace formulas, Dubrovin-type equations describing the dynamics of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions. Before we enter a description of the contents of each section, it seems appropriate to comment on existing treatments of this subject and to justify the addition of yet another detailed account on this topic. The theory of commuting matrix-valued differential expressions and, more generally, the algebro-geometric approach to matrix hierarchies of soliton equations has been developed in great generality by Dubrovin and Krichever. Corresponding authoritative accounts can be found, for instance, in [5, Chapters 3, 4] [14, 15, 16, 30, 31, 35, 36, 44], and the references therein. In contrast to these references, our own approach relies on two basic ingredients, an 345 Reviews in Mathematical Physics, Vol. 10, No. 3 (1998) 345–391 c World Scientific Publishing Company
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elementary polynomial approach to Lax pairs (or zero-curvature pairs) of the AKNS hierarchy and its explicit connection with a fundamental meromorphic quantity φ (cf. (3.10), (4.23)) which allows for a unified algebro-geometric treatment of the entire AKNS hierarchy. In Sec. 2 we describe a recursive approach to Lax pairs (and zero-curvature pairs) of the AKNS hierarchy following Al’ber’s treatment of the KdV and nonlinear Schr¨ odinger hierarchies [1, 2, 3] and establish its connection with the Burchnall– Chaundy theory [7, 8, 9] and hence with hyperelliptic curves. Combining the recursive formalism of Sec. 2 with a polynomial approach to represent positive divisors of degree n of a hyperelliptic curve of genus n originally developed by Jacobi [33] and applied to the KdV case by Mumford [41, Section III.a.1] and McKean [38] (see also [17, 45]), a detailed analysis of the stationary AKNS hierarchy is provided in Sec. 3. This includes, in particular, the theta function representation of algebro-geometric solutions of the stationary AKNS hierarchy. The corresponding time-dependent formalism is then developed in detail in Sec. 4. Appendix A collects the relevant material for hyperelliptic Riemann surfaces and their theta functions. Appendix B contains a simple illustration of the Riemann– Roch theorem (cf. Theorem B.1). We emphasize that our treatment comprises, in particular, the important special case of the nonlinear Schr¨ odinger (NS) equation (cf. (3.87)), whose algebro-geometric solutions have been studied, for instance, in [5, Ch. 4] [15, 30, 31, 37, 39, 44]. Similarly, the case of the modified Korteweg–de Vries (mKdV) equation (cf. (3.89)), whose algebro-geometric solutions have been studied, for instance, in [21, 22], are included as a special case of our formalism. Moreover, the present elementary approach is not at all restricted to the AKNS hierarchy but applies quite generally to 1+1-dimensional hierarchies of soliton equations. In fact, the KdV case has been treated in [27], the case of the Toda and Kac–van Moerbeke hierarchies in [6], and the case of the Boussinesq hierarchy in [13]. Finally, we mention that a combination of the AKNS formalism developed in this paper and the Picard-type techniques introduced in a recent explicit characterization of all elliptic solutions of the KdV hierarchy [26] (see also [25]) are expected to yield a similar characterization of all elliptic solutions of the AKNS hierarchy, a topic that continues to attract considerable interest (see, e.g. [5, Ch. 7] [10, 47]). 2. The AKNS Hierarchy, Recursion Relations, and Hyperelliptic Curves In this section we briefly review the construction of the AKNS hierarchy using a recursive approach advocated by Al’ber [1, 2, 3] (see also [12, Ch. 12] [20, 23, 24, 27]) and outline its connection with the analog of the Burchnall–Chaundy polynomial [7, 8, 9], and associated hyperelliptic curves. Suppose p, q ∈ C ∞ (R) (or meromorphic on C) and introduce the Dirac-type matrix-valued differential expression d −q (2.1) D = i dx d , x ∈ R (or C) . p − dx
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
347
In order to explicitly construct higher-order matrix-valued differential expressions En+1 , n ∈ N0 (= N ∪ {0}) commuting with D, which will be used to define the stationary AKNS hierarchy later, one can proceed as follows. Pick n ∈ N0 and define {f` (x)}0≤`≤n , {g` (x)}0≤`≤n+1 , {h` (x)}0≤`≤n recursively by f0 (x) = −iq(x) , f`+1 (x) =
g0 (x) = 1 ,
h0 (x) = ip(x) ,
i f`,x (x) − iq(x)g`+1 (x) , 2
0 ≤ ` ≤ n−1, 0 ≤ ` ≤ n,
g`+1,x (x) = p(x)f` (x) + q(x)h` (x) ,
i h`+1 (x) = − h`,x (x) + ip(x)g`+1 (x) , 2
(2.2)
0 ≤ ` ≤ n−1.
Explicitly, one computes f0 = −iq , f1 =
1 qx + c1 (−iq) , 2
i i f2 = qxx − pq 2 + c1 4 2
1 qx 2
+ c2 (−iq) ,
g0 = 1 , g 1 = c1 , g2 =
(2.3)
1 pq + c2 , 2
i g3 = − (px q − pqx ) + c1 4
1 pq + c3 , 2
h0 = ip , h1 =
1 p + c1 (ip) , 2 x
i i h2 = − pxx + p2 q + c1 4 2
1 p 2 x
+ c2 (ip) ,
etc., where {c` }1≤`≤n+1 are integration constants. Given (2.2), one defines the matrixvalued differential expression En+1 by En+1 = i
n+1 X `=0
−gn+1−` −hn−`
fn−` gn+1−`
and verifies [En+1 , D] =
0 2ihn+1
D` ,
n ∈ N0 ,
−2ifn+1 0
f−1 = h−1 = 0 ,
(2.4)
n ∈ N0
(2.5)
,
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F. GESZTESY and R. RATNASEELAN
( [ . , . ] the commutator symbol). The pair (En+1 , D) represents the celebrated Lax pair for the AKNS hierarchy. Varying n ∈ N0 , the stationary AKNS hierarchy is then defined by the vanishing of the commutator of En+1 and D in (2.5), that is, by (2.6) [En+1 , D] = 0 , n ∈ N0 , or equivalently, by fn+1 = hn+1 = 0 ,
n ∈ N0 .
(2.7)
Explicitly, one obtains for the first few equations of the stationary AKNS hierarchy, ( −px + c1 (−2ip) = 0 , −qx + c1 (2iq) = 0 , i 2 2 pxx − ip q + c1 (−px ) + c2 (−2ip) = 0 , i − qxx + ipq 2 + c1 (−qx ) + c2 (2iq) = 0 , 2 i 1 3 2 pxxx − ppx q + c1 p − ip q + c2 (−px ) + c3 (−2ip) = 0 , 4 2 2 xx 3 i 1 qxxx − pqqx + c1 − qxx + ipq 2 + c2 (−qx ) + c3 (2iq) = 0 , 4 2 2
(2.8)
etc . By definition, solutions (p(x), q(x)) of any of the stationary AKNS Eqs. (2.8) are called algebro-geometric stationary finite-gap solutions associated with the AKNS hierarchy. If (p, q) satisfies the nth equation (n ∈ N0 ) of (2.8) one also calls (p, q) a stationary n-gap solution. Next, we introduce polynomials Fn , Gn+1 , Hn with respect to z ∈ C, Fn (z, x) =
n X
fn−` (x)z ` ,
f0 (x) = −iq(x) ,
`=0
Gn+1 (z, x) =
n+1 X
gn+1−` (x)z ` ,
g0 (x) = 1 ,
(2.9)
`=0
Hn (z, x) =
n X
hn−` (x)z ` ,
h0 (x) = ip(x)
`=0
and note that (2.6) respectively, (2.7) become Fn,x (z, x) = −2izFn (z, x) + 2q(x)Gn+1 (z, x) , Gn+1,x (z, x) = p(x)Fn (z, x) + q(x)Hn (z, x) , Hn,x (z, x) = 2izHn (z, x) + 2p(x)Gn+1 (z, x) .
(2.10) (2.11) (2.12)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
(2.10)–(2.12) yield that G2n+1 − Fn Hn
x
...
=0
349
(2.13)
and hence Gn+1 (z, x)2 − Fn (z, x)Hn (z, x) = R2n+2 (z) ,
(2.14)
where the integration constant R2n+2 (z) is a monic polynomial in z of degree 2n+2. Thus one may write R2n+2 (z) =
2n+1 Y
(z − Em ) ,
{Em }0≤m≤2n+1 ⊂ C .
(2.15)
m=0
Explicitly, one obtains for the first few polynomials in (2.9), F0 = −iq , 1 F1 = −iqz + qx + c1 (−iq) , 2
1 i i 1 F2 = −iqz 2 + qx z + qxx − pq 2 + c1 −iqz + qx + c2 (−iq) , 2 4 2 2 G1 = z + c1 , 1 G2 = z 2 + pq + c1 z + c2 , 2
i 1 1 2 G3 = z + pqz − (px q − pqx ) + c1 z + pq + c2 z + c3 , 2 4 2
(2.16)
3
H0 = ip , 1 H1 = ipz + px + c1 (ip) , 2
1 i i 2 1 H2 = ipz + px z − pxx + p q + c1 ipz + px + c2 (ip) , 2 4 2 2 2
etc. One can use (2.10)–(2.12) and (2.14) to derive differential equations for Fn and Hn separately by eliminating Gn+1 . One obtains for Fn , qx 1 2 qx 2 Fn Fn,xx − Fn Fn,x − Fn,x + 2z − 2iz − 2pq Fn2 q 2 q = −2q 2 R2n+2 (z)
(2.17)
and upon dividing (2.17) by q 2 and differentiating the result with respect to x, qxx qx2 qx qx 2 + 3 2 Fn,x Fn,xxx − 3 Fn,xx + 4z − 4iz − 4pq − q q q q q2 qxx qx + 2pqx − 2px q Fn = 0 . + −4z 2 + 6iz x2 − 2iz (2.18) q q q
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Similarly one obtains for Hn , px 1 2 px 2 Hn Hn,xx − Hn Hn,x − Hn,x + 2z + 2iz − 2pq Hn2 p 2 p = −2p2 R2n+2 (z) ,
(2.19)
pxx p2 px px Hn,xx + 4z 2 + 4iz − 4pq − + 3 x2 Hn,x p p p p p2 pxx px − 6iz x2 + 2iz + 2px q − 2pqx Hn = 0 . + −4z 2 p p p
Hn,xxx − 3
(2.20)
(2.17) and (2.19) can be used to derive recursion relations for f` and h` in the homogeneous case where all c` = 0, ` ∈ N (cf. Lemma 4.5). This has interesting applications to the high-energy expansion of the Green’s matrix of D as briefly discussed in Remark 4.6. Next, we consider the kernel (i.e. the formal null space in a purely algebraic sense) of (D − z), z ∈ C, ψ1 (z, x) , z∈C (2.21) (D − z)Ψ = 0 , Ψ(z, x) = ψ2 (z, x) and, taking into account (2.6), that is, [En+1 , D] = 0, compute the restriction of En+1 to Ker(D − z). Using ψ1,x = −izψ1 + qψ2 , ψ2,x = izψ2 + pψ1 ,
etc.,
(2.22)
in order to eliminate higher-order derivatives of ψj , j = 1, 2, one obtains from (2.2), (2.4), (2.7), (2.9), and (2.10)–(2.12), Fn (z, x) −Gn+1 (z, x) =i . (2.23) En+1 −Hn (z, x) Gn+1 (z, x) Ker(D−z) Ker(D−z) Still assuming fn+1 = hn+1 = 0 as in (2.7), [En+1 , D] = 0 in (2.6) yields an algebraic relationship between En+1 and D by a celebrated result of Burchnall and Chaundy [7, 8, 9] (see also [45, 48]). The following theorem details this relationship. Theorem 2.1. Assume fn+1 = hn+1 = 0, that is, [En+1 , D] = 0 for some n ∈ N0 . Then the Burchnall–Chaundy polynomial Fn (D, En+1 ) of the pair (D, En+1 ) explicitly reads (cf. (2.15)) 2 + R2n+2 (D) = 0 , Fn (D, En+1 ) = En+1
R2n+2 (z) =
2n+1 Y
(z − Em ) ,
m=0
z ∈ C.
(2.24)
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...
351
Proof. [En+1 , D] = 0, (2.14), and (2.23) imply 2 En+1
Ker(D−z)
= En+1
Ker(D−z)
2
Ker(D−z)
0 G2n+1 − Fn Hn 0 G2n+1 − Fn Hn 1 0 = −R2n+2 (z) 0 1 Ker(D−z) = −R2n+2 (D) . =−
Ker(D−z)
(2.25)
Ker(D−z)
Since z ∈ C is arbitrary one infers (2.24).
Remark 2.2. Equation (2.24) naturally leads to the (possibly singular) hyperelliptic curve Kn , Kn :
Fn (z, y) = y − R2n+2 (z) = 0 , 2
R2n+2 (z) =
2n+1 Y
(z − Em ) ,
n ∈ N0
m=0
(2.26) of (arithmetic) genus n. Next, introducing a deformation parameter tn ∈ R in (p, q) (i.e., (p(x), q(x)) → (p(x, tn ), q(x, tn ))), the time-dependent AKNS hierarchy (cf., e.g. [42, Chapters 3 and 5 and the references therein]) is defined as the collection of evolution equations (varying n ∈ N0 ), d D(tn ) − [En+1 (tn ), D(tn )] = 0 , dtn
(x, tn ) ∈ R2 ,
n ∈ N0 ,
(2.27)
or equivalently, by ( AKNSn (p, q) =
ptn (x, tn ) − 2hn+1 (x, tn ) = 0 , qtn (x, tn ) − 2fn+1 (x, tn ) = 0 ,
(x, tn ) ∈ R2 ,
n ∈ N0 , (2.28)
that is, by ( AKNSn (p, q) =
ptn + iHn,x + 2zHn − 2ipGn+1 = 0 , qtn − iFn,x + 2zFn + 2iqGn+1 = 0 ,
(x, tn ) ∈ R2 ,
n ∈ N0 . (2.29)
Explicitly, one obtains for the first few equations in (2.28) or (2.29),
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F. GESZTESY and R. RATNASEELAN
( AKNS0 (p, q) = ( AKNS1 (p, q) = ( AKNS2 (p, q) =
pt0 − px + c1 (−2ip) = 0 , qt0 − qx + c1 (2iq) = 0 , pt1 + 2i pxx − ip2 q + c1 (−px ) + c2 (−2ip) = 0 ,
(2.30)
qt1 − 2i qxx + ipq 2 + c1 (−qx ) + c2 (2iq) = 0 ,
pt2 + 14 pxxx − 32 ppx q + c1 ( 2i pxx − ip2 q) + c2 (−px ) + c3 (−2ip) = 0 , qt2 + 14 qxxx − 32 pqqx + c1 (− 2i qxx + ipq 2 ) + c2 (−qx ) + c3 (2iq) = 0 ,
etc. Remark 2.3. We chose to start by postulating the recursion relation (2.2) and then developed the whole formalism based on (2.2), (2.4)–(2.6). Alternatively one could have started from (D − z)Ψ(P ) = 0 ,
(En+1 − iy(p))Ψ(P ) = 0 ,
P = (z, y) ∈ Kn \{∞± }
(2.31)
and obtained the recursion relation (2.2) and the remaining stationary results of this section as a consequence of (2.9) and (2.23). Similarly, starting with ∂ − En+1 Ψ(P, tn ) = 0 , tn ∈ R , (2.32) (D − z)Ψ(P, tn ) = 0 , ∂tn one infers the time-dependent results (2.27)–(2.30). Remark 2.4. Define −iz q(x) U (z, x) = , p(x) iz
Vn+1 (z, x) = i
−Gn+1 (z, x) Fn (z, x) −Hn (z, x) Gn+1 (z, x)
. (2.33)
Then (2.23) implies 1 0 = {−Vn+1,x (z) + [U (z), Vn+1 (z)]} −i [En+1 , D] 0 −1 Ker(D−z) Ker(D−z) (2.34) and the stationary part of this section, being a consequence of [En+1 , D] = 0, can equivalently be based on the equation −Vn+1,x + [U, Vn+1 ] = 0 .
(2.35)
In particular, the hyperelliptic curve Kn in (2.26) is then obtained from the characteristic equation for Vn+1 (z, x), det[yI − iVn+1 (z, x)] = y 2 − det[Vn+1 (z, x)] = y 2 − Gn+1 (z, x)2 + Fn (z, x)Hn (z, x) = y 2 − R2n+2 (z) = 0.
(2.36)
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353
Similarly, the time-dependent part (2.28)–(2.30), being based on the Lax Eq. (2.27), can equivalently be developed from the zero-curvature equation Utn − Vn+1,x + [U, Vn+1 ] = 0 .
(2.37)
In fact, since the latter approach (2.37) is almost universally adopted in the contemporary literature on the AKNS hierarchy, we decided to recall its proper origin in connection with the Lax pair [En+1 , D] and based our treatment on matrix-valued differential expressions instead. 3. The Stationary AKNS Formalism In this section we continue our discussion of the AKNS hierarchy and concentrate on the stationary case. Following [27], where the analogous treatment of the stationary KdV hierarchy can be found, we outline the connections between the polynomial approach described in Sec. 2 and a fundamental meromorphic function φ(P, x) defined on the hyperelliptic curve Kn . Moreover, we discuss in detail the associated stationary Baker–Akhiezer function Ψ(P, x, x0 ), the common eigenfunction of D and En+1 (we recall that [En+1 , D] = 0), and associated positive divisors of degree n on Kn (which should be considered as the analogs of Dirichlet and Neumann divisors in the KdV context). We recall the hyperelliptic curve (2.26), Kn :
Fn (z, y) = y 2 − R2n+2 (z) = 0 ,
R2n+2 (z) =
2n+1 Y
(z − Em ) ,
(3.1)
m=0
where n ∈ N0 will be fixed throughout this section and denote its compactification (adding the points ∞± ) by the same symbol. Thus Kn becomes a (possibly singular) two-sheeted hyperelliptic Riemann surface of arithmetic genus n in a standard manner. We shall introduce a bit more notation in this context (see Appendix A for more details). Points P on Kn are represented as pairs P = (z, y) satisfying (3.1) together with ∞± , the points at infinity. The complex structure on Kn is defined in the usual way by introducing local coordinates ζP0 : P → (z − z0 ) near points P0 ∈ Kn which are neither branch nor singular points of Kn , ζ∞± : P → 1z near ∞± , and similarly at branch and/or singular points of Kn . The holomorphic sheet exchange map (involution) ∗ is defined by ( ∗:
Kn → Kn P = (z, y) 7→ P ∗ = (z, −y) .
(3.2)
A detailed description of Kn and its complex structure in the two most frequently discussed cases in applications where either Em ∈ R, 0 ≤ m ≤ 2n + 1 or {Em }0≤m≤2n+1 = {E2m0 , E2m0 }0≤m0 ≤n is provided at the end of Appendix A.
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F. GESZTESY and R. RATNASEELAN
Finally, positive divisors on Kn of degree are denoted by Kn → N0 ( m if P occurs m times in {Q1 , . . . , Qn } , DQ : Q = (Q1 , . . . , Qn ) . P 7→ DQ (P ) 0 if P ∈ / {Q1 , . . . , Qn } , (3.3) Given these preliminaries, let Ψ(P, x, x0 ) denote the common normalized eigenfunction of D and En+1 , whose existence follows from the commutativity of D and En+1 (cf., e.g. [7, 8] in the case of scalar differential expressions), that is, due to [En+1 , D] = 0
(3.4)
for a given n ∈ N0 , or equivalently, due to the requirement, fn+1 = hn+1 = 0 .
(3.5)
Explicitly, this yields DΨ(P, x, x0 ) = zΨ(P, x, x0 ) , En+1 Ψ(P, x, x0 ) = iy(P )Ψ(P, x, x0 ) , ψ1 (P, x, x0 ) Ψ(P, x, x0 ) = , P = (z, y) ∈ Kn \{∞± } , x ∈ R ψ2 (P, x, x0 )
(3.6)
for some fixed x0 ∈ R with the assumed normalization, P ∈ Kn \{∞± } .
ψ1 (P, x0 , x0 ) = 1 ,
(3.7)
Ψ(P, x, x0 ) is called the Baker–Akhiezer (BA) function. Closely related to Ψ(P, x, x0 ) is the following meromorphic function φ(P, x) on Kn , defined by φ(P, x) =
ψ2 (P, x, x0 ) , ψ1 (P, x, x0 )
P ∈ Kn ,
x ∈ R.
(3.8)
Since φ(P, x) will be the fundamental object for the stationary AKNS hierarchy, we next seek its connection with the recursion formalism of Sec. 2. Recalling (2.23), one infers Fn ψ2 − Gn+1 ψ1 ψ1 = iy (3.9) En+1 Ψ = i Gn+1 ψ2 − Hn ψ1 ψ2 and hence by (3.8), φ(P, x) =
−Hn (z, x) y(P ) + Gn+1 (z, x) = , Fn (z, x) y(P ) − Gn+1 (z, x) P = (z, y) ∈ Kn .
(3.10)
By (2.9) we may write, Fn (z, x) = −iq(x)
n Y
(z − µj (x)) ,
(3.11)
j=1
Hn (z, x) = ip(x)
n Y j=1
(z − νj (x)) .
(3.12)
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355
Defining µ ˆj (x) = (µj (x) , Gn+1 (µj (x), x)) ∈ Kn , 1 ≤ j ≤ n, x ∈ R ,
(3.13)
νˆj (x) = (νj (x) , −Gn+1 (νj (x), x)) ∈ Kn , 1 ≤ j ≤ n, x ∈ R ,
(3.14)
one infers from (3.10) that the divisor (φ(P, x)) of φ(P, x) is given by (φ(P, x)) = Dνˆ(x) (P ) − Dµˆ (x) (P ) + D∞+ (P ) − D∞− (P ) ,
(3.15)
ν1 (x), . . . , νˆn (x)) , µ ˆ (x) = (ˆ µ1 (x), . . . , µ ˆn (x)) . νˆ(x) = (ˆ Here we used our convention (3.3) and additive notation for divisors. Equivalently, ˆ1 (x), . . . , µ ˆn (x), its ∞+ , νˆ1 (x), . . . , νˆn (x), are the n + 1 zeros of φ(P, x) and ∞− , µ n + 1 poles. Clearly µj (x) and νj (x) play the analogous role of Dirichlet and Neumann eigenvalues when comparing to the KdV case. In particular, Dµˆ (x) and Dνˆ(x) represent the corresponding analogs of Dirichlet and Neumann divisors. Next we summarize a variety of properties of φ(P, x) and Ψ(P, x, x0 ). Lemma 3.1. Assume (3.4)–(3.8), P = (z, y) ∈ Kn \{∞± }, and let (z, x, x0 ) ∈ C × R2 . Then (i)
(ii)
Ψ(P, x, x0 ) satisfies the first-order system (cf. (2.33)) Ψx (P, x, x0 ) = U (z, x)Ψ(P, x, x0 ) ,
(3.16)
iy(P )Ψ(P, x, x0 ) = Vn+1 (z, x)Ψ(P, x, x0 ) .
(3.17)
φ(P, x) satisfies the Riccati-type equation φx (P, x) + q(x)φ(P, x)2 − 2izφ(P, x) = p(x) .
(iii) φ(P, x)φ(P ∗ , x) =
Hn (z, x) . Fn (z, x)
(iv) φ(P, x) + φ(P ∗ , x) =
(v)
(3.18)
φ(P, x) − φ(P ∗ , x) = Z
(3.19)
2Gn+1 (z, x) . Fn (z, x)
(3.20)
2y(P ) . Fn (z, x)
(3.21)
x
(vi) ψ1 (P, x, x0 ) = exp
dx [−iz + q(x )φ(P, x )] 0
0
0
(3.22)
x0
Fn (z, x) = Fn (z, x0 )
12
(vii) ψ1 (P, x, x0 )ψ1 (P ∗ , x, x0 ) =
Z exp y(P )
x
0
0
0 −1
dx q(x )Fn (z, x )
.
(3.23)
x0
Fn (z, x) . Fn (z, x0 )
(3.24)
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F. GESZTESY and R. RATNASEELAN
(viii) ψ2 (P, x, x0 )ψ2 (P ∗ , x, x0 ) =
(ix)
Hn (z, x) . Fn (z, x0 )
ψ1 (P, x, x0 )ψ2 (P ∗ , x, x0 ) + ψ1 (P ∗ , x, x0 )ψ2 (P, x, x0 ) =
(3.25) 2Gn+1 (z, x) . Fn (z, x0 ) (3.26)
Proof. (i) is an immediate consequence of (2.33), (3.6), and (3.10). (ii) follows from (i), (2.22) and (3.8). (iii)–(v) are clear from (3.10). (3.22) follows from (2.22) and (3.8). (3.23) is a consequence of (iv), (v), (2.10), (3.22), and 1 1 [φ(P ) + φ(P ∗ )] + [φ(P ) − φ(P ∗ )] 2 2 y 1 Fn,x y Gn+1 + = + iz + . = Fn Fn q Fn Fn
φ(P ) =
(3.27)
(vii) is clear from (3.23) and (viii) is a consequence of (3.8), (iii), and (vii). Finally, (ix) is a consequence of (3.8), (3.20), and (3.24). In order to motivate our introduction of the basic quantity φ(P, x) we started with the common eigenfunction ψ(P, x, x0 ) of D and En+1 . However, given (2.14) ψ1 we could have defined φ(P, x) as in (3.10) and then verified that Ψ = defined ψ2 by (3.8) and (3.22) satisfies (3.6) and (3.7). Concerning the dynamics of the zeros µj (x) and νj (x) of Fn (z, x) and Hn (z, x) one obtains the following Dubrovin-type equations. Lemma 3.2. Assume (3.4)–(3.8), (3.11), (3.12) and let x ∈ R. Then (i)
−2iy(ˆ µj (x)) , k=1 (µj (x) − µk (x))
µj,x (x) = Qn
1 ≤ j ≤ n.
(3.28)
k6=j
(ii)
−2iy(ˆ νj (x)) , k=1 (νj (x) − νk (x))
νj,x (x) = Qn
1 ≤ j ≤ n.
(3.29)
k6=j
Proof. Combine (2.10), (3.11), and (3.13) and (2.12), (3.12), and (3.14) in order to arrive at (3.28) and (3.29), respectively. Combining the polynomial approach of Sec. 2 with (3.11) and (3.12) readily yields trace formulas for the AKNS invariants. We indicate the first few of these below.
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357
Lemma 3.3. Assume (3.4)–(3.8) and let x ∈ R. Then (i)
i
n X px (x) − 2c1 = 2 νj1 (x) , p(x) j =1 1
n X i px (x) 1 pxx (x) 1 − p(x)q(x)+c1 νj1 (x)νj2 (x) , −c2 = − 4 p(x) 2 2 p(x) j ,j =1
(3.30)
1 2 j1 <j2
etc . (ii)
n X qx (x) + 2c1 = −2 µj1 (x) , i q(x) j =1 1
n X i qx (x) 1 qxx (x) 1 − p(x)q(x)+c1 − −c2 = − µj1 (x)µj2 (x) , (3.31) 4 q(x) 2 2 q(x) j1 ,j2 =1 j1 <j2
etc . Here c1 = −
2 n+1 1 X Em1 , 2 m =0 1
1 c2 = 2
2X n+1
Em1 Em2
m1 ,m2 =0
1 − 8
2X n+1
!2 Em1
,
(3.32)
m1 =0
m1 <m2
etc .
Proof. (3.30) and (3.31) follow by comparison of powers of z substituting (3.11) into (2.9) taking into account (2.3). (3.32) follows in exactly the same way from (2.3), (2.9), and (3.12). Finally, we shall provide an explicit representation of Ψ, φ, p, and q in terms of the Riemann theta function associated with Kn . We freely employ the notation established in Appendix A. In order to avoid the trivial case n = 0 (considered in Example 3.8) we assume n ∈ N for the remainder of this argument. Assuming Kn to be nonsingular, that is, Em 6= Em0
for
0 ≤ m, m0 ≤ 2 n + 1 ,
(3.33)
we choose, without loss of generality, the base point P0 = (E0 , 0) and denote by ˆ , the vector AP0 (.), αP0 (.) the Abel maps as defined in (A.30)–(A.32), and define Ξ P0 of Riemann constants, by
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F. GESZTESY and R. RATNASEELAN
ˆ ΞP0 = Ξ P0
(mod Ln ) , n X ˆ P0 ,j = 1 + τj,j − Ξ 2
ˆ P0 ,1 , . . . , Ξ ˆ P0 ,n ) , ˆ = (Ξ Ξ P0
Z
k=1 k6=j
ak
(3.34)
AˆP0 ,j ωk .
(3)
Next, consider the normal differential of the third kind ω∞+ ,∞− , which has simple poles at ∞+ and ∞− , corresponding residues +1 and −1, vanishing a-periods, and is holomorphic otherwise on Kn . Hence we have (cf. (A.36), (A.37)) Qn π − λj )d˜ π j=1 (˜ (3) (3) (3) , ω∞ = −ω∞ , (3.35) ω∞+ ,∞− = − ,∞+ + ,∞− y Z (3) ω∞ = 0, 1 ≤ j ≤ n, (3.36) + ,∞− aj
(3)
U (3) = (U1 , . . . , Un(3) ) , Z 1 (3) = ω (3) Uj 2πi bj ∞+ ,∞− = Aˆ∞− ,j (∞+ ) = 2AˆP0 ,j (∞+ ) , Z
P P0
(3) ω∞ = ±[ln(ζ) − ln(ω0 ) + O(ζ)] , + ,∞− ζ→0
1 ≤ j ≤ n, P = (ζ −1 , y) near ∞± ,
(3.37) (3.38)
where the numbers {λj }1≤j≤n are determined by the normalization (3.36). The (2) Abelian differential of the second kind ω∞± ,0 (cf. (A.34), (A.35)) are chosen such that ω∞± ,0 = [ζ −2 + O(1)] dζ near ∞± , (2)
ζ→0
(3.39)
Z (2)
aj (2) U0
Z
= P
(2) (2) (U0,1 , . . . , U0,n ) ,
(2)
Ω0 P0
ω∞± ,0 = 0 , 1 = 2πi
(3.40)
Z (2)
(2) (2) Ω0 = ω ∞ − ω∞ , (3.41) +,0 −,0
= ∓[ζ −1 + e0,0 + e0,1 ζ + O(ζ 2 )] ,
P = (ζ −1 , y) near ∞± . (3.42)
ζ→0
(2) U0,j
1 ≤ j ≤ n,
Ω0 , bj
(2)
Next, we formulate the following auxiliary result. Lemma 3.4. Let ψ(., x), x ∈ R be meromorphic on Kn \{∞+ , ∞− } with essen˜ x) defined by tial singularities at ∞± such that ψ(., # " Z P (2) ˜ Ω0 (3.43) ψ(., x) = ψ(., x) exp −i(x − x0 ) P0
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359
is multi-valued meromorphic on Kn and its divisor satisfies ˜ x)) ≥ −Dµˆ (x) . (ψ(.,
(3.44)
˜ x)) = D0 (x) − Dµˆ (x) . (ψ(.,
(3.45)
Define a divisor D0 (x) by
Then D0 (x) ∈ σ n Kn ,
D0 (x) > 0 ,
deg(D0 (x)) = n .
(3.46)
Moreover, if D0 (x) is nonspecial for all x ∈ R, that is, if i(D0 (x)) = 0 ,
x ∈ R,
(3.47)
then ψ(., x) is unique up to a constant multiple (which may depend on x). Proof. By the Riemann–Roch theorem (see (A.42)) there exists at least one such function ψ(., x). Suppose ψj (., x), j = 1, 2 are two such functions satisfying (3.45) with corresponding divisors D0,j (x), j = 1, 2. Then (ψ1 (., x)/ψ2 (., x)) = D0,1 (x) − D0,2 (x) .
(3.48)
Since i(D0,2 (x)) = 0, deg(D0,2 (x)) = n by hypothesis, the multi-valued version of (A.42) yields r(−D0,2 (x)) = 1, x ∈ R and hence ψ1 (., x)/ψ2 (., x) is a constant on Kn . (For simplicity of notation, we did not prescribe the path of integration in (3.43). This in turn forced us to use the multi-valued form of the Riemann–Roch theorem, see, e.g. [18, Sec. III.9.]) Assuming DQ to be nonspecial, that is, i(DQ ) = 0, Q = (Q1 , . . . , Qn ), a special case of Riemann’s vanishing theorem yields that θ(ΞP0 − AP0 (P ) + αP0 (DQ )) = 0 if and only if P ∈ {Q1 , . . . , Qn } .
(3.49)
Hence the divisor (3.15) of φ(P, x) suggests considering expressions of the type θ(ΞP0 − AP0 (P ) + αP0 (Dνˆ (x) )) exp C(x) θ(ΞP0 − AP0 (P ) + αP0 (Dµˆ (x) ))
"Z
#
P
P0
(3) ω∞ + ,∞−
,
(3.50)
where C(x) is independent of P ∈ Kn . In fact, abbreviating z(P, Q) = AP0 (P ) − αP0 (DQ ) − ΞP0 , z ± (Q) = z(∞± , Q) ,
Q = (Q1 , . . . , Qn ) ,
(3.51) (3.52)
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F. GESZTESY and R. RATNASEELAN
one obtains the following theta function representation for φ, Ψ and (p, q) (the analog of the celebrated Its–Matveev formula [32] in the KdV context). Theorem 3.5. Let P ∈ Kn \{∞+ , ∞− }, (x, x0 ) ∈ R2 , and assume Kn to be nonsingular, that is, Em 6= Em0 for m 6= m0 , 0 ≤ m, m0 ≤ 2n+1. Moreover, suppose Dµˆ (x) , or equivalently, Dνˆ (x) to be nonspecial, that is, i(Dµˆ (x) ) = i(Dνˆ(x) ) = 0. Then φ(P, x) =
µ(x0 ))) θ(z + (ˆ µ(x))) θ(z(P, νˆ(x))) 2i θ(z − (ˆ q(x0 )ω0 θ(z + (ˆ µ(x0 ))) θ(z − (ˆ ν (x))) θ(z(P, µ ˆ (x))) # "Z
(3.53)
P
× exp P0
(3) ω∞ + ,∞−
− 2i(x − x0 )e0 ,
" !# Z P θ(z + (ˆ µ(x0 ))) θ(z(P, µ ˆ (x))) (2) exp i(x − x0 ) e0 + Ω0 , ψ1 (P, x, x0 ) = θ(z(P, µ ˆ (x0 ))) θ(z + (ˆ µ(x))) P0 (3.54) ψ2 (P, x, x0 ) =
θ(z − (ˆ µ(x0 ))) θ(z(P, νˆ(x))) 2i q(x0 )ω0 θ(z(P, µ ˆ (x0 ))) θ(z − (ˆ ν (x))) "Z
Z
P
× exp P0
(3) ω∞ + i(x − x0 ) −e0 + + ,∞−
!#
P
(2)
Ω0
(3.55) .
P0
Moreover, one derives p(x) = p(x0 )
θ(z − (ˆ ν (x0 ))) θ(z + (ˆ ν (x))) −2i(x−x0 )e0 e , θ(z + (ˆ ν (x0 ))) θ(z − (ˆ ν (x)))
(3.56)
q(x) = q(x0 )
θ(z + (ˆ µ(x0 ))) θ(z − (ˆ µ(x))) 2i(x−x0 )e0 e , θ(z − (ˆ µ(x0 ))) θ(z + (ˆ µ(x)))
(3.57)
p(x0 )q(x0 ) =
µ(x0 ))) ν (x0 ))) θ(z − (ˆ 4 θ(z + (ˆ , 2 ω0 θ(z − (ˆ ν (x0 ))) θ(z + (ˆ µ(x0 )))
(3.58)
and (2)
(3.59)
(2)
(3.60)
αP0 (Dµˆ (x) ) = αP0 (Dµˆ (x0 ) ) − i(x − x0 )U 0 , αP0 (Dνˆ(x) ) = αP0 (Dνˆ(x0 ) ) − i(x − x0 )U 0 .
−
RP
ω (3)
Proof. Since φ(P, x)e P0 ∞+ ,∞− is meromorphic on Kn with divisor Dνˆ(x) − Dµˆ (x) , Dµˆ (x) is nonspecial if and only if Dνˆ (x) is nonspecial (cf. the comment
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
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361
following (A.40)). Combining (3.7), (3.8), (3.10), (3.11), (3.12), (3.19), and (3.24) yields the asymptotic behavior i 2 p(x)ζ + O(ζ ) , P near ∞+ , 2 (3.61) φ(P, x) = 2i −1 ζ→0 ζ + O(1) , P near ∞ , − q(x) −i(x−x0 )ζ −1 +O(ζ) , e ψ1 (P, x, x0 ) = q(x) i(x−x0 )ζ −1 +O(ζ) ζ→0 , q(x ) + O(ζ) e 0
P near ∞+ , P near ∞− ,
i p(x)ζ + O(ζ 2 ) e−i(x−x0 )ζ −1 +O(ζ) , 2 ψ2 (P, x, x0 ) = ζ→0 2i −1 −1 ζ + O(1) ei(x−x0 )ζ +O(ζ) , q(x0 )
(3.62)
P near ∞+ , (3.63) P near ∞− .
The asymptotic behavior (3.62) (near ∞+ ), (3.24) and (3.49) together with Lemma 3.4 then directly yield the theta function representation (3.54) for ψ1 . (3.49) also immediately yields that φ(P, x) equals (3.53) which, together with (3.61), implies p(x) =
ν (x))) 2C(x) θ(z + (ˆ , iω0 θ(z + (ˆ µ(x)))
(3.64)
q(x) =
µ(x))) 2i θ(z − (ˆ . C(x)ω0 θ(z − (ˆ ν (x)))
(3.65)
On the other hand (3.62) (near ∞− ) and (3.54) yield (3.57). A comparison of (3.57) and (3.65) determines C(x) and p(x) as in (3.56), (3.58). Given C(x), one determines φ in (3.53) from (3.50) and hence ψ2 as in (3.55) from ψ2 = φψ1 . By (3.28) and a special case of Lagrange’s interpolation formula, n X j=1
one infers
µk−1 j
n Y
(µj − µ` )−1 = δk,n ,
µj ∈ C ,
1 ≤ j , k ≤ n,
(3.66)
`=1 `6=j
d α (ˆ µ(x))) = −2icn , dx P0
where ωj =
n X k=1
cj,k
cn = (c1,n , . . . , cn,n ) ,
π ˜ k−1 d˜ π , y
1≤j≤n
(3.67)
(3.68)
abbreviates the basis of holomorphic differentials on Kn . By (A.35) this yields d (2) α (ˆ µ(x))) = −iU 0 dx P0 and hence (3.59) (respectively, (3.60)).
(3.69)
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F. GESZTESY and R. RATNASEELAN
For completeness we also mention another theta function representation for the product p(x)q(x), originally due to [30]. Corollary 3.6. Assume the hypotheses of Theorem 3.5. Then p(x)q(x) = −e0,1 −
d2 ln(θ(z + (ˆ µ(x)))) . dx2
(3.70)
Proof. Eliminating ψ2 (z, x) in (2.22) results in ψ1,xx (z, x) =
qx (x) qx (x) ψ1,x (z, x) + (p(x)q(x) + iz − z 2 )ψ1 (z, x) . q(x) q(x)
(3.71)
Next, using ψ1 (P, x, x0 ) = e−i(x−x0 )(ζ
−1
+e0,1 ζ+O(ζ 2 ))
ζ→0
(1 + c1 (x)ζ + c2 (x)ζ 2 + O(ζ 3 )) , (3.72)
one infers 0 = −ψ1,xx (P, x, x0 ) +
qx (x) qx (x) −1 ψ1,x (P, x, x0 ) + (p(x)q(x) + i ζ q(x) q(x)
−ζ −2 )ψ1 (P, x, x0 ) = e−i(x−x0 )(ζ
−1
+e0,1 ζ+O(ζ 2 ))
ζ→0
(e0,1 + 2ic1,x (x) + p(x)q(x) + O(ζ)) . (3.73)
By the uniqueness of ψ1 (P, x, x0 ) as discussed in Lemma 3.4 one concludes p(x)q(x) = −e0,1 − 2ic1,x(x) .
(3.74)
It remains to determine c1,x (x). First we recall from (A.24) that ω = (c(n) + O(ζ))dζ near ∞+ ζ→0
(3.75)
and hence 1 (2) AP0 (P ) = AP0 (∞+ ) + c(n)ζ + O(ζ 2 ) = AP0 (∞+ ) + U 0 ζ + O(ζ 2 ) , (3.76) ζ→0 ζ→0 2 where we combined (3.41) and (A.35) in the second equality. Since p(x)q(x) only depends on c1,x (x) as opposed to c1 (x) itself, it suffices to consider the following expansion near ∞+ . (2)
d n µ(x)) + w)|w=0 ˆ (x))) θ(z(P, µ 1 Σj=1 U0,j dwj θ(z + (ˆ = 1− ζ + O(ζ 2 ) θ(z + (ˆ µ(x))) ζ→0 2 θ(z + (ˆ µ(x)))
1 d ln(θ(z + (ˆ µ(x))))ζ + O(ζ 2 ) . (3.77) 2i dx Here we used (3.59) to arrive at the last equality in (3.77). A comparison of (3.54), (3.73), and (3.77) then yields = 1+
c1,x (x) = −
i d2 ln(θ(z + (ˆ µ(x)))) , 2 dx2
which finally yields (3.70) employing (3.74).
(3.78)
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363
We note that the free constant q(x0 ) in (3.56) (and in (3.57) using (3.58)) cannot be determined by this formalism since the AKNS Eqs. (2.28) are invariant with respect to scale transformations. More precisely, one has Lemma 3.7. Suppose (p, q) satisfies one of the AKNS Eqs. (2.28) for some n ∈ N0 , (3.79) AKN Sn (p, q) = 0 . Consider the scale transformation p(x, tn ), q˘(x, tn )) = (Ap(x, tn ), A−1 q(x, tn )) , (p(x, tn ), q(x, tn )) → (˘
A ∈ C\{0} . (3.80)
Then p, q˘) = 0 . AKN Sn (˘
(3.81)
˘ E ˘n+1 ) be associated with (p, q) and (˘ p, q˘), Proof. Let (D, En+1 ) and (D, respectively and defined according to (2.1) and (2.4). Defining the matrix T in C2 by ! 1 (A 2 )−1 0 T = (3.82) 1 0 A2 1
(fixing a particular square root branch A 2 ) one computes ˘, T DT −1 = D T En+1 T
−1
=i
(3.83)
n+1 X
−gn+1−`
A−1 fn−`
`=0
−Ahn−`
gn+1−`
! ˘` = E ˘n+1 . D
(3.84)
A comparison of (3.84) with ˘n+1 = i E
n+1 X `=0
−˘ gn+1−` ˘ n−` −h
f˘n−`
!
g˘n+1−`
˘` D
(3.85)
yields f˘n−` = A−1 fn−` ,
g˘n+1−` = gn+1−` ,
˘ n−` = A−1 hn−` , h
0 ≤ ` ≤ n + 1 (3.86)
and hence (3.81), taking into account (2.28) and (3.80).
In the particular case of the nonlinear Schr¨ odinger (NS) hierarchy, where p(x, tn ) = ±q(x, tn ) ,
n ∈ N0 ,
(3.87)
(3.80) further restricts A to be unimodular, that is, |A| = 1 .
(3.88)
364
F. GESZTESY and R. RATNASEELAN
In the special case of the modified Korteweg–de Vries (mKdV) hierarchy, where p(x, tn ) = ±q(x, tn ) ,
n ∈ 2N0 ,
c2`+1 = 0 ,
` ∈ N0 ,
(3.89)
(3.80) implies the additional restriction A ∈ {−1, +1} .
(3.90)
Next, we briefly consider the trivial case n = 0 excluded in Theorem 3.5. Example 3.8. Assume n = 0. Then Q1 F0 (z, y) = y 2 − R2 (z) = y 2 − m=0 (z − Em ), c1 = −(E0 + E1 )/2, p(x) = p(x0 ) exp[−2ic1 (x − x0 )], q(x) = q(x0 ) exp[2ic1 (x − x0 )], p(x)q(x) = (E0 − E1 )2 /4, φ(P, x) =
ip(x) y(P ) + z + c1 = , −iq(x) y(P ) − z − c1
ψ1 (P, x, x0 ) = exp[i(x − x0 )(y(P ) + c1 )], ψ2 (P, x, x0 ) =
y(P ) + z + c1 exp[i(x − x0 )(y(P ) − c1 )]. −iq(x0 )
Finally, we mention an interesting characterization of all algebro-geometric AKNS potentials due to De Concini and Johnson [11] in the special case where D generates a self-adjoint operator in L2 (R) ⊗ C2 . In this case the algebro-geometric potentials are characterized by the fact that the corresponding spectrum consists of finitely many intervals and the Lyapunov exponent vanishes a.e. on the spectrum. In this context we might also point out that a detailed study of Floquet theory for periodic and self-adjoint AKNS operators (not necessarily of algebro-geometric type) generated by D can be found in [28]. 4. The Time-Dependent AKNS Formalism In our final section we indicate how to generalize the polynomial approach of Secs. 2 and 3 to the time-dependent AKNS hierarchy. Our starting point is a stationary n-gap solution (p(0) (x), q (0) (x)), associated with Kn , i
n (0) X px (x) (0) = 2c + 2 νj1 (x) , 1 p(0) (x) j =1 1
n (0) X qx (x) (0) = −2c1 − 2 µj1 (x) , i (0) q (x) j =1 1
(4.1)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
satisfying
( AKNSn (p
(0)
,q
(0)
)=
...
−2hn+1 = 0
365
(4.2)
−2fn+1 = 0
for some fixed n ∈ N0 and a given set of integration constants {c` }1≤`≤n+1 . Our principal aim is to construct the rth AKNS flow
AKNSr (p, q) =
=
pt − 2 h ˜ r+1 = 0 r qt − 2f˜r+1 = 0 r pt + i H ˜ r,x + 2z H ˜ r − 2ipG ˜ r+1 = 0 , r qt − iF˜r,x + 2z F˜r + 2iq G ˜ r+1 = 0 , r
(4.3)
x∈R
(p(x, t0,r ), q(x, t0,r )) = (p(0) (x), q (0) (x)) ,
for t0,r ∈ R and some fixed r ∈ N0 . In terms of Lax pairs this amounts to solving d ˜r+1 (tr ), D(tr )] = 0 , D(tr ) − [E dtr
tr ∈ R ,
(4.4)
[En+1 (t0,r ), D(t0,r )] = 0 .
(4.5)
As a consequence one obtains that [En+1 (tr ), D(tr )] = 0 ,
tr ∈ R ,
En+1 (tr )2 = −R2n+2 (D(tr )) = −
(4.6) 2n+1 Y
(D(tr ) − Em ) ,
tr ∈ R
(4.7)
m=0
since the AKNS flows are isospectral deformations of D(t0,r ). We emphasize that the integration constants {˜ c` } in E˜r+1 and {c` } in En+1 , in general, are independent of each other (even if r = n). Hence we shall employ the ˜ l , c˜l , etc., in order to distinguish ˜ r , f˜l , g˜l , h ˜ r+1 H notation E˜r+1 , V˜r+1 , F˜r , G it from En+1 , Vn+1 , Fn , Gn+1 , Hn , fl , gl , hl , cl , etc. In addition, we followed a more elaborate notation inspired by Hirota’s τ - function approach and indicated the individual rth AKNS flow by a separate time variable tr ∈ R. (The latter notation suggests considering all AKNS flows simultaneously by introducing t = (t0 , t1 , t2 , ........).) Instead of working directly with (4.4), (4.5) and (4.6), it is more convenient to take the zero-curvature Eq. (2.37) as our point of departure, that is, we start from Utr − V˜r+1,x + [U, V˜r+1 ] = 0 ,
(x, tr ) ∈ R2 ,
(4.8)
−Vn+1,x + [U, Vn+1 ] = 0 ,
(x, tr ) ∈ R2 ,
(4.9)
366
F. GESZTESY and R. RATNASEELAN
where (cf. (2.9)) −iz q(x, tr ) iz p(x, tr )
U (z, x, tr ) =
V˜r+1 (z, x, tr ) = i
Vn+1 (z, x, tr ) = i
Fn (z, x, tr ) =
! , !
˜ r+1 (z, x, tr ) −G ˜ r (z, x, tr ) −H
F˜r (z, x, tr ) ˜ r+1 (z, x, tr ) G
−Gn+1 (z, x, tr )
Fn (z, x, tr )
−Hn (z, x, tr )
Gn+1 (z, x, tr )
n X
,
(4.10)
! ,
f0 (x, tr ) = −iq(x, tr ) ,
fn−` (x, tr )z ` ,
`=0
Fn (z, x, t0,r ) =
Fn(0) (z, x)
=
n X
(4.11) (0) fn−` (x)z `
,
(0) f0 (x)
= −iq
(0)
(x) ,
`=0
Gn+1 (z, x, tr ) =
n+1 X
gn+1−` (x, tr )z ` ,
g0 (x, tr ) = 1 ,
`=0 (0)
Gn+1 (z, x, t0,r ) = Gn+1 (z, x) =
n+1 X
(4.12) (0)
gn+1−` (x)z j ,
(0)
g0 (x) = 1 ,
`=0
Hn (z, x, tr ) =
n X
hn−` (x, tr )z ` ,
h0 (x, tr ) = ip(x, tr ) ,
`=0
Hn (z, x, t0,r ) = Hn(0) (z, x) =
n X
(4.13) (0) hn−` (x)z `
,
(0) h0 (x)
= ip(0) (x)
`=0 (0)
for fixed t0,r ∈ R, n ∈ N0 , r ∈ N0 . Here f` (x, tr ) and f` (x) are defined as in (2.2) with (p(x), q(x)) replaced by (p(x, tr ), q(x, tr )) and (p(0) (x), q (0) (x)), respectively. Explicitly, (4.8) and (4.9) are equivalent to ˜ r,x − 2z H ˜ r + 2ipG ˜ r+1 , ptr = −iH
(4.14)
˜ r+1 , qtr = iF˜r,x − 2z F˜r − 2iq G
(4.15)
˜r ˜ r+1,x = pF˜r + q H G
(4.16)
and (cf. (2.10)–(2.12)) Fn,x = −2izFn + 2qGn+1 , Gn+1,x = pFn + qHn , Hn,x = 2izHn + 2pGn+1 ,
(4.17) (4.18) (4.19)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
367
respectively. In particular, (2.14) holds in the present tr -dependent setting, that is, G2n+1 − Fn Hn = R2n+2 .
(4.20)
In analogy to (3.11) and (3.12) we write Fn (z, x, tr ) = −iq(x, tr )
n Y
(z − µj (x, tr )) ,
(4.21)
j=1
Hn (z, x, tr ) = ip(x, tr )
n Y
(z − νj (x, tr ))
(4.22)
j=1
and define in analogy to (3.10) the following meromorphic function φ(P, x, tr ) on Kn , the fundamental ingredient for constructing algebro-geometric solutions of the time-dependent AKNS hierarchy, φ(P, x, tr ) =
−Hn (z, x, tr ) y(P ) + Gn+1 (z, x, tr ) = , Fn (z, x, tr ) y(P ) − Gn+1 (z, x, tr ) P = (z, y) ∈ Kn .
(4.23)
As in (3.13) and (3.14) one introduces µ ˆ j (x, tr ) = (µj (x, tr ), Gn+1 (µj (x, tr ), x, tr )) ∈ Kn , 1 ≤ j ≤ n,
(x, tr ) ∈ R2 ,
(4.24)
νˆj (x, tr ) = (νj (x, tr ), −Gn+1 (νj (x, tr ), x, tr )) ∈ Kn , 1 ≤ j ≤ n,
(x, tr ) ∈ R2 ,
(4.25)
and infers that the divisor (φ(P, x, tr )) of φ(P, x, tr ) is given by (φ(P, x, tr )) = Dνˆ(x,tr ) (P ) − Dµˆ (x,tr ) (P ) + D∞+ (P ) − D∞− (P ) . Next we define the time-dependent BA-function Ψ(P, x, x0 , tr , t0,r ) by ! ψ1 (P, x, x0 , tr , t0,r ) , Ψ(P, x, x0 , tr , t0,r ) = ψ2 (P, x, x0 , tr , t0,r ) Z
x
ψ1 (P, x, x0 , tr , t0,r ) = exp
(4.26)
(4.27)
dx0 [−iz + q(x0 , tr )φ(P, x0 , tr )]
x0
Z
tr
+i
) ˜ ˜ ds[Fr (z, x0 , s)φ(P, x0 , s) − Gr+1 (z, x0 , s)] , (4.28)
t0,r
ψ2 (P, x, x0 , tr , t0,r ) = φ(P, x, tr )ψ1 (P, x, x0 , t, t0,r ) , P ∈ Kn \{∞± } ,
(x, tr ) ∈ R2 ,
(4.29)
368
F. GESZTESY and R. RATNASEELAN
with fixed (x0 , t0,r ) ∈ R2 . The following Lemma records properties of φ(P, x, tr ) and Ψ(P, x, x0 , tr , t0,r ) in analogy to the stationary case discussed in Lemma 3.1. Lemma 4.1. Assume (4.14)–(4.20), P = (z, y) ∈ Kn \{∞± }, and let (z, x, x0 , tr , t0,r ) ∈ C × R4 . Then (i) φ(P, x, tr ) satisf ies φx (P, x, tr ) + q(x, tr )φ(P, x, tr )2 − 2izφ(P, x, tr ) = p(x, tr ) ,
(4.30)
˜ r+1 (z, x, tr )] , [q(x, tr )φ(P, x, tr )]tr = i∂x [F˜r (z, x, tr )φ(P, x, tr ) − G
(4.31)
˜ r+1 (z, x, tr )φ(P, x, tr ) + φtr (P, x, tr ) = 2iG
1 ˜ r+1,x (z, x, tr ) [−iG q(x, tr )
+iF˜r (z, x, tr )φx (P, x, tr ) + 2z F˜r (z, x, tr )φ(P, x, tr )] .
(4.32)
(ii) ψj (P, x, x0 , tr , t0,r ), j = 1, 2 satisf y ψ1,x (P, x, x0 , tr , t0,r ) = [q(x, tr )φ(P, x, tr ) − iz]ψ1 (P, x, x0 , tr , t0,r ) ,
(4.33)
˜ r+1 (z, x, tr )] ψ1,tr (P, x, x0 , tr , t0,r ) = i[F˜r (z, x, tr )φ(P, x, tr ) − G ×ψ1 (P, x, x0 , tr , t0,r ) ,
(4.34)
ψ2,x (P, x, x0 , tr , t0,r ) = [p(x, tr )φ(P, x, tr )−1 + iz]ψ1 (P, x, x0 , tr , t0,r ) , (4.35) ˜ r (z, x, tr )φ(P, x, tr )−1 − G ˜ r+1 (z, x, tr )] ψ2,tr (P, x, x0 , tr , t0,r ) = −i[H ×ψ2 (P, x, x0 , tr , t0,r ) ,
(4.36)
or equivalently, Ψx (P, x, x0 , tr , t0,r ) = U (z, x, tr )Ψ(P, x, x0 , tr , t0,r ) , iy(P )Ψ(P, x, x0 , tr , t0,r ) = Vn+1 (z, x, tr )Ψ(P, x, x0 , tr , t0,r ) ,
(4.37)
Ψtr (P, x, x0 , tr , t0,r ) = V˜r+1 (z, x, tr )Ψ(P, x, x0 , tr , t0,r ) ˜r+1 Ψ). (i.e. (D − z)Ψ = 0, (En+1 − iy)Ψ = 0, Ψtr = E (iii) φ(P, x, tr )φ(P ∗ , x, tr ) =
Hn (z, x, tr ) . Fn (z, x, tr )
(4.38)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
369
(iv) φ(P, x, tr ) + φ(P ∗ , x, tr ) =
2Gn+1 (z, x, tr ) . Fn (z, x, tr )
(4.39)
(v) φ(P, x, tr ) − φ(P ∗ , x, tr ) =
2y(P ) . Fn (z, x)
(4.40)
Proof. (4.30) follows from (4.9) and (4.23). (4.31) can be proven as follows. Using (4.3) and (4.30) one infers by a straight forward (but rather lengthy) calculation that qx ˜ r+1 )x ] = 0 . (4.41) [(qφ)t − i(F˜r φ − G ∂x + 2qφ − 2iz − q Thus ˜ r+1 )x = Ce (qφ)t − i(F˜r φ − G
Rx
dx0 [2iz+
q 0 x q
−2qφ]
,
(4.42)
where C is independent of x (but may depend on P and tr ). By inspection of (4.23), the left-hand side of (4.42) is meromorphic on Kn while the right-hand side of (4.42) is not meromorphic at ∞+ and ∞− unless C = 0. Hence one infers C = 0 and thus (4.31). (4.32) is then an immediate consequence of (4.3) (i.e., the AKNS equation for qtr ) and (4.31). (4.33) is clear from (4.28) and (4.35) is obvious from (4.29), (4.30), and (4.33). (4.34) follows from (4.28), and (4.36) is a straightforward consequence of (4.29), and (4.34). Finally, (iii)–(v) are proved as in Next we consider the tr -dependence of Fn (z, x, tr ), Gn+1 (z, x, tr ), and Hn (z, x, tr ). Lemma 4.2. Assume (4.14)–(4.20) and let (z, x, tr ) ∈ C × R2 . Then ˜ r+1 (z, x, tr )] . (i) Fn,tr (z, x, tr ) = 2i[F˜r (z, x, tr )Gn+1 (z, x, tr ) − Fn (z, x, tr )G (4.43) ˜ r (z, x, tr )] . (ii) Gn+1,tr (z, x, tr ) = i[F˜r (z, x, tr )Hn (z, x, tr ) − Fn (z, x, tr )H (4.44) ˜ r (z, x, tr )Gn+1 (z, x, tr ) − Hn (z, x, tr )G ˜ r+1 (z, x, tr )] . (iii) Hn,tr (z, x, tr ) = −2i[H (4.45) In particular, (i)–(iii) are equivalent to −Vn+1,tr + [V˜r+1 , Vn+1 ] = 0 .
(4.46)
Proof. By (4.23), (4.32), (4.39), and (4.40) one infers φtr (P ) − φtr (P ∗ ) = −
2y(P )Fn,tr 4iy(P ) ˜ = (Gr+1 Fn − F˜r Gn+1 ) , Fn2 Fn2
(4.47)
which proves (4.43). Similarly, differentiating (4.39) with respect to tr , using (4.30), (4.32), (4.38)–(4.40), and (4.16), proves (4.44). (4.45) finally follows from (G2n+1 − Fn Hn )tr = 0 (cf. (4.20)), (4.43), and (4.44).
370
F. GESZTESY and R. RATNASEELAN
The remaining items (vi)–(ix) of Lemma 3.1 in the present time-dependent setting then read Lemma 4.3. Assume (4.14)–(4.20), P = (z, y) ∈ Kn \{∞}, and let (z, x, x0 , tr , t0,r ) ∈ C × R4 . Then (i) ψ1 (P, x, x0 , tr , t0,r )ψ1 (P ∗ , x, x0 , tr , t0,r ) =
Fn (z, x, tr ) . Fn (z, x0 , t0,r )
(4.48)
(ii) ψ2 (P, x, x0 , tr , t0,r )ψ2 (P ∗ , x, x0 , tr , t0,r ) =
Hn (z, x, tr ) . Fn (z, x0 , t0,r )
(4.49)
1 Fn (z, x, tr ) 2 (iii) ψ1 (P, x, x0 , tr , t0,r ) = Fn (z, x0 , t0,r ) Z x × exp y(P ) dx0
x0
Z
tr
ds
+ t0,r
F˜r (z, x0 , s) Fn (z, x0 , s)
q(x0 , tr ) Fn (z, x0 , tr ) !) .
(4.50)
(iv) ψ1 (P, x, x0 , tr , t0,r )ψ2 (P ∗ , x, x0 , tr , t0,r ) + ψ1 (P ∗ , x, x0 , tr , t0,r )ψ2 (P, x, x0 , tr , t0,r ) =
2Gn+1 (z, x, tr ) . Fn (z, x0 , t0,r )
(4.51)
Proof. (4.48) follows from (4.17), (4.28), (4.39), and (4.43). (4.49) is clear from (4.29), (4.38), and (4.48). (4.50) follows from (4.17), (4.28), (4.39), (4.40), and (4.43) by splitting φ(P ) = 12 [φ(P ) + φ(P ∗ )] + 12 [φ(P ) − φ(P ∗ )] in (4.28). Finally, (4.51) is clear from (4.29), (4.39), and (4.48). The dynamics of the zeros µj (x, tr ) and νj (x, tr ) of Fn (z, x, tr ) and Hn (z, x, tr ), in analogy to Lemma 3.2, is then described in Lemma 4.4. Assume (4.14)–(4.22) and let (x, tr ) ∈ R2 . Then (i)
−2iy(ˆ µj (x, tr )) , (µ (x, tr ) − µk (x, tr )) j k=1
µj,x (x, tr ) = Qn
1 ≤ j ≤ n,
(4.52)
k6=j
µj,tr (x, tr ) =
2F˜r (µj (x, tr ), x, tr )y(ˆ µj (x, tr )) Qn , q(x, tr ) k=1 (µj (x, tr ) − µk (x, tr )) k6=j
1 ≤ j ≤ n.
(4.53)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
(ii)
−2iy(ˆ νj (x, tr )) , k=1 (νj (x, tr ) − νk (x, tr ))
νj,x (x, tr ) = Qn
...
1 ≤ j ≤ n,
371
(4.54)
k6=j
νj,tr (x, tr ) =
˜ r (νj (x, tr ), x, tr ))y(ˆ −2H νj (x, tr )) Qn , p(x, tr ) k=1 (νj (x, tr ) − νk (x, tr ))
1 ≤ j ≤ n.
(4.55)
k6=j
Proof. (4.52) and (4.54) are proved as in Lemma 3.2 and follow from (4.21), (4.22), (4.17), (4.24), and (4.25). Similarly, (4.53) and (4.55) follow from (4.21), (4.24), (4.43) and (4.22), (4.25), (4.45), respectively. The initial condition (p(x, t0,r ), q(x, t0,r )) = (p(0) (x), q (0) (x)), x ∈ R
(4.56)
in (4.3) is taken care of by (0)
ˆj (x) , µ ˆ j (x, t0,r ) = µ
(0)
νˆj (x, t0,r ) = νˆj (x) ,
1 ≤ j ≤ n,
x∈R
(4.57)
(cf. (4.11), (4.13) and (4.21), (4.22)). The trace relations in Lemma 3.3 extend in a one-to-one manner to the present time-dependent setting by substituting (p(x), q(x)) → (p(x, tr ), q(x, tr )) , (4.58) (µj (x), νj (x)) → (µj (x, tr ), νj (x, tr )) ,
1 ≤ j ≤ n,
keeping {c` }1≤`≤n as in (3.32) since Kn is tr -dependent. It remains to provide the explicit theta function representation of Ψ, φ, p, and q. We rely on the notation established in Sec. 3 and Appendix A in the following, assuming Kn to be nonsingular as in (3.33). As in Sec. 3 we assume n ∈ N for the remainder of this argument. In addition to (3.34)–(3.42) we need to introduce the Abelian differentials of the (2) second kind ω∞± ,r (cf. (A.34), (A.35)) defined by (2) = [ζ −2−r + O(1)] dζ near ∞± , ω∞ ± ,r
r ∈ N0 ,
ζ→0
(4.59)
Z aj
(2) ω∞ = 0, ± ,r
1 ≤ j ≤ n,
˜ (2) , . . . , U ˜ (2) ), U ˜ (2) = 1 ˜ (2) = (U U r r,n r,1 r,j 2πi ˜ (2) = Ω r
r X q=0
(4.60) Z ˜ (2) , Ω r bj
(2) (2) (q + 1)˜ cr−q (ω∞ − ω∞ ), + ,q − ,q
(4.61)
372
F. GESZTESY and R. RATNASEELAN
Z
P
˜ (2) = ∓ Ω r ζ→0
P0
" r X
# c˜r−q ζ −1−q + e˜r,0 + O(ζ) ,
P = (ζ −1 , y) near ∞± ,
q=0
(4.62) with {˜ c` }1≤`≤r , c˜0 = 1 the integration constants in F˜r . Moreover, writing ! ! ∞ ∞ X X m m dj,m (∞± )ζ dj,m (∞+ )ζ dζ = ± dζ near ∞± , ωj = m=0
(4.63)
m=0
relation (A.35) yields ˜ (2) = 2 U r,j
r X
c˜r−q dj,q (∞+ ) ,
1 ≤ j ≤ n.
(4.64)
q=0
Before we can prove the main result of this section we need the following auxiliary result which is of independent interest due to its implications for the Green’s matrix of the differential expression D. Lemma 4.5. Let P ∈ Kn \{∞+ , ∞− } and (x, tr ) ∈ R2 . Then, for P = (ζ −1 , y) near ∞± , one obtains the asymptotic expansions ∞
X Fn (ζ −1 , x, tr ) = ∓ζ fˆk (x, tr )ζ k , ζ→0 y(P )
fˆ0 (x, tr ) = −iq(x, tr ) ,
(4.65)
ˆ 0 (x, tr ) = ip(x, tr ) , h
(4.66)
k=0 ∞
X Hn (ζ −1 , x, tr ) ˆ k (x, tr )ζ k , = ∓ζ h ζ→0 y(P ) k=0
hk (x, tr ) denote the homogeneous coefficients fk (x, tr ) and where fˆk (x, tr ) and ˆ hk (x, tr ) in (4.11) and (4.13) (i.e., the ones satisfying (2.2) with all integration ˆ k (x, tr ) can be computed from constants c` = 0, ` ∈ N). Explicitly, fˆk (x, tr ) and h the recursion relations, 1 fˆ0 = −iq , fˆ1 = qx , 2 k−2 X i iqx ˆ ˆ i ˆ ˆ ip ˆ ˆ ˆ ˆ ˆ − f` fk−2−`,xx + 2 f` fk−2−`,x + f`,x fk−2−`,x + f` fk−2−` fk = 4q 4q 8q 2 `=0
k−1 k−1 i Xˆˆ qx X ˆ ˆ f` fk−1−` − f` fk−` , − 2 2q 2q `=0
k≥2
(4.67)
`=1
and ˆ 0 = ip , h ˆk = h
k−2 X `=0
−
ˆ 1 = 1 px , h 2 i ˆ ˆ ipx ˆ ˆ i ˆ ˆ iq ˆ ˆ h` hk−2−`,xx − 2 h h`,x hk−2−`,x − h ` hk−2−`,x − ` hk−2−` 4p 4p 8p 2
k−1 k−1 i Xˆ ˆ px X ˆ ˆ + h h h` hk−` , ` k−1−` 2p2 2p `=0
`=1
k ≥ 2.
(4.68)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
Proof. Define
then
...
Fn (ζ −1 , x, tr ) Fˆ (P, x, tr ) = y(P )
373
(4.69)
qx 1 qx Fˆ Fˆxx − Fˆ Fˆx − Fˆx2 + 2 ζ −2 − iζ −1 − pq Fˆ 2 = −2q 2 q 2 q
(4.70)
by (2.17). Moreover, Pn ` `=0 f` (x, tr )ζ ˆ F (P, x, tr ) = ∓ζ Q2n+1 1 , [ m=0 (1 − Em )ζ] 2 = ∓ζ
ζ→0
∞ X
P ∈ Π±
fˆk (x, tr )ζ k near ∞± ,
(4.71)
k=0
f0 (x, tr ) = fˆ0 (x, tr ) = −iq(x, tr ) , where we chose the branch # 12 "2n+1 Y 1 (1 − Em ζ) = 1− ζ→0 2 m=0
2n+1 X
(4.72) !
Em
ζ + O(ζ 2 ) .
(4.73)
m=0
Insertion of (4.72) into (4.70) yields the recursion relation (4.67) which represents the homogeneous solutions for the f` due to the lack of any possible integration constants in (4.67). (4.66) and (4.68) are proved in the same manner using (2.19). The recursion technique in Lemma 4.5 represents the AKNS analog of the recursive KdV approach in Secs. 2–4 of [46]. Lemma 4.5 has interesting consequences for the asymptotic high-energy expansion of the Green’s matrix G(z, x, x0 ) of D (i.e., the integral kernel of the resolvent (D − z)−1 ) as described in the following remark. Remark 4.6. Let D be given by (2.1) (p, q not necessarily algebro-geometric coefficients) and assume that for some z ∈ C, and all x0 ∈ R, ψ1,± (z, .), ψ2,± (z, .) ∈ L2 ((x0 , ±∞)) satisfy (2.22), that is, ψ1,±,x = −izψ1,± + qψ2,± , ψ2,±,x = izψ2,± + pψ1,± . Then the Green’s matrix G(z, x, x0 ), x 6= x0 of ψ1,+ (z, x)ψ2,− (z, x0 ) 0 i ψ2,+ (z, x)ψ2,− (z, x ) 0 G(z, x, x ) = W (z) ψ1,− (z, x)ψ2,+ (z, x0 ) ψ2,− (z, x)ψ2,+ (z, x0 )
(4.74)
D is given by ψ1,+ (z, x)ψ1,− (z, x0 ) ψ2,+ (z, x)ψ1,− (z, x0 ) ψ1,− (z, x)ψ1,+ (z, x0 ) ψ2,− (z, x)ψ1,+ (z, x0 )
, x > x0 , , x < x0 , (4.75)
374
F. GESZTESY and R. RATNASEELAN
where the Wronskian ψ (z, x) ψ (z, x) 1,+ 1,− W (z) := = ψ1,− (z, x)ψ2,+ (z, x) − ψ2,− (z, x)ψ1,+ (z, x) ψ2,− (z, x) ψ2,+ (z, x) (4.76) is x-independent. Note that G(z, x, x0 ) is continuous at x = x0 in its off-diagonal elements but discontinuous on the diagonal. In the special algebro-geometric context we may replace ψj,+ (z, x) by ψj (P, x, x0 ), ψj,− (z, x) by ψj (P ∗ , x, x0 ), j = 1, 2, P = (z, y) (4.77) and (cf. (3.7), (3.8), and (3.21)) 1 1 2y(P ) (4.78) W (z) by W (P ) = = φ(P, x0 )−φ(P ∗ , x0 ) = ∗ φ(P , x0 ) φ(P, x0 ) Fn (z, x0 ) since W is x-independent. Substituting (4.77) and (4.78) into (4.75), denoting the result by G(P, x, x0 ), then yields 1 1 [G(P, x, x + 0) + G(P, x, x − 0)] = [G(P, x − 0, x) + G(P, x + 0, x)] 2 2 Gn+1 (z, x) Fn (z, x) i = 2y(P ) Hn (z, x) Gn+1 (z, x) ˆ i G(P, x) Fˆ (P, x) , P = (z, y) , = ˆ ˆ 2 H(P, x) G(P, x) (4.79) where (cf. (4.69)) Fn (z, x) , Fˆ (P, x) = y(P )
Gn+1 (z, x) ˆ G(P, x) = , y(P )
Hn (z, x) ˆ H(P, x) = y(P )
(4.80)
denote homogeneous quantities encountered in Lemma 4.5. Since G(P, x, x0 ) is discontinuous at x = x0 , we introduced the arithmetic mean of the corresponding one-sided limits following the usual treatment of first-order systems (see, e.g., [4, Sec. 9.4]). In fact, the arithmetic mean in (4.79) leads to the characteristic function of D (in the terminology of [4, Sec. 9.5]), the fundamental object for studying spectral properties of D. The asymptotic expansions (4.65) and (4.66) for Fˆ (P, x) and ˆ H(P, x) as P → ∞± then determine the off-diagonal asymptotic high-energy expansions of the arithmetic mean of the diagonal Green’s matrix in (4.79). Similarly, using (2.10) or (2.12), ˆ G(P, x) =
1 1 ˆ ˆ x(P, x)] (4.81) [Fˆ (P, x) + 2iz Fˆx(P, x)] = [H(P, x) − 2iz H 2q(x) 2p(x)
= ∓1 + O(ζ −2 ) near ∞± ,
ζ→0
(4.82)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
375
one obtains the asymptotic high-energy expansion of (4.79) as P → ∞± for its diagonal elements. Even though (4.65), (4.66), (4.79), and (4.81) were derived in the special algebro-geometric context, we emphasize, however, that the asymptotic expansion of (4.79) as P → ∞± only involves the homogeneous coefficients fˆk (x), ˆ k (x) which are universal differential polynomials in (p(x), q(x)). Thus, identifying h Ψ± (z, x) and Ψ(P, x, x0 ), Ψ(P ∗ , x, x0 ) as in (4.77) yields the universal high-energy expansion of the arithmetic mean of the diagonal Greens matrix G(z, x, x0 ) of D as z → ∞ in the general (not necessarily algebro-geometric) case. The recursive and hence systematic approach to this high-energy expansion, based on (4.65)–(4.68), appears to be new. The theta function representations for φ, Ψ, and (p, q) then finally read as follows. Theorem 4.7. Let P ∈ Kn \{∞+ , ∞− }, (x, x0 , tr , t0,r ) ∈ R4 , and assume Kn to be nonsingular, that is, Em 6= Em0 for 0 ≤ m, m0 ≤ 2 n + 1. Moreover, suppose Dµˆ(x,t) , or equivalently, Dνˆ(x,t) to be nonspecial, that is, i(Dµˆ(x,t) ) = i(Dνˆ(x,t) ) = 0. Then φ(P, x, tr ) =
θ(z − (ˆ µ(x0 , t0,r ))) θ(z + (ˆ µ(x, tr ))) 2i q(x0 , t0,r )ω0 θ(z + (ˆ µ(x0 , t0,r ))) θ(z − (ˆ ν (x, tr ))) ×
θ(z(P, νˆ(x, tr ))) θ(z(P, µ ˆ(x, tr ))) "Z
#
P
× exp P0
ψ1 (P, x, x0 , tr , t0,r ) =
(3) ω∞ − 2i(x − x0 )e0 − 2i(tr − t0,r )˜ er , (4.83) + ,∞−
θ(z + (ˆ µ(x0 , t0,r ))) θ(z(P, µ ˆ (x, tr ))) θ(z(P, µ ˆ(x0 , t0,r ))) θ(z + (ˆ µ(x, tr ))) ! " Z P
× exp i(x − x0 ) e0 +
(2)
Ω0 P0
Z
!#
P
˜ (2) Ω r
+i(tr − t0,r ) e˜r +
,
(4.84)
P0
ψ2 (P, x, x0 , tr , t0,r ) =
θ(z − (ˆ µ(x0 , t0,r ))) θ(z(P, νˆ(x, tr ))) 2i q(x0 , t0,r )ω0 θ(z(P, µ ˆ(x0 , t0,r ))) θ(z − (ˆ ν (x, tr ))) ! "Z Z P
× exp P0
P
(3) ω∞ + i(x − x0 ) −e0 + + ,∞−
Z
!#
P
˜ (2) Ω r
er + +i(tr − t0,r ) −˜ P0
(2)
Ω0 P0
.
(4.85)
376
F. GESZTESY and R. RATNASEELAN
Moreover, one derives p(x, tr ) = p(x0 , t0,r )
θ(z − (ˆ ν (x0 , t0,r ))) θ(z + (ˆ ν (x, tr ))) θ(z + (ˆ ν (x0 , t0,r ))) θ(z − (ˆ ν (x, tr )))
er ] , × exp[−2i(x − x0 )e0 − 2i(tr − t0,r )˜ q(x, tr ) = q(x0 , t0,r )
p(x0 , t0,r )q(x0 , t0,r ) =
(4.86)
θ(z + (ˆ µ(x0 , t0,r ))) θ(z − (ˆ µ(x, tr ))) θ(z − (ˆ µ(x0 , t0,r ))) θ(z + (ˆ µ(x, tr )))
er ] , × exp[2i(x − x0 )e0 + 2i(tr − t0,r )˜
(4.87)
µ(x0 , t0,r ))) ν (x0 , t0,r ))) θ(z − (ˆ 4 θ(z + (ˆ , ω02 θ(z − (ˆ ν (x0 , t0,r ))) θ(z + (ˆ µ(x0 , t0,r )))
(4.88)
and (2) ˜ (2) , αP0 (Dµˆ (x,tr ) ) = αP0 (Dµˆ(x0 ,t0,r ) ) − i(x − x0 )U 0 − i(tr − t0,r )U r
(4.89)
(2) ˜ (2) αP0 (Dνˆ(x,tr ) ) = αP0 (Dνˆ(x0 ,t0,r ) ) − i(x − x0 )U 0 − i(tr − t0,r )U r .
(4.90)
Proof. We first prove the θ-function representation (4.84) for ψ1 . Without loss of generality it suffices to treat the homogeneous case cˆ0 = 1, cˆq = 0, 1 ≤ q ≤ r. Define the left-hand side of (4.84) to be ψ˜1 (P, x, x0 , tr , t0,r ); we need to prove ψ1 = ψ˜1 with ψ1 given by (4.28). For that purpose we first investigate the local zeros and poles of ψ1 . Since they can only come from zeros of Fn (z, x0 , s), Fn (z, x0 , tr ) in (4.28), we note that q(x0 , tr )φ(P, x0 , tr )
=
P →ˆ µj (x0 ,tr )
q(x0 , tr )
2y(ˆ µj (x0 , tr )) Q n −iq(x0 , tr ) k=1 (µj (x0 , tr ) − µk (x0 , tr )) k6=j
× =
−µj,x0 (x0 , tr ) + O(1) , z − µj (x0 , tr )
=
2iFˆr (z, x0 , s)y(ˆ µj (x0 , s)) Qn −iq(x0 , s) k=1 (µj (x0 , s) − µk (x0 , s))
P →ˆ µj (x0 ,tr )
iFˆr (z, x0 , s)φ(P, x0 , s)
1 + O(1) z − µj (x0 , tr )
P →ˆ µj (x0 ,s)
(4.91)
k6=j
× =
P →ˆ µj (x0 ,s)
1 + O(1) z − µj (x0 , s)
−µj,s (x0 , s) + O(1) z − µj (x0 , s)
(4.92)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
377
using (4.23), (4.24), (4.52), and (4.53). Thus for P near µ ˆj (x, tr ) 6= µ ˆj (x0 , t0,r ) , (z − µj (x, tr ))O(1) ˆj (x0 , t0,r ) , for P near µ ˆj (x, tr ) = µ ψ1 (P, x, x0 , tr , t0,r ) = O(1) (z − µj (x0 , tr ))−1 O(1) for P near µ ˆj (x0 , t0,r ) 6= µ ˆj (x, tr ) , (4.93) ˜ with O(1) 6= 0 and hence ψ1 and ψ1 have identical zeros and poles on Kn \{∞+ , ∞− } which are all simple. It remains to study the behavior of ψ1 near ∞± . One infers from (4.21)–(4.23) that i 2 2 p(x, tr )ζ + O(ζ ) , P near ∞+ , (4.94) φ(P, x, tr ) = ζ→0 2i −1 ζ + O(1) , P near ∞ . − q(x, t ) r Thus (4.23), (4.43), (4.94), and Lemma 4.5 yield Z x dx0 [−iζ −1 + q(x, tr )φ(P, x, tr )] x0
Z
tr
+
˜ r+1 (ζ −1 , x0 , s)] ds[F˜r (ζ −1 , x0 , s)φ(P, x0 , s) − G
t0,r
(
" ∓i(x − x0 )ζ
=
ζ→0
Z
−1
+
O(ζ),
P → ∞+
O(1),
P → ∞−
)#
"
iF˜r (ζ −1 , x0 , s)y(P ) 1 Fn,tr (ζ −1 , x0 , s) + ds + Fn (ζ −1 , x0 , s) 2 Fn (ζ −1 , x0 , s) t0,r tr
(
" ∓i(x − x0 )ζ
=
ζ→0
−1
+
O(ζ),
P → ∞+
O(1),
P → ∞−
#
)#
# ˜` (x0 , s)ζ ` f q 1 (x , s) t 0 r + O(ζ) ds ∓iζ −r−1 P`=0 + + ∞ ˜ ` 2 q(x0 , s) t0,r `=0 f` (x0 , s)ζ Z
tr
"
Pr
(
" =
ζ→0
∓i(x − x0 )ζ Z
tr
+
−1
+
" ds ∓iζ
t0,r
−r−1
O(ζ), O(1),
P → ∞+ P → ∞−
# if˜r+1 (x0 , s) 1 qtr (x0 , s) + O(ζ) + ± 2 q(x0 , s) f˜0 (x0 , s) (
= ∓i(x − x0 )ζ −1 ∓ i(tr − t0,r )ζ −r−1 +
ζ→0
where we used f˜0 = −iq and
)#
qtr = 2f˜r+1
O(ζ),
P → ∞+ ,
O(1),
P → ∞− ,
(4.95)
(4.96)
378
F. GESZTESY and R. RATNASEELAN
(cf. (4.3)) in the homogeneous case c˜0 = 1, c˜q = 0, 1 ≤ q ≤ r. (4.95) yields the correct essential singularity structure of ψ˜1 near ∞± . Moreover, (3.42), (4.62), and the O(ζ)-term in (4.95) as P → ∞+ also prove that ψ1 and ψ˜1 are identically analogR of Lemma 3.4 normalized (near ∞+ )Rand hence coincide by the R P t-dependent P P ˜ (2) (2) (2) (replacing −i(x − x0 ) P0 Ω0 by −i(x − x0 ) P0 Ω0 − i(tr − t0,r ) P0 Ω r ). This proves (4.84). The expression (4.26) for the divisor of φ then yields φ(P, x, tr ) = C(x, tr )
θ(z(P, νˆ(x, tr ))) e θ(z(P, µ ˆ(x, tr )))
RP P0
(3) ω∞
+ ,∞−
,
(4.97)
where C(x, tr ) is independent of P ∈ Kn . Thus (4.94) implies p(x, tr ) =
ν (x, tr ))) 2C(x, tr ) θ(z + (ˆ , iω0 θ(z + (ˆ µ(x, tr )))
(4.98)
q(x, tr ) =
θ(z − (ˆ µ(x, tr ))) 2i . C(x, tr )ω0 θ(z − (ˆ ν (x, tr )))
(4.99)
Re-examining the asymptotic behavior (4.95) of ψ1 near ∞− , taking into account (3.62), yields ψ1 (P, x, x0 , tr , t0,r ) =
ζ→0
q(x, tr ) exp[i(x − x0 )ζ −1 + O(ζ)] q(x0 , tr ) ×
=
ζ→0
q(x0 , tr ) exp[i(tr − t0,r )ζ −1−r + O(ζ)] q(x0 , t0,r )
q(x, tr ) exp[i(x − x0 )ζ −1 q(x0 , t0,r ) +i(tr − t0,r )ζ −1−r + O(ζ)] for P near ∞− .
(4.100)
A comparison of (4.84), (4.99), and (4.100) then proves (4.87). A further comparison of (4.87) and (4.99) then determines C(x, tr ) and hence yields (4.86) and (4.88). Given C(x, tr ) one determines φ in (4.83) from (4.97) and hence ψ2 in (4.85) from ψ2 = φψ1 . Finally, the linearization property of the Abel map in (4.89) and (4.90) can be proved directly using Lagrange interpolation as in the proof of Theorem 3.5. However, for increasing values of r this method becomes exceedingly cumbersome and it is simpler to resort to a standard investigation (cf., e.g., [43, pp. 141–144]) of the differential Ω1 (x, x0 , tr , t0,r ) = d ln ψ1 (., x, x0 , tr , t0,r ) (respectively, Ω2 (x, x0 , tr , t0,r ) = d ln ψ2 (., x, x0 , tr , t0,r ) in order to prove (4.89) (respectively, (4.90)). Since Corollary 3.6 extends to the present time-dependent setting in a straightforward manner we record the corresponding result without proof, p(x, tr )q(x, tr ) = −e0,1 −
d2 ln(θ(z + (ˆ µ(x, tr )))) . dx2
(4.101)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
379
The open constant q(x0 , t0,r ) in (4.86)–(4.88) is inherent to the AKNS formalism as discussed in Lemma 3.7. In analogy to Example 3.8, the special case n = 0 (excluded in Theorem 4.7) yields solutions (p(x, tr ), q(x, tr )) as in (4.86), (4.87) replacing the theta quotients by 1. We note again that the results for Ψ and (p, q) in Theorem 4.7 are known and can be found, for instance, in [5, Chap. 4] [14, 15, 30, 31], and [44]. Our main new contribution to this circle of ideas is the elementary alternative derivation of Theorem 4.7 based on the fundamental meromorphic function φ on Kn and its connection with the polynomial recursion formalism of Sec. 2. Appendix A. Hyperelliptic Curves and Theta Functions We briefly summarize our notation and some of the basic facts on hyperelliptic curves and their theta functions as employed in Secs. 3 and 4. For background information on this standard material we refer, for instance, to [18, Chaps. I–III, IV], [19] [29, Chap. 2] [34, Chap. X] [40, Chap. 2]. Consider the points {Em }0≤m≤2n+1 ⊂ C, n ∈ N0
(A.1)
and introduce an appropriate set of n + 1 (nonintersecting) cuts Cj joining Ej and Ek(j) , where Ek(j) = Ej for some j is permitted in order to include singular curves. Denote [ Cj , Cj ∩ Ck = ∅ for j 6= k , (A.2) C= j∈J
where the finite index set J ⊂ {0, 1, . . . , 2n + 1} has cardinality n + 1 and define the cut plane Π, Π := C \ C . (A.3) Next, introduce the holomorphic function ( Π→C 1/2 : R2n+2 (.) Q2n+1 z 7→ [ m=0 (z − Em )]1/2
(A.4)
on Π with a definite choice of the square root branch (A.4). Given the holomorphic function (A.4) one defines the set Mn = {(z, σR2n+2 (z)1/2 ) | z ∈ C, σ ∈ {+, −}} ∪ {∞+ , ∞− }
(A.5)
Bs = {(Em , 0)}0≤m≤2n+1 ,
(A.6)
and the set of branch and/or singular points. Mn becomes a Riemann surface upon introducing appropriate charts (UP0 , ζP0 ) defined in a standard manner. Let P0 = (z0 , σ0 R2n+2 (z0 )1/2 ) or P0 = ∞± , P = (z, σR2n+2 (z)1/2 ) ∈ UP0 ⊂ Mn , VP0 = ζP0 (UP0 ) ⊂ C .
(A.7)
380
F. GESZTESY and R. RATNASEELAN
P0 ∈ / {Bs ∪ {∞+ , ∞− }}: UP0 = {P ∈ Mn |z − z0 | < Cz0 , σR2n+2 (z)1/2 the branch obtained by straight line analytic continuation starting from z0 } , VP0
Cz0 = min|z0 − Em | ,
= {ζ ∈ C |ζ| < Cz0 } , ( ζP0 :
(
UP0 → VP0 P 7→ (z − z0 ) ,
ζP−1 0
:
m
VP0 → UP0 ζ 7→ (z0 + ζ, σR2n+2 (z0 + ζ)1/2 ) .
(A.8)
P0 = ∞± : −1 UP0 = {P ∈ Mn |z| > C∞ }, C∞ = max|Em |, VP0 = {ζ ∈ C |ζ| < C∞ }, m
ζP0
U → VP0 P0 : P 7→ z −1 ∞ 7→ 0 ,
ζP−1 0
±
[Πm (1 − Em ζ)]
1/2
V → UP0 P0 : ζ 7→ (ζ −1 , ∓[Πm (1 − Em ζ)]1/2 ζ −n−1 ) 0 7→ ∞ ,
(A.9)
±
1 =1− 2
X
! Em ζ + O(ζ 2 ) .
m
Similarly, local coordinates for branch and/or singular points P0 ∈ Bs are defined as ζP0 (P ) = (z − z0 )r/2 for appropriate r = 1 or 2. For the reader’s convenience we provide a detailed treatment of branch points in the nonsingular case (where Em 6= Em0 for m 6= m0 ) for the two most frequently occurring situations, the self-adjoint case where {Em }0≤m≤2n+1 ⊂ R and the case where {Em }0≤m≤2n+1 = {` , ` }0≤`≤n consists of complex conjugate pairs at the end of this appendix. In addition, it is useful to consider the subsets Π± ⊂ Mn (i.e., upper and lower sheets) Π± = {(z, ±R2n+2 (z)1/2 ) ∈ Mn | z ∈ Π} and the associated charts
( ζ± :
Π± → Π P 7→ z
.
(A.10)
(A.11)
(A.8), (A.9), and the corresponding charts for P0 ∈ Bs define a complex structure on Mn . We shall denote the resulting Riemann surface by Kn . In general, Kn is a (possibly singular) curve of (arithmetic) genus n. Next, consider the holomorphic sheet exchange map (involution) Kn → Kn ∗ : (z, σR2n+2 (z)1/2 ) 7→ (z, σR2n+2 (z)1/2 )∗ = (z, −σR2n+2 (z)1/2 ) ∞ 7→ ∞∗ = ∞ ±
±
∓
(A.12)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
381
and the two meromorphic projection maps Kn → C ∪ {∞} Kn → C ∪ {∞} 1/2 (z, σR2n+2 (z)1/2 ) 7→ z , R2n+2 : (z, σR2n+2 (z)1/2 ) 7→ σR2n+2 (z)1/2 , π ˜: ∞ 7→ ∞ . ∞ 7→ ∞ , ± ± (A.13) π ˜ has poles of order 1 at ∞± and R2n+2 (z)1/2 has poles of order n + 1 at ∞± . Moreover, 1/2 1/2 ˜ (P ), R2n+2 (P ∗ ) = −R2n+2 (P ) , P ∈ Kn . (A.14) π ˜ (P ∗ ) = π Thus Kn is a two-sheeted ramified covering of the Riemann sphere C∞ (∼ = C ∪ {∞}), ˜ is open and C∞ is compact), and Kn is hyperelliptic (since Kn is compact (since π it admits the meromorphic function π ˜ of degree two). In the following we abbreviate P = (z, y), P ∈ Kn \{∞+ , ∞− } ,
(A.15)
1/2
(i.e. we define y(P ) = R2n+2 (P ), see (A.13)). Next we turn to nonsingular curves Kn where Em 6= Em0 , for m 6= m0 , 0 ≤ m, m0 ≤ 2n + 1 .
(A.16)
One infers that for n ∈ N, d˜ π /y is a holomorphic differential on Kn with zeros of order n − 1 at ∞± and hence ηj =
π ˜ j−1 d˜ π , y
1≤j≤n
(A.17)
form a basis for the space of holomorphic differentials on Kn . Next we introduce a canonical homology basis {aj , bj }1≤j≤n for Kn where the cycles are chosen such that their intersection matrix reads aj ◦ bk = δj,k ,
1 ≤ j, k ≤ n .
(A.18)
Introducing the invertible matrix C in Cn , Z C = (Cj,k )1≤j,k≤n , Cj,k =
ηj , ak
(A.19) c(k) = (c1 (k), . . . , cn (k)), cj (k) = (C −1 )j,k , the normalized differentials ωj , 1 ≤ j ≤ n, ωj =
n X `=1
Z cj (`)η` ,
ωj = δj,k , ak
1 ≤ j, k ≤ n
(A.20)
382
F. GESZTESY and R. RATNASEELAN
form a canonical basis for the space of holomorphic differentials on Kn . The matrix τ in Cn of b-periods, Z ωj (A.21) τ = (τj,k )1≤j,k≤n , τj,k = bk
satisfies τj,k = τk,j , Im(τ ) =
1 ≤ j, k ≤ n ,
(A.22)
1 (τ − τ ∗ ) > 0 . 2i
(A.23)
π near ∞± one infers In the charts (U∞± , ζ∞± ≡ ζ) induced by 1/˜ ω=±
n X j=1
(
c(j)
ζ n−j dζ [Πm (1 − Em ζ)]1/2 "
# ) 2n+1 X 1 2 c(n) = ± c(n) + Em + c(n − 1) ζ + O(ζ ) dζ . 2 m=0
(A.24)
Associated with the homology basis {aj , bj }1≤j≤n we also recall the canonical ˆ n of the dissection of Kn along its cycles yielding the simply connected interior K ˆ n given by fundamental polygon ∂ K ˆ n = a1 b1 a−1 b−1 a2 b2 a−1 b−1 · · · a−1 b−1 . ∂K n n 1 1 2 2
(A.25)
The Riemann theta function associated with Kn is defined by X exp[2πi(n, z) + πi(n, τ n)], z = (z1 , . . . , zn ) ∈ Cn , θ(z) =
(A.26)
n∈Zn
where (u, v) = properties
Pn j=1
uj vj denotes the scalar product in Cn . It has the fundamental
θ(z1 , . . . , zj−1 , −zj , zj+1 , . . . , zn ) = θ(z) , θ(z + m + τ n) = exp[−2πi(n, z) − πi(n, τ n)]θ(z) ,
m, n ∈ Zn .
(A.27)
A divisor D on Kn is a map D : Kn → Z, where D(P ) 6= 0 for only finitely many P ∈ Kn . The set of all divisors on Kn will be denoted by Div(Kn ). With Ln we denote the period lattice Ln := {z ∈ Cn | z = m + τ n, m, n ∈ Zn }
(A.28)
and the Jacobi variety J(Kn ) is defined by J(Kn ) = Cn /Ln .
(A.29)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
The Abel maps AP0 (.) respectively αP0 (.) are defined by K → J(Kn ) n Z P AP0 : (P ) = ω mod (Ln ) , P → 7 A P0
...
383
(A.30)
P0
αP0
Div(Kn ) → J(Kn ) X : D(P )AP0 (P ) , D 7→ αP0 (D) =
(A.31)
P ∈Kn
with P0 ∈ Kn a fixed base point. (In the main text we agree to fix P0 = (E0 , 0) for convenience.) In connection with (A.25) we shall also need the maps (cf. (3.34)) n ˆ Div(Kn ) → Cn K → C n X Z P α ˆ P0 : (A.32) AˆP0 : D 7 → D(P )AˆP0 (P ) , ω, P 7→ ˆ P ∈K n
P0
ˆn. with path of integration lying in K 1 Let M(Kn ) and M (Kn ) denote the set of meromorphic functions (0-forms) and meromorphic differentials (1-forms) on Kn . The residue of a meromorphic differential ν ∈ M1 (Kn ) at a point Q0 ∈ Kn is defined by Z 1 ν, (A.33) res(ν) = Q0 2πi γQ0 where γQ0 is a counterclockwise oriented smooth simple closed contour encircling Q0 but no other pole of ν. Holomorphic differentials are also called Abelian differentials of the first kind (dfk). Abelian differentials of the second kind (dsk) ω (2) ∈ M1 (Kn ) are characterized by the property that all their residues vanish. They are normalized, for instance, by demanding that all their a-periods vanish, that is, Z ω (2) = 0 ,
1 ≤ j ≤ n.
(A.34)
aj (2) ˆ n with principal part ζ −n−2 dζ, If ωP1 ,n is a dsk on Kn whose only pole is P1 ∈ K ∞ m n ∈ N0 near P1 and ωj = (Σm=0 dj,m (P1 )ζ ) dζ near P1 , then Z 2πi (2) dj,n (P1 ) . ωP1 ,n = (A.35) n+1 bj
Any meromorphic differential ω (3) on Kn not of the first or second kind is said to be of the third kind (dtk). A dtk ω (3) ∈ M1 (Kn ) is usually normalized by the vanishing of its a-periods, that is,
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Z ω (3) = 0 ,
1 ≤ j ≤ n.
(A.36)
aj (3) ˆ n , P1 6= P2 by definition A normal dtk ωP1 ,P2 associated with two points P1 , P2 ∈ K has simple poles at P1 and P2 with residues +1 at P1 and −1 at P2 and vanishing (3) ˆ n , holomorphic on a-periods. If ωP,Q is a normal dtk associated with P , Q ∈ K Kn \{P, Q}, then Z P Z (3) ωP,Q = 2πi ωj , 1 ≤ j ≤ n , (A.37) bj
Q
ˆ n (i.e. does not touch any of the cycles aj , bj ). where the path from Q to P lies in K We shall always assume (without loss of generality) that all poles of dsk’s and ˆ n (i.e., not on ∂ K ˆ n ). dtk’s on Kn lie on K 1 For f ∈ M(Kn )\{0}, ω ∈ M (Kn )\{0} the divisors of f and ω are denoted by (f ) and (ω), respectively. Two divisors D, E ∈ Div(Kn ) are called equivalent, denoted by D ∼ E, if and only if D − E = (f ) for some f ∈ M(Kn )\{0}. The divisor class [D] of D is then given by [D] = {E ∈ Div(Kn ) | E ∼ D}. We recall that deg((f )) = 0, deg((ω)) = 2(n − 1), f ∈ M(Kn )\{0}, ω ∈ M1 (Kn )\{0} ,
(A.38)
where the degree deg(D) of D is given by deg(D) = ΣP ∈Kn D(P ). It is custom to call (f ) (respectively, (ω)) a principal (respectively, canonical) divisor. Introducing the complex linear spaces L(D) = {f ∈ M(Kn ) | f = 0 or (f ) ≥ D}, r(D) = dimC L(D) ,
(A.39)
L1 (D) = {ω ∈ M1 (Kn ) | ω = 0 or (ω) ≥ D}, i(D) = dimC L1 (D) , (A.40) (i(D) the index of specialty of D) one infers that deg(D), r(D), and i(D) only depend on the divisor class [D] of D. Moreover, we recall the following fundamental facts. Theorem A.1. Let D ∈ Div(Kn ), ω ∈ M1 (Kn )\{0}. Then (i) i(D) = r(D − (ω)) ,
n ∈ N0 .
(A.41)
(ii) (Riemann–Roch theorem). r(−D) = deg(D) + i(D) − n + 1 ,
n ∈ N0 .
(A.42)
(iii) (Abel’s theorem). D ∈ Div(Kn ), n ∈ N is principal if and only if deg(D) = 0 and αP0 (D) = 0 .
(A.43)
(iv) (Jacobi’s inversion theorem). Assume n ∈ N, then αP0 : Div(Kn ) → J(Kn ) is surjective.
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
385
For notational convenience we agree to abbreviate Kn → {0, 1} 1, P = Q DQ : P → 7 0, P 6= Q
(A.44)
and, for Q = (Q1 , . . . , Qn ) ∈ σ n Kn (σ n Kn the nth symmetric power of Kn ), Kn → {0, 1, . . . , n} k if P occurs k times in {Q1 , . . . , Qn } DQ : P 7→ 0 if P ∈ / {Q1 , . . . , Qn } .
(A.45)
Moreover, σ n Kn can be identified with the set of positive divisors 0 < D ∈ Div(Kn ) of degree n. Lemma A.2. Let DQ ∈ σ n Kn , Q = (Q1 , . . . , Qn ). Then 1 ≤ i(DQ ) = s(≤ n/2)
(A.46)
if and only if there are s pairs of the type (P, P ∗ ) ∈ {Q1 , . . . , Qn } (this includes, of course, branch points for which P = P ∗ ). Finally, still assuming the nonsingular case (A.16) for simplicity, we consider two frequently encountered special cases, namely Case I: The self-adjoint case, where {Em }0≤m≤2n+1 ⊂ R ,
E0 < E1 < · · · < E2n+1
(A.47)
and Case II: Complex conjugate pairs of branch points, that is, {Em }0≤m≤2n+1 = {` , ` }0≤`≤n .
(A.48)
Without loss of generality we assume Re(` ) < Re(`+1 ), 0 ≤ ` ≤ n − 1, Im(` ) < Im(` ), 0 ≤ ` ≤ n .
(A.49)
We start with Case I: Define Cj = [E2j , E2j+1 ] ,
0 ≤ j ≤ n,
(A.50)
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and extend R2n+2 (.)1/2 in (A.4) to all of C by R2n+2 (λ)1/2 = lim R2n+2 (λ + i)1/2 , ↓0
λ∈C,
(A.51)
with the sign of the square root chosen according to −1, λ ∈ (E2n+1 , ∞) , (−1)n+j+1 , λ ∈ (E2j+1 , E2j+2 ), 0 ≤ j ≤ n − 1 , R2n+2 (λ)1/2 = |R2n+2 (λ)1/2 | λ ∈ (−∞, E0 ) , (−1)n , i(−1)n+j+1 , λ ∈ (E , E 2j 2j+1 ), 0 ≤ j ≤ n . (A.52) In this case (A.8) and (A.9) are supplemented as follows. P0 = (Em0 , 0):
UP0 = P ∈ Mn |z − Em0 | < Cm0 , Cm0 = min |Em0 − Em | , VP0
( ζP0 :
ζP−1 0
m6=m0
1/2 = ζ ∈ C |ζ| < Cm , 0
UP0 → VP0 P 7→ σ(z − Em0 )1/2 ,
(z − Em0 )1/2 = |(z − Em0 )1/2 |e(i/2) arg(z−Em0 ) , [0, 2π) , m0 even , arg(z − Em0 ) ∈ (−π, π] , m0 odd ,
VP0 → UP0 1/2 Y : ζ 7→ Em0 + ζ 2 , (Em0 − Em + ζ 2 ) ζ , m6=m0
1/2
Y
(Em0 − Em + ζ 2 )
= (−1)n i−m0 −1
m6=m0
1/2 Y X 1 × (Em0 − Em ) 1 + (Em0 − Em )−1 ζ 2 + O(ζ 4 ) . 2 m6=m0
m6=m0
(A.53) Case II: Define C` = {z ∈ C | z = ` + t(` − ` ), 0 ≤ t ≤ 1} ,
0≤`≤n
(A.54)
and extend R2n+2 (.)1/2 in (A.4) to all of C by R2n+2 (z)1/2 = lim R2n+2 (z + (−1)n+` )1/2 , ↓0
z ∈ C` , 0 ≤ ` ≤ n ,
(A.55)
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
...
387
with the sign of the square root chosen according to
R2n+2 (λ)1/2
Re(λ) ∈ (n , ∞) , −1, 1/2 n+`+1 = |R2n+2 (λ) | (−1) , λ ∈ (Re(` ), Re(`+1 )), 0 ≤ ` ≤ n − 1 , (−1)n , λ ∈ (−∞, (Re( )) . 0
(A.56) In this case (A.8) and (A.9) are supplemented as follows. P0 = (Em0 , 0): UP0 = P ∈ Mn |z − Em0 | < Cm0 , Cm0 = min |Em0 − Em | , VP0 ( ζP0 :
m6=m0
1/2 = ζ ∈ C |ζ| < Cm , 0
UP0 → VP0
(z − Em0 )1/2 = |(z − Em0 )1/2 |e(i/2) arg(z−Em0 ) ,
P 7→ σ(z − Em0 )1/2 ,
(A.57) π 5π , 2 2 , arg(z − ` ) ∈ π 5π 2, 2 , π 5π , 2 2 , arg(z − ` ) ∈ π 5π , 2 2 ,
ζP−1 0
` even , ` odd ,
` even , ` odd ,
π 3π − , , 2 2 arg(z − ` ) ∈ π 3π , − , 2 2 π 3π − , , 2 2 arg(z − ` ) ∈ π 3π − , , 2 2
` even , n even , ` odd ,
` even , n odd , ` odd ,
VP0 → UP0 1/2 Y : ζ 7→ Em0 + ζ 2 , (Em0 − Em + ζ 2 ) ζ , m6=m0
Y
1/2 (Em0 − Em + ζ 2 )
=e
(i/2)
P m6=m0
arg(Em0 −Em )
m6=m0
1/2 Y X 1 × (Em0 − Em ) 1 + (Em0 − Em )−1 ζ 2 + O(ζ 4 ) , 2 m6=m m6=m0 0
where exp[(i/2)Σm6=m0 arg(Em0 − Em )] can be determined from (A.56) by analytic continuation.
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Cases I and II are of course compatible with our general choice of " # ! 2n+1 1 X 1/2 Em ζ + O(ζ 2 ) ζ −n−1 as P → ∞± . y(P ) = R2n+2 (P ) = ∓ 1 − ζ→0 2 m=0 (A.58)
Appendix B. An Explicit Illustration of the Riemann Roch Theorem We provide a brief illustration of the Riemann–Roch theorem in connection with nonsingular hyperelliptic curves Kn of the type (2.26) and explicitly determine a basis for the vector space L(−kD∞− −m(k)D∞+ −Dµˆ (x0 ) ), where m(k) = max (0, k− 2) and k ∈ N0 . (The corresponding case of hyperelliptic curves Kn branched at infinity has been discussed in Appendix B of [27].) We freely use the notation introduced in Appendix A and refer, in particular, to the definition (A.39) of L(D) and the Riemann–Roch theorem stated in Theorem (ii). In addition, we use the short-hand notation kD∞− + m(k)D∞+ + Dµˆ (x0 ) =
k X `=1
X
m(k)
D∞− +
`=1
D∞+ +
n X
Dµˆj (x0 ) ,
j=1
(B.1) ˆ(x0 ) = (ˆ µ1 (x0 ), . . . , µ ˆn (x0 )) k ∈ N0 , µ and recall that L(−kD∞− − m(k)D∞+ − Dµˆ (x0 ) ) = f ∈ M(Kn ) f = 0 or (f ) + kD∞− + m(k)D∞+ + Dµˆ (x0 ) ≥ 0 ,
k ∈ N0 . (B.2)
With φ(P, x), ψj (P, x, x0 ), j = 1, 2 defined as in (3.8), (3.10), (3.22) we obtain the following result. Theorem B.1. Assume Dµˆ (x0 ) to be nonspecial (i.e., i(Dµˆ (x0 ) ) = 0) and of degree n ∈ N. For k ∈ N0 , a basis for the vector space L(−kD∞− − m(k)D∞+ − Dµˆ (x0 ) ) is given by {1},
k = 0,
{˜ π ` φ(., x0 )}0≤`≤k−1 , k ∈ N . π ` }0≤`≤m(k) ∪ {˜
(B.3)
Proof. The elements in (B.3) are easily seen to be linearly independent and belonging to L(−kD∞− − m(k)D∞+ − Dµˆ (x0 ) ). It remains to be shown that they are maximal. Since i(Dµˆ (x0 ) ) = i(kD∞− + m(k)D∞+ + Dµˆ (x0 ) ) = 0, the Riemann–Roch
AN ALTERNATIVE APPROACH TO ALGEBRO-GEOMETRIC SOLUTIONS
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389
theorem (A.42) implies r(−kD∞− − m(k)D∞+ − Dµˆ (x0 ) ) = k + m(k) + 1 proving (B.3). Replacing φ by φ−1 one can discuss L(−kD∞+ − m(k)D∞− − Dνˆ(x0 ) ), k ∈ N0 in an analogous fashion. Acknowledgment We thank Gerald Teschl for discussions. References [1] S. I. Al’ber, “Investigation of equations of Korteweg–de Vries type by the method of recurrence relations”, J. London Math. Soc. 19 (2) (1979) 467–480 (Russian). [2] S. I. Al’ber, “On stationary problems for equations of Korteweg–de Vries type”, Commun. Pure Appl. Math. 34 (1981) 259–272. [3] S. I. Al’ber and M. S. Al’ber, “Hamiltonian formalism for nonlinear Schr¨ odinger equations and sine-Gordon equations”, J. London Math. Soc. 36 (2) (1987) 176–192. [4] F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. [5] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its, and V. B. Matveev, Algebrogeometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994. [6] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, “Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac–van Moerbeke hierarchies”, Memoirs Amer. Math. Soc., to appear. [7] J. L. Burchnall and T. W. Chaundy, “Commutative ordinary differential operators”, Proc. London Math. Soc. 21 (2) (1923) 420–440. [8] J. L. Burchnall and T. W. Chaundy, “Commutative ordinary differential operators”, Proc. Roy. Soc. London A118 (1928) 557–583. [9] J. L. Burchnall and T. W. Chaundy, “Commutative ordinary differential operators II. The identity P n = Qm ”, Proc. Roy. Soc. London A134 (1932) 471–485. [10] P. L. Christiansen, J. C. Eilbeck, V. Z. Enolskii, and N. A. Kostov, “Quasi-periodic solutions of the coupled nonlinear Schr¨ odinger equations”, Proc. Roy. Soc. London A451 (1995) 685–700. [11] C. De Concini and R. A. Johnson, “The algebraic-geometric AKNS potentials”, Ergod. Th. and Dynam. Sys. 7 (1987) 1–24. [12] L. A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapore, 1991. [13] R. Dickson, F. Gesztesy, and K. Unterkofler, “A new approach to the Boussinesq hierarchy”, Math. Nachr., to appear. [14] B. A. Dubrovin, “Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties”, Funct. Anal. Appl. 11 (1977) 265–277. [15] B. A. Dubrovin, “Matrix finite-zone operators”, Revs. Sci. Tech. 23 (1983) 20–50. [16] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems. I”, in Dynamical Systems IV, eds. V. I. Arnol’d and S. P. Novikov, Springer, Berlin, 1990, pp. 173–280. [17] N. M. Ercolani and H. Flaschka, “The geometry of the Hill equation and of the Neumann system”, Phil. Trans. Roy. Soc. London A315 (1985) 405–422. [18] H. M. Farkas and I. Kra, Riemann Surfaces, 2nd ed., Springer, New York, 1992. [19] J. D. Fay, “Theta functions on Riemann surfaces”, Lecture Notes in Math. 352, Springer, Berlin, 1973.
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[20] I. M. Gel’fand and L. A. Dikii, “Integrable nonlinear equations and the Liouville theorem”, Funct. Anal. Appl. 13 (1979) 6–15. [21] F. Gesztesy, “Quasi-periodic, finite-gap solutions of the modified Korteweg–de Vries equation”, in Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, Volume 1, eds. S. Albeverio, J. E. Fenstad, H. Holden, and T. Lindstrøm, Cambridge Univ. Press, Cambridge, 1992, pp. 428–471. [22] F. Gesztesy and R. Svirsky, “(m)KdV solitons on the background of quasi-periodic finite-gap solutions”, Memoirs Amer. Math. Soc. 118 (No. 563) (1995). [23] F. Gesztesy and R. Weikard. “Spectral deformations and soliton equations”, in Differential Equations with Applications to Mathematical Physics, eds. W. F. Ames, E. M. Harrell II, and J. V. Herod, Academic Press, Boston, 1993, pp. 101–139. [24] F. Gesztesy and R. Weikard, “Lam´e potentials and the stationary (m)KdV hierarchy”, Math. Nachr. 176 (1995) 73–91. [25] F. Gesztesy and R. Weikard, “Treibich–Verdier potentials and the stationary (m)KdV hierarchy”, Math. Z. 219 (1995) 451–476. [26] F. Gesztesy and R. Weikard, “Picard potentials and Hill’s equation on a torus”, Acta Math. 176 (1996) 73–107. [27] F. Gesztesy, R. Ratnaseelan, and G. Teschl, “The KdV hierarchy and associated trace formulas”, in Proc. International Conference on Applications of Operator Theory, eds. I. Gohberg, P. Lancaster, P. N. Shivakumar, Operator Theory: Advances and Applications, Vol. 87, Birkh¨ auser, 1996, pp. 125–163. [28] B. Grebert and J. C. Guillot, “Gaps of one dimensional periodic AKNS systems”, Forum Math., to appear. [29] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978. [30] A. R. Its, “Inversion of hyperelliptic integrals and integration of nonlinear differential equations”, Vestnik Leningrad Univ. Math. 9 (1981) 121–129. [31] A. R. Its, “On Connections between solitons and finite-gap solutions of the nonlinear Schr¨ odinger equation”, Sel. Math. Sov. 5 (1986) 29–43. [32] A. R. Its and V. B. Matveev, Schr¨ odinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg–de Vries equation”, Theoret. Math. Phys. 23 (1975) 343–355. ¨ [33] C. G. T. Jacobi, “Uber eine neue Methode zur Integration der hyperelliptischen Differentialgleichungen und u ¨ber die rationale Form ihrer vollst¨ andigen algebraischen Integralgleichungen”, J. Reine Angew. Math. 32 (1846) 220–226. [34] A. Krazer, Lehrbuch der Thetafunktionen, Chelsea, New York, 1970. [35] I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry”, Funct. Anal. Appl. 11 (1977) 12–26. [36] I. M. Krichever, “Nonlinear equations and elliptic curves”, Revs. Sci. Tech. 23 (1983) 51–90. [37] Y.-C. Ma and M. J. Ablowitz, “The periodic cubic Schr¨ odinger equation”, Stud. Appl. Math. 65 (1981) 113–158. [38] H. P. McKean, “Variation on a theme of Jacobi”, Commun. Pure Appl. Math. 38 (1985) 669–678. [39] J. Mertsching, “Quasi periodic solutions of the nonlinear Schr¨ odinger equation”, Fortschr. Phys. 85 (1987) 519–536. [40] D. Mumford, Tata Lectures on Theta I, Birkh¨ auser, Boston, 1983. [41] D. Mumford, Tata Lectures on Theta II, Birkh¨ auser, Boston, 1984. [42] A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985. [43] S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons, Consultants Bureau, New York, 1984. [44] E. Previato, “Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schr¨ odinger equation”, Duke Math. J. 52 (1985) 329–377.
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QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS F. MONTI and H. R. JAUSLIN Centre de Dynamique des Syst` emes Complexes and Laboratoire de Physique, CNRS, Universit´ e de Bourgogne BP 400, 21011 Dijon CEDEX France Received 21 July 1997 A quantum version of Nekhoroshev estimates for Floquet Hamiltonians associated to quasi-periodic time dependent perturbations is developped. If the unperturbed energy operator has a discrete spectrum and under finite Diophantine conditions, an effective Floquet Hamiltonian with pure point spectrum is constructed. For analytic perturbations, the effective time evolution remains close to the original Floquet evolution up to exponentially long times. We also treat the case of differentiable perturbations.
Introduction Effective stability estimates, valid for long but finite times, are well-established tools in classical dynamics [20, 4, 19, 21, 10]. These results based on the Nekhoroshev Theorem provide efficient tools for applications in, for example, celestial mechanics and particle accelerators. In the present work, we develop some corresponding concepts for quantum mechanics. We consider quantum systems driven by periodic or quasi-periodic external fields. Such systems can be described by a family of time dependent Hamiltonians and indexed by a phase Hθ (t) = H(θ + ωt) acting on a fixed Hilbert space vector θ ∈ Td , the d-dimensional torus. A time-independent Hamiltonian can be associated to Hθ (t) by adding d degrees of freedom to the system [1]. This is done into = L2 (Td , ) ' ⊗ L2 (Td ). The by extending the Hilbert space family of propagators Uθ (t, t0 ) between time t0 and t associated to Hθ (t) are lifted , which acts as a multiplicative opto a time-dependent unitary U (t, t0 ; θ) on erator with respect to θ. Composing it with the translation operator Tt defined given by (Tt ψ)(θ) = ψ(θ + ωt), we get a one parameter unitary group on −˚ ı (t−t0 ) K(θ) . Its infinitesimal generator is the Floquet by Tt U (t, t0 ; θ) T−t0 = e Hamiltonian or quasi-energy operator K(θ) = −˚ ı ω·∂θ + H(θ) associated to H. Detailed accounts of this approach can be found in [1] or [15]. The spectral properties of the Floquet Hamiltonian can be used to characterize the stability of the motion of periodically or quasi-periodically driven quantum systems (see [12, 15]). It has been proven that, if the Floquet spectrum is pure point, then the energy of a state remains uniformly bounded along all trajectories.
H
H
K
H
H
K
K
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Moreover, quantities such as the mean time correlations behave almost-periodically in time [2]. On the other hand, if there is a countinuous component in the Floquet spectrum, the motion is unstable [2, 5]. Floquet Hamiltonians are used as a theoretical tool to analyze the interaction between an atom or a molecule and a laser beam. The associated quasi-energy operator is given by K(θ) = −˚ ı ω·∂θ + H0 + V (θ), where H0 is the free molecular Hamiltonian, V (θ) represents the interaction between the laser field and the particle, and ω = (ω1 , . . . , ωd ) are the laser frequencies. Adding an extra time dependence in the interaction term V , describing, for example, the variation of the intensity, allows to treat the case of laser pulses. Very intense short laser pulses with tunable frequencies have been used to control molecular processes. Some examples are selective population of energy levels, coherent photo-dissociation or photo-ionization, laser controlled tunneling or selective bond excitation. To describe and predict these phenomena, numerical simulations interpreted with a combination of Floquet theory and adiabatic principles have been used extensively. See for example [6, 14] and the references therein. Most of these considerations are based on the hypothesis that the relevant part of the Floquet spectrum is pure point at each instant. The applications described above motivate the interest to understand under what conditions the (instantaneous) spectrum of a Floquet Hamiltonian is pure point. The free quasi-energy operator K0 = −˚ ı ω·∂θ + H0 has in general a dense point spectrum. This leads to small denominator problems in perturbation theory. The usual way to overcome them is to apply quantum KAM techniques which allow, under suitable hypothesis, to show that the point spectrum is stable under perturbations [2, 3, 8, 9, 5, 11]. To be able to apply these KAM techniques, one needs to impose some Diophantine conditions on the frequencies. These are however very restrictive: the admissible frequencies form a Cantor set. As a consequence, this prevents the possibility of tuning continuously the frequencies within this set. Here, we will take another point of view. Since in applications the laser pulses are of finite duration, we need only to control the nature of the Floquet dynamics during this time interval. This leads to the concept of effective stability. We will prove a quantum version of the well-known Nekhoroshev Theorem of classical mechanics [20]. A related problem in finite dimensions has been treated in [16]. The aim is to construct iteratively an effective Floquet Hamiltonian whose spectrum is pure point and whose dynamics coincide with high precision with the original one for a very long time. This time is proportional to the size of the inverse of the difference between the effective and original Floquet Hamiltonians. If the perturbation is sufficiently differentiable, then this time is polynomially small in the coupling. In case of an analytic perturbation, it is exponentially small in the coupling. An advantage of Nekhoroshev estimates with respect to KAM results is that the set of admissible frequencies contains open parts. Hence, it becomes possible to describe laser controlled molecular processes which require a time dependent sweep of the frequency [7, 13].
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In the next section, we will introduce the needed notations and state the main results. We will also explain the general idea of the proof. The subsequent sections are devoted to the proof of the results. 1. Nekhoroshev Estimates for Floquet Hamiltonians In this section we will state the main results and their consequences. We are interested in quasi-periodic Floquet Hamiltonian describing the interaction between a laser and an atom or a molecule. The molecular energy operator H0 is defined . The interaction between the laser on an (infinite dimensional) Hilbert space and the molecule is given by a family V (θ + tω) of bounded self-adjoint operators . Here θ ∈ Td represents the initial phase vector, i.e. characterizes the value on of the interaction at t = 0, Td is the d-dimensionnal torus (i.e. the set (R/2πZ)d or equivalently the d-fold product of circles of length 2π), and ω ∈ Rd is the vector frequency of the electric field. By extending the Hilbert space, we can decouple the field time dependence from the rest of the interaction. The enlarged Hilbert space = L2 (Td , ), which is naturally isomorphic to ⊗ L2 (Td ). The space Td is is is endowed with its natural normalized Haar measure: dd θ/(2π)d . A vector in satisfying a function ψ : Td −→ Z d 2 d θ 2 kψ(θ)kH < ∞, kψk = (2π)d Td
H
H
K
H
H
K
H
K
simply by k·k and where k·kH is the norm of where we have denoted the norm of . The same symbols will also be used to denote the norm of a bounded operator or respectively. The Hilbert space is naturally embedded in over by considering a vector of as a constant -valued function. Moreover, as the . This inclusion measure of Td is equal to 1, we have that kϕkH = kϕk for any ϕ ∈ gives us the embedding of ( ) in ( ), and implies that kM k = kM kH for any M ∈ ( ). The quasi-energy operator or Floquet Hamiltonian KF associated to the family H0 + V (θ + tω) is given by
H
K
H
BH
H
BH
H
H
BK
ı ω·∂ + H0 + V , KF = K0 + V = −˚
K
H
(1.1)
where (ω·∂ ψ)(θ) = Σj ωj ∂θj ψ(θ), (H0 ψ)(θ) = H0 ψ(θ) and (V ψ)(θ) = V (θ) ψ(θ). Within this model, we can easily describe a laser pulse by adding an extra time dependence in the interaction term: (t, θ) 7−→ V (t; θ). A special case which is well suited to describe a pulse is given by a potential of the form V (t; θ) = a(t) V (θ), where a : R −→ R is a positive function with compact support, describing the envelope of the pulse amplitude. The aim of this paper is to prove that we can perform a unitary transformation , such that KF is transformed into a new Floquet Hamiltonian which is the on sum of a quasi-energy operator with pure point spectrum and a rest which is very small with respect to the original perturbation. In case of an extra time dependence we think of the instantaneous spectrum, i.e. for each fixed t. Moreover the evolutions associated respectively to the original Floquet Hamiltonian and the one with pure point spectrum remain near for a time which is of the order of the inverse of the size
K
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of the rest. These results are a formulation in quantum mechanics of the well-known Nekhoroshev effective stability theorems of classical mechanics. To get these Nekhoroshev estimates, we will need some hypothesis on the spectrum of the initial Hamiltonian H0 , on the set of admissible frequencies and on the smoothness of the interaction potential. 1.2. Spectral Hypothesis. We will suppose that the spectrum of H0 is discrete and that it satisfies the following separation condition: let J be a set of consecutive numbers in Z which labels the eigenvalues of H0 , sp H0 = {zj }j∈J
and
zj+1 − zj ≥ g .
In particular, the multiplicity of each eigenvalue is finite but otherwise arbitrary.a We want to show that the quasi-energy operator is close to a Floquet Hamiltonian with pure point spectrum. In general this is not true for all frequencies due to resonances. We will prove it for a set satisfying a sufficient “finite” Diophantine condition: 1.3. “Finite” Diophantine Hypothesis. For a positive integer p• , we define the set of admissible frequencies as Ωp• (γ, τ ) = ω ∈ Rd ; |ω·m + zj − zk | > γ(|m|1 + |j − k|)−τ for all j, k ∈ J, m ∈ Zd , 0 < |m|1 < p• (1.4) for some fixed positive constants γ and τ > d. Here | · |1 designates the 1-norm of Rd: |x|1 = Σj |xj |. Notice that Ωp• contains open subsets, at least if γ is sufficiently small. In order to treat the small denominator problems arising in the KAM algorithm, we need the perturbation to satisfy some smoothness condition. We will consider two cases of regularity: θ 7−→ V (t; θ) will be either sufficiently differentiable or analytic. Stronger regularity leads to stronger estimates. We need also a regularity condition in the variable t, in order to be able to define a unitary propagator of the associated evolution which is strongly differentiable on the domain of K0 . 1.5. Regularity Hypothesis. We will suppose that the interaction term V is self-adjoint and satisfies one of the following conditions: (i) differentiability condition t 7−→ V (t; θ) is strongly 1 for all θ ∈ Td ;
C
a This spectral hypothesis can be slightly improved. We may assume that the spectrum of H0 consists of blocks Σj made of eigenvalues of finite multiplicity and separated by gaps: dist(Σj+1 , Σj ) ≥ g > 0. The number of different eigenvalues in each Σj must be bounded by an integer n which is independent of j. We will not consider this case, as it complicates the notations.
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
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C
θ 7−→ V (t; θ) and θ 7−→ ∂t V (t; θ) are strongly d(r+2) for some r > τ + 1/2 and all t; V (t; θ), ∂t V (t; θ) and all their θ-derivatives up to order d(r + 2) are uniformly bounded: sup d
θ∈T ,t 0≤|a|1 ≤d(r+2)
k∂θa V (t; θ)kH < ∞ and
sup d
θ∈T ,t 0≤|a|1 ≤d(r+2)
k∂θa ∂t V (t; θ)kH < ∞ ;
(ii) analyticity condition t 7−→ V (t; ϑ) is strongly 1 for all ϑ in a neighbourhood D of the strip Dr = {ϑ = θ +˚ ıy; θ ∈ Td , |y|2 ≤ r}, for some r > 0; ϑ 7−→ V (t; ϑ) and ϑ 7−→ ∂t V (t; ϑ) are holomorphic in D for all t and
C
sup kV (t; ϑ)kH < ∞ and
ϑ∈D,t
sup k∂t V (t; ϑ)kH < ∞ .
ϑ∈D,t
For example, this hypothesis will be satisfied if V (t; θ) = a(t) V (θ), where t 7−→ a(t) is 1 and is uniformly bounded together with its derivative, and θ 7−→ V (θ) is bounded and either strongly d(r+2) for some r > τ + 1/2 or holomorphic in a neighbourhood of the set Dr for some r > 0. In particular those conditions are verified for V independent of t and sufficiently smooth (i.e. by taking a(t) ≡ 1 above). We will begin by stating the Nekhoroshev estimates as a function of the Diophantine threshold parameter p• .
C
C
1.6. Theorem. Let p• ≥ 1 be given and suppose that the frequency ω belongs to Ωp• . Suppose that (t, θ) 7−→ V (t; θ) verifies one of the regularity hypothesis 1.5 for some r. Then we can find an p• such that for any ≤ p• , there exists a unitary transformation U∗ (t) leaving the domain of K0 invariant and satisfying U∗ (t) (K0 + V (t)) U∗ (t)−1 = K0 + Y∗ (t) + R∗ (t) ,
BH
where Y∗ (t) ∈ ( ) and commutes with H0 and with K0 . This implies that the instantaneous spectrum of K0 + Y∗ (t) is pure point. Moreover, we have the following estimates: −(r−δ) 1 + p• (differentiable case) , CV 21/δ kY∗ (t)kH ≤ cV and kR∗ (t)k ≤ r (analytic case) , CV exp − p• 2 where δ = τ + 1/2, and cV , CV are constants that do not depend on p• or . The bound p• for the coupling is proportional to γ g τ +1/2 , where γ, τ are the Diophantine parameters of Hypothesis 1.3 and g is the minimal gap in the spectral condition 1.2. Remark. As we will see explicitly in the proof, at each time t, the eigenfamily of K0 + Y∗ (t) splits: the eigenvalues are given by ω·m + z∗ j (t), where the z∗ j (t)
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H
are the instantaneous eigenvalues of H0 + Y∗ (t) acting on . The corresponding is an eigenvectors are simple products of the form ϕ∗ j (t) χm , where ϕ∗ j (t) ∈ instantaneous eigenvector of H0 + Y∗ (t) corresponding to z∗j (t) and χm (θ) = e˚ı m·θ is the eigenvector of −˚ ı ω·∂ associated to ω·m.
H
1.7 Corollary. If K∗ (t) = U∗ (t)−1 (K0 + Y∗ (t)) U∗ (t) and W (t, s), W∗ (t, s) are the evolutions associated respectively to K0 + V (t) and to K∗ (t), then we have under the hypothesis of the theorem:
kW∗ (t, s) − W (t, s)k ≤ η
r−δ η 1 + p• C 21/δ V for any |t − s| ≤ r η CV exp 2 p•
(differentiable case) , (analytic case) .
These kind of results are interesting when the frequency vector ω is known only up to a certain precision. This is always the case when one wants to compare the theory to experiments or to discuss the results of computer-based simulations. Indeed, in those cases the frequency can only be known up to a certain number of decimals and there is also an uncertainty on the exact value, due to technical difficulties in experiments or rounding errors in computer simulations. In this theorem we can adjust the Diophantine threshold parameter p• to the known precision of the frequency. The next kind of results shows that, if the perturbation is analytic (sufficiently differentiable) and if the frequency satisfies a suitable “finite” Diophantine condition, we can find a unitary transform such that the size of the rest is exponentially (polynomially) small in the size of the perturbation. 1.8. Theorem. Suppose that (t, θ) 7−→ V (t; θ) satisfies one of the regularity hypothesis 1.5 for some r. Then, we can find an ∗ > 0 such that for any ≤ ∗ there is a p for which, if ω ∈ Ωp , there exists a unitary transformation U∗ (t) leaving the domain of K0 invariant and satisfying U∗ (t) (K0 + V (t)) U∗ (t)−1 = K0 + Y∗ (t) + R∗ (t) ,
BH
where Y∗ (t) ∈ ( ) commutes with H0 and with K0 , the instantaneous spectrum of K0 + Y∗ (t) is pure point and we have the following estimates:
kY∗ (t)kH ≤ cV
and
2r 2τ +1 C V ∗ kR∗ (t)k ≤ 2 ∗ (2τ +1) CV exp −
(differentiable case) ,
(analytic case) .
cV , CV are constants that do not depend on , and ∗ is proportional to γ g τ +1/2 , where γ, τ are the Diophantine parameters of Hypothesis 1.3 and g is the minimal gap in the spectral condition 1.2.
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
399
In this result again, the instantaneous eigenfamily of K0 + Y∗ (t) splits for each value of t. 1.9. Corollary. If K∗ (t) = U∗ (t)−1 (K0 + Y∗ (t)) U∗ (t) and W (t, s), W∗ (t, s) are the evolutions associated respectively to K0 + V (t) and K∗ (t), then we have under the hypothesis of the theorem:
kW∗ (t, s) − W (t, s)k ≤ η
for any |t − s| ≤
r η ∗ 2τ2+1 CV 2 η ∗ (2τ +1) exp CV
(differentiable case) ,
(analytic case) .
These results are the quantum analogues of the classical Nekhoroshev Theorem, which is usually only stated in the analytic case. As we will see in the proof, the Diophantine threshold is given by p = ar (∗ /)2/(2τ +1) for some constant ar (where bxc denotes the integer part of x). As goes to zero, the estimates are valid for longer times. But in order to obtain them, also p tends to infinity. This means that one imposes stronger Diophantine conditions on the set of admissible frequencies. The idea of the proofs is the following: we perform iteratively a finite number of unitary transforms which bring the initial quasi-energy operator to a better approximate normal form at each step. In this context a normal form (with respect to 0 ) is a Floquet Hamiltonian K0 + Y where Y ∈ ( ), i.e. it is independent of θ, and commutes with H0 and K0 . Noticing that initially, we start with K (0) = K0 +Y (0) +R(0) where Y (0) = 0 and (0) R = V . A concrete step of the iterative process is described as follows. Given a quasi-energy operator K (n) = K0 + Y (n) + R(n) , where K0 + Y (n) is in normal form, we look for a unitary e X , such that the transformed Floquet Hamiltonian (1.10) K (n+1) = K0 + Y (n+1) + R(n+1) = e X K0 + Y (n) + R(n) e −X
K
BH
is in better approximate normal form. This means that K0 + Y (n+1) is in normal form and the size of R(n+1) is smaller than the size of R(n) . To find a solution of Eq. (1.10), we consider its linearized part (n) adK X + Z = R
and where R
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impose the spectral condition 1.2 and the Diophantine hypothesis 1.3. To overcome the smallness of the denominators, R(n) has to satisfy one of the regularity conditions 1.5. Indeed, the denominators add some polynomial growth to the Fourier series of R(n+1) . By converting part of the regularity of R(n) , we can ensure that the Fourier coefficients of R(n+1) decrease fast enough to be absolutely summable. To keep control of the size of the successive rests R(n) and to take into account the loss of regularity at each iterative step, we introduce a family of norms which reflects the behaviour of the Fourier coefficients of the R(n) . If the size of the initial perturbation is sufficiently small, we can show that the size of the rests R(n) decreases at each step. Using these norms, we can complete the proof by optimizing the size of the rest R(n) with respect to the coupling or the Diophantine threshold p• . This can be done by choosing adequately the total number of steps of the iteration and some other parameters such as the loss of regularity or the contraction factor of the size of the succesive rests. The spectral condition 1.2 and the Diophantine hypothesis 1.3 are also necessary for the spectrum of the optimal normal form K0 + Y∗ to be pure point. The paper is organized as follows. For notational simplicity, we will first give the proofs of the results for the case where V is time independent. In the last section, we will discuss the minor differences necessary to treat the case when the interaction has an extra time dependence. In Sec. 2, we solve Eq. (1.10) at a formal level and we introduce the notion of a . Fourier transform of a bounded multiplicative operator on Sec. 3 is devoted to the proof that the formal solution obtained in Sec. 2 is an actual solution of (1.10). In Sec. 4, we introduce the family of norms, and their associated ∗-subalgebras of ( ), which will allow us to control the size of the rest. We also prove that, if V satisfies the regularity condition 1.5 for some r, then it belongs to one of these subalgebras. Sec. 5 develops the quantum KAM algorithm to compute iteratively the unitary transforms and to get the estimates on the norms of the successive rests. In Sec. 6, we minimize the size of the rest with respect to the parameters to obtain the exponential or polynomial estimates stated in the theorems. Sec. 7 shows that the normal form K0 + Y∗ obtained in Sec. 6 has indeed a pure point spectrum. It contains also the proof of the corollaries in the time independent case. Finally, Sec. 8 treats the case when there is an extra time dependence in the perturbation.
K
BK
2. The Formal Algorithm In this section we will describe the algorithm at a formal level, which we will justify in later sections. The idea is to transform the initial Hamiltonian iteratively in order to put it in a normal form with a rest that is smaller than the initial perturbation. To perform the unitary transforms we will need to write the perturbation as a Fourier series.
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
401
Fourier series of a bounded multiplicative operator
K BH
which acts by mutiplication in the θ Consider a bounded operator L on . Then variable: (Lψ)(θ) = L(θ)ψ(θ) with L(θ) ∈ ( ) for all θ ∈ Td and ψ ∈ we can define, at least formally, its Fourier series by X Lm T m L∼
K
m∈Td
BH
where Lm ∈ ( ) for each m and (T m ψ)(θ) = e˚ı m·θ ψ(θ). If such a series exists we must have Z dd θ L(θ) e −˚ı m·θ . Lm = (2π)d Td
(2.1)
This can easily be seen by computing the matrix coefficients of L and of Σm∈Td Lm T m in a basis of of the form {δ j χm }, where δ j represents any orthonorm and χ (θ) = e˚ı m·θ is the canonical basis of L2 (Td ). Now, if mal basis of Σm∈Td Lm T m defines a bounded operator, with Lm satisfying (2.1), it must be equal to the operator L, because they are bounded and they coincide on a dense set. This leads us to the following definition:
H
K
K
. The 2.2. Definition. Let L be a bounded multiplicative operator on m is the operator Σm∈Td Lm T with Lm given by (2.1), if it Fourier series of . We will then say that L admits a Fourier exists as a bounded operator on expansion.
L
K
We will use the following notation: 2.3. Notation. If L = Σm∈Td Lm T m defines a bounded operator, we introduce the following X X Lm T m and L≥p• = Lm T m = L − L
m:|m|1
In later sections we will give some conditions which will guarantee the existence of the Fourier Series of the involved bounded operators. Normal forms and Lie series In this part we will describe an iterative process that transforms the initial Floquet Hamiltonian K0 + V into normal form. We will describe the algorithm at a formal level, proving in the next section that we get a well-defined solution. We will suppose that V admits a Fourier expansion. The iterative procedure will transform the initial Hamiltonian into the approximate normal form K0 + Y + R, where R is small and Y commutes with K0 . We will also suppose that R admits a Fourier expansion and that Y ∈ ( ), i.e. is independent of θ. Since this is indeed the case for the original Hamiltonian K0 + V , we will assume in this section that the Floquet
BH
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Hamiltonian KF is given in the form K0 + Y + R, and we will try to find a unitary transform e X that yields an improved approximation to normal form. We will now describe one step of the iterative procedure at a formal level, justifying the computations in subsequent sections. Starting with a Floquet Hamiltonian in approximate normal form, we look for a bounded anti-hermitian operator X such that e e X (K0 + Y + R) e −X = K0 + Ye + R where the resulting Hamiltonian is a better approximate normal form with Ye = e admits a Fourier expansion. To determine the infinitesimal Y + Z ∈ ( ) and R generator X of the unitary transform and Z, we develop the conjugation by e X and try to solve the linear part of the obtained equality. Using the notation adW (·) = [W, · ], we get
BH
e X (K0 + Y + R) e −X =
∞ X 1 adlX (K0 + Y + R) l! l=0
= K0 + Y + R
∞ ∞ X X 1 1 adX l (Y + R) + adX l (K0 ) . l! l! l=1
l=2
We look for a bounded solution (X, Z) of the linearized equation: adK X + Z = R
ad Z = 0 K0
(2.4)
If such a pair exists, we find that e e X (K0 + Y + R) e −X = K0 + Ye + R with e = R≥p• + R
∞ ∞ X X 1 1 adX l (Y + R) + adX l (Z − R
(2.5)
l=1
and
Ye = Y + Z .
In order to solve (2.4), we can formally express this system of equations in terms P of Fourier coefficients and try to find solutions of the form X = m Xm T m and P Z = m Zm T m and then verify that we get an actual solution of (2.4). Noticing that formally X X [H0 , Xm ] T m −˚ ı Xm [ω·∂, T m ] adK0 X = m
=
X
m
(adH0 Xm + ω·m Xm ) T m
m
and equating the coefficients corresponding to T m for each m, we get the systems of equations
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
403
adH X0 + Z0 = R0 0 ad H0 Z0 = 0 adH Xm + ω·m Xm + Zm = Rm 0 ad H0 Zm + ω·m Zm = 0 adH0 Xm + ω·m Xm + Zm = 0
if 0 < |m|1 < p• ,
(2.6)
if |m|1 ≥ p• .
ad H0 Zm + ω·m Zm = 0
In order to find a solution for this system of equations, we compute adH0 W using the spectral decomposition of H0 = Σj zj Pj : X X X zj (Pj W − W Pj ) = (zj − zk ) Pj W Pk . adH0 W = j
j
k:k6=j
If ω·m + zj − zk 6= 0, a formal solution of (2.6) is given by X X Pj R0 Pk if m = 0 j k:k6=j zj − zk X X Pj Rm Pk Xm = if 0 < |m|1 < p• ω·m + zj − zk j k 0 if |m|1 ≥ p• X Pj R0 Pj if m = 0 j Zm = 0 otherwise
(2.7)
We now have to prove that the formal solutions X = Σm Xm T m and Z = Z0 obtained with the help of (2.6) are bounded solutions of (2.4). We must show that with Z ∈ ( ), X, Z: dom K0 −→ dom K0 X and Z are bounded operators on and that adK0 X + Z = R
K
BH
3. Existence of Bounded Solutions of (2.4) We will first show that X and Z defined by their Fourier coefficients in (2.7) are , under the hypothesis 1.2, 1.3 and 1.5. Notice that if W bounded operators in then kW k = kW kH , since is a bounded operator on Z Z d d 2 d θ 2 2 d θ 2 2 kW ψ(θ)kH ≤ kW k kψ(θ)k = kW kH kψk2 , kW ψk = H H d d (2π) (2π) d d T T
K
H
H
and on the other hand for any ε > 0, there exists a ϕε ∈ such that kW ϕε kH ≥ , we find that kW ϕε k ≥ (kW kH − (kW kH − ε) kϕε kH . Viewing ϕε as a vector of ε) kϕε k, proving the equality between kW k and kW kH .
K
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3.1. Lemma. Suppose that Rm are bounded operators on
H . Then
(i) Z is bounded and kZkH ≤ kR0 kH ; X 1 (ii) if sup < ∞, then X0 is bounded and |zl − zk |2 l k:k6=l
kX0 kH ≤ sup l
(iii) if sup l
X k
X
1/2 |zl − zk |−2
kR0 kH ;
k:k6=l
1 < ∞, then Xm is bounded and |ω·m + zl − zk |2 kXm kH ≤
sup l
X
!1/2 |ω·m + zl − zk |−2
kRm kH .
k
As a consequence X is a bounded operator on for all |m|1 < p• .
K , if (ii) and (iii) are satisfied
Proof. Consider Z0 and Xm given by Eq. (2.7).
H
, that (i) By the Pythagorean Theorem, we have for any ϕ ∈ X X 2 2 2 2 2 2 kPj R0 Pj ϕkH ≤ kR0 kH kPj ϕkH = kR0 kH kϕkH . kZ0 ϕkH = j
j
(ii) The proof is a consequence of the following assertion, which is easily obtained using the properties of the norm and the Cauchy–Schwartz inequality. Assertion. For any sequences of complex numbers {aj } and vectors {ζj } in a Hilbert space, we have that kΣj aj ζj k2 ≤ Σj |aj |2 Σk kζk k2 . By the Pythagorean Theorem and the assertion, we get kX0 ϕkH = 2
≤
X X Pj R0 Pk 2
ϕ H zj − zk j k:k6=j
X j
≤ sup l
≤ sup l
X
|zj − zk |−2
k:k6=j
X k:k6=l
X k:k6=l
|zl − zk |−2
X n:n6=j
X n,j
kPj R0 Pn ϕkH
kPj R0 Pn ϕkH
|zl − zk |−2 kR0 kH kϕkH 2
2
2
2
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
405
Using the same kind of arguments, we can show (iii). , it is sufficient to notice that Next, to see that X is bounded on X X kXm kH kT m k = kXm kH . kXk ≤
K
m:|m|1
m:|m|1
Which ends the proof of Lemma 3.1.
Before considering sufficient conditions on H0 , ω and R so that the hypothesis of Lemma 3.1 are satisfied, we will prove that (X, Z) is a solution of (2.4). 3.2. Lemma. Let R be a bounded self-adjoint operator admitting a Fourier expansion and let X and Z be defined by their Fourier components given in Eq. (2.7) and suppose that the conditions of Lemma 3.1 are satisfied. Then, (i) Z is a bounded self-adjoint operator which commutes with K0 , −˚ ı ω·∂ and H0 . (ii) X is a bounded anti self-adjoint operator, such that X: dom K0 −→ dom K0 , and adK0 X = R
BH
Proof. First notice that, if W = Σm Wm T m is a bounded operator admitting a ∗ T m , since (T m )∗ = Fourier expansion, then its adjoint is given by W ∗ = Σm W−m −m . T The spectral projectors Qm associated with −˚ ı ω·∂ are given by Z dd θ m def χ = ψm χm ; where χm (θ) = e˚ı m·θ . Qm ψ = ψ(θ) e −˚ı m·θ d (2π) d T If {Pj } is the spectral resolution associated to H0 then Pj and Qm commute. So ı ω·∂ and K0 : we have the following domains for H0 , −˚ ( ) X 2 2 dom H0 = ψ ∈ ; |zj | kPj ψk < ∞ ;
K
j
( ψ∈
dom ω·∂ =
) |ω·m| kQm ψk < ∞ ; 2
2
m
( dom K0 =
K;
X
ψ∈
K;
X
) |ω·m + zj | kQm Pj ψk < ∞ . 2
2
j,m
(i) To prove the first assertion, notice that Z : dom H0 −→ dom H0 and [Z, H0 ] = 0:
406
F. MONTI and H. R. JAUSLIN
H0 Zψ =
X
zj Pj R0 Pj ψ
j
hence,
kH0 Zψk2 ≤ kR0 kH
2
X
|zj |2 kPj ψk2 .
j
BH
Moreover, for any ψ ∈ dom H0 , we have H0 Zψ −ZH0 ψ = 0. Now, as Z ∈ ( ), it commutes with Qm and Z sends dom(ω·∂) into itself. The same is true for dom K0 , as Z commutes also with H0 . This shows that [Z, K0 ] = [Z, ω·∂] = 0. The selfadjointness of R implies easily that of Z, and we have already proven that Z is bounded in Lemma 3.1. (ii) The proof is a little more delicate in this case. By Lemma 3.1, we know that X is a bounded operator, and an easy computation shows that X is anti selfadjoint. Next, if we denote by PF = Σj∈F Pj and by QG = Σm∈G Qm for any finite sets F ⊂ J and G ⊂ Zd , then the operators X X zj Pj = PF H0 = H0 PF and (ω·m + zj ) Pj Qm = PF QG K0 = K0 PF QG j∈F
j∈F m∈G
are bounded and converge strongly respectively to H0 and K0 on their respective domains as F % J and G % Zd . We compute now for a fixed m 6= 0, X Pj Rm Pk (zj + ω·n) Qn QG PF K0 Xm PF 0 = ω·m + zj − zk 0 j∈F,k∈F n∈G
=
X j∈F,k∈F 0 n∈G
zk + ω·(n − m) Qn Pj Rm Pk + ω·m + zj − zk
X
Qn Pj Rm Pk
j∈F,k∈F 0 n∈G
= PF Xm PF 0 QG K0 − ω·m PF Xm PF 0 QG + PF Rm PF 0 QG . Hence, if ψ ∈ dom K0 , we have for any finite sets F, F 0 ⊂ J and G ⊂ Zd that kQG PF K0 Xm PF 0 ψk ≤ kXm k kK0ψk + |ω·m| kXm k kψk + kRm k kψk which gives a uniform bound in the graph norm kψk2 + kK0 ψk2 . Now we have that PF Xm K0 PF 0 QG ψ − ω·m PF Xm PF 0 QG ψ + PF Rm PF 0 QG ψ −→ Xm K0 ψ − ω·m Xm ψ + Rm ψ as F, F 0 % J and G % Zd . This proves that QG PF K0 Xm PF 0 ψ converges to K0 Xm ψ as F, F 0 % J and G % Zd , that Xm: dom K0 −→ dom K0 , and that K0 Xm = Xm K0 − ω·m Xm + Rm
on dom K0 .
The same arguments are true for X0 with obvious changes of notations (see 2.7). Whence, X0 : dom K0 −→ dom K0 and K0 X0 = X0 K0 + R0 − Z on dom K0 . Next, notice that Qn T m = T m Qn−m and that T m commutes with each Pj . So for any ψ ∈ dom K0 , X X (zj + ω·n)Pk Qn T m ψ = (zj + ω·(n + m))T m Pk Qn ψ . K0 T m ψ = j,n
j,n
407
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
This implies that T m: dom K0 −→ dom K0 and that K0 T m = T m K0 + ω·m T m . As X is equal to Σm:|m|1
on dom K0 ,
and
adK0 X + Z = R
(3.3)
K
by extending the operator adK0 X to the whole of . (iii) We will first prove that e ±X : dom K0 −→ dom K0 . This is a consequence of the following result. Assertion. Let L be a selfadjoint operator on a Hilbert space. If M : dom L −→ dom L is a bounded solution of adL M + bM = N with b ∈ C and N a bounded operator, then e sM : dom L −→ dom L for any s ∈ C. Indeed, on dom L we have that LM = M L − bM + N , and for any positive integer r, we have that M r : dom L −→ dom L. Hence, for ψ ∈ dom L we get that L
n X sk k=0
k!
M kψ =
n X sk k=0
k!
M k Lψ +
n k−1 X sk X k=0
k!
M j [L, M ] M k−j−1 ψ .
j=0
This implies that for any n we have that n n
X sk k
X |s|k M ψ ≤ kLM k ψk
L k! k! k=0
k=0
≤
n X |s|k k=0
≤e
k!
|s| kMk
kM kk kLψk + |s|
n−1 X k=0
kLψk + |s| e
|s| kMk
|s|k kM kk kN − b M k kψk k!
kN − b M k kψk < ∞ .
Taking the limit n −→ ∞ shows that e sM : dom L −→ dom L. We have already proven that X : dom K0 −→ dom K0 , and that adX K0 = k Z − R
K
We will now find sufficient conditions on the eigenvalues of H0 and the set of admissible frequencies in order that the hypothesis of Lemma 3.1 are satisfied.
408
F. MONTI and H. R. JAUSLIN
These conditions are exactly the hypothesis made on the spectrum of H0 and the set of admissible frequencies in the first section. 3.4. Lemma. (1) If the spectrum {zj } of H0 is discrete and satisfies zn+1 −zn ≥ g > 0 for all n, then X |zl − zk |−2 ≤ f0 2 < ∞ where f0 = f0 (g) . k:k6=l
(2) Moreover if ω satisfy the following condition for a given m, |ω·m + zl − zk | > γ (|m|1 + |l − k|)−τ
for all l, k ,
then X
|ω·m + zl − zk |−2 ≤ f1 2 |m|1 2τ +1
where f1 = f1 (g, τ, γ, |ω|2 ) .
(3.5)
k
We introduced the Diophantine condition on ω for |m|1 < p• to obtain resonable bounds on kXm k. Those bounds are choosen to allow us to control the size of the e later. rest R Proof. (1) We first notice that |zk − zl | ≥ g |l − k|. It is sufficient to consider the case k > l (the other one is obtained by exchanging the role of the indices), then zk − zl = Σk−1 j=l (zj+1 − zj ) ≥ g (k − l). Hence, we have that, X
|zl − zk |−2 ≤
k:k6=l
∞ 1 X 2 X −2 π2 −2 |l − k| ≤ j = = f0 2 . g2 g 2 j=1 3g 2 k:k6=l
Remark. For X0 to be bounded, the gap hypothesis on sp H0 in 1.2 is essential. Indeed, if zj+1 − zj = g j −a for some a > 0 and all j > 0, then taking Pj R0 Pk = δk,j Pj + δk,j−1 Pj−1 , we have kX0 ϕkH 2 =
=
X
j
X j
≥2
2 Pj ϕ Pj−1 ϕ
+ zj − zj+1 zj − zj−1 H
kPj−1 ϕkH kPj ϕkH + 2 |zj − zj+1 | |zj − zj−1 |2 2
2
!
X kPj ϕk 2 H = 2 X j 2a kPj ϕk 2 . H |zj+1 − zj |2 g2 j≥2
j≥2
H
, but kX0 ϕkH = ∞. So if ϕ satisfies kPj ϕkH = j −1−2a , we have that ϕ ∈ (2) We fix an arbitrary l. We define the set Rm of “resonant indices” as 2
Rm = {k; g |k − l| ≤ 1 + |ω·m|} .
2
409
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
In Rm , we have that |k − l| ≤
1 + |ω|2 1 (1 + |ω·m|) ≤ |m|2 = nω |m|2 ≤ nω |m|1 . g g
This shows that the cardinality of Rm ≤ 2nω |m|1 + 1. On Rm , we will use the Diophantine condition to get an upper bound for the summand: γ 2 (|m|1 + |l − k|)−2τ < |ω·m + zl − zk |2 .
(3.6)
The complementary set NRm of Rm is the set of “non-resonant integers”. In NRm , we have the following lower bound: 1 + |ω·m| < g |l − k| ≤ |zl − zk |, and setting jm = |ω·m|/g, we obtain 1 < g 2 (|l − k| − jm )2 ≤ (|zl − zk | − |ω·m|)2 ≤ |ω·m + zl − zk |2 .
(3.7)
We will now find a bound on the sum in (3.5) by using (3.6) and (3.7): X X X |ω·m + zl − zk |−2 = |ω·m + zl − zk |−2 + |ω·m + zl − zk |−2 k∈Rm
k
≤
k∈NRm
1 X 1 (|m|1 + |l − k|)2τ + 2 γ2 g k∈Rm
X
(|l − k| − jm )−2 .
k∈NRm
The first sum can be estimated by X
nω |m|1
(|m|1 + |l − k|)
2τ
X
≤
X
(nω +1)|m|1
(|m|1 + |j|)
≤2
2τ
j=−nω |m|1
k∈Rm
Z ≤2
j=|m|1
(nω +1)|m|1 +1
j 2τ dj |m|1
=
2 2 ((nω + 1)|m|1 + 1)2τ +1 − |m|1 2τ +1 2τ + 1 2τ + 1
≤
2 (nω + 2)2τ +1 |m|1 2τ +1 2τ + 1
=
2 (|ω|2 + 3)2τ +1 |m|1 2τ +1 . 2τ + 1 g 2τ +1
The second sum can be bounded by X k∈NRm
j 2τ
(|l − k| − jm )−2 ≤ 2
X
j −2 ≤ 2 g 2 +
j>1/g
Z
! j −2 dj
= 2 (g 2 + g) .
j>1/g
Hence, putting together theses estimates and using that |m|1 ≥ 1, we obtain the upper bound X 2 (|ω|2 + 3)2τ +1 + g 2τ (g + 1)γ 2 |m|1 2τ +1 |ω·m + zl − zk |−2 ≤ 2τ +1 2 g γ 2τ + 1 k
= f1 2 |m|1 2τ +1 .
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F. MONTI and H. R. JAUSLIN
We get hence that f1 ≤ B(τ ) g −τ −1/2 γ −1 with B(τ ) finite (it is always possible to impose an upper bound on g, γ without loss of generality). 4. Algebras and Norms
BK
In this section we will introduce some norms on ( ) which take into account the behaviour of the Fourier components of bounded multiplicative operators. These norms will define Banach ∗-subalgebras of ( ) whose elements admit a Fourier series. These algebras are “natural” sets where Nekhoroshev’s estimates are valid.
BK
4.1. Lemma. Let ν : Zd −→ R be a positive function such that ν(m + n) ≤ ν(m) ν(n) and ν(−m) = ν(m). If ν 6≡ 0, then ν(m) ≥ 1, and the application X X ν(m) kLm kH , if L ∼ Lm T m ||| L |||ν = m
m
K
defines a ∗-algebraic norm on the set of bounded multiplicative operators on . Moreover kLk ≤ ||| L |||ν , in particular the set ν = {L = Σm Lm T m ; ||| L |||ν < ∞} is a ∗-subalgebra of ( ). This shows that each element L of ν admits a Fourier expansion.
S
BK
S
Proof. Choose m such that ν(m) > 0. Then ν(m) = ν(0 + m) ≤ ν(0) ν(m), which implies that ν(0) ≥ 1. Then 1 ≤ ν(0) = ν(n − n) ≤ ν(n)2 , and therefore ν(n) ≥ 1 for any n ∈ Zd . Let L, M be bounded multiplicative operators and a ∈ C, then we have that X X aL ∼ a Lm T m L∗ ∼ L∗−m T m m
M +L ∼
X
m
(Lm + Mm ) T
m
LM ∼
m
X X m
! Ln Mm−n
Tm
n
From these expressions and the fact that k·kH is a ∗-algebra norm, it is easy to see that ||| · |||ν is a ∗-algebra norm. Indeed, ||| L |||ν ≥ 0 and if it is equal to 0, then Lm = 0 for all m which implies that L = 0. Next, ||| aL |||ν = |a| ||| L |||ν and ||| L + M |||ν ≤ ||| L |||ν + ||| M |||ν . Using the fact that ν(−m) = ν(m), we get that ||| L∗ |||ν = ||| L |||ν . Finally, X X
X ν(m) Ln Mm−n ≤ ν(m+n) kLm kH kMn kH ≤ ||| L |||ν ||| M |||ν ||| L M |||ν = m
H
n
m,n
S
This implies that ν is a ∗-algebra. Moreover, using that kLm k = kLm kH and kT mk = 1, X
X
X
Lm T m ≤ kLm kH ≤ ν(m) kLm kH = ||| L |||ν . m
S
m
m
B K ) and that each element of Sν admits a Fourier expanSν can be proven the same way as the proof of the fact
This shows that ν ⊂ ( sion. The completeness of
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
411
that L2 (X, dµ) is a Banach algebra (c.f. Proposition 2 of Sec. I.9 in the book of K. Yosida [22]). Examples of such functions ν that we will use, are ν(m) = 1 for any m r ≥ 0, 0 < s ≤ 1
ν(m) = (1 + |m|∗ s )r ν(m) = e r |m|∗
s
r ≥ 0, 0 < s ≤ 1
where | · |∗ is any norm on Rd . Notice that (1 + |m|∗ )s r ≤ (1 + |m|∗ s )r ≤ (1 + |m|∗ )r for 0 < s ≤ 1. We will now find some conditions on families θ 7−→ V (θ) of bounded operator which imply that their embedding in ( ) belongs to one of the algebras on . ν
S
H
BK
4.2. Lemma. Let V : Td −→ θ ∈ Td . Then
B(H ) be a family of operators parametrized by
(i) if the application θ 7−→ V (θ) is holomorphic in a neighbourhood D of the ıy; θ ∈ Td , |y|2 ≤ r}, then V ∈ ν with ν(m) = e r |m|1 ; strip Dr = {ϑ = θ +˚ (ii) if the application θ 7−→ V (θ) is strongly dbrc + 2d differentiable then V ∈ r ν , where ν(m) = (1 + |m|1 ) .
S
S
Proof. (i) By Hypothesis, there exists a u > r such that Du ⊂ D. Since V is holomorphic in D, we have that cu = supϑ∈Du kV (ϑ)kH < ∞. Next, for any ϑ = θ +˚ ıy with |y| ≤ u, Cauchy’s Theorem implies that Z Z d dd θ −˚ ı m·ϑ d ϑ V (θ) e −˚ı m·θ = V (ϑ) e . Vm = (2π)d (2π)d Td Td +˚ ıy This gives us the bound kVm kH ≤ cu e −u |m|1 , by choosing yj = − sgn(mj ) u. Hence, we have that d X X 2 e r |m|1 kVm kH ≤ cu e −|m| (u−r) = cu < ∞ as r < u . 1 − e r−u m m
C
(ii) Suppose that θ 7−→ V (θ) is strongly s , then As = supθ,a:|a|1≤s k∂θa V (θ)kH < Q a ∞, where ∂θa denotes j ∂θjj . Using integration by parts, we get if |a|1 = Σj |aj | ≤ s, Z Vm =
V (θ) e Td
−˚ ı m·θ
dd θ = (2π)d
Z
Y
Td j=1,...,d: mj 6=0
d (−˚ ı)aj aj −˚ ı m·θ d θ ∂ V (θ) e . θ (mj )aj j (2π)d
This gives us the following bound: Z Y
Y aj
1 dd θ
∂ V (θ) kVm kH ≤ θ j H (2π)d ≤ As |mj |aj Td j:mj 6=0
j:mj 6=0
Y j:mj 6=0
1 . |mj |aj
412
F. MONTI and H. R. JAUSLIN
Noting that (1 + |m|1 )r ≤ (d + 1)r X m
Q j:mj 6=0
|mj |r if m 6= 0, we obtain that
(1 + |m|1 )r kVm kH ≤ (d + 1)r
X Y m j:mj 6=0
≤ As (d + 1)r
|mj |r kVm kH
X Y
|mj |r−aj
m j:mj 6=0
= As (d + 1)r
d Y
1 +
= As (d + 1)
|mj |r−aj
mj :mj 6=0
j=1
r
X
1+2
∞ X
!d k
r−aj
k=1
The last term is finite if aj > r + 1. Hence, we need to take aj = brc + 2, for each j. This is possible if s ≥ dbrc + 2d. There is a reciprocal result which allows to assign to each element of smooth application from Td to ( ):
BH
Sν
a
S
4.3. Lemma. For each V = Σm Vm T m ∈ ν , let us define V (θ) = Σm Vm e˚ı m·θ as a family of operators on ( ) depending parametrically on θ ∈ Td .
BH
(i) If ν(m) = e r |m|1 , then the application θ 7−→ V (θ) is holomorphic in the strip {ϑ; <e ϑ ∈ Td , | =m ϑ|2 < r}. s (ii) If ν(m) = e r |m|1 for some r > 0 and 0 < s ≤ 1, then θ 7−→ V (θ) is ∞ . (iii) If ν(m) = (1 + |m|1 )r , then θ 7−→ V (θ) is r−1 in norm.
C
C
Proof. (i) Suppose that V = Σm Vm T m satisfies Σm e r |m|1 kVm kH < ∞. Then, , for any ϑ such we can define the family of operator V (ϑ) = Σm Vm e˚ı m·ϑ on that | =m ϑ|2 ≤ r. Indeed, we have that
H
kV (ϑ)kH ≤
X m
e |m·=m ϑ| kVm kH ≤
X m
e |m|1 r kVm kH < ∞ ,
BH
which shows that ϑ 7−→ V (ϑ) ∈ ( ) on the set Dr = {ϑ; <e ϑ ∈ Td , | =m ϑ|2 ≤ r}, because by definition: V (ϑ + n) = V (ϑ) for any n ∈ Zd . Moreover, if | =m ϑ|2 < r, we have for any a = (a1 , . . . , ad ), ∂ϑa V (ϑ) =˚ ı |a|1
X m
ma Vm e˚ı m·ϑ
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
where ma =
Q j
413
a
mj j . Hence,
k∂ϑa V (ϑ)kH ≤
X m
≤
|ma | e |m|1 | =m ϑ|2 kVm kH
|a|1 e (r − | =m ϑ|2 )
|a|1 X m
e |m|1 r kVm kH < ∞ ,
|a|
where we have used that |ma | ≤ |m|1 1 . (ii) The same arguments as above shows that V (θ) = Σm Vm e˚ı m·θ is a bounded operator and that k∂θa V
(θ)kH ≤
2 |a|1 esr
|a|s 1 X m
e |m|1 r/2 kVm kH < ∞ . s
C
Showing that θ 7−→ V (θ) is ∞ . (iii) As in the previous point, we know that V (θ) is a bounded operator on and if |a|1 ≤ r, X X |ma | kVm kH ≤ (1 + |m|1 )|a|1 kVm kH < ∞ . k∂θa V (θ)kH ≤ m
To show that θ 7−→ V (θ) is k∂θa V
(θ) −
∂θa V
H,
m
C r−1, we consider 0
(θ )kH ≤ ≤
X m
X m
≤
X
|m | kVm kH a
|ma | kVm kH |a|1 +1
|m|1
m
d X
0
|e˚ı mj θj − e˚ı mj θj |
j=1 d X
| mj | |θj − θj0 |
j=1
kVm kH |θ − θ0 |2 .
Thus, if |a|1 + 1 ≤ r, we have that θ 7−→ ∂θa V (θ) is norm continuous.
In the following, we will restrict the discussion to the cases ν(m) = (1 + |m|1 )r s where r ≥ 0 or ν(m) = e r |m|1 with r ≥ 0, 0 < s ≤ 1. This motivates the following notations: 4.4. Definition. We introduce the following two families of Banach ∗-subalgebras of ( ) indexed by r ≥ 0:
BK
(i) the differentiable class, when ν(m) = 1 + |m|1 : ( ) X L; ||| L |||r = ν(m)r kLm kH < ∞ ; r =
D
m
414
F. MONTI and H. R. JAUSLIN
(ii) the quasi-analytic class, when ν(m) = e |m|1 for some fixed 0 < s ≤ 1:b ( ) X r L; ||| L |||r = ν(m) kLm kH < ∞ . r = s
A
m
The analytic case corresponds to s = 1. Notice that if r0 < r, then ||| L |||r0 ≤ ||| L |||r , and hence
Dr ⊂ Dr
0
and
Ar ⊂ Ar . 0
5. Iterative Lemma and KAM Algorithm In this section we will describe one iterative step of the procedure used to reduce the size of the rest. We will obtain estimates of the norms of the rest after the transformation. We will study the iterative procedure with a Floquet Hamiltonian of the form K0 + Y + R, where Y belongs to ( ) and commutes with H0 and K0 , and where R ∈ r . The symbol r denotes either r or r . We recall that this implies that R admits a Fourier expansion and, as we have shown in Lemmas 3.1 and 3.4, the solution X of Eq. (2.4) admits a Fourier expansion whose components τ +1/2 kRm kH for 0 < |m|1 < p• satisfy kX0 kH ≤ f0 kR0 kH and kXm kH ≤ f1 |m|1 and Xm = 0 if |m|1 ≥ p• . In order to be able to treat simultaneously the differentiable and quasi-analytic cases, we will need some notations which will be motivated in the proof of the next result.
S
S
BH
D
A
Consider the bounding constants f0 and f1 obtained in
5.1. Notations. Lemma 3.4. Then
(i) for the differential case, we denote by δ = τ + 1/2 and fδ = max{f0 , f1 }; (ii) in the quasi-analytic case, we will choose δ > 0, to be fixed later, and fδ = max{f0 , f1 (b/e)b } δ −b , where b = (2τ + 1)/2s. (iii) We will also make a slight abuse of notation in this section by writing s ν(p• ) = 1 + p• , respectively ν(p• ) = e p• , for p• ∈ N in the differential, respectively quasi-analytic, case. 5.2. Lemma. Consider the Floquet Hamiltonian K0 + Y + R, where K0 = −˚ ı ω·∂ + H0 , Y ∈ ( ) commutes with H0 and R ∈ r . Let (X, Z) be the solution of the equation adK X + Z = R
BH
S
ad Z = 0 K0 e Then, for any r0 ≤ r − δ, given by (2.7), and e X (K0 + Y + R) e −X = K0 + Ye + R. (i) Ye = Y + Z ∈
B(H ) commutes with H0 and kZkH ≤ kR0 kH ≤ ||| R |||r ;
b This class is equivalent to the Gevrey class of order 1/s.
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
(ii) X ∈
Sr
0
415
and its norm can be estimated by ||| X |||r0 ≤ fδ ||| R
e∈ (iii) R
Sr
0
and e |||r0 ≤ ||| R
3 ν(p• )−δ + (e 2 ||| X |||r0 − 1) ||| R |||r0 +δ 2 1 + (e 2 ||| X |||r0 − 1) (kY kH + kZkH ) 2
Proof. (i) This is a direct consequence of Lemmas 3.2 and 3.1. (ii) We have obtained the following estimates in Lemmas 3.1 and 3.4: τ +1/2
kX0 kH ≤ f0 kR0 kH
and kXm kH ≤ f1 |m|1
Hence for any r0 ≤ r − δ, we have X 0 ν(m)r kXm kH ≤ f0 kR0 kH + f1 ||| X |||r0 = m
kRm kH , if 0 < |m|1 < p• .
X
0
τ +1/2
ν(m)r |m|1
kRm kH .
m:0<|m|1
Now, in the differentiable case, ν(m) = 1 + |m|1 and we took δ = τ + 1/2, this gives us X 0 ν(m)r +δ kRm kH = fδ ||| R
In the quasi-analytic case, ν(m) = e |m|1 and we have taken a δ ∈ (0; 1) satisfiying s δ < r. Using the inequality xa e −δx ≤ (a/(e δs))a/s , we find that X 0 s τ +1/2 ||| X |||r0 ≤ f0 kR0 kH + f1 e r |m|1 |m|1 kRm kH s
m:0<|m|1
≤ f0 kR0 kH + f1 ≤ max{f0 , f1
b eδ
b
−b b b e }δ
X
e (r
0
+δ) |m|s1
kRm kH
m:0<|m|1
X
ν(m)r
0
+δ
kRm kH = fδ ||| R
m:|m|1
Where we have set b = (2τ + 1)/2s. e is given by the following expression: (iii) Recall that R e = R≥p• + R
∞ ∞ X X 1 1 adlX (Y + R) + adlX (Z − R
l=1
Using that ||| · |||r0 is a ∗-algebra norm, we get inductively that M |||r0 ≤ · · · ≤ 2l ||| X |||lr0 ||| M |||r0 ||| adlX M |||r0 ≤ 2 ||| X |||r0 ||| adl−1 X for any bounded operator M . On the other hand using the expression of this norm, we find that
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F. MONTI and H. R. JAUSLIN
X
||| R≥p• |||r0 =
0
m:|m|1 ≥p•
≤ ν(p• )−δ
ν(p• )r kRm kH X
0
m:|m|1 ≥p•
ν(p• )r +δ kRm kH = ν(p• )−δ ||| R≥p• |||r0 +δ .
Putting together theses estimates, and using the inequality e x −x−1 ≤ x (e x −1)/2, we obtain, e |||r0 ≤ ν(p• )−δ ||| R≥p• |||r0 +δ + ||| R
∞ X 2l ||| X |||l 0 r
l=1
+
∞ X 2l−1 ||| X |||l−1 0 r
l=2
l!
l!
(||| Y |||r0 + ||| R |||r0 )
(||| Z |||r0 + ||| R
≤ ν(p• )−δ ||| R |||r0 +δ + (e 2 ||| X |||r0 − 1) (||| Y |||r0 + ||| R |||r0 ) +
e 2 ||| X |||r0 − 2 ||| X |||r0 − 1 (||| Z |||r0 + ||| R |||r0 ) 2 ||| X |||r0
≤ ν(p• )−δ ||| R |||r0 +δ + +
3 2 ||| X |||r0 (e − 1) ||| R |||r0 2
e 2 ||| X |||r0 − 1 (2 ||| Y |||r0 + ||| Z |||r0 ) . 2
Finally, using the fact that ||| Y |||r0 = kY kH and ||| Z |||r0 = kZkH , we obtain the e asserted estimate on the r0 -norm of R. 5.3. Remark. The choice of the norm and of the Diophantine condition on ω allows us to treat the problems related to the apparition of small denominators in the definition of X. But this is done at the cost of a loss in regularity. Indeed, as we have seen in the preceeding section, the parameter r of the norm ||| · |||r is related to the degree of smoothness of the operator. After one step, this degree is diminished by δ. We will now iterate the procedure described by the above Lemma. We will also show that the size of the rest decreases at each step if the size of the initial perturbation is small enough. We introduce the following notation: K (n) = K0 + Y (n) + R(n) denotes the Floquet Hamiltonian obtained after n KAM transformations. We start with K (0) = K0 + V , i.e. Y (0) = 0 and R(0) = V , and we suppose that V belongs to r for some r large enough. Then we have the following iterative result.
S
5.4. Lemma. Suppose that p• satisfies the inequality ν(p• )δ ≥ 2a, r0 ≤ r and that V ∈ r with 1 1 for some a ≥ 2 . log 1 + ||| V |||r ≤ 2fδ 4a
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QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
417
Then, for any positive integer q satisfying r0 +qδ ≤ r, we can find a unitary transform −1 = K (q) = U (q) preserving the domain of K0 and such that U (q) (K0 + V )U (q) (q) (q) (q) + R , where Y ∈ ( ) commutes with H0 and K0 + Y
BH
kY (q) kH ≤
a − a−q+1 ||| V |||r0 +qδ a−1
and ||| R(q) |||r0 ≤
1 ||| V |||r0 +qδ . aq
Proof. The proof of the Lemma will be done inductively. The conclusion is true for q = 0, because the initial conditions are Y (0) = 0 and R(0) = V . Suppose that for some n satisfying (n + 1)δ ≤ r − r0 , we have found a unitary transform U (n) −1 such that K (n) = U (n) (K0 + V )U (n) = K0 + Y (n) + R(n) where Y (n) and R(n) satisfy the conclusion of the Lemma with q replaced by n. Then by Lemma 5.2, we can find a solution (X (n) , Z (n) ) of Eq. (2.4) such that Z (n) ∈ ( ) commutes with H0 and which satisfies the estimates:
BH
||| X (n) |||r0 ≤ fδ ||| R(n) |||r0 +δ ≤ kZ (n) kH ≤ ||| R(n) |||r0 ≤
fδ ||| V |||r0 +(n+1)δ an
1 ||| V |||r0 +(n+1)δ an
by induction hypothesis. Setting e X K (n) e −X = K0 + Y (n+1) + R(n+1) , we (n+1) (n) (n) =Y +Z belongs to ( ) and commutes with H0 , and have that Y (n)
(n)
BH
kY (n+1) kH ≤ kY (n) kH + kZ (n) kH ≤
a − a−n+1 1 ||| V |||r0 +nδ + n ||| V |||r0 +(n+1)δ a−1 a
≤
a − a−n ||| V |||r0 +(n+1)δ . a−1
Using point (iii) of Lemma 5.2 and the inductive hypothesis, we get the following estimate on the r0 -norm of R(n+1) : (n) 3 ||| R(n+1) |||r0 ≤ ν(p• )−δ + (e 2||| X |||r0 − 1) ||| R(n) |||r0 +δ 2 (n) 1 + (e 2||| X |||r0 − 1)(kY (n) kH + kZ (n) kH ) 2
≤
ν(p• )−δ ||| V |||r0 +(n+1)δ an +(e
≤
1 an
2fδ /an ||| V |||r0 +(n+1)δ
− 1)
n ν(p• )−δ + (e 2fδ /a ||| V
|||r
2 a − a−n+1 + n a−1 a − 1)
an+1 + a − 2 a−1
||| V |||r0 +(n+1)δ ||| V |||r0 +(n+1)δ .
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F. MONTI and H. R. JAUSLIN
This shows that ||| R(n+1) |||r0 ≤ a−n−1 ||| V |||r0 +(n+1)δ , if we have ν(p• )−δ ≤
1 2a
and (e 2fδ ||| V |||r /a − 1) n
1 an+1 + a − 2 ≤ . a−1 2a
This implies the following bounds on the size of p• and V : an a−1 . log 1 + ν(p• )δ ≥ 2a and ||| V |||r ≤ 2fδ 2a (an+1 + a − 2)
(5.5)
Using the inequalities x/(x + 1) ≤ log(1 + x) ≤ x for x ≥ 0, it can be readily seen that the function a−1 n 7−→ ga (n) = an log 1 + 2a (an+1 + a − 2) is increasing if a ≥ 2. Hence the inequalities (5.5) will be satisfied for any a ≥ 2, if 1 1 1 . ν(p• ) ≥ 2a and ||| V |||r ≤ ga (0) = log 1 + 2fδ 2fδ 4a δ
(5.6)
So, provided that p• and V satisfy the inequalities (5.6) that r0 +(n+1)δ ≤ r, we (n) have shown that we can take U (n+1) = e X U (n) , to obtain K (n+1) = U (n+1) (K0 + −1 V )U (n+1) = K0 +Y (n+1) +R(n+1) where Y (n+1) and R(n+1) satisfy the conclusion of the Lemma with q replaced by n + 1. This concludes the induction step and the proof. 6. Estimates of the Size of the Rest In this section we will minimize the size of the rest R(q) , obtained after q KAM iterations as described in the proof of Lemma 5.4, viewed as a function of the Diophantine threshold p• or of the size of the perturbation ||| V |||r . The differentiable case
D
We begin by considering the differentiable case, i.e. V ∈ r for some r ≥ r0 + τ + 1/2 where r0 ≥ 0 is a fixed constant which represents the degree of regularity in θ that will be left for the rest R∗ . In the formulation of Theorems 1.6 and 1.8 we took r0 = 0, i.e. we exploited all what is possible from the regularity of V in order to diminish the size of the rest. But this is not a necessity, as we will see it in the proofs below. We have shown in Lemma 4.2, that V ∈ r if the family of bounded operator θ 7−→ V (θ) is strongly k for a k ≥ dbrc + 2d. Recall also that in the differentiable case V ∈ r , we have set δ = τ + 1/2 and ν(m) = 1 + |m|1 . We will denote fδ = max{f0 , f1 } simply by f . As we have seen in Lemma 5.4, we can perform at most q∗ iterative steps to diminish the size of the rest, where q∗ is the greatest integer such that r0 + qδ ≤ r. We will then minimize the size of R∗ = R(q∗ ) by maximizing the stretching factor a of Lemma 5.4 with respect to p• or ε = ||| V |||r .
D
C
D
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
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6.1. Lemma. Suppose that we are given an integer p• ≥ 1 and that V belongs to r for some r ≥ r0 + δ and satisfies ||| V |||r ≤ 1/2f log(1 + 1/2(1 + p• )δ ), where δ = τ + 1/2. Then, for any frequency ω ∈ Ωp• , there exists a unitary transform U∗ , preserving the domain of K0 , such that U∗ (K0 + V )U∗ −1 = K0 + Y∗ + R∗ , where Y∗ ∈ ( ) commutes with H0 ,
D
BH
kY∗ kH ≤ 2 ||| V |||r
and kR∗ k ≤ ||| R∗ |||r0 ≤ ||| V |||r
1 + p• 21/δ
−r+r0 +δ .
Proof. Suppose first that p• ≥ 41/δ − 1. Choosing q∗ as b(r − r0 )/δc ≥ 1, Lemma 5.4 implies the existence of a unitary operator U (q∗ ) = U∗ transforming K0 + V in K0 + Y∗ + R∗ with Y∗ ∈ ( ) commuting with H0 under the condition that (1 + p• )δ ≥ 2a and ||| V |||r ≤ log(1 + 1/4a) for some a ≥ 2. Such an a always exits by the lower bound on p• . Choosing ap• as the maximal real such that a ≤ (1 + p• )δ /2, we get that ap• ≥ 2 and the asserted condition on ||| V |||r . The bounds obtained in Lemma 5.4 give
BH
kY∗ kH ≤ ||| R∗ |||r0 ≤
ap• ||| V |||r ≤ 2 ||| V |||r ap• − 1 ∗ a−q p•
||| V |||r =
(1 + p• )δ 2
0 +1 − r−r δ
||| V |||r ,
as q∗ ≥ (r − r0 )/δ − 1. . Then, Now, if p• < 41/δ − 1, we can take for U∗ the identity operator on we have that Y∗ = 0 and R∗ = V and it is easy to see that the stated bounds are satisfied because we have that (1 + p• )/21/δ > 1.
K
The second result relates the size of the rest with the r-norm of V .
D
6.2. Lemma. Suppose that V ∈ r for some r ≥ r0 + δ and that ε = ||| V |||r ≤ ε∗ , where δ = τ + 1/2 and ε∗ = log(9/8)/2f. Let pε = b(2 ε∗ /ε)1/δ c. Then for any frequency in the set Ωε = {ω; |ω·m + zl − zk | ≥ γ (|m|1 + |l − k|)−τ for 0 < |m|1 < pε } , there exist a unitary transform U∗ such that U∗ (K0 + V )U∗ −1 = K0 + Y∗ + R∗ , where Y∗ ∈ ( ) commutes with H0 ,
BH
kY∗ kH ≤ 2 ε and
kR∗ k ≤ ||| R∗ |||r0 ≤ ε∗
ε ε∗
r−r0 δ
.
6.3. Remark. Notice that the set of admissible frequencies dimishes as ε & 0 because pε % ∞. Proof. Again we use Lemma 5.4 with q∗ = b(r − r0 )/δc, and set U∗ = U (q∗ ) , Y∗ = Y (q∗ ) and R∗ = R(q∗ ) . To obtain the relation between the size of ε = ||| V |||r
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F. MONTI and H. R. JAUSLIN
and ||| R∗ |||r0 , we choose a as a function of ε by setting aε = ε∗ /ε. We will now determine the best value for ε∗ . We have that e 2f ε − 1 ≤ e 2f ε∗ − 1 ≤
1 1 ≤ 8 4aε
for ε ≤ ε∗ .
Using the condition aε ≥ 2, we see that we must take ε∗ = log(9/8)/2f . This choice gives then that kY∗ kH
aε ε ≤ 2ε ≤ aε − 1
and ||| R∗ |||r0 ≤
∗ a−q ε
ε = ε∗
ε ε∗
q∗ +1
≤ ε∗
ε ε∗
r−r0 δ
.
This imposes the condition that p• must satisfy (1 + p• )δ ≥ 2aε ≥ 2ε∗ /ε. The least such integer pε is given by b(2ε∗ /ε)1/δ c. The quasi-analytic case
A
In this case we have V ∈ r with ||| V |||r = Σm ν(m)r kV kH , where ν(m) = e |m|1 for some fixed 0 < s ≤ 1. In Lemma 4.2, we have shown that if the family V : Td −→ ( ) is holomorphic, then V belongs to r for some r > 0 and s = 1. Recall also that in this case we have fδ = max{f0 , f1 (b/e)b } δ −b = f δ −b , where b = (2τ + 1)/2s and δ > 0 is a parameter not yet choosen, apart that it must satisfy δ ≤ min{1; r}. To obtain the relations between the size of the rest and p• or ε = ||| V |||r , we will in this case optimize the number of iterations of the KAM algorithm and δ. s
BH
A
6.4. Lemma. Given a p• ≥ 1, suppose that ω ∈ Ωp• and that V is in some r > r0 ≥ 0 and satisfies ||| V |||r ≤
Ar for
1 def log(4)b log(9/8) p• −sb = εp• , 2f
where b = (2τ + 1)/2s. Then, there exists a unitary U∗ such that U∗ (K0 + V )U∗ −1 = K0 + Y∗ + R∗ with Y∗ ∈ ( ) commuting with H0 . Moreover we have the following estimates:
BH
kY∗ kH ≤ 2 ||| V |||r
and
||| R∗ |||r0 ≤ 2 ||| V |||r e −
r−r0 p s • 2
.
In this lemma r0 represents again the degree of quasi-analycity left for the rest R∗ . In the statement of Theorem 1.6, we put r0 = 0. Proof. We will use Lemma 5.4 with a = 2. The idea is to optimize the number of iterative steps q in order to minimize the size of R(q) . To be able to apply the KAM algorithm, we need e δ p• ≥ 4 and 2f δ −b ||| V |||r ≤ log(9/8) . s
We first consider the case when p• s ≥ log(4)/(r − r0 ). Then, we can take δ = (r − r0 )/q where q ≥ 1 is the number of iterations we will perform. This integer is bounded by r − r0 s r − r0 ≤ p• . q= δ log(4)
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
421
Taking q∗ as the greatest integer satisfying this inequality, we have that r − r0 s r − r0 s p• ≥ p• − 1 . q∗ = log(4) log(4) We need to satisfy the bound on ||| V |||r so, if we impose that q∗ b ||| V |||r ≤
(r − r0 )b sb (r − r0 )b log(9/8) , p• ||| V |||r ≤ b log(4) 2f
we get the wanted upper estimate on 2f δ −b ||| V |||r . This gives us the following bound: 1 log(4)b log(9/8) p• −sb = εp• . ||| V |||r ≤ 2f If these bounds are satisfied and setting Y∗ = Y (q∗ ) , R∗ = R(q∗ ) , we obtain the following estimates kY∗ kH ≤ 2 ||| V |||r
and ||| R∗ |||r0 ≤ 2−q∗ ||| V |||r ≤ 2 e −k p• ||| V |||r , s
where k = (r − r0 )/2. If p• s < log(4)/(r − r0 ), we take U∗ = 1lK , and hence Y∗ = 0 and R∗ = V . It is not difficult to see that the estimates announced in the Lemma are satisfied. s Indeed, we have that 2 e −k p• > 1 in this case. 6.5. Lemma. Let V ∈
Ar for some r > r0 ≥ 0 satisfy
ε = ||| V |||r ≤
1 1 (r − r0 ) bs log(9/8) = ε∗ . 2f
Then for any frequency in the set Ωε = {ω; |ω·m + zl − zk | > γ (|m|1 + |l − k|)−τ for 0 < |m|1 < pε where pε = (log(4)/(r − r0 ))1/s (ε∗ /ε)1/bs , we can find a unitary transform U∗ such that U∗ (K0 + V )U∗ −1 = K0 + Y∗ + R∗ with Y∗ ∈ ( ) commutes with H0 ,
BH
kY∗ kH ≤ 2 ε
and
||| R∗ |||r0 ≤ 2 ε 2−(ε∗/ε)
1/b
≤ e ε e −(ε∗ /ε)
1/b
,
where b = (2τ + 1)/2s. Notice that as in the differential case, the set of admissible frequencies diminishes with the size ε of V when it tends to 0. Proof. To be able to perform q KAM iterations with δ = (r − r0 )/q, we need that 1 b b b = ε∗ . q ||| V |||r = q ε ≤ 2f (r − r0 ) log 1 + 4a s
So we can do at most q∗ = b(ε∗ /ε)1/b c steps. By the inequality e δ p• ≥ 2a, we see that this choice imposes the following restriction on p• :
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F. MONTI and H. R. JAUSLIN
p• ≥
log(2a) q∗ r − r0
1s .
The least such integer satisfying this inequality is 1 1 log(2a) s ε∗ bs +1. pε = r − r0 ε
(6.6)
For these choices of pε and q∗ , we get when setting U∗ = U (q∗ ) , Y∗ = Y (q∗ ) and R∗ = R(q∗ ) the wanted estimates kY∗ kH ≤
a ε a−1
and ||| R∗ |||r0 ≤ a−q∗ ε = a ε a−(ε∗ /ε)
1/b
.
Taking a = 2 in the estimates above minimizes the size of R∗ . Remark. If we inject the value pε obtained in (6.6) in the preceeding lemma, we get the same estimate for ||| R∗ |||r0 . 7. The Spectrum of
K0 +Y∗ and the Comparison of the Evolutions
In this section we will prove that the spectrum of K0 + Y∗ is pure point and that the evolutions associated respectively to K0 + V and U∗ −1 (K0 + Y∗ )U∗ do not differ too much for times of the order of the inverse of the size of R∗ . This will complete the proofs for the case when the perturbation does not depend on time. ı ω·∂. Moreover We have shown that Y∗ is in ( ) and hence commutes with −˚ the construction showed that Y∗ commutes also with H0 (Lemma 5.4): we have that X zk Pk + Pk Y∗ Pk H0 + Y∗ =
BH
k
H
which is an (unbounded) self-adjoint operator on composed as an orthogonal sum of finite dimensional blocks of matrices (by the hypothesis on the spectrum of H0 ). Hence, H0 + Y∗ can be diagonalized by blocks, which shows that its spectrum ı ω·∂. We have is pure point. Moreover, as H0 + Y∗ ∈ ( ), it commutes with −˚ proved the following result, which shows the splitting of the eigenfamily associated to K0 + Y∗ .
LH
7.1. Lemma. The spectrum of K0 + Y∗ is pure point and its eigenvalues are given by ω·m + z∗j for m ∈ Zd and j ∈ J∗ . Here, z∗j are the eigenvalues of H0 + Y∗ viewed as an operator on (degenerate eigenvalues are labeled by different indices). The corresponding eigenvectors are simply given by products: ψm,j (θ) = ϕ∗ j e˚ı m·θ , is the eigenvector corresponding to z∗j . where ϕ∗ j ∈
H
H
7.2. Remark. The fact that the eigenprojectors associated to H0 are finite dimensional is essential. Indeed, if one of the projectors were infinite dimensional, then it would be very simple to construct an example which admits a continuous spectrum by taking, in the eigenspace associated to this projector, a perturbation
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
423
which is unitarily equivalent to the discrete Laplacian in `2 (Z). This perturbation is independent of θ and commutes with H0 , and thus the KAM algorithm leaves the Floquet Hamiltonian unchanged.c Now we will compare the two evolutions associated to K∗ = U∗ −1 (K0 + Y∗ )U∗ and K0 + V . Notice that we need to transform back K0 + Y∗ via U∗ in order to be able to compare them. In case of time independent potential, the evolutions associated to K∗ and K0 + V are given by W∗ (t, s) = e −˚ı(t−s) K∗ and W (t, s) = e −˚ı(t−s) (K0 +V ) respectively. 7.3. Lemma. For any η > 0, we have ke −˚ıt (K0 +V ) − e −˚ıt K∗ k ≤ η
for |t| <
η . ||| R∗ |||r0
Proof. In order to prove this lemma we will need the following result. Assertion. If L is a self-adjoint operator on a Hilbert space and M is a bounded symmetric operator on dom L, then L + M is a self-adjoint operator on dom L and for any t ∈ R, we have the following estimates for their evolutions: ke −˚ıt (L+M) − e −˚ıt L k ≤ |t| kM k . Assuming the assertion, the proof can be done as follows: if U∗ : dom K0 −→ dom K0 is the unitary transform obtained in the last section, then U∗ (K0 +V )U∗ −1 = K0 + Y∗ + R∗ . Using the assertion we have ke −˚ıt (K0 +V ) − e −˚ıt K∗ k = ke −˚ıt (K0 +V ) − U∗ −1 e −˚ıt (K0 +Y∗ ) U∗ k = kU∗ −1 (U∗ e −˚ıt (K0 +V ) U∗ −1 − e −˚ıt (K0 +Y∗ ) )U∗ k = k(e −˚ıt (K0 +Y∗ +R∗ ) − e −˚ıt (K0 +Y∗ ) )U∗ k ≤ |t| kR∗ k kU∗ k ≤ |t| ||| R∗ |||r0 ≤ η , if |t| ≤ η/||| R∗ |||r0 . We now prove the assertion. By Theorem V.4.3, p. 287 in [18], we know that L + M is self-adjoint on dom L. By Stone’s Theorem, we obtain the existence of the associated evolutions e −˚ıt (L+M) and e −˚ıt L which are differentiable on dom L and leave dom L invariant. So for any ϕ ∈ dom L and any t ∈ R, we get t (e −˚ıt (L+M) − e −˚ıt L ) ϕ = (e −˚ıs (L+M) e −˚ı(t−s) L ) ϕ s=0
Z
t
d −˚ıs (L+M) −˚ı(t−s) L (e e ϕ)ds ds
= 0
Z
=−
t
(e −˚ıs (L+M)˚ ıM e −˚ı(t−s) L ϕ)ds .
0 c More generally, the same kind of arguments show that the spectrum of H cannot admit an 0 accumulation point, in order to guarantee that sp(K0 + Y∗ ) is pure point.
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F. MONTI and H. R. JAUSLIN
The differentiation is justified because ϕ and e −˚ı(t−s) L ϕ ∈ dom L and the integral exists because the integrands are continuous functions. Taking norm inequalities, we easily obtain that k(e −˚ıt (L+M) − e −˚ıt L ) ϕk ≤ |t| kM k kϕk for any t ∈ R, ϕ ∈ dom L . Now, dom L being a dense set, an /3-argument allows to extend this inequality to the whole Hilbert space. 8. Nekhoroshev Estimates for Time-Dependent Perturbations In this section we will extend the previous results to the case when the perturbation admits an extra time dependence: (t, θ) 7−→ V (t; θ). Under the regularity hypothesis 1.5, t −→ V (t) is strongly 1 with a bounded derivative. The notasatisfying (L(t) ψ)(θ) = tion L(t) will be given for time-dependent operators on L(t; θ) ψ(θ). Hence, we can construct the unitary propagator W (t, s) which gives the evolution associated to K0 + V (t), by Theorem XIV.4.1 in [22] or Remark 6.2 in [17]. Moreover, under those regularity hypothesis, V (t) admits a Fourier expansion V (t) = Σm Vm (t) T m . Applying, for fixed t, the KAM algorithm to K0 + V (t) as described in the preceeding sections, we obtain that
C
K
U∗ (t)(K0 + V (t))U∗ (t)−1 = K0 + Y∗ (t) + R∗ (t)
BH
where Y∗ (t) ∈ ( ) commutes with H0 and Y∗ (t), R∗ (t) satisfy the same estimates as in Lemma 6.1, 6.2, 6.4 or 6.5 for each t. But in order to be able to prove that a strongly differentiable unitary propagator associated to K0 + Y∗ (t) + R∗ (t) or K∗ (t) = U∗ (t)−1 (K0 + Y∗ (t))U∗ (t) exists, we need to show that U∗ (t), Y∗ (t) and R∗ (t) remain strongly 1 in t with a bounded derivative and that the time derivative of U∗ (t) leaves also dom K0 invariant. To prove this, we will show that each step of the iterative procedure maintains the differentibility properties in t and that the Fourier components are also differentiable and uniformly bounded. We have the following result.
C
8.1. Lemma. If (t, θ) − 7 → V (t; θ) satisfies one of the regularity hypothesis 1.5 for some r, then V and ∂t V belong to r,∞ = {L(t) ∈ r ; Σm ν(m)r supt kLm (t)kH < ∞}.
S
S
S
The proof that V ∈ r,∞ can be done by adapting that of Lemma 4.2. By con, we can prove, using the same kind of arguments, sidering ∂t V (t)ψ for any ψ ∈ that ∂t V ∈ r,∞ .
S
K
S
C
8.2. Lemma. Suppose that t 7−→ R(t) ∈ r,∞ is strongly 1 and that its derivative belongs to r,∞ . Then each of its Fourier component is strongly 1 with a uniformly bounded strong derivative. Moreover, the Fourier series associated to R(t) can be differentiated term by term, and
S
C
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
∂t R(t) = ∂t
X m
∂t R(t) ∈
Sr,∞
Rm (t) T m =
X
425
∂t Rm (t) T m
m
and ||| ∂t R(t) |||r ≤
X m
ν(m)r sup k∂t Rm (t)kH . t
The proof of the strong differentiability of each Fourier component and of the term by term differentiability is an application of Lebesgue’s dominated convergence theorem or equivalently of Weierstraß’ Theorem on uniform convergence of series. We will now prove that the regularity properties in t of the perturbation V remain stable during the iterative procedure, if the general hypothesis made in Sec. 1 about the spectrum of H0 and the “finite” Diophantine condition on the frequency ω are satisfied.
S
8.3. Lemma. Suppose that Y (t) and R(t) ∈ r0 ,∞ are strongly continuously differentiable with bounded derivatives in r0 ,∞ and that Y (t) ∈ ( ) commutes with H0 . Then, we can find a pair (X(t), Z(t)) of solutions of the equation adK X(t) + Z(t) = R
S
BH
ad Z(t) = 0 K0
C
S
which are strongly 1 and belong together with their derivative to r0 −δ,∞ . Moreover, Z(t) ∈ ( ) and commutes with H0 , and X(t), ∂t X(t): dom K0 −→ dom K0 .
BH
Proof. Recall that the solutions of this equation are given by their Fourier components: X X Pj R0 (t)Pk if m = 0 zj − zk j k:k6 = j X X Pj Rm (t)Pk Xm (t) = if 0 < |m|1 < p• ω·m + zj − zk j k (8.4) 0 if |m|1 ≥ p• X Pj R0 (t)Pj if m = 0 j Zm (t) = 0 otherwise where the Pj are the eigenprojectors associated to H0 . The Lebesgue’s dominated convergence Theorem allows us to conclude the differentiability of the solution (X(t), Z(t)) and the uniform boundness of its derivative. The rest of the properties and the existence of these operators can be justified as in the time independent case. Using this, the Lebesgue’s dominated convergence Theorem and the same arguments as in the time independent case, it is not difficult to prove the following.
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8.5. Lemma. Let (X(t), Z(t)) the solution of Eq. (8.4) and Y (t), R(t) satisfy the same hypothesis as in the preceeding lemma. Then, e ±X(t) : dom K0 −→ dom K0 and e , e X(t) (K0 + Y (t) + R(t))e −X(t) = K0 + Ye (t) + R(t)
BH
where Ye (t) = Y (t) + Z(t) ∈ ( ) and commutes with H0 . Moreover e ±X(t) , e Ye (t) and R(t) are strongly 1 on dom K0 with a uniformly bounded derivative, e e ∈ r0 −δ,∞ . ∂t R(t) ∂t e ±X(t) : dom K0 −→ dom K0 and R(t),
C
S
S
This proves that, if V (t) and ∂t V (t) are in r,∞ , then we can apply the same arguments as in the time independent case to obtain a unitary transform U∗ (t) : dom K0 −→ dom K0 such that U∗ (t)(K0 + V (t))U∗ (t)−1 = K0 + Y∗ (t) + R∗ (t)
BH
with Y∗ (t) ∈ ( ) commuting with H0 which again implies that the instantaneous spectrum of K0 + Y∗ (t) is pure point with a splitted eigenfamily for each t. Moreover, U∗ (t)±1 , Y∗ (t) and R∗ (t) belong to r0 ,∞ and are strongly 1 with bounded derivatives in r0 ,∞ , with U∗ (t)±1 , ∂t U∗ (t)±1 : dom K0 −→ dom K0 . By Remark 6.2 in [17], these conditions assert that the unitary propagator W∗ (t, s) associated to K∗ (t) = U∗ (t)−1 (K0 + Y∗ (t))U∗ (t) exists, leaves dom K0 invariant and is strongly differentiable on dom K0 :
S
S
d W∗ (t, s) ψ = K∗ (t) W∗ (t, s) ψ dt
and
C
d W∗ (t, s) ψ = W∗ (t, s) K∗ (s) ψ . ds
Now, to compare the evolutions associated to K0 + V (t) and K∗ (t), we can consider the following computation: for any t, s ∈ R and ϕ ∈ dom K0 , we have (W (t, s) − W∗ (t, s)) ϕ Z t d (W (t, r)W∗ (r, s)) ϕ dr =− dr s Z t W (t, r)(K0 + V (r) − K∗ (r))W∗ (r, s) ϕ dr = −˚ ı s
Z t W (t, r)U∗ (r)−1 (U∗ (r)(K0 + V (r))U∗ (r)−1 − K0 − Y∗ (r)) = −˚ ı s
×U∗ (r)W∗ (r, s) ϕ dr Z t W (t, r)U∗ (r)−1 R∗ (r)U∗ (r)W∗ (r, s) ϕ dr . = −˚ ı s
Taking the norm and using its properties and the unitarity of U∗ (r), W (t, r) and W∗ (r, s), we find
(W (t, s) − W∗ (t, s)) ϕ ≤ |t − s| sup R∗ (r) kϕk . r∈[s;t]
QUANTUM NEKHOROSHEV THEOREM FOR QUASI-PERIODIC FLOQUET HAMILTONIANS
427
As in the proof of the time independent case we can extend this bound to the whole by density of dom K0 and an /3-argument. of
K
Acknowledgements The authors would like to thank M. Flato for his constant support and interest during this work and also A. Delshams, L. H. Eliasson, A. Jorba and P. Guti´errez for helpful discussions. Supports from the EC contract ERB-CHRX-CT-94-0460 for the project “Stability and Universality in Classical Mechanics” and from a grant of the Conseil R´egional de Bourgogne are acknowledged. References [1] J. Bellissard, “Stability and chaotic behaviour of quantum rotators” in Stochastic Processes in Classical and Quantum Systems, Ascona, eds. S. Albeverio, G. Casati and D. Merlini, Springer-Verlag, 1986. [2] J. Bellissard, “Stability and instability in quantum mechanics” in Trends and Developments in the Eighties, eds. S. Albeverio and Ph. Blanchard, World Scientific, (1985) 1–106. [3] J. Bellissard, “Small divisors in quantum mechanics” in Chaotic Behaviour in Quantum Systems, ed. G. Casati, Nato series B120 Plenum, 1985. [4] G. Benettin and G. Gallavotti, “Stability of motions near resonances in quasiintegrable Hamiltonian systems”, J. Stat. Phys. 44 (1986) 293–338. [5] P. M. Blekher, H. R. Jauslin and J. L. Lebowitz, “Floquet spectrum for two-level systems in quasi-periodic time-dependent fields”, J. Stat. Phys. 68 (1992) 271–310. [6] H. P. Breuer and M. Holthaus, “Quantum phases and Landau–Zener transitions in oscillating fields”, Phys. Lett. A140 (1989) 507–512. [7] S. Chelkowski, A. D. Bandrauk and P. B. Corkum, “Efficient molecular dissociation by a chirped ultrashort infrared laser pulse”, Phys. Rev. Lett. 65 (1990) 2355–2358. [8] M. Combescure, “The quantum stability problem for time-periodic perturbations of the harmonic oscillator”, Ann. Inst. Henri Poincar´e 47 (1987) 63–83; (1987) 451–454. [9] M. Combescure, “The quantum stability problem for some class of time-dependent Hamiltonians”, Ann. Physics 185 (1988) 86–110. [10] A. Delshams and P. Guti´errez, “Effective stability and KAM Theory”, J. Diff. Eq. 128 (1996) 415–490. ˇˇtov´ıˇcek, “Floquet Hamiltonians with pure point spectrum”, [11] P. Duclos and P. S Commun. Math. Phys. 177 (1996) 327–347. [12] V. Enss and K. Veseli´c, “Bound states and propagating states for time-dependent Hamiltonians”, Ann. Inst. Henri Poincar´e A39 (1983) 159–191. [13] S. Gu´erin, “Complete dissociation by chirped laser pulses designed by adiabatic Floquet analysis”, Phys. Rev. A56 (1997) 1458–1462. [14] S. Gu´erin and H. R. Jauslin, “Laser-enhanced tunneling through resonant intermediate levels”, Phys. Rev. A55 (1997) 1262–1275. [15] H. R. Jauslin and J. L. Lebowitz, “Spectral and stability aspects of quantum chaos”, Chaos 1 (1991) 114–121. ` Jorba, R. Ram´ırez-Ros and J. Villanueva, “Effective reducibility of quasi-periodic [16] A. linear equations close to constant coefficients”, SIAM J. Math. Anal. 28 (1997) 178–188. [17] T. Kato, “Linear evolution equations of “Hyperbolic” type”, J. Fac. Sci. Univ. T¯ oky¯ o Sect. IA Math. 17 (1970) 241–258.
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[18] T. Kato, “Perturbation theory for linear operators”, reprint of the 2nd ed., SpringerVerlag, 1995. [19] P. Lochak, “Canonical perturbation theory via simultaneous approximation”, Russian Math. Surveys 47 (1992) 57–133. [20] N. N. Nekhoroshev, “An exponential estimate of the time of stability of nearlyintegrable Hamiltonian systems”, Russian Math. Surveys 32 (1977) 1–65. [21] J. P¨ oschel, “Nekhoroshev estimates for quasi-convex Hamiltonian systems”, Math. Z. 213 (1993) 187–216. [22] K. Yosida, “Functional analysis”, reprint of the 6th ed., Springer-Verlag, 1995.
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS VIA THE GENERATING FUNCTION: A DIRECT METHOD ˜ A. BRAGA and PAULO C. LIMA GASTAO Departamento de Matem´ atica–UFMG Caixa Postal 1621 30161–970 Belo Horizonte MG, Brazil
MICHAEL L. O’CARROLL Departamento de F´ısica–UFMG Caixa Postal 702 30161–970 Belo Horizonte MG, Brazil Received 17 April 1997 Revised 8 July 1997 We consider statistical mechanics lattice models where the external field dependent partition function can be represented as a standard polymer system. Using this polymer representation and elementary complex analytic arguments, we obtain upper bounds and give a simple proof on the uniform (in n) exponential decay of the n-point truncated correlation function. We illustrate the method by applying it to the high and low temperature Ising model and to contour models.
1. Introduction For many high temperature or weakly coupled spin, gauge-matter lattice models the finite volume partition function can be represented as a standard polymer system, e.g. [1, 2, 3, 4, 5]. Also low temperature discrete spin, discrete gauge group and contour models have a formulation in terms of a standard polymer system, e.g. [1, 2, 3, 4, 6, 7, 8]. Here we show that once we have the external field dependent finite volume partition function represented as a standard polymer system (as in [3, 4]) then bounds and exponential decay (uniform in n) of the n-point truncated correlation function (cf) can be obtained by elementary complex analytic methods thus giving a simple proof of some well-known results. In contrast with the methods of [4, 6, 9], we believe our method is simpler because no new expansions nor new polymers associated with subsets of the n points need to be introduced, and no relationships between spin correlations and contour correlations need to be used. Basically, the idea is the following: truncated cf are obtained as field derivatives of the log of the partition function, which, under certain hypothesis, is an analytic function of the fields. These derivatives are then represented by contour integrals in complex field plane. From Cauchy estimates for derivatives and the maximum 429 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 429–438 c World Scientific Publishing Company
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modulus theorem, we obtain upper bounds which, under certain hypothesis, will give the uniform exponential decay. We illustrate the method by applying it to the high and low temperature Ising Model and to low temperature contour models in the formulation of [8]. For contour models, we consider only the case where the site variables take on a finite number of real values. In these models, contour correlation functions (cf) can be obtained rather easily but their relation to spin cf is complicated, even in the simplest case of the Ising model, e.g. [4, 6]. To apply these methods to models with discrete or continuous unbounded spins or to continuous space models further analysis is needed. In particular, a generalization of Lemma 1 (see below) would be required for those models. We emphasize that the idea of getting decay of truncated correlation functions by Cauchy estimates is not a new one, see [9, 12]. However, our approach is simpler than the one in [9] and the approach in [12] cannot handle the region of parameters where we have a phase transition, in particular, they do not prove any type of decay for truncated correlation functions at low temperature and h = 0, although such a behavior is well known, e.g. [3, 4], and is easily handled by our approach. This paper is organized as follows: In Sec. 2 we state Theorems 1, 2 and 3 which give the decay of the truncated correlation functions for the Ising model at high, low temperatures and for contour models, respectively. In this section we also prove Lemma 1 which is crucial to control h-dependent partition functions and ratios of partition functions needed in the proof of the Theorems. In Secs. 3, 4 and 5 we prove the Theorems 1, 2 and 3, respectively. 2. Results For the high temperature Ising model in Λ ⊂ Z d we take as the h-dependent partition function with free boundary conditions Z P βsi sj dµ(s) , (1) Z(Λ, h) = ehs e hi,ji P Q where dµ(s) = k∈lΛ dν(sk ), dν(sk ) = 12 [δ(sk −1)+δ(sk +1)]dsk and hs = k hk sk . The low temperature Ising model for ± boundary conditions in Λ ⊂ Z d is taken as Z P P h (s −1)+ β(si sj −1) hi,ji dµ± (s) , (2) Z± (Λ, h) = e k k k Q where dµ± (s) = k∈Λ dν± (sk ), dν± (sk ) is like the above-defined dν measure if k is an interior point and dν± (sk ) = δ(sk ∓ 1)dsk if k is a boundary point. We give bounds for the truncated m-point correlation function (cf) ˜ ∂ m ln Z(Λ, h) (3) S(x1 , . . . , xm ) = ∂h1 . . . hm h=0 where the points {xi } are distinct and Z˜ can be Z of (1) or Z± of (2), respectively. In Theorems 1, 2 and 3 below there will appear ri which is a radius in the complex hi plane which in our method for obtaining bounds is used in Cauchy estimates.
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS
...
431
We state our high temperature result as Theorem 1. With r = (r1 , . . . , rm ), assume ri > 0 ∀i and |r|∞ ≤ 1. Set (β) ≡ e−(| ln β|−β−3|r|∞ −ln 2d) , and let βo such that (βo ) < 16 . Then for all β < βo ! m Y β 1 |Sm | ≤ H()e−| ln( βo )|τm , rk k=1
where τm is the minimal tree length on {x1 , . . . , xm } and 2 −1 1 5 5 + . H() = 1− 1− 8 1− 1− ± Sm
For the d ≥ 2 dimensional Ising Model at low temperatures we have, denoting the m-point truncated cf with ± boundary conditions: Theorem 2. With r = (r1 , . . . , rm ), assume ri > 0 ∀i and |r|1 ≤ 1. Set (β) ≡ e−(2β−3|r|1 −ln 2d) .
and let βo such that (βo ) < 16 . Then for all β > βo ! m Y 1 ± |Sm | ≤ L()e−2(β−βo)dp rk k=1
where dp the defined in Eq. (23) and L() =
(1 + ) 1 + (1 − )3 8
5 1−
2 −2 5 . 1− 1−
Remark. For high temperatures the complex external field can be bounded (such as |h|∞ ≤ 1), for low temperatures we require h to be small and summable, i.e., |h|1 ≤ 1. Exponential tree decay is obtained for high temperature while for low temperatures we state our exponential decay in terms of the separation distance of two subsets of points. Other forms of exponential decay are given in [4, 9]. We point out that exact decay rates for the 2-point function are given in [10], where convergent expansions are obtained for the correlation length using results of [11]. The tree decay stated in [2] does not have a complete proof (see also [9] for the same conclusion). For our contour model we assume that at each site i ∈ Λ ⊂ Z d , d ≥ 2, the spin si takes values in a finite set Ω ⊂ R. We assume there are N stable phases labelled by q = 1, 2, . . . , N . For more details on these models see [6, 7, 8]. The q phase partition function is Zq =
0 X s
where
P0 s
e−βH(s)
Y
ehk sk ,
k
is a sum over spin configurations with boundary condition q.
(4)
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
We assume that Zq has the standard polymer representation XY z(γk , h) . Zq =
(5)
{γk } k
The sum is over subsets of disjoint polymers or contours with outer boundary condition q. We emphasize that implicit in this notation is a specification of the phases m on the interior of the γ contours. The size of a polymer is denoted by |γ|, |γ| ≥ 1 and we assume that the number of polymers of size |γ| passing through a 0 point is bounded by ed |γ| where d0 depends on the dimension d and the number of stable phases N . Concerning the activities z(γ, h) we assume Y (6) z(γ, h) = ρ(γ, h) (Zm (Intm γ, h)/Zq (Intm γ, h)) , m
where ρ, Zr depend on h under suppγ, Intm γ, respectively. Intm γ is the region enclosed by the inner boundary of γ with boundary condition m. Furthermore with ρ(γ, h) analytic in h, |h|1 ≤ 1 we assume the bounds |ρ(γ, h)| ≤ e−(β−b|hγ |1 )|γ| , Zm (γ 0 ) cm |γ 0 | , Zq (γ 0 ) ≤ e
(7) (8)
where hγ means h restricted to γ. We give our result for |si | ≤ 1, from which the case |si | ≤ B can be obtained by scaling the h0i s. Denoting the m-point truncated cf with q-boundary conditions q we have, with L() and dp as in Theorem 2. by Sm Theorem 3. With r = (r1 , . . . , rm ), assume ri > 0 ∀i and |r|1 ≤ 1. Set P c = m cm where cm , 1 ≤ m ≤ N, is the constant appearing in (8) and set 0
(β) ≡ e−(β−(b+3)|r|1−c−d ) . Let βo such that (βo ) < 16 . Then, for all β > βo ! m Y 1 q |Sm |≤ L()e−(β−βo )dp rk k=1
with L() and dp as in Theorem 2. We will need to control h-dependent partition functions and ratios of them. The following lemma will be useful for the case of bounded spins or fields. R P Lemma 1. Let I = eh s dρ(s), hs = i hi si , where dρ(s) is a probability measure on [−1, 1]|Ω| , Ω ⊂ Z d . If |h|1 ≤ 1, then e−2|h|1 ≤ |I| ≤ e|h|1 .
(9)
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS
...
433
R Proof. Writting h = u + iv, we have I = eus (cos(vs) + i sin(vs))dρ(s). Setting φ = vs we have |φ| ≤ |v|1 ≤ |h|1 ≤ 1, and since for |φ| ≤ 1, cos φ ≥ e−|φ| > 0, it follows that |I| ≥ e−|u|1 e−|v|1 ≥ e−2|h|1 . Remark. If dρ(−s) = dρ(s) then the first term of I is Z Z cosh (us)cos (vs) dρ(s) ≥ cos (vs) dρ(s) ≥ e−|v|1 ≥ e−|h|1 for |h|1 ≤ 1, so we can write the lower bound as ≥ e−|h|1 . 3. High Temperature Ising Model The partition function Z(Λ, h), defined by (1) can be written as Z ∗ ehs dµ(s) , Z(Λ, h) = Z (Λ, h) which upon Mayer expanding
Q
hi,ji ((e
(10)
− 1) + 1) and noting that
βsi sj
Z
1
eβsi sj − 1 = βsi sj
etβsi sj dt , 0
Z ∗ (Λ, h), has the standard polymer representation given by (see [3]) XY X z(γi , h) = a(A)z A , Z ∗ (Λ, h) = {γi } i
where
Z " Y z(γ, h) = β |γ|b
hi,ji⊂γ
(11)
A
Z
1
si sj Z
e 0
βtij si sj
P
e
i∈γ
# P h s dtij e i∈γ i i dµ(sγ )
hi si
.
(12)
dµ(sγ )
Here the polymer γ is identified with the sites of a connected subset of nearest neighbours bonds. On the right-hand side of (11) we use the multi-index notation of [3] and the sum is over disjoint subsets of polymers, A. From (12) we have the representation X aT (A)z A . (13) ln Z ∗ (Λ, h) = A
The sum in (13) is over connected subsets of polymers A = {γ1 , . . . , γk } (aT (A) = 0 unless {γ1 , . . . , γk } is k-connected). Denote supp A by ∪i suppγi . We let |γ| (|γ|b ) denote the number of sites (bonds) in γ. R When calculating the truncated cf (3) the factor ehs dµ(s) doesn’t contribute; in what follows we will be analyzing Z ∗ (Λ, h) and we will drop the superscript ∗. We now turn to a bound for Sm . When calculating the h-derivatives in (3) at h = 0 only those terms will appear in (13) such that {x1 , . . . , xm } ⊂ supp A. As a
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device to obtain decay we extend z(γ, h) to z(γ, h, w) analytic in w in |w| < βo ≡ wo by replacing the factor β |γ|b by w|γ|b . Analyticity of ln Z 0 (Λ, h) in the h’s follows from [3]. Substituting Z by Z˜ and using the Cauchy representation for the h derivatives, Eq. (3) becomes Z m 00 Y X 1 dh j aT (A)z(h, w)A |w=β , (14) Sm = 2πi |hj |=rj h2j j=1 A
P00
where A means {xi } ⊂ supp A. Noting that each term in (14) has, at least, a factor of wτm where τm is the minimum tree length of {xi }, multiplying and dividing by ( wwo )τm , using Cauchy estimates for the integrals and the maximum modulus theorem in w we have, for β < βo , " # 00 m Y X τm 1 T 0 A w |a (A)||z(h, w )| |Sm | ≤ sup{h,w0 } r wo w=β j=1 j A
≤
m Y 1 0 −| ln ββ |τm o Ce r j=1 j
(15)
where we set hi = 0 for a i such that xi is not one of the points of the correlation function and the sup is taken over {hi : |hi | = ri > 0, |r|∞ ≤ 1}, {w0 : |w0 | = wo = βo }. In (15) 0 X |aT (A)||z(h, w0 )|A , (16) C 0 ≡ sup{h,w0 } P0
A
means only x1 ∈ supp A. where The sum in (16) can be bounded as in the standard free energy bound in [3]. Let us recall the -value from [3]. If (a) the bound for N (x, k), the number of polymers of size |γ| = k passing through x, can be written ec|γ| and (b) the bound for |z(γ)| can be written as e−E|γ| then the -value is ec−E . If < 16 the bound for the sum in (16) is (see [3]) 2 −1 1 5 5 + . (17) 1− H() = 1− 8 1− 1− In our case N (x, k) ≤ (2d)k and using Lemma 1 for each factor in the denominator of (12) gives the bound |z(γ, h, w0 )| ≤ e−(− ln β−β−3|h|∞ )|γ| ,
(18)
where we have used that |h|1 ≤ |h|∞ |γ| and that |h|1 is evaluated on γ. This leads to the -value (19) (β) = e−(| ln β|−β−3|h|∞ −ln 2d) .
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS
...
435
Thus, if |h|∞ ≤ 1, for all β < βo , βo sufficiently small, we have (β) < (βo ) < 16 and C 0 ≤ H(), which proves Theorem 1. 4. Low Temperature Ising Model We treat explicitly the case of dimension d = 2, the extension for d > 2 is straightforward. We take the + boundary partition function, with Λ ⊂ Z 2 , defined by Eq. (2). As in [2], we obtain the polymer system representation Z+ (Λ, h) =
XY
z(γα , h) =
{γα } α
X
a(A)z(h)A ,
(20)
A
where z(γ, h) = e−2β|γ|
Z− (Intγ, h) , Z+ (Int γ, h)
(21)
where the γ are Peierls contours and |γ| equals the perimeter length. We identify γ with the bonds where si sj = −1 or the midpoint of the bond. We extend z(γ, h) to z(γ, h, w) analytic in w in |w| ≤ wo ≡ e−2βo by replacing e−2β|γ| by w|γ| . Thus the m-point truncated function is Sm =
Z 00 m Y 1 dhj X T a (A)z(h, w)A |w=e−2β , 2 2πi h |hj |=rj j j=1
(22)
A
where 00 means {xi } ⊂ ∪i Int γi ≡ v(A). In words, {xi } are contained in the “volume” enclosed by {γi }. As in the analysis for high temperature, analyticity of ln Z+ (Λ, h) in the h’s follows from [3]. Note that each term in (22) has at least a factor wdp , where for any integer p, 1 ≤ p < m, dp = min dist {xk1 , . . . , xkp } , {xkp+1 , . . . , xkm } , k1 ,...,km
(23)
where the minimum is taken over all permutations {k1 , . . . , km } of {1, . . . , m}. Multiplying and dividing by ( wwo )dp , using Cauchy estimates and the maximum modulus theorem in w we have, for |w| ≤ wo
" # 00 m Y X dp 1 T 0 A w |Sm | ≤ |a (A)||z(h, w )| sup{h,w0 } r wo w=e−2β j=1 j A
≤
m Y 1 0 −2(β−βo )dp . Ce r j=1 j
(24)
In (24) 0
0
C ≡ sup
X
{h,w 0 } A
|aT (A)||z(h, w0 )|A ,
(25)
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G. A. BRAGA, P. C. LIMA and M. L. O’CARROLL
P0 where only means that x1 ∈ V (A) and sup is taken over {h : |hi | = ri > 0, |r|1 ≤ 1}, {w0 : |w0 | = wo ≡ e−2βo }. Concerning the bound C 0 we first consider the bounds for z(γ, h, w0 ) (for values 2 x, k) (here x ˆ ∈ Z + 12 and of w0 on the disc of radius e−2β < e−2βo ) and N (ˆ x ˆ ∈ γ). The ratio of partition functions is bounded using Lemma 1, noting that 0 0 − (Λ ,h)/Z− (Λ ,0) 0 0 Z− (Λ0 , h)/Z+ (Λ0 , h) = Z Z+ (Λ0 ,h)/Z+ (Λ0 ,0) and Z+ (Λ , 0) = Z− (Λ , 0). This leads to the bound |z(γ, h, w0 )| ≤ e−2β|γ|e3|hInt γ |1 ≤ e−(2β−3|hInt γ |1 )|γ| since |γ| ≥ 1 (actually |γ| ≥ 4). A bound for N (ˆ x, k) is 4k . Thus the -value we take, with |h|1 ≤ 1, (β) = e−(2β−3|h|1−ln 4) .
(26)
The bound of the sum in (25) differs from the corresponding high temperature ˆ1 ∈ {γi } with x ˆ1 fixed. one, since x1 ∈ V (A). The free energy bound requires only x 0 where C10 is the sum over the linear terms, i.e. Write C 0 = C10 + C>2 sup{h,w0 }
X
|z(γ, h, w0 )| .
γ:x1 ∈V (γ)
For a given size |γ| = k only contours γ will contribute that lie within a square of side length k centered at x1 . Thus for < 1 C10
≤
∞ X k=1
k 2 k =
(1 + ) ≡ L1 () . (1 − )3
(27)
0 is obtained analogously to the “sum over trees” bound for The bound for C≥2 the free energy in Chapter V of [3] with the difference that x1 ∈ V (γ1 ). Summing over ordered sequences of polymers (γ1 , . . . , γn ) with sizes (k1 , . . . , kn ) following [3] the bound is given by
0 C≥2
∞ X (n − 2)! 1 X Q ≤ n! l (dl − 1)! n=2 {dj }
X (k1 ,...,kn )
k12
n Y
kr (dr −1) kr .
(28)
r=1
The sum is over ordered sequences (k1 , . . . , kn ), ki ≥ 1, and differs from the free energy bound by the appearance of the factor k12 as in the case of linear terms only. {. . .} is Cayley’s tree formula for the number of trees with n vertices, dj , Pn 1 ≤ dj ≤ n − 1, j=1 dj = 2(n − 1). Using Lemma V.7.5 of [3] for the {ki } sums with < Lemma V.7.8 for the {dj } sums we obtain the bounds
1 6,
d1 ≤ n − 1 and
EXPONENTIAL DECAY OF TRUNCATED CORRELATION FUNCTIONS
0 C≥2 ≤
...
437
n 5 1 X d1 + 1)d1 ( n−1 4(1 − ) n=2 ∞ X
{dj }
∞ X ≤ n n=2
1 ≤ 8
5 4(1 − )
5 1−
n X
1
{dj }
2 −2 5 1− 1−
≡ L≥2 () .
(29)
Thus, with |h|1 ≤ 1, if β > βo and βo is sufficiently large, we have (β) < (βo ) < 16 and C 0 ≤ L1 () + L≥2 () ≡ L(), which proves Theorem 2. 5. Contour Model The proof of Theorem 3 follows along the lines of the proof of Theorem 2, Sec. 4. Here we extend the contour activities to the complex w-plane and, using assumptions (7) (h-dependent Peierls condition) and (8) (q-phase stability), we find the -value for these modified activities. Thus, z(γ, h), defined by Eq. (6), is extended to z(γ, h, w) by extending ρ(γ, h) to ρ(γ, h, w) =
w|γ| ρ(γ, h) , e−β|γ|
where ρ(γ, h, w) is analytic in w, in particular for |w| ≤ e−βo ≡ wo , and w = e−β is the physical value. Using the Lemma 1 and (8) we bound Y Zm (h) Y Zm (h)/Zm (0) Zm (0) = Zq (h) Zq (h)/Zq (0) Zq (0) m m Q by m e3|hIntγ |1 +cm |γ| . Since for |w| < e−β , the Ineq. (7) gives the bound |ρ(γ, h, w)| P ≤ e−(β−b)+|rγ |1 )|γ| with c = m cm , we have the bound (again for |w0 | < e−β )
|z(γ, h, w0 )| ≤ e−(β−(b+3)|r|1 −c)|γ| . Thus, the -value is 0
= e−(β−(b+3)|r|1 −c−d ) . The rest of the proof is as in Sec. 4.
6. Acknowledgments This work has been partially supported by the Brazilian agencies CNPq, FAPEMIG and PROPES-UFMG. Gast˜ ao A. Braga and Paulo C. Lima thank CAPES for the financial support while visiting the Institute for Advanced Study and Rutgers University, respectively.
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References [1] E. Seiler, “Gauge theories as a problem of constructive quantum field theory and statistical mechanics”, Lecture Notes in Phys., 159 (1982) Berlin, New York, Springer. [2] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd edition, New York, Springer, 1986. [3] B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press, 1993. [4] V. A. Malyshev and R. A. Minlos, Gibbs Random Fields: Cluster Expansions, Boston, Kluwer Academic Publ., 1991. [5] R. Kotecky and D. Preiss, “Cluster expansion for abstract polymer models”, Commun. Math. Phys. 103 (1986) 491–498. [6] Y. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, 1982. [7] M. Zahradnik, “An alternate version of Pirogov–Sinai theory”, Commun. Math. Phys. 93 (1984) 559–581. [8] C. Borgs and J. Imbrie, “A unified approach to phase diagrams in field theory and statistical mechanics”, Commun. Math. Phys. 123 (1989) 305–328. [9] R. L. Dobrushin, “Estimates of semiinvariants for the Ising Model at low temperatures”, The Erwin Schroedinger Inst. for Math. Phys. preprint ESI 125 (1994). [10] M. L. O’Carroll, “Analyticity properties and a convergent expansion for the inverse correlation length of the low temperature d-dimensional Ising model”, J. Stat. Phys. 34 (1984) 609–614. [11] R. S. Schor, “The particle structure of ν-dimensional Ising model at low temperature”, Commun. Math. Phys. 59 (1978) 213–233. [12] M. Duneau, D. Iagolnitzer and B. Souillard, “Strong cluster properties for classical systems with finite range interactions,” Commun. Math. Phys. 35 (1974) 307–320.
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF THEIR CONTINUOUS FUNCTIONS ELISA ERCOLESSI Dipartimento di Fisica, Universit` a di Bologna and INFM Via Irnerio 46, I-40126, Bologna, Italy
GIOVANNI LANDI The E. Schr¨ odinger International Institute for Mathematical Physics Pasteurgasse 6/7, A-1090 Wien, Austria Dipartimento di Scienze Matematiche, Universit` a di Trieste P. le Europa 1, I-34127, Trieste, Italy and INFN, Sezione di Napoli, Napoli, Italy
PAULO TEOTONIO-SOBRINHO The E. Schr¨ odinger International Institute for Mathematical Physics Pasteurgasse 6/7, A-1090 Wien, Austria Department of Physics, University of Illinois at Chicago 60607-7059 Chicago, IL, USA and Universidade de Sao Paulo, Instituto de Fisica - DFMA Caixa Postal 66318, 05389-970, Sao Paulo, SP, Brasil Received 14 December 1996 Received 15 May 1997 Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for their use in quantum physics. Posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure ole spaces of noncommutative C ∗ -algebras. These noncommutative algebras play the same rˆ as the algebra of continuous functions C(M ) on a Hausdorff topological space M and can be thought of as algebras of operator valued functions on posets. In this article, we will review some mathematical results that establish a duality between finite posets and a certain class of C ∗ -algebras. We will see that the algebras in question are all postliminal approximately finite dimensional (AF) algebras.
1. Introduction It is well known that the standard discretization methods used in quantum physics (where a manifold is replaced by a lattice of points with the discrete topology) are not able to describe any significant topological attribute of the continuum, this being equally the case for both the local and global properties. For example, 439 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 439–466 c World Scientific Publishing Company
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there is no nontrivial concept of winding number and hence no way to formulate theories with topological solitons or instantons on these lattices. A new kind of finite approximation to continuum topological spaces has been introduced in [1], with the name of posets or partially ordered sets. As we will see in Sec. 3, posets are also T0 topological spaces and can reproduce important topological properties of the continuum, such as the homology and the homotopy groups, with remarkable fidelity [1, 2]. This ability to capture topological information has been the main motivation for their use in quantum physics in place of the ordinary discrete lattices. In [3], quantum mechanics has been formulated on posets and it has been proved that it is possible to study nontrivial topological configurations, such as θ-states for particles on the poset approximations to a circle. Some promising results have also been obtained in the formulation of solitonic field theories [3] as well as of gaouge field theories [4]. In [5], the poset approximation scheme has been developed in a novel direction. Indeed, it has been observed that posets are truly noncommutative spaces, or noncommutative lattices, since they can be realized as structure spaces (spaces of irreducible representations) of noncommutative C ∗ -algebras. These noncommutative algebras play the same rˆ ole as the algebra of continuous functions C(M ) on a manifold M and can be thought of as algebras of operator valued functions on posets. This naturally leads to the use of noncommutative geometry [6] (see also [7]) as the tool to rewrite quantum theories on posets and gives a remarkable connection between topologically meaningful finite approximations to quantum physics and noncommutative geometry. The duality relation between Hausdorff topological spaces and commutative C ∗ -algebras is provided by the Gel’fand–Naimark theorem. There is no analogue of this theorem in the noncommutative setting. In this article, we will review how it is possible to establish a relation between finite posets and a particular class of noncommutative C ∗ -algebras. For such class of algebras the situation is very similar to the commutative case. We will see that the algebras in question are all approximately finite dimensional (AF) postliminal algebras [8, 9], i.e. C ∗ -algebras that can be approximated in norm by a sequence of finite dimensional matrix algebras and whose irreducible representations are completely characterized by the kernels. This is exactly what makes them of some interest in mathematics: they present virtually all the attributes and complications of other infinite dimensional algebras, but many techniques and results valid in the finite dimensional case can be used in their study. Thus, for example, a complete classification of AF C ∗ -algebras is available [10]. These algebras were first extensively studied by Bratteli [8], who also introduced a diagrammatic representation which is very useful for the study of their algebraic properties. In particular we will see how to use the Bratteli diagram of an AF algebra to construct its structure space. Then we will see how, given any finite poset P , it is possible to construct the Bratteli diagram of an AF algebra whose structure space is the given poset P . Being noncommutative, this AF C ∗ -algebras is far from being unique. Indeed there is a whole family of AF algebras that have
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P as structure space and that can be classified by means of results due to Behncke and Leptin [11]. In this article, we will not present the classification of AF C ∗ -algebras that can be formulated in terms of their algebraic K-theory [10]. In view of their relation with posets, this would represent also a first step in the construction of bundles and characteristic classes over noncommutative lattices. This is indeed the content of [12], to which we refer the interested reader for a detailed analysis of the K-theory of AF algebras. This article is organized as follows. In Sec. 2 we review some elementary algebraic concepts as well as the Gel’fand Naimark theorem in order to clarify the notation and the terminology used in the sequel. In Sec. 3 we briefly describe the topological approximation of continuous spaces that leads to partially ordered sets (posets). In Sec. 4, AF C ∗ -algebras are introduced and the connection between Bratteli diagrams and posets is discussed in detail. Finally, in Sec. 5 we present the classification theorem of the AF C ∗ -algebras that have a poset as structure space due to Behncke and Leptin. Several interesting examples will be examined throughout the article. 2. C∗ -algebras and Structure Spaces Let us start with some elementary algebraic preliminaries [13, 14] that will be also useful to establish notation. In the sequel, A will always denote a C ∗ -algebra over the field of complex numbers C. We remind that this means that A is equipped with a norm of algebra k · k : A → C (with respect to which A is complete) and an involution ∗ : A → A, satisfying the identity: (2.1) ka∗ ak = kak2 , ∀ a ∈ A . The following are examples of commutative and noncommutative C ∗ -algebras which will be used in the article: (1) the (noncommutative) algebra M(n, C) of n × n complex matrices T , with T ∗ given by the hermitian conjugate of T and the squared norm k T k2 being equal to the largest eigenvalue of T ∗ T ; (2) the (noncommutative) algebra B(H) of bounded operators B on a separable (infinite-dimensional) Hilbert space H as well as its subalgebra K(H) of compact operators. Now ∗ is given by the adjoint and the norm is the operator norm: kBk = supkξk≤1 kBξk (ξ ∈ H); (3) the (commutative) algebra C(M ) of continuous functions on a compact Hausdorff topological space M , with * denoting complex conjugation and the norm given by the supremum norm, k f k∞ = supx∈M |f (x)|. If M is not compact but only locally compact, then one can consider the algebra C0 (M ) of functions vanishing at infinity. Notice that K(H) and C0 (M ) (with M only locally compact) are examples of C ∗ -algebras without unit I, in contrast to M(n, C) and B(H).
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2.1. Commutative C∗ -algebras: the Gel’fand Naimark Theorem In the third example above we have seen how it is possible to associate a commutative C ∗ -algebra with (without) unit, namely C(M ) (C0 (M )), to any Hausdorff compact (locally compact) topological space M . Vice versa, given any commutative C ∗ -algebra C with (without) unit, it is possible to construct a Hausdorff compact (locally compact) topological space M such that C is isometrically ∗-isomorphic to the algebra of continuous functions C(M ) (C0 (M )). This is precisely the content of the Gel’fand–Naimark theorem [14] that will be discussed in this paragraph. For simplicity we will consider only the case when C is a commutative C ∗ -algebra with unit. Given such a C, we let Cb denote the structure space of C, namely the space of equivalence classes of irreducible ∗ -representations (IRR’s) of C. The trivial IRR given by C → {0} is not included in M and will therefore be ignored here and hereafter. Since the C ∗ -algebra C is commutative, every IRR is one-dimensional, i.e. is a (non-zero) linear functional φ : C → C satisfying φ(ab) = φ(a)φ(b) and b The space Cb is made φ(a∗ ) = φ(a), ∀a, b ∈ C. It follows that φ(I) = 1, ∀φ ∈ C. into a topological space by endowing it with the Gel’fand topology, namely with the topology of pointwise convergence on C. Then C can be proved to be a Hausdorff compact topological space. For a commutative C ∗ -algebra, two-irreducible representations are unitarily equivalent if and only if they have the same kernel. Thus one can consider also the space of kernels of IRR’s, the so called primitive spectrum PrimC. Now, these kernels are maximal ideals of C and, vice versa, any maximal ideal is the kernel of b then Ker(φ) is of an irreducible representation [14]. Indeed, suppose that φ ∈ C, codimension 1 and so is a maximal ideal of C. Conversely, suppose that I is a maximal ideal of C, then the natural representation of C on C/I is irreducible, hence one-dimensional. It follows that C/I ∼ = C, so that the quotient homomorphism b Clearly, I =Ker(φ). Thus PrimC C → C/I can be identified with an element φ ∈ C. can be thought of as the space of maximal ideals. As such, PrimC is equipped with the Jacobson or hull kernel topology, that will be described in the next paragraph. The map that to each class of unitary representations associates its kernel gives a map Cb → PrimC, which turns out to be a homeomorphism of the two topological spaces so that we may equivalently talk of the structure space or of the primitive spectrum of a commutative C ∗ -algebra. If c ∈ C, the Gel’fand transform cˆ of c is the complex-valued function on Cb given by cˆ(φ) = φ(c) , ∀φ ∈ Cb . (2.2) It is clear that cˆ is continuous for each c. We thus get the interpretation of elements b The Gel’fand–Naimark theorem states in C as C-valued continuous functions on C. b that all continuous functions on C are indeed of the form (2.2) for some c ∈ C [14]: Proposition 2.1. Let C be a commutative C ∗ -algebra with unit. Then the Gel’fand transform map c 7→ cˆ is an isometric ∗-isomorphism of the C ∗ -algebra C b (equipped with the supremum norm k · k∞ ). onto the C ∗ -algebra C(C)
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Suppose now that M is a compact Hausdorff topological space. We have a natural C ∗ -algebra, C(M ), associated to it. It is then natural to ask what is the \) and M itself. It turns out that this two relation between the Gel’fand space C(M spaces can be identified both setwise and topologically. We notice first that each \) through the evaluation map: m ∈ M gives a complex homomorphism φm ∈ C(M φm : C(M ) → C ,
φm (f ) = f (m) .
(2.3)
Let Im denote the kernel of φm , namely the maximal ideal of C(M ) consisting of all functions vanishing at m. We have the following theorem [14]: Proposition 2.2. The map Φ : m 7→ φm given by (2.3) is a homeomorphism of \), namely M ∼ \). Moreover, every maximal ideal of C(M ) is of M onto C(M = C(M the form Im for some m ∈ M . In conclusion, the previous two theorems set up a one-to-one correspondence between the ∗-isomorphism classes of commutative C ∗ -algebras and the homeomorphism classes of locally compact Hausdorff spaces. If C has a unit, then Cˆ and PrimC are compact. 2.2. Noncommutative algebras and associated spaces The scheme described in the previous section cannot be directly generalized to noncommutative C ∗ -algebras. There is more than one candidate for the analogue of the topological space M . In particular, since non-equivalent unitary transformations may now have the same kernel, we have to distinguish even setwise between: (1) the structure space Ab of A or the space of all unitary equivalence classes of irreducible ∗ -representations;a (2) the primitive spectrum PrimA of A or the space of kernels of irreducible ∗ -representations. Any element of PrimA is automatically a two-sided ∗ ideal of A. One can define a natural topology on both Ab and PrimA. While for a commutative C ∗ -algebra the resulting topological spaces are homeomorphic, this is no longer true in the noncommutative case. For instance, in Sec. (4.1) we will describe a C ∗ -algebra A associated to the Penrose tiling of the plane [6], whose structure space Ab consists of an infinite set of points, whereas PrimA consists of a single point. The topology one puts on Ab is called regional topology [14] and is a generalization of the pointwise convergence we have used in the previous paragraph, to which it a b A is often referred to as the spectrum of the algebra A (see for example [13]). Here we prefer to
call it structure space (as it is done for example in [14]) to avoid any confusion with the concept of primitive spectrum.
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reduces in the commutative case. This topology is constructed by defining a basis of neighborhoods for the points (classes of representations) of Ab as follows. Given b let us denote with HT the Hilbert space of the representation T . Then an a T ∈ A, open neighborhood of T is identified by a finite sequence ξ1 , ξ2 , . . . , ξn of vectors in HT , a positive number and a finite not void set F ⊂ A by means of: U (T ; ; ξ1 , ξ2 , . . . , ξn ; F ) =: {T 0 ∈ Ab : ∃ ξ10 , ξ20 , . . . , ξn0 ∈ HT 0 with |(ξi0 , ξj0 )HT 0 − (ξi , ξj )HT | < , |(T 0 (a)ξi0 , ξj0 )HT 0 − (T (a)ξi , ξj )HT | < for i, j = 1, 2, . . . , n
and ∀a ∈ F } .
(2.4)
On PrimA we instead define a closure operation as follows [13, 14]. Given any subset W of PrimA, the closure W of W is by definition the set of all elements in PrimA containing the intersection of the elements of W , namely \ W = {I ∈ Prim A : W ⊂ I} . (2.5) This “closure operation” satisfies the Kuratowski axioms [15] and thus defines a topology on PrimA, which is called Jacobson or hull-kernel topology. With respect to this topology we have: Proposition 2.3. Let W be a subset of PrimA. Then W is closed if and only if W is exactly the set of primitive ideals containing some subset of A. Proof. If W is closed, by 2.5, W is the set of primitive ideals containing W ⊂ I. Conversely, let V ⊆ A. If W is the set of primitive ideals of A containing T V , then V ⊆ W ⊂ I, for all I ∈ W , so that W ⊂ W , and W = W . T
In general Ab and PrimA fail to be Hausdorff (or T2 ). Recall [15] that a topological space is called T0 if for any two distinct points of the space there is a neighborhood of one of the two points which does not contain the other. It is called T1 if any point of the space is closed. It is called T2 if there exist disjoint neighborhoods of any two points. Whereas nothing can be said concerning the separation properties b it turns out that PrimA is always a T0 space and that it is T1 if and only if all of A, primitive ideals in A are maximal, as it is established by the following propositions [13, 14]. Proposition 2.4. The space PrimA is a T0 space. Proof. Suppose I1 and I2 are two distinct points of PrimA so that say I1 6⊂ I2 . Then the set of those I ∈ PrimA which contain I1 is a closed subset W (by Proposition 2.3) such that I1 ∈ W and I2 6∈ W . Then its complement W c is an open set containing I2 and not I1 . Proposition 2.5. Let I ∈ PrimA. Then the point {I} is closed in PrimA if and only if I is maximal among primitive ideals.
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Proof. Indeed {I} is just the set of primitive ideals of A containing I. As in the commutative case, both Ab and PrimA are locally compact topological spaces. In addition, if A has a unit, then they are compact. Notice that, in general, Ab being compact does not imply that A has a unit. For instance, the algebra K(H) of compact operators on an infinite dimensional Hilbert space H has no unit but its structure space consists of a single point. Let us now come to a comparison between the space Ab and PrimA. There is a canonical surjection of Ab onto PrimA, given by the map that to each IRR π associates its kernel kerπ. The pull-back of the Jacobson topology from PrimA to Ab defines another topology on the latter that turns out to be equivalent to the regional topology defined above [14]. But Ab and PrimA are homeomorphic only under the hypotheses stated below [14]. Proposition 2.6. Let A be a C*-algebra, then the following conditions are equivalent : (i) Ab is a T0 space. (ii) Two irreducible representations of Ab with the same kernel are equivalent. (iii) The canonical map Ab −→ PrimA is a homeomorphism. Proof. By construction, a subset S ∈ Ab will be closed if and only if it is of the form {π ∈ Ab : kerπ ∈ W } for some W closed in PrimA . As a consequence, given b the representation π1 will be in the any two (classes of) representations π1 , π2 ∈ A, closure of π2 if and only if ker π1 is in the closure of ker π2 , or, by Proposition 2.5 if and only if kerπ2 ⊂ kerπ1 . In turn, π1 and π2 are one in the closure of the other if and only if kerπ2 = kerπ1 . Therefore, π1 and π2 will not be distinguished by the topology of Ab if and only if they have the same kernel. On the other side, if Ab is T0 one is able to distinguish points. It follows that (i) implies (ii), namely, that if Aˆ is a T0 space, two representations with the same kernel must be equivalent. The other implications are obvious. 3. Noncommutative Lattices For convenience, we will review in this section the content of [1, 2, 3], where it is shown how it is possible to approximate a continuum topological space by means of a finite or countable set of points P [1] which, being equipped with a partial order relation, is a partially ordered set or a poset. As explained there, these approximating spaces are able to reproduce important topological properties of the continuum. Moreover, in Sec. 4.1 we will see that any of these spaces can be identified with the space Ab = PrimA of primitive ideals of some (noncommutative) AF algebra A, which thus plays the rˆ ole of the algebra of continuous functions on P [5]. This fact will make any poset a truly noncommutative space [1], hence also the name noncommutative lattice. This is the reason why, in this article, we will consider only a special class of algebras, namely postliminal approximately finite (AF) algebras. In Sec. 4 we will
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see in detail that AF algebras are approximated in norm by direct sums of finite dimensional matrices [8, 9]. As for postliminal we refer to [13] for the exact definition. For what concerns this article, we need just to know that, as a consequence of general theorems, this implies that Ab and PrimA are homeomorphic. In other words, in the following we will have to deal only with structure spaces (or primitive spectrum spaces) which are T0 locally compact topological spaces. 3.1. The finite topological approximation Let M be a continuum topological space. Experiments are never so accurate that they can detect events associated with points of M ; rather they only detect events as occurring in certain sets Oλ . It is therefore natural to identify any two points x, y of M if they can never be separated or distinguished by the sets Oλ . Let us assume that each Oλ is open and that the family {Oλ } covers M : [ Oλ . (3.1) M= λ
We also assume that {Oλ } is a topology for M [15]. This implies that both 0λ ∪ 0µ and 0λ ∩ Oµ are in U if Oλ,µ ∈ U. This hypothesis is physically consistent because experiments can isolate events in Oλ ∪ Oµ and Oλ ∩ Oµ if they can do so in Oλ and Oµ separately, the former by detecting an event in either Oλ or Oµ , and the latter by detecting it in both Oλ and Oµ . Given x and y in M , we write x ∼ y if every set Oλ containing either point x or y contains the other too: x ∼ y means x ∈ Oλ ⇔ y ∈ Oλ for every Oλ .
(3.2)
Then ∼ is an equivalence relation, and it is reasonable to replace M by M/ ∼ ≡ P (M ) to reflect the coarseness of observations. We assume that the number of sets Oλ is finite when M is compact so that P (M ) is an approximation to M by a finite set in this case. When M is not compact, we assume instead that each point has a neighborhood intersected by only finitely many Oλ , so that P (M ) is a “finitary” approximation to M [1]. In the notation we employ, if P (M ) has N points, we sometimes denote it by PN (M ). The space P (M ) inherits the quotient topology from M [15], i.e. a set in P (M ) is declared to be open if its inverse image for Φ is open in M , Φ being the map from M to P (M ) obtained by identifying equivalent points. The topology generated by these open sets is the finest one compatible with the continuity of Φ. Let us illustrate these considerations for a cover of M = S 1 by four open sets as in Fig. 1(a). In that figure, O1 , O3 ⊂ O2 ∩ O4 . Figure 1(b) shows the corresponding discrete space P4 (S 1 ), the points xi being images of sets in S 1 . The map Φ : S 1 → P4 (S 1 ) is given by O1 → x1 ,
O2 \ [O2 ∩ O4 ] → x2 ,
O3 → x3 ,
O4 \ [O2 ∩ O4 ] → x4 .
(3.3)
''$$ &&%%
ss s s
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O1
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x2
Φ
-
x1
O4
x3
x4
Fig. 1. The covering of S 1 that gives rise to the poset P4 (S 1 ).
The quotient topology for P4 (S 1 ) can be read off from Fig. 1, the open sets being {x1 } , {x3 } , {x1 , x2 , x3 } , {x1 , x4 , x3 } ,
(3.4)
and their unions and intersections (an arbitrary number of the latter being allowed as P4 (S 1 ) is finite). Notice that our assumptions allow us to isolate events in certain sets of the form Oλ \ [Oλ ∩ Oµ ] which may not be open. This means that there are in general points in P (M ) coming from sets which are not open in M and therefore are not open in the quotient topology. This implies that in general P (M ) is neither Hausdorff nor T1 . However, it can be shown [1] that it is always a T0 space. For example, iven the points x1 and x2 of P4 (S 1 ), the open set {x1 } contains x1 and not x2 , but there is no open set containing x2 and not x1 . We will see now how the topological properties of P (M ) can also be encoded in a combinatorial structure, namely a partial order relation, that can be defined on it. Since P (M ) is finite (finitary), its topology is generated by the smallest open neighborhoods Ox of its points x. It is possible to introduce a partial order relation [2, 16] by declaring that: x y ⇔ Ox ⊂ Oy .
(3.5)
In this way P (M ) becomes a partially ordered set or a poset. Later, we will write x ≺ y to indicate that x y and x 6= y. A point x ∈ P such that there exists no y ∈ P with x ≺ y(x y) is said to be maximal (minimal). In addition, a set {x1 , x2 , . . . , xk } of points in P is said to be a chain if xj+1 covers xj (j = 1, . . . , k − 1). A chain is maximal if x1 and xk are respectively a minimal and a maximal point. It is easy to read the topology of P (M ) once the partial order is given. It is not difficult to check that Ox = {y ∈ P (M ) : y x} . Indeed, one can even prove a stronger result [1, 2], namely that any finite set P on which a partial order is defined can be made into a finite T0 topological space by declaring that the smallest open neighborhood Ox containing x is given exactly by the above set.
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Throughout this article, we will use “finite poset” and “finite T0 space” interchangeably. It is convenient to graphically represent a poset by a diagram, the Hasse diagram, constructed by arranging its points at different levels and connecting them using the following rules [1, 16]: (1) if x ≺ y, then x is at a lower level than y; (2) if x ≺ y and there is no z such that x ≺ z ≺ y, then x is at a level immediately below y and these two points are connected by a line called a link. Let us consider P4 (S 1 ) again. The partial order reads x1 x2 , x1 x4 , x3 x2 , x3 x4 ,
(3.6)
where we have omitted writing the relations xj xj . The corresponding Hasse diagram is shown in Fig. 2. In the language of partially ordered sets, the smallest open set Ox containing a point x ∈ P (M ) consists of all y preceding x : Ox = {y ∈ P (M ) : y x}. In the x4
x1
s s s s s
@@
@@ @
@
@
s s s s s
x2
@x
3
Fig. 2. The Hasse diagram for the circle poset P4 (S 1 ).
x6
x4
x1
@@
@
@ @@
@@
@
@ @@
x5
@@
x2
@@
x3
Fig. 3. The Hasse diagram for the sphere poset P6 (S 2 ).
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Hasse diagram, it consists of x and all points we encounter as we travel along links from x to the bottom. In Fig. 2, this rule gives {x1 , x2 , x3 } as the smallest open set containing x2 , just as in (3.4). As one example of a three-level poset, consider the Hasse diagram of Fig. 3 for a finite approximation P6 (S 2 ) of the two-dimensional sphere S 2 derived in [1]. Its open sets are generated by {x1 }, {x3 }, {x1 , x2 , x3 }, {x1 , x4 , x3 } , {x1 , x2 , x3 , x4 , x5 }, {x1 , x2 , x3 , x4 , x6 } ,
(3.7)
by taking unions and intersections. One of the most remarkable properties of a poset is its ability to accurately reproduce the homology and the homotopy groups of the Hausdorff topological space it approximates. For example, as for S 1 , the fundamental group of PN (S 1 ) is Z whenever N ≥ 4 [1]. Similarly, as for S 2 , π1 (P6 (S 2 )) = {0} and π2 (P6 (S 2 )) = Z. This has been widely discussed in our previous work [3, 5], where we argued that global topological information relevant for quantum physics can be captured by such discrete approximations. Furthermore, the topological space being approximated can be recovered by considering a sequence of finer and finer coverings, the appropriated framework being that of projective systems of topological spaces. We refer to [1, 17] for details. In this article we are however mostly concerned with the algebraic properties of a poset, i.e. with the fact that any finite poset can be regarded as the structure space of a C ∗ -algebra. This will be extensively discussed and proved in the following sections, but let us first illustrate a simple example. Consider the following C ∗ algebra: (3.8) A = {λ1 I1 + λ2 I2 + k12 : λj ∈ C, k12 ∈ K12 } L acting on the direct sum of two Hilbert spaces H = H1 H2 and generated by multiples of the identity I1 on H1 , multiples of the identity I2 on H2 and compact operators K12 on the whole Hilbert space H. This algebra admits only three classes of irreducible representations, two finite dimensional ones and an infinite dimensional one: (1) π1 : λ1 I1 + λ2 I2 + k12 7→ λ1 , (2) π2 : λ1 I1 + λ2 I2 + k12 7→ λ2 , (3) ρ : λ1 I1 + λ2 I2 + k12 7→ λ1 I1 + λ2 I2 + k12 . Hence the primitive spectrum consists of only three points p1 = kerπ1 , p2 = kerπ2 , q = kerρ, corresponding respectively to the three representations given above. This space has to be given the Jacobson topology as explained in the previous section. This is easily done if one notices that, the space being finite, this amounts to give a partial order relation on the set {p1 , p2 , q} [14]. Indeed one can show that, on any finite primitive spectrum PrimA of a C ∗ -algebra A, the Jacobson topology is equivalent to the following partial order relation: pj ≺ pk ⇔ kerπj ⊂ kerπk ,
(3.9)
s
450
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
p1
@
@ @@
s
p2
@
q
Fig. 4. The
W
poset, primitive spectrum of A = CI1 + CI2 + K12 .
where pj is the point in PrimA corresponding to the IRR πj of A. Thus in our example, since kerρ ⊂ kerπ1 and kerρ ⊂ kerπ2 , the set {p1 , p2 , q} is equipped with the order relations q ≺ p1 , q ≺ p2 and therefore corresponds to the poset of Fig. 4, W which will be referred to as the poset from now on. 4. AF Algebras 4.1. Bratteli diagrams A C ∗ -algebra A is said to be approximately finite dimensional (AF) [8, 9] if there exists an increasing sequence I0
I1
I2
I3
In−1
In
A0 ,→ A1 ,→ A2 ,→ A3 ,→ · · · ,→ An ,→ · · ·
(4.10)
of finite dimensional subalgebras of A, such that A is the norm closure of ∪n An . Here the maps In are injective ∗ -homomorphisms. In other words, A is the direct limit in the category of C ∗ -algebras with morphisms given by *-algebra maps (not S isometries) of the sequence (An )n∈N . As a set, n An is made of coherent sequences, [ An = {a = (an )n∈N , an ∈ An |∃N0 : an+1 = In (an ), ∀ n > N0 } . (4.11) n
Now the sequence (kan kAn )n∈N is eventually decreasing, since kan+1 k ≤ kan k (the maps In are norm decreasing) and therefore convergent. One writes for the norm k(an )N k = lim kan kAn . n→∞
(4.12)
Since the maps In are injective, the expression (4.12) gives directly a true norm and not simply a seminorm and there is no need to divide out by the zero norm elements. Each subalgebra An , being a finite dimensional C ∗ -algebra, is a matrix algebra L n (n) (dk , C) where M(n) (dk , C) is the and therefore can be written as An = N k=1 M algebra of dk × dk matrices with complex coefficients. Given any two such matrix L 1 L 2 (1) (2) (dj , C) and A2 = N (dk , C) with A1 ,→ A2 , one algebras A1 = N j=1 M k=1 M can always choose suitable bases in A1 and A2 such that A1 is identified with a subalgebra of A2 of the following form [8]: N2 N1 M M Nkj M(1) (dj , C) . (4.13) A1 ' k=1
j=1
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
...
451
Here, for any nonnegative integers p and q, the symbol p M(q, C) stands for M(q, C)⊗ Ip . In (4.13), the coefficients Nkj represent the multiplicity of the partial embedding of M(1) (dj , C) in M(2) (dk , C) for each k and j and satisfy the condition N1 X
Nkj dj = dk .
(4.14)
j=1
A useful way to represent the algebras A1 , A2 and the embedding A1 ,→ A2 is by means of a diagram, the Bratteli diagram [8], which can be constructed out of the dimensions, dj (j = 1, . . . , N1 ) and dk (k = 1, . . . , N2 ), of the diagonal blocks of the two algebras and the numbers Nkj that describe the partial embeddings. To construct the diagram, we draw two horizontal rows of vertices, the top (bottom) one representing A1 (A2 ) and consisting of N1 (N2 ) vertices, labeled by the corresponding dimensions d1 , . . . , dN1 (d1 , . . . , dN2 ). Then for each j = 1, . . . , N1 and k = 1, . . . , N2 , we draw Nkj edges between dj and dk . We will also write (1) (2) (1) (2) dj &Nkj dk to denote the fact that M(dj , C) is embedded in M(dk , C) with multiplicity Nkj . By repeating the procedure at each level, we obtain a semi-infinite diagram denoted by D(A) which completely defines A up to isomorphisms. Notice that the diagram D(A) depends not only on A but also on the particular sequence {An }n∈N which generates A. However, it is possible to show [8] that all diagrams corresponding to AF algebras which are isomorphic to A can be obtain from the chosen D(A) by means of an algorithm. As an example of an AF algebra, let us consider the subalgebra A of the algebra B(H) of bounded operators on H = H1 ⊕ H2 given in (3.8). This C ∗ -algebra can be obtained as the direct limit of the following sequence of finite dimensional algebras: A0 = M(1, C) A1 = M(1, C) ⊕ M(1, C) A2 = M(1, C) ⊕ M(2, C) ⊕ M(1, C) .. . An = M(1, C) ⊕ M(2n − 2, C) ⊕ M(1, C) .. . where, for n ≥ 1, An is embedded in An+1 as the subalgebra M(1, C) ⊕ [M(1, C) ⊕ M(2n − 2, C) ⊕ M(1, C)] ⊕ M(1, C):
λ1 an = 0 0
0 m2n−2×2n−2 0
λ1
0 0 0 ,→ 0 λ2 0 0
0
0
0
λ1 0
0 m2n−2×2n−2
0 0
0 0
0 0
λ2 0
0
. 0 λ2 0 0
(4.15)
452
ss ss
s ss s
ss ss
E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
1
@
@
@1 @ @ 1 @ @2 1 @ 1 @ @4 1 @ 1 @ @6 1 1
.. .
.. .
Fig. 5. The Bratteli diagram corresponding to A = CI1 + CI2 + K12 .
It is therefore described by the diagram of Fig. 5. As a second example, consider the C ∗ -algebra of the Penrose tiling. This is an example of an AF algebra which is not postliminal, since this algebra admits an infinite number of nonequivalent representations all with the same kernel. At each level, the finite dimensional algebra is given by [6] An = M(dn , C) ⊕ M(d0n , C) , with inclusion An ,→ An+1 : A 0 A 0 ,→ 0 B 0 B 0 0
0 0 ; A ∈ M(dn , C) , A
ss s ss ss s s
n ≥ 1,
B ∈ M(d0n , C) ,
1
@
@ HH @ 1 HHH 2H HHHH 1 3 HH HH 2 HHH H 5H H 3 1
.. .
Fig. 6. The Bratteli diagram of the Penrose tiling.
(4.16)
(4.17)
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
...
453
so that dn+1 = dn + d0n and d0n+1 = dn . The corresponding Bratteli diagram is shown in Fig. 6. To conclude this section we remark that an AF algebra is commutative if and only if all its factors M(n) (dk , C) are one dimensional, i.e. they are just C. Thus the corresponding diagram has the property that for each M(n) (dk , C) with n ≥ 1 there is exactly one M(n−1) (dj , C) and M(n−1) (dj , C) &pkj M(n) (dk , C) with pkj = 1. An interesting example is given in Fig. 7, which corresponds to the AF C ∗ -algebra of continuous functions on the Cantor set [10].
s
s s ss s s s s s s s s s s HH HH 1
1 ZZ ZZ 1 1
JJ
JJ
J@1
1 J@1 1
..@ .. ..@ .. .
.
.
.
HHH HZ1 ZZ Z1 1
J
JJ
J
J@1 1
1J 1
..@ ..@ ..@ .. .
.
.
.
Fig. 7. The Bratteli diagram corresponding to the AF C ∗ -algebra of continuous functions on the Cantor set.
4.2. From Bratteli diagrams to posets The Bratteli diagram D(A) of an AF algebra A is useful not only because it gives the finite approximations of the algebra explicitly, but also because it is possible to read the ideals and the primitive ideals of the algebra (hence the topological properties of PrimA) out of it very easily. Indeed one can show that the following proposition holds [8]: Proposition 4.1. (1) There is a one-to-one correspondence between the proper ideals I of A and the subsets Λ = ΛI of the Bratteli diagram satisfying the following two properties: (i) if M(n) (dk , C) ∈ Λ and M(n) (dk , C) & M(n+1) (dj , C) then necessarily M(n+1) (dj , C) belongs to Λ; (ii) if all factors M(n+1) (dj , C) (j = {1, 2, . . . , Nn+1 }), for which M(n) (dk , C) & M(n+1) (dj , C), belong to Λ, then M(n) (dk , C) ∈ Λ. (2) A proper ideal I of A is primitive if and only if the associated subdiagram ΛI satisfies: (iii) ∀n there exists an M(m) (dj , C), with m > n, not belonging to ΛI such that, for all k ∈ {1, 2, . . . , Nn } with M(n) (dk , C) not in ΛI , one can find
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a sequence M(n) (dk , C) & M(n+1) (dα , C) & M(n+2) (dβ , C) & · · · & M(m) (dj , C). For example, consider the diagram of Fig. 5, representing the AF C ∗ -algebra A = CI1 + CI2 + K12 already discussed in Sec. 3.1. This algebra contains only three nontrivial ideals, whose diagrams are represented in Figs. 8(a), (b), (c). In these pictures the points belonging to the ideals are marked with a “♣”. It is not difficult to check that only I1 and I2 are primitive ideals, since I3 does not satisfy property (iii) above.
ss ss s
@I @@ ♣ @ @@ @@ ♣ ♣ @ @@ ♣ ♣ @ @♣ ♣ .. .
1
.. .
ss sss ss ss ss s s s
@I @@ ♣ @ @@ ♣ @@ ♣ @ ♣ @@ ♣ ♣ @ @♣
(a)
@I @@
2
.. .
.. .
(b)
3
@ @@ @@ ♣ @ @@ ♣ @ @♣ .. .
.. .
(c)
Fig. 8. The representation of the ideals of A = CI1 + CI2 + K12 in the corresponding Bratteli diagram.
We remark the following here: (1) The whole A is an ideal, which by definition is not primitive since the trivial representation A → 0 is not irreducible. (2) The set {0} ⊂ A is an ideal, which is primitive if and only if A has one irreducible faithful representation. This can also be understood from the Bratteli diagram in the following way. The set {0} is not a subdiagram of D(A), being represented by the element 0 of the matrix algebra of each finite level, so that there is at least one element a ∈ A not belonging to the ideal {0} at any level. Thus to check if {0} is primitive, i.e. to check property (iii) above, we have to examine whether all the points at a given level, say n, can be connected to a single point at a level m > n. For example this is the case for the diagram of Fig. 5 and not for that of Fig. 7. Proposition 4.1 above allows us to understand the topological properties of PrimA at once. This becomes particularly simple if the algebra admits only a finite number of nonequivalent irreducible representations. In this case PrimA is a T0 topological space with only a finite number of points, hence a finite poset P . To reconstruct the latter we just need to draw the Bratteli diagram D(A) and find
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
...
455
the subdiagrams that, according to properties (i), (ii), (iii), correspond to primitive ideals. Then P has so many points as the number of primitive ideals and the partial order relation in P that determines the T0 topology is simply given by the inclusion relations that exist among the primitive ideals. As an example consider again Fig. 5. We have seen that the corresponding AF algebra has only three primitive ideals: the {0} ideal and the ideals I1 , I2 W represented in Fig. 8(a), (b). Clearly {0} ⊂ I1 , I2 so that PrimA is the poset of Fig. 4. Figure 7 leads to another interesting topological space. As we have mentioned, such a diagram corresponds to a commutative AF algebra C and hence to a Hausdorff PrimC, which is homeomorphic to the Cantor set. 4.3. From posets to Bratteli diagrams In the preceding subsection we have described the properties of the Bratteli diagram D(A) of an AF algebra A and in particular we have seen how, out of it, it is possible to read the primitive spectrum of A, in particular when the latter is a finite poset. In the following we will see that, under some rather mild hypotheses which are always verified in the cases of interest to us, it is possible to reverse the construction and thus build the AF algebra that corresponds to a given (finite) T0 topological space. Such a reconstruction rests on the following theorem of Bratteli and Elliott [18, 19], which specifies a class of topological spaces which are the primitive spectra of AF algebras: Proposition 4.2. A topological space Y is the primitive spectrum PrimA of an AF algebra A if it has the following properties: (i) Y is T0 ; (ii) Y contains at most a countable number of closed sets; (iii) if {Fn }n∈Λ , Λ being any direct set, is a decreasing sequence of closed subsets of Y, then ∩n Fn is an element in {Fn }n∈Λ ; (iv) if F ⊂ Y is a closed set which is not the union of two proper closed subsets, then F is the closure of a one-point set. It is not difficult to check that all the above conditions hold true if Y is a T0 topological space with a finite number of points, so that we have the corollary: Corollary. A finite poset P is the primitive spectrum PrimA for some AF algebra A. Here we will not report the proof of Proposition 4.2, which can be found in [18]. Also a more general characterization of spaces arising as the primitive spectrum of a separable AF algebra has been given in [19]. Here, starting from the techniques used in such a proof, we want to show only how one can explicitly find an AF algebra A whose primitive spectrum is a given finite poset P . First we will give the general construction and then discuss an example.
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Let {K1 , K2 , K3 , . . .} be the collection of all closed sets in P , where K1 = P . To construct the nth level of the Bratteli diagram D(A), we consider the subcollection of closed sets Kn ≡ {K1 , K2 , . . . , Kn } and denote with Kn0 the smallest collection of closed sets in P that contains Kn and is closed under union and intersection. The collection Kn determines a partition of the topological space P , by taking intersections and complements of the sets Kj ∈ Kn (j = 1, . . . , n). We denote with Y (n, 1), Y (n, 2), . . ., Y (n, kn ) the sets of such partition. Also, we write F (n, j) for the smallest closed set which contains Y (n, j) and belongs to the subcollection Kn0 . Then we can construct a Bratteli diagram following the rules: (1) the nth level of D(A) has kn points, one for each set Y (n, j); (2) the point at the level n of the diagram corresponding to Y (n, α) is linked to the point at the level n + 1 corresponding to Y (n + 1, β) if and only if Y (n, α) ∩ F (n + 1, β) 6= ∅. In this case, the multiplicity of the embedding is always 1. To illustrate this construction, let us consider the {p1 , p2 , q}. Now there are four closed sets:
W
poset of Fig. 4: P =
K1 = {p1 , p2 , q} , K2 = {p1 } , K3 = {p2 } , K4 = {p1 , p2 } . Thus it is not difficult to check that: K1 = {K1 }
K10 = {K1 }
Y (2, 1) = {p1 , p2 , q} ⊂ F (1, 1) = K1
K2 = {K1 , K2 }
K20 = {K1 , K2 }
Y (2, 1) = {p1 } Y (2, 2) = {p2 , q}
K3 = {K1 , K2 , K3 }
K30 = {K1 , K2 , K3 , K4 } Y (3, 1) = {p1 } Y (3, 2) = {q} Y (3, 3) = {p2 }
⊂ F (3, 1) = K2 ⊂ F (3, 2) = K1 ⊂ F (3, 3) = K3
K4 = {K1 , K2 , K3 , K4 } K40 = {K1 , K2 , K3 , K4 } Y (4, 1) = {p1 } Y (4, 2) = {q} Y (4, 3) = {p2 } .. .
⊂ F (4, 1) = K2 ⊂ F (4, 2) = K1 ⊂ F (4, 3) = K3
⊂ F (2, 1) = K2 ⊂ F (2, 2) = K1
Notice that, since P has only a finite number of points and hence a finite number of closed sets, the partition of P we have to consider at each level n repeats itself after a certain point (n = 3 in this case). Figure 9 shows the corresponding diagram, obtained through rules (1) and (2) above. Recalling then that the first matrix algebra that gives origin to an AF algebra is C and using the fact that all the embeddings have multiplicity one, we eventually obtain the sequence of finite dimensional algebras shown by the Bratteli diagram of Fig. 5. As we have said previously, such a diagram corresponds to the AF algebra A = CI1 + CI1 + K12 . It is a general fact that the Bratteli diagram describing any finite poset “stabilizes”, i.e. repeats itself, after a certain level n0 , when the family Kn0 of closed sets
ss ss
s ss s
ss ss
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
Y21 Y31 Y41 Y51
Y11
@@
@@ @ @@ Y @ @@ Y @ @Y .. .
32
42
51
.. .
...
457
@Y
22
Y33 Y43 Y53
Fig. 9. The construction of the Bratteli diagram of the AF algebra corresponding to the of Fig. 4.
W
poset
we choose is such that it determines a partition of the poset which distinguishes each point of the poset itself. In particular, this is the case if we choose n0 in such a manner that Kn0 contains all closed sets. Then, each Y (n0 , j) will contain a single point of the poset and F (n0 +1, j) will be the smallest closed set containing Y (n0 , j). It is only this stable part of the diagram which is relevant for the inductive limit and hence for the determination of the AF algebra it represents. Indeed, diagrams (or sequences of finite dimensional algebras) that differ only for a finite numbers of initial levels give different finite approximations to the same AF algebra [8, 9]. To conclude this section, we want to describe the AF algebras whose primitive spectra are the poset approximations of the circle, P4 (S 1 ), and of the sphere, P6 (S 2 ). As for P4 (S 1 ), given in Fig. 2, the Bratteli diagram repeats itself for n > n0 = 4 and the stable partition is given by Y (n0 , 1) = {x2 }
F (n0 + 1, 1) = {x2 }
Y (n0 , 2) = {x1 } Y (n0 , 3) = {x3 }
F (n0 + 1, 2) = {x1 , x2 , x4 } F (n0 + 1, 3) = {x2 , x3 , x4 }
Y (n0 , 4) = {x4 }
F (n0 + 1, 4) = {x4 } .
(4.18)
The corresponding Bratteli diagram is in Fig. 10. The set {0} is not an ideal. The limit algebra A turns out to be a subalgebra of bounded operators on the Hilbert space H = H1 ⊕ · · · ⊕ H4 , with Hi , i = 1, . . ., 4 infinite dimensional Hilbert spaces: (4.19) A = CI13 ⊕ CI24 ⊕ K12 ⊕ K34 . Here Iij and Kij denote the identity operator and the algebra of compact operators on Hi ⊕ Hj respectively. For the poset P6 (S 2 ) for the two-dimensional sphere, given in Fig. 3, n0 = 6 and the stable partition is given by
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
Y (n0 , 1) = {x5 } Y (n0 , 2) = {x2 }
F (n0 + 1, 1) = {x5 } F (n0 + 1, 2) = {x2 , x5 , x6 }
Y (n0 , 3) = {x1 } Y (n0 , 4) = {x3 }
F (n0 + 1, 3) = {x1 , x2 , x4 , x5 , x6 , } F (n0 + 1, 4) = {x2 , x3 , x4 , x5 , x6 }
Y (n0 , 5) = {x4 } Y (n0 , 6) = {x6 }
F (n0 + 1, 5) = {x4 , x5 , x6 } F (n0 + 1, 6) = {x6 } .
(4.20)
ss ss ss ss ssss
The corresponding Bratteli diagram is in Fig. 11.
The set {0} is not an ideal. The inductive limit is a subalgebra of bounded operators on the Hilbert space H = (H1 ⊕ H3 ) ⊗ (H5 ⊕ H6 ) ⊕ (H2 ⊕ H4 ) ⊗ (H7 ⊕ H8 ) .. .
H@H@HH @HHH H@H@ H @HHH @ H @ .. .
Fig. 10. The stable part of the Bratteli diagram for the circle poset P4 (S 1 ).
s s s s s s s s s s s s s s s s s s .. .
H PPHPHP XXXXX @ H @ @H H H@ HP X P H X P X @X H P X @ X P H X H XX @ HP @ H X X H P X @ HP HP P @ X H XHXXHXX P @@ HHHP @ P PPPHHXXXX H @ H PPHPHPH XXXXX @ H @ H PHP XX @ H @ X P H X H P X @H P X @ H X P H X H P X @@HHP @ H X PX X H P X H P @ X H X P @ HHH P @PPHXPXHXHXXX @@ HH@ PPHPHPHXXXXX H @ H@H PHPHP XXXX @ X P H X H P X @@HP PPXX @@HH HX .. . Fig. 11. The stable part of the Bratteli diagram for the sphere poset P6 (S 2 ).
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
...
459
with Hi , i = 1, . . ., 8 infinite dimensional Hilbert spaces, given by: A = CIH1 ⊗(H5 ⊕H6 )⊕H2 ⊗(H7 ⊕H8 ) ⊕ CIH3 ⊗(H5 ⊕H6 )⊕H4 ⊗(H7 ⊕H8 ) ⊕ (KH1 ⊕H3 ⊗ CIH5 ⊕H6 ) ⊕ (KH2 ⊕H4 ⊗ CIH7 ⊕H8 ) ⊕ KH5 ⊗(H1 ⊕H3 )⊕H7 ⊗(H2 ⊕H4 ) ⊕ KH6 ⊗(H1 ⊕H3 )⊕H8 ⊗(H2 ⊕H4 ) . 5. The Behncke Leptin Construction Given a poset P , there is always an AF algebra A such that Aˆ = P . A particular procedure to find such an algebra has been described in the previous section, but it is known that there exists more than one C ∗ -algebra A whose primitive spectrum is P . For example, if P consists of a single point, we can take for A any of the C ∗ -algebras M(n, C) of all n × n matrices valued in C. It is natural then to ask what are all the algebras associated to a given finite T0 topological space P . This problem was solved by Behncke and Leptin in 1973. In [11] they give a complete classification of all separable C ∗ -algebras A with finite primitive spectra. Such classification requires the definition of a function d on P , called defector, valued in IN = {∞, 0, 1, 2, . . .}. Given P and d, the Behncke– ˆ d) = P . Leptin construction gives a separable C ∗ -algebra A(P, d) such that A(P, Furthermore, any separable C ∗ -algebra A satisfying Aˆ = P is isomorphic to A(P, d) for some d. A defector d on the poset P is an IN -valued function on P such that d(x) > 0 if x is maximal.
(5.22)
Two defectors d and d0 are declared to be equal if there exists an automorphism ϕ of P such that d0 = d ◦ ϕ. They are called immediately equivalent if d(x) = d(x0 ) for all x ∈ P with the exception of at most one nonmaximal y ∈ P , such that d(y) = d0 (y) + d0 (z) or d0 (y) = d(y) + d(z) for some z coveringc y, if d(z) = d0 (z) < ∞. In the case d(z) = ∞, d(y) and d0 (y) may be arbitrary. Then two defectors are defined to be equivalent (d ∼ d0 ) if there exists a finite sequence of immediately equivalent defectors connecting them. We will start by describing the Behncke–Leptin construction for a special class of posets called forests. Then we will give the generalization for an arbitrary finite poset. 5.1. The Behncke Leptin construction for a forest A forest is a poset F such that {x, y, z ∈ F, x z, y z} ⇒ {x y or y x} . c We
say that y covers x if x ≺ y and there is no z such that x ≺ z ≺ y.
(5.23)
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
Given a forest F and a defector d on F , the Behncke–Leptin construction consists of the following steps. First we introduce a Hilbert space H(F, d) associated to the whole forest F . Second, for each point x ∈ F , we introduce a subspace H(x) ⊆ H(F, d) and a set of operators Rx acting on H(x). Actually, Rx can be thought of as acting on the whole H(F, d) by defining its action on the complement of H(x) to be zero. Then the C ∗ -algebra A associated to the forest is the one generated by the Rx ’s as x varies in F . Now we explain how to determine H(F, d), H(x) and Rx . The Hilbert space H(F, d) can be obtained using an auxiliary forest F 0 constructed from F in the following way. The forest F 0 contains a point x(1) for each maximal point x ∈ F (1) (2) and a pair of points xi and xi for each non maximal point xi ∈ F . Then on (2) (1) F 0 we introduce a partial order by declaring that xi is covered by both xj and (2)
s
s s
s
xj if and only if xi is covered by xj . Figure 12 shows an example of F and the corresponding F 0 . x3
@
@@
s s
(1)
x4
@@ x
x3
@
(1) 1
x1
ss ss
@@ @@ x A x AA AA x x (1) 2
2
(1)
x4
(2) 2
(2) 1
F0
F
Fig. 12. An example of a forest F and the auxiliary forest F 0 .
In F 0 we consider the maximal chains Cα = {x1 be seen to be necessarily of the form
(p1 )
(2)
(2)
(2)
(p2 )
, x2
(p )
, . . . , xk k }, which can
(1)
Cα = {x1 , x2 , . . . , xk−1 , xk } .
(5.24)
For example, in F 0 of Fig. 12, the maximal chains are {x1 }, {x1 , x2 }, {x1 , x2 , (1) (2) (2) (1) x3 }, {x1 , x2 , x4 }. To each maximal chain Cα we associate the Hilbert space (1)
h(Cα ) = lx1 ⊗ lx2 ⊗ . . . ⊗ lxk−1 ⊗ Cd(xk ) ,
(2)
(1)
(2)
(2)
(5.25)
where d(xk ) is the value of the defector d at the point xk ∈ F and lxi can be realized as the Hilbert space `2 of sequences (f1 , f2 , . . .) of complex numbers with P 2 n |fn | < ∞. We then define the total Hilbert space H(F, d) associated to F to be M h(Cα ) , (5.26) H(F, d) = α
where we sum over all maximal chains Cα in F 0 .
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In a similar way, we introduce the subspaces H(xi ) associated to a single point xi ∈ F by M (p) h(Cβ ) for all Cβ such that xi ∈ Cβ . (5.27) H(xi ) = β
Notice that if we consider the subforest Fx of F given by Fx = {y ∈ F |x y} ,
(5.28)
we can construct the Hilbert space H(Fx , dx ), where dx is the restriction of d to Fx . An important property of H(x) defined in (5.27) is that it satisfies H(x) = Hx ⊗ H(Fx , dx ) where Hx =
O
lxi for all xi ≺ x
(5.29)
(5.30)
i
and Hx = C if x is a minimal point. Now we are ready to define the C ∗ -algebra A(F, d). First, let us introduce the algebra of operators Rx , acting on H(x), given by Rx = CIHx ⊗ K(H(Fx , dx )) ,
(5.31)
IHx being the identity operator on Hx and K(H(Fx , dx )) being the algebra of compact operators on H(Fx , dx ). In other words, Rx acts as multiples of the identity on the Hilbert space Hx determined by the points xi ≺ x which precede x, as in (5.30), and as compact operators on the Hilbert space H(Fx , dx ) determined by the points xj x which follow x. Then A(F, d) is the algebra of operators on H(F, d) generated by all Rx as x varies in F . The algebras Rx , with x ∈ F , satisfy: Rx Ry ⊂ Rx if x y and Rx Ry = 0 if x and y are incomparable .
(5.32)
One of the major results of [11] is the following theorem, which establishes that the primitive spectrum of the C ∗ -algebra A(F, d) constructed according to the rules given above is homeomorphic to the forest F : Proposition 5.1. Let F be a finite forest with defector d and A(F, d) the algebra of operators on H(F, d) defined as above. Then we have: N (i) if E is a closed subset of F with complement U, then IE = x∈U Rx is a N closed two-sided ideal of A(F, d), and AE = x∈E Rx is a closed subalgebra of A(F, d); (ii) every two-sided ideal of A(F, d) is of the form IE for some closed E ⊂ F ˆ d) = F . and IE is primitive iff E = {x}. In particular, A(F,
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
s
s
Let us illustrate the Behncke–Leptin construction for a very simple forest, W namely the poset of Fig. 4. The correspondent associated forest P 0 is illustrated in Fig. 13.
ss
(1)
p1
@@ @
(1)
p2
@@
q (1)
q (2)
Fig. 13. The forest associated to the
W
poset.
We consider a generic defector d. From the diagram of P 0 in Fig. 13 we can write down all its maximal chains: (1)
(1)
{q (2) , p1 }, {q (2) , p2 }, {q (1) } and following (5.25) and (5.26) we see that H(F, d) is given by H(F, d) = lq ⊗ Cd(p1 ) ⊕ lq ⊗ Cd(p2 ) ⊕ Cd(q) .
(5.33)
The subspaces H(xi ) can also be determined from the diagram of P 0 : H(p1 ) = lq ⊗ Cd(p1 ) H(p2 ) = lq ⊗ Cd(p2 ) H(q) = H(F, d) .
(5.34)
Notice that the factorization expressed in (5.29) is satisfied, where now Hp1 = lq , H(Fp1 , dp1 ) = Cd(p1 ) Hp2 = lq , H(Fp2 , dp2 ) = Cd(p2 ) Hq = C, H(Fq , dq ) = H(F, d) .
(5.35)
The C ∗ -algebra A(F, d) is generated by all Rx , x ∈ F . The latter reads Rp1 = CIHp1 ⊗ K(Cd(p1 ) ) Rp2 = CIHp2 ⊗ K(Cd(p2 ) ) Rq = K(H(F, d)) .
(5.36)
Notice that for the defector d(p1 ) = d(p2 ) = 1 and d(q) = 0 we get H(F, d) = W H1 ⊕ H2 and A = CI1 + CI2 + K12 and thus recover the algebra we got for the poset via the Bratteli construction in Sec. 4.1.
NONCOMMUTATIVE LATTICES AND THE ALGEBRAS OF
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5.2. The Behncke Leptin construction for posets To generalize the procedure of the last section to an arbitrary poset P with defector d, we have first to introduce a forest P , uniquely determined by P . Let P be a finite poset. A rope r of P is a (not necessarily maximal) chain in P starting from a minimal element and ending at some x ∈ P . The set P of all ropes of P ordered by inclusion is a poset. One can show that P is in fact a forest. Let ϕ : P → P denote the surjective map which assigns to each rope r ∈ P its end point ϕ(r) ∈ P . Following [11], we will call the pair (P , ϕ) the covering forest of P . An example is given in Fig. 14, which shows the covering forest of the circle poset P4 (S 1 ) of Fig. 2.
s
{x1 , x2 }
@@ @@
s
s s
{x1 , x4 }
{x3 , x2 }
@@ @@
@
{x1 }
s
s
{x3 , x4 }
@
{x3 }
Fig. 14. The covering forest of the circle poset P4 (S 1 ).
Given a defector d on P we define a defector d on P in a natural way via the pull-back: d = d ◦ ϕ. (5.37) Then, since P is a forest, we can construct the algebra A(P , d) following Sec. 5.1. Finally, to identify the C ∗ -algebra A(P, d) associated to the poset P and the defector d, we proceed to realize A(P, d) as a subalgebra of A(P , d). In order to do so, we need to point out a simple property of the covering forest (P , ϕ). Let r, s ∈ P be in the inverse image ϕ−1 (x) of x ∈ P . Then, the subforest (P )r (see (5.28)) is naturally isomorphic to (P )s . Indeed, (P )r and (P )s consist of all extensions of the rope r and s respectively. By hypothesis, r and s have the same end point x ∈ P , so that (P )r ∼ (P )s . Thus K H((P )r , dr ) ' K H((P )s , ds ) ≡ Kx , so that the algebras Rs , Rr ∈ A(P , d) are given by Rr = CIHr ⊗ Kx , Rs = CIHs ⊗ Kx . For each x ∈ P we define the algebra Ax M Ax = r∈ϕ−1 (x)
Rr
(5.38)
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E. ERCOLESSI, G. LANDI and P. TEOTONIO-SOBRINHO
and a subalgebra Rx ⊂ Ax given by all elements a ∈ Ax of the form a = (λr1 IHr1 ⊗ k) + (λr2 IHr2 ⊗ k) + · · · + (λrn IHrn ⊗ k) , where ri ∈ ϕ−1 (x), λrj ∈ C and k ∈ Kx . Thus Rx = {a ∈ Ax |a =
M
(λr IHr ⊗ k), λr ∈ C and k ∈ Kx } .
(5.39)
r∈ϕ−1 (x)
The C ∗ -algebra A(P, d) that satisfies ˆ d) = P A(P,
(5.40)
is then generated by all Rx with x ∈ P . There is an intuitive interpretation for (5.39). The poset P can be obtained from P by identifying any two ropes r and s that have the same ending point. Equation (5.39) simply expresses this identification at an algebraic level. For example, for the circle poset P4 (S 1 ) these rules give the following algebras, acting on H = H1 ⊕ H2 ⊕ H3 ⊕ H4 : Ax4 = CI1 + CI3 Ax2 = CI2 + CI4 Ax1 = CI1 + CI2 + K12 Ax3 = CI3 + CI4 + K34 ,
(5.41)
if one chooses the defector d(x1 ) = d(x2 ) = 1, d(x3 ) = d(x4 ) = 0. Thus the algebra associated to P4 (S 1 ) is A = CI1 + CI2 + CI3 + CI4 + K12 + K34 .
(5.42)
As before this is the algebra one gets for P4 (S 1 ) by means of the Bratteli construction explained in Sec. 4.3. Equivalent defectors give rise to isomorphic C ∗ -algebras, whereas by choosing different non-equivalent defectors one can construct non-isomorphic C ∗ -algebras that all have P as primitive spectrum. In this way one can obtain all C ∗ -algebras A whose (finite) spectrum Aˆ is homeomorphic to the poset P , as it is established in [11], which we quote to conclude this section: Proposition 5.2. (i) Every separable C ∗ -algebra A with finite dual Aˆ = P is isomorphic to some A(P, d). (ii) A(P, d) is isomorphic to A(P, d0 ) if and only if d and d0 are equivalent.
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6. Final Remarks In this article, we have seen how a finite poset is truly a “noncommutative space” or “noncommutative lattice”, since it can be described as the primitive spectrum of a noncommutative C ∗ -algebra A, which turns out to be always a postliminal AF algebra. We have also seen that this correspondence is not one-to-one, more than one non-isomorphic C ∗ -algebra leading to the same poset. This relation between posets and C ∗ -algebras was used in [17] to give a dualization of the approximation method for topological spaces introduced in [1]. In our previous work [5] we have showed how it is possible to construct a quantum theory on posets, by making use of the corresponding C ∗ -algebra. We have also seen how important topological properties of the continuum, such as homotopy, can be captured by the poset approximation and manifest themselves in the corresponding quantum mechanics. We are thus naturally led to examine how one can construct further geometric structures on posets, as is suggested by Connes’ noncommutative geometry [6]. First of all, we are interested in the construction of bundles and characteristic classes over a poset and, as a first step in this direction, one should examine the K-theory of these noncommutative lattices. This is the topic discussed in [12], where we present a study of the algebraic K-theory of AF algebras associated to a poset. Then one would like to construct bundles, and notably nontrivial bundles, over a poset, and consider, for instance, the analogue of the monopole bundle over the lattice approximating the two-dimensional sphere and of nontrivial “topological charges”. Work in this direction is in progress. Acknowledgements This work was initiated while the authors were at Syracuse University. We thank A. P. Balachandran, G. Bimonte, F. Lizzi e G. Sparano for many fruitful discussions and useful advice. The final version was written while G. L. and P. T-S were at ESI in Vienna. They would like to thank G. Marmo and P. Michor for the invitation and all people at the Institute for the warm and friendly atmosphere. We thank the “Istituto Italiano per gli Studi Filosofici” in Napoli for partial support. The work of P. T-S. was also supported by the Department of Energy, U.S.A. under contract number DE-FG-02-84ER40173. The work of G. L. was partially supported by the Italian “Ministero dell’ Universit`a e della Ricerca Scientifica”. References [1] R. D. Sorkin, Int. J. Theor. Phys. 30 (1991) 923. [2] P. S. Aleksandrov, Combinatorial Topology, Vols. 1-3, Greylock, 1960. [3] A. P. Balachandran, G. Bimonte, E. Ercolessi and P. Teotonio-Sobrinho, Nucl. Phys. B418 (1994) 923. [4] A. P. Balachandran, G. Bimonte, G. Landi, F. Lizzi and P. Teotonio-Sobrinho, “Lattice Gauge Fields and noncommutative geometry” (preprint ESI 299, 1995, hepth/9604012), to appear in J. Geom. Phys. [5] A. P. Balachandran, G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, Nucl. Phys. B 37C Proc. Suppl., 20 (1995); J. Geom. Phys. 18
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(1996) 163. [6] A. Connes, Noncommutative Geometry, Academic Press, 1994; G. Landi, An introduction to Noncommutative Spaces and their Geometries, Springer-Verlag, 1997. [7] J. C. V´ arilly and J. M. Gracia-Bond´ia, J. Geom. Phys. 12 (1993) 223. [8] O. Bratteli, Trans. Amer. Math. Soc. 171 (1972) 195. [9] K. R. Goodearl, Notes on Real and Complex C ∗ -algebras , Shiva Publishing Limited. [10] G. A. Elliott, J. Alg. 38 (1976) 29. E. G. Effros, Dimension and C ∗ -algebras, Amer. Math. Soc., 1981. [11] H. Behncke and H. Leptin, J. Functional Analysis 14 (1973) 253; 16 (1974) 241. [12] E. Ercolessi, G. Landi and P. Teotonio-Sobrinho, “K-theory of noncommutative lattices” (preprint, 1995); (9-alg/9607017). [13] J. Dixmier, C ∗ -algebras , North-Holland, 1982. [14] J. M. G. Fell and R. S. Doran, Representations of ∗ -Algebras, Locally Compact Groups and Banach ∗ -Algebraic Bundles, Academic Press, 1988. [15] J. L. Kelley, General Topology, Springer-Verlag, 1955. J. Hocking, G. Young, Topology, Dover, 1988. [16] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wordsworth and Brooks/Cole Advanced Books and Software, 1986. [17] G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. Teotonio-Sobrinho, “Lattices and their continuum limits”, J. Geom. Phys. 20 (1996) 318; (hepth/9507147). G. Bimonte, E. Ercolessi, G. Landi, F. Lizzi, G. Sparano and P. TeotonioSobrinho, “Noncommutative lattices and their continuum limits”, J. Geom. Phys. 20 (1996) 329 (hep-th/9507148). [18] O. Bratteli, J. Functional Analysis 16 (1974) 192. [19] O. Bratteli and G. A. Elliott, J. Functional Analysis 30 (1979) 74.
ON THE SUPER-UNITARITY OF DISCRETE SERIES REPRESENTATIONS OF ORTHOSYMPLECTIC LIE SUPERALGEBRAS AMINE M. EL GRADECHI D´ epartement de Math´ ematiques, Facult´ e des sciences Jean Perrin Universit´ e d’Artois, rue Jean Souvraz S.P. 18, 62307 Lens, France and U.R.A. C.N.R.S. 0751 D E-mail : [email protected] E-mail : [email protected] Received 5 April 1997 Revised 2 July 1997 1991 Mathematical Subject Classification: 17A70, 22E43, 22E45, 46C05, 47A05, 47A67, 47B25 We investigate the notion of super-unitarity from a functional analytic point of view. For this purpose we consider examples of explicit realizations of a certain type of irreducible representations of low rank orthosymplectic Lie superalgebras which are super-unitary by construction. These are the so-called superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) which we recently constructed using a Z2 –graded extension of the orbit method. It turns out here that super-unitarity of these representations is a consequence of the self-adjointness of two pairs of anticommuting operators which act in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su(1, 1) such that the difference of the respective lowest weights is 12 . At an intermediate stage, we show that the generators of the considered orthosymplectic Lie superalgebras can be realized either as matrix-valued first order differential operators or as first order differential superoperators. Even though the former realization is less convenient than the latter from the computational point of view, it has the advantage of avoiding the use of anticommuting Grassmann variables, and is moreover important for our analysis of super-unitarity. The latter emphasizes the fundamental role played by the atypical (or degenerate) superholomorphic discrete series representations of osp(2/2, R) for the super-unitarity of the other representations considered in this work, and shows that the anticommuting (unbounded) self-adjoint operators mentioned above anticommute in a proper sense, thus connecting our work with the analysis of supersymmetric quantum mechanics. Keywords: Orthosymplectic Lie superalgebra, discrete series representation, superunitarity, first order differential operator, self-adjoint operator, anticommuting self-adjoint operators.
1. Introduction Our recent successful extension of geometric quantization to certain coadjoint orbits of low rank non-compact orthosymplectic Lie supergroups, has led to explicit constructions of infinite-dimensional super-unitary irreducible representations of the corresponding Lie superalgebras [1, 2]. In analogy with the non-graded 467 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 467–497 c World Scientific Publishing Company
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situation, we coined the obtained representations superholomorphic discrete series representations. These are concrete realizations of the abstract discrete series representations recently studied, from different points of view using purely algebraic methods, by Furutsu and Hirai [3], Nishiyama [4], and Jakobsen [5]. Up to our knowledge, except for the superholomorphic realizations considered here, the only known explicit realizations of super-unitary representations of orthosymplectic Lie superalgebras are the so-called oscillator representations previously obtained by Nishiyama [6] (see also [7]); these are very special realizations. In the present work, we investigate from the functional analytic and operatorial theoretic points of view the structure of the superholomorphic discrete series representations of the Lie superalgebras osp(1/2, R) and osp(2/2, R). In particular, we present here an original interpretation of the notion of super-unitarity. Our interpretation relies on simple Hilbert space theoretic considerations, even though super-unitarity is consistently defined on a super-Hilbert space. The results of our previous contributions [1, 2] (see also [8]) and those described here put Kostant’s program within reach. Initiated in [9], this program which is partly at the origin of our interest in this subject is aimed at developing a harmonic analysis on Lie supergroups and on their homogeneous spaces (see [10] for an explicit statement). Quite remarkably, our present analysis of the super-unitarity of the superholomorphic discrete series representations under consideration combines in a unique way both classical and modern notions and techniques in functional analysis and operator theory. Indeed, on one hand this analysis relies on methods analogous to those developed by Bargmann in his celebrated work on the holomorphic realizations of unitary irreducible representations of two different Lie algebras: the bosonic C.C.R. (or Weyl–Heisenberg Lie algebra) [11] and the su(1, 1) Lie algebra [12]. This last realization was the first holomorphic discrete series representation to be constructed (see [13] for a modern perspective). Methods relevant to its study naturally intervene in our present work since the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) decompose into a direct sum of such su(1, 1) representations [2, 4], su(1, 1) ' sp(2, R) being a subalgebra of osp(1/2, R) and osp(2/2, R). Moreover, methods pertaining to the study of the holomorphic realization of the C.C.R. appear here to be of relevance in the analysis of the properties of the operators representing the odd generators of osp(1/2, R) and osp(2/2, R). We recall that the holomorphic realization of the C.C.R. was originally devised by Fock [14], his partial results were subsequently completed and generalized by Dirac [15], Bargmann [11], and Segal [16]. (The description in [11] is sufficient for our purpose.) On the other hand, the recently introduced notion of anticommuting (unbounded) self-adjoint operators plays here an important role as it turns out that the superunitarity of the considered representations is a consequence of the self-adjointness of two pairs of anticommuting operators acting in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su(1, 1) such that their respective lowest weights differ by 12 . The notion of anticommuting self-adjoint operators was first introduced by Vasilescu [17], it was then further
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
469
developed by Samoilenko [18] and Pedersen [19]. Several interesting applications of this theory were considered by Arai [20] (see also references therein), these range from supersymmetric quantum mechanics to the analysis of operators of Dirac’s type. Hence our study shows that representation theory of orthosymplectic Lie superalgebras provides a new field of applications for this theory. In fact, we will show how one of Arai’s characterizations of proper anticommutativity of self-adjoint operators apply to the above pairs of operators. As already mentioned before, our considerations are restricted here to the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) as obtained by geometric quantization in [1, 2]. In the case of osp(2/2, R) both the typical and the atypical (or degenerate) superholomorphic discrete series representations are studied [2]. (It is well known that only the former have non-graded analogs.) It turns out from our analysis that the latter play a fundamental role regarding super-unitarity of all the other representations considered in this work. Indeed, the operators mentioned in the previous paragraph belong to the odd part of the Lie superalgebra osp(2/2, R) in the atypical superholomorphic discrete series representations. Our analysis of super-unitarity requires a preliminary step which consists in rewriting the results of [1, 2] in a more appropriate form. Indeed, thanks once again to the special structure of the carrier spaces of the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R), the first order differential superoperators representing the generators of these Lie superalgebras obtained in [1, 2] can be rewritten in the form of matrix-valued first order differential operators. The superoperator realization is expressed in terms of anticommuting variables that belong to a complex Grassmann algebra, while in the alternative realization the matrix form replaces the dependence in such variables. Very much in the spirit of the present journal, this paper gathers original results (Secs. 4 and 5), an overview of the central theme which is representation theory of orthosymplectic Lie superalgebras (Sec. 3), mathematical preliminaries intended to make the content as self contained as possible (Sec. 2) and an up to date (though not exhaustive) bibliography. It is organized as follows. In Sec. 2 we set our notations by giving the definitions of the main notions used throughout. Section 3 starts with an overview of the recent progress made in the construction of both abstract and explicit realizations of discrete series representations of orthosymplectic Lie superalgebras. After that, we give a description of the superholomorphic discrete series representations of the Lie superalgebras osp(1/2, R) and osp(2/2, R), and we display the first order differential superoperators representing their generators as obtained by geometric quantization. Finally, we derive the alternative realization of the latter in terms of matrix-valued first order differential operators. Section 4 is devoted to a rigorous Hilbert space analysis of the notion of super-unitarity which is shown to follow from the self-adjointness of two pairs of (naively) anticommuting operators. In Sec. 5 we show that the latter anticommute in a proper sense. Concluding remarks and future directions of investigation are presented in Sec. 6.
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2. Preliminaries In this section we give basic definitions and we introduce the main notations for the ingredients needed throughout. More precisely, we give the definitions of an orthosymplectic Lie superalgebra, a Grassmann algebra, super-unitarity and anticommuting self-adjoint operators, together with a brief account on the holomorphic discrete series representations of su(1,1). 2.1. Orthosymplectic Lie superalgebras Let V = V0 ⊕ V1 be a direct sum of two vector spaces over R or C ; V will be called a Z2 -graded vector space. A homogeneous vector a ∈ V is a vector which belongs either to V0 or to V1 . If a ∈ Vj , then its degree is (a) = j, for j ∈ Z2 . Note that in what follows we implicitly assume that those elements which appear in an equation together with their degree of homogeneity are homogeneous elements of the Z2 -graded vector space they belong to. Moreover, “super”, “Z2 –graded” or simply “graded” are used interchangeably throughout. Definition 2.1. A Lie superalgebra is a Z2 -graded algebra (g = g0 ⊕ g1 , [·, ·]), i.e. [gi , gj ] ⊂ g(i+j)mod 2 , such that: (i) [a, b] = (−1)(a)(b) [b, a] , ∀ a, b ∈ g , (ii) (−1)(a)(c)[a, [b, c]]+(−1)(b)(a)[b, [c, a]]+(−1)(c)(b)[c, [a, b]] = 0 , ∀ a, b, c ∈ g . Starting from an associative Z2 -graded algebra (g = g0 ⊕ g1 , ·), i.e. gi · gj ⊂ g(i+j)mod 2 , one can equip g with a Lie superalgebra structure using the Lie superbracket: (2.1) [a, b] = ab − (−1)(a)(b) ba , ∀ a, b ∈ g . The restriction of this superbracket to g0 × g0 and g0 × g1 is a commutator, while its restriction to g1 × g1 is an anticommutator. We denote the latter [·, ·]− and [·, ·]+ , respectively. Now, let V = V0 ⊕ V1 be a real Z2 -graded vector space, and let V0 and V1 be, respectively, m and 2n-dimensional. The algebra gl(V, R) of linear operators on V is naturally a Z2 -graded algebra. Indeed, gl(V, R) = gl(V, R)0 ⊕ gl(V, R)1 , where gl(V, R)i = A ∈ gl(V ) | AVj ⊂ V(i+j)mod 2 for i = 0, 1. We designate by gl(m/2n, R) the matrix representation of gl(V, R) in a basis of V , ordered in such a way that its first m elements form a basis of V0 . Let (·, ·) be the non-degenerate bilinear form on V , defined by: u, v ∈ V ,
(u, v) = t uBv , where
Im
B= 0
0
0 0 −In
0
In . 0
(2.2)
(2.3)
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Definition 2.2. The orthosymplectic Lie superalgebra osp(m/2n, R) is the Z2 –graded subalgebra of gl (m/2n, R) given by: osp(m/2n, R) = {X ∈ gl(m/2n, R) (Xu, v) + (−1)(X)(u) (u, Xv) = 0 ,
∀ u, v ∈ V } .
(2.4)
In what follows osp(m/2n, R) will be simply denoted osp(m/2n). For more detail about these (and other) Lie superalgebras we refer the reader to the basic references [21, 22, 23]. In the remaining of this paper we will only consider the low rank cases, namely osp(1/2) and osp(2/2). We now display their defining relations in a specific basis of the superalgebra. Let {K0 , K± , F± } be the Cartan–Weyl basis of osp(1/2). The defining relations of the latter are as follows: [K0 , K± ]− = ±K± , [K0 , F± ]− = ± 12 F± ,
[K+ , K− ]− = −2K0 ,
[K± , F± ]− = 0 ,
[F± , F± ]+ = K±
[K± , F∓ ]− = ∓F± ,
and [F+ , F− ]+ = K0 .
(2.5) (2.6) (2.7)
As a Z2 -graded algebra osp(1/2) = osp(1/2)0 ⊕ osp(1/2)1 , where osp(1/2)0 is the simple Lie algebra sp(2, R) ' su(1, 1), with its usual Cartan–Weyl basis {K0 , K± }, and osp(1/2)1 is a 2-dimensional irreducible su(1, 1)-module. Similarly, let {B, K0 , K± ; V± , W± } be the Cartan–Weyl basis of osp(2/2). The defining relations of the latter are as follows: [K0 , K± ]− = ±K± ,
[K+ , K− ]− = −2K0 ,
(2.8a)
[B, K± ]− = 0,
[B, K0 ]− = 0 ,
(2.8b)
[K0 , V± ]− = ± 12 V± ,
[K0 , W± ]− = ± 12 W± ,
(2.8c)
[K± , V± ]− = 0,
[K± , W± ]− = 0 ,
(2.8d)
[K± , V∓ ]− = ∓V± ,
[K± , W∓ ]− = ∓W± ,
(2.8e)
[B, V± ]− = 12 V± ,
[B, W± ]− = − 21 W± ,
(2.8f)
[V± , V± ]+ = 0,
[W± , W± ]+ = 0 ,
(2.8g)
[V± , V∓ ]+ = 0,
[W± , W∓ ]+ = 0 ,
(2.8h)
[V± , W± ]+ = K± ,
[V± , W∓ ]+ = K0 ∓ B .
(2.8i)
As a Z2 -graded algebra osp(2/2) = osp(2/2)0 ⊕ osp(2/2)1 , where osp(2/2)0 = so(2) ⊕ su(1, 1), with its Cartan–Weyl basis {K0 , K± , B}, and osp(2/2)1 is the direct sum of two irreducible 2-dimensional su(1, 1)-modules spanned, respectively, by {V+ , V− } and {W+ , W− }.
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2.2. Grassmann algebras Definition 2.3. A real (complex) Grassmann algebra BN is a 2N -dimensional associative unital algebra over R (C) generated by I, ξ1 , . . . , ξN which satisfy the relations: I ξi = ξi I = ξi ,
and ξi ξj = −ξj ξi ,
∀ i, j = 1, . . . , N .
(2.9)
Clearly, all the generators are nilpotent, i.e. ξi2 = 0, ∀ i = 1, . . . , N . The example of a complex Grassmann algebra we will be using throughout is V that of the complex exterior algebra over CN , i.e. BN ≡ CN . As a Z2 -graded V V 0 1 0 1 ⊕ BN , where BN = ⊕r even r CN and BN = ⊕r odd r CN . algebra, BN = BN 0 1 (ξ1 ∈ BN ) Hence, any element ξ ∈ BN decomposes as ξ = ξ0 + ξ1 , where ξ0 ∈ BN is called the even (odd) component of ξ. Moreover, the component of ξ ∈ BN that V0 N C ≡ C is called the body of ξ and the remaining part is called its belongs to soul. Grassmann algebras were introduced in physics in order to provide physical models with anticommuting variables that are used in the mathematical description of observables which obey to Fermi statistics. Usual analysis has been extended to accommodate (super)functions of both commuting and anticommuting variables [24]. Because of the nilpotent character of the anticommuting variables, such functions are straightforward generalizations of usual functions. For instance, let z ∈ C 1 , then the superfunction and θ ∈ BN f (z, θ) ≡ f0 (z) + θ f1 (z) ,
(2.10)
where f0 and f1 are usual functions on C. In particular, a notion of integration was introduced. This is a functional procedure, simply defined by the following rules: Z Z 1 . (2.11) θ dθ = 1 and dθ = 0 , ∀θ ∈ BN More about this and other graded extensions of usual analytical, algebraic and geometric notions can be found in [9, 24]. 2.3. Super-unitarity In order to extend representation theory to the Z2 –graded context, one needs to define extensions of the notions of a Hermitian structure, a Hilbert space and unitarity. So far, different definitions of such extensions have been proposed. Here we only consider those we used in [1, 2]. They are based on the general definition of a graded Hermitian structure given in [25]. They moreover agree with the definition of super-unitarity used in [4, 6]. In [1, 2] we went one step further. We explicitly defined the notion of a super-Hilbert space. This was a consequence of our explicit construction of irreducible super-unitary representations of osp(1/2) and osp(2/2). Below, the definitions are given in the following order: super-Hermitian structure, super-Hilbert space and then super-unitarity.
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Definition 2.4. Let V = V0 ⊕ V1 be a Z2 -graded complex vector space. A super-Hermitian structure on V is a sesquilinear form hh·, ·ii : V × V → C, such that hhu, vii = (−1)(u)(v) hhv, uii ,
∀ u, v ∈ V .
(2.12)
Such a form is said to be an even super-Hermitian form, if hhu, vii = 6 0 only when u and v have the same degree. In the present work we will consider the following type of even super-Hermitian forms: hhu, vii = hu0 , v0 i0 + i hu1 , v1 i1 ,
(2.13)
where h·, ·ii is a Hermitian form on Vi , (i = 0, 1), and u = u0 + u1 , v = v0 + v1 ∈ V . Such an even super-Hermitian form is said to be positive definite if both h·, ·i0 and h·, ·i1 are positive definite. Definition 2.5. A super-Hilbert space is a Z2 -graded complex vector space V equipped with a non-degenerate even positive definite super-Hermitian structure hh·, ·ii, such that (V0 , h·, ·i0 ) and (V1 , h·, ·i1 ) are Hilbert spaces. In particular, v = v0 + v1 ∈ V is said to be super square integrable if Re(hhv, vii) = kv0 k20 < ∞ and Im(hhv, vii) = kv1 k21 < ∞, where “Re” and “Im” designate, respectively, the real and the imaginary parts, and k · ki is the L2 -norm on Vi (for i = 0, 1). For g a given Lie superalgebra, assume that V is a g-module equipped with a positive definite super-Hermitian form. Definition 2.6. The representation ρ of g in V is said to be super-unitary if hhρ(X)u, vii = (−1)(X)(u) hhu, ρ(X)vii ,
∀ u, v ∈ V and ∀ X ∈ g, .
(2.14)
This simply means that ρ(X) has to be super-Hermitian (or super symmetric). This equation differs by a sign from the one used in [4], simply because there ρ(X) is required to be super-skew-Hermitian. Note also that if ρ(X) is an unbounded linear differential superoperator on a super-Hilbert space, then (2.14) only makes sense for u and v in the domain of ρ(X). Moreover, for X ∈ g0 such that ρ(X) is unbounded, (2.14) does not imply that ρ(X) is self-adjoint. Hence, by Stone’s theorem [26] a self-adjoint extension of ρ(X) needs to be found in order to be able to associate to X a one parameter group of unitaries on V . We are mentioning these subteleties because all the (super)operators that appear throughout are unbounded and hence domain considerations are mandatory. Let us also mention that the choice of sign made in (2.13) (i.e. “+i” instead of “−i”) is the one that allows super-unitarization of the lowest weight modules of the Lie superalgebras considered in this work. Super-unitarization of the highest weight modules requires the opposite sign. For more details we refer to [4, 2]; in the second reference a geometric interpretation of this fact is given. Finally, it is important to note that even though the notion of a super-Hilbert space given in Definition 2.5 is the most appropriate from the Z2 -grading point of view, it is nevertheless possible to use a more conventional concept [22]. More
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precisely, one can define a super-Hilbert space as in Definition 2.5, but with the super-Hermitian structure hh·, ·ii replaced with a usual Hermitian one h·, ·i (i.e. replacing the ‘i’ in (2.13) by ‘1’). The super-Hilbert space turns then into a Hilbert sum. Accordingly, one needs to modify Definition 2.6 in the following way: (2.14) has to be replaced by hρ(X)u, vi = (−i)(X) hu, ρ(X)vi. Our notion of a super-unitary representation becomes then the notion of a star representation introduced in the physics literature (see [5, 22] for more details). This equivalence will be used in Sec. 4 in the investigation, by means of pure Hilbert space analytic methods, of properties of certain odd operators in terms of which the super-unitarity of the superholomorphic diecrete series representations of osp(1/2) and osp(2/2) is expressed. 2.4. Holomorphic discrete series of su(1,1) Here we give a brief description of the holomorphic discrete series representations of SU(1,1). More precisely, we will concentrate our attention on their infinitesimal realizations, namely the holomorphic discrete series of su(1, 1) that are derived representations of SU(1, 1). (For more details we refer the reader to Bargmann’s original work [12] and to Knapp’s book [13].) These representations are lowest weight unitary irreducible representations, denoted by D(k), where the lowest weight k is such that: k ∈ 12 N and k > 12 . They are explicitly realized in the Hilbert space Hk defined by: n o (2.15) Hk = ψ : D(1) → C, ψ holomorphic kψk2k < ∞ , where D(1) = {z ∈ C | |z| < 1} is the unit disc, and k · kk is the L2 -norm associated with the inner product h , ik given by: Z 2k − 1 dz d¯ z φ(z) ψ(z) , ∀ φ, ψ ∈ Hk . (2.16) hφ, ψik = 2 )2−2k π (1 − |z| (1) D The normalization is chosen here in such a way that kψk2k = 1 for ψ(z) = 1. The Hilbert space Hk is separable. The following set of holomorphic functions on D(1) is a complete orthonormal basis of Hk : ) ( 1/2 Γ(m + 2k) (k) m z , m∈N . (2.17) um (z) = m! Γ(2k) (k)
The superscript (k) in um (z) refers to the discrete series D(k). The generators K0 and K± of su(1, 1) which satisfy (2.5), are represented by the following first order differential operators on D(1) : b + = z 2 d + 2kz , K dz
b0 = z d + k , K dz
b− = d . K dz
(2.18)
These are unbounded linear operators on Hk . On a properly defined dense domain b 0 is self-adjoint, while K b ± is the adjoint of K b ∓ [12]. in Hk , one can show that K More precisely, one has
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ORTHOSYMPLECTIC LIE SUPERALGEBRAS
b 0 ψik = hK b 0 φ, ψik hφ, K
b ± ψik = hK b ∓ φ, ψik , and hφ, K
(
where Uk =
∀ φ, ψ ∈ Uk ⊂ Hk , (2.19) )
∞ X 2 ψ ∈ Hk m2 |hu(k) m , ψik | < ∞
,
(2.20)
m=0
and moreover DK0 = DK † = DK± = DK † = Uk . Here DA denotes the domain of ∓
0
the operator A in Hk while A† denotes the adjoint of A. The above realizations of the discrete series representations of su(1, 1) can be explicitly constructed using the general method of geometric quantization [27, 28] which associates these representations to the elliptic coadjoint orbits of SU(1, 1). The latter are K¨ ahler manifolds represented by the unit disc D(1) ∼ = SU(1, 1)/U(1) z. equipped with its SU(1, 1)-invariant K¨ ahler form ω = −2ik (1 − |z|2 )−2 dz ∧ d¯ 2.5. Anticommuting self-adjoint operators In this section we give Pedersen’s definition of the anticommutativity of two non necessarily bounded self-adjoint operators in a Hilbert space [19]. Then, following Arai [20], we give one characterization of such a notion that will be useful in Sec. 5. Proofs of the results cited below can be found in [20]. Definition 2.7. Two (unbounded) self-adjoint operators A and B in a Hilbert space H are said to be anticommuting if eitA B ⊂ Be−itA ,
∀t ∈ R.
(2.21)
As shown in [19] this definition is symmetric in A and B. A characterization of the anticommutativity of self-adjoint operators devised by Arai [20, Theorem 6.3] in the context of supersymmetric quantum mechanics is now given. Proposition 2.8 [20]. Let Q1 and Q2 be self-adjoint operators in a Hilbert space H with inner product (· , ·) such that Q21 = Q22
and
(Q1 ψ, Q2 φ) + (Q2 ψ, Q1 φ) = 0 ,
ψ, φ ∈ DQ1 ∩ DQ2 .
(2.22)
Then Q1 and Q2 anticommute. 3. Superholomorphic Discrete Series and First Order Differential Operators The first part of this section is devoted to a qualitative review of the recent progress made in the extension of the theory of discrete series representations to orthosymplectic Lie superalgebras. We first describe different abstract constructions of such representations, then their explicit realizations through geometric methods. The second part of this section contains a detailed description of the structure of the superholomorphic realization of the discrete series representations of osp(1/2) and osp(2/2), and a derivation of their alternative matrix realization.
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3.1. An overview The abstract representation theory of Lie superalgebras started with Kac’s seminal work [21] and culminated recently in Jakobsen’s systematic classification of all unitarizable highest weight modules of basic classical Lie superalgebras [5]. Its gradual development was strewed by several significant contributions in which both mathematical physicists and pure mathematicians investigated specific examples. The list of these contributions is too long to be included here. For a good account on the evolution of the theory of finite and infinite-dimensional representations of Lie superalgebras and a list of references, we refer the interested reader to [5, 29]. Here we concentrate our presentation on the representation theory of the two orthosymplectic Lie superalgebras of interest to us. Abstract irreducible representations of osp(1/2) and osp(2/2) have been considered in [3] and [4, 5] from two different points of view. In the first reference the authors develop a formalism aimed at classifying the irreducible representations of a Lie superalgebra g = g0 ⊕ g1 starting from the known irreducible representations of its even Lie subalgebra g0 . They define a notion of super-unitarity which allows them to decide which of the irreducible representations found are superunitarizable. The example of osp(1/2) is then fully considered. It turns out that the only non-trivial irreducible representations of osp(1/2) which are super-unitarizable are those which are irreducible extensions of the discrete series representations of su(1,1) ' osp(1/2)0 . It seems then well justified to call the obtained super-unitary irreducible representations discrete series representations of osp(1/2). (Note that the first abstract construction of the latter was obtained in [30] using the shift operator technique which was developed in the mathematical physics literature. Moreover, [30] contains the first oscillator representation of an ortosymplectic Lie superalgebra. A generalization of this type of representations to all orthoymplectic Lie superalgebras has been obtained in [6, 7].) In [4] a more conventional approach is adopted. More precisely, a generalization of the standard procedure of induction from a parabolic sub-(super)algebra (with a compact reductive part) is considered, and is subsequently applied to osp(2/2). Provided that a notion of super-unitarity is defined, such a method leads, as in the non-graded case, to only a subset of the set of all super-unitary irreducible representations of the considered Lie superalgebra. For osp(2/2), only discrete series representations are obtained in this way [4]. More precisely, the representations so obtained are identified as elements of the discrete series because they share with the usual discrete series representations several of their known properties [4]. These include the facts that the eigenvalues of the Laplace–Casimir operator are discretely distributed and that the representation space is an extremal weight module. Note that this method has recently been extended by Jakobsen to all basic classical Lie superalgebras and has led to a classification of all their unitarizable highest weight modules [5]. The different approaches described here are based on equivalent notions of super-unitarity. It would be interesting to probe their equivalence in general. It is worth mentioning in connection with what precedes that the point that is of importance to us in this work is that a discrete series representation of an
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477
orthosymplectic Lie superalgebra decomposes into a direct sum of discrete series representations of the even part of the superalgebra (see [4] for the proof). This can be viewed as the reverse procedure to the extension technique of [3] applied to discrete series. In the case of osp(2/2), there appear two types of super-unitary irreducible representations that belong to the discrete series [4] (see also [2]). We coined them in [2] the typical and the atypical discrete series. We borrowed this terminology from the theory of finite-dimensional irreducible representations of Complex Basic Classical (CBC) Lie superalgebras [21, 23]. In order to explain the situation in that context, let us recall a known result: any finite-dimensional reducible representation of a complex semi-simple Lie algebra is completely reducible. This result does not hold in the graded case. Indeed, there are CBC Lie superalgebras which admit reducible but not completely reducible finite-diemensional representations. The irreducible quotients obtained from the latter are precisely the so-called atypical representations. They have no counterparts in the non-graded case. On the other hand, the typical representations are the irreducible summands of the direct sum decomposition of a completely reducible representation. (Notice that apart from osp(1/2n,C) all CBC Lie superalgebras admit both types of representations.) As shown in [2, 4], the same situation holds for the infinite-dimensional representations of osp(2/2) which is a real form with non-compact even part of the CBC Lie superalgebra of type I osp(2/2,C). According to [4], the typical discrete series representations are “generic” lowest weight super-unitary irreducible modules constructed through parabolic induction. For some “specific” values of the lowest weight, there appears in the preceding osp(2/2)-module a primitive vector which breaks the irreduciblity by generating a submodule. The atypical discrete series representations appear then as the super-unitary irreducible quotients. (Note that the abstract atypical discrete series representations of orthosymplectic Lie superalgebras were previously coined in the mathematical physics literature shortened multiplets or shortened representations [31].) As explained in the introduction, our contribution to this field originated from our desire to pursue Kostant’s program [9, 10]. We recall that harmonic analysis is based on explicit unitary irreducible representations of Lie (super)groups or Lie (super)algebras. In the non-graded context, Kirillov’s orbit method [32] which is nothing but geometric quantization [33] applied to a special type of symplectic manifolds, allows one to associate in an explicit and constructive way unitary irreducible representations of a Lie group G to those of its coadjoint orbits which admit an invariant polarization (see [34] for an up-to-date general description of the orbit method). The obtained representation space is the space of those L2 -sections of a certain complex line bundle-with-connection over the orbit which are moreover covariantly constant along the polarization. In order to construct explicit representations of Lie supergroups, one needs to extend the orbit method to Z2 –graded coadjoint orbits, or more generally, geometric quantization to supersymplectic supermanifolds. This last point was partially achieved (only prequantization) by
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Kostant in [9]. The extension of the full procedure was lacking the crucial notion of a polarization. Using Z2 –graded extensions of coherent states techniques [35] and the available abstract representation theory, we were able in [8, 1, 2] to overcome this difficulty by producing in a systematic way naturally polarized Z2 –graded coadjoint orbits. The complete orbit method was then successfully applied to examples of this kind. More precisely, our approach in [8, 1, 2] consists of two main steps. In the first one, we identify the coadjoint orbits associated to the abstract discrete series representations of osp(1/2) and osp(2/2). This is done using a Z2 –graded extension of Berezin’s dequantization procedure [35]: a method based on the notion of coherent states. It yields a coordinatization of the coadjoint orbits, an explicit expression of their invariant supersymplectic forms, and a locally equivariant moment map. A detailed geometric study of these results leads then to a natural definition of a super-K¨ ahler supermanifold, the considered orbits being particular examples of this notion. The second step consists in applying Kostant’s prequantization [9] to the obtained supersymplectic supermanifolds, and then completing the quantization procedure using the invariant super-K¨ ahler polarizations uncovered in the first step. As a result we obtain superholomorphic discrete series representations as realizations of the abstract discrete series representations we started from. In particular, the generators of the Lie superalgebras are represented by first order superoperators acting in a super-Hilbert space. For the considered examples, namely osp(1/2) and osp(2/2), it turns out, as expected, that the obtained representations are Z2 –graded extensions of the known holomorphic discrete series of su(1,1). For a detailed description of the above mentioned geometric constructions we refer to [8, 1, 2]. A summary of the main results is displayed in the first parts of the following subsections. In the second part of each of them we derive the alternative matrix realization of the considered representations of osp(1/2) and osp(2/2). 3.2. Discrete series representations of osp(1/2) The defining relations of osp(1/2) are given in (2.5)–(2.7). This is a rank one Lie superalgebra. Very much like its Lie subalgebra su(1, 1) (see (2.5)), its Cartan subalgebra is generated by K0 . Its so-called superholomorphic discrete series representations are lowest weight super-unitary irreducible representations, denoted by V (τ ), where the lowest weight τ is such that: τ ∈ 12 N and τ > 12 (as for k in Sec. 2.4). Let O(D(1) ) denotes the space of holomorphic functions on the unit disc D(1) , then O(D(1) ) ⊗ B1 is the holomorphic superstructure sheaf of the super unit disc D(1|1) . The latter is a natural graded extension of the unit disc D(1) . It is a realization of the OSp(1/2)-coadjoint orbit OSp(1/2)/U(1) which extends the K¨ ahler elliptic SU(1, 1)-coadjoint orbit SU(1, 1)/U(1) (see [1] for more details). Sections of O(D(1) ) ⊗ B1 are called superholomorphic functions on D(1|1) . These are functions Ψ(z, θ), where z ∈ D(1) and θ is the complex anticommuting variable generating B1 .
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The super-Hilbert space carrying V (τ ) is: n o Vτ = Ψ ∈ O(D(1) ) ⊗ B1 Re (hhΨ, Ψiiτ ) < ∞ and Im (hhΨ, Ψiiτ ) < ∞ , (3.1) where the super-Hermitian structure hh·, ·iiτ is given by: Z i dz d¯ z dθ dθ¯ hhΦ, Ψiiτ = 2 ¯ 1−2τ Φ(z, θ) Ψ(z, θ) , π D(1|1) (1 − |z| − iθθ)
∀Ψ, Φ ∈ Vτ .
(3.2)
Here θ¯ designates the complex conjugate of θ. In Vτ the osp(1/2) generators (2.5)–(2.7) are represented by the following first order differential superoperators, b0 = z ∂ + θ ∂ + τ , K ∂z 2 ∂θ
(3.3)
b + = z 2 ∂ + zθ ∂ + 2τ z , K ∂z ∂θ ∂ i ∂ b −z + 2τ θ , F+ = − √ zθ ∂z ∂θ 2
b− = ∂ , K ∂z i Fb− = − √ 2
(3.4) ∂ ∂ − . θ ∂z ∂θ
(3.5)
The super-unitarity of V (τ ) is expressed by the following equalities, b 0 Ψiiτ , b 0 Φ, Ψiiτ = hhΦ, K hhK
b ± Φ, Ψiiτ = hhΦ, K b ∓ Ψiiτ , hhK
hhFb± Φ, Ψiiτ = i (−1)(Φ) hhΦ, Fb∓ Ψiiτ .
(3.6) (3.7)
These equations differ from those in (2.14) because we are working in the Cartan– Weyl basis. Note also that the notation used here for osp(1/2) differs √ slightly from that in [8, 1]. Indeed, the anticommuting variable θ in [8, 1] is 2 times the one used here. As observed in [1], and in agreement with the description of the abstract discrete series in [3], the representation V (τ ) decomposes in the following way in terms of the discrete series D(k) (see Sec. 2.4) of su(1, 1) ⊂ osp(1/2): V (τ ) = D(k = τ ) ⊕ D(k = τ + 12 ) .
(3.8)
Concretely, writing Ψ(z, θ) = ψ0 (z) + θ
√ 2τ ψ1 (z) ,
∀ Ψ ∈ Vτ ,
(3.9)
and performing Berezin’s integration (see (2.11)) over θ and θ¯ in (3.2), one obtains [1] (3.10) hhΦ, Ψiiτ = hφ0 , ψ0 ik=τ + ihφ1 , ψ1 ik=τ + 12 , where h·, ·ik is given in (2.16). Combining this result with (3.1), (2.16) and (2.13) one immediately sees that as a vector space Vτ = Hk=τ ⊕ Hk=τ + 12 .
(3.11)
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(This reflects at the same time the Z2 -gradation of Vτ .) One can then represent any Ψ ∈ Vτ (of the form (3.9)) as a 2-components column vector t (ψ0 , ψ1 ), with ψ0 ∈ Hτ and ψ1 ∈ Hτ + 12 . One can easily show that in this alternative realization of Vτ , the first order differential superoperators given in (3.3)–(3.5) are represented by the following matrix-valued first order differential operators: e0 = K
e+ = K
d z dz +τ
!
0 d +τ + z dz
0
,
1 2
!
d z 2 dz + 2τ z
0
0
d + 2(τ + 12 )z z 2 dz
−i Fe+ = √ 2 τ
0
−2τ z
d z dz + 2τ
0
(3.12)
! ,
,
e− = K −i Fe− = √ 2 τ
d dz
0
0
d dz
! ,
0
−2τ
d dz
0
(3.13) ! . (3.14)
The main interesting feature of this new realization of osp(1/2) generators lies in its independance on anticommuting variables. It shows that when dealing with super structures, at the representation theoretic level, it is not necessary to introduce anticommuting variables. As a consequence of (3.6) and (3.10), super-unitarity of V (τ ) on the even b ± , reflects usual unitarity of D(k = τ ) and D(k = τ + 12 ). b 0 and K superoperators K However, for the odd superoperators Fb± , super-unitarity expresses interesting new features that involve simultaneously both D(k = τ ) and D(k = τ + 12 ). This point will be discussed in detail in Sec. 4. 3.3. Typical discrete series representations of osp(2/2) The defining relations of osp(2/2) are given in (2.8a)–(2.8i). This is a rank two Lie superalgebra. Its Cartan subalgebra is generated by K0 and B. Its so-called typical superholomorphic discrete series representations are lowest weight superunitary irreducible representations, denoted by V (τ, b), where the lowest weight (τ, b) (associated with the pair (K0 , B)) is such that: τ ∈ 12 N, τ > 12 , b ∈ 12 Z and |b| < τ . (The atypical superholomorphic discrete series representations which will be considered in the next subsection appear for the limiting values b = ±τ of the last inequality.) Let O(D(1) ) denotes as before the space of holomorphic functions on the unit disc D , then O(D(1) ) ⊗ B2 is the holomorphic superstructure sheaf of the super unit disc D(1|2) . The latter is another natural graded extension of the unit disc D(1) ; it is a realization of the OSp(2/2)-coadjoint orbit OSp(2/2)/U(1)×U(1) which extends the K¨ ahler elliptic SU(1, 1)-coadjoint orbit SU(1, 1)/U(1) (see [2] for more details). Sections of O(D(1) ) ⊗ B2 are called superholomorphic functions on D(1|2) . These are functions Ψ(z, θ, χ), where z ∈ D(1) and θ, χ are the complex anticommuting variables generating B2 . (1)
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The super-Hilbert space carrying V (τ, b) is: n o Vτ,b = Ψ ∈ O(D(1) ) ⊗ B2 Re (hhΨ, Ψiiτ,b ) < ∞ and Im (hhΨ, Ψiiτ,b ) < ∞ , (3.15) where the super-Hermitian structure hh·, ·iiτ,b is given by: Z 2τ dz d¯ z dθ dθ¯ dχ dχ ¯ hhΦ, Ψiiτ,b = π(b2 − τ 2 ) D(1|2) 2 2 ¯ χχ ¯ θθ − b −τ Φ(z, θ, χ) Ψ(z, θ, χ) e 2τ 1−|z|2 × −b−τ , ¯ b−τ 1 − |z|2 − iχχ ¯ 1 − |z|2 − iθθ
(3.16)
¯ designate the complex conjugates of θ and χ, respec∀ Ψ, Φ ∈ Vτ,b . Here θ¯ and χ tively. In Vτ,b the osp(2/2) generators (2.8a)–(2.8i) are represented by the following first order differential superoperators, b = θ ∂ − χ ∂ +b, B 2 ∂θ 2 ∂χ
(3.17)
b0 = z ∂ + θ ∂ + χ ∂ + τ , K ∂z 2 ∂θ 2 ∂χ
(3.18)
b + = z 2 ∂ + zθ ∂ + zχ ∂ + 2τ z , K ∂z ∂θ ∂χ
(3.19)
b− = ∂ , K ∂z
(3.20)
∂ ∂ + i (z + κ− χθ) − 2iτ κ− θ , Vb+ = −iκ− zθ ∂z ∂χ
(3.21)
c− = −iκ+ χ ∂ + i ∂ , W ∂z ∂θ
(3.22)
∂ ∂ +i , Vb− = −iκ− θ ∂z ∂χ
(3.23)
c+ = −iκ+ zχ ∂ + i (z − κ+ χθ) ∂ − 2iτ κ+ χ , W ∂z ∂θ where κ± = equalities:
τ ±b 2τ .
(3.24)
The super-unitarity of V (τ, b) is expressed through the following b τ,b , b Ψiiτ,b = hhΦ, BΨii hhBΦ,
b 0 Ψiiτ,b , b 0 Φ, Ψiiτ,b = hhΦ, K hhK c± Φ, Ψiiτ,b = i (−1)(Φ) hhΦ, Vb∓ Ψiiτ,b , hhW
b ± Φ, Ψiiτ,b = hhΦ, K b ∓ Ψiiτ,b , hhK
(3.25) (3.26)
c∓ Ψiiτ,b . hhVb± Φ, Ψiiτ,b = i (−1)(Φ) hhΦ, W (3.27)
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A. M. EL GRADECHI
Note that our notation for the Cartan–Weyl basis of osp(2/2) is misleading. Indeed, the pairs V+ versus V− and W+ versus W− do not correspond to the root space decomposition: “positive root” versus its opposite “negative root”. This is clearly reflected in the above equations. Our notation stresses the role played by the pairs (V+ , V− ) and (W+ , W− ) as basis of the two irreducible su(1, 1)-modules intervening in the definition of osp(2/2) as a Lie superalgebra (see (2.8a)–(2.8i)). As for osp(1/2) the representation V (τ, b) decomposes in the following way in terms of discrete series D(k) of su(1, 1) ⊂ osp(1/2) [2, 4], V (τ, b) = D(k = τ ) ⊕ 2 · D(k = τ + 12 ) ⊕ D(k = τ + 1) ,
(3.28)
where the factor 2 in front of D(k = τ + 12 ) indicates that this representation has multiplicity 2 in the decomposition. Concretely, writing p p p Ψ(z, θ, χ) = ψ1 (z)+θ 2τ κ− ψ2 (z)+χ 2τ κ+ ψ3 (z)+χθ 2τ (2τ + 1)κ+ κ− ψ4 (z) , (3.29) ¯ in ∀ Ψ ∈ Vτ,b , and performing Berezin’s integration (see (2.11)) over θ, χ, θ¯ and χ (3.16) one obtains [2], hhΦ, Ψiiτ,b = hφ1 , ψ1 ik=τ + ihφ2 , ψ2 ik=τ + 12 + ihφ3 , ψ3 ik=τ + 12 + hφ4 , ψ4 ik=τ +1 , (3.30) where h·, ·ik is given in (2.16). Hence, one has Vτ,b = Hk=τ ⊕ 2 · Hk=τ + 12 ⊕ Hk=τ +1 .
(3.31)
Here the Z2 -gradation is less transparent than in the case of osp(1/2): the even subspace of Vτ,b is the direct sum of the first and the last Hilbert spaces of the above direct sum decomposition, the odd subspace consists of the two copies of Hk=τ + 12 . Now, one can represent any Ψ ∈ Vτ,b (of the form (3.29)) as a 4-components column vector t (ψ1 , ψ2 , ψ3 , ψ4 ), with ψ1 ∈ Hτ , ψ2 , ψ3 ∈ Hτ + 12 and ψ4 ∈ Hτ +1 . The first order differential superoperators given in (3.17)–(3.24) can then be represented by the following matrix-valued first order differential operators: b 0 e B = 0 0
0 b+
0 1 2
0
0
b−
0
0
1 2
0 0 , 0 b
(3.32)
d z dz + τ 0 e0 = K 0
0
0 d z dz
+τ + 0 0
0 1 2
0
0 d z dz
+τ + 0
0 1 2
0 d +τ +1 z dz
,
(3.33)
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ORTHOSYMPLECTIC LIE SUPERALGEBRAS
d z 2 dz + 2τ z 0 e+ = K 0 0
0 d z 2 dz
0
0
0
0
0
d + 2(τ + 12 )z z 2 dz
0
0
0
d + 2(τ + 1)z z 2 dz
+ 2(τ +
1 2 )z
(3.34)
d
dz 0 e K− = 0 0
0
0
d dz
0
0
d dz
0
0
0 0 , 0
(3.35)
d dz
0
q −i κ− z d + 2τ 0 2τ dz Ve+ = 0 0 0 0
√ i 2τ κ+ z
0
q i
κ− 2τ +1
0
p 0 i (2τ + 1)κ+ z , 0 0 d 0 z dz + 2τ + 1 (3.36)
0 q0 f− = W −i κ+ d 2τ dz 0
√ i 2τ κ− 0
0 0
0
0
−i
q
0 q
−i κ− e 2τ V− = 0 0
d dz
κ+ d 2τ +1 dz
0
√ i 2τ κ+
0
0
0
q 0
0
i
κ− d 2τ +1 dz
0
p , −i (2τ + 1)κ− 0 0 0
(3.37)
0
p i (2τ + 1)κ+ , 0 0 √ i 2τ κ− z
0 0 0 q f+ = W κ+ d 0 −i 2τ z dz + 2τ q + d + 2τ + 1 z dz 0 −i 2τκ+1
(3.38)
0 0 0 0
0 p . −i (2τ + 1)κ− z 0 0
(3.39)
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A. M. EL GRADECHI
We see here that compared to its superoperator counterpart, the matrix realization starts to become less convenient for practical uses. This situation worsens for higher rank orthosymplectic Lie superalgebras. For instance, the generic discrete series representations of osp(N/2) will admit a matrix realization in terms of 2N × 2N matrices. 3.4. Atypical discrete series representations of osp(2/2) When the lowest weight (τ, b) of the previous section is such that b = ±τ , a primitive vector (i.e. a null vector generating a submodule) occurs in Vτ,b=±τ viewed simply as an osp(2/2)-module (not as a super-Hilbert space). The quotient module is irreducible and can be super-unitarized, leading thus to the so-called atypical superholomorphic discrete series representations of osp(2/2) [2, 4]. We denote the latter A± (τ ) for b = ±τ , respectively. As in [2], here we only consider the case b = −τ , the other situation is perfectly symmetric. The explicit realization of A− (τ ) given below follows [2]. Most of the results of this section can be obtained as the limit when b → −τ of those of the previous one. When this limit doesn’t exist (as it is the case for the inner product defining the super-Hilbert space) one has to rederive the needed expressions. However, this can be avoided using the similarities between the geometry of the coadjoint orbits to which are associated the atypical superholomorphic discrete series of osp(2/2), and the geometry of the OSp(1/2) coadjoint orbits of Sec. 3.2. It turns out that the super-Hilbert space carrying A− (τ ) is exactly the super-Hilbert space Vτ of (3.1) equipped with the same super-Hermitian structure (3.2). This is a consequence of the fact that the super unit disc D(1|1) , already encountered in Sec. 3.2, is also a realization of the OSp(2/2)-coadjoint orbit OSp(2/2)/U(1/1) whose geometric quantization leads to A± (τ ) (see [2] for more details). In Vτ the osp(2/2) generators (2.8a)–(2.8i) are represented by the following first order differential superoperators, b0 = z ∂ + θ ∂ + τ , K ∂z 2 ∂θ
b = θ ∂ −τ, B 2 ∂θ
(3.40)
b + = z 2 ∂ + zθ ∂ + 2τ z , K ∂z ∂θ
b− = ∂ , K ∂z
(3.41)
∂ − 2iτ θ , Vb+ = −izθ ∂z
c− = i ∂ , W ∂θ
(3.42)
∂ , Vb− = −iθ ∂z
c+ = iz ∂ . W ∂θ
(3.43)
The super-unitarity of A− (τ ) is reflected by the same equations as in the previous section, namely (3.25)–(3.27), rewritten now in terms of the superHermitian structure (3.2) instead of (3.16). Using the same arguments as in Sec. 3.2 which were based on (3.9)–(3.11), one obtains the alternative realization of the above operators in terms of the following matrix-valued first order differential operators:
485
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
e0 = K
e+ = K
d z dz +τ
!
0 d +τ + z dz
0
1 2
d z 2 dz + 2τ z
0
0
d + (2τ + 1)z z 2 dz
−i Ve+ = √ 2τ −i Ve− = √ 2τ
0
0
d z dz + 2τ
0
0
0
d dz
0
!
! ,
,
e =− B
, ! ,
e− = K
τ
0
0
τ−
d dz
0
0
d dz
! , (3.44)
1 2
! ,
√ f− = i 2τ W
0 1
√ f+ = i 2τ W
0
z
0
0
(3.45)
!
0 0
,
(3.46)
.
(3.47)
!
4. Super-unitarity Here we analyze the notion of super-unitarity defined in Sec. 2.3 in the light of the explicit representations described in the preceding section. In Sec. 4.1 we prove a few propositions that lead to a rigorous functional analytic interpretation of super-unitarity. We then discuss the latter and its consequences in Sec. 4.2. 4.1. A Hilbert space analysis Since the superholomorphic discrete series representations of the Lie superalgebras considered here decompose into direct sums of holomorphic discrete series representations of su(1, 1), it is not hard to see that the restriction of super-unitarity of the former to the even part of the Lie superalgebras simply reflects usual unitarity of the latter (separately within each direct summand). However, for the odd part of these Lie superalgebras super-unitarity reveals “something” new worth to be analyzed. Note that these two behaviors are encoded in the form of the matrixvalued operators of Sections. 3.2, 3.3 and 3.4: the even operators are diagonal while the odd ones are not. We start by studying the case of osp(1/2) using results from Sec. 3.2: (a) Even part of osp(1/2) — From (3.12)–(3.13) one immediately sees that (3.6) expresses simply the fact that the holomorphic discrete series representations D(k = τ ) and D(k = τ + 12 ) of su(1, 1) are unitary (see Sec. 2.4). In other words, the first order operators representing K0 (resp. K+ and K− ) within the superholomorphic discrete series representation V (τ ) = D(k = τ ) ⊕ D(k = τ + 12 ) of osp(1/2) is selfadjoint (resp. are each other adjoints) on Uk=τ ⊕ Uk=τ + 12 ⊂ Vτ = Hk=τ ⊕ Hk=τ + 12 , where Uk is defined by (2.20). As mentioned in Sec. 3.2, the previous direct sums are not Hilbert sums, but simply vector spaces direct sums. Note however that when V (τ ) is restricted to su(1, 1) = osp(1/2)0 , (3.6) makes also sense in Vτ = Hk=τ ⊕ Hk=τ + 12 considered now as a Hilbert sum. This is a direct consequence of the fact that the even superoperators are transparent to the super-Hilbert space structure of Vτ .
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A. M. EL GRADECHI
(b) Odd part of osp(1/2) — From (3.7), with Fb+ in the left-hand side and Fb− in the right hand side, and with Ψ, Φ ∈ Vτ of the form (3.9), a straightforward computation leads to the equalities: √ 1 d φ1 , ψ0 2τ hzφ1 , ψ0 ik=τ = √ , (4.1) dz 2τ k=τ + 1 2
√ 1 2τ hφ0 , ψ1 ik=τ = √ 2τ
d + 2τ φ0 , ψ1 . z dz k=τ + 1
(4.2)
2
The inner products appearing here are those introduced in Sec. 2.4 for the holomorphic discrete series representations of su(1, 1). Note that the operators involved in (4.1)–(4.2) are unbounded. The necessary domain considerations together with further analysis of (4.1)–(4.2) will be considered shortly. Before that we briefly discuss points (a) and (b) above in the case of the typical and the atypical superholomorphic discrete series representations of osp(2/2). Similarly to (a), one easily sees that for both typical and atypical superholomorphic discrete series representations of Secs. 3.3 and 3.4, the restriction of super-unitarity to the even part of osp(2/2) simply reflects unitarity of the holomorphic discrete series representations of su(1, 1) ⊂ osp(2/2) which appear in the direct sum decompositions of V (τ, b) and A± (τ ), respectively. Hence, once again nothing new arises from the even part of the Lie superalgebra. However, as in (b) above, the restriction of super-unitarity to the odd part of osp(2/2), for both typical and atypical superholomorphic discrete series representations, turns out to lead to exactly the two equalities exhibited above. More precisely, in the atypical case one obtains (4.1)–(4.2), while in the typical case one obtains two pairs of equalities, namely (4.1)–(4.2) and their shifted version where τ is replaced by τ + 12 . These facts confer to (4.1)–(4.2) a fundamental role. The rest of this section is devoted to a rigorous and detailed analysis of their validity. Before that, let us verify their formal validity by comparing both sides of (4.1)–(4.2) using results from the theory of holomorphic discrete series representations of su(1, 1) as described in Sec. 2.4. We start by displaying easy to prove and useful identities involving the complete orthonormal basis (2.17) of Hk and the operators intervening in (4.1)–(4.2): Proposition 4.1. √ √ (k=τ + 12 ) (k=τ ) 2τ z um = m + 1 um+1 , 1 1 d (k=τ ) √ (k=τ + 1 ) √ um = m um−1 2 , √ dz 2τ 2τ
(k)
√ √ (k=τ + 12 ) (k=τ ) 2τ um = m + 2τ um ,
(4.3)
√ d (k=τ + 12 ) (k=τ ) + 2τ um = m + 2τ um . z dz (4.4)
(The z-dependance in um (z) has been suppressed for convenience.) Now, using Proposition 4.1, the expansions
487
ORTHOSYMPLECTIC LIE SUPERALGEBRAS
Hk=τ 3 ψ0 (z) =
∞ X
(k=τ ) (k=τ ) hum , ψ0 ik=τ um (z) ,
(4.5)
m=0
Hk=τ + 12 3 ψ1 (z) =
∞ X
(k=τ + 12 )
hum
(k=τ + 12 )
, ψ1 ik=τ + 12 um
(z) ,
(4.6)
m=0
and their analogs for φ0 and φ1 , one can evaluate both sides of (4.1) and (4.2), respectively obtaining the following formal equalities: ∞ X √ √ (k=τ + 12 ) (k=τ ) 2τ hzφ1 , ψ0 ik=τ = m + 1 hum , φ1 ik=τ + 1 hum+1 , ψ0 ik=τ 2
m=0
1 = √ 2τ √
2τ hφ0 , ψ1 ik=τ =
d φ1 , ψ0 , dz k=τ + 1
∞ X √ m=0
1 = √ 2τ
(4.7)
2
(k=τ )
m + 2τ hum
(k=τ + 12 )
, φ0 ik=τ hum
, ψ1 ik=τ + 12
d + 2τ φ0 , ψ1 z . dz k=τ + 1
(4.8)
2
In what follows we provide the necessary domain considerations that will establish the above formal results on a firm ground. More precisely, we will seek for a pure Hilbert space theoretic interpretation of (4.1)–(4.2), since they both involve two holomorphic discrete series representations of su(1, 1), namely D(k = τ ) and D(k = τ + 12 ), without any reference to super-Hilbert spaces (despite their origin). More precisely, even though (4.1)–(4.2) are consequences of the decomposition (3.11) (or (3.31)) which is not a Hilbert sum but simply a direct sum of Hilbert spaces forming a super-Hilbert space, it is nevertheless possible to rewrite them as equations involving operators acting in a Hilbert sum. This is not surprising, it is in fact in a perfect agreement with the remark we made at the end of Sec. 2.3. Concretely, consider the Hilbert sum Wτ = Hk=τ ⊕ Hk=τ + 12 ,
(4.9)
with the inner product of Φ = (φ0 , φ1 ) and Ψ = (ψ0 , ψ1 ) ∈ Wτ given by: (Φ, Ψ)τ = hφ0 , ψ0 ik=τ + hφ1 , ψ1 ik=τ + 12 .
(4.10)
We denote the corresponding norm by k · kτ (there will be no possible confusion with the norms in Hk=τ and Hk=τ + 12 , since these are denoted k · kk=τ and k · kk=τ + 12 , respectively). Consider now the following two pairs of linear operators acting in Wτ ; ! ! √ 0 0 0 z 1 , (4.11) , Q− = 2τ Q+ = √ d 2τ dz 0 0 0 P
+
1 = √ 2τ
0
0
d z dz + 2τ
0
! ,
P
−
√ = 2τ
0
1
0
0
! .
(4.12)
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A. M. EL GRADECHI
It is not hard to see that the equalities (4.1) and (4.2) can be respectively rewritten in the following form: (Q− Φ, Ψ)τ = (Φ, Q+ Ψ)τ ,
(4.13)
(P − Φ, Ψ)τ = (Φ, P + Ψ)τ .
(4.14)
The first (resp. second) equation is only valid for Ψ ∈ DQ+ (resp. Ψ ∈ DP + ) and Φ ∈ DQ− (resp. Φ ∈ DP − ). Proposition 4.2. The domains DQ± and DP ± are such that : DQ+ = DP +
and DQ− = DP − .
(4.15)
Proof. We start with the first equality in (4.15). Let Ψ = (ψ0 , ψ1 ) ∈ Wτ . From (4.9), (4.10) and (4.11) one immediately sees that kQ
+
Ψk2τ
2 d 1
ψ0 = 2τ dz k=τ + 1
and kP
+
Ψk2τ
2
2
d 1
z + 2τ ψ0 = .
2τ dz k=τ + 12 (4.16)
Using (4.4), (4.5) and (4.16) one obtains kQ+ Ψk2τ =
∞ X
(k=τ ) m|hum , ψ0 ik=τ |2 ,
∞ X
kP + Ψk2τ =
m=0
(k=τ ) (m + 2τ )|hum , ψ0 ik=τ |2 .
m=0
(4.17) Hence, kP + Ψk2τ = kQ+ Ψk2τ + 2τ kψ0 k2k=τ
(4.18)
which can be rewritten in the form:
2 2
d 1 1
z d + 2τ ψ0
ψ0 = + 2τ kψ0 k2k=τ .
2τ dz 2τ dz k=τ + 1 k=τ + 1 2
(4.19)
2
This equality is to be interpreted in the following way [11]: either both sides are infinite or they are both finite and then have the same finite value. This proves the first part of (4.15). The second part can be proven similarly. Indeed, repeating the same computations for Q− and P − , one arrives to the following: kQ− Ψk2τ =
∞ X
(k=τ + 12 )
(m + 1)|hum
, ψ1 ik=τ + 12 |2 ,
m=0
kP
−
Ψk2τ
=
∞ X
(k=τ + 12 )
(m + 2τ )|hum
, ψ1 ik=τ + 12 |2 .
(4.20)
m=0
Hence, kP − Ψk2τ = kQ− Ψk2τ + (2τ − 1)kψ1 k2k=τ + 1 2
(4.21)
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ORTHOSYMPLECTIC LIE SUPERALGEBRAS
which can be rewritten in the form: 2τ kψ1 k2k=τ = 2τ kzψ1 k2k=τ + (2τ − 1)kψ1 k2k=τ .
(4.22)
The second part of (4.15) follows from the same arguments used for the first part. An exact characterization of the above domains can be deduced from the proof just given. Indeed, from (4.17) and (4.20) one can easily see that the following takes place: Proposition 4.3. DQ+ = DP + = Kk=τ ⊕ Hk=τ + 12 ⊂ Wτ and DQ− = DP − = Hk=τ ⊕ Kk=τ + 12 ⊂ Wτ , where ( ) ∞ X (k) 2 m|hum , ψik | < ∞ (4.23) Kk = ψ ∈ Hk m=0
is a dense domain in Hk . Moreover, one has: Proposition 4.4. Q+ and Q− (resp. P + and P − ) are each other adjoints, i.e. † † resp. P ± = P ∓ . (4.24) Q± = Q∓
Proof. Recall that (4.13) and (4.14) (and thus (4.1) and (4.2)) are rigorously true provided that Ψ and Φ belong to the appropriate domains (see Proposition 4.2 and 4.3). Hence, in order to prove (4.24) one only needs to prove that D(Q± )† = DQ∓ (resp. D(P ± )† = DP ∓ ). It is not hard to see that these are straightforward consequences of (4.7) (resp. (4.8)). Finally, from each of the two pairs, (Q+ , Q− ) and (P + , P − ), one can construct another pair of operators, namely Q1 = Q+ + Q− , Q2 = i(Q+ − Q− ) and P1 = P + + P − , P2 = i(P + − P − ) (4.25) which are such that: Proposition 4.5. (a) DQi = DPi = DQ+ ∩ DQ− = Kk=τ ⊕ Kk=τ + 12 ⊂ Wτ , for i = 1, 2, and (b) Qi and Pi for i = 1, 2 are self-adjoint on their common dense domain given in (a). Proof. (a) Direct computations based on (4.11), (4.12) and (4.25) lead to: kQ1 Ψk2τ = kQ+ Ψk2τ + kQ− Ψk2τ = kQ2 Ψk2τ ,
(4.26)
kP1 Ψk2τ = kP + Ψk2τ + kP − Ψk2τ = kP2 Ψk2τ .
(4.27)
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A. M. EL GRADECHI
Arguments analogous to those used in proving Proposition 4.2 (see the reasoning following (4.19)), show that indeed DQi = DQ+ ∩ DQ− = DPi , for i = 1, 2. The explicit form of the intersection follows from Proposition 4.3. Part (b) is a direct consequence of Proposition 4.4, (4.25) and part (a). Observe that the first equality in Proposition 4.5(a) can be easily seen as a direct consequence of the formula kP1 Ψk2τ = kQ1 Ψk2τ + 2τ kψ0 k2k=τ + (2τ − 1)kψ1 k2k=τ + 1
(4.28)
2
which follows from (4.18), (4.21), (4.26) and (4.27). 4.2. Discussion and consequences Proposition 4.5 provides an alternative interpretation of the fundamental relations (4.1) and (4.2). More precisely, the validity of the latter is equivalent to the self-adjointness of the operators Qi and Pi . Further properties of these operators will be investigated in the next section. Now, we conclude this section by closing the loop. More precisely, since (4.1) and (4.2) are the fundamental equations at the origin of the super-unitarity of the considered representations (and not only of their restrictions to the odd part of the Lie superalgebras as we will shortly show), we should be able to write the matrix-valued first order operators of Secs. 3.2, 3.3 and 3.4 in terms of the Q’s and the P ’s studied in the present section. In an increasing order of difficulty, one finds: A. Atypical discrete series of osp(2/2) — Comparing (3.46)–(3.47) with (4.13)– (4.14) one immediately sees that: f− = iP − , W
Ve+ = −iP + ,
Ve− = −iQ+
f+ = iQ− . and W
(4.29)
B. Discrete series of osp(1/2) — Here also the expressions are not hard to find. Indeed, comparing (3.14) with (4.13)–(4.14) one gets: −i −i and Fe− = √ Q+ − P − . (4.30) Fe+ = √ P + − Q− 2 2 C. Typical discrete series of osp(2/2) — In this case one needs to introduce two copies of the Q’s and P ’s. More precisely, if we denote those used above Q± (τ ) and ± ± ± P(τ ) , we need now to consider also Q(τ + 1 ) and P(τ + 1 ) (a simple shift of τ by 12 ). 2 2 One can then rewrite (3.36)–(3.39) in the following form: ! ! + 0 0 P(τ Q− ) (τ ) √ √ −iϕ −1 , + i κ+ e Rϕ Rϕ Ve+ = −i κ− + − 0 −P(τ + 1 ) 0 −Q(τ + 1 ) 2
2
(4.31) f− = W
√ i κ−
!
− P(τ )
0
0
− −P(τ +1) 2
√ − i κ+ eiϕ Rϕ
!
Q+ (τ )
0
0
−Q+ (τ + 1 )
−1 , Rϕ
2
(4.32)
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ORTHOSYMPLECTIC LIE SUPERALGEBRAS
√ Ve− = −i κ−
!
Q+ (τ )
0
0
−Q+ (τ + 1 )
√ + i κ+ e−iϕ Rϕ
!
− P(τ )
0
0
− −P(τ +1)
2
−1 , Rϕ
2
(4.33) f+ = W
√ i κ−
!
Q− (τ )
0
0
−Q− (τ + 1 )
√ − i κ+ eiϕ Rϕ
2
!
+ P(τ )
0
0
+ −P(τ +1)
−1 , Rϕ
2
(4.34) where
1 0 Rϕ = 0 0
0
0
0
−eiϕ
e−iϕ
0
0
0
0 0 , 0 1
0 ≤ ϕ < 2π .
(4.35)
The action of Rϕ on t (ψ1 , ψ2 , ψ3 , ψ4 ) ∈ Vτ,b (given by (3.31)) simply interchanges ψ2 and ψ3 (up to a phase factor). This is a well-defined action, since ψ2 and ψ3 belong to the same Hilbert space Hk=τ + 12 . The above formulae give the impression that we have a one parameter family of matrix-valued operators. In fact, the parameter ϕ which is clearly absent in (3.36)–(3.39), is artificial. (Moreover, note that one can use a different ϕ for each of the above formulae.) The appearance of this parameter is reminiscent of an SU(2) symmetry hidden in the structure of the typical discrete series representations of osp(2/2). A discussion of this interesting point is beyond the scope of the present work, we will come back to it in a forthcoming publication [36]. Finally, let us discuss further on points A, B and C, and some of their consequences: I. As already mentioned at the end of Sec. 2.3, instead of working in a super-Hilbert space one can directly work in an associated Hilbert sum. For the matrix-valued operators of Secs. 3.2, 3.3 and 3.4 this correspondance is obvious from the relations obtained in A, B and C above. Note however that the superoperator version in case C is less obvious to obtain. II. From A, B and C, and Propositions 4.2–4.5, one can determine the domains of all the matrix-valued operators considered in this work. Indeed, for the odd operators one finds: II.A — DVe = Kk=τ ⊕ Hk=τ + 12 ⊂ Vτ and DW e ± = Hk=τ ⊕ Kk=τ + 12 ⊂ Vτ , where ± now Vτ is the super-Hilbert space of Sec. 3.2. This follows directly from A and Proposition 4.3. II.B — DFe = DQ+ ∩ DQ− = Kk=τ ⊕ Kk=τ + 12 ⊂ Vτ . This follows from B and ± Proposition 4.5.
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II.C — DVe = Kk=τ ⊕ Hk=τ + 12 ⊕ Kk=τ + 12 ⊕ Kk=τ +1 ⊂ Vτ,b and DW e± = ± Kk=τ ⊕ Kk=τ + 12 ⊕ Hk=τ + 12 ⊕ Kk=τ +1 ⊂ Vτ,b . This follows directly from C and Proposition 4.3. Here one needs to be careful about the ordering of the Hilbert spaces that occur with multiplicity 2 in the decomposition (3.31), since Rϕ interchanges their roles. Domains of the even operators follow from II.A, II.B and II.C, and the oddodd part of the defining relations of the Lie superalgebras under consideration, namely (2.7) and (2.8i). Straightforward computations based on the evaluation of domains of anticommutators lead to results in perfect agreement with the domain considerations of Sec. 2.4. Let n us for example considerothe first relation in (2.7). One e finds that DK e± = D(Fe± )2 = Ψ ∈ DFe± | F± Ψ ∈ DFe± , where the second equality is a definition. Using II.B one gets: DK e± = Uk=τ ⊕ Uk=τ + 12 ⊂ Kk=τ ⊕ Kk=τ + 12 ⊂ Vτ , where Uk was defined in (2.20). III. Similar arguments to those used in the preceding paragraph show that superunitarity of the superholomorphic discrete series representations considered here requires only the validity of (2.14) for X in the odd part of the corresponding Lie superalgebras. This originates from the fact that one can choose for both osp(1/2) and osp(2/2) a system of simple roots which is purely odd. In fact, for osp(1/2) this is the only possible choice. This is not so for osp(2/2) (see [2]). IV. Now, we come back to the fundamental role played by the relations (4.1)–(4.2) (or equivalently (4.13)–(4.14)) uncovered at the beginning of the present section. The expressions found in A, B and C not only confirm this fundamental role, but also show that (4.1)–(4.2) simply expresses super-unitarity of the atypical superholomorphic discrete series representations of osp(2/2). This fact follows from A and III. Hence, super-unitarity of the other two superholomorphic discrete series representations follows from the super-unitarity of the atypical superholomorphic discrete series representations of osp(2/2). This can be viewed as the quantum theoretical counterpart of a similar fact observed in [2] at the level of classical theory. Indeed, the supergeometry of the OSp(1/2) coadjoint orbits and of the OSp(2/2) typical coadjoint orbits, whose quantization leads, respectively, to the superholomorphic discrete series representations of Secs. 3.2 and 3.3, turns out to be completely determined by the supergeometry of the OSp(2/2) atypical coadjoint orbits (see [2] for more details). These observations deserve further investigations; we will come back to this point in a forthcoming publication. 5. Anticommutativity We have just shown that the self-adjointness of the two pairs of operators (Q1 , Q2 ) and (P1 , P2 ) ensures the super-unitarity of the superholomorphic discrete series representations considered in this work. Here we provide a functional analytic and operatorial theoretic analysis of special algebraic properties of these operators. This reveals an interesting connection between representation theory of
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orthosymplectic Lie superalgebras and that of the superalgebra underlying N = 2 supersymmetric quantum mechanics (SSQM). One can easily check that Q1 and Q2 (resp. P1 and P2 ) anticommute in a naive sense. Do they anticommute in the proper sense defined in Sec. 2.5? Proposition 5.1. Q1 and Q2 (resp. P1 and P2 ) are self-adjoint anticommuting operators in Wτ . Proof. The self-adjointness was already proven in Proposition 4.5. The proper anticommutativity follows from Proposition 2.8. Indeed, straightforward computations show that Q1 and Q2 (resp. P1 and P2 ) satisfy both conditions in (2.22) on their common dense domain Kk=τ ⊕ Kk=τ + 12 ⊂ Wτ . Arai’s characterization of the anticommutativity of self-adjoint operators we just used was originally devised in the context of supersymmetric quantum mechanics or more precisely in the context of the representation theory of the superalgebra underlying SSQM (see [20] for a precise definiton of SSQM). Our result shows then that super-unitarity of the superholomorphic discrete series representations of osp(1/2) and osp(2/2) follows from the super-unitarity of a specific holomorphic representation of the superalgebras underlying two N = 2 SSQM. The latter are essentially generated by the triplets (Q1 , Q2 , H1 ) and (P1 , P2 , H2 ), where (Q1 , Q2 ) (resp. (P1 , P2 )) play the role of supercharges while H1 = Q21 = Q22 (resp. H2 = P12 = P22 ) plays the role of the supersymmetric Hamiltonian. In the Hilbert space Wτ , the latter are represented by the following self-adjoint matrix-valued differential operators: ! ! d d 0 + 2τ 0 z dz z dz and H2 = . (5.1) H1 = d d +1 + 2τ 0 z dz 0 z dz There are other interesting connections of our results with another of Arai’s characterizations of proper anticommutativity, we will report on them in a forthcoming publication. 6. Conclusion Our present study of the functional analytic and the operatorial theoretic meaning of super-unitarity of the superholomorphic discrete series representations of osp(1/2, R) and osp(2/2, R) shows that super-unitarity is a consequence of the selfadjointness of two pairs of anticommuting operators (Q1 , Q2 ) and (P1 , P2 ) which act in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su(1, 1) such that the respective lowest weights differ by 1/2. Our analysis exhibits the fundamental role played by the atypical superholomorphic discrete series representations of osp(2/2, R) regarding super-unitarity of the other discrete series of osp(1/2, R) and osp(2/2, R). Direct consequences of these results were discussed in detail at the end of Sec. 4. Now, we would like to discuss possible generalizations of our work.
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(a) Higher rank orthosymplectic Lie superalgebras — It would be interesting to generalize our results to orthosymplectic Lie superalgebras of higher rank. This requires applying to the latter the entire program carried out for osp(1/2, R) and osp(2/2, R) in our previous work [8, 1, 2] as well as in the present contribution. Such a generalization would proceed in two stages involving, first, a classical mechanical (or geometric) part which consists in determining all the super-K¨ahler coadjoint orbits of the Lie supergroup corresponding to the considered Lie superalgebra, and second, a quantum mechanical (or a representation theoretic) part which consists in applying geometric quantization to these orbits. So far, only the first stage has been partly achieved in [37]. Indeed, let us recall that the N = 1 and N = 2 super unit discs studied in [8, 1, 2] are Z2 –graded extensions of the usual unit disc which is the simplest of the Cartan domains of type II. Cartan domains are irreducible symmetric Hermitian spaces of non–compact type which are K¨ ahler homogeneous spaces for simple Lie groups [38]. They are particular coadjoint orbits of the latter. The type II Cartan domains correspond to the homogeneous spaces Sp(2n, R)/U(n). (The unit disc occurs for n = 1.) The super-K¨ ahler structure of their Z2 –graded extensions was determined in [37], except for its geometric interpretation ` a la Rothstein as in [8, 2]. Note that in [37] another type of quantization was applied to these Cartan superdomains. This is the so-called non-perturbative quantization. Unlike geometric quantization, this procedure is not aimed at producing irreducible representations of the considered Lie (super)algebras. Hence, the analysis carried out here cannot be applied to the results obtained in [37], unless the second stage of our program (geometric quantization) is extended to the Cartan superdomains of type II. This would lead to an explicit construction of superholomorphic discrete series representations of OSp(N/2n, R) which would be concrete realizations (other than the oscillator representations considered in [6, 7]) of some of the abstract representations constructed in [5]. Finally, it would be interesting to apply our program to the Z2 –graded extensions of the other types of Cartan domains which are also described in [37]. (b) Matrix-valued operators versus superoperators, and Clifford algebras — Kostant made the following very interesting observation in [9]: applying prequantization to the simplest supersymplectic supermanifold which is just a supermanifold built over a zero-dimensional manifold, leads to a representation of a Clifford algebra which is a quantization of the exterior algebra defining the supermanifold. If one considers a less trivial supersymplectic supermanifold (by definition the latter is built over a symplectic manifold), and succeeds in applying geometric quantization to it, one would then naturally expect the outcome of this procedure to be in the form of a nontrivial combination of the quantization of the base symplectic manifold and the representation of a Clifford algebra. This is very much the case for the examples treated in [1, 2]. Spin structures and Clifford algebras appear at different stages of our program: they appear at the algebraic, the geometric and the representationtheoretic levels. For instance, the matrix realizations derived in Secs. 3.2, 3.3,
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and 3.4 give a flavor of this fact (see the comment at the end of Sec. 3.3). The matrix and the superoperator realizations correspond to two different realizations of a representation of a Clifford algebra. An understanding of these underlying structures will certainly help answering the questions raised above in (a) in an efficient way. (c) Algebraic quantization versus geometric quantization — Recently, we studied the interplay between two types of quantizations: algebraic quantization and geometric quantization [39]. We showed that these two procedures are complementary when used in the construction of holomorphic discrete series representations. It would be interesting to extend these results to the Z2 –graded context by exploring and devising the Z2 –graded algebraic ingredients needed for a superalgebraic quantization. Acknowledgements The author is indebted to S. T. Ali for very stimulating conversations. He thanks C. Duval, G. G. Emch, J. Harnad, G. Tuynman, P. Winternitz, and T. Wurzbacher for their interest in his work and for their much appreciated encouragements. This work was supported by a CRM–ISM fellowship, it was initiated while the author was a postdoctoral fellow at both the Centre de Recherches Math´ematiques of Universit´e de Montr´eal, and the Department of Mathematics and Statistics of Concordia University. The author thanks these institutions for their hospitality. References [1] A. M. El Gradechi, “Geometric quantization of an OSp(1/2) coadjoint orbit”, Lett. Math. Phys. 35 (1995) 13–26. [2] A. M. El Gradechi and L. M. Nieto, “Supercoherent states, super-K¨ ahler geometry and geometric quantization”, Commun. Math. Phys. 175 (1996) 521–563. [3] H. Furutsu and T. Hirai, “Representations of Lie superalgebras. I. Extensions of representations of the even part”, J. Math. Kyoto Univ. 28 (1988) 695–749. [4] K. Nishiyama, “Characters and super-characters of discrete series representations for orthosymplectic Lie superalgebras”, J. Algebra 141 (1991) 399–419. [5] H. P. Jakobsen, “The full set of unitarizable highest weight modules of basic classical Lie superalgebras”, Mem. Amer. Math. Soc. 111 (1994) (532). [6] K. Nishiyama, “Oscillator representations for orthosymplectic algebras”, J. Algebra 129 (1990) 231–262; “Decomposing oscillator representations of osp(2n/n; R) by a super dual pair osp(2/1; R)×so(n), Compositio Math. 80 (1991) 137–149; “Super dual pairs and highest weight modules of orthosymplectic algebras”, Adv. Math. 104 (1994) 66–89. [7] H. Furutsu and K. Nishiyama, “Realization of irreducible unitary representations of osp(M/N;R) on Fock spaces”, in The Proceedings of Fuji-Kawaguchiko Conference on Representation Theory of Lie Groups and Lie Algebras, eds. T. Kawazoe et al. 1–21, World Scientific, Singapore, 1992. [8] A. M. El Gradechi, “On the supersymplectic homogeneous superspace underlying the OSp(1/2) coherent states”, J. Math. Phys. 34 (1993) 5951–5963. [9] B. Kostant, “Graded manifolds, graded Lie theory and prequantization”, in Lecture Notes in Math., Vol. 570, Springer-Verlag, Berlin, 1977, 177–306.
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[10] B. Kostant, “Harmonic analysis on graded (or super) Lie groups”, in Lecture Notes in Phy., Vol. 79, 47–50, Springer-Verlag, Berlin, 1978. [11] V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform – Part I”, Commun. Pure Appl. Math. 14 (1961) 187–214. [12] V. Bargmann, “Irreducible unitary representations of the Lorentz group”, Ann. Math. 48 (1947) 568–640. [13] A. W. Knapp, Representation Theory of Semisimple Groups – An Overview Based on Examples, Princeton Univ. Press, Princeton, New Jersey, 1986. [14] V. Fock, “Verallgemeinerung und l¨ osung der Diracschen statistischen gleichung”, Z. Physik 49 (1928) 339–357. [15] P. A. M. Dirac, “La seconde quantification”, Ann. Inst. H. Poincar´ e 11 (1949) 15–47. [16] I. E. Segal, “Mathematical characterization of the physical vacuum for a linear Bose– Einstein field”, Illinois J. Math. 6 (1962) 500–523. [17] F.-H. Vasilescu, “Anticommuting self-adjoint operators”, Rev. Roumaine Math. Pures Appl. 28 (1983) 77–91. [18] Yu. S. Samoilenko, Spectral Theory of Families of Self-Adjoint Operators, Kluwer Academic Publishers, Dordrecht, 1991. [19] S. Pedersen, “Anticommuting Selfadjoint Operators”, J. Funct. Anal. 89 (1990) 428– 443. [20] A. Arai, “Analysis on Anticommuting Self-Adjoint Operators”, in Advanced Studies in Pure Math., Vol. 23, 1–15, North-Holland, Amsterdam, 1994. [21] V. Kac, “Representations of classical Lie superalgebras”, in Lecture Notes in Math., Vol. 676, 597–626, Springer-Verlag, Berlin, 1978. [22] M. Scheunert, “The Theory of Lie Superalgebras – An Introduction”, in Lecture Notes in Math., Vol. 716, Springer-Verlag, Berlin, 1979. [23] J. F. Cornwell, Group Theory in Physics Vol. 3 – Supersymmetries and Infinite Dimensional Algebras, Academic Press, London, 1989. [24] F. A. Berezin, Introduction to Superanalysis, Reidel, Dordrecht, 1987. [25] S. Sternberg and J. Wolf, “Hermitian Lie algebras and metaplectic representations”, Trans. Amer. Math. Soc. 238 (1978) 1–43. [26] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I, Academic Press, New York, 1972. [27] P. Renouard, Vari´et´es symplectiques et quantification, Thesis, Orsay, 1969. [28] S. De Bi`evre and A. M. El Gradechi, “Quantum mechanics and coherent states on the anti-de Sitter spacetime and their Poincar´e contraction”, Ann. Inst. H. Poincar´ e 57 (1992) 403–428. [29] H. Furutsu and K. Nishiyama, “Classification of irreducible super-unitary representations of su(p,q/n)”, Commun. Math. Phys. 141 (1991) 475–502. [30] J. W. B. Hughes, “Representations of Osp(2, 1) and the metaplectic representation”, J. Math. Phys. 22 (1981) 245–250. [31] J. Van der Jeugt, “Representations of N=2 extended supergravity and unitarity conditions in Osp(N,4)”, J. Math. Phys. 28 (1987) 758–764. ´ ements de la th´eorie des repr´esentations, Editions ´ [32] A. A. Kirillov, El´ Mir, Moscou, 1974. [33] N. M. J. Woodhouse, Geometric Quantization, Clarendon Press, Oxford, 1980. [34] D. A. Vogan, “The orbit method and unitary representations for reductive Lie groups”, in Perspectives in Math., Vol. 17, Academic Press, 1997. [35] F. A. Berezin, “General concept of quantization”, Commun. Math. Phys. 40 (1975) 153–174. [36] A. M. El Gradechi, in preparation. [37] D. Borthwick, S. Klimek, A. Lesniewski, and M. Rinaldi, “Matrix Cartan superdomains, super Toeplitz operators, and quantization”, J. Funct. Anal. 127 (1995) 456–510.
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[38] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. [39] S. T. Ali, A. M. El Gradechi, and G. G. Emch, “Modular algebras in geometric quantization”, J. Math. Phys. 35 (1994) 6237–6243.
ON NAMBU POISSON MANIFOLDS NOBUTADA NAKANISHI Department of Mathematics Gifu Keizai University 5-50 Kitagata-cho, Ogaki-city Gifu, 503, Japan E-mail : [email protected] Received 12 May 1997
1. Introduction In 1973, Y. Nambu [6] gave a generalization of classical Hamiltonian mechanics. Originally he considered his mechanics on R3 . The equation of motion of an observable f ∈ C ∞ (R3 ) is defined by df = {H1 , H2 , f } , dt where H1 , H2 ∈ C ∞ (R3 ) are two Hamiltonians. The bracket in the right-hand side is precisely defined by ∂(H1 , H2 , f ) , {H1 , H2 , f } = ∂(x, y, z) where (x, y, z) are the standard coordinates on R3 . About twenty years later, from the viewpoint of the generalization of classical Poisson brackets, Takhtajan [7] introduced so-called Nambu–Poisson brackets. Let M be a C ∞ -manifold. Then a Nambu–Poisson bracket is an n-linear skew-symmetric mapping from n-copies of C ∞ (M ) into C ∞ (M ), which satisfies the Leibniz rule and the Fundamental Identity: {f1 , . . . , fn−1 , {g1 , . . . , gn }} = {{f1 , . . . , fn−1 , g1 }, g2 , . . . , gn } + {g1 , {f1 , . . . , fn−1 , g2 }, g3 , . . . , gn } + · · · + {g1 , . . . , gn−1 , {f1 , . . . , fn−1 , gn }} for all f1 , . . . , fn−1 , g1 , . . . , gn ∈ C ∞ (M ). We should note that (f1 , . . . , fn−1 ) acts on {g1 , . . . , gn } as a derivation. If n = 2, we have usual Poisson manifolds. But if n ≥ 3, there appear some aspects which are different from the case of usual Poisson manifolds. More precisely, Nambu–Poisson structure should be more rigid than usual Poisson structure. (For example, see Theorem 5.5.) P. Gautheron [3] also proved the same result as ours in a completely different way. Using the Fundamental Identity, we know that the flow of the equation of motion induces an automorphism of a Nambu–Poisson bracket. 499 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 499–510 c World Scientific Publishing Company
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For each Nambu–Poisson bracket, there corresponds the n-tensor (or the Vn T M by {f1 , . . . , fn } = η(df1 , . . . , dfn ). Such an n-tensor n-vector) η : M → η must, of course, satisfy the Fundamental Identity. We call this n-tensor Nambu– Poisson tensor . After geometrical formulations have been done by Takhtajan’s work, there can be found several papers on Nambu–Poisson geometry. (See, for example, [1, 2, 3].) In this paper, we shall define a kind of Poisson bracket on some function space, and construct the basic theory similar to the classical Poisson geometry. In particular, we shall also study the normal form of Nambu–Poisson tensors. 2. Nambu–Poisson Manifolds Let M be an m-dimensional C ∞ -manifold and denote by F the algebra of C ∞ functions on M . We shall define a Nambu–Poisson bracket and a Nambu–Poisson manifold following Takhtajan’s formalism [7]. Definition 2.1. A Nambu–Poisson bracket of order n, m ≥ n, on M is an n-linear skew-symmetric map from F n to F such that (1) (Leibniz rule) {f1 , . . . , fn−1 , g1 · g2 } = {f1 , . . . , fn−1 , g1 } · g2 + g1 · {f1 , . . . , fn−1 , g2 } , (2) (Fundamental Identity) {f1 , . . . , fn−1 , {g1 , . . . , gn }} = {{f1, . . . , fn−1 , g1 }, g2 , . . . , gn } + {g1 , {f1 , . . . , fn−1 , g2 }, g3 , . . . , gn } + · · · + {g1 , . . . , gn−1 , {f1 , . . . , fn−1 , gn }} for all f1 , . . . , fn−1 , g1 , . . . , gn ∈ F. Vn
To each Nambu–Poisson bracket, there corresponds an n-vector field η : M → T M such that {f1 , . . . , fn } = η(df1 , . . . , dfn ) ,
which satisfies the Fundamental Identity. Then η is called a Nambu–Poisson tensor of order n. We should remark that the Fundamental Identity implies strong constraints on n-tensor η [7]. Definition 2.2. Let η be a Nambu–Poisson tensor of order n on M . Then the pair (M, η) is called a Nambu–Poisson manifold. Let Rn = (x1 , . . . , xn ) be the n-dimensional Euclidean space. Then we can define a Nambu–Poisson bracket of order n by {f1 , . . . , fn } =
∂(f1 , . . . , fn ) , ∂(x1 , . . . , xn )
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for all f1 , . . . , fn ∈ C ∞ (Rn ). This is just the example which Nambu gave in his paper [6]. The corresponding Nambu–Poisson tensor η is given by η=
∂ ∂ ∧··· ∧ . ∂x1 ∂xn
We shall call this η the standard Nambu–Poisson tensor . And the pair (Rn , η) is said to be the standard Nambu–Poisson manifold . Moreover for a Nambu–Poisson manifold (M, η), it is said to be trivial if η = 0 on M . For classical Poisson manifolds, some fundamental notions were already defined [9]: Poisson brackets, Casimir functions, Hamiltonian vector fields, Poisson vector fields, etc. In the case of Nambu–Poisson manifolds, we also define the same notions as classical Poisson manifolds. Let (M, η) be a Nambu–Poisson manifold, where η is a Vn−1 P F, Nambu–Poisson tensor of order n. For an element A = fi1 ∧· · ·∧fin−1 ∈ we define a vector field XA by XA (h) =
X
{fi1 , . . . , fin−1 , h}
for all h ∈ F. Then XA is called a Hamiltonian vector field corresponding to Vn−1 Vn−1 F . If XA (h) = 0 for all h ∈ F, A ∈ F is called a Casimir function. A∈ We denote by C the space of Casimir functions. We denote by L the Lie algebra of infinitesimal automorphisms of (M, η). That is, L = {X ∈ χ(M )|L(X)η = 0} , where L(X) denotes the Lie derivative along X. Then we can easily prove the following: Proposition 2.3. A vector field X is contained in L if and only if it satisfies X · {f1 , . . . , fn } = {X · f1 , f2 , . . . , fn } + {f1 , X · f2 , . . . , fn } + · · · + {f1 , f2 , . . . , X · fn } for all f1 , . . . , fn ∈ F. Moreover we denote by H the Lie algebra of Hamiltonian vector fields. Then we shall prove in the next section that H is an ideal of L. 3. Structure of Poisson Brackets P Vn−1 F . Let A = fi1 ∧ · · · ∧ fin−1 First we define Poisson bracket [ , ] on P Vn−1 and B = gj1 ∧ · · · ∧ gjn−1 be any elements of F . Then Poisson bracket of A and B is defined by
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[A, B] =
X
{fi1 , . . . , fin−1 , gj1 } ∧ gj2 ∧ · · · ∧ gjn−1
+ gj1 ∧ {fi1 , . . . , fin−1 , gj2 } ∧ gj3 ∧ · · · ∧ gjn−1
+ · · · + gj1 ∧ · · · ∧ gjn−2 ∧ {fi1 , . . . , fin−1 , gjn−1 } . Vn−1 F is obtained by exAs is easily seen, this definition of Poisson bracket on Vn−1 F . We may also write this bracket as L(XA )B. tending the action of XA to If n = 2, our definition of Poisson bracket agrees with the usual Poisson bracket. Hence it follows that [A, B] = −[B, A]. If n ≥ 3, the situation is quite different. In fact, in this case the bracket operation is not generally skew-symmetric. But we can prove Vn−1 F , [A, B] + [B, A] is a Casimir Lemma 3.1. Let n ≥ 3. For all A, B ∈ function. In particular, if C is a Casimir function, then [C, A] = 0 and [A, C] ∈ C. Proof. By using the Fundamental Identity, we have for all h ∈ F: X[A,B] (h) =
X {{fi1 , . . . , fin−1 , gj1 }, gj2 , . . . , gjn−1 , h} + {gj1 , {fi1 , . . . , fin−1 , gj2 }, gj3 , . . . , gjn−1 , h} + · · · + {gj1 , . . . , gjn−2 , {fi1 , . . . , fin−1 , gjn−1 }, h}
=
X {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}} − {gj1 , . . . , gjn−1 , {fi1 , . . . , fin−1 , h}}
=
X {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}} − {{gj1 , . . . , gjn−1 , fi1 }, fi2 , . . . , fin−1 , h} − · · · − {fi1 , . . . , fin−2 , {gj1 , . . . , gjn−1 , fin−1 }, h} − {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}}
= −X[B,A] (h) . This implies that [A, B] + [B, A] ∈ C. By the definition of Poisson bracket, it is obvious that if C ∈ C, then [C, A] = 0, and hence we have [A, C] ∈ C. Remark. Even in the case of n = 2, since C contains zero, we can say that [A, B] + [B, A] ∈ C. Thus for every n ≥ 2, Lemma 3.1 is valid. Lemma 3.2. [XA , XB ] = X[A,B] , for all A, B ∈
Vn−1
F.
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Proof. For all h ∈ F, we have [XA , XB ](h) = XA (XB (h)) − XB (XA (h)) X {fi1 , . . . , fin−1 , {gj1 , . . . , gjn−1 , h}} = − {gj1 , . . . , gjn−1 , {fi1 , . . . , fin−1 , h}} =
X {{fi1 , . . . , fin−1 , gj1 }, gj2 , . . . , gjn−1 , h} + · · · + {gj1 , . . . , gjn−2 , {fi1 , . . . , fin−1 , gjn−1 }, h}
= X[A,B] (h) . This completes the proof.
V V Lemma 3.3. Let π : n−1 F → ( n−1 F )/C be the natural projection. Put ¯ B] ¯ = ¯ Then the bracket operation on (Vn−1 F )/C can be defined by [A, π(A) = A. Vn−1 [A, B] for all A, B ∈ F. Proof. By Lemma 3.1, we know that [A + C1 , B + C2 ] = [A, B] + [A, C2 ] for all Vn−1 F )/C C1 , C2 ∈ C. This implies that we can define the bracket operation on ( ¯ ¯ by [A, B] = [A, B]. Vn−1 F )/C Poisson bracket , and denote it by the We also call this bracket on ( Vn−1 F with Poisson bracket [ , ] does not same symbol. If n ≥ 3, recall that admit Lie algebra structure, because of lack of skew-symmetry. Hence we should V move to ( n−1 F )/C to make use of the theory of Lie algebras. Vn−1 F )/C has a structure of a Lie algebra, and Proposition 3.4. The space ( it is isomorphic to H as Lie algebras. Proof. By Lemma 3.1, it is clear that ¯ B] ¯ + [B, ¯ A] ¯ = [A, B] + [B, A] = 0 . [A, ¯ A] ¯ = −[A, ¯ B]. ¯ Next we prove Jacobi identity. Since X[F,G] (H) = [XF , XG ] Thus [B, Vn−1 F by Lemma 3.2, we have (H) for all F , G, H ∈ [[F, G], H] = [F, [G, H]] − [G, [F, H]] . Hence ¯ H] ¯ = [F¯ , [G, ¯ H]] ¯ − [G, ¯ [F¯ , H]] ¯ . [[F¯ , G], Vn−1 F )/C is skew-symmetric, we obtain Since Poisson bracket on ( ¯ H] ¯ + [[G, ¯ H], ¯ F¯ ] + [[H, ¯ F¯ ], G] ¯ = 0. [[F¯ , G],
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Vn−1 A linear mapping F → H, (A 7→ XA ) is surjective and its kernel is C. ComVn−1 F )/C ∼ bining this with Lemma 3.2, we obtain that ( = H as Lie algebras. For any A =
P
fi1 ∧ · · · ∧ fin−1 ∈
Vn−1
F and g1 , . . . , gn ∈ F, it is clear that
XA · {g1 , . . . , gn } = {XA · g1 , g2 , . . . , gn } + · · · + {g1 , . . . , gn−1 , XA · gn } . (This is just the Fundamental Identity.) This implies H ⊂ L. Moreover for any Y ∈ L and h ∈ F, we have X X {fi1 , . . . , fin−1 , h} − {fi1 , . . . , fin−1 , Y · h} [Y, XA ](h) = Y · =
X
{Y · fi1 , . . . , fin−1 , h} + · · · + {fi1 , . . . , Y · fin−1 , h}
= XB (h) , P
where B = L(Y )(fi1 ∧ · · · ∧ fin−1 ) ∈ proved the following:
Vn−1
F . Hence [Y, XA ] ∈ H. Thus we have
Proposition 3.5. H is an ideal of L. 4. The Spaces L/H of Nambu Poisson Manifolds In the theory of Poisson manifolds, it is well known that the notion of Poisson cohomologies is a matter of great importance. But unfortunately it is difficult to calculate them, when, in particular, Poisson tensors have singularities. (See [5, 8].) For a Nambu–Poisson manifold (M, η), any suitable generalizations of the usual Poisson cohomology have not been found yet. In this section, the spaces L/H are calculated for some Nambu–Poisson manifolds. In the case of Poisson manifolds, these spaces L/H are nothing but the spaces of the first Poisson cohomology groups. As the first step for the cohomology theory, it may be interesting to determine the space L/H even if some conditions are imposed. In the first place, we assume that a manifold M is paracompact and that the dimension of M is equal to the order of η. Recall that any n-vector field η on an n-dimensional manifold becomes a Nambu–Poisson tensor by the result of P. Gautheron [3]. Moreover η is assumed to be nowhere vanishing on M . Thus in our case, η is non-vanishing Nambu–Poisson tensor of order n. The volume form ω corresponding to η, can be defined as follows [3]: X1 ∧ · · · ∧ Xn = (−1)n−1 ωx (X1 , . . . , Xn )ηx , for all X1 , . . . , Xn ∈ Tx (M ). Using this volume form ω, the following facts are easily proved: Lemma 4.1. For a vector field X, it satisfies L(X)η = 0 if and only if L(X) ω = 0.
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Lemma 4.2. Let A = ∧ · · · ∧ dfin−1 .
P
fi1 ∧ · · · ∧ fin−1 ∈
Vn−1
F . Then i(XA )ω =
P
dfi1
Since ω is non-degenerate, the following mapping n−1 (M ) , τ : L → HdR
(X 7→ [i(X)ω])
is surjective, where [i(X)ω] denotes the cohomology class of a closed (n − 1)-form i(X)ω. By Lemma 4.2, it holds that [i(XA )ω] = 0. Thus we have H ⊂ ker τ . Conversely let X be an element of ker τ . First note that any (n − 2)-form β P fi1 dfi2 ∧ · · · ∧ dfin−1 . on a paracompact C ∞ -manifold M can be written as β = Since i(X)ω is an exact (n − 1)-form, there exists an (n − 2)-form β such that X i(X)ω = dβ = dfi1 ∧ · · · ∧ dfin−1 . Then by Lemma 4.2, we have X = XA
for A =
X
fi1 ∧ · · · ∧ fin−1 ∈
^
n−1
F.
This means that ker τ ⊂ H. Thus we have proved: Theorem 4.3. Let (M, η) be an n-dimensional Nambu–Poisson manifold with n−1 (M ). non-vanishing η of order n. Then L/H is isomorphic to HdR Secondly let us consider the case that η is a tensor of order n on Rn+1 which ∂ ∧ · · · ∧ ∂x∂n+1 on Rn+1 . is defined by the standard Nambu–Poisson tensor η0 = ∂x 1 More precisely, η is defined by {f1 , . . . , fn }η = {f1 , . . . , fn , f }η0 , where f1 , . . . , fn are arbitrary C ∞ -functions on Rn+1 , and f ∈ C ∞ (Rn+1 ) is a fixed function. Then it is easy to see that η actually becomes a Nambu–Poisson tensor. We denote by Ω the standard volume form on Rn+1 . Under these notations, we prove: Lemma 4.4. X ∈ L if and only if d(Xf ) = (divΩ X) · (df ). Proof. For any f1 , . . . , fn ∈ C ∞ (Rn+1 ) and any vector field X, we have (L(X)η)(df1 , . . . , dfn ) = X · ({f1 , . . . , fn }η ) − {Xf1, f2 , . . . , fn }η − · · · − {f1 , . . . , fn−1 , Xfn }η = X · ({f1 , . . . , fn , f }η0 ) − {Xf1 , f2 , . . . , fn , f }η0 − · · · − {f1 , . . . , fn−1 , Xfn , f }η0 − {f1 , . . . , fn , Xf }η0 + {f1 , . . . , fn , Xf }η0 = (L(X)η0 )(df1 , . . . , dfn , df ) + η0 (df1 , . . . , dfn , d(Xf )) = −(divΩ X) · η0 (df1 , . . . , dfn , df ) + η0 (df1 , . . . , dfn , d(Xf )) .
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Thus if X ∈ L, then η0 (df1 , . . . , dfn , d(Xf )) = (divΩ X) · η0 (df1 , . . . , dfn , df ). This implies that d(Xf ) = (divΩ X) · (df ). The converse is clear. Since η is defined by the standard Nambu–Poisson tensor η0 , it is quite easy to show the following lemma. Lemma 4.5. For XA ∈ H, where A ∈ divΩ XA = 0.
Vn−1
C ∞ (Rn+1 ), we have XA f = 0 and
If we take f = 12 (x21 + x22 − x23 − · · · − x2n+1 ), a linear Nambu–Poisson tensor η on Rn+1 is defined by {f1 , . . . , fn }η = {f1 , . . . , fn , f }η0 . Let m(t) be a C ∞ -function of one variable which is flat at the origin and is zero if t ≤ 0. Using these functions, let us define a vector field X by x2 ∂ ∂ x1 + 2 . X = m(f ) x21 + x22 ∂x1 x1 + x22 ∂x2 Then X satisfies d(Xf ) = (divΩ X)(df ), hence X ∈ L by Lemma 4.4. On the contrary, since Xf = m(f ) 6= 0, X is not contained in H by Lemma 4.5. Thus we can easily conclude as follows: Proposition 4.6. If η is a linear Nambu–Poisson tensor on Rn+1 induced from f = 12 (x21 + x22 − x23 − · · · − x2n+1 ), then the space L/H is infinite dimensional. Next if we take f = 12 (x21 + x22 + · · · + x2n+1 ), we have another linear Nambu– Pn+1 ∂ , n ≥ 3, be an element of L. Poisson tensor π of order n. Let X = i=1 fi ∂x i Pn+1 Then by using the same method as [4], we can obtain that i=1 xi fi = 0, and divΩ X = 0. By Lemma 4.5, these are necessary conditions for X to be contained in H. If n = 2, π is just a linear Poisson tensor on so(3, R)∗ , and it is well known that L = H. So it may be natural that we conjecture as follows: Conjecture. For a linear Nambu–Poisson tensor π of order n, (n ≥ 3), it also holds that L = H. 5. Canonical Local Coordinates of Nambu Poisson Manifolds Let (M, η) be a Nambu–Poisson manifold of order n. A point x0 ∈ M is called regular if η(x0 ) 6= 0. Then the main theorem of this section states that around a regular point every Nambu–Poisson manifold with a Nambu–Poisson tensor η of order n ≥ 3 is locally written as the product of a standard Nambu–Poisson manifold and a trivial Nambu–Poisson manifold. Note that the product of two Nambu–Poisson manifolds is, in general, no longer a Nambu–Poisson manifold. But one of them is, in our case, a “trivial” Nambu– Poisson manifold. So the product manifold is also a Nambu–Poisson manifold in a natural manner.
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Suppose that η 6= 0 at x0 . Then there are n-local functions x1 , . . . , xn−1 , x0n such that {x1 , . . . , xn−1 , x0n }(x0 ) 6= 0. Since a Hamiltonian vector field Xx1 ∧···∧xn−1 (x0 ) 6= 0, there exists a system of local coordinates (z1 , . . . , zm ) around x0 such that Xx1 ∧···∧xn−1 = ∂z∂ 1 . Rewriting z1 = xn , we have {x1 , . . . , xn−1 , xn } = 1. We shall define n-Hamiltonian vector fields {Yi }1≤i≤n by Yi = (−1)n−i Xx1 ∧···∧ˆxi ∧···∧xn , where the symbol x ˆi denotes the absence of the corresponding factor. Lemma 5.1. Y1 , . . . , Yn are n-linearly independent vector fields which commute each other around x0 . Proof. Since Yi (xj ) = δij , it is clear that Y1 , . . . , Yn are linearly independent. For proving the commutativity of {Yi }, it suffices to show the case i > j. If i > j, one has [x1 ∧ · · · ∧ xˆi ∧ · · · ∧ xn , x1 ∧ · · · ∧ xˆj ∧ · · · ∧ xn ] ˆj ∧ · · · ∧ xi−1 ∧ 1 ∧ xi+1 ∧ · · · ∧ xn . = (−1)n−i x1 ∧ · · · ∧ x Hence [Yi , Yj ] = (−1)2n−i−j [Xx1 ∧···∧ˆxi ∧···∧xn , Xx1 ∧···∧ˆxj ∧···∧xn ] = (−1)n−j Xx1 ∧···∧ˆxj ∧···∧xi−1 ∧1∧xi+1 ∧···∧xn = 0,
and this proves lemma.
By virtue of Lemma 5.1 and the theorem of Frobenius, one can find local coordinates (a1 , . . . , an , b1 , . . . , bs ), (n + s = m) with Yi =
∂ , ∂ai
(i = 1, 2, . . . , n) .
Each Yi clearly satisfies Yi (bj ) = 0 ,
(1 ≤ i ≤ n, 1 ≤ j ≤ s) .
Lemma 5.2. (x1 , . . . , xn , b1 , . . . , bs ) is a system of local coordinates around x0 . Proof. Since
∂xi ∂aj
= Yj (xi ) = δij , we have ∂(x1 , . . . , xn , b1 , . . . , bs ) 6= 0 . ∂(a1 , . . . , an , b1 , . . . , bs )
Lemma 5.3. With respect to the new local coordinates (x1 , . . . , xn , b1 , . . . , bs ), it holds
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N. NAKANISHI
∂ (a) Yi = ∂x , (1 ≤ i ≤ n), i (b) {xi1 , . . . , xin−1 , bj } = 0, (c) {xi1 , . . . , xik , bj1 , . . . , bjl } = 0, (k + l = n, k ≥ 1) .
Proof. Note that Yi (xj ) = δij and Yi (bj ) = 0. Then (a) is clear. For the proof of (b), it suffices to show that {x1 , . . . , xn−1 , bj } = 0. (Other cases can be shown in the same manner.) In fact we have {x1 , . . . , xn−1 , bj } = Yn (bj ) = 0. It will be enough to prove (c) in the case where {x1 , . . . , xk , bj1 , . . . , bjl } = 0 only. 1 k−1 2 (−1) x1 , x2 , . . . , xn−1 , {x2 , . . . , xk , xn , bj1 , . . . , bjl } 2 using the Fundamental Identity 2 1 k−1 x1 , x2 , . . . , xn−1 , xn , bj1 , . . . , bjl = x2 , . . . , xk , (−1) 2 = x2 , . . . , xk , (−1)k−1 x1 , bj1 , . . . , bjl = x1 , . . . , xk , bj1 , . . . , bjl on the other hand, by the Leibniz rule = (−1)k−1 x1 {x1 , x2 , . . . , xn−1 , {x2 , . . . , xk , xn , bj1 , . . . , bjl }} = (−1)k−1 x1 {x2 , . . . , xk , 1, bj1 , . . . , bjl } = 0. Hence we have {x1 , . . . , xk , bj1 , . . . , bjl } = 0.
Lemma 5.4. Nambu–Poisson brackets {bj1 , . . . , bjn } are functions of b1 , . . . , bs only. Proof. By Lemma 5.3 (a), we have for any xi (1 ≤ i ≤ n) ∂ {bj , . . . , bjn } = Yi {bj1 , . . . , bjn } ∂xi 1 ˆi , . . . , xn , {bj1 , . . . , bjn }} = (−1)n−i {x1 , . . . , x using Lemma 5.3 (b) and the Fundamental Identity = 0.
This completes the proof.
Assume that η 6= 0 at x0 ∈ M . Then by Lemma 5.3 (b) and (c), we can find a system of local coordinates (x1 , . . . , xn , b1 , . . . , bs ) such that η=
X ∂ ∂ ∂ ∂ ∧ ···∧ + Pj1 ...jn ∧ ···∧ . ∂x1 ∂xn j <···<j ∂bj1 ∂bjn 1
n
ON NAMBU–POISSON MANIFOLDS
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This expression of η looks like that of the usual Poisson tensors (Weinstein’s splitting theorem [9]). But, as the final result, we shall prove that the second term of the right-hand side of η vanishes in the case of Nambu–Poisson tensors. This will be caused by the Fundamental Identity which has stronger constraints than the usual Jacobi identity. Here the condition n ≥ 3 is essential. In fact in the proof of Theorem 5.5 below, if n = 2, we cannot conclude that {bj1 , bj2 } = 0. We can only say that {bj1 , bj2 } is a function of b1 , . . . , bs . And this is just the case of a usual Poisson manifold [9]. Theorem 5.5. Let (M, η) be a Nambu–Poisson manifold with a Nambu–Poisson tensor η of order n ≥ 3. Assume that η 6= 0 at x0 . Then there exists a system of local coordinates (x1 , . . . , xn , b1 , . . . , bs ) with η=
∂ ∂ ∧··· ∧ . ∂x1 ∂xn
In other words, there is a neighborhood U of a regular point x0 such that U is isomorphic to the product of a standard Nambu–Poisson manifold {S(x1 , . . . , xn ), ∂ ∧ · · · ∧ ∂x∂ n } and a trivial Nambu–Poisson manifold {N (b1 , . . . , bs ), ηS = ∂x 1 ηN = 0}. Proof. If s < n, then ∂b∂j ∧ · · · ∧ ∂b∂j = 0, and we are done. If s ≥ n, we shall n 1 show that Pj1 ...jn = 0. In fact, using the Fundamental Identity and the Leibniz rule, we have 0 = {x1 bj1 , x2 , . . . , xn−1 , {xn , bj2 , . . . , bjn }} = {{x1 bj1 , x2 , . . . , xn−1 , xn }, bj2 , . . . , bjn } + {xn , {x1 bj1 , x2 , . . . , xn−1 , bj2 }, bj3 , . . . , bjn } + · · · + {xn , bj2 , . . . , bjn−1 , {x1 bj1 , x2 , . . . , xn−1 , bjn }} = {bj1 {x1 , . . . , xn }, bj2 , . . . , bjn } = {bj1 , bj2 , . . . , bjn } = Pj1 ...jn . Hence we obtain that η =
∂ ∂x1
∧ ···∧
∂ ∂xn .
Acknowledgements The author wishes to express his gratitude to Professor M. Ichiyanagi for having introduced Nambu mechanics to him. He would also like to thank Professor H. Sato for helpful discussions. The latter half of Sec. 4 was improved in the simpler form through the discussion with Professor J. P. Dufour. The author wishes to thank him for wonderful hospitality at Universit´e de Montpellier II.
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References [1] R. Chatterjee and L. Takhtajan, “Aspects of classical and quantum Nambu mechanics”, Lett. Math. Phys. 37 (1996) 475–482. [2] G. Dito, M. Flato, D. Sternheimer and L. Takhtajan, “Deformation quantization and Nambu mechanics”, Commun. Math. Phys. (1996). [3] P. Gautheron, “Some remarks concerning Nambu mechanics”, Lett. Math. Phys. 37 (1996) 103–116. [4] N. Nakanishi, “On the structure of infinitesimal automorphisms of linear Poisson manifolds II”, J. Math. Kyoto Univ. 31 (1991) 281–287. [5] N. Nakanishi, “Poisson cohomology of plane quadratic Poisson structures”, Publ. RIMS. Kyoto Univ. (1997). [6] Y. Nambu, “Generalized Hamiltonian mechanics”, Phys. Rev. D7 (1973) 2405–2412. [7] L. Takhtajan, “On foundation of the generalized Nambu mechanics”, Commun. Math. Phys. 160 (1994) 295–315. [8] I. Vaisman, Lectures on the geometry of Poisson manifolds, 1994, Birkhauser. [9] A. Weinstein, “The local structure of Poisson manifolds”, J. Diff. Geom. 18 (1983) 523–557.
INTRODUCTION TO QUANTUM GROUPS ´∗ P. PODLES Department of Mathematical Methods in Physics, Faculty of Physics University of Warsaw Hoz˙ a 74, 00–682 Warszawa Poland E-mail : [email protected]
¨ E. MULLER Graduiertenkolleg “Mathematik im Bereich ihrer Wechselwirkung mit der Physik” Department of Mathematics Munich University Theresienstraße 39, 80333 M¨ unchen Germany E-mail : [email protected] Received 27 March 1997 We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (∗-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq (N )-groups and quantum Lorentz groups.
Contents 1. Introduction and Physical Motivations 2. Polynomials on Classical Groups of Matrices 3. Examples of Quantum Groups 4. ∗-Structures 5. Compact Hopf ∗-algebras 6. Actions on Quantum Spaces 7. Quantum Lorentz Groups References
511 515 523 533 536 544 548 550
1. Introduction and Physical Motivations What are quantum groups? Let G be a group in the usual sense, i.e. a set satisfying the group axioms, and C be a field of complex numbers. With this group one can associate a commutative, associative C-algebra of functions from G to C with pointwise algebra ∗ Supported
by Graduiertenkolleg “Mathematik im Bereich ihrer Wechselwirkung mit der Physik”, Dept. of Mathematics, Munich University and by Polish KBN grant No. 2 P301 020 07
511 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 511–551 c World Scientific Publishing Company
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structure, i.e. for any two elements f and f 0 , for any scalar α ∈ C, and g ∈ G we have (f + f 0 )(g) := f (g) + f 0 (g) , (αf )(g) := αf (g) , (f f 0 )(g) := f (g)f 0 (g) . If G is a topological group, usually only continuous functions are considered, and for an algebraic group the functions are normally polynomial functions. These algebras are called “algebras of functions on G”. These algebras inherit some extra structures and axioms for those structures from the group structure and its axioms on G. Locally compact groups can be reconstructed from this algebra. Now the algebra is deformed or quantized , i.e. the algebra structure is changed so that the algebra is not commutative any more, but the extra structures and axioms for them remain the same. This algebra is called “algebra of functions on a quantum group”, where “quantum group” is just an abstract object “described” by the deformed algebra. This process can be summarized as follows: classical group G axioms of a group ↓ commutative algebra of functions on G with corresponding extra axioms
forget about group
—–−→
quantum group (abstract object) l non-commutative algebra with same extra axioms; “algebra of functions on a quantum group”
There is a similar concept of “quantum spaces”: If G acts on a set X (e.g. a vector space), there is a corresponding so-called coaction of the commutative algebra of functions on G on the commutative algebra of functions on X satisfying certain axioms. The latter algebra can often be deformed/quantized into a non-commutative algebra, called the “algebra of functions on a quantum space” with a similar coaction. There are three ways of considering algebras of functions on a group and their deformations: (a) polynomial functi ons Poly(G) (developed by Woronowicz and Drinfel’d), (b) continuous functions C(G), if G is a topological group (developed by Woronowicz), (c) formal power series (developed by Drinfel’d). Only the first two approaches will be dealt with in the sequel. They include representation theory, Peter–Weyl theory, Tannaka–Krein theory, and actions on quantum spaces. There is a second approach to quantum groups. If G is a connected, simply connected Lie group, G can be reconstructed from the universal enveloping algebra U (g) of the corresponding Lie algebra g. The algebra U (g) again inherits some extra structures and axioms and can be deformed. The deformed universal enveloping
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algebra can be regarded as universal enveloping algebra corresponding to a quantum group. One can consider (d) the quantized universal enveloping algebra Uq (g) (developed by Jimbo), (e) formal power series (to be more precise, the ring of formal power series in h over a free algebra, subject to certain relations which are the same as for U (g) in the case h = 0. From this ring the algebra Uq (g) can be extracted. This approach has been developed by Drinfel’d). This approach will not be used in the sequel. Physical motivations There are some physical motivations for quantum groups including 1. integrable models — handled with approach (e), 2. conformal field theory — handled with approach (e), 3. physical models based on quantized space-time — handled with approaches (a), (b), and (e). The last motivation shall be explained in more detail. One of the main problems in Quantum Field Theory (QFT) is to join QFT and General Relativity Theory in a consistent way. It seems that in such a new theory it would be impossible to study the geometry of the space when very small volumes are considered. If you consider a cube in space, each vertex of it having Planck’s length or less, and measure simultaneously the three coordinates x, y, and z of a particle in it, then the uncertainty of the measurement, i.e. the errors ∆x, ∆y, and ∆z are very small, whence by Heisenberg’s uncertainty relation the errors of the coordinates of the momentum are big and therefore the uncertainty of the energy ∆E is big, too. Since the energy is positive, the expected value hEi of the energy is big, and the smaller the cube the bigger the energy, which at a certain stage generates a black hole. Therefore the observation of the geometry of the space gives it a different geometry, which makes this observation useless (we have used here the arguments by Professor W. Nahm). Quantum mechanics says that physical quantities such as momentum and position, which can be measured, correspond to self-adjoint operators on a Hilbert space. Its elements describe possible states of a physical system. When a quantity is measured, the state is projected onto an eigenvector of the operator, and the result of the measurement is the corresponding eigenvalue. Two quantities can be measured simultaneously if and only if the corresponding operators commute. In usual quantum mechanics the operators corresponding to the three coordinates of space commute and can be measured simultaneously, which leads to the problem with the black hole. Thus it is reasonable to assume that the operators corresponding to the coordinates x, y, and z do not commute (whence they cannot be measured simultaneously). Hence the commutative algebra generated by the operators corresponding to x, y, and z, which is isomorphic to the algebra of polynomials on R3 , is
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replaced by a non-commutative algebra on a quantum space. In order to give sense to self-adjoint operators, this algebra should be a ∗-algebra. Definition 1.1. (a) A ∗-algebra is a C-algebra A equipped with an antilinear, antimultiplicative involution ∗ : A → A, i.e. for all a, b ∈ A and λ ∈ C the following holds: (a + b)∗ = a∗ + b∗ , ¯ ∗, (λa)∗ = λa (ab)∗ = b∗ a∗ , (a∗ )∗ = a . (b) Let A, B be ∗-algebras. An algebra homomorphism φ : A → B is called ∗-homomorphism, if φ(a∗ ) = φ(a)∗ for all a ∈ A. Physical experiments should be comparable and reproducible, i.e. the same experiment performed at different places and times ought to give the same result. Therefore the theory should be invariant with respect to certain symmetry groups (containing translations in time and space). But the classical (symmetry) groups do not fit well to quantum spaces, so they have to be changed to quantum groups, too. (Example: The group SO3 (R) of rotations in three-dimensional space acts on the sphere S 2 . When the algebra of functions on S 2 is properly deformed such that the algebra becomes non-commutative, then there is no reasonable coaction of the usual algebra of functions on SO3 (R) any more. [8 Remark 2]) There is another motivation — deformation of an existing physical theory may help to understand the theory in a better way. It can reveal why the theory works, what is a consequence, and what is just a coincidence. Example [11]: After looking at deformations of standard Dirac theory, the covariance of the Dirac equation can be seen more directly — on the level of groups ¯ = Ψ† γ 0 , rather than Lie algebras. For the wave vector Ψ there is the equation Ψ 0 where γ also appears in the Dirac equation. In the deformed theory there is ¯ = Ψ† A with A 6= γ 0 in general, so that A = γ 0 is just a coincidence, and the Ψ condition A = γ 0 is not really important for the theory. In physics symmetry groups are usually groups of matrices, therefore the case of matrix groups is considered. Acknowledgments These lecture notes were written down by E. M. after the lectures by P. P. given at the Department of Mathematics, Munich University. The first author is very grateful to Professor Dr. Hans-J¨ urgen Schneider for his warm hospitality at Munich University. We thank him for his useful comments. An earlier version of these lectures was given by P. P. at Kyoto University in 1990–91. The first author would like to express his gratitude to Professor Huzihiro Araki for his kind hospitality and encouragement.
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2. Polynomials on Classical Groups of Matrices Notations In the sequel the base field of all vector spaces and algebras is the field C of complex numbers. A unital algebra is an (associative) algebra with a unit element, and a unital mapping is a mapping between unital algebras which sends the unit element to the unit element. Let N, N0 and R denote the sets of positive integers, non-negative integers and real numbers respectively and fix M , N ∈ N. Let A be a unital algebra and let MM×N (A) be the vector space of M × N -matrices with entries in A. If M = N , MN (A) := MN ×N (A) is a unital algebra. For each matrix M ∈ MM×N (A) let Mij be the entry at the ith row and jth column of M . Let B be another algebra, φ : A → B a map and M ∈ MM×N (A). Then φ(M ) is shorthand for the matrix in MM×N (B) with entries φ(Mij ). The group GL(N, C) of invertible N × N -matrices with complex entries is equipped with a topology inherited from the norm topology 2 of the vector space MN (C) ∼ = CN . The neutral element of a group is denoted by e. Let CN denote the space of row vectors and N C the space of column vectors. Using matrix multiplication, CN can be regarded as dual space of N C. If {e1 , . . . , eN } is a basis of N C, then there is a dual basis {e01 , . . . , e0N } of CN such that e0i ej = δij for all i, j ≤ N . In a similar way there are dual bases of the k-fold tensor products (N C)⊗k and (CN )⊗k . In the sequel the indices i, j, i0 , j 0 , k denote positive integers less or equal to N . Let 11N denote the identity matrix with N rows and columns or the identity endomorphism of CN or N C. Functions on groups Let G be an arbitrary subgroup of the group GL(N, C). Let Fun(G) be the algebra of complex valued functions on G. This algebra is unital with unit element 1 : G → C, g 7→ 1 and is a ∗-algebra, where for all f ∈ Fun(G) the function f ∗ is defined by f ∗ (g) := f (g) for all g ∈ G. For all i and j, the coefficient functions uij : G → C, g 7→ gij
−1 and u−1 )ij ij : G → C, g 7→ (g
belong to Fun(G). Then the matrices u := (uij )1≤i,j≤N and u−1 := (u−1 ij )1≤i,j≤N belong to MN (Fun(G)) and are inverses of each other in MN (Fun(G)). This justifies the notation u−1 . Definition 2.1. Let Pol(G) be the subalgebra of Fun(G) generated by the elements uij and u−1 ij for all i and j. Remark. This algebra is automatically unital because of the relation 1 = PN −1 k=1 u1k u k1 . The algebra is called “algebra of holomorphic polynomials on G”, too.
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Lemma 2.2. elements uij .
If G ⊂ SL(N, C) then Pol(G) is already generated by the
Proof. By the usual formula for the inverse of a matrix, (g −1 )ij = (−1)i+j det g˜j,i / det(g) for all g ∈ G, where the (N − 1) × (N − 1)-matrix g˜j,i is obtained from g by deleting the jth row and the ith column. But det(g) = 1, whence also u−1 ij is a polynomial in the functions ui0 j 0 . Definition 2.3. Let Poly(G) be the ∗-subalgebra of Fun(G) generated by the elements uij and u−1 ij . Usually the algebra Poly(G) is considerably bigger than Pol(G). Lemma 2.4. If G is a compact subgroup of GL(N, C), then Poly(G) is generated by the elements uij as ∗-subalgebra. Proof. The map φ : G → R+ , g 7→ | det(g)| is a group homomorphism from G into the multiplicative group of positive real numbers. Since φ is continuous and G is compact, the image of φ is a compact subgroup of R+ . But {1} is the only compact subgroup of R+ , whence φ(g) = 1 for all g ∈ G. Therefore 1 = det(g)det(g) = det(g) det((¯ gij )1≤i,j≤N ) . Thus det(u) is invertible in Poly(G) with inverse det((u∗ij )1≤i,j≤N ), whence the ∗ elements u−1 ij can be expressed by the ui0 j 0 and ui0 j 0 . Q Remark 2.5. Let I be an index set and let G be a subgroup of α∈I GL(Nα , C). Each element g of this group can be written as g = (gα )α∈I with gα ∈ GL(Nα , C) for all α ∈ I and define α −1 uα ij : G → C , ij , (u )
uα ij (g) := (gα )ij ,
(uα )−1 ij := (gα−1 )ij
for all g ∈ G. The algebras Pol(G) and Poly(G) are generated by the elements uα ij and (uα )−1 ij as algebras or ∗-algebras, respectively. This generalization covers all compact groups G, because the group homomorphism Y G→ GL(dim(π), C), g 7→ (π(g))π∈G b, b π∈G b where G is the set of finite dimensional irreducible representations of G, is injective if G is compact (cf. Tannaka–Krein duality). The multiplication, unit, and the inverse on G lead to the following extra structures on Fun(G): ∆ : Fun(G) → Fun(G × G) , (∆f )(g, h) := f (gh) for all g, h ∈ G (Comultiplication) , : Fun(G) → C , (f ) := f (e) (Counit) , S : Fun(G) → Fun(G) ,
(Sf )(g) := f (g −1 ) for all g ∈ G (Antipode) .
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These maps are unital ∗-homomorphisms. The (algebraic) tensor product Fun(G) ⊗ Fun(G) is the vector subspace of Fun(G × G) generated by elements u ⊗ v, where u, v ∈ Fun(G), by defining (u ⊗ v)(g, h) := u(g)v(h) for all g, h ∈ G. Equality only holds if G is finite. The axioms for the group structure on G are reflected by certain axioms for the extra structures on Fun(G). Let f be an element of Fun(G) such that ∆(f ) ∈ Fun(G) ⊗ Fun(G). Since the multiplication in G is associative, we have (∆ ⊗ id)∆(f ) = (id ⊗ ∆)∆(f ) .
(1)
The property of the neutral element, namely ge = eg = g for all g ∈ G, leads to the equation ( ⊗ id)∆(f ) = (id ⊗ )∆(f ) = f . (2) (Here the usual identification C ⊗ V ∼ =V ⊗C ∼ = V for all C-vector spaces is used.) Let the linear map µ : Fun(G) ⊗ Fun(G) → Fun(G), f ⊗ f 0 → f f 0 be induced by the multiplication in Fun(G). Then the properties gg −1 = g −1 g = e of the inverse can be expressed as µ(S ⊗ id)∆(f ) = µ(id ⊗ S)∆(f ) = (f )1 .
(3)
Definition 2.6. A unital algebra H is called Hopf algebra, if there are unital algebra homomorphisms ∆ : H → H ⊗ H and : H → C and a linear map S : H → H satisfying axioms (1) – (3) for all f ∈ H. The following lemma gives examples of Hopf algebras and shows why Pol(G) and the elements u−1 ij are interesting. Lemma 2.7. Pol(G) is a Hopf algebra satisfying ∆uij =
N X
uik ⊗ ukj ,
∆u−1 ij =
k=1
N X
−1 u−1 kj ⊗ u ik ,
k=1
(uij ) = (u−1 ij ) = δi,j ,
S(uij ) = u−1 ij ,
S(u−1 ij ) = uij .
If G is finite, also Fun(G) is a Hopf algebra. Proof. For all g, h ∈ G, ∆uij (g, h) = uij (gh) = (gh)ij =
N X k=1
gik hkj =
N X k=1
uik (g)ukj (h) =
N X
(uik ⊗ ukj )(g, h) .
k=1
A similar computation yields the formula for ∆(u−1 ij ). Therefore the image of Pol(G) under ∆ is contained in Pol(G) ⊗ Pol(G). The values of the counit can be computed: (uij ) = (u−1 ij ) = eij = δi,j . The equations for the antipode follow from
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(S(uij ))(g) = uij (g −1 ) = u−1 ij (g) for all g ∈ G. The Hopf algebra axioms are clearly satisfied, because Pol(G) is a subalgebra of Fun(G). If G is finite, then Fun(G) is a Hopf algebra because Fun(G) ⊗ Fun(G) = Fun(G × G). The following general theorem for Hopf algebras can be inferred from [1]. Theorem 2.8. Let H be a Hopf algebra with unit element 1. (a) The maps and S are unique if ∆ is fixed. (b) S is a unital antihomomorphism (c) If τ : H ⊗ H → H ⊗ H, x ⊗ y 7→ y ⊗ x denotes the flip automorphism, then ∆S = τ (S ⊗ S)∆ ,
S = .
(d) Let S 0 : H → H be a C-linear map. Then the following are equivalent: (i) µ(id ⊗ S 0 )τ ∆(f ) = µ(S 0 ⊗ id)τ ∆(f ) = (f )1 for all f ∈ H, (ii) S ◦ S 0 = S 0 ◦ S = id. Remark 2.9. (a) In general, the antipode of a Hopf algebra is not invertible. (b) A map S 0 such as in part (d) of Theorem 2.8 is called skew antipode, and there is another Hopf algebra structure on H with comultiplication τ ∆, counit and antipode S 0 . (c) A motivation for the fact, that the counit, but not the antipode is an algebra homomorphism, if H is not commutative: Since ∆ and the identity are algebra homomorphisms, there is no reason following from axiom (2) that should not be an algebra homomorphism. But the map µ in axiom (3) is an algebra homomorphism if and only if H is commutative. Therefore it should not be expected that S is an algebra homomorphism. For all f ∈ Fun(G) satisfying ∆(f ) ∈ Fun(G) ⊗ Fun(G), the following equation holds: ∆(f ∗ )(x, y) = f ∗ (xy) = f (xy) = ∆f (x, y) = =
X
X
f1 (x)f2 (y)
f1∗ (x)f2∗ (y) = (∗ ⊗ ∗)∆(f )(x, y)
for all x, y ∈ G. This motivates the following definition. Definition 2.10. A unital algebra H is called a Hopf ∗-algebra, if H is both a Hopf algebra and a ∗-algebra such that ∆(f ∗ ) = (∗ ⊗ ∗)∆f for all f ∈ H. From the definitions and Lemma 2.7 follows immediately Lemma 2.11. Poly(G) is a Hopf ∗-algebra, and if G is finite, also Fun(G) is a Hopf ∗-algebra.
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Proposition 2.12. Let H be a Hopf ∗-algebra. Then (a) For all x ∈ H, (x∗ ) = (x), i.e. is a ∗-homomorphism. (b) S ◦ ∗ ◦ S ◦ ∗ = id, in particular, S is bijective. Proof. (a) Since the map H → C, x 7→ (x∗ ) satisfies the properties of the counit, both are equal by Theorem 2.8, part (a), whence the assertion follows. (b) The map ∗◦S◦∗ satisfies all properties of the skew antipode. By Theorem 2.8, part (d) it is equal to it. This implies the two equivalent equalities ∗ ◦ S ◦ ∗ ◦ S = idH = S ◦ ∗ ◦ S ◦ ∗. Elements of representation theory Let H be a Hopf algebra. Definition 2.13. Let k be a positive integer. A matrix v ∈ Mk (H) is called corepresentation, if the entries satisfy the following relations for all indices a and b. P (a) ∆vab = kc=1 vac ⊗ vcb , (b) (vab ) = δa,b The number dim v := k is called the degree of the corepresentation, and the elements vab are called the matrix elements of the corepresentation. Remark 2.14. (a) Let v be a corepresentation of a Hopf algebra H. Then S(vab ) = (v −1 )ab for all indices a, b. Thus condition (b) of Definition 2.13 can be equivalently replaced by invertibility of v (note that condition (a) implies (v)v = v). (b) Let G be a classical group of matrices and H one of the Hopf algebras Pol(G) or Poly(G). Let v : G → Mk (C) ,
g 7→ (vab (g))1≤a,b≤k
be a map such that all functions vab are contained in H. Then (vab )1≤a,b≤k is a corepresentation if and only if v is a representation of G. Proof. (a) This follows from the axioms for the antipode of a Hopf algebra. (b) For all x, y ∈ G the following equations hold. (∆vab )(x, y) = vab (xy) = (v(xy))ab , ! k k X X vac ⊗ vcb (x, y) = vac (x)vcb (y) = (v(x)v(y))ab . c=1
c=1
Therefore condition (a) in Definition 2.13 is equivalent to v(xy) = v(x)v(y). A computation of (vab ) shows that condition (b) is equivalent to v(e) = 1k . Now fix a Hopf algebra H.
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Definition 2.15. Let v and w be two corepresentations of H. (a) Then v ⊕ w and v ⊗ w are corepresentations of H, where v ⊕ w is a matrix with dim(v) + dim(w) rows and columns given by v 0 , 0 w and the matrix of v ⊗ w has dim(v) dim(w) rows and columns and entries given by (v ⊗ w)ij,kl := vik wjl , where the indices i, k take values between 1 and dim(v) and the indices j, l between 1 and dim(w). (b) A dim(w) × dim(v) matrix A over C intertwines v with w, if Av = wA. Define Mor(v, w) as vector space of intertwining matrices between v and w. The elements of Mor(v, w) can be regarded as C-linear maps from Cdim v to Cdim w . The corepresentations v and w are said to be equivalent (v ∼ = w) if dim(v) = dim(w) and there is an invertible element in Mor(v, w). Definition-Lemma 2.16. Let w be a corepresentation of degree N and V ⊂ N C a subspace of dimension l. Then the following are equivalent: (a) For each % ∈ Hom(H, C) the statement %(w)V ⊆ V holds. (b) There is a corepresentation v and a basis a1 , . . . , al of V such that for the N × l-matrix Al := (a1 . . . al ) the equation wAl = Al v holds. This is equivalent to the condition that Al is an injective intertwiner of v with w. (c) There is a corepresentation v of degree l and an invertible matrix A, the first l columns of which are a basis of V and such that v ∗ wA = A . 0 ∗ If one of the equivalent conditions holds, then V is called “w-invariant subspace”, and the corepresentation v in part (b) and (c) is called “subcorepresentation of w and we write v = w|V . (Note that v depends on the chosen basis of V.)” Proof. (a) ⇒ ((b) ⇐⇒ (c)). Let a1 , . . . , al be a basis of V and extend it to a basis a1 , . . . , aN of N C. Then let Al be the N × l-matrix (a1 . . . al ) and A be the N × N -matrix (a1 . . . aN ). Then A is invertible and let B := A−1 wA. Let % ∈ Hom(H, C). Then A−1 %(w)A = %(B). Now condition (a) means that there is a matrix C% ∈ Ml (C) such that %(w)Al = Al C% , whence %(B) looks like %(v) ∗ , 0 ∗ where v is the submatrix of B consisting of the first l rows and columns. Since this holds for all linear forms, there is the matrix equation v ∗ v ∗ A−1 wA = ⇐⇒ wA = A 0 ∗ 0 ∗ or, equivalently, by restriction wAl = Al v.
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(b) ⇒ (a). From (b) it follows for all % ∈ Hom(H, C) that %(w)Al = %(v)Al , which gives %(w)V ⊆ V . Definition 2.17. Let w a corepresentation. (a) w is said to be irreducible if w 6= 0 and there is no subcorepresentation v such that 0 < dim(v) < dim(w). (b) w is called completely reducible if w is equivalent to a direct sum of irreducible subcorepresentations. Lemma 2.18. The intersection of invariant subspaces is an invariant subspace. Proof. This follows directly from Definition-Lemma 2.16, part (a). Lemma 2.19. Let A ∈ Mor(v, w). Then Ker(A) is v-invariant and Im(A) is w-invariant. Proof. Use Definition-Lemma 2.16, part (a). For each % ∈ Hom(H, C) the equation A%(v) = %(w)A follows. If x ∈ Ker(A) then A%(v)x = %(w)Ax = 0, whence %(v)x ∈ Ker(A) and the kernel is v-invariant. If y ∈ Im(A), say y = Az, then %(w)y = %(w)Az = A%(v)z is in the image of A, too. Lemma 2.20. (Schur). Let v, w be irreducible corepresentations. If v and w are not equivalent, then Mor(v, w) = {0}. If v is irreducible, then Mor(v, v) = C1, where 1 is the identity. Proof. Let A ∈ Mor(v, w) \ {0}. Since v and w are irreducible, by Lemma 2.19, A must be injective and surjective, whence v and w are equivalent. Now let w = v and λ be an eigenvalue of A ∈ Mor(v, v). Then A − λ1 ∈ Mor(v, v) is not injective and therefore vanishes. Remark 2.21. There is a relationship between finite dimensional right comodules of H and corepresentations. Theorem 2.22. Let H be a Hopf algebra. (a) The matrix elements of corepresentations span H.a (b) The matrix elements of a set of non-equivalent irreducible corepresentations are linearly independent. (c) The following are equivalent: (i) There is a set T of non-equivalent irreducible corepresentations such that the matrix elements of them form a basis of H.
a This
result is related to the fact that each element of a Hopf algebra is contained in a finite dimensional subcoalgebra.
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(ii) Each corepresentation is completely reducible.b Moreover if (i) holds then T contains all non-equivalent irreducible corepresentations. Proof. (a) Let x ∈ H. Then there is a number N ∈ N, linearly independent PN elements x1 , . . . , xN and y1 , . . . , yN in H such that ∆(x) = j=1 xj ⊗ yj . By PN PN coassociativity, j=1 ∆(xj ) ⊗ yj = j=1 xj ⊗ ∆(yj ), whence there are elements vij of H such that N X xi ⊗ vij ∆(xj ) = i=1
for all j. Using coassociativity and the properties for the counit, from these equations it follows that the elements vij are matrix elements of a corepresentation and P xj = i (xi )vij for all j. But then
x=
N X
xj (yj ) =
j=1
N X
(yj )(xi )vij
i,j=1
is a linear combination of matrix elements. (b) Use the arguments in the proof of [21, Proposition 4.7]. (c) The conclusion (ii) ⇒ (i) is now obvious, because by (a), the Hopf algebra is spanned by matrix elements of irreducible corepresentations, which are linearly independent by (b). The conclusion (i) ⇒ (ii) is proved in [9, Appendix]. The last remark follows from (b). Proposition 2.23. Let {vα | α ∈ I} and {vβ0 | β ∈ J} be sets of irreducible corepresentations such that M M vα ∼ vβ0 . = α∈I
β∈J
Then the multiplicities of equivalence classes of irreducible corepresentations are the same on both sides.c L L Proof. The set Mor( vα , vβ0 ) can be computed using Schur’s lemma (Lemma 2.20). But this set must contain an invertible element, since both direct sums are equivalent. Definition-Lemma 2.24. (a) Let w be a corepresentation of a Hopf algebra H. c := S(wji ) is a corepresentation, Then also the matrix wc with matrix elements wij the contragradient corepresentation to w. b In c cf.
the language of Hopf algebras this means that H is cosemisimple. Krull–Remak–Schmidt theorem
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(b) Let w be a corepresentation of a Hopf ∗-algebra H. Then also the matrix w ¯ ∗ is a corepresentation. Define w∗ to be the transpose with matrix elements w ¯ij := wij of w. ¯ (c) A corepresentation w of a Hopf ∗-algebra is called unitary if w ¯ = wc or equivalently ww∗ = w∗ w = 1dim w . Proof. (a) and (b) follow from the identities ∆ ◦ S = τ (S ⊗ S)∆, S = , ∆ ◦ ∗ = (∗ ⊗ ∗)∆. 3. Examples of Quantum Groups Quantum SL(2)-groups The simplest Lie group over the complex numbers, which is interesting and important in physics, is SL(2, C). We want to find quantum analogues of Pol(SL(2, C)). The corepresentations of this Hopf algebra have the following properties: (1) (2) (3) (4)
The irreducible corepresentations are wα , where 2α ∈ N0 . dim(wα ) = 2α + 1 for all α, wα ⊗ wβ ∼ = w|α−β| ⊕ w|α−β|+1 ⊕ · · · wα+β (Clebsch Gordan), Each corepresentation is completely reducible, or equivalently, the matrix α span the Hopf algebra. elements wij
Remark. The fundamental corepresentation is w := w1/2 given by g 7→ (gij )1≤i,j≤2 for g ∈ SL(2, C), and w0 is the identity. Definition 3.1. properties (1)–(4).
A quantum SL(2)-group is a Hopf algebra satisfying the
Theorem 3.2. Up to isomorphism there are the following quantum SL(2)groups H. The Hopf algebra H is generated by the matrix elements wij (1 ≤ i, j ≤ 2) of the fundamental corepresentation w := w1/2 and relations (w ⊗ w)E = E ,
E 0 (w ⊗ w) = E 0 ,
where the base field C is canonically embedded into H and there is the following extra relation between the row vector E 0 ∈ C2 ⊗ C2 and the column vector E ∈ 2 C ⊗ 2 C : Let {e1 , e2 } be a basis of 2 C and {e01 , e02 } be a dual basis of C2 . There is the following presentation: 2 2 X X 0 0 Eij ei ⊗ ej , E 0 = Eij ei ⊗ e0j . E= i,j=1
i,j=1
0 are inverses. There is a basis Then the 2 × 2 matrices with entries Eij and Eij 2 {e1 , e2 } of C such that 0 1 1 1 or E = e1 ⊗ e2 − e2 ⊗ e1 + e1 ⊗ e1 = , E = e1 ⊗ e2 − qe2 ⊗ e1 = b b −q 0 −1 0
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where q ∈ C \ {0} must not be a non-real root of unity. In the first case the quantum group is called the standard deformation SLq (2), in the second case it is called the non-standard deformation SLt=1 (2). The non-standard deformation SLt=1 (2) is not isomorphic to any of the standard deformations, and two standard deformations SLq (2) and SLq0 (2) are isomorphic if and only if q = q 0 or qq 0 = 1. Remark 3.3. (a) There is a set of non-standard deformations SLt (2) indexed by a parameter t ∈ C\{0} corresponding to the vector Et = e1 ⊗e2 −e2 ⊗e1 +te1 ⊗e1 , but they are all equivalent to the deformation for t = 1, because if the basis vector e1 is replaced by e01 = e1 t then b 1. tEt = e01 ⊗ e2 − e2 ⊗ e01 + e01 ⊗ e01 =E Since the relations remain the same when E is multiplied by a non-zero scalar, the Hopf algebras are isomorphic. (b) For t → 0, the vector Et tends to the vector for q = 1. (c) Parts of the proof of Theorem 3.2 can be found e.g. in [4, 23, 6]. To prepare the proof of Theorem 3.2, some extra definitions and lemmas are useful. Definition 3.4. Let q be a complex number. Then a Hecke algebra of degree n is a unital algebra generated by elements σ1 , . . . , σn−1 subject to the relations σk σk+1 σk = σk+1 σk σk+1 for 1 ≤ k ≤ n − 2 , (σk − 1)(σk + q 2 1) = 0 , σk σl = σl σk for |k − l| ≥ 2 . From these relations follows an important property of Hecke algebras and quotients of them: Definition-Lemma 3.5. Let A be a Hecke algebra as in Definition 3.4. Let π be an element of the symmetric group Πn of degree n, i.e. a permutation of the set I := {1, 2, . . . , n}. Then π can be written as the composition of transpositions tj (where tj interchanges the elements j and j + 1 of I). The minimal number of such transpositions is called length of π and is denoted by l(π). Let π = tk1 . . . tkl be a decomposition of π into a minimal number l = l(π) of transpositions. Then σk1 . . . σkl does not depend on the actual choice of transpositions as far as their number is minimal. Therefore σπ := σk1 . . . σkl is well defined. Definition-Lemma 3.6. Let A be a Hecke algebra as in Definition 3.4. Then define the element X q −2l(π) σπ ∈ A . Sn := π∈Πn
This element satisfies the property (σk − 1)Sn = 0 for 1 ≤ k ≤ n − 1.
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Proof. Let k be an integer between 1 and n − 1. Let π ∈ Πn be a permutation such that π(k) > π(k + 1) and let π 0 := tk π. If tk1 . . . tkl is a decomposition of π 0 into a minimal number of transpositions then tk tk1 . . . tkl = tk π 0 is a decomposition of π into a minimal number of transpositions and l(π) = l(π 0 ) + 1. Therefore for all k Sn =
X
π∈Πn π(k)>π(k+1)
π∈Πn π(k)<π(k+1)
−2 = σk q
X
q −2l(π) σπ +
X
= (q −2 σk + 1)
X
X
q −2l(π) σπ +
π∈Πn π(k)<π(k+1)
q −2l(π) σπ
π∈Πn π(k)<π(k+1)
q −2l(π) σπ
q −2l(π) σπ
π∈Πn π(k)<π(k+1)
and hence (σk − 1)Sn = (σk − 1)(1 + q −2 σk ) {z } | =0 (Hecke algebra)
X
q −2l(π) σπ = 0 .
π∈Πn π(k)<π(k+1)
Remark 3.7. The Hecke algebra is a generalization of the symmetric group, and for q = 1 the Hecke algebra relations are just the relations between the transpositions of the symmetric group. Let V be a vector space. The symmetric group acts on V ⊗n by permutations of the tensor factors. The operator σk corresponding to a transposition tk has the eigenvalues 1 and −1. The intersection of the kernels of all σk −1 or of the kernels of all σk +1 are called “totally symmetric vectors” or “totally antisymmetric vectors”, respectively. When a Hecke algebra (or a quotient of it) acts on V ⊗n , then the eigenvalues are 1 and −q 2 due to the second Hecke algebra relation. The intersection of the kernels of all σk − 1 or of the kernels of all σk + q 2 is called the space of “totally q-symmetric vectors” or “totally q-antisymmetric vectors”. The element Sn is called “symmetrization operator”, which is justified by Definition-Lemma 3.6, which also explains the factor q −2l(π) in the definition of Sn . Proof of Theorem 3.2. Let K = 2 C be the space of column vectors and K 0 = C2 the dual space of row vectors. w0 has always the matrix element 1 (because ∆(1) = 1 ⊗ 1). Let w be the fundamental corepresentation w1/2 . Then w⊗w ∼ = w0 ⊕ w1 , which is equivalent to the matrix 1 0 , 0 w0 where w0 ∈ M3 (H). This matrix has the column eigenvector E = (1 0 0 0)T and the row eigenvector E 0 = (1 0 0 0). Therefore there are the relations E 0 (w ⊗ w) = E 0 = w0 E 0 ,
(w ⊗ w)E = E = Ew0 .
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Thus the vectors E and E 0 , considered as 4 × 1 or 1 × 4 matrices, intertwine w ⊗ w and w0 . Moreover (4) E 0 E 6= 0 . Now (E 0 ⊗ 12 )(12 ⊗ E) can be regarded as an intertwiner of w with w, because w∼ = w0 ⊗ w and = w ⊗ w0 ∼ (E 0 ⊗12 )(12 ⊗E)(w⊗w0 ) = (E 0 ⊗12 )(w⊗w⊗w)(12 ⊗E) = (w0 ⊗w)(E 0 ⊗12 )(12 ⊗E) . Since w is irreducible, by Schur’s Lemma 2.20 (E 0 ⊗ 12 )(12 ⊗ E) is a multiple of the identity, say λ times the identity. Using the coordinate representation of E and E 0 with respect to a basis {ei ⊗ ej | 1 ≤ i, j ≤ 2} of K ⊗ K and the dual basis {e0i ⊗ e0j | 1 ≤ i, j ≤ 2} of K 0 ⊗ K 0 , X X 0 0 E= Eij ei ⊗ ej , E 0 = Eij ei ⊗ e0j , i,j
this condition becomes
i,j 2 X
0 Eik Ekj = λδij .
k=1
b 0 with entries E 0 satisfy b with entries Eij and E Therefore the matrices E ij b0 E b = λ12 . E b must have rank 1, because if it has rank 2, then E 0 = 0 and if it If λ = 0 then E bij = xi yj for some x1 , x2 , has rank 0 then E = 0 in contradiction to (4). Hence E y1 , y2 ∈ C and E has the form E = x ⊗ y, where x = x1 e1 + x2 e2 , y = y1 e1 + y2 e2 . From (w ⊗ w)E = E it follows that wx ⊗ wy = x ⊗ y ,
x ⊗ wy = w−1 x ⊗ y .
Both sides are in K ⊗ K ⊗ A. Applying φ ⊗ idK ⊗ idA , where φ is a linear form on K such that φ(x) = 1, we get wy = y ⊗ (φ ⊗ idA )(w−1 x) . Therefore Cy is an w-invariant subspace in contradiction to the fact that w is irreducible. Thus λ 6= 0, and by scaling of E 0 which does not change the relations, b −1 . The vector E in K ⊗ K can be written as sum of a symmetric b0 = E one gets E tensor E sym , i.e. an element of K ⊗ K which is invariant with respect to the flip automorphism τ of K ⊗ K, mapping x otimesy to y ⊗ x, and an antisymmetric tensor E asym satisfying τ (E asym ) = −E asym , defined by E sym = 12 (E + τ (E)) and P E asym = 12 (E − τ (E)). Symmetric tensors i,j aij ei ⊗ ej in K ⊗ K, where aij = aji P for all i, j, can be identified with quadratic forms Q on K 0 , namely Q( i vi e0i ) = P sym has one of the following i,j aij vi vj . In particular there are bases such that E presentations: (a) E sym = e1 ⊗ e2 + e2 ⊗ e1 if Q has rank 2, (b) E sym = e1 ⊗ e1 if Q has rank 1, (c) E sym = 0 if Q has rank 0.
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With respect to any basis {˜ e1 , e˜2 } of K an antisymmetric tensor E asym is a scalar multiple of e˜1 ⊗ e˜2 − e˜2 ⊗ e˜1 . Therefore E has one of the following presentations: b has rank 2, both (a) E = (1 + c)e1 ⊗ e2 + (1 − c)e2 ⊗ e1 with c ∈ C. Since E coefficients must not vanish. Therefore E is a scalar multiple of e1 ⊗ e2 − qe2 ⊗ e1 , where q = c−1 c+1 and q ∈ C \ {0, 1}. b has rank 2. (b) E = e1 ⊗ e1 + c(e1 ⊗ e2 − e2 ⊗ e1 ), where c ∈ C \ {0}, because E Therefore E is a scalar multiple of e1 ⊗ e2 − e2 ⊗ e1 + te1 ⊗ e1 , where t = 1c . According to Remark 3.3 this is equivalent to the vector for SLt=1 (2). In this case let q := 1. (c) E = c(e1 ⊗ e2 − e2 ⊗ e1 ), where c ∈ C \ {0}. This is the case q = 1 which is not included in (a). Now let the associative, unitary algebra H0 be generated by the elements α, β, γ, δ subject to the relations (v ⊗ v)E = E, E 0 (v ⊗ v) = E 0 for v = ( αγ βδ ). There are uniquely determined comultiplication, counit and antipode such that this algebra becomes a Hopf algebra and v is a corepresentation (see Proposition 3.8 and Proposition 3.9 below). Since the relations between the generators of H0 are satisfied in H, there is a Hopf algebra map ψ : H0 → H, vij 7→ wij . We shall study the corepresentation theory of H0 . Consider the 4 × 4 matrix σ := 14 + qE · E 0 (where E and E 0 are again 4 × 1 and 1 × 4 matrices, respectively). Then σ is an element of the vector space Mor(v ⊗ v, v ⊗ v). It satisfies the relations (σ − 14 )(σ + q 2 14 ) = 0 , (σ ⊗ 12 )(12 ⊗ σ)(σ ⊗ 12 ) = (12 ⊗ σ)(σ ⊗ 12 )(12 ⊗ σ) . Fix an integer n ≥ 2 and define for integers k satisfying 0 < k < n: σk = 12 ⊗ · · · ⊗ 12 ⊗σ ⊗ 12 ⊗ · · · ⊗ 12 . | {z } | {z } k−1
n−k−1
These are operators on the n-fold tensor product K ⊗n and intertwine v ⊗n with v ⊗n . They satisfy the Hecke algebra relations (cf. Definition 3.4). Now define the operators σπ as in Definition-Lemma 3.5 and the symmetrization operator as in Definition-Lemma 3.6: Sn :=
X
q −2l(π) σπ .
π∈Πn
Due to Definition-Lemma 3.6 it takes values in K n/2 := {x ∈ K ⊗n | ∀ k : σk (x) = x} =
\ k
Ker(σk − 1) .
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The dimension of the space K n/2 is n + 1. (Proof: analyze relations on coordinates of elements of K n/2 or see [23].) The space K n/2 is v ⊗n -invariant as intersection of the kernels of the intertwiners σk − 1 by Lemma 2.18 and Lemma 2.19 and a right comodule. Let v n/2 denote the corresponding subcorepresentation of v ⊗n as in Definition-Lemma 2.16. Then v n/2 is a corepresentation of dimension n + 1. By definition, v = v 1/2 , v 0 is the trivial one-dimensional corepresentation. At this moment we assume that q is not a non-real root of unity. For all s ∈ 12 N0 , one has (A) v k is irreducible for all k ≤ s + 12 , 1 1 (B) v k ⊗ v ∼ = v k+ 2 ⊕ v k− 2 , where by definition v −1/2 := 0, for all k ≤ s. These statements will be proved by induction: The case s = 0 follows from the result on monomials below. Suppose the statements are true for s replaced by s − 12 . We want to decompose v s ⊗ v and consider the map 1
φ : K s− 2 → K s ⊗ K, x 7→ (S2s ⊗ 12 )(x ⊗ E) (antisymmetrization-symmetrization procedure). Note that E ∈ K ⊗ K. The map is well defined due to the property of the symmetrization operator and intertwines v ⊗(2s−1) ∼ = v ⊗(2s−1) ⊗ v 0 with v ⊗2s ⊗ v. By inspection (σ2s − 1)φ (e1 ⊗ · · · ⊗ e1 ) 6= 0 | {z } 2s−1 factors 1
if q is not a non-real root of unity. Therefore φ(e1 ⊗ · · · ⊗ e1 ) ∈ / K s+ 2 and 1 Ker(φ) 6= K s− 2 . By induction hypothesis, there is no proper non-trivial v 2s−1 1 invariant subspace of K s− 2 and the kernel of φ is invariant by Lemma 2.19. Con1 sequently φ is injective and Im(φ) corresponds to v s− 2 . Moreover 1
Im(φ) ∩ K s+ 2 6= Im(φ) . 1
1
Since v s− 2 is irreducible, there is no proper non-trivial v s− 2 -invariant subspace of 1 Im(φ) and Im(φ) ∩ K s+ 2 = {0}. Thus 1 1 1 K s− 2 ⊕ K s+ 2 ∼ = Im(φ) ⊕ K s+ 2 ⊂ K s ⊗ K .
Equality follows by dimension arguments (dim K t = 2t + 1) and yields (B). By definition of the tensor product of representations, the monomials in α, β, γ, δ of degree smaller or equal to 2s + 1 are linear combinations of the matrix elements of v ⊗(2r+1) , r ≤ s. Using result (B) yields that they are linear combinations of 1 1 matrix elements of v 0 , v 2 , . . . , v s+ 2 . The space of these monomials has dimenP2s+2 2 sion k=1 k , as in the classical case for Pol(SL(2, C)). This has been shown in [23, 26]. Since v t has (2t + 1)2 matrix elements for all t ∈ 12 N0 , the space spanned 1 by the matrix elements of v 0 , v 2 , . . . , v s+1/2 has this dimension if and only if all matrix elements are linearly independent. Hence (A) follows. Now it is easy to prove that H0 is a quantum SL(2)-group. The matrix ψ(v s ) is a corepresentation of H because ψ respects ∆ and .
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Then ψ(v s ) ∼ = ws for all s ∈ 12 N0 . Proof by induction: The assertion is trivial for s = 0 and s = 12 . Suppose the statement is true for all non-negative half 1 integers smaller than s. Then by part (B), v s− 2 ⊗ v ∼ = v s−1 ⊕ v s . Since ψ is an algebra homomorphism, by the definitions of direct sum and tensor product of corepresentations the following holds: 1 ws−1 ⊕ ψ(v s ) ∼ = ψ(v s−1 ) ⊕ ψ(v s ) ∼ = ψ(v s−1 ⊕ v s ) ∼ = ψ(v s− 2 ⊗ v) 1 1 = ψ(v s− 2 ) ⊗ ψ(v) ∼ = ws− 2 ⊗ w ∼ = ws−1 ⊕ ws .
Due to condition (4) for the quantum SL(2)-group, the corepresentation ψ(v s ) is completely reducible, whence by Proposition 2.23 ψ(v s ) ∼ = ws . Thus ψ is an isomorphism and H can be identified with H0 . Now we consider the case when q is a non-real root of unity (see e.g. [6]). Let q be a non-real root of unity of order N . Define ( N0 :=
N N/2
if N is odd , if N is even .
Then H0 has a corepresentationd z=
αN0
β N0
γ N0
δ N0
! .
1 1 Then v k ⊗ v ∼ = v k+ 2 ⊕ v k− 2 for k < (N0 − 1)/2, v k is irreducible for k ≤ 12 (N0 − 1), and 1 v 2 N0 −1 ∗ ∗ ∼ . v (N0 −1)/2 ⊗ v = 0 z ∗ (N0 −1)/2 0 0 v
It is possible to show ψ(v k ) ∼ = wk for k ≤ hand,
1 2 (N0
− 1) as before, but on the other
1 1 1 1 w 2 N0 −1 ⊕ w 2 N0 ∼ = ψ(v 2 (N0 −1) ⊗ v) = w 2 (N0 −1) ⊗ w ∼ 1 w 2 N0 −1 ∗ ∗ ∼ 0 ψ(z) ∗ = 1 0 0 w 2 N0 −1 1 1 ∼ = w 2 N0 −1 ⊕ ψ(z) ⊕w 2 N0 −1 ,
because corepresentations in H are completely reducible. But this is a contradiction to Proposition 2.23. d cf.
[19, part 5.2]
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Let q1 and q2 be two values such that SLq1 (2) ∼ = SLq2 (2) (q1 , q2 ∈ C \ Y ∪ {t = 1}, where the subset Y contains 0 and all non-real roots of unity), i.e. that the Hopf algebras are isomorphic. Then the fundamental representation w1 is mapped to w2 , i.e. they are equivalent: w1 = Qw2 Q−1 . Let E1 , E2 be the corresponding eigenvectors. Then (w1 ⊗ w1 )E1 = E1 , (Qw2 Q−1 ⊗ Qw2 Q−1 )E1 = E1 ⇒ (w2 ⊗ w2 )((Q−1 ⊗ Q−1 )E1 ) = (Q−1 ⊗ Q−1 )E1 = λE2 with λ ∈ C \ {0}, because the space of eigenvectors of w2 ⊗ w2 for the eigenvalue 1 is one-dimensional. There are symmetric tensors E1sym , E2sym and antisymmetric tensors E1asym , E2asym such that E1 = E1sym + E1asym and E2 = E2sym + E2asym . Therefore E1 = λ(Q ⊗ Q)E2 ⇒ E1sym = λ(Q ⊗ Q)E2sym ,
E1asym = λ(Q ⊗ Q)E2asym
(5)
and E1sym and E2sym have the same rank. If the rank is 0 or 1, it is the same deformation, and if the rank is 2, one can use (5) and the fact that the rank of (Qe1 ⊗ Qe2 ) is one, to get q1 = q2 or q1 q2 = 1 (in the last case the isomorphism is given by e1 ↔ e2 ). Quantum SL(N)-groups Let N be a positive integer greater than 1. The Hopf algebra H of the group SL(N, C) corresponds to the commutative unital algebra generated by the matrix elements wij for 1 ≤ i, j ≤ N of a fundamental corepresentation w subject to the relations (6) w⊗N E = E , E 0 w⊗N = E 0 , where E and E 0 are classical completely antisymmetric elements of (N C)⊗N and (CN )⊗N respectively, i.e. with respect to a basis {e1 , . . . , eN } of N C and a dual basis {e01 , . . . , e0N } of CN , they can be presented as X E= (−1)l(π) eπ(1) ⊗ · · · ⊗ eπ(N ) , π∈Πn
E0 =
X
(−1)l(π) e0π(1) ⊗ · · · ⊗ e0π(N ) .
(7)
π∈Πn
Then the relations just mean (assuming commutativity) that the determinant of the matrix w is one. For SL(2) this is just e1 ⊗ e2 − e2 ⊗ e1 and e01 ⊗ e02 − e02 ⊗ e01 , which is changed to e1 ⊗ e2 − qe2 ⊗ e1 and up to a non-zero factor to e01 ⊗ e02 − qe02 ⊗ e01 in the standard deformation SLq (2). Therefore it is natural to define X Eq = (−q)l(π) eπ(1) ⊗ · · · ⊗ eπ(N ) , π∈Πn
Eq0 =
X π∈Πn
(−q)l(π) e0π(1) ⊗ · · · ⊗ e0π(N )
(8)
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and to consider the relations w⊗N Eq = Eq ,
Eq0 w⊗N = Eq0 .
(9)
For q not being a non-real root of unity they imply (cf. [22]) w⊗2 σ = σw⊗2 where
qej ⊗ ei σ(ei ⊗ ej ) := qej ⊗ ei + (1 − q 2 )ei ⊗ ej e i ⊗ ei
(10) if i < j , if i > j , if i = j ,
for i, j = 1, . . . , N . Now SLq (N ) is introduced as the unital algebra generated by wij for 1 ≤ i, j ≤ N subject to the relations (9), (10) (cf. [9]). One can check that this definition coincides with the standard one (cf. [3, 15]). The following proposition shows that all unital algebras with relations defined by intertwiners are bialgebras. If the intertwiners are chosen badly, the bialgebras can be small and uninteresting. For each matrix w and each n ∈ N define the matrix w⊗n as for corepresentations in Definition 2.15 and let w⊗0 := 11 . Proposition 3.8. Let H be the universal unital algebra generated by elements wij for 1 ≤ i, j ≤ N, which are the entries of a matrix w subject to relations Em w⊗sm = w⊗tm Em
(11)
for m in an index set I, sm , tm ∈ N0 and Em ∈ MN tm ×N sm (C). Then there exist a unique comultiplication and counit such that H is a bialgebra and w is a corepresentation of H. P Proof. (a) Uniqueness: We must have ∆wij = N k=1 wik ⊗ wkj and (wij ) = δij for all i and j. Since ∆ and are unital algebra homomorphisms, they are uniquely determined if they exist. P (b) Existence: Define w bij := N k=1 wik ⊗C wkj ∈ H⊗H for all i and j. The matrix b⊗n = w⊗n ⊗C w⊗n w b with entries w bij also satisfies the relations (11), because w follows from the rule (a ⊗ b)(c ⊗ d) = (ac ⊗ bd) and b⊗sm = Em (w⊗sm ⊗C w⊗sm ) = w⊗tm Em ⊗C w⊗sm Em w = w⊗tm ⊗C Em w⊗sm = w⊗tm ⊗C w⊗tm Em = w b⊗tm Em , eij := δij for all because the entries of Em are just complex numbers. Define w i, j. Then the matrix w e with entries w eij satisfies the properties e⊗sm = Em , Em w
w e⊗tm Em = Em ,
whence it satisfies relations (11). Now the universality of H gives the existence of bij and (wij ) = w eij . It is enough unital homomorphisms ∆, such that ∆(wij ) = w
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to check conditions (1) and (2) for bialgebras (cf. Definition 2.6) for elements f = wij when they are obvious. Proposition 3.9. Let the conditions of Proposition 3.8 be satisfied. Let {e1 , . . . , eN } be a basis of N C and {e01 , . . . , e0N } be a dual basis of CN . Moreover assume that there exist positive integers s and t and elements E ∈ Mor(11 , w⊗t ) and E 0 ∈ Mor(w⊗s , 11 ) such that E=
N X
ek ⊗ fk ,
E0 =
k=1
N X
fk0 ⊗ e0k
k=1 ⊗t−1
such that the elements fk ∈ ( C) and ∈ (CN )⊗s−1 are linearly independent. −1 Then the matrix w exists and there is a uniquely determined antipode S such that the bialgebra H is a Hopf algebra. N
fk0
Proof. From the relation w⊗t E = E it follows that (w ⊗ w⊗(t−1) )E = E ⇒
N X
wek ⊗ w⊗(t−1) fk =
k=1
N X
ek ⊗ fk .
(12)
k=1
Since the elements fk of (N C)⊗(t−1) are linearly independent, there are elements gk0 of the dual space (CN )⊗(t−1) such that gi0 fj = δij . Apply e0i ⊗ gj0 to Eq. (12): N X
e0i wek ⊗ gj0 w⊗(t−1) fk =
k=1
N X
e0i ek ⊗ gj0 fk = 1 ⊗ gj0 fi = δij .
k=1
Therefore the matrix G with entries Gkj := gj0 w⊗(t−1) fk is a right inverse to w. From the second condition it follows in a similar way that there is a left inverse of w. Thus w−1 exists. Finally, when to the relation Em w⊗sm = w⊗tm Em , (w⊗sm )−1 = (w−1 )⊗ sm is applied to the right and (w⊗tm )−1 = (w−1 )⊗ tm to the left (the tensor product “⊗op ” is ⊗ with respect to the algebra Hop with opposite multiplication), then op
op
(w−1 )⊗
op
tm
Em = Em (w−1 )⊗
op
sm
.
Therefore there is a unital algebra homomorphism S : H → Hop such that S(w) = w−1 . Equivalently, S : H → H is a unital antihomomorphism. It is enough to check condition (3) for the antipode (cf. Definition 2.6) for f = wij when it is obvious. Uniqueness of S follows from Theorem 2.8. Remark 3.10. For the quantum SL(N ) group take I = {1, 2, 3}, E1 = Eq , t1 = N , s1 = 0, E2 = Eq0 , t2 = 0, s2 = N , E3 = σ, t3 = s3 = 2. Then the algebras SLq (N ) are Hopf algebras. Remark 3.11. For 0 < q ≤ 1 the corepresentation theory of SLq (N ) is the same as for the classical SL(N ) (cf. [22, 9]). If q is transcendental, see [15, 5]. If
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q ∈ C \ {0} is not a non-real root of unity, see [12]. There are deformations of the orthogonal and symplectic groups [16, 18] (cf. [9]). 4. ∗-Structures In the classical theory there exist ∗-structures on Pol(SL(2)) which give the Hopf ∗-algebras Poly(SU (2)), Poly(SU (1, 1)) and Poly(SL(2, R)). We will classify the Hopf ∗-algebra structures on the quantum SL(2)-groups H described in Theorem 3.2. Firstly recall that H is generated as an algebra by the matrix elements of a 2 × 2 matrix w subject to the relations (w ⊗ w)E = E ,
E 0 (w ⊗ w) = E 0
or equivalently X j,l
wij wkl Ejl = Eik ,
X
0 0 Eik wij wkl = Ejl .
(13)
i,k
Lemma 4.1. Let ψ be an (anti-)linear comultiplicative algebra (anti-)automorphism of a quantum SL(2)-group H. Then (a) there exists a matrix Q ∈ GL(2, C) such that ψ(w) = QwQ−1 . (b) If and only if the matrix Q ∈ GL(2, C) satisfies the conditions (Q−1 ⊗ Q−1 )E = cE ,
E 0 (Q ⊗ Q) = c0 E 0
(14)
for some numbers c, c0 ∈ C \ {0}, there is a Hopf algebra automorphism ψ of H such that ψ(w) = QwQ−1 . Moreover, all Hopf algebra automorphisms of H can be described in this way. ¯ and E ¯ 0 denote (c) Let τ denote the linear twist (interchanging factors) and let E 2 2 2 2 the elements of C ⊗ C and C ⊗ C with conjugate complex coefficients with respect to the bases ei ⊗ ej , e0i ⊗ e0j . Then if and only if the matrix Q ∈ GL(2, C) satisfies the conditions (Q−1 ⊗ Q−1 )τ E¯ = cE ,
¯ 0 τ (Q ⊗ Q) = c0 E 0 E
(15)
for some c, c0 ∈ C \ {0}, there is an antilinear, comultiplicative algebra antiautomorphism ψ of H such that ψ(w) = QwQ−1 . (d) Let the antilinear involutive comultiplicative algebra antiautomorphisms ψ, ψˆ ˆ be defined as in (a). Then the and the corresponding matrices Q and Q Hopf algebra H equipped with ∗-structures ψ and ψˆ gives isomorphic Hopf ∗algebras if and only if ψˆ is equivalent to ψ up to a Hopf algebra automorphism ˆ = cA¯−1 QA, where c ∈ C \ {0} and φ (i.e. ψˆ = φψφ−1 ) if and only if Q A ∈ GL(2, C) corresponds to φ via (b). Proof. (a) Since ψ is comultiplicative, the matrix ψ(w) is a corepresentation. The following conclusions follow from the fact that ψ is bijective: w is irreducible
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if and only if the matrix elements wij are linearly independent if and only if the matrix elements ψ(wij ) are linearly independent if and only if ψ(w) is irreducible. But there is only one irreducible corepresentation of dimension 2 up to isomorphism, therefore there is a matrix Q ∈ GL(2, C) such that ψ(w) = QwQ−1 .
(16)
(b) Since the trivial corepresentation appears in the direct sum decomposition w⊗w ∼ = w1 ⊕ w0 only once, by Lemma 2.20 the space of intertwiners in Mor(w ⊗ 0 w, w ) is one-dimensional. Thus condition (14) is equivalent to the condition that (Q−1 ⊗ Q−1 )E intertwines w ⊗ w with w0 and E 0 (Q ⊗ Q) intertwines w0 with w ⊗ w: (w ⊗ w)(Q−1 ⊗ Q−1 )E = (Q−1 ⊗ Q−1 )E ⇐⇒ (QwQ−1 ⊗ QwQ−1 )E = E and E 0 (Q ⊗ Q)(w ⊗ w) = E 0 (Q ⊗ Q) ⇐⇒ E 0 (QwQ−1 ⊗ QwQ−1 ) = E 0 .
) (17)
Let ψ be a Hopf algebra automorphism of H. Then by part (a), there is a matrix Q ∈ GL(2, C) such that ψ(w) = QwQ−1 . The automorphism ψ must map the relations between the generators of H to relations in H, therefore Eq. (17) holds. Conversely, let Eq. (17) be satisfied. Let F be the free associative unital algebra generated by the matrix elements of w and let I be the two-sided ideal generated by the relations (13). Then the map ψ can be defined as unital algebra homomorphism on F such that ψ(w) = QwQ−1 . Equation (17) shows that ψ maps I to I, therefore it induces a unital algebra homomorphism on H = F /I. Such ψ preserves the Hopf algebra structure of H. Moreover, replacing Q by Q−1 (Eq. (14) still holds for c−1 and (c0 )−1 ) we get ψ −1 . (c) The proof is similar as in part (b). The only changes arise from the fact that ψ should be an antilinear algebra antiautomorphism instead of a linear algebra automorphism. Therefore, ψ applied to relations (13) yields X X ¯jl = E ¯ik , ¯0 ¯ 0 ψ(wkl )ψ(wij ) = E ψ(wkl )ψ(wij )E E ik
j,l
jl
i,k
or shortly τ (ψ(w) ⊗ ψ(w))τ E¯ = E¯ ,
¯0 . ¯ 0 τ (ψ(w) ⊗ ψ(w))τ = E E
Using τ 2 = idH , we get the desired results. ¯ = A¯−1 QAwA−1 Q−1 A, ¯ while (d) φψφ−1 (w) = φψ(A−1 wA) = φ(A¯−1 QwQ−1 A) −1 −1 ¯−1 ˆ ˆ ˆ ˆ ψ(w) = QwQ . The left-hand sides are equal if and only if Q A QA ∈ Mor(w, w) = C12 . Remark. It is easy to check that the second condition in (14) (and also the second condition in (15)) is redundant. Theorem 4.2. All non-equivalent Hopf ∗-algebra structures on the quantum SLq (2)-groups H are defined by w ¯ = QwQ−1 , where ¯ = w. This algebra is called Poly(SLq (2, R)). (a) Q = ( 10 01 ), |q| = 1. Then w
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0 (b) Q = ( 01 0q ), q ∈ R \ {0}. Then w∗ Bw = wBw∗ = B, for B := ( 10 −1 ). This algebra is called Poly(SUq (1, 1)). Then w is unitary. This algebra is called (c) Q = ( 01 −q 0 ), q ∈ R \ {0}. Poly(SUq (2)).
The only equivalence among them is Poly(SL1 (2, R)) ∼ = Poly(SU1 (1, 1)). For the non-standard deformation SLt=1 (2) there is only one Hopf ∗-algebra 0 ). structure (up to equivalence), namely for Q = ( 10 −1 Except for (c), the above corepresentations w are not equivalent to unitary ones (the above examples were given in [20, 16, 23]). Ideas of the proof. Since the map “∗” is an antilinear comultiplicative algebra antiautomorphism, by Lemma 4.1, part (a) there is a matrix Q ∈ GL(2, C) such that w ¯ = QwQ−1 . By part (c) of Lemma 4.1, the map “∗” can be an algebra antiautomorphism if and only if Q satisfies the condition (Q−1 ⊗ Q−1 )τ E¯ = cE for some c ∈ C \ {0}. The equation ∗2 = idH is equivalent to ¯ = d12 QQ with d ∈ C \ {0}. Q is determined up to the equivalence relation as in Lemma 4.1, part (d). Consider the standard quantum deformations SLq (2), q 6= 1, first. From the relations (13) it follows that there are only the following characters (algebra homomorphisms) χ : H → C: a 0 0 a 0 , and in addition to that for q = −1 : χ (w) = χa (w) = a 0 a−1 a−1 0 where a ∈ C \ {0} (Relations (13) are equivalent to w11 w12 = qw12 w11 ,
w11 w21 = qw21 w11 ,
w21 w22 = qw22 w21 ,
w12 w22 = qw22 w12 ,
w12 w21 = w21 w12 ,
w11 w22 − qw12 w21 = w22 w11 − q −1 w12 w21 = 1 , and the numbers χ(wij ) should satisfy the same relations). Now the following trick can be used in order to compute all possible ∗-structures: If χ is a character, then the map χ# : x 7→ χ(x∗ ) is also a character, because C is commutative. Then for any a ∈ C \ {0} there exists b ∈ C \ {0} such that χ# a = χb or (for 0 = χ . Applying both sides to w, we get that Q is a diagonal or q = −1) χ# a b antidiagonal matrix. Similarly (use χ 7→ χ ◦ φ), isomorphisms φ of Hopf algebras are given by diagonal or (q = −1) antidiagonal matrices. Then we use the other conditions for Q and part (d) of Lemma 4.1.
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For the non-standard deformation SLt=1 (2) split E into E sym and E asym as in the proof of Theorem 3.2. Then consider Q with respect to both. For q = 1, equivalent Q’s can be regarded as matrices of the same antilinear mapping j such that j 2 = d · id (j is equivalent to kj for some k ∈ C \ {0}). Then d = 1 corresponds to (a), (b) while d = −1 to (c). Remark 4.3. [16, 9] (a) There exist the following ∗-structures on SLq (N ): (i) For |q| = 1 you can choose w ¯ = w. The corresponding quantum group is called SLq (N, R). (ii) If q is real then for 1 , . . . , N ∈ {±1} there are ∗-structures such that w∗ Bw = wBw∗ = B, where B is a diagonal matrix with diagonal elements 1 , . . . , N . The corresponding quantum group is called SUq (N ; 1 , . . . , N ). For 1 = · · · = N = 1 we get the quantum group SUq (N ), in which w is a unitary corepresentation. (b) There are also ∗-structures on the orthogonal and symplectic quantum groups. 5. Compact Hopf ∗-algebras In this chapter we follow [21, 22, 7]. Let A be a Hopf ∗-algebra. Definition 5.1. A is called compact if there are unitary corepresentations such that their matrix elements generate A as algebra. Example. The fundamental corepresentation of Poly(SUq (N )) is unitary and generates Pol(SLq (N )) as algebra. Lemma 5.2. Let A be a compact Hopf ∗-algebra. (a) The matrix elements of unitary corepresentations span A. (b) Let v be a unitary corepresentation. Then v is equivalent to a direct sum of irreducible unitary corepresentations. (c) The matrix elements of non-equivalent irreducible unitary corepresentations form a linear basis of A. (d) Each irreducible corepresentation is equivalent to a unitary one. (e) Each corepresentation is completely reducible (into irreducible ones). Since the irreducible corepresentations are equivalent to unitary corepresentations, each corepresentation is equivalent to a unitary corepresentation. Proof. (a) By definition, A is spanned by matrix elements of tensor products of unitary corepresentations, but tensor products of unitary corepresentations are unitary. (b) Proof by induction with respect to the dimension d of corepresentations. If d = 1 or the corepresentation is irreducible, then there is nothing to do. Now assume that the corepresentation v is not irreducible. Then choose an orthonormal
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basis of an invariant proper subspace L and add some more orthonormal elements in order to get an orthonormal basis B of Cdim v . The transition from the standard basis to B is unitary and intertwines v with a unitary corepresentation A B =: w , 0 C where A, B, C are matrices of suitable size and with at least one entry. Since w is unitary, w ¯ = wc or S(w) = w∗ or equivalently ! A∗ 0 S(A) S(B) = . B∗ C ∗ 0 S(C) Therefore B = 0, moreover A and C are unitary and w a direct sum of them (notice that L⊥ is also invariant and C = w|L⊥ ). By induction hypothesis, the corepresentations A and C of dimensions less than d are direct sums of irreducible unitary corepresentations, whence w is a direct sum of irreducible unitary corepresentations. (c) This follows from (a), (b), and Theorem 2.22, part (b). (d) and (e) follow from Theorem 2.22, part (c). Remark. All irreducible corepresentations can be obtained by decomposition of tensor products of those unitary corepresentations which generate A as algebra (cf. Lemma 5.2, part (a)). Peter–Weyl Theory and Haar measure Let A be a compact Hopf ∗-algebra. Let I be an index set and let {uα | α ∈ I} be a complete set of non-equivalent irreducible unitary corepresentations. Let I := u0 be the one dimensional corepresentation. Then the elements uα mn form a basis of A (Lemma 5.2, part (c)). Definition 5.3. The Haar measure is a linear functional on A defined by h(uα mn ) = δα,0 . Since the uα mn are matrix elements of corepresentations, for all x ∈ A the Haar measure satisfies the equations (h ⊗ idA )∆(x) = (idA ⊗ h)∆(x) = h(x)1, h(1) = 1 .
(18)
(By definition, also h(S(x)) = h(x) holds for all x ∈ A.) In order to compute h on products, some preparation is necessary. Lemma 5.4. For each α ∈ I there is a strictly positive definite matrix Fα such that (uα )cc = Fα uα Fα−1 . Proof. For each α ∈ I, the matrix uα is also a corepresentation and equivalent β β c β to a unitary one, say uβ : Qα uα Q−1 α = u . Then u = (u ) and
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(uα )cc = (uα )c β c T β c −1 T = (Q−1 α u Qα ) = Qα (u ) (Qα ) T T α −1 −1 T = QTα uβ (Q−1 α ) = Qα Qα u Qα (Qα )
and therefore (uα )cc = Fα uα Fα−1 , where Fα = QTα (QTα )∗ is a strictly positive definite matrix. Fix an irreducible corepresentation v and let n := dim(v). Since S(v) is the inverse of v, there are intertwiners AI = (v ⊗ v c )A , where A =
Pn
k=1 ek
⊗ ek and B =
Pn
B(v c ⊗ v) = IB ,
0 k=1 ek
⊗ e0k .
Lemma 5.5. Let v and w be irreducible corepresentations of dimensions n and m respectively. Then (a) Mor(v c ⊗ w, I) ∼ = Mor(w, v), Mor(v c ⊗ v, I) = CB. c ∼ (b) Mor(I, w ⊗ v ) = Mor(v, w), Mor(I, v ⊗ v c ) = CA. Proof. (a) If X intertwines v c ⊗ w with I then X(v c ⊗ w) = IX and (1n ⊗ X)(A ⊗ 1m )(I ⊗ w) = (1n ⊗ X)(v ⊗ v c ⊗ w)(A ⊗ 1m ) = (v ⊗ I)(1n ⊗ X)(A ⊗ 1m ) . Since I ⊗ w ∼ = w and v ⊗ I ∼ = v, (1n ⊗ X)(A ⊗ 1n ) can be regarded as intertwiner of w and v. Conversely, let Y ∈ Mor(w, v). Then Y w = vY and B(1n ⊗ Y )(v c ⊗ w) = B(v c ⊗ v)(1n ⊗ Y ) = IB(1n ⊗ Y ) . Therefore B(1n ⊗Y ) intertwines v c ⊗w with I. The maps between Mor(v c ⊗w, I) and Mor(w, v) are inverses of each other because (1n ⊗B)(A⊗1n ) = (B ⊗1n )(1n ⊗A) = 1n . The second statement follows from the first with Schur’s Lemma 2.20. (b) is proved in a similar way. Now the Haar measure is computed on certain products of basis elements: Theorem 5.6. The Haar measure satisfies the Peter–Weyl–Woronowicz relations: (Fα )ln δmj β ∗ (19) = δα,β h uα mn ujl T r(Fα ) and
∗ (Fα−1 )mj δln h uβjl uα mn = δα,β T r(Fα−1 )
for all α, β ∈ I, 1 ≤ m, n ≤ dim(uα ), 1 ≤ j, l ≤ dim(uβ ).
(20)
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Proof. Let w be any corepresentation (of dimension N ). Application of h ⊗ id and id ⊗ h to ∆wij yields together with Eq. (18) h(w)w = wh(w) = h(w)1 . This matrix equation means N X
h(w)ik wkj = h(w)ij 1 =
k=1
N X
wik h(w)kj
k=1
or equivalently that for each i the row vector with coordinates h(w)ij for j = 1, . . . , N intertwines w with I and for each j the column vector with coordinates h(w)ij for i = 1, . . . , N intertwines I with w. These facts will be applied to the sets Mor(I, uα ⊗ uβ c ) and Mor(uα cc ⊗ uα c , I) for α, β ∈ I. βc Therefore for w = uα ⊗uβ c and for fixed indices k, l, the element (h(uα ik ujl ))1≤i,j≤N is in Mor(I, uα ⊗ uβ c ). By Lemma 5.5 it vanishes for α 6= β and is a multiple of A for α = β. Thus there are numbers λα kl ∈ C such that β h uα ik ujl
c
= δα,β λα kl δij
(21)
for all i, j, k, l. Similarly, for w = uα cc ⊗ uα c and for fixed indices i, j, the element cc α c α cc ⊗ uα c , I) = CB. Therefore there are numbers (h(uα ik ujl ))1≤k,l≤N is in Mor(u α %ij ∈ C such that cc α c α (22) h(uα ik ujl ) = %ij δkl . But from Lemma 5.4, uα = Fα−1 (uα )cc Fα , which yields by linearity and Eq. (22) the equation X cc α c αc h(uα (Fα−1 )mi h(uα mn ujl ) = ik ujl )(Fα )kn i,k
= (Fα )ln
X
(Fα−1 )mi %α ij .
i c
Comparison with Eq. (21) and uβjl = (uβjl )∗ yields β h uα mn ujl
∗
= cα δα,β (Fα )ln δmj
for some cα ∈ C. These constants can be evaluated using the unitarity of uα : X X α ∗ 1 = h(1) = h(uα (Fα )nn = cα T r(Fα ) . mn umn ) = cα n
n
The trace of Fα is positive because Fα is positive definite. This proves Eq. (19). The other equation is proved in a similar way. Remark 5.7. (a) Since the matrices Fα can be scaled by a positive number, we normalize them by the condition T r(Fα ) = T r(Fα−1 ). After normalization they are uniquely determined.
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(b) Example: In the standard deformation SUq (2) for q ∈ R \ {0}, ! |q|−1 0 F0 = (1) , F1/2 = . 0 |q| −1
δ −q Proof. w1/2 = w = ( αγ βδ ), and S(w) = ( −qγ α
wcc = S 2 (w) =
α q2 γ
q −2 β δ
β
). Then
! −1 = F1/2 wF1/2
where F1/2 is as desired. Note that the absolute value of q must be used, because the eigenvalues of a positive definite matrix must be positive. Theorem 5.8 (Positivity of the Haar measure). For all x ∈ A, h(x∗ x) ≥ 0, and equality only holds for x = 0. α Proof. Since A has a basis {uα mn | 1 ≤ m, n ≤ dim(u ), α ∈ I}, a general element a of A can be written as X α aα a= mn umn . m,n,α
By the second Peter–Weyl–Woronowicz relation (20) h(a∗ a) =
−1 X (aα ¯α mp (Fα )mn a np ) , T r(F ) α α,m,n,p
P −1 in which the sums m,n aα ¯α mp (Fα )mn a np are strictly positive unless all coefficients α amp for fixed α, p vanish, because the matrices Fα−1 are strictly positive definite for all α. Corollary 5.9 (Scalar product). There is a scalar product on A defined by (a|b) := h(a∗ b) for all a, b ∈ A. Proof. This inner product is antilinear in the first argument and linear in the second argument by definition and positive definite by Theorem 5.8. Corollary 5.10 (Modular Homomorphism). There is a uniquely determined algebra automorphism σ of A such that h(ab) = h(bσ(a)) for all a, b ∈ A. It is defined on elements of the basis as α σ(uα mn ) = (Fα u Fα )mn .
Proof. Uniqueness: Let a be an element of A and let a0 , a00 ∈ A such that for all b ∈ A the equation h(ab) = h(ba0 ) = h(ba00 )
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holds. Then h(b(a0 − a00 )) = 0 for all b ∈ A, whence a0 = a00 by Corollary 5.9. Existence: From the second Peter–Weyl–Woronowicz relation it follows that ∗ β h(uα jl σ(umn )) =
δα,β (Fα )ln δmj ∗ = h uβmn uα . jl T r(Fα )
Therefore by linearity h(ab) = h(bσ(a)) for all a, b ∈ A. Moreover F0 = (1) implies σ(1) = 1, and for all a, b, c ∈ A, h(aσ(bc)) = h(bca) = h(caσ(b)) = h(aσ(b)σ(c)) . Therefore σ is a unital algebra homomorphism. Since Fα is invertible, also σ is invertible. C∗ -structure For any Hilbert space H let (.|.)H denote the inner product and B(H) the set of bounded linear operators on H. Then B(H) is a ∗-algebra. Let A be a compact Hopf ∗-algebra and consider the set Π := {π : A → B(H) | H Hilbert space, π unital ∗ -homomorphism} (it is enough to consider some fixed H with dim(H) ≥ dim(A) as cardinal numbers, thus Π is actually a set). Fix π ∈ Π and let H be the corresponding Hilbert space. Let uα be a unitary corepresentation of A. Then π(uα ) is a unitary matrix in Mdim uα (B(H)) and X
∗ α π(uα mn ) π(umn ) = 1
m
for all n ≤ dim uα . Therefore for all x ∈ H and k ≤ dim uα X ∗ α (x | x)H = (π(uα mn ) π(umn )x | x)H m
=
X
α α α (π(uα mn )x | π(umn )x)H ≥ (π(ukn )x | π(ukn )x) ,
m
whence the operator norm kπ(uα kn )k is at most 1, and for each a = ∈ A there is the inequality X |aα kπ(a)k ≤ mn | < ∞ .
P
α α α,m,n amn umn
α,m,n
Therefore the following definition is possible: Definition-Lemma 5.11. There is a norm k.kC ∗ on A such that for all a ∈ A, kakC ∗ = sup kπ(a)k . π∈Π
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Moreover this norm satisfies the equations kabkC ∗ ≤ kakC ∗ kbkC ∗ , ka∗ kC ∗ = kakC ∗ , ka∗ akC ∗ = kak2C ∗ for all a, b ∈ A. Proof. The main problem is to show kakC ∗ = 0 ⇒ a = 0 for a ∈ A. The inner product (. | .) on A induces a norm k.k(.|.) (cf. Corollary 5.9). For each a ∈ A let π0 (a) denote the operator of left multiplication by x on A. Then for all x ∈ A ! X ∗ X α ∗ α 2 kπ0 (uα umn umn x mn )(x) k(.|.) = h x m m {z } | =1
= h(x∗ x) = kxk2(.|.) , whence the operator norm k.k0(.|.) of π0 (uα mn ) is at most 1. For all a = α α amn umn ∈ A X |aα kπ0 (a)k0(.|.) ≤ mn | .
P α,m,n
α,m,n
Therefore for each a ∈ A the operator π0 (a) is bounded on A and can be extended to the completion H of A with respect to the norm k.k(.|.) as a bounded linear operator π0 (a)k0(.|.) := kπ0 (a)k0(.|.) . Therefore π ¯0 ∈ Π, and π ¯0 (a) with same operator norm k¯ kakC ∗ = 0 ⇒ kπ0 (a)k0(.|.) = 0 ⇒ kπ0 (a)1k(.|.) = 0 ⇒ kak(.|.) = 0 ⇒ a = 0 . The other properties of this norm follow from the corresponding properties of the operator norms of the representations in Π. Definition 5.12. Let A be the closure of A with respect to the norm k.kC ∗ . Then A is a C ∗ -algebra by Definition-Lemma 5.11. The following properties of C ∗ -algebras are useful: Proposition 5.13. Let A be a C ∗ -algebra. Then (a) There is a Hilbert space H such that A can be embedded as closed ∗-subalgebra into B(H) [2, 2.6.1]. (b) Let B be another C ∗ -algebra. Then each ∗-homomorphism from A to B is continuous [2, 1.3.7]. The comultiplication of A can be extended to a ∗-homomorphism from A to b where A⊗A b denotes the (topological) tensor product of C ∗ -algebras, defined A⊗A, as follows: Let H be a Hilbert space and let ι : A → B(H) be an embedding of b is identified with the closure of (ι ⊗ ι)(A ⊗ A) in B(H ⊗H), b C ∗ -algebras. Then A⊗A ∗ b is the (topological) tensor product of Hilbert spaces. The C -algebra where H ⊗H b does not depend (up to isomorphisms) on the embedding ι [2, 2.12.15]. The A⊗A map ∆ b A −→ A ⊗ A ,→ B(H ⊗H)
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b is a Hilbert space, π1 belongs to Π is a ∗-homomorphism called π1 . Since H ⊗H and can be extended to a ∗-homomorphism on A. It is again called ∆. Definition 5.14. A compact matrix quantum group is a pair (A, ∆) or shortly A where: (a) A is a unital C ∗ -algebra generated by some elements uij ∈ A for 1 ≤ i, j ≤ N and some positive integer N , P b is a unital ∗-homomorphism such that ∆(uij ) = N (b) ∆ : A → A⊗A k=1 uik ⊗ ukj for all i, j, (c) the matrices u and u¯ are invertible. Remark 5.15. (a) Let A be a Hopf ∗-algebra generated as unital algebra by matrix elements of one unitary corepresentation u or (equivalently) generated as unital ∗-algebra by matrix elements of a corepresentation v such that v and v¯ are equivalent to unitary corepresentations. Then the C ∗ -algebra constructed as above is a compact matrix quantum group. (b) For all positive integers N the compact Hopf ∗-algebra of SUq (N ) gives rise to a compact matrix quantum group. (c) The general example of a compact matrix quantum group comes from C ∗ algebras A as in (a) after dividing by closed two-sided ideals I ⊂ {x ∈ A : h(x∗ x) b = 0} such that ∆ induces a ∗-homomorphism A/I → A/I ⊗A/I. Theorem 5.16. Let A be a compact matrix quantum group constructed as in Remark 5.15, part (c). (a) Then |h(x)| ≤ kxkC ∗ for all x ∈ A, therefore h can be extended to a (positive) continuous functional on A, which will be denoted by h again. (b) The algebra A is embedded into A (because for all x ∈ A \ {0} the inequality h(x∗ x) > 0 holds). P (c) Any corepresentation of A (in the sense ∆vab = c vac ⊗ vcb , v −1 exists) has matrix elements in A and thus A can be recovered from A as the span of matrix elements of corepresentations. Remark 5.17. For I1 := {x ∈ A : h(x∗ x) = 0} (it is a closed two-sided ideal due to [21, p. 656]), h is faithful on A/I1 (i.e. h(x∗ x) = 0 ⇒ x = 0), while for I2 := {0}, is continuous on A/I2 ∼ = A. In the case of SUq (2), I1 and I2 coincide, cf. [10, Remark 6]. The notion of compact matrix quantum groups generalizes that of algebras of continuous functions on compact groups of matrices. To be more precise: Let G be a compact group of matrices. Then there is a Haar measure µ on G. There is an inner product on C(G) given by Z χψdµ ¯
(χ, ψ) := G
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for χ, ψ ∈ C(G). The algebra Poly(G) as in Definition 2.3 is a compact Hopf ∗-algebra (cf. proof of Lemma 2.4). The inner product as above can also be expressed as h(χ∗ ψ). Therefore the completion of Poly(G) with respect to the norm k.k(.|.) is the same as L2 (G), and the completion of Poly(G) with respect to the norm k.kC ∗ is ∼ b the same as C(G). Here the comultiplication ∆ : C(G) → C(G)⊗C(G) = C(G × G) is given by ∆(χ)(g, h) = χ(gh) for all g, h ∈ G and χ ∈ C(G) (cf. Chap. 2). In the following, each compact topological space is by definition a Hausdorff space. There are one-to-one correspondences induced by Gel’fand’s theorem: compact topological spaces X ←→ unital commutative C ∗ -algebras C(X) continuous mappings λ : X → Y ←→ unital ∗ -homomorphisms λ∗ : C(Y ) → C(X) b cartesian product X × Y ←→ topological tensor product C(X)⊗C(Y ) compact group of matrices G ←→ compact matrix quantum group C(G) for commutative A = Poly(G)
6. Actions on Quantum Spaces Definition and spectral decomposition [10, Sec. 1] This chapter deals with a topological counterpart of right comodule algebras. Let V be a topological vector space and Z ⊂ V a subset. Then hZi denotes the closure of the linear span of the elements of Z in V . Definition 6.1. Let (A, ∆) be a compact matrix quantum group and B a unital b is called a coaction for A C ∗ -algebra. The unital ∗-homomorphism Γ : B → B ⊗A on B if (a) (Γ ⊗ idA )Γ = (idB ⊗ ∆)Γ, (b) B ⊗ A = h(idB ⊗ y)Γ(x)|x ∈ B, y ∈ Ai. Remark 6.2. (a) Let G be a compact group of matrices, X a compact topological space and X × G → X, (x, g) 7→ xg for x ∈ X and g ∈ G, an action. Then there is a coaction Γ : C(X) → C(X × G) given by Γ(χ)(x, g) = χ(xg) for all χ ∈ C(X), g ∈ G, x ∈ X. The properties x(gh) = (xg)h and xe = x for all g, h ∈ G, x ∈ X correspond to conditions (a) and (b) in Definition 6.1, respectively. Given a coaction as in Definition 6.1 for commutative A and B, the group action can be recovered by Gel’fand’s theorem. (b) Quantum analogues of left actions are considered in [10, Remark 7]. Theorem 6.3. Let A be a compact matrix quantum group, B a unital C ∗ -algebra and Γ a coaction. Then there exists a maximal ∗-subalgebra B of B such that B is dense in B and an A right comodule algebra, i.e. for γ := Γ|B : γ(B) ⊂ B ⊗ A ,
(γ ⊗ id)γ = (id ⊗ ∆)γ ,
(id ⊗ )γ = id .
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For each α ∈ I there is a set Iα such that the algebra B has a basis eαrk for α ∈ I, r ∈ Iα , 1 ≤ k ≤ dim(uα ) such that X eαrs ⊗ uα Γ(eαrk ) = sk . s
Idea of proof (cf. [10, Theorem 1.5]). From the Peter–Weyl–Woronowicz relation (20) it follows that there are elements xα sm ∈ A which span A such that the continuous linear functionals α %α sm : A → C, x 7→ h(xsm x) β satisfy %α sm (ukr ) = δα,β δsk δmr . Then the operators α Esm = (idB ⊗ %α sm )Γ : B → B P α have properties of matrix units. The traces s Ess are projections onto subspaces Wα ⊆ B which contain all elements x ∈ B such that
Γ(x) ⊂ B ⊗ (⊕ik Cuα ik ) . Construction of the basis: For each α ∈ I let {eαr1 | r ∈ Iα } be a basis of the vector α α ) and eαrs := Es1 (eαr1 ). Let B denote the linear span of all elements space Im(E11 eαrs . Then the closure of B is α α hEsm (x) | x ∈ B, α, s, mi = hEsm (x) | x ∈ B, α, s, mi
= h(id ⊗ h)(id ⊗ xα sm )Γ(x) | x ∈ B, α, s, mi = h(id ⊗ h)h(id ⊗ y)Γ(x) | y ∈ A, x ∈ Bii b = h(id ⊗ h)(B ⊗A)i =B. Definition 6.4. Let a compact matrix quantum group A coact by Γ on a quantum space B. (a) For each α ∈ I, the number cα denotes the cardinality of Iα as in Theorem 6.3 and is called “multiplicity of uα in the spectrum of Γ”. (b) For each α ∈ I let Wα be the linear span of the elements eαrs as in Theorem 6.3. Quantum spheres [8] Since the quantum groups SUq (2) and SU1/q (2) are isomorphic by Theorem 3.2, we can restrict ourselves to the case q ∈ [−1, 1]\{0}. For the quantum SU (2) groups, I is the set of non-negative half integers and uk = wk for k ∈ I. We want to classify coactions Γ of SUq (2) such that ( 1 if k ∈ N0 (1) ck = 0 if k ∈ N0 + 12 ,
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(2) the subspaces W0 and W1 generate B as a C ∗ -algebra. The pairs (B, Γ) are called “quantum spheres” (cf. the case q = 1 in Theorem 6.5 below). For convenience, the matrix elements of the unitary irreducible corepresentations of SUq (2) will be indexed by numbers in the index set Nα := {−α, −α + 1, . . . , α} instead of the index set {1, . . . , 2α + 1} for each α ∈ 12 N0 . Theorem 6.5 [8]. In the case q = 1 there is only one object B = C(S 2 ) and the coaction is induced by the standard right action of SU (2) on the sphere S 2 . Here W0 = C 1 and W1 = Cx + Cy + Cz. Then condition (2) means that the coordinates x, y, z separate the points of S 2 by the Stone–Weierstrass theorem. In the case q = −1 there is only one object B−1,0 with coaction Γ−1,0 . In the case −1 < q < 1 and q 6= 0 there are — up to isomorphisms — the ∗ following quantum spaces Bqc for c ∈ R+ 0 ∪ {∞}. The C -algebra Bqc is generated by the elements e−1 , e0 , e1 of W1 subject to the relations e∗i = e−i f or i ∈ {−1, 0, 1} , (1 + q 2 )(e−1 e1 + q −2 e1 e−1 ) + e20 = %1 , e e − q 2 e e = λe 0 −1
−1 0
−1
(1 + q 2 )(e−1 e1 − e1 e−1 ) + (1 − q 2 )e20 = λe0 , e1 e0 − q 2 e0 e1 = λe1 , where
( λ=
1 − q2
if c ∈ R
0
if c = ∞
( and % =
(23)
(1 + q 2 )2 q −2 c + 1 if c ∈ R (1 + q 2 )2 q −2
if c = ∞ .
The coaction Γqc is given by Γ(ei ) =
1 X
ej ⊗ u1ji
j=−1
for i ∈ {−1, 0, 1}. Here we choose a non-unitary form δ2 −(q 2 + 1)δγ −qγ 2 αγ . u1 = −q −1 βδ 1 + (q + q −1 )βγ (q + q −1 )βα α2 −q −1 β 2 Ideas of proof. Due to Theorem 6.3 and condition (1), the algebra B has the linear basis {eαk | α ∈ N0 , k ∈ Nα } such that X Γ(eαk ) = eαs ⊗ uα sk for α ∈ N0 , k ∈ Nα . s∈Nα
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Therefore the eαk ’s are analogues of spherical harmonics. One has (u1lk )∗ = u1−l,−k . Then X X e∗−l ⊗ (u1−l,−k )∗ = e∗−l ⊗ u1lk . Γ(e∗−k ) = l
l
From the irreducibility of u it follows that there is a constant c such that e∗−k = cek for all k. Moreover the modulus of c is one because of ek = (e∗k )∗ = (ce−k )∗ = c¯ cek . Thus it is possible to achieve c = 1 by scaling the elements ek with a suitable complex number of modulus one. Now consider products of the generators: Because of the Clebsch–Gordan relation u1 ⊗ u1 ∼ = u0 ⊕ u1 ⊕ u2 there are injective intertwiners Gα ∈ Mor(uα , u1 ⊗ u1 ) for α ∈ {0, 1, 2}. From the equation 1
Γ(ek el ) =
X
em er ⊗ u1mk u1rl
m,r
it follows for the elements e˜α,t := Γ(˜ eα,t ) =
P
α k,l ek el Gkl,t :
X
em er ⊗ u1mk u1rl Gα kl,t
k,l,m,r
=
X X n
|
m,r
! ⊗uα nt .
em er Gα mr,n {z = e˜α,n
}
Therefore the elements e˜α,t satisfy the same relations for the coaction as the elements ek . Since the corepresentations uα are irreducible, there are constants λα ∈ C such that e˜α,t = λα eα,t . For α ∈ {0, 1} this gives relations for the generators: X
ek el G1rl,t = λet (here λ = λ1 ) ,
k,l
X
ek el G0rl,0 = %1 (here % = λ0 ) .
k,l
These are the relations (23) for the quantum spheres. Applying “∗” to both sides, we obtain that λ and % are real. There is still the freedom of scaling the ek ’s by a non-zero real number. Consider the case 0 < |q| < 1. If λ does not vanish, it can be scaled to the value λ = 1 − q 2 . Then define c by % = (1 + q 2 )2 q −2 c + 1 . The existence of a faithful C ∗ -norm on B implies that c is a non-negative number. It remains λ = 0, % positive (B is a C ∗ -algebra). Then % can be scaled to the value (1 + q 2 )2 q −2 . These (B, Γ)’s are indeed quantum spheres. No extra relation can be imposed, because then we would get a coaction for a quantum subspace. But c0 = 1 means
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that the space is homogeneous (cf. [10, Definition 1.8]), and from the facts that h is faithful (i.e. h(x∗ x) = 0 ⇒ x = 0) and the counit is continuous (cf. Remark 5.17) it follows here that the homogeneous space corresponding to B has no non-trivial homogeneous subspaces (this idea stands behind the proof in the paper [8]). The case q = 1 can be handled similarly, and the case q = −1 reduces to q = 1. Remark 6.6. (a) If the first condition for the quantum spheres is weakened to c0 = c1 = 1, there are some more homogeneous spaces for c ∈ {c(2), c(3), . . . }, 0 < |q| < 1, where c(n) = −q 2n /(1 + q 2n )2 for all n ∈ N . These objects satisfy the conditions ( ck =
1
if k = 0, 1, . . . , n − 1
0
otherwise .
There exist analogues of these objects in the case q = 1. They correspond (cf. [8]) to the adjoint action of SU (2) on U (su(2)) taken in its n-dimensional irreducible ∗-representation (X ∗ = −X for X ∈ su(2)). 2 (b) For 0 < |q| < 1, c ∈ R+ 0 ∪ {∞} ∪ {c(2), c(3), . . . } the quantum sphere Sqc = (Bqc , Γqc ) is a quotient space if and only if c = 0, embeddable (i.e. can be regarded as a non-zero C ∗ -subalgebra of A, where Γ is induced by the comultiplication) if and only if c ∈ [0, ∞], and homogeneous for all considered c (for the compact groups of matrices these three notions coincide). (c) An algebraic version of Theorem 6.5 can be found in [17]. 7. Quantum Lorentz Groups (cf. [26]) The algebra A = Poly(SL(2, C)) is called the algebra of polynomials on the Lorentz group. Its corepresentations have the following properties (cf. Chap. 3): (1) There are irreducible corepresentations wα for α ∈ 12 N0 such that all nonequivalent irreducible corepresentations are wα ⊗ wβ for α, β ∈ 12 N0 . (2) dim(wα ) = 2α + 1 for all α, (3) wα ⊗ wβ ∼ = w|α−β| ⊕ w|α−β|+1 ⊕ · · · wα+β (Clebsch Gordan), (4) Each corepresentation is completely reducible, or equivalently, the matrix β ∗ α (wkl ) give a basis of A. elements wij 1 (5) For all α, β ∈ 2 N0 the corepresentations wα ⊗wβ and wβ ⊗wα are equivalent. Definition 7.1. A quantum Lorentz group is a Hopf ∗-algebra A satisfying properties (1) – (5). Theorem 7.2. Up to isomorphisms, all quantum Lorentz groups A are given as follows: The Hopf ∗-algebra A is generated by the matrix elements wij (1 ≤ i, j ≤ 2) of the fundamental corepresentation w := w1/2 and relations
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(i) (w ⊗ w)E = E, (ii) E 0 (w ⊗ w) = E 0 , (iii) X(w ⊗ w) ¯ = (w¯ ⊗ w)X, where the base field C is canonically embedded into A, the vectors E 0 ∈ C2 ⊗ C2 and E ∈ 2 C ⊗ 2 C are the same as in Theorem 3.2 and X ∈ M4 (C) satisfies the properties: (iv) X is invertible, ¯ = cX, (v) there is a scalar factor c ∈ C \ {0} such that τ Xτ ¯ w ¯ ⊗ w ⊗ w) (vi) the intertwiners 12 ⊗ E and (X ⊗ 12 )(12 ⊗ X)(E ⊗ 12 ) in Mor(w, 0 0 ∼ ∼ ¯=w ¯ ⊗ w ). are proportional (note that w ¯ =w ⊗w Idea of proof. Necessity of relations: Restrict attention to the corepresentations wα first. Their matrix elements give a basis of a quantum SL(2)-group H as in Theorem 3.2. This shows conditions (i) and (ii) and gives E and E 0 . From assertions (1) and (4) it follows that there is a linear isomorphism A∼ = H · H∗ ∼ = H ⊗ H∗ , α β α β α β (wmn )∗ 7→ wkl · (wmn )∗ 7→ wkl ⊗C (wmn )∗ , wkl
where “·” denotes multiplication. Assertion (5) for α = β = 12 shows that there is a bijective intertwiner X ∈ Mor(w ⊗ w, ¯ w ¯ ⊗ w), which gives conditions (iii) and (iv). Apply the map “∗” to (iii) and use the formula v ⊗ w = τ (w¯ ⊗ v¯)τ as in the proof of Lemma 4.1, part (c): ¯ ⇒ Xτ ¯ (w ⊗ w)τ ¯ ¯ ⊗ w) ¯ = (w ¯ ⊗ w)X ¯ = τ (w¯ ⊗ w)τ X X(w ¯ (w ⊗ w) ¯ . ε ⇒ τ Xτ ¯ = (w ¯ ⊗ w)τ Xτ ¯ must be Since w ¯ ⊗ w and w ⊗ w ¯ are irreducible, the intertwiners X and τ Xτ proportional, which gives condition (v). The last condition follows, because both 12 ⊗ 12 and X(12 ⊗ 12 ⊗ E 0 )(12 ⊗ X ⊗ 12 )(E ⊗ 12 ⊗ 12 ) are elements of Mor(w ¯ ⊗ w). Existence: We set A := H ⊗ H∗ with H as in Theorem 3.2 and laborously introduce the Hopf ∗-algebra stucture on A by means of (iii) – (vi). β ∗ α (wkl ) linearly Sufficiency of relations: More relations would make the elements wij dependent. Remark 7.3. (a) Possible matrices X have been found (up to isomorphisms of the corresponding Hopf ∗-algebras) in [26]. (b) There is also a topological structure for two examples of A ([13, 25]) which uses the notion of affiliated elements [24]. (c) Quantum Poincar´e groups arise by adding translations [14]. (d) Quantum analogues of Poly(SL(N, C)) were considered in [9] (cf. [27]).
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References [1] E. Abe, Hopf Algebras, Cambridge Univ. Press, 1980. [2] J. Dixmier, C ∗ -Algebras, North Holland, 1977. [3] V. G. Drinfeld, “Quantum groups”, Proc. ICM-1986, Berkeley, AMS, 1987, pp. 798– 820. [4] M. Dubois-Violette and G. Launer, “The quantum group of a non-degenerate bilinear form”, Phys. Lett. B 245 (1990), 175–177. [5] T. Hayashi, “Quantum deformation of classical groups”, Publ. RIMS, Kyoto 28 (1992) 57–81. [6] P. Kondratowicz and P. Podle´s, “On representation theory of quantum SLq (2) groups at roots of unity”, hep-th/9405079, Quantum Groups and Quantum Spaces, eds. R. Budzy´ nski et al., Banach Center Publ. 40 (1997), 223–248, Inst. of Math., Polish Acad. Sci. [7] T. H. Koornwinder, “General compact quantum groups, a tutorial”, hep-th/9401114, in Representations of Lie Groups and Quantum Groups, eds. V. Baldoni and M. A. Picardello, Pitman Res. Notes Math. Ser. 311, Longman Scientific & Technical, 1994, pp. 46–128. [8] P. Podle´s, “Quantum spheres”, Lett. Math. Phys. 14 (1987) 193–202. [9] P. Podle´s, “Complex quantum groups and their real representations”, Publ. RIMS, Kyoto 28 (1992) 709–745. [10] P. Podle´s, “Symmetries of Quantum Spaces. Subgroups and Quotient Spaces of Quantum SU (2) and SO(3) Groups”, Commun. Math. Phys. 170 (1995) 1–20. [11] P. Podle´s, “The Dirac operator and gamma matrices for quantum Minkowski spaces”, q-alg/9703014, J. Math. Phys. 38 (1997) 4474–4491. [12] B. Parshall and J. P. Wang, “Quantum linear groups”, Mem. AMS 439 (1991) AMS. [13] P. Podle´s and S. L. Woronowicz, “Quantum Deformation of Lorentz group”, Commun. Math. Phys. 130 (1990) 381–431. [14] P. Podle´s and S. L. Woronowicz, “On the classification of Quantum Poincar´e Groups”, Commun. Math. Phys. 178 (1996) 61–82 and references therein. [15] M. Rosso, “Alg`ebres enveloppantes quantifi´ees, groupes quantiques compacts de matrices et calcul diff´erentiel non commutatif”, Duke Math. J. 61(1) (1990) 11–40. [16] N. Yu. Reshetikhin, L. A. Takhtadzyan and L. D. Faddeev, “Quantization of Lie groups and Lie algebras”, Leningrad Math. J. 1(1) (1990) 193–225. [17] J. Apel and K. Schm¨ udgen, “Classification of three-dimensional covariant differential calculi on Podle´s’ quantum spheres and on related spaces”, Lett. Math. Phys. 32 (1994) 25–36. [18] M. Takeuchi, “Quantum orthogonal and symplectic groups and their embedding into quantum GL”, Proc. Japan Acad. 65 (1989), Series A, No. 2, 55–58. [19] M. Takeuchi, “Some Topics on GLq (n)”, J. Algebra 147 (1992) 379–410. [20] S. L. Woronowicz, “Twisted SU (2) group. An example of a non-commutative differential calculus”, Publ. RIMS, Kyoto 23 (1987) 117–181. [21] S. L. Woronowicz, “Compact matrix pseudogroups”, Commun. Math. Phys. 111 (1987) 613–665. [22] S. L. Woronowicz, “Tannaka–Krein duality for compact matrix quantum groups. Twisted SU (N ) groups”, Invent. Math. 93 (1988) 35–76. [23] S. L. Woronowicz, “New quantum deformation of SL(2, C). Hopf algebra level”, Rep. Math. Phys. 30 (1991) 259–269; cf. also Yu. I. Manin, Le¸cons Coll´ege de France (1989); E. E. Demidov, Yu. I. Manin, E. E. Mukhin and D. V. Zhdanovich, “Nonstandard quantum deformations of GL(n) and constant solutions of the Yang–Baxter equation”, Prog. Theor. Phys. Suppl. 102 (1990) 203–218; S. Zakrzewski, “A Hopf star-algebra of polynomials on the quantum SL(2, R) for a ‘unitary’ R-matrix”,
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[25] [26] [27]
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Lett. Math. Phys. 22 (1991) 287–289; H. Ewen, O. Ogievetsky, J. Wess, “Quantum matrices in two dimensions”, Lett. Math. Phys. 22 (1991) 297–305. S. L. Woronowicz, “Unbounded elements affiliated with C ∗ -algebras and non-compact quantum groups”, Commun. Math. Phys. 136 (1991) 399–432; S. L. Woronowicz, “C ∗ -algebras generated by unbounded elements”, Rev. Math. Phys. 7(3) (1995) 481– 521. S. L. Woronowicz and S. Zakrzewski, “Quantum Lorentz group having Gauss decomposition property”, Publ. RIMS, Kyoto 28 (1992) 809–824. S. L. Woronowicz and S. Zakrzewski, “Quantum deformations of the Lorentz group. The Hopf ∗-algebra level”, Comp. Math. 90 (1994) 211–243. S. Zakrzewski, “Realifications of complex quantum groups”, in Groups and Related Topics, eds. R. Gielerak et al., Kluwer Academic Publ., 1992, pp. 83–100.
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION∗ MANFRED SALMHOFER Mathematik, ETH Z¨ urich, CH-8092 Z¨ urich Switzerland E-mail : [email protected] Received 16 April 1997 Revised 11 May 1997 The naive perturbation expansion for many-fermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion self-energy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, we have shown sufficient differentiability to solve the FSR equation for a class of models with a non-nested, non-spherical Fermi surface. I will first motivate the problem and give a definition of FSR, and then describe the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the fourpoint function. They imply that only ladder diagrams can give singular contributions to the four-point function.
1. Overview In this paper, I give a pedagogical outline of the foundations of the improved power counting bounds developed in [7], and show how to apply them to solve the problem of Fermi surface renormalization for systems with a nonspherical Fermi surface, such as electrons in a crystal. All the results that will be discussed are perturbative (this means, analysis of all orders of perturbation theory), but some of them directly concern the singularities of the four-point function and thus have non-perturbative consequences. All results stated without an explicit reference are derived in collaboration with Joel Feldman (UBC, Vancouver) and Eugene Trubowitz (ETH Z¨ urich) [7, 8, 9]. Before going into the details, I want to discuss briefly the motivation for studying these systems. The discussion about the unusual normal-state properties of the high-temperature superconductors revealed that it is not known precisely under which circumstances many-fermion systems behave as Fermi liquids. The question was not so urgent for three-dimensional Fermi systems because Fermi liquid behaviour was observed experimentally in three dimensions a long time ago. In one dimension, Fermi liquid behaviour does not hold; it is replaced by Luttinger ∗ Based
on two talks given at the workshop on Field Theoretic Methods for Fermions at the Erwin–Schr¨ odinger Institute for Mathematical Physics, Jan. 21 – Feb. 3, 1996.
553 Reviews in Mathematical Physics, Vol. 10, No. 4 (1998) 553–578 c World Scientific Publishing Company
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liquid behaviour, which means, roughly, that the discontinuity of the Fermi distribution at the Fermi vector disappears. Because of certain singularities in the self-energy that are absent in three, but present in two dimensions, the question arose if two-dimensional many-fermion systems can exhibit Fermi liquid behaviour at all, or if they instead behave similarly to a Luttinger liquid. Feldman, Kn¨ orrer, Lehmann and Trubowitz [3] show that in a system with a dispersion relation ε(p) that is not T -symmetric, i.e. where ε(−p) 6= ε(p) for all except a discrete set of momenta p, the familiar Cooper instability is absent because of this asymmetry and the system behaves like a Fermi liquid. More precisely, they prove nonperturbatively that the perturbation expansion in which two-legged insertions are removed is analytic in a disk uniformly in the volume and the temperature. Thus this system is a Fermi liquid if the perturbation expansion can be renormalized and if renormalization does not change the model. In [7, 8, 9], we prove that renormalization is possible and that it does not change the model, and thus provide a crucial ingredient in the proof of the existence of that Fermi liquid. A byproduct of the analysis of [7, 8, 9] is that for the first time, nonspherical Fermi surfaces have been treated in a mathematically rigorous way. In the filling region relevant for high-Tc theory, the Fermi surface is nonspherical, in fact, quite near to nesting [13]. Although we will assume that no perfect nesting takes place in the sense that the Fermi surface is not allowed to have identically flat sides, one can learn from the way our bounds behave how things change as half-filling is approached. In other words, one can understand nesting effects in a fairly easy way using improved power counting. Precisely at half-filling, things work a bit differently, but one can also renormalize the expansion. Since the subject of many-body fermion theory goes back to the fifties, an obvious question is whether all this has not been done before. The answer is no, because, as mentioned, the arguments provided in the literature came with some unproven, but seemingly plausible assumptions “on the side”. This is not unusual in the physics literature, but some of these assumptions turned out to be very nontrivial. They concern the regularity (i.e. differentiability) of the fermion self-energy, which is assumed without proof. Whoever tends to believe that such things are no big deal and uninteresting for any real physics may be reminded that a common feature of several proposals for non-Fermi liquid behaviour, e.g. [1], or the marginal fermi liquid scenario, is a fermion self-energy that is not a C 1 function of momentum and frequency. What we have done in [7, 8, 9] is to show that for systems in any dimension d ≥ 2 with a sufficiently rapidly decaying interaction (e.g. |x|−d−3 will do), the fermion self-energy is indeed smooth enough such that non-Fermi liquid behaviour can only come from singularities in the four-point function. More precisely, this means that non-Fermi-liquid behaviour cannot be seen in any fixed order in perturbation theory. In contrast, the situation considered in [1] is that of fermions interacting with transverse gauge fields, which means long-range interactions, and 1/3 there the first order self-energy correction already behaves as p0 instead of p0 , so it changes the behaviour of the fermion propagator completely. The problem with a long-range interaction is much more singular than the one we have solved, but the
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case of a short-range interaction is already nontrivial from second order on because of the singularity of the fermion propagator. 2. Fermi Surface Renormalization I will first describe a lattice system that falls into the class of models we consider, mainly to introduce the notation, and then give the more general hypotheses under which our theorems hold. Let d ≥ 2 be the spatial dimension and Λ ⊂ Zd be finite (any lattice Γ of maximal rank can replace Zd here). The case d = 1 has been treated by Benfatto, Gallavotti and coworkers [11, 12]. Let F be the Fock space generated by the fermion operators satisfying the anticommutation relations + 0 0 cα (x)c+ α0 (x ) + cα0 (x )cα (x) = δαα0 δxx0 .
(2.1)
The Hamiltonian H(c, c+ ) = H0 + λV has a free part X
H0 = −
X
tx−y
x,y∈Λ
c+ α (x)cα (y)
(2.2)
α∈{↑,↓}
which describes hopping from a site y to another site x with an amplitude tx−y = ty−x , and an interaction part with a small coupling constant λ. The interaction is a four-fermion interaction X n(x)v(x − y)n(y) , (2.3) V (c, c+ ) = x,y∈Λ
where n(x) =
X
c+ α (x)cα (x)
(2.4)
α∈{↑,↓}
is the number operator at x. For instance, the simplest Hubbard model is given by λ = U2 , where U is the usual Hubbard-U , and by v(x − y) = δxy , and the hopping term is tx−y = t if |x − y| = 1 and zero otherwise. It will turn out soon that to do renormalization properly, we have to deal with a whole class of free Hamiltonians H0 . One general condition that we impose is that both the hopping term and the interaction have a spatial decay that makes their Fourier transform a C 2 function. This is fulfilled if the second moments are finite: X |tx−y | |x − y|2 < ∞ y∈Zd
X
(2.5) |v(x − y)| |x − y|2 < ∞
y∈Zd
At temperature T and chemical potential µ, the grand canonical partition function is given by (2.6) ZΛ = tr e−β(H−µN )
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P 1 with N = x n(x), and β = kB T . The trace is over Fock space. We want to determine whether observables, typically given by expectation values of polynomials in the c and c+ , 1 tr e−β(H−µN ) O(c, c+ ) (2.7) hOi = ZΛ have a finite thermodynamic limit and whether an expansion in λ can be used to get their behaviour at small or zero temperature T . For instance, one would like to expand the two-point function G2 (x, y) = hc+ (x)c(y)i =
∞ X
λr G2,r (x, y)
(2.8)
r=0
It is by now well known that at T = 0, lim G2,r = ∞ for all r ≥ 3 [5, 6]. Λ→∞
This infrared divergence makes renormalization necessary. It will turn out that renormalization is merely a method to do the expansion in a cleverer way. For T > 0, the unrenormalized expansion is convergent, but the radius of convergence shrinks to zero as T → 0 so that the result is not uniform in the temperature. To study this problem, we use the standard path integral representation for the generating functional for the correlation functions. It is obtained by partitioning the interval [0, β] into L intervals of length β/L, and applying the Trotter product formula to the trace for ZΛ and hOi. In the limit L → ∞, the trace becomes a ¯ x) with a new euclidean functional integral over Grassmann fields ψ(τ, x) and ψ(τ, time τ varying from 0 to β. We also temporarily introduce an infrared cutoff ε > 0 to make the infinite volume model well defined, by forbidding values e(p) < ε. The generating functional for the amputated connected correlation functions, Z ¯ ¯ ¯ −λV (ψ,ψ) = dµCε (χ − ψ, χ ¯ − ψ)e , (2.9) eGε (χ,χ) is then a convolution of the Grassmann Gaussian measure dµC , given by the propagator 1 h((e(p)/ε)2 ) , (2.10) Cε (p) = ip0 − e(p) with the exponential of V . Here the Matsubara frequency p0 is the Fourier variable dual to τ , and p = (p0 , p). For T > 0, p0 ∈ πT (2Z + 1). For T = 0, p0 ∈ R. The function h is a C ∞ cutoff function that is zero for x ≤ 1/4 and one for x ≥ 1. Without the infrared cutoff ε, G(χ, χ) ¯ would not be defined because of the infrared divergences mentioned above. The cutoff removes these divergences because it cuts off the singularity of the propagator C. When ε → 0, the coefficients in the perturbation expansion of Gε in powers of λ diverge because they are not yet renormalized. Renormalization will remove these divergences and we can then take the limit ε → 0. In the following, I set T = 0 and Λ = Zd to discuss the singularities that give rise to the infrared divergences. If λ = 0, the electrons are independent and one can calculate the correlation functions simply by doing a Fourier transform. For the lattice systems discussed above, Fourier space is given by B = Rd /Γ# where
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Γ# is the lattice dual to the position space lattice, e.g. Γ# = 2πZd for Γ = Zd . The Fourier transform of the hopping term gives the band structure (or dispersion relation) X tx e−ip·x − µ (2.11) e(p) = − x
Pd which, for the Hubbard case, reduces to e(p) = −2t i=1 cos pi − µ. At zero temperature, all states with e(p) < 0 are filled. The boundary of the occupied region in k-space is the Fermi surface S = {p ∈ B : e(p) = 0} .
(2.12)
1 N is given by the volume enclosed by S (in two dimensions The density ρ = |Λ| by the area inside S). In the example of the Hubbard model, the function e has its minimum at p = 0, and it is strictly convex (and analytic) near this minimum. Consequently, the Fermi surface is strictly convex for µ slightly larger than −2td. This is the generic behaviour of systems in solid state models: the band function is strictly convex around a minimum, so a small (but macroscopic) occupation of electrons in that conduction band gives rise to a strictly convex, curved Fermi surface. As the filling increases (which happens if µ is increased), the shape of the Fermi surface changes and it even becomes diamond-shaped at µ = 0 (half-filling) for the H0 of the Hubbard model when d = 2. Before discussing renormalization, I state our hypotheses more precisely. We assume that the Fourier transform vˆ of the two-body potential v is vˆ ∈ C 2 (R×B, C), that (2.13) vˆ(−p0 , p) = vˆ(p0 , p) ,
and that all derivatives of vˆ up to second order are bounded functions on R × B. Since λ and vˆ appear only in the combination λˆ v , we may assume that |ˆ v2 | ≤ 1, P where |f |2 = |α|≤2 kDα f k∞ . Note that the interaction potential may depend on p0 as well. To show convergence at large p0 we need only that vˆ approaches a finite limit as p0 → ±∞ (see [8, Hypothesis (H1)] for details). These assumptions about the behaviour at large |p0 | are satisfied in all the models discussed above, they are in fact much weaker than the usual analyticity assumptions (if the interaction is instantaneous, as in the Hamiltonian used in the above motivation, its Fourier transform vˆ is even independent of p0 ). The physically relevant assumption is the regularity of vˆ because it requires sufficient decay in position space. We assume that e ∈ C 2 (B, R) and that for all p ∈ S, ∇e(p) 6= 0. This implies that the Fermi surface is a C 2 -submanifold of B. Moreover, we assume one of the following: (A) S has no identically flat sides (for a precise definition see [7, Assumption A3]) (B) S is strictly convex with strictly positive curvature (for a precise definition, see [8, Hypotheses (H3)–(H5)]) These assumptions exclude half-filling (µ = 0) because there the Fermi surface has flat sides and because the gradient vanishes at the corners of the diamond. I will
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not discuss this case further here; the renormalization problem is actually simpler in that case because the Fermi surface stays fixed by the particle–hole symmetry. Obviously, assumption (B) implies assumption (A). The connection between the singularity of C and the infrared divergences in the model is easy to see: the functional Gε has an expansion in the coupling and the fields, ¯ = Gε (ψ, ψ)
X r≥1
r
λ
2(r+1) Z 2m X Y m=0
dpi δ(p1 + · · · + pm − pm+1 − · · · − p2m )
i=1
×Gε,mr (p1 , . . . , p2m )
m Y
¯ i )ψ(pm+i ) ψ(p
(2.14)
i=1
with the kernels Gε,mr given by a sum over values of Feynman diagrams. Every such contribution is a finite-dimensional integral. The integrand consists of various combinations and powers of C given by the Feynman rules. However, powers of C are in general not locally integrable: introducing variables ρ transversal and ω tangential to S, the integral Z |ip0 −e(p)|<ε
dp0 dd p |ip0 − e(p)|α
Z =
e(p)=ρ
dp0 dρ |ip0 − ρ|α
Z
Zε dωJ(ρ, ω) =
rdr f (r) rα
(2.15)
0
diverges for α ≥ 2. (Here J is the Jacobian of the change of variables and f its integral over ω, so f (0) is nonzero). By momentum conservation and the Feynman rules, graphs that contain a string of two-legged insertions produce arbitrarily high powers of C and thus divergences (see [7, Sec. 1.4] for a detailed explanation). The cutoff ε removes these divergences, but the values of these graphs become large when ε becomes small. To renormalize, we perform subtractions of the insertions at the singularity on S. The details of this subtraction can be found in [7, Sec. 2.2]. We will discuss in detail below what these subtraction terms really are and give an alternative characterization which is less technical. The physical intuition behind these divergences is very simple and has been well known for a long time. One expects that the interaction produces a self-energy σ(λ, p) of the fermion such that the propagator Z G(p) =
dτ
X
¯ x)ieip0 τ −ip·x hψ(0, 0)ψ(τ,
(2.16)
x
behaves essentially as G(p) = (ip0 − e(p) − σ(λ, p))−1 .
(2.17)
If σ is a reasonable function, then the integrability properties of G will be the same as those of the free propagator C, but the singularity is at a different place, namely
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e(p) + σ(λ, 0, p) = 0. Thus the Fermi surface moves when the interaction is turned on. An expansion Z
∞
dp
X 1 → ip0 − e(p) − σ(λ, p) r=0
Z dp
σr (ip0 − e(p))r+1
(2.18)
just introduces artificial divergences. In other words, if one can put σ into the denominator, the divergences should disappear. There are two problems with putting σ into the denominator. First, σ is not known but has to be calculated itself. This is not a problem in perturbation P theory because the first order contribution σ1 to σ = r≥1 σr λr is finite and one can therefore proceed recursively. The real problem is the assumption that σ is a “reasonable function”. One has to show some regularity of σ to verify that the interacting propagator indeed has the same integrability properties as the free one. One cannot simply assume this because σ itself is determined by the free model and the interaction. The same argument that shows finiteness of σ1 also suggests that ∂ σ2 = ∞, even after renormalization. This is the behaviour suggested by already ∂p naive power counting bounds. These bounds are not sharp, however, and in [7, 8] we have improved power counting sufficiently to show that σ ∈ C 2− for all > 0 if d = 2 and σ ∈ C 2 for d ≥ 3. We now proceed to do renormalization using counterterms instead of putting the σ into the denominator. For the purposes of the following discussion, putting counterterms in the action makes the concepts clearer in this problem. Another reason to do that is that we can show more regularity of the counterterm function K which essentially restricts σ to the Fermi surface than of the self-energy σ itself: we have shown that K is C 2 for all d ≥ 2. If one prefers to change the propagator, one should put K in there instead of σ to use our bounds. One way to motivate putting counterterms is as follows: since turning on λ makes the Fermi surface move, and since this movement causes all the trouble with the expansion, one can try to add a function K(λ, p) to the bilinear part of the action such that the Fermi surface S stays fixed. In other words, K compensates all self-energy corrections that would move the Fermi surface under the interaction. Theorem 2.1. Assume (A). There is a formal power series K (ε) (λ, p) =
∞ X
λr Kr(ε) (p)
(2.19)
r=1
such that the model defined as Z R ¯ ¯ G ren (ψ,ψ) ¯ −λV (χ,χ)− = dµCε (χ − ψ, χ ¯ − ψ)e e
(ε) dpχ(p)K ¯ (λ,p)χ(p)
(2.20)
ren has Fermi surface fixed to S = {p : e(p) = 0}. Moreover, the kernels Gren m,r of G (ε) all have finite limits as ε → 0, and K (λ, p) has a finite limit K(λ, p). The Borel transform in λ of G has a positive radius of convergence uniformly in ε.
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Thus, fixing the Fermi surface indeed removes all infrared divergences. It is interesting to note that the counterterms are finite. This theorem [7, Theorem 1.2] is a nontrivial extension of the statements proven in [5] because the counterterms are momentum dependent. The dependence of K on p is really there in absence of rotational symmetry, and it leads to substantial technical complications. K is also a functional of e, so K = K(λ, e, p). Using (2.13), one can show that the self-energy σ satisfies σ(−p0 , p) = σ(p0 , p). This implies that K(λ, p) ∈ R for all p because K is constructed from the self-energy by evaluating at p0 = 0 and p ∈ S. More technically speaking, the graphs contributing to K are the two-legged one-particle irreducible graphs that also contribute to σ, but they are evaluated at p0 = 0 and p ∈ S (see [7, Sec. 2.3]). Although we have now removed the infrared divergences, we have done so at the price of changing the model. Because of the counterterm function K, the quadratic part of the action is now Z ¯ (2.21) A0 = dpψ(p)(ip 0 − e(p) − K(λ, p))ψ(p) and corresponds to a free Hamiltonian with dispersion relation e + K, which is λ-dependent, instead of e. Thus, if e is the free dispersion relation, Theorem 2.1 makes a statement not about the original model but about a changed model. To do the renormalized expansion for a prescribed free model with band structure E and interaction V , one has to solve the equation E(p) = e(p) + K(λ, e, p)
(2.22)
for e. Equation (2.22) is the central equation of the problem. I first explain how we solve it and then show how to renormalize without changing the model by using the solution e = R(E, λ) of (2.22) (e also depends on the two-body potential vˆ, but all bounds are uniform in vˆ for the set of vˆ specified above, so I suppress that dependence in the notation). Let K (R) (λ, p) =
R X
λr Kr (p)
(2.23)
r=1
be the function K up to order R in perturbation theory. Crudely speaking, the right-hand side of (2.22) is the identity plus a small term, because K (R) is of order λ, so an iteration is the natural strategy to get a solution. However, because of the various dependences of K (R) on e and p one has to be very careful what one means by small (no matter how small λ is chosen, the properties of the sum f + λg will differ very much from those of f if g is more singular than f ). Since e ∈ C 2 is the basic condition for all our bounds, we need at least K (R) ∈ C 2 , because in every step of the iteration, e gets replaced by e + K. Also, to use a fixed (R) point theorem, one needs control over δKδe . But since 1 1 δ ∼ , δe ip0 − e (ip0 − e)2
(2.24)
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
561
taking such derivatives seems to lead to new divergences. Nonetheless we have the following theorem [7, Theorems 1.2 and 1.6]. Theorem 2.2. Assume (A). Then for all R ∈ N, K (R) (λ, ·) ∈ C 1 (B, R), and is also C 1 in e. Denote the Fr´echet derivative of K with respect to e by K (R) δK ∈ L(C 1 , C 0 ) and the sup norm on C 0 by |·|0 . Then for all h ∈ C 1 (B, R) δe δK (R) (2.25) δe (h) ≤ const |λ| |h|0 . 0 (R)
(R)
Because of (2.25), δKδe extends uniquely to a bounded linear operator from C 0 to C 0 . The set of e satisfying (A) is open, so if e1 and e2 are close enough, (A) holds for all e on the line connecting e1 and e2 . Then e1 + K (R) (e1 ) = e2 + K (R) (e2 ) implies R1 (R) by Taylor expansion that (1+L)(e2 −e1 ) = 0, where L = 0 dt δKδe ((1−t)e1 +te2 ). (R)
Since R is fixed and δKδe is a bounded operator for all t, 1 + L is invertible for λ small enough. Thus, we have [7, Theorem 1.7]: Theorem 2.3 Assume (A). For all R ∈ N, there is λR > 0 such that for all λ ∈ (−λR , λR ), the map e 7→ e + K (R) is locally injective. This implies uniqueness of the solution under the quite general conditions (A). The existence proof requires the stronger assumptions (B) because for that, we need to show that K is even in C 2 . It is a priori not clear that K must have the same differentiability properties as e. One might be in the situation that one always loses some regularity, i.e. that e ∈ C k only implies K ∈ C k−1 , or that even e ∈ C ∞ leads only to K ∈ C k0 for some fixed k0 . It took us some time and optimal bounds to prove that for k = 2, there is no loss of regularity. Theorem 2.4. There is an open set E ⊂ C 2 of dispersion relations e fulfilling (B) and e(−p) = e(p) such that K (R) ∈ C 2 for all e ∈ E. There is an open subset E 0 ⊂ E and for all R ∈ N, there is λR > 0 and a map R : (−λR , λR ) × E 0 → E such that for all (λ, E) ∈ (−λR , λR ) × E 0 , e = R(λ, E) solves (2.22), with K replaced by its truncation to order R, K (R) . The regularity statement is [9, Theorem 1.1]. The inversion statement is proven in [10]. To state a similar theorem for the nonsymmetrical case requires introducing older continuity of the second spaces of C 2+ -functions (the extra is meant as -H¨ derivative), because in the nonsymmetric case, one needs to prove more regularity to bound the particle–particle ladders. This was done in [8], and the regularity statements of the above theorems in the C 2+ class of functions are proven in [8, 9]. For conciseness, I do not state all the details here; they are provided in [8]. Using Theorem 2.4, we can now do renormalization without changing the model, as follows. Let the model be given by a potential V and by a dispersion relation E for the independent electrons, with E ∈ E 0 . Let R ∈ N and λ ∈ (−λR , λR ). Use Theorem 2.4 to determine e = R(E, λ), and set κ(E, λ, p) = K(λ, R(E, λ), p).
562
M. SALMHOFER
Then E = e + κ = e + K(e). Denoting the propagator with E by C(E) and the one with e by C(e), we have, by standard shift formulas for Gaussian measures, the identity Z ¯ ¯ −λV (χ,χ) dµC(E) (ψ − χ, ψ¯ − χ)e =
Ze ZE
Z
¯
¯ −(ψ−χ,K(ψ−χ)) ¯ dµC(e) (ψ − χ, ψ¯ − χ)e ¯ −λV (χ,χ) e
Ze −(ψ,Kψ) ¯ = e ZE
Z
(2.26)
¯ ¯ χ,Kχ) ¯ ¯ dµC(e) (ψ − χ, ψ¯ − χ)e ¯ −λV (χ,χ)−( e(ψ,Kχ)+(χ,Kψ) .
This is an identical rewriting of the generating functional for the model given by E and V in terms of the quantities e and K that appear in the renormalized expansion, obtained by moving the K from the propagator to the interaction. Since E = e+K, this leaves e in the propagator. The change in normalization factor is irrelevant for any correlation function, and the extra source terms in the integrand just modify the external legs in a trivial way. This identity also holds if a cutoff ε > 0 is in place. In that case, all bounds are uniform in ε and Theorem 2.1 implies that the (R) kernels Gm,ε converge as ε → 0 (here the superscript R indicates G up to order R in λ, similarly to the definition of K (R) ). Physically, this procedure means the following. Applying the map R, i.e. going from E to e, shifts the Fermi surface from the free surface S(E) to the interacting Fermi surface S(e). Thus, in this step, the deformation of the surface caused by the interaction is taken into account. The renormalized expansion is then done at fixed interacting Fermi surface S(e), and it can be used to calculate other self-energy effects, and the other correlation functions. As mentioned in the statement of Theorem 2.1, the bounds that we prove for the kernels are not sufficient (and shouldn’t be sufficient) to show convergence of the perturbation series in λ. This is the reason for the explicit restriction to a finite order in perturbation theory in the other theorems. The general bound we obtain is the standard de Calan–Rivasseau bound, e.g. for the two-point function G2 (p) =
∞ X
G2,r (p)λr
(2.27)
r=0
it reads |G2,r (p) ≤ r! Qr ,
(2.28)
where Q is some constant. If this bound is saturated, the perturbation series has convergence radius zero, and more precisely, it means that the λR of the above theorems behaves as 1 (2.29) λR ∝ . R Even if the perturbation series diverges, it is conceivable that the map from the interacting to the free dispersion relation is invertible for some range of coupling
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
563
constants of the form (0, λ0 ), although analyticity does not hold. The investigation of the reasons for such a nonanalyticity is, of course, very important. The renormalization method yields statements about how and why these factorials can appear: (1) In order r, there are so many graphs that bounding the sum of graphs by the sum of their absolute values already gives an r factorial. This bound is not sharp because in it, the Pauli principle is ignored. For fermions, one may expect sign cancellations, such as in determinant bounds, to be useful. Determinant bounds do not work uniformly in the cutoff in these models, however, and it is a hard problem to implement the Pauli principle to show that the number of graphs does not produce a factorial. This was done by Feldman, Magnen, Rivasseau, and Trubowitz [4] for d = 2. A similar result is expected to hold for fermions in any dimension. (2) Singularities in values of individual four-legged diagrams can produce r factorials as well. The best-known example of this are the BCS ladders which produce symmetry breaking [6]. For item (2), the improved power counting method provides the following theorem. Theorem 2.5. Assume (A). The only graphs that can produce r factorials are generalized ladder graphs. For details and the proof, see [8, Theorem 2.46] and the next section. This theorem holds for any d ≥ 2 and for the very general class of Fermi surfaces satisfying only the condition (A) of non-flatness. No resummation, and hence no condition on the sign of the coupling is required. For the special case of spherical Fermi surfaces, a similar theorem was stated in [11]. The meaning of the term “generalized ladder” is explained in detail in [7, Sec. 2.4], and also below. The generalized ladders (called dressed bubble chains in [7, Definition 2.24]) are non-overlapping four-legged diagrams. By [7, Lemma 2.26], any non-overlapping four-legged graph is a generalized ladder. It is constructed from the usual ladder graphs by replacing the bare vertices by effective vertices of a higher scale. The non-overlapping four-legged graphs emerge in a natural way in the renormalization flow because their scale behaviour is marginal, which produces the factorials. All contributions to the four-point function from overlapping graphs are bounded, or, in renormalization group language, irrelevant. The importance of this is that convergence of perturbation theory can now be checked by looking at the ladders only: if they have singularities, then perturbation theory does not converge. If they don’t, the expansion in λ converges. The structure of the ladders is so simple that their properties essentially only depend on the fermion propagator, in particular, the Fermi surface. In absence of nesting (such as takes place at half-filling in the Hubbard model), the particle–hole ladders have no singularities. The particle–particle ladders always have a singularity at zero transfer momentum if the Fermi surface is symmetric, i.e. if e(−p) = e(p). The
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M. SALMHOFER
Fermi liquid of [3] has a Fermi surface for which the particle–particle ladders are uniformly bounded as well because there is no symmetry of e under p → −p. 3. Improved Power Counting In this section, I discuss the reasons behind the improved power counting bounds. I have written the present section so that it can be read as an easy introduction to the technical parts (Chaps. 2 and 3) of [7] and to the regularity analysis done in [8]. I shall discuss two examples of graphs to bring out the main point. After that, it should be obvious to generalize it to all graphs, given the graph classification of [7, Sec. 2.4]. To do estimates we need some definitions from scale analysis. As in all modern treatments of renormalization, we decompose (“slice”) the propagator around its singularity. There is a lot of freedom in doing this, but the decomposition is chosen such that the propagator has very simple behaviour on each slice. In the previous section, we introduced an infrared cutoff ε. Since all quantities scale like powers of ε and logε, we take ε of the form ε = MI ,
with M > 1 fixed and I a negative integer.
(3.1)
Removing the cutoff thus means taking the limit I → −∞. Moreover, we will now trace back the behaviour when the energy scale varies by looking at the contributions from energy shells M j−2 ≤ |e(p)| ≤ M j , for I ≤ j < 0. This decomposition is natural because it is adapted to the singularity. For definiteness, here are the details (readers interested only in the main features of the decomposition can skip this paragraph). Let r0 > 0 be chosen such that in an r0 -neighbourhood of the Fermi surface the coordinates ρ and ω of (2.15) can be used. Let M ≥ max{43 , r10 } (then |e(p)| < M −1 implies |ρ| < r0 ), and let −4 a ∈ C ∞ (R+ , a(x) = 1 for x ≥ M −2 , 0 , [0, 1]) such that a(x) = 0 for 0 ≤ x ≤ M and a0 (x) > 0 for all x ∈ (M −4 , M −2 ). Set 0 a(x)
if x ≤ M −4
if M −4 ≤ x ≤ M −2 x x = f (x) = a(x) − a M2 if M −2 ≤ x ≤ 1 1−a 2 M 0 if x ≥ 1,
(3.2)
so that, for all x > 0, f (x) ≥ 0 and 1 − a(x) =
−1 X
f (M −2j x) .
(3.3)
j=−∞
Calling fj (x) = f (M −2j x), supp fj = [M 2j−4 , M 2j ] .
(3.4)
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
565
The decomposition of C is + X eip0 0 a(p20 + e(p)2 ) eip0 0 = + eip0 0 Cj (p) , ip0 − e(p) ip0 − e(p) j<0 +
+
where
(3.5)
fj (|ip0 − e(p)| ) f (M −2j |ip0 − e(p)| ) = . (3.6) ip0 − e(p) ip0 − e(p) The first term in (3.5) is bounded and therefore cannot give rise to infrared singularities. A bit of care is required, however, to treat it in the ultraviolet (large p0 ). This is not difficult and done in [8, Appendix D]. For the purposes of the present discussion, I just discard this term. Then all scale sums go from j = I to j = −1, where I < 0 is the infrared cutoff, and all propagators are supported in a small + neighbourhood of the Fermi surface. In particular, the eip0 0 in front of the sum in (3.5) can be omitted. So much for the details which I included for definiteness. In the following we shall need only the following two facts: firstly, the cutoff propagator is a sum 2
2
Cj (p) =
C(p) =
−1 X
Cj (p) ,
(3.7)
j=I
and secondly, the propagator on scale j, Cj , has simple properties: it is easy to prove (see [7, Lemmas 2.1 and 2.3]) that |Dα Cj (p)| ≤ Ws M −(s+1)j 1(|ip0 − e(p)| ≤ M j ) ,
(3.8)
where Dα is a derivative with respect to p (α is a multiindex with |α| = s, 0 ≤ s ≤ k) of order s. The indicator functions take the value 1(X) = 1 if X is true and 1(X) = 0 otherwise. In words: on “slice” number j, the propagator is for all (p0 , p) of almost constant absolute value, at most M 2−j , (s = 0 in (3.8); W0 = M 2 ), and each derivative produces another large factor M −j . The constant Ws depends on |e|s and on g0 (g0 is the lower bound on |∇e| near the Fermi surface). Moreover, the support of Cj is contained in the product of an interval of length 2M j in p0 and a thin shell Rj of thickness const M j around the Fermi surface S. An example of such an Rj is drawn for the two-dimensional case in Fig. 1(a). By our choice of M , M −1 < r0 , so for all j < 0 this shell is contained in the region where the variables ρ and ω, introduced in (2.15), can be used.
(a)
(b)
(c)
Fig. 1. Intersection of a shell around the Fermi surface with its translates.
566
M. SALMHOFER
3.1. Volume improvement bounds The scale decomposition is a natural way to understand power counting because it allows us to weight the growth of the propagator in the vicinity of its singularity S against the smallness of the volume of shells around the Fermi surface, where it becomes large. In our scale decomposition, momentum space is cut into shells around the Fermi surface, as sketched in Fig. 1(a). Because we have assumed that the gradient of e does not vanish on the Fermi surface, the p-volume of a shell, in which M j−1 ≤ |e(p)| ≤ M j (j < 0), is bounded by a constant times M j . Similarly, the support condition of the propagator Cj restricts p0 to the interval [−M j , M j ]. One can deduce the integrability properties of the infrared part (1 − a)C of the propagator C discussed above by weighting this volume against the growth of |C| in a summation over shells: Z 2 X Z 1 d 2 2 d fj (|ip0 − e(p)| ) dp0 d p (1 − a(p + e(p) )) = dp d p . 0 0 |ip0 − e(p)|α |ip0 − e(p)|α j<0 R×B
R×B
(3.9) By (3.4), and because the size of |Cj | is bounded by M −j+2 , this is bounded by Z Z X (−j+2)α j dp0 1(|p0 | ≤ M ) M dd p 1j (p) , (3.10) j<0
where 1j (p) = 1(|e(p)| ≤ M j ). Since the volume of the d-dimensional shell Rj = {p ∈ B : |e(p)| ≤ M j }
(3.11)
is bounded by const M j , this is ≤ M 2α
X
M −jα 2M j const M j
j<0
= const
X
M j(2−α)
(3.12)
j<0
= const
1 1 − M −(2−α)
if α < 2. So, for instance the first order correction to the self-energy is a sum over scales j < 0 of Z v (q − p) . (3.13) σ1,j (q) = dd+1 p Cj (p)ˆ By the above bound with α = 1, and since kvk∞ ≤ 1, |σ1,j (q)| ≤ const M j
(3.14)
so the sum converges. The scaling bound M j is the typical bound for self-energy contributions at a scale j. Note that this bound is not good enough to show convergence of a derivative of the self-energy because the derivative of the propagator Cj
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
567
contains an additional factor M −j . If these bounds were saturated, the derivative of the self-energy would already be logarithmically divergent. For the first order, this is not so because the Cj in (3.13) does not depend on the external momentum q, and because the interaction potential is C 2 by assumption. But in second or higher order, some of the propagators in the integral do depend on the external momentum and bounding derivatives becomes a problem, as I shall discuss below.
Q− q
p
k
q
Q− p
Q− k
Fig. 2. The particle–particle bubble.
Up to this point, the scale decomposition introduced only a rewriting of (2.15) as a convergence statement for series instead of integrals. However, the geometry of the shells around the Fermi surface has important consequences for nontrivial graphs, which I discuss now. In the following, I impose the cutoff I so that all scale sums run from I to −1. I first start with a discussion of the contribution of the particle–particle bubble B (shown in Fig. 2) to the four-point function. By momentum conservation, the sum of the ingoing momenta must equal that of the outgoing ones, so only three of the external momenta are independent. We choose them as indicated in Fig. 2. Then Z dd+1 p C(p) vˆ(p − k) C(−p + Q) vˆ(q + p − Q) . (3.15) Val(B)(k, Q, q) = R×B
Writing both propagators as scale sums one obtains X V al(Bjh )(k, Q, q) Val(B)(k, Q, q) = lim I→−∞
with
(3.16)
I≤j,h<0
Z dd+1 p Cj (p) vˆ(p − k) Ch (−p + Q) vˆ(q + p − Q)
Val(Bjh )(k, Q, q) =
(3.17)
R×B
and therefore (since |ˆ v |0 ≤ |ˆ v |2 ≤ 1) Z 2−j 2−h dp0 1(|p0 | ≤ M j ) 1(| − p0 + Q0 | ≤ M h ) |Val(Bjh )| ≤ M M Z ×
dd p 1j (p)1h (−p + Q) .
(3.18)
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M. SALMHOFER
This bound is invariant under j ↔ h because the integral is invariant under a change of integration variable p → −p + Q. Therefore, it suffices to discuss convergence properties of the sum over I ≤ j ≤ h < 0. We bound the p0 -integral and get (−)
|Val(Bjh )| ≤ 2M 2−j M 2−h M j V1 where
(Q, j, h) ,
(3.19)
Z (±)
V1
dd p 1j (p)1h (±p + Q)
(Q, j, h) = B
= vol (Rj ∩ (±Rh ∓ Q))
(3.20)
is the volume of the intersection of shell Rj with the translate of ±Rh by Q (see Figs. 1(b) and (c)). The only estimate uniform in Q is to bound this volume by that of a single shell, (±) (3.21) V1 (Q, j, h) ≤ const M j . I took M j here since j ≤ h implies M j ≤ M h . With these bounds XX X M j−h ≤ const M −k |Val(B)| ≤ const j≥I k≥0
I≤j≤h<0
≤ const
X 1 1 M 0j ≤ const |I| −1 1−M 1 − M −1
(3.22)
j≥I
= const |log M I | . The marginal behaviour M 0j is the standard power counting behaviour of fourlegged graphs on scale j. This lack of decay causes a logarithmic divergence of the bound as the infrared cutoff ε = M I → 0. In Fig. 1(b), one can see, however, that for those values of Q where the two surfaces intersect transversally, one has a much better bound because vol Rj ∩ (±Rh ∓ Q) is only ∝ M j+h , instead of M j for an entire shell. But this volume gain is not uniform in Q: it is certainly absent for Q = 0. For the surface drawn in Fig. 1, there are additional external momenta for which the intersection leads to no gain over the normal volume factor M j , such as the one shown in Fig. 1(c). For strictly convex Fermi surfaces, one can prove: Lemma 3.1. If e(−p) = e(p) and if the Fermi surface has everywhere strictly positive curvature, then for j ≤ h kF min M j , 1 M j+h if |Q| ≤ (±) |Q| 2 . (3.23) V1 (Q, j, h) ≤ const j h/2 otherwise M M Here kF = inf{|p| : p ∈ S}.
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
569
The extra factor M h/2 comes from the curvature of the Fermi surface, which provides a volume gain for any Q 6= 0. Lemma 3.1 implies the simpler bound ( 1 if |Q| ≤ M h/2 (±) j V1 (Q, j, h) ≤ const M . (3.24) M h/2 otherwise Inserting (3.24) into (3.19), and proceeding as in (3.22), one sees that for any |Q| 6= 0, the sum converges. More precisely, we get Val(B) ≤ const | log |Qk .
(3.25)
The logarithm can be understood from (3.24): it is the scale h0 = max{h < 0 : M h/2 ≤ |Q|}. Below scale h0 , the convergent behaviour M h/2 takes over. Above scale h0 , there is no decay in the sum, and that gives a logarithm in the same way as it appeared in (3.22), only that now h0 replaces the cutoff scale I. It is not hard to verify explicitly here that if e(−p) = e(p), this bound, obtained by looking only at volume effects, is sharp: Val(B)(k, (0, Q), q) diverges logarithmically at Q = 0. For the particle–hole bubble, where the lower propagator is in the other direction, the bound is not saturated because of a sign cancellation, and the function is finite at zero. This follows for Q = 0 by [7, Lemma 2.42] and for Q 6= 0 by a Taylor expansion argument (see [3]). Physically, both these facts are important. The singularity of the particle– particle ladder causes Cooper pairing if the interaction is attractive. The absence of a singularity in the particle–hole ladder implies that there is no competition to Cooper pairing from the particle–hole channel. Note that Lemma 3.1 requires positive curvature. If the curvature vanishes, the particle–hole bubble can have singularities. This is, for instance, the case in the half-filled Hubbard model. As one approaches half-filling, the curvature tends to zero and a singularity builds up. It is believed to lead to antiferromagnetism exactly at half-filling. Note also that although the particle–hole bubble is bounded, its derivatives with respect to external momenta are not. If e(−p) 6= e(−p), Lemma 3.1 gets replaced by a different statement in the (−) case of V1 , because there are nonzero translation momenta q for which the Fermi surface S intersects its translate S +q tangentially. The detailed geometrical picture and the bounds are given in [8, Appendix C]. k− p+q
k q
p
q
Fig. 3. A second-order graph contributing to the self-energy.
570
M. SALMHOFER
An analogous function, the polarization bubble, plays an important role in the second-order contribution to the self-energy. The polarization bubble is bounded, but not C 1 (actually, not even continuous) in the external momentum. Therefore it is a nontrivial question whether the value of the second-order skeleton self-energy correction shown in Fig. 3 is differentiable or not. Its value is given by Z Z Val(G)(q) = dd+1 pC(p)ˆ v (q − p)2 dd+1 kC(k)C(k − p + q) (3.26) The subintegral over k is the polarization bubble mentioned above. Note that this time a derivative with respect to q will act on one of the propagators. Since the absolute square of the propagator is not integrable, we have to do the bounds carefully. Inserting scale sums as before, and bounding as before, we have for j≤h≤i X M −i−j−h M j+h V2 (q, j, h, i) |Val(G)(q)| ≤ const j≤h≤i<0
from |Cj |, etcd p0 -integral spatial volume
(3.27)
where V2 (q, j, h, i) = max
u,v,w=±1
vol{p, k : |e(p)| ≤ M j , |e(k)| ≤ M h , |e(up + vk + wq)| ≤ M i } (3.28)
is the integration volume for this two-loop graph. The restriction to j ≤ h ≤ i is allowed because permutations of the scales only change signs in the linear combinations of the momenta, which we took into account by taking the maximum over signs u, v, w in the definition of V2 . Again, the easiest bound for V2 is obtained dropping the condition |e(up + vk + wq)| ≤ M h , which gives V2 (q, j, h, i) ≤ const M j+h .
(3.29)
From this, one can see that the sum for Val(G) converges since it is majorized by const
X
M j+h−i = const
−1 X j=I
j≤h≤i<0
Mj
−1 −1 X X
−1 X j=I
Mj
−1 X h=j
(3.30)
h=j i=h
The sum over i is a convergent geometric series bounded by continues with ≤ const
M −(i−h) P k≥0
M −k , so (3.30)
−1 X X M ≤ const |j|M j ≤ const k M −k ≤ const. M −1 j=I
k≥0
(3.31)
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
571
A derivative with respect to q acts on the propagator Ci , and by (3.8), it produces another factor M −i . Proceeding as above, one the series for Dq Val(G)(q) is majorized by const
X
M
j+h−2i
= const
−1 −1 X X
M
−(h−j)
j=I h=j
j≤h≤i<0
−1 X
M −2(i−h) .
(3.32)
i=h
The sums over i and h are convergent, but there is no decay left in the sum over j, and this suggests a divergence of the first derivative as |I| when I → −∞. However, the actual behaviour of Val(G) is much better because of the following estimate for V2 . Written as an integral, Z (v) V2 (q, j, h, i) = max dp 1j (p) V1 (up + wq, h, i) . (3.33) u,v,w=±1
Inserting the bound (3.24), we get two contributions, Z dp 1j (p) V2 (q, j, h, i) ≤ const M h max u,w=±1
i i i × M 2 1 |up + wq| > M 2 + 1 |up + wq| ≤ M 2
(3.34)
In the first term, we drop the q-dependent indicator function, so that the p-integral gives vol Rj ≤ const M j . Thus o n i h j+ 2i 2 . (3.35) V2 (q, j, h, i) ≤ const M M + max vol Rj ∩ p : |p + sq| ≤ M s=±1
h
2M 2 2M i
q
Fig. 4. The intersection of Ri with the ball of radius M
h 2
around q.
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M. SALMHOFER
The second term is the d-dimensional volume of the intersection of the shell Rj of i thickness M j with a ball of radius M 2 around ±q, sketched in Fig. 4. This volume is bounded by the diameter of the ball times the thickness of the shell, which is at most const M j+i/2 , for all d ≥ 2. For d ≥ 3, the bound is even const M j+i . We have thus shown the following volume improvement estimate. Lemma 3.2. There is a constant QV > 0 such that for all q ∈ B and all j ≤ h ≤ i < 0, i (3.36) V2 (q, j, h, i) ≤ QV M j+h+ 2 . (±)
Note that, unlike the estimate for V1 , this estimate is uniform in the external momentum q. If the symmetry e(−p) = e(p) does not hold, the above proof does not work, but a different argument shows the same bound (in fact, a better bound; see [8, Theorem 1.1 and Appendix B]). Using this bound in the sum for V al(G), we get a majorizing series const
X
3
M j+h− 2 i = const
j≤h≤i<0
−1 X
j
M2
j=I
≤ const
−1 X
−1 X h=j
j
M2
M −(h−j)/2
−1 X
M −3(i−h)/2
i=h
(3.37)
j=I
which converges. So, Val(G) is C 1 , and in fact the remaining decay in the sum over j can be used to show that its derivative is H¨older continuous in q of order 12 − δ for any δ > 0. Above, I have only discussed boundedness of the scale sums uniformly in the cutoff I. Convergence of the functions as I → −∞ follows from the above bounds by an application of the dominated convergence theorem. For details, see [7, Theorem 2.46 (iv)]. To summarize, we have shown for the second-order graph of Fig. 3 by some elementary volume estimates that for a strictly convex Fermi surface, the convergence of the scale sums is sped up such that its value is C 1 in the external momentum q. We call this faster convergence improved power counting. It was shown in [7] that improved power counting holds for much more general Fermi surfaces than the strictly convex ones. They only have to satisfy the assumption (A) mentioned above. The only difference is that instead of an improvement factor M i/2 one gets in general only M i , with > 0 depending on the surface. For example, the surface drawn in Fig. 1 also has > 0. 3.2. Non-overlapping graphs In this section, I generalize the improved power counting estimates to arbitrary graphs. This is possible because the entire argument leading to Lemma 3.2 used only very simple properties of the graph G, namely that G has two loops that have a fermion line in common. We call such graphs overlapping. Whenever a graph is
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
573
overlapping, the spatial volume integral appearing in the power counting bounds contains a subintegral bounded by V2 (q, j, h, i). For a large graph, the momentum q may be a very complicated combination of other loop momenta and the external momenta, but this is irrelevant because the volume improvement bound Lemma 3.2 is uniform in q. Therefore, any graph that contains two overlapping loops has an additional speedup of the convergence of scale sums. In general, we have [7, Theorem 2.40]: if a two–legged graph is overlapping, its scale behaviour improves from M j to M j(1+) with 0 < ≤ 1. if a four–legged graph is overlapping, its scale behaviour improves from M 0j to M j with 0 < ≤ 1. The value of depends on the geometry of the Fermi surface. I have discussed only strictly convex surfaces here, but the volume gain from overlapping loops holds for much more general surfaces (see [7]), for example also for the one shown in Fig. 1. The statements about the gain are true for overlapping graphs with an arbitrary number of external legs, but we need them only for two- and four-legged ones. Obviously, now we only have to see which graphs are not overlapping, because all others have an extra gain that allows us to take a derivative in the two-legged case, and that gives us an extra decay that makes the value of G converge, and free of singularities in the limit, in the four-legged case. For graphs with more than four legs, standard power counting is sufficient — we need only deal with the relevant (E = 2) and the marginal (E = 4) terms. Since we are only concerned with loops overlapping on fermion lines, we may replace interaction lines by vertices for the purposes of this graph classification. This replacement is indicated in Fig. 5.
Fig. 5. Replacing intersection lines by vertices.
The classification of all non-overlapping graphs is done in [7, Sec. 2.4]. That section of [7] is elementary and should be rather easy to read, therefore I will not discuss this any further here. The result is, as already anticipated in Theorem 2.5, the following. Theorem 3.3. Let G be a non-overlapping and one-particle irreducible graph, with vertices that have an even number of legs. If G is two-legged, G is a generalized Hartree–Fock graph. If G is four–legged, G is a dressed bubble chain. The precise definition of a generalized Hartree–Fock graph is given in [7, Definition 2.21] (there called GST graphs). The dressed bubble chains are defined in [7, Definition 2.26]. Roughly speaking a generalized Hartree–Fock graph is built
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M. SALMHOFER
(a)
(b)
Fig. 6. (a) A generalized Hartree–Fock graph. (b) A dressed bubble chain.
from a two-legged graph consisting of one vertex with self-contractions by (possibly recursively) replacing lines by other Hartree–Fock graphs. Similarly, dressed bubble chains are built from simple bubble chains. These notions are probably easiest understood by looking at Fig. 6. An ordinary Hartree–Fock graph would have only vertices with incidence number four. The essential difference between bubble chains and dressed bubble chains is that the dressed ones may have twolegged insertions of generalized Hartree–Fock graphs, and that the vertices may be effective vertices instead of bare ones. The restriction to one-particle irreducible (1PI) graphs in Theorem 3.3 is justified because one-particle reducible graphs are strings of 1PI graphs in the two-legged case and, in the four-legged case, they are obtained by attaching a string of two-legged diagrams to each of the external legs of a 1PI four-legged graph (see [7, Remark 2.23]). As is obvious from Fig. 6, the structure of these graphs is simple enough for an explicit analysis. In particular, the differentiability problem for the self-energy is easy since the external momentum does not enter any line of a generalized Hartree–Fock graph, so that there is no problem of it acting on a line without an accompanying volume improvement factor arising at the same scale. Note that this holds only because all effective vertices have an even incidence number. For graphs that have vertices with odd incidence numbers, this statement would be false; see [7, Remark 2.30]. For the four-legged graphs, the volume improvement bound gives a rigorous argument why one-loop summations take into account the leading behaviour: all other contributions are from overlapping graphs, hence nonsingular (factorials are tied to logarithmic singularities in the four-point function because a subgraph that of length n contributes a factor (log |p|)n to the integral, and Ris a bubble chain n dp| log |p| | ∼ n!). In [7, (2.126)], the keeping track of these subgraphs is formalized by introducing a number nf counting the number of times a non-overlapping four-legged subgraph can appear. The factorials in values of graphs are given by nφ ! where φ indicates the lowest scale in the graph (the root of the associated tree). See [7, Theorem 2.47]. Some overlapping four-legged graphs are shown in Fig. 7. Two remarks are appropriate, however. First, the bubble chains of Theorem 3.3 are not the bare ladders of the theory: the vertices in the bubble chains can, and
IMPROVED POWER COUNTING AND FERMI SURFACE RENORMALIZATION
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Fig. 7. Examples of four-legged overlapping graphs.
will, be effective vertices, i.e. their vertex functions are in general values of subdiagrams. The one-loop resummation justified by Theorem 3.3 is one where this scale dependence is taken into account, e.g. by defining suitable scale-dependent couplings. Similarly, the notion that a graph with scales associated to its lines is overlapping depends on the associated Gallavotti–Nicol` o (GN) tree. All this is dealt with in [7]. Our graph classification does not require the vertices to be bare vertices. In particular, there is no requirement that each vertex have, e.g., at most four legs. The combination of graph classification with the scale structure leads to a natural decomposition of the GN trees, constructed in Sec. 2.5 of [7]. The number nf appearing in [7, (2.126) and Theorem 2.46] is the number of bubble chains that appear in a graph associated to a particular GN tree, and it determines whether or not the value of a given rth order graph can have a value of order r! Second, the bounds implying that the ladders give the leading behaviour involve the curvature of the Fermi surface in the following way. The constant QV in Lemma 3.2 depends on the curvature of the Fermi surface, and it diverges if the curvature of the Fermi surface vanishes. If one is given a Fermi surface with a strictly positive, but small, curvature, such as the Fermi surface of the Hubbard model near to half-filling, QV will be large, and consequently, Lemma 3.2 will provide an improvement QV M i/2 over the ordinary power counting bound M j+h only if QV M i/2 is very small (the “improved” bound is worse than the naive one if QV M i/2 > 1). Thus the curvature of the Fermi surface sets an intrinsic energy scale ε0 , and the above statement that the ladder diagrams dominate holds only below this scale. One gets to this scale by integrating over the fields with momenta further away from the Fermi surface, which gives the effective action at scale ε0 . In the Hubbard model, one-loop calculations suggest that even for a repulsive initial interaction, the effective action at scale ε0 (which is not local any more) has attractive nearest-neighbour interactions. However, the above discussion shows that volume effects do not justify an approximation of the effective action at scale ε0 by a one-loop resummation, so finding a controlled approximation above scale ε0 requires further investigations. 3.3. Further remarks I end with a few remarks that relate the results stated in Sec. 2 to the bounds motivated in Sec. 3. I start with the discussion of the regularity of the self-energy. The above bounds were only sufficient to prove that the self-energy and the counterterm function K are C 1 in the external momentum. How does one get from that
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M. SALMHOFER
to the statement that K is C 2 ? It turns out that getting C 2 is still a much trickier problem which cannot be done by the above volume estimates alone. The main points leading to its solution are: (1) For the set of Fermi surfaces satisfying (B), the optimal volume improvement exponent (shown to be ≥ 12 above) is actually “almost equal to one”, more precisely, the gain in Lemma 3.2 is not just M h/2 but |h|M h in d = 2, and M h in d ≥ 3 [8, Theorem 1.1]. Inserted into the bounds, this shows that the second derivative of Σ and K is at most logarithmically divergent, so one may hope for convergence by doing still more careful bounds. (2) In second order, this is possible by a very detailed analysis of the singularities that goes well beyond volume bounds, and one can even show that K is C k if e is C k , for any k ≥ 2 [8, Theorem 1.2]. For d ≥ 3, we show that the selfenergy σ is C 2 if e is C 2 [8, Theorem 1.2]. In two dimensions, our bounds for the second derivative of σ are logarithmically divergent, i.e. σ ∼ k0 2 log k0 for k0 → 0. Explicit calculations [2] show that there is indeed such a logarithm, so our bounds do not overestimate the actual behaviour. Here one sees that the function K, which is C 2 also for d = 2, is more regular than σ. (3) The analysis of the second-order K is quite tricky, and it depends on too many details specific to the second order situation to be done for general graphs. However, by a new graph classification involving double overlaps [9, Sec. 2], we can show that volume estimates are sufficient to prove finiteness of the second derivative for all graphs except the second order and two related graphs (which can also be treated explicitly [8, Sec. 4]). This also provides a classification of all graphs that can contribute to the k0 2 log k0 -behaviour of σ. The graphs that contribute can be obtained by a generalized RPA resummation. The combination of these items into a common strategy is done in [8, 9]. The reason why the asymmetric system is a Fermi liquid is now rather easily dis(−) cussed: The asymmetry of S implies that the function V1 obeys a bound M j+h/3 uniformly in Q [8, Lemma 4.8]. In other words, at Q = 0 there is no singularity because S and −S are transversal to one another. There is also no singularity at nonzero Q. Curvature effects alone are not sufficient for this; for details, see [8, Appendix C]. Therefore there are no singularities in the ladder diagrams. By Theorem 3.3, no other contribution to the four-point function can be singular, so the four-point function is bounded. Converting these statements, which we proved in any finite order in perturbation theory, to non-perturbative ones requires a combination of the methods of [7, 8, 9] with those of [4, 3]. 4. Summary In summary, improved power counting is a method for proving improved regularity of Green functions, and thus of the self-energy, the vertex function etc., to all orders of perturbation theory. It is based on elementary volume estimates that follow directly from the geometry of the Fermi surface. These bounds do not
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require any explicit calculations, such as doing frequency sums or angular integrals exactly, but they give the correct behaviour of the self-energy and the particle– particle Green function (possibly up to logarithms). One can see this by comparing our bounds (which apply to arbitrary graphs) to explicit calculations in low orders. The only case known to me where the bound overestimates the actual behaviour is that of the particle–hole bubble. In that integral there is an extra sign cancellation, which is well-understood (also in presence of a cutoff, see [9, Lemma 2.42]. The notion of overlapping graphs makes precise in which situations certain one-loop resummations are justified. The condition that overlaps, or even multiple overlaps, occur in large graphs provides a rigorous meaning to the statement that “more loop integrations smooth out the singularities”. The graph classification of [7, 9] determines for which graphs this smoothing really occurs: there is no smoothing for non-overlapping graphs. The geometric conditions we make on the Fermi surface are not only sufficient, but also necessary for these regularity properties. For surfaces with flat sides, the volume improvement effect, and thus the power counting gain discussed above, is absent, and indeed, the singularities suggested by naive power counting are really there. For instance, the self-energy fails to be C 1 . Acknowledgements I would like to thank Joel Feldman for carefully reading this text and for various helpful suggestions. I also thank Walter Metzner for pointing out reference [2] to me, Volker Bach and the Erwin–Schr¨ odinger Institute for the invitation the workshop, and the ESI for its support during the workshop. References [1] B. L. Altshuler, L. B. Ioffe and A. J. Millis, “Low-energy properties of Fermions with singular interactions”, Phys. Rev. B50 (1994) 14048. [2] S. Fujimoto, “Anomalous damping of quasiparticles in two-dimensional Fermi Systems”, J. Physical Soc. Japan 60 (1991) 2013. [3] J. Feldman, H. Kn¨ orrer, D. Lehmann and E. Trubowitz, “Fermi liquids in two space dimensions”, in Constructive Physics, ed. V. Rivasseau, Springer Lecture Notes in Physics, 1995, and to appear. [4] J. Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, “An infinite volume expansion for many fermion Green’s Functions”, Helvetica Physica Acta 65 (1992) 679. [5] J. Feldman and E. Trubowitz, “Perturbation theory for many-fermion systems”, Helvetica Physica Acta 63 (1990) 156. [6] J. Feldman and E. Trubowitz, “The flow of an electron–phonon system to the superconducting state”, Helvetica Physica Acta 64 (1991) 213. [7] J. Feldman, M. Salmhofer and E. Trubowitz, “Perturbation theory around non-nested Fermi surfaces I. Keeping the Fermi surface fixed”, J. Stat. Phys. 84 (1996) 1209. [8] J. Feldman, M. Salmhofer and E. Trubowitz, “Regularity of the moving Fermi surface: RPA contributions”, to appear in Commun. Pure Appl. Math. [9] J. Feldman, M. Salmhofer and E. Trubowitz, “Regularity of interacting nonspherical Fermi surfaces: The full selfenergy”, to appear in Commun. Pure Appl. Math. [10] J. Feldman, M. Salmhofer and E. Trubowitz, in preparation.
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[11] G. Benfatto and G. Gallavotti, “Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems”, J. Stat. Phys. 59 (1990) 541. [12] G. Benfatto, G. Gallavotti, A. Procacci and B. Scoppola, “Beta function and Schwinger functions for a many-Fermion system in one dimension. Anomaly of the Fermi Surface”, Commun. Math. Phys. 160 (1994) 93. [13] D. M. King et al., Phys. Rev. Lett. 73 (1994) 3298; K. Gofron et al., Phys. Rev. Lett. 73 (1994) 3302.
STABILITY OF BIFURCATING SOLUTIONS FOR THE GINZBURG LANDAU EQUATIONS CATHERINE BOLLEY Ecole Centrale de Nantes BP 92101 44321 Nantes Cedex 03, France
BERNARD HELFFER D´ epartement de Math´ ematiques Universit´ e Paris-Sud F-91405 Orsay, France Received 19 May 1997 This paper is concerned with superconducting solutions of the Ginzburg–Landau equations for a film. We study the structure and the stability of the bifurcating solutions starting from normal solutions as functions of the parameters (κ, d), where d is the thickness of the film and κ is the Ginzburg–Landau parameter characterizing the material. Although κ and d play independent roles in the determination of these properties, we will exhibit the dominant role taken up by the product κd in the existence and uniqueness of bifurcating solutions as much as in their stability. Using the semi-classical analysis developed in our previous papers for getting the existence of asymmetric solutions and asymptotics for the supercooling field, we prove in particular√that the symmetric bifurcating solutions are stable for (κ, d) such that κd is small and d ≤ 5 − η (for any η > 0) and unstable for (κ, d) such that κd is large. We also show the existence of an explicit critical value Σ0 such that, for κ ≤ Σ0 − η and κd large, the asymmetric solutions are unstable, while, for κ ≥ Σ0 + η and κd large, the asymmetric solutions are stable. Finally, we also analyze the symmetric problem which leads to other stability results.
1. Introduction The states of a superconducting material, submitted to an external magnetic field, are described by the superconductivity theory introduced by V. L. Ginzburg and L. D. Landau in [16]. In the case when the material is a film and when the ~ e , is parallel to its surface, these authors consider a oneexternal magnetic field, H dimensional functional, known as the Ginzburg–Landau functional for a film Z (∆G)h (f, A) =
d 2
−d 2
1 κ−2 f 02 − f 2 + f 4 + A2 f 2 + (A0 − h)2 dx 2
(1.1)
which is defined for (f, A) ∈ (H 1 (] − d/2, d/2[))2 . Here, f is a real “wave” function whose modulus gives the density of super-electrons in the material and, by a choice of a divergence free gauge, A is the only non-zero component of the inner magnetic potential vector. The states of the film are determined as the critical points (f, A) of 579 Reviews in Mathematical Physics, Vol. 10, No. 5 (1998) 579–626 c World Scientific Publishing Company
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C. BOLLEY and B. HELFFER
the functional (∆G)h . The properties of these states depend on three characteristic parameters: κ, called the Ginzburg–Landau parameter, d > 0, the thickness of the ~ e |. film, and h > 0 which is proportional to |H The critical points of the functional (∆G)h satisfy the following equations, called Ginzburg–Landau equations on ] − d/2, d/2[, (a) (b) (GL)d (c) (d)
−κ−2 f 00 − f + f 3 + A2 f = 0 in ] − d/2, d/2[ , f 0 (±d/2) = 0 , −A00 + f 2 A = 0 in ] − d/2, d/2[ ,
(1.2)
A0 (±d/2) = h ,
where f and A are in H 2 (] − d/2, d/2[). These equations will often be denoted by (GL)d in reference to the width d. For given κ > 0, h ≥ 0 and e ∈ R, it is immediate to see that (f, A) = (0, h(x+e)) is a solution of (1.2). These solutions are called normal solutions. The other solutions of (GL)d for which f is not identically 0 are called superconducting solutions. In preceding papers, mainly in [2, 7], (see also the older study of F. Odeh [22]), we have proved, using a bifurcation study, the existence of curves of superconducting solutions starting from particular normal solutions. We define bifurcating solutions for fixed κ and d. In such a case, we actually consider (∆G)h as a functional defined on a set of triples (f, A; h) and “solutions” as triples (f, A; h) in H 2 (] − d/2, d/2[) × H 2 (] − d/2, d/2[) × R satisfying (GL)d and we write ∆G(f, A; h) = (∆G)h (f, A). If we now consider a continuous curve → (f (., ), A(., ); h()) of solutions, we shall say that it is a bifurcating curve of solutions starting from a normal solution (0, h0 (x + e); h0 ), with h0 > 0 and e ∈ R, if these solutions are superconducting solutions and if there exists 0 > 0 such that the solutions (f (., ), A(., ); h()) exist for || ≤ 0 and if the map → (f (., ), A(., ); h()) is continuous from [−0 , 0 ] to H 2 (] − d/2, d/2[) × H 2 (] − d/2, d/2[) × R and satisfies (f (., 0), A(., 0); h(0)) = (0, h0 (x + e); h0 ) . When this map is C 1 , we speak about a C 1 -bifurcation. ˆ of (GL)d is a locally stable solution for ˆ h) We also say that a solution (fˆ, A; ˆ (GL)d if it gives, at fixed h, a local minimum of the GL functional with respect to (f, A). Otherwise, it will be called an unstable solution. We recall (see [4, Proposition 0.1]) that, for any κ > 0, d > 0 and e ∈ R, ¯ = h(κ, ¯ d, e) such that the normal solutions (0, h(x + e); h) there exists a unique h ¯ d, e) and unstable when 0 < h < h(κ, ¯ d, e). The are locally stable when h > h(κ, ¯ d, e) of h will be characterized in (2.4). critical value h(κ, We adopt the following definitions.
STABILITY OF BIFURCATING SOLUTIONS
...
581
Definition 1.1.a Let κ > 0 and d > 0. 1. A bifurcating curve of solutions for (GL)d of the form (2.6) starting from a normal solution (0, h0 (x+e)) is called subcritical at (0, h0 (x+e)) if there exists 0 (κ, d) such that the bifurcating solutions are unstable for 0 < || ≤ 0 (κ, d). 2. A bifurcating curve (2.6) is called supercritical at (0, h0 (x + e)), if there exists 0 (κ, d) s.t. the bifurcating solutions are locally stable solutions for 0 < || ≤ 0 (κ, d). 3. When the bifurcating solutions are locally stable for 0 < ≤ 0 (κ, d) and unstable for ¯0 (κ, d) ≤ < 0 (or unstable for > 0 and locally stable for < 0), the bifurcation is called transcritical at (0, h0 (x + e)). In previous papers, mainly in [3] and [4], we have studied the bifurcating solutions starting from normal solutions by considering a new scaling which will be recalled later. We have then deduced existence and uniqueness results for any fixed value of the parameter κ and when d tends to 0 or to +∞. We prove, in this paper, that the domain of validity of our results is improved by considering as main parameter the product κd, in the limit κd → 0 as well as in the limit κd → +∞. In these asymptotic regimes, we study the structure of the superconducting solutions starting from normal solutions. One part of this work concerns the sign 1 of ∂h ∂ (0) when h is C. Another part concerns the local stability of the bifurcating solutions. The plan of this study is the following. In Sec. 2 we recall results on a spectral problem attached to a linearization of the GL equations at a normal solution. In Sec. 3, we extend, in new theorems, some previous existence and uniqueness results on bifurcating solutions to regims when κd tends to 0 or to +∞. Section 4 presents the main results on the structure of the bifurcating solutions. In Sec. 5, we study the sign of ∂h ∂ (0) in the symmetric case (e = 0), first in Subsec. 5.1, when κd tends to +∞, then in Subsec. 5.2, when κd tends to 0. In Sec. 6, we study the sign of ∂h ∂ (0) in the asymmetric case (e 6= 0) when κd tends to +∞. Section 7 is devoted to the stability results. We determine, in Subsec. 7.1, the two first eigenvalues of the spectral problem attached to a linearization of the GL equations at a bifurcating solution. In Subsec. 7.2, we study the stability or instability of the symmetric bifurcating solutions, first when κd tends to +∞, then when κd tends to 0, and finally in the case when one restricts the problem to the symmetric solutions. The stability or instability of the asymmetric bifurcating solutions is treated in Subsec. 7.3 when κd tends to +∞. 2. General Results on Bifurcating Solutions The starting point of this study is the observation that the rigorous results established in our previous articles can also be used to determine the structure and the stability of bifurcating solutions starting from normal solutions. Before going a Some authors adopt other definitions.
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further, we think that there is a need for a short summary of these results. This leads us, in many cases, to actually improve their domain of validity. 2.1. Necessary and sufficient conditions for the existence of bifurcating solutions In [2] and [7], we have given necessary conditions and sufficient conditions on the parameters e and h = h0 for the existence of bifurcating solutions starting from a normal solution (0, h(x + e); h). One necessary condition is the existence of a double eigenvalue λ equal to zero for the spectral problem attached to a linearization of the GL equations at the normal solution (0, h(x + e); h). For all κ > 0, d > 0, h > 0 and e ∈ R, this spectral problem is the following: ( −κ−2 φ00 + h2 (x + e)2 φ − φ = λφ in ] − d/2, d/2[ (a) φ0 (±d/2) = 0 , ( (2.1) −v 00 = λv in ] − d/2, d/2[ (b) v 0 (±d/2) = 0 with (φ, v) ∈ (H 2 (] − d/2, d/2[))2 and λ ∈ R. This is a diagonal system which always admits φv = 0c , with c ∈ R, as an eigenvector associated to the eigenvalue λ = 0. We consequently get that a necessary condition for the existence of a bifurcation is that λ = 0 is also an eigenvalue for the problem (2.1)(a). This leads us to consider, with τ = 1 + λ, the spectral problem −2 00 2 2 −κ φ + h (x + e) φ = τ φ in ] − d/2, d/2[ (2.2) φ0 (±d/2) = 0 , 2 φ ∈ H (] − d/2, d/2[) , with the normalization condition kφkL2 (]−d/2,d/2[) = 1 .
(2.3)
The previous necessary condition is now that 1 is an eigenvalue of the spectral problem (2.2). If we add the conditionb that φ > 0, and if we denote by τ = τ (κ, d, e, h) the principal eigenvalue of the Neumann problem, the necessary condition becomes that τ (κ, d, e, h) = 1 . We have proved in [4, Proposition 0.1] that, for all κ > 0, d > 0 and e ∈ R, there ¯ d, e) such that exists a unique h = h(κ, ¯ d, e)) = 1 . τ (κ, d, e, h(κ,
(2.4)
b This condition is not justified mathematically but seems to us natural if one want to analyze the most stable solutions.
STABILITY OF BIFURCATING SOLUTIONS
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583
The condition (2.4) is not sufficient for bifurcations to appear and we gave in [7] a more precise study. We have, indeed, proved the following existence theorem of bifurcating solutions. Theorem 2.1 [Theorem 2.1 in [7] (see also [2])]. Let (κ, d) ∈]0, +∞[2 and let (e, h0 ) ∈ R×]0, +∞[ satisfying ¯ d, e) , (a) h0 = h(κ, ∂τ (κ, d, e, h0 ) = 0 , (2.5) ∂e ∂2τ (c) (κ, d, e, h0 ) 6= 0 . ∂e2 Then, there exists a constant ˜0 = ˜0 (κ, d) > 0 and a C ∞ curve, in a neighborhood of 0, → (f (., ), A(., ); h()) of superconducting solutions such that, for 0 < || ≤ ˜0 , (b)
f (x, ) = f0 (x) + 3 f1 (x) + o(3 )
in H 2 (] − d/2, d/2[) ,
A(x, ) = A0 (x) + 2 A1 (x) + o(2 )
in H 2 (] − d/2, d/2[) ,
(2.6)
h() = h0 + 2 h1 + o(2 ) , where f0 is the principal normalized positive eigenfunction φ defined by (2.2), h0 = ¯ d; e) and A0 = h0 (x + e). h(κ, Moreover, there exist constants 0 = 0 (κ, d) > 0 and γ0 = γ0 (κ, d) > 0 such that, for 0 < || ≤ 0 , the solution (f (., ), A(., ); h()) is the unique solution of (GL)d such that (i)
kf (., )kH 2 (]−d/2,d/2[) ≤ γ0 ,
(ii)
kA(., ) − A0 kH 2 (]−d/2,d/2[) ≤ γ0 ,
(iii)
|h() − h0 | ≤ γ0 ,
(iv)
(f (., ), f0 )L2 (]−d/2,d/2[) = .
Condition (2.5)(b) means that 1 is a critical value for τ , and (2.5)(c) is a condition of non degeneracy. The two conditions (2.5)(a)–(b) are necessary conditions for the existence of bifurcating solutions starting from (0, h0 (x + e); h0 ) (see [2] and [7]). ¯ d, 0) verifying (2.4) satisfies automatically When e = 0, a value of h = h(κ, (2.5)(b). We know that condition (c) is at least satisfied when d is small enough or large enough and that there is at least one point for which condition (c) is not satisfied (see [3]). We also know the existence of values of e, different from zero giving bifurcating solutions. These results are extended in Sec. 3. 2.2. The spectral problem 2.2.1. Scalings We have used, in preceding papers, two different scalings for studying (2.2) and we shall refer to them often.
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(i) The first one, is given by y=
√ κh(x + e) ,
d√ κh , a= 2
¯ = (κh)−1/4 φ(x) , f(y)
√ c = e κh ,
(2.7)
κτ . µ= h
It will be used in this paper when κd is large. The spectral problem (2.2)–(2.3) becomes, for a > 0 and c ∈ R, ¯ ¯ P f = µf in ] − a + c, a + c[ , 0 ¯ f (±a + c) = 0 , ¯ f ∈ H 2 (] − a + c, a + c[) ,
(2.8)
kf¯kL2 (]−a+c,a+c[) = 1 ,
(2.9)
with f¯ = f¯(y; a, c) and
where P is the harmonic oscillator P ≡−
d2 + y2 . dy 2
(2.10)
(ii) The second scaling gives an interval independent of d. It is defined by x ¯ , φ(u) = d1/2 φ(x) , d e d√ ¯ = κ2 d2 τ . κh , c˜ = , λ a= 2 d
u=
(2.11)
It will be used when κd is small. The spectral problem becomes 4 2¯ ¯¯ ¯00 −φ + 16a (u + c˜) φ = λφ in ] − 1/2, 1/2[ , 0 ¯ φ (±1/2) = 0 , ¯ φ ∈ H 2 (] − 1/2, 1/2]) ,
(2.12)
¯ a, c˜) and with φ¯ = φ(u; kφ¯ kL2 (]−1/2,1/2[) = 1 .
(2.13)
(iii) We remark that (2.8) and (2.12) are linked by the change of variables and parameters ¯ λ c. (2.14) y = 2au + c , µ = 2 , c = 2a˜ 4a
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2.2.2. The principal eigenvalue ¯ and φ¯ depending only on Considering (2.12) with c˜ = 0 gives eigen-elements λ a and verifying ( ¯ φ¯ in ] − 1/2, 1/2[ , −φ¯00 + 16a4 u2 φ¯ = λ(a) (2.15) φ¯0 (±1/2) = 0 , √ where, as in (2.7), a = d2 κh. ¯ are simple and are C ∞ This is a Sturm–Liouville Problem whose eigenvalues λ functions on ]0, +∞[ with respect to the parameter a (see [20]). By differentiating this equation with respect to a, we get ¯ ∂λ (a) = 64a3 ∂a
Z
1 2
u2 φ¯2 (u)du ,
(2.16)
− 12
so that ¯ a → λ(a) is a strictly increasing function of a on ]0, +∞[ .
(2.17)
Moreover, it satisfies ¯ ¯ λ(0) = 0 and λ(+∞) = +∞ ,
(2.18)
with, when a → 0, (a)
4 ¯ λ(a) = a4 + O(a8 ) , 3
(b)
¯ a) = 1 + O(a4 ) , φ(u,
(2.19)
and, when a → +∞, 5 ¯ . λ(a) = 4a2 + O exp − a2 8
(2.20)
(other asymptotic results, when a → +∞, are recalled in Subsec. 5.1). Using now (2.16) and (2.19(b), we get ¯ 16 3 ∂λ (a) = a + O(a7 ) as a → 0 . ∂a 3
(2.21)
Let us now consider the scaling (2.7) with c = 0. We get, using (2.14), (2.19) and (2.21), for a > 0 small enough, a2 + O(a6 ) , 3
(a)
µ(a) =
(b)
2 ∂µ (a) = a + O(a5 ) . ∂a 3
(2.22)
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¯ d, e) 2.2.3. Universal lower bounds for h(κ, ¯ d, e). Let us now give lower bounds for h(κ, Lemma 2.2. (a) Let e = 0 and (κ, d) ∈]0, +∞[2 , then ¯h(κ, d, 0) > κ .
(2.23)
(b) Let (κ, d) ∈]0, +∞[2 and e ∈ R. If ¯h(κ, d, e) satisfies (2.5)(b) then, e ∈ [−d/2, d/2] . (c) Let (κ, d) ∈]0, +∞[2 , then, (i)
when e ∈ [−d/2, d/2] ,
(ii)
when e = 0 ,
¯ d, e) ≥ dh(κ,
√ 3.
√ ¯ d, 0) ≥ 2 3 . dh(κ,
(2.24)
Part (a) is proved in [3, Proposition 2.5)] Part (b) in [4] (Formula (2.7)). Part (c)–(i) results immediatly from [4, relation 2.14]. Part (c)–(ii) results from Corollary 2.2 and Formula (2.15) in [4]. 2.2.4. Heilman–Feynman relations We shall also use the following relations between the derivatives of τ and those ¯ of h, Lemma 2.3. (a) Let (κ, d) ∈]0, +∞[2 and e ∈ R, then, (i)
(ii)
Z d2 ¯ ∂τ ¯ d, e) · ∂ h (κ, d, e) (κ, d, e, ¯ h(κ, d, e)) = −2h(κ, (x + e)2 φ(x)2 dx , d ∂e ∂e −2 Z d2 ∂τ ¯2 (κ, d, e, ¯ h(κ, d, e)) = 2h (x + e)φ(x)2 dx . ∂e −d 2
¯ d, e)) satisfying (2.5)(a, b), then, (b) Let (κ, d) ∈]0, +∞[2 and (e, h0 = h(κ, Z d2 2¯ ∂2τ ¯ ¯ d, e)· ∂ h (κ, d, e) (i) (κ, d, e, h(κ, d, e)) = −2 h(κ, (x+e)2 φ(x)2 dx. d ∂e2 ∂e2 −2 # " d Z 2 ∂φ ∂2τ 2 ¯ (κ, d, e) 1 + 2 (κ, d, e, ¯ h(κ, d, e)) = 2h (x + e) (x)φ(x)dx . (ii) ∂e2 ∂e −d 2 The lemma is proved by differentiating (2.4) with respect to e, and (2.2), when τ = 1, with respect to h. 2.3. Formulas for h1 The computation of the Ginzburg–Landau functional ∆G(f (., ), A(., ); h()) on a bifurcating solution gives:
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Proposition 2.4 [see Formula (2.20) in [7]]. Let κ > 0 and d > 0. Let (e, h0 ) satisfying (2.5). Then, there exists 0 > 0 such that for || ≤ 0 , !# " Z d 2
∆G(f (., ), A(., ); h()) = h1 /
−d 2
4 + O(6 ) .
h20 (x + e)2 f02 dx
(2.25)
This formula shows that the energy ∆G, when calculated at a bifurcating solution, has, for small enough, the same sign as the constant h1 (when h1 6= 0). Now, with the notations of [7], we split A1 defined in (2.6) as A1 = a1 + A1,0 + h1 x ,
with a1 ∈ R ,
where A1,0 is the unique solution in H 2 (] − d/2, d/2[) of the following problem 00 2 −A1,0 + f0 A0 = 0 in ] − d/2, d/2[ , 0 A1,0 (±d/2) = 0 , A1,0 (0) = 0 ,
(2.26)
and get the following formula (see [7, Sec. 2] Sec. 2 or [21]): 2
h1 h0
Z
d 2
−d 2
Z (A0 (x))2 (f0 (x))2 dx = −
d 2
−d 2
Z (f0 (x))4 dx + 2
d 2
−d 2
(A01,0 (x))2 dx ,
(2.27)
which will be useful for the determination of the sign of h1 . An analogous formula is studied in S. J. Chapman [9] and in C. Bolley-B. Helffer [7] in two limiting problems associated with d = ∞: the symmetric case (e = 0) and the asymmetric one (e 6= 0). The first case gives the condition h1 (κ − 2−1/2 ) > 0 and exhibits consequently the well-known critical value κ = 2−1/2 (between type 1 and type 2 superconductors). In [9], the author also gives a formal study of the stability of bifurcating solutions for this limiting problem, but the splitting of the first eigenvalue of (2.1), which has a multiplicity two at the bifurcation, is not considered (see our study in Sec. 7). The sign of ∆G is also studied in the paperc by S. P. Hastings and W. C. Troy [17], when κ is large, for particular asymmetric solutions, by using the limiting problem d = +∞. They also show the existence of stable asymmetric superconducting solutions for κ large and suitable h’s, but they don’t exhibit them. 2.4. The critical value Σ0 The analysis of the second case (the asymmetric case) leads to the introduction of another limiting critical value for κ, denoted by Σ0 , which can be defined as follows. c In a first version of their paper, these authors assert that our study of bifurcating solutions starting from normal solutions, given in previous papers [2, 3, 5, 4], concern only the case κ → 0. We emphasize that these results are true for any κ.
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We consider the problem P φ = µφ
in R ,
(2.28)
2
d 2 where P is, as in (2.10), defined by P ≡ − dy 2 + y . For every parameter µ, we can choose a basis {φ1 (., µ), φ2 (., µ)} for the set of the solutions of (2.28), such that, when y → +∞: 2 y · y (µ−1)/2 · (1 + O(y −2 )) , (2.29) φ1 (y, µ) = exp − 2
φ2 (y, µ) = exp
y2 2
· y −(µ+1)/2 · (1 + O(y −2 )) ,
(2.30)
and where the O are locally uniform with respect to µ (see Y. Sibuya [24]). The function φ1 (., µ) is the solution of P − µ on R whose behavior, when y → +∞, is such that lim φ1 (y, µ) = 0 . y→+∞
Let α > 0, and let µ1 (α) be the first eigenvalue of the harmonic oscillator P in ] − α, +∞[ with the Neumann condition at −α. Then, the first eigenfunction is necessary given by γφ1 (., µ1 (α)), where γ is a constant. It is proved in [12] that the function [0, +∞[3 α → µ1 (α) has a minimum µ01 which is reached at a unique α0 > 0. Moreover, µ01 ≤ µ1 (α) ≤ 1 ,
with µ01 = µ1 (α0 ) .
(2.31)
Computations in [4], Appendix 2, give µ1 (α0 ) ≈ 0, 59 and α0 ≈ 0, 73. Let us now introduce, for α > 0, the two functions, Z ∞ φ1 (y, µ1 (α))4 dy ρ(α) =
(2.32)
for any α > 0 ,
−α
and
Z Σ0 (α) = 2
∞
−α
Z
−α
Then, we define Σ0 by
Σ0 =
2
y
t · φ1 (t, µ1 (α))2 dt
σ(α0 ) ρ(α0 )
dy .
(2.33)
12 .
(2.34)
The constant Σ0 is computed in [5, Subsec. 9.4.2]. We got, Σ0 ≈ 0, 4 . We shall meet, in Theorems 4.2 and 6.2, the two critical values 2−1/2 and Σ0 of κ, when the parameter κd tends to +∞. 3. Existence and Uniqueness Theorems for Bifurcating Solutions Existence theorems are given in earlier papers, but we give, in this subsection, more general and new results by considering the natural parameter κd.
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The following uniqueness theorem extends Theorem 0.7 in [4]. Theorem 3.1. (a) There exists a constant a0 > 0 such that, ∂τ ¯ (κ, d, e); (κ, d, e, h(κ, d, e)) = 0 and 0 < κd ≤ a0 ∂e = {(κ, d, 0); 0 < κd ≤ a0 } . (b) Let us suppose that e = 0. There exists a0 > 0 such that, (i) (ii)
∂2τ ¯ d, 0)) > 0 , (κ, d, 0, h(κ, ∂e2 ∂2τ ¯ d, 0)) < 0 . for κd ≥ a−1 (κ, d, 0, h(κ, 0 , ∂e2 for κd ≤ a0 ,
The first part was proved in [4, Theorem 0.7], in the case when κ is fixed and d small, and the second part was given in [5], for also fixed κ, (see (1.23)2,3 ) using estimates of [3]. Proof of Theorem 3.1. In [4, Theorem 0.7], we have established, using ¯ 1/2 small enough, the relation the scaling (2.7) with e = 0, that for a = d2 (κh) ∂τ ¯ ∂e (κ, d, e, h(κ, d, e)) = 0 implies e = 0. So, for proving (a), we only need a control ¯ d, 0) as κd tends to 0. This is given by the following lemma: of dh(κ, Lemma 3.2. There exists a constant a0 > 0 such that, for (κ, d) satisfying 0 < κd ≤ a0 , then √ ¯ d, 0) = 2 3(1 + O(κ2 d2 )) , (3.1) dh(κ, This lemma completes (2.24)(ii). Proof of the lemma. We take back the proof of Corollary 7.6 in [4], and deduce a more accurate asymptotic formula than thatpgiven in this corollary. ¯ d, 0) is small enough, It is proved in [4] that for (κ, d) such that a = d2 κh(κ, ∂τ ¯ the unique h solution of ∂e = 0 is given by, ¯ = 4u , h κd2
(3.2)
√ κ2 d2 , Ψ(u) ≡ uµ( u) = 4
(3.3)
where u is the solution of
2
d 2 and were µ(a) is the first eigenvalue of the harmonic oscillator P ≡ − dy 2 + y , for the Neumann problem on the interval ] − a, a[ (or on ]0, a[). The function ]0, +∞[3 a → µ(a) can be extended as a positive C ∞ even function on R verifying (2.22) when a ∈] − a0 , a0 [ with a0 small enough. Consequently, Ψ
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can be extended as a C ∞ positive eigenfunction and can be seen as the square of a C ∞ function χ which, using (2.22) satisfies u χ(u) = √ + O(u3 ) as u → 0 . 3
(3.4)
Equation (3.3) can then be written, χ(u) =
κd , 2
with χ invertible in a neighborhood of u = 0. The implicit function Theorem gives, for κd small, the existence of a unique u = u(κd), with κd u √ = + O((κd)3 ) . 2 3 Using (3.2), (3.1) follows for a and κd small enough. 1/2 ¯ , we deduce that Now, from (2.17), (2.19)(a) and the relation κd = (λ(a))
a → 0 as κd → 0 . Consequently, Lemma 3.2 is proved and implies, for κd small enough, a=
1 1 1 1 d ¯ (κh(κ, d, e)) 2 = 3 4 2− 2 (κd) 2 (1 + O(κ2 d2 )) , 2
(3.5)
Proof of (a) in Theorem 3.1. This results from (3.5) and from Theorem 3.4 in [4]. Proof of (b)(i) in Theorem 3.1. According to Lemma 3.2 and (3.5), the proof of (b)(i) is the same as in [3, Proof of Proposition 2.18].d We reproduce it after rescaling for completeness. 2
We shall use the expression of ∂∂eτ2 given by Lemma 2.3 (b)(ii) with e = 0 in the scaling (2.11). We get, for all κ > 0, d > 0 and for c = 0 (or equivalently e = 0) ¯ d, 0), and h = h(κ, # " Z 12 ∂ φ¯ ¯ ∂ 2τ 2 ¯ · φ(u)du . (3.6) (κ, d, 0, h) = 2h 1 + 2 u· ∂e2 ∂˜ c − 12 We are going to prove that the integral term tends to 0 as κd → 0. ¯
Let us first prove that ∂∂φc˜ → 0 in H2 (] − 1/2, 1/2[), as κd → 0. ¯ We denote ∂∂˜φc by ψ. The equations satisfied by ψ are given by differentiating ¯ (2.12) and (2.13). Using that ∂∂˜λc (κ, d, 0, ¯h) = 0 (see Lemma 2.3 (a)(ii)), we get with p ¯ d, 0), when e = 0 and h = h(κ, ¯ d, 0), a = d2 κh(κ, d In the right-hand side of the last relation in [3, p. 268] one has to read 1 a4 h2 κ2 and not 2 (λ1 −a2 κ2 ) a4 h2 1 . 2 (λ1 −a2 κ2 )
In this proof, a = d and x is changed in
x . a
STABILITY OF BIFURCATING SOLUTIONS
...
−ψ 00 + 16a4 u2 ψ − κ2 d2 ψ = −32a4 uφ¯ in ] − 1/2, 1/2[ ψ 0 (±1/2) = 0 , ¯ L2 (]−1/2,1/2[) = 0 , (ψ, φ) ψ ∈ H 2 (] − 1/2, 1/2[) .
591
(3.7)
We define the operator T from H 2 (] − 1/2, 1/2[) to L2 (] − 1/2, 1/2[) (see Eq. (2.12)) by T : ξ → T ξ ≡ −ξ 00 + 16a4 u2 ξ . The spectral problem attached to this operator, with the Neumann conditions at ¯ d, 0), the first ±1/2, is a Sturm–Liouville Problem. With the choice h = h(κ, eigenvalue for this operator, with the Neumann condition at ±1/2, is ¯ = κ2 d2 , λ0 = λ and the second eigenvalue is given by λ1 =
inf
ξ∈H 1 (]−1/2,1/2[) ¯ (ξ,φ) =0 L2 (]−1/2,1/2[)
(T ξ, ξ)L2 (]−1/2,1/2[) . kξk2L2 (]−1/2,1/2[)
¯ we get Therefore, with the choice ξ = ψ (ψ is orthogonal to φ), (T ψ, ψ)L2 (]−1/2,1/2[) ≥ λ1 kψk2L2 (]−1/2,1/2[) .
(3.8)
Now, (3.7) implies ¯ ψ)L2 . (T ψ, ψ)L2 − κ2 d2 kψk2L2 = 16a4 (uφ, So (3.8) and the Cauchy–Schwarz inequality give ¯ L2 · kψkL2 . (λ1 − κ2 d2 )kψk2L2 ≤ 16a4 kuφk Using now (2.13) and the strict inequality λ1 > λ0 = κ2 d2 , we get kψkL2 ≤
16a4 1 · . 2 (λ1 − κ2 d2 )
(3.9)
Using the continuity of the eigenvalues with respect to the coefficients, we get that λ1 tends, when a → 0, to the second eigenvalue of the Neumann problem, 00 −ψ = µψ in ] − 1/2, 1/2[ , ψ 0 (±1/2) = 0 , ψ ∈ H 2 (] − 1/2, 1/2[) , which is equal to 4π 2 . Consequently, lim ψ = 0
a→0
in L2 (] − 1/2, 1/2[) .
(3.10)
A bootstrap argument implies the convergence in H 2 (] − 1/2, 1/2[). It results from (3.5) that ψ tends to 0 in H 2 (] − 1/2, 1/2[) as κd tends to 0.
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C. BOLLEY and B. HELFFER
2 Let us prove that ∂∂e2τ (κ, d, 0, ¯ h(κ, d, 0)) > 0 for κd small. We apply the Cauchy–Schwarz inequality to the integral appearing in (3.6). We get, using (2.13), Z 1
1 2
∂ φ¯ ∂ φ¯ ¯
(u) · φ(u)du ≤ u· . 2 ∂˜ −1 ∂˜ c c L2 (]−1/2,1/2[) 2 2
Therefore, using now (3.10), ∂∂eτ2 (κ, d, 0, h) is positive for κd small enough, and the relation (b)(i) of Theorem 3.1 follows. Proof of (ii). It is a consequence of the proof of [3, Proposition 2.21] where, √ ¯ using the scaling (2.7) with e = 0 (or c = 0), we have established that, for a = d2 κh large enough, then ∂2τ (κ, d, 0, ¯h(κ, d, 0)) < 0 . ∂e2 For getting (ii), we use√once again (2.17)–(2.18) or we simply remark, using ¯ > κd when e = 0. Therefore, a tends to +∞ as Lemma 2.2 (a), that, d2 κh 2 κd tends to +∞ and the assertion (ii) follows. We have studied in [3, 4] the existence of pairs (e, h0 ) with e = e(κ, d) 6= 0 ¯ d, e(κ, d)) such that (2.5) is satisfied. The results can be extended and h0 = h(κ, as follows. Theorem 3.3. There exists a constant a1 > 0 and a function (κ, d) → e¯(κ, d) defined for (κ, d) satisfying κd > a1 , such that ∂τ (κ, d, e, ¯h(κ, d, e)) = 0 and κd ≥ a1 (κ, d, e); ∂e [ = {(κ, d, 0); κd ≥ a1 } {(κ, d, e¯(κ, d)); κd ≥ a1 } [
{(κ, d, −¯ e(κ, d)); κd ≥ a1 } .
Proof. When κ > 0 is fixed and d is large, the result is proved in [4, Theorem 0.5] using the existence of e¯(κ, d) > 0 established in [3, Proposition 2.25]. Let us prove Theorem 3.3. We proceed as in [3], but for large κd instead of d large. Theorem 3.1 (ii) gives 2 that for κd ≥ a1 , with a1 large enough, ∂∂eτ2 (κ, d, 0) is strictly negative. Then, from 2¯ Lemma 2.3 (b)(ii), ∂∂eh2 (κ, d, 0) is strictly positive. Moreover, in [3, Proposition 2.24] we have established that for κ > 0 and d > 0 fixed, lim ¯h(κ, d, e) = 0 , e→+∞
Therefore, by continuity and differentiability of e → ¯h(κ, d, e), the existence for κd ¯ large enough of some e¯(κ, d) > 0 such that ∂∂eh (κ, d, e¯(κ, d)) = 0 follows from the Rolle Theorem. Lemma 2.3 (a)(i), gives then the equivalence with the condition
STABILITY OF BIFURCATING SOLUTIONS
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∂τ ¯ d, e¯(κ, d))) = 0 . (κ, d, e¯(κ, d), h(κ, ∂e Let us prove the uniqueness of e¯(κ, d) > 0 for large κd. In the scaling (2.7), we have proved in [4, Theorem 3.3] the uniqueness, for large a, of a value c¯ of c such that ∂µ ∂c (a, c) = 0. Using that, from (2.7), h3/2 ∂µ ∂τ (κ, d, e, h) = 1/2 (a, c) , ∂e ∂c κ ¯ d, e) the only point is to verify that κd large implies a large for any e such that h(κ, satisfies (2.5)(b). This is given by Lemma 2.2 (b) and (c)(i) which imply that d ¯ 1/2 ≥ 31/4 (κd)1/2 . We then get the uniqueness of e¯(κ, d) for large κd and 2 (κh) 2 eventually Theorem 3.3. In the following, we will say that a bifurcating solution is symmetric when f (., ) is even and A(., ) is odd and that it is asymmetric otherwise. When e = 0, an eigenfunction f0 = φ of (2.2) is then an even function and A0 is odd. We observe also that, if (f (x, ), A(x, ), h()) is a solution of the GL equations, then (f (−x, ), −A(−x, ), h()) and (−f (x, −), A(x, −), h(−)) are also solutions and that these solutions are equal when = 0. By uniqueness of the curve of bifurcation, we conclude, in the symmetric case, that, ( f (x, ) = f (−x, ) = −f (x, −) , (3.11) A(x, ) = −A(−x, ) = A(x, −) , so that f is even and A is odd in a neighborhood of = 0. We summarize the existence results of bifurcating solutions by the following theorem. Theorem 3.4. (i) There exists a constant a0 > 0 such that, for (κ, d) satisfying κd ≤ a0 , there exists a unique C ∞ curve → (f (., ), A(., ); h()) of bifurcating solutions starting from all the normal solutions (0, h0 (x + e); h0 ). These solutions ¯ d, 0) are starting from the particular normal solutions (0, h0 x; h0 ), where h0 = h(κ, satisfies (2.5), and are symmetric solutions. (ii) There exists a constant a1 > 0 such that for (κ, d) verifying κd ≥ a1 , there exist exactly three C ∞ curves → (f (., ), A(., ); h()) of bifurcating solutions starting from normal solutions (0, h0 (x + e); h0 ). One curve of solutions, starting from ¯ d, 0), is composed with symmetric solutions, the other (0, h0 x; h0 ) with h0 = h(κ, two ones are composed with asymmetric solutions corresponding to e = e¯(κ, d) and e = −¯ e(κ, d) and deduced from each other by symmetry. Proof. Theorem 3.1 (a) gives the uniqueness of a value of e (which is e = 0) leading to a curve of bifurcating solutions when κd is small enough. Theorem 2.1, combined with Theorem 3.1 (b), gives, for κd small or large enough, the uniqueness of the curve of bifurcating solutions when e = 0. Their symmetry follows from this uniqueness. We get (i).
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Theorem 3.3 implies the existence of exactly three values of e (e = 0, e = e¯(κ, d) and e = −¯ e(κ, d)) such that the conditions (2.5)(a, b) are satisfied when κd is large enough. The value e = 0 leads as in (i) to symmetric solutions. The condition (2.5)(c) was not proved to be satisfied in earlier papers when e = e¯(κ, d), but will result from the relations (7.16) and (7.8) established in Sec. 7. The second equation in (2.6) shows that, in that case, the first term for A is not symmetric so that the bifurcating solutions are asymmetric solutions. We get (ii). Remark 3.5. The existence of asymmetric solutions for κ fixed and d large enough was given by [2, Theorem 3.1] (which proves that the two conditions (2.5)(a, b) are sufficient for getting the existence of bifurcating solutions satisfying (2.6)) combined with [3, Proposition 2.25 (i)] (which proves that (2.5)(a, b) is satisfied for d large enough). 4. Main Results on the Structure of the Bifurcating Solutions Let us now give the main results proved in this paper concerning the bifurcating solutions. It is generally admitted in the literature (see for example [11] or [1]) that the bifurcating solutions starting from a normal solution are supercritical for all d when κ is large, and that they are supercritical for small d and subcritical for large d when κ is small. But it seems that no proof of any part of these results is available, except in the case of a bounded interval with d small which is studied in [7, Sec. 2]. The situation is in fact more complicated as we will see later. The a main purpose of this paper is to study two properties of the bifurcating solutions constructed in Theorem 2.1, which were not analyzed in our previous papers. The first one is to calculate the sign of h1 as function of the parameters, near the bifurcating points. The second one is to analyze the stability of the bifurcating solutions. The two studies are performed as function of κd and κ, or of κd and d according to the asymptotic regime in study. When considering the sign of h1 , we distinguish between the symmetric case (associated to e = 0) and the asymmetric case (associated to e = e¯(κ, d)). We first analyze the symmetric case. We prove in particular the two following theorems. Theorem 4.1 [κd large]. Let e = 0. For any η > 0, there exists a constant a1 > 0 such that, for (κ, d) satisfying κd ≥ a1 and |κ − 2−1/2 | > η, and for h0 satisfying (2.5), there exists a constant 1 = 1 (κ, d) > 0 s.t. the curve of superconducting solutions (f (., ), A(., ); h()) starting from the normal solution (0, h0 x; h0 ) satisfies for 0 < || ≤ 1 , (a) (b)
(2−1/2 − κ) · [h() − h0 ] > 0, (2−1/2 − κ) · ∆G(f (., ), A(., ); h()) > 0.
When d is fixed and κ is large, we have a more precise description for the asymptotic result.
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595
Theorem 4.2 [d fixed, κ large]. Let d > 0 and e = 0. There exists a constant κ0 = κ0 (d) > 2−1/2 such that, for κ ≥ κ0 and for h0 satisfying (2.5), there exists 1 = 1 (κ, d) > 0 s.t. the curve of bifurcating solutions (f (., ), A(., ); h()), starting from the normal solution (0, h0 x; h0 ) verifies, for 0 < || ≤ 1 , (a) (b)
h() < h0 , ∆G(f (., ), A(., ); h()) < 0.
There exists a constant C > 0 such that κ0 can be chosen s.t. 1 1 √ < κ0 (d) = √ + O(exp(−Cd2 )) 2 2
when d → +∞ .
(4.1)
The relation (4.1), together with Theorem 4.1, gives that κ = √12 is the limiting critical value for κ, when the thickness d of the film tends to +∞, deliminating different behaviors for the solutions. This asymptotic value is often used (as an approximation) for distinguishing a superconductor of type 1 and a superconductor of type 2. This critical value has also been discussed in [7]. We then study the asymptotics when κd √ tends to 0 and verify that, as in the limiting f − constant model, the value d = 5 is the theoretical value determining the locally stable solutions and the unstable ones. In the asymmetric case, we get similar results to Theorems 4.1 and 4.2 when κd tends to +∞, and we recover the critical value Σ0 , for κ, defined in (2.34). The stability of the bifurcating solutions is then studied by first considering the behavior of the spectral problem attached to a linearization of the GL equations at a bifurcating solution when is small. For = 0 (problem (2.1)), the lowest eigenvalue is of multiplicity two. When is small, this double eigenvalue splits in general into two distinct eigenvalues that we compute in Subsec. 7.1. The sign of one of them is opposite to the sign of h1 and the sign of the other is given by the 2 sign of ∂∂eτ2 (κ, d, e, h0 ); this value is linked to the existence or not of asymmetric solutions and can be positive or negative in function of (κ, d, e) (but independently of h1 ). We then study the sign of these two eigenvalues in the different asymptotic cases considered before. We prove, in particular: Theorem 4.3 [Symmetric bifurcating solutions, κd small]. √ Let e = 0. For any η > 0, there exists a0 > 0 such that for κd ≤ a0 and |d − 5| ≥ η, and for h0 satisfying (2.5), there exists 1 > 0 such that, for 0 < || ≤ 1 , the following properties are satisfied. √ (i) When d < 5, the bifurcating solutions (f (., ), A(., ); h()) starting from the normal solution (0, h0 x; h0 ) are locally stable and consequently supercritical. √ (ii) When d > 5, the bifurcating solutions starting from the normal solution (0, h0 x; h0 ) are unstable and consequently subcritical. Remark 4.4. We emphasize that the assumptions of Theorem √ 4.3 allow us, in particular, to treat the case κ > 0 and d → 0, and the case d 6= 5 and κ → 0.
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We complete the study of the symmetric case by an asymptotic instability result for large κd. Theorem 4.5 [Symmetric bifurcating solutions, κd large]. There exists a1 > 0 such that for (κ, d) satisfying κd ≥ a1 and for h0 satisfying (2.5), there exists 1 > 0 such that, for 0 < || ≤ 1 , the bifurcating solutions starting from the normal solution (0, h0 x; h0 ) are unstable and consequently subcritical. We remark that, to our knowledge, the instability property presented in this theorem in the case of a fixed large κ, does not seem to be proved nor even be mentioned in the literature, but it corresponds probably to the linear instability with respect to a two-dimensional perturbation introduced by S. J. Chapman in [9]. However, when we restrict our study to the research of symmetric solutions, as was done in another context in [6] and in previous articles [15] or [9], we consider another Ginzburg–Landau functional for which the symmetric bifurcating solutions are locally stable for κd and κ large enough (see Theorem 7.5). Concerning the asymmetric solutions, we prove Theorem 4.6 [Asymmetric bifurcating solutions]. Let a1 be the positive constant defined in Theorem 3.3 and Σ0 > 0 defined in (2.34). For any η > 0, there exists a constant a2 ≥ a1 such that for (κ, d) satisfying κd ≥ a2 and |κ−Σ0 | ≥ η and for e = e¯(κ, d) and h0 satisfying (2.5), there exists 1 > 0 such that, for 0 < || ≤ 1 , the following properties are satisfied: (i) When κ < Σ0 , the bifurcating solutions starting from the normal solution (0, h0 (x + e¯(κ, d)); h0 ) are subcritical. (ii) When κ > Σ0 , the bifurcating solutions starting from the normal solution (0, h0 (x + e¯(κ, d)); h0 ) are supercritical. Remark 4.7. Again, the assumptions of Theorem 4.6 allow us to treat, in particular, the case d > 0 and κ → 0 or +∞, or the case κ > Σ0 and d → +∞. Let us first consider h1 in the symmetric case. 5. The Sign of h1 in the Symmetric Case We have established in [3, Proposition 2.9] in the case when e = 0 and in [4, Proposition 5.3] in the case when e 6= 0, asymptotics for the eigenfunctions of the linearized problem (2.2)–(2.3), when the length d of the interval tends to +∞. For this purpose, we have used the scaling (2.7) which reduces the problem to the study of the Neumann realization of the harmonic oscillator: P ≡−
d2 + y2 dy 2
on some bounded interval ] − α, β[ .
In this section, we use these results for the study of h1 in the case when e = 0, with a symmetric interval ] − a, a[. The parameters are here κd and κ with κd large. Another part, Subsec. 5.2, concerns the study of h1 as function of κd and d, when κd is small.
STABILITY OF BIFURCATING SOLUTIONS
...
597
5.1. The asymptotics for large κd For studying the asymptotics of the principal normalized eigenfunction f0 when d tends to +∞, we have studied, in [3], the problem (2.8) with c = 0. In the spirit of B. Helffer–J. Sj¨ ostrand [18], we have constructed an approximation of f¯ and of µ valid when a tends to +∞. For this purpose, we have considered the problem (2.28) with µ = 1, that is Pf = f
in R ,
and the basis {φ1 (., 1), φ2 (., 1)}, defined by (2.29)–(2.30), of the solutions of this equation. In that particular case, we denote these functions more simply by φ1 (.) and φ2 (.). They satisfy 2 y for y ∈ R φ1 (y) = exp − 2 1 1 y2 · · 1+O as y → +∞ . 2 y y2 We have proved, using techniques from [18], the following result:
and
φ2 (y) = exp
(5.1)
Proposition 5.1 [see Proposition 2.9 in [3]]. There exist positive constants C1 , C2 , δ1 , δ2 (with δ1 and δ2 > 12 ) and a0 such that for a ≥ a0 , (i)
|µ(a) − 1| ≤ C1 exp(−δ1 a2 ) .
(5.2)
(ii)
supy∈[−a,a] |f¯(y) − f¯a (y)| ≤ C2 exp(−δ2 a2 ) ,
(5.3)
where f¯a is an even approximation of f¯ in the form 2 y y a ¯ + ρ(a) · φ2 (|y|) · Ξ , f (y) = β(a) exp − 2 a
(5.4)
with β(a) = π −1/4 + O(a exp(−a2 )) as a → +∞ , 1 −1/4 2 , a exp(−a ) · 1 + O ρ(a) = π a2
(5.5) (5.6)
and where Ξ is an even C ∞ (R) cutoff function s.t. Ξ(x) = 0if |x| ≤ 1/2 ;
Ξ(x) = 1 if |x| ≥ 3/4 .
Remark 5.2. The proof shows that we can choose δ1 =
11 16
and δ2 = 58 .
The idea of the proof was that the positive function φ1 could be a good approximation of a positive solution of P f¯ = µf¯ on ] − a, a[, when a is large enough, but φ1 does not satisfies the Neumann conditions at ±a. Therefore, using the cutoff function Ξ, we have added in (5.4), for large y > 0, a term ρ(a)φ2 (y)Ξ( ya ) in such a way that the boundary condition at x = a is satisfied. Then, we use the symmetry for y = −a.
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The constant β is chosen such that kf¯a kL2 (]a,a[) = 1. ¯ d, 0) as d tends to +∞ In [3], we have deduced asymptotics for f0 and h0 = h(κ, with κ fixed. Here, we keep the scaling (2.7) for the study of h1 by rewriting the relation (2.27) in these new variables and parameters. We get, with p y = κh0 x 2
h1 h0
Z
a
−a
¯0 (y)2 f¯(y)2 dy B
Z p = κh0 · −
a
f¯(y)4 dy + 2κ−2
−a
Z
a −a
0 2 ¯ B1,0 (y) dy ,
(5.7)
¯1,0 ∈ H 2 (] − a, a[) are defined by ¯0 and B where B ( ¯0 (y) = A0 (x) p B with y = κh0 x , ¯1,0 (y) = κA1,0 (x) , B ¯1,0 satisfies In particular, B 00 ¯1,0 (y) = y f¯(y)2 B ¯ 0 (±a) = 0 , B 1,0 ¯ B1,0 (0) = 0 . If we define Λ by
Z Λ≡ −
a
for y ∈] − a, a[ , (5.8)
f¯(y)4 dy + 2κ−2
−a
Z
a
−a
0 2 ¯ B1,0 (y) dy ,
(5.9)
and the function sign by, sign(x) = +1 when x > 0 ,
sign(x) = −1 when x < 0 .
(5.10)
then, from (5.7), sign(h1 ) = sign(Λ) .
(5.11)
Let us show Lemma 5.3. Let e = 0. For any η > 0, there exists a constant a1 such that for a ≥ a1 and |κ − 2−1/2 | ≥ η, we have 1 √ −κ ·Λ > 0. 2 Proof. We compute the two integrals appearing in the right-hand side of (5.9). Using (5.3) with δ2 = 58 , we get Z a Z a 5 2 4 a 4 ¯ ¯ f (y) dy = 2 f (y) dy + O a exp − a 8 −a 0 as a tends to +∞.
STABILITY OF BIFURCATING SOLUTIONS
Let us now compute
Z
...
599
a
f¯a (y)4 dy , 0
by using (5.4) and the binomial formula. We remark that, for a large enough, the product φ1 φ2 is bounded, and that by the choice of the cutoff function and the variations of the function φ2 , we have, for some constants C > 0 and C˜ > 0, 2 y a φ2 (y) ≤ C exp , ∀y ∈ [0, a] , 0 ≤ Ξ a 2 and ∀y ∈ [0, a] ,
2 a ˜ 0≤Ξ φ1 (y) ≤ C exp − . a 4 y
Therefore, as a → +∞, Z a Z f¯a (y)4 dy = β(a)4 0
a
0
2
5 , exp(−2y 2 )dy + O a exp − a2 4
so that, using (5.5): Z a f¯a (y)4 dy = 2−3/2 π −1/2 + O(a exp(−a2 )) ,
as a tends to + ∞ .
(5.12)
0
Let us now compute the last integral in (5.7). ¯1,0 verifies, for y ∈] − a, a[, The function B Z y 0 ¯ t(f¯(t))2 dt . B1,0 (y) = −a
Using again (5.3), we get, as a tends to +∞: Z y 5 2 0 2 2 ¯ t exp(−t )dt + O a exp − a B1,0 (y) = β(a) 8 −a 5 2 −1 −1/2 2 . exp(−y ) + O a exp − a = −2 π 8 Therefore, as a tends to +∞, Z a Z 0 2 −2 −1 ¯ B1,0 (y) dy = 2 π −a
5 2 2 exp(−2y )dy + O a exp − a 8 −a 5 . = 2−5/2 π −1/2 · 1 + O a2 exp − a2 8 a
2
We then get that Λ is equal, as a tends to +∞, to 5 2 −3/2 −1/2 −2 2 2 2 Λ=2 1 − 2κ + (1 + κ )O a exp − a . π κ 8
(5.13)
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Consequently, for any η > 0, |κ − 2−1/2 | ≥ η, and then for a ≥ a1 for a1 large enough, sign(Λ) = sign(1 − 2κ2 ) . Lemma 5.3 follows. Coming back to the initial units (see (2.7)), the estimate (5.2) gives the existence ˜0 > 0 such that we have, of positive constants C > 0, δ3 and a p κ δ1 . (5.14) for d κh0 ≥ a ˜0 , − 1 ≤ C exp(−δ3 d2 κh0 ) with δ3 = h0 4 Actually, δ1 > 12 gives δ3 > 18 . √ Using Lemma 2.2 (a), we have d κh0 ≥ κd. Therefore, from (5.14), when κd ≥ a ˜0 , h0 = κ + O(exp(−δ3 (dκ)2 )) . and Λ=2
−3/2 −1/2 −2
π
κ
2 2 d κ 2 2 . 1 − 2κ + (1 + κ )O exp − 8
(5.15)
We get Proposition 5.4 [κd large]. Let e = 0. For any η > 0, there exists a ˜0 > 0 ˜0 , and for h0 satisfying (2.5), then, such that for (κ, d) s.t. |κ − √12 | ≥ η and κd ≥ a
1 √ − κ · h1 > 0 . 2
Proof. The proposition results immediately from (5.15) and (5.7) or (5.11). Theorem 4.1 follows then using (2.6) and (2.25). We shall use (5.15) in the particular situation when d is keeped fixed, and κ tends to +∞. We get the following result: Proposition 5.5 [d fixed, κ large]. Let d > 0 and e = 0. Then, there exists a constant κ0 = κ0 (d) > 2−1/2 such that, for κ ≥ κ0 and for h0 satisfying (2.5), we have h1 < 0. At last, using (2.6) we get (a) and (b) in Theorem 4.2 and using once again (5.15), we easily get (4.1) and the limiting value 2−1/2 for κ. 5.2. The asymptotics for small κd 5.2.1. Main results for κd small When e = 0, we have proved in [7, Lemma 2.6] the following result: ¯ 2 (κ, d, 0) < 32, then h1 < 0. Lemma 5.6. Let e = 0. If (κ, d) satisfies d4 h
STABILITY OF BIFURCATING SOLUTIONS
Using that ¯ h(κ, d, 0)d tends to
√
...
601
12 as κd tends to 0 (see Lemma 3.2), we get
Proposition 5.7 [d small]. Let e = 0. There exists constants C0 and d0 > 0 such that for (κ, d) s.t. d ≤ d0 and κd ≤ C0 , and for h0 satisfying (2.5), then h1 < 0. The following result, which was suggested by the formal computations and numerical results of [15, 13, 5], improves Proposition 5.7 when κd is small. Theorem 5.8 [κd small]. Let e = 0. For any η > 0, there exists a0 such that √ for (κ, d) satisfying |d − 5| ≥ η, and 0 < κd ≤ a0 , and for h0 satisfying (2.5), then (i) √ ( 5 − d) · h1 < 0 . (ii) There exists 0 = 0 (κ, d) > 0 s.t. the curve of bifurcating solutions (f (., ), A(., ); h()) starting from (0, h0 x; h0 ) verifies for 0 < || ≤ 0 , √ ( 5 − d) · (h() − h0 ) < 0 . Theorem 5.8 is proved in Subsec. 5.2.3. In [8, Sec. 2.2] is studied (with more details given in the preliminary [5]) the behavior of the bifurcating solutions when κ and tend to 0. We got, in that paper, that, for fixed d, for κ ≤ κ1 and ≤ 0 (with 0 = 0 (d) and κ1 = κ1 (d) small), these solutions are uniquely defined, in a neighborhood of the bifurcating point (0, h0 x; h0 ) by the parameters κ and . The limiting model is the f − constant model studied in Subsec. 3.1 of [7]. In Theorem 10 of [8] is also studied the case when d and tend to 0, where the limiting problem is also the f − constant model. We first prove an analogous result when κd tends to 0. Here, we consider the behavior of the bifurcating solutions when both and κd tend to 0. 5.2.2. Bifurcations when and κd tend to 0 We consider, as in the proof of Theorem 3.1 (b)(i), the scaling (2.11), with g(u) = f (x) ;
V (u) = A(x)
for u =
x ∈ [−1/2, 1/2] , d
(5.16)
which gives a study interval independent of d. The Ginzburg–Landau equations become, for κ > 0, d > 0 and h > 0, 00 −g + σ(−1 + g 2 + V 2 )g = 0 in ] − 1/2, 1/2[ , 0 g (±1/2) = 0 , −V 00 + d2 g 2 V = 0 in ] − 1/2, 1/2[ , V 0 (±1/2) = η , (g, V ) ∈ (H 2 (] − 1/2, 1/2[))2 , with σ = κd and η = hd.
(5.17)
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C. BOLLEY and B. HELFFER
Let us prove, using the new parameter ¯ = d−1/2 , √ √ Proposition 5.9 [Bifurcation from the solution (0, 12u; 12)]. There , σ)) for (5.17), exists a C ∞ family of bifurcating solutions (g(., ¯, σ), V (., ¯, σ); η(¯ parametrized by ¯ and σ = κd (with κ > 0 and d > 0), which bifurcates from √ √ (0, 12u; 12). | ≤ ¯0 and for 0 < σ ≤ There exist constants ¯0 , σ0 and γ0 such that for 0 < |¯ , σ)) is defined and is the unique solution in σ0 , the solution (g(., ¯, σ), V (., ¯, σ); η(¯ H 2 (] − 1/2, 1/2[) × H 2 (] − 1/2, 1/2[) × R of (5.17) s.t. (i) (ii) (iii)
kg(., ¯, σ)kH 2 (]−1/2,1/2[) ≤ γ0 , √ kV (., ¯, σ) − 12ukH 2 (]−1/2,1/2[) ≤ γ0 , Z 12 g(u, ¯ , σ)g0 (u)du = ¯ . − 12
The function g0 (., η, σ) is the first eigenfunction of the Neumann problem (2.12) ¯ with g0 = φ. Proof. In our asymptotic study of the bifurcating solutions when κ tends to 0, in Sec. 2.2 of [8], we have considered all the starting normal solutions and proved that the only possible bifurcating solutions are the symmetric ones. Here, we write, when κd tends to 0, a simpler proof by using Theorem 3.4 which still gives that, for small κd, the bifurcating solutions are symmetric. Let us define the spaces, H 2,N e (] − 1/2, 1/2[) = {g ∈ H 2 (] − 1/2, 1/2[); g 0(±1/2) = 0} , ( H 2,od (] − 1/2, 1/2[) =
V ∈ H 2 (] − 1/2, 1/2[); V 0 (−1/2) = V 0 (1/2),
Z
)
1 2
V (u)du = 0
.
− 12
We consider the map Ψ from H 2,N e (] − 1/2, 1/2[) × R × H 2,od(] − 1/2, 1/2[) × R into L2 (] − 1/2, 1/2[) × R × L2 (] − 1/2, 1/2[) × R3 , defined by, (g, g+ , V, σ) → Ψ(g, g+ , V, v+ , σ) = (z, ¯, B, η, ρ) , and
(a) (b) (c) (d) (e)
z = −g 00 + σ(−g + g 3 + V 2 g) + φ¯1 g+ , Z 12 g0 gdu , ¯ = − 12
B = −V 00 + d2 g 2 V , η = V 0 (−1/2) , ρ =σ.
(5.18)
STABILITY OF BIFURCATING SOLUTIONS
...
603
We verify that Ψ is a local diffeomorphism in a neighborhood of (g, g+ , V, σ) = (0, 0, ηu, 0) by proving that the derivative of Ψ at (0, 0, ηu, 0) is invertible. We have, indeed, at (0, 0, ηu, 0), with V0 = ηu, 00 2 ¯ δz = −δg + σ(−1 + V0 )δg + φ1 δg+ , 1 Z 2 g0 δg du , δ¯ = −1 2
δB = −δV 00 , δη = δV 0 (−1/2) , δρ = δσ . So, the injectivity and the surjectivity are easy. If we denote by Φ = (Φi )i=1,...,4 the local inverse of Ψ, the solutions of the Ginzburg–Landau equations are given by the solutions of Φ2 (0, ¯, 0, η, ρ) = 0 . Let ζ(¯ , η, ρ) = Φ2 (0, ¯, 0, η, ρ) , 2 be the function defined from R ×√[0, +∞[ √ to R. We know that (g, V ; η) = (0, 12u; 12) is a solution of (5.17), so that
ζ(0, η, ρ) = 0 . We search a function ηˆ(¯ , ρ) such that ( ζ(0, ηˆ(¯ , ρ), ρ) = 0 , √ ηˆ(0, 0) = 12 . We shall then get a solution of the GL equations as function of ¯ and σ. For this, we apply the implicit function theorem to the function ˜ , η, ρ) = 1 ζ(¯ , η, ρ) . ζ(¯ ρ¯ We remark, indeed, that ∂g+ ¯, (0, 0, 0, η, ρ) = −λ ∂¯ ¯ is the principal eigenvalue of (2.12). where λ ¯ = 0 and ∂g+ (0, 0, 0, η, 0) = 0. When σ = 0, then λ ∂¯ We get, √ ∂ζ (0, 12, 0) = 0 . ∂¯ ¯ Moreover, we have λ = σ[τ (σ, 1, 0, η) − 1], where τ (κ, d, e, h) is the principal eigenvalue of (2.2). Therefore, ¯ λ lim = 0 , σ→0 σ
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C. BOLLEY and B. HELFFER
and then
√ ∂2ζ (0, 12, 0) = 0 . ∂σ∂¯ In order to apply the implicit function theorem, we have now to prove that √ 1 ∂2 ζ (0, 12, 0) 6= 0 . ∂η∂¯ σ But, ∂2 ∂η∂¯
√ Z 12 √ ∂ 3 1 1¯ 2 2 ζ (0, 12, 0) = − λ = 2η , u g0 (u) du = σ ∂η σ 3 − 12
where we have used that η = Proposition 5.9 follows.
√ 12 and g0 (u) ≡ 1 when σ = 0.
5.2.3. Proof of Theorem 5.8 Let us now study the bifurcation starting from the normal solution (0, h0 x; h0 ) uniformly with respect to κd in ]0, a1 ]. We calculate, for fixed small σ, the first terms of the partial expansion in powers of at (σ, 0), of the C ∞ solutions given by this theory. If we let σ = κd, the bifurcating solutions (f (., σ, ), A(., σ, ); h(σ, )) satisfy, in a neighborhood of (0, A0 (., σ); h0 ) with A0 (., σ) = h0 x and h0 = h0 (σ), an expansion in powers of . In the scaling (5.16) and with ¯ = d−1/2 we get a partial expansion in powers of ¯ at (σ, 0), of (g(., σ, ¯), V (., σ, ¯); η(σ, ¯)) with η = hd (see Proposition 5.9) g(., σ, ¯) = ¯g0 (., σ) + ¯g˜(., σ, ¯) , 3 ) , V (., σ, ¯) = V0 (., σ) + ¯2 V1 (., σ) + O(¯ V 0 (±1/2, σ, ¯) = dh(σ, ¯) = dh0 + ¯2 dh1 (σ) + O(¯ 3 ) , with (g0 , g˜)L2 (]−1/2,1/2[) = 0 , g˜(., σ, ¯) =
p X
¯2j gj (., σ) + O(¯ 2p+1 ) ,
j=1
and where the elements gi are in H 2 (]− 1/2, 1/2[), Vi ∈ H 2 (]− 1/2, 1/2[) and hi ∈ R for the various indices i. They depend on σ and d, but we omit in the following the d-dependence. Using the regularity of the problem, we get immediately that the O are uniform in σ for 0 < σ ≤ a1 . We determine the terms of these expansions by equating powers of ¯ in the Ginzburg–Landau equations (5.17). First using the uniqueness of the bifurcating solutions, we get that the functions gi are even functions and that the Vi are odd functions (see (3.11)). We remark also that the expansion is actually in powers of κ2 d2 .
STABILITY OF BIFURCATING SOLUTIONS
...
605
We have, in particular, see (3.1), √ dh0 = 2 3 + O(σ 2 ) .
(5.19)
The cancellation of the ¯ terms give, in the first GL equation, ( −g000 + σ 2 (−1 + V02 )g0 = 0 in ] − 1/2, 1/2[ , g00 (±1/2) = 0 , therefore, g0 (u, σ) = g0 (u, 0) + O(σ 2 ) with g0 (u, 0) ≡ const. Moreover, the normalization (2.13) implies that g0 (u, σ) ≡ 1 + O(σ 2 ) .
(5.20)
The ¯2 -terms give (
−V100 + d2 g02 V0 = 0 in ] − 1/2, 1/2[ , V10 (±1/2) = dh1 (σ) .
We get, by integration, using (5.19) and (5.20), ! ! √ √ 3 3 2 2 2 3 2 2 + O(σ ) u + dh1 (σ) − d + d O(σ ) u , V1 (u, κ) = d 3 4
(5.21)
The term h1 (σ) will be determined later. Now, the ¯3 terms give ( −g100 + σ 2 (−1 + V02 )g1 + σ 2 [g03 + 2V0 V1 ]g0 = 0 , g10 (±1/2) = 0 . So, the compatibility condition implies Z 12 1 2 4 4 2 2 4 g0 + h0 d (1 + O(σ ))g0 u du 3 − 12 Z + 2h0 d
1 2
− 12
d3 h0 (1 + O(σ 2 )) u2 g02 du = 0 . dh1 − 8
We then get, using (5.19) and (5.21), √ d2 2 2 2 + (1 + d )O(κ d ) . dh1 (κ) = − 3 1 − 5 √ So that, when |d − 5| ≥ η and κd is small enough, √ sign (h1 ) = sign (d − 5) . When ¯ is small enough, we have √ d2 2 2 2 + (1 + d )O(κ d ) ¯2 + Oκ,d (¯ 3 ) . dh(κ, ¯) = dh0 (κ) − 3 1 − 5
(5.22)
(5.23)
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C. BOLLEY and B. HELFFER
With the parameter , we get, √ d2 3 + (1 + d2 )O(κ2 d2 ) 2 + Oκ,d (3 ) . h(κ, ) = h0 (κ) − 2 1 − d 5
(5.24)
Theorem 5.8 follows. 6. The Sign of h1 in the Asymmetric Case Let us now study the asymmetric case. We consider bifurcating solutions starting from normal solutions (0, h0 (x + e); h0 ) when e is different from zero and ¯ d, e). h0 = h(κ, 6.1. Main results It results from Theorem 3.3 and Theorem 3.4 the existence, for κd large, of a unique positive value of e = e¯(κ, d) defining a bifurcation point. We shall show, in that section, the following theorems: Theorem 6.1 [κd large]. Let a1 be the constant defined in Theorem 3.3 and Σ0 > 0 defined in (2.34). For any η > 0, there exists a constant a2 ≥ a1 , such that for (κ, d) satisfying |κ − Σ0 | ≥ η and κd ≥ a2 , for e = e¯(κ, d) and h0 satisfying (2.5), we have: (a) (Σ0 − κ) · h1 > 0 . (b) There exists 0 = 0 (κ, d) such that, if (f (., ), A(., ); h() is a curve of bifurcating solutions starting from the normal solution (0, h0 (x+ e¯(κ, d)); h0 ) then, for 0 < || ≤ 0 , (Σ0 − κ) · [h() − h0 ] > 0 ,
(6.1)
(Σ0 − κ) · ∆G(f (), A(); h()) > 0 .
(6.2)
and
When d is fixed, we get Theorem 6.2 [d fixed, κ large]. Let d > 0. There exists a constant κ ˜0 = ˜ 0 (d), for e = e¯(κ, d) and for h0 satisfying (2.5), κ ˜ 0 (d) > Σ0 such that, for κ ≥ κ we have: (i) h1 < 0. (ii) There exists 0 = 0 (κ, d) such that, if (f (., ), A(., ); h() is a curve of e(κ, d)); h0 ) bifurcating solutions starting from the normal solution (0, h0 (x+¯ then, for 0 < || ≤ 0 , h() < h0
and
∆G(f (), A(); h()) < 0 .
There exists a constant C > 0 such that κ ˜ 0 can be chosen verifying κ ˜ 0 (d) = Σ0 + O(exp(−Cd2 )
when
d → +∞ .
(6.3)
STABILITY OF BIFURCATING SOLUTIONS
...
607
Remark 6.3. This suggests that, for large d, there exists a curve d → κ(d), whose asymptot is Σ0 , determining the change of sign of h1 (κ, d) for large κd. Before giving the proofs, let us recall some previous results on the linearized problem around a normal asymmetric solution. 6.2. Previous results on the principal eigenfunction We used, in [4], the scaling (2.7) and then considered, for κ > 0, d > 0, e > 0, ¯ d, e), the problem (2.8)–(2.9). h0 = h(κ, In [4], we have constructed quasimodes for this problem when a tends to +∞ and when c verifies (6.4) c > −ρ2 · a for some fixed ρ2 satisfying 0 < ρ2 < 1. Let us recall the principal ideas for this construction. We consider the basis φ1 (., µ), φ2 (., µ), defined in (2.29) and (2.30), of the set of the solutions of the problem (2.28). Then, for α > 0, we consider, as in Subsec. 2.4, the first eigenvalue µ1 (α) of the harmonic oscillator P in ] − α, +∞[ with the Neumann condition at −α. The associated normalized eigenfunction is given by γφ1 (., µ1 (α)), where γ is a constant. Let us now consider the spectral problem P f¯ = µf¯ on the bounded interval ] − α, β[, with α = a − c, β = a + c. We have obtained the following result: Proposition 6.4 [see Proposition 5.3 in [4]]. There exists a0 > 0 such that, if (a, c) verifies (6.4) with a > a0 , the principal eigenvalue µ = µ(a − c, a + c) for the Neumann problem satisfies: 11 (6.5) |µ − µ1 (a − c)| ≤ C(a0 ) exp − (a + c)2 , 16 and the corresponding eigenfunction f¯ verifies 5 kf¯ − ΨkC 1 ([−a+c,a+c]) ≤ C(a0 ) exp − (a + c)2 , 8 where
˜ Ψ(y, a, c) = γ(a, c)φ1 (y, µ1 (a − c)) + δ(a, c)φ2 (y, µ1 (a − c))Ξ
y (a + c)
(6.6) ,
(6.7)
with: (a)
δ(a, c) = γ(a, c) exp(−(a + c)2 ) · (a + c)µ1 · (1 + O(a−2 ))
(b)
γ(a, c) = g(a − c) + O exp − 14 (a + c)2 ,
(6.8)
˜ verifies (uniformly for (a, c) verifying (6.4) with a > a0 ), where the cutoff function Ξ ˜ Ξ(y) =0
if y ≤
1 , 2
˜ Ξ(y) =1
if y ≥
3 , 4
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C. BOLLEY and B. HELFFER
and where α → g(α) = kφ1 (y, µ1 (α))k−1 L2 (]−α,+∞[) is a bounded, continuous, strictly positive function such that: (6.9) g(0) = 21/2 · π −1/4 . ˜ the second term with φ2 , in the definition By the choice of the cutoff function Ξ, of Ψ in (6.7), appears as a corrective term for large y which makes the Neumann condition at y = a + c satisfied. The Neumann condition at y = −a + c is also satisfied due to the choice of the function φ1 which is an eigenfunction for the Neumann problem on ] − a + c, +∞[. It results immediately from (6.5) and from (2.31), that, for a ≥ a0 and c satisfying (6.4), we have, because h0 = µκ (see (2.7)), h0 =
11 κ + O exp − (a + c)2 . µ1 (a − c) 16
(6.10)
Let us now consider the problem (2.8) when the normal solution is a bifurcation point. As we recalled above, the positive parameter e = e¯(κ, d) is then uniquely determined (in the initial units) as a function of κ and d (for κd large enough). It is in fact proved in [4] (see Lemma 3.2), that, in the scaling (2.7), the positive value of c for which we have a bifurcation is uniquely determined as a function c¯ of a. Our study, in [4], shows also that α(a) = a − c¯(a) and µ1 (a − c¯(a)) tend exponentially fast respectively to the constants α0 and µ01 , when a tends to +∞, where α0 and µ01 are defined in Subsec. 2.4 (see (2.31)). Lemma 6.5 in [4] gives 1 2 , (6.11) |a − c¯(a) − α0 | ≤ C · exp − a 2 and we have
1 |µ1 (a − c¯(a)) − µ01 | ≤ C · exp − a2 2
as a → +∞ .
Therefore, when a tends to +∞, κ 1 2 . h0 = 0 + O exp − a µ1 2
(6.12)
We have also verified in that paper that, for all a > 0, h0 >
κ . µ01
(6.13)
6.3. Structure of the bifurcating solutions Our purpose, here, is to prove Theorem 6.1 and Theorem 6.2 which give sign(h1 ) as functions of κ and d. Several steps are needed for the proofs. The first three ones are dealing with the scaling (2.7) and only ask for a large and c satisfying (6.4)
STABILITY OF BIFURCATING SOLUTIONS
...
609
Step 1. Rewriting of the relation (2.27) giving h1 We rewrite the relation (2.27) which gives h1 , in the new variables introduced in (2.7). We get, as in (5.7), Z h1 a+c ¯ ¯ 2 dy 2 B0 (y)2 f(y) h0 −a+c Z a+c Z a+c p 4 −2 0 2 ¯ ¯ (6.14) f (y) dy + 2κ B1,0 (y) dy , = κh0 − −a+c
−a+c
√ ¯ 1,0 (y) = κA1,0 (x) with y = κh0 (x + e). ¯0 (y) = A0 (x) and B where B ¯1,0 verifies (cf. (5.8)) The function B −1 00 ¯ 2 ¯ 00 B1,0 (y) = h0 A1,0 (x) = y · f (y) for y ∈] − a + c, a + c[ , ¯ 0 (±a + c) = 0 , B 1,0 B ¯1,0 (0) = 0 . ˜ where Λ ˜ is defined by From (6.14), sign(h1 ) will be given by sign(Λ), Z a+c Z a+c 4 −2 0 2 ¯ ˜ ¯ Λ(a) ≡ − f (y) dy + 2κ B1,0 (y) dy . −a+c
(6.15)
(6.16)
−a+c
Let us prove, for c verifying (6.4), the following lemma: Lemma 6.5. There exists a0 > 0 such that for (a, c) verifying a ≥ a0 and (6.4), and for κ > 0, ˜ Λ(a) = (g(a − c))4 · −ρ(α(a)) + κ−2 σ(α(a)) 1 , + (1 + κ−2 )O exp − (a + c)2 4
(6.17)
where ρ(α) and σ(α), defined resp. in (2.32) and (2.33), depend only on α. The proof of Lemma 6.5 is given in the following two steps, corresponding to ˜ the analysis of the two terms appearing in the right-hand side of (6.16) defining Λ. ˜ Step 2. Analysis of the first integral in Λ 4 Using (6.6) and the decomposition of Ψ given by (6.7), we have, for a > a0 , Z a+c Z a+c 5 4 4 4 2 ¯ , (6.18) φ1 (y) dy + O a exp − (a + c) f(y) dy = γ(a, c) 8 −a+c −a+c uniformly for c verifying (6.4). Here, we have used thate the product φ1 φ2 is ˜ and ˜ and that the terms (δ(a, c)3 φ22 Ξ) bounded, for a large, on the support of Ξ, 4 4˜ (δ(a, c) φ2 Ξ) are small in comparison with the preceding error given by (6.6). e In the following, we omit the reference to the parameter µ for φ and φ . It is given by µ = µ (α). 1 2 1
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C. BOLLEY and B. HELFFER
Then, (6.8)(b) implies Z a+c Z 4 4 ¯ (f (y)) dy = g(a − c)
1 2 . φ1 (y) dy + O a exp − (a + c) 4 −a+c −a+c (6.19) We need however to verify that the last term in (6.19) is small in comparison with the preceding integral, when a tends to +∞. This is given by the following lemma: a+c
4
Lemma 6.6. There exists a0 > 0 s.t. for (a, c) verifying (6.4) with a ≥ a0 , Z a+c g(a − c)4 φ1 (y)4 dy ≥ Ca−1 , −a+c
for some strictly positive constant C. Proof of Lemma 6.6. Let us first recall that the function g is defined by Z +∞ 2 φ1 (y)2 dy = 1 . g(a − c) −a+c
Therefore, for every η > 0, there exists a0 > 0 s.t. for a ≥ a0 , Z a+c ∀ a > a0 , 1 − η ≤ g(a − c)2 · φ1 (y)2 dy ≤ 1 . −a+c
We then get, using the Cauchy–Schwarz inequality, Z g(a − c)
4
a+c
1 φ1 (y) dy ≥ 2a −a+c 4
≥
Z g(a − c)2
2
a+c 2
−a+c
φ1 (y) dy
1 (1 − η)2 , 2a
which gives Lemma 6.6. Now, using that for large β (see (2.29) and the upper bound µ ≤ 1 in (2.31)), Z Z ∞ 2 ∞ 1 exp(−2β 2 ) , φ1 (y)4 dy ≤ y exp(−2y 2 )dy = β 2β β β we can approximate in the right-hand side of (6.19) the integral on the bounded interval [−a + c, a + c] by an integral on [−a + c, +∞[. Therefore, (6.19) becomes Z a+c Z ∞ 1 . (6.20) φ1 (y)4 dy + O exp − (a + c)2 f¯(y)4 dy = g(a − c)4 4 −a+c −a+c ˜ Step 3. Analysis of the second integral in Λ ˜ as above, we have, Using (6.15) and the same properties for φ1 , φ2 and Ξ Z y 5 0 2 2 2 ¯ . t · φ1 (t) dt + O (a + c) exp − (a + c) B10 (y) = γ(a, c) 8 −a+c
STABILITY OF BIFURCATING SOLUTIONS
Therefore, using also that Z
a+c
−a+c
R a+c −a+c
...
611
γ 2 φ21 dy = O(1) for large a, we get
¯ 0 (y)2 dy = γ(a, c)4 B 1,0
Z
a+c
Z
2
y
t · φ1 (t) dt 2
−a+c
−a+c
dy
5 . + O (a + c)2 exp − (a + c)2 8
(6.21)
¯ 0 are only defined on [−a + c, a + c], but the functions ¯ 1,0 and B The functions B 1,0 φ1 and φ2 are defined on the interval [−a + c, ∞[, with, Z y 1 2 2 . t · φ1 (t) dt ≤ C exp − (a + c) for y > a + c, 2 a+c Therefore, Z
a+c
−a+c
¯ 0 (y)2 dy B 1,0 "Z
= γ(a, c) · 4
∞
−a+c
Z
2
y
t · φ1 (t) dt 2
−a+c
# 1 2 , dy + O exp − (a + c) 2
and then, using again (6.8)(b), Z
a+c −a+c
¯ 0 (y)2 dy = g(a − c)4 · B 1,0
Z
∞
−a+c
Z
2
y
−a+c
t · φ1 (t)2 dt
1 . + O exp − (a + c)2 4
dy (6.22)
Lemma 6.5 results from (6.22) and (6.20). ˜ and of h1 Step 4. The sign of Λ We suppose now that c = c¯(a) (or equivalently e = e¯(κ, d)). Using the regularity C 1 of φ1 and the strict positivity of the function g (see Lemma 4.1 in [7]) we get that g(α), σ(α) and ρ(α) tend respectively to g(α0 ), σ(α0 ) and ρ(α0 ) as α tends to α0 . More precisely, the function g(α), σ(α) and ρ(α) are lipschitzian in a neighborhood of α0 , so that the fast convergence of α(a) to α0 when a → +∞ (see (6.11)), gives that g(α(a)), σ(α(a)) and ρ(α(a)), tend respectively to g(α0 ), σ(α0 ) and ρ(α0 ) like exp(− 21 a2 ) when a tends to +∞. These results are completely independent of κ. Therefore, we get, from (6.17) and the definition (2.34) of Σ0 , the existence of some a2 > 0, independent of κ, such that, for a ≥ a2 , 1 ˜ Λ(a) = g(α0 )4 ρ(α0 ) · κ−2 (Σ20 − κ2 ) + (1 + κ2 )O exp − a2 . 4
612
C. BOLLEY and B. HELFFER
We have proved: Lemma 6.7. For any η > 0, there exists a constant a2 > 0 such that, for a ≥ a2 , and for |κ − Σ0 | ≥ η, ˜ = sign(Σ2 − κ2 ) . sign(Λ) 0 Let us now come back to the initial units. Using (6.12) and (6.13), we get κd , therefore, for κd large a + c¯(a) ≥ √ 0 2
µ1
1 κ2 d2 4 −2 2 2 2 ˜ Σ0 − κ + (1 + κ )O exp − . Λ(a) = g(α0 ) ρ(α0 ) · κ 4 µ01
(6.23)
We get, combining (6.14), (6.16) and (6.23), the following proposition: Proposition 6.8. Let a1 be the constant defined in Theorem 3.3 and Σ0 > 0 defined in (2.34). For any η > 0, there exists a constant a ˜2 > a1 such that, for (κ, d) ˜2 , and for h0 satisfying (2.5) with e = e¯(κ, d), satisfying |κ − Σ0 | ≥ η and κd ≥ a then h1 · (Σ0 − κ) > 0 . Consequently, using Theorem 3.4, we get Theorem 6.1. Theorem 6.2 follows from Proposition 6.8 and Theorem 3.4. The asymptotic relation (6.3) results from (6.23). The critical value κ ˜ 0 (d) plays, for the asymmetric bifurcating solutions, the same role as the constant κ0 (d) in Theorem 4.2 for the symmetric solutions. 7. Stability of the Bifurcating Solutions 7.1. The spectral problem when is small The local stability of a bifurcating solution (f (., ), A(., ), h()), given by Theorem 2.1, is obtained by studying for h = h() the hessian of (∆G)h calculated at the point (f (., ), A(., )). It will be denoted by Hessf (.,),A(.,)(∆G)h or more shortly by Q . It is defined on (H 1 (] − d/2, d/2[))2 × (H 1 (] − d/2, d/2[))2 by g (g, b)|Q | b L2 (]−d/2,d/2[) =κ
−2
Z
d 2
02
Z
d 2
g dx + −d 2
Z
−d 2
Z
d 2
+ 4 −d 2
(A20 − 1 + 22 A0 A1 + 32 f02 )g 2 dx
A0 f0 gbdx +
d 2
−d 2
b02 dx + 2
Z
d 2
−d 2
f02 b2 dx + o(2 ) .
(7.1)
The study of the local stability is then reduced to the analysis of the spectrum of the self-adjoint operator attached to this quadratic form. Because the resolvent is
STABILITY OF BIFURCATING SOLUTIONS
...
613
compact, the spectrum is discrete and we shall deduce the local stability from the analysis of the sign of the lowest eigenvalue. This leads us to study the corresponding linearized GL equations at (f (., ), A(., ); h()): ( (a) ( (b)
−κ−2 g 00 + (A(., )2 + 3f (., )2 − 1)g + 2A(., )f (., )b = λ()g , g 0 (±d/2) = 0 , −b00 + f (., )2 b + 2A(., )f (., )g = λ()b
(7.2)
in ] − d/2, d/2[) ,
b0 (±d/2) = 0 .
¯ d, e), with e ∈ R, the lowest eigenvalue for this problem By the choice h(0) = h(κ, when = 0 is equal to zero with multiplicity two (see (2.2)). The corresponding eigenspace is generated by f0 0 and u2 = . u1 = 0 1 When → 0, (7.2) can be written in the form g g = λ() , (M0 + M1 + 2 M2 + O(3 )) b b with
2 −2 d 2 −κ + A − 1 0 0 dx2 ; M0 = 2 d 0 − 2 dx 2A0 A1 + 3f02 0 . M2 = 0 f02
M1 =
0 2A0 f0
2A0 f0 0
;
Because the problem (7.2) is self-adjoint and is regular with respect to as obtained in Theorem 2.1, we can apply the general theory of perturbation for self-adjoint operators (see T. Kato [20]). Therefore, there exist two C 3 functions λ of , denoted (1) = λ(2) (0) = 0 and such that (7.2) has non-zero λ(1) and λ(2) , such that λ (0) g 3 solutions b which are also C with respect to . It is then sufficient to make a formal study in order to compute explicitly the first terms. The eigen-elements (g(., ), b(., ), λ()) will then be calculated such that ! ! 0 g(., ) f0 g2α,β g1α,β 2 + + O(3 ) , +β + =α (7.3) α,β 0 1 b(., ) bα,β b 1 2 and λ() = λα,β + 2 λα,β + O(3 ) . 1 2 with (α, β) ∈ R2 − {(0, 0)}, giα,β , bα,β in H 2 (] − d/2, d/2[) and λα,β in R for i = 1, 2. i i Let us prove
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C. BOLLEY and B. HELFFER
Proposition 7.1. Let d > 0, κ > 0 and (e, h0 ) satisfying (2.5). Then, there exists 0 s.t. for 0 < || ≤ 0 , the eigen-elements corresponding to the two lowest eigenvalues of (7.2) can be chosen in the set described by (1) 2 (1) 3 λ () = λ2 + O( ) (7.4) (a) g (1) (x, ) = f0 + O(2 ) b(1) (x, ) = 2 A1,0 + O(3 ) and
(2) 2 (2) 3 λ () = λ2 + O( ) g (2) (x, ) = f0 + O(3 ) b(2) (x, ) = 1 + O(2 ) ,
(b)
(7.5)
with (1) λ2
h1 = −4 h0
Z
d 2
−d 2
A20
f02
dx ;
(2) λ2
1 = d
2 1+ h0
Z
!
d 2
−d 2
A0 f0 ψ0 dx
and where ψ0 is the unique solution in H 2 (] − d/2, d/2[) of −2 00 2 −κ ψ0 + (A0 − 1) ψ0 = −2 h0 A0 f0 ψ00 (±d/2) = 0 , (ψ0 , f0 )L2 (]−d/2,d/2[) = 0 .
,
(7.6)
f
in ] − d/2, d/2[ , (7.7)
Remark 7.2. We remark that, using Lemma 2.3 (b) (ii), (2)
λ2 =
1 ∂2τ ¯ d, e)) , (κ, d, e, h(κ, 2 d ¯h2 ∂e2
(7.8)
where τ is defined in (2.2). Proof of Proposition 7.1. We calculate the first terms of the expansion in powers of (whose existence is proved in [20]). We expand the equations in (7.2) by using (7.3), (2.2) and (2.6) and equal the corresponding powers of . Cancellation of the terms. −κ−2 (g1α,β )00 + (A20 − 1) g1α,β + 2β A0 f0 = α λα,β f0 in ] − d/2, d/2[ , 1 (g α,β )0 (±d/2) = 0 , 1 α,β 00 α,β 2 −(b , 1 ) + 2A0 f0 = β λ1 α,β 0 (b ) (±d/2) = 0 . 1 f The function ψ is the partial derivative of f with respect to the parameter e (see [3]). 0 0
(7.9)
STABILITY OF BIFURCATING SOLUTIONS
...
615
Compatibility equations give then the necessary conditions for the existence of a solution, Z d2 2β A0 f02 dx = α λα,β , 1 −d 2
Z
d 2
2α −d 2
A0 f02 dx = β λα,β . 1
¯ d, e) can be written (see But, the condition (2.5)(b) which is satisfied by h0 = h(κ, Lemma 2.3 (a) (ii)), Z d2 A0 f02 dx = 0 . −d 2
Therefore, we get
(
α λα,β = 0, 1 β λα,β = 0, 1
and then, because (α, β) 6= (0, 0), = 0. λα,β 1 We then solve the system (7.9) in such a way that all the elements g1 and b1 are normalized by Z d2 Z d2 g1α,β f0 dx = 0 and bα,β dx = 0 . 1 −d 2
−d 2
We get β ψ0 and bα,β = 2α A1,0 , 1 h0 is defined in (2.26). The constants α and β remain free at this stage. g1α,β =
where A1,0
Cancellation of the 2 terms −κ−2 (g2α,β )00 + (h20 (x + e)2 − 1) g2α,β + 2α A0 A1 f0 + 3α f03 + 2A0 f0 bα,β = α λα,β f0 1 2 α,β 0 (g2 ) (±d/2) = 0 , α,β β 00 2 −(bα,β , 2 ) + βf0 + 2 h0 A0 f0 ψ0 = β λ2 α,β 0 (b2 ) (±d/2) = 0 .
in ] − d/2, d/2[ , (7.10)
Using the relation (2.27), the compatibility conditions give then the new necessary conditions, Z d2 αλα,β = α 2A0 A1 f02 dx 2 −d 2
Z
d 2
+ 3α −d 2
Z f04 dx + 2α
d 2
−d 2
A0 A1,0 f02 dx .
(7.11)
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C. BOLLEY and B. HELFFER
and
Z βλα,β 2 d =β
d 2
−d 2
f02 dx + 2
β h0
Z
d 2
A0 f0 ψ0 dx ,
−d 2
(7.12)
If α 6= 0, the condition (7.11) then becomes Z (1) λ2
d 2
=4 −d 2
Z A0 A1,0 f02
and using (2.27), (1) λ2
h1 = −4 h0
Z
d 2
dx + 2
d 2
−d 2
−d 2
f04 dx
(7.13)
A20 f02 dx .
(7.14)
When β 6= 0, the condition (7.12) gives, (2) λ2
1 = d
2 1+ h0
Z
!
d 2
−d 2
A0 f0 ψ0 dx
.
(7.15)
We will see that in most of the limiting problems these two quantities are not equal. In any case, we get the first solution with β = 0, the second with α = 0. In particular, g11,0 ≡ 0 ; b0,1 1 = 0. and, if as before (g2α,β , f0 )L2 = 0 and (bα,β 2 , 1)L2 (]−d/2,d/2[) = 0, g20,1 ≡ 0 ;
b1,0 2 = 0.
Proposition 7.1 follows. 7.2.
The stability and instability of symmetric solutions
7.2.1. Proof of Theorems 4.3 and 4.5 For showing the stability or the instability of the bifurcating symmetric solutions, (1) (2) we only have to calculate the two first eigenvalues λ2 and λ2 . Let us first remark, using (7.8) and Theorem 3.1 (b), that (2)
Lemma 7.3. (i) There exists a constant a0 such that, for κd ≤ a0 ,λ2 > 0. (2) (ii) We suppose e = 0. There exists a constant a2 such that, for κd ≥ a2 , λ2 < 0. (2)
The positivity of λ2 for κd small can also be proved as follows. Starting from the expansions of g0 (u) = f0 (x) and V0 (u) = A0 (x) in powers of σ = κ2 d2 written, in the scaling u = xd (see Subsec. 5.2), we search an expansion for ψ¯0 (u) = ψ0 (x), solution of (7.7), as ψ¯0 = ψ¯0,0 + κ2 d2 ψ¯0,1 + O(κ4 d4 ) . We can easily justify this expansion by regular perturbation theory.
STABILITY OF BIFURCATING SOLUTIONS
...
617
The cancellation of the κ0 -terms in (7.7) gives ¯00 ψ0,0 = 0 , 0 ψ¯0,0 (±1/2) = 0 , (ψ¯ , g ) 2 0,0 0 L (]−1/2,1/2[) = 0 . Therefore, ψ¯0,0 ≡ 0, and, from (7.15), (2)
λ2 =
1 (1 + O(κ2 d2 )) . d
Let us prove the first part of Theorem 4.3 which concerns κd small. From (1) (2) Theorem 7.1, it is sufficient to verify that the both λ2 and λ2 are positive. (1) From Proposition 5.8, we get that, for any η > 0, λ2 is positive when κd ≤ a0 √ (2) and d ≤ 5 − η. From Lemma 7.3 (i), we obtain λ2 > 0 in the same case. The stability of the bifurcating solutions, for 6= 0 in a neighborhood of 0, follows. Proof the first part. When √ of Theorem 4.3 (ii). We proceed as in the proof of (1) d > 5 and κd small, we use Theorem 5.8, which gives that λ2 is stricly negative. Part (ii) in Proposition 4.3 follows. 7.2.2. The instability for large κd: proof of Theorem 4.5 Theorem 4.5 results from Lemma 7.3 (ii), because, for large κd, one eigenvalue (2) (here λ2 ) is strictly negative. The instability of the symmetric bifurcating solutions, for 6= 0 in a neighborhood of 0 is then proved for κd large. Remark 7.4. When e = 0, we have shown (see Proposition 5.4) that h1 < 0 when κ > 2−1/2 and κd is large, and similarly that h1 > 0 when κ < 2−1/2 and κd is large. Theorem 4.3 (ii) shows that the local stability of the symmetric solutions is not always given by the sign of h1 as generally admitted. (1)
(2)
The asymptotics given in Sec. 5 permit to analyze the behavior of λ2 and λ2 when e = 0 and κd tends to +∞. (2) The asymptotics for d f¯(a)−2 λ2 , calculated in [3, Proof of Proposition 2.21] when e = 0 and κd tends to +∞, give in fact more information than those of Lemma 7.3. We get 2 2 1 d κ (2) 1 + O(d−2 κ−2 ) . (7.16) λ2 = − π −1/2 d2 κ3 exp − 2 4 (1)
Moreover, the coefficient λ2 is given by (7.6) and, with the notations of Subsec. 5.1, (1) by λ2 = −2 κ Λ. Therefore, using the computations of Sec. 5 (see (5.15)), we get when κd tends to +∞, 2 2 d κ (1) −1/2 −1/2 −1 2 2 . (7.17) 1 − 2κ + (1 + κ ) O exp − π κ λ2 = −2 8
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7.2.3. Local stability for a reduced symmetric problem In the study [6], and as was done in previous articles of [15] or [9], we have restricted the research of the solutions of the Ginzburg–Landau equations to the set of the symmetric solutions. This assumption leads us to restrict the domain of the GL functional to a subset Hsym of (H 1 (] − d/2, d/2[))2 corresponding to these solutions. This subset is defined by Hsym = {(f, A) ∈ (H 1 (] − d/2, d/2[))2 ; ∀ x ∈] − d/2, d/2[ (f (−x), A(−x)) = (f (x), −A(x))} . This symmetric problem is equivalent to a problem restricted to the half interval ]0, d/2[), where we consider a functional Φh defined on {(f, A) ∈ (H 1 (]0, d/2[))2 ; A(0) = 0} by Z d/2 1 κ−2 f 02 − f 2 + f 4 + A2 f 2 + (A0 − h)2 dx . Φh (f, A) = 2 0 This last point of view was used in particular in [6]. We remark that, by considering the symmetric problem, the stability of a GL solution can be different than in the initial problem. This is in particular the case for the symmetric bifurcating solutions when κd is large. We have proved in Theorem 4.5 that the symmetric bifurcating solutions are unstable for κd large enough (with respect to the GL functional (∆G)h ). We prove now the following theorem. Theorem 7.5. Let e = 0. For any η > 0, there exists a constant a1 > 0 such that, for (κ, d) satisfying κd ≥ a1 and |κ − 2−1/2 | ≥ η, and for h0 satisfying (2.5), there exists 1 > 0 such that for 0 < || ≤ 1 , the following properties are verified. (i) When κ > 2−1/2 , the bifurcating solutions (f (., ), A(., ); h()) starting from (0, h0 x; h0 ) are locally stable by respect to the symmetric problem. (ii) When κ < 2−1/2 , they are unstable. Proof. It is sufficient to prove that the bifurcating solutions give local minima for (∆G)h in restriction to Hsym , when h is fixed. So, as in the proof of Theorem 4.5, we study the hessian of the functional at a bifurcating solution, but we restrict this hessian to Hsym × Hsym . Because (f (.; ), A(.; )) ∈ Hsym , the operator defined by the left-hand side of (7.2) leaves Hsym stable, so that the new spectral problem is a classical one. The restriction to Hsym eliminates the eigenvalue λ(2) () and the lowest eigenvalue µ(1) () of the hessian reduced to the symmetric solutions is simple and equal to λ(1) (). We get, for small, (7.18) λ(1) () = µ(1) () . For proving Theorem 7.5, it is then sufficient to apply Proposition 5.4 which gives that h1 < 0 when κ > 2−1/2 and h1 > 0 when κ < 2−1/2 , then to apply Proposition 7.1 which proves that for small enough, λ(1) has the sign of h1 .
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7.3. The stability or instability of asymmetric solutions (2)
In view of proving Theorem 4.6, we first estimate λ2 (2)
7.3.1. The study of λ2
when κd tends to +∞.
for κd large
Let us show: Proposition 7.6. Let a1 > 0 be defined as in Theorem 3.3. There exists a2 ≥ ¯ d, e¯(κ, d)) satisfying a1 such that, for (κ, d) satisfying κd ≥ a2 and for h0 = h(κ, (2.5)(a, b), 1 1 2 2 (2) 2 2 . (7.19) λ2 = α0 (g(α0 )) (φ1 (−α0 )) + O exp − d κ d 16 (2)
Proof. According to (7.8), the sign of λ2
is given by the sign of
∂2τ ¯ d, e¯)) . (κ, d, e¯(κ, d), h(κ, ∂e2 For getting an estimate of this term, we use a characterization of the variation of the eigenvalues of the operator P with respect to the boundaries of the domain which is given in [12]. In the scaling used in Sec. 6 and with µ ˜(a, c) = µ(a − c, a + c), we get ∂µ ˜ (a, c) = Φ(c) − Φ(−c) , ∂c
(7.20)
with Φ(c) = ((a + c)2 − µ ˜(a, c)) · (f¯(a + c, a, c))2 , and τ (κ, d, e, h) =
h d µ ˜ (κh)1/2 , (κh)1/2 e . κ 2
Formula (7.20) has also been used for proving Lemma 7.3 in the symmetric case. We remark that 2 ∂2τ ˜ ¯ d, e¯)) = h ¯2 ∂ µ (κ, d, e¯(κ, d), h(κ, (a, c) , 2 ∂e ∂c2
(7.21)
and compute the last term. Using the property ∂∂cµ˜ (a, c) = 0 when c = c¯(a), the symmetry properties and the boundary conditions f¯0 (±a + c, a, c) = 0, we get by differentiation of (7.20), ˜ ∂2µ (a, c) = 2(a + c)(f¯(a + c, a, c))2 ∂c2 ∂ f¯ (a + c, a, c) + 2[(a + c)2 − µ ˜ (a, c)] · f¯(a + c, a, c) · ∂c + 2(a − c)(f¯(−a + c, a, c))2 ∂ f¯ (−a + c, a, c) . (7.22) − 2[(a − c)2 − µ ˜ (a, c)] · f¯(−a + c, a, c) · ∂c
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In the symmetric case, the functions f¯(., a, c), ∂2µ ˜ ∂c2 (a, c)
∂ f¯ ∂c (., a, c)
(with c = 0) and then
are exponentially small near x = ±a, when a is large, so that we needed accurate estimates on each term in order to get the result. Let us prove, in the asymmetric case: Proposition 7.7. There exists a constant A > 0 such that for a ≥ A and c = c¯(a), 1 ˜ ∂2µ 2 2 2 (a + c ¯ (a)) . (7.23) (a, c ¯ (a)) = 2α (g(α )) (φ (−α )) + O exp − 0 0 1 0 ∂c2 4 Proof. We first deduce from Proposition 6.4 (using also (2.29) and (2.30)) and from (6.11) that, when a is large and c = c¯(a), 1 2 ¯ , f (a + c, a, c) = O exp − (a + c) 2 1 2 2 ((a − c) − µ . ˜ (a, c)) = O exp − (a + c) 2 Let us estimate f¯(−a + c, a, c) when c = c¯(a) and a tends to +∞. Using (6.6) and (6.7), 5 2 ¯ f (−a + c, a, c) = γ(a, c)φ1 (−a + c) + O exp − (a + c) , 8 then, from (6.8)(b) and (6.11)
1 2 ¯ f (−a + c, a, c) = g(−α0 )φ1 (−α0 ) + O exp − (a + c) , 4
(7.24)
with g(−α0 )φ1 (−α0 ) > 0 . ∂2 µ ˜ ¯(a)) ∂c2 (a, c
when a is large, we only need a control of the So, in order to estimate ∂ f¯ term ∂c (±a + c, a, c). We will use the following lemma. Lemma 7.8. There exist constants C > 0 and A > 0 such that for a ≥ A and c = c¯(a), ∂ f¯ (±a + c, a, c) ≤ C(a + c)4 . ∂c Proof. Let us use the following translated function: v(x, a, c) = f¯(x + c, a, c) for x ∈ [−a, a] , in order to differentiate more easily f¯. Then, v is solution of −v 00 + (x + c)2 v = µv in ] − a, a[ 0 v (±a) = 0 , kvkL2 (]−a,a[) = 1 , v ∈ H 2 (] − a, a[) .
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Its first partial derivative w ≡ ∂c v with respect to c satisfies ∂µ 00 2 −w + (x + c) w − µw = −2(x + c)v + ∂c v in ] − a, a[ w0 (±a) = 0 , (w, v)L2 (]−a,a[) = 0 , w ∈ H 2 (] − a, a[) , and it is studied at a point where
∂µ ∂c
= 0. Moreover,
∂c f¯(x + c, a, c) = ∂c v(x, a, c) − ∂x v(x, a, c)
for x ∈] − a, a[ .
Using the boundary conditions, we now have ∂c f¯(±a + c, a, c) = ∂c v(±a, a, c) . So, we can estimate ∂c v instead of ∂c f¯. We prefer to calculate this term with the variable y = x + c. The function z defined by z(y, a, c) = ∂c v(x, a, c) satisfies −z 00 + y 2 z − µz = −2y f¯ in ] − a + c, a + c[ z 0 (±a + c) = 0 , (7.25) ¯ L2 (]−a+c,a+c[) = 0 , (z, f) z ∈ H 2 (] − a + c, a + c[) , If we introduce the spaces, Fa,c = {v ∈ L2 (] − a + c, a + c[) ; (v, f¯)L2 (]−a+c,a+c[) = 0 \ Ga,c = {v ∈ H 2 (] − a + c, a + c[) Fa,c ; v 0 (±a + c) = 0} then (P − µ) is an isomorphism from Ga,c onto Fa,c . More precisely, we get, as in [4], using classical estimates, k(P − µ)−1 kL(Fa,c ,Ga,c ) ≤ C(a + c)2 . Consequently, it results from (7.25) that, kzkH 2 (]−a+c,a+c[) ≤ C(a + c)3 . Using the control with respect to a of the norm of the continuous injection from H 1 in C 0 , we get the lemma. In the initial units, it results from (7.8) and (7.21), that (2)
λ2 = Then, from (7.23) with a = (6.13)), we get (7.19).
d 2
1 ∂2µ ˜ (a, c¯(a)) . 2d ∂c2
√ κh0 and using once again that ¯h ≥ µ01 > 0 (see
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As a consequence, we obtain the following corollary: Corollary 7.9. There exists a constant a2 > 0 such that for (κ, d) satisfying κd ≥ a2 and for h0 satisfying (2.5)(a, b) with e = e¯(κ, d), then (2)
λ2 > 0 . We then get that (2.5)(c) is satisfied. This result, combined with (7.8), has been announced in the proof of Theorem 3.4 and permits to complete the proof of the theorem. 7.3.2. Stability or instability for κd large Let us prove Theorem 4.6 which gives the stability of the asymmetric bifurcating solutions when κd is large. Part (i) results from Theorem 6.1 because, under the hypothesis of the theorem, the eigenvalue λ(1) of (7.2) is negative. The second part of Theorem 4.6 results from Theorem 6.1 which gives that λ(1) is strictly positive for κ > Σ0 , and from Corollary 7.9 which gives that the eigenvalue λ(2) is then also strictly positive. 8. Conclusion Let us summarize the stability results in three pictures as function of the parameters κ and d. We distinguish different domains. For each of them we give, when they are known, the sign of the first eigenvalues λ(1) and λ(2) , and write S when the bifurcating solutions are stable and U when they are unstable. 8.1. Stability of symmetric solutions The parameters d and κd are the main parameters of this study (see Theorems 4.3 and 4.5). From Proposition 4.3, Lemma 7.3, (7.16) and (7.17), we √ get 7 , κd = a , d = {(κ, d) ; d = 5−η}, different domains limited by the curves κd = a 0 1 1 √ d2 = {(κ, d) ; d = 5 + η}, κ1 = {(κ, d) ; κ = 2−1/2 − η} and κ2 = {(κ, d) ; κ = 2−1/2 + η} for any η > 0 and unknown constants a0 and a1 . Domain Domain Domain Domain Domain Domain Domain
(1) (2) (3) (4) (5) (6) (7)
: : : : : : :
S, λ(1) > 0, λ(2) > 0. unknown unless λ2 > 0. U, λ(1) < 0, λ(2) > 0. unknown. U, λ(1) < 0, λ(2) < 0. U, λ(1) unknown, but λ(2) < 0. U, λ(1) > 0, λ(2) < 0.
One can probably replace, changing also a0 and a1 , the domains (2) and (6) by curves separating the domains (1) and (3) on one part and (5) and (7) on the other part.
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Fig. 1. Stability of symmetric bifurcating solutions: theoretical results.
8.2. Stability of asymmetric solutions (Fig. 2) We recall (see Theorem 3.3), that in that case, the asymmetric solutions exist when κd ≥ a1 (for some positive constant a1 large enough). Theorem 6.2 gives the stability results. We get 5 different domains limited by the curves κd = a1 , κd = a2 , κ1 = {(κ, d); κ = Σ0 − η} and κ2 = {(κ, d); κ = Σ0 + η} for any η > 0 and some constant a2 verifying a2 ≥ a1 . Domain Domain Domain Domain Domain
(1) (2) (3) (4) (5)
: : : : :
no asymmetric solutions. unknown. U, λ(1) < 0, λ(2) > 0. unknown, unless λ(2) > 0. S, λ(1) > 0, λ(2) > 0.
One can also probably replace the domain (4) by a curve separating the domains (3) and (5), with other a1 and a2 . 8.3. Stability for the symmetric problem (Fig. 3) Using Subsec. 7.2.3 and formula (7.18), we only need to consider, in this case, the sign of the eigenvalue µ(1) , or of the eigenvalue λ(1) for (7.2). From Theorems 7.1, and 7.5, we get, as in the first limited by curves κd = a0 , √ case, 7 different domains √ κd = a1 , d1 = {(κ, d); d = 5 − η}, d2 = {(κ, d); d = 5 + η}, κ1 = {(κ, d) κ =
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Fig. 2. Stability of asymmetric bifurcating solutions: theoretical results.
Fig. 3. Stability of bifurcating solutions for the symmetric problem: theoretical results.
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2−1/2 − η} and κ2 = {(κ, d); κ = 2−1/2 + η} for any η > 0 and unknown constants a0 and a1 . We get Domain Domain Domain Domain Domain Domain Domain
(1) (2) (3) (4) (5) (6) (7)
: : : : : : :
S, µ(1) > 0. unknown. U, µ(1) < 0. unknown. U, µ(1) < 0. unknown. S, µ(1) > 0.
The picture representing these results (see Fig. 3) appears similar as Fig. 1 with possibly different constants. But it is clear that the domain where the solutions are stable are no more the same as before. 8.4. Epilogue This study justifies mathematically a great part of the stability results, in various asymptotics, which were generally admitted in the literature but not proved before. But, the instability of the symmetric bifurcating solutions, when κd is large, is also established. Because in that case the normal solutions are also unstable, this result means that there exists another superconducting solution which is stable. We conjecture that, for κd large and for h near ¯h(κ, d, 0) with h < ¯h(κ, d, 0), bifurcating ¯ d, 0)), solutions belonging to the two curves starting, one from (0, ¯h(κ, d, 0)x; h(κ, ¯ ¯ the other from (0, h(κ, d, e¯(d))(x + e¯(d)); h(κ, d, e¯(d))) exist simultaneously and that the stable one is the asymmetric solution. On the other hand, we have proved the stability of the symmetric bifurcating solutions for κd and κ large enough when we restrict the problem to symmetric solutions. Acknowledgements This study was partially motivated by questions and numerical computations of J. Chapman and private discussions with him a few years ago. We were also motivated by the paper by S. P. Hastings–W. C. Troy [17] and correspondence with S. P. Hastings on the existence of asymmetric solutions. References [1] H. Berestycki, A. Bonnet and S. J. Chapman, “A semi-elliptic system arising in the theory of type-II superconductivity, ” Comm. App. Nonlinear Anal. 1 (3) (1994) 1–21. [2] C. Bolley, “Familles de branches de bifurcations dans les ´ equations de Ginzburg– Landau,” M2 AN 25 (3) (1991) 307–335. [3] C. Bolley, “Mod´elisation du champ de retard a ` la condensation d’un supraconducteur par un probl`eme de bifurcation,” M2 AN 26 (2) (1992) 235–287. [4] C. Bolley and B. Helffer, “An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material,” Annales de l’Institut Henri Poincar´e (Section Physique Th´eorique) 58 (2) (1993) 189–233.
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[5] C. Bolley and B. Helffer, “Rigorous results on the Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field,” preprint Ecole Centrale de Nantes, 1993. [6] C. Bolley and B. Helffer, “Rigorous results for the Ginzburg–Landau equations associated to a superconducting film in the weak κ-limit, Rev. Math. Phys. 8 (1) (1996) 43–83. [7] C. Bolley and B. Helffer, “Rigorous results on the Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field. Part I,” Nonlinear Studies 3 (1) (1996) 1–29. [8] C. Bolley and B. Helffer, “Rigorous results on the Ginzburg–Landau models in a film submitted to an exterior parallel magnetic field. Part II.” Nonlinear Studies 3 (2) (1996) 1–32. [9] S. J. Chapman, “Nucleation of superconductivity in decreasing fields. I and II. “European J. Appl. Math. 5 (4) (1994) 449–468, 469–494. [10] S. J. Chapman, S. D. Howison, J. B. McLeod and J. R. Ockendon, “Normalsuperconducting transitions in Ginzburg–Landau theory,” Proc. Roy. Soc. Edin. 119A (1991) 117–124. [11] S. J. Chapman, S. D. Howison and J. R. Ockendon, “Macroscopic models for superconductivity,” SIAM review, 344 (1992) 529–560. [12] M. Dauge and B. Helffer, “Eigenvalues variation I, Neumann Problem for Sturm– Liouville operators,” J. Differential Eqs., 104 (2) (1993) 243–262. [13] B. Dugnoille, “Etude th´eorique et exp´erimentale des propri´et´es magn´etiques des couches minces supraconductrices de type 1 et de κ faible,” thesis, Mons, 1978. [14] V. L. Ginzburg “On the theory of superconductivity,” Nuovo Cimento 2, (1995) 1234. [15] V. L. Ginzburg, “On the destruction and the onset of superconductivity in a magnetic field,” Soviet Phy. JETP 7 (1958) 78. [16] V. L. Ginzburg and L. D. Landau, “On the theory of superconductivity,” Zh. Eksperim. i teor. Fiz. 20 (1950) 1064–1082; English translation L. D. Landau Men of Physics ed. D. Ter Haar, Pergamon Oxford, (1965) 138–167. [17] S. P. Hastings and W. C. Troy, “There are asymmetric minimizers for the onedimensional Ginzburg–Landau model of superconductivity,” preprint, 1996. [18] B. Helffer and J. Sj¨ ostrand, “Multiple wells in the semiclassical limit I, Comm. in P.D.E., 9 (4) (1984) 337–408. [19] D. St. James and P. G. de Gennes, “Onset of superconductivity in decreasing fields, Phys. Lett. 7 (1963) 306. [20] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980. [21] M. H. Millman and J. B. Keller, “Perturbation theory of nonlinear boundary-value problems,” J. Math. Phys. 10 (2) (February 1969). [22] F. Odeh, “Existence and bifurcation theorems for the Ginzburg–Landau equations,” J. Math. Phys. 8 (12) (December 1967). [23] F. Odeh, “A bifurcation problem in superconductivity,” in Bifurcation Theory and Nonlinear Eigenvalue Problems, eds. J. B. Keller, S. Antman and W. A Benjamin, Inc. (1969) 99–112. [24] Y. Sibuya, “Global theory of a second linear differential equation with a polynomial coefficient,” North-Holland (1975).
CLUSTER PROPERTIES OF ONE PARTICLE ¨ SCHRODINGER OPERATORS. II V. KOSTRYKIN Institut f¨ ur Reine and Angewandte Mathematik Rheinisch-Westf¨ alische Technische Hochschule Aachen D-52056 Aachen, Germany
R. SCHRADER Institut f¨ ur Theoretische Physik Freie Universit¨ at Berlin Arnimallee 14, D-14195 Berlin, Germany Received 17 May 1997 We continue the study of cluster properties of spectral and scattering characteristics of Schr¨ odinger operators with potentials given as a sum of two wells, begun in our preceding article [Rev. Math. Phys. 6 (1994) 833–853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separation of the wells tends to infinity. In this article we determine the explicit form of the subleading contributions, which in particular show strong oscillatory behaviour. Also we apply our methods to the critical and subcritical double well problems.
1. Introduction We consider Schr¨odinger operators with double well potentials when the distance between the wells tends to infinity. More precisely the main subject of our study is the asymptotic behaviour as |d| → ∞ of Hamiltonians in L2 (R3 ) of the form H(d) = −∆ + Vd ,
d ∈ R3 ,
(1.1)
with Vd = V1 + V2 (· − d) .
(1.2)
Here ∆ is the Laplace operator, and V1 and V2 are real valued functions in the Rollnik class R, acting as multiplication operators on L2 (R3 ). Recall that V ∈ R iff Z |V (x)kV (y)| dxdy < ∞ . |x − y|2 The present work is a direct continuation of our previous paper [27], where we established the limiting interrelations (as |d| → ∞) between the resolvent, scattering matrix and the spectral shift function of H(d) and those of the single well Hamiltonians Hi = −∆ + Vi , i = 1, 2 . (1.3) 627 Reviews in Mathematical Physics, Vol. 10, No. 5 (1998) 627–683 c World Scientific Publishing Company
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Following the terminology of many particle scattering theory [16, 19, 20] we call the aforementioned relations cluster properties. The main aim of the present article is to exhibit the explicit structure of the subleading terms in the asymptotic expansion of the spectral shift function and the scattering amplitude when |d| → ∞. Cluster properties of the discrete spectrum of H(d) have been the subject of intensive study above all due to the method of Born and Oppenheimer in the theory of molecules. This property can be described as follows [15]: every eigenvalue of H(d) (1.1) in the limit |d| → ∞ tends to some eigenvalue of H1 or H2 . If H1 and H2 have a common eigenvalue (say E0 ) then H(d) has a pair of eigenvalues E± (d), which are asymptotically degenerated, i.e. E± (d) → E0 for |d| → ∞. The splitting between asymptotically degenerated eigenvalues has been studied in [15, 25, 9]. Cluster properties of the discrete spectrum for long-range potentials which are not in R (in particular, Coulomb potentials, Vi (x) = |x|−1 ) were discussed in [26, 30, 15, 3, 8, 31]. It is interesting to note that for such potentials the limit |d| → ∞ is essentially equivalent to the semiclassical (large coupling constant) limit, since in the case of the Coulomb potential, say, H(d) is unitarily equivalent (up to a scaling factor |d|−2 ) to the Hamiltonian ! 1 1 , (1.4) + −∆ − |d| ˆ |x| |x − d| with dˆ = d|d|−1 . The study of the so-called critical double well problem was initiated by Ovchinnikov and Sigal [34] in order to investigate the Efimov effect [12] and by Klaus and Simon [24]. These authors considered the potential Vd (1.2) where Vi have compact support and the Hamiltonians Hi (1.3) have no negative bound states, but have zero energy resonances (see below for the precise definition of this notion). Under these assumptions Klaus and Simon [24] showed that for large |d| the operator H(d) has the only eigenvalue E(d) = −α2 /d2 + O(|d|−3 ) ,
(1.5)
where α is the (unique) real solution of the equation e−α = α. Later Høegh–Krohn and Mebkhout [17] found an infinite sequence of resonances En (d), n = 1, 2, . . ., tending to zero as |d| → ∞, such that En (d) = −γn2 /d2 + O(|d|−3 ). The constants γn do not depend on the potentials Vi and are the complex solutions of e−α = α. On the other hand Tamura [43] extended the results of [24] to the case of potentials with noncompact supports satisfying |Vi (x)| ≤ C(1 + |x|)−2− , > 0 (however with a lack of uniqueness, i.e. the proof in [43] does not prohibit the existence of other eigenvalues different from E(d) (1.5) and tending to zero as |d| → ∞). The low-energy scattering properties of the Hamiltonian H(d) (1.1) as |d| → ∞ were studied in [18], displaying the connection between (1.1) and Hamiltonians with point interaction. Let h0 be the Hamiltonian of the two-fixed-center point interaction, which is formally ˆ . h0 = −∆ + ν1 δ(x) + ν2 δ(x − d)
(1.6)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
629
The rigorous meaning of h0 and a detailed investigation of its properties can be found in [2]. Let Wr (r > 0) be the unitary scaling in L2 (R3 ) given by (Wr ψ)(x) = −1 = |d|−2 h , where r3/2 ψ(rx). Then W|d| H(d)W|d| h = −∆ + −2
V1
x
+ V2
! x − dˆ ,
with = |d|−1 . When → 0 for compactly supported Vi ∈ R, h converges to h0 (1.6) in the norm sense [2]. The limit is, however, very delicate since it depends very crucially on detailed properties of the spectral point zero for one center operators (1.3). One can expect that all zero-energy characteristics of H(d) in the limit |d| → ∞ are well described by the Hamiltonian h0 (1.6). Indeed, it is proven in [18] for compactly supported Vi ’s that Sd (E/d2 ) = s (E) , and s (E) = s0 (E) + O() ,
(1.7)
−1
again with = |d| . Here the operators Sd (E), s (E), and s0 (E) are the on-shell scattering matrices at energy E of H(d), h , and h0 respectively. A result of another type was proved by the present authors in [27]. Let S1 (E), S2 (E; d) and Sd (E) denote the on-shell scattering matrices of energy E for the pairs (H1 , H0 ), (H2 (d) = H0 + V2 (· − d), H0 ) and (H(d), H0 ) respectively. It was shown that both Sd (E) − S1 (E)S2 (E; d) and Sd (E) − S2 (E; d)S1 (E) tend to zero in Hilbert–Schmidt norm as |d| → ∞ uniformly in E on compact sets in (0, +∞). For Vi ∈ R ∩ L1 (R3 ) the result holds also in the trace norm sense. Since s0 (E) does not possess the cluster property, relation (1.7) shows that in general uniform convergence cannot hold on compact sets in [0, +∞). We turn to a description of the main results of the present article. For d ∈ R3 , let U (d) denote the unitary translation operator (U (d)f )(x) = f (x − d) ,
f ∈ L2 ,
(1.8)
such that U (d)V2 U (d)−1 = V2 (· − d) .
(1.9)
H2 (d) = U (d)H2 U (d)−1 = −∆ + V2 (· − d) .
(1.10)
Let We denote by R(z; d), R1 (z), R2 (z; d) and R0 (z) the resolvents of H(d), H1 , H2 (d) and H0 respectively and with the sign convention R0 (z) = (H0 − z)−1 , etc. We note that obviously R2 (z; d) = U (d)R2 (z)U (d)−1 , where R2 (z) is the resolvent of H2 . It is well known (see [41]) that for V1 and V2 in R ∩ L1 the differences R(z; d) − R0 (z), R1 (z) − R0 (z) and R2 (z; d) − R0 (z) are trace class for Im z 6= 0 and for all d ∈ R3 . Let Ei , i = 1, 2 be the set of z ∈ C for which the homogeneous equation φ = 1/2 1/2 Vi R0 (z)|Vi |1/2 φ has a solution in L2 (R3 ), where Vi (x) = |Vi (x)|1/2 sign Vi (x)
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1/2
(−)
such that Vi = Vi |Vi |1/2 . Note that Ei = Ei ∩ (−∞, 0) represents the negative (+) discrete spectrum of Hi . The sets Ei = Ei ∩ (0, +∞) are bounded closed sets of Lebesgue measure zero. More precisely, they are the unions of two subsets, belonging to the upper and lower lips of the cut (0, +∞) respectively. Let Π0 = C \ [0, +∞) and let Π0 be the closure of Π0 with the two lips of the cut added. We write E = E1 ∪ E2 . For real valued V1 and V2 in R ∩ L1 Krein’s spectral shift functions ξ1 (E), ξ2 (E; d), and ξ(E; d) for the pairs (H1 , H0 ), (H2 (d), H0 ) and (H(d), H0 ) respectively exist, satisfy (1 + |·|2 )−1 ξ ∈ L1 (R) and for each of them the following trace relations hold: Z φ0 (E)ξ1 (E)dE , tr φ(H1 ) − φ(H0 ) = R
tr φ(H2 (d)) − φ(H0 ) = tr φ(H(d)) − φ(H0 ) =
Z
R
Z
φ0 (E)ξ2 (E; d)dE ,
(1.11)
φ0 (E)ξ(E; d)dE
R
with φ being a function in a suitable class of continuously differentiable functions. For instance we may take φ to be in C0∞ (R), also φ(E) = e−tE , t > 0, φ(E) = (E − z)−1 , Im z 6= 0 are in this class. Note that ξ2 (E; d) is independent of d such that ξ2 (E; d) = ξ2 (E), where ξ2 (E) is spectral shift function for the pair (H2 , H0 ). The spectral shift functions are not uniquely defined by (1.11). They may be changed by an additive constant. We normalize the spectral shift functions by the conditions that ξ(E; d), ξ1 (E), and ξ2 (E) are identically zero for E below the spectra of H(d), H1 and H2 respectively (H(d) is bounded below uniformly in d by the Kato inequality). With this normalization for E < 0 one has ξi (E − 0) = −Ni (E) ,
ξ(E − 0; d) = −N (E; d) ,
where Ni (E) and N (E; d) are the counting functions for the Hamiltonians Hi and H(d) respectively. By the Birman–Krein theorem [4] the spectral shift function ξ(E) for the pair (H0 + V , H0 ) is related to the scattering matrix S(E) for fixed energy E: ξ(E) = −
1 log detS(E) , 2πi
E > 0.
(1.12)
The normalization of the specral shift function introduced above leads to the relation (see e.g. [32, 7]) ( ) √ Z E ξ(E) − 2 V (x)dx = 0 , (1.13) lim E→+∞ 4π which fixes the branch of the logarithm in (1.12) uniquely. Theorem 1. Let V1 and V2 be in L1 ∩ R. Let the spectral shift functions be normalized as above. Then
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
Z lim
|d|→∞
R
ξ(E; d) − ξ1 (E) − ξ2 (E) φ(E) dE = 0
631
(1.14)
for arbitrary φ ∈ C0∞ (R). This result extends results in [27], where we only established that ξ 0 (E; d) − − ξ20 (E) tends to zero in the sense of distributions as |d| → ∞. For potentials satisfying slightly more restrictive conditions we will provide a local information on the convergence of ξ(E; d)−ξ1 (E)−ξ2 (E) in the limit |d| → ∞. We impose the following additional condition on the potentials Vi : ξ10 (E)
Property A. The potential V satisfies (1 + | · |)2 V ∈ L1 (R3 ) ∩ R. There are constants R0 > 0 and C > 0 such that |V (x)| ≤ C for all |x| ≥ R0 and |xkV (x)| → 0 for |x| → ∞. The assumption (1 + | · |)2 V ∈ L1 (R3 ) ∩ R implies that V is an Agmon potential, and therefore σsc (H) = ∅ [38]. The fact that V is bounded outside some ball and |xkV (x)| → 0 for |x| → ∞ guarantees that σp (H) ∩ (0, ∞) = ∅ [38]. Hence for Vi ’s having Property A the sets Ei satisfy Ei ∩ (0, ∞) = ∅. Let Ai (ω, ω 0 ; E), i = 1, 2; ω, ω 0 ∈ S2 , E ∈ R+ be the scattering amplitude for the Hamiltonian Hi . We recall that it can be expressed in terms of the integral kernel Si (ω, ω 0 ; E) of the on-shell S-matrix Si (E) (see e.g. [37]): 2πi Ai (ω, ω 0 ; E) = − √ (Si (ω, ω 0 ; E) − δ(ω − ω 0 )) . E We recall the notation dˆ = d|d|−1 ∈ S2 . Theorem 2. Let the potentials Vi , i = 1, 2 have Property A. Then the function ξ12 (E; d) = ξ(E; d) − ξ1 (E) − ξ2 (E)
(1.15)
for sufficiently large |d| is jointly continuous in d and E ∈ (0, +∞) and has the asymptotic representation √ √ cos(2 E|d|) ˆ + sin(2 E|d|) Re a(E; d) ˆ + o(|d|−2 ) , (1.16) Im a(E; d) ξ12 (E; d) = |d|2 |d|2 where
ˆ = − 1 A1 (d, ˆ −d; ˆ E)A2 (−d, ˆ d; ˆ E) . a(E; d) π
The error term o(|d|−2 ) is uniform in E on compact sets in (0, +∞). Remarks. (1) For C0∞ -potentials V1 and V2 the functions ξ(E; d), ξ1 (E) and ξ2 (E) are infinitely differentiable in E on (0, +∞) [39]. Moreover, one can show that for any E ∈ (0, +∞) ξ 0 (E; d) → ξ10 (E) + ξ20 (E) as |d| → ∞. The statement of Theorem 2 shows in particular that the second derivative ξ 00 (E; d) does not converge pointwise as |d| → ∞.
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(2) Under additional restrictions on the potentials (for instance, the Vi ’s have compact support) the error term in (1.16) can be replaced by O(|d|−3 ). (3) The formulas (1.15), (1.16) remain valid for the Hamiltonians with point interactions centered at x = 0 and x = d. The behaviour of ξ12 (E; d) for E = 0 as |d| → ∞ is more delicate and strongly depends on the properties of the spectral point zero for H1 and H2 . We recall that zero is said to be a resonance for H, if the differential equation Hu = 0 has a nontrivial solution u ∈ L2s (R3 ) for some s such that −3/2 ≤ s < −1/2 and u ∈ / L2 (R3 ). Here L2s (R3 ) = {u ∈ L2loc(R3 )|(1 + x2 )s/2 u ∈ L2 (R3 )}. A detailed discussion of zero energy resonances can be found in [21]. Following Jensen and Kato [21] we call the point E = 0 regular for H if it is neither an eigenvalue nor a resonance of H. If E = 0 is a resonance but not an eigenvalue, it is said to be an exceptional point of the first kind. We note that the last case is in no way pathological: even for a square well potential there are isolated values of the coupling constant for which E = 0 is an exceptional point of the first kind. Also one can construct examples of square wells for which E = 0 is an eigenvalue but not a resonance (see e.g. [38]). In this case E = 0 is said to be an exceptional point of the second kind . If E = 0 is both a resonance and an eigenvalue, it is said to be an exceptional point of the third kind . We note that if H has no negative eigenvalues but has a resonance at zero energy, then E = 0 cannot be an eigenvalue of H (see [44]). Let c0 (V ) denote the scattering length of the Hamiltonian H0 + V (see Sec. 5). Then we have Theorem 3. Let the potentials V1 and V2 satisfy Property A. (i) Let E = 0 be either a regular point or an exceptional point of the first kind for both H1 and H2 . Then ξ12 (0+; d) = 0 for all sufficiently large |d|. (ii) Let E = 0 be a regular point for one of the Hamiltonians Hi , i = 1, 2 (say H1 ) and an exceptional point of the first kind for H2 . Then ξ12 (0+; d) = −signc0 (V1 )/2 for all sufficiently large |d|. We note that our definition of the scattering length (5.2) agrees with that used by Kato and Jensen [21] and differs by a sign from the definition customary used in physical literature [1, 6]. Now we apply Theorem 3 to an analysis of the discrete spectrum of H(d). The value of the spectral shift function at zero energy gives information on the multiplicity of the non-positive point spectrum of the Hamiltonian. By the generalized Levinson theorem [32, 10, 11, 6] ξ(0+) equals minus the number of nonpositive eigenvalues counting their multiplicities minus 1/2, if zero is a resonance for H = H0 +V . Thus, we have: Corollary 4. Let the potentials V1 and V2 have Property A. Let E = 0 be either an exceptional point of the first kind for both H1 and H2 , or a regular point for one of the Hamiltonians Hi with positive scattering length and an exceptional point
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
633
of the first kind for the other one. Moreover, let the negative spectrum of both H1 and H2 be empty. Then for all sufficiently large |d| the operator H(d) has a unique (nonpositive) eigenvalue. Now we comment on the case when E = 0 is an exceptional point of the first kind for (say) H2 and the regular one for H1 . For the case of compactly supported potentials we can easily prove that E(d) satisfies the equation √
√ e−2 −E|d| −E = c0 (V1 ) + O(|d|−3 ) + O(−E) . |d|2
(1.17)
Without explicitly pointing out the connection with the scattering length the Eq. (1.17) was proved by Klaus and Simon [24] for the case when V1 ≤ 0 and H1 is subcritical (i.e. E = 0 is a regular point for H1 and the operator H0 + (1 + )V for all −1 ≤ < δ has no bound states). It is easy to show (see Appendix A below) that if V1 ≥ 0 and is not identically zero then the scattering length c0 (V1 ) < 0. If V1 ≤ 0 and the discrete spectrum of H0 + V1 is empty then c0 (V1 ) > 0. In this case Eq. (1.17) has a solution. For stronger attractive potentials V1 which bind several states the scattering length can be positive or negative (for rotationally symmetric potentials see the discussion in [33]). The existence of the solution to (1.17) is controlled by signc0 (V1 ) in agreement with the statement of Theorem 3. Neither Eq. (1.17) nor Theorem 3 give an information on the low-energy spectrum of H(d) if c0 (V1 ) = 0. To avoid a possible reader’s question whether this is possible or not, we show in the Appendix A how to construct the potentials V for which c0 (V ) = 0. Consider now potentials Vi satisfying the inequality |Vi (x)| ≤ C(1 + |x|)−5− for some C > 0 and > 0. Combining Corollary 4 with the results of Tamura [43] we can conclude that for the case of critical double well potentials (the negative spectra of both H1 and H2 are empty and E = 0 is an exceptional point of the first kind for both H1 and H2 ) the Hamiltonian H(d) has a unique negative eigenvalue with asymptotics for |d| → ∞ given by formula (1.5). We note that our methods also can be applied to study other characteristics of H(d) in the limit |d| → ∞. As an example we consider the scattering amplitude 2πi Ad (ω, ω 0 ; E) = − √ (Sd (ω, ω 0 ; E) − δ(ω − ω 0 )) , E where now Sd (ω, ω 0 ; E) (ω, ω 0 ∈ S2 , E ∈ R+ ) is the integral kernel of the on-shell S-matrix Sd (E) for the pair of Hamiltonians (H(d), H0 ). Let us note that for Vi ∈ R ∩ L1 and for all E ∈ R+ \ E the scattering amplitude is a bounded function of its arguments. Let S2 (E; d) be the on-shell S-matrix for the Hamiltonian H2 (d). By (1.10) we obviously have S2 (E; d) = U (d)S2 (E)U (d)−1 . Note that since U (d) commutes with H0 , it has a corresponding spectral decomposition inducing translations in the spaces of fixed energy of H0 , also denoted by U (d). Hence for the corresponding scattering amplitude A2 (ω, ω 0 ; E; d) we have the relation
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V. KOSTRYKIN and R. SCHRADER
√
A2 (ω, ω 0 ; E; d) = e−i
Ehω−ω 0 ,di
A2 (ω, ω 0 ; E) .
(1.18)
Property B. The potential V has Property A and satisfies (1+|·|)4 V ∈ L1 (R3 ). We will prove the following analogue of Theorem 2. Theorem 5. Let the Vi ’s have Property B. Then the function A12 (ω, ω 0 ; E; d) = Ad (ω, ω 0 ; E) − A1 (ω, ω 0 ; E) − A2 (ω, ω 0 ; E; d) for all ω, ω 0 ∈ S2 and for sufficiently large |d| is continuous with respect to d and has the asymptotic representation A12 (ω, ω 0 ; E; d) =
1 ˆ E; d)A1 (d, ˆ ω 0 ; E) + A1 (ω, −d; ˆ E)A2 (−d, ˆ ω 0 ; E; d)] + o(|d|−1 ) . (1.19) [A2 (ω, d; |d|
The error term o(|d|−1 ) is uniform in ω, ω 0 ∈ S2 and E on compact sets in R+ . Intuitively this result is clear (see Fig. 1). By construction of A12 to A12 , both scattering centers have to contribute. Therefore to leading order in |d|−1 , the parˆ if it first has hit center 1, in order to hit center 2 ticle has to move in direction d, (and in direction −dˆ if it first hits center 2 and then center 1). Below we will also give (besides a rigorous proof) a formal proof based on this geometric picture and the Born series for the scattering amplitude. The scattering amplitude A12 also exhibits strong oscillations in the limit |d| → ∞, since by means of (1.18) we can rewrite (1.19) in the form 0
A12 (ω, ω ; E, d) =
√ E|d| h
ei
|d|
√
e−i
√ ˆ Ehω 0 ,di
+ ei
1
i ˆ E)A2 (−d, ˆ ω 0 ; E) + o(|d|−1 ) . A1 (ω, −d; 2
u
SS SS SSd SoS SS SSu
ˆ E)A1 (d, ˆ ω 0 ; E) A2 (ω, d;
S SS
2
ω
ˆ Ehω,di
u
ω0
SS SwS−d SS S
ω0
Fig. 1. Dominant contributions to A12 .
SS u ω 1
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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We remark that the present results can be easily generalized to potentials which are the sum of n > 2 terms by considering the situation where all the separations of their respective centers tend to infinity. The paper is organized as follows. In Sec. 2 we prove several auxiliary lemmas. The Secs. 3, 4 and 5 are devoted to the proofs of Theorems 2, 3 and 5, respectively. All the proofs can be made much easier, when considering the case of compactly supported potentials. The major efforts are made to accomodate the case of potentials having Properties A and B. Throughout the paper we freely do not distinguish between the formal expressions of the form V α R0 (z)|V |1−α defined on D(V 1−α ) and their closures defined on the whole L2 (R3 ). All formal manipulations with the operators of this form can be easily justified. Here after we will use the notations Jp (p ≥ 1) for the trace ideals of p-summable compact operators. In particular, J1 denotes the ideal of trace class operators, and J2 stands for the set of all Hilbert–Schmidt operators. The corresponding norms are denoted by k · kJp . 2. Auxiliary Results Hereafter we assume that Vi ∈ R, i = 1, 2. We start with the formulation of some technical results from our preceeding paper [27] (Theorem 1.1 and Lemma 2.2), which we recall here for the reader’s convenience. Theorem T. Let V1 , V2 ∈ R. Then 1/2
(i) the operator K12 (z; d) = V1 R0 (z)|V2 |1/2 (· − d) tends to zero in Hilbert– Schmidt norm uniformly in z ∈ Π0 as |d| → ∞, (ii) there is a constant c0 > 0 such that the Hamiltonians H1 , H2 , and H(d) defined in the form sense are bounded below by the constant −c0 for all d ∈ R3 . Moreover, if V1 , V2 ∈ R ∩ L1 the operator R(z; d) − R1 (z) − R2 (z; d) + R0 (z)
(2.1)
tends to zero as |d| → ∞ in trace norm uniformly in z on compact subsets of Π0 \ E. We note that a result similar to (ii) for H0 -form compact potentials, for which the convergence of (2.1) to zero holds in the operator norm, was proved earlier by Klaus [25, Appendix]. Lemma 2.1. Let V1 , V2 ∈ R. Then for all z ∈ Π0 the operator K12 (z; d) is continuous in d ∈ R3 in Hilbert–Schmidt norm. Proof. We recall that the translation operator U (d) is strongly continuous in all Lp -spaces with 1 ≤ p < ∞. By means of simple limiting arguments and by the inequality 2/3
1/3
kV kR ≤ 31/2 (2π)1/3 kV kL2 kV kL1 ,
V ∈ L1 (R3 ) ∩ L2 (R3 )
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V. KOSTRYKIN and R. SCHRADER
(see [41]) one can easily prove that U (d) is also strongly continuous in the Rollnik norm, Z 1/2 |V (x)kV (y)| dxdy . kV kR = |x − y|2 We recall that with this norm R is a complete normed vector space. For arbitrary d1 and d2 we estimate 1/2
kK(z; d1 ) − K(z; d2 )kJ2 ≤ kK(z; d1 ) − Vd1 R0 (z)|Vd2 |1/2 kJ2 1/2
+ kVd1 R0 (z)|Vd2 |1/2 − K(z; d2 )kJ2 ≤ kVd1 kR k(|Vd1 |1/2 − |Vd2 |1/2 )2 kR 1/2
1/2
+ kVd2 kR k(Vd1 − Vd2 )2 kR .
(2.2)
Now we use the inequality (|a + b|1/2 − |a|1/2 )2 ≤ |b| ,
(2.3)
which is valid for all a, b ∈ R and which is a simple consequence of |a + b|1/2 ≤ |a|1/2 + |b|1/2 . Due to (2.3) we have (|Vd1 (x)|1/2 − |Vd2 (x)|1/2 )2 = (|Vd1 (x) + V2 (x − d2 ) − V2 (x − d1 )|1/2 − |Vd1 (x)|1/2 )2 ≤ |V2 (x − d2 ) − V2 (x − d1 )| . Therefore the r.h.s. of (2.2) can be bounded by (kVd1 kR + kVd2 kR )kV2 (x − d2 ) − V2 (x − d1 )kR . Since the shift operator U (d) is strongly continuous in Rollnik norm this completes the proof of the lemma. Now for potentials having Property A we provide a more detailed information on the decay of K12 (z; d) as |d| → ∞. For z ∈ Π0 we set √ ˆ = |V1 (x)|1/2 exp{−i zhx, di} ˆ , Φ1 (x; z, d) (2.4) √ ˆ = |V2 (x)|1/2 exp{−i zhx, di} ˆ . (2.5) Φ2 (x; z, d) ˆ are in L2 (R3 ). However for non-compactly supported The functions Φi (·; z, d) loc potentials they are generally not in L2 (R3 ). This is a main obstacle for extending the results of [24, 25] to the case of non-compactly supported potentials. For E ≥ 0 we define √ (±) ˆ = |Vi (x)|1/2 exp ∓ i Ehx, di ˆ , i = 1, 2, (2.6) Φi (x; E, d) ˆ ∈ ˆ is the inner product of x and dˆ = d|d|−1 in R3 . Obviously Φ(±) (·; E, d) where hx, di i 2 3 (±) ˆ defined by (2.6) are the limiting values L (R ), i = 1, 2. The functions Φ (·; E, d) of (2.4), (2.5) on the upper and lower lips of the cut [0, +∞) respectively.
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Lemma 2.2. Let the potentials Vi have Property A. Then there is a constant C > 0 such that kK12 (z; d)kJ2 ≤
C , |d|
kK21 (z; d)kJ2 ≤
C |d|
for all sufficiently large |d| and for all z ∈ Π0 . Moreover for all real E ≥ 0 the operators √
e±i E|d| (±) ˆ (±) (· − d; E, d), ˆ ·) , sign V1 Φ1 (·; E, d)(Φ |d|K12 (E ± i0; d) − 2 4π
(2.7)
√
e±i E|d| (∓) ˆ (∓) (·; E, d), ˆ ·) (2.8) sign V2 (· − d)Φ2 (· − d; E, d)(Φ |d|K21 (E ± i0; d) − 1 4π tend to zero in Hilbert–Schmidt norm as |d| → ∞. The convergence is uniform in E on compact sets in (0, +∞). Remark. For compactly supported potentials Vi ∈ R ∩ L1 the second claim of Lemma 2.2 can be extended to the complex plane. For instance, one can then easily prove that the operator √
|d|K12 (z; d) −
ei
z|d|
4π
ˆ 2 (· − d; z, d), ˆ ·) → 0 sign V1 Φ1 (·; z, d)(Φ
(2.9)
in the Hilbert–Schmidt norm uniformly in z on compact sets in Π0 . Moreover, there is a constant C > 0 such that
√
ei z|d| C
ˆ ˆ sign V1 Φ1 (·; z, d)(Φ2 (· − d; z, d), ·) ≤
|d|K12 (z; d) −
4π |d| J2
uniformly in d for all large |d| and in z on compact sets in Π0 . Formula (2.9) was used by Klaus and Simon [24, 25] to study the critical double well problem. Our approach heavily uses the formula Z
1 f (x)g(y) dxdy = 2 |x − y| 4π
Z b f (p)b g (p) dp , |p|
(2.10)
which is valid for all f, g ∈ R. Here the hat b denotes the Fourier transform, b = f(p)
Z
e−ihp,xi f (x) dx .
The proof of the formula (2.10) can be found in [41].
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V. KOSTRYKIN and R. SCHRADER
To prove Lemma 2.2 we use the following asymptotics: Lemma 2.3. Let f ∈ C 2 (R3 ) and | · |−1 ∂ α f ∈ L1 (R3 ) for all |α| ≤ 2 (α is multiindex). Then for all E ≥ 0 and all large d ∈ R3 one has Z 4πf (0) f (p) −ihd,pi e dp = + o(|d|−2 ) , (2.11) (i) |p| |d|2 3 R Z (ii) R3
√ 2π 2 ±i√E|d| f (p)e−ihd,pi ˆ + o(|d|−1 ) , dp = e f (∓ E d) 2 p − E ∓ i0 |d|
(2.12)
where the error term is uniform in E on compact sets in (0, ∞). Remark. Here (x ∓ i0)−1 is understood in the sense of distributions, such that (x ∓ i0)−1 = v.p.
1 ± iπδ(x) . x
Since the distributions (x ∓ i0)−1 have order 1 they can be extended to linear continuous functionals on C k (R) with k ≥ 1. Proof. The claim (i) follows from the results of [40]. Asymptotics (2.12) is also well known. Its proof for analytic f is quite elementary. However, since we could not find a proof for the case f ∈ C 2 (R3 ) in the literature, we give it in Appendix B. Proof of Lemma 2.2. We consider only the operator K12 (z; d), since K21 (z; d) can be considered similarly. By means of the formula (2.10) one can easily show that Z |d|2 |V1 (x)kV2 (y − d)| dxdy |d|2 kK12 (z; d)k2J2 ≤ 2 (4π) |x − y|2 =
|d|2 (4π)3
Z d d |V1 |(p)|V2 |(p)e−ihd,pi dp . |p|
2 3 α d d d The Assumption A guarantees that |V i |(p) ∈ C (R ) and | · |∂ (|V1 |(p)|V2 |(p)) ∈ 1 3 L (R ). Applying Lemma 2.3 (i) we find that
|d|2 (4π)3
Z d d |V1 |(p)|V2 |(p)e−ihd,pi dp |p|
=
1 d d |V1 |(0)|V2 |(0) + o(1) (4π)2
=
1 kV1 kL1 kV2 kL1 + o(1) , (4π)2
(2.13)
which proves the first part of the claim. Now let z = E + i0. To prove the second part of the claim we consider the squared Hilbert–Schmidt norm of the operator (2.7), which is given by
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Z 1 |V1 (x)kV2 (y − d)| |V1 (x)kV2 (y − d)|dxdy dxdy + |x − y|2 (4π)2 Z |d| −i√E|d| |V1 (x)kV2 (y − d)| i√E|x−y| i√Ehx−y+d,di ˆ e − e e dxdy (4π)2 |x − y| Z |d| i√E|d| |V1 (x)kV2 (y − d)| −i√E|x−y| −i√Ehx−y+d,di ˆ e − e e dxdy . (2.14) (4π)2 |x − y|
|d|2 (4π)2
Z
We have already shown that the first term coincides asymptotically with the second one. Now we prove that for all real fi ∈ R ∩ L1 (R3 ) and all real E ≥ 0 1 4π
Z
f1 (x)f2 (y) ±i√E|x−y| e dxdy = |x − y|
Z
fb1 (p)fb2 (p) dp . p2 − E ∓ i0
(2.15)
First we note that the integral on the l.h.s. of (2.15) is well defined. Indeed, Z |f1 (x)kf2 (x)| dxdy ≤ k |f1 |1/2 R0 (0)|f2 |1/2 kJ2 k|f1 |1/2 kL2 k |f2 |1/2 kL2 4π|x − y| 1/2
1/2
= k |f1 |1/2 R0 (0)|f2 |1/2 kJ2 kf1 kL1 kf2 kL1 .
(2.16)
Now consider the integral 1 4π
Z
f1 (x)f2 (y) i√z|x−y| e dxdy |x − y|
(2.17)
for z ∈ Π0 . First let us suppose that fi ∈ L2 ∩ L1 . Then by the convolution formula (see e.g. [41]) Z Z 1 fb(p)b h(p)b g (p)dp , f (x)h(x − y)g(y)dxdy = (2π)3 where f, h ∈ L2 , g ∈ L1 we have that (2.17) equals Z b f1 (p)fb2 (p) dp . p2 − z Since the integrand in (2.17) is dominated by |f1 (x)kf2 (x)kx−y|−1 , by the Lebesgue dominated convergence theorem the limit Imz → ±0 exists. Thus (2.15) is proven for f ∈ L2 ∩L1 . A simple limiting procedure and (2.16) proves (2.15) for fi ∈ R∩L1 . Consider now the third and fourth terms of (2.14). By (2.15) we have Z √ |d| ∓i√E|d| |V1 (x)kV2 (y − d)| ±i√E|x−y| ±i√Ehx,di ˆ ∓i Ehy−d,di ˆ e e e e dxdy (4π)2 |x − y| =
|d| 1 ∓i√E|d| e (4π)2 2π 2
√ √ Z d d ˆ |V ˆ −ihd,pi |V1 |(p ± E d) E d)e 2 |(p ± dp . 2 p − E ∓ i0
(2.18)
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V. KOSTRYKIN and R. SCHRADER
Applying Lemma 2.3 (ii) we see that the r.h.s. of (2.18) has the asymptotics 2 −1 d d kV1 kL1 kV2 kL1 + o(1) , (4π 2 )−1 |V 1 |(0)|V2 |(0) + o(1) = (4π )
where the error term is uniform in E on compact sets in R+ . Summing up all contributions we obtain the claim of the lemma. The case z = E − i0 can be considered in exactly the same way. To proceed further we need the following technical Lemma 2.4. Let the Vi ’s have Property A. Then for every α ∈ (0, 1) both integrals Z |V1 (x)|α |V2 (x − d)|1−α |V1 (y)| dxdy |x − y|2 and
Z
|V1 (x)|α |V2 (x − d)|1−α |V2 (y − d)| dxdy |x − y|2
are o(|d|−2 ) as |d| → ∞. Proof. First we note that Vi ∈ R, i = 1, 2 implies |V1 |α |V2 |1−α ∈ R for any α ∈ (0, 1) (see [27]). Moreover by H¨older inequality k(1 + | · |)2 |V1 |α |V2 |1−α (· − d)kL1 ≤ k(1 + | · |)2α |V1 |α kL1/α · k(1 + | · |)2(1−α) |V2 |1−α (· − d)kL1/(1−α) 1−α 2 = k(1 + | · |)2 |V1 | kα L1 · k(1 + | · |) |V2 |(· − d)kL1 .
Therefore if the potentials Vi have Property A, then |V1 |α |V2 |1−α (· − d) has Property A also for all α ∈ (0, 1) and all d ∈ R3 . By means of formula (2.10) we obtain |d|2 4π
Z
= |d|2 4π
|V1 (x)|α |V2 (x − d)|1−α |V1 (y)| dxdy |x − y|2 |d|2 (4π)2
Z
d F1 (p; d)|V 1 |(p) dp , |p|
(2.19)
Z
|V1 (x)|α |V2 (x − d)|1−α |V2 (y − d)| dxdy |x − y|2 Z |d|2 |V1 (x + d)|α |V2 (x)|1−α |V2 (y)| = dxdy 4π |x − y|2
=
|d|2 (4π)2
Z
d F2 (p; d)|V 2 |(p) dp , |p|
(2.20)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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where Z F1 (p; d) = =e
e−ihp,xi |V1 (x)|α |V2 (x − d)|1−α dx
−ihp,di
e−ihp,xi |V1 (x + d)|α |V2 (x)|1−α dx , (2.21)
Z F2 (p; d) =
Z
e
−ihp,xi
Z
|V1 (x + d)| |V2 (x)| α
1−α
dx
e−ihp,xi |V1 (x)|α |V2 (x − d)|1−α dx .
= eihp,di
Consider first the r.h.s. of (2.19). With the help of Lemma 2.3 we get that in the limit |d| → ∞ |d|2 (4π)2
Z
d 1 d F1 (p; d)|V 1 |(p) dp = |V1 |(0) |p| 4π
Z |V1 (x + d)|α |V2 (x)|1−α dx + o(1) .
In the case α = 1/2 by Riemann–Lebesgue lemma we have Z Z \ \ 1/2 (p)|V 1/2 (p) = o(1) . |V1 (x + d)|1/2 |V2 (x)|1/2 dx = e−ihd,pi |V 1| 2|
(2.22)
Consider now the general case α ∈ (0, 1). Without loss of generality we can assume that α < 1/2. Then by H¨ older inequality Z |V1 (x + d)|α |V2 (x)|1−α dx Z =
|V1 (x + d)|α |V2 (x)|α |V2 (x)|1−2α dx Z
≤
|V1 (x + d)|
1/2
|V2 (x)|
1/2
2α Z 1−2α |V2 (x)|dx dx ,
which is o(1) by (2.22). The r.h.s. of (2.20) can be considered similarly.
3. Proof of Theorem 1 As already mentioned in the introduction the spectra of H(d), H1 and H2 are bounded below by a common constant [27]. Pick a real positive number c0 such that −c0 < min{inf σ(H(d)), inf σ(H1 ), inf σ(H2 )}. Then for Vi ∈ R∩L1 the differences Vd = R(−c0 ; d) − R0 (−c0 ) , Vi = Ri (−c0 ) − R0 (−c0 ) , V2 (d) = R2 (−c0 ; d) − R0 (−c0 ) with V2 (d = 0) = V2 are trace class for all d ∈ R3 .
i = 1, 2,
(3.1)
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V. KOSTRYKIN and R. SCHRADER
By Krein’s theorem [28] (see also [5]) for self-adjoint operators A2 and A1 such that A2 − A1 is trace class the spectral shift function ξ(λ; A2 , A1 ) exists and is given by ξ(λ; A2 , A1 ) = π −1 lim arg det(I + (A2 − A1 )(A1 − λ − i)−1 ) . (3.2) →+0
For all z ∈ C with Imz 6= 0 it has the property Z ξ(λ; A2 , A1 ) dλ = log det(I + (A2 − A1 )(A1 − z)−1 ) . λ−z R The branch of the logarithm is uniquely fixed by the condition log det(I + (A2 − A1 )(A1 − z)−1 ) → 0 , when Imz → ∞. We have Z R
ξ(λ; A2 , A1 )dλ = tr(A2 − A1 ) , (3.3)
Z R
|ξ(λ; A2 , A1 )|dλ = kA2 − A1 kJ1 .
In particular this guarantees the existence of the spectral shift functions ˜ d) = ξ(λ; R(−c0 ; d), R0 (−c0 )) , ξ(λ; ξ˜1 (λ) = ξ(λ; R1 (−c0 ), R0 (−c0 )) , ξ˜2 (λ; d) = ξ(λ; R2 (−c0 ; d), R0 (−c0 )) . Note that ξ˜2 (λ; d) = ξ˜2 (λ) = ξ(λ; R2 (−c0 ), R0 (−c0 )) by the unitarity of the translations (1.8). By the invariance principle (see e.g. [5, 22]) we can define the spectral shift functions ξ(E; d), ξ1 (E) and ξ2 (E; d) for the pairs (H(d), H0 ), (H1 , H0 ), and (H2 (d), H0 ) by ˜ ξ(E; d) = −ξ((E + c0 )−1 ; d) , ξ1 (E) = −ξ˜1 ((E + c0 )−1 ) ,
(3.4)
ξ2 (E; d) = −ξ˜2 ((E + c0 )−1 ; d) . Obviously, ξ(E; d) = ξ1 (E) = ξ2 (E; d) = 0 for any E ≤ −c0 and for all d ∈ R3 . Remark. Note that from (3.3) it follows that if A2 ≥ A1 then ξ(λ; A2 , A1 ) ≥ 0 for all λ ∈ R. Therefore by the monotonicity of the resolvent and by the invariance principle for arbitrary real W1 and W2 in R ∩ L1 satisfying W1 ≥ W2 pointwise we have ξ(E; H0 + W1 , H0 ) ≥ ξ(E; H0 + W2 , H0 ) for all E ∈ R. This yields an elementary proof of Kato’s monotonicity theorem [23] (see also [13]).
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Lemma 3.1. The limit relation Z ˜ Z ˜ ξ(λ; d) ξ1 (λ) + ξ˜2 (λ) dλ = dλ lim λ−z |d|→∞ R λ − z R holds for every z ∈ C with Imz 6= 0. Proof. Using the relation between the spectral shift function and the perturbational determinant [5] we have Z ˜ ξ(λ; d) dλ = log det(I + Vd r0 (z)) , R λ−z Z R
ξ˜i (λ) dλ = log det(I + Vi r0 (z)) , λ−z
(3.5) i = 1, 2,
where Vd and Vi , i = 1, 2 are defined by (3.1), and r0 (z) = [R0 (−c0 ) − z]−1 . In the representations (3.5) the branch of the logarithm is fixed uniquely by the conditions log det(I + Vd r0 (z)) → 0 , log det(I + Vi r0 (z)) → 0 ,
(3.6) i = 1, 2,
when Imz → ∞. Since ξ2 (λ) = ξ2 (λ; d) it suffices to prove that o n lim log det (I + Vd r0 (z))(I + V2 (d)r0 (z))−1 (I + V1 r0 (z))−1 = 0
(3.7)
for all nonreal z. If we now show that o n det (I + Vd r0 (z))(I + V2 (d)r0 (z))−1 (I + V1 r0 (z))−1 → 1
(3.8)
|d|→∞
as |d| → ∞ and that the convergence is uniform in Imz ∈ [, +∞) (or Imz ∈ (−∞, −]) for all > 0, then the conditions (3.6) guarantee (3.7). We note that the operators I + V2 (d)r0 (z) and I + V1 r0 (z) are invertible for nonreal z. The operator norms of (I + V1 r0 (z))−1 and (I + V2 (d)r0 (z))−1 can be bounded uniformly in Imz ∈ [, +∞) (Imz ∈ (−∞, −]) for all > 0. Therefore relation (3.8) will be proved once we show that kI + Vd r0 (z) − (I + V1 r0 (z))(I + V2 (d)r0 (z))kJ1 → 0 when |d| → ∞ uniformly in z ∈ Π0 . By Theorem T we have that Vd −V1 −V2 (d) → 0 in trace norm. Thus, it suffices to prove that lim kV1 r0 (z)V2 (d)kJ1 = 0 .
|d|→∞
(3.9)
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To establish (3.9) we note that by the second resolvent identity and due to the obvious relation zr0 (z)R0 (−c0 ) = −R0 (z −1 − c0 ) , we have V1 r0 (z)V2 (d) = [R1 (−c0 ) − R0 (−c0 )]r0 (z)[R2 (−c0 ; d) − R0 (−c0 )] = R1 (−c0 )V1 R0 (−c0 )r0 (z)R0 (−c0 )V2 (· − d)R2 (−c0 ; d) = R1 (−c0 )V1 R0 (−c0 )V2 (· − d)R2 (−c0 ; d) + zR1 (−c0 )V1 r0 (z)R0 (−c0 )V2 (· − d)R2 (−c0 ; d) = R1 (−c0 )V1 R0 (−c0 )V2 (· − d)R2 (−c0 ; d) − R1 (−c0 )V1 R0 (z −1 − c0 )V2 (· − d)R2 (−c0 ; d) .
(3.10)
Consider the first term on the r.h.s. of (3.10). We decompose R1 (−c0 )V1 R0 (−c0 )V2 (· − d)R2 (−c0 ; d) 1/2
= R1 (−c0 )|V1 |1/2 V1
1/2
R0 (−c0 )|V2 |1/2 (· − d)V2
(· − d)R2 (−c0 ; d) . 1/2
Note that the Hilbert–Schmidt norms of R1 (−c0 )|V1 |1/2 and V2 are uniformly bounded in d. Moreover, (i) of Theorem T yields 1/2
k V1
(· − d)R2 (−c0 ; d)
R0 (−c0 )|V2 |1/2 (· − d)kJ2 → 0
when |d| → ∞. The second term on the r.h.s. of (3.10) can be discussed in a similar way. In this case Theorem T says that the last term tends to zero in Hilbert– Schmidt norm as |d| → ∞ uniformly in z ∈ Π0 , thus proving (3.9), and by the previous remark also Lemma 3.1. Now we use the continuity of the Stieltjes transform (see e.g. [35, Appendix A]): Lemma 3.2. Let the sequence of measurable functions µn (t) on R (n ∈ N) satisfy the following properties: Z |µn (t)| dt < ∞ , R 1 + |t| (3.11) Z |µn (t)| dt = 0 . lim sup c→∞ n≥1 |t|≥c |t| R (t)dt If the sequence µnt−z converges to a function f (z) for every nonreal z, then there exists a measurable function µ(t) such that µn (t) → µ(t) in the sense of distributions and f is the Stieltjes transform of µ, i.e. Z µ(t)dt . f (z) = t−z
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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˜ d). Obviously, We show that the condition (3.11) is satisfied for ξ(λ; Z |t|≥c
˜ d)| 1 |ξ(t; dt ≤ |t| c
Z |t|≥c
˜ d)|dt ≤ |ξ(t;
1 kVd kJ1 . c
The norm kVd kJ1 is uniformly bounded in d since kR(−c0 ; d) − R0 (−c0 )kJ1 ≤ kR(−c0 ; d) − R1 (−c0 ) − R2 (−c0 ; d) + R(−c0 )kJ1 + kR1 (−c0 ) − R0 (−c0 )kJ1 + kR2 (−c0 ) − R0 (−c0 )kJ1 ≤ sup {kR(−c0 ; d) − R1 (−c0 ) − R2 (−c0 ; d) + R(−c0 )kJ1 } d∈R3
+ kR1 (−c0 ) − R0 (−c0 )kJ1 + kR2 (−c0 ) − R0 (−c0 )kJ1 . By Theorem T the norm kR(−c0 ; d) − R1 (−c0 ) − R2 (−c0 ; d) + R0 (−c0 )kJ1 can be bounded by a constant which is independent of d. Now, it follows from Lemmas 3.1 and 3.2 that Z ˜ d) − ξ˜1 (λ) − ξ˜2 (λ) ψ(λ)dλ = 0 lim ξ(λ; |d|→∞
R
for all ψ ∈ C0∞ (R). Using (3.4) we get Z (ξ(E; d) − ξ1 (E) − ξ2 (E))ψ((E + c0 )−1 ) lim |d|→∞
R
dE =0 (E + c0 )2
for arbitrary ψ ∈ C0∞ (R). Let us consider functions ψ ∈ C0∞ (R) with supp ψ ⊂ (0, +∞). Then obviously ψ((E + c0 )−1 ) (3.12) φ(E) = (E + c0 )2 is infinitely differentiable and has compact support lying in (−c0 , +∞). Conversely, an arbitrary C0∞ -function φ with suppφ ⊂ (−c0 , +∞) can be represented in the form (3.12) with ψ ∈ C0∞ (R). Since c0 can be taken arbitrary large this proves Theorem 1. 4. Proof of Theorem 2 In the proof of Theorem 2 we will use the representation of the spectral shift function for the pair of Hamiltonians H = H0 + V and H0 in terms of regularized Fredholm determinants. We recall that for an arbitrary Hilbert–Schmidt operator A the regularized Fredholm determinant det2 (I + A) is defined as the product Q −λj (A) , where the λj (A) are the eigenvalues of A. j (1 + λj (A))e
646
V. KOSTRYKIN and R. SCHRADER
Lemma 4.1. For V ∈ R ∩ L1 the spectral shift for the pair (H = H0 + V, H0 ) function can be represented in the form ξ(E; H, H0 )
" ( )# √ Z i E 1 det2 (I + V 1/2 R0 (E + i0)|V |1/2 ) log exp θ(E) V dx , (4.1) = 2πi 2π R det2 (I + V 1/2 R0 (E − i0)|V |1/2 )
where θ(E) is the Heaviside unit step function, θ(t) = 1, t > 0, and θ(t) = 0 otherwise. The branch of the logarithm is chosen so that lim|Imz|→∞ log det2 (I + V 1/2 R0 (z)|V |1/2 ) = 0. This formula has previously appeared in [32, 14, 7]. Since its proof for V ∈ R ∩ L1 still seems to be unpublished, for the reader’s convenience it will be given in Appendix C. Remark. We note that the operators V 1/2 R0 (E±i0)|V |1/2 are continuous in the Hilbert–Schmidt norm in E ∈ R+ \E [37]. Since the determinant det2 A is continuous with respect to A (see e.g. [42]), the functions det2 (I + V 1/2 R0 (E ± i0)|V |1/2 ) are both continuous in E ∈ R+ \ E. Since the operators I + V 1/2 R0 (E ± i0)|V |1/2 are invertible for all E ∈ R+ \ E and due to (4.1) the spectral shift function is continuous in E ∈ R+ \ E. For the potentials having Property A the intersection E ∩ (0, +∞) = ∅. Therefore in this case ξ(E; H, H0 ) is continuous on (0, +∞). Let 1/2
Ki (z) = Vi K2 (z; d) =
R0 (z)|Vi |1/2 ,
1/2 V2 (·
i = 1, 2,
− d)R0 (z)|V2 |1/2 (· − d) ,
1/2
R0 (z)|V2 |1/2 (· − d) ,
1/2
(· − d)R0 (z)|V1 |1/2 ,
1/2
R0 (z)|Vd |1/2 .
K12 (z; d) = V1 K21 (z; d) = V2
K(z; d) = Vd
We recall that for arbitrary Hilbert–Schmidt operators A, B, C the following identity holds (see e.g. [42]) det2 ((I + A)(I + B)(I + C)) = det2 (I + A)det2 (I + B)det2 (I + C) · exp{−tr[AB + AC + BC + ABC]} . (4.2) We can write I + K(z; d) = (I + K1 (z))(I + K2 (z; d))(I + L(z; d)) ,
(4.3)
with L(z; d) = (I + K2 (z; d))−1 (I + K1 (z))−1 · (K(z; d) − K1 (z) − K2 (z; d) − K1 (z)K2 (z; d)) .
(4.4)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Applying the formula (4.2) to (4.3) we get det2 (I + K(z; d)) = det2 (I + K1 (z)) · det2 (I + K2 (z; d))det2 (I + L(z; d))e−t(z;d) , where t(z; d) = trK1 (z)K2 (z; d) + trK1 (z)L(z; d) + trK2 (z; d)L(z; d) + trK1 (z)K2 (z; d)L(z; d) .
(4.5)
Now by means of Lemma 4.1 we get the representation ξ(E; d) = ξ1 (E) + ξ2 (E) + ξ12 (E; d) , where ξ12 (E; d) =
1 det2 (I + L(E + i0; d)) log exp{−t(E + i0; d) + t(E − i0; d)} . (4.6) 2πi det2 (I + L(E − i0; d))
Due to (1.13) the branch of the logarithm can be fixed uniquely by the condition lim ξ12 (E; d) = 0 .
E→+∞
By Theorem T L(z; d) tends to zero in Hilbert–Schmidt norm uniformly in z on compact sets in Π0 when |d| → ∞. Therefore the operator I + L(E ± i0; d) is invertible for all sufficiently large |d|. From Lemma 2.1 it follows that the operator K2 (z; d) is J2 -continuous in d (special case V1 = 0). Thus the operator L(z; d) and the function t(z; d) are continuous in d. By the continuity of the determinant det2 , ξ12 (E; d) is continuous in d for all sufficiently large |d| and fixed E. Combining these arguments with the remark after Lemma 4.1 we can easily prove the joint continuity of ξ12 (E; d) in E and d. As an easy consequence of Lemma 2.4 we get: Lemma 4.2. Let the potentials Vi have Property A. Then for all z ∈ Π0 kK1 (z)K2 (z; d)kJ1 = o(|d|−2 ) . as |d| → ∞ Proof. It is easy to see that 1/2
kK1 (z)K2 (z; d)kJ1 ≤ kV1
R0 (z)|V1 |1/4 |V2 |1/4 (· − d)kJ2
· k |V1 |1/4 |V2 |1/4 (· − d)R0 (z)|V2 |1/2 (· − d)kJ2 ≤
1 (4π)2 ·
Z
1 (4π)2
Z
|V1 (x)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy |x − y|2
1/2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
which is o(|d|−2 ) due to Lemma 2.4.
1/2 ,
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V. KOSTRYKIN and R. SCHRADER
e d) as We define the operator K(z; e d) = K(z; d) − K1 (z) − K2 (z; d) − K12 (z; d) − K21 (z; d) . K(z; e d)kJ2 tends Lemma 4.3. Let the potentials Vi have Property A. Then |d| kK(z; to zero as |d| → ∞ uniformly in z on compact subsets of Π0 . e d) is identically Remark. For compactly supported potentials the operator K(z; zero for sufficiently large |d|. Proof. Obviously,
1/2 1/2 1/2 e d)kJ2 ≤ kK(z;
Vd − V1 − V2 (· − d) R0 (z)|Vd |1/2
J2
1/2
+ V1 R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)
J2
1/2
+ V2 (· − d)R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)
J2
. (4.7)
Due to Lemma 2.3 of [27]
1/2
Vd
1/2
(x) − V1
1/2
(x) − V2
2 (x − d) ≤ 4|V1 (x)|1/2 |V2 (x − d)|1/2 .
(4.8)
Also Vi ∈ R ∩ L1 implies |V1 |1/2 |V2 |1/2 (· − d) ∈ R ∩ L1 . Let us consider the first term on the r.h.s. of (4.7). The two other terms can be discussed in a completely similar way. It is easy to see that
2
1/2
1/2 1/2 |d|2 Vd − V1 − V2 (· − d) R0 (z)|Vd |1/2
J2
≤
|d|2 4π 2
≤
|d|2 4π 2
Z
|V1 (x)|1/2 |V2 (x − d)|1/2 |Vd (y)| dxdy |x − y|2
Z
|V1 (x)|1/2 |V2 (x − d)|1/2 |V1 (y)| dxdy |x − y|2 Z |d|2 |V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy . + 2 4π |x − y|2
Due to Lemma 2.4 both these integrals are o(1), which completes the proof of the lemma. We can represent the operator L(z; d) (4.4) in the form L(z; d) = (I + K2 (z; d))−1 (I + K1 (z))−1 e d) , · [K12 (z; d) + K21 (z; d)] + L(z;
(4.9)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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where e d) = (I + K2 (z; d))−1 (I + K1 (z))−1 L(z; e d) − K1 (z)K2 (z; d) , · K(z; Due to Lemma 4.2 and Lemma 4.3 for all z ∈ C \ E and all large |d| the operator e d) can be estimated as L(z; e d)kJ2 = o(|d|−1 ) . kL(z; We estimate now the square of the operator L(z; d). Lemma 4.4. For all z ∈ C \ E and sufficiently large |d|, L2 (z; d) = (I + K1 (z))−1 K12 (z; d)(I + K2 (z))−1 K21 (z; d) + (I + K2 (z; d))−1 K21 (z; d)(I + K1 (z))−1 K12 (z; d) + o(|d|−2 ) , where the error term is understood in the sense of trace norm und uniform in z on compact sets in C \ E. Proof. Due to (4.9) and Lemma 4.3 for all z ∈ C \ E and all sufficiently large |d| we have L2 (z; d) = (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) + (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) + (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) + (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) + o(|d|−2 ) ,
(4.10)
where the error term o(|d|−2 ) is understood in the sense of trace norm. Consider the first term in the r.h.s. of (4.10) (the third term can be considered in exactly the same way). Let us estimate the operator |d|2 K12 (z; d)(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) . For the case of compactly supported potentials this operator is identically zero for all large |d|. Obviously, for all z in compact sets in C \ E we have
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|d|2 kK12 (z; d)(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d)kJ1 2 = |d|2 k(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d)kJ1 2 ≤ C|d|2 kK12 (z; d)kJ1 1/2
≤ CkV1
1/4
R0 (z)|V1 |1/4 |V2 |1/4 (· − d)kJ2 kV1
|V2 |1/4 (· − d)R0 (z)|V2 |1/2 (· − d)kJ2 . (4.11)
for some C > 0. The r.h.s. of (4.11) can be bounded by C (4π)2 Z ·
Z
|V1 (x)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy |x − y|2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
1/2
1/2 ,
which is o(|d|−2 ) by Lemma 2.4. We turn to the discussion of the second term in (4.10) (the fourth term can be considered in exactly the same way). We show that |d|2 {(I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 (I + K1 (z))−1 K21 (z; d) − (I + K1 (z))−1 K12 (z; d)(I + K2 (z; d))−1 K21 (z; d)}
(4.12)
tends to zero in trace norm as |d| → ∞. Again for the case of compactly supported potentials the expression (4.12) is identically zero for all large |d|. First we express the difference (4.12) in the following form |d|2 (I + K2 (z; d))−1 (I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 {(I + K1 (z))−1 − I}K21 (z; d) + |d|2 {(I + K2 (z; d))−1 − I}(I + K1 (z))−1 K12 (z; d) · (I + K2 (z; d))−1 K21 (z; d) . We show that |d|2 k{(I + K1 (z))−1 − I}K21 (z; d)kJ1 → 0 as |d| → ∞. Clearly, (I + K1 (z))−1 − I = −(I + K1 (z))−1 K1 (z) . Thus, it suffices to prove that |d|2 kK1 (z)K21 (z; d)kJ1 → 0
(4.13)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
651
as |d| → ∞. To this end we use the estimate kK1 (z)K21 (z; d)kJ1 1/2
≤ kV1
R0 (z)|V1 |1/4 |V2 |1/4 (· − d)kJ2
1/4
· kV1 ≤
1 (4π)2
1/4
V2 Z
(· − d)R0 (z)|V1 |1/2 kJ2
|V1 (x)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy . |x − y|2
This integral is o(|d|−2 ) due to Lemma 2.4. Similarly we can show that |d|2 kK21 (z; d){(I + K2 (z; d))−1 − I}kJ1 → 0
(4.14)
as |d| → ∞. Consider the second term in (4.13). Obviously, |d|2 k{(I + K2 (z; d))−1 − I}(I + K1 (z))−1 · K12 (z; d)(I + K2 (z; d))−1 K21 (z; d)kJ1 = |d|2 kK21 (z; d){(I + K2 (z; d))−1 − I} · (I + K1 (z))−1 K12 (z; d)(I + K2 (z; d))−1 kJ1 . Due to (4.14) this tends to zero as |d| → ∞.
Consider now the function t(z; d) given by (4.5). Lemma 4.5. For all z ∈ C \ E and sufficiently large |d|, t(z; d) = o(|d|−2 ) uniformly in z on compact sets in C \ E. Remark. large |d|.
For Vi ’s with compact supports t(z; d) = 0 for all sufficiently
Proof. From (4.5) by Lemma 4.2 it follows that t(z; d) = trK1 (z)L(z; d) + trK2 (z; d)L(z; d) + o(|d|−2 ) . Consider trK1 (z)L(z; d). By (4.4) and again by Lemma 4.2 we have |trK1 (z)L(z; d)| = |trL(z; d)K1 (z)| = |tr[(I + K2 (z; d))−1 (I + K1 (z))−1 {K(z; d) − K1 (z)}K1 (z)]| + o(|d|−2 ) ≤ CkK2 (z; d)K1 (z)kJ1 + CkK12 (z; d)K1 (z)kJ1 e d)K1 (z)kJ1 + o(|d|−2 ) . + CkK21 (z; d)K1 (z)kJ1 + CkK(z;
652
V. KOSTRYKIN and R. SCHRADER
In the course of the proof of Lemma 4.4 we have already shown that kK1 (z)K21 (z; d)kJ1 = o(|d|−2 ) when d → 0. In exactly the same way we can show that kK12 (z; d)K1 (z)kJ1 is o(|d|−2 ) also. The norm kK2 (z; d)K1 (z)kJ1 is o(|d|−2 ) by Lemma 4.2. We consider e d)K1 (z)kJ1 . As in the course of the proof of Lemma 4.3 using (4.8) we now kK(z; can estimate e d)kJ1 kK1 (z)K(z;
1/2 1/2 1/2 ≤ K1 (z) Vd − V1 − V2 (· − d) R0 (z)|Vd |1/2
J1
1/2
+ V1 R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d) K1 (z)
J1
1/2
+ V2 R0 (z) |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d) K1 (z)
J1
1/2
≤ 2kV1
R0 (z)|V1 |3/8 |V2 |1/8 (· − d)kJ2 · kV1 |3/8 |V2 |1/8 (· − d)R0 (z)|Vd |1/2 kJ2
1/2
R0 (z)|V1 |3/8 |V2 |1/8 (· − d)kJ2 · kV1
1/2
(· − d)R0 (z)|V1 |3/8 |V2 |1/8 (· − d)kJ2
+ 2kV1 + 2kV2 3/8
· kV1
3/8
|V2 |1/8 (· − d)R0 (z)|V1 |1/2 kJ2
|V2 |1/8 (· − d)R0 (z)|V1 |1/2 kJ2 .
Due to Lemma 2.4 this is o(|d|−2 ). In exactly the same way one can estimate the remaining term trK2 (z; d)L(z; d), thus completing the proof of the lemma. To proceed further with the proof of Theorem 2 we use the obvious estimate log det2 (I + A) + 1 trA2 ≤ kA3 kJ1 , (4.15) 2 where A is an arbitrary Hilbert–Schmidt operator with operator norm satisfying kAk < 1/2. (This estimate easily follows from the definition of the modified Fredholm determinant.) We apply (4.15) to the operator L(E ± i0; d). Let us note that kL3 (E ± i0; d)kJ1 ≤ kL(E ± i0; d)k kL2 (E ± i0; d)kJ1 ≤ kL(E ± i0; d)k3J2 . From (4.9), Lemmas 2.2–2.4 and 4.3 it follows that kL(E ± i0; d)k3J2 ≤
C |d|3
for some C > 0. Therefore, it follows from (4.6) and Lemma 4.5 that ξ12 (E; d) = −
1 tr[L2 (E + i0; d) − L2 (E − i0; d)] + o(|d|−2 ) . 4πi
(4.16)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
653
Applying Lemmas 4.4 and 2.2 we get that √
e±2i E|d| |d| trL (E ± i0; d) − 2 (4π)2 (∓) ˆ (I + K1 (E ± i0))−1 sign V1 Φ(±) (·; E, d) ˆ · Φ1 (·; E, d), 1 2
2
(±) ˆ (I + K2 (E ± i0; d))−1 sign V2 (· − d)Φ(∓) (· − d; E, d) ˆ · Φ2 (· − d; E, d), 2 tends to zero when |d| → ∞. By translation invariance (±) ˆ (I + K2 (E ± i0; d))−1 sign V2 (· − d)Φ(∓) (· − d; E, d) ˆ Φ2 (· − d; E, d), 2 (±) ˆ (I + K2 (E ± i0))−1 sign V2 Φ(∓) (·; E, d) ˆ . = Φ2 (·; E, d), 2 Therefore to leading order in |d|−1 tr[L2 (E + i0; d) − L2 (E − i0; d)] √
2 e2i E|d| (−) ˆ (I + K1 (E + i0))−1 sign V1 Φ(+) (·; E, d) ˆ Φ (·; E, d), = 1 1 (4π)2 |d|2 (+) ˆ (I + K2 (E + i0))−1 sign V2 Φ(−) (·; E, d) ˆ · Φ2 (·; E, d), 2 √
2 e−2i E|d| (+) ˆ (I + K1 (E − i0))−1 sign V1 Φ(−) (·; E, d) ˆ Φ1 (·; E, d), − 1 2 2 (4π) |d| (−) ˆ (I + K2 (E − i0))−1 sign V2 Φ(+) (·; E, d) ˆ + o(|d|−2 ) . · Φ2 (·; E, d), 2 With the help of the well-known representation for the scattering amplitude (see e.g. [37, 1]) √ 0 1 i√Ehω,·i 1/2 e |Vi |1/2 , (I + Ki (E + i0))−1 Vi ei Ehω ,·i , (4.17) Ai (ω, ω 0 ; E) = − 4π the symmetry relation √ √ ˆ ˆ 1/2 |Vi |1/2 e−i Ehd,·i , (I + Ki (E − i0))−1 Vi ei Ehd,·i √ √ 1/2 ˆ ˆ , (I + Ki (E + i0))−1 Vi e−i Ehd,·i = |Vi |1/2 ei Ehd,·i and (2.6) one gets tr[L2 (E + i0; d) − L2 (E − i0; d)] √
e2i E|d| ˆ −d; ˆ E)A2 (−d, ˆ d; ˆ E) A1 (d, =2 |d|2 √
e−2i E|d| ˆ −d; ˆ E)A2 (d, ˆ −d; ˆ E) + o(|d|−2 ) , −2 A1 (d, |d|2 which together with (4.16) gives (1.16).
654
V. KOSTRYKIN and R. SCHRADER
5. Proof of Theorem 3 Let E = 0 be a regular point for both H1 and H2 . Hence the operators I +Ki (0), i = 1, 2 are invertible. Therefore all the arguments used to prove Theorem 2 can be repeated verbatim in the case E = 0, thus yielding ξ12 (0+; d) = where
1 ˆ + o(|d|−2 ) , Ima(0; d) |d|2
(5.1)
ˆ −d; ˆ 0)A2 (−d, ˆ d; ˆ 0) . ˆ = − 1 A1 (d, a(0; d) π
We recall that c0 (Vi ) := Ai (ω, ω 0 ; 0) = −
1 1/2 1/2 |Vi | , (I + Ki (0))−1 Vi , 4π
(5.2)
ˆ = 0. From is the scattering length, which is obviously real. Therefore Ima(0; d) (5.1) and the fact that ξ12 (0+; d) is integer or half-integer, it follows that for all sufficiently large |d| ξ12 (+0; d) = 0 . Now we consider the case where E = 0 is an exceptional point of the first kind at least for one of the operators H1 and H2 . It is well known (see e.g. [1]) that E = 0 is an exceptional point of the first kind for Hi iff the equation Ki (0)ϕi = −ϕi ,
(5.3)
has a unique solution ϕi ∈ L2 (R3 ) and (|Vi |1/2 , ϕi ) 6= 0. The corresponding eigenprojector we denote by Pi , i.e. Pi =
ϕi (ϕ˜i , ·) , (ϕ˜i , ϕi )
where ϕ˜i = (sign Vi )ϕi . Obviously ϕ˜i satisfies the equation Ki (0)∗ ϕ˜i = −ϕ˜i . It is / easy to see that (ϕ˜i , ϕi ) 6= 0 (see [2, pp. 21–22]). The function ψi = R0 (0)|Vi |1/2 ϕi ∈ L2 is called a zero energy resonance wave function and satisfies Hi ψi = 0 in the sense of distributions. For potentials having Property A we can expand the operators Ki (z) in the √ powers of z: √ i z 1/2 V (|Vi |1/2 , ·) + zNi + o(z) , Ki (z) = Ki (0) + (5.4) 4π i where Ni is a Hilbert–Schmidt operator with integral kernel Ni (x, y) = −
1 1/2 V (x)|x − ykVi |1/2 (y) . 8π i
The low-energy expansion obtained in [1] yields the operator relation 4π (I + Ki (z))−1 = √ Qi + Mi (z) , i z
(5.5)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
with
ϕi (ϕ˜i , ·) , |(|Vi |1/2 , ϕi )|2
Qi =
655
(5.6)
being a rank 1 operator and Mi (z) is a bounded operator in a neighborhood of z = 0. For z → 0 Mi (z) has a representation ϕi (ϕ˜i , Vi )(Ti∗ |Vi |1/2 , ·) Ti Vi (|Vi |1/2 , ϕi )(ϕ˜i , ·) − |(|Vi |1/2 , ϕi )|2 |(|Vi |1/2 , ϕi )|2 1/2
Mi (z) = Ti −
1/2
1/2
+
ϕi (|Vi |1/2 , Ti Vi )(ϕ˜i , ·) ϕi (ϕ˜i , Ni ϕi )(ϕ˜i , ·) + (4π)2 + o(1) , (5.7) 1/2 2 |(|Vi | , ϕi )| |(|Vi |1/2 , ϕi )|4
where 1/2
Ti = n − lim (I + Vi →0+
R0 (0)|Vi |1/2 + )−1 (I − Pi ) .
The error term in (5.7) is understood in the sense of the operator norm. Below we will use the formula 4π (I + Ki (z))Mi (z) = I − √ (I + Ki (z))Qi i z 1/2
= I − Vi
√ (|Vi |1/2 , ·)Qi + O( z) .
(5.8)
To prove this we multiply (5.5) by Ki (z) thus obtaining 4π Ki (z)(I + Ki (z))−1 = √ Ki (z)Qi + Ki (z)Mi (z) . i z
(5.9)
On the other hand Ki (z)(I + Ki (z))−1 = I − (I + Ki (z))−1 4π = − √ Qi + I − Mi (z) . i z
(5.10)
Comparing (5.9) and (5.10) proves the first part of (5.8). Now expanding Ki (z) √ around z = 0 in z and using Ki (0)Qi = −Qi we obtain (I + Ki (z))Qi =
√ i z 1/2 V (|Vi |1/2 , ·)Qi + O(z) , 4π i
thus proving (5.8). First let us consider the case when E = 0 is a regular point for H1 and an exceptional one of the first kind for H2 . In this case we represent the operator L(z; d) (4.4) in the form 4π L(z; d) = √ L(0) (z; d) + L(1) (z; d) , i z
656
V. KOSTRYKIN and R. SCHRADER
where L(0) (z; d) = Q2 (d)(I + K1 (z))−1 · (K12 (z; d) e d) − K1 (z)K2 (z; d)) , + K21 (z; d) + K(z; L(1) (z; d) = M2 (z; d)(I + K1 (z))−1 · (K12 (z; d) + K21 (z; d)
(5.11)
e d) − K1 (z)K2 (z; d)) . + K(z; Here Q2 (d) = U (d)Q2 U (d)−1 and M2 (z; d) = U (d)M2 (z)U (d)−1 . We note that the operator L(0) (z; d) has rank 1. Due to Theorem T for all sufficiently large |d| and all small z one has kL(1) (z; d)k < 1. Therefore I + L(1) (z; d) is then invertible. Hence, det2 (I + L(z; d))
4π (0) (1) −1 √ L (z; d)(I + L (z; d)) = det2 (I + L (z; d)) det2 I + i z 4π (0) (1) −1 (1) · exp − √ trL (z; d)(I + L (z; d)) L (z; d) i z 4π (0) (1) (1) −1 = det2 (I + L (z; d)) det I + √ L (z; d)(I + L (z; d)) i z 4π · exp − √ trL(0) (z; d) . i z (1)
Since for any rank 1 operator A, det(I + A) = 1 + trA, using (5.6) we can easily calculate 4π det I + √ L(0) (z; d)(I + L(1) (z; d))−1 i z 4π = 1 + √ trL(0) (z; d)(I + L(1) (z; d))−1 i z 4π e d) = 1 + √ tr Q2 (d)(I + K1 (z))−1 · (K12 (z; d) + K21 (z; d) + K(z; i z − K1 (z)K2 (z; d))(I + L(1) (z; d))−1 4π = 1 + √ (ϕ˜2 (· − d), (I + K1 (z))−1 · K12 (z; d) + K21 (z; d) i z e d) − K1 (z)K2 (z; d) · (I + L(1) (z; d))−1 ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 . + K(z; Further we calculate t(z; d) defined in (4.5), 4π t(z; d) = √ t(0) (z; d) + t(1) (z; d) , i z
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
657
where t(0) (z; d) = trL(0) (z; d)(K1 (z) + K2 (z; d) + K1 (z)K2 (z; d)) = (ϕ˜2 (· − d), (I + K1 (z))−1 [K12 (z; d) + K21 (z; d) e d) − K1 (z)K2 (z; d)] · [K1 (z) + K2 (z; d) + K(z; + K1 (z)K2 (z; d)]ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 , t(1) (z; d) = trL(1) (z; d)[K1 (z) + K2 (z; d) + K1 (z)K2 (z; d)] + trK1 (z)K2 (z; d) . Now we study the limit E → +0 of the expression 4π 4π ± √ trL(0) (E ± i0; d) ± √ t(0) (E ± i0; d) . i E i E
(5.12)
Simple calculations shows that (5.12) equals 4π ± √ (ϕ˜2 (· − d), (I + K1 (E ± i0))−1 [K12 (E ± i0; d) + K21 (E ± i0; d) i E e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 4π ± √ (ϕ˜2 (· − d), (I + K1 (E ± i0))−1 [K12 (E ± i0; d) + K21 (E ± i0; d) i E e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d] · [K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d)]ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 . (5.13) Now we prove Lemma 5.1. Let the Vi ’s have Property A. Then for E → +0 1 √ [(K1 (E ± i0) + K2 (E ± i0; d) E + K1 (E ± i0)K2 (E ± i0; d))ϕ2 (· − d) + ϕ2 (· − d)] √ i 1/2 (I + K1 (0))V2 (· − d)(|V2 |1/2 , ϕ2 ) + O( E) =± 4π
(5.14)
in L2 -norm uniformly in d ∈ R3 .
in
Proof. We expand the operators K1 (E ± i0) and K2 (E ± i0; d) in Taylor series √ E at E = 0: √ i E 1/2 V (|V1 |1/2 , ·) + O(E) , (5.15) K1 (E ± i0) = K1 (0) ± 4π 1 √ i E 1/2 V (· − d)(|V2 |1/2 (· − d), ·) + O(E) , (5.16) K2 (E ± i0; d) = K2 (0; d) ± 4π 2
658
V. KOSTRYKIN and R. SCHRADER
where the error terms O(E) are understood in the sense of Hilbert–Schmidt norm. Equation (5.14) follows immediately from (5.15) and (5.16). Remark. The low energy expansion for the operator K12 (z; d) √ i z 1/2 V (|V2 |1/2 (· − d), ·) + zN12 (d) + o(z) K12 (z; d) = K12 (0; d) + 4π 1 is not uniform in d. Here N12 (d) is the Hilbert–Schmidt operator with integral kernel 1 1/2 N12 (x, y; d) = − V1 (x)|x − ykV2 |1/2 (y − d) . 8π It is easy to show that kN12 kJ2 increases linearly with d. For this reason the limit E → +0 of ξ12 (E; d) is not uniform in d. √ Now by Lemma 5.1 the r.h.s. of (5.13) equals C(d) + O( E) (independent of the sign in (5.13)), where C(d) is a d-dependent constant. Therefore 4π lim √ trL(0) (E + i0; d)− trL(0) (E − i0; d)+ t(0) (E + i0; d)− t(0) (E − i0; d) = 0 E→+0 i E for all fixed d ∈ R3 . Since the operator I + L(1) (0; d) is invertible for all sufficiently large |d|, due to (4.6) we have (" 1 4π lim log 1 + √ (ϕ˜2 (· − d), ξ12 (+0; d) = 2πi E→+0 i E e d) − K1 (0)K2 (0; d) (I + K1 (0))−1 K12 (0; d) + K21 (0; d) + K(0; # · (I + L(1) (0; d))−1 ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2 "
4π · 1 − √ (ϕ˜2 (· − d), (I + K1 (0))−1 i E e d) − K1 (0)K2 (0; d) · K12 (0; d) + K21 (0; d) + K(0; #) · (I + L(1) (0; d))−1 ϕ2 (· − d))|(|V2 |1/2 , ϕ2 )|−2
.
(5.17)
To proceed further with our calculations of ξ12 (+0; d) we prove the following: Lemma 5.2. For all sufficiently large |d| (ϕ˜2 (· − d), (I + K1 (0))−1 [K12 (0; d) + K21 (0; d) e d) − K1 (0)K2 (0; d)](I + L(1) (0; d))−1 ϕ2 (· − d)) + K(0; =
|(|V2 |1/2 , ϕ2 )|2 c0 (V1 ) + o(|d|−2 ) , 4π|d|2
where c0 (V1 ) is the scattering length (5.2).
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
659
Proof. First we show that k(I + K1 (0)∗ )−1 ϕ˜2 (· − d) − ϕ˜2 (· − d)kL2 = o(|d|−2 ) for sufficiently large |d|. To this end we use K2 (0; d)∗ ϕ˜2 (· − d) = −ϕ˜2 (· − d) and write (I + K1 (0)∗ )−1 ϕ˜2 (· − d) − ϕ˜2 (· − d) = −(I + K1 (0)∗ )−1 K1 (0)∗ ϕ˜2 (· − d) = (I + K1 (0)∗ )−1 K1 (0)∗ K2 (0; d)∗ ϕ˜2 (· − d) . Its L2 -norm can be bounded by CkK1 (0)K2 (0; d)kJ1 kϕ2 kL2 , which is o(|d|−2 ) by Lemma 4.2. Therefore it suffices to consider e d) − K1 (0)K2 (0; d) (ϕ˜2 (· − d), K12 (0; d) + K21 (0; d) + K(0; · (I + L(1) (0; d))−1 ϕ2 (· − d))
= ϕ˜2 (· − d), K12 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d) e d)(I + L(1) (0; d))−1 ϕ2 (· − d) + ϕ˜2 (· − d), K(0;
− ϕ˜2 (· − d), K1 (0)K2 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d) + ϕ˜2 (· − d), K21 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d) .
(5.18)
Here the first summand is o(|d|−2 ) since K12 (0; d)∗ ϕ˜2 (· − d) = −K12 (0; d)∗ K2 (0; d)∗ ϕ˜2 (· − d) 1/4
= −|V2 |1/2 (· − d)R0 (0)V1
1/4
V2
(· − d)
· |V1 |1/4 |V2 |1/4 (· − d)R0 (0)|V2 |1/2 (· − d)ϕ˜2 (· − d) . Its L2 -norm can be bounded by 1 (4π)2 Z ·
Z
|V2 (x − d)kV1 (y)|1/2 |V2 (y − d)|1/2 dxdy |x − y|2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
which indeed is o(|d|−2 ) by Lemma 2.4.
1/2
1/2 kϕ2 kL2 ,
660
V. KOSTRYKIN and R. SCHRADER
Consider the second term on the r.h.s. of (5.18). Since L(1) (0; d) → 0 in Hilbert– e d)ϕ2 (· − d)) only. Schmidt norm as |d| → ∞, it suffices to consider (ϕ˜2 (· − d), K(0; Mimicking the idea of the proof of Lemma 4.3 we can write e d)ϕ2 (· − d)) (ϕ˜2 (· − d), K(0; 1/2
= −(ϕ˜2 (· − d), K2 (0; d)(Vd
1/2
+ (ϕ˜2 (· − d), K2 (0; d)V1
1/2
− V1
1/2
− V2
(· − d))R0 (0)|Vd |1/2 ϕ2 (· − d))
R0 (0)(|Vd |1/2 − |V1 |1/2
− |V2 |1/2 (· − d)) · K2 (0; d)ϕ2 (· − d)) 1/2
− (ϕ˜2 (· − d), V2
(· − d)R0 (0)(|Vd |1/2 − |V1 |1/2
− |V2 |1/2 (· − d)) · K2 (0; d)ϕ2 (· − d)) .
(5.19)
With the help of inequality (4.8) the first term on the r.h.s. of (5.19) can be bounded by Z 1/2 2 |V2 (x − d)kV2 (y − d)|3/4 |V1 (y)|1/4 dxdy 4π 2 |x − y|2 Z ·
|V1 (x)|1/4 |V2 (x − d)|3/4 |Vd (y)| dxdy |x − y|2
1/2 kϕ2 k2L2 .
By Lemma 2.4 this is o(|d|−2 ). This completes the proof of that the first term on the r.h.s. of (5.19) is o(|d|−2 ). The two other terms can be treated analogously. Hence the second term on the r.h.s. of (5.18) is o(|d|−2 ). The third term on the r.h.s. of (5.18) is o(|d|−2 ) due to Lemma 4.2. Finally consider the fourth term on the r.h.s. of (5.18). We replace (I + L(1) (0; d))−1 by the first two terms of its Neumann series expansion. It follows from (5.11) and Lemma 2.2 that (I + L(1) (0; d))−1 = I − L(1) (0; d) + O(|d|−2 ) . Hence (ϕ˜2 (· − d), K21 (0; d)(I + L(1) (0; d))−1 ϕ2 (· − d)) = (ϕ˜2 (· − d), K21 (0; d)ϕ2 (· − d)) − (ϕ˜2 (· − d), K21 (0; d)L(1) (0; d)ϕ2 (· − d)) + O(|d|−3 ) . Obviously, |(ϕ˜2 (· − d), K21 (0; d)ϕ2 (· − d))| = |(ϕ˜2 (· − d), K21 (0; d)K2 (0; d)ϕ2 (· − d))| ≤
1 (4π)2 ·
|V2 (x − d)kV2 (y − d)|1/2 |V1 (y)|1/2 dxdy |x − y|2
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
1/2
1/2 kϕk2L2 ,
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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which is o(|d|−2 ) due to Lemma 2.4. Now applying Lemma 2.2 we get − (ϕ˜2 (· − d), K21 (0; d)L(1) (0; d)ϕ2 (· − d)) =−
1 1/2 (ϕ˜2 , V2 )(|V1 |1/2 , L(1) (0; d)ϕ2 (· − d)) + o(|d|−2 ) . 4π|d|
(5.20)
Using the representation (5.11) we find that (|V1 |1/2 , (I+K1 (0))−1 K12 (0; d)ϕ2 (·−d)) describes the leading term of the asymptotics of (|V1 |1/2 , L(1) (0; d)ϕ2 (· − d)) as |d| → ∞ (here we omit the corresponding calculations). Applying Lemma 2.2 once more we obtain that (5.20) is given by −
1 1/2 1/2 (ϕ˜2 , V2 )(|V1 |1/2 , (I + K1 (0))−1 V1 )(|V2 |1/2 , ϕ2 ) + o(|d|−2 ) (4π)2 |d|2 =
|(|V2 |1/2 , ϕ2 )|2 c0 (V1 ) + o(|d|−2 ) , 4π|d|2
which proves the lemma. Now we apply Lemma 5.2 to (5.17). Then for sufficiently large |d| we have 1 lim log ξ12 (+0; d) = 2πi E→+0
1 · 1− √ i E
(
1 1+ √ i E
c0 (V1 ) −2 + o(|d| ) |d|2
−1 ) c0 (V1 ) −2 . + o(|d| ) |d|2
Noting that for E > 0 and c0 (V1 ) 6= 0 c0 (V1 ) √ sign c0 (V1 ) < 0 , arg 1 + i E we calculate the limit E → +0 thus obtaining that for all sufficiently large |d| ξ12 (+0; d) =
1 1 log exp{−iπsign c0 (V1 )} = − sign c0 (V1 ) . 2πi 2
Thus the claim (ii) of Theorem 3 is proved. Next let us consider the case when E = 0 is an exceptional point of the first kind for both H1 and H2 . In this case we decompose the operator L(z; d) into its singular and regular parts, L(z; d) = L(0) (z; d) + L(1) (z; d) ,
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L(0) (z; d) = −
(4π)2 Q2 (d)Q1 z
e d) − K1 (z)K2 (z; d) · K12 (z; d) + K21 (z; d) + K(z; 4π + √ Q2 (d)M1 (z) i z e d) − K1 (z)K2 (z; d) · K12 (z; d) + K21 (z; d) + K(z; 4π + √ M2 (z; d)Q1 i z e d) − K1 (z)K2 (z; d) , · K12 (z; d) + K21 (z; d) + K(z;
(5.21)
L(1) (z; d) = M2 (z; d)M1 (z) e d) − K1 (z)K2 (z; d) . · K12 (z; d) + K21 (z; d) + K(z; We note that the operator L(0) (z; d) has rank 2. Due to Theorem T for all sufficiently large |d| and all small z one has kL(1) (z; d)k < 1. Therefore the operator I + L(1) (z; d) is invertible. Hence, det2 (I + L(z; d)) = det2 (I + L(1) (z; d)) · det2 [I + L(0) (z; d)(I + L(1) (z; d))−1 ] · exp{−trL(0) (z; d)(I + L(1) (z; d))−1 L(1) (z; d)} = det2 (I + L(1) (z; d)) · det[I + L(0) (z; d)(I + L(1) (z; d))−1 ] · exp{−trL(0) (z; d)} .
(5.22)
It is easy to show that kM2 (z; d)ϕ1 − ϕ1 kL2 → 0 , (ϕ2 (· − d), ϕ1 ) → 0 as |d| → ∞. We note that M2 (z; d)ϕ1 and ϕ2 (· − d) for sufficiently large |d| depend on d continuously (in L2 -norm). Therefore for sufficiently large |d| the set {M2 (z; d)ϕ1 , ϕ2 (·− d)} forms a basis (in general non-orthogonal) in Ran(L(0) (z; d)). Let P be the projector onto Ran(L(0) (z; d)), such that P = M2 (z; d)ϕ1 (ψ1 (z; d), ·) + ϕ2 (· − d)(ψ2 (z; d), ·) , where ψ1 (z; d) = c11 (d)M2 (z; d)ϕ1 + c12 (z; d)ϕ2 (· − d) , ψ2 (z; d) = c21 (z; d)M2 (z; d)ϕ1 + c22 (z; d)ϕ2 (· − d) is the basis in Ran(L(0) (z; d)) dual with respect to {M2 (z; d)ϕ1 , ϕ2 (· − d)}. The coefficients cij are given by
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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c11 (d) = (ϕ2 , ϕ2 )D−1 , c12 (z; d) = −(ϕ2 (· − d), M2 (z; d)ϕ1 )D−1 , c21 (z; d) = −(M2 (z; d)ϕ1 , ϕ2 (· − d))D−1 , c22 (z; d) = (M2 (z; d)ϕ1 , M2 (z; d)ϕ1 )D−1 , D = (ϕ2 , ϕ2 )(M2 (z; d)ϕ1 , M2 (z; d)ϕ1 ) − (ϕ2 (· − d), M2 (z; d)ϕ1 )(M2 (z; d)ϕ1 , ϕ2 (· − d)) . We note that D is nothing but Gram’s determinant of the vectors ϕ2 (· − d) and M2 (z; d)ϕ1 . Since for sufficiently large |d| these vectors are linear independent, we have that D 6= 0. Due to the identity det(I + AB) = det(I + BA) one has det I + L(0) (z; d)(I + L(1) (z; d))−1 = det I + P L(0) (z; d)(I + L(1) (z; d))−1 = det I + P L(0) (z; d)(I + L(1) (z; d))−1 P 1 + a11 a12 , = det a21 1 + a22 where ai1 = ψi (z; d), L(0) (z; d)(I + L(1) (z; d))−1 M2 (z; d)ϕ1 , ai2 = ψi (z; d), L(0) (z; d)(I + L(1) (z; d))−1 ϕ2 (· − d) for i = 1, 2. Now elementary calculations give α1 (z; d) α1/2 (z; d) + det I + L(0) (z; d)(I + L(1) (z; d))−1 = +1, z iz 1/2
(5.23)
where the functions αi (z; d) (i = 1/2, 1) are regular at z = 0 and are given by α1 (z; d) =
(4π)2 |(|V1 |1/2 , ϕ1 )|2 |(|V2 |1/2 , ϕ2 )|2 e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) · (ϕ˜1 , [K12 + K21 + K e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) · (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) − (ϕ˜1 , [K12 + K21 + K e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) · (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K
e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) − (ϕ˜2 (· − d), ϕ1 )(ϕ˜1 , [K12 + K21 + K
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V. KOSTRYKIN and R. SCHRADER
and α1/2 (z; d) =
4π |(|V1 |1/2 , ϕ1 )|2 +
e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) (ϕ˜1 , [K12 + K21 + K
4π |(|V2 |1/2 , ϕ2 )|2
e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) . (ϕ˜2 (· − d), [K12 + K21 + K
e K1 , K2 and L(1) for K12 (z; d), Here we have used the abbreviations K12 , K21 , K, (1) e K21 (z; d), K(z; d), K1 (z), K2 (z; d) and L (z; d) respectively. Lemma 5.3. For sufficiently large |d| α1 (0; d) = |d|−2 + o(|d|−2 ) . Proof. The calculations of the expression e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) (ϕ˜1 , [K12 + K21 + K
(5.24)
are very similar to those used in the course of the proof of Lemma 5.2. First one can easily show that the dominant contribution to (5.24) is given by (ϕ˜1 , K12 (0; d)ϕ2 (·− d)). Its asymptotic can be calculated by means of Lemma 2.2 finally giving 1 (ϕ1 , |V1 |1/2 )(|V2 |1/2 , ϕ2 ) + o(|d|−1 ) . 4π|d| In almost the same way we can calculate the expression e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) , (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K thus obtaining 1 (ϕ2 , |V2 |1/2 )(|V1 |1/2 , ϕ1 ) + o(|d|−1 ) . 4π|d| Also the expressions e − K1 K2 ](I + L(1) )−1 M2 (z; d)ϕ1 ) (ϕ˜1 , [K12 + K21 + K and
e − K1 K2 ](I + L(1) )−1 ϕ2 (· − d)) (ϕ˜2 (· − d), M1 (z)[K12 + K21 + K
are O(|d|−2 ). Finally we estimate (ϕ˜2 (· − d), ϕ1 ). To this end we write (ϕ˜2 (· − d), ϕ1 ) = (K2 (0; d)∗ ϕ˜2 (· − d), K1 (0)ϕ1 ) = (ϕ˜2 (· − d), K2 (0; d)K1 (0)ϕ1 ) , which is o(|d|−2 ) by Lemma 4.2.
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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Hence we have lim log
E→+0
det[I + L(0) (E + i0; d)(I + L(1) (E + i0; d))−1 ] =0 det[I + L(0) (E − i0; d)(I + L(1) (E − i0; d))−1 ]
for all sufficiently large |d|. Now we turn to the estimate of the function ξ12 (E; d) as E → +0. By (4.6) and (5.22) we have ξ12 (E; d) =
1 det2 (I log 2πi det2 (I 1 det(I log · 2πi det(I ·
+ L(1) (E + i0; d)) + L(1) (E + i0; d))
+ L(0) (E + i0; d)(I + L(1) (E + i0; d))−1 ) + L(0) (E − i0; d)(I + L(1) (E − i0; d))−1 )
1 [−t(E + i0; d) + t(E + i0; d) 2πi
− trL(0) (E + i0; d) + trL(0) (E − i0; d)] .
(5.25)
Since the operator I + L(1) (0; d) is invertible for all large |d|, the first term on the r.h.s. of (5.25) equals zero in the limit E → +0 for all sufficiently large |d|. By our previous discussion the second term is also zero for E → +0 and for all sufficiently large |d|. Consider the third term of (5.25). Obviously t(E ± i0; d) + trL(0) (E ± i0; d) = trK1 (E ± i0)K2 (E ± i0; d) + trK1 (E ± i0)L(1) (E ± i0; d) + trK2 (E ± i0; d)L(1) (E ± i0; d) + trK1 (E ± i0)K2 (E ± i0; d)L(1) (E ± i0; d) + tr[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I]L(0) (E ± i0; d) . Here the first four terms are regular at E = 0. Consider the fifth term. Lemma 5.4. For every fixed d ∈ R3 lim tr[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I]
E→+0
· L(0) (E ± i0; d) = C(d) , where C(d) is a d-dependent constant.
(5.26)
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V. KOSTRYKIN and R. SCHRADER
Proof. By means of (5.21) we can write tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I]L(0) (E ± i0; d)} =−
(4π)2 tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] E
· Q2 (d)Q1 · [K12 (E ± i0; d) + K21 (E ± i0; d) e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]} 4π ± √ tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] i E · Q2 (d)M1 (E ± i0) · [K12 (E ± i0; d) + K21 (E ± i0; d) e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]} 4π ± √ tr{[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] i E · M2 (E ± i0; d)Q1 · [K12 (E ± i0; d) + K21 (E ± i0; d) e + K(E ± i0; d) − K1 (E ± i0)K2 (E ± i0; d)]} .
(5.27)
By Lemma 5.1 the second term on the r.h.s. of (5.27) is e d) − K1 (0)K2 (0; d)] (ϕ˜2 (· − d), M1 (0)[K12 (0; d) + K21 (0; d) + K(0; 1/2
· (I + K1 (0))V2
√ (· − d))(ϕ2 , |V2 |1/2 )−1 + O( E) .
Consider the last term on the r.h.s. of (5.27). Using the identity (5.8) we can write [K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] · M2 (E ± i0; d)Q1 = (I + K1 (E ± i0))(I + K2 (E ± i0; d))M2 (E ± i0; d)Q1 = (I + K1 (E ± i0))Q1 4π ∓ √ (I + K1 (E ± i0))(I + K2 (E ± i0; d))Q2 (d)Q1 . i E Now we use the identity (I + Ki (z))Qi =
√ i z 1/2 V (|Vi |1/2 , ·)Qi + zNi Qi + o(z) , 4π i
which is a direct consequence of (5.4) and of Ki (0)Qi = −Qi . Thus we obtain
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
667
[K1 (E ± i0) + K2 (E ± i0; d) + K1 (E ± i0)K2 (E ± i0; d) + I] · M2 (E ± i0; d)Q1 1/2
= −(I + K1 (0))V2 (· − d)(|V2 |1/2 (· − d), ·)Q2 (d)Q1 √ i E 1/2 V (|V1 |1/2 , ·)Q1 ± 4π 1 √ i E 1/2 1/2 V (|V1 |1/2 , V2 (· − d))(|V2 |1/2 (· − d), ·)Q2 (d)Q1 ∓ 4π 1 √ √ ±4πi E(I + K1 (0))N2 (d)Q2 (d)Q1 + o( E) , where N2 (d) = U (d)N2 U (d)−1 . Therefore the last term on the r.h.s. of (5.27) equals 4π e d) − K1 (0)K2 (0; d)] ∓ √ (ϕ˜1 , [K12 (0; d) + K21 (0; d) + K(0; i E 1/2
· (I + K1 (0))V2
(· − d))(ϕ˜2 (· − d), ϕ1 )(ϕ2 , |V2 |1/2 )−1 |(ϕ1 , |V1 |1/2 )|−2
+ C3 (d) + o(1) , where C3 (d) is a d-dependent constant. Now we consider the first term on the r.h.s. of (5.27). By Lemma 5.1 this term equals 4π e d) + K1 (0)K2 (0; d)] ± √ (ϕ˜1 , [K12 (0; d) + K21 (0; d) + K(0; i E 1/2
· (I + K1 (0))V2
(· − d))(ϕ˜2 (· − d), ϕ1 )(ϕ2 , |V2 |1/2 )−1 |(ϕ1 , |V1 |1/2 )|−2
+ C1 (d) + o(1) , Summing up all the contibutions completes the proof of the lemma.
Thus from (5.26) and Lemma 5.4 it follows that lim
E→+0
t(E + i0; d) + trL(0) (E + i0; d) − t(E − i0; d) − trL(0) (E − i0; d) = 0
for all d ∈ R3 . This completes the proof of Theorem 3. 6. Proof of Theorem 5 Before we present the rigorous proof of Theorem 5 we give a simple derivation based on the Born series for the scattering amplitudes (see e.g. [37]). Here we restrict ourselves to the leading terms in the Born series since the consideration of higher order terms is essentially the same.
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The first contribution in the Born series of the scattering amplitude Ad (ω, ω 0 ; E) involving both potentials V1 and V2 (· − d) is given by the Born approximation 0 ABorn 12 (ω, ω ; E; d)
√ Z b √ V1 ( Eω − q)Vb2 (q − Eω 0 ) −ihd,qi e dq q 2 − E − i0 √ √ Z 2π 2 −i√Ehd,ωi Vb2 ( Eω − q)Vb1 (q − Eω 0 ) ihd,qi e e dq . + (2π)6 q 2 − E − i0
2π 2 i√Ehd,ω0 i = e (2π)6
Applying now Lemma 2.3 we find that 0 ABorn 12 (ω, ω ; E; d)
=
√ √ (2π 2 )2 −1 b √ ˆ 0i ˆ Vb2 ( E(−ω 0 − d))e ˆ i Ehd,d+ω |d| V1 ( E(ω + d)) 6 (2π)
+
√ √ (2π 2 )2 −1 b √ ˆ ˆ Vb1 ( E(−ω 0 + d))e ˆ i Ehd,d−ωi |d| V2 ( E(ω − d)) + o(|d|−1 ) . 6 (2π)
Noting that the first term in the Born series for the amplitudes A1 and A2 is given by the Born approximation (ω, ω 0 ; E) = − ABorn 1 ABorn (ω, ω 0 ; E; d) = − 2
2π 2 b √ V1 ( E(ω − ω 0 )) , (2π)3 √ 0 2π 2 b √ V2 ( E(ω − ω 0 ))e−i Ehd,ω−ω i , 3 (2π)
we arrive at the claim of Theorem 5 in the Born approximation. Theorem 5 then by Ai . “follows” by replacing ABorn i We now turn to the rigorous proof and start with: Lemma 6.1. Let the Vi ’s have the Property A. Then |d| (I + K(z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 + (I + K1 (z))−1 K12 (z; d)(I + K2 (z; d))−1 + (I + K2 (z; d))−1 K21 (z; d)(I + K1 (z))−1 → 0 as |d| → ∞ in Hilbert–Schmidt norm uniformly in z on compact sets in Π0 \ E. Proof. We start with a remark on the invertibility of the operator I + K(z; d). Let us denote by E(d) the set of all z ∈ Π0 for which I + K(z; d) has no bounded inverse. We proved in Lemma 2.7 in [27] that for an arbitrary compact set I ⊂ Π0 \E there is d(I) > 0 such that for all |d| > d(I) the set E(d) has an empty intersection with I, E(d) ∩ I = ∅. Therefore we fix a compact set I ⊂ Π0 \ E and suppose that |d| > d(I).
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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First we show that |d| k(I + K1 (z) + K2 (z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 kJ2 → 0 (6.1) as |d| → ∞. This follows immediately from the equality (I + K1 (z) + K2 (z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 = (I + K1 (z))−1 (I + K2 (z; d))−1 K2 (z; d)K1 (z)(I + K1 (z))−1 · [I − (I + K2 (z; d))−1 K2 (z; d)K1 (z)(I + K1 (z))−1 ]−1 (I + K2 (z; d))−1 and Lemma 4.2. Consider the difference (I + K(z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 = (I + K(z; d))−1 − (I + K1 (z) + K2 (z; d))−1 + (I + K1 (z) + K2 (z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 . Due to (6.1) the second term is o(|d|−1 ) in the sense of Hilbert–Schmidt norm. Due to the second resolvent identity the first term can be written as follows: (I + K(z; d))−1 − (I + K1 (z) + K2 (z; d))−1 = −(I + K1 (z) + K2 (z; d))−1 [K(z; d) − K1 (z) − K2 (z; d)](I + K(z; d))−1 e d)](I + K(z; d))−1 = −(I + K1 (z) + K2 (z; d))−1 · [K12 (z; d) + K21 (z; d) + K(z; = −(I + K1 (z) + K2 (z; d))−1 [K12 (z; d) + K21 (z; d)](I + K1 (z) + K2 (z; d))−1 + (I + K1 (z) + K2 (z; d))−1 [K12 (z; d) + K21 (z; d)](I + K1 (z) + K2 (z; d))−1 e d)](I + K(z; d))−1 · [K12 (z; d) + K21 (z; d) + K(z; e d)(I + K(z; d))−1 . − (I + K1 (z) + K2 (z; d))−1 K(z;
(6.2)
Due to Lemmas 2.2 and 4.3 the second and third terms on the r.h.s. of (6.2) are o(|d|−1 ) in the sense of Hilbert–Schmidt norm. The relation (6.1) shows that the first term on the r.h.s. of (6.2) up to corrections of order o(|d|−1 ) is given by −(I +K1 (z))−1 (I +K2 (z; d))−1 [K12 (z; d)+K21 (z; d)]·(I +K1 (z))−1 (I +K2 (z; d))−1 . Hence |d| k(I + K(z; d))−1 − (I + K1 (z))−1 (I + K2 (z; d))−1 + (I + K1 (z))−1 (I + K2 (z; d))−1 [K12 (z; d) + K21 (z; d)] · (I + K1 (z))−1 (I + K2 (z; d))−1 kJ2 → 0 . To complete the proof of the lemma it remains to apply once more the arguments used in the proof of Lemma 4.4.
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Due to Lemma 6.1 and the representation (4.17) for the scattering amplitude to prove Theorem 5 it suffices to show that h √ √ 0 1/2 |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i i + 4πA1 (ω, ω 0 ; E) + 4πA2 (ω, ω 0 ; E; d) → 0 ,
(6.3)
ˆ E)A2 (−d, ˆ ω 0 ; E; d) 4πA1 (ω, −d; √ − |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 K12 (E + i0; d) √ 1/2 i Ehω 0 ,·i
· (I + K2 (E + i0; d))−1 Vd
e
→ 0,
(6.4)
ˆ E; d)A1 (d, ˆ ω 0 ; E) 4πA2 (ω, d; √ − |d| ei Ehω,·i |Vd |1/2 , (I + K2 (E + i0; d))−1 K21 (E + i0; d) √ 1/2 i Ehω 0 ,·i
· (I + K1 (E + i0))−1 Vd
e
→0
(6.5)
as |d| → ∞ uniformly in ω, ω 0 ∈ S2 and in E on compact sets in R+ . First we prove (6.3). We represent the l.h.s. of (6.3) h √ √ 0 1/2 |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i i + 4πA1 (ω, ω 0 ; E) + 4πA2 (ω, ω 0 ; E; d) h √ √ 0 1/2 = |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i √ √ 0 1/2 − ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 V1 ei Ehω ,·i i √ √ 0 1/2 (6.6) − ei Ehω,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 V2 (· − d)ei Ehω ,·i as the sum of the following terms: √ |d| ei Ehω,·i (|Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)) , √ 1/2 i Ehω 0 ,·i
(I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 Vd
e
√ + |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 i √ h 1/2 1/2 i Ehω 0 ,·i e · (I + K2 (E + i0; d))−1 Vd − V1 √ h + |d| ei Ehω,·i |V2 |1/2 (· − d), (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 1/2
·Vd
− (I + K2 (E + i0; d))−1 V2
1/2
i √ 0 (· − d) ei Ehω ,·i .
(6.7)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
671
Since (I + K1 (E + i0))−1 and (I + K2 (E + i0; d))−1 are uniformly norm bounded in E in any compact set in R+ (say, by a constant C) and due to the Schwarz inequality the first term of (6.7) can be bounded by C 2 k |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)kL2 kVd kL1 ≤ C 2 k |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)kL2 (kV1 kL1 + kV2 kL1 ) . Due to the inequality (4.8) we have 1/2
k |Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)kL2 ≤ 2k |V1 |1/2 |V2 |1/2 (· − d)kL1 Z =2
\ \ 1/2 (p)|V 1/2 (p)e−ipd dp |V 1| 2|
1/2 . (6.8)
\ 1/2 ∈ L2 (R3 ) ∩ C 2 (R3 ). For the potentials Vi having the Property B one has that |V i| Now we use the following asymptotics: Lemma 6.2. Let f ∈ L1 (R3 ) ∩ C 2 (R3 ). Then Z f (p)e−ipd dp = o(|d|−2 ) as |d| → ∞. This asymptotics can be directly derived from the asymptotic formula (2.11). Now due to Lemma 6.2 the first term of (6.7) is o(1). Consider the second term of (6.7). First we show that |d| k(I + K2 (E + i0; d))−1 Vd
1/2
− (I + K2 (E + i0; d))−1 V2
1/2
1/2
(· − d) − V1
kL2 (R3 ) → 0
(6.9)
as |d| → ∞. For this aim we represent the L2 -function on the l.h.s. of (6.9) in the form (I + K2 (E + i0; d))−1 Vd
1/2
− (I + K2 (E + i0; d))−1 V2
1/2
= (I + K2 (E + i0; d))−1 (Vd
1/2
1/2
− V1
1/2
− V2
− (I + K2 (E + i0; d))−1 K2 (E + i0; d)V1
1/2
(· − d) − V1
(· − d))
1/2
.
(6.10)
The first term on the r.h.s. of (6.10) is o(|d|−1 ) in the sense of L2 (R3 )-norm due to (6.8) and Lemma 6.2. The second term on the r.h.s. of (6.10) can be represented in the form (I + K2 (E + i0; d))−1 · V2
1/2
1/4
· V1
|V2 |1/4 (· − d)) .
(· − d))R0 (E + i0)|V1 |1/4 |V2 |1/4 (· − d))
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V. KOSTRYKIN and R. SCHRADER
Therefore its L2 -norm is bounded by 1/2
1/4
1/2
1/2
(· − d))R0 (E + i0)V1 |V2 |1/4 (· − d))kJ2 kV1 |V2 |1/2 (· − d))kL1 Z 1/2 C |V2 (x − d)kV1 (y)|1/2 |V2 (y − d)|1/2 ≤ dxdy 4π |x − y|2
CkV2
1/2
· kV1
1/2
|V2 |1/2 (· − d)kL1 ,
which is o(|d|−2 ) due to Lemma 2.4, (6.8) and Lemma 6.2. Thus (6.9) is proven. Now to prove that the second term of (6.7) is o(1), it suffices to show that √ |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 · (I + K2 (E + i0; d))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
→0
as |d| → ∞ uniformly in ω, ω 0 ∈ S2 . The proof of this fact is similar to that of Lemma 2.9 in [27]. First we show that h √ |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 · (I + K2 (E + i0; d))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
√ − ei Ehω,·i |V1 |1/2 (I + K2 (E + i0; d))−1 · (I + K1 (E + i0))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
i
→0
as |d| → ∞. This follows immediately from the identity (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 − (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 = (I + K1 (E + i0))−1 (I + K2 (E + i0; d))−1 · [K1 (E + i0)K2 (E + i0; d) − K2 (E + i0; d)K1 (E + i0)] · (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 and Lemma 4.2. Now we show that h √ |d| ei Ehω,·i |V1 |1/2 , (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 1/2
· V2
√
(· − d)ei
Ehω 0 ,·i
√ i √ 0 1/2 − ei Ehω,·i |V1 |1/2 , V2 (· − d)ei Ehω ,·i → 0
(6.11)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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as |d| → ∞. To this end we can represent the l.h.s. of (6.11) in the form √ − |d| ei Ehω,·i |V1 |1/2 , K2 (E + i0; d)(I + K2 (E + i0; d))−1 · (I + K1 (E + i0))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
√ √ 0 1/2 − |d| ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 K1 (E + i0)V2 (· − d)ei Ehω ,·i . Here the second term tends to zero since 1/2
|d| kK1 (E + i0)V2
1/2
(· − d)kL2 ≤ |d| kV1
R0 (E + i0)|V1 |1/4 |V2 |1/4 (· − d)kJ2 1/2
· k |V1 |1/2 |V2 |1/2 (· − d)kL1 . The first term can be rewritten in the form: √ 1/4 − |d| ei Ehω,·i |V1 |1/4 V2 (· − d), |V1 |1/4 |V2 |1/4 (· − d)R0 (E + i0)|V2 |1/2 (· − d) · (I + K2 (E + i0; d))−1 (I + K1 (E + i0))−1 V2
1/2
√
(· − d)ei
Ehω 0 ,·i
.
Therefore its absolute value can be bounded by 1/2
|d|C 2 kV1
1/2
V2
1/4
· k |V1 |1/4 V2
|d|C 2 ≤ 4π
1/2
(· − d)R0 (E + i0)|V2 |1/2 (· − d)k kJ2
Z
1/2
· kV1
1/2
(· − d)kL1 kV2 kL1
|V1 (x)|1/2 |V2 (x − d)|1/2 |V2 (y − d)| dxdy |x − y|2
1/2
V2
1/2
1/2
1/2
(· − d)kL1 kV2 kL1 ,
which is again o(1) as |d| → ∞. Now we note that √ √ 0 1/2 |d| ei Ehω,·i |V1 |1/2 , V2 (· − d)ei Ehω ,·i 1/2
≤ |d| kV1
1/2
V2
(· − d)kL1 ,
which is again o(1) due to Lemma 6.2. This estimate and (6.11) completes the proof of that the second term in (6.7) tends to zero as |d| → ∞. The third term in (6.7) can be considered in exactly the same way. Thus (6.3) is proven. Now we prove (6.4). (6.5) can be treated in exactly the same way. Due to Lemma 2.2 √ |d| ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 K12 (E + i0; d) √ 1/2 i Ehω 0 ,·i
· (I + K2 (E + i0; d))−1 Vd −
e
√ 1 i√Ehω,·i ˆ 1/2 e |Vd |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i 4π
√ √ 0 ˆ 1/2 · e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i
674
V. KOSTRYKIN and R. SCHRADER
tends to zero when |d| → ∞. Therefore to prove (6.4) it suffices to show that both √ √ ˆ 1/2 ˆ E) ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i + 4πA1 (ω, −d; √ √ ˆ 1/2 = ei Ehω,·i |Vd |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i √ √ ˆ 1/2 − ei Ehω,·i |V1 |1/2 , (I + K1 (E + i0))−1 V1 e−i Ehd,·i
(6.12)
and √ √ 0 ˆ 1/2 ˆ E; d) e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i + 4πA2 (ω, d; √ √ 0 ˆ 1/2 = e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 Vd ei Ehω ,·i √ √ 0 ˆ 1/2 − e−i Ehd,·i |V2 |1/2 (· − d), (I + K2 (E + i0; d))−1 V2 (· − d)ei Ehω ,·i tend to zero. We give the proof of (6.12) only. First we represent (6.12) in the form √ √ ˆ 1/2 ei Ehω,·i (|Vd |1/2 − |V1 |1/2 − |V2 |1/2 (· − d)), (I + K1 (E + i0))−1 V1 ei Ehd,·i √ √ ˆ 1/2 + ei Ehω,·i |V2 |1/2 (· − d)), (I + K1 (E + i0))−1 V1 ei Ehd,·i . The first term is o(1) due to inequality (6.8) and Lemma 6.2. To consider the second term we represent it in the form √ √ ˆ 1/2 − ei Ehω,·i |V2 |1/2 (· − d)), K1 (E + i0)(I + K1 (E ± i0))−1 V1 ei Ehd,·i √ √ ˆ 1/2 + ei Ehω,·i |V2 |1/2 (· − d)), V1 ei Ehd,·i . We have already shown that the second term is o(1). The first term can be written as √ 1/4 − ei Ehω,·i V1 |V2 |1/4 (· − d), |V1 |1/4 |V2 |1/4 (· − d)R0 (E + i0)|V1 |1/2 √ ˆ 1/2 i Ehd,·i
· (I + K1 (E + i0))−1 V1
e
.
The absolute value of this expression can be bounded by 1/2
1/2
CkV1 kL1 kV1 1/4
· kV1
≤
1/2
|V2 |1/2 (· − d)kL1
|V2 |1/4 (· − d)R0 (E + i0)|V1 |1/2 kJ2 C 1/2 1/2 1/2 kV1 kL1 kV1 |V2 |1/2 (· − d)kL1 4π Z 1/2 |V1 (x)|1/2 |V2 (x − d)|1/2 |V1 (y)| · dxdy , |x − y|2
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
675
which is o(|d|−2 ) again due to Lemma 2.4, (6.8) and Lemma 6.2. This completes the proof of Theorem 5. Appendix A Here we prove the following statements: Theorem A.1. Consider the Hamiltonians H± = H0 ±V with V ∈ R∩L1 (R3 ) and V is nonnegative. Moreover suppose that E = 0 is a regular point for H− , which in addition has no negative bound states. Then the scattering lengths c0 (±V ) = ∓
1 1/2 V , (I ± V 1/2 R0 (0)V 1/2 )−1 V 1/2 4π
satisfy ∓c0 (±V ) > 0. Proof. The operator V 1/2 R0 (0)V 1/2 is self-adjoint compact and non-negative. Therefore (I +V 1/2 R0 (0)V 1/2 )−1 > 0. By the Birman–Schwinger Principle the total multiplicity of the discrete spectrum of H− equals the number of eigenvalues λk of −V 1/2 R0 (0)V 1/2 such that λk ≤ −1. Since by assumption the discrete spectrum is empty, we have that λk > −1 for all k, and therefore I − V 1/2 R0 (0)V 1/2 > 0. Hence (I − V 1/2 R0 (0)V 1/2 )−1 exists and is a positive operator. Theorem A.2. Let V satisfy the following conditions: (i) (ii) (iii) (iv)
1 V R ∈R∩L , V dx = 0, V is not identically zero, kV kR < 2π.
Let W ≥ 0 satisfy (i), (iii), and (iv). Then for any sufficiently small λ > 0 there is a ∈ (0, 1) such that c0 (λ(V + aW )) = 0. Proof. Denote Ua = V + aW . The assumptions of the theorem guarantee that the Neumann expansion (I + λUa1/2 R0 (0)|Ua |1/2 )−1 = I − λUa1/2 R0 (0)|Ua |1/2 + O(λ2 ) converges for all λ ∈ [0, 1] and a ∈ [0, 1]. Therefore Z
λ2 |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ3 ) 4π Z λ2 λa W dx + |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ3 ) . (A.1) =− 4π 4π
c0 (λUa ) = −
We note that
λ 4π
Ua dx +
|Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 > 0 .
676
V. KOSTRYKIN and R. SCHRADER
Indeed,
Z U (x)U (y) a a dxdy . |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 = 4π|x − y|
By (2.15) we have that Z
Ua (x)Ua (y) dxdy = 4π|x − y|
Z
ba (p)|2 |U dp > 0 . p2
Now consider the equation c0 (λUa ) = 0, which by (A.1) can be written in the form: Z λ a W dx + |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ2 ) = 0 . f (a, λ) := − 4π 4π Now fix λ0 > 0 so small that 0<
Z 1 λ |Ua |1/2 , Ua1/2 R0 (0)|Ua |1/2 · Ua1/2 + O(λ2 ) < W dx 4π 4π
for all a ∈ [0, 1] and all 0 < λ < λ0 . For every λ with 0 < λ < λ0 we have f (0, λ) > 0 and f (1, λ) < 0. Since f (a, λ) is continuous in a ∈ [0, 1], there is a = a(λ) such that f (a, λ) = 0. Appendix B This appendix is devoted to a proof of relation (2.12). Let us denote Z f (p)e−ihd,pi dp . I(d) = 2 R3 p − E ∓ i0
(B.1)
√ We define the function ηδ (q) ∈ C0∞ (R+ ) such that ηδ (q) ≡ 1 for |q − E| < δ and √ ηδ (q) ≡ 0 for |q − E| > 2δ. We represent the integral (B.1) as follows: Z I(d) =
f (p)ηδ (|p|)e−ihd,pi dp + p2 − E ∓ i0
Z
f (p)(1 − ηδ (|p|))e−ihd,pi dp . p2 − E ∓ i0
(B.2)
The integrand in the second term is in C 2 ∩ L1 . Therefore due to (2.11) the second integral is o(|d|−2 ). Consider the first integral on the r.h.s. of (B.2) for which we preserve the notation I(d). Also we write fδ (p) for f (p)ηδ (|p|). Integration by parts for an arbitrary differentiable function g(ω) on the unit sphere S2 gives Z o 2π n ˆ −i|p||d| ˆ i|p||d| g(d)e g(ω) exp{−i|p|hd, ωi}dω = − − g(−d)e i|p||d| S2 Z 1 ˆ ∇ig(ω)dω . (B.3) exp{−i|p|hd, ωi}hd, + i|p||d| S2 ˆ ∇i we mean the differential operator ∂/∂θ, if (θ, φ) are the polar coordinates By hd, of ω ∈ S2 with dˆ as polar axis.
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
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First we consider the case E > 0. Let us fix δ such that E > 4δ 2 . With the help of the identity (B.3) we represent I(d) as the sum of two terms 2π I1 (d) = − i|d|
√
Z
√
E+2δ
E−2δ
and 1 I2 (d) = i|d|
√
Z
Z
ˆ −iq|d| qdq 2π fδ (q d)e + 2 q − E ∓ i0 i|d|
√
E+2δ
E−2δ
qdq q 2 − E ∓ i0
Z S2
√ E+2δ
√
E−2δ
ˆ iq|d| qdq fδ (−q d)e , q 2 − E ∓ i0
dωe−iqhd,ωi f1,δ (qω) ,
(B.4)
ˆ ∇ifδ (qω) ∈ C 1 (R3 ) by the assumption. Since (q 2 − E ∓ i0)−1 where f1,δ (qω) = hd, is the distribution of order 1, the integral (B.4) is well defined. Due to the distributional identities (q ∈ R) 1 1 = v.p. 2 ± iπδ(q 2 − E) , q 2 − E ∓ i0 q −E √ √ 1 δ(q − E) = (δ(q − E) + δ(q + E)) , 2|q|
(B.5)
2
one has (1)
(2)
I1 (d) = I1 (d) + I1 (d) , (1) I1 (d)
2π v.p. =− i|d|
Z
2π v.p. + i|d| (2)
I1 (d) = ∓
√ E+2δ
ˆ −iq|d| qdq fδ (q d)e q2 − E
√ E−2δ
Z
√ E+2δ
√ E−2δ
ˆ iq|d| qdq fδ (−q d)e , 2 q −E
√ √ π 2 √ ˆ −i√E|d| ˆ i E|d| . f ( E d)e − f (− E d)e |d|
(1)
Now we rewrite I1 (d) in the following way: (1) I1 (d)
√ ˆ − ηδ (q)f ( E d)]e ˆ −iq|d| qdq [fδ (q d) √ 2 q −E E−2δ √ √ Z E+2δ ˆ − ηδ (q)f (− E d)]e ˆ iq|d| qdq 2π [fδ (−q d) + √ i|d| E−2δ q2 − E
2π =− i|d|
Z
√
E+2δ
√
Z E+2δ 2π √ ˆ ηδ (q)e−iq|d| qdq f ( E d) v.p. √ − i|d| q2 − E E−2δ √ 2π ˆ v.p. f (− E d) + i|d|
Z
√
√
E+2δ
E−2δ
ηδ (q)eiq|d| qdq . q2 − E
(B.6)
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V. KOSTRYKIN and R. SCHRADER
Due to the fact that the functions √ ˆ − ηδ (q)f (± E d) ˆ fδ (±q d) 2 q −E have compact support and are bounded, by Riemann–Lebesgue lemma the first two terms on the r.h.s. of (B.6) are o(|d|−1 ). The integrands in the third and √ fourth terms on the r.h.s. of (B.6) are analytic for complex q in a small vicinity of E. Therefore due to the well-known property of integrals in the sense of principal value we have Z v.p.
√
√
E+2δ
E−2δ
Z
ηδ (q)e∓iq|d| qdq = q2 − E
(∓)
γ
√ iπ ηδ (q)e∓iq|d| qdq ∓ e∓i|d| E , 2 q −E 2
√ √ (±) , (0 < < 2δ) consists of the intervals [ E − 2δ, E − ], where the contour γ √ √ √ [ E + , E + 2δ] and the half circle |q − E| = , ∓Imq ≥ 0 respectively. (±) Since ηδ (q) and all its derivatives are zero at the ends of the contour γ and (±) since |e±i|d|q | ≤ 1 on γ respectively, by means of an integration by parts we can show that Z ηδ (q)e∓iq|d| qdq = O(|d|−∞ ) . 2−E (∓) q γ Thus we arrive at the asymptotic formula I1 (d) =
√ √ 2π 2 ˆ ±i E|d| + o(|d|−1 ) , f (∓ E d)e |d|
where the error term o(|d|−1 ) is uniform in E on compact sets in (0, +∞). Now we turn to the discussion of the integral I2 (d) given by (B.4). To complete the proof of (2.12) it suffices to show that √
Z
√
E+2δ
E−2δ
qdq 2 q − E ∓ i0
Z S2
e−iqhω,di f1,δ (qω)dω → 0
(B.7)
uniformly in E on compact sets in (0, ∞) as |d| → ∞. Let us note that f1,δ (p) = f1 (p)ηδ (|p|) ,
ˆ ∇if (p) . f1 (p) = hd,
Therefore the integral in (B.7) can be represented as the sum of two terms Z
√
E+2δ
√ E−2δ
Z +
qηδ (q)dq 2 q − E ∓ i0
√ E+2δ
√
E−2δ
Z S2
qηδ (q)dq 2 q − E ∓ i0
h i √ e−iqhω,di f1 (qω) − f1 ( Eω) dω
Z S2
√ e−iqhω,di f1 ( Eω)dω .
(B.8)
(B.9)
¨ CLUSTER PROPERTIES OF ONE PARTICLE SCHRODINGER OPERATORS. II
679
Since f1 is differentiable, the integrand in (B.8) is regular, and hence by Riemann– Lebesgue lemma the integral tends to zero as |d| → ∞. By (B.5) the integral (B.9) can be represented in the form Z v.p.
√
√
±
E+2δ
E−2δ
iπ 2
Z
S2
qηδ (q)dq q2 − E
Z S2
√ e−iqhω,di f1 ( Eω)dω
√ e−iqhω,di f1 ( Eω)dω ,
where the second term tend to zero again by Riemann–Lebesgue lemma. Consider the first term, Z √E+2δ qηδ (q) F (q, d)dq , (B.10) v.p. √ 2 E−2δ q − E where
Z F (q, d) = S2
√ e−iqhω,di f1 ( Eω)dω .
By (B.3) we have F (q, d) = F1 (q, d) + F2 (q, d) , o √ 2π n √ ˆ −iq|d| ˆ iq|d| , f1 ( E d)e F1 (q, d) = − − f1 (− E d)e iq|d| Z √ 1 e−iqhω,di f2 ( Eω)dω , F2 (q, d) = iq|d| S2 √ √ where f2 ( Eω) = hd, ∇if1 ( Eω). The arguments already used above allow one to calculate the contribution from F1 (q, d) thus giving Z v.p.
√
√
E+2δ
E−2δ
qηδ (q) F1 (q, d)dq q2 − E
√ √ π 2 √ ˆ −i|d|√E ˆ i|d| E + O(|d|−∞ ) . + f1 (− E d)e f1 ( E d)e = √ E|d|
Consider now the contribution from F2 (q, d), 1 v.p. i|d|
Z
√ E+2δ
√ E−2δ
ηδ (q)dq q2 − E
Z S2
√ e−iqhω,di f2 ( Eω)dω .
By the obvious estimate Z Z a ϕ(x) a ϕ(x) − ϕ(0) v.p. dx = dx ≤ 2a max |ϕ0 (x)| , x x∈[−a,a] −a x −a
(B.11)
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we have that the absolute value of (B.11) can be bounded by 4δ
max√
√ q∈[ E−2δ, E+2δ]
Z √ 1 ∂ ηδ (q) e−iqhω,di f2 ( Eω)dω . |d| ∂q q + E S2
Note that 1 ∂ |d| ∂q
Z S2
Z √ √ ˆ 2 ( Eω)dω , e−iqhω,di f2 ( Eω)dω = −i e−iqhω,di hω, dif S2
which is o(1) as |d| → ∞ by Riemann–Lebesgue lemma. This completes the proof of (B.7). The proof of (2.12) in the case E = 0 follows from the results of [40]. Appendix C Here we give the proof of Lemma 4.1. According to (3.2) the spectral shift function for the pair of operators H = H0 + V and H0 can be written as ξ(E) = −ξ((E + c0 )−1 ; R(−c0 ), R0 (−c0 )) =− =
det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − (E + c0 )−1 − i)−1 ] 1 lim log 2πi →+0 det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − (E + c0 )−1 + i)−1 ]
det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − ζ)−1 ] 1 lim log , 2πi →+0 det[I + (R(−c0 ) − R0 (−c0 ))(R0 (−c0 ) − ζ)−1 ]
(C.1)
where ζ = (E + i + c0 )−1 . Due to the resolvent equation we have R(−c0 ) − R0 (−c0 ) = −R0 (−c0 )|V |1/2 (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (−c0 ) . Obviously R0 (−c0 )|V |1/2 and V 1/2 R0 (−c0 ) are Hilbert–Schmidt operator. We recall that for an arbitrary trace class operator A the modified Fredholm determinant is given by det2 (I + A) = det(I + A)e−trA . Therefore det[I + (R(−c0 ) − R(−c0 ))(R0 (−c0 ) − ζ)−1 ] = det2 [I − (I + V 1/2 R0 (−c0 )|V |1/2 )−1 · V 1/2 R0 (−c0 )(R0 (−c0 ) − ζ)−1 R0 (−c0 )|V |1/2 ] · exp{−tr[(I + V 1/2 R0 (−c0 )|V |1/2 )−1 · V 1/2 R0 (−c0 )(R0 (−c0 ) − ζ)−1 R0 (−c0 )|V |1/2 ]} .
(C.2)
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Now we note that R0 (−c0 )(R0 (−c0 ) − ζ)−1 R0 (−c0 ) = R0 (−c0 ) + ζ(R0 (−c0 ) − ζ)−1 R0 (−c0 ) = R0 (−c0 ) − R0 (ζ −1 − c0 ) = R0 (−c0 ) − R0 (E + i) . Therefore the r.h.s. of (C.2) equals n det2 I − (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (−c0 )|V |1/2 + (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (E + i)|V |1/2
o
· exp − tr(I + V 1/2 R0 (−c0 )|V |1/2 )−1
· (V 1/2 R0 (−c0 )|V |1/2 − V 1/2 R0 (E + i)|V |1/2 ) .
Since (I + V 1/2 R0 (−c0 )|V |1/2 )−1 V 1/2 R0 (−c0 )|V |1/2 = I − (I + V 1/2 R0 (−c0 )|V |1/2 )−1 , (C.3) we get det[I + (R0 (−c0 ) − R0 (−c0 ))(R0 (−c0 ) − ζ)−1 ] = det2 (I + V 1/2 R0 (−c0 )|V |1/2 )−1 (I + V 1/2 R0 (E + i)|V |1/2 ) · exp − tr(I + V 1/2 R0 (−c0 )|V |1/2 )−1 · (V 1/2 R0 (−c0 )|V |1/2 − V 1/2 R0 (E + i)|V |1/2 ) . (C.4) It is easy to see that for arbitrary Hilbert–Schmidt operators A and B such that I + A is invertible det2 [(I + A)−1 (I + B)] =
det2 (I + B) tr(I+A)−1 A(B−A) e . det2 (I + A)
(C.5)
Therefore using (C.3) and (C.5) the expression (C.4) can be represented as det2 (I + V 1/2 R0 (E + i)|V |1/2 ) det2 (I + V 1/2 R0 (−c0 )|V |1/2 ) · exp tr(V 1/2 R0 (E + i)|V |1/2 − V 1/2 R0 (−c0 )|V |1/2 ) . Finally we get det[I + (R(−c0 ) − R0 (−c0 )(R0 (−c0 ) − ζ)−1 )] det[I + (R(−c0 ) − R0 (−c0 )(R0 (−c0 ) − ζ)−1 )] =
det2 (I + V 1/2 R0 (E + i)|V |1/2 ) det2 (I + V 1/2 R0 (E − i)|V |1/2 ) · exp tr(V 1/2 R0 (E + i)|V |1/2 − V 1/2 R0 (E − i)|V |1/2 ) .
(C.6)
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Taking the limit → 0 in (C.6) and using the relation lim tr(V 1/2 R0 (E + i)|V |1/2 − V 1/2 R0 (E − i)|V |1/2 )
→0
√ Z i E V dx , E ≥ 0 2π R3 = 0, E<0 leads to (4.1). References [1] S. Albeverio, F. Gesztesy and R. Høegh-Krohn, “The low energy expansion in nonrelativistic scattering theory”, Ann. Inst. Henri Poincar´e, Physique th´eorique 37 (1982) 1–28. [2] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer, Berlin, 1988. [3] P. Aventini and R. Seiler, “On the electronic spectrum of the diatomic molecular ion”, Commun. Math. Phys. 41 (1975) 119–134. [4] M. Sh. Birman and M. G. Krein, “On the theory of wave operators and scattering operators”, Sov. Math. – Dokady 3 (1962) 740–744. [5] M. Sh. Birman and D. R. Yafaev, “The spectral shift function. Work by M. G. Krein and its further development”, St. Petersburg Math. J. 4 (1993) 833–870. [6] D. Boll´e and S. F. J. Wilk, “Low-energy sum rules for two-particle scattering”, J. Math. Phys. 24 (1983) 1555–1563. [7] Y. Colin de Verdi`ere, “Une formule de traces pour l’op´erateur de Schr¨ odinger dans ´ Norm. Sup. 14 (1981) 27–39. R3 ”, Ann. scient. Ec. [8] J. M. Combes and R. Seiler, “Regularity and asymptotic properties of the discrete spectrum of the electronic Hamiltonians”, Int. J. Quantum Chem. 14 (1978) 213–229. [9] E. B. Davies, “The twisting trick for double well Hamiltonians”, Commun. Math. Phys. 85 (1982) 471–479. [10] T. Dreyfus, “The determinant of the scattering matrix and its relation to the number of eigenvalues”, J. Math. Anal. Appl. 64 (1978) 114–134. [11] T. Dreyfus, “The number of states bound by non-central potentials”, Helv. Phys. Acta 51 (1978) 321–329. [12] V. Efimov, “Energy levels arising from resonant two-body forces in a three-body system”, Phys. Lett. B33 (1970) 563–564. [13] R. Geisler, V. Kostrykin and R. Schrader, “Concavity properties of Krein’s spectral shift function”, Rev. Math. Phys. 7 (1995) 161–181. [14] L. Guillop´e, “Asymptotique de la phase de diffusion pour l’op´erateur de Schr¨ odinger avec potentiel”, C. R. Acad. Sc. Paris, S´erie I 293 (1991) 601–603. [15] E. M. Harrell, “Double wells”, Commun. Math. Phys. 75 (1980) 239–261. [16] K. Hepp, “Spatial cluster decomposition properties of the S-matrix”, Helv. Phys. Acta 37 (1964) 659–662. [17] R. Høegh-Krohn and M. Mebkhout, “The 1r expansion for the critical multiple well problem”, Commun. Math. Phys. 91 (1983) 65–73. [18] H. Holden, R. Høegh-Krohn and M. Mebkhout, “The short-range expansion for multiple well scattering theory”, J. Math. Phys. 26 (1985) 145–151. [19] W. Hunziker, “Cluster properties of multiparticle systems”, J. Math. Phys. 6 (1965) 6–10.
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[20] W. Hunziker, “Mathematical theory of multi-particle quantum systems”, in Quantum Theory and Statistical Physics”, eds. A. O. Barut and W. E. Brittin, Lectures in Theoretical Physics, Gordon and Breach, New York, 1968, pp. 1–48. [21] A. Jensen and T. Kato, “Spectral properties of Schr¨ odiger operators and time-decay of the wave functions”, Duke Math. J. 46 (1979) 583–611. [22] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966. [23] T. Kato, “Monotonicity theorem in scattering theory”, Hadronic J. 1 (1978) 134–154. [24] M. Klaus and B. Simon, “Binding of Schr¨ odinger particles through conspirancy of potential wells”, Ann. Inst. H. Poincar´ e, Phys. Theor. 30 (1979) 83–87. [25] M. Klaus, “Some remarks on double – wells in one and three-dimensions”, Ann. Inst. H. Poincar´e, Phys. Theor. 34 (1981) 405–417. [26] I. V. Komarov and S. Yu. Slavyanov, “The two Coulomb problem at large center separation”, J. Phys. B: Atom. Molec. Phys. 1 (1968) 1066–1072. [27] V. Kostrykin and R. Schrader, “Cluster properties of one particle Schr¨ odinger operators”, Rev. Math. Phys. 6 (1994) 833–853. [28] M. G. Krein, “On a trace formula in perturbation theory”, Matem. Sbornik 33 (1953) 597–626 (in Russian). [29] M. G. Krein, “Perturbation determinants and a formula for the traces of unitary and self-adjoint operators”, Sov. Math. – Doklady 3 (1962) 707–710. [30] E. H. Lieb and B. Simon, “Monotonicity of the electronic contribution to the BornOppenheimer energy”, J. Phys. B: Atom. Molec. Phys. 11 (1978) L537–L542. [31] J. D. Morgan III and B. Simon, “Behavior of molecular potential energy curves for large nuclear separations”, Int. J. Quantum Chem. 17 (1980) 1143–1166. [32] R. G. Newton, “Noncentral potentials: The generalized Levinson theorem and the structure of the spectrum”, J. Math. Phys. 18 (1977) 1348–1357. [33] R. G. Newton, Scattering Theory of Waves and Particles, Springer, New York, 1982. [34] Yu. N. Ovchinnikov and I. M. Sigal, “Number of bound states of three-body systems and Efimov’s effect”, Ann. Phys. 123 (1979) 274–295. [35] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992. [36] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [37] M. Reed and B. Simon, Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, New York, 1979. [38] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. [39] R. Schrader, “High energy behaviour for non-relativistic scattering by stationary external metrics and Yang-Mills potentials”, Z. Phys. C: Particles and Fields 4 (1980) 27–36. [40] P. N. Shivakumar and R. Wong, “Asymptotic expansion of multiple Fourier transforms”, SIAM J. Math. Anal. 10 (1979) 1095–1104. [41] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton Univ. Press, Princeton, 1971. [42] B. Simon, Trace Ideals and Their Applications, Cambridge Univ. Press, Cambridge, 1979. [43] H. Tamura, “Existence of bound states for double well potentials and the Efimov effect”, in Functional-Analytic Methods for Partial Differential Equations, eds. H. Fujita, T. Ikebe, and S. T. Kuroda, pp. 173–186. Lecture Notes in Mathematics, Vol. 1450, Springer, Berlin, 1990. [44] H. Tamura, “The Efimov effect of three-body Schr¨ odinger operators: Asymptotics for the number of negative eigenvalues”, Nagoya Math. J. 130 (1993) 55–83.
GENERALISED DEFORMATIONS, KOSZUL RESOLUTIONS, MOYAL PRODUCTS FRANC ¸ OIS NADAUD Laboratoire de Physique Math´ ematique Universit´ e de Bourgogne UFR Sciences et Techniques 9, Avenue Alain Savary B.P. 400, 21011 Dijon Cedex, France E-mail : [email protected] Received 6 June 1997 We generalise Gerstenhaber’s theory of deformations, by dropping the assumption that the deformation parameter should commute with the elements of the original algebra. We give the associated cohomology and construct a Koszul resolution for the polynomial algebra C[X] in the “homogeneous” case. We then develop examples in the case of C[x, y] and find some Moyal-like products of a new type. Finally, we show that, for any field K, matrix algebras with coefficients in K and finite degree extensions of K are rigid, as in the commutative case.
Introduction Given an algebra A over a field K, a deformation of A, in classical Gerstenhaber’s theory [4], is a K[[t]]-bilinear associative product ∗ on A[[t]] such that A[[t]]/A[[t]]t ≈ A as algebras. If we suppose that A has a unit, then any deformation owns a unit too and there exists an equivalent deformation which has the same unit as A. In this context, the K[[t]]-bilinearity implies t ∗ a = (1t) ∗ a = (1 ∗ a)t = at and a ∗ t = a ∗ (1t) = (a ∗ 1)t = at. So t is a new generator, necessarily central. There are several reasons [8] to believe that this theory might be too restrictive. To generalise, we shall no longer assume that t commutes with the elements of A. For instance, if A is a super-algebra, it can be natural to impose that t supercommutes with the elements of A: t ∗ a = (−1)a a ∗ t. So we shall not only look for products that are K[[t]]-bilinear but for those which satisfy some weaker conditions, such as: (ta) ∗ b = t(a ∗ b), (at) ∗ b = a ∗ (tb), a ∗ (bt) = (a ∗ b)t, without assuming that ta and at are equal but rather that they are related by ta = σ(a)t, where σ is an endomorphism of the algebra A. We will call this products t-bilinear. Such a theory was developed for the first time in [8], where the basic definitions are given and the corresponding cohomology described. In Sec. 1, we recall the needed structure on A[[t]] and the definitions of t-multilinear maps and σ-deformations. We make this in the framework of an algebraic theory; that is, we don’t call for any topology, even when dealing with infinite dimensional algebras. 685 Reviews in Mathematical Physics, Vol. 10, No. 5 (1998) 685–704 c World Scientific Publishing Company
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In Sec. 2, we recall the cohomological frame adapted to this type of deformations. There are two possible approaches. The first one involves an infinite number of σ-invariant Hochschild cohomologies, the Hσ∗ (A, AσN ). The second one leads to the K[[t]]-relative Hochschild cohomology of A[[t]], the trivial σ-deformation of A; cochains are precisely t-multilinear maps. The link between these two approaches is very simple: the last cohomology is the infinite free product of the first ones. These spaces have the same kind of interpretation as the usual Hochschild cohomology spaces in Gerstenhaber’s theory, with a new difficulty: the space involved is different at each “step” (i.e. in each power of t). For instance, the N th obstruction to the integrability of a 2-cocycle lie in the space Hσ3 (A, AσN ) and if all the Hσ2 (A, AσN ) are 0, then A cannot be non-trivially σ-deformed. This was studied in [8], so we give the results without proof. At this point, we introduce some topology to overcome some difficulties that arise when dealing with infinite dimensional algebras. In Sec. 3, we construct a Koszul resolution for C[X] which gives the desired cohomology, in the particular case where σ is homogeneous of degree zero (this is necessary for the construction). The new point, compared to the classical construction, is to check that our boundaries and homotopies are σ-invariant. This is needed for our last resolution, the one of C[X][[t]], to have boundaries that are morphisms of C[X][[t]]-bimodules and homotopies, morphisms of C[[t]]-bimodules. Unfortunately, when considering the cohomology of this resolution, the coboundary is no longer zero, unlike in the classical theory (when σ = id); so the computation of the cohomology spaces can lead to tricky calculations. In Sec. 4, we give the complete cohomology of C[x, y], when σ is diagonal (or equivalent to a diagonal matrix). Then we construct a q-Moyal product: it is a σ-deformation of C[x, y] for σ(x) = qx and σ(y) = q −1 y, q ∈ C being fixed. This product is explicitly given: it looks like the usual Moyal product (which is the case q = 1) except that q-exponentials are involved instead of the usual exponential. We then show that this algebra is generated by x, y and t with relations: x∗y−y∗x=t t ∗ x = qx ∗ t t ∗ y = q −1 y ∗ t so it is a Gerstenhaber deformation, with scalar parameter q, of the enveloping algebra of the Heisenberg Lie-algebra [x, y] = t. In Sec. 5, we show that, for any field K, finite degree field extensions of K and matrix algebras are rigid in this wider theory. To conclude, let us insist on the fact that this new theory is not only a “game” consisting of an abstract generalisation of the usual theory of deformations. As shown in [8], it gives some light to the meaning of rigidity (the Weyl algebra is rigid in Gerstenhaber theory, but it is no more in super-commutative theory) and it provides new links between some physical theories (e.g. quantum mechanics and supersymetry). On the other hand, it is not too wild a theory, because “very”
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rigid structures (as matrix algebras and finite degree extensions — see Sec. 5) keep their rigidity in the enlarged theory. The theory can sometimes lead to many different cases (see Sec. 4.1) and here we think that some interpretations need to be given; this will be done later. Acknowledgements I am very grateful to G. Pinczon for guiding and helping me through this work. I also thank people of the laboratory, and especially M. Flato for welcoming me and sharing many interesting ideas with me. Notations In all the paper, K is a field of characteristic 0. ⊗ denotes the tensorial product over the field K and ⊗t the one over the ring K[[t]]. If A and B are K-vector N spaces, LN (A, B) = L(A⊗ , B), or just LN (A) if A = B, denotes the space of K-multilinear maps of order N from A to B. Given a fixed σ ∈ L(A), LN σ (A) N stands for the subspace of L (A) consisting of σ-invariant maps (i.e. f such that N f ◦ σ ⊗ = σ ◦ f ). If A is an algebra and B and C are A-bimodules, then HomA−A (B, C) denotes the space of morphisms of A-bimodules from B to C. 1. σ-Deformation Theory 1.1. σ-Deformation of a vector space, t-multilinear maps A is a K-vector space. We give A[[t]] its usual (free) right K[[t]]-module structure: X X X a n tn · λp tp = λp an tn+p an ∈ A, λp ∈ K . n≥0
p≥0
n,p≥0
We endow A[[t]] with a left K[[t]]-module structure, on which we impose two conditions: t · A ⊆ A · t and the two actions commute, i.e. A[[t]] is a K[[t]]-bimodule. Then there exists a K-linear map σ on A satisfying: t · a = σ(a) · t
∀a ∈ A
Conversely, given any σ ∈ L(A), there exists a unique left K[[t]]-module structure of such type on A[[t]]: X X X λp tp · a n tn = λp σ p (an )tn+p . p≥0
n≥0
n,p≥0
Definition 1. The obtained K[[t]]-bimodule is called the σ-deformation of the vector space A. ˜ is a We can extend σ into a map σ ˜ on A[[t]] by σ ˜ (Σ an tn ) = Σ σ(an )tn and σ morphism of K[[t]]-bimodule. Similarly, we can extend any ϕ ∈ L(A) to a map ϕ˜ on
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A[[t]]; ϕ˜ is a morphism of right K[[t]]-module but will not respect the left structure, unless ϕ is σ-invariant. More generally, given a family (ϕn )n≥0 of linear maps on A, the map ϕ˜ defined by ϕ(Σ ˜ an tn ) = Σ ϕn (an )tn is a morphism of K[[t]]-bimodule if, and only if, all the ϕn are σ-invariant. We obtain in this way all the morphisms of the K[[t]]-bimodule A[[t]]: Y Lσ (A)tn . HomK[[t]]−K[[t]](A[[t]]) = n≥0
Definition 2. A t-multilinear map on A[[t]], of order N, is a K-multilinear ˜·a ˜ · ϕ(˜ ˜λ ˜1 , a ˜2 , . . . , a ˜N ) = λ ˜ a1 , a ˜2 , . . . , a ˜N ), map ϕ˜ : A[[t]]N → A[[t]] such that ϕ( ˜ a ˜·a ˜ = ˜i · λ, ˜i+1 , . . . , a ˜n ) = ϕ(˜ ˜ a1 , . . . , a ˜i , λ ˜i+1 , . . . , a ˜N ), ϕ(˜ ˜ a1 , . . . , a ˜N · λ) ϕ(˜ ˜ a1 , . . . , a ˜ for all i ∈ {1, . . . , N − 1}, λ ˜ ∈ K[[t]] and a ˜N ) · λ, ˜i ∈ A[[t]]. We note ϕ(˜ ˜ a1 , . . . , a LN t (A[[t]]) the space of these maps. For N = 1, we get the morphisms of K[[t]]-bimodules; but for N ≥ 2, t-multilinear maps are generaly not K[[t]]-multilinear! In terms of tensorial products, we have ⊗N t , A[[t]]) . LN t (A[[t]]) = HomK[[t]]−K[[t]] (A[[t]] A t-multilinear map ϕ˜ is σ ˜ -invariant and determined by its restriction to AN . N ˜ AN ∈ L (A), where πn (Σap tp ) = an . We have ϕ(a ˜ 1 , . . . , aN ) = We set ϕn = πn ◦ ϕ| n Σϕn (a1 , . . . , aN )t for ai ∈ A and the ϕn are σ-invariant. Conversely, given a family (ϕn )n≥0 of σ-invariant multilinear maps on A, there exists a unique t-multilinear map ϕ˜ on A[[t]] such that ϕ| ˜ AN = Σϕn tn . This means Y n LN LN t (A[[t]]) = σ (A)t . n≥0
1.2. σ-Deformations of an algebra We are looking for K-algebra structures on A[[t]] that are compatible with the previous K[[t]]-bimodule structure, i.e. for associative t-bilinear maps ∗ on A[[t]]. ˜ to be a We note At,∗ the algebra obtained (or just At ). The associativity forces σ K-algebra morphism. Now At ·t is a two-sided ideal of At (because a∗(b·t) = (a∗b)·t and (b · t) ∗ a = (b ∗ σ(a)) · t) so At /At · t, which is isomorphic to A as a K-vector space, inherits a K-algebra structure and for this structure σ is a morphism. This leads to the definition: Definition 3. A being a K-algebra and σ an endomorphism of A, a σdeformation of A is a t-bilinear associative map on A[[t]], such that At /At · t = A as K-algebras. The product ∗ is determined by the Cn ∈ L2σ (A) defined by a ∗ b = Σ Cn (a, b)tn for all a, b ∈ A. Conversely, given such a family and defining ∗ by this formula,
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we only miss the associativity to get a σ-deformation. It is enough to check that a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ A. This is equivalent to X Cn (Cp (a, b), σ p (c)) − Cn (a, Cp (b, c)) = 0 ∀N ≥ 0 . n+p=N
For N = 0, the condition is just the associativity of the product of A; for N ≥ 1, we can write it in the equivalent way — the right-hand side of the equality being the coboundary of CN for the σ N -cohomology introduced in Sec. 2.1: X Cn (Cp (a, b), σ p (c)) − Cn (a, Cp (b, c)) n+p=N n,p≥1
= aCN (b, c) − CN (ab, c) + CN (a, bc) − CN (a, b)σ N (c)
(1)
1.3. Trivial product, equivalence Given a σ-deformation ∗ of A, the canonical inclusion i : A → At,∗ is a K-linear map, but not a morphism of algebra, except if a ∗ b = ab, i.e. X X X b p tp = an σ p (bp )tn+p . a n tn ∗ Definition 4. This algebra is called the trivial σ-deformation of A and is noted A[[t]]. Definition 5. Two σ-deformations ∗ and ∗0 are equivalent if there exists a K-algebra isomorphism φ˜ : At,∗ → At,∗0 that is also a morphism of K[[t]]-bimodule and such that π0 ◦ φ˜ = π0 (i.e. φ˜ = idA + φ1 t + · · · ). Proposition 6. A σ-deformation ∗ of A is equivalent to the trivial one if, and ˜ ◦ τ and only if, there exists a K-algebra morphism τ : A → At such that τ ◦ σ = σ π0 ◦ τ = idA . Proof. If φ˜ is an equivalence between ∗ and the trivial product, define τ = φ˜ ◦ i. ˜ an tn ) = Σ τ (an )tn . Conversely, given the section τ , define φ(Σ 1.4. A[[t]]-bimodules We assume that A has a unit and σ(1) = 1; A[[t]] is the trivial σ-deformation of A. Let P be an A-bimodule; P is a K-vector space (thanks to the unit of A) so, given an α ∈ L(P ), we can consider P [[t]], the α-deformation of the vector space P . P [[t]] is a K[[t]]-bimodule and an A-bimodule. We define, for pn ∈ P and ap ∈ A: X X X a p tp = pn σ n (ap )tn+p p n tn · X
a p tp ·
X
p n tn =
X
ap αp (pn )tn+p
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It is straightforward to check that this defines a right A[[t]]-module action (we only use the fact that σ is a morphism). If we want the second formula to define a left A[[t]]-module action, then α must satisfy: α(ap) = σ(a)α(p) for all a ∈ A and p ∈ P . Moreover, if we want the two actions to commute, then we must also have α(pa) = α(p)σ(a); so: Proposition 7. If σ and α verify α(apb) = σ(a)α(p)σ(b) for all a, b ∈ A and p ∈ P, then P [[t]] is an A[[t]]-bimodule. In this situation, given a family (ϕn )n≥0 of A-bimodule morphisms of P that are α-invariant, there exists a unique A[[t]]-bimodule morphism of P [[t]] defined by ϕ˜ = Σϕn tn . 2. Cohomological Tools Adapted to σ-Deformation Theory A is a K-algebra with unit and σ an endomorphism of A (σ(1) = 1). 2.1. The Hσ∗ (A,Aσn ) point of view We note Aσn the K-vector space A endowed with the following A-bimodule structure: a · b · c = abσ n (c), the product in the right-hand side being the one of A. H ∗ (A, Aσn ) denotes the Hochschild cohomology of A with coefficients in this bimodule. Cochains are the same for all n: C p = C p (A, Aσn ) = Lp (A). The coboundary is given by δσn (f )(a1 , . . . , ap+1 ) = a1 f (a2 , . . . , ap+1 ) +
p X
(−1)i f (a1 , . . . , ai ai+1 , . . . , ap+1 )
i=1
+ (−1)p+1 f (a1 , . . . , ap )σ n (ap+1 ) , where f ∈ C p and a1 , . . . , ap+1 ∈ A. This coboundary is well adapted to σdeformations — see formula (1); but we are not dealing with all multilinear maps, only with σ-invariant ones. Now, they form a subcomplex of the preceding one: if we denote Cσp = Lpσ (A) ⊆ C p then δσn (Cσp ) ⊆ Cσp+1 . The cohomology obtained is denoted Hσ∗ (A, Aσn ). In general, we do not have Hσ∗ (A, Aσn ) ⊆ H ∗ (A, Aσn ). However, it is true if σ is an automorphism that generates a finite group [8]. Now, given a family (Cn )n≥0 of maps in Cσ2 , we define a ∗ b = Σ Cn (a, b)tn ; the associativity of ∗ is equivalent to having, for all N ≥ 1: X Cn (Cp (a, b), σ p (c)) − Cn (a, Cp (b, c)) = δσN (CN )(a, b, c) . n+p=N n,p≥1
The following results are proved in [8], using G-algebra structures [4]: Lemma 8. Given N0 ≥ 1, if the preceding relation is true for all N ≤ N0 , then for N = N0 + 1 the left-hand side lies in Zσ3 (A, AσN0 +1 ).
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Corollary 9. Given ϕ ∈ Zσ2 (A, AσN ), N ≥ 1, if Hσ3 (A, Aσn ) = 0 for all n ≥ N + 1, then there exists a σ-deformation such that a ∗ b = ab + ϕ(a, b)tN + · · · . Likewise, the obstructions to the integrability of elements in Zσ1 (A, AσN ) (i.e. σinvariant σ N -derivations) into formal automorphisms of A[[t]] (i.e. t-linear algebra morphisms φ of the trivial σ-deformation, verifying φ0 = idA ) lie in the spaces Hσ2 (A, Aσn ). Concerning rigidity, we have, following [8]: Lemma 10. If a ∗ b = ab + Cn (a, b)tn + · · · is a σ-deformation, n ≥ 1, then Cn ∈ Zσ2 (A, Aσn ). Moreover, if Cn is a coboundary, then ∗ is equivalent to a σ0 (a, b)tn+1 + · · · . deformation ∗0 such that a ∗0 b = ab + Cn+1 Corollary 11. If Hσ2 (A, Aσn ) = 0 for all n ≥ 1, then A is σ-rigid (i.e. every σ-deformation of A is equivalent to the trivial one). 2.2. The H ∗ (A[[t]],K[[t]];A[[t]]) point of view A[[t]] is the trivial σ-deformation of A. K[[t]] is a central sub-algebra of A[[t]], so we can consider the K[[t]]-relative Hochschild cohomology [4] of A[[t]]: Ht∗ = Q H ∗ (A[[t]], K[[t]]; A[[t]]). Cochains are t-multilinear maps: Ctp = n≥0 Cσp tn . The ˜i ∈ A[[t]]): coboundary is (f˜ ∈ Ctp , a ˜ f˜)(˜ ˜p+1 ) = a ˜1 f˜(˜ ˜p+1 ) a2 , . . . , a δ( a1 , . . . , a +
p X
(−1)i f˜(˜ ˜i a ˜i+1 , . . . , a ˜p+1 ) a1 , . . . , a
i=1
˜p )˜ ap+1 a1 , . . . , a + (−1)p+1 f˜(˜ Q Proposition 12. δ˜ = Σn≥0 δσn tn and Htp = n≥0 Hσp (A, Aσn )tn . Proof. Let us write f˜ = Σ fn tn with fn ∈ Cσp and restrict the formula to the case ai ∈ A, this leads to: δt (f )(a1 , . . . , ap+1 ) = Σ δσn (fn )(a1 , . . . , ap+1 )tn . The cohomology Ht∗ is given by a Bar resolution [4]: p+2
· · · → A[[t]]⊗t
∂˜p
∂˜
0 −→ · · · → A[[t]] ⊗t A[[t]] −→ A[[t]] → 0
a0 ⊗ t · · · ⊗ t a ˜p+1 ) = ∂˜p (˜
p X i=0
(−1)i a ˜ 0 ⊗t · · · ⊗t a ˜i a ˜i+1 ⊗t · · · ⊗t a ˜p+1
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∂˜ is a morphism of A[[t]]-bimodules and there exists a contracting homotopy ˜ x ) = 1 ⊗t x ˜. The complex is a K[[t]]which is a morphism of K[[t]]-bimodules: h(˜ relative projective resolution of the A[[t]]-bimodule A[[t]]. When we apply the functor HomA[[t]]−A[[t]](−, A[[t]]), we retrieve the cohomology Ht∗ : p+2
HomA[[t]]−A[[t]](A[[t]]⊗t
p
, A[[t]]) = HomK[[t]]−K[[t]] (A[[t]]⊗t , A[[t]]) = Ctp
with the coboundary we want. 2.3. Continuous cohomology So far, we have only considered the “algebraic” case. Now, for infinite dimensional algebras, algebraic Hochschild cohomology has to be replaced by continuous or differential cohomology (i.e. where cochains are multi-differential operators, on algebras of smooth functions or distributions on a manifold, for instance — see [1, 7]). Moreover, for infinite dimensional spaces, A ⊗ K[[t]] and A[[t]] are not equal, the first being strictly included in the second. Let us introduce some topology to overcome these difficulties. We could choose the so-called t-adic topology, which does not require any topology on A, but it has a great defect: it does not behave well under dualisation. We prefer to follow [1] and suppose that A is endowed with a well-behaved topology, i.e. a complete, Hausdorff, locally convex, complex vector space topology which is moreover nuclear and Fr´echet or dual of Fr´echet. Spaces of polynomials, formal series and smooth functions on a manifold have such a topology. It satisfies: ˆ A⊗C[[t]] = A[[t]], where the hat denotes completion with respect to the projective topology on the tensorial product [10] and A[[t]] is given the product topology. When σ is continuous, a σ-deformation is a topological C[[t]]-bimodule and we call it a topological σ-deformation. By topological equivalence we mean a continuous equivalence φ˜ (i.e. such that all the φp are continuous). Proposition 7 is still valid: if we suppose that P is a topological A-bimodule and α a continuous map, then P [[t]] is a topological A[[t]]-bimodule. ∗ (A, Aσn ) the cohomology given by continuous cochains inside We note Hσ,cont (the coboundary of a continuous cochain being continuous). Results of 2 (A, Aσn ) = 0 Sec. 2.1 are still valid in the topological case. For instance, if Hσ,cont for all n ≥ 1, then every σ-deformation of A is topologicaly equivalent to the trivial one. ∗ the continuous cohomology inside Ht∗ . Now, f˜ = Σ fn tn ∈ Ctp is We note Ht,cont Q p p = n≥0 Hσ,cont (A, continuous if, and only if, all the fn are continuous; so: Ht,cont ∗ can be deduced from a topological Bar resolution, Aσn ). The cohomology Ht,cont which admits a continuous and C[[t]]-bilinear contracting homotopy:
Hσ∗ (A, Aσn )
ˆ p+2
· · · → A[[t]]⊗t
∂˜p
∂˜
0 ˆ t A[[t]] −→ −→ · · · → A[[t]]⊗ A[[t]] → 0
(2)
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ˆ p+2
HomA[[t]]−A[[t]],cont(A[[t]]⊗t
693
, A[[t]]) ˆp
= HomC[[t]]−C[[t]],cont(A[[t]]⊗t , A[[t]]) p = Ct,cont p
p
ˆt ˆ Proposition 13. A[[t]]⊗ = A⊗ [[t]] as topological A[[t]]-bimodules, when the p structure of the right-hand side is given by the map α = σ ⊗ (which clearly satisfies the condition of Proposition 7).
Proof. The identification as vector spaces follows from Proposition 1.5 of [9] ˆ and from the fact that A[[t]] = A⊗C[[t]]. Moreover, the action of A[[t]] is formally the same in both cases. So the Bar resolution can be writen: ˆ p+2
· · · → A⊗
∂˜p
∂˜
0 ˆ [[t]] −→ · · · → (A⊗A)[[t]] −→ A[[t]] → 0 .
The boundary and the homotopy are, in restriction to A, the usual ones and are extended in the only possible way, as they are σ-invariant. 3. A Koszul Resolution of C [X] In this section we study the case of P = C[X] = C[x1 , . . . , xN ], the algebra of polynomials in N indeterminates with coefficients in C; it is a well-behaved algebra and all multilinear maps are continuous [1, 10], so algebraic and continuous cohomology coincide. We note Λp = Λp (x1 , . . . , xN ) the pth exterior space of the vector space generated by the xi . As we are dealing with σ-invariant maps, we must p assume σ ⊗ (Λp ) ⊆ Λp for all p, i.e. that σ is homogeneous of degree 0. 3.1. A Koszul resolution of C We consider the following complex (P ∈ P, 1 ≤ p ≤ N ): ∂p
∂
∂
∂
N 1 0 · · · → P ⊗ Λp −→ · · · −→ P −→ C→0 0 → P ⊗ ΛN −→
∂p (P ⊗ xi1 ∧ · · · ∧ xip ) =
p X
(3)
(−1)j+1 xij P ⊗ xi1 ∧ · · · ∧ xc i1 ∧ · · · ∧ xip
j=1
∂0 (P ) = P (0) C is given the trivial P-module structure by the augmentation ∂0 : P ·λ = P (0)λ. Let us define hp : P ⊗ Λp−1 → P ⊗ Λp by hp (P ⊗ ω) =
N Z X k=1
0
1
tp
∂P (tx)dt ⊗ xk ∧ ω , ∂xk
where ω ∈ Λp−1 , and h0 : C → P the unit of P.
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Proposition 14. hp is a contracting homotopy: ∂0 h0 = idC and ∂p hp + hp−1 ∂p−1 = idP⊗Λp−1 . So the complex (3) is a projective resolution of the P-module C. The proof consists in a classical and easy computation. The choice of the homotopy results from the fact that the complex is the transposition (up to the place of the integral) of the de Rham cohomology complex of exterior forms with coefficients in the algebra C[[X]]. Proposition 15. ∂p and hp are σ-invariant. Proof. It is clear for ∂p , let us show it for hp . We note σ(xl ) = Σσkl xk . Let n P = xn1 1 · · · xp p with ni ≥ 1. hp (σ(P ) ⊗ σ ⊗ =
N Z X
1
tp
N Z X
1
tp 0
k=1
=
p Z X l=1
(ω))
0
k=1
=
p−1
p−1 ∂σ(x1 )n1 · · · σ(xp )np (tx)dt ⊗ xk ∧ σ ⊗ (ω) ∂xk
p X
nl
l=1
1
tp n l σ
0
=σ⊗σ
⊗p
p+1
P xl
p Z X l=1
= σ⊗
p−1 ∂σ(xl ) σ(P ) (tx) (tx)dt ⊗ xk ∧ σ ⊗ (ω) σ(xl ) ∂xk
0
1
(tx)dt ⊗
N X
σkl xk ∧ σ ⊗
k=1
P t nl (tx)dt ⊗ xl ∧ ω xl
p−1
(ω)
!
p
(hp (P ⊗ ω))
3.2. A Koszul resolution of C [X] (2)
We note P (2) = P ⊗ P = C[X, Y ]. Let PE be the vector space P (2) endowed with the left P (2) -module structure given by the algebra product and the right Pmodule structure defined from the map E : P → P (2) given by E(P ) = P (X − Y ). (2) We apply the functor PE ⊗P -to the Koszul resolution of C: ∂¯p
∂¯
∂¯
∂¯
N 1 0 · · · → P (2) ⊗ Λp −→ · · · −→ P (2) −→ P →0 0 → P (2) ⊗ ΛN −→
∂¯p (P ⊗ xi1 ∧ · · · ∧ xip ) =
p X
(4)
(−1)j+1 (xij − yij )
j=1
× P ⊗ xi1 ∧ · · · ∧ xc ij ∧ · · · ∧ xip ∂¯0 (P ) = m(P ) = P (X, X)
(5)
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Proposition 16. (4) is a free resolution of P by P-bimodules.
Proof. [3], Chapter X, Theorem 6.1, p. 194.
This is a somewhat pedantic way to describe the transition. An easier description is as follows: apply − ⊗ P to the resolution (i.e. add an indeterminate Y ) and then use the change of variables (X, Y ) → (X − Y, Y ). Now, what has happened to (2) the homotopy? We cannot apply PE ⊗P − to it, because it is not a morphism of P-module. Guided by the change of variable, we set ¯ p (P ⊗ ω) = h
N Z X k=1
0
1
tp
∂P (tx + (1 − t)y, y)dt ⊗ xk ∧ ω . ∂xk
(6)
¯ p is a contracting homotopy for (4). So we recover the result Proposition 17. h of Proposition 16. ¯ p are σ-invariant. Proposition 18. ∂¯p and h The proofs are the same as those of Propositions 14 and 15. 3.3. A Koszul resolution of C [X][[t]] ˆ Now the final step: we apply −⊗C[[t]] to the last resolution. P (2) ⊗ Λp is a ⊗p+2 is a C-linear map on it, that immediatly satisfies P-bimodule, and α = σ ˆ = (P (2) ⊗ Λp )[[t]] is a P[[t]]the condition of Proposition 7; so (P (2) ⊗ Λp )⊗C[[t]] bimodule. Therefore, we obtain the following resolution: ∂˜p
∂˜
N · · · → (P (2) ⊗ Λp )[[t]] −→ · · · 0 → (P (2) ⊗ ΛN )[[t]] −→
(7)
∂˜
0 P[[t]] → 0 · · · → P (2) [[t]] −→
The boundary is (5), extended in the only possible way as to be a morphism of P[[t]]-bimodules; likewise for the homotopy (6) which extends to a morphism of C[[t]]-bimodules. So: Proposition 19. (7) is a C[[t]]-relative projective resolution of the P[[t]]bimodule P[[t]]. 3.4. The cohomology given by the Koszul resolution We apply the functor HomP[[t]]−P[[t]](−, P[[t]]); since HomP[[t]]−P[[t]](P[[t]], P[[t]]) = Zσ (P[[t]]) = {˜ a ∈ P[[t]]/˜ σ(˜ a) = a ˜ and a ˜˜b = ˜b˜ a ∀˜b}
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and HomP[[t]]−P[[t]]((P (2) ⊗ Λp )[[t]], P[[t]]) = Lσ (Λp , P[[t]]) the cohomological complex is δ˜
δ˜
δ˜
0 1 N (P[[t]])σ −→ Lσ (Λ1 , P[[t]]) → · · · −→ Lσ (ΛN , P[[t]]) → 0 , 0 → Zσ (P[[t]]) −→
a)(xi ) = xi a ˜−a ˜xi and for p ≥ 1: with δ˜0 the inclusion, δ˜1 (˜ δ˜p (f˜)(xi1 ∧ · · · ∧ xip ) =
p X
(−1)j+1 (xij f˜(xi1 ∧ · · · ∧ xc ij ∧ · · · ∧ xip )
j=1
− f˜(xi1 ∧ · · · ∧ xc ij ∧ · · · ∧ xip )xij ) Proposition 20. Htp = Ker(δ˜p+1 )/Im(δ˜p ).
Proof. It is a corollary of Proposition 19.
The coboundary is generally not zero, unlike in the case σ = id. Since Lσ (Λp , Q P[[t]]) = n≥0 Lσ (Λp , P)tn as vector spaces, this complex is the product of an infinite number of complex: δ0,σn
δ1,σn
δ2,σn
δN,σn
0 → πn (Zσ (P[[t]])) −→ Pσ −→ Lσ (Λ1 , P) −→ · · · −→ Lσ (ΛN , P) → 0 δp,σn (f )(xi1 ∧ · · · ∧ xip ) =
p X
(−1)j+1 (xij − σ n (xij ))
j=1
× f (xi1 ∧ · · · ∧ xc ij ∧ · · · ∧ xip ) Proposition 21. Hσp (P, Pσn ) = Ker(δp+1,σn )/Im(δp,σn ). 4. σ-Deformations of the Algebra C [x,y] From now on, P denotes the algebra C[x, y]. First, we compute its second σcohomology when σ is homogeneous of degree 0, by using the preceding Koszul resolution. Then we give explicit deformations for σ defined by σ(x) = qx and σ(y) = q −1 y. 4.1. The second σ-cohomology spaces of C [x,y] Any linear endomorphism σ of Λ1 = Vect(x, y) can be extended to a unique morphism of the algebra P. It is shown in [8] that two equivalent σ give the same cohomology (and thus lead to the same σ-deformation theory); so we can assume that σ is a Jordan matrix. For simplicity, we will study the case σ diagonal:
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σ(x) = λx, σ(y) = µy. So, given φ ∈ Lσ (Λ1 , P), we have (δσn (φ))(x ∧ y) = (1 − λn )xφ(y) + (1 − µn )φ(x)y . In the following discussion, we consider that λ0 = 1 for all λ ∈ C (even for λ = 0). Proposition 22. The space Hσ2 (P, Pσn ) is isomorphic to Vect{xk y l /λk µl = λµ} Vect{(1 − λn )xk+1 y l /λk µl = µ} ∪ {(1 − µn )xk y l+1 /λk µl = λ} Proof. Proposition 21 tells us that this space is Lσ (Λ2 , P) . δσn (Lσ (Λ1 , P)) Now, L(Λ2 , P) ≈ P by φ ↔ φ(x ∧ y) and φ is σ-invariant if σ(φ(x ∧ y)) = φ(σ(x) ∧ σ(y)) = λµφ(x ∧ y). So by the previous isomorphism Lσ (Λ2 , P) ≈ Ker(σ − λµid) = Vect{xk y l /λk µl = λµ}. We also have L(Λ1 , P) ≈ P × P and by this isomorphism Lσ (Λ1 , P) = Ker (σ − λid) × Ker(σ − µid) = Vect{xk y l /λk µl = λ} × Vect{xk y l /λk µl = µ}, so that δσn (Lσ (Λ1 , P) ≈ Vect{(1 − λn )xk+1 y l /λk µl = µ} ∪ {(1 − µn )xk y l+1 / λk µl = λ}. Corollary 23. The dimension of Hσ2 (P, Pσn ) as a C-vector space is:
µn = 1 µ=0 µn 6= 1 µ 6= 0
λn = 1
λ=0
∞ ∞
∞ 2
0
∞
λn 6= 1 λ 6= 0 0 ∞ 1 if λµ = 1 0 if λµ 6= 1
We will now examine the case λµ = 1, because this gives deformations compatible with the Heisenberg relation x ∗ y − y ∗ x = t. In this case the Hσ2 are one-dimensional, except the Hσ2 (P, PσkN ), k ∈ N, if λ in an N th root of 1, which are infinite dimensional. 4.2. Some q-calculus formulae Let q ∈ C \ {0}, we set, for all n ∈ N, (n)q = 1 + q + · · · + q n−1 and (n)q ! = (1)q (2)q · · · (n)q . When q is not a root of 1, we can define, for 0 ≤ k ≤ n: (n)q · · · (n − k + 1)q n . = k q (k)q !
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n Lemma 24. 1. If 1 ≤ k ≤ n then k−1 + nk q q k = n+1 k . q 2. If a and b are elements of an algebra such that ba = qab then (a + b)n = Σnk=0 nk q ak bn−k . When q is an N th-root of unity, N ≥ 2, nk q is defined only for n ≤ N − 1, but we may define it by induction for all n, assuming n0 q = nn q = 1 and using formula 1 of Lemma 24 and then statement 2 is true. The q-exponential is defined by X an . expq (a) = (n)q ! n≥0
When q is an N th-root of unity, it is still defined if aN = 0 (then the terms vanish for n ≥ N ). We recall that (a and b are elements of an algebra): Lemma 25. 1. If ba = qab, then expq (a + b) = expq (a) expq (b). 2. For all a : expq (a) expq−1 (−a) = id. 4.3. q-Moyal products Let q ∈ C \ {0}, we define on P = C[x] ⊗ C[y] the endomorphism σ = σx ⊗ σy , with σx (f )(x) = f (qx) , σy (g)(y) = g(q −1 y) . We extend σx and σy to P, by σx ⊗ id and id ⊗ σy respectively. So σ = σx ◦ σy = σy ◦ σx . When q 6= 1, we define ∆x and ∆y on C[x] and C[y] by ∆x (f ) =
f (qx) − f (x) , (q − 1)x
∆y (g) =
g(q −1 y) − g(y) . (q −1 − 1)y
We extend ∆x and ∆y to P, by ∆x ⊗ id and id ⊗ ∆y . For the limit case q = 1, we have ∆x = ∂/∂x and ∆y = ∂/∂y. Note that ∆x (xn ) = (n)q xn−1 and ∆y (y n ) = (n)q−1 y n−1 . Let m denote the abelian product of P: Lemma 26. 1. ∆x ◦ m = m ◦ (∆x ⊗ id + σx ⊗ ∆x ) = m ◦ (∆x ⊗ σx + id ⊗ ∆x ), i.e. ∆x is a σx -derivation. 2. ∆x ◦ σx = qσx ◦ ∆x . The same holds for y (when replacing q by q −1 in 2). Let us define three bilinear operators on P by: η(f, g) = ∆x (f )∆y (σx (g)) , ω(f, g) = −∆y (f )∆x (σy (g)) , π(f, g) =
1 [∆x (f )∆y (σx (g)) − ∆y (f )∆x (σy (g))] . 2
GENERALISED DEFORMATIONS, KOSZUL RESOLUTIONS, MOYAL PRODUCTS
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Proposition 27. η, ω and π are equivalent non trivial σ-invariant 2-cocycles, i.e. represent the same non zero element in Hσ2 (P, Pσ ) (we recall that, when q 6= 1, this space has dimension 1). Proof. Let us show that η is a σ-invariant cocycle: η(σ(f ), σ(g)) = ∆x (σ(f ))∆y (σx (σ(g))) = qσ(∆x (f ))q −1 σ(∆y (σx (g))) = σ(η(f, g)) (δσ η)(f, g, h) = f ∆x (g)∆y (σx (h)) − [f ∆x (g) + ∆x (f )σx (g)]∆y (σx (h)) + ∆x (f )[σx (g)∆y (σx (h)) + ∆y (σx (g))σ(h)] − ∆x (f )∆y (σx (g))σ(h) =0 The same can be done for ω and therefore for π. Now, η(x, y) − η(y, x) = 1; if η was a coboundary: η = δσ (f ), with f ∈ Lσ (P), this would imply (1 − q)xf (y) − (1 − q −1)f (x)y = 1, which is impossible. Finally, consider φ = 12 ∆x ∆y ∈ Lσ (P); an easy computation shows that π + δσ φ = ω and π − δσ φ = η. Now, Ht3 = 0 (thanks to the existence of a Koszul resolution and from the fact that Λ3 = 0), so these cocycles are integrable into σ-deformations — not unique: at each step Hσ2 (P, Pσn ) is not 0 (its dimension is 1 if q n 6= 1 and ∞ either). Let us define three t-bilinear products on P[[t]] by (f, g ∈ P): f ∗η g =
X n≥0
f ∗ω g =
1 ∆n (f )(∆y σx )n (g)tn (n)q ! x
X (−1)m ∆m (f )(∆x σy )m (g)tm (m)q−1 ! y
m≥0
f ∗π g =
X n,m≥0
(−1)m ∆n ∆m (f )(∆y σx )n (∆x σy )m (g)tn+m 2n+m (n)q !(m)q−1 ! x y
These products can also be written (t denotes the multiplication by t in C[[t]]): ∗η = (m ⊗ idt ) ◦ expq (∆x ⊗ ∆y σx ⊗ t) ∗ω = (m ⊗ idt ) ◦ expq−1 (−∆y ⊗ ∆x σy ⊗ t) 1 1 ∆x ⊗ ∆y σx ⊗ t ◦ expq−1 − ∆y ⊗ ∆x σy ⊗ t ∗π = (m ⊗ idt ) ◦ expq 2 2
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Definition 28. ∗π is called the q-Moyal product. ∗η and ∗ω are called respectively its right form and its left form in normal ordering. For q = 1, we retrieve the usual Moyal product and its forms in normal ordering — see [6]. Proposition 29. ∗η , ∗ω and ∗π are associative; therefore they are σ-deformations of C[x, y], with respective leading cocycles η, ω and π. Proof. Let us check that ∗η ◦ (id ⊗t ∗η ) = ∗η ◦ (∗η ⊗t id). ∗η ◦ (id ⊗t ∗η ) = (m ⊗ idt ) ◦ expq (∆x ⊗ ∆y σx ⊗ t) ◦ (id ⊗ m ⊗ idt ) ◦ expq (id ⊗ ∆x ⊗ ∆y σx ⊗ t) = (m ◦ (id ⊗ m) ⊗ idt ) ◦ expq (∆x ⊗ (∆y σx ⊗ σx + σ ⊗ ∆y σx ) ⊗ t) ◦ expq (id ⊗ ∆x ⊗ ∆y σx ⊗ t) = (m ◦ (id ⊗ m) ⊗ idt ) ◦ expq ((∆x ⊗ ∆y σx ⊗ σx + ∆x ⊗ σ ⊗ ∆y σx + id ⊗ ∆x ⊗ ∆y σx ) ⊗ t) = (m ◦ (m ⊗ id) ⊗ idt ) ◦ expq ((∆x ⊗ ∆y σx ⊗ σx + ∆x ⊗ σy ⊗ ∆y σx + σx ⊗ ∆x ⊗ ∆y σx ) ⊗ t) We used successively Lemma 26, Lemma 25, associativity of m and again Lemma 26. We make an analogous calculation for the right-hand side: ∗η ◦ (∗η ⊗t id) = (m ⊗ idt ) ◦ expq (∆x ⊗ ∆y σx ⊗ t) ◦ (m ⊗ id ⊗ idt ) ◦ expq (∆x ⊗ ∆y σx ⊗ σ ⊗ t) = (m ◦ (m ⊗ id) ⊗ idt ) ◦ expq ((∆x ⊗ id ⊗ ∆y σx + σx ⊗ ∆x ⊗ ∆y σx + ∆x ⊗ ∆y σx ⊗ σ) ⊗ t) = (m ◦ (m ⊗ id) ⊗ idt ) ◦ expq ((∆x ⊗ ∆y σx ⊗ σx + ∆x ⊗ σy ⊗ ∆y σx + σx ⊗ ∆x ⊗ ∆y σx ) ⊗ t) The same proof can be done for ∗ω . For ∗π , it takes a little more time, but the computation is of same kind. Anyway, thanks to the next statement, associativity for one of the products implies associativity for the others. Proposition 30. ∗η , ∗ω and ∗π are equivalent σ-products. Proof. Let us define φt ∈ Lt (P[[t]]) by φt = expq ( 12 ∆x ∆y ⊗ t). We will show that ∗η ◦ (φt ⊗t φt ) = φt ◦ ∗π . This is equivalent to, using Lemma 25 thrice:
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1 ∆x ∆y ⊗ σ ⊗ t 2 1 1 id ⊗ ∆x ∆y ⊗ t ◦ expq ∆y ⊗ ∆x σy ⊗ t ◦ expq 2 2 1 (∆x ∆y ⊗ σx + ∆x σy ⊗ ∆y = (m ⊗ idt ) ◦ expq 2 1 ∆x ⊗ ∆y σx ⊗ t + ∆y ⊗ ∆x σy + σx ⊗ ∆x ∆y ) ⊗ t ◦ expq 2
(m ⊗ idt ) ◦ expq (∆x ⊗ ∆y σx ⊗ t) ◦ expq
Using Lemma 25 again, we can collect, on both side, all terms in a unique qexponential. Then we must use Lemma 26 to conclude. An analogous calculation −1 −1 −1 is expq−1 (− 12 ∆x ∆y ⊗ t).) Finally, the leads to ∗ω ◦ (φ−1 t ⊗t φt ) = φt ◦ ∗π . (φt 2 equivalence between ∗η and ∗ω is φt . 4.4. Generators and relations Let q ∈ C \ {0} and consider the algebra Hq generated by x, y and t, with relations: x∗y−y∗x=t t ∗ x = qx ∗ t t ∗ y = q −1 y ∗ t Proposition 31. The underlying vector space of Hq is P[t] = C[x, y, t]. Proof. We construct Hq from C[t] by two successive Ore extensions (see [5]). We consider first B = C[t][y, α, 0] with α(t) = qt. It is the algebra generated by t and y, with relation yt = qty; its underlying vector space is C[t][y] = C[y, t]. We set β(y) = y and β(t) = q −1 t; this defines a unique endomorphism β of the algebra B. Moreover, we define a map d on B by: d(y s tu ) = (s)q−1 y s−1 tu+1 . Since d(F ∗ G) = d(F ) ∗ G + β(F ) ∗ d(G), we can consider the Ore extension B[x, β, d]. It is the algebra generated by x, y and t with the desired relations and its underlying vector space is C[y, t][x] = C[x, y, t]. Let us consider two particular isomorphisms P[t] → Hq : φη (xr y s tu ) = y ∗s ∗ x∗r ∗ t∗u φω (xr y s tu ) = x∗r ∗ y ∗s ∗ t∗u Proposition 32. For all F, G ∈ P[t] : φη (F ∗η G) = φη (F ) ∗ φη (G) and φω (F ∗ω G) = φω (F ) ∗ φω (G).
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Proof. x ∗ y ∗s ∗ x∗r ∗ t∗u = (y ∗s ∗ x + (r)q−1 y ∗r−1 ∗ t) ∗ x∗r ∗ t∗u = y ∗s ∗ ∗ t∗u + (r)q−1 y ∗r−1 ∗ q r x∗r ∗ t∗u+1 ; so, for all G ∈ P[t]: φη (x) ∗ φη (G) = x φη (xG + ∆y σx (G)t) = φη (x ⊗ idt + ∆y σx ⊗ t)(G), where x denotes the abelian product by x. Now, (∆y σx ⊗ t) ◦ (x ⊗ idt ) = q(x ⊗ idt )(∆y σx ⊗ t), so we can use Lemma 24: ! n X n n n−k k (x ⊗ idt ) (∆y σx ⊗ t) (G) = φη (xn ∗η G) . φη (x) φη (G) = φη k q ∗r+1
k=0
For ∗ω , we can make an analogous calculation or just show that φω = φη φ2t , where φ2t is the equivalence between ∗η and ∗ω (see Proposition 30). What about φπ = φη φt = φω φ−1 t ? When q = 1, we know that it is the symetrisation of ∗ with respect to the decomposition into linear factors [6]. Unfortunately, this is no longer true when q 6= 1 and we did not find such an easy expression for φπ . Finally, remember that the Weyl algebra is the quotient of H1 by the relation t = 1. If q 6= 1, such a quotient of Hq is trivial, because the relation t = 1 is not compatible with tx = qxt (and ty = q −1 yt). But when q is an N th-root of 1, tN = 1 is compatible and we have: Proposition 33. Wq = Hq /(tN = 1) is isomorphic to the subalgebra of L(C[x]) generated by x, ∆x and σx (x denotes the abelian product by x). When q = 1, we recover the usual Weyl algebra W1 generated by x, ∂/∂x and id. Proof. Indeed, for all P ∈ C[x]: x∆x (P ) − ∆x (xP ) = −σx (P ), σx (xP ) = qxσx (P ) and σx (∆x (P )) = q −1 ∆x (σx (P )). 5. Two Classical Rigid Algebras 5.1. Algebras of matrices In this subsection, we study the case of M = L(V ), when V is a finite dimensional K-vector space. Let α : V ⊗V → V ⊗V be the flip of V ⊗V : α(v⊗w) = w⊗v; α is an invertible element of M ⊗ M = L(V ⊗ V ) and α−1 = α. Lemma 34. For all f, g ∈ M : α−1 ◦ (f ⊗ g) ◦ α = g ⊗ f ; i.e. the inner automorphism associated to α is the flip of M ⊗ M. Proof. α◦(f ⊗g)◦α(v⊗w) = α◦(f ⊗g)(w⊗v) = α(f (w)⊗g(v)) = g(v)⊗f (w) = (g ⊗ f )(v ⊗ w). Let (ei ) be a basis of V and (ei ) its dual basis; we note eij = ej ⊗ei ∈ V ⊗V ∗ = M (i.e. eij (ek ) = δki ej where δki is 1 if i = k and 0 either).
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Lemma 35. α = Σ eij ⊗ eji and m(α) = dim(V )idV ; where m : M ⊗ M → M is the product of M. Proof. Σ (eij ⊗ eji )(ek ⊗ el ) = Σ δki δlj ej ⊗ ei = el ⊗ ek and m(α) = Σ eij ◦ eji = Σ (ej ⊗ ei ) ◦ (ei ⊗ ej ) = Σ ei (ei )ej ⊗ ej = dim(V )Σ ej ⊗ ej = dim(V )idV . We define e = α/ dim(V ), so m(e) = idV and (f ⊗ g)◦ e = e ◦ (g ⊗ f ) for all f, g ∈ M. This implies that τ : M → M⊗M defined by τ (f ) = (f ⊗id)◦e = e◦(id⊗f ) is a morphism of M-bimodules that satisfies m ◦ τ = id. So M is a sub-M-bimodule of the free M-bimodule M ⊗ M and hence it is a projective M-bimodule. Lemma 36. (σ ⊗ σ)(e) = e, therefore τ is σ-invariant: τ ◦ σ = (σ ⊗ σ) ◦ τ. Proof. All morphisms of the algebra M are inner, so there exists an M ∈ GL(V ) such that σ(f ) = M −1 ◦ f ◦ M . σ ⊗ σ is the inner automorphism associated to M ⊗M ∈ GL(V ⊗V ): (σ⊗σ)(f ⊗g) = σ(f )⊗σ(g) = (M −1 ◦f ◦M )⊗(M −1 ◦g◦M ) = (M ⊗ M )−1 ◦ (f ⊗ g) ◦ (M ⊗ M ); now, e is a central element of M ⊗ M. So τ can be extended to a morphism of M[[t]]-bimodules τ˜ : M[[t]] → M[[t]] ⊗t M[[t]] that satisfies (m ⊗ idt ) ◦ τ = id. Thus [4] M[[t]] is a K[[t]]-relative projective M[[t]]-bimodule, which implies: Proposition 37. The cohomology spaces H p (M[[t]], K[[t]]; M[[t]]) are zero for p ≥ 1. In particular, M is a rigid algebra under σ-deformations. 5.2. Finite degree extensions of K Let L be an extension of the field K, i.e. L is a field and a K-vector space. The endomorphisms of the K-algebra L are precisely the elements of the Galois Group of the extension. We assume that the extension is finite, i.e. L is a finite dimensional K-vector space, then this group is finite and so [8]: Hσ∗ (L, Lσ ) ⊆ H ∗ (L, Lσ ). Let us recall why this last cohomology is trivial: Lemma 38. L ⊗K L is a semi-simple K-algebra. Proof. Indeed L ⊗K L is the direct sum of a finite number of fields (see [2, Sec. 7.3, Corollary 1, p. 86]). Corollary 39. H p (L, M ) = 0 for all p ≥ 1 and L-bimodule M. Proof. [3] p. 176. Corollary 40. L is rigid in any σ-deformation theory.
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References [1] P. Bonneau, M. Flato, M. Gerstenhaber and G. Pinczon, “The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations”, Commun. Math. Phys. 161 (1994) 117–126. [2] N. Bourbaki, Modules et Anneaux Semi-simples, Hermann, 1958. [3] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, 1956. [4] M. Gerstenhaber and S. D. Schack, Algebraic Cohomology and Deformation Theory, Kluwer Academic Publ., 1988. [5] C. Kassel, Quantum Groups, Springer-Verlag, 1995. [6] A. A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, 1976. [7] G. Pinczon, “On the equivalence between continuous and differentiable deformation theory”, Lett. Math. Phys. 39 (1997) 143–156. [8] G. Pinczon, “Non commutative deformation theory”, Lett. Math. Phys. 41 (1997) 101–117. [9] J. L. Taylor, “Homology and cohomology for topological algebras”, Advances in Math. 9 (1972) 137–182. [10] F. Tr`eves, Topological vector spaces, distributions and kernels, Academic Press, 1967.
DETERMINANTS, PFAFFIANS AND QUASI-FREE REPRESENTATIONS OF THE CAR ALGEBRA MAURO SPERA Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universit` a di Padova Via Belzoni 7, 35131 Padova, Italy E-mail : [email protected]
TILMANN WURZBACHER Institut de Recherche Math´ ematique Avanc´ ee Universit´ e Louis Pasteur et C.N.R.S. 7, Rue Ren´ e-Descartes, F-67084 Strasbourg, France E-mail : [email protected] Received 5 June 1997 Revised 28 July 1997 1991 AMS Classification: 46L55, 58F06, 32H02, 14F25, 81R30 In this paper we apply the theory of quasi-free states of CAR algebras and Bogolubov automorphisms to give an alternative C ∗ -algebraic construction of the Determinant and Pfaffian line bundles discussed by Pressley and Segal and by Borthwick. The basic property of the Pfaffian of being the holomorphic square root of the Determinant line bundle (after restriction from the Hilbert space Grassmannian to the Siegel manifold, or isotropic Grassmannian, consisting of all complex structures on an associated Hilbert space) is derived from a Fock–anti-Fock correspondence and an application of the Powers–Størmer purification procedure. A Borel–Weil type description of the infinite dimensional Spinc -representation is obtained, via a Shale–Stinespring implementation of Bogolubov transformations.
1. Introduction In this paper we give an alternative, functorial construction of the Determinant and Pfaffian line bundles over the Hilbert space Grassmannian, discussed by Pressley and Segal [19] and by Borthwick [4], in a C ∗ -algebraic fashion. We shall employ the theory of quasi-free states of CAR algebras [17, 18], see Secs. 2 and 3, which have already proved to be very useful in deriving the explicit Pl¨ ucker type equations governing the projective embedding of the Hilbert space Grassmannian [23]. Alternatively, our constructions can be viewed as a geometrization of the theorems of Powers–Størmer and Shale–Stinespring [18, 21]. Roughly speaking, and partially referring to the notation introduced further on in the paper, we exploit the basic equivariance property shared by the two theories. The (Sato–Segal–Wilson) Hilbert space Grassmannian Gr(K, K+ ) [19] coincides with the Ures (K, K+ )-orbit of the quasi-free state ω+ pertaining to the reference subspace K+ (see e.g. [23] and below) and, by Powers–Størmer, it parametrizes the 705 Reviews in Mathematical Physics, Vol. 10, No. 5 (1998) 705–721 c World Scientific Publishing Company
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quasi-free states yielding irreducible representations which are unitarily equivalent to the GNS representation induced by ω+ . Upon realizing these representations in the same GNS space H+ , the complex lines therein corresponding to the different GNS vectors build up the fibres of the Determinant line bundle. This yields at ∼ ∼ (K, K+ )-equivariance (Ures (K, K+ ) being a the same time local triviality and Ures central U (1)-extension of Ures (K, K+ )): the module of “square integrable” holomorphic sections of Det∗ carries a projective unitary representation of Ures (K, K+ ), in a Borel–Weil–Bott fashion (see e.g. [5]). At the Lie algebra level one gets a non-trivial cocycle yielding the K¨ ahler structure of Gr(K, K+ ). The same portrait can be depicted for the Pfaffian line bundle. Here the relevant geometric object is the restricted isotropic Grassmannian Jres (H) (see Sec. 5) describing the Ores (HC , B)-orbit of the anti-Fock states unitarily equivalent to a fixed one (in the sense of their GNS representations): this is the geometrical version of the Shale–Stinespring theorem on the implementation of Bogolubov automorphisms induced by elements in the restricted rotation group Ores (HC , B). The lines generated by the GNS anti-Fock vectors yield the fibres of the Pfaffian line bundle. Again this ∼ (HC , B)-equivariance: the module of “square gives at once local triviality and Ores integrable” holomorphic sections of P f ∗ carries a projective unitary representation of Ores (HC , B). This is exactly the infinite dimensional Spin Representation (or, better, Spinc Representation) of Shale and Stinespring, again displayed a` la Borel– ∼ (HC , B) Weil. One has a central U (1)-extension of Ores (HC , B), denoted by Ores which, in contrast to the finite dimensional situation, cannot be reduced to Z2 . This interpretation of the Spin representation is close in spirit to E. Cartan’s approach to finite dimensional spinors (see e.g. [11], and also [7]). The two bundles and their section modules both appear as outcomes of the recipe of (holomorphic) geometric quantization applied to the K¨ ahler manifolds involved, in an infinite dimensional context (see [8] for a recent treatment of the subject). The basic relation Det|Jres (H) = P f ⊗2 is naturally understood in terms of a Fock–anti-Fock correspondence and a Powers–Størmer purification argument [18]. The latter also has a vivid geometric counterpart in terms of a (generalized) Segre embedding (cf. [12]) of a product of two “small” Grassmannians into a “big” one. In the final section we derive an explicit expression of Calabi’s diastasis function [10, 9, 22] on the Siegel manifold and point out the relation to coherent state quantization [15]. The paper is divided into several sections for the sake of readability. However, in order to keep it short, we have to refer the reader to the quoted literature for background and further motivation. We especially point out [19] for the geometric point of view, and [6, 14] for C ∗ -algebra theory. 2. CAR and Clifford Algebras We briefly review some basic facts concerning the CAR algebra and the theory of quasi-free states and Clifford algebras, just to fix notation. We refer to [3, 17, 18, 16, 21] for more details.
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Given a complex, separable Hilbert space K (called the one particle space), the CAR (Canonical Anticommutation Relations) algebra A(K) is the universal unital C ∗ -algebra generated by creation operators a∗ (f ) (depending linearly on f ∈ K) and their adjoints a(f ), annihilation operators (depending antilinearly on f ∈ K), fulfilling the following relations: a∗ (f )a(g) + a(g)a∗ (f ) = hf, gi · 1 (2.1) a(f )a(g) + a(g)a(f ) = 0 with f , g ∈ K, h , i denoting the scalar product in K, depending linearly on the first factor, and 1 being the identity element of A(K) (slightly different conventions are often used in the literature). The unitary group U (K), i.e. the natural symmetry group of K, acts on A(K) through C ∗ -automorphisms αU defined by αU (a(f )) := a(U f ) for U ∈ U (K) and f ∈ K. Similarly, let (H, g = (, )) be a real Hilbert space and HC = H ⊗R C its complexification, endowed with the C-linear extension B = g C of the metric and the canonical hermitian structure hu, vi = B(u, v), where h ⊗ λ = h ⊗ λ denotes the canonical conjugation of HC . The Clifford algebra is the real (universal) Banach algebra with unit 1 generated by operators γ(u), u ∈ H fulfilling the anticommutation relation (u, v ∈ H) γ(u)γ(v) + γ(v)γ(u) = g(u, v) · 1 (2.2) (the present convention adheres to quantum field theorists’ usage). The complex Clifford algebra C(HC , B) is then by definition C(HC , B) = C(H, g) ⊗R C .
(2.3)
It turns out to be a unital C ∗ -algebra isomorphic to A(W ), where W is any Bisotropic subspace of HC such that W = W ⊥ . Any such W gives rise to a complex structure J on H, that is J ∈ J (H), where J (H) = {J : H → H|J is g − isometric and J 2 = −1}
(2.4)
(the Siegel manifold, or isotropic Grassmannian). Indeed J yields W = Eig(J C , +i) (J C being the complexification of J) and conversely the operator J C = i · (EW − EW ⊥ )
(2.5)
preserves H and yields a J ≡ JW : J = i · (EW − EW ⊥ )|H
(2.6)
(here and in the sequel we denote the orthoprojector onto a subspace Y ⊂ HC by EY ). For finite dimensional Clifford algebras see [2, 19]. In analogy to the unitary group case, O in O(HC , B), the orthogonal group associated with B, acts on C(HC , B) by a C ∗ -automorphism group αO , extending αO (γ(v)) = γ(Ov). The same holds, of course, for A(W ). These particular automorphisms are called Bogolubov automorphisms (or transformations, see e.g. [1, 3, 6, 24]).
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Remark. If H is a complex Hilbert space, regarded as a real one, the relationship between A(H) and C(H) is displayed by the formula γ(f ) = 2− 2 (a(f ) + a∗ (f )) 1
(2.7)
(see e.g. [3]). The assignment f 7→ γ(f ) is called Majorana field, in quantum field theory. 3. Quasi-free States of the CAR Algebra and the Fock anti-Fock Correspondence We shall consider the state ωE associated to a hermitian projection operator E on K (that is, E = E ∗ = E 2 ). See for instance [17, 18], and also [16, 23, 24]. Such a state is defined by ωE (a∗ (f1 ) . . . a∗ (fn )a(g1 ) . . . a(gm )) = δn,m det(hfi , Egj i)
(3.1)
for fi , gj ∈ K and is already determined by its 2-point functions, the non-trivial ones reading: (3.2) ωE (a∗ (f )a(g)) = hf, Egi for f , g ∈ K. These states are called gauge-invariant since they are invariant under the U (1)-action given by the multiplication with unit complex numbers on K and quasi-free since they relate to free fermionic quantum fields. The GNS-construction now associates a representation πE of A(K) by bounded operators on a complex Hilbert space (HE , h , iHE ) and a cyclic unit vector ξE (unique up to a phase) such that (3.3) ωE (a) = hπE (a)ξE , ξE iHE for all a in A(K). Since the state ωE is pure, the GNS-theory tells us in addition that πE is an irreducible representation of A(K). The cases E = 0 and E = I yield the so-called Fock and anti-Fock representation, respectively. The GNS vector ξE is characterized (up to a phase) by the property: πE (a∗ (f ))ξE = 0, πE (a(g))ξE = 0,
∀ f ∈ EK ∀ g ∈ (I − E)K .
(3.4)
Let K 0 denote the dual of K; we have hf 0 , g 0 iK 0 = hg, f iK (f 7→ f 0 denotes the natural antilinear duality operator. We shall need a special instance of the Powers– Størmer “purification map” [18]: the state ωE on A(K) = A(K ⊕0) is the restriction of the state ωE⊕F 0 on A(K ⊕ K 0 ), where F = I − E, and F 0 is the corresponding adjoint operator on K 0 . The space (E ⊕ F 0 )(K ⊕ K 0 ) is the isotropic subspace in K ⊕ K 0 corresponding to EK (with the complex bilinear form induced by the scalar product in K ⊕ K 0 , given by evaluation). This has a natural counterpart in terms of Grassmannians, see below. For the sequel we need the following simple but crucial
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Proposition 3.1. (i) There exists a canonical complex linear C ∗ -algebra isomorphism χ : A(K) → A(K 0 ) induced by the map a(f ) 7→ a∗ (f 0 ). (ii) One has ωF 0 ◦ χ = ωE . (iii) χ induces a map (denoted by the same symbol) which intertwines the representations (πE , HE , ξE ) and (πF 0 , HF 0 , ξF 0 ) of A(K) and A(K 0 ), respectively. In particular, χ intertwines the corresponding Fock and anti-Fock representations. We shall call it the Fock–anti-Fock correspondence, or FaF, in short. We omit the easy proof. 4. The Hilbert Space Grassmannian and the Powers Størmer Theorem A special case of the Powers–Størmer theorem [18] states that two quasi-free representations of A(K) (πE , HE , ξE ) and (πF , HF , ξF ) (corresponding to hermitian projections E and F ) are unitarily equivalent if and only if E −F ∈ I2 (K) (Hilbert– Schmidt operators). A relatively easy and constructive proof thereof can be found in [24] (where it is referred to as the “Shale–Stinespring criterion”). We recall, in view of future use, that given two irreducible GNS representations (of the CAR algebra), they can be realized on the same space if and only if they are unitarily equivalent; this being the case, the corresponding states are of course vector states pertaining to (cyclic) vectors in the common representation space. The above problem is tightly related to the problem of unitary implementation (in a representation space) of a symmetry group of automorphisms, in our case the unitary group U (K). Specifically, let us consider a polarization K = K+ ⊕ K− , with both K± infinite dimensional. The typical example is provided by the Fourier decomposition polarization of L2 (S 1 , C). Let E± denote the orthogonal projections onto K± , respectively. Fix the GNS representation (πE+ , HE+ , ξE+ ), or, in short (π+ , H+ , ξ+ ). ˜ ∈ U (H+ ) such that Then the existence question of an implementation U ˜ ◦ π+ (a) ◦ U ˜ −1 π+ ◦ αU (a) = U
(4.1)
for a ∈ A(K), U ∈ U (K), is settled by the theorem of Powers and Størmer. Indeed, setting Y = U K+ , we have the following result (cf. also [23]). Proposition 4.1. The following are equivalent: (i) (ii) (iii) (iv)
U ∈ U (K) is implemented on H+ . πY is unitarily equivalent to π+ . EY − E+ ∈ I2 (K). [U, E+ − E− ] ∈ I2 (K).
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Remark. Writing an operator on the polarised Hilbert space K = K+ ⊕ K− as a (2 × 2)-matrix ac db with a a map from K+ to K+ etc., the restricted unitary
group Ures (K, K+ ) of K with respect to the polarization K+ ⊕K− is defined as {U ∈ U (K)|b, c ∈ I2 (K)}. Thus (iv) is tantamount to saying that U is in Ures (K, K+ ). Applying Schur’s lemma to the irreducible A(K)-representations π+ and π+ ◦αU (with U in Ures (K, K+ )), Proposition 4.1 provides us with a projective unitary representation: Ures (K, K+ ) → PU (H+ ) .
(4.2)
The corresponding non-trivial Lie algebra 2-cocycle on Lie Ures (K, K+ ) is also known as the “Schwinger term” because of its relation to anomalous commutation relations in 1 + 1-dimensional Quantum Electrodynamics. Now the orbit of the projector E+ under the conjugation action of Ures (K, K+ ) yields exactly the Hilbert space Grassmannian Gr(K, K+ ) (see e.g. [23], Proposiahler manifold: tion 2.1), which turns out to be a Ures (K, K+ )-homogeneous K¨ Gr(K, K+ ) ∼ = Ures (K, K+ )/(U (K+ ) × U (K− )) ,
(4.3)
ahsince the Schwinger term descends to it yielding a Ures (K, K+ )-invariant K¨ ler structure. In view of Proposition 4.1, Gr(K, K+ ) also coincides with the Ures (K, K+ )-orbit of ω+ , in the state space of A(K), Ures (K, K+ ) acting via α. See also below, Corollary 8.2. ucker Embedding `a la Kodaira into a The Grassmannian Gr(K, K+ ) allows a Pl¨ projective Hilbert space. Before giving ex novo a construction of the Determinant line bundle and this embedding in Sec. 8, we recall here the approach of Pressley– Segal–Wilson given e.g. in [19] and [20]. In order to be as brief as possible we shall concentrate on the connected component of the Grassmannian, i.e. the subspaces of K having virtual dimension equal to zero. A bounded linear map w : K+ 7→ K is called an admissible basis of W if w is an isomorphism between K+ and Im(w) = W and E+ ◦ w has a determinant, i.e. differs from the identity by a trace class operator. Any two such bases w and w0 of a fixed W fulfill w0 = w ◦ L, where L is an element of GL1 (K+ ), the group of invertible endomorphisms of K+ having a determinant. The Determinant bundle Det∼ of [19] is now given as the quotient by the following action of GL1 (K+ ) on the set {(λ, w)|λ ∈ C, w an admissible basis for a W ∈ Gr(K, K+ )}: L · (λ, w) = (det(L) · λ, w ◦ L−1 ) .
(4.4)
Let us fix now a Hilbert basis {z j |j ∈ Z} of K such that K+ respectively K− are the closures of the span of {z j |j ≥ 0} respectively {z j |j < 0}. Given an element of S = {S ⊂ Z|N \ S and S \ N are finite} (and assuming that they have equal cardinality since we concentrate here on the connected component of the Grassmannian) we can define a subspace KS = (({z j |j ∈ S})) and the projector ES = EKS onto it. (Observe that KN = K+ ).
DETERMINANTS, PFAFFIANS AND QUASI-FREE REPRESENTATIONS OF
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Furthermore we can define a map π ˆS : Det∼ → C by π ˆS ([λ, w]) = λ · det(DS,N ◦ ES ◦ w) ,
(4.5)
where DS,N is the canonical isomorphism from KS to K+ given by ordering S = ˆS is {z sj |j ∈ N} lexicographically (sj < sj+1 ) and setting DS,N (z sj ) = z j . Since π ∼ well-defined and fiberwise linear on Det , it corresponds to a holomorphic section of its dual. Recalling furthermore from [19 p. 110] that X |ˆ πS ([1, w])|2 = det(w∗ w) < ∞ , (4.6) 0< S∈S
one concludes that ˆ ([λ, w]) = π ˆ : Det∼ → `2 (S)\{0}, π
X
π ˆS ([λ, w])eS ,
(4.7)
S∈S
where eS is the indicator function of the the point S in S, is well defined. The map π ˆ descends to an application π = P l∼ : Gr(K, K+ ) → P(`2 (S)) ,
(4.8)
the Pl¨ ucker embedding a` la Pressley–Segal–Wilson. ˆ ([1, (DS,N )−1 ]) one arrives at Since eS = π π ([1, w]), π ˆ ([1, (DS,N )−1 ])i`2 (S) , π ˆS ([1, w]) = hˆ
(4.9)
which we will also refer to as πS (W ) = hW, KS i`2 (S) = hW, KS i
(4.10)
in the sequel, keeping in mind that these are projective Pl¨ ucker coordinates (cf. [23, Theorem 3.1]). 5. The Isotropic Grassmannian and the Shale Stinespring Theorem The automorphism group implementation problem can be posed for the rotation group in the Clifford algebra context as well. With the notations of Sec. 2, we consider the anti-Fock state ωI on A(W ) and its corresponding GNS-representation (πI , HI , ξI ). Each U in U (W ) obviously preserves the projector I, so by Powers– Størmer it is implemented, but in fact we have a larger symmetry group O(HC , B) coming from the Clifford point of view. Since these automorphisms αO (for O in O(HC , B)) of A(W ) are not induced from operators on W in general, we need the Shale–Stinespring theorem to decide when O is implemented, which we formulate in the following guise: ˜ ∈ U (HI ) such that πI ◦ Proposition 5.1. For O in O(HC , B) there exists O −1 ˜ ˜ for all a ∈ A(W ) if and only if O ∈ Ores (HC , B) := αO (a) = O ◦ πI (a) ◦ O O(HC , B) ∩ Ures (HC , H+ ).
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The original source being [21], its proof can also be found in the recent textbook [16]. Again we get a projective unitary representation Ores (HC , B) → PU (HI ) having a non-trivial Lie algebra cocycle, yielding the K¨ ahler structure of the restricted isotropic Grassmannian Jres (H) (see below). Now, in close analogy to the previous case, the Ores (HC , B)-orbit of the projector onto W in HC yields by definition the (restricted) isotropic Grassmannian, or restricted Siegel manifold Jres (H). Equivalently, in view of formula (2.6), it is easy to see that Jres (H) describes all isometric complex structures on H differing from JW by a Hilbert–Schmidt operator. We have, specifically Jres (H) ∼ = Ores (HC , B)/U (W ) ∼ = {J ∈ J (H)|J − JW ∈ I2 (H)} .
(5.1)
Using the discussion in Sec. 2, it follows easily that there is a Ores (HC , B)-equivariant embedding (5.2) i : Jres (H) ,→ Gr(HC , W ) sending J in Jres (H) to Y = Eig(J C , i) and having as its diffeomorphic image {Y ∈ Gr(HC , W )|Y is B-isotropic and Y ⊥ = Y }. Notice that, in view of Proposition 5.1, Jres (H) also parametrizes all anti-Fock states ω := ωI ◦ αO yielding GNS representations of the CAR algebra A(W ) unitarily equivalent to the reference one πI (see also Corollary 9.3). Working in the reference GNS space HI , the complex lines generated by their GNS vectors will furnish the fibres of the Pfaffian line bundle (see Sec. 9). Remark. We recall, for future use, how a Bogolubov automorphism αO associated to O in Ores (HC , B) acts on A(W ): an element u of HC considered as γ(u) in C(HC , B) ∼ = A(W ) is mapped, up to a factor, to a∗ (EW Ou) + a(EW Ou) (cf. [3]). 6. Geometry of the Fock anti-Fock Correspondence We begin by pointing out the following proposition, which, though not explicitly stated in the literature, to our knowledge, is widely used and easily proved: it can be extracted, for instance, from the discussion of the Shale–Stinespring criterion given in [24]. It will be relevant for the subsequent developments. Let us denote ˆ the various topological tensor products appearing henceforth by ⊗. Proposition 6.1. With the above notation ˆ 0 (A(W )) . πEW (A(HC )) ∼ = πIW (A(W ))⊗π W
(6.1)
Remark. Observe that, in order to enforce the correct anticommutation relations, one has to twist operators in one of the two spaces involved in the r.h.s. of Eq. (6.1). More precisely, let V be a unitary of HIW with V 2 = I implementing the C ∗ -automorphism induced by the map f 7→ −f on W . This operator exists and is unique up to a sign by the irreducibility of the representation πIW of A(W ).
DETERMINANTS, PFAFFIANS AND QUASI-FREE REPRESENTATIONS OF
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Moreover V can be uniquely fixed by requiring V ξI = ξI . Let α respectively β be monomials in the creation and annihilation operators in A(W ) respectively A(W ). Their action is now given by πIW (α) ⊗ 1 and V |β| ⊗ π0W (β), respectively. Here |β| denotes the degree of β (defined in a natural way). See e.g. [17]. The Fock–anti-Fock correspondence has a geometrical counterpart which will also be relevant in our discussion of the Segre embedding. Let us take a complex polarized Hilbert space H = H+ ⊕ H− , which we regard as a real one. Let us set H − := H− and, correspondingly, denote its projector by E − . Upon complexification, any subspace W1 ∈ Gr(H, H+ ) goes to an isotropic subspace Y = W1 ⊕ W1⊥ ∈ Jres (H) ⊂ Gr(HC , W ), where W = H+ ⊕ H − : this is a geometric version of Powers–Størmer purification. The above map is equivariant with respect to the natural embedding Ures (H, H+ ) ,→ Ores (HC , B). Let us notice that, for Y ∈ Jres (H), A(Y ) and A(Y ) are canonically isomorphic by the Fock– anti-Fock correspondence, since Y can be identified C-linearly, via B, with Y 0 . Let HE+ respectively HE − denote the GNS-spaces carrying the GNS representations of A(H) respectively A(H) induced by ωE+ respectively ωE − . Let S0 denote the set of all S 0 = Z − S with S in S. We have the following theorem, whose proof is straightforward; in particular, (i) is a reformulation of (part of) Proposition 6.1. Notations are those of [19] and [23]. Theorem 6.2. ˆ 2 (S0 ) ∼ (i) One has HE+ ∼ = = `2 (S), HE − ∼ = `2 (S0 ), and HE+ ⊕E − ∼ = `2 (S)⊗` ˆ E . HE+ ⊗H − (ii) The Fock–anti-Fock correspondence reads as follows, in terms of the natural orthonormal bases in HE+ and HE − (with a slight abuse of notation) χ : HS 7→ HS⊥ = H S 0 .
(6.2)
(iii) It also gives rise to a K¨ ahler isometry Gr(H, H+ ) ∼ = Gr(H, H − )
(6.3)
(which allows identification of the two manifolds). ucker coor(iv) The embedding Gr(H, H+ ) ,→ Gr(HC , W ) reads, in terms of Pl¨ dinates: π(S,T 0 ) (Y ) = hW1 ⊕ W1⊥ , HS ⊕ HT⊥0 i = hW1 , HS ihW1 , HT i = πS (W1 )πT (W1 ) .
(6.4)
π(S,S 0 ) (W1 ⊕ W1⊥ ) = πS (W )2 .
(6.5)
In particular
Let us notice, for future use, the following consequence of FaF and Proposition 6.1., again obtained by using a “twisting” unitary map as in the remark after the latter:
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M. SPERA and T. WURZBACHER
Proposition 6.3. One has ˆ IW (A(W )) . πEW (A(HC )) ∼ = πIW (A(W ))⊗π
(6.6)
7. An Equivariance Lemma In order to lay down our C ∗ -algebraic construction of the Determinant and Pfaffian line bundles we need the following general Lemma 7.1. Let G be a Banach Lie group and ρ : G → P GL(V ) be a projective representation of G by bounded, linear, invertible operators on a complex topological vector space V. Let G/H be a G-homogeneous complex manifold. Furthermore let F ⊂ G/H × V be a closed analytic subset such that {gH} × FgH = (pr1 |F )−1 (gH) ⊂ G/H × V gives a complex vector subspace FgH of V. Assume further that at the projective level we have, for [v] ∈ P(Fg0 H ) ⊂ P(V ), g(g 0 H, [v]) = (gg 0 H, [ρ(g)v]) ,
(7.1)
and this is in P(Fgg0 H ). Then F is biholomorphic to a holomorphic vector bundle on G/H with typical fibre FeH . Proof. We have the following commutative diagram:
π
ind
G ↑ ˜ G
ρ →
P GL(V ) ↑ π
(7.2)
GL(V )
→ ρ ˜
˜ = {(g, A) ∈ G × GL(V )|ρ(g) = π(A)} denotes the induced central C∗ where G ˜ acts on G/H with isotropy H ˜ = (π ind )−1 (H) and on V . It extension of G. G follows easily that the map ˜ ˜ FeH G× H ↓ ˜ ˜ G/H
→ ∼ =
∼ =
F ↓ G/H
(7.3)
defined by ˜ ˜ −1 , ρ˜(h)v] ˜ ρ˜(˜ 7→ (˜ g H, g )v) [˜ g , v] = [˜ g(h) yields the desired isomorphism.
(7.4)
Let L → X be a holomorphic line bundle over a complex manifold X, and let L∗ be its dual bundle. The space of holomorphic sections of L∗ will be denoted by ΓO (X, L∗ ). One has ΓO (X, L∗ ) ∼ = {f : L → C | f is holomorphic and linear in each fibre } .
(7.5)
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In our infinite dimensional setting extra care is needed, but the calculations involved characterize the “square integrable” holomorphic sections (cf. [19]). In our case all fibres will be complex lines in a fixed Hilbert space, and we shall confine ourselves to those sections which can be induced by a continuous linear functional on the Hilbert space, and hence, via the Riesz theorem, by a vector in it. Explicitly, we have the following maps, inverse to each other: f 7→ σf , with σf (x) := f |Lx ∈ Hom (Lx , C)
(7.6)
and σ 7→ fσ , with fσ (lx ) = (σ(x))(lx ) for
lx ∈ L x .
(7.7)
(We denote here and below the fibre of L over x by Lx .) This yields — in absence of base points, where all sections vanish — a commutative diagram (setting V = ΓO (X, L∗ )) L − {0} → V ∗ − {0} (7.8) ↓ ↓ X → P(V ∗ ) by lx 7→ (f 7→ f (lx )), and x 7→ (f 7→ f (lx )) for lx ∈ Lx − {0} or σ 7→ σ(x). This works for any “sufficiently big” submodule W of V as well: X → P(W ∗ ) ([12]). 8. The Determinant Line Bundle In this section we shall give a C ∗ -algebraic construction of the Determinant line bundle, equivalent to Pressley–Segal–Wilson’s one, as we shall see. Notations are those of Secs. 2 and 3. Let us fix the GNS-representation (π+ , H+ , ξ+ ). We drop the suffix + in order to ease notation. Any other GNS-representation corresponding to Y ∈ Gr(K, K+ ) is unitarily equivalent to it and can thus be realized on H. The GNS state becomes a vector state and gives rise to a complex line in H, which will be denoted by DetY . In view of Ures (K, K+ )-equivariance, one has, if Y = U K+ , U ∈ Ures (K, K+ ): ˜ DetK+ DetY = DetUK+ = U
(8.1)
˜ denotes its implementing unitary up to a phase in H; one gets a natural where U ∼ (K, K+ ). In view of the central extension of Ures (K, K+ ) by U (1) denoted by Ures equivariance lemma, this gives rise to a holomorphic hermitian line bundle Det → Gr(K, K+ ). Theorem 8.1. ∼ (K, K+ ) (i) Det → Gr(K, K+ ) is a holomorphic hermitian line bundle and Ures acts equivariantly on the total space, and specifically we have DetUH+ = ˜ in U ∼ (K, K+ ) lies over U in Ures (K, K+ ). ˜ + , where U C · Uξ res ∗ (ii) H is canonically realized inside ΓO (Gr(K, K+ ), Det∗ ). (iii) The bundle Det is isomorphic to the Det-bundle of [19].
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M. SPERA and T. WURZBACHER
Proof. (i) One has, in view of the characterization of the quasi-free GNS representations given by (3.4) Det = {(Y, ξ) ∈ Gr(K)×H|a∗ (y)ξ = 0 ∀y ∈ Y and a(y ⊥ )ξ = 0 ∀y ⊥ ∈ Y ⊥ } . (8.2) Operators are meant to act in the representation π+ . The assertion follows from the equivariance lemma. (ii) Since Det ⊂ Gr(K) × H and DetY ⊂ H, each vector in H defines a function f u : Det → C by f u ((Y, v)) := hv, uiH (v ∈ DetY ⊂ H), namely, a holomorphic section of the dual Det∗ . By virtue of the discussion in Sec. 7, we shall restrict to sections of the above form. This yields a C-linear map H∗ → ΓO (X, Det∗ ) ucker map and therefore H = (H∗ )∗ gives, as the dual of a section module, a Pl¨ P l : Gr(K) → P(H). (iii) Let again Det∼ denote the Pressley–Segal–Wilson Determinant line bundle ˜ with H ˜ = l2 (S) (compare Sec. 4). In view ucker embedding into P(H), and P l∼ its Pl¨ of Theorem 3.1 of [23], we have two unitarily equivalent irreducible representations ˜ thereby yielding a unique projective unitary θ˜ : P(H) → of A(K) on H and H, ˜ P(H). A lengthy but elementary calculation using the Hilbert basis reviewed in Sec. 4, an explicit form of the isomorphism between the two representation spaces and (4.10) proves for any Y ∈ Gr(K) θ˜ ◦ P l(Y ) = P l∼ (Y )
(8.3)
i.e. the commutative diagram Gr(K) P l∼ &
Pl →
P(H) . θ˜
(8.4)
˜ P(H) Let O(−1)V denote the tautological line bundle over P(V ), for an arbitrary C-vector space. Since both maps are Ures (K, K+ )-equivariant embeddings, we have: ˜ ∗ (O(−1) ˜ ) = P l∗ (O(−1)H ) ∼ Det∼ ∼ = Det . = (P l∼ )∗ (O(−1)H˜ ) = P l∗ (θ) H
(8.5)
We also observe that Det∼ and Det are equivalent not only as holomorphic line ∼ (K, K+ )-bundles with hermitian metric (induced by the bundles, but also as Ures embeddings). Remark. In quantum field theoretic terms DetY corresponds to the Dirac vacuum filled with the antiparticles relative to Y (“Dirac sea”). The preceding theorem admits the following partial reformulation, which we state as a corollary (see also Sec. 4). Corollary 8.2. The Hilbert space Grassmannian Gr(K, K+ ) is diffeomorphic, via the Pl¨ ucker map and the GNS construction, to the (projective) Ures (K, K+ )-orbit of the GNS-vector ξ+ associated to the reference quasi-free state ω+ in H+ , the latter group acting via the projectively unitarily implemented automorphism group α.
DETERMINANTS, PFAFFIANS AND QUASI-FREE REPRESENTATIONS OF
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9. The Pfaffian Line Bundle and the Spin Representation We start by giving the following: A(W )
of the CAR Definition 9.1. Let S be the GNS representation space HIW ∼ (HC , B) the central U (1)-extension of Ores (HC , B) implealgebra A(W ) and Ores menting the Bogolubov transformations αO as unitary endomorphisms of S. This ∼ (HC , B) on S is called the Spinc -representation of Ores (HC , B). linear action of Ores We recall that in contrast with the finite dimensional case, one cannot render it two-valued only [19, 4]. We shall realize this representation in a Borel–Weil fashion by resorting to the Pfaffian line bundle. Referring to Secs. 2 and 5, taking Lemma 7.1 into due account and in view of the remark on Bogolubov transformations at the end of Sec. 5, the Pfaffian line bundle in infinite dimensions is given by: A(W )
P f = {(Y, ξ) ∈ Jres (H) × HI + a(yW ⊥ ))ξ = 0
|(a∗ (yW )
∀y = yW + yW ⊥ ∈ Y } ,
(9.1)
where yW and yW ⊥ are the projections onto W and W ⊥ respectively, πI is the GNSrepresentation of A(W ) associated to the anti-Fock state ωIW and the operators act in the anti-Fock representation πI := πIW . More precisely, we have the following: Theorem 9.2. (i) P f → Jres (H) is a holomorphic hermitian line bundle. ∼ (HC , B) acts equivariantly on P f . (ii) Ores A(W ) ∗ ∗ ) is canonically realized inside the holomorphic section mod(iii) S = (HI ∗ ule of P f , i.e. we have a Borel–Weil description of the infinite dimensional spinor module. (iv) Using the embedding i : Jres (H) ,→ Gr(HC , W ) given in (5.2), we have i∗ Det = P f ⊗2 . Proof. It is similar to the determinant case, in view of the equivariance lemma and the Shale–Stinespring theorem, and by recalling that a Bogolubov automorphism mixes creation and annihilation operators. See the remarks at the end of Sec. 5. We only need to explain (iii) and (iv) in detail. As for (iii), it is enough to notice that the submodule S ∗ ⊂ ΓO (Jres (H), P f ∗ ) is base-point free and thus yields ∼ (HC , B)-equivariant embedding of Jres (H) into P(S). Let us a holomorphic, Ores A(W ) prove (iv). Since P fW = CξIW ⊂ HIW , we get, by virtue of Propositions 6.1 and 6.3 A(W ) ˆ IA(W ) P fW ⊗ P fW ∼ = CξIW ⊗ CξIW ⊂ HIW ⊗H W A(W ) ∼ ˆ 0A(W ) = CξIW ⊗ Cξ0W ⊂ HIW ⊗H W
(9.2)
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M. SPERA and T. WURZBACHER
and this entails, at the level of GNS vectors A(H ) P fW ⊗ P fW ∼ = CξIW ⊕0W = CξEW ⊂ HEW C
(9.3)
which is the line DetW . The conclusion now follows from Ores (HC , B)-equivariance. Remarks. (i) The Pfaffian line bundle is uniquely defined (see [4]). (ii) In quantum field theoretic terms, the passage from Det to Pf amounts to considering a Majorana theory (a particle is identified with its antiparticle), instead of a Dirac one (where the two are regarded as distinct). Let Pf denote the projective embedding of Jres (H) into P(S) by means of the vector space of sections S ∗ of the line bundle P f ∗ . We have, in close analogy to Corollary 8.2, the following Corollary 9.3. The restricted isotropic Grassmannian Jres (H) is diffeomorphic, via the map Pf, to the (projective) Ores (HC , B)-orbit in P(S) of the anti-Fock state vector ξIW . 10. The Segre Embedding Upon referring to Sec. 6, we are in a position to state the following Theorem 10.1. (i) Upon restriction to the “small” Grassmannian Gr(H) = Gr(H, H+ ), P f |Gr(H) = Det .
(10.1)
(ii) There exists a natural Segre embedding S : Gr(H) × Gr(H) → Gr(HC ) = Gr(HC , W ) defined by S((W1 , W2 )) = W1 ⊕ W2⊥
(10.2)
(projective embeddings understood). The embedding is realized a ` la Kodaira via the bundle Det∗Gr(H) Det∗Gr(H) (“box” product). Proof. (i) stems from an application of purification and FaF. In view of equivA(H) ariance we may set W1 = H+ . The Determinant line is DetH+ = CξE+ ⊂ HE+ , A(W )
whereas the Pfaffian line is P fW =H+ ⊕H − = CξIW ⊂ HIW A(W )
HIW
. Hence
A(H ) A(H ⊕H ) A(H) ∼ ˆ IA(H − ) ∼ = HIH+ + ⊗H = HE+ + − = HE+
(10.3)
H−
identifying P fW with DetH+ . As for (ii), one has, with notation as in Sec. 6: π(S,T 0 ) (W1 ⊕ W2⊥ ) = h W1 ⊕ W2⊥ , HS ⊕ HT⊥0 i = h W1 , HS ih W2 , HT i = πS (W1 )πT (W2 ) .
(10.4)
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11. Concluding Remarks and Outlook (i) Comparison with Borthwick’s work Borthwick [4] has given explicitly the transition functions of Det. They can also be obtained by recalling that Det is the pull-back to Gr(H) of the tautological ucker embedding and Gr(H) is described by the line bundle O(−1)H under the Pl¨ Pl¨ ucker coordinates (4.10). The Pfaffian is given in [4] by taking the square root of the transition functions, after restriction to the Siegel manifold, by using the well-known fact that the determinant of a finite dimensional antisymmetric matrix, as a function of its entries, is the square of the Pfaffian polynomial (see e.g. [19]). (ii) Finite dimensional case recovered Our C ∗ -algebraic construction can be perfectly adapted to the finite (even) dimensional case as well. The infinite dimensional case we discussed corresponds, roughly speaking, to a limiting case of Grassmannians Gr(2k, k), for k → ∞. (iii) Central extension ∼ (HC , B) In our treatment, an explicit construction of the central extension Ores is lacking; however, at the Lie algebra level, it is induced by the K¨ahler form on Jres (H), which equals one-half of the pull-back of the K¨ahler form on Gr(HC ). The fairly detailed realization of the group extension in [4], based on [19], employs the Jaffe–Lesniewski–Weitsmann theory of the relative Pfaffian [13]. (iv) Naturality of the Hilbert–Schmidt ideal We stress the fact that the C ∗ -algebraic approach to the Pfaffian and Determinant bundles, uncovering the geometric nature of the Powers–Størmer and Shale– Stinespring theories, shows the “naturality” of the Hilbert–Schmidt operators with respect to more general Schatten ideals (cf. [4]). (v) Calabi Diastasis function Formula (6.6) can also be interpreted in terms of diastatic formulae [22, 23]. In particular, it appears as a consequence of the naturality of the Calabi diastasis A(H ) function under pull-backs [10, 9]. Furthermore, denoting HEW C by V , we have a Segre map τ : P(S) × P(S) → P(V ) (11.1) fulfilling τ ◦ (Pf × Pf) = P l .
(11.2)
Det
induced by P l on Gr(HC ) equals twice the Thus the induced K¨ahler structure ω structure ω P f induced by the map Pf on Jres (H). We recall the formula for the diastasis function DDet given in [22]. For Y1 , Y2 in a single coordinate chart (given by graphs) of Gr(HC ) one has DDet (Y1 , Y2 ) = 2kE1 − E2 k2I2 (H
C)
,
(11.3)
where Ej is the projector onto Yj , j = 1, 2. By naturality, and in view of (2.5) and (2.6) we find, if Yj ∈ Jres (H), for j = 1, 2 DP f (Y1 , Y2 ) =
1 Det 1 D (Y1 , Y2 ) = kJY1 − JY2 k2I2 (H) 2 4
(11.4)
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(vi) Coherent state quantization The formulae relating to scalar products in Sec. 12.3 of [19] and its generalizations to infinite dimensions in [4] should be easily interpreted in terms of coherent states, in the spirit of [22]. Thus, in accordance with the general coherent state philosophy (see e.g. [9, 15]), a geometric quantization procedure without using a measure is possible. (vii) The square map Formula (6.4) corresponds, in finite dimensions, to the square map of Pressley and Segal (see p. 243 in [19]); therefore the latter map gives only part of the coordinates (this can be ascertained within their formalism as well). (viii) Outlook Our results provide a preliminary step in understanding spinors on loop spaces of Riemannian manifolds. The C ∗ -algebraic approach we propose is functorial and intrinsically related to the underlying geometry, and, in many respects, generalizes and clarifies earlier treatments. In principle, it could be extended to explore differential geometric aspects of other C ∗ -algebras. Acknowledgements The first-named author wishes to thank the I.R.M.A. at Universit´e Louis Pasteur, Strasbourg, where our collaboration began, for the excellent hospitality and scientific atmosphere. We are grateful to P. Gosselin, R. L´eandre and M. Slupinski for useful discussions. We also thank the referee for his/her critical remarks. References [1] H. Araki, “On quasi-free states of the CAR and Bogolubov automorphisms”, Publ. Res. Inst. Math. Sci. Kyoto University, Ser. A6 (1970) 385–442. [2] M. F . Atiyah, R. Bott and A. Shapiro, Clifford modules Topology 3, Suppl. 1 (1964) 3–38. [3] E. Balslev, J. Manaceau and A. Verbeure, “Representations of anticommutation relations and Bogolubov transformations”, Commun. Math. Phys. 8 (1968) 315–326. [4] D. Borthwick, “The Pfaffian line bundle”, Commun. Math. Phys. 149 (1992) 463–493. [5] R. Bott, “Homogeneous vector bundles”, Ann. Math. 57 (1957) 203–248. [6] O. Bratteli and D. W. Robinson, Operator algebras and Quantum Statistical Mechanics I, II, Springer Verlag, New York, 1979 and 1981. [7] R. Brauer and H. Weyl, “Spinors in n-dimensions”, Amer. J. Math. 57 (1935) 425– 449. [8] J. L. Brylinski, Loop Spaces, Geometric Quantization, and Characteristic Classes, Birkh¨ auser, Basel, 1993. [9] M. Cahen, S. Gutt and J. Rawnsley, “Quantization of K¨ ahler manifolds I, II”, J. Geom. Phys. 7 (1990) 45–62 and Trans. Amer. Math. Soc. 337 (1993) 73–98. [10] E. Calabi, “Isometric imbeddings of complex manifolds”, Ann. Math. 58 (1953) 1–23. [11] M. Dubois-Violette, Complex structures and the Elie Cartan approach to the theory of spinors, in “Spinors, Twistors, Clifford Algebras and Quantum Deformations”, eds. Oziewicz et al., Kluwer Acad. Press, (1993) 17–23. [12] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.
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[13] A. Jaffe, A. Lesniewski and J. Weitsman, “Pfaffians on Hilbert space”, J. Funct. Anal. 83 (1989) 348–363. [14] G. J. Murray, C ∗ -algebras and operator theory, Academic Press, Boston, 1990. [15] A. M. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin, Heidelberg, New York, 1986. [16] R. J. Plymen and P. L. Robinson, Spinors in Hilbert space, Cambridge Univ. Press, 1994. [17] R. T. Powers, Representations of the Canonical Anticommutation Relations, Thesis, Princeton, 1967. [18] R. T. Powers and E. Størmer, “Free states of the canonical anticommutation relation”, Commun. Math. Phys. 16 (1970) 1–33. [19] A. Pressley and G. Segal, Loop Groups, Oxford Univ. Press, Oxford, 1986. [20] G. Segal and G. Wilson, “Loop groups and equations of KdV type”, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985) 5–65. [21] D. Shale and W. F. Stinespring, “Spinor representations of infinite orthogonal groups”, J. Math. Mech. 14 (1965) 315–322. [22] M. Spera and G. Valli, “Remarks on Calabi’s diastasis function and coherent states”, Quart. J. Math. Oxford 44 (1993) 497–512. [23] M. Spera and G. Valli, “Pl¨ ucker embedding of the Hilbert space Grassmannian and the CAR algebra”, Russian J. Math. Phys. 2 (1994) 383–392. [24] B. Thaller, The Dirac Equation, TMP, Springer Verlag, Berlin, Heidelberg, New York, 1984.
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES SERGIO ALBEVERIO and SHAO-MING FEI Institute of Mathematics Ruhr-University Bochum D-44780 Bochum, Germany Received 21 July 1997 PACS numbers: 02.50.-r, 64.60.Cn, 05.20.-y A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with An symmetry and the related Temperley– Lieb algebraic structures and representations are discussed. It is shown that corresponding to these An symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains with transition matrices (resp. intensity matrices) having spectra which coincide with the ones of the corresponding integrable models.
1. Introduction Integrable chain models have played significant roles in statistical and condensed matter physics. Some of them have been obtained and investigated using an algebraic “Bethe Ansatz method” [1], see e.g. [2] for periodic boundary conditions and [3] for fixed boundary conditions. The intrinsic symmetry of these integrable chain models plays an essential role in finding complete sets of eigenstates of the systems. On the other hand, stochastic models like stochastic reaction-diffusion models, models describing coagulation/decoagulation, birth/death processes, pair-creation and pair-annihilation of molecules on a chain, have attracted considerable interest due to their importance in many physical, chemical and biological processes [4]. For example, the simplest models for diffusion-reaction processes describe particles stochastically hopping on a lattice [5]. These models have been connected to spin1/2 Heisenberg quantum spin chain [6], and then further developed and generalized to SU (2) symmetric spin-s chains [7]. It was shown in [8] that the Uq SU (p/m) invariant models [9] also naturally appear as time-evolution operators of chemical systems and the Uq SU (3/0), Uq SU (1/2) and Uq SU (2/1) symmetric chains were discussed from a similar point of view in [10]. The theoretical description of stochastic reaction-diffusion systems is given by the “master equation” which describes the time evolution of the probability ∗ Alexander
von Humboldt-Stiftung fellow. On leave from Institute of Physics, Chinese Academy of Sciences, Beijing 723
Reviews in Mathematical Physics, Vol. 10, No. 6 (1998) 723–750 c World Scientific Publishing Company
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distribution function [8, 11]. This equation has the form of a heat equation with potential (i.e. a Schr¨ odinger equation with “imaginary time”). For some reactiondiffusion processes the “Hamiltonians” in the “master equation” coincide with the generators of the Hecke algebra [12]. If an integrable system with open boundary condition can be transformed into a stochastic reaction-diffusion system, e.g., by a unitary transformation between their respective Hamiltonians, looked upon as self-adjoint operators acting in the respective Hilbert spaces, then the stochastic model so obtained is exactly soluble with the same energy spectrum as the one of the integrable system [8, 12, 13]. In this paper, by using the coproduct properties of bi-algebras, we present a general procedure for the construction of open chain models having a certain Lie algebra or quantum Lie algebra symmetry with nearest or/and non-nearest neighbours interactions. The models obtained in this way can be reduced to integrable ones via a detailed representation of the symmetry algebras involved. As an example we discuss integrable models with An symmetry. These models turn out to have an additional Temperley–Lieb (TL) algebraic structure, in the sense that the Hamiltonians give rise to unitary representations of the TL algebra and can be expressed by the elements of the TL algebra. Above all, we find that all these models can be transformed into both stationary discrete-time (discrete reaction-diffusion models) and stationary continuous-time Markov chains with transition matrices resp. intensity matrices having the same spectra as the ones of these An invariant integrable models. The stationary discrete-time An symmetric Markov chain model obtained in our paper describes a chain with L + 1 sites and n + 1 possible states, say, τi1 , . . . , τin+1 , at site i, i = 1, . . . , L+1. The allowed stochastic processes for any nearest neighbour β α ) → (τiβ , τi+1 ), α, β ∈ {1, . . . , n + pair are the interchanges of their states: (τiα , τi+1 1}. For n = 1, there are two possible states at every site: empty or occupied by one particle. The A1 Markov chain describes then stochastic hopping of particles into left and right vacancies, which is the well-known SU (2) random chain. For n = 2, i.e. the SU (3) case, there are three possible states at every site, say, empty, one spin up and one spin down particle states. This model describes stochastic hopping of spin up and down particles into left and right vacancies, interchanging of one spin up and one spin down particle at the nearest neighbours. In Sec. 2 we first recall the basic properties of bi-algebras and then describe their use in the construction of open chain models with a Lie algebra symmetry resp. a quantum Lie algebra symmetry. In Sec. 3 we discuss the An symmetric integrable models in the fundamental representation of An . We give the representation of a TL algebra in these models. In Sec. 4 we prove that these An symmetric integrable chain models can be transformed into both continuous-time and discrete-time Markov chains. Some conclusions and remarks are given in Sec. 5. 2. Chain Models with Algebraic Symmetry 2.1. Bi-algebra Let A be an associative algebra. A is said to be a bi-algebra if it contains two linear operators, the multiplication m and the coproduct ∆.
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
725
The operation of multiplication m is defined by m : A ⊗ A → A,
m(a ⊗ b) = ab ,
∀ a, b ∈ A .
Let {ei } be the set of base elements of the algebra A. Then (1) means X mkij ek . m(ei ⊗ ej ) =
(1)
(2)
k
The tensor mkij determines the properties of m completely. The multiplication m is associative, m(m ⊗ id) = m(id ⊗ m) , i.e.
X
mkij mnkl =
k
X
(3)
mnik mkjl ,
(4)
k
where id denotes the identity transformation, id : A → A ,
id(a) = a .
However in general m is not commutative, i.e. we have m ◦ p 6= m ,
(5)
or equivalently, mkij 6= mkji , where p is the transposition operator, p : A ⊗ A → A⊗ A,
p(a ⊗ b) = (b ⊗ a) ,
∀ a, b ∈ A .
The coproduct operator ∆ maps A into A ⊗ A: X jk ∆ : A → A ⊗ A , ∆(ei ) = µi ej ⊗ ek .
(6)
(7)
jk
The properties of the coproduct are described by the tensor µjk i . ∆ is an algebraic homomorphism, X X pq k µrs mrkp mstq µkt ∆(ab) = ∆(a)∆(b), ∀ a, b ∈ A , k mij = i µj , k
(8)
ktpq
where ∆(a) and ∆(b) belong to A ⊗ A and the multiplication of tensors is defined by (a1 ⊗ a2 )(b1 ⊗ b2 ) = a1 b1 ⊗ a2 b2 , a1 , a2 , b1 , b2 ∈ A . The coproduct is associative (∆ ⊗ id)∆ = (id ⊗ ∆)∆ ,
X
tp µrs t µi =
t
X
rt µsp t µi ,
(9)
t
but in general not co-commutative p ◦ ∆ 6= ∆ ,
sr µrs j 6= µj .
(10)
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The operation ∆ preserves all the algebraic relations of the algebra A. It gives a way to find representations of the algebra A in the direct product of spaces. If a bi-algebra has in addition unit, counit and antipode operators, then it is called a Hopf algebra. Lie algebras are Hopf algebras with ∆ co-commutative. Quantum algebras are Hopf algebras that are not co-commutative, see e.g. [14] and references therein. 2.2. Chain models with Lie algebraic symmetry Let A be a Lie algebra with basis e = {eα }, α = 1, 2, . . . , n, satisfying the Lie commutation relations γ eγ , (11) [eα , eβ ] = Cαβ γ where Cαβ are the structure constants with respect to the base e. Let ∆ (resp. C(e)) be the coproduct operator (resp. Casimir operator) of the algebra A. We have (12) [C(e), eα ] = 0 , α = 1, 2, . . . , n .
The coproduct operator action on the Lie algebra elements is given by ∆eα = eα ⊗ 1 + 1 ⊗ eα ,
(13)
1 stands for the identity operator. It is easy to check that γ ∆eγ . [∆eα , ∆eβ ] = Cαβ
From the properties of the coproduct, ∆C(e) is a rank two tensor satisfying [∆C(e), ∆eα ] = 0 ,
α = 1, 2, . . . , n .
(14)
Let F denote an entire function defined on the (L + 1)th tensor space A ⊗ A ⊗ · · · ⊗ A of the algebra A. From (14) we have [F(∆C(e)), ∆eα ] = 0 ,
α = 1, 2, . . . , n .
(15)
We consider “a chain with L + 1 sites”, i.e. the set {1, 2, . . . , L + 1}. We call it an algebraic chain in the following. To each point i of the chain we associate a (finite dimensional complex) Hilbert space Hi . We can then associate to the whole chain the tensor product H1 ⊗ H2 ⊗ · · · ⊗ HL+1 . For simplicity we use subindices i, j, k, . . . for the points in the chain sites. The generators of the algebra A acting on the Hilbert space H1 ⊗H2 ⊗· · ·⊗HL+1 associated with the above chain are given by Eα = ∆L eα ,
α = 1, 2, . . . , n ,
(16)
where we have defined ∆m = (id ⊗ · · · ⊗ id ⊗∆) . . . (id ⊗ id ⊗ ∆)(id ⊗ ∆)∆ , {z } | m times
∀ m ∈ N.
(17)
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SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
Eα also generates the Lie algebra A, γ Eγ . [Eα , Eβ ] = Cαβ
We call H=
L X
F(∆C(e))i,i+1
(18)
i=1
the (quantum mechanics) Hamiltonian associated with the chain {1, 2, . . . , L + 1} and given by the real entire function F. Here F(∆C(e))i,i+1 means that the ranktwo tensor element F(∆C(e)) is associated with the sites i and i + 1 of the chain, i.e. F(∆C(e))i,i+1 = 11 ⊗ · · · ⊗ 1i−1 ⊗ F(∆C(e)) ⊗ 1i+2 ⊗ · · · ⊗ 1L+1 . Theorem 1. The Hamiltonian H is a self-adjoint operator acting in H1 ⊗ H2 ⊗ · · · ⊗ HL+1 and is invariant under the algebra A. Proof. That H is self-adjoint is immediate from the definition. To prove the invariance of H it suffices to prove [H, Eα ] = 0, α = 1, 2, . . . , n. From formula (13) Eα in (16) is simply L X Eα = (eα )i , (19) i=1
where (eα )i = 11 ⊗ · · · ⊗ 1i−1 ⊗ (eα ) ⊗ 1i+1 ⊗ · · · ⊗ 1L+1 . Obviously [F(∆C(e))i,i+1 , (eα )j ] = 0, ∀ j 6= i, i + 1 .
(20)
By using formula (20) and (13) we have, for all α = 1, 2, . . . , n: L L X X F(∆C(e))i,i+1 , (eα )j [H, Eα ] = i=1
=
L X i=1
=
L X
F(∆C(e))i,i+1 ,
j=1 i−1 X
(eα )j +
j=1
L X
(eα )j + (eα )i + (eα )i+1
j=i+2
[F(∆C(e))i,i+1 , (eα )i + (eα )i+1 ]
i=1
=
L X
[F(∆C(e))i,i+1 , (∆eα )i,i+1 ] = 0 ,
i=1
where in the last step we used formula (15).
The Hamiltonian system given by (18) describes nearest neighbours interactions. Systems with A-invariant Hamiltonians and non-nearest neighbours interactions can be constructed by iterating the application of the coproduct operator to the
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S. ALBEVERIO and S.-M. FEI
Casimir operator. For instance, the following Hamiltonian describes a system with N + 1(N ≤ L) sites interactions: HN =
L−N X+1
F(∆N C(e))i,i+1,...,i+N ,
(21)
i=1
with ∆N as in definition (17). H N commutes with the generators Eα , α = 1, 2, . . . , n, of the algebra A on the Hilbert space H1 ⊗ · · · ⊗ HL+1 since L−N L X+1 X [H N , Eα ] = F(∆N C(e))i,i+1,...,i+N , (eα )j i=1
=
L−N X+1
j=1
F(∆N C(e))i,i+1,...,i+N ,
i=1
=
i+N X
(eα )j
j=i
L−N X+1
F(∆N C(e)), (∆N eα ) i,i+1,...,i+N = 0 ,
i=1 N
where the relation ∆ ([F(C(e)), eα ]) = [∆N (F(C(e))), ∆N (eα )] = 0 has been used. 2.3. Chain models with quantum Lie algebraic symmetry Let e = {eα , fα , hα }, α = 1, 2, . . . , n, be the Chevalley basis of a Lie algebra A with rank n. Let e0 = {e0α , fα0 , h0α }, α = 1, 2, . . . , n, be the corresponding elements of the quantum (q-deformed) Lie algebra Aq . We denote by rα the simple roots of the Lie algebra A. The Cartan matrix (aαβ ) is then aαβ =
1 (rα · rβ ) , dα
dα =
1 (rα · rα ) . 2
(22)
We introduce a complex quantum parameter q, such that q dα 6= ±1, 0. The quantum algebra generated by {e0α , fα0 , h0α } is defined by the following relations [14]: [h0α , h0β ] = 0 , [h0α , e0β ] = aαβ e0β , (23) dα h0α −dα h0α [h0α , fβ0 ] = −aαβ fβ0 , [e0α , fβ0 ] = δα,β q qdα −q −d α −q together with the quantum Serre relations " # 1−aαβ X γ 1 − aαβ (−1) (e0α )γ e0β (e0α )1−aαβ −γ = 0 , γ γ=0 q dα " # 1−aαβ X 1 − aαβ (−1)γ (fα0 )γ fβ0 (fα0 )1−aαβ −γ = 0 , γ d γ=0 α
i 6= j , (24) i 6= j ,
q
where for m ≥ n ∈ N, " # m [m]q ! , = [n]q ![m − n]q ! n q
[n]q ! = [n]q [n − 1]q . . . [2]q [1]q ,
[n]q =
q n − q −n . q − q −1
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SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
The coproduct operator ∆0 of the quantum algebra Aq is given by ∆0 h0α = h0α ⊗ 1 + 1 ⊗ h0α , 0
(25) 0
∆0 e0α = e0α ⊗ q −dα hα + q dα hα ⊗ e0α ,
(26)
−dα h0α
(27)
∆0 fα0 = fα0 ⊗ q
+q
dα h0α
⊗ fα0 .
It is straightforward to check that ∆0 preserves all the algebraic relations in (23) and (24). Let Cq (e0 ) be the Casimir operator of Aq , i.e. [Cq (e0 ), a] = 0, ∀ a ∈ Aq . For any entire function F of Cq (e0 ), we have
and
[F(Cq (e0 )), a] = 0, ∀ a ∈ Aq
(28)
[∆0 F(Cq (e0 )), ∆0 a] = 0, ∀ a ∈ Aq .
(29)
Especially, by formula (25) one gets 0
0
0
∆0 q ±dα hα = q ±dα hα ⊗ q ±dα hα .
(30)
Hence 0
0
0
[∆0 F(Cq (e0 )), ∆0 q ±dα hα ] = [∆0 F(Cq (e0 )), q ±dα hα ⊗ q ±dα hα ] = 0 .
(31)
The generators of Aq on a chain with (L + 1)-sites are given by Hα0 = ∆0L h0α =
L+1 X
(h0α )i
i=1
Eα0
= =
∆0L e0α L+1 X
, 0
0
0
0
0
0
0
0
(q dα hα )1 ⊗ · · · ⊗ (q dα hα )i−1 ⊗ (e0α )i ⊗ (q −dα hα )i+1 ⊗ · · · ⊗ (q −dα hα )L+1 ,
i=1
Fα0
= ∆0L fα0 =
L+1 X
(q dα hα )1 ⊗ · · · ⊗ (q dα hα )i−1 ⊗ (fα0 )i ⊗ (q −dα hα )i+1 ⊗ · · · ⊗ (q −dα hα )L+1 .
i=1
(32) Theorem 2. The chain model defined by the following Hamiltonian acting in H1 ⊗ · · · ⊗ HL+1 is invariant under the quantum algebra Aq : Hq =
L X i=1
(∆0 F(Cq (e0 )))i,i+1 .
(33)
730
S. ALBEVERIO and S.-M. FEI
Proof. Using (32) and (25) we get L L+1 X X (∆0 F(Cq (e0 )))i,i+1 , (h0α )j [Hq , Hα0 ] = i=1
=
L X
j=1
[(∆0 F(Cq (e0 )))i,i+1 , (h0α )i + (h0α )i+1 ]
i=1
=
L X
[(∆0 F(Cq (e0 ))), ∆0 (h0α )]i,i+1 = 0 .
i=1
From formulae (32) and (31) we have " L X 0 (∆0 F(Cq (e0 )))i,i+1 , [Hq , Eα ] = i=1
×
L+1 X
0
0
(q dα hα )1 ⊗ · · · ⊗ (q dα hα )j−1 ⊗ (e0α )j
j=1
⊗ (q
=
#
−dα h0α
)j+1 ⊗ · · · (q
−dα h0α
)L+1
L h i X 0 0 (∆0 F(Cq (e0 )))i,i+1 , (e0α )i ⊗ (q −dα hα )i+1 + (q dα hα )i ⊗ (e0α )i+1 . i=1
Using formulae (26) and (29) we get [Hq , Eα0 ] =
L X
[(∆0 F(Cq (e0 ))), ∆0 (e0α )]i,i+1 = 0 .
i=1
Similarly we have " L X 0 (∆0 F(Cq (e0 )))i,i+1 , [Hq , Fα ] = i=1
×
L+1 X
0
0
(q dα hα )1 ⊗ · · · ⊗ (q dα hα )j−1
j=1
⊗
=
(fα0 )j
# ⊗ (q
−dα h0α
)j+1 ⊗ · · · (q
−dα h0α
)L+1
L h i X 0 0 (∆0 F(Cq (e0 )))i,i+1 , (e0α )i ⊗ (q −dα hα )i+1 + (q dα hα )i ⊗ (fα0 )i+1 i=1
=
L X
[(∆0 F(Cq (e0 ))), ∆0 (fα0 )]i,i+1 = 0 .
i=1
Therefore Hq commutes with the generators of Aq for the chain.
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
731
The Hamiltonian (33) stands for a system with nearest neighbours interactions. Generally by using the coproduct operator ∆0 we can construct models with N + 1(N ≤ L) sites interactions: L−N X+1
HqN =
F(∆0N Cq (e0 ))i,i+1,...,i+N .
(34)
i=1
Taking into account the relation 0
0
0
∆0N ([F(Cq (e0 )) , q ±dα hα ]) = [∆0N F(Cq (e0 )), q ±dα hα ⊗ · · · ⊗ q ±dα hα ] = 0 , | {z } N times
HqN
has the symmetry of the algebra Aq . In we can prove that the Hamiltonian fact, L−N L+1 X+1 X (∆0N F(Cq (e0 )))i,i+1,...,i+N , (h0α )j [HqN , Hα0 ] = i=1
=
j=1
L−N X+1
(∆0N F(Cq (e0 )))i,i+1,...,i+N ,
i=1
=
i+N X
(h0α )j
j=i
L−N X+1
0N (∆ F(Cq (e0 ))), ∆0N (h0α ) i,i+1,...,i+N = 0 ;
i=1
" L−N +1 X N 0 [Hq , Eα ] = (∆0N F(Cq (e0 )))i,i+1,...,i+N , i=1
×
i+N X
0
0
(q dα hα )i ⊗ · · · ⊗ (q dα hα )j−1 ⊗ (e0α )j
j=i
⊗ (q
=
#
−dα h0α
L−N X+1
)j+1 ⊗ · · · ⊗ (q
−dα h0α
)i+N
0N (∆ F(Cq (e0 ))), ∆0N (e0α ) i,i+1,...,i+N = 0 ;
i=1
" L−N +1 X N 0 [Hq , Fα ] = (∆0N F(Cq (e0 )))i,i+1,...,i+N , i=1
×
i+N X
0
0
(q dα hα )i ⊗ · · · ⊗ (q dα hα )j−1
j=i
#
⊗ (fα0 )j ⊗ (q
=
L−N X+1 i=1
−dα h0α
)j+1 ⊗ · · · ⊗ (q
−dα h0α
)i+N
0N (∆ F(Cq (e0 ))), ∆0N (fα0 ) i,i+1,...,i+N = 0 .
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S. ALBEVERIO and S.-M. FEI
The Hamiltonian system (33) is expressed by the quantum algebraic generators e0 = (h0α , e0α , fα0 ). Assume now that e → e0 (e) is an algebraic map from A to Aq (we remark that for rank one algebras, both classical and quantum algebraic maps can be discussed in terms of the two dimensional manifolds related to the algebras, see [15]). We then have HqN =
L−N X+1
F(∆0N Cq (e0 (e)))i,i+1,...,i+N .
(35)
i=1
In this way we obtain Hamiltonian systems having quantum algebraic symmetry but expressed in terms of the usual Lie algebraic generators {eα }. 3. Integrable Models with An Symmetry 3.1. Quantum Yang Baxter equation The quantum Yang–Baxter equation (QYBE) [16] is the “master equation” for integrable models in statistical mechanics. It plays an important role in a variety of problems in theoretical physics such as the study of exactly soluble models (like the six and eight vertex models) in statistical mechanics [17], of integrable model field theories [18], of exact S-matrix theoretical models [19], as well as in the investigation of two dimensional field theories involving fields with intermediate statistics [20], in conformal field theory and in the study of quantum groups [14]. In this section we will investigate the integrability of the chain models having a certain algebraic symmetry constructed in Sec. 2. We also present a series of solutions of the QYBE from the construction of integrable models. Let V be a complex vector space and R be the solution of QYBE without spectral parameters, see e.g. [14]. Then R takes values in EndC (V ⊗ V ). The QYBE is R12 R13 R23 = R23 R13 R12 .
(36)
Here Rij denotes the matrix on the complex vector space V ⊗ V ⊗ V , acting as R on the ith and the jth components and as the identity on the other components. ˇ = Rp, p as in (6). Then the QYBE (36) becomes Let R ˇ 23 R ˇ 23 R ˇ 12 = R ˇ 12 R ˇ 23 , ˇ 12 R R
(37)
ˇ ⊗ 1, R ˇ 23 = 1 ⊗ R ˇ and 1 is the identity operator on V . ˇ 12 = R where R In the following we say that a chain model with nearest neighbours interactions having a (quantum mechanical) Hamiltonian of the form H=
L X
(H)i,i+1
(38)
i=1
is integrable in the sense that the rank-two tensor operator H satisfies the QYBE relation (37), i.e. (39) (H)12 (H)23 (H)12 = (H)23 (H)12 (H)23 , where (H)12 = H ⊗ 1 and (H)23 = 1 ⊗ H. Here H is a solution of the Yang–Baxter equation without spectral parameters. After Baxterization the Hamiltonian system
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
733
(38) satisfying relation (39) can in principle be exactly solved by the algebraic Bethe Ansatz method, see e.g. [1]. 3.2. Integrable An symmetric chain models The integrability of the models having a certain algebraic symmetry presented in Sec. 2 depends on the detailed representation of the corresponding symmetry algebra. In the following we investigate the integrability of chain models with nearest neighbours interactions and Lie algebraic symmetry An . Let (aαβ ) be the Cartan matrix of the An algebra. In the Chevalley basis the algebra An is spanned by the generators {hα , eα , fα }, α = 1, 2, . . . , n, with the following algebraic relations: [hα , hβ ] = 0 ,
[hα , eβ ] = aαβ eβ ,
[hα , fβ ] = −aαβ fβ ,
[eα , fβ ] = δαβ hα , (40)
together with the generators with respect to non-simple roots, eα...βγ = [eα , . . . , [eβ , eγ ] . . .] ,
fα...βγ = [fα , . . . , [fβ , fγ ] . . .] .
(41)
Let Eαβ be an (n + 1) × (n + 1) matrix such that (Eαβ )γδ = δαγ δβδ , i.e. the only non-zero element of the matrix Eαβ is 1 at row α and column β. Hence Eαβ Eγδ = δβγ Eαδ , [Eαβ , Eγδ ] = δβγ Eαδ − δδα Eβγ .
(42)
For the fundamental representation we take the basis of the algebra An as hα = Eαα − Eα+1,α+1 , α = 1, 2, . . . , n ) e = {Eαβ } β > α = 1, 2, . . . , n f = {Eβα }
(43)
Both {eα } and {fα } have a total of n(n + 1)/2 generators. With respect to the basis (43), the Casimir operator of the algebra An is given by X
n(n+1)/2
CAn = (n + 1)
(eα fα + fα eα ) +
α=1
+
n n−α X X
n X
α(n + 1 − α)h2α
α=1
2α(n + 1 − α − β)hα hα+β − a ,
(44)
α=1 β=1
where a is an arbitrary real constant. The coproduct operator ∆ is given by ∆(hα ) = hα ⊗ 1 + 1 ⊗ hα , α = 1, 2, . . . , n ) ∆(eβ ) = eβ ⊗ 1 + 1 ⊗ eβ β = 1, 2, . . . , n(n + 1)/2 , ∆(fβ ) = fβ ⊗ 1 + 1 ⊗ fβ ∆(1) = 1 ⊗ 1 ,
where the identity operator 1 is the (n + 1) × (n + 1) identity matrix.
(45)
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S. ALBEVERIO and S.-M. FEI
By (44) and (45) we have ∆CAn = CAn ⊗ 1 + 1 ⊗ CAn − a1 ⊗ 1 X
n(n+1)/2
+ (n + 1)
(eα ⊗ fα + fα ⊗ eα ) +
α=1
+
n n−α X X
n X
α(n + 1 − α)hα ⊗ hα
α=1
α(n + 1 − α − β)(hα ⊗ hα+β + hα+β ⊗ hα ) .
(46)
α=1 β=1
It is easy to check that under the representation (43) CAn is equal to n(n + 2)1. Therefore the sum of the first two terms on the right-hand side of (46) is 2n(n + 2)1 × 1. In the following we take a in (46) to be 2n(n + 2) so that the terms that are proportional to the (n + 1)2 × (n + 1)2 identity matrix will disappear in (46). From (43) and (46) we have ∆CAn = (n + 1)
n+1 X
Eαβ ⊗ Eβα
α6=β=1
+
n X
α(n + 1 − α)(Eαα − Eα+1,α+1 ) ⊗ (Eαα − Eα+1,α+1 )
α=1
+
n n−α X X
α(n + 1 − α − β)[(Eαα − Eα+1,α+1 )
α=1 β=1
⊗ (Eα+β,α+β − Eα+β+1,α+β+1 ) + (Eα+β,α+β − Eα+β+1,α+β+1 ) ⊗ (Eαα − Eα+1,α+1 )] .
(47)
∆CAn in (47) is an (n + 1)2 × (n + 1)2 matrix. Its matrix representation is (∆CAn )αβ = δαβ [(n + 1)δα,l(n+1)+l+1 − 1] + (n + 1)[δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1) + δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] ,
(48)
where α, β = 1, 2, . . . , (n + 1)2 , l = 0, 1, . . . , n, j = 0, 1, . . . , n − 1, k = 0, 1, . . . , n − j − 1, δα,j(n+2)+k+2 = 0 if α 6= j(n + 2) + k + 2 for all possible values of j and k. Here we give explicitly, as examples, the matrix representations of ∆CAn for n = 1, 2, 3: 1 · · · · −1 2 · (49) ∆CA1 = , · 2 −1 · · ·
· 1
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
∆CA2
2 · · · = · · · · ·
· · −1 · · 3 · · · ·
∆CA3
· · · · −1 · · · 3 · · · −1 · · · · · · · 2 · · · · , · · · −1 · 3 · 3 · · · −1 · · · · · 3 · −1 ·
·
3 · · · · · · −1 · · 4 · · · −1 · · · · · · −1 · · · 4 · · −1 · · · · · · 3 · · · · · · · · · · · · = · · 4 · · · · · · · · · · · · · · · · · · · · · · · · 4 · · · · · · · · · · · · · · · ·
·
·
· · · 3 · ·
·
· · ·
· ·
·
(50)
· 2
· · · · · · 4 · · · · · · · · · · · · · 4 · · · · · · · · · · · · · · · · · · · · · · · −1 · · 4 · · · · · · · −1 · · · · · 4 · · , · · −1 · · · · · · · 4 · · −1 · · · · · · · · · · 3 · · · · · · · · · · −1 · · 4 · · · · · · · −1 · · · · 4 · · · · · −1 · · · · · · · 4 · · −1 · · ·
· · ·
· ·
·
· ·
·
735
· · · · · ·
· · ·
· ·
·
· ·
·
(51)
· 3
where for simplicity · stands for 0. Lemma 1. ∆CAn satisfies the following relation: (∆CAn )2 + 2∆CAn − n(n + 2)1 ⊗ 1 = 0 . Proof. From (48) we have (n+1)2 2
[(∆CAn ) ]αγ =
X
(∆CAn )αβ (∆CAn )βγ
β=1 (n+1)2
=
X
[δαβ ((n + 1)δα,l(n+1)+l+1 − 1)
β=1
+ (n + 1)(δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1)
(52)
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S. ALBEVERIO and S.-M. FEI
+ δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) )] · [δβγ ((n + 1)δβ,l0 (n+1)+l0 +1 − 1) + (n + 1)(δβ,j0 (n+2)+k0 +2 δγ,(j 0 +1)(n+2)+k0 (n+1) + δγ,j 0 (n+2)+k0 +2 δβ,(j 0 +1)(n+2)+k0 (n+1) )] = δαγ [(n + 1)2 δα,l(n+1)+l+1 δγ,l0 (n+1)+l0 +1 ] − (n + 1)(δα,l(n+1)+l+1 + δγ,l0 (n+1)+l0 +1 ) + 1] − 2(n + 1)[δα,j(n+2)+k+2 δγ,(j+1)(n+2)+k(n+1) + δγ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] + (n + 1)2 [δα,j(n+2)+k+2 δγ,j(n+2)+k+2 + δα,(j+1)(n+2)+k(n+1) δγ,(j+1)(n+2)+k(n+1) ] = δαγ [(n − 1)2 δα,l(n+1)+l+1 ] − 2(n + 1)[δα,j(n+2)+k+2 δγ,(j+1)(n+2)+k(n+1) + δγ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] + (n + 1)2 δαγ [δα,j(n+2)+k+2 + δα,(j+1)(n+2)+k(n+1) ] = − 2(∆CAn )αγ + (n + 1)2 δαγ [δα,j(n+2)+k+2 + δα,(j+1)(n+2)+k(n+1) ] + (n + 1)2 δαγ δα,l(n+1)+l+1 − δαγ = − 2(∆CAn )αγ + (n + 1)2 δαγ − δαγ = − 2(∆CAn )αγ + n(n + 2)δαγ , where the identity δα,l(n+1)+l+1 + δα,j(n+2)+k+2 + δα,(j+1)(n+2)+k(n+1) = 1 , l = 0, 1, . . . , n, j = 0, 1, . . . , n − 1, k = 0, 1, . . . , n − j − 1, has been used.
(53)
Lemma 2. The coproduct of the An Casimir operator ∆CAn has the following properties: (∆CAn ⊗ 1)(1 ⊗ ∆CAn )(∆CAn ⊗ 1) − n[(1 ⊗ ∆CAn )(∆CAn ⊗ 1) + (∆CAn ⊗ 1)(1 ⊗ ∆CAn )] + (n2 − 1)(∆CAn ⊗ 1) + n2 (1 ⊗ ∆CAn ) + n(1 − n2 )1 ⊗ 1 ⊗ 1 = 0
(54)
737
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
and (1 ⊗ ∆CAn )(∆CAn ⊗ 1)(1 ⊗ ∆CAn ) − n[(1 ⊗ ∆CAn )(∆CAn ⊗ 1) + (∆CAn ⊗ 1)(1 ⊗ ∆CAn )] + (n2 − 1)(1 ⊗ ∆CAn ) + n2 (∆CAn ⊗ 1) + n(1 − n2 )1 ⊗ 1 ⊗ 1 = 0 .
(55)
Proof. By using the representation of ∆CAn in (48) we have (∆CAn ⊗ 1)αβ = (∆CAn )(α−γ)/(n+1)+1,(β−γ)/(n+1)+1 = δαβ [(n + 1)δα−γ,l(n+1)(n+2) − 1] + (n + 1)[δα−γ,(n+1)(j(n+2)+k+1)) δβ−γ,(n+1)(j(n+2)+(k+1)(n+1)) + δβ−γ,(n+1)(j(n+2)+k+1) δα−γ,(n+1)(j(n+2)+(k+1)(n+1)) ]
(56)
and (1 ⊗ ∆CAn )αβ = (∆CAn )α−(n+1)2 (γ 0 −1),β−(n+1)2 (γ 0 −1) = δαβ [(n + 1)δα−(n+1)2 (γ 0 −1),l(n+1)+l+1) − 1] + (n + 1)[δα−(n+1)2 (γ 0 −1),j(n+2)+k+2)) δβ−(n+1)2 (γ 0 −1),(j+1)(n+2)+k(n+1) + δβ−(n+1)2 (γ 0 −1),j(n+2)+k+2 δα−(n+1)2 (γ 0 −1),(j+1)(n+2)+k(n+1) ] , (57) where α, β = 1, . . . , (n + 1)3 , l = 0, 1, . . . , n, j = 0, 1, . . . , n − 1 and k = 0, 1, . . . , n − j − 1 as in formula (48), γ = 1, . . . , n + 1 such that (α − γ)/(n + 1) and (β − γ)/(n + 1) in (56) are integers and γ 0 = 1, . . . , n + 1 in (57). Using the formulae (56) and (57) one can get (54) and (55) from straightforward calculations. From Theorem 1 we know that the following Hamiltonian is invariant under An H=
L X
F(∆CAn )i,i+1 .
(58)
i=1
For the given representation (43) of An the integrability of (48) depends on the form of the entire function F. Due to the relation (52) in Lemma 1, (∆CAn )l , l ≥ 2, can be expressed as c∆CAn + c0 1 ⊗ 1 for some real constants c and c0 . Therefore F(∆CAn ) is a polynomial in ∆CAn up to powers of order two. Theorem 3. The following An invariant Hamiltonian is integrable HA n =
L X i=1
(H)i,i+1 =
L X i=1
(∆CAn + 1)i,i+1
738
S. ALBEVERIO and S.-M. FEI
=
L X
((eα )i (fα )i+1 + (fα )i (eα )i+1 )
α=1
i=1
+
X
n(n+1)/2
(n + 1)
n X
α(n + 1 − α)(hα )i (hα )i+1
α=1
+
n n−α X X
α(n + 1 − α − β)((hα )i (hα+β )i+1 + (hα+β )i (hα )i+1 ) + L ,
(59)
α=1 β=1
where H = ∆CAn + 1 and the number 1 should be understood as the identity operator, 1 ⊗ 1, on the tensor space H1 ⊗ · · · ⊗ HL+1 (L + 1 is the number of lattice sites of the chain). Proof. What we have to prove is that H satisfies the QYBE (39), i.e. (H)12 (H)23 (H)12 = (H)23 (H)12 (H)23 , where (H)12 = (∆CAn + 1) ⊗ 1 and (H)23 = 1 ⊗ (∆CAn + 1). We have (H)12 (H)23 (H)12 = (1 ⊗ ∆CAn + ∆CAn ⊗ 1 + (∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + 1 ⊗ 1 ⊗ 1)(H)12 = (1 ⊗ ∆CAn )(∆CAn ⊗ 1) + (∆CAn ⊗ 1)(∆CAn ⊗ 1) + (∆CAn ⊗ 1)(1 ⊗ ∆CAn )(∆CAn ⊗ 1) + (∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + 2∆CAn ⊗ 1 + 1 ⊗ ∆CAn + 1 ⊗ 1 ⊗ 1 and (H)23 (H)12 (H)23 = (H)23 (1 ⊗ ∆CAn + ∆CAn ⊗ 1 + (∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + 1 ⊗ 1 ⊗ 1) = (1 ⊗ ∆CAn )(1 ⊗ ∆CAn ) + (1 ⊗ ∆CAn )(∆CAn ⊗ 1) + (1 ⊗ ∆CAn )(∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + (∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + 2(1 ⊗ ∆CAn ) + (∆CAn ⊗ 1) + 1 ⊗ 1 ⊗ 1 . Hence (H)12 (H)23 (H)12 − (H)23 (H)12 (H)23 = I + II + III , where I = (∆CAn ⊗ 1)(∆CAn ⊗ 1) − (1 ⊗ ∆CAn )(1 ⊗ ∆CAn ) , II = (∆CAn ⊗ 1)(1 ⊗ ∆CAn )(∆CAn ⊗ 1) − (1 ⊗ ∆CAn )(∆CAn ⊗ 1)(1 ⊗ ∆CAn ) , III = ∆CAn ⊗ 1 − 1 ⊗ ∆CAn .
(60)
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
739
Using (52) we have I = (∆CAn )2 ⊗ 1 − 1 ⊗ (∆CAn )2 = −(2∆CAn − n(n + 2)1 ⊗ 1) ⊗ 1 + 1 ⊗ (2∆CAn − n(n + 2)1 ⊗ 1) = −2(∆CAn ⊗ 1 − 1 ⊗ ∆CAn ) . Hence I + III = 1 ⊗ ∆CAn − ∆CAn ⊗ 1 .
(61)
By Lemma 2 we get II = ∆CAn ⊗ 1 − 1 ⊗ ∆CAn . Therefore (H)12 (H)23 (H)12 − (H)23 (H)12 (H)23 = I + II + III = 0 .
3.3. Temperley Lieb algebraic structures and representations An L-state TL algebra is described by the elements ei , i = 1, 2, . . . , L, satisfying the TL algebraic relations [21], ei ei±1 ei = ei ,
ei ej = ej ei ,
if |i − j| ≥ 2 ,
(62)
and e2i = βei ,
(63)
where β is a complex constant and i = 1, 2, . . . , L. A quantum chain with a certain algebraic symmetry constructed in our way as described in Sec. 2 does not necessarily relate to a TL algebra, when it contains non-nearest neighbours interactions and is not in the fundamental representation of the symmetry algebra. Nevertheless for our example of the An chain, we believe that the usual well-known way (taking two representations of a given algebra on two successive sites such that their product contains the scalar representation) to write quantum chains in terms of the TL algebra can be applied, though this way does not always lead to stochastic processes. In this section we simply use the results of the last section to show that there is a TL algebraic structure related to the integrable chain model (59), in the sense that the model gives a representation of the TL algebra. We suppose that the representation of an L-state TL algebra on an L + 1 chain is of the following form: ei = 11 ⊗ 12 ⊗ · · · ⊗ 1i−1 ⊗ E ⊗ 1i+2 ⊗ · · · ⊗ 1L+1 ,
(64)
where 1 is the (n + 1) × (n + 1) identity matrix as in Sec. 3.2 and E is a (n + 1)2 × (n + 1)2 matrix. According to formulae (63) and (62) E should satisfy E 2 = βE . (E ⊗ 1)(1 ⊗ E)(E ⊗ 1) = E ⊗ 1 , (1 ⊗ E)(E ⊗ 1)(1 ⊗ E) = 1 ⊗ E .
(65) (66)
740
S. ALBEVERIO and S.-M. FEI
It is direct to check the following conclusion: Theorem 4. For a given representation of the TL algebra of the form (64) with E satisfying (65) and (66), p −β ± β 2 − 4 ˇ 1⊗1 (67) R=E+ 2 is a solution of the QYBE (37). Hence from the representations of the TL algebra one can construct integrable chain models (for the construction of the TL algebraic representations associated with the quantum A1 , Bn , Cn and Dn algebras, see [22]). However in general the ˇ of the QYBE (37), converse of Theorem 4 does not hold, i.e. for a given solution R there does not necessarily exist a TL algebraic representation of the form (64) with ˇ + b satisfying (65) and (66) for any constants a and b (a chain model with E = aR a certain algebraic symmetry constructed in our way as described in Sec. 2 even ˇ matrix). Nevertheless the solutions H of does not necessarily associate with a R the QYBE in our An symmetric integrable model (59) do give rise to TL algebraic representations in the following sense: Theorem 5. The following (n + 1)2 × (n + 1)2 matrix E=−
H +1⊗1 n+1
(68)
gives the L-state TL algebraic representation (64) with β = 2. Proof. What we should check is that E in (68) satisfies Eqs. (65) and (66). By Lemma 1 we have 2 2 H ∆CAn + 1 ⊗ 1 +1⊗1 = − +1⊗1 E2 = − n+1 n+1 =
(∆CAn )2 − 2n∆CAn + n2 1 ⊗ 1 (−2(n + 1)∆CAn + 2n(n + 1)1 ⊗ 1 = (n + 1)2 (n + 1)2
= βE = 2E , i.e. β = 2. From Lemma 1 and (54) in Lemma 2 we get 1⊗H H⊗1 1⊗1⊗1− (E ⊗ 1)(1 ⊗ E)(E ⊗ 1) = 1 ⊗ 1 ⊗ 1 − n+1 n+1 H⊗1 × 1⊗1⊗1− n+1
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
=
741
−1 [(∆CAn ⊗ 1)(1 ⊗ ∆CAn )(∆CAn ⊗ 1) (n + 1)3 − n((∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + (1 ⊗ ∆CAn )(∆CAn ⊗ 1)) + 2n(n + 1)∆CAn ⊗ 1 + n2 1 ⊗ ∆CAn − 2(n3 + n2 )1 ⊗ 1 ⊗ 1]
=
−1 [(n + 1)2 ∆CAn ⊗ 1 − n(n + 1)2 1 ⊗ 1 ⊗ 1] (n + 1)3
=
n1 ⊗ 1 ⊗ 1 − ∆CAn ⊗ 1 = E ⊗1. (n + 1)
By using Lemma 1 and formula (55) in Lemma 2 we conclude that 1⊗H H⊗1 (1 ⊗ E)(E ⊗ 1)(1 ⊗ E) = 1 ⊗ 1 ⊗ 1 − 1⊗1⊗1− n+1 n+1 1⊗H × 1⊗1⊗1− n+1 =
−1 [(∆CAn ⊗ 1)(1 ⊗ ∆CAn )(∆CAn ⊗ 1) (n + 1)3 − n((∆CAn ⊗ 1)(1 ⊗ ∆CAn ) + (1 ⊗ ∆CAn )(∆CAn ⊗ 1)) + 2n(n + 1)1 ⊗ ∆CAn + n2 ∆CAn ⊗ 1 − 2(n3 + n2 )1 ⊗ 1 ⊗ 1]
=
−1 [(n + 1)2 1 ⊗ ∆CAn − n(n + 1)2 1 ⊗ 1 ⊗ 1] (n + 1)3
=
n1 ⊗ 1 ⊗ 1 − 1 ⊗ ∆CAn =1⊗E. (n + 1)
From (68) we see that the Hamiltonian of the An symmetric integrable chain model (59) can be expressed by the TL algebraic elements HA n =
L X i=1
(H)i,i+1 =
L X i=1
(n + 1)(−E + 1)i,i+1 =
L X
(n + 1)ei + (n + 1)L ,
(69)
i=1
with ei as in (64) and E as in (68). Hence instead of the algebraic Bethe Ansatz method, the energy spectrum of HAn can also be studied by using the properties of the TL algebra [23] (for the case of the Heisenberg spin chain model, n = 1, see [24]). The spectra of HAn and their degeneracies for different n will be discussed elsewhere.
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S. ALBEVERIO and S.-M. FEI
4. Integrable Models and Stationary Markov Chains 4.1. Stationary Markov chains We first briefly recall some concepts of the theory of Markov chains (for a detailed mathematical description of Markov chains, we refer to [25]). Let Ω denote the sample space (the set of all possible outcomes of an experiment). If Ω is a finite or countably infinite sample space and if P is a probability measure defined on the σ-algebra of all subsets of Ω, then the pair (Ω, P ) is called a probability space. A subset A of Ω is then called an event with probability P (A). A function X ≡ X(ω), ω ∈ Ω, that maps a sample space into the real numbers is called a random variable. A stochastic process is a family (Xt )t∈I , I a certain index set, of random variables defined on some sample space Ω. If I is countable, i.e. I ∈ N, the process is denoted by X1 , X2 , . . . and called a discrete-time process. If I = R+ , then the process is denoted by {Xt }t≥0 and called a continuous-time process. The range of X (a subset of real numbers) is called the state space. In what follows we consider the case where the state space S is countable or finite. In this case the related stochastic process is called a (stochastic or random) chain. Let (Ω, P ) be a probability space in above sense and E, F be two subsets of Ω. We denote by P (E|F ) the (conditional) probability of E given that F has occurred. A discrete-time stochastic process {Xi }, i = 1, 2, . . . with state space S = N is said to satisfy the Markov property if for every l and all states i1 , i2 , . . . , il it is (a.s.) true that P [Xl = il |Xl−1 = il−1 , Xl−2 = il−2 , . . . , X1 = i1 ] = P [Xl = il |Xl−1 = il−1 ] , i.e. the values of Xl−2 , . . . , X1 in no way affect the value of Xl , given the value of Xl−1 . Such a discrete-time process is called a Markov chain. It is said to be stationary if the probability of going from one state to another is independent of the time at which the transition is being made. That is, for all states i and j, P [Xl = j|Xl−1 = i] = P [Xl+k = j|Xl+k−1 = i] for k = −(l − 1), −(l − 2), . . . , −1, 0, 1, 2, . . . . In this case we set pij ≡ P [Xl = j|Xl−1 = i] and call pij the transition probability for going from state i to j. For a discrete time stationary Markov chain {Xi }, i ∈ N, with a finite state space S = {1, 2, 3, . . . , m}, there are m2 transition probabilities {pij }, i, j = 1, 2, . . . , m. P = (pij ) is called the transition matrix corresponding to the discrete-time stationary Markov chain {Xi }. The transition matrix P has the following properties: pij ≥ 0 ,
m X
pij = 1 ,
i, j = 1, 2, . . . , m .
(70)
i=1
Any square matrix that satisfies condition (70) is called a stochastic matrix. A continuous-time stochastic process, {Xt }t∈R+ is said to satisfy the Markov property if for all times t0 < t1 < · · · < tl < t and for all l it is true that P [Xt = j|Xt0 = i0 , Xt1 = i1 , . . . , Xtl = il ] = P [Xt = j|Xtl = il ] .
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
743
Such a process is called a continuous-time Markov chain. It is said to be stationary if for every i and j the transition function, P [Xt+h = j|Xt = i], is independent of t. In this case P (t) = P (Xt = j|X0 = i) is a semigroup (e.g. on l2 (S)), called transition semigroup associated with the Markov chain. Its generator Q = (qij ) has the properties: X qij , i, j = 1, 2, . . . , m (71) qij ≥ 0 , i 6= j , qii = − i6=j
and is called an intensity matrix. Vice versa, any Q (satisfying (71) and properly defined as a closed operator when S is infinite) gives rise to a unique continuous-time transition semigroup, P (t) = eQt , t ≥ 0, which can be interpreted as a transition semigroup associated to a certain Markov chain (with state space S) [25]. The properties of Markov chains are determined by the transition matrix P resp. intensity matrix Q. If the eigenvalues and eigenstates of P resp. Q are known, then exact results related to the stochastic processes, such as time-dependent averages and correlations, can be obtained. Now we consider a chain (in the algebraic sense of Secs. 1–3) with L + 1 sites. To every site i of the chain we associate n + 1 states described by the variable τi taking n + 1 integer values, τi ≡ (τi0 = 0, τi1 = 1, τi2 = 2, . . . , τin = n) ,
(72)
(conventionally a vacancy at site i is associated with the state 0). We associate to the lattice site i of the algebraic chain a Hilbert space of dimension n + 1. The state space of the algebraic chain is then finite and has a total of (n + 1)L+1 states. By definition, for an integrable chain model with Hamiltonian H one has exact solutions for the eigenvalues and eigenstates of H. The system remains integrable if one adds to H a constant term c and multiplies H by a constant factor c0 . Moreover the eigenvalues of H will not be changed if one changes the local basis, i.e. the following Hamiltonian H 0 , defined by H 0 = BHB −1 ,
B = ⊗L+1 i=1 Bi ,
(73)
where Bi ≡ b and b is an (n + 1) × (n + 1) non-singular matrix, has the same eigenvalues as H. Therefore if an integrable chain model with Hamiltonian H can be transformed by B (modulo constants c, c0 in C) into a matrix M = P resp. Q, in the sense that (74) M = B(c0 H + c11)B −1 , where 11 is the (n + 1)L+1 × (n + 1)L+1 identity matrix, B as in (73), and P resp. Q as in (70) resp. (71) (with m = (n + 1)L+1 ), then P resp. Q defines a discrete-time resp. continuous-time Markov chain and the related stochastic process can be studied by using the properties of the corresponding integrable model with Hamiltonian H. In the following we discuss the question of whether the integrable models obtained in the way presented in this paper could be transformed into stationary
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Markov chains through transformations of the forms (74). We remark that in general P and Q have different spectra, hence it is necessary to discuss the cases M = P and M = Q separately. 4.2. Discrete-time Markov Chains related to An symmetric integrable models We first note that for an integrable chain model with Hamiltonian H = PL i=1 hi,i+1 and (n + 1) states at every site i, i = 1, 2, . . . , L + 1, if the sum of the elements in any row of the (n + 1)2 × (n + 1)2 matrix h is 1/L, then the sum of the elements in any row of the matrix H is 1. Hence if under the following transformation h → h0 given by h0 = (b ⊗ b)(c0 h + c1 ⊗ 1)(b−1 ⊗ b−1 ) ,
(75)
the sum of the elements in any row of h0 is 1/L and (h0 )α,β ≥ 0, α, β = 1, 2, . . . , (n + 1)2 , for some real constants c0 , c and a non-singular (n + 1) × (n + 1) matrix b, P 0 then P = L i=1 hi,i+1 defines a stationary discrete-time Markov chain. P has the same eigenvalue spectrum (shifted by a constant) as the spectrum of the integrable model with Hamiltonian H. If P is invariant under a certain algebra A, we call the Markov chain A symmetric. Theorem 6. The following matrix: X 1 1 HA n = (∆CAn + 1 ⊗ 1)i,i+1 L(n + 1) L(n + 1) i=1 L
PAn =
n(n+1)/2 L X X 1 = ((eα )i (fα )i+1 + (fα )i (eα )i+1 ) (n + 1) L(n + 1) i=1 α=1 +
n X
α(n + 1 − α)(hα )i (hα )i+1
α=1
+
n n−α X X
α(n + 1 − α − β)((hα )i (hα+β )i+1 + (hα+β )i (hα )i+1 ) +
α=1 β=1
1 (n + 1) (76)
defines a stationary discrete-time An symmetric Markov chain. Proof. I. Set h0 ≡
1 L(n+1) (∆CAn
+ 1 ⊗ 1). Then
PAn =
L X i=1
h0i,i+1 .
(77)
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SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
From formula (48) we have (h0 )αβ = =
1 (∆CAn + 1 ⊗ 1)αβ L(n + 1) 1 [δαβ [(n + 1)δα,l(n+1)+l+1 − 1] L(n + 1) + (n + 1)[δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1) + δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] + δαβ ]
=
1 [δαβ δα,l(n+1)+l+1 + δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1) L + δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] ≥ 0 .
(78)
Therefore (PAn )αβ ≥ 0, α, β = 1, 2, . . . , (n + 1)2 . II. By using the identity (53), we get (n+1)2
X
β=1
2
0
(h )αβ
(n+1) X 1 = [(n + 1)δαβ δα,l(n+1)+l+1 L(n + 1) β=1
+ (n + 1)[δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1) + δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ]] =
1 1 (n + 1) = . L(n + 1) L
Hence the sum of the elements of any row of the matrix PAn is one, i.e. P(n+1)L+1 PL (PAn )αβ = β=1 ( i=1 h0i,i+1 )αβ = 1.
P(n+1)L+1 β=1
H
An is obviously invariant under III. As HAn is invariant under An , PAn = L(n+1) An and has the same spectrum as HAn . By the definition (70) PAn is the transition matrix of a stationary discrete-time An symmetric Markov chain.
We give some discussions on the properties of the stationary discrete-time An symmetric Markov chain associated with the stochastic matrix PAn . The state space of this Markov chain is S = (1, 2, . . . , (n + 1)L+1 ), which corresponds to (n + 1)L+1 states, (79) (τ0 ⊗ τ1 ⊗ · · · ⊗ τn ) , τi as in (72), of the algebraic chain with L + 1 lattice sites. The properties of a Markov chain are determined by the transition matrix P = (pij ). A subset C of the state space S is called closed if pij = 0 for all i ∈ C and j 6∈ C. If a closed set consists of a single state, then that state is called an absorbing state. A Markov chain is called irreducible if there exists no nonempty closed set other than S itself. A non-irreducible Markov chain is said to be reducible.
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S. ALBEVERIO and S.-M. FEI
From formula (78) we have (h0 )αα = (h0 )(n+1)2 ,(n+1)2 =
1 L
,
(h0 )αβ = (h0 )βα = 0 ,
α = l(n + 1) + l + 1 ,
β 6= α ,
l = 0, 1, . . . , n .
(80)
Let S0 =
α|α = l
(n + 1)((n + 1)L − 1) + n +1 n
,
l = 0, 1, . . . , n ,
(81)
be a subset of the state space S. From formula (77), with (h0 )i,i+1 = 11 ⊗ 12 ⊗ · · · ⊗ 1i−1 ⊗ h0 ⊗ 1i+2 ⊗ · · · ⊗ 1L+1 , we get (PAn )αα = 1 ,
(PAn )βα = (PAn )αβ = 0 ,
β 6= α ,
α ∈ S0 .
(82)
Therefore the n + 1 states in S0 are absorbing states of the Markov chain PAn . This chain is by definition reducible. For a reducible Markov chain the “long time” probability distribution, if it exists, may depend on the initial conditions, i.e. liml→∞ (PAn )lγβ may depend on γ. From the properties (82) of PAn , we see that if the Markov chain PAn is initially in one of the states α ∈ S0 , it will remain in that state α forever. These n + 1 absorbing states correspond to the states of the algebraic chain through (79). For instance, the states 1 and (n + 1)L+1 in S correspond to the states (0, 0, . . . , 0) (all the sites of the algebraic chain are at state 0) and (n, n, . . . , n) (all the sites of the algebraic chain are at state n). 4.3. Continuous-time Markov Chains related to An symmetric integrable models PL For an integrable chain model with Hamiltonian H = i=1 hi,i+1 and with (n + 1) states at every site of the chain, if the sum of the elements in any column of the matrix h is 0, then the sum of the elements in any column of the matrix H is also 0. Hence if under the following transformation h → h00 with h00 = (b ⊗ b)(c0 h + c1 ⊗ 1)(b−1 ⊗ b−1 ) ,
(83)
the sum of the elements in any column of h00 is 0 and (h00 )α,β ≥ 0, α 6= β = 1, 2, . . . , (n + 1)2 , for some real constants c0 , c and a non-singular (n + 1) × (n + 1) P 00 matrix b, then Q = L i=1 hi,i+1 is the intensity matrix for some stationary continuous-time Markov chain. Q has the same eigenvalue spectrum (shifted by a constant) as the spectrum of the Hamiltonian H. We call the Markov chain A symmetric if Q is invariant under the algebra A. Theorem 7. The following matrix Q is the intensity matrix of a stationary continuous-time Markov chain:
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SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
QAn = HAn − (n + 1)L =
L X
(∆CAn − n1 ⊗ 1)i,i+1
i=1
=
"
L X
X
n(n+1)/2
(n + 1)
+
((eα )i (fα )i+1 + (fα )i (eα )i+1 )
α=1
i=1 n X
α(n + 1 − α)(hα )i (hα )i+1
α=1
+
n n−α X X
# α(n + 1 − α − β)((hα )i (hα+β )i+1 + (hα+β )i (hα )i+1 )
α=1 β=1
− nL .
(84)
Proof. Set h00 = (∆CAn − n1 ⊗ 1). Then Q An =
L X
h00i,i+1 .
(85)
i=1
From (48) we observe that, for α 6= β, h00α6=β = (n + 1)[δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1) + δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] ≥ 0 . Therefore (QAn )α6=β ≥ 0, α, β = 1, 2, . . . , (n + 1)L+1 . Again by (48) the sum of the elements in any given column β of the matrix h00 is (n+1)2
X
(n+1)2
h00αβ
=
α=1
X
(n + 1)[δαβ (δα,l(n+1)+l+1 − 1)
α=1
+ (n + 1)[δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1) + δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] (n+1)2
X
=
(n + 1)[−δαβ + δα,j(n+2)+k+2 δβ,(j+1)(n+2)+k(n+1)
α6=l(n+1)+l+1
+ δβ,j(n+2)+k+2 δα,(j+1)(n+2)+k(n+1) ] = (n + 1)(−1 + δβ,(j+1)(n+2)+k(n+1) |α=j(n+2)+k+2 + δβ,j(n+2)+k+2 |α=(j+1)(n+2)+k(n+1) ) = 0, i.e. the sum of the elements in any given column β, β = 1, 2, . . . , (n + 1)2 , of the matrix h00 is zero. Therefore the sum of the elements in any given column β,
748
S. ALBEVERIO and S.-M. FEI
P(n+1)L+1 β = 1, 2, . . . , (n + 1)L+1 , of the matrix QAn is also zero, α=1 (QAn )αβ = 0. At last QAn = HAn − (n + 1)L is obviously An symmetric with the same spectrum (shifted by a constant) as HAn . The long run distribution of the Markov chain described in Theorem 7 is given by the vector π = (π1 , π2 , . . .), where πi represents the “long time” probability of the state i ∈ S, satisfying (n+1)L+1
X
(n+1)L+1
(QAn )αβ πα = 0, ∀ β ∈ S ,
α=1
X
πα = 1 .
(86)
α=1
However as this Markov chain is reducible, the solution of the Eq. (86) is not unique but depends on the initial conditions. From (48) we see that (h00 )αβ = (h00 )βα = 0 ,
∀ β,
α = l(n + 1) + l + 1 ,
l = 0, 1, . . . , n .
Hence from (85) we get (QAn )αβ = (QAn )βα = 0 ,
α ∈ S0 , ∀ β ,
with S0 as in (81). Therefore if this Markov chain is initially in a given state α ∈ S0 , it will remain in that state. The states β 6∈ S0 form a closed subset of S. From (48) and (85) one also learns that the absolute value of all the non-zero elements of any column of the intensity matrix QAn are equal. Let S 0 be a closed subset of S with l elements. If the Markov chain is initially in the closed set S 0 , then it will remain in S 0 and the long run distribution is π = (π1 , π2 , . . . , π(n+1)L+1 ), where πi = 1/l for i ∈ S 0 and πi = 0 if i 6∈ S 0 . 5. Conclusion and Remark Using the Casimir operators and coproduct operations of algebras, we have given a general way to construct chain models with a certain algebraic symmetry and nearest or/and non-nearest neighbours interactions. We discussed integrable chain models with nearest neighbours interactions with the symmetries provided by the fundamental representation of the classical Lie algebra An . We have shown that the stochastic processes correspond to these An symmetric integrable chain models are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains with the transition matrices (resp. intensity matrices) which coincide with those of the corresponding integrable models. Other symmetric integrable models (e.g. with Bn , Cn , Dn symmetry) and Markov chains can be investigated in a similar way. The discussion of integrable models related to higher dimensional representations of the algebras and the integrability of chain models with non-nearest neighbours interactions is postponed to further work.
SYMMETRY, INTEGRABLE CHAIN MODELS AND STOCHASTIC PROCESSES
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Acknowledgements We are very grateful to Professor V. Rittenberg and Dr. G. M. Sch¨ utz for interesting comments, remarks and useful references. We would like to thank the A. V. Humboldt Foundation for the financial support given to the second-named author. References [1] V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, 1993; L. D. Faddeev and L. A. Takhtadzhyan, Russ. Math. Surv. 34 (1979) 11. [2] H. J. de Vega, Int. J. Mod. Phys. A4 (1989) 2371; N. Reshetikhin, Theor. Math. Phys. 63 (1986) 555; H. J. de Vega and M. Karowski, Nucl. Phys. B280 (1987) 225; N. Reshetikhin and P. B. Wiegman, Phys. Lett. B18 (1987) 125. [3] E. K. Sklyanin, J. Phys. A2 (1988) 2375; A. Foerster and M. Karowski, Nucl. Phys. B408 (1993) 512; H. J. de Vega and A. G. Ruiz, J. Phys. A26 (1993) L519; Nucl. Phys. B417 (1994) 553; L. Mezincescu and R. I. Nepomechie, Nucl. Phys. B372 (1992) 597; S. Artz, L. Mezincescu and R. I. Nepomechie, Int. J. Mod. Phys. A10 (1995) 1937; J. Phys. A2 (1995) 5131. [4] For a review and an extensive literature list concerning physical aspects of the models, see e.g., J. W. Evans, Rev. Mod. Phys. 65 (1993) 1281. [5] F. Spitzer, Adv. Math. 5 (1970) 246; T. Liggett, Interacting Particle Systems, New York, Springer, 1985. [6] S. Alexander and T. Holstein, Phys. Rev. B18 (1978) 301. [7] G. Sch¨ utz and S. Sandow, Phys. Rev. E49 (1994) 2726. [8] F. C. Alcaraz, M. Droz, M. Henkel and V. Rittenberg, Ann. Phys. 230 (1994) 250. [9] B. Sutherland, Phys. Rev. B12 (1975) 3795; J. H. H. Perk and C. L. Schultz, Non-Linear Integrable Systems, Classical Theory and Quantum Theory, eds. M. Jimbo and T. Miwa, Singapore, World Scientific, 1981. [10] S. R. Dahmen, J. Phys. A28 (1995) 905. [11] M. Doi, J. Phys. A9 (1996) 1465; P. Grassberger and M. Scheunert, Fortschr. Phys. 28 (1980) 547; L. P. Kadanoff and J. Swift, Phys. Rev. 165 (1968) 310. [12] F. C. Alcaraz and V. Rittenberg, Phys. Lett. B314 (1993) 377. [13] I. Peschel, V. Rittenberg and U. Schultze, Nucl. Phys. B430 (1995) 633; H. Simon, J. Phys. A28 (1995) 6585; M. Henkel, E Orlandini and G. M. Sch¨ utz, J. Phys. A28 (1995) 6335; G. M. Sch¨ utz, J. Stat. Phys. A28 (1995) 243. [14] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, 1994; Z. Q. Ma, Yang-Baxter Equation and Quantum Enveloping Algebras, World Scientific, 1993; C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995; S. Majid, Foundations of Quantum Group Theory, Cambridge Univ. Press, 1995. [15] S.-M. Fei, J. Phys. A24 (1991) 5195; S. Albeverio and S.-M. Fei, “Integrable Poisson Algebras and Two Dimensional Manifolds”, J. Phys. A31 (1998) 1211–1218; S.-M. Fei and H. Y. Guo, Commun. Theor. Phys. 20 (1993) 299. [16] C. N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [17] R. J. Baxter, Exactly Solved Models in Statistical Physics, Academic Press, New York, 1982.
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[18] E. K. Sklyanin, J. Soviet Math. (1982) 19; P. P. Kulish and E. K. Sklyanin, J. Soviet Math. 19 (1982) 1956; P. P. Kulish and N. Yu. Reshetikhin, J. Soviet Math. 23 (1983) 2435; H. J. de Vega, H. Eichenherr and J. M. Maillet, Nucl. Phys. B240 (1984) 377. [19] A. B. and Al. B Zamolodchikov, Ann. Phys. 120 (1979) 253. [20] J. Fr¨ ohlich, Statistics of Fields, The Yang-Baxter Equation,and Theory of Knots and Links, Carg`ese Lectures, 1987. [21] H. N. V. Temperley and E. Lieb, Proc. Roy. Soc. London A (1971) 251. [22] M. T. Batchelor and A. Kuniba, J. Phys. A24 (1991) 2599. [23] D. Levy, Phys. Rev. Lett. 64 (1990) 499; 67 (1991) 1971. [24] B. Y. Hou, B. Y. Hou and Z. Q. Ma, J. Phys. A24 (1991) 2847. [25] D. L. Isaacson and R. W. Madsen, Markov Chains, Theory and Applications, Wiley Series in Probability and Mathematical Statistics, 1976; K. L. Chung, Markov Chains with Stationary Transition Probabilities, Springer, Berlin, 1967; S. Ross, Introduction to Probability Models, Academic Press, New York, 1972; M. Iosifescu, Finite Markov Processes and Applications, J. Wiley, Chichester, 1980. [26] S.-M. Fei, H. Y. Guo and H. Shi, J. Phys. A25 (1992) 2711; J. Hietarinta, Phys. Lett. A165 (1992) 245.
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY IN DILUTED ISING MODELS ON THE CAYLEY TREE J. C. A. BARATA∗ and D. H. U. MARCHETTI† Instituto de F´ısica Universidade de S˜ ao Paulo P. O. Box 66 318 S˜ ao Paulo, SP, Brazil E-mail: [email protected] E-mail: [email protected] Received 2 May 1997 Revised 16 September 1997 Some results on the two-point function and on the analytic structure of the momenta of the effective fugacity at the origin for a class of diluted ferromagnetic Ising models on the Cayley tree are presented. Keywords: Lee–Yang singularities, Griffiths’ singularities, two-point functions, susceptibility.
1. Introduction and Previous Results In a previous work [1] a detailed analysis of the presence of Griffiths’ singularities has been performed in a class of diluted ferromagnetic Ising models on the Cayley tree. This involved the study of the analytic structure of the quenched magnetization at the origin, as a function of the fugacity, as well as the study of its differentiability for real values of the magnetic field. A detailed proof has been presented of the infinite differentiability of the quenched magnetization for real magnetic fields under suitable choices of disorder. The set of Lee–Yang zeros of the deterministic ferromagnetic Ising model on the Cayley tree has also been studied and detailed results have been presented, both in the ferromagnetic and in the paramagnetic phases, showing that this set becomes dense in subsets of the unit circle when the thermodynamic limit is taken. In the ferromagnetic phase, in particular, the set of Lee–Yang zeros becomes dense in the whole unit circle when this limit is performed. These facts are responsible for the emergence of Griffiths’ singularities and for the absence of meta-stable states in the diluted models considered. In this note we extend our previous results on the diluted models considered. Our main intention here is to present some results on the momenta of the effective ∗ Partially † Partially
supported by CNPq. supported by CNPq and FAPESP. 751
Reviews in Mathematical Physics, Vol. 10, No. 6 (1998) 751–773 c World Scientific Publishing Company
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J. C. A. BARATA and D. H. U. MARCHETTI
fugacity (defined below) produced on the spin at the origin in the random model introduced in [1], as well as an explicit expression for the two-point functions of that model. Some conclusions on the quenched susceptibility at the origin are drawn. As in [1], we consider a random Ising ferromagnet in a homogeneous rooted Cayley tree of order 2, C2 , described by the Hamiltonian X X H(σ; ξ) := − Jxy σx σy − H σx , (1.1) hxyi
x
where σ : C2 3 x 7→ σx ∈ {1, −1} is a configuration of spins. The coupling constants Jxy are random variables whose distribution is described as follows. To each generation M of C2 a Bernoulli random variable ξM ( 1 , with probability pM ξM = (1.2) 0 , with probability qM = 1 − pM is assigned and we then set Jxy = ξM if hxyi ≡ b is a bond at the generation M and 0 otherwise. See Fig. 1.
0 Q Q Q .
ξ1
ξ2
BB ξ3 B ξ3 BB B
QQ
ξ1
QQ QQ @ @ @@ ξ2 @@ ξ2 ξ2 @ @ @ @ BB BB BB ξ3 B ξ3 ξ3 B ξ3 ξ3 B ξ3 BB BB BB B B B
. . . . .
. . . . .
. . . . .
Fig. 1. The homogeneous rotted Cayley tree of order 2 and the couplings ξM .
For any event A, which is an element of the σ-algebra generated by the space of spin configurations, we define the quenched expected value of A as A ≡ E[A] = Eξ hAi(ξ) ,
(1.3)
where Eξ is the expectation with respect to the disorder variables and hAi(ξ) =
1 X A(σ) e−βH(σ;ξ) Z(ξ) σ
(1.4)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
753
is the thermal average with Z(ξ) =
X
e−βH(σ;ξ)
σ
being the partition function. The expectation (1.3) is a function of the inverse temperature β, the external magnetic field H and the sequence of Bernoulli parameters p = {pj }∞ j=1 . We let π = {p : 0 < pj ≤ 1, j = 1, 2, . . .} and for 0 ≤ a ≤ 1, let πa = {p ∈ π : limn→∞ p1 . . . pn = a}. Also, for convenience, we write ζ = e−2β and z = e−2βH , the so-called fugacity. We first take the expectation in a finite tree C2,M , M = 1, 2, . . ., with free boundary conditions and study the function A : [0, 1] × C × π 7→ A(ζ, z, p) ∈ C in the limit as M goes to infinite. In [1] we were concerned, following [2], with the magnetization at the origin m := E[σ0 ] = Eξ hσ0 i(ξ) which can be written as m=F +I, where F :=
∞ X
(1.5)
aM hσ0 iM (1)
M=0
and I := lim p1 . . . pN hσ0 iN (1) N →∞
give the contribution to m due to finite clusters and the (unique) infinite cluster, respectively. Here, for any observable A, hAiM is the thermal average (1.4) with C2 replaced by the finite tree C2,M and aM , M ∈ N, is given by aN := p1 . . . pN (1 − pN +1 ) ,
N ≥1
P∞ with a0 := 1 − p1 . Clearly, {an }∞ n=0 is a summable sequence with n=0 an = 1 − a for p ∈ πa . Equation (1.5) is a consequence of the fact that hσ0 i(ξ) = hσ0 iM (1) for all ξ such that ξ1 = · · · = ξM = 1 and ξM+1 = 0 with the finite volume one–point function given as follows (see Sec. 2 for details): (M)
hσ0 iM (1) = (n)
where τz
1 − z τz 1+z
(1)
(M) τz (1)
,
(1.6) (n)
denotes the nth composition of τz with itself: τz
:= τz ◦ · · · ◦ τz and, | {z } n-times
for z ∈ C, τz : C 7→ C is a rational map given by τz (u) := h(zu), with h(w) :=
ζ +w 1 + ζw
2 .
(1.7)
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J. C. A. BARATA and D. H. U. MARCHETTI
The analytic properties of F and I have been studied in [1] based on the following facts. Let us denote by D<1 the open unit disk in the Riemann sphere S 2 and by D>1 the complement of its closure. In [1] it was shown that, for the whole interval 0 ≤ ζ < 1, the function h maps D<1 , D>1 and S 1 into itself. This implies that, (n) as function of z, τz (1) maps each of the sets D<1 , S 1 and D>1 into itself for all n ∈ N and all ζ ∈ [0, 1). In addition, each of the sets [0, 1) and (1, ∞) ⊂ R+ is also (n) mapped into itself and, for z = 1 one has τ1 (1) = 1, for all n ∈ N. The description of the Lee–Yang singularities and the smoothness of the quenched magnetization (n) are also obtained from these and further properties of the orbit τz (1), n ∈ N, of the dynamical system induced by τz . For the present analysis these properties will play an important role as well. (n) The quantity τz (1) will be called the effective fugacity at the origin for a tree with n generations. The reason for this nomenclature comes from the analogy between (1.6) and the expression for the magnetization of a lattice containing a single spin: m = 1−z 1+z . In general, for the Ising model on an arbitrary lattice, the effective fugacity at a site x is defined as the quantity given by 1 − hσx i(z) , tx (z) := z −1 1 + hσx i(z) so that, as in (1.6), we may always write hσx i = (1 − z tx (z))/(1 + z tx (z)). Denote τj,z (u) := hj (zu) ,
(1.8)
where hj is given by (1.7) with ζ replaced by ζj = e−2βξj . Define the family of random variables {∆j }M j=1 through ∆j−1 = τj,z (∆j ) ,
j ∈ {1, . . . , M } ,
(1.9)
with ∆M = 1. Since the couplings are random, so are the quantities ∆j ’s. Note that ∆j depends on the generation level M of the Cayley tree through its initial condition. The distribution of ∆0 , when M → ∞, is worth studying for the following reason. The magnetization hσ0 iM (ξ) can be expressed as 1 − z∆0 (ξ) . 1 + z∆0 (ξ)
hσ0 iM (ξ) =
(1.10)
For |z| < 1 this can be expressed by the absolutely converging series representation hσ0 iM (ξ) = 1 + 2
∞ X
(−z)a (∆0 (ξ))a .
(1.11)
a=1
Hence, the quenched magnetization at the origin may be expressed as ! ∞ X a a m = lim 1 + 2 (−z) Eξ [(∆0 (ξ)) ] , M→∞
a=1
(1.12)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
755
where the infinite sum also converges for |z| < 1, by the mentioned properties of h and τz . The study of the analytic structure of ∆0 (ξ) and its moments Eξ [(∆0 (ξ))a ] may provide indications on whether the functions on the right-hand side of (1.10) and (1.12) can be analytically continued through the unit circle S 1 , the locus of the Lee–Yang singularities of the ferromagnetic Ising model. In the next theorem the analytic properties of the moments Ms = Eξ [∆0 (ξ)s ], s ∈ N, of ∆0 will be described. We begin by noting that they can be written as Ms = Fs + Is ,
(1.13)
where Fs := limM→∞ Fs,M with Fs,M :=
M−1 X
s an τz(n) (1)
(1.14)
n=0
and Is := limM→∞ Is,M with s Is,M := p1 · · · pM τz(M) (1) .
(1.15)
Our first result is in the following theorem, which will be proven in Sec. 3. The proof involves a careful analysis of the dynamical system induced by τz on the complex plane. Theorem 1.1 (Moments). For fixed s ∈ N, let Ms : C×[0, 1]×π 3 (z, ζ, p) 7→ Ms (z, ζ, p) ∈ C be given by (1.13). Then Ms is an analytic function in z for |z| < 1. In addition, for ζ ∈ [0, 1], Ms is a holomorphic function of z in the open set {z ∈ C; |z| > 1}\ Z, where Z is the closure of Z :=
[
Za ,
a∈N
with Za being the set of the 2a+1 − 1 solutions of the equation 1 = 0. τz ◦ · · · ◦ τz (1) + | {z } ζz a-times
Furthermore, Za are self-conjugated sets (i.e., z ∈ Za if z ∈ Za ) satisfying Za ∩ [0, ∞) = ∅ and Z 0 ≡ Z \Z ⊂ S 1 . There exists a ζ0 ∈ [0, 1/3] with ζ0 ' 0.29559a such that, for ζ ∈ [0, ζ0 ), we have the inclusion o n Za ⊂ z ∈ C : 1 < |z| < (1/ζ)1/(a+1) . We conjecture that Z 0 = S 1 for ζ ∈ [0, 1/3]. a See also Remark 3.13.
(1.16)
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J. C. A. BARATA and D. H. U. MARCHETTI
z=1
Fig. 2. The location of the set of poles Z and the unit circle S 1 . The points of Z accumulate on subsets Z 0 of the unit circle. Numerical computations indicate that Z 0 = S 1 in the ferromagnetic phase. (a+1)
The elements of the set Za are poles of τz (1). These singularities are removable in the magnetization (1.6). Theorem 1.1 states that the set of singular points of Ms is located in D>1 and accumulate on a subset Z 0 of S 1 . Figure 2 describes the singular set Z. We conjecture that the accumulation set Z 0 coincides with the accumulation set of the Lee–Yang singularities of the magnetization at the origin, studied in [1]. Numerical computations seem to confirm this idea. The inclusion (1.16) indicates how fast the sets Za tend to accumulate on S 1 for ζ ∈ [0, ζ0 ). As a consequence of the recurrence relation (1.9), we are also able to compute the two-point function explicitly. The next result shows that the fluctuation on the Cayley tree is bounded by that of the one-dimensional lattice model. Theorem 1.2 (Two-Point Function). The truncated two-point function hσ0 ; σx iN (ξ) := hσ0 σx iN (ξ) − hσ0 iN (ξ) hσx iN (ξ) can be written as hσ0 ; σx iN (ξ) = s(z∆0 )
n Y
t(ζj , z∆j )
j=1
where n = n(x) = dist(0, x) is the generation of the site x, s(x) := 1 −
1−x 1+x
2
(1.17)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
and t(x) = t(ζ, x) :=
x−1
...
757
ζ −1 − ζ . + x + ζ −1 + ζ
For all ζ ∈ [0, 1], z ∈ R+ and p ∈ πa , 0 ≤ a ≤ 1, the following identity holds for the expected value Eξ hσ0 ; σx i(ξ) = limN →∞ Eξ hσ0 ; σx iN (ξ): Eξ hσ0 ; σx i(ξ) =
∞ X
ak s(wk )
n Y
t(wk−j ) + a s(w)[t(w)]n ,
(1.18)
j=1
k=n
(n)
where w = w(ζ, z) is the limit point of the sequence wn = z τz (1), n ∈ N.
Remark 1.3. A simple upper bound on (1.18) for the paramagnetic phase can be obtained by using Perez’s self-avoiding random walk estimate [3]. Only one term, corresponding to the single self-avoiding path connecting 0 to x, contributes to the correlation functions in Ck . This fact shows that the Ising model on the Cayley tree behaves, on what concerns the asymptotic behavior of the correlations, as a one dimensional system. As a consequence, the correlation length is always finite, even at the transition point. (Note that |t(ζ, x)| ≤ 1/2 if ζ ∈ [1/3, 1] and x ∈ R+ .) Our last results shows that the quenched susceptibility at the origin χ := limN →∞ χN given by X Eξ hσ0 ; σx i(ξ) (1.19) χN := x∈C2,N
diverges as |ζ − ζc |γ at the critical point ζc = 1/3 with an exponent given by the classical theory γ = 1. More precisely, Theorem 1.4 (Susceptibility). The quenched susceptibility at origin χ = χ(ζ, z) is finite for all ζ ∈ [0, 1], z ∈ R+ \{1}. and p ∈ π. Moreover, limz→1 χ is also finite provided ζ 6= ζc . In addition, for p ∈ πa with 0 < a ≤ 1, the following asymptotic behavior χ(ζ, 1) ∼ Cη |η|γ
as
η := ζ − ζc → 0
(1.20)
holds with γ = 1, limη&0 Cη = 4a/3 and limη%0 Cη = 2a/3 provided the condition ∞ X
n an < ∞
n=0
is satisfied. The proofs of Theorems 1.2 and 1.4 will be given in Sec. 2.
(1.21)
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J. C. A. BARATA and D. H. U. MARCHETTI
2. The Two-Point Correlation Function This section is dedicated to the proof of Theorems 1.2 and 1.4. We shall consider the truncated two-point function hσ0 ; σx iM (ξ) = hσ0 σx iM (ξ) − hσ0 iM (ξ)hσx iM (ξ) , where hσ0 σx iM (ξ) =
X 1 σ0 σx e−βH(σ;ξ) ZM (ξ) σ
(2.1)
(2.2)
for all x ∈ C2, M with M large enough. 2.1. The one-point function Let us first recall the iteration procedure leading to expression (1.6) for the magnetization at the origin. We start by computing the partition function ZM (ξ) in a finite tree with M generations. Let Zj = (Zj+ , Zj− ), j = 0, 1, . . . , M , be a sequence of two-component vectors defined recursively by σ := Zj−1
!2
X
0
0
eβξj σσ eβhσ Zjσ
0
σ0 =±1
2 (1−σ)/2 + (1+σ)/2 = (ζj z)−1 ζj Zj + ζj z Zj−
(2.3)
+ − = ZM = 1. with ZM If the spin configurations are summed starting from the branches towards the root, the partition function ZM (ξ) can be written as
ZM (ξ) = z −1/2 Z0+ + z 1/2 Z0− .
(2.4)
To compute the one-point function hσ0 iM (ξ) =
X 1 σ0 e−βH(σ;ξ) , ZM (ξ) σ
(2.5)
we repeat the procedure leading to (2.4) for the numerator in (2.5). Except by the last summation on the spin at the root, all remaining ones give exactly the previous expressions. We thus have hσ0 iM (ξ) =
z −1/2 Z0+ − z 1/2 Z0− 1 − z∆0 , + − = −1/2 1/2 1 + z∆0 z Z0 + z Z0
where ∆j := Zj− /Zj+ . From (2.3), we have ∆j−1 =
ζj + z∆j 1 + ζj z∆j
2 = τj,z (∆j )
(2.6)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
759
with ∆M = 1. Recall that ∆j is a random variable since ζj = e−2βξj with ξj as given by (1.2). 2.2. The two-point function To compute the numerator of (2.2), we repeat the steps in the calculation of the partition function ZM (ξ). Our aim is to derive an expression analogous to (2.4). Let Z˜j = (Z˜ + , Z˜ − ), j = 0, . . . , M , be a sequence of two-component vectors j
j
defined recursively as in the following. For j = n0 + 1, . . . , M , with n0 =dist(0, x), we have Z˜jσ = Zjσ , i.e., Z˜jσ satisfy + − Eq. (2.3) with initial conditions Z˜M = Z˜M = 1. For j ≤ n0 , we consider a linear transformation of the form σ σ = Zj−1 Z˜j−1
ζ (1−σ)/2 Z˜j+ + ζ (1+σ)/2 z Z˜j− ζ (1−σ)/2 Zj+ + ζ (1+σ)/2 z Zj−
,
(2.7)
with Z˜nσ0 = σZnσ0 . We now observe that the sum over all spin configurations in the numerator of (2.2), except by spin at the origin, is determined by (2.7) and the sum over all spin configurations in the denominator is determined by (2.3). The two-point function (2.2) can thus be written in the following form: hσ0 σx iM (ξ) =
Z˜0+ − z Z˜0− . Z0+ + z Z0−
(2.8)
In the following lemma Eq. (2.8) will be reorganized and reexpressed in terms of the quantities Zjσ , j = 0, 1, . . . , M and σ = +, −. Lemma 2.1. The sequence of vectors Z˜n , n = 1, . . . , n0 , defined by (2.7), can be written as σ σ = (An + σBn )Zn−1 , (2.9) Z˜n−1 where An = An Bn+1 + · · · + An0 −1 Bn0 + An0
(2.10)
Bj = Bj Bj+1 . . . Bn0 ,
(2.11)
and with j = n, . . . , n0 . Here Aj = Aj (ζj , z) and Bj = Bj (ζj , z), j = 1, . . . , n0 , are random variables given by Aj = and Bj =
(z∆j )−1 − z∆j (z∆j )−1 + z∆j + ζj−1 + ζj ζj−1 − ζj (z∆j )−1 + z∆j + ζj−1 + ζj
(2.12)
.
(2.13)
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J. C. A. BARATA and D. H. U. MARCHETTI
Proof. We shall prove Lemma 2.1 by induction. We let j = n0 and observe that, −(1−σ)/2 (z∆n0 )−1 + by multiplying the numerator and the denominator of (2.7) by (ζj −(1+σ)/2
ζj
)/Zn+0 , it can be written as (z∆n0 )−1 − z∆n0 + ζj−σ − ζjσ Z˜nσ0 −1 = Znσ0 −1 (z∆n0 )−1 + z∆n0 + ζj−σ + ζjσ =
(z∆n0 )−1 − z∆n0 + σ(ζj−1 − ζj ) (z∆n0 )−1 + z∆n0 + ζj−1 + ζj
= An0 + σBn0 .
(2.14)
Now, let j = n + 1. Assuming (2.9) valid, (2.7) can be written as (1−σ)/2 (1+σ)/2 ζj − ζj z∆n+1 Z˜nσ = A + Bn+1 . n+1 σ (1−σ)/2 (1+σ)/2 Zn ζj + ζj z∆n+1
(2.15)
Multiplying the numerator and the denominator of the second term on the right−(1−σ)/2 −(1+σ)/2 (z∆n+1 )−1 + ζj , gives hand side of this equation by ζj (z∆n+1 )−1 − z∆n+1 + σ(ζj−1 − ζj ) Z˜nσ = An+1 + Bn+1 (z∆n+1 )−1 + z∆n+1 + ζj−1 + ζj Zˆnσ = An+1 + (An + σBn )Bn+1 which, in view of (2.10) and (2.11), concludes the proof of Lemma 2.1.
Now we proceed with the proof of Theorem 1.2. Proof of Theorem 1.2. Using (2.9) to compute (2.8), gives hσ0 σx iM (ξ) =
(A1 + B1 )Z0+ − z(A1 − B1 )Z0− Z0+ + zZ0−
= A1
1 − z∆0 + B1 . 1 + z∆0
(2.16)
The one-point function at x can be computed by a procedure analogous to one described at the beginning of this subsection. The difference between this onepoint function and the two-point function is the spin at origin. As before, (2.7) with j = 0, . . . , n0 is of relevance for the description of the numerator of hσx i. The iteration gives an expression of the form (2.8) with the minus sign replaced by plus. The one–point function can thus be written as hσx iM (ξ) =
1 − z∆0 Z˜0+ + z Z˜0− . + − = A1 + B1 1 + z∆0 Z0 + z Z0
(2.17)
THE TWO-POINT FUNCTION AND THE EFFECTIVE FUGACITY
...
761
Inserting (2.16) and (2.17) into (2.1) and taking into account (2.6), yields " 2 # 1 − z∆0 B1 . (2.18) hσ0 ; σx iM (ξ) = 1 − 1 + z∆0 When the thermodynamic limit, M → ∞, has been taken, the random variables ∆j ’s in (2.18) can be replaced by their limits limM→∞ ∆j (recall that B1 depends 0 , each of which defined by a recursion relation with initial condition on {∆j }nj=0 ∆M = 1 dependent on the generation M ). Since ∆j is bounded from above and below, the convergence in distribution is guaranteed by the convergence of their moments [4]. It follows from Theorem 1.1 that limM→∞ Eξ ∆j exists and is a real analytic function of z in z ∈ R+ \{1}. In order to take the expectation value of (2.18) we shall use the following property: for any bounded function f (ξ) = f (ξj , ξ 0 ) of the random variables ξ = (ξj , ξ 0 ), we have ζ −1 − ζ (2.19) Eξ f (ξ)Bj (ξ) = pj Eξ0 f (1, ξ 0 ) (z∆j )−1 + z∆j + ζ −1 + ζ since ζj−1 − ζj = e2βξj − e−2βξj = 0 if ξj = 0. Define 2 1−x s(x) := 1 − 1+x and ζ −1 − ζ . t(x) = t(ζ, x) := −1 x + x + ζ −1 + ζ Using (2.19) in the expected value of (2.18) gives n0 Y Eξ hσ0 ; σx iM (ξ) = p1 · · · pn0 Eξ0 s(z∆0 ) t(z∆j )
(2.20)
(2.21)
j=1
=
M−1 X k=n0
ak s(wk )
n0 Y
j=1
t(wk−j ) + p1 · · · pM s(wM )
n0 Y
t(wM−j ) ,
j=1
(2.22) (n) zτz (1), n
∈ N. Here, we have first taken partial expectations with where wn = respect to the variables ξ1 , . . . , ξn0 and have used, in the sequel, for all remaining expectations, that the sequence ∆j , j ∈ N, satisfies the recurrence relation z∆j−1 = zhj (z∆j ) with hj (1) = 1. Equation (1.18) then follows since s(x) and t(x) are continuous in R+ and wn converges to the solution w = w(ζ, z) of the fixed point equation w = zh(w) in this domain provided z ∈ R+ . This concludes the proof of Theorem 1.2. We now turn to the proof of Theorem 1.4 on the quenched susceptibility at origin χ. We note the following facts on the function s and t (the proof will be omitted):
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J. C. A. BARATA and D. H. U. MARCHETTI
Proposition 2.2. The function s : w ∈ R+ 7→ s(w) ∈ R+ and t : (ζ, w) ∈ [0, 1] × R+ 7→ t(ζ, w) ∈ R+ given by (2.20) and (2.21), respectively, have a maximum value at w = 1 with s(1) = 1 and t(ζ, 1) = (1 − ζ)/(1 + ζ), are monotonically increasing function of w in [0, 1] and satisfy s(w) = s(w−1 ) and t(ζ, w) = t(ζ, w−1 ). For z ∈ [0, 1) we recall that wn , n ∈ N, is a monotonically decreasing sequence with wn < 1 and for each n, wn = wn (z) is monotonically decreasing function of z in this domain. So, in view of Proposition 2.2, 2t(ζ, wn ) ≤ 2
1−ζ <1 1+ζ
(2.23)
holds for all ζ ∈ (1/3, 1] and z ∈ R+ \{1}. In addition, for ζ ∈ [0, 1/3) (ferromagnetic phase) and z ∈ [0, 1), wn converges to a nontrivial solution w(ζ, z) of w = zh(w) (see [1]) with w % w as z % 1 with 1 − 2ζ − ζ 2 1 − ζ2 − w= 2ζ 2 2ζ 2
1 − 3ζ 1+ζ
1/2 .
This and Proposition 2.2 implies that there exist finite number n0 = n0 (ζ, z) ∈ N such that for n > n0 2ζ <1 (2.24) 2t(ζ, wn ) ≤ 2t(ζ, w) = 1−ζ holds for all ζ ∈ [0, 1/3) and z ∈ R+ \{1}. Substituting Eq. (1.18) into (1.19) and taking into account the inequalities (2.23) and (2.24), gives ∞ n ∞ ∞ X X Y X 2n ak s(wk ) t(wk−j ) + a s(w) [2t(w)]n χ= n=0
=
∞ X
ak s(wk )
k X
2n
n=0
k=0
Note that lim
z→1
n=0
j=1
k=n
n Y
t(wk−j ) +
j=1
a s(w) ∼ Cη |η| 1 − 2t(w)
a s(w) . 1 − 2t(w)
(2.25)
as η := ζ − ζc → 0
holds with limη&0 Cη = 4a/3 and limη%0 Cη = 2a/3. Note also that, for p ∈ πa , P 0 < a ≤ 1, such that n n an < ∞, the first term on the second line of (2.25) does not diverge at z = 1 for any ζ ∈ [0, 1] since, by the dominated convergence theorem, lim
x%1
∞ X 1 − xk ak < ∞ . 1−x k=0
A non-classical exponent γ = γ(p) may appear if (1.21) is violated. This concludes the proof of Theorem 1.4.
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763
3. The Effective Fugacity and the Analyticity Domain of Mr In this section we will prove Theorem 1.1 concerning the analyticity domain of Mr . We split the proof in two parts, the first, and simplest, dedicated to the function Ir and the second, and more elaborated, to the function Fr . The study of the singularities of Fr involves a careful analysis of the dynamical system induced by τz on the complex plane. Let us first establish the analyticity results on the functions Ir (z), r ∈ N. Let us denote by D<1 the open unit disk in S 2 and by D>1 the complement of its closure. More generally, for a > 0 call D
(3.1)
D>a := {w ∈ S 2 : |w| > a} .
(3.2)
For further purposes define also for a, b ∈ R+ , a < b, Da,b := {w ∈ S 2 : a < |w| < b} .
(3.3)
The following theorem has been proven in [1]: (n)
Theorem 3.1. The sequence τz (1), n ∈ N, of analytic functions on D<1 converges uniformly to an analytic function τ = τ (z) on the whole set D<1 . For all ζ ∈ (0, 1] the function τ = τ (z) fulfills the fixed point equation τ = h(zτ ) on the whole set D<1 . As a consequence, τ (z) has no zeros on D<1 . Analyticity of Ir (z) on D<1 is, actually, the statement of Theorem 3.1. Let us now consider Ir (z) on D>1 . For r ∈ N, p ∈ πa and z ∈ D>1 , we have h h ir i−r −r (n) = a τ (z −1 ) . Ir (z) = a lim τz(n) (1) = a lim τz−1 (1) n→∞
n→∞
(3.4)
Therefore, in view of Theorem 3.1 which says that τ (w) has no zeros for w ∈ D<1 , Ir (z) is an analytic function on D>1 . 3.1. The analyticity domain of Fr In this subsection we will establish the analyticity results on the functions Fr (z), r ∈ N. The proof will be given in two theorems, according to whether z ∈ D<1 or z ∈ D>1 . Let us prove our first result on the analyticity of Fr (z). Theorem 3.2 (The First Analyticity Theorem). The function Fr (z) defined in (1.14) is an analytic function of z in the open set D<1 for all p ∈ π and ζ ∈ [0, 1).
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J. C. A. BARATA and D. H. U. MARCHETTI
Proof. The partial sums Fr,N (z) :=
N X
r an τz(n) (1) ,
r, N ∈ N ,
(3.5)
n=0
are analytic in D<1 since the pole of h lies in D>1 . Moreover, one has |τz (1)| < 1 for z ∈ D<1 and so, for any ε > 0 one has |Fr,N (z) − Fr,M (z)| <
M X
an ≤ ε ,
n=N +1
for all N and M , N < M , large enough, since the sequence {an , n ∈ N} is summable. So, the partial sums form a uniform Cauchy sequence of analytic functions whose limit exists and is analytic in D<1 . We will now prove the analyticity of the function Fr (z) in D>1 \Z. We start by studying the location of the singularities of Fr (z). First we prove the following lemma. (n)
Lemma 3.3. Let B ⊂ C be an open set such that τz (1), n ∈ N, are analytic on B and (n) −1 > 0. (3.6) inf inf τz (1) − z∈B n∈N zζ Then Fr (z) is an analytic function in B. Remark 3.4. Note that h given by (1.7) has a unique double pole at w = −1/ζ. Moreover, h(w) is bounded on C\O, where O is any open set containing the pole. Remark 3.5. Note that inf n∈N |τz (1) − (−z −1 /ζ)| = 0 is not a necessary (n) condition for Fr to be singular at z. For instance, for z = −1, one has |τz (1) − −1 (−z /ζ)| = 1/ζ − 1 for all n ∈ N. However, as we will see below, Fr (z) is not analytic at z = −1 because −1 is an accumulation point of poles of Fr . (n)
Proof of Lemma 3.3. If condition (3.6) holds for a given B it implies that there exists an open neighborhood O of −1/ζ such that, for all z ∈ B and for all (n) (n+1) (n) (1) = h(zτz (1)), this means that there exists n ∈ N, zτz (1) ∈ C \ O. Since τz (n) a positive constant M , such that for all z ∈ B and all n ∈ N, |τz (1)| ≤ M . Hence, by the same argument used in the proof of Theorem 3.2, the partial sums Fr,N (z) form a uniform Cauchy sequence of analytic functions in B and consequently, their limit exists and is analytic in B. Our task now is to localize the points z ∈ C where the condition (n) −1 =0 inf τz (1) − n∈N zζ
(3.7)
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holds. The next proposition shows that, for each n ∈ N, there exist at least one (n) point z ∈ C such that |τz (1) − (−z −1 /ζ)| = 0. Proposition 3.6. For each n ∈ N there is at least one and at most 2n+1 − 1 distinct solutions in C of the equation τz(n) (1) −
−1 zζ
= 0.
(3.8)
(n)
Proof. We begin by observing that, τz (1) is the ratio of two polynomials of (n) degree 2n+1 − 2 : τz (1) = Pn (z)/Qn (z). This is a consequence of the fact that the function h is the ratio of two polynomials in z of degree 2. To be more explicit, Pn and Qn are defined recursively through Pn+1 (z) := (ζQn (z) + zPn (z))2 ,
n ∈ N,
Qn+1 (z) := (Qn (z) + ζzPn (z))2 ,
n∈N
(3.9)
with P0 (z) = Q0 (z) := 1. Therefore, (3.8) means that zPn (z) +
1 Qn (z) = 0 , ζ
(3.10)
which has at least one and at most 2n+1 − 1 possible distinct solutions in C, since the left-hand side is a polynomial of degree 2n+1 − 1. Clearly, for n = 0, the unique solution is the pole of h, z = −1/ζ. Let us denote by Zn the finite set of all z’s satisfying (3.8) for n ∈ N. Define Z :=
[
Za
a∈N
and let Z be the closure of Z. The set difference Z 0 := Z \Z is the set of accumulation points of Z. To establish that Fr is holomorphic in C\Z we need to describe where the elements of Z and Z 0 are located. Theorem 3.7. For all ζ ∈ (0, 1) and k ∈ N one has Zk ⊂ D>1 with the accumulation points of Z lying in the unit circle, i.e., Z 0 ⊂ S 1 . In addition, Zk ∩ [1, ∞) = ∅, for all k ∈ N. The sets Zk are also self-conjugated in the sense that z ∈ Zk if z ∈ Zk and, for all k ∈ N, one has Zk ∩ (−∞, −1) 6= ∅. This last fact together with Z 0 ⊂ S 1 implies that −1 ∈ Z 0 .
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Proof. Firstly, observe that the equation τz(k) (1) = −
1 zζ
(3.11) (k)
has no solutions for z ∈ D1 := D<1 ∪ S 1 , since τz (1) maps D1 into itself and ζ < 1. This equation has no solutions for |z| ≥ 1/ζ either, since in this region, the left-hand side of (3.11) lies in D>1 , but the right-hand side in D1 . We conclude that Z ⊂ D1,1/ζ . Analogously, (3.11) has no solutions in (1, ∞), since there its left-hand side is positive while the right-hand side is negative. To prove that Z 0 ⊂ S 1 we proceed as follows. The elements of Zn are the poles (n+1) (n+1) (1) in D>1 . Therefore, if w ∈ Zn , w−1 is a zero of τz (1) in D<1 . of τz (n) The sequence of functions τz converges uniformly in D<1 to an analytic function τ = τ (z) (Theorem 3.1). According to a theorem by Hurwitz (see, e.g., [5]) the fact that τ has no zeros on D<1 (Theorem 3.1) implies that the set of zeros of the (n) functions τz has no limit points on D<1 and therefore, if they exist, they must lie on S 1 . The sets Zk are all self-conjugated by the definition of h, since the left-hand side of (3.10) is a polynomial with real coefficients. This polynomial is also of odd degree, namely 2k+1 −1. Both facts together imply that this polynomial has at least one real root. However, according to the previous remarks, the real roots cannot lie in [−1, ∞). So, they must be negative and lower than −1. From the fact that Z 0 ⊂ S 1 it follows that these negative roots must converge to −1 when k → ∞ and so −1 ∈ Z 0 . This shows, in particular, that Z 0 6= ∅. Remark 3.8. Numerical computations seem to indicate that Z 0 = S 1 for the whole ferromagnetic region 0 ≤ ζ < 1/3. Unfortunately we were not able to prove this conjecture. (n)
After these considerations on the location of the singularities of τz the proof of the analyticity statement on Fr .
we turn to
Lemma 3.9. Let B be an open set such that B ⊂ D>1 \Z, where B is the (n) closure of B. Then the sequence τz (1), n ∈ N, converges to the function 1/τ (z −1 ) uniformly on B. Proof. For z ∈ B call w ≡ z −1 the “image” of z, and let the open set Bi given by Bi := {w ∈ D<1 such that w−1 ∈ B} be the “image set” of B. From Theorem 3.1 we know that τ (w) has no zeros on D<1 . Hence, for w ∈ Bi , |τ (w)| > ρ1 , for some positive ρ1 . Next, we argue that there is also a positive (n) constant ρ2 such that |τw (1)| > ρ2 for all n ∈ N and all w ∈ Bi . For, notice that (n) no element of the sequence of functions τw (1) has a zero in Bi . Since the sequence
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converges uniformly and the limit function has no zeros in Bi , there must be a (n) positive constant such that |τw (1)| > ρ2 for all w ∈ Bi and n ∈ N. The desired uniformity follows now from (n) (n) 1 1 τ (w) − τw (1) τz − = |τ (w) − τw(n) (1)| < ε (3.12) ≤ −1 (n) τ (w)τw (1) ρ1 ρ2 τ (z ) for any prescribed ε > 0, uniformly on Bi and for n large enough since, on Bi , the (n) sequence τw (1) converges uniformly to τ (w). Lemma 3.10. Let B be as in Lemma 3.9. Then the condition −1 =0 inf inf τz(n) (1) − zζ z∈B n∈N
(3.13)
is impossible for any ζ ∈ (0, 1]. (n)
Proof. By the definition of B, the condition τz (1) + 1/(zζ) = 0 does not hold for any finite n. Therefore, condition (3.13) says that 1 −1 − = 0. inf −1 ) zζ z∈B τ (z Since B is closed and 1/τ (z −1 ) + 1/(zζ) is a continuous function of z on B, this implies the existence of a point u ∈ B such that τ (u−1 ) = −u ζ. Since τ (w) fulfills the fixed point equation τ˜ = h(w τ˜) for w ∈ D<1 , we conclude that −u ζ = h(−ζ) ≡ 0. But this is impossible for ζ > 0 and u ∈ D>1 . Theorem 3.11 (The Second Analyticity Theorem). The function Fr (z) defined in (1.14) is a holomorphic function of z in the open set D>1 \Z for all p ∈ π and ζ ∈ (0, 1]. If an 6= 0 the singularities of Fr at the points of Zn are poles of order not greater than (2n+1 − 1)r. Proof of Theorem 3.11. Pick an open set B such that B ⊂ D>1 \Z, where B is the closure of B. Then, analyticity of Fr on B follows from Lemmas 3.3 and 3.10. Since D>1 \Z can be covered by such open sets the theorem is proven. (n)
3.2. Estimates on the location of the poles of τz
(1)
To study of how fast the sets Zk accumulate on S 1 we will make use of a contraction theorem, described below, on the inverse mappings of h. First, some definitions. For w ∈ C, w 6= 1/ζ, define g(w) :=
ζ −w ζw − 1
(3.14)
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and let u1/2 be some branch of the square root function in C. Define 1/2 ), h−1 + (u) := g(u
1/2 h−1 ). − (u) := g(−u
(3.15)
Then one has (h ◦ h−1 ± )(z) = z, ∀ z ∈ C. Theorem 3.12 (Contraction Theorem). There exists a number ζ0 ∈ [0, 1/3), whose approximate value is ζ0 ' 0.29559, such that for ζ ∈ [0, ζ0 ) there exists a strictly positive function e(ζ) such that for all u ∈ D1,a(ζ) with a(ζ) := ζ −1 + e(ζ) one has (3.16) |h−1 ± (u)| < |u| . Remark 3.13. Numerical computations indicate to be impossible to improve the region of validity of Eq. (3.16) to ζ ≥ ζ0 . Remark 3.14. As already observed, the inequality (3.16) becomes an equality in S 1 , which is the internal boundary of D1,a(ζ) . It is important to note also that the set D1,a(ζ) contains the pole z = −1/ζ, of h. We will present the proof of the Contraction Theorem in Appendix A. Let us now explore some of its consequences. Theorem 3.15. Let ζ0 as in Theorem 3.12. For all ζ ∈ (0, ζ0 ) and k ∈ N Zk ⊂ D1,rk ,
(3.17)
−1
holds with rk = ζ k+1 . This theorem illustrates explicitly that the accumulation points Z 0 lie in the unit circle and shows how fast the sets Zk converge to it, at least for ζ ∈ (0, ζ0 ). Proof. The proof of Theorem 3.15 makes use of the Contraction Theorem which requires the following technical lemma. We note that, from Eq. (3.11), there exists a finite sequence of signals {s(l) ∈ {−, +}, 1 ≤ l ≤ k} such that 1 1 1 −1 1 −1 1 hs(k) hs(k−1) · · · h−1 · · · = 1. (3.18) − z z z z s(1) zζ Lemma 3.16. Given z ∈ Zk , k ∈ N, k ≥ 1, consider a sequence of signals {s(l) ∈ {−, +}, 1 ≤ l ≤ k} satisfying (3.18) and define 1 −1 1 1 1 −1 1 h · · · h · · · , l ∈ {1, . . . , k} . − wl := h−1 z s(l) z s(l−1) z z s(1) zζ Then all wl ’s belong to D1,a(ζ) , except, of course, wk which is equal to 1.
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Proof. To prove Lemma 3.16 we take, without loss of generality, k > 1 and −1 /ζ) ∈ D1,1/ζ , by the Contraction note that, since z ∈ D1,1/ζ , one has h−1 s(1) (−z Theorem. Hence, w1 ∈ Dζ,1/ζ . On the other hand w1 cannot belong to Dζ,1 ∪ S 1 for the following reason: h−1 ± maps D<1 ∪ S 1 into itself and, since 1/|z| < 1, this implies w2 ∈ D<1 , which, in turn, would imply that all subsequent wl ’s would also be in D<1 and so they could not reach the point 1 after a finite number of steps. Therefore, w1 ∈ D1,1/ζ and we can apply again the Contraction Theorem and claim that h−1 s(2) (w1 ) ∈ D1,1/ζ . So, we conclude again that w2 ∈ Dζ,1/ζ . If k = 2 then, by the hypotheses, w2 = 1. Otherwise, if k > 2, repeating the previous arguments, we conclude again that w2 ∈ D1,1/ζ . This argument can be repeated a finite number of times, thus proving the lemma. Lemma 3.16 implies that all wl , 1 ≤ l < k, lie in the region of validity of the Contraction Theorem. Therefore, applying the inequality (3.16) repeatedly in (3.18) we get ζ|z|k+1 < 1, which proves (3.17). The fact that Z 0 ⊂ S 1 follows easily from (3.17). This completes the proof of Theorem 3.15. The Contraction Theorem has another consequence. Theorem 3.17. For the sets Zk defined above we have, in the region of validity of the Contraction Theorem, that Zl ∩ Zm = ∅, for all l, m ∈ N, l 6= m. Proof. Let m > l and pick a z ∈ Zl ∩ Zm . Then one has 1 1 =− . τz(m−l) − zζ zζ This means also that there exists a finite sequence {s(a) ∈ {−, +}, 1 ≤ a ≤ m − l} such that 1 1 −1 1 −1 1 1 −1 1 − = hs(m−l) h · · · hs(1) − ··· . (3.19) zζ z z s(m−l−1) z z zζ Applying the Contraction Theorem to the last equality we get 1 1 < m−l+1 , |z|ζ |z| ζ what means |z|m−l < 1, a contradiction since Zl ∩ Zm ⊂ D1,1/ζ .
A. The Contraction Theorem This appendix is devoted to the proof of Theorem 3.12 which is implied by the following theorem. Theorem A.1. There exists a ζ0 ∈ [0, 1/3), whose approximate value is ζ0 ' 0.29559, such that there is a strictly positive function f (ζ) in the interval ζ ∈ [0, ζ0 )
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such that, in the open set defined by 1 < |w| < √1 + f (ζ), w ∈ C, one has |g(w)| < ζ
|w|2 . In view of (3.14), (3.15) and Theorem A.1, for u ∈ C such that 2 1 1 1 < |u| < √ + f (ζ) =: + (ζ) , ζ ζ 1/2 one has |h−1 )| < |u|. This concludes the proof of Theorem 3.12. ± (u)| = |g(±u
Proof of Theorem A.1. Writing w = x + iy, with x, y ∈ R, a simple computation shows that (ζ 2 + |w|2 ) − 2ζx . (A.1) |g(w)|2 = (1 + ζ 2 |w|2 ) − 2ζx To analyze the right-hand side, consider the following lemma. Lemma A.2. Let t : R 7→ R, be given by t(a) := (α − βa)(γ − βa), for α, β and γ ∈ R with α > γ > 0 and β ≥ 0. Consider the open interval J := (−∞, γ/β). Then, t is a continuous, strictly positive, nowhere constant (for β > 0) and increasing function in J. Proof. Continuity on J is obvious. A computation shows that t0 (a) = β(α−γ)/ (γ − βa)2 . All claims follow immediately from continuity and from this relation. Looking at (A.1) we can make the following identifications: α ≡ ζ 2 + |w|2 ;
β ≡ 2ζ ;
γ ≡ 1 + ζ 2 |w|2 ;
a ≡ x.
(A.2)
Note that, in view of this α − γ = (1 − ζ 2 )(|w|2 − 1) > 0 ,
(A.3)
for all w ∈ D>1 . Moreover, the condition a < γ/β means x<
1 + ζ 2 |w|2 . 2ζ
This condition is always satisfied if |w| < 1/ζ, since x ≤ |w| and since the stronger condition 1 + ζ 2 |w|2 |w| < 2ζ is equivalent to the condition (1 − ζ|w|)2 > 0. In view of Lemma A.2 we conclude that, for w ∈ D1,1/ζ , |g(w)|2 ≤
(ζ − |w|)2 ζ 2 + |w|2 − 2ζ|w| = . 2 2 1 + ζ |w| − 2ζ|w| (1 − ζ|w|)2
Finally, note that D1,1/√ζ+f (ζ) ⊂ D1, 1/ζ for a convenient f . Now we come to the technically most important result of this appendix.
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Theorem A.3. Under the assumptions of Theorem A.1 one has (ζ − |w|)2 < |w|4 . (1 − ζ|w|)2
(A.4)
Proof. Condition (A.4) is equivalent to the condition that the polynomial P (t) := (ζ − t)2 − t4 (1 − ζt)2
(A.5)
is strictly negative for 1 < t < √1 + f (ζ). Writing it explicitly one has ζ
P (t) = −ζ 2 t6 + 2ζt5 − t4 + t2 − 2ζt + ζ 2 = ((ζ − t) + t2 (1 − ζt))((ζ − t) − t2 (1 − ζt)) . Using the fact that ±1 is a root of the polynomial (ζ − t) ± t2 (1 − ζt), we can write P (t) = (t − 1)(t + 1)(−ζt2 + (1 − ζ)t − ζ)(ζt2 − (1 + ζ)t + ζ) . This last relation allows us to find explicitly the roots of P (t): they are the elements of the set {−1, +1, a−, a+ , b− , b+ }, where a± := and
b± :=
1−ζ 2ζ
1+ζ 2ζ
±
1p (1 + ζ)(1 − 3ζ) 2ζ
±
1p (1 − ζ)(1 + 3ζ) . 2ζ
The following lemma gives the possible signs of the polynomial P (t) on the real axis. Lemma A.4. For a± and b± defined above as functions of ζ, one has the following results: (1) (2) (3) (4)
For 0 ≤ ζ < 1/3 one has 0 ≤ a− ≤ 1. For 0 ≤ ζ < 1 one has 0 ≤ b− ≤ 1. For 0 ≤ ζ < 1/3 one has b+ − a+ ≥ 1. √ For 0 ≤ ζ < ζ0 one has a+ − 1/ ζ > 0, where ζ0 ∈ [0, 1/3), ζ0 ' 0.29559. This in particular says that a+ > 1 and, by (3), b+ > 1.
Before we prove this lemma, let us finish the proof of Theorem A.3 and, hence, of the Theorem A.1. According to Lemma A.4, P (t) < 0 if 1 < t < a+ . This follows from the localization of the roots and from the fact that P is a polynomial of even degree with a negative leading term. See Fig. 3.
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P(t)
1
-1
a+
0
b+ t
a_ or b_
Fig. 3. The graphic of P (t).
√ The proof is completed by defining f (ζ) := a+ (ζ) − 1/ ζ, which, according with item 4 of Lemma A.4, is strictly positive for 0 ≤ ζ < ζ0 . This completes also the proof of the Contraction Theorem. Proof of Lemma A.4. We will prove each of the items of Lemma A.4 separately. Proof of item (1). The hypothesis that a− ≥ 0 is equivalent to (1 − ζ)2 ≥ (1 + ζ)(1 − 3ζ) for 0 ≤ ζ < 1/3. This last relation means 4ζ 2 ≥ 0, which is obviously verified. The hypothesis that a− ≤ 1 is equivalent to p 1 − (1 + ζ)(1 − 3ζ) ≤ 3ζ , which is equivalent to (1 − 3ζ)2 ≤ 1 − 2ζ − 3ζ 2 . This means 4ζ(3ζ − 1) ≤ 0, what is true for 0 ≤ ζ < 1/3, the equality holding only if ζ = 0. Proof of item (2). The hypothesis that b− ≥ 0 is equivalent to (1 + ζ)2 ≥ (1 − ζ)(1 + 3ζ), which is equivalent to 4ζ 2 ≥ 0, which is, of course, always true. The hypothesis that b− ≤ 1 is equivalent to (1 − ζ)2 ≤ (1 − ζ)(1 + 3ζ). This means that 4ζ(ζ − 1) ≤ 0, which is always true for 0 ≤ ζ ≤ 1. Proof of item (3). Since b + − a− = 1 +
p 1 p 1 + 2ζ − 3ζ 2 − 1 − 2ζ − 3ζ 2 , 2ζ
item (3) is proven, provided the term between parenthesis above is positive. This is implied by the inequality 1 + 2ζ − 3ζ 2 ≥ 1 − 2ζ − 3ζ 2 , which is always true for ζ ≥ 0.
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√ Proof of item (4). The condition a+ > 1/ ζ means that p p (1 + ζ)(1 − 3ζ) > ζ + 2 ζ − 1 . (A.6) √ The right-hand side is strictly negative for 0 ≤ ζ < ( 2 − 1)2 ' 0.171. So, in this region the condition above is automatically satisfied. On the other hand, if the right-hand side of (A.6) is positive, we can square both sides and arrive to the equivalent condition (A.7) 4s(s3 + s2 + s − 1) < 0 √ where s = ζ. The polynomial s3 + s2 + s − 1 has one real root at s0 ' 0.543689 and two complex roots at s± ' −0.77 ± 1.115i. We call ζ0 := s20 , which gives ζ0 ' 0.295597. Thus, condition (A.7) is satisfied for 0 < ζ < ζ0 . With this the proof of Lemma A.4 is complete. References [1] J. C. A. Barata and D. H. U. Marchetti, Griffiths’ singularities in diluted Ising models on the Cayley Tree, J. Stat. Phys. 88 (1997) 231–268. [2] R. B. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23 (1969) 17–19. [3] J. F. Perez, Controlling the effect of Griffiths’ singularities in random ferromagnets: smoothness of the magnetization, Brazilian J. Phys. 23 (1993) 356–362. [4] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, John Wiley and Sons, second ed., 1971. [5] E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, second ed., 1939.
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS OF THE SIMPLEST QUANTUM FILTERING EQUATION VASSILI N. KOLOKOL’TSOV Department of Mathematics, Statistics and Operational Research Nottingham Trent University Burton St., Nottingham, NG1 4BU, UK Received 20 November 1996 Revised 29 September 1997 The paper deals with the quantum Langevin equation describing a quantum particle with continuously observed position. Special Banach spaces of entire analytic functions are introduced and studied (including a theorem of Paley–Wiener type for them), which comprise all solutions of this equation and in which the uniform convergence (as time tends to infinity) of the solutions to the Gaussian function with a fixed dispersion (selflocalisation or continuous collapse) is proved. The asymptotic behavior at infinity of the mean position and momentum of the limit Gaussian wave packet (which satisfy classical Langevin equations) is also investigated.
1. Introduction 1.1. Quantum filtering equation The main equation of the theory of continuous quantum measurement (in the case of a measurement of diffusion type) has the form 1 ? i H + R R χ dt = Rχ dQ , (1.1) dχ + h 2 where χ is the unknown aposterior (non-normalized) wave function of the given continuously observed quantum system in a Hilbert space H, h is the Planck constant, the selfadjoint operator H = H ? in H is the Hamiltonian of a free (non-observed) quantum system, the vector-valued operator R = (R1 , . . . , Rd ) in H stands for the observed physical values, and Q is the standard (input) d-dimensional Brownian motion. In this general form this equation was first obtained by V. P. Belavkin in the framework of his quantum stochastic filtering theory [7, 8]. The Belavkin ˆ (which is observed in the theorem states also that the output diffusion process Q 2 process of measurement) has the density kχk with respect to the standard Wiener measure P of the input process Q, i.e. the mean EQˆ of some functional f (χ) of the ˆ has the form state χ over all realization of the output process Q EQˆ f (χ) = EQ (f (χ)kχk2 ) , 801 Reviews in Mathematical Physics, Vol. 10, No. 6 (1998) 801–828 c World Scientific Publishing Company
(1.2)
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V. N. KOLOKOL’TSOV
where EQ denotes the mean with respect to the Wiener measure P of the process Q. In particular, averaging the mean value hAi = hAiϕ = (Aϕ, ϕ) = (Aχ, χ)/(χ, χ)
(1.3)
of some operator A in H with respect to the normalized wave function ϕ = χ/kχk we have (1.4) EQˆ hAi = EQˆ (Aϕ, ϕ) = EQ (Aϕ, ϕ)(χ, χ) = EQ (Aχ, χ) . Rewriting Eq. (1.1) in terms of the so-called innovating process W , defined by the stochastic equation dW = dQ − 2hRidt , (1.5) and the normalised state vector ϕ = χ/kχk we obtain the equation [7] 1 i ? (H − hhReRiImR) + (R − hReRi) (R − hReRi) ϕ dt dϕ + h 2 = (R − hReRi)ϕ dW ,
(1.6)
where the symmetric operators ReR = (R + R? )/2 and ImR = (R − R? )/2i are the real and imaginary parts of R. For most natural physical examples, the operator R is selfadjoint and (1.1) and (1.6) reduce respectively to the stochastic equations 1 2 i H + R χ dt = Rχ dQ (1.7) dχ + h 2
and dϕ +
1 i H + (R − hRi)2 ϕ dt = (R − hRi)ϕ dW . h 2
(1.8)
A particular case of the last equation, when the Hamiltonian H in (1.8) is equal to zero, was proposed previously by N. Gisin [35], who followed the earlier ideas from [15, pp. 2 and 3]. In the theory of continuous quantum measurement, Eq. (1.6) was discovered in an attempt to avoid the so-called quantum Zeno paradox [28, 56, 16], which appears when trying to describe a continuous measurement as a limit of sequential discrete measurements with the interval of time between the latter tending to zero. In the first deduction of (1.6) [7–9], this equation was obtained as the equation of stochastic filtering applied to the Hudson–Parthasarathy [42] quantum stochastic evolution under the nondirect continuous nondemolition measurement of diffusion type, the nondemolition condition [10, 29] being simply the commutativity relation [X(t), Y (s)] = 0 ∀t ≥ s for the Heisenberg operators of the input and output process X(t) and Y (t) respectively. Another deduction proposed first in [22] for a particular case (1.14) below was based on the previous ideas from [5, 6, 34] and on the so-called master equation, which was obtained from different points of view in [20, 21] and [54] and was used in [5] to solve quantum Zeno’s paradox in the framework of the theory of nonideal measurements. The deduction of (1.14) from the theory of instruments see in [4] and [41]. In recent years there appeared many
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physical papers that present equation (1.6) (or some its modification, where, for instance, dW is considered to be a complex Brownian motion) as a new fundamental model of classical and quantum mechanics (state diffusion model), showing how this model can be deduced from some general physical principles, and discussing possible experimental justifications. Let us point out the papers [38, 39, 57] as more theoretically oriented and papers [40, 30, 37] as more experimentally oriented. In these papers one can find also a complete bibliography. Let us mention specially the papers [36, 33], where the connection with relativity is discussed, and the papers [24, 25, 32] dedicated to the possible applications to quantum gravity. Similar ideas were proposed also in [17, 18, 55]. The discussion of various points of view on Eq. (1.14) can be found in Proceedings [12], see also [61, 3, 53]. The important property of the model (1.1), (1.6) (and also one of the important motivation for its appearence) is the possibility to describe by means of it the process of sponteneous collapse (stochastically continuous selflocalization or reduction) of quantum systems. In fact, if H = 0 in (1.7), then its solution is χ(t) = exp{−tR2 + RQ(t)}χ0 and it tends (continuously collapses), as t → ∞, to the spectral subspace of the operator R2 corresponding to the minimum of its spectrum. In particular, if R2 has nondegenerate lowest eigenvalue, then χ/kχk tends to its lowest eigenfunction. When H 6= 0, the situation is surely more complicated, see discussion and some results in [11, 39, 57, 60]. In the present paper we consider an important nontrivial case of this situation, when, on the one hand, there is a collapse of the form of the solution (dispersion of its position and other central moments of higher orders tend to a fixed limit) and, on the other hand, there is scattering of position and momentum (the latter tend to infinity according to a classical Langevin equation). From the mathematical point of view, the linear Eq. (1.1) is distinguished from the general linear stochastic equation dχ + Aχ dt = Bχ dQ (with some linear operators A, B in a Hilbert space) by its norm conservation property. Namely, applying formally Ito’s formula to the squared norm kχk2 of the solution of (1.1), we get dkχk2 = 2(Rχ, χ) dQ = 2hRikχk2 dQ , which implies (again formally) that Z t Z t 2 2 kχk = exp 2 hRi dQ − 2 hRi dt . 0
(1.9)
(1.90 )
0
Rt Consequently, if for a solution of (1.1) almost surely 0 hRi2 dt is finite (with Rt respect to the measure defined by the Wiener process Q), then 0 hRi dQ is well defined and is a local martingale, which in turn implies that kχk2 is a positive supermartingale and a local martingale. In particular, EQ kχk2 (t) ≤ 1 for all t and EQ kχk2 (min(t, τk )) = 1 for a (so-called localizing) sequence of random Markov moments τk such that almost surely τk → ∞, as k → ∞. If some additional regularity properties are satisfied (see, for instance, [46]), which insure that a positive local martingale is, in fact, a martingale (and a proof of these properties for all physically meaningful situations is an important mathematical problem, whose
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V. N. KOLOKOL’TSOV
importance in the state diffusion model, or in quantum filtering theory, can be compared with that of the proof of the selfadjointness of a formally symmetric operator in the standard quantum mechanics), then EQ kχk2 (t) = 1
(1.10)
for all times t. Therefore, equation of type (1.1) describes natural stochastic generalization of the unitary evolution (it defines an evolution that preserves the expectation of the norm squared of solutions). The measures defined by the input process Q and the innovating process W are connected by the famous Girsanov formula. Namely, the Girsanov theorem states that if (1.10) holds for all times t, i.e. if kχk2 is a positive martingale (with respect to the Wiener measure of the process Q), then the innovating process W is a Wiener martingale with respect to the measure P˜ that has the density kχk2 with respect to the measure P : (1.11) dP˜ = kχk2 dP . Consequently, the mean (1.2) is equal to Z Z EQˆ f (χ) = EQ (f (χ)kχk2 ) = f (χ)kχk2 dP = f (χ)dP˜ = EW f (χ) .
(1.12)
In the sequel, when we speak that some property of the measured quantum process is satisfied for almost all (a.a.) Q, we mean the measure P , i.e. we consider the process Q to be the standard Brownian motion, and when we speak that some property is satisfied for a.a. W , we mean the measure P˜ , i.e. we consider the process W to be the standard Brownian motion. Let us stress that the difference in these two notions is quite essential, because although the measures generated by processes W and Q are equivalent on any finite interval of time (if (1.10) holds), they are not equivalent as measures defined on the space of continuous functions on [0, ∞) (see [52]). Due to the equation EQˆ = EW , the consideration of the innovating process is equivalent to the consideration of the output process, and consequently, from the point of view of the theory of measurement, all results should be formulated in terms of W (so, the process Q and linear equation (1.1) play only auxiliary role). 1.2. Plan of the paper and results This paper is mostly devoted to the investigation of the Belavkin equation dχ +
1 −ih∆ + x2 χ dt = xχ dQ , 2
(1.13)
describing the evolution of a “free” quantum particle with continuously observed coordinate. Equation (1.13) is a particular case of (1.7), where H = L2 (R), H = ˆ is the operator of the multiplication on x. As we have already −h2 ∆/2 and R = x mentioned, the corresponding normalized equation of form (1.8) dϕ +
1 −ih∆ + (x − hˆ xi)2 ϕ dt = (x − hˆ xi)ϕ dW , 2
(1.14)
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for this particular case was obtained also by L. Diosi [22] (in the same year that Belavkin’s paper [7] appeared with the general equations (1.1), (1.6)). In [22], L. Diosi has also written the system for the mean coordinate hˆ xi and momentum hˆ pi of the wave functions satisfying (1.14): (
xi2 ) dW , dhˆ xi = hˆ pi dt + 2(hˆ x2 i − hˆ dhˆ pi = (hˆ pxˆ + x ˆpˆi − 2hˆ xihˆ pi) dW .
(1.15)
These are classical Langevin equations for a free Newton particle disturbed by a random martingale force. Having this in mind, L. Diosi has referred to Eq. (1.14) as the quantum Langevin equation. For simplicity, we consider in Secs. 1–4 one-dimensional x, although all results and proofs hold for any dimensions. Only for the results of the last section the dimension is essential, which will be discussed therein. It turns out that analytic properties of the solution of the Cauchy problem for (1.13) are similar to those of the equation of the oscillator diffusion process ∂χ = ∂t
h∆ mΩ2 2 − x χ 2m 2h
(1.16)
with complex mass m and frequency Ω such that Re(mΩ) > 0 and ReΩ > 0. In Sec. 2 we describe in detail these analytic properties. For this purpose, we prove a theorem of Paley–Wiener type which seems to be of independent interest. Namely, we introduce a family of spaces of analytic functions (belonging to the Schwartz space S) that is invariant under Fourier transform (the classical Paley– Wiener duality between finite functions and their analytic Fourier transforms breaks the nice invariance property of the Schwartz space) and which comprises all solutions of (1.14) and (1.16). In Sec. 3 we prove the crucial asymptotic property of Eq. (1.14), namely, the uniform convergence (continuous collapse) in the spaces of entire analytic functions introduced in Sec. 2 of any solution to a Gaussian form with some fixed finite dispersion, as the time tends to infinity and for a.a. realizations of the innovating Wiener process W . In Sec. 4 we sketch another proof of this result based on improved arguments from [52], where this convergence was proved in L2 -norm (in fact, with a small error: roughly speaking, Lemma 4.1, see Sec. 4 below, was absent in [52]). The proven convergence property implies, in particular, that the solutions of (1.14) are not spreading at infinity (do not tend to zero in the uniform topology, as t → ∞), as are the solutions of the classical free Schr¨ odinger equation, which gives an explanation (in the framework of the stochastic theory of continuous quantum measurement) of the so-called watchdog effect (see, for instance, physical discussion in [14]). Moreover, it implies that the limit of the dispersion of the position for the wave functions solving (1.14) exists for a.a. realisations of the innovating process W (and does not depend on the initial function). This limit was called in [49, 50] the coefficient of the quality of measurement (the finiteness of this coefficient stands for the watchdog effect, and its positivity stands for the uncertainty principle).
806
V. N. KOLOKOL’TSOV
In Sec. 5, we study the dynamics (1.15) of the means of position and momentum of the solution. We prove that for dimensions more than 2 the means of the position and of the momentum tend to infinity and give the estimates of its growth. This result allows one to speak about the (some special sort of) scattering described by dynamics (1.14). On the other hand, this result give interesting information about the behavior of the Brownian motion at infinity and seems to be interesting by itself. Let us note now that the well-posedness theorem for a natural generalization of Eq. (1.14) (when a bounded deterministic potential is also present) was given in [51, 52], and the ergodic properties of finite dimensional Eq. (1.1) (describing continuously observed spin systems) was investigated in [49, 50]. Some applications of the latter results in the study of computer graphic systems appeared afterwards in [44, 45]. 1.3. Local properties of the quantum Langevin equation We discuss here some known properties of Eqs. (1.13) and (1.16) that we need further. Proposition 1.1. For any χ0 ∈ L2 (R) there exists a unique solution χ of equation (1.13), which tends to χ0 in L2 , as t → 0. Moreover, kχk2 is a positive martingale (and therefore, the fundamental Eqs. (1.10) and (1.12) hold). This was proved by different methods in [31], and [13, 48] (see also [52]). In the paper [13], the explicit integral representation for the solution was also given (which allows one, in particular, to construct its solutions for rather general initial data). Namely, the following result holds. Proposition 1.2. The solution G(t, x, ξ) of (1.13) with initial data G(0, x, ξ) = δ(x − ξ) ,
(1.17)
(i.e. the Green function of the Cauchy problem for (1.13)) exists and has the form o n ω G (x2 + ξ 2 ) + βG xξ + aG x + bG ξ + γG , (1.18) G = CG exp − 2 where the coefficients CG , ωG , βG are deterministic (they do not depend on Q) and are equal to −1 −1/2 2t 2t 2t 2π sinh ωG = α coth , βG = α sinh , CG = , (1.19) α α α α and other coefficients are −1 Z t 2t 2τ dQ(τ ) , aG = sinh sinh α α 0 Z bG = ihα 0
t
aG (τ ) dτ , sinh(2τ /α)
γG =
ih 2
Z
t
a2G (τ ) dτ . 0
(1.20)
(1.21)
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There are two natural ways to obtain this result, one of them (presented in [13, 52],) is based on the investigation of Gaussian solutions of Eq. (1.13) (see below) and another being an application of the stochastic WKB method developed simultaneously by A. Truman, H. Zhao in [62, 63] (following some previous ideas from [27]) and by the author in [51] (following some ideas from [13]). Let us point out that these two approaches can be also used to obtain fifth and sixth proofs of the Mehler formula (see formula (1.35) below) for the Green function of the oscillator Eq. (1.16), in addition to the four methods of the proof described in [19]. An important role in the theory of Eqs. (1.13) and (1.14) is played by the Gaussian functions of the form ω i ω 2 (1.22) χ = cgq,p = c exp − (x − q) + px 2 h with some real q and p (being respectively the mean values of the operators R = x ˆ of the multiplication on the coordinate x and of the momentum operator pˆ = −ih∂/∂x on the Gaussian function (1.22)) and complex ω and c such that c 6= 0 and Reω > 0. It is easy to derive (see, [31, 14, 52]) the following Proposition 1.3. If the initial function for the Cauchy problem of Eq. (1.13) has the Gaussian form (1.22), then the solution will also have the Gaussian form (1.22), whose parameters will satisfy the system of ordinary stochastic differential equations dω = (2 − ihω 2 ) dt , 1 2q dt + dQ , dq = p − Reω (Reω
(1.23) (1.24)
Imω (2q dt − dQ) , Reω " ! 2 c 2 i Imω Imω dc = − q 1 + + p2 + 2i 2 Reω Reω h
dp = h
−ω
1 − ih (Reω)2
dt + cq
ω dQ . Reω
(1.25)
(1.26)
Remark. Similar results for the equation 1 dχ + (−ih∆ + x2 )χ dt = ixχ dQ 2 (which differs from (1.13) by the coefficient i in the r.h.s. and which defines with probability one a unitary evolution) was obtained in [62]. In terms of the complex variable z = ωq + ip/h Eqs. (1.24) and (1.25) can be rewritten in the equivalent form dz = −ihωzdt + dQ .
(1.27)
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V. N. KOLOKOL’TSOV
Equations (1.23) and (1.27) can be easily solved explicitly: ω0 + α tanh(2t/α) , ω0 tanh(2t/α) + α Z t Z τ Z t ω(s)ds} z|t=0 + exp{ih ω(s) ds}dQ(τ ) . z = exp{−ih ω(t) = α
0
0
(1.28) (1.29)
0
In particular, for any initial ω0 with Reω0 > 0, the solution of Eq. (1.23) tends to the same limit (1.30) lim ω(t) = α = h−1/2 (1 − i) , t→∞
and therefore the limits of the dispersions Dxˆ = (2Reω)−1 ,
Dpˆ = h2 |ω|2 (2Reω)−1
of the coordinate and momentum respectively for any solution of form (1.22) are given by formulas √ (1.31) lim Dxˆ = h/2 , lim Dpˆ = h3/2 . t→∞
t→∞
Remark. Approximating the Dirac δ-function by Gaussian functions χ0 and taking limit of the corresponding Gaussian solutions χ , one can obtain Proposition 1.2. At the end of the introduction we collect (for the convenience of the future references) some well-known properties (see, for instance, [19]) of the operator H =−
mΩ2 x2 h2 ∆+ . 2m 2
(1.32)
Note however that in the standard textbooks these properties are given for the case of positive mass m and frequency Ω (when in particular, the operator H is positive and selfadjoint) and we need them in a slightly more general situation, for which these properties are still valid (though H is not more selfadjoint). Proposition 1.4. Let h > 0 and let Ω, m be complex constants such that ReΩ > 0 and Re(Ωm) > 0. Then (i) the operator (1.32) has discrete spectrum {λn = hΩ(n + 12 ), n = 0, 1 . . . , } (ii) the corresponding eigenfunctions are √ 1 ω 1/4 −ωx2 /2 e Hn ( ωx) , ψn (x) = √ n π 2 n!
(1.33)
where ω = mΩ/h and Hn are the Hermite polynomials, defined by the equation n 2 d 2 Hn (y) = (−1)n ey e−y , (1.34) n dy
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
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809
(iii) the operator exp{−tH/h} is a compact integral operator with the integral kernel s mΩ mΩ exp − ((x2 + y 2 ) cosh(Ωt) − 2xy) . (1.35) 2πh sinh(Ωt) 2h sinh(Ωt) 2. Analytic Properties of the Solutions of Equations (1.13) and (1.16) and a Paley Wiener Type Theorem The aim of this section is to describe natural spaces of analytic functions, which comprise the solutions of the quantum Langevin equation (1.13). In order first to make clear the main ideas, we start with more simple Eq. (1.16) with real positive frequency Ω and mass m. Definition 1. Let A and B be real numbers such that A + B ≥ 0. We denote by SA,B the space of entire analytic functions f such that 1 2 1 2 Ax + By |f (x + iy)| ≤ C exp 2 2 for some positive constant C and all x, y. It is easy to see that the infimum of those C for which the latter inequality holds, defines a norm on SA,B , and that SA,B is a Banach space with respect to this norm. Note also, that the restriction A + B ≥ 0 in the definition is essential, because for A + B < 0 the corresponding space SA,B would comprise only identically vanishing functions (which is not difficult to prove). Our first statement describes the action of the Fourier transform Φ on the family of spaces SA,B . It turns out that though the space of analytical functions of the classical Paley–Wiener theorem (which are the Fourier images of finite smooth functions) are not invariant with respect to Φ, the family of spaces SA,B is invariant with respect to Φ, as is its comprising Schwartz space S. Propositon 2.1. For any B > A > 0, the Fourier transform Φ (as well as its inverse Φ−1 ) is an isomorphism of the Banach spaces S−A,B 7→ S−B −1 ,A−1 with the norm A−1/2 . In particular, S−A,A−1 is invariant under the action of Φ, and Φ is a norm preserving isomorphism of the space S−1,1 . Proof. Let B > A > 0 and f ∈ S−A,B . It is clear that Φf is an entire analytic function. Further, Z 1 e−i(µ+iλ)(x+iy) f (x + iy) dx (Φf )(µ + iλ) = √ 2π for any y ∈ R (due to the Cauchy theorem). It follows that Z A 2 B 2 1 √ dx exp µy + λx − x + y |(Φf )(µ + iλ)| ≤ 2 2 2π 2 B 2 λ −1/2 =A + µy + y . exp 2A 2
810
V. N. KOLOKOL’TSOV
Taking minimum over all y ∈ R we obtain
1 2 1 2 µ + λ . |(Φf )(µ + iλ)| ≤ A−1/2 exp − 2B 2A
We have proved that Φ takes S−A,B in S−B −1 ,A−1 and its norm does not exceed A−1/2 . Let us now consider the function f (z) = (1 − )e−Az
2
/2
+ e−Bz
2
/2
,
z = x + iy ,
which belongs to S−A,B and has the norm equal to one for any ∈ (0, 1). Clearly 2 2 1− Φf (z) = √ e−z /2A + √ e−z /2B A B
belongs to S−B −1 ,A−1 and has the norm
1− √ + √ . A B −1/2
conclude that the norm of Φ is equal to A
Since ∈ (0, 1) is arbitrary, we
, which completes the proof.
ˆ ω denote the integral operator in L2 (R) with the Now, let ω > β ≥ 0 and let K β integral kernel n ω o Kβω (x, ξ) = exp − (x2 + ξ 2 ) + βxξ . (2.1) 2 ˆω In particular, the resolving operator for the Cauchy problem of Eq. (1.16) is K β
(up to a constant coefficient) with ω = coth(Ωt) and β = sinh−1 (Ωt). It turns out ˆ ω almost coincides (up to a “subtle fiber”) with some SA,B . that the image of K β Namely: ˆ ω is a continuous linear Proposition 2.2. For any p ≥ 1, the operator K β 2π 1/2q mapping Lp 7→ S−(ω− β2 ),ω with the norm that does not exceed ( qω ) (respectively, ω
1) for p > 1, where p−1 + q −1 = 1 (respectively, for p = 1). Further, for any real 2 ˆ ω (L∞ ) a, b such that ω − βω ≤ a ≤ b < ω, the space S−a,b belongs to the image of K β and there exists a continuous inverse operator S−a,b 7→ S−(
β2 β2 ω−a −ω), ω−b −ω
.
Proof. The first part of the theorem is a direct consequence of the H¨ older ˆ ω in the product of four inequality and to prove the second part we decompose K β operators ˆ ω = M−ωx2 /2 ◦ Liβ ◦ Φ ◦ M−ωx2 /2 , K β where Mf (x) denotes the multiplication on exp f (x), and Liβ f (x) = f (iβx) is the rotation with dilatation in the complex plane. Then we use Proposition 1 and the trivial remarks that M−ωx2 /2 is a (norm preserving) isomorphism SA,B 7→ SA−ω,B+ω and Liβ is a (also norm preserving) isomorphism SA,B 7→ Sβ 2 B,β 2 A . ˆ ω not only on Lp but also on other It is natural to consider the action of K β functions that do not increase very fast at infinity. Therefore, analogously to the previous result we obtain the following
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
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Proposition 2.3. For any A, B such that A + B ≥ 0 and A < ω, the operator ˆ ω is an isomorphism K β SA,B 7→ S β2 −ω,ω− β2 , ω−A
B+ω
ˆ ω )−1 is and conversely, for any a, b such that a + b ≥ 0 and b < ω, the operator (K β a continuous mapping Sa,b 7→ Sω− β2 , β2 −ω . ω+a ω−b
We proceed now to the case of the resolvent operator for the Cauchy problem of Eq. (1.16) with complex frequency or mass. More precisely, we shall consider ˆ ω , where constants β, ω should be complex with the only the integral operator K β condition Reω > |Reβ|. To consider that case one should introduce a generalization of the spaces SA,B . Moreover, it is convenient (in order to obtain a simple formula for the linear change of variables) to reparametrize them also. Definition 2. Let µ > 0 and let Γ, a be any complex constants. We denote by Sµ,Γ,a the space of entire analytic functions f (z), z = x + iy, such that µ 2 1 |z| − Re(Γz) + Re(az) (2.2) |f (z)| ≤ exp 2 2 for all z, or equivalently, µ − ReΓ 2 µ + ReΓ 2 x + y + xyImΓ + xRea − yIma . |f (x + iy)| ≤ exp 2 2 Clearly SA,B = S B+A , B−A ,0 , so that the spaces SA,B are the particular cases of 2 2 Sα,Γ,a corresponding to a = ImΓ = 0. We write down first the generalization of Proposition 2.1. Proposition 2.10 . If µ < ReΓ, the Fourier transform Φ is an isomorphism Φ : Sµ,Γ,a 7→ SµD−1 ,ΓD −1 ¯ −1 ,−i(µ¯ ¯ a+Γa)D with the norm a2 ))} , (ReΓ − µ)−1/2 exp{(2D)−1 (|a|2 + Re(Γ¯ where D = |Γ|2 − µ2 . Proof. It is quite similar to Proposition 2.1 and we drop it. We now give the law of the transformation of the spaces Sµ,Γ,a under the linear change of variables. Simple calculations give the following Lemma 2.1. The change of variable Lβ f (z) = f (βz) with a complex β is a norm preserving isomorphism Sµ,Γ,a 7→ Sµ|β|2 ,Γβ 2 ,aβ , and the shift (Tq f )(z) = f (z + q) is an isomorphism Sµ,Γ,a 7→ Sµ,Γ,a+µ¯q −Γq with the norm µ 2 1 |q| − Re(Γq 2 ) + Re(aq) . exp 2 2
812
V. N. KOLOKOL’TSOV
Now we obtain the generalization of Propositions 2.2 and 2.3 on the case of the resolving operator of the Cauchy problem for general Eq. (1.16) and also for (our main object of investigation) Eq. (1.13). Proposition 2.20 . Let Reω > |Reβ| and let a, b, γ be any complex constants. Then the integral operator with the kernel n ω o exp − (x2 + ξ 2 ) + βxξ + ax + bξ (2.3) 2 is a continuous (injective) linear mapping Lp 7→ S |β|2
β2 Reb 2Reω ,ω− 2Reω ,a+ Reω β
whose norm does not exceed
2π qReω
1/2q
exp
(Reb)2 2Reω
and
exp
(Reb)2 2Reω
for p > 1 and p = 1 respectively. Proposition 2.30 . Let the assumptions of the previous Proposition hold. If Re(Γ + ω) > µ, then the integral operator with the integral kernel (2.3) is a bounded linear operator Sµ,Γ,γ 7→ S µ|β|2 ,(Γ+¯ ¯ ω ) β 2 +ω,β(µ(¯ ¯ ω ))D −1 +a , γ +¯ b)+(γ+b)(Γ+¯ D
D
where D = |Γ + ω|2 − µ2 . On the other hand, the inverse operator is defined when Re((ω − Γ)β¯2 ) > µ|β|, as a continuous operator Sµ,Γ,γ 7→ S
¯ ω ¯ ω µ γ ¯ −¯ a µ ¯ Γ− ¯ γ−a ,− Γ− ¯ + β ¯2 D −ω, |β|2 β ¯2 β −b β |β|2 D
,
where D = (|Γ − ω|2 − µ2 )|β|−4 . Proof. The proof of both these Propositions is the same as that of Propositions 2.2 and 2.3: one should represent our integral operator in the product of multiplications operators, change of variable and the Fourier transform, then use Proposition 2.10 and Lemma 2.1. We conclude this section with the following simple Proposition 2.4. For any complex Γ, γ, the space S0,Γ,γ is one-dimensional and is generated by the function exp{γz − Γz 2 /2}. Proof. In fact, if f ∈ S0,Γ,γ , then f (z) exp{−γz + Γz 2 /2} is a bounded entire analytic function and thus it is a constant.
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3. The Convergence of the Solutions of the Quantum Langevin Equation to the Gaussian Form ˜ −γt ) denote a function that is of We now formulate the main theorems. Let O(e −t order O(e ) for any < γ, as t → ∞. We use here (as previously) the notation (1.22) for Gaussian functions and the letter α for the complex constant h−1/2 (1 − i). Moreover, Mf and Tq will denote respectively the operator of the multiplication on the function exp f and the shift Tq f (x) = f (x + q). The symbols k.k and k.kµ,Γ,a will denote respectively the L2 -norm of a function and its norm in the Banach space Sµ,Γ,a introduced above. Theorem 3.1. Let ϕ be the solution of the Cauchy problem for Eq. (1.14) with any initial function ϕ0 ∈ L2 , kϕ0 k = 1. Then for a.a. trajectories of the innovating Wiener process W, √ √ α ˜ − ht ) , (3.1) kϕ − (π h)1/4 ghˆ ˆ ϕ k = O(e xiϕ ,hpi as t → ∞, and moreover, there exist real constants q˜(W ), p˜(W ) such that √ √ Rt ˜ − ht ) hˆ xiϕ = q˜(W ) + p˜(W )t + hW + h 0 W (s) ds + O(e . √ ˜ − ht ) hˆ piϕ = p˜(W ) + hW + O(e
(3.2)
Theorem 3.2. With the assumptions of Theorem 3.1, the asymptotic formula (3.1) for solutions of Eq. (1.14) is still valid if instead of the L2 -norm we put there the norm of the uniform convergence in the space C k of k times continuously differentiable functions (k is arbitrary). Moreover, the same is true for the spaces of entire functions introduced in the previous section, if we centralize the position of the limit Gaussian wave packet. More precisely, for any positive √ √ 1/4 α ˜ − ht ) k ghˆxiϕ ,hpi = O(e (3.10 ) kM−ihpi ,α,0 ˆ ϕ x/h Thˆ xiϕ ϕ − (π h) ˆϕ for a.a. W as t → ∞. This section is devoted to the complete proof of these theorems. In the next section we sketch another proof. The plan of the proof is the following. First of all, due to Proposition 1.2, we represent the resolving operator for the Cauchy problem (1.13) as the composition ˆ ωG ◦ MbG x , CG exp γG MaG x ◦ K βG
(3.3)
where all the coefficients with the index G are given by formulas (1.19)–(1.21) and ˆ ωG denotes the integral operator with the kernel of the form (2.1). the operator K βG ˆ ωG = exp{− t H}, where Therefore, due to Proposition 1.4, CG K βG h H=h
h ∆ + x2 2i
,
(3.30 )
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V. N. KOLOKOL’TSOV
√ is the operator of form (1.32) with Ω = (1 + i) h = 2/α and m = −i. Equivalently, ˆ ωG is the resolving operator of the Cauchy problem for Eq. (1.16) with the CG K βG same Ω, m. The crucial step in our proof of the theorem will be the statement that the real coefficients a1 , a2 defined by the equation aG = ω G a1 +
i a2 h
(3.4)
have asymptotics (3.2) with some real q˜(W ), p˜(W ). It will imply almost immediately that the coefficient bG has a finite limit b∞ = b∞ (W ) for a.a. W . Next important step is to prove that the projection of the function Mb∞ x ϕ0 on the first eigenfunction ψ0 (x) =
α 1/4 π
n α o exp − x2 2
(3.5)
of the operator (3.30 ) should not vanish for a.a. W . It implies that exp{−tH/h} MbG x ϕ0 tends asymptotically to the Gaussian function (3.5). Using the fact that aG does not tend to infinity very fast, one shows at last that the application of the multiplication operator MaG x reduces asymptotically to the desired shift of (3.5). We proceed now with details. Note first that due to the result of the previous section, we can and will consider ϕ0 to belong to some space SA,B . In particular, |ϕ0 (x)| ≤ C exp{−Ax2 /2}
(3.6)
for all real x and some positive constants A, C. Lemma 3.1. For a.a. W, f (t) ≡ hˆ xiϕ −
√ ReaG = O(e−t h ) . ReωG
(3.7)
=ReaG is the mean coordinate of the Green function Remark. Note that a1Reω G (1.18) for ξ = 0, i.e. the lemma states that the difference between the mean coordinate of each solution of Eq. (1.14) and the mean coordinate of the Green function vanishes rapidly at infinity almost surely with respect to the measure defined by the innovating process W .
R Proof. Let χ(t, x) = G(t, x, ξ)ϕ0 (ξ) dξ be the solution of Eq. (1.13) with initial function χ(0, x) = ϕ0 (x). Let us recall that then ϕ = χ/kχk and (χ, xχ) = hˆ xiϕ (χ, χ). Therefore we obtain by direct calculations that f (t)kχk2 is equal to Z r 2(ReaG )2 βG ξ + β¯G η π dξdηϕ0 (ξ)ϕ¯0 (η) |CG |2 exp 2ReγG + ReωG ReωG 2ReωG 2 2 1 βG 1 β¯G |βG |2 2 2 ωG − ω ¯G − ξ − η + × exp − ξη 2 ReωG 2 ReωG ReωG βG ReaG β¯G ReaG ¯ ξ + bG + 2 η . × exp + bG + 2 ReωG ReωG
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
815
The inequality |ξη| ≤ 12 (ξ 2 + η 2 ) implies that |f (t)|kχk2 does not exceed r 2(ReaG )2 π 2 exp 2ReγG + |CG | ReωG ReωG Z |βG |(|ξ| + |η|) g(ξ)g(η) , × dξdη|ϕ0 (ξ)||ϕ0 (η)| 2ReωG where
2ReβG 2ReβG ReaG 2 u + RebG + u . g(u) = exp − ReωG − ReωG ReωG
(3.8)
Estimating now g(ξ)g(η) ≤ 12 (g 2 (ξ) + g 2 (η)) and then using the symmetry between ξ and η we get r 2(ReaG )2 π exp 2ReγG + |f (t)|kχk2 ≤ |CG |2 ReωG ReωG Z |βG |(|ξ| + |η|) 2 g (ξ) . (3.9) × dξdη|ϕ0 (ξ)||ϕ0 (η)| 2ReωG The key point in the proof is the remark that for any ξ Z MarQ (ξ) = |G(t, x, ξ)|2 dx is a positive martingale with respect to the measure defined by the (input) Brownian motion Q (more precisely, it is a martingale for t ≥ t0 for any t0 > 0). It follows from Proposition 1.1. A simple calculation gives r 2(ReaG )2 π g 2 (ξ) exp 2ReγG + MarQ (ξ) = |CG |2 ReωG ReωG and therefore, (3.9) can be rewritten as Z |βG |(|ξ| + |η|) 2 |f (t)|kχk ≤ dξdη|ϕ0 (ξ)||ϕ0 (η)| MarQ (ξ) . 2ReωG
(3.10)
Due to (3.6), we conclude that Z |f (t)|kχk ≤ C|βG | 2
e−ξ
2
/
MarQ (ξ) dξ
(3.11)
for > 0, C > 0. As the integral in the r.h.s. of (3.11) is obviously also some positive martingale, say MarQ , with respect to Q, we have |f (t)| ≤ |βG |
MarQ . kχk2
By the theorem on the transformation of the martingale property by means of the Girsanov transformation (see, for instance [46]), we conclude that MarW = MarQ /kχk2 is a positive martingale with respect to the Wiener measure defined
816
V. N. KOLOKOL’TSOV
by the innovating process W (as MarQ was a martingale with respect to Q), and therefore, due to the Doob convergence theorem, for a.a. W the limit exists of MarW as t → ∞ and this limit is integrable with respect to the Wiener measure of the process W . In particular, MarW is bounded for a.a. W . It implies that |f (t)| = O(|βG |) and (3.7) follows at last from (1.19). Lemma 3.2. The real coefficients a1 , a2 defined by (3.4) satisfy (3.2), i.e. for a.a. W, there exist q˜(W ), p˜(W ) such that √ √ R ˜ − ht ) a1 = q˜(W ) + p˜(W )t + hW + h 0t W (s) ds + O(e . (3.12) √ ˜ − ht ) a2 = p˜(W ) + hW + O(e Proof. Since a1 and a2 are the mean position and momentum respectively of the Gaussian solution G(t, x, 0) (see (1.18) of Eq. (1.13), it follows from Proposition 1.3 that they satisfy Eqs. (1.24) and (1.25) (it can be also directly verified from the explicit formula (1.20) for aG ). Let us rewrite these equations in terms of the innovating process W : ( da1 = (a2 + 2(ReωG )−1 f (t)) dt + (ReωG )−1 dW , (3.13) da2 = −h(ImωG )(ReωG )−1 (2f (t) dt + dW ) where f (t) is defined in (3.7). Due to (3.7) and to the exponentially fast convergence of ωG to α, as t → ∞, we first obtain the second equation in (3.12) from the second equation in (3.13). Then, substituting it in the first Eq. (3.13), we obtain the first Eq. (3.12). Lemma 3.3. The coefficient bG in (1.21) tends to some finite limit b∞ for a.a. W, as t → ∞, and moreover, (also almost surely) ˜ − bG = b∞ + O(e
√ ht
),
˜ kebG x ϕ0 − eb∞ x ϕ0 k = O(e
√
(3.14) ht
).
(3.15)
Proof. Due to (3.12), aG = O(t2 ) for a.a. W , as t → ∞, and therefore (3.14) follows from (1.21). In his turn, (3.15) follows from (3.14) and (3.6). We shall show now that if a function a(t) does not tend to infinity very fast (as t → ∞), then the application of the operator Ta Max does not “spoil” the asymptotic structure of the image of the operator exp{−tH/h} with H of the form (1.32). P Lemma 3.4. Let ψ ∈ L2 (R) and let ψ = µn ψn be its Fourier decomposition with respect to the basis of the eigenfunctions (1.33) of the operator (1.32) such that the assumptions of Proposition 1.4 hold. Let a = a(t) be any real function on time t ≥ 0 such that |a(t)| ≤ Ctκ for some positive C, κ. Then ∞ X 2 t Ta Maωx exp − H ψ = ea ω/2 e−Ωt(k+1/2) µ ˜k ψk , (3.16) h k=0
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
817
where ˜ −Ωt )ek/2 µ ˜k = µk + O(e
(3.17)
˜ −Ωt ) depends only on C, κ, kψk. and the function O(e Proof. We have t Maωx exp − H ψ (x) h =
∞ π 1/4 X
ω
√
n=1
√ 2 2 1 e−Ωt(n+1/2) ea ω/2 µn Hn ( ωx)e−ω(x−a) /2 . 2n n!
Due to the well-known formula Hn0 = 2nHn−1 for the Hermite polynomials (1.34), we also have n X n! Hk (y) , 2n−k bn−k Hn (y + b) = k!(n − k)! k=0
and therefore r n ∞ X X √ t n! ψk a2 ω/2 −Ωt(n+1/2) n−k e µn ( 2ωa) Ta Maωx exp − H ψ = e h k! (n − k)! n=0
k=0
=e
=e
a2 ω/2
a2 ω/2
√ ∞ ∞ X ψk X −Ωt(n+1/2) n! √ √ ( 2ωa)n−k e µn (n − k)! k! k=0 n=k ∞ X
e−Ωt(k+1/2) µ ˜k ψk
(3.18)
k=0
with
∞ 1 X −Ωnt e µ ˜k = √ k! n=0
p (n + k)! √ ( 2ωa)n µn+k , n!
or equivalently µ ˜ k = µk + e
−Ωt
∞ X √ √ 1 1 + k 2ωaµk+1 + √ e−Ωt δn k! n=2
p (n + k)! µn+k , n!
(3.19)
where
√ δn = e−Ω(n−1)t ( 2ωa)n . √ We choose now t0 such that e−Ωt/2 2ω|a| < 1 for all t > t0 . Then δn < 1 for all P t > t0 and n ≥ 2. Noting that kψk2 = |µn |2 we obtain consequently that the last term in (3.19) does not exceed in magnitude v u∞ u X (n + k)! 1 √ kψke−Ωt t . k!(n!)2 k! n=2
(3.20)
818
V. N. KOLOKOL’TSOV
Due to the well-known Stirling formula, there exists a constant C1 such that √ √ C1−1 2πnnn e−n < n! < C1 2πnnn e−n for all n > 1. This implies that p 2π(n + k) exp{(n + k) log(n + k) − (n + k)} (n + k)! 4 √ . < C1 k!(n!)2 2πn 2πk exp{k log k − k + 2n log n − 2n}
(3.21)
Estimating here the r.h.s. by means of the trivial inequality (n + k) log(n + k) ≤ n log n + k log k + n + k , we obtain that (3.20) does not exceed in magnitude C2 kψke−Ωt ek/2 for some con˜k follows. stant C2 . Now from (3.19) the desired estimate (3.17) for µ The last important step in the proof of the theorem is the following. Lemma 3.5. For a.a. W, the function eb∞ x ϕ0 (x) has a nonvanishing projection on the Gaussian function ψ0 . Proof. If the assertion of the lemma does not hold, then, using (3.3), (3.15), and applying Lemma 3.4 in the case of operator (3.30 ) we conclude that the following estimate holds for the norm of the solution of (1.13) with the initial function χ0 = ϕ0 : ( √ ) √ h α − ht 2 ˜ a + Reγ − t . (3.22) ) exp kχk = O(e 2 1 2 Now we use the same trick as at the end of the proof of Lemma 3.1. Namely, the squared norm of the solution G(t, x, 0) (see (1.18)) is a positive martingale with respect to the measure of the (input) Wiener process Q. We denote it by MarQ . Then by the martingale transformation theorem we conclude that MarW = MarQ /kχk2 is a martingale with respect to the innovating process W and thus it should be bounded for a.a. W . But the latter property contradicts (3.22) and the √ obvious remark that MarQ is of order exp{αa21 + 2Reγ − ht}. Proof of Theorem 3.1. It follows directly from formula (3.3), Lemmas 3.3 and 3.5, and Lemma 3.4 applied to operator (3.30 ). Proof of Theorem 3.2. We prove only (3.10 ). Let −1 −Reγ−ωG a1 /2 χ(t) ˜ = CG e Ta1 M−ia2 x/h χ(t) . 2
Then
Z χ ˜=
o n ω G (x2 + ξ 2 ) + βG xξ + (βG a1 + bG )ξ χo (ξ) dξ , exp − 2
where, due to Lemmas 3.2 and 3.4, and formula (1.19), ˜ − βG a1 + bG = b∞ + O(e
√ ht
).
(3.23)
(3.24)
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819
Using the trivial inequality |eν2 − eν1 | ≤ |eν1 ||ν2 − ν1 |e|ν2 −ν1 |
(3.25)
(which holds for any complex ν1 , ν2 ), we can estimate (for t > τ ) |(χ(t) ˜ − χ(τ ˜ ))(x + iy)| Z ωG (τ ) ((x + iy)2 + ξ 2 ) + β(τ )(x + iy)ξ + b∞ ξ ≤ exp Re − 2 ˜ − × O(e
√ ht
˜ − ) exp{O(e
√
ht
)(x2 + y 2 + ξ 2 + 1)}|χ0 (ξ)| dξ ,
which implies (due to Proposition 2.20 , or by direct application of H¨ older inequality) that χ(t) ˜ is a Cauchy family, as t → ∞, in the space S,α,0 for any > 0. Therefore, χ(t) ˜ tends in S,α,0 to some (automatically also entire) function F . Further, due ˜ ∈ Sg(t),α,0 with a function g(t) of order to Lemma 2.1 and Proposition 2.20 , χ(t) √ − ht ˜ ), and moreover, the set of norms kχ(t)k ˜ O(e g(t),α,0 is bounded. This obviously implies that the limit F belongs to S0,α,0 and consequently, due to Proposition 2.4, F = A exp{−α(x + iy)2 /2} with some constant A. Noting now that the solution of (1.14) is equal to ˜, ϕ = CG eReγ+ωa1 /2 kχk−1 χ 2
and (as it was shown in the proof of Lemma 3.5) the coefficient before χ ˜ in this formula is bounded, we conclude that A 6= 0 (otherwise, 1 = kϕk would tend to zero), which implies the assertion of the theorem. 4. A Sketch of Another Proof of Theorem 3.2 Due to the results of Sec. 2, we can consider the initial function for the Cauchy problem of Eqs. (1.13) or (1.14) to belong to some space SA,B . In particular, we can present it in the form of “infinite” linear combination of Gaussian functions 2 Z i x (4.1) χ0 = ϕ0 = c(µ) exp − + µx dµ h with some > 0 and some function c(µ) also belonging to some SA,B so that the integral Z 2 (4.2) c(µ)2 eAµ dµ converges for some A > 0. Propositions 1.1 and 1.3 imply that the solution of (1.13) with such initial function can be represented in the form Z i ω(t) 2 (x − q(µ, t)) + p(µ, t)x dµ , (4.3) χ = c(µ, t) exp − 2 h where ω, q, p, c satisfy (1.23)–(1.26) with initial values ω0 = 2−1 ,
c(µ, 0) = c(µ) ,
p(µ, 0) = µ ,
q(µ, 0) = 0 .
(4.4)
820
V. N. KOLOKOL’TSOV
The first step in the second proof of Theorem 3.1 (and just this step was missing in the original exposition of this proof in [52]) is the following. Lemma 4.1. For a.a. W and for any µ0 ˜ − |hˆ xiϕ − q(µ0 , t)| = (|µ0 | + 1)O(e
√ ht
).
Proof. We calculate r Z |zµν (t)|2 π 2 c(µ, t)¯ c(ν, t) exp − kχk = Reω 4Reω(t) i (p(µ, t) − p(ν, t))(q(µ, t) + q(ν, t)) , + 2h where
(4.5)
i zµν (t) = ω(t) q(µ, t) − q(ν, t)) + (p(µ, t) − p(ν, t) . h
Due to (1.29)), zµν (t) =
(4.6)
Z t i (µ − ν) exp −ih ω(s) ds . h 0
(4.7)
By (1.28), we conclude that ˜ e− |zµν (t)| = |µ − ν|O
√ ht
.
(4.8)
Calculating analogously hˆ xiϕ = (χ, xχ) we obtain (hˆ xiϕ − q(µ0 , t))kχk2 Z r 1 π dµdν c(µ, t)¯ c(ν, t)(zµµ0 (t) + z¯νµ0 (t)) = 2 (Reω)3 i |zµν (t)|2 + (p(µ, t) − p(ν, t))(q(µ, t) + q(ν, t)) . × exp − 4Reω 2h Using the obvious inequality 2|c(µ, t)¯ c(ν, t)| ≤ (|c(µ, t)|2 eAµ + |c(ν, t)|2 eAν )e−Aµ 2
2
2
/2 −Aν 2 /2
e
,
the symmetry in µ, ν, and (4.8) we get the estimate Z r √ 2 π 2 − ht ˜ |c(µ, t)|2 eAµ dµ . ) |hˆ xiϕ − q(µ0 , t)|kχk ≤ (|µ0 | + 1)O(e Reω
(4.9)
We can finish the proof of the lemma analogously to the end of the proof of Lemma 3.1, using the fact that the integral in the r.h.s. of (4.9) is a positive martingale with respect to the Wiener measure of the process Qp (which follows from the existence of the integral (4.2), the martingale property of π/Reω|c(µ, t)|2 for any µ, and Fubbini’s theorem).
LOCALIZATION AND ANALYTIC PROPERTIES OF THE SOLUTIONS
...
Lemma 4.2. For a.a. W, (3.2) holds and also for each µ √ √ R ˜ − ht ) q(µ, t) = q˜(W ) + p˜(W )t + hW + h 0t W (s) ds + (|µ| + 1)O(e
˜ − p(µ, t) = p˜(W ) + hW + (|µ| + 1)O(e
√
ht
821
,
(4.10)
)
˜ − where the random constants q˜(W ) and p˜(W ) and a random function O(e not depend on µ.
√ ht
) do
Proof. It follows from the previous lemma in the same way as Lemma 3.2 follows from Lemma 3.1. Lemma 4.3. For all µ, ν such that c(µ) 6= 0 and c(ν) 6= 0, the limit lim (log c(µ, t) − log c(ν, t)) = −(µ2 − ν 2 )A + (µ − ν)B[W ]
t→∞
(4.11)
exists, where A is some (explicitly calculated) complex constant (depending only on ω0 = 2−1 ) with positive real part and B[W ] is almost surely finite random variable (not depending on µ, ν). Proof. By direct calculations, we get d(log c(µ, t)) − log c(ν, t)) = −(µ2 − ν 2 )f (t) dt + (µ − ν)(g1 (t, [W ]) dt + g2 (t) dW ) ,
(4.12)
where the functions f, g1 , g2 are exponentially small, as t → ∞, and moreover, f does not depend on W and −2
Ref (t) = (hReω)
2 Z t Z t exp{h Imω(s) ds} sin{h Reω(s) ds} > 0 , 0
0
which implies the assertion of the lemma. Lemma 4.4. Let us fix any µ0 such that c(µ0 ) 6= 0. Then for a.a. W, the limit g(µ) = lim (c(µ, t)/c(µ0 , t)) t→∞
exists, the limit function g(µ) is fast decreasing so that Z 2 |g(µ)|eδµ dµ < ∞
(4.13)
for some δ > 0, and also Z g(µ) dµ 6= 0 .
(4.14)
Proof. The first two statements follow directly from the previous lemma. To prove (4.14) we should again use the “martingale” trick of Lemma 3.1. Namely, since
822
V. N. KOLOKOL’TSOV
p MartQ = π/Reω|c(µ0 , t)|2 is a positive martingale with respect to the Wiener process Q, we conclude that MartW = M artQ /kχk2 is a positive martingale with respect to the innovating process W and thus it is bounded almost surely. Using (4.5) and Lemma 4.2, we get Z −2 g(µ) dµ . lim MartW = t→∞
Since it is bounded, we get (4.14). Second proof of Theorem 3.2. Due to (4.14), we get √ 1/4 α ghˆxiϕ ,hpi M−ihpi ˆ ϕ x/h Thˆ xiϕ (ϕ − (π h) ˆ ϕ ) (x + iy) Z c(µ, t) ω α g ≤C δq,δp − g0,0 (x + iy) dµ . c(µ0 , t)
(4.15)
Further, using estimate (3.25), we shall have ω α ˜ − − g0,0 )(x + iy)| ≤ O(e |(gδq,δp
√ ht
)(x2 + y 2 + (|µ| + 1)(|x| + |y|) + (|µ| + 1)2 )
˜ − × exp{O(e
√ ht
)(x2 + y 2 + (|µ| + 1)(|x| + |y|)
1 + (|µ| + 1)2 ) − Re(α(x + iy)2 )} . 2 Making now the integration over µ in (4.15) we conclude that the r.h.s. in (4.15) does not exceed ˜ − O(e
√ ht
1 )O(x2 + y 2 + 1) exp{− Re(α(x + iy)2 )} , 2
which obviously implies (3.10 ). 5. Long Time Asymptotics of the Brownian Motion and its Integral Rt We give here the estimates of growth of the integral V (t) = 0 W (s) ds of the standard d-dimensional Brownian motion W (t), as t → ∞, and as a consequence, obtain the estimates of growth of the mean position and momentum (satisfying (3.2)) of the solutions of Eq. (1.14). In this section the dimension of the problem is essential. Theorem 5.1. Suppose g(t) be an increasing positive function on R+ and d > 1. R∞ Then, if the integral 0 (g(t)t−3/2 )d dt is convergent, then lim inf (|V (t)|/g(t)) = ∞ t→∞
for a.a. W. In particular, for any β <
3 2
−
1 d
lim inf (|V (t)|/tβ ) = ∞ t→∞
almost surely.
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823
We give the proof at the end of the section. Let us note now that the same method can be used to obtain a simple proof of the estimate of growth of the Brownian motion itself, due to Dvoretski–Erd¨ os (see [26, 43]), namely the following result. √ Rt Proposition 5.1. If d > 2 and the integral 0 (g(t)/ t)d dt is convergent, then lim inf t→∞ (|W (t)|/g(t)) is almost surely infinite. In particular, lim inf t→∞ |Wtβ(t)| = ∞ for any β < 12 − 1d . From Theorem 5.1, Proposition 5.1, and representation (3.2) for the mean position and momentum of the solution of Eq. (1.14), it follows immediately: Theorem 5.3. Let d > 2 and let ϕ be any solution of the Cauchy problem for Eq. (1.14). Then |hˆ xiϕ | |hˆ piϕ | = ∞ , lim inf β−1 = ∞ lim inf t→∞ t→∞ t tβ almost surely for any β <
3 2
− d1 .
As another application of Theorem 5.1, let us mention the proof (given in [1, 2]) of the existence and completeness of the wave operators in the scattering theory for the Newton equation describing a particle driven by the force that is the sum of a deterministic scattering force and the white noise, namely for the system ( x˙ = v dv = K(x) dt + dW with a bounded and fast decreasing at infinity function K(x). Proof of Theorem 5.1. For any event B on the Wiener space we shall denote by P (B) the probability of B with respect to the standard Wiener measure. The following technical result is the key point in the proof of Theorem 1. t is the event on the Lemma 5.1. Let A be a fixed positive constant and BA,g Wiener space, which consists of all trajectories W such that the set {V (s) : s ∈ [t, t + 1]} has nonempty intersection with the cube [−Ag(t), Ag(t)]d . Then t ) = (O(g(t)t−3/2 ) + O(t−1 ))d . P (BA,g
(5.1)
Let us show first that Theorem 1 is a direct consequence of this statement. In P 3 1 n fact (5.1) implies that ∞ n=1 P (BA,g ) < ∞, if d > 1 and β < 2 − d . Then by the n can hold. It first Borell–Cantelli lemma only a finite number of the events BA,β means the existence of a constant T such that V (t) ∈ / [−Ag([t]), Ag([t])]d for t > T , where [t] denotes the integer part of t. This obviously implies the statement of Theorem 5.1.
824
V. N. KOLOKOL’TSOV
Proof of Lemma 5.1. Obviously, it is enough to consider the case d = 1. The density pt (x, y) of the joint distribution of W (t) and V (t) is well known (see, [47]): √ 2 2 6 6 2 3 pt (x, y) = 2 exp − x + 2 xy − 3 y . πt t t t In particular,
√ 2 x 3 . pt (x, y) ≤ 2 exp − πt 2t
(5.2)
It is clear that Z t P (BA,g ) = P (V (t) ∈ [−Ag(t), Ag(t)]) + 2
×P
Z
τ
min (y + τ x +
0≤τ ≤1
+∞
dy Ag(t)
Z
∞
−∞
pt (x, y)
W (s) ds) < Ag(t) dx .
(5.3)
0
The first term here is equal to √
y2 exp − 3 dy 2t −Ag(t)
Z
1 2πt3
Ag(t)
and is of order O(g(t)t−3/2 ). The second term can be estimated from above by the integral Z +∞ Z ∞ dy pt (x, y)P min τ x + min W (τ ) < Ag(t) − y dx . (5.4) 2 Ag(t)
0≤τ ≤1
−∞
0≤τ ≤1
We decompose this integral in the sum I1 +I2 +I3 of three integrals, whose domain of integration in the variable x are {x ≥ 0}, {Ag(t) − y ≤ x ≤ 0}, and {x < Ag(t) − y} respectively. We shall show that the integrals I1 and I2 are of order O(t−3/2 ) and the integral I3 is of order O(t−1 ), which will complete the proof of the Lemma. It is clear that Z ∞ Z ∞ dy pt (x, y)P min W (τ ) < Ag(t) − y dx . I1 = 2 Ag(t)
0≤τ ≤1
0
Enlarging the domain of integration over x to the whole line, integrating over x, and using the well-known distribution for the minimum of the Brownian motion we obtain 2 Z ∞ Z ∞ 2 y2 z dz . exp − 3 dy exp − I1 ≤ √ 3 2t 2 π t Ag(t) y−Ag(t) Changing the order of integration we can rewrite the last expression in the form 2 Z Ag(t)+z Z ∞ z y2 2 √ dz exp − exp − 3 dy . 2 2t π t3 0 Ag(t) Consequently, 2 I1 ≤ √ π t3
Z 0
∞
z2 z exp − 2
dz = O(t−3/2 ) .
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825
We proceed with I2 . Making the change of the variable x 7→ −x we obtain Z ∞ Z y−Ag(t) I2 = 2 dy pt (−x, y)P min < Ag(t) − y + x dx . Ag(t)
0≤τ ≤1
0
Making the change of the variable s = y − Ag(t) and using the distribution of the minimum of the Brownian motion we get further that r Z ∞ 2 Z ∞ Z s z 2 dz . ds pt (−x, s + Ag(t)) dx exp − I2 = 2 π s−x 2 0 0 Estimating pt (x, y) by (5.2) and changing the order of integration we get √ Z ∞ 2Z ∞ 2 Z x+z 4 z x 3 I2 ≤ √ dz exp − dx exp − ds . 2 2 2t 2π πt 0 0 x The last integral is obviously of order O(t−3/2 ). It remains to estimate the integral I3 . We have Z ∞ Z ∞ I3 = 2 dy pt (−x, y) dx Ag(t)
Z
Z
∞
Ag(t)+x
pt (−x, y) dx
=2 0
≤
y−Ag(t)
dy Ag(t)
√ Z 2 2 3 ∞ x dx x exp − πt2 0 2t
= O(t−1 ) . The proof is complete. Acknowledgments I am very thankful to Profs. Albeverio, V. P. Belavkin, and A. Ponosov for useful discussions. References [1] S. Albeverio, A. Hilbert and V. Kolokoltsov, “Transience for stochastically perturbed Newton systems”, Ruhr Universit¨ at Bochum, SFB 237, preprint 269 (1995), to appear in Stochastics and Stochastics Reports. [2] S. Albeverio, A. Hilbert and V.Kolokoltsov, “Estimates uniform in time for the transition probability of diffusions with small drift and for stochastically perturbed Newton equations”, Ruhr Universit¨ at Bochum, SFB 237, preprint 291 (1995). [3] S. Albeverio, V. Kolokoltsov and O. Smolyanov, “Continuous quantum measurement: local and global approaches”, to appear in Rev. Math. Phys. [4] A. Barchielli and V. P. Belavkin, “Measurements continuos in time and a posteriori states in quantum mechanics”, J. Phys. A: Math. Gen. 24 (1991) 1495–1514. [5] A. Barchielli, L. Lanz and G. M. Prosperi, “Statistics of continuous trajectories in quantum mechanics: Operator valued stochastic processes”, Found. Phys. 13 (1983) 779–812.
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[6] A. Barchielli, L. Lanz, G. M. Prosperi, “A model for macroscopic description and continuous observation in quantum mechanics”, Nuovo Cimento, 72B (1982) 79– 121. [7] V. P. Belavkin, “Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes”, in: Modelling and Control of Systems, Proc. Bellman Continuous Workshop, Sophia-Antipolis 1988, Lect. Notes in Contr. and Inform. Sci., 121 (1988) 245–265. [8] V. P. Belavkin. “A new wave equation for a continuous nondemolition measurement, Phys. Let. A 140 (1989) 355–358. [9] V. P. Belavkin, “A posterior Schr¨ odinger equation for continuous nondemolition measurement”, J. Math. Phys. 31 (1990) 2930–2934. [10] V. P. Belavkin, “The reconstruction theorem for quantum stochastic process”, Teor. Mat. Fis. 62 (1985) 409–431. English translation in Theor. Math. Phys. [11] V. P. Belavkin, “Quantum continual measurements and a posteriori collapse on CCR, Comm. Math. Phys. 146 (1992) 611–635. [12] V. P. Belavkin, O. Hirota and R. L. Hudson eds, “Quantum communications and measurement”, Proc. Int. Conf. held on July 11-16, 1994, Nottingham, Plenum Press, N. Y., 1995. [13] V. P. Belavkin and V. N. Kolokoltsov, “Quasy-classical asymptotics of quantum stochastic equations”, Teor. i Mat. Fis. 89 (1991) 163–178. English translation in Theor. Math. Phys. [14] V. P. Belavkin and P. Staszewski, “A stochastic solution of Zeno paradox for quantum Brownian motion”, Phys. Rev. A 45 (3) (1992) 1347–1356. [15] D. Bohm and J. Bub, “A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory”, Rev. Mod. Phys. 38 (1966) 453–469. [16] Ph. Blanchard and A. Jadczyk, “Strongly coupled quantum and classical systems and Zeno’s effect”, Phys. Lett. A 183 (1993) 272–276. [17] Ph. Blanchard and A. Jadczyk, “On the interaction between classiacal and quantum systems”, Phys. Lett. A 175 (1993) 157–164. [18] Ph. Blanchard and A. Jadczyk, “Evant-enhanced formalism of quantum theory or Columbus solution to the quantum measurement problem. Univ. Bielefeld, preprint BiBoS 655/7/94 (HEP-TH 9408021), 1994. [19] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon. Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, 1987. [20] E. B. Davies, “Quantum stochastic processes”, Commun. Math. Phys. 15 (1969) 277– 304. [21] E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976. [22] L. Diosi, “Continuous quantum measurement and Ito formalism”, Phys. Lett. A 129 (1988) 419–423. [23] L. Diosi, “Localized solution of a simple nonlinear quantum Langevin equation”, Phys. Lett. A 132 (1988) 233–236. [24] L. Diosi, “Models for universal reduction of macroscopic quantum fluctuations”, Phys. Rev. A 40 (1989) 1165–1173. [25] L. Diosi, “Quantum measurement and quantum gravity for each other”, in Quantum Chaos, Quantum Measurement; NATO ASI Series C: Math. Phys. Sci. 357 (1992) 299–304. Dordrecht, Kluwer. [26] A. Dvoretski and P. Erd¨ os, “Some problems on random walk in space”, Second Berkeley Simposium in Probability, Univ. of California Press, (1951) 353–367. [27] K. D. Elworthy and A. Truman, “The diffusion equation and classical mechanics: An elementary formula”, in Stochastic Processes in Quantum Physics LNP 173 (1982) 136–146. [28] Ch. N. Friedman, “Semigroup product formulas, compressions, and continuous observation in quantum mechanics”, Indiana Univ. Math. J. 21 (1972) 1001–1011.
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[51] V. N. Kolokoltsov, “Stochastic Hamilton–Jacobi equation and stochastic method WKB”, preprint 236 (November 1994), Ruhr Univ. Bochum, SFB 237, to appear in Proc. Int. Conf. “Idempotency” held on October 1994 in Bristol, Cambridge Univ. Press, 1996. [52] V. N. Kolokoltsov, “Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate”, J. Math. Phys. 36 (6) (1995) 2741– 2760. [53] V. N. Kolokoltsov, “Short deduction and mathematical properties of the main equation of the theory of continuous quantum measurement”, in GROUP21. Physical Applications and Mathematical aspects of Geometry, Groups, and Algebras, Proc. XXI Int. Colloq. on Group Theoretical Methods in Physics 15–20 July 1996 in Goslar, Germany, (eds. H. D. Doebner, P. Nattermann and W. Scherer, World Scientific, 1997, v. 1, 326–330. [54] G. Lindblad, “On the generators of quantum dynamical semigroups”, Commun. Math. Phys. 48 (1976) 119–130. [55] G. J. Milburn, “Intrinsic decoherence in quantum mechanics”, Phys. Rev. A 44 (1991) 5401–5406. [56] B. Misra and E. C. Sudarshan, “Quantum Zeno paradox”, J. Math. Phys. 18 (1977) 756–763. [57] I. C. Percival, “Localization of wide open quantum systems”, J. Phys. A 27 (1994) 1003–1020. [58] P. Pearle, “Reduction of the state vector by a nonlinear Schr¨ odinger equation”, Phys. Rev. D 13 (1976) 857–868. [59] P. Pearle, “Towards explaining why events occur”, Int. J. Theor. Phys. 18 (1979) 489–518. [60] P. Pearle, “Combining stochastic dynamical state-vector reduction with spontaneous localisation”, Phys. Rev. A 39 (1989) 2277–2289. [61] P. Staszewski, “Quantum mechanics of continuously observed systems”, Habilitation thesis, Nicholas Copernicus Univ. Press, Torun, 1993. [62] A. Truman and H. Zhao, “The stochastic Hamilton Jacobi equation, stochastic heat equation and Scr¨ odinger equation”, Swansea Univ. preprint 1994, to appear in Stochastic Analysis and Applications. eds. A. Truman, I. M. Davies and K. D. Elworthy, World Scientific Publ. Co. (1996), 441–464. [63] A. Truman and H. Zhao, “The stochastic Hamilton Jacobi theory and related topics”, in Stochastic Partial Differential equations, London Math. Soc. Lecture Notes Series 276, Cambridge Univ. Press, 1995, 287–303.
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN? H. NEIDHARDT Fachbereich Mathematik Universit¨ at Potsdam Postfach 60 15 53 14 415 Potsdam Germany E-mail : [email protected]
V. A. ZAGREBNOV D´ epartment de Physique Universit´ e de la M´ editerran´ ee (Aix-Marseille II) CPT-Luminy Case 907 13288 Marseille Cedex 9 France E-mail : [email protected] Received 27 September 1996 Revised 25 September 1997 Mathematics Subject Classification: 47A05, 47B25, 81C10, 81C12 We show that any symmetric operator H has a dense maximal b-stability domain Ds (i.e. H|Ds ≥ bI, b ∈ R1 ) if and only if H is unbounded from above. This abstract result allows an application to singular perturbed Schr¨ odinger operators which are not semi-bounded from below, i.e., to the so-called “fall to the center problem”. It turns out that in this case the regularization problem is always ill-posed which implies that there is no unique “right Hamiltonian” for corresponding perturbed system. We give an example of singular perturbed Schr¨ odinger operator for which stability domains are described explicitly. Keywords: singular perturbation, symmetric operator, self-adjoint extension, regularization, Schr¨ odinger operator, fall to the center.
1. Introduction If a self-adjoint operator H on separable Hilbert space H is Hamiltonian of some quantum system of finitely many particles, then for stability of the system it is natural to demand that it has to be semi-bounded from below. Otherwise particles would occupy states with lower and lower energies, which finally leads to collapse. However, having some semi-bounded self-adjoint operator A and some singular perturbation W of A it can happen that perturbed operator H, Hf = Af + W f ,
f ∈ dom(H) = D ⊆ dom(A) ∩ dom(W ) ,
(1.1)
where D is some linear dense subset of dom(A) ∩ dom(W ), is not essentially selfadjoint and, moreover, it is not semi-bounded from below. If H admits self-adjoint 829 Reviews in Mathematical Physics, Vol. 10, No. 6 (1998) 829–850 c World Scientific Publishing Company
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H. NEIDHARDT and V. A. ZAGREBNOV
extensions then none of them can be regarded as the Hamiltonian of the perturbed quantum system because each extension is not semi-bounded from below. Example 1.1. Let A = −d2 /dx2 be the Laplace operator on H = L2 (R1 ). Let W be the multiplication operator corresponding to W (x) = −
1 1 , 4 |x|2
x ∈ R1 ,
(1.2)
and H κ1 f = Af +
1 Wf , κ
f ∈ dom(H κ1 ) = D = C0∞ (R1 \ {0}) ,
0 < κ < ∞ . (1.3)
If κ ≥ 1, then the operator H κ1 is semi-bounded from below (see [?, ?]) while for κ ∈ (0, 1) not: “fall to the center of attraction”. An instructive discussion of this phenomenon can be found in [?, ?, ?]. In the latter paper it is shown how the “fall to the center” is related to an infinite family of self-adjoint extensions of H1/κ , see also Sec. 4. Since the operator H κ1 commutes with the complex conjugation, it has equal deficiency indices (von Neumann’s theorem, see e.g. [?, Th. X3]). Hence H κ1 has several self-adjoint extensions and, moreover, if κ ∈ (0, 1), then each of these extensions is not semi-bounded from below. Therefore, in the last case there is no self-adjoint extension which would be a candidate for a stable Hamiltonian. To find a way out of this situation one can take the point of view that the domain dom(H) is too large and, hence, even the symmetric operator H is badly defined. In other words, it might be possible that in H there is a dense domain Ds ⊆ dom(H) such that H|Ds is semi-bounded from below. Definition 1.2. Let H be a (closed or not closed) symmetric operator. A dense linear subset Ds ⊆ dom(H) is called a stability domain of H if H|Ds is semi-bounded from below. Stability domain Ds of H is called a b-stability domain, b ∈ R1 , if inf
f ∈B1 (Ds )
(Hf, f ) ≥ b ,
(1.4)
where B1 (Ds ) = {f ∈ Ds : kf k ≤ 1} .
(1.5)
A b-stability domain Ds is called maximal if for each b-stability domain Ds0 obeying Ds ⊆ Ds0 one gets Ds = Ds0 . Therefore, the problem of constructing a semi-bounded from below Hamiltonian is reduced to existence of stability domain of H. However, this leads to a general question: whether each symmetric operator has a stability domain? The general answer is, of course, no.
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DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
Definition 1.3. Let H be a densely defined unbounded symmetric operator. We say H is unbounded from above if def
λ+ 1 (H) =
sup
(Hf, f ) = +∞ .
(1.6)
f ∈B1 (dom(H))
The operator is called unbounded from below if (−H) is unbounded from above. If H is unbounded from below but semi-bounded from above, then evidently H has no stability domain. Otherwise the existence of a stability domain Ds of H would imply that H is bounded. So the problem reduces to the question: whether each symmetric operator, which is unbounded from above, has a stability domain? The answer to this question is yes. The aim of this paper is to prove this as well as to describe them and corresponding stable Hamiltonians. Of course, the question is trivial if the operator H is unbounded from above but semi-bounded from below. So we face a real problem if H is unbounded from below and above. A partial solution of this problem is obtained in [?]. There it is proven that two (closed or unclosed) densely defined unbounded symmetric operators H1 and H2 which satisfy ∞ \ dom(Hs ) = dom(Hsn ) , s = 1, 2 , (1.7) n=1
possess unitarily equivalent densely defined restrictions if and only if both operators are strongly unbounded from below or from above. Definition 1.4. Let H be a densely defined unbounded symmetric operator such that the condition H(dom(H)) ⊆ dom(H) is satisfied and let k, n ∈ N, where N = {1, 2, . . .}. We set def
Bn (dom(H)) = {f ∈ dom(H) : |(H j f, f )| ≤ 1
for
j = 0, . . . , n − 1}
(1.8)
and def
2k−1 λ+ f, f ) 2k−1 (H) = supf ∈B2k−1 (dom(H)) (H def λ− 2k−1 (H) =
inf f ∈B2k−1 (dom(H)) (H
2k−1
(1.9)
f, f ) .
The operator H is called strongly unbounded from above (below) if λ+ 2k−1 (H) = +∞ (H) = −∞) for all k ∈ N. (λ− 2k−1 In application to our situation this means that if the symmetric operator H satisfies conditions: H(dom(H)) ⊆ dom(H) and dom(H) =
∞ \
dom(H n ) ,
(1.10)
n=1
and it is strongly unbounded from above, then H possesses a stability domain. To see this we set H1 = H and choose for H2 a restriction of an arbitrary unbounded def T∞ non-negative self-adjoint operator K to the domain D∞ = n=1 dom(K n ), i.e. H2 = K|D∞ which is essentially self-adjoint. One can easily prove that H1 and H2
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H. NEIDHARDT and V. A. ZAGREBNOV
obey (??) and, moreover, that H2 is strongly unbounded from above. Hence H1 and H2 have densely defined restrictions which are unitarily equivalent. In particular, this means that H has densely defined non-negative restriction, i.e., H has a stability domain with b = 0. However, the condition (??) and the strong unboundedness from above are too limited for our purpose. The paper is organized as follows. In the next section we give necessary and sufficient conditions (Theorem ??) that a symmetric operator has a maximal b-stability domain. In Sec. 3 we discuss applications of the main Theorem ?? to the problem of the unique, stable “right Hamiltonian” for non-positive singular perturbations. Our principal observation is a kind of “no-go” Theorem ?? which says that if a naturally defined perturbed operator is not semi-bounded from below, then there is no unique “right Hamiltonian”. We demonstrate these above abstract statements by an instructive quantum mechanical example in Sec. 4. Concluding remarks and discussions are postponed to the last Sec. 5. 2. Stability The aim of this section is to prove the assertion made above about the existence of a stability domain for an unbounded from above symmetric operator. Lemma 2.1. Let H be a densely defined unbounded symmetric operator which is unbounded from above. Let F ⊆ dom(H) be a finite dimensional subspace. Then sup
(Hf, f ) = +∞
(2.1)
f ∈B1 (dom(H) F )
Proof. Let PFH be the orthogonal projection from H onto F . Since F ⊆ dom(H) and dim(F) < ∞ there is a constant C > 0 such that ||HPFH f || ≤ C||f ||, f ∈ H. Hence |(H(I − PFH )f, (I − PFH )f ) − (Hf, f )| ≤ 2C||f ||2 ,
f ∈ dom(H) .
(2.2)
Since H is unbounded from above, for each n ∈ N there is a fn ∈ dom(H), ||fn || ≤ 1, such that (Hfn , fn ) ≥ n. We set gn = (I − PFH )fn ,
n ∈ N.
(2.3)
Then gn ∈ dom(H) F, ||gn || ≤ 1 and by (??) one gets (Hgn , gn ) = (H(I − PFH )fn , (I − PFH )fn ) ≥ n − 2C for n ∈ N which proves (??).
(2.4)
Next we need some facts from linear algebra. Lemma 2.2. Let H be a symmetric operator and {gl }m l=1 be a sequence of linearly independent elements gl ∈ dom(H). Let Gm be the subspace spanned by
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
833
{gl }m l=1 . Then one has (Hf, f ) > 0 for any non-trivial element f ∈ Gm if and only if the determinants (Hg1 , g1 ) · · · (Hgk , g1 ) (Hg1 , g2 ) · · · (Hgk , g2 ) (2.5) D(g1 , . . . , gk ) = det ...................... (Hg1 , gk ) · · · (Hgk , gk ) obey D(g1 , . . . , gk ) > 0, for each k = 1, 2, . . . , m. Proof. We set def
Let f =
1 ≤ l, n ≤ m .
aln = (Hgn , gl ) ,
Pm
(2.6)
n=1 cn gn .
Obviously one has ! m m m m m X m X X X X X H cn g n , cl g l = cn cl (Hgn , gl ) = aln cn cl . n=1
l=1
n=1 l=1
Let Am = kaln km l,n=1
(2.7)
n=1 l=1
(Hg1 , g1 ) · · · (Hgm , g1 )
(Hg1 , g2 ) · · · (Hgm , g2 )
=
........................
(Hg , g ) · · · (Hg , g ) 1
m
m
(2.8)
m
Note that the matrix Am is symmetric, i.e. anl = aln , 1 ≤ l, n ≤ m. Furthermore, the kernel of Am would be non-trivial if {gl }m l=1 are not linearly independent. Introducing the vector c1 · (2.9) ~c = · · cm one gets (see (??)) (Hf, f ) = hAm~c, ~c i ,
(2.10)
where h· , ·i is the usual scalar-product in Cm . Therefore one has (Hf, f ) > 0 for each f ∈ Gm , f 6= 0, if and only if the matrix Am is positive. However, by a wellknown criterion (Theorem 27, [?]) this is equivalent to the fact the determinants D(g1 , . . . , gk ) are positive for each k = 1, 2, . . . , m. In the following we use decomposition of determinants into minors. One gets D(g1 , . . . , gm+1 ) = am+1,1 (−1)m+2 M (g2 , . . . , gm+1 ) + · · · + am+1,m+1 (−1)2m+2 M (g1 , . . . , gm ) ,
(2.11)
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H. NEIDHARDT and V. A. ZAGREBNOV
where am+1,j = (Hgj , gm+1 ) ,
j = 1, 2, . . . , m + 1 .
(2.12)
The minors M (g1 , g2 , . . . , gj−1 , gj+1 , . . . , gm+1 ) are given by M (g1 , . . . , gj−1 , gj+1 , . . . , gm+1 ) (Hg1 , g1 ) · · · (Hgj−1 , g1 ) (Hgj+1 , g1 ) · · · (Hgm+1 , g1 ) (Hg1 , g2 ) · · · (Hgj−1 , g2 ) (Hgj+1 , g2 ) · · · (Hgm+1 , g2 ) = det . (2.13) ........................................................ (Hg1 , gm ) · · · (Hgj−1 , gm ) (Hgj+1 , gm ) · · · (Hgm+1 , gm ) Note that D(g1 , . . . , gm ) = M (g1 , . . . , gm ). If g1 , . . . , gm are fixed elements, then there are constants Cj (g1 , . . . , gm ) such that |M (g1 , . . . , gj−1 , gj+1 , . . . , gm+1 )| ≤ Cj (g1 , . . . , gm )kgm+1 k ,
j = 1, 2, . . . , m . (2.14)
This comes from the estimate |(Hgm+1 , gn )| ≤ kHgn k kgm+1k, 1 ≤ n ≤ m. Theorem 2.3. Let H be a densely defined symmetric operator. Then for any b ∈ R1 this operator has a maximal b-stability domain if and only if H is unbounded from above. Proof. Let b = 0 and {ξn }∞ n=1 be an orthonormal basis in H such that ξn ∈ dom(H) for each n = 1, 2, . . . . Such basis always exists. On the other hand, by Lemma ?? there is a sequence {fm }∞ m=1 , fm ∈ dom(H), kfm k ≤ 1, such that (i) fm ⊥ f1 , . . . , fm−1 , m = 2, 3, . . . , (ii) fm ⊥ ξ1 , . . . , ξ[m]+1 , m = 1, 2, . . . , (iii) (Hfm , fm ) > γm > 0 , m = 1, 2, . . . , where {γm }∞ m=1 is a sequence of positive numbers such that limm→∞ γm = +∞, which will be defined in the following, and n(n + 1) def < m , m ∈ N, (2.15) [m] = max n ∈ N0 : 2 where N0 = {0, 1, 2, . . .} and N = {1, 2, . . .}. We introduce the function p(·) : N → N, [m]([m] + 1) def (2.16) p(m) = m − 2 and the elements def
gm = fm + q(m)ξp(m) ,
m ∈ N,
def
q(m) = ([m] − p(m) + 2) .
(2.17)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
835
Few first terms of the sequence {gm }∞ m=1 are given by g1 = f1 + 1 · ξ1 g2 = f2 + 2 · ξ1 g3 = f3 + 1 · ξ2 g4 = f4 + 3 · ξ1
(2.18)
g5 = f5 + 2 · ξ2 g6 = f6 + 1 · ξ3 g7 = f7 + 4 · ξ1 ..............
Note that the sequence {gm }∞ m=1 consists of linearly independent elements. To see this let us assume that L X cl g l = 0 . (2.19) l=1
By the properties (i) and (ii) one has fL ⊥ f1 , . . . , fL−1 and fL ⊥ ξ1 , . . . , ξ[L]+1 . Therefore ! L X 0= cl gl , fL = cL kfL k2 . (2.20) l=1
PL−1 Hence l=1 cl gl = 0. Repeating this reasoning one gets c1 = c2 = · · · = cL = 0. Let us prove that there is a suitable sequence {γm }∞ m=1 such that the sequence obeys D(g , . . . , g ) > 0 for each m = 1, 2, . . . . The existence of {γm }∞ {gm }∞ 1 m m=1 m=1 we prove by induction. We choose γ1 ≥ 3kHξ1 k .
(2.21)
Then we obtain D(g1 ) = (Hg1 , g1 ) = (H(f1 + ξ1 ), (f1 + ξ1 )) ≥ (Hf1 , f1 ) − 3kHξ1 k > 0 .
(2.22)
Let us assume that there are γ1 , . . . , γm such that the corresponding g1 , . . . , gm satisfy the condition D(g1 , . . . , gl ) > 0 for l = 1, 2, . . . , m. We have to show that there is a γm+1 such that D(g1 , . . . , gm , gm+1 ) > 0. Let γm+1 be an arbitrary positive number which satisfies the condition γm+1 ≥ ([m + 1] + 1)([m + 1] + 3)
sup
kHξk k
1≤k≤[m+1]+1
+
m 1 + ([m + 1] + 1)2 X kHgj kCj (g1 , . . . , gm ) , D(g1 , . . . , gm ) j=1
(2.23)
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H. NEIDHARDT and V. A. ZAGREBNOV
where Cj (g1 , . . . , gm ) is given by (??). Taking into account (??) and (??) one gets D(g1 , . . . , gm+1 ) ≥ (Hgm+1 , gm+1 )D(g1 , . . . , gm ) −
m X
kHgj k kgm+1k |M (g1 , . . . , gj−1 , gj+1 , . . . , gm+1 )| . (2.24)
j=1
Using (??) we obtain D(g1 , . . . , gm+1 ) ≥ (Hgm+1 , gm+1 )D(g1 , . . . , gm ) − kgm+1 k2
m X
kHgj kCj (g1 , . . . , gm ) .
(2.25)
j=1
Since D(g1 , . . . , gm ) > 0 by assumption, one gets D(g1 , . . . , gm+1 ) > 0 if kgm+1 k2 X kHgj kCj (g1 , . . . , gm ) > 0 D(g1 , . . . , gm ) j=1 m
def
∆ = (Hgm+1 , gm+1 ) −
(2.26)
From (??) and (??) we find kgm+1 k2 ≤ 1 + q(m + 1)2 ≤ 1 + ([m + 1] + 1)2
(2.27)
and (Hgm+1 , gm+1 ) ≥ (Hfm+1 , fm+1 ) − 2q(m + 1)kHξp(m+1) k − q(m + 1)2 kHξp(m+1) k .
(2.28)
Hence (Hgm+1 , gm+1 ) ≥ (Hfm+1 , fm+1 ) − ([m + 1] + 1)([m + 1] + 3)
sup
kHξk k .
1≤k≤[m+1]+1
(2.29) Therefore, the estimate (??) gets the form: ∆ ≥ (Hfm+1 , fm+1 ) − ([m + 1] + 1)([m + 1] + 3)
sup
kHξk k
1≤k≤[m+1]+1
−
m 1 + ([m + 1] + 1)2 X kHgj kCj (g1 , . . . , gm ) . D(g1 , . . . , gm ) j=1
(2.30)
Using condition (??) one finally obtains ∆ ≥ (Hfm+1 , fm+1 ) − γm+1 .
(2.31)
Applying (iii) we immediately obtain ∆ > 0 which yields D(g1 , . . . , gm+1 ) > 0. So, by Lemma ?? the proof of stability follows by induction. It remains to show that the linear span G of {gm }∞ m=1 is dense in H. To this end we note that by (??) def
ηm =
1 fm + ξp(m) ∈ G , q(m)
m = 1, 2, . . . .
(2.32)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
837
If m = n(n+1) + k, 1 ≤ k ≤ n + 1. Then by definitions (??) and (??) we have 2 [m] = n and p(m) = k. Hence q(m) = n − k + 2. Therefore, (??) transforms into η n(n+1) +k = 2
1 f n(n+1) +k + ξk . 2 n−k+2
(2.33)
Fixing k and tending n → ∞ one gets limn→∞ η n(n+1) +k = ξk . However, by con2 struction {ξk }∞ k=1 forms an orthonormal basis in H. Hence G is dense in H. def
If b 6= 0, then we consider instead of H the operator Hb = H − bI. Applying the first part of the proof we obtain a stability domain Ds such that Hb |Ds ≥ 0. Hence H|Ds ≥ bI. It remains to show that there is always a maximal stability domain. To prove this we introduce the set Xb of all dense domains Ds ⊆ dom(H) such that H|Ds ≥ bI. The set Xb is partially ordered with respect to the set-inclusion relation. Moreover, for every linearly ordered subset Yb of Xb there is a upper bound Dp . The upper S bound is given by Dp = Ds ∈Yb Ds . Obviously, Dp is a linear dense subset of dom(H) such that H|Dp ≥ bI. Then by the Zorn’s lemma (see e.g. [?, Th. I.2]) Xb contains at least one maximal element Dm , i.e., such that from Ds0 ⊇ Dm , Ds0 ∈ Xb , it follows Ds0 = Dm . Note that the Zorn lemma does not say that the maximal element Dm is unique. We shall illustrate this in Sec. 4 by Example. 3. Singular Perturbations Let us relate the above result to our previous papers [?, ?]. There we associated with two self-adjoint operators A ≥ 0 and W ≤ 0 having a common dense domain D ⊆ dom(A) ∩ dom(W ) the symmetric operator Hα f = Af + αW f ,
dom(Hα ) = D ,
α > 0.
(3.1)
A dense linear subset Ds ⊆ D was called a stability domain of the pair {A, W } if there are constants 0 < a < 1, b ≥ 0, such that the estimate (−W f, f ) ≤ a(Af, f ) + b(f, f ) ,
f ∈ Ds ,
(3.2)
takes place. In general, however, such a stability domain Ds ⊆ D might not exist for given constants 0 < a < 1, b ≥ 0. So, a natural question arises whether there is always a stability domain of the pair {A, W } with respect to D ? Moreover, does such stability domain exist for any constants 0 < a < 1, b ≥ 0 ? Finally, is there always a maximal stability domain for given constants a, b ? This means a domain Dm such that for any other stability domain Ds , Dm ⊂ Ds ⊆ D, for which (??) holds, one gets Dm = Ds . Theorem 3.1. Let {A ≥ 0, W ≤ 0} be a pair of self-adjoint operators with common dense domain D ⊆ dom(A) ∩ dom(W ). For each 0 < a < 1, and b ≥ 0 there is a maximal stability domain Da,b ⊆ D of the pair {A, W } if and only if for α = 1/a the symmetric operator H1/a given by (??) is unbounded from above.
838
H. NEIDHARDT and V. A. ZAGREBNOV
Proof. Let H1/a be unbounded from above. Applying Theorem ?? to the operator H1/a we find (− ab )-stability domain Da,b ⊆ D of H1/a . This means that 1 b − (f, f ) ≤ (Af, f ) + (W f, f ) , a a
f ∈ Da,b ,
(3.3)
which immediately yields that Da,b is a stability domain of {A, W }. Moreover, by Theorem ?? the domain Da,b can be chosen maximal for H1/a which implies that Da,b is a maximal stability domain of {A, W } for constants a, b. The converse is obvious. Consequently, constants a, b obeying 0 < a < 1, b ≥ 0 exist if and only if at least for one α > 1 the operator Hα (??) is unbounded from above. However, this seems to be a natural assumption from the physical point of view. To proceed further we have to recall briefly main results of [?, ?]. Assume that D itself is a stability domain of {A, W }, i.e. Ds = D, and that symmetric def
operators A0 = A|D and Hα , 0 < α ≤ 1, are not essentially self-adjoint on this stability domain. Hence the problem arises to find out a right Hamiltonian for the perturbed system which is expected to be a semi-bounded from below self-adjoint extension of Hα . To obtain this extension one associates with Hα an approximating sequence of ˜ α,n }∞ self-adjoint operators {H n=1 of the form ˜ + αWn f , ˜ α,n f = Af H
˜ α,n ) = dom(A) ˜ , f ∈ dom(H
(3.4)
where {Wn }∞ n=1 is a regularizing sequence of bounded operators, which converges in the strong resolvent sense to W , and A˜ is a self-adjoint extension of A0 . If the ˜ α,n }∞ sequence {H n=1 tends in the strong resolvent sense to some self-adjoint operator ˜ α and H sup kWn f k < ∞ , f ∈ D , (3.5) n
˜ α is a self-adjoint extension of Hα and it is usually adopted as the then indeed H right Hamiltonian of the perturbed system. An essential ingredient of our approach is a maximality of the Friedrichs extension Aˆ of A0 with respect to the perturbation W . This means that if for some semi-bounded self-adjoint extension A˜ we have √ (3.6) dom(ˆ ν ) ⊆ dom(˜ ν ) ⊆ dom( −W ) , ˜ respectively, then where νˆ and ν˜ are quadratic forms associated with Aˆ and A, ˜ ˆ A coincides with √ the Friedrichs extension A. Note that by (??) one always has dom(ˆ ν ) ⊆ dom( −W ). Under the assumption that Aˆ is maximal with respect to the perturbation W , that the regularizing sequence {Wn }∞ n=1 satisfies the condition p √ √ |Wn | + If = −W + If , f ∈ dom( −W ) , lim n→∞
(3.7)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
839
˜ α,n }∞ obeys and that the approximating sequence {H n=1 p p ˜ α,n − z)−1 |Wn |k < ∞ , =m(z) 6= 0 , sup k |Wn |(H
(3.8)
n
we obtain that if convergence of the sequence (??) takes place in the strong resolvent sense to some semi-bounded self-adjoint operator, then the limit is an extension of Hα , which is canonical in the sense that it necessarily coincides with the form-sum . ˆ α = Aˆ + αW . Conversely, for each semi-bounded self-adjoint extension A˜ of A0 H there is a regularizing sequence {Wn }∞ n=1 such that the corresponding approximatˆ α and the condition (??) is satisfied. So, ing sequence (??) tends to the form-sum H ˆ α or we have no either we have convergence to the unique canonical form-sum H convergence at all. That is why the case, when the Friedrichs extension is maximal with respect to the perturbation, was called in [?] a well-posed regularization problem. We have shown (see e.g. [?, ?]) that the well-posed regularization problem takes place if and only if √ (3.9) dom( −W ) ∩ ker(A∗0 − ηI) = {0} for some η < 0. √ The opposite case when dom( −W ) ∩ ker(A∗0 − ηI) 6= {0} for some η < 0 and, hence for each η < 0, was called in [?] an ill-posed regularization problem. In this √ case ν ) ⊆ dom( −W ) for each semi-bounded self-adjoint extension A˜ of A0 obeying dom(˜ the approximating sequence (??) converges in the strong resolvent sense to the form . ˜ α = A˜ + αW at least for sufficiently small coupling constants α. Moreover, sum H the condition (??) is satisfied. Consequently, in the case of the ill-posed problem we have as a disadvantage the lost of the uniqueness of the approximating procedure. ˜ α for physical Hamiltonian of the perturbed Hence, there are a lot of candidates H system. In other words, the approximating procedure is not helpful in finding a right Hamiltonian because it gives a variety of possible Hamiltonians. To single out in this variety a right Hamiltonian one has to apply some additional physical or mathematical arguments, for instance, to restrict the class of allowed semi-bounded self-adjoint extensions A˜ of A0 . Let us show that the ill-posed case realizes if D itself is not a stability domain of the pair {A, W }. Lemma 3.2. Let ν ≥ 0 and γ ≥ 0 be two closed quadratic forms such that dom(ν) ∩ dom(γ) is dense. Then we have a dense domain D ⊆ dom(ν) ∩ dom(γ) and constants α, a, b, 0 < α ≤ 1, 0 < a < 1, b ≥ 0, such that αγ(f, f ) ≤ aν(f, f ) + b(f, f ) ,
f ∈ D,
(3.10)
if and only if there is a densely defined closed restriction νˆ of ν such that dom(ˆ ν) ⊆ dom(γ). Proof. If (??) is satisfied, then one obviously has that the closure νˆ of ν|D obeys dom(ˆ ν ) ⊆ dom(ν) and dom(ˆ ν ) ⊆ dom(γ). Conversely, let us assume that
840
H. NEIDHARDT and V. A. ZAGREBNOV
there is a closed restriction νˆ of ν such that dom(ˆ ν ) ⊆ dom(γ). Let us introduce a scalar product (3.11) (f, g)ν = ν(f, g) + (f, g) , f, g ∈ dom(ν) . Since ν is a closed form, the domain dom(ν) endowed with the scalar product (??) def
ν ) is a closed subspace forms a Hilbert space Hν = {dom(ν), (· , ·)ν }. The set dom(ˆ ν ), then dom(γ) contains the closed subspace of dom(ˆ ν ). of Hν . If dom(γ) ⊇ dom(ˆ Applying the closed graph principle (see e.g. [?, III Sec. 5.4]) one gets a constant C > 0 such that γ(f, f ) ≤ C(ˆ ν (f, f ) + (f, f )) ,
f ∈ dom(ˆ ν) .
Setting D = dom(ˆ ν ) we find constants α, a, b such that (??) holds.
(3.12)
Theorem 3.3. Let {A ≥ 0, W ≤ 0} be a pair of self-adjoint operators with common dense domain D ⊆ dom(A) ∩ dom(W ). If D is not a stability domain of {A, W }, then for any stability domain Ds ⊆ D of {A, W } the Friedrichs extension Aˆ of A|Ds is not maximal with respect to W . Proof. Let us denote closed quadratic forms associated with A and −W and the Friedrichs extension Aˆ of A0 = A|Ds by ν, γ and νˆ, respectively. Since Ds ⊆ D is a stability domain of {A, W } there are constants 0 < a < 1, b ≥ 0, such that (??) is satisfied. By Lemma ?? this ν ) is a closed subspace of Hν obeying √ yields that dom(ˆ ν ) ⊆ dom(ν) ∩ dom(γ). dom(ˆ ν ) ⊆ dom(γ) = dom( −W ). Hence one gets dom(ˆ Note that D ⊆ dom(ˆ ν ) is impossible. Otherwise D would be a stability domain of {A, W }. Hence there is an element f ∈ D such that f 6∈ dom(ˆ ν ). Since f ∈ dom(ν) the element f admits an orthogonal decomposition in Hν of the form f = g +h,
g ∈ dom(ˆ ν) ,
h ⊥ dom(ˆ ν) .
(3.13)
Since f , g ∈ dom(γ), one obviously has h ∈ dom(γ). Hence h ∈ dom(ν) ∩ dom(γ) and h ⊥ dom(ˆ ν ) in Hν , i.e. ν(h, f ) + (h, f ) = 0 ,
∀f ∈ Ds .
(3.14)
Since Ds ⊆ dom(A), one gets (h, (A + I)f ) = (h, (A0 + I)f ) = 0 , ∀f ∈ Ds . (3.15) √ √ Therefore, h ∈ ker(A∗0 + I). Since h ∈ dom( −W ), one gets dom( −W )∩ker(A∗0 + I) 6= {0}. Hence Aˆ is not maximal with respect to W . If D is not a stability domain, by Theorem ?? there is always a smaller stability domain Ds ⊆ D of {A, W } provided Hα is unbounded from above at least for one α > 1. Hence results of [?, ?] can be generalized. However, this has the price that the ill-posed case inevitably emerges even if the stability domain Ds of {A, W } is maximal for given constants 0 < a < 1, b ≥ 0. In particular this means that approximating method fails in definition of the right Hamiltonian.
841
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
Note that the D is not a stability domain of {A, W } if and only if Hα is unbounded from below for any α > 1. Hence conclusions of Theorem ?? are meaningful if for any α > 1 the operator Hα is unbounded from below (D is no stability domain) and at least for one α > 1 the operator Hα is unbounded from above (existence of stability domain Ds ⊆ D). 4. Example Let us illustrate Theorem ?? by Example ?? when κ ∈ (0, 1): “fall to the center problem”. Since we fix κ ∈ (0, 1), in the following we omit the κ indices for def simplicity, i.e., we set H = H κ1 . As far as the operator H is direct sum of two symmetric operators H ± which corresponds to the positive and negative half axis, i.e. (H ± f )(x) = −
d2 1 1 f (x) − f (x) , dx2 4κ x2
f ∈ dom(H ± ) = C0∞ (R1± ) ,
x ∈ R1± , (4.1)
we can concentrate our considerations only on H + . Since H + is unbounded from above, by Theorem ?? one gets that for each b ∈ R1 there is a maximal b-stability domain Db ⊆ C0∞ (R1+ ) for H + . Below we describe this domain in explicit form. Let us introduce an auxiliary operator: (H(a)f )(x) = −
1 1 d2 f (x) − f (x) , dx2 4κ (x + a)2
dom(H(a)) = {f ∈
W22 ([0, ∞))
: f (0) = 0} ,
(4.2) a > 0.
The operator H(a) is self-adjoint and semi-bounded from below for each a > 0. Let def
λ(a) = inf σ(H(a)) ,
a > 0,
(4.3)
where by σ(X) we denote the spectrum of an operator X. Then the function λ(·) is continuous. Since the family of operators {H(a)}a>0 is non-decreasing, one gets that λ(·) is non-decreasing too. A straightforward computation shows that lima→∞ λ(a) = 0. Since H + is unbounded from below one has that inf a>0 λ(a) = −∞. Let b < 0. Then we set def
N (b) = {a > 0 : λ(a) ≥ b} ,
b < 0.
(4.4)
def
Let a0 = inf N (b). Then by continuity of λ(·) one gets λ(a0 ) = b. Furthermore, a < a0 yields λ(a) < b. Further let us introduce a self-adjoint operator (K(r)f )(x) = −
1 1 d2 f (x) − f (x) , 2 dx 4κ (x + r)2
dom(K(r)) = {f ∈
W22 ([0, 1])
: f (0) = f (1) = 0} ,
(4.5) r > 0.
and def
µ(r) = inf σ(K(r)) .
(4.6)
842
H. NEIDHARDT and V. A. ZAGREBNOV
First of all we note that µ(r) is continuous and non-decreasing function of r > 0 and one has (4.7) lim µ(r) = inf σ(K(∞)) , r→∞
where (K(∞)f )(x) = −
d2 f (x) , dx2
dom(K(∞)) = {f ∈
W22 ([0, 1])
(4.8) : f (0) = f (1) = 0} .
Since inf σ(K(∞)) = π 2 , one gets lim µ(r) = π 2 > 0 .
(4.9)
r→∞
As above one can prove that inf r>0 µ(r) = −∞. Next we introduce the set a2 b ≥0 , M(a) = r ∈ (0, ∞) : µ(r) − (1 + r)2 2
a > 0.
(4.10)
2
a b a b 2 Since limr→0 (1+r) 2 = a b < 0 and limr→∞ (1+r)2 = 0, the set M(a) is always not empty. Let a0 be chosen as above. For a = a0 we set def
r1 = inf M(a0 ) .
(4.11)
By the continuity of µ(·) one always has a20 b . (1 + r1 )2
(4.12)
r1 < a0 . 1 + r1
(4.13)
µ(r1 ) = Let a1 = a 0
def
Now we consider the set M(a1 ). Then we define r2 = inf M(a1 ) and set a2 = a1
r2 < a1 < a0 . 1 + r2
(4.14)
Moreover, one has
r2 r1 . (4.15) 1 + r2 1 + r1 By this manner we work out the sequence of numbers {rn }∞ n=1 which defines the sequence {an }∞ n=1 , where n Y rk an = a 0 . (4.16) 1 + rk a2 = a0
k=1
Let us show that the sequence {an }∞ n=0 tends to zero as n → ∞. To this end we def
note that rj ≤ r0 , j = 1, 2, . . . , where r0 = inf M(0). Since r0 rj ≤ < 1, 1 + rj 1 + r0
j = 1, 2, . . . ,
(4.17)
843
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
one gets an = a0
n Y k=1
rk ≤ 1 + rk
r0 1 + r0
n a0 ,
n = 1, 2, . . . ,
(4.18)
which immediately implies limn→∞ an = 0. We set ∞ [ {an } . Γ=
(4.19)
n=0
and define domain Db = {f ∈ C0∞ (R1+ ) : f (an ) = 0 , an ∈ Γ} which is obviously dense in L2 (R1+ ). To prove that Db is a stability domain for H + we use the following lemma. Lemma 4.1. The inequality Z +∞ Z +∞ Z +∞ 1 1 2 |f 0 (x)|2 dx − |f (x)| dx ≥ b |f (x)|2 dx , 2 4κ x a a a
(4.20)
holds for f ∈ C0∞ (R1+ ), f (a) = 0, 0 < a < +∞ and b < 0 if and only if λ(a) ≥ b. Furthermore, one has Z
β
|f 0 (x)|2 dx −
α
1 4κ
Z
β
α
1 |f (x)|2 dx ≥ b x2
Z
β
|f (x)|2 dx ,
(4.21)
α
for f ∈ C0∞ (R1+ ), f (α) = f (β) = 0, 0 < α < β < +∞ and b < 0 if and only if µ(r) ≥
β2 (1+r)2 b
def
where r =
α β−α .
Proof. Setting x = t + a and g(t) = f (t + a) one gets that (??) is equivalent to Z
+∞ 0
|g 0 (t)|2 dt −
1 4κ
Z 0
+∞
1 |g(t)|2 dt ≥ b (t + a)2
Z
+∞
|g(t)|2 dt ,
(4.22)
0
g ∈ C ∞ (R1+ ), g(0) = 0. Since (??) is equivalent to H(a) ≥ bI one gets that (??) is valid if and only if λ(a) ≥ b. By x = αr (y+r) and h(y) = f ( αr (y +r)), y ∈ [0, 1], we find that (??) is equivalent to Z 1 Z 1 Z 1 1 α2 1 2 |h0 (y)|2 dy − |h(y)| dy ≥ b |h(y)|2 dy , (4.23) 4κ 0 (y + r)2 r2 0 0 α2 r 2 bI.
h ∈ C ∞ ([0, 1]), h(0) = h(1) = 0. However, (??) is equivalent to K(r) ≥ Therefore, the inequality (??) is satisfied if and only if µ(r) ≥
α2 r2 b
=
2
β (1+r)2 b.
Let f ∈ C0∞ (R1+ ) and f (a0 ) = 0. Then we have to verify that Z ∞ Z ∞ Z ∞ 1 1 2 |f 0 (x)|2 dx − |f (x)| dx ≥ b |f (x)|2 dx . 4κ a0 x2 a0 a0
(4.24)
By Lemma ?? this condition is satisfied if λ(a0 ) ≥ b which is right due to (??).
844
H. NEIDHARDT and V. A. ZAGREBNOV
Let f ∈ C0∞ (R1+ ) and f (an ) = f (an−1 ) = 0, n = 1, 2, . . . . Then one has to prove that Z an−1 Z an−1 Z an−1 1 1 0 2 2 |f (x)| dx − |f (x)| dx ≥ b |f (x)|2 dx , n = 1, 2, . . . 4κ an x2 an an (4.25) a2
n−1 Applying Lemma ?? we obtain that (??) is valid if µ(rn ) ≥ (1+r 2 b, n = 1, 2, . . . . n) Since rn = inf M(an−1 ), this immediately follows from (??). Next let us prove that the domain Db is maximal. In fact the reasoning below is the proof of a constrained variational principle. To this end we assume that Db is not maximal. Therefore, there is at least one function f0 ∈ C0∞ (R1+ ) such that / Db but f0 ∈ (4.26) (H + f0 , f0 ) ≥ b(f0 , f0 )
and f0 (ap ) 6= 0 for at least one ap ∈ Γ. Otherwise f0 ∈ Db . We set Γ0 = {an ∈ Γ : f0 (an ) 6= 0} ⊆ Γ .
(4.27)
Denoting by Db0 the linear span of the linear set Db and the element f0 we find that ) ( ) = 0 a ∈ Γ \ Γ g(a n n 0 (4.28) Db0 = g ∈ C0∞ (R1+ ) : f0 (am )g(an ) = f0 (an )g(am ) am , an ∈ Γ0 In other words, if for some am ∈ Γ0 the value g(am ), g ∈ Db0 , is given, then g(an ) is given for all an ∈ Γ0 . Since f0 ∈ C0∞ (R1+ ) there is an m such that inf Γ0 = am , i.e. f0 (an ) = 0 for n = m + 1, m + 2, . . . . For a given a00 ∈ (a1 , a0 ) we define the sequence {a0n }∞ n=1 by a0n = a00
n Y k=1
rk , 1 + rk
n = 1, 2, . . . .
(4.29)
Choosing a suitable a00 ∈ (a1 , a0 ) we satisfy conditions an+1 < a0n < an ,
n = 1, 2, . . . , m .
(4.30)
Since a00 < a0 one has λ(a00 ) < λ(a0 ) = b. Otherwise, we have λ(a00 ) = λ(a0 ) which implies a00 = a0 . Hence by Lemma ?? there is a b00 < b, λ(a00 ) < b00 < b, and a non-trivial element g0 ∈ C0∞ (R1+ ), g0 (a00 ) = 0, such that Z ∞ Z ∞ Z ∞ 1 1 2 0 |g00 (x)|2 dx − |g (x)| dx ≤ b |g0 (x)|2 dx . (4.31) 0 0 4κ a00 x2 a00 a00 Since µ(rn ) ≥ Z
a0n−1
a0n
a2n−1 (1+rn )2 b
|g 0 (x)|2 dx −
1 4κ
0
2 an−1 (1+rn )2
= Z
a0n−1
a0n
an−1 a0n−1
2 b, by Lemma ?? we get that
1 |g(x)|2 dx ≥ x2
an−1 a0n−1
2 Z b
a0n−1
a0n
|g(x)|2 dx ,
(4.32)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
845
for g ∈ C0∞ (R1+ ), g(a0n ) = g(a0n−1 ) = 0, n = 1, 2, . . . , m. Fixing a sequence {b0n }m n=1 , which satisfies the condition 2 an−1 b < b0n < b < 0 , n = 1, 2, . . . , m , (4.33) a0n−1 ∞ 1 0 by Lemma ?? we find a sequence of elements {gn }m n=1 , gn ∈ C0 (R+ ), gn (an ) = gn (a0n−1 ) = 0, n = 1, 2, . . . , m, such that
Z
a0n−1
a0n
|gn0 (x)|2 dx
1 − 4κ
Z
a0n−1
1 |gn (x)|2 dx ≤ b0n x2
a0n
Z
a0n−1
a0n
|gn (x)|2 dx
(4.34)
Let us show that gn (an ) 6= 0 ,
an ∈ Γ 0 .
(4.35)
If gn (an ) = 0 for some an ∈ Γ0 , then Z
a0n−1
|gn0 (x)|2 dx
an
and
Z
an
a0n
1 − 4κ
Z
|gn (x)|2 dx −
a0n−1
an
1 4κ
Z
1 |gn (x)|2 dx ≥ b x2
an
a0n
which contradicts (??). We set g=
Z
|gn (x)|2 dx
(4.36)
|gn (x)|2 dx
(4.37)
an
1 |gn (x)|2 dx ≥ b x2
m X
a0n−1
Z
an
a0n
αn gn ,
(4.38)
n=0
where
0 1 αn = f0 (an ) gm (am ) f0 (am ) gn (an )
an ∈ Γ \ Γ0 n=m
(4.39)
n 6= m, an ∈ Γ0 .
Then g ∈ C0∞ (R1 ) and, moreover, it satisfies f0 (am )g(an ) = f0 (an )g(am ) for an ∈ Γ0 . Hence g ∈ Db0 . Moreover, by (??) one gets that Z ∞ Z ∞ 1 1 |g 0 (x)|2 dx − |g(x)|2 dx 4κ 0 x2 0 Z ∞ Z ∞ ≤ sup b0n |g(x)|2 dx < b |g(x)|2 dx . (4.40) n=0,1,...,m
0
0
which shows that Db0 is not a b-stability domain. The proof of the maximality gives an idea how to find other maximal domains. To this end one chooses a number a = a000 > a0 (see (??)). After that we find numbers a001 , a002 , . . . following the formula (??). Assume that a001 < a0 < a000 which can be arranged by the choice of a000 . Then we set Db00 = {f ∈ C0∞ (R1+ ) : f (a0m ) = 0 ,
m = 0, 1, 2, . . .} .
(4.41)
846
H. NEIDHARDT and V. A. ZAGREBNOV
Repeating the above line of reasoning for a = a000 we obtain that the domain Db00 is also maximal but Db00 6= Db . Note that by construction the symmetric b-stable restriction H + |Db has (as one should anticipate) infinite deficiency indices {∞, ∞} which corresponds to infinite cardinality of the set Γ. Furthermore, Example ?? allows to describe explicitly the Friedrichs extension Aˆ+ of A+ 0 = A|Db . Denoting by ∆n the intervals ∆0 = [a0 , ∞) and ∆n = [an , an−1 ], n = 1, 2, . . ., one can decompose the Hilbert space H = L2 (R1+ ) into H=
∞ M
Hn ,
Hn = L2 (∆n ) .
(4.42)
n=0
ˆ n , n = 0, 1, 2, . . ., the operators Denoting by L ˆ 0 f )(x) = − (L
d2 f (x) , dx2
ˆ 0 ) = {f ∈ W22 (∆0 ) : f (a0 ) = 0} f ∈ dom(L
(4.43)
and d2 ˆ n ) = {dom(W22 (∆n ) : f (an ) = f (an−1 ) = 0} , f (x) , f ∈ dom(L dx2 (4.44) n = 1, 2, . . ., one gets ∞ M ˆn . (4.45) L Aˆ+ = ˆ n f )(x) = − (L
n=0
To illustrate Theorem ?? we show that the Friedrichs extension Aˆ+ is not maximal with respect to W . The element h belongs to the deficiency subspace Nη = ker(A+∗ 0 − ηI) if and only if h is continuous and admits the representation h(x) = b0 e−
√ −ηx
,
x ∈ ∆0 ,
b0 ∈ C ,
(4.46)
and h(x) = bn e−
√ −ηx
+ cn e
√ −ηx
,
x ∈ ∆n ,
bn ∈ C ,
n = 1, 2, . . . .
(4.47)
The continuity of h imposes some relations between the coefficients bn and cn , n = 0, 1, 2, . . .. Obviously there are elements h ∈ Nη such that h(x) = 0 for x ∈ ∆n for n greater than some natural number N > 1. For instance, if N = 2 the function √ − −ηx b e x ∈ ∆0 0 √ √ sinh( −η(x − a )) 1 √ x ∈ ∆1 h(x) = b0 e− −ηa0 (4.48) sinh( −η(a − a 0 1 )) 0 x ∈ ∆n , n = 2, 3, . . . belongs to Nη and has the property that √ h ∈ dom( −W ) ,
(4.49)
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
which means that
Z
∞
0
1 |h(x)|2 dx = x2
Z
∞ a1
1 |h(x)|2 dx < +∞ . x2
847
(4.50)
√
On the other hand, the element h(x) =√be− −ηx , x ∈ R1+ , also belongs to Nη , √ but h ∈ / dom( −W ). Therefore, dom( −W ) ∩ ker(A+∗ 0 − ηI) 6= {0}, i.e., the regularization problem is ill-posed. . Note that by construction (??) of Aˆ+ , the spectrum of form sum H + = A+ + 1 κ W , 0 < κ < 1, for b < 0 has in addition to a discrete part an absolutely continuous part which results from the infinite interval ∆0 . The situation changes, if the b is chosen non-negative, i.e. b ≥ 0. Since the set N (b) (??) is empty for b ≥ 0 (see e.g. [?]), we proceed as follows. First we choose an arbitrary a = a0 > 0. Then as above we define with this a0 the sequence {an }∞ n=1 and we can prove that limn→∞ an = 0. After that we introduce the set a2 b def (4.51) M− (a) = r ∈ (0, ∞) : µ(r) − 2 ≥ 0 , a > 0 . r 2
2
Since limr→0 ar2b = ∞ (or zero if b = 0) and limr→∞ ar2b = 0, (??) implies that the set M− (a) is always not empty. Furthermore, one has inf M− (a) > 0. So we can define (4.52) r−1 = inf M− (a0 ) and a−1 = a0
1 + r−1 > a0 . r−1
(4.53)
Then we pass to r−2 = inf M− (a1 ) and a−2 = a1
1 + r−2 1 + r−2 1 + r−1 = a0 > a−1 > a0 . r−2 r−2 r−1
(4.54)
∞ Finally, we obtain a sequence {r−n }∞ n=1 which defines the sequence {a−n }n=1 , where
a−n = a0
n Y 1 + r−k . r−k
(4.55)
k=1
Let us show that limn→∞ a−n = ∞. By monotonicity limn→∞ a−n = a−∞ exist. Assume that a−∞ < +∞. We set r−∞ = inf M− (a−∞ ). Obviously we have r−j ≤ r−∞ which yields 1< Since a−n = a0
1 + r−j 1 + r−∞ ≤ , r−∞ r−j
j = 1, 2, . . . .
n n Y 1 + r−k 1 + r−∞ ≥ a0 , r−k r−∞
n = 1, 2, . . . ,
(4.56)
(4.57)
k=1
one gets limn→∞ a−n = a−∞ = ∞. Therefore, the sequence {a−n }∞ n=0 tends to infinity as n → ∞.
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Setting −
Γ =
∞ [
{an }
(4.58)
n=−∞
one defines as above Db = {f ∈ C0∞ (R1+ ) : f (an ) = 0 , an ∈ Γ− }. Similarly to the case b < 0 one can show that Db is a maximal b-stability domain for H + . Moreover, starting with a = a00 6= a0 one can construct another maximal b-stability domain Db0 for b ≥ 0. In contrast to the case b < 0, when limn→∞ an = 0, it turns out that for . b ≥ 0, when limn→∞ a−n = +∞, the spectrum of the form sum H + = Aˆ + κ1 W , 0 < κ < 1, is purely point one, i.e., absolutely continuous part of the spectrum is absent. 5. Conclusions This paper is initiated by a general problem of definition and construction of the right Hamiltonian for non-positive singular perturbations, see [?, ?, ?, ?]. The theory developed there for a pair of self-adjoint operators {A ≥ 0, W ≤ 0} includes as an essential ingredient a stability domain of the pair {A, W }, i.e., a dense domain Ds ⊆ dom(A) ∩ dom(W ) such that (??) is satisfied. A priori the existence of Ds is not evident. Our abstract Theorem ?? gives a simple criterion for that: symmetric operator Hα = A + αW with domain dom(Hα ) = dom(A) ∩ dom(W ) should be unbounded from above at least for one α > 1. From the physical point of view this is a very reasonable condition which is, of course, very often satisfied in physical systems. Naturally this leads to a simple idea to handle non-positive singular perturbations as follows: we choose a stability domain of the pair {A, W } and a regularizing sequence for the non-positive singular perturbation W , with this we can construct the approximating Hamiltonian sequence and determine the limit of this sequence which we call the “right Hamiltonian” of the perturbed system, i.e., unique, self-adjoint, semi-bounded from below operator. However, this simple idea is false. The main reason for that is that in general the “right Hamiltonian” is not uniquely determined in this way. In order to guarantee uniqueness of the “right Hamiltonian” one has to verify maximality of Friedrichs extension Aˆ of A|Ds with respect to the singular perturbation W [?, ?]. If one has maximality, then the “right Hamiltonian” for {A, W } is unique and coincides with the form-sum of Friedrichs extension Aˆ and the perturbation W . Since this case is very satisfactory it was called in [?, ?] the “well-posed regularization problem”. In the opposite case, i.e., if there is no maximality, one gets a variety of “right Hamiltonians”. So, the pleasant and important from the physical point of view uniqueness is lost. Consequently, the regularization problem was called ill-posed. In [?] it was mentioned that in some sense an ill-posed regularization problem is an incomplete well-posed one and that it can be reduced to the latter but not in a unique way. Consequently, in order to guarantee the uniqueness of the “right Hamiltonian” it is necessary to introduce besides the regularization method some supplementary physical principles to select it.
DOES EACH SYMMETRIC OPERATOR HAVE A STABILITY DOMAIN?
849
Unfortunately the ill-posed case always realizes if the set dom(A) ∩ dom(W ) is not a stability domain of {A, W }. This means, that the Theorem ??, which guarantees the existence of a stability domain Ds ⊆ dom(A) ∩ dom(W ) of {A, W }, does not save the situation: the ill-posed case necessarily appears due to Theorem ??. Hence the regularization method of [?, ?, ?, ?] is unsuitable to find a “right Hamiltonian” if we have no canonical stability domain from the beginning. In other words, the regularization problem is well-posed for the pair {A, W } only if dom(A) ∩ dom(W ) is a stability domain of {A, W } and, moreover, the Friedrichs extension Aˆ of A|dom(A) ∩ dom(W ) is maximal with respect to the singular perturbation W . This is the physical quintessence of Sec. 3. Besides this we hope that Theorem ?? has an independent meaning since it gives a criterion when a symmetric operator can be restricted to semi-bounded one. In particular it implies that well-known quantum momentum (or coordinate) operator on the real axis can be restricted to a non-negative one.a Finally, in Sec. 4 we present a solution of the quantum mechanics “fall to the center” problem for the Hamiltonian H κ1 , κ ∈ (0, 1), of Example ??. Theorem ?? suggests the idea that H κ1 is defined on a domain which is too large. Hence one has to restrict H κ1 in a suitable manner such that it becomes semi-bounded. In Sec. 4 it is shown how this can be done. From the construction of (maximal) stability domains it follows that the physical interpretation of the way to prevent the fall to the center is to introduce a suitable sequence of point barriers of infinite height which come closer and closer to the center. How to choose the sequence of point barriers is indicated in by Γ (??) and Γ− (??). However, it is easy to see that there are several possibilities to do this. Finally, to illustrate our “no-go” Theorem ?? we show that the Friedrichs extension Aˆ+ is not maximal with respect to the perturbation W . Acknowledgments We would like to thank the referee for numerous useful remarks and especially for his help in finding out of a flaw in the first proof of maximality of the domain Db in our Example (Sec. 4). This flaw is corrected in the present version of the paper. We are grateful to Charles Radin who attracted our attention to the papers [?, ?, ?]. The first author (H. N.) thanks the Center de Physique Th´eorique, Universit´e de Toulon et du Var and Universit´e de Provence for hospitality during his visits to Marseille–Luminy and for financial support. Note added in Proof We would like to thank Werner Timmermann who attracts our attention to the von Neumann’s remark in Math. Ann. 102 (1929) 49–131 (“Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren”) about “patalogies” of unbounded symmetric operators. There von Neumann discovered the same phenomenon as a N. Neidhardt and V. A. Zagrebnov, “On semi-bounded restrictions of self-adjoint operators”, preprint CPT-97/P. 3512, Marseille-Luming, 1997.
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we discuss in Sec. 2 of the present paper. References [1] K. M. Case, “Singular potentials”, Phys. Rev. 80 (5) (1950) 797–806. [2] H. van Haeringen, “Bound states for r −2 -like potentials in one and more dimensions”, J. Math. Phys. 19 (10) (1978) 2171–2179. [3] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. [4] H. Neidhardt and V. A. Zagrebnov, “Regularization and convergence for singular perturbations”, Commun. Math. Phys. 149 (1992) 573–586. [5] H. Neidhardt and V. A. Zagrebnov, “Singular perturbations, regularization and extension theory”, in Operator Theory: Advances and Applications 70, 299–305, Birkh¨ auser Verlag, Basel, 1994. [6] H. Neidhardt and V. A. Zagrebnov, “On the right Hamiltonian for singular perturbations: General Theory”, Rev. Math. Phys. 9 (5) (1997). [7] H. Neidhardt and V. A. Zagrebnov, “Towards the right Hamiltonian for singular perturbations via regularization and extension theory”, Rev. Math. Phys. 8 (5) (1996) 715–740. [8] E. Nelson, “Feynman integrals and Schr¨ odinger equation”, J. Math. Phys. 5 (3) (1964) 332–343. [9] C. Radin, “Some remarks on the evolution of a Schr¨ odinger particle in an attractive 1/r2 potential”, J. Math. Phys. 16 (3) (1975) 544–547. [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, 1972. [11] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. [12] K. Schm¨ udgen, “On restrictions of unbounded symmetric operators”, J. Operator Theory. 11 (1984) 379–393. [13] G. E. Shilov, An Introduction to the Theory of Linear Spaces, Prentice-Hall, Inc., N. J., 1961.
CONSTRUCTION OF KINK SECTORS FOR TWO-DIMENSIONAL QUANTUM FIELD THEORY MODELS AN ALGEBRAIC APPROACH DIRK SCHLINGEMANN II. Institut f¨ ur Theoretische Physik Universit¨ at Hamburg Germany and Erwin Schr¨ odinger International Institute for Mathematical Physics (ESI) Boltzmanngasse 9, A-1090 Wien, Austria Several two-dimensional quantum field theory models have more than one vacuum state. Familiar examples are the Sine-Gordon and the φ42 -model. It is known that in these models there are also states, called kink states, which interpolate different vacua. A general construction scheme for kink states in the framework of algebraic quantum field theory is developed in a previous paper. However, for the application of this method, the crucial condition is the split property for wedge algebras in the vacuum representations of the considered models. It is believed that the vacuum representations of P (φ)2 -models fulfill this condition, but a rigorous proof is only known for the massive free scalar field. Therefore, we investigate in a construction of kink states which can directly be applied to a large class of quantum field theory models, by making use of the properties of the dynamics of a P (φ)2 and Yukawa2 models.
1. Introduction Studying 1 + 1-dimensional quantum field theories from an axiomatic point of view shows that kink sectors naturally appear in the theory of superselection sectors [21, 22, 58]. This paper is concerned with the construction of kink sectors for concrete quantum field theory models, like P (φ)2 and Yukawa2 models.a Our subsequent analysis is placed into the framework of algebraic quantum field theory which has turned out to be a successful formalism to describe physical concepts like observables, states, superselection sectors (charges) and statistics. These notions can be appropriately described by mathematical concepts like C ∗ -algebras, positive linear functionals and equivalence classes representations. For the convenience of the reader, we shall state the relevant definitions and assumptions here. Let O ⊂ R1,s be a region in space-time. We denote by A(O) the algebra generated by all observables which can be measured within O. For technical reasons we always suppose that A(O) is a C ∗ -algebra and O is a double cone, i.e. a bounded and causally complete region. Motivated by physical principles, we make the following assumptions: a Parts
are extracted from the PhD thesis [61]. 851
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D. SCHLINGEMANN
(1) The assignment A : O 7→ A(O) is an isotonous net of C ∗ -algebras, i.e. if O1 is contained in O2 , then A(O1 ) is a C ∗ -sub-algebra of A(O2 ). The isotony encodes the fact that each observable which can be measured within O can also be measured in every larger region. Furthermore, the C ∗ -inductive limit C ∗ (A) of the net A can be constructed since the set of double cones is directed. We refer to [57] for this notion. (2) Two local operations which take place in space-like separated regions should not influence each other. The principle of locality is formulated as follows: If the regions O1 and O2 are space-like separated, then the elements of A(O1 ) commute with those of A(O2 ). (3) Each operator a which is localized in a region O should have an equivalent counterpart which is localized in the translated region O + x. The principle of translation symmetry is encoded by the existence of an automorphism group {αx ; x ∈ R1,s } which acts on the C ∗ -algebra C ∗ (A) such that αx maps A(O) onto A(O + x). A net of C ∗ -algebras which fulfills conditions (1) to (3) is called a translationally covariant Haag–Kastler net . In order to discuss particle-like concepts, we select an appropriate class S of normalized positive linear functionals, called states, of C ∗ (A). We require the states ω ∈ S to fulfill the conditions: (1) There exists a strongly continuous unitary representation of the translation group U : x 7→ U (x) on the GNSb -Hilbert space H which implements the translations in the GNS-representation π, i.e. π(αx a) = U (x)π(a)U (−x) for each a ∈ C ∗ (A). (2) The stability of a physical system is encoded in the spectrum condition (positivity of the energy), i.e. the spectrum (of the generator) of U (x) is contained in the closed forward light cone. These conditions are also known as the Borchers criterion. States which satisfy the Borchers criterion and which are, in addition, translationally invariant are called vacuum states. a state ω ∈ S, we obtain via GNS-construction a Hilbert space H, a ∗ -representation π of C ∗ (A) on H and a vector Ω ∈ H such that hΩ, π(a)Ωi = ω(a) for each a ∈ C ∗ (A). The triple (H, π, Ω) is called the GNS-triple of ω.
b Given
CONSTRUCTION OF KINK SECTORS FOR
...
853
Kinks already appear in classical field theories and the typical systems in which they occur are 1 + 1-dimensional. Familiar examples are the Sine-Gordon and the φ42 -model. We briefly describe the latter: The Lagrangian density of the model is given by L(φ, x) =
1 ∂µ φ(x)∂ µ φ(x) − U (φ(x)) , 2
where the potential U is given by U (z) := λ/2 (z 2 − a)2 . The energy of a classical field configuration φ is Z 1 1 (∂0 φ(0, x))2 + (∂1 φ(0, x))2 + U (φ(0, x)) . E(φ) = dx 2 2 With the choice of U , given above, the absolute minimum value of U is zero and thus the energy functional E : φ 7→ E(φ) is positive. There are two configurations φ± with zero energy E(φ± ) = 0: φ± : (t, x) 7→ ± a . These configurations are invariant under space-time translations and represent the vacua of the classical system. There are two further configurations φs , φs¯ which are stationary points of the energy functional E. They are given by √ √ φs : (t, x) 7→ a tanh( λax) and φs¯ : (t, x) 7→ −a tanh( λax) . These configurations represent the kinks of the classical system which interpolate the vacua φ± . Indeed, we have for the kink φs lim φs (t, x) = φ± (t, x) = ±a .
x→±∞
(1)
The configuration φs¯, which interpolates the vacua φ± in the opposite direction, represents the anti-kink of φs . Both of them have the same energy, namely E(φs ) = E(φs¯) =
4√ 3 λa . 3
From the classical example above, we see that the crucial properties of a kink are to interpolate vacuum configurations as well as to be a configuration of finite energy. Motivated by these properties, in quantum field theory a kink state ω is defined as follows: The interpolation property: For each observable a, the limits lim ω(α(t,x) (a)) = ω± (a)
x→±∞
(2)
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D. SCHLINGEMANN
exist and ω± are vacuum states. Note that Eq. (2) is the quantum version of the interpolation property (1). Positivity of the energy: ω fulfills the Borchers criterion. In the literature the concept of kink as described above is often called soliton (see [27, 28]) or more seldom lump (see [12]). In the following, we shall use the word kink. In [59], a construction scheme for kink states has been developed which is based on general principles. In order to make the comprehension of the subsequent sections easier we shall state the main ideas here. The construction of an interpolating kink state is based on a simple physical idea: Let A be a Haag–Kastler net of W ∗ -algebras in 1 + 1-dimensions. Each double cone O splits our system into two infinitely extended laboratories, namely the laboratory which belongs to the left space-like complement OLL , and the laboratory ORR which belongs the right spacelike complement ORR . In order to prepare an interpolating kink state, we wish to prepare one vacuum state ω1 in the left laboratory OLL , and another vacuum state ω2 in the right laboratory ORR . This can only be done if the preparation of ω1 does not disturb the preparation procedure of ω2 . In other words, the physical operations which take place in the laboratory on the left side OLL should be statistically independent of those which take place in ORR . Therefore, we require that there exists a vacuum representation π0 such that the W ∗ -tensor product Aπ0 (OLL ) ⊗ Aπ0 (ORR ) is unitarily isomorphic to the von Neumann algebra Aπ0 (OLL ) ∨ Aπ0 (ORR ) , where Aπ0 is the net in the vacuum representation π0 .c This condition is equivalent to the existence of a type I factor N which sits between Aπ0 (ORR ) and Aπ0 (OR ): Aπ0 (ORR ) ⊂ N ⊂ Aπ0 (OR ) . Here OR is the space-like complement of OLL . In other words, the inclusion Aπ0 (ORR ) ⊂ Aπ0 (OR )
(3)
is split . A detailed investigation of standard split inclusions of W ∗ -algebras has been carried out by S. Doplicher and R. Longo [19]. We also refer to the results of D. Buchholz [9], C. D’Antoni and R. Longo [14] and C. D’Antoni and K. Fredenhagen [13]. an unbounded region U , Aπ0 (U ) denotes the von Neumann algebra which is generated by all local algebras Aπ0 (O) with O ⊂ U .
c For
CONSTRUCTION OF KINK SECTORS FOR
...
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Let ω1 and ω2 be two inequivalent vacuum states whose restrictions to each local algebra A(O) are normal. Using the isomorphy Aπ0 (OLL ) ⊗ Aπ0 (ORR ) ∼ = Aπ0 (OLL ) ∨ Aπ0 (ORR ) we conclude that the map ab 7→ ω1 (a)ω2 (b), a is localized in OLL and b is localized in ORR , defines a state of the algebra C ∗ (A, OLL ∪ ORR ) which, by the Hahn–Banach theorem, can be extended to a state ω of the C ∗ -algebra of all observables. The state ω interpolates the vacua ω1 and ω2 correctly, but for an explicit construction of an interpolating state which satisfies the Borchers criterion, some technical difficulties have to be overcome. The condition that the inclusion (3) is split is sufficient to develop a general construction scheme for interpolating kink states. We shall give a brief description of it here. Step 1 : We consider the W ∗ -tensor product of the net A with itself: A ⊗ A : O 7→ A(O) ⊗ A(O) . The map αF which is given by interchanging the tensor factors, αF : a1 ⊗ a2 7→ a2 ⊗ a1 is called the flip automorphism. Since the inclusion (3) is split, the flip automorphism is unitarily implemented on Aπ0 ⊗ Aπ0 (ORR ) by a unitary operator θ which is contained in Aπ0 ⊗ Aπ0 (OR ) [14]. The adjoint action of θ induces an automorphism β := (π0 ⊗ π0 )−1 ◦ Ad(θ) ◦ π0 ⊗ π0 which maps local algebras into local algebras. Here we have assumed that the representation π0 is faithful in order to build the inverse π0−1 . For each observable a which is localized in the left space-like complement of O we have β(a) = a, and for each observable b which is localized in the right space-like complement of O we have β(b) = αF (b). Note that β may depend on the choice of the vacuum representation π0 . Step 2 : It is obvious that the state ω := ω1 ⊗ ω2 ◦ β|C ∗ (A)⊗1 interpolates ω1 and ω2 . Let π1 and π2 be the GNS-representations of ω1 and ω2 respectively. Then the GNS-representation π = π1 ⊗ π2 ◦ β|C ∗ (A⊗1) of ω is translationally covariant because the automorphism αx ◦ β ◦ α−x ◦ β
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is implemented by a cocycle γ(x) of local operators in C ∗ (A). The positivity of the energy can be proven by showing the additivity of the energy-momentum spectrum for automorphisms like β. This together implies that ω is an interpolating kink state. In comparison to previous constructions, in particular the work of J. Fr¨ ohlich in which the existence of kink states for the φ42 and the Sine-Gordon model is proven [27, 29, 28], our construction scheme has the following advantages: ⊕ It is independent of specific details of the considered model because the split property (3), which is the crucial condition for applying the construction scheme, can be motivated by general principles. ⊕ It can be applied to pairs of vacuum sectors which are not related by a symmetry transformation, whereas the techniques of J. Fr¨ohlich rely on the existence of a symmetry transformation connecting different vacua. Indeed, according to J. Z. Imbrie [44], there are examples for P (φ)2 models possessing more than one vacuum state, but where the different vacua are not related by a symmetry. We also mention here the papers of K. Gawedzki [35] and S. J. Summers [65]. Unfortunately, there is one disadvantage which is the price we have to pay for using a model independent analysis. The split property for wedge algebras (3) has to be proven for the vacuum states of the model under consideration if we want to apply our construction scheme to it. It is believed that the vacuum states of the P (φ)2 - and Yukawa2 models fulfill this condition, but a rigorous proof is only known for the massive free Bose and Fermi field [13, 9, 64]. In the present paper, we investigate an alternative construction of kink states which can directly be applied to models. It is convenient to formulate our setup in the time slice formulation of a quantum field theory. The time slice-formulation has two main aspects. First, the Cauchy data with respect to a given space-like plane Σ which describes the boundary conditions at time t = 0. Second, the dynamics which describes the time evolution of the quantum fields. The Cauchy data of a quantum field theory are given by a net of v Neumannalgebras M := {M(I) ⊂ B(H0 ) ; I is open and bounded interval in Σ} represented on a Hilbert-space H0 . This net has to satisfy the following conditions: (1) The net is isotonous, i.e. if I1 ⊂ I2 , then M(I1 ) ⊂ M(I2 ). (2) The net is local, i.e. if I1 ∩ I2 = ∅, then M(I1 ) ⊂ M(I2 )0 . (3) There exists a unitary and strongly continuous representation U : x ∈ R 7→ U (x) ∈ U(H0 ) of the spatial translations in Σ ∼ = R, such that αx := Ad(U (x)) maps M(I) onto M(I + x).
CONSTRUCTION OF KINK SECTORS FOR
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A one-parameter group of automorphisms α = {αt ∈ Aut(M); t ∈ R} (Aut(M) denotes the automorphisms of C ∗ (M)) is called a dynamics of the net M if the following conditions are fulfilled: (a) The automorphism group α has propagation speed ps(α) ≤ 1, where ps(α) is defined by ps(α) := inf{β 0 |αt M(I) ⊂ M(Iβ 0 |t| ) ; ∀ t, I} . Here Is := I + (−s, s) denotes the interval, enlarged by s > 0. (b) The automorphisms {αt ∈ Aut(M); t ∈ R} commute with the automorphism group of spatial translations {αx ∈ Aut(M); x ∈ R}, i.e. αt ◦ αx = αx ◦ αt ; ∀ x, t . The set of all dynamics of M is denoted by dyn(M). For our purposes it is crucial to distinguish carefully the C ∗ -inductive limit C ∗ (M) of the net M and the corresponding C ∗ - and W ∗ -algebras, which belong to an unbounded region J ⊂ Σ. They are denoted by C ∗ (M, J ) :=
[
k·k
M(I)
and M(J ) :=
I⊂J
_
M(I) respectively .
I⊂J
We claim here that the Cauchy data of the P (φ)2 - and the Yukawa2 model are given by the nets of the corresponding free fields at time t = 0. Before we continue to discuss our methods for constructing kink states, we briefly give here a review of methods and techniques which has been applied in previous papers. During the 70s, examples for interacting quantum field theory models were constructed. It was proven by J. Glimm, A. Jaffe and T. Spencer that two-dimensional models with P (φ)2 -interaction exist, and their vacuum states satisfy the Wightman axioms [36, 40]. Interactions between fermions and bosons have also been studied, in particular the Yukawa2 interactions [36, 37, 62, 63]. Furthermore, an investigation of the Sine-Gordon model has been carried out by J. Fr¨ ohlich an E. Seiler [34]. A few years later, a great deal of attention has been paid to the construction of new superselection sectors which are different from vacuum sectors. In 1976, the existence of kink sectors for the (Φ · Φ)22 -and the Sine-Gordon model was established by J. Fr¨ ohlich [27, 29] (compare also [30, 31]). To illustrate the ideas and techniques which have been used in [27], we give a short review of the construction of the kink sectors of the (Φ · Φ)22 -model. The basic ingredients for the construction of kink sectors of the (Φ · Φ)22 -model have been taken from the work of J. Glimm, A. Jaffe and T. Spencer [36]. They have proven that there are two inequivalent vacuum states ω± for the (Φ·Φ)22 model which are related by a symmetry χ ∈ Aut(M) αt ◦ χ = χ ◦ αt
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in the following way: ω+ ◦ χ = ω− . The construction proceeds in several steps: Step 1 : Let s be a smooth test function with the property: There exists a bounded interval I ⊂ Σ such that π if x ∈ IRR , s(x) = 0 if x ∈ ILL where IRR is the right and ILL is the left complement of I. The O(2)-valued function cos(s(x)) sin(s(x)) ∈ O(2) gs : x 7→ gs (x) = − sin(s(x)) cos(s(x)) induces a Bogoliubov automorphism ρs which is defined on the Weyl operators by ρs : exp(iΦ(f1 ) + iΠ(f2 )) 7→ exp(iΦ(gs f1 ) + iΠ(gs f2 )) where Φ = (Φ1 , Φ2 ) is a massive free two-component Bose field and Π its canonically conjugate, acting as operator valued distributions on the Fock space H0 . Since −12 if x ∈ IRR , gs (x) = 12 if x ∈ ILL the automorphism ρs acts trivially on operators which are localized in ILL and as the symmetry χ on those which are localized in IRR . ¯ s := ω+ ◦ ρ−1 Obviously, the states ωs := ω− ◦ ρs and ω s fulfill the interpolation condition for kink states. Step 2 : The explicit knowledge of the dynamics α can be used to prove the existence of a strongly continuous function γ : (t, x) 7→ γ(t, x) , where γ(t, x) is a unitary operator, localized in a sufficiently large interval I(t,x) . It implements the automorphism α(t,x) ◦ ρs ◦ α(−t,−x) ◦ ρ−1 s = Ad(γ(t, x)) and satisfies the cocycle condition: γ(t1 + t, x1 + x) = α(t,x) (γ(t1 , x1 ))γ(t, x) .
(4)
The operators γ(t, x) describe the translation by (t, x) of the kink charge [ω− ◦ ρs ]. It follows from the properties of γ that ωs is translationally covariant and satisfies the spectrum condition. The same holds for the state ω ¯ s := ω+ ◦ ρ−1 s . This implies ¯ s are kink states. that ωs and ω In 1977 J. Fr¨ohlich proved the existence of the kink states of the one-component φ42 -model [28] by using, in comparison to [27], an alternative method. The technical
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difficulties which arise here are due to the fact that one has to deal with a onecomponent Bose field. Therefore, there is no a priori choice for a Bogoliubov transformation ρs . We shall give a brief summary of the results of [28] to illustrate the main differences to the construction of the (Φ · Φ)22 -kinks. The construction of the vacuum sectors of the φ42 -model, which is presented in [28], uses the methods of field theory. The vacuum states of the φ42 -model can be obtained from two measures µ± on S 0 (R2 ) which satisfy the Osterwalder–Schrader axioms. We briefly explain how the measures µ± are constructed as limits of perturbations of the Gaussian measure µ0 . Step 1 : Let µ0 be the Gaussian measure on the space of tampered distributions S 0 (R2 ) with mean zero and covariance C where the integral kernel of C is Z C(x − y) = d2 p (p2 + m2 )−1 eip(x−y) . The regularized interaction part of the Euclidean action is Z S1 (g, φ) = d2 x g(x) (λ : φ(x)4 :µ0 −σ : φ(x)2 :µ0 ) , where : · :µ0 is the normal ordering with respect to the Gaussian measure µ0 and g is a smooth test function. The action S1 (g, φ) is invariant under the substitution φ 7→ −φ. To approximate one of the measures µ± the Z2 symmetry has to be broken explicitly by introducing appropriate boundary terms. The test function g can be chosen in such a way that it is one in the region IT × IL and zero outside a slightly larger region. Here the interval Is is defined by Is := (−s/2, s/2). For L1 < L the region IL \IL1 has two connected components I± and there are two possibilities (corresponding to µ+ or µ− ) to choose boundary conditions with respect to each of the regions IT × I± This gives four different boundary terms {δSj,± (φ) = φ(gj,± ) + cj,± ; j = ±} , where gj,± are suitable test functions which have support in a neighborhood of IT × I± and cj,± are appropriate constants. The regularized interaction part of the Euclidean action with boundary terms is Sij (g, φ) = S1 (g, φ) + δSi,+ (φ) + δSj,− (φ) . Step 2 : To approximate the measure µ± , we perturb µ0 by a positive L1 -function dµT,L,± (φ) := Z(T, L, ±) dµ0 (φ) exp(−S±± (g, φ)) , where the constant Z(T, L, ±) is for normalization. According to J. Glimm, A. Jaffe and T. Spencer [40], the limits Z Z dµT,L,± (φ) exp(φ(f )) , dµ± (φ) exp(φ(f )) = lim lim L→∞ T →∞
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D. SCHLINGEMANN
which determine the measures µ± , exist for each test function f . Since the different choices for the boundary terms are related by the the map φ 7→ −φ, i.e. φ(g+,± ) = −φ(g−,± ) , the measures µ+ and µ− satisfy the relation dµ+ (−φ) = dµ− (φ) . Step 3 : There are four Hamilton operators {Hij (L); i, j = ±} acting on the Fock space H0 of the massive free scalar field. They are related to the unnormalized measures dµT,L,ij (φ) := dµ0 (φ) exp(−Sij (g, φ)) by Nelson’s Feynman–Kac formula: Z dµT,L,ij (φ) = hΩ0 , exp(−T Hij (L))Ω0 i . Here Ω0 is the bare vacuum vector in H0 . Let M : I 7→ M(I) be the net of Cauchy data for the massive free scalar field. The dynamics of the φ42 -model can be obtained by the prescription αt (a) := lim eitHij (L) a e−itHij (L) , L→∞
where the limit is independent of the choice of the boundary conditions. Finally, by using the Osterwalder–Schrader reconstruction theorem, two vacuum states ω± with respect to the dynamics α can be constructed from the measures µ± . The crucial property which allows us to carry through the analysis of [28] is the following: Let I be a bounded interval, then the observables which are localized in the left complement ILL of I statistically independent of those which are localized in the right complement IRR . This means, formulated in the language of operator algebras, that the W ∗ -tensor product M(ILL ) ⊗ M(IRR ) is unitarily isomorphic to the W ∗ -algebra M(ILL ) ∨ M(IRR ) . The statistical independence for half-line algebras has been proven according to [9, 64, 59]. We now describe the main steps of the construction of the kink sectors of the φ42 -model. Step 1 : According to [14, 19], the statistical independence of M(ILL ) and M(IRR ) implies the existence of a unitary operator uI which has the following properties: Let a and b be operators which are localized in ILL and IRR respectively. Then the relations uI a u∗I = a and uI b u∗I = χ(b) hold .
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Here χ is the Bogoliubov automorphism which is induced by the map φ 7→ −φ. Step 2 : According to the results of [27], it can be shown that for each t the limit γ 0 (t) := lim exp(itH++ (L))uI exp(−itH−+ (L))u∗I L→∞
exists and that the operator γ 0 (t) is localized in a sufficiently large interval It . Note that the Hamiltonian H−+ (L) belongs to the following interpolating boundary conditions: The left boundary term is chosen with respect to the boundary conditions for the vacuum ω− and the right boundary term is chosen with respect to the boundary conditions for the vacuum ω+ . Finally, the charge transporters are given by γ(t, x) := αx (γ 0 (t)uI )u∗I and the corresponding interpolating automorphism ρ can be obtained from γ by the uniform limit ρ(a) = lim γ(t, x) a γ(t, x)∗ . x→−∞
It follows from its construction that ρ acts trivially on the observables which are localized in ILL and acts as the symmetry χ on those which are localized in IRR . The kink sector and its anti-kink sector are θ = [ω+ ◦ ρ] and θ¯ = [ω+ ◦ ρ−1 ] respectively . This result is in complete analogy to the result for the (Φ · Φ)22 -model, i.e. in both models the same four irreducible sectors appear. At this point, we shall mention here some further treatments of kink sectors: (i) In [27, Chap. 5], the existence of kink states in general P (φ)2 -models is discussed. However, this construction leads only to kink states which interpolate vacua which are connected by the internal symmetry transformation φ 7→ −φ. We shall see later that we achieve a generalization of this result. (ii) In the late 80s, J. Fr¨ ohlich and P. A Marchetti developed a quantization of kinks in terms of Euclidean functional integrals which has been applied to several lattice field theories [32, 51, 33]. (iii) Recently, a construction of kink sectors for a lattice version of the XY model has been carried out by H. Araki [1] and for XXZ models by T. Matsui [52, 53]. (iv) Moreover, by using euclidean techniques (compare [28]), an estimate for the mass of the (λφ4 )2 soliton has been established in [3]. In the following, we briefly explain how the construction of kink states can be generalized to a larger class of quantum field theory models for which the conditions, listed below, hold: (i) The dynamics of the model satisfies an appropriate extendibility condition which we shall explain later. (ii) The vacuum states are local Fock states which is automatically satisfied for P (φ)2 and Yukawa2 models [36, 63].
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D. SCHLINGEMANN
Step 1 0 : We consider the two-fold net M ⊗ M : I 7→ M(I) ⊗ M(I) . Like in Step 1 of our previous construction scheme, the split property implies that on M(IRR ) ⊗ M(IRR ), the flip automorphism is implemented by a unitary operator θI [14]. The adjoint action of θI is an automorphism β I which has the following properties: (i) The automorphism β I acts trivially on observables which are localized in the left complement of I and it acts like the flip on observables which are localized in the right complement of I. (ii) The automorphism β I maps local algebras into local algebras. Note that the automorphism β I does not depend on the dynamics α. Step 2 0 : Let ω1 , ω2 be two vacuum states with respect to a given dynamics α. The state ω := ω1 ⊗ ω2 ◦ β I |C ∗ (M)⊗C1 interpolates the vacua ω1 and ω2 . Moreover, it is covariant under spatial translations since for each x the operator γ(0, x) = (αx ⊗ αx )(θI )θI is localized in a sufficiently large bounded interval. Indeed, the unitary operators U (0, x) := (U1 (0, x) ⊗ U2 (0, x)) (π1 ⊗ π2 )(γ(0, −x)) implement the spatial translations in the GNS-representation of ω where U1 and U2 implement the translations in the GNS-representations π1 , π2 of ω1 and ω2 respectively. Step 3 0 : It remains to be proven that ω is translationally covariant with respect to the dynamics α. For this purpose, we wish to construct a cocycle γ(0, t) such that the operators U (t, 0) := (U1 (t, 0) ⊗ U2 (t, 0)) (π1 ⊗ π2 )(γ(−t, 0)) implement the dynamics α in the GNS-representation of ω. The operator γ(t, 0) := (αt ⊗ αt )(θI )θI is a formal solution. Unfortunately, the flip implementer θI is not contained in any local algebra and the term (αt ⊗ αt )(θI ) has no mathematical meaning unless α is the free dynamics. However, it can be given a meaning in some cases. We shall see that for an interacting dynamics there exists a suitable cocycle of the operators γ(t, 0) such that γ(t, 0) is localized in a bounded interval whose size depends linearly on |t|.
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In order to formulate a sufficient condition for the existence of γ(t, 0), we conˆ to be the von Neumann struct an extension of the net M ⊗ M. We define M(I) algebra which is generated by M(I) ⊗ M(I) and the operator θI . The net ˆ : I 7→ M(I) ˆ M is an extension of M ⊗ M which does not fulfill locality. This is due to the nontrivial implementation properties of θI . We shall call a dynamics α extendible if ˆ which is an extension of α ⊗ α. Indeed, there exists a dynamics α ˆ of M t 7→ γ(t, 0) := α ˆ t (θI )θI is a cocycle which has the desired properties. Finally, we conclude like in Step 3 of our previous construction scheme that the state ω := ω1 ⊗ ω2 ◦ β I |C ∗ (M)⊗C1 is a kink state where ω1 , ω2 are vacuum states with respect to the dynamics α. Since the extendibility condition is rather technical one might worry that it is only fulfilled for few exceptional cases. Fortunately, this is not true. There is a large class of quantum field theory models whose dynamics are extendible. We shall prove that the extendibility holds for the following models: (i) P (φ)2 -models. (ii) Yukawa2 models. (iii) Special types of Wess–Zumino models. Note that a Dirac spinor field contributes to the field content of the Yukawa2 and Wess–Zumino models, and the nets of Cauchy data fulfill twisted duality instead of Haag duality [64]. According to recent results which have been established by M. M¨ uger [54], our results remain true for these cases also. Wess–Zumino models have been studied in several papers. We refer to the work of A. Jaffe, A. Lesniewski, J. Weitsman and S. Janowsky [45, 48, 49, 46, 47]. It has been proven in [46] that some Wess–Zumino models possess more than one vacuum sector. An application of our construction scheme proves the existence of kink states for these models. 2. Preliminaries In the first part (Sec. 2.1) of this preliminary section, we briefly describe how to construct a Haag–Kastler net from a given net of Cauchy data and a given dynamics. Examples for physical states with respect to an interacting dynamics are given in the second part (Sec. 2.2). 2.1. From Cauchy data to Haag Kastler nets We denote by U (M) the group of unitary operators in C ∗ (M). Let G(R, M) be the group which is generated by the set {(t, u)|t ∈ R modulo the following relations:
and u ∈ U (M)}
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D. SCHLINGEMANN
(1) For each u1 , u2 ∈ U (M) and for each t1 , t2 , t ∈ R, we require (t, u1 )(t, u2 ) = (t, u1 u2 ) and (t, 1) = 1 (2) For u1 ∈ M(I1 ) and u2 ∈ M(I2 ) with I1 ⊂ (I2 + [−|t|, |t|])c we require for each t1 ∈ R: (t1 + t, u1 )(t1 , u2 ) = (t1 , u2 )(t1 + t, u1 ) . We conclude from relation (1) that (t, u) is the inverse of (t, u∗ ). Furthermore, a localization region in R×Σ can be assigned to each element in G(R, M). A generator (t, u), u ∈ M(I) is localized in O ⊂ R × Σ if {t} × I ⊂ O. The subgroup of G(R, M) which is generated by elements which are localized in the double cone O, is denoted by G(O). We easily observe that relation (2) implies that group elements commute if they are localized in space-like separated regions. The translation group in R2 is naturally represented by group-automorphisms of G(R, M). They are defined by the prescription β(t,x) (t1 , u) := (t + t1 , αx u) . Thus the subgroup G(O) is mapped onto G(O + (t, x)) by β(t,x) . To construct the universal Haag–Kastler net, we build the group C ∗ -algebra B(O) with respect to G(O). For convenience, we shall describe the construction of B(O) briefly. In the first step we build the ∗ -algebra B0 (O) which is generated by all complex valued functions a on G(O), such that a(u) = 0 for almost each u ∈ G(O) . We write such a function symbolically as a formal sum, i.e. X a(u) u a= u ∗
The product and the -relation is given as follows: X X XX 0 0 −1 0 ab = a(u) u · b(u ) u = a(u)b(u u ) u0 u
u0
a∗ =
u0
X
u
a ¯(u−1 ) u
u
It is well known, that the algebra B0 (O) has a C ∗ -norm which is given by kak := sup kπ(a)kπ , π
where the supremum is taken over each Hilbert space representation π of B0 (O). Finally, we define B(O) as the closure of B0 (O) with respect to the norm above. The C ∗ -algebra which is generated by all local algebras B(O) is denoted by ∗ C (B). By construction, the group isomorphisms β(t,x) induce a representation of the translation group by automorphisms of C ∗ (B).
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Observation. The net of C ∗ -algebras B := {B(O)|O is a bounded double cone in R2 } is a translationally covariant Haag–Kastler net. We have to mention that the universal net B is not Lorentz covariant. The universal properties of the net B are stated in the following proposition: Proposition 2.1. Each dynamics α ∈ dyn(M) induces a C ∗ -homomorphism ια : C ∗ (B) → C ∗ (M) such that ια ◦ β(t,x) = α(t,x) ◦ ια , for each (t, x) ∈ R2 . In particular, Aα : O 7→ Aα (O) := ια (B(O))00 is a translationally covariant Haag–Kastler net. Proof. Given a dynamics α of M. We conclude from ps(α) ≤ 1 that the prescription (t, u) 7→ αt u defines a C ∗ -homomorphism ια : C ∗ (B) → C ∗ (M) . In particular, ια is a representation of C ∗ (B) on the Hilbert space H0 . This statement can be obtained by using the relations, listed below. (a) ια ((t, u1 )(t, u2 )) = αt u1 αt u2 = αt (u1 u2 ) = ια (t, u1 u2 ) (b) If (t1 , u1 ) and (t1 + t, u2 ) are localized in space-like separated regions, then we obtain from ps(α) ≤ 1: [ια (t1 , u1 ), ια (t1 + t, u2 )] = αt1 [u1 , αt u2 ] = 0 (c) ια (β(t,x) (t1 , u)) = ια (t + t1 , αx u) = α(t,x) αt1 u
In general we expect that for a given dynamics α the representation ια is not faithful. Hence each dynamics defines a two-sided ideal ∗ J(α) := ι−1 α (0) ∈ C (B)
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D. SCHLINGEMANN
in C ∗ (B) which we call the dynamical ideal with respect to α and the quotient C ∗ -algebras B(O)/J(α) ∼ = Aα (O) may depend on the dynamics α. Indeed, if O is a double cone whose base is not contained in Σ, then for different dynamics α1 , α2 the algebras Aα1 (O) and Aα2 (O) are different. On the other hand, if the base of O is contained in Σ, then we conclude from the fact that the dynamics α has finite propagation speed and from Proposition 2.1. Corollary 2.2. If I ⊂ Σ is the base of the double cone O, then the algebra Aα (O) is independent of α. In particular, the C ∗ -algebra C ∗ (M) =
[
k·k
M(I)
I
=
[
k·k
Aα (O)
O
is the C ∗ -inductive limit of the net Aα . From the discussion above, we see that two dynamics with the same dynamical ideal induces the same quantum field theory. 2.2. Examples for physical states Let us consider the set S of all locally normal states on C ∗ (M), i.e. for each state ω ∈ S and for each bounded interval I, the restriction ω|M(I) is a normal state on M(I). As mentioned in the introduction, we are interested in states with vacuum and particle-like properties, i.e. states satisfying the Borchers criterion (See the Introduction for this notion). Notation. Given a dynamics α ∈ dyn(M). We denote the corresponding set of all locally normal states which satisfies the Borchers criterion by S(α) and analogously the set of all vacuum states by S0 (α). Moreover, we write for the set of vacuum sectors sec0 (α) := {[ω]|ω ∈ S0 (α)} ,
(5)
where [ω] denotes the unitary equivalence class of the the GNS-representation of ω. Examples. Examples for vacuum states are the vacua of the P (φ)2 -models [36, 37]. The interacting part of the cutoff Hamiltonian is given by a Wick polynomial of the time zero field φ0 , i.e. H1 (I) = H1 (χI ) =: P (φ0 ) : (χI ) ,
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where χI is a test function with χI (x) = 1 for x ∈ I and χI (y) = 0 on the complement of a slightly larger region Iˆ ⊃ I. It is well known that H1 (I) is a self-adjoint operator, which has a joint core with the free Hamiltonian h0 , and is ˆ The operator h1 (I) induces a automorphism group αI which affiliated with M(I). is given by αI,t (a) := eiH1 (I)t a e−iH1 (I)t . Consider the inclusion of intervals I0 ⊂ I1 ⊂ I2 . Then we have for each a ∈ M(I0 ): αI1 ,t (a) = αI2 ,t (a) Hence, there exists a one-parameter automorphism group {α1,t ∈ Aut(M); t ∈ R} such that α1,t acts on a ∈ M(I) as follows: α1,t (a) = αI,t (a) ; ∀ t ∈ R . The automorphism group {α1,t ∈ Aut(M); t ∈ R} is a dynamics of M with zero propagation speed, i.e. ps(α1 ) = 0. Since H1 (I) has a joint core with the free Hamiltonian H0 , we are able to define the Trotter product of the automorphism groups α0 and α1 which is given for each local operator a ∈ M(I) by αt (a) := (α0 × α1 )t (a) = s − lim (α0,t/n ◦ α1,t/n )n (a) . n→∞
The limit is taken in the strong operator topology. Furthermore, the propagation speed is sub-additive with respect to the Trotter product [36], i.e. ps(α0 × α1 ) ≤ ps(α0 ) + ps(α1 ) and we conclude that α ∈ dyn(M) is a dynamics of M. We call the dynamics α interacting. It is shown by Glimm and Jaffe [36] that there exist vacuum states ω with respect to the interacting dynamics α. We have to mention, that there is no vector ψ in Fock space H0 , such that the state a 7→ hψ, aψi is a vacuum state with respect to an interacting dynamics α, but there is a sequence of vectors (Ωn ) in H0 such that the weak∗ limit ω = w∗ − limhΩn , ·Ωn i n
is a vacuum state with respect to the dynamics α.
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3. On the Existence of Kink States The main theorem (Theorem 3.2) of this paper is formulated in the first part (Sec. 3.1) of the present section. In order to prepare the proof of Theorem 3.2, we need some technical preliminaries which are given in Sec. 3.2. In the last part (Sec. 3.3), we prove a criterion for the existence of kink states (the extendibility of the dynamics) which turns out to be satisfied by the P (φ)2 and Yukawa2 models. 3.1. The main result We now reformulate the definition (see Introduction) of a kink state within the time-slice formulation. Definition 3.1. Let α ∈ dyn(M) be a dynamics of M. A state ω of M is called a kink state, interpolating vacuum states ω1 , ω2 ∈ S0 (α) if (a) ω satisfies the Borchers criterion (b) and there exists a bounded interval I, such that ω fulfills the relations: π|C ∗ (M,ILL ) ∼ = π1 |C ∗ (M,ILL )
and π|C ∗ (M,IRR ) ∼ = π2 |C ∗ (M,IRR )
where the symbol ∼ = means unitarily equivalent and (H, π, Ω), (Hj , πj , Ωj ) are the GNS-triples of the states ω ∈ S(α) and ωj ∈ S0 (α); j = 1, 2 respectively. The set of all kink states which interpolate ω1 and ω2 is denoted by S(α|ω1 , ω2 ). As already mentioned in the Introduction, a criterion for the existence of an interpolating kink state, can be obtained by looking at the construction method of [59]. In our context, we have to select a class of dynamics which are equipped with good properties. Such a selection criterion is developed in Sec. 3. We shall show that each dynamics of a P (φ)2 -model satisfies this criterion which leads to the following result: Theorem 3.2. If α ∈ dyn(M) is a dynamics of a model with P (φ)2 plus Yukawa2 interaction, then for each pair of vacuum states ω1 , ω2 ∈ S0 (α) there exists an interpolating kink state ω ∈ S(α|ω1 , ω2 ). We shall prepare the proof of Theorem 3.2 during the subsequent sections. 3.2. Technical preliminaries Definition 3.3. Let M be a net of Cauchy data. We denote by G(M) the group of unitary operators u ∈ B(H0 ) whose adjoint actions χu := Ad(u) commute with the spatial translations, i.e. χu ◦ αx = αx ◦ χu .
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Let α ∈ dyn(M) be a dynamics of the net M. Then we define the following subgroup of G(M): G(α, M) := {u ∈ G(M)|χu ◦ αt = αt ◦ χu for each t ∈ R} . Remark. Each operator u ∈ G(α, M) induces a symmetry of the Haag–Kastler net Aα . We make the following assumptions for the net of Cauchy data M: Assumption. (a) The net M fulfills duality, i.e. M(I)0 = M(ILL ) ∨ M(IRR )
(6)
(b) There exists a dynamics α0 and a normalized vector Ω0 in H0 , such that ω0 = hΩ0 , (·)Ω0 i is a vacuum state with respect to the dynamics α0 . (c) For each bounded interval I, the inclusion (M(IRR ), M(IR )) is split. (d) The net fulfills weak additivity. According to our assumption, we conclude from the Theorem of Reeh and Schlieder that Ω0 is a standard vector for the inclusion (M(IRR ), M(IR )) which implies that Λ(I) := (M(IRR ), M(IR ), Ω0 )
(7)
is a standard split inclusion for each interval I, and hence (see [19]) there exists a unitary operator wI : H0 ⊗ H0 → H0 such that for a ∈ M(ILL ) and b ∈ M(IRR ) we have wI (a ⊗ b)wI∗ = ab . Thus there is an interpolating type I factor N ∼ = B(H0 ), i.e. M(IRR ) ⊂ N ⊂ M(IR ) which is given by N := wI (1 ⊗ B(H0 ))wI∗ . Hence we obtain an embedding of B(H0 ) into the algebra M(IR ): ΨI : F ∈ B(H0 ) 7→ wI (1 ⊗ F )wI∗ ∈ M(x, ∞) This embedding is called the universal localizing map.
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Remark. We shall make a few remarks on the assumptions given above. (i) The results, which we shall establish in the following, remain correct if the net of Cauchy data fulfills twisted duality instead of duality [54, 64]. (ii) For the application of our analysis to quantum field theory models, like P (φ)2 - or Yukawa2 models, we can choose as Cauchy data tensor products of the time-zero algebras of the massive free Bose or Fermi field. The timezero algebras of the massive free Bose field fulfill the assumptions (a) and (b) and it has been shown [60, Appendix] (compare also [9]) that (c) is also fulfilled. Replacing duality by twisted duality, the assumptions (a) to (c) hold for the massive free Fermi field, too [64]. In addition to that, we claim that the weak additivity (d) is also fulfilled in these cases. (iii) The state ω0 plays the role of a free massive vacuum state, called the bare vacuum. Proposition 3.4. Let u ∈ G(M) be an operator and let I be a bounded interval. Then there exists a canonical automorphism χIu with the properties: (1) The relations χIu |C ∗ (M,ILL ) = idC ∗ (M,ILL )
and χIu |C ∗ (M,IRR ) = χu |C ∗ (M,IRR )
(8)
hold. 1 : Σ → C ∗ (M) such that : (2) There exists a strongly continuous map γ(u,I) (i) 1 (x)) = αx ◦ χIu ◦ α−x ◦ (χIu )−1 . Ad(γ(u,I)
(ii) The cocycle condition is fulfilled : 1 1 1 (x + y) = αx (γ(u,I) (y))γ(u,I) (x) . γ(u,I)
Proof. (1) In the same manner as in [59], we show that ˆ ⊂ M(I) ˆ Ad(ΨI (1 ⊗ u))(M(I)) if the interval Iˆ contains I. This implies that χIu := Ad(ΨI (1 ⊗ u)) is a well defined automorphism of C ∗ (M). By using the properties of the universal localizing map ΨI , we conclude that χIu fulfills Eq. (8). (2) By a straight forward generalization of the proof of [59, Proposition 4.2], we 1 (x) is given by conclude that the statement (2) holds where γ(u,I) 1 γ(u,I) (x) = ΨI+x (1 ⊗ u)ΨI (1 ⊗ u∗ ) .
Let ω be a vacuum state with respect to the dynamics α and let u ∈ G(α, M), then the state ωuI := ω ◦ χIu
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seem to be a good candidate for an interpolating kink state. Indeed, it follows from the construction of χIu that ωuI |C ∗ (M,IRR ) = ω ◦ χu |C ∗ (M,IRR ) ωuI |C ∗ (M,ILL ) = ω|C ∗ (M,ILL ) . Hence ωuI interpolates ω and ω ◦ χu . To decide whether ωuI is a positive energy state, we investigate in the subsequent section, how χIu is transformed under the action of a dynamics α. 3.3. When does a theory possess kink states? Let α be a dynamics and G ⊂ G(α, M) be a finite subgroup. By using the universal localizing map ΨI , we obtain for each bounded interval I a unitary representation of G UI : G 3 g 7→ UI (g) := ΨI (1 ⊗ g) ∈ M(IR ) . In the previous section it has been shown that UI (g) implements an automorphism χIg which is covariant under spatial translations (Proposition 3.4). For a dynamics α ∈ dyn(M), we wish to construct a cocycle γ(g,I) in order to show that χIg is an interpolating automorphism. The formal operator γ(g,I) (t, x) := α(t,x) (UI (g))UI (g)∗ seems to be a useful Ansatz since it formally implements the automorphism α(t,x) ◦ χIg ◦ α(−t,−x) ◦ (χIg )−1 . Unfortunately, the operators UI (g) are not contained in C ∗ (M) and the term α(t,x) (UI (g)) has no well-defined mathematical meaning. To get a well-defined solution for γ(g,I) , we construct an extension of the net M which contains the operators UI (g) (compare also [54]). Definition 3.5. Let G ⊂ G(M) be a compact sub-group. The net M o G is defined by the assignment M o G : I 7→ (M o G)(I) := M(I) ∨ UI (G)00 . Proposition 3.6. Let I be a bounded interval, then the map π I : M(I) o G 3 a · g 7→ a UI (g) ∈ M(I) ∨ UI (G)00 is a faithful representation of the crossed product M(I) o G. Proof. First, we easily observe that π I is a well-defined representation of M(I) o G. According to [43, Theorem 2.2, Corollary 2.3], we conclude that the crossed product M(I)oG is isomorphic to the von Neumann algebra M(I)∨UI (G)00 and π I is a W ∗ -isomorphism.
872
D. SCHLINGEMANN
Definition 3.7. A one parameter automorphism group α, which satisfies the conditions, listed below, is called a G-dynamics of the extended net M o G. (a) α is a dynamics of the net M o G (See Introduction). (b) The automorphisms αt commute with the automorphisms χg , i.e. αt ◦ χg = χg ◦ αt ; for each t ∈ R and for each g ∈ G . The set of all G-dynamics of M o G is denoted by dynG (M o G). Proposition 3.8. Let α ∈ dynG (M o G) be a G-dynamics and I be a bounded interval. Then the operator 0 (t) := αt (UI (g))UI (g)∗ γ(g,I)
is contained in M(I|t| ) where I|t| denotes the enlarged interval I + (−|t|, |t|) and the operator γ(g,I) (t, x) := α(t,x) (UI (g))UI (g)∗ fulfills the cocycle condition: γ(g,I) (t + t0 , x + x0 ) = α(t,x) (γ(g,I) (t0 , x0 ))γ(g,I) (t, x) . Proof. For a ∈ C ∗ (M, I|t|,RR ), the operator α−t (a) is contained in C ∗ (M, IRR ) which implies a αt (UI (g))UI (g)∗ = αt (α−t (a)UI (g))UI (g)∗ = αt (UI (g)χg α−t (a))UI (g)∗ = αt (UI (g)α−t χg (a))UI (g)∗ = αt (UI (g))χg (a)UI (g)∗ = αt (UI (g))UI (g)∗ a and we conclude αt (UI (g))UI (g)∗ ∈ C ∗ (M, I|t|,RR )0 = M(I|t|,L ) . By a similar argument, αt (UI (g))UI (g)∗ is contained in M(I|t|,R ) and we conclude from duality that it is contained in M(I|t| ). The cocycle condition for γ(g,I) is obviously fulfilled and the proposition follows. Definition 3.9. Let α ∈ dyn(M) be a dynamics and G ⊂ G(M) be a compact subgroup. We shall call α G-extendible if there exists a G-dynamics α ˆ of the extended net M o G, such that α ˆ t |C ∗ (M) = αt for each t ∈ R.
CONSTRUCTION OF KINK SECTORS FOR
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We are now prepared to prove one of our key results: Theorem 3.10. Let α ∈ dyn(M) be a G-extendible dynamics and let χIg be the automorphism which can be constructed by Proposition 3.4. Then for each vacuum state ω with respect to α the state ωgI := ω ◦ χIg is a kink state which interpolates ω and ω ◦ χg . Proof. As postulated, there exists an extension α ˆ ∈ dynG (M o G) of α. We show that for each g ∈ G the operator 0 (t) := α ˆt (UI (g))UI (g)∗ γ(g,I)
implements the automorphism αt ◦ χIg ◦ α−t ◦ (χIg )−1 on C ∗ (M). Indeed, we have for each a ∈ C ∗ (M): 0 Ad(γ(g,I) (t))a = α ˆ t (UI (g))UI (g)∗ a UI (g)ˆ αt (UI (g))∗
=α ˆ t (UI (g)) (χIg )−1 (a) α ˆ t (UI (g))∗ −1 =α ˆ t UI (g)α−t χIg (a) UI (g)∗ −1 = αt UI (g)α−t χIg (a) UI (g)∗ = αt ◦ χIg ◦ α−t ◦ (χIg )−1 (a) . Finally we conclude from Proposition 3.8 that γ(g,I) (t, x) := α(t,x) (UI (g))UI (g)∗ is a cocycle where γ(g,I) (t, x) is localized in a sufficiently large bounded interval. By a straight forward generalization of the proof of [59, Proposition 5.5] we conclude that ωuI is a positive energy state which implies the result. The dynamics α of P (φ)2 - and Yukawa2 models are locally implementable by unitary operators. More precisely, for each bounded interval I and for each positive number τ > 0, there exists a unitary operator u(I, τ |t) with the properties: (1) If |t1 |, |t2 |, |t1 + t2 | < τ , then we have u(I, τ |t1 + t2 ) = u(I, τ |t1 )u(I, τ |t2 ) . (2) For |t| < τ , the operator u(I, τ |t) implements αt on M(I), i.e. αt (a) = u(I, τ |t) a u(I, τ |t)∗ ; for each a ∈ M(I) .
(9)
874
D. SCHLINGEMANN
Let G ⊂ G(α, M) be a compact sub-group. In order to show that α is G-extendible, it is sufficient to prove that the operators u(I1 , τ |t)UI (g)u(I1 , τ |t)∗ , which are the obvious candidates for α(U ˆ I (g)), are independent of I1 for I1 ⊃ I and |t| ≤ τ . Lemma 3.11. If for each I ⊂ I1 , for each τ < τ1 and for each g ∈ G the equation u(I, τ |t)UI (g)u(I, τ |t)∗ = u(I1 , τ1 |t)UI (g)u(I1 , τ1 |t)∗
(10)
holds, then the dynamics α is G-extendible. Here u(I, τ |t) are unitary operators which fulfill Eq. (9). Proof. Let (In , τn )n∈N be a sequence, such that limn In = R and limn τn = ∞. We conclude from our assumption (Eq. (10)) that the uniform limit α ˆ t (a) := lim Ad(u(In , τn |t))(a) n→∞
exists. Thus α ˆ : t 7→ α ˆ t is a well-defined one-parameter automorphism group, extending the dynamics α. It remains to be proven that α ˆ has propagation speed ps(ˆ α) ≤ 1. Since α ˆ is an extension of α and ps(α) ≤ 1, we conclude for each a ∈ C ∗ (M, I|t|,RR ) and for each b ∈ C ∗ (M, I|t|,LL ): ab α ˆ t (UI (g)) = α ˆ t (α−t (a)α−t (b)UI (g)) =α ˆ t (UI (g)α−t χg (a)α−t (b)) =α ˆ t (UI (g)) χg (a)b Thus the operator α ˆt (UI (g)) is contained in M(I|t|,R ) and implements χg on M(I|t|,RR ). This finally implies α ˆt (UI (g))UI|t| (g)∗ ∈ M(I|t| )
and the lemma follows.
Let us consider the two-fold W ∗ -tensor product of the net of Cauchy data, i.e. M ⊗ M : I 7→ M(I) ⊗ M(I) . Observation. (i) If the net M fulfills the conditions (a) to (c) of the previous section, then the net M ⊗ M fulfills them, too. (ii) Let α ∈ dyn(M) be a dynamics of M, then α⊗2 is a dynamics of M ⊗ M. Note that the flip operator uF , which is given by uF : H0 ⊗ H0 → H0 ⊗ H0 ; ψ1 ⊗ ψ2 7→ ψ2 ⊗ ψ1
CONSTRUCTION OF KINK SECTORS FOR
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is contained in G(α⊗2 , M ⊗ M). Hence uF induces an embedding of Z2 into G(α⊗2 , M ⊗ M). (iii) According to Definition 3.5, we can construct a non-local extension ˆ := (M ⊗ M) o Z2 M of the two-fold net M ⊗ M. Let ΨI be the universal localizing map of the standard split inclusion Λ(I) ⊗ Λ(I) = (M(IRR )
⊗ 2
, M(IR )
⊗ 2
, Ω0 ⊗ Ω0 )
ˆ is simply given by and define θI := ΨI (1 ⊗ uF ). Then the algebra M(I) ˆ M(I) = ((M ⊗ M) o Z2 )(I) = (M(I) ⊗ M(I)) ∨ {θI }00 . (iv) By Proposition 3.4, there exists a canonical automorphism β I := Ad(θI ) ,
(11)
associated with the pair (uF , I). Notation. Let α be a dynamics of M. In the sequel, we shall call α extendible if α⊗2 is Z2 -extendible. We conclude this section by the following corollary which can be derived by a direct application of Theorem 3.10: Corollary 3.12. Let α ∈ dyn(M) be an extendible dynamics, then for each pair of vacuum states ω1 , ω2 ∈ S0 (Aα ), the state ω = µβ I (ω1 ⊗ ω2 ) is a kink state. 4. Application to Quantum Field Theory Models We show that a sufficient condition for the existence of interpolating automorphisms, i.e. the extendibility of the dynamics, is satisfied for the P (φ)2 , the Yukawa2 and special types of Wess–Zumino models. 4.1. Kink states in P (φ)2 -models We shall show that the dynamics of P (φ)2 -models are extendible. As described in Sec. 2.2 the dynamics of a P (φ)2 -model consists of two parts. (1) The first part is given by the free dynamics α0 , with propagation speed ps(α0 ) = 1, α0,t (a) = eiH0 t a e−iH0 t , where (H0 , D(H0 )) is the free Hamiltonian which is a self-adjoint operator on the domain D(H0 ) ⊂ H0 .
876
D. SCHLINGEMANN
(2) The second part is a dynamics α1 with propagation speed ps(α1 ) = 0, i.e. α1,t maps each local algebra M(I) onto itself. The interaction part of the full Hamiltonian is given by a Wick polynomial of the time-zero field φ: Z H1 (I) = H1 (χI ) =
dx : P (φ(x)) : χI (x)
where χI is a smooth test function which is one on I and zero on the complement of a slightly lager region Iˆ ⊃ I. The unitary operator exp(itH1 (I)) implements the dynamics α1 locally, i.e. for each a ∈ M(I) we have α1,t (a) := eiH1 (I)t a e−iH1 (I)t . Definition 4.1. An operator valued distribution v : S(R) → L(H0 ) is called an ultra local interaction, if the following conditions are fulfilled: (1) For each real valued test function f ∈ S(R), v(f ) is self-adjoint and has a common core with H0 . (2) Let f ∈ S(R) be a real valued test function with support in a bounded interval I, then the spectral projections of v(f ) are contained in M(I). (3) For each pair of test functions f1 , f2 ∈ S(R), the spectral projections of v(f1 ) commute with the spectral projections of v(f2 ). Remark. It has been proven in [36], that the Wick polynomials of the time zero fields are ultra local interactions. Furthermore, each ultra local interaction v induces a dynamics αv ∈ dyn(M) with propagation speed ps(αv ) = 0. Let I be a bounded interval and let χI ∈ S(R) be a positive test function with χI (x) = 1 for each x ∈ I. Indeed, by an application of J. Glimm’s and A. Jaffe’s analysis [36], we conclude that the automorphisms αvt : M(I) → M(I) ; a 7→ Ad(exp(itv(χI ))) a define a dynamics with zero propagation speed. In the sequel, we shall call a dynamics αv ultra local if it is induced by an ultra local interaction v. In order to prove that a dynamics α, which is given by the Trotter product α = α0 × αv of a free and an ultra local dynamics, is extendible, we show that each part of the dynamics can be extended separately. Since the free part of the dynamics can be extended to the algebra B(H0 ) of all bounded operators on the Fock space H0 , it is obvious that α0 is extendible. Thus it remains to be proven the following: Lemma 4.2. Each ultra local dynamics αv ∈ dyn(M) is extendible.
CONSTRUCTION OF KINK SECTORS FOR
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Proof. Let us consider any ultra local interaction v. For each test function f ∈ S(R), we introduce the unitary operator u(f |t) := eitv(f ) ⊗ eitv(f ) . Let I be a bounded interval and denote by I , > 0, the enlarged interval I+(−, ). We choose test functions χ(I,) ∈ S(R) such that 1 x∈I . χ(I,) (x) = 0 x ∈ Ic = I \R For an interval Iˆ ⊃ I , the region Iˆ \I consists of two connected components (Iˆ \I )± and there exist test functions χ± ∈ S(R) with supp(χ− ) ⊂ (Iˆ \I )− ⊂ ILL supp(χ+ ) ⊂ (Iˆ \I )+ ⊂ IRR ˆ
χ(I,) − χ(I,) = χ+ + χ− . Let us write u(I, |t) := u(χ(I,) |t) and u± (t) := u(χ± |t) . Since we have [u(f1 |t), u(f2 |t)] = 0 for any pair of test functions f1 , f2 ∈ S(R), we ˆ obtain for each > 0 and for I ⊂ I: ˆ |t) = u(I, |t)u− (t)u+ (t) u(I,
(12)
If we make use of the fact that u+ (t) is αF -invariant and localized in IRR , we conclude that θI and u± (t) commute. Thus we obtain ˆ |t))θI = Ad(u(I, |t))θI Ad(u(I,
(13)
which depends only of the localization interval I since > 0 can be chosen arbitrarily small. According to Lemma 3.11, the automorphisms ˆ ˆ α ˆ vt : M(I) 3 a 7→ Ad(u(I, |t))a ∈ M(I) ˆ whose restriction to M ⊗ M is αv ⊗ αv . Thus αv is define a dynamics of M extendible. ˆ ˆv If α ˆ 0 denotes the natural extension of the free dynamics α⊗2 0 to M and let α v v be the extension of the ultra local dynamics α ⊗ α then, by using the Trotter product, we conclude that the dynamics α ˆ := α ˆ0 × α ˆv ˆ This leads to the following result: is an extension of the dynamics (α0 ×αv )⊗2 to M. Proposition 4.3. Each dynamics of a P (φ)2 -model is extendible.
878
D. SCHLINGEMANN
Proof. The statement follows from Lemma 4.2 and from the fact that each dynamics of a P (φ)2 -model is a Trotter product of the free dynamics α0 and an ultra local dynamics α1 . The existence of interpolating kink states in P (φ)2 -models is an immediate consequence of Proposition 4.3. Corollary 4.4. Let α ∈ dyn(M) be a dynamics of a P (φ)2 -model. Then for each pair of vacuum states ω1 , ω2 ∈ S0 (α, M) there exists an interpolating kink state ω ∈ S(ω1 , ω2 ). Proof. By Proposition 4.3 each dynamics of a P (φ)2 -model is extendible and we can apply Corollary 3.12 which implies the result. 4.2. The dynamics of the Yukawa2 model Since the dynamics of a Yukawa2 -like model cannot be written as a Trotter product which consists of a free and an ultra local dynamics, it is a bit more complicated to show that these dynamics are extendible. We briefly summarize here the construction of the Yukawa2 dynamics which has been carried out by J. Glimm and A. Jaffe [36]. We also refer to the work of R. Schrader [62, 63]. Let Ms and Ma be the nets of Cauchy data for the free Bose and Fermi field, represented on the Fock spaces Hs and Ha respectively. The Cauchy data of the Yukawa2 model are given by the W ∗ -tensor product M := Ms ⊗ Ma of the nets Ms and Ma . Moreover, we set H0 := Hs ⊗ Ha . Step 1 : In the first step, a Hamiltonian, which is regularized by an UV-cutoff c0 > 0 and an IR-cutoff c1 > 1, c0 c1 , is constructed. For this purpose, one chooses test functions δc0 , χc1 ∈ S(R) with the properties: (a) Z dx δc0 (x) = 1 supp(δc0 ) ⊂ (−c0 , c0 ) and (b) supp(χc1 ) ⊂ (−c1 − 1, c1 + 1) and χc1 (x) = 1 for each x ∈ (−c1 , c1 ) . The UV-regularized fields are given by φ(c0 , x) := (φ ∗ δc0 )(x)
and ψ(c0 , x) := (ψ ∗ δc0 )(x) ,
(14)
where φ is a massive free Bose field and ψ a free Dirac spinor field at t = 0. The fields, defined by Eq. (14), act on H0 via the operators Φ(c0 , x) := φ(c0 , x) ⊗ 1Ha
and Ψ(c0 , x) := 1Hs ⊗ ψ(c0 , x) .
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The regularized Hamiltonian H(c0 , c1 ) can be written as a sum of three parts: (1) The free Hamiltonian H0 which is given by H0 = H0,s ⊗ 1Ha + 1Hs ⊗ H0,a , where H0,s and H0,a are the free Hamilton operators of the Bose and the Fermi field respectively. (2) The regularized Yukawa interaction term: Z ¯ 0 , x)Ψ(c0 , x) . HY (c0 , c1 ) = dx χc1 (x) Φ(c0 , x) : Ψ(c (3) The counterterms: HC (c0 , c1 ) =
N X
Z zn (c0 )
dx χc1 (x) : Φ(x)n ,
n=0
where zn (c0 ) are suitable renormalization constants. The following statement has been established by J. Glimm and A. Jaffe [36, 38]: Theorem 4.5. The counterterms HC (c0 , c1 ) can be chosen in such a way that (1) the cutoff Hamiltonian H(c0 , c1 ) = (H0 + HY (c0 , c1 ) + HC (c0 , c1 ))∗∗ is a positive and self adjoint operator with domain D(H0 ). (2) The uniform limit R(c1 , ζ) = lim (H(c0 , c1 ) − ζ)−1 c0 →0
is the resolvent of a self adjoint operator H(c1 ). (3) H(c1 ) is the limit of H(c0 , c1 ) in the strong graph topology. Notation. In the sequel, we shall use the following notation: u(c0 , c1 , t) := exp(itH(c0 , c1 )) and u(c1 , t) := exp(itH(c1 )) . Remark. The aim is to show that H(c1 ) induces a dynamics of M, given locally by the equation αt |M(I) = Ad(u(c1 , t)) for I|t| := I + (−|t|, |t|) ⊂ (−c1 , c1 ) . However, in comparison to the P (φ)2 -models, there are some more technical difficulties which have to be overcome. (i) The Hamiltonian H(c1 ) is only defined as a limit of the Hamiltonians H(c0 , c1 ) and it has no mathematical meaning when written as a sum H0 + HY (c1 ) + HC (c1 ) . Thus the construction scheme for a dynamics, as it has been used for P (φ)2 models, does not apply.
880
D. SCHLINGEMANN
(ii) On the other hand, one might try to apply P (φ)2 -like methods to the Hamiltonian H(c0 , c1 ), for which the UV-cutoff is not removed. For this purpose, one wishes to write H(c0 , c1 ) as a sum H(c0 , c1 ) = H1 (c0 , c1 ) + H2 (c0 , c1 ) where H1 (c0 , c1 ) induces a dynamics α1 with propagation speed ps(α1 ) ≤ 1 and H2 (c0 , c1 ) induces a dynamics α2 with propagation speed ps(α2 ) = 0. The difficulty with writing such a decomposition for H(c0 , c1 ) arises from the fact that the Yukawa interaction term HY (c0 , c1 ) induces an automorphism group with infinite propagation speed. Step 2 : In the next step, one introduces test functions χ(I,s,c0 ) (see Fig. 1), depending on a bounded interval I, a real number s > 0 and the UV-cutoff c0 , fulfilling the conditions supp(χ(I,s,c0 ) ) ⊂ I2c0 +|s|+ \I|s|−
and (15)
χ(I,s,c0 ) (x) = 1 if x ∈ I2c0 +|s| \I|s| .
Here c0 is any sufficiently small positive number. The Hamiltonian H(c0 , c1 ) is replaced by the operator H(I, s, c0 , c1 ) := H0 + HC (c0 , c1 ) + HY (I, s, c0 , c1 )
(16)
depending additionally on I and s, where HY (I, s, c0 , c1 ) is given by Z ¯ 0 , x)Ψ(c0 , x) : (χc1 (x) − χ(I,s,c ) (x)) . HY (I, s, c0 , c1 ) := dx Φ(c0 , x) : Ψ(c 0 In order to construct from these data a c1 -independent approximation of the dynamics which maps M(I) onto M(I|t| ), one defines the unitary operators n Y t w(I, c0 , c1 , t) := exp i H(I, (n − j)n−1 t, c0 , c1 ) , n j=1 where n is equal to the integral part of |c−1 0 t|. The lemma, given below, has been established in [36].
1
2c0
|s|
I
|s|
Fig. 1. The figure shows the graph of the function χ(I,s,c0 ) .
2c0
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Lemma 4.6 [36, Lemma 9.1.2]. The adjoint action of w(I, c0 , c1 , t) induces an automorphism (I,c0 )
αt
:= Ad(w(I, c0 , c1 , t)) : M(I) → M(I|t| )
which is independent of c1 . Step 3 : For technical reasons, to control convergence as c0 tends to zero, the length of time propagation is scaled, and one defines for λ ∈ [0, 1] the c1 -independent automorphism (I,c0 ,λ)
αt
:= Ad(w(I, c0 , c1 , λ, t)) : M(I) → M(I|t| )
where w(I, c0 , c1 , λ, t) is given by w(I, c0 , c1 , λ, t) :=
λ·t H(I, (n − j)n−1 t, c0 , c1 ) . exp i n j=1 n Y
The final approximation is given by averaging over λ: Z (I,c0 ,`) (I,c ,λ) (a) := dλ f` (λ) αt 0 (a) , αt where f` is a positive continuous function such that Z dλ f` (λ) = 1 and supp(f` ) ⊂ [1 − `, 1], ` ≤ 1 . Finally, J. Glimm and A. Jaffe have established the result: Theorem 4.7 [36, Theorem 9.1.3]. There exists a function c : ` 7→ c` with lim`→0 c` = 0 such that (I,c` ,`)
αYt (a) := w − lim αt `→0
(a) = u(c1 , t) a u(c1 , t)∗
(17)
for each a ∈ M(I) and for each sufficiently large c1 . 4.3. Kink states in models with Yukawa2 interaction We shall use an analogous strategy as above (Steps 1–3) in order to show that the dynamics αY , which is given due to Theorem 4.7 is extendible. Theorem 4.8. The dynamics αY of the Yukawa2 model is extendible. Let us prepare the proof. First, we give a few comments on the notation to be used.
882
D. SCHLINGEMANN
Notation. (a) In the sequel, we write w(· ˆ · · ) = w(· · · )⊗2 and uˆ(· · · ) = u(· · · )⊗2 for the corresponding quantities of the two-fold theory. As in Step 3 above, we also define the automorphism (I,c0 ,λ)
α ˆt
:= Ad(w(I, ˆ c0 , c1 , λ, t))
and the average
Z (I,c0 ,`)
α ˆt
(I,c0 ,λ)
(a) =
dλ f` (λ) α ˆt
(a) .
(b) Let ω0 be the vacuum state with respect to the free dynamics which is induced by H0 . We denote by ΨI the universal localizing map of the standard split inclusion Λ(I) ⊗ Λ(I) and we define θI := ΨI (1 ⊗ uF ). Lemma 4.9. The adjoint action of w(I, ˆ c0 , c1 , t) induces an automorphism (I,c0 )
ˆ ˆ |t| ) : M(I) → M(I
α ˆt which is independent of c1 .
Proof. By Lemma 4.6, it is sufficient to prove that Ad(w(I, ˆ c0 , c1 , t))θI is c1 -independent. Indeed, following the arguments in the proof of Proposition 4.3, we conclude that θI0 := exp(iτ H(I, s, c0 , c1 ))⊗2 θI exp(−iτ H(I, s, c0 , c1 ))⊗2 ˆ |s|+|τ | ). Composing is c1 -independent for |τ | ≤ c0 and that θI0 is contained in M(I n such maps, we obtain the lemma. In complete analogy to Theorem 4.7 we have Lemma 4.10. (I,c` ,`)
ˆt α ˆ Y (a) := w − lim α `→0
(a) = u ˆ(c1 , t) a u ˆ(c1 , t)∗ .
ˆ For each a ∈ M(I) and for each sufficiently large c1 . Proof. By Theorem 4.7, we conclude that the lemma holds for each a ∈ M(I) ⊗ M(I). Hence it remains to be proven that (I,c` ,`)
w − lim α ˆt `→0
(θI ) = uˆ(c1 , t) θI u ˆ(c1 , t)∗ .
The Corollary 9.1.9 of [36] states: Z w − lim dλ (w(I, ˆ c` , c1 , λ, t) − u ˆ(c` , c1 , λt))f` (λ) = 0 . `→0
CONSTRUCTION OF KINK SECTORS FOR
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We define Z (I,c` ,`)
θI (`, t) := α ˆt
(θI )
and θ¯I (`, t) :=
dλ f` (λ) Ad(ˆ u(c` , c1 , λt))θI .
The Schwarz’s inequality implies for each ψ ∈ H0 ⊗ H0 : |hψ, θI (`, t) − θ¯I (`, t)ψi| 1/2 Z 2 ˆ c` , c1 , λ, t) − u ˆ(c` , c1 , λt))ψk ≤ 2kψk · dλ f` (λ) k(w(I, Since k(v − u)ψk2 = 2 · Re(h(v − u)ψ, uψi), we obtain |hψ, θI (`, t) − θ¯I (`, t)ψi| Z ˆ c` , c1 , λ, t) ≤ 4kψk · dλ f` (λ)Re (w(I, 1/2 −ˆ u(c` , c1 , λt))ψ, u ˆ(c` , c1 , λt)ψ which proves the lemma.
Proof of Theorem 4.8. We conclude from Lemmas 4.10 and 3.11 that the ˆ whose restriction to automorphism group α ˆ Y is a dynamics of the extended net M Y Y Y M ⊗ M is α ⊗ α . Thus α is extendible. Remark. According to [63], each dynamics αY +P of a quantum field theory model with Yukawa2 plus P (φ)2 boson self-interaction is extendible. Finally, we conclude from Theorem 4.8: Corollary 4.11. Let αY +P be a dynamics of a quantum field theory model with Yukawa2 plus P (φ)2 boson self-interaction. For each pair ω1 , ω2 of vacuum states with respect to αY +P , there exists a kink state ω in S(αY +P |ω1 , ω2 ). 4.4. Wess Zumino models One interesting class of quantum field theory models which possess more than one vacuum sector are the N = 2 Wess–Zumino models in two-dimensional spacetime. Their properties have been studied in several papers [45, 48, 49, 46, 47] and we summarize the main results which are established there in order to setup our subsequent analysis. The field content of these models with a finite volume cutoff c > 0 consists of one complex Bose field φc and one Dirac spinor field ψc , acting as operator valued distributions on the Fock spaces
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Ha (c) :=
∞ M
L2 (Tc , C2 )⊗a ,
n=0
Hs (c) :=
∞ M
L2 (Tc , C)⊗s ,
n=0
where a, s stands for symmetrization or anti-symmetrization of the tensor product and L2 (Tc , Ck ) (k = 1, 2) denotes the Hilbert space of Ck -valued and square integrable functions, living on the circle Tc of length c. The net of Cauchy data for the finite volume theory is given by Mc : (−c, c) ⊃ I 7→ Mc (I) = Mc,s (I) ⊗ Mc,a (I) , where the nets Mc,s and Mc,a are defined by the assignments: 00 Mc,s : (−c, c) ⊃ I 7→ Mc,s (I) := ei(φc (f1 )+πc (f2 )) supp(fj ) ⊂ I , Mc,a
00 ¯ : (−c, c) ⊃ I → 7 Mc,a (I) := ψc (f1 ), ψc (f2 ) supp(fj ) ⊂ I ,
where πc is the canonically conjugate of φc . Let M := Mc=∞ be the net of Cauchy data in the infinite volume limit, then the map φc (f11 ) πc (f12 ) φ(f11 ) π(f12 ) → 7 ; supp(fij ) ⊂ (−c, c) ιc : ¯ 22 ) ψ(f21 ) ψ(f ψc (f21 ) ψ¯c (f22 ) is a W ∗ -isomorphism which identifies the nets M and Mc for those regions I which are contained in (−c, c). The interaction part of the formal Hamiltonian consists of two parts. (a) A P (φ)2 -like part: Z HP (v, c) = dx : |v 0 (Φc )|2 : − : |Φc |2 Tc
(b) A Yukawa2 -like part:
Z ¯c dx : Ψ
HY (v, c) := Tc
v 00 (Φc ) − 1 0
0 00 v (Φc )∗ − 1
Ψc
where v is a polynomial of degree deg(v) = n, called superpotential , and the fields Φc and Ψc are given by Φc := φc ⊗ 1Ha (c)
and Ψc := 1Hs (c) ⊗ ψc .
According to the results of [45, 47, 48, 49], it has been shown that, there is a self-adjoint Fredholm operator Q(v, c), called supersymmetry generator , on H0 (c) := Hs (c) ⊗ Ha (c). The Fredholm index of Q(v, c) ind(Q(v, c)) = dim ker(Q(v, c)) − dim coker(Q(v, c))
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has been computed in [48]. The result is |ind(Q(v, c))| = deg(v) − 1 . The space H0 (c) may be decomposed H0 (c) = H+ (c)⊕H− (c) into the eigenspaces of the fermion parity operator Γ := (−1)Na , where Na is the fermion number operator. With respect to this decomposition, the operator Q(v, c) has the form Q(v, c) =
0 Q− (v, c)
Q+ (v, c) 0
.
The full Hamiltonian of the finite volume model is given by H(v, c) = Q(v, c)2 which implies dim ker(H(v, c)) = |dim ker(Q+ (v, c)) − dim ker(Q− (v, c))| = deg(v) − 1 . The Hamiltonian H(v, c) induces a dynamics α(v,c) of the finite volume model and we conclude from the results of [45]: Theorem 4.12 [45, Theorem 1]. There exists at least deg(v) − 1 vacuum sectors with respect to the dynamics αv := α(v,c=∞) of the model in the infinite volume limit. 4.5. Kink states in Wess Zumino models In order to prove the existence of kink sectors, we now apply the results which have been established in Secs. 4.1 and 4.3 to N = 2 Wess–Zumino Models. The case deg(v) = 3: Let us have a closer look at the simplest non-trivial case deg(v) = 3. We let v 0 (z) = λ2 z 2 + λ1 z + λ0 . As in the previous sections (Eq. (14)), we introduce the UV-regularized fields: Φ(c0 , x) := (Φ ∗ δc0 )(x)
and Ψ(c0 , x) := (Ψ ∗ δc0 )(x) ,
where δc0 is a smooth test function with support in (−c0 , c0 ). We obtain for the P (φ)2 -like part of the regularized interaction Hamiltonian Z HP (v; c0 , c1 ) =
2 2 2 dxχc1 (x) : |λ2 Φ(c0 , x) + λ1 Φ(c0 , x) + λ0 | : − : |Φ(c0 , x)| :
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and for the Yukawa2 -like part: HY (v; c0 , c1 ) Z = dxχc1 (x) ¯ 0 , x) × : Ψ(c
0 2λ2 Φ(c0 , x) + λ1 − 1 ¯ 2 Φ(c0 , x)∗ + λ ¯1 − 1 0 2λ
Ψ(c0 , x) :
Using the same techniques as in Secs. 4.1 and 4.3, we obtain the corollary (see also Corollary 4.11): Corollary 4.13. Let v be a superpotential of degree deg(v) = 3. Then the following statements are true: (1) The dynamics αv ∈ dyn(M) of the model in the infinite volume limit is extendible. (2) There exists two different vacuum sectors e1 , e2 ∈ sec0 (αv , M) and two different kink sectors θ ∈ sec(e1 , e2 ), θ¯ ∈ sec(e2 , e1 ). The case deg(v) > 3: We close this section by discussing the remaining case. In order to show the extendibility of αv ∈ dyn(M), we can try to proceed in the same manner as for the case deg(v) = 3. According to Sec. 4.3 (Steps 2 and 3), we construct an approximation Z (v;I,c0 ,`) (v;I,c0 ,λ) ˆt (a) := dλ f` (λ) α ˆt (a) M(I) ⊗ M(I) 3 a 7→ α of the dynamics αv ⊗ αv of the two-fold theory. Provided that the corresponding (v;I,c0 ,`) result of Lemma 4.9 is true for the case deg(v) > 3 also, the linear maps α ˆt ˆ can be extended to the algebra M(I). For the generalization of Theorem 4.8, it seems that the most difficult part is to show that there exists a function c : ` 7→ c` with lim`→0 c` = 0 such that (v;I,c` ,`)
αvt (a) := w − lim αt `→0
(a) .
(18)
The regularized Yukawa-like part of the Hamilton density contains terms of the form : Ψ(i) (c0 , x)Ψ(j) (c0 , x) : : Φ(c0 , x)k : i, j ∈ {0, 1}, i 6= j
and k ≤ deg(v) − 2 ,
where Ψ(j) denotes the j-component of the Dirac spinor field Ψ. Since there are contributions with k > 1, the proof of Theorem 4.7 does not directly apply. Provided that for each superpotential v the dynamics αv is extendible, we conclude that for each pair of vacuum sectors e1 , e2 ∈ sec0 (αv , M) there exists a kink state ω ∈ S(e1 , e2 ). Then the model possesses at least deg(v)(deg(v) − 1) different non-trivial kink sectors.
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5. Conclusion and Outlook In the present paper, a construction scheme for kink sectors has been developed which can be applied to a large class of quantum field theory models. Most of the techniques which are used, except those in the proof of the extendibility of the dynamics, concern operator algebraic methods. They are model independent in the sense that they can be derived from first principles. There are still some interesting open problems and we shall make a few remarks on them here. Some further remarks on kink states: Let us consider a quantum field theory model (P (φ)2 , Y2 ), possessing vacua ω1 , ω2 which are related by a symmetry χ. According to Theorem 3.10, there exists an automorphism χI which induces a kink state ω = ω1 ◦ χI . Note that ω is a pure state in this case. Alternatively, we obtain a kink state ω ˆ by passing to the two-fold tensor product of the theory with itself first and then by restricting the αF -interpolating automorphism β I , whose existence follows also from Theorem 3.10, to the first tensor factor, i.e. ω ˆ = ω1 ⊗ ω2 ◦ βC ∗ (A)⊗1 . Provided the split property for wedge algebras holds for the interacting vacua [59], then, by applying a recent result of M. M¨ uger [55], we conclude that [ˆ ω ] is nothing else but the infinite multiple of [ω]. The problem of reducibility: The problem of reducibility arises if the vacua under consideration are not related by a symmetry since then our construction scheme leads to kink representations of the form π = π1 ⊗ π2 ◦ β|C ∗ (A)⊗1 , where β is an automorphism and π1 , 4π2 are vacuum representations. The representation π is not irreducible and whether π can be decomposed into irreducible sub-representations is still an open problem. Some of our results [60, Theorem 4.4.3] suggest that π is, in non exceptional cases, an infinite multiple of one irreducible component. Kink sectors in d > 1+1 dimensions: It would be desirable to apply our program to quantum field theories in higher dimensions. Let us suppose a theory, given by a net of W ∗ -algebras A, possesses two locally normal vacuum states ω1 , ω2 . As a sensible generalization of a kink states to d > 1 + 1, we propose to consider locally normal states ω which fulfill the interpolation condition: ω|C ∗ (A,S1 ) = ω1 |C ∗ (A,S1 )
and ω|C ∗ (A,S20 ) = ω2 |C ∗ (A,S20 ) ,
(19)
where S1 , S2 , S1 ⊂ S2 , are space-like cones. The state ω describes the coexistence of two phases which are separated by the phase boundary ∂S := S10 ∩ S2 . Let us assume duality for space-like cones in the vacuum representations under consideration. Furthermore, we presume that the inclusion
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Λ = (Aπ1 (S1 ), Aπ1 (S2 ), Ω1 ) is standard split. Here (H1 , π1 , Ω1 ) is a GNS-triple with respect to ω1 . Unfortunately, for d > 1+1 the phase boundary ∂S is not compact and therefore our construction scheme can not directly be generalized to higher dimensions. In order to overcome this difficulties, we consider a sequence of standard split inclusions Λn := (Aπ1 (O1n ), Aπ1 (O2n ), Ω1 ) , where O1n ⊂⊂ O2n are bounded double cones such that Ojn tends to Sj for n → ∞. As in the 1 + 1-dimensional case we pass now to the two-fold tensor product of the theory with itself. Denote by ΨΛn ⊗Λn the universal localizing map with respect to the inclusion Λn ⊗ Λn . Since the operators θn := ΨΛn ⊗Λn (1 ⊗ uF ) are localized in a bounded region, we may define the following automorphisms of C ∗ (A): βn := (π1 ⊗ π1 )−1 ◦ βn ◦ (π1 ⊗ π1 ) . We obtain a sequence of states {ωn , n ∈ N} where ωn is given by ωn := ω1 ⊗ ω2 ◦ βn |C ∗ (A)⊗1 . For large n the states ωn have almost the correct interpolation property, namely for each pair of local observables a, b where a is localized S20 and b is localized in S1 , there exists a sufficiently large N such that ωn (a) = ω1 (a) and ωn (b) = ω2 (b) for each n > N . Note that each state ωn fulfills the Borchers criterion since ωn belongs to the vacuum sector [ω1 ]. In order to obtain generalized kink states, we propose to investigate weak∗ -limit points of the sequence {ωn , n ∈ N}. Note that each weak*-limit ωι point of the sequence {ωn , n ∈ N} fulfills the interpolation condition (19). It remains to be proven that the weak∗ -limit points are locally normal. Acknowledgment I am grateful to Prof. K. Fredenhagen for supporting this investigation with many ideas. I am also grateful to Dr. K. H. Rehren for many hints and discussion. This investigation is financially supported by the Deutsche Forschungsgemeinschaft which is also gratefully acknowledged. References [1] H. Araki, “Soliton sectors of the XY-model”, preprint from Research Inst. for Mathematical Sciences, Kyoto Univ., Japan 1995. [2] H. Araki and R. Haag, “Collision cross sections in terms of local observables”, Commun. Math. Phys. 4 (1967) 77–91.
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[3] J. Bellissard, J. Fr¨ ohlich and B. Gidas, “Soliton mass and surface tension in the (Λ|φ|4 )2 quantum field model”, Commun. Math. Phys. 60 (1978) 37–72. [4] H.-J. Borchers, “Energy and momentum as observables in quantum field theory”, Commun. Math. Phys. 2 (1966) 49. [5] H.-J. Borchers, “CPT-Theorem in the theory of local observables”, Commun. Math. Phys. 143 (1992) 315–332. [6] H.-J. Borchers, “On the converse of the Reeh–Schlieder theorem”, Commun. Math. Phys. 10 (1968) 269–273. [7] H.-J. Borchers, Commun. Math. Phys. 4 (1967) 315–323. [8] O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Berlin, Heidelberg, New York, Springer, 1979. [9] D. Buchholz, “Product states for local algebras”, Commun. Math. Phys. 36 (1974) 287–304. [10] D. Buchholz, S. Doplicher and R. Longo, “On Noether’s theorem in quantum field theory”, Ann. Phys. 170 (1) (1986). [11] D. Buchholz and K. Fredenhagen, “Locality and the structure of particle states”, Commun. Math. Phys. 84 (1982) 1–54. [12] S. Coleman, Aspects of Symmetry, Cambridge Univ. Press, 1985. [13] C. D’Antoni and K. Fredenhagen, “Charges in spacelike cones”, Commun. Math. Phys. 94 (1984) 537–544. [14] C. D’Antoni and R. Longo, “Interpolation by Type I factors and the flip automorphism”, J. Funct. Anal. 51 (1983) 361–371. [15] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations I”, Commun. Math. Phys. 13 (1969) 1–23. [16] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations II”, Commun. Math. Phys. 15 (1969) 173–200. [17] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics I”, Commun. Math. Phys. 23 (1971) 199–230. [18] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics II”, Commun. Math. Phys. 35 (1971) 49–58. [19] S. Doplicher and R. Longo, “Standard split inclusions of von Neumann algebras”, Invent. math. 75 (1984) 493–536. [20] W. Driessler, “On the type of local algebras in quantum field theory”, Commun. Math. Phys. 53 (1977) 295. [21] K. Fredenhagen, “Generalization of the theory of superselection sectors”, The Algebraic Theory of Superselection Sectors, World Scientific, 1989. [22] K. Fredenhagen, “Superselection sectors in low dimensional quantum field theory”, J. Geom. Phys. 11 (1993) 337–348. [23] K. Fredenhagen, “On the existence of antiparticles”, Commun. Math. Phys. 79 (1981) 141–151. [24] K. Fredenhagen, “A Remark on the Cluster Theorem”, Commun. Math. Phys. 97 (1985) 461. [25] K. Fredenhagen, K. H. Rehren and B. Schroer, “Superselection sectors with braid group statistics and exchange algebras I”, Commun. Math. Phys. 125 (1989) 201. [26] K. Fredenhagen, K. H. Rehren and B. Schroer, “Superselection sectors with braid group statistics and exchange algebras II”, Rev. Math. Phys. Special Issue, (1992) 111–154. [27] J. Fr¨ ohlich, “New superselection sectors (soliton states) in two-dimensional bose quantum field models”, Commun. Math. Phys. 47 (1976) 269–310. [28] J. Fr¨ ohlich, Quantum Theory of Nonlinear Invariant Wave (Field) Equations, Erice, Sicily, Summer 1977.
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[52] T. Matsui, “On ground states of the one-dimensional ferromagnetic XXZ model”, Lett. Math. Phys. 37 (1996) 397–403. [53] T. Matsui, “On the spectra of the kink for ferromagnetic XXZ models”, to appear in Lett. Math. Phys. (1997). [54] M. M¨ uger, “Quantum double actions on operator algebras and orbifold quantum field theories”, DESY-96-117, June 1996, to appear in Commun. Math. Phys. [55] M. M¨ uger, “Superselection structure of massive quantum field theories in 1 + 1 dimensions”, to be published. [56] H. Roos, “Independence of local algebras in quantum field theory”, Commun. Math. Phys. 16 (1970) 238–246. [57] S. Sakai, C ∗ -Algebras and W ∗ -Algebras, Berlin, Heidelberg, New York, Springer, 1971. [58] D. Schlingemann, “On the algebraic theory of soliton and anti-soliton sectors”, Rev. Math. Phys. 8 (2) (1996) 301–326. [59] D. Schlingemann, “On the existence of kink (soliton) states”, Rev. Math. Phys. 8 (8) (1996) 1187–1203. [60] D. Schlingemann, Kink States in P (φ)2 -Models, DESY 96-051, Apr. 1996. pp. 37. [61] D. Schlingemann, “On the algebraic theory of kink sectors: application to quantum field theory models and collision theory”, PhD thesis, DESY 96-228, Oct. 1996. pp. 163. [62] R. Schrader, “A remark on Yukawa Plus boson self-interaction in two space-time dimensions”, Commun. Math. Phys. 21 (1971) 164–170. [63] R. Schrader, “A Yukawa quantum field theory in two space-time dimensions without cutoffs”, Ann. Phys. 70 (1972) 412–457. [64] S. J. Summers, “Normal product states for fermions and twisted duality for CCR and CAR type algebras with application to Yukawa2 quantum field model”, Commun. Math. Phys. 86 (1982) 111–141. [65] S. J. Summers, “On the phase diagram of a P (φ)2 quantum field model”, Ann. Inst. Henri Poincar´e 34 (1981) 173–229.
SYMMETRIES OF THE QUANTUM STATE SPACE AND GROUP REPRESENTATIONS∗ GIANNI CASSINELLI Dipartimento di Fisica Universit` a di Genova, I.N.F.N. Sezione di Genova, Via Dodecaneso 33 16146 Genova, Italy E-mail : [email protected]
ERNESTO DE VITO Dipartimento di Matematica Universit` a di Modena via Campi 213/B, 41100 Modena Italy and I.N.F.N., Via Dodecaneso 33 16146 Genova, Italy E-mail : [email protected]
PEKKA LAHTI Department of Physics University of Turku FIN-20014 Turku, Finland E-mail : [email protected]
ALBERTO LEVRERO Dipartimento di Fisica Universit` a di Genova, I.N.F.N. Sezione di Genova, Via Dodecaneso 33 16146 Genova, Italy E-mail : [email protected] Received 17 November 1997 The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the R-multipliers of the universal covering G∗ of G and the characters of G∗ . As an application, the Poincar´e group and the Galilei group, both in 3 + 1 and 2 + 1 dimensions, are considered.
1. Introduction In the standard framework of Quantum Mechanics, the physical properties of a quantum system are described in terms of a Hilbert space. In particular, the one dimensional projectors are the pure states of the system and the physical content of the theory is given by the transition probabilities between such states. ∗ Dedicated
to Francesca. 893
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In this context, a symmetry is a bijective map from the set of pure states onto itself preserving the transition probabilities; the set S of all symmetries is a group under the composition of maps. A group G is a symmetry group for a quantum system if there exists a group homomorphism from G to S. We call such homomorphisms symmetry actions of G. Given a symmetry group G (for example, the covariance group of space-time is a symmetry group for the free particles), one can pose the mathematical problem of describing all possible symmetry actions of G. For the problem to be well-posed, one has to specify a suitable notion of equivalence between symmetry actions. Taking into account the physical meaning of the theory, a symmetry action α1 , acting on a Hilbert space H1 , is equivalent to another symmetry action α2 , acting on a Hilbert space H2 , if there exists a bijective map β from the set of pure states of H1 onto the set of pure states of H2 such that β preserves the transition probabilities and intertwines α1 and α2 . Using the Wigner theorem on the characterisation of symmetries in terms of unitary operators [1], one can analyse the symmetry actions of G in the framework of projective representations, i.e., maps g 7→ Ug from G to the group of unitary operators satisfying Ug Uh = µ(g, h)Ugh g, h ∈ G , where µ(g, h) belongs to the circle group T and the map (g, h) 7→ µ(g, h) is called a multiplier of G. However, the natural notion of unitary equivalence among projective representations does not correspond to the one among symmetry actions. The classification of projective representations for finite groups, up to unitary equivalence, was given by Schur [2, 3] who first introduced the concept of projective representation. In particular, he showed the existence of a finite group with the property that there is a one to one correspondence between its irreducible unitary representations and the irreducible projective representations of G and this correspondence preserves the unitary equivalence, (see, for example, [4, Sec. 14.2] for a modern exposition of these results). The first to solve this problem for an infinite group was Wigner in his celebrated paper [5]. He classified the projective representations of the Poincar´e group considering the unitary representations of its universal covering group. Bargmann, in his seminal paper [6], considered the case of a connected Lie group G, reducing the problem of classifying its projective representations to the one of classifying the projective representations of the universal covering G∗ . He proved that the multipliers of G∗ can always be chosen smooth and that its projective representations are in one to one correspondence with a family of unitary representations of a set of Lie groups associated with G∗ . The fundamental contribution of Mackey to this problem was a complete analysis in the case of G being a locally compact topological group [7]. In particular, he showed that there is a one to one correspondence between the projective representations of G and the unitary representations of a family of locally compact topological groups, namely the central extensions of T by G, parametrised by the group H 2 (G, T) of multipliers of G.
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The structure of H 2 (G, T) was exploited completely by Moore in the context of the cohomology of locally compact topological groups [8–11]. In particular, he proved that there exists a central extension H of an abelian group by G, called splitting group, such that there is a surjective map from the set of irreducible representations of H and the set of irreducible projective representations of G, preserving again the notion of unitary equivalence [9]. The construction of the splitting group given by Moore requires a careful analysis of the cohomology groups H 2 (G, Z) and H 2 (G, R) (see Sec. 2 of [9]). Since G, in general, is not simply connected, one cannot use linear methods to study these cohomology groups. A clear and complete exposition of the theory of projective representations and multipliers can be found in Varadarajan’s book [12], which we use in the following as a standard reference. In this paper we consider the case of a connected Lie group G and we prove that there exist a Lie group G, namely the universal extension of G, and a notion of equivalence among unitary representations of G, called physical equivalence, such that there is a one to one correspondence between the equivalence classes of irreducible unitary representations of G (with respect to physical equivalence) and the equivalence classes of irreducible symmetry actions of G. The explicit construction of G, which is a splitting group in the sense of Moore, requires the knowledge of H 2 (G∗ , R), where G∗ is the universal covering group of G. Moreover, G is a central extension of an abelian Lie group K by G and H 2 (G, T) b , where V is a is isomorphic, as a topological group, to the quotient group K/V b of K and V is completely subgroup (not necessarily closed) of the dual group K defined in terms of H 1 (G∗ , T), i.e. the group of characters of G∗ . Since G∗ is simply connected, the study of H 2 (G∗ , R) and H 1 (G∗ , T) can be done by the use of linear methods. Finally, we show that every symmetry action of G is induced by a representation of G satisfying an admissibility condition and vice versa. This admissibility condition is at the root of the existence of superselection rules in the case of reducible representations (this topic has been recently considered also by Divakaran [13]). In the paper we assume that G is connected, otherwise one can consider only the connected component of the identity of G. We do not consider the problem of discrete symmetries. 2. Preliminary Results In this section we introduce the notations and we review some results on the theory of multipliers for simply connected Lie groups. 2.1. Notations 1. By Hilbert space we mean a complex separable Hilbert space with scalar product h·, ·i linear in the second argument. 2. If H is a Hilbert space, we denote by PH (or P) the set of one dimensional projectors on H.
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3. We use the word representation to mean a unitary representation of a topological group, acting on a Hilbert space and continuous with respect to the strong operator topology. 4. If α is a Lie group homomorphism we denote by α˙ the differential of α at the identity. Let H be the Hilbert space associated with a quantum system. Then P is the set of pure states of the system and the transition probability between two pure states P1 , P2 ∈ P is given by tr P1 P2 , where tr · denotes the trace. Definition 1. A map α : P → P is a symmetry of P if α is bijective and tr P1 P2 = tr α(P1 )α(P2 ) , P1 , P2 ∈ P . We denote by S the set of symmetries of P. The set S is a group with respect to the usual composition of maps and it is a topological space with respect to the initial topology given by the family of functions S 3 α 7→ tr P1 α(P2 ) ∈ R , labelled by P1 , P2 ∈ P. Using the Wigner theorem [1], we can obtain a useful characterisation of S. In fact, let U denote the set of unitary operators on H, U the set of antiunitary operators on H, and let T = {z ∈ C : |z| = 1} denote the circle group. We identify T with {zI : z ∈ T} ⊂ U. The set U ∪ U is a group under the usual composition of operators and it is a topological space with respect to the restriction of the strong operator topology. Then we have the following results, whose proofs can be found, for example, in [12] (for a review see also [14]). Proposition 1. With the above notations: 1. the group U ∪ U is a second countable metrisable topological group, U is the connected component of the identity and T is its centre; 2. the group S is a second countable metrisable topological group; 3. the map π : U ∪ U → S defined as π(U )(P ) = U P U −1 ,
P ∈P,
is a continuous surjective open group homomorphism and its kernel is the group T; 4. there exists a measurable map s from S0 , the connected component of the identity of S, to U such that s(I) = I, π(s(α)) = α , We call such a map a section for π.
α ∈ S0 .
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We are now in a position to define a symmetry action. Let G be a connected Lie group. Definition 2. A symmetry action of G on H is a continuous group homomorphism α : G → S. Two symmetry actions α1 and α2 of G acting, respectively, on the Hilbert spaces H1 and H2 are equivalent if there is a bijective map β from PH1 onto PH2 such that tr P Q H = tr β(P )β(Q) H , 1
βα1g
2
=
α2g β
,
for all P, Q ∈ PH1 and for all g ∈ G. A symmetry action α is irreducible if for all P1 , P2 ∈ P there is g ∈ G satisfying tr P2 αg (P1 ) 6= 0 . Since G is connected all symmetry actions take values in S0 . Moreover, if α is an irreducible symmetry action, then every symmetry action equivalent to α is irreducible too. Finally, the condition on the continuity of the symmetry actions can be weakened, using the standard result that a group homomorphism α : G → S is continuous if and only if it is measurable (see, for example, Lemma 5.28 of [12]). 2.2. Multipliers for Lie groups: A review In this section we give a brief review of the theory of multipliers for a connected, simply connected Lie group. For this class of groups the problem of the classification (up to equivalence) of multipliers can be reduced to a finite-dimensional linear problem on the Lie algebra of the group. We stress that the theory works only in the case of simply connected Lie groups; nevertheless, as we are going to show in the next section, this is enough to study the symmetry actions also for Lie groups which are not simply connected. All the proofs of this section can be found, for example, in Chap. 7 of [12] where a systematic study of multipliers is presented. Let H be a connected Lie group and A an abelian Lie group (for the moment we do not assume that H is simply connected). We denote by e and 1 the unit elements of H and A, respectively. Definition 3. An A-multiplier of H is a measurable map τ from H × H to A such that τ (e, g) = τ (g, e) = 1 , τ (g1 , g2 g3 )τ (g2 , g3 ) = τ (g1 , g2 )τ (g1 g2 , g3 ) ,
g∈H, g1 , g2 , g3 ∈ H .
Two A-multipliers τ1 and τ2 of H are equivalent if there is a measurable map b from H to A such that τ2 (g1 , g2 ) =
b(g1 g2 ) τ1 (g1 , g2 ) , b(g1 )b(g2 )
g1 , g2 ∈ H .
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An A-multiplier τ is exact if it is equivalent to the multiplier 1, that is, τ (g1 , g2 ) =
b(g1 g2 ) , b(g1 )b(g2 )
g1 , g2 ∈ H
for some measurable map b : H → A. The set of A-multipliers is an abelian group under the pointwise multiplication and the set of exact A-multipliers is a subgroup. We denote by H 2 (H, A) the corresponding quotient group. Remark 1. There is a natural topology on H 2 (H, A) converting it into a locally compact group (in general not Hausdorff), see [11]. If A = T it coincides with the (quotient) topology of uniform convergence on compact sets defined on the set of T-multipliers (see Theorem 6 of [11]). We will always consider H 2 (H, T) endowed with this topology. From now on, assume that H is simply connected. If A is the group T, we have the following result. Lemma 1. Each T-multiplier of H is similar to one of the form eiτ , where τ is a smooth R-multiplier of H. Moreover, τ is exact if and only if eiτ is exact. If A is the vector group Rn , we denote it additively. The set of Rn -multipliers is a real vector space under the pointwise operations and the set of exact Rn -multipliers is a subspace of it, so that H 2 (H, Rn ) is a vector space. Moreover, we have: Lemma 2. Any Rn -multiplier of H is equivalent to a smooth one. Let Lie (H) be the Lie algebra of H. Definition 4. A bilinear skew symmetric map F from Lie (H) × Lie (H) to Rn such that F (X, [Y, Z]) + F (Z, [X, Y ]) + F (Y, [Z, X]) = 0 ,
X, Y, Z ∈ Lie (H) ,
is called a closed Rn -form. A closed Rn -form F is exact if there is a linear map q from Lie (H) to Rn such that F (X, Y ) = q([X, Y ]) ,
X, Y ∈ Lie (H) .
The set of closed Rn -forms is a finite dimensional real vector space and the set of exact Rn -forms is a subspace. We denote by H 2 (Lie (H), Rn ) the corresponding quotient space.
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Theorem 1. The vector spaces H 2 (H, Rn ) and H 2 (Lie (H), Rn ) are canonically isomorphic. We exhibit explicitly the above isomorphism. Let F be a closed Rn -form, and denote by Rn ⊕F Lie (H) the Lie algebra defined by the following Lie bracket: (v1 , X1 ), (v2 , X2 ) := F (X1 , X2 ), X1 , X2 , for all X1 , X2 ∈ Lie (H) and v1 , v2 ∈ Rn . Let α : Rn → Rn ⊕F Lie (H) be the natural injection and β : Rn ⊕F Lie (H) → Lie (H) be the natural projection. Both these maps are Lie algebra homomorphisms and Ker β = Im α. By the general theory of Lie groups, there exist a unique (up to an isomorphism) connected, simply connected Lie group HF , such that Lie (HF ) = Rn ⊕F Lie (H), and two Lie group homomorphisms a : Rn → HF , b : HF → H such that a˙ = α and b˙ = β. Moreover, one can prove that a is a homeomorphism from Rn onto a(Rn ) and HF /a(Rn ) is isomorphic to H. By a lemma of Malˇcev (see, for example, Lemma 7.26 of [12]) there exists a smooth map c from H to HF such that c(e) = e and b(c(h)) = h for all h ∈ H. If we define τF (h1 , h2 ) = c(h1 )c(h2 )c(h1 h2 )−1 ,
h1 , h2 ∈ H ,
then τF is a smooth Rn -multiplier. The equivalence class of τF is the image of the equivalence class of F under the isomorphism of the above theorem. Since τF is smooth, one can easily check that HF is isomorphic, as a Lie group, to Rn ×τF H, which is a Lie group with respect to the product (v1 , g1 )(v2 , g2 ) = (v1 + v2 + τF (g1 , g2 ), g1 g2 ) ,
v1 , v2 ∈ Rn , g1 , g2 ∈ H .
3. Main Results In this section we introduce the notion of universal central extension G for a connected Lie group G and the notions of physical equivalence and admissibility for representations of G. By the use of these concepts, we then state the main results of the paper. Moreover, we discuss the physical equivalence for induced representations in the case that G is a semidirect product with abelian normal factor. 3.1. Universal central extension Let G be a connected Lie group. We denote by G∗ its universal covering group and by δ the corresponding covering homomorphism. Let H 2 (G∗ , R)δ be the set of equivalence classes [τ ] ∈ H 2 (G∗ , R) such that τ (k, g ∗ ) = τ (g ∗ , k) ,
k ∈ Ker δ, g ∗ ∈ G∗ .
(1)
Since Ker δ is central in G∗ , Eq. (1) holds for all R-multipliers equivalent to τ and, hence, the definition of H 2 (G∗ , R)δ is well-posed. Moreover, H 2 (G∗ , R)δ is a subspace of H 2 (G∗ , R), so that, due to Theorem 1, it has finite dimension N .
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By Lemma 2, we can fix N smooth R-multipliers of G∗ , τ1 , . . . , τN , such that the equivalence classes [τ1 ], . . . , [τN ] form a basis of H 2 (G∗ , R)δ . Let τ : G∗ × G∗ → RN be defined as τ (g1∗ , g2∗ )i = τi (g1∗ , g2∗ ) ,
g1∗ , g2∗ ∈ G∗ , i = 1, . . . , N ,
then τ is a smooth RN -multiplier of G∗ . The restriction of τ to Ker δ×Ker δ is an RN -multiplier of the discrete group Ker δ, hence it is exact (see Proposition 2, Sec. 4, Chap. 1 of [15]). Then, without loss of generality, we can always assume that τ is smooth and that τ (k1 , k2 ) = 0 ,
k1 , k2 ∈ Ker δ .
(2)
Definition 5. Let G = RN ×τ G∗ be the product manifold. Since τ is smooth, G is a Lie group with the product (v1 , g1∗ )(v2 , g2∗ ) = (v1 + v2 + τ (g1∗ , g2∗ ), g1∗ g2∗ ) ,
g1∗ , g2∗ ∈ G∗ , v1 , v2 ∈ Rn .
We call it the universal central extension of G and we denote by σ the smooth map from G to G given by σ(v, g ∗ ) = δ(g ∗ ) ,
v ∈ RN , g ∗ ∈ G∗ .
We denote by K the closed subgroup of G defined as K = {(v, k) ∈ G : v ∈ RN ,
k ∈ Ker δ} .
The main properties of G are stated in the following lemma. Lemma 3. Let G be the universal central extension of G. 1. The restriction of a character of G to the subgroup RN is the identity character. Any character of G∗ extends naturally to a character of G. 2. The map σ is a surjective group homomorphism whose kernel is K, which is central in G. Moreover, the group K is the direct product of RN and Ker δ. 3. There is a measurable map c : G → G such that c(e) = e and σ(c(g)) = g for all g ∈ G (we call such a map a section for σ). 4. Given a section c for σ, let Γc from G × G to K be the map Γc (g1 , g2 ) = c(g1 )c(g2 )c(g1 g2 )−1 ,
g1 , g2 ∈ G,
then Γc is a K-multiplier of G and its equivalence class does not depend on the choice of the section c. 5. If we consider RN as a subgroup of K in a natural way, then the K-multiplier Γc ◦ (δ × δ) of G∗ is equivalent to τ .
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Proof. 1. Let χ be a character of G. The restriction of χ to RN is of the form χ(v, e∗ ) = eiw·v
v ∈ RN ,
for some w ∈ RN . Then, if g1∗ , g2∗ ∈ G∗ , χ((0, g1∗ ))χ((0, g2∗ )) = χ((τ (g1∗ , g2∗ ), e∗ ))χ((0, g1∗ g2∗ )) ∗
∗
= eiw·τ (g1 ,g2 ) χ((0, g1∗ g2∗ )) . Hence, by Lemma 1, the R-multiplier w · τ is exact, so that w = 0. The other statement is evident. 2. The fact that K is central in G follows taking into account that, by definition of H 2 (G∗ , R)δ , τ (k, g ∗ ) = τ (g ∗ , k) for all k ∈ Ker δ and g ∗ ∈ G∗ . Using Eq. (2), one has K = RN × Ker δ. The other facts are evident. 3. By the previous item, G is isomorphic, as a Lie group, to the quotient G/K. The existence of a section is thus a standard result (see, for example, Theorem 5.11 of [12]). 4. If g1 , g2 ∈ G, then σ(Γc (g1 , g2 )) = e, so that Γc (g1 , g2 ) ∈ K. By direct computation one checks that Γc is a K-multiplier. Let c0 be another section, then, for all g ∈ G, c(g) = b(g)c0 (g) for some measurable map b from G to K. Hence, for all g1 , g2 ∈ G b(g1 g2 ) Γc (g1 , g2 ) . Γc0 (g1 , g2 ) = b(g1 )b(g2 ) 5. Let i : G∗ → G be the natural immersion and a be the measurable map from G to G a(g ∗ ) = c(δ(g ∗ ))i(g ∗ )−1 , g ∗ ∈ G∗ . ∗
Since σ(a(g ∗ )) = e, then a takes values in K. Then, if g1∗ , g2∗ ∈ G∗ , Γc (δ(g1∗ ), δ(g2∗ )) = c(δ(g1∗ ))c(δ(g2∗ ))c(δ(g1∗ )δ(g2∗ ))−1 = a(g1∗ )i(g1∗ )a(g2∗ )i(g2∗ )i(g1∗ g2∗ )−1 a(g1∗ g2∗ )−1 = a(g1∗ )a(g2∗ )a(g1∗ g2∗ )−1 i(g1∗ )i(g2∗ )i(g1∗ g2∗ )−1 = a(g1∗ )a(g2∗ )a(g1∗ g2∗ )−1 (τ (g1∗ , g2∗ ), e∗ ), i.e., Γc ◦ (δ × δ) is equivalent to τ .
The following theorem describes the group H 2 (G, T) in terms of the characters of K and the characters of G∗ . We first observe that: Lemma 4. Let c : G → G be a section for σ and Γc the corresponding Kmultiplier of G defined in statement 4 of Lemma 3. Let χ be a character of K, then the map µχ from G × G to T defined as µχ (g1 , g2 ) = χ (Γc (g1 , g2 )) ,
g1 , g2 ∈ G ,
is a T-multiplier of G and its equivalence class [µχ ] does not depend on the choice of the section c.
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Proof. It is a simple consequence of the properties of Γc given in statement 4 of Lemma 3. b be the dual group of K (with the topology of uniform convergence on compact Let K sets) and V the subgroup of characters that extend to characters of G, which, due to the statement 1 of Lemma 3, can be identified with characters of G∗ . Theorem 2. The mapping b 3 χ 7→ [µχ ] ∈ H 2 (G, T) K is a surjective homomorphism whose kernel is V. Moreover, H 2 (G, T) is isomorphic, b as a topological group, to the quotient group K/V. Proof. By direct computation, one can check that χ 7→ [µχ ] is a group homomorphism. To show its surjectivity, we notice that, since the equivalence class [µχ ] does not depend on the specific form of the section c, we can choose for c the particularly simple form c(g) = (0, c˜(g)) g ∈ G , c(g)) = g for all where c˜ : G → G∗ is measurable and satisfies c˜(e) = e∗ and δ(˜ g ∈ G. With this choice, a straightforward calculation shows that c(g1 ), c˜(g2 )) − τ (γ(g1 , g2 ), c˜(g1 g2 )) , γ(g1 , g2 )) , Γc (g1 , g2 ) = (τ (˜
(3)
where g1 , g2 ∈ G and c(g2 )˜ c(g1 g2 )−1 ∈ Ker δ . γ(g1 , g2 ) = c˜(g1 )˜ Let now µ be a T-multiplier of G and µ∗ the T-multiplier of G∗ µ∗ (g1∗ , g2∗ ) = µ(δ(g1∗ ), δ(g2∗ )) ,
g1∗ , g2∗ ∈ G∗ .
Applying Lemma 1 to µ∗ , we have that µ∗ (g1∗ , g2∗ ) =
a(g1∗ g2∗ ) iτ (g1∗ ,g2∗ ) e , a(g1∗ )a(g2∗ )
g1∗ , g2∗ ∈ G∗ ,
(4)
for some smooth R-multiplier τ of G∗ and some measurable function a from G∗ to T. We claim that τ (k, g ∗ ) = τ (g ∗ , k) k ∈ Ker δ , g ∗ ∈ G∗ . In fact, let k ∈ Ker δ and g ∗ ∈ G∗ , since µ∗ (k, g ∗ ) = µ∗ (g ∗ , k) = 1, then eiτ (k,g
∗
)
=
∗ a(k)a(g ∗ ) a(k)a(g ∗ ) = = eiτ (g ,k) . ∗ ∗ a(kg ) a(g k)
(5)
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Hence τ (k, g ∗ ) = τ (g ∗ , k) + 2πn(k, g ∗ ) where n(k, g ∗ ) is an integer. By continuity of τ (k, ·) and since G∗ is connected, the map n(·, ·) depends only on k, and, choosing g ∗ = k, we conclude that n(k, g ∗ ) = 0 for all k ∈ Ker δ, g ∗ ∈ G∗ . Due to (5), the equivalence class of τ belongs to H 2 (G∗ , R)δ and, by definition of τ , there is w ∈ RN such that, up to equivalence, τ = w · τ . Hence (4) becomes µ∗ (g1∗ , g2∗ ) =
a(g1∗ g2∗ ) iw·τ (g1∗ ,g2∗ ) e , a(g1∗ )a(g2∗ )
g1∗ , g2∗ ∈ G∗ .
(6)
The previous equality implies that the map χ from K to T χ(v, k) := eiw·v a(k) ,
v ∈ RN , k ∈ Ker δ ,
is, in fact, a character of K. Hence, by Lemma 4, χ defines a T-multiplier µχ of G. We will show that µχ is equivalent to µ. In fact, using Eq. (3), one has µχ (g1 , g2 ) = χ(Γc (g1 , g2 )) = eiw·(τ (˜c(g1 ),˜c(g2 ))−τ (γ(g1 ,g2 ),˜c(g1 g2 ))) a(γ(g1 , g2 )) Using twice Eq. (6) we obtain eiw·τ (˜c(g1 ),˜c(g2 )) =
a(˜ c(g1 ))a(˜ c(g2 )) µ(g1 , g2 ) a(˜ c(g1 )˜ c(g2 ))
e−iw·τ (γ(g1 ,g2 ),˜c(g1 g2 )) =
a(˜ c(g1 )˜ c(g2 )) a(γ(g1 , g2 ))a(˜ c(g1 g2 ))
so that µχ (g1 , g2 ) =
a(˜ c(g1 ))a(˜ c(g2 )) µ(g1 , g2 ) , a(˜ c(g1 g2 ))
which shows the equivalence of µ and µχ . Suppose now that χ is a character of K that extends to a character of G (still denoted by χ). Then µχ (g1 , g2 ) = χ(c(g1 )c(g2 )c(g1 g2 )−1 ) = χ(c(g1 ))χ(c(g2 ))χ(c(g1 g2 )−1 ), showing that µχ is exact. Conversely, assume that µχ (g1 , g2 ) =
a(g1 g2 ) a(g1 )a(g2 )
for some measurable function a : G → T. Observe that, for all h ∈ G, hc(σ(h))−1 ∈ K and define χ0 : G → T as χ0 (h) = χ(hc(σ(h))−1 )a(σ(h))−1
h ∈ G.
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Then χ0 is a character of G. Indeed, χ0 is measurable, and if h1 , h2 ∈ G, χ0 (h1 )χ0 (h2 ) =
χ(h1 c(σ(h1 ))−1 h2 c(σ(h2 ))−1 ) a(σ(h1 ))a(σ(h2 ))
=
χ(h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 )µχ (g1 , g2 ) a(σ(h1 h2 ))
=
χ(h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 c(σ(h1 ))c(σ(h2 ))c(σ(h1 h2 ))−1 ) a(σ(h1 h2 ))
= χ(h1 h2 c(σ(h1 h2 ))−1 )a(σ(h1 h2 ))−1 = χ0 (h1 h2 ) . Moreover, since a(e) = 1, χ0 (k) = χ(k) for all k ∈ K. Hence, H 2 (G, T) is isomorphic, as an abstract group, to the quotient group K/V . This completes the proof but to observe that G is a splitting group for T in the sense of Moore (see Definition 3 of [11]) so that, using Theorem 6 of [11], H 2 (G, T) is isomorphic, as a topological group, to the quotient group K/V . As a consequence of the previous result, one has that H 2 (G, T) is Hausdorff if and b The following example, inspired by Moore [9], shows that only if V is closed in K. this is not always the case. Example 1. Let G = T2 × R2 be the Lie group with product 0 0 0 0 2π (z, ζ, x, y)(z 0 , ζ 0 , x0 , y 0 ) = zz 0 ei α (xy −yx ) , ζζ 0 ei2π(xy −yx ) , x + x0 , y + y 0 , where α ∈ R, α 6= 0. The universal covering group G∗ of G is G∗ = R2 × R2 with product (v, w, x, y)(v 0 , w0 , x0 , y 0 ) = (v + v 0 + (xy 0 − yx0 ), w + w0 + (xy 0 − yx0 ), x + x0 , y + y 0 ) , and with covering homomorphism δ from G∗ to G 2π δ(v, w, x, y) = ei α v , ei2πw , x, y . We have Ker δ = {(αn, m, 0, 0) : n, m ∈ Z} . A simple algebraic calculation shows that any R-multiplier of G∗ is equivalent to one of the form τ ((v, w, x, y), (v 0 , w0 , x0 , y 0 )) = Q((v, w), (x0 , y 0 )) , where Q(·, ·) is a bilinear form on R2 × R2 . It follows that H 2 (G∗ , R)δ is {0} so that b is isomorphic the universal central extension of G is G∗ and K = Ker δ. Then K to T × T.
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Moreover, one checks that the characters of G∗ are of the form (v, w, x, y) 7→ eia(v−w)+bx+cy , where a, b, c ∈ R. Hence we have V = (eiαa , e−ia ) ∈ T × T : a ∈ R , so that V is closed in T × T if and only if α is rational. If V is closed, we can give a better description of H 2 (G, T). Define K0 = {(v, k) ∈ K : b(k) = 1 for any character b of G∗ } , then K0 is an abelian closed subgroup of K. Since V is closed, a standard result on abelian locally compact groups (see, for example, Theorem 4.39 of [16]) shows that c0 b c0 of K0 . In particular, any element χ ∈ K K/V is isomorphic to the dual group K b extends to an element χ b ∈ K and χ b is uniquely defined by χ, up to an element of V . Let µχ be the T-multiplier of G defined by b(Γc (g1 , g2 )) , µχ (g1 , g2 ) = χ
g1 , g2 ∈ G ,
where Γc is defined in statement 4 of Lemma 3. As a consequence of Theorem 2, the equivalence class [µχ ] depends only on χ and not on the particular extension chosen. Corollary 1. If V is closed , the mapping c0 3 χ 7→ [µχ ] ∈ H 2 (G, T) K is an isomorphism of topological groups. We now turn to the representations of G. Definition 6. A representation U of G satisfying the condition Uh ∈ T
for all h ∈ K ,
(7)
is called admissible. Definition 7. Let U be an admissible representation, then its restriction to K is a character of K and, due to statement 2 of Lemma 3, is of the form U(v,k) = eiw·v (k) ,
v ∈ RN , k ∈ Ker δ ,
for some w ∈ RN and some character of Ker δ. We call w the algebraic charge of U and the topological charge.
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In the above definition we have followed the terminology of Divakaran [13]. Definition 8. Let U and U 0 be two representations of G acting respectively in H and H0 . We say that U and U 0 are physically equivalent if there exist a unitary or antiunitary operator B : H → H0 and a measurable map b : G → T such that BUh = b(h)Uh0 B ,
h ∈ G.
(8)
We notice the following facts concerning these definitions. 1. The notion of physical equivalence preserves condition (7) and the usual notion of irreducibility of representations. The case of unitary equivalence is a particular instance of the physical equivalence. 2. Since K is central in G, every irreducible representation of G is admissible. 3. Since in (8) U and U 0 are representations, the map b is, in fact, a character of G and, by statement 1 of Lemma 3, a character of G∗ . We are now in a position to state the main property of G. Given an admissible representation U of G, define, for all g ∈ G, αU g = π(Uh ) ,
(9)
where π is defined in Proposition 1 and h ∈ G is such that σ(h) = g. The following theorem is then obtained. Theorem 3. With the above notations, αU is a symmetry action of G and the correspondence [U ] 7→ [αU ] between the physical equivalence classes of admissible representations of G and the equivalence classes of symmetry actions of G is a bijection. The representation U of G is irreducible if and only if αU is an irreducible symmetry action of G. Proof. In the following we fix a section c : G → G for σ (cf. item 3 of Lemma 3) and a section s : S0 → U for π : U → S0 (cf. item 4 of Proposition 1). Due to condition (7), if h1 , h2 ∈ G are such that σ(h1 ) = σ(h2 ) = g, then π(Uh1 ) = π(Uh2 ), showing that αU g is well-defined. In particular, we have αU g = π(Uc(g) ) g ∈ G . First we show that g 7→ αU g is a symmetry action of G. Indeed, if g1 , g2 ∈ G then U αU g1 αg2 = π Uc(g1 ) π Uc(g2 ) = π Uc(g1 ) Uc(g2 ) = π Uc(g1 )c(g2 )c(g1 g2 )−1 π Uc(g1 g2 ) = π Uc(g1 g2 ) = αU g1 g2 ,
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where we used the fact that c(g1 )c(g2 )c(g1 g2 )−1 ∈ K and condition (7). Since c is U measurable, αU is measurable too, and αU e = I. Hence α is a symmetry action of G. Now let U and U 0 be two physically equivalent admissible representations of G acting on H and H0 , respectively, then the corresponding symmetry actions αU and 0 αU are equivalent too. Indeed, in this case BUh = b(h)Uh0 B,
h ∈ G,
for some unitary or antiunitary B : H → H0 and a character b : G → T. Define β from PH to PH0 as β(P ) = BP B −1 . Then β is bijective, preserves the transition U0 U probabilities and satisfies βαU g = αg β for all g ∈ G, which is just to say that α 0 U U and α are equivalent. This shows that the map [U ] 7→ [α ] is well-defined. We now show its surjectivity. Let α be a symmetry action of G, define µ : G × G → U as µ(g1 , g2 ) := s(αg1 )s(αg2 )s(αg1 g2 )−1 ,
g1 , g2 ∈ G .
Since π(µ(g1 , g2 )) = I then µ(g1 , g2 ) ∈ T. Moreover, µ is measurable and by a direct computation one confirms that µ is, in fact, a T-multiplier of G. By Theorem 2, there are a character χ of K and a measurable function a : G → T such that µ(g1 , g2 ) =
a(g1 g2 ) µχ (g1 , g2 ) a(g1 )a(g2 )
g1 , g2 ∈ G .
Define a map U α : G → U as Uhα := χ(hc(σ(h))−1 )a(σ(h))s(ασ(h) ) ,
h ∈ G.
Then U α is a representation of G. Indeed, 1. U α is measurable as a composition of measurable maps; α 2. U(0,e ∗ ) = I, since a(e) = 1 and s(I) = I; 3. for any h1 , h2 ∈ G, Uh1 Uh2 = χ(h1 c(σ(h1 ))−1 h2 c(σ(h2 ))−1 ) ×a(σ(h1 ))a(σ(h2 ))s(ασ(h1 ) )s(ασ(h2 ) ) = χ h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 ×a(σ(h1 ))a(σ(h2 ))µ(g1 , g2 )s(ασ(h1 h2 ) ) = χ h1 h2 c(σ(h2 ))−1 c(σ(h1 ))−1 ×χ c(σ(h1 ))c(σ(h2 ))c(σ(h1 h2 ))−1 ×a(σ(h1 h2 ))s(ασ(h1 h2 ) ) = χ h1 h2 c(σ(h1 h2 ))−1 a(σ(h1 h2 ))s(ασ(h1 h2 ) ) = U h1 h2 .
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Since π ◦ s = id S0 and σ ◦ c = id G , one readily verifies that αU = α, proving the surjectivity of the map [U ] 7→ [αU ]. 0 Assume next that αU and αU are equivalent symmetry actions. By the definition of equivalence of symmetry actions, there is a bijective map β, preserving the transition probabilities, such that, for all g ∈ G, 0 β = βπ Uc(g) . π Uc(g) Applying the Wigner theorem [1], we deduce that for some unitary or antiunitary operator B and for some measurable map b : G → T, 0 = b(g)BUc(g) B −1 . Uc(g)
Let h ∈ G, g = σ(h), and k = hc(g)−1 , then k ∈ K and 0 Uh0 = Uk0 Uc(g)
= Uk0 b(c(g))BUc(g) B −1 = Uk0 b(c(g))BUk−1 Uh B −1 = ˆb(h)BUh B −1 , taking into account that, due to (7), Uk0 and Uk−1 are phase factors that we have collected in ˆb. This shows that U and U 0 are physically equivalent representations of G, proving the injectivity of the map [U ] 7→ [αU ]. Finally, due to (7), an admissible representation U is irreducible if and only if
φ1 , Uc(g) φ2 = 0 ∀ g ∈ G =⇒ (φ1 = 0 or φ2 = 0) . This last condition is equivalent to the fact that αU is irreducible.
This theorem shows that the equivalence classes of admissible representations of G classify the different (with respect to the given symmetry group G) quantum systems. In particular, the irreducible representations of G are always admissible and describe the elementary systems. Let us now consider the case of reducible representations of G. For the sake of simplicity, let U = U1 ⊕ U2 where U1 and U2 are irreducible representations. Since the representations Ui are irreducible, they are admissible and we denote by wi and i the corresponding algebraic and topological charges. However, in general, U is not admissible. A simple calculation shows that U is admissible if and only if w1 = w2 and 1 = 2 . This fact is at the root of the existence of superselection rules for non-elementary systems, as it will be discussed in more detail in the examples. Furthermore, the relation between the decomposition into irreducible representations and the notion of physical equivalence requires some special care. One can easily show that, if b is a nontrivial character of G that is 1 on K, then U1 ⊕ U2 and U1 ⊕ b U2 are physically inequivalent admissible representations, even though U2 and bU2 are physically equivalent. In the same way, if the algebraic charge w of
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U1 and U2 is zero and their topological charge is such that 2 ∈ V , then U1 ⊕ U2 and U1 ⊕ bBU2 B −1 (where b is any character of G that extends 2 and B is any antiunitary operator) are physically inequivalent admissible representations, even though U2 and bBU2 B −1 are physically equivalent. This kind of phenomenon does not occur if one considers the unitary equivalence instead of the physical one. We add some comments about the relation between the admissible representations of G and the projective representations of G and G∗ . Let U be an admissible representation of G, w and its algebraic and topological charges. 1. The map G∗ 3 g ∗ 7→ U(0,g∗ ) ∈ U is a projective representation of G∗ with ∗ ∗ T-multiplier µ∗ (g1∗ , g2∗ ) = eiw·¯τ (g1 ,g2 ) . 2. If c : G → G is a section for σ then the map G 3 g 7→ Uc(g) ∈ U is a projective representation of G and its T-multiplier is µχ where χ(v, k) = eiw·v (k) and µχ is defined in Lemma 4. As a consequence of statement 5 of Lemma 3, µ∗ and µχ ◦ (δ × δ) are equivalent, nevertheless, even if µ∗ is exact, µχ could be nonexact (see remark 1 in Sec. 4.1 and 1 in Sec. 4.2). 3.2. The physical equivalence for semidirect products According to Theorem 3, the irreducible inequivalent symmetry actions of a group G are completely described by the irreducible physically inequivalent representations of its universal central extension G. In the examples we consider in the next section, the universal central extension is a regular semidirect product with abelian normal subgroup, so that any irreducible representation is unitarily equivalent to some induced one [17]. In this way, the problem of characterising physically inequivalent irreducible representations is reduced to the analogous problem for the induced ones. In the present section we describe the solution in terms of properties of the orbits in the dual space and of the inducing representations. Let G = A ×0 H be a Lie group with A an abelian normal closed subgroup and H a closed subgroup. We denote by Aˆ the dual group of A and by (g, ·) 7→ g[·] ˆ If x ∈ A, ˆ let Gx both the inner action of G on A and the dual action of G on A. be the stability subgroup of G at x and G [x] the corresponding orbit. We assume that each orbit in Aˆ is locally closed (i.e. the semidirect product is regular) and, to simplify the exposition, that it has a G-invariant σ-finite measure. Moreover, given x ∈ Aˆ and a representation D of Gx ∩ H acting in a Hilbert G (xD) the representation of G unitarily induced space K, we denote by U = IndG x
by the representation xD of Gx , (xD)ah = xa Dh ,
a ∈ A, h ∈ Gx ∩ H .
Explicitly, let ν be a G-invariant σ-finite measure on G [x] and c a measurable map from G [x] to G such that c(x) = e and c(y)[x] = y for all y ∈ G [x] (we call such a map a section for G [x]), then U acts on the Hilbert space L2 ( G [x], ν, K) as (Ug f )(y) = (xD)(c(y)−1 gc(g−1 [y])) f (g −1 [y]) , where y ∈ G [x], f ∈ L2 ( G [x], ν, K), and g ∈ G.
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We are now in a position to classify all the equivalence classes (with respect to the notion of physical equivalence) of irreducible representations of G in the case of regular semidirect products. ˆ i.e., Let Aˆs be the set of singleton G-orbits in A, Aˆs = y ∈ Aˆ : g[y] = y ,
g∈G .
Define for all x ∈ Aˆ the orbit class ex := yg[x ] : y ∈ Aˆs , O
g ∈ G, = ±1 .
ex , G [x0 ] ⊂ O ex and O ex = O ex0 , so that we can choose a Obviously, for all x0 ∈ O ˆ ˆ exi . family {xi }i∈I of elements in A such that A is the disjoint union of the sets O Theorem 4. Let G = A ×0 H be a regular semidirect product. 1. Every irreducible representation of G is physically equivalent to one of the (xi D) for some index i and some irreducible representation D of form Ind G G xi
Gxi ∩ H. 2. If i 6= j and D, D0 are two representations of Gxi ∩ H and Gxj ∩ H, (xi D) and Ind G (xj D0 ) are physically inequivalent. respectively, then Ind G G G xi
xj
(xD) 3. Let x ∈ Aˆ and D, D0 be two representations of Gx ∩ H. Then Ind G G x
(xD0 ) are physically equivalent if and only if one of the following and Ind G Gx two conditions is satisfied: (a) there exist y ∈ Aˆs , a character χ of H, and a unitary operator M such that G [x] = yG [x] , 0 −1 , Dhsh −1 = χs M Ds M
s ∈ Gx ∩ H ,
where h ∈ H is such that x = yh[x]; (b) there exist y ∈ Aˆs , a character χ of H, and an antiunitary operator M such that G [x] = yG [x]−1 , 0 −1 , Dhsh −1 = χs M Ds M
s ∈ Gx ∩ H ,
where h ∈ H is such that x = yh[x−1 ]. Motivated by the above theorem, if U is a representation of G physically equiv(xD) we say, with slight abuse of teralent to some induced representation Ind G Gx e minology, that U lives on the orbit class Ox . The proof of the theorem is based on the following lemma.
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ˆ Let D be a representation of Gx ∩ H acting in K and Lemma 5. Let x, x0 ∈ A. D a representation of Gx0 ∩ H acting in K0 . The induced representations Ind G G 0
x
(xD) and Ind G (x0 D0 ) are physically equivalent if and only if there exist h ∈ G, G x0
a character χ e of G and a unitary or antiunitary operator M from K onto K0 such that 1. Gx0 = hGx h−1 ; eg M (xD)g M −1 for all g ∈ Gx . 2. (x0 D0 )hgh−1 = χ Moreover, every character of G is of the form (a, h) 7→ χ ˆ a χh
a ∈ A, h ∈ H
where χ ˆ ∈ Aˆs and χ is a character of H. e is a character Proof. First we prove the statement on the characters of G. If χ ˆ and χ be its restrictions to A and H, respectively. Then χ is a character of G, let χ of H and, by definition of dual action, χˆ ∈ Aˆs . The proof of the converse implication is similar. We now turn to the first statement. To simplify the notations, denote U = (xD) and U 0 = Ind G (x0 D0 ). The representations U and U 0 are physically Ind G G G x
x0
equivalent if and only if there exist a character χ e of G and a unitary or antiunitary operator B such that eB −1 U B . U0 = χ As a first step we define in terms of U and χ e two induced representations U + and − U of G such that eW±−1 U W± , U± = χ where W+ [resp. W− ] is unitary [resp. antiunitary]. In particular, U + and U − are physically equivalent to U . By the previous result χ e = χχ, ˆ where χ ˆ ∈ Aˆs and χ is a character of H. ˆ ˆ ˆ ±1 . The maps ψ± are Define the maps ψ+ and ψ− from A onto A as ψ± (x) := χx measurable isomorphisms that commute with the action of G, so that ψ± maps the orbit G [x] onto the orbit G [ψ± (x)] and one has Gx = Gψ± (x) . If ν is an invariant measure on G [x], the image measure ν ± with respect to ψ± is an invariant measure −1 is a section on G [ψ± (x)] and if c is a section for the orbit G [x], then c± = c ◦ ψ± for the action of G on the orbit G [ψ± (x)]. Fix a unitary operator L+ and an antiunitary operator L− on K. Consider the representations of Gx , g 7→ χ eg L± (xD)g L−1 ± , and observe that their restriction to A are exactly the elements x± := ψ± (x). Since Gx± = Gx we can define the induced representations of G, U ± := Ind G G acting in L2 ( G [x± ], ν ± , K).
x±
(e χL± xDL−1 ± ),
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Moreover, define the operators W± from L2 ( G [x± ], ν ± , K) onto L2 ( G [x], ν, K) −1 e±1 (W± f )(y) = χ c(y) L± f (ψ± (y)) ,
y ∈ G [x] .
It is easy to show that W+ [resp. W− ] is unitary [resp. antiunitary]. We have eW±−1 U W± . U± = χ In fact, let g ∈ G, f ∈ L2 ( G [x± ], ν ± , K), and y ∈ G [x± ] −1 eg χ e−1 χ eg W±−1 Ug W± f (y) = χ c± (y) L± (Ug W± f ) (ψ± (y)) −1 −1 ± e−1 [ψ± (y)]) =χ eg χ c± (y) L± (xD)γ (g,y) (W± f )(g −1 −1 ± e−1 e±1 [y]) =χ eg χ c± (y) L± (xD)γ (g,y) χ c± (g−1 [y]) L± f (g −1 [y]) = (e χL± xDL−1 ± )γ ± (g,y) f (g
= (Ug± f )(y) , −1 −1 (y))−1 gc(g −1 [ψ± (y)]). where γ ± (g, y) = c± (y)−1 gc± (g −1 [y]) = c(ψ± To conclude the proof of the lemma, observe first that there always exist a unitary operator V such that either B = W+ V or B = W− V , according to the fact that B is unitary or antiunitary. Hence U and U 0 are physically equivalent if and only if U 0 is unitarily equivalent either to U + or to U − . Due to a theorem of Mackey (see, for example, Theorem 6.42 of [16]), this is possible if and only if there exist h ∈ G such that Gx0 = hGx h−1 and a unitary or antiunitary operator M (depending on the fact that B is unitary or antiunitary) such that (x0 D0 )hgh−1 = χ eg M (xD)g M −1 for all g ∈ Gx .
We turn to the proof of Theorem 4. Proof of Theorem 4. 1. Since the semidirect product is regular, a theorem of Mackey (see, for example, Theorem 6.42 of [16]) implies that each irreducible unitary representation of G is unitarily (hence physically) equivalent to one of the (xD0 ) for some x ∈ Aˆ and some irreducible representation D0 of Gx ∩ H. form Ind G Gx exi and, by definition of orbit class, there exist There is an index i such that x ∈ O ˆ y ∈ As and h ∈ G such that x = yh[x ] where = ±1. Hence Gx = hGxi h−1 and i
we can define a representation D of Gxi ∩ H either as Dg = Dh0 −1 gh ,
g ∈ Gxi ∩ H ,
if = 1, or as Dg = M Dh0 −1 gh M −1 ,
g ∈ Gxi ∩ H ,
(xD0 ) if = −1, where M is a fixed antiunitary operator. Then, by Lemma 5, Ind G G x
(xi D). is physically equivalent to Ind G G xi
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2. If Ind G (xi D) and Ind G (xj D0 ) are physically equivalent, the condition (2) G G xi
xj
of Lemma 5 with the choice g = a ∈ A implies that xj = yh[xi ] for some y ∈ Aˆs and = ±1, so that, by definition of xi , i = j. 3. Apply Lemma 5 with x = x0 , taking into account the form of the characters of G. We observe that if D0 is unitarily equivalent to D, the conditions (a) of item 3 of Theorem 4 are satisfied with y = 1, χ e = 1, and h = e and this is exactly the case of unitary equivalence of the induced representations. However, in general, there are other possibilities apart from the unitary equivalence. There are even situations in which both conditions (a) and (b) hold. 4. Examples In this section we give a brief review of the classification of the free quantum particles for the Poincar´e group and for the Galilei group using the framework introduced in the previous section. We consider the case of the Galilei group in 2 + 1 dimensions and we confront our results with the ones obtained by Bose [19]. 4.1. The Poincar´ e group Let G be the connected component of the Poincar´e group, which is the semidirect product of A = R4 and the connected component H of the Lorentz group. The covering group G∗ of G is the semidirect product of A = R4 and SL(2, C). It is a standard result (see, for example, [12]) that each multiplier of G∗ is exact. Hence, the universal central extension of G is its universal covering group and our Theorem 3 reduces, in this case, to Theorem 7.40 of [12]. The classification of relativistic free quantum particles is thus traced back to the problem of classifying the irreducible representations of G∗ . This problem was first solved by Wigner [5], in terms of two parameters: the mass, labelling the orbits in the dual group of A, and the spin, labelling the irreducible representations of the stability group at the origin of each orbit. We add some comments. 1. Since every multiplier of G∗ is exact, we have that K = Ker δ = Z2 . Moreover, G∗ has only the trivial character, since the only singleton orbit in Aˆ is the origin and the semisimple group SL(2, C) has no nontrivial characters. b ' Z2 . Explicitly, any Hence, by Theorem 2, H 2 (G, T) is isomorphic to K T-multiplier of G is either exact or equivalent to (g, g 0 ) 7→ (c(g)c(g 0 )c(gg 0 )−1 ) , where c is a section for the covering homomorphism δ and is the nontrivial character of K.
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2. Since G∗ has only the trivial character, two representations of G∗ are physically equivalent if and only if they are either unitarily or antiunitarily equivalent. 3. For reducible representations the admissibility condition (7) gives rise to the superselection rule that does not allow the superposition among fermions and bosons. 4.2. The Galilei group We discuss this case in more details since it presents a nontrivial application of the notion of universal central extension. Let V := (R3 , +) be the group of velocity transformations and let SO(3) be the rotation group in R3 . The group SO(3) acts on V in a natural way and we can consider the corresponding semidirect product, which is the homogeneous Galilei group, G0 := V ×0 SO(3) . The elements of G0 are denoted by (v, R). Let Ts := (R3 , +) be the group of space translations and Tt := (R, +) the group of time translations; we denote the group of space-time translations by T := Ts × Tt and its elements by (a, b). The action of G0 on T is defined by (v, R)[(a, b)] := (Ra + bv, b) ,
(v, R) ∈ G0 , (a, b) ∈ T ,
and the corresponding semidirect product G := T ×0 G0 is the Galilei group. For any g ∈ G we write g = (a, b, v, R). The covering group of G is G∗ = T ×0 (V ×0 SU (2)) , where SU (2) acts on V and V ×0 SU (2) acts on T in a natural way using the covering homomorphism δ from SU (2) onto SO(3). We denote again by δ the covering homomorphism G∗ → G (this is a small abuse of notation that does not cause any confusion) and we notice that Ker δ = {(0, 0, 0, ±I)}. The corresponding Lie algebra is, as a vector space, Lie (G∗ ) = Lie (T ) ⊕ Lie (V) ⊕ Lie (SU (2)) = R4 ⊕ R3 ⊕ su (2) , and we denote its elements by (a, b, v, A), with b ∈ R, a, v ∈ R3 and A ∈ su(2). We apply the results of Sec. 2.2 to compute the multipliers of G∗ . A classical result of Bargmann [6], shows that H 2 (Lie (G∗ ), R) is one dimensional and a nonexact closed R-form is given by F ((a1 , b1 , v1 , A1 ), (a2 , b2 , v2 , A2 )) = v1 · a2 − v2 · a1 ,
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where · denotes the scalar product on R3 . To compute the R-multiplier τF corresponding to F , we have to exhibit the simply connected Lie group G∗F such that Lie (G∗F ) = R ⊕F Lie (G∗ ) . Denote the elements of Lie (G∗F ) by (c, a, b, v, A). By a direct computation one can confirm that {(v, A) ≡ (0, 0, 0, v, A) : (v, A) ∈ Lie (V) ⊕ su (2)} is a subalgebra of Lie (G∗F ) isomorphic to Lie (V ×0 SU (2)), and that {(c, a, b) ≡ (c, a, b, 0, 0) : (c, a, b) ∈ R ⊕ Lie (T )} is an abelian ideal of Lie (G∗F ) isomorphic to Lie (R × T ). Hence, Lie (G∗F ) is isomorphic to the semidirect sum of Lie (R × T ) and Lie (V ×0 SU (2)). Explicitly, if (v, A) ∈ Lie (V ×0 SU (2)) and (c, a, b) ∈ Lie (R × T ), one has ˙ [(v, A), (c, a, b)] = (v · a, δ(A)a + bv, 0, 0, 0) =: ρ(v, ˙ A)(c, a, b) , with ρ(v, ˙ A) denoting the 5 × 5 real matrix 0 v 0 ˙ ρ(v, ˙ A) = 0 δ(A) v, 0 0 0 which acts on the (column) vector (c, a, b) ∈ Lie (R × T ) ' R × T . Let ρ be the representation of V ×0 SU (2) on R × T such that its differential at the identity is ρ. ˙ Then G∗F is the semidirect product of R × T and V ×0 SU (2) with respect to the action defined by ρ. We denote the elements of G∗F by (c, a, b, v, h) where b, c ∈ R, a, v ∈ R3 and h ∈ SU (2). To compute explicitly ρ, if A ∈ su (2), then ˙ ρ(0, eA ) = eρ(0,A)
=
∞ X 1 ρ(0, ˙ A)n n ! n=0
1
0
0
˙ = 0 eδ(A) 0 . 0 0 1 Thus, for all h ∈ SU (2)
1 0 0 ρ(0, h) = 0 δ(h) 0 . 0 0 1
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In a similar way, if v ∈ V, one gets 1 2 v 2 ρ(v, I) = 0 I v . 0 0 1
1 v
Hence, the action of V ×0 SU (2) on R × T is explicitly given by 1 ρ(v, h)[(c, a, b)] = (c + v · δ(h)a + bv2 , δ(h)a + bv, b) , 2 with (v, h) ∈ V ×0 SU (2) and (c, a, b) ∈ R × T , and the multiplication law in G∗F is 1 g1 g2 = (c1 +c2 +v1 ·δ(h1 )a2 + b2 v2 , a1 +δ(h1 )a2 +b2 v1 , b1 +b2 , v1 +δ(h1 )v2 , h1 h2 ) , 2 for all gi = (ci , ai , bi , vi , hi ) ∈ G∗F , i = 1, 2. The corresponding R-multiplier τF for G∗ is 1 τF (g1∗ , g2∗ ) = v1 · δ(h1 )a2 + b2 v12 g1∗ , g2∗ ∈ G∗ . 2 We notice that the usual way to deduce τF from F is not so direct and requires more computations. It is evident that, if k ∈ Ker δ and g ∗ ∈ G∗ , τF (k, g ∗ ) = τF (g ∗ , k) , so that dim H 2 (G∗ , R)δ = 1. Since τF (k1 , k2 ) = 0 k1 , k2 ∈ Ker δ , we can choose τ = τF , and we have G = G∗F and K = R × Z2 . Since G is the semidirect product of A := R × T and H = V ×0 SU (2), we can use the results of Sec. 3.2 to classify the physically inequivalent irreducible representations of G. We identify the dual group Aˆ of A with R × R3 × R by the pairing h(m, p, E), (c, a, b)i = mc − p · a + Eb . With this identification, the dual action of G on Aˆ is 1 2 g[(m, p, E)] = m, δ(h)p + mv, E + mv + v · δ(h)p , 2 with g = (c, a, b, v, h) ∈ G. With respect to this action Aˆ splits into three kinds of orbits: 1. for each E0 ∈ R, m ∈ R, m 6= 0, p2 3 G [(m, 0, E0 )] = (m, p, E) : p ∈ R , E = E0 + ; 2m
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2. for each r ∈ R, r > 0, G [(0, pr , 0)] = {(0, p, E) : p ∈ R3 , p2 = r2 , E ∈ R} , where pr = (0, 0, r); 3. for each E0 ∈ R, G [(0, 0, E0 )] = {(0, 0, E0 )} . ˆ the semidirect product is regular and we can Since these orbits are closed in A, apply Theorem 4. To do this, we observe that the set of singleton orbits is Aˆs = {(0, 0, E0 ) : E0 ∈ R} , and the orbit classes are the following: 1. for any m > 0, e(m,0,0) = O
[
G [(m, 0, E)] ∪ G [(−m, 0, E)] ;
E∈R
2. for any r > 0, e(0,0,0) = 3. O
S E∈R
e(0,(0,0,r),0) = G [(0, (0, 0, r), 0)] ; O G [(0, 0, E)] .
e(m,0,0) , that have a direct physical interWe consider only the set of orbit classes O pretation. Define, for each m > 0, e(m,0,0) , pm := (m, 0, 0) ∈ O then the stability subgroup at pm is A ×0 SU (2) and the irreducible unitary representations of SU (2) are unitarily equivalent to those of the form Dj acting on the Hilbert space C2j+1 , with 2j ∈ N. Moreover, 1. if y ∈ Aˆs , y 6= 0, then yG [pm ] 6= G [pm ]; 2. G [pm ] 6= G [pm ]−1 3. H has only the character 1, 0 4. the representations Dj and Dj , with j 6= j 0 , act on Hilbert spaces with different dimension, so that they are unitarily inequivalent.
Applying Theorem 4, we conclude that every irreducible representation of G living epm is physically equivalent to an induced representation of the on an orbit class O form j U m,j := Ind G A×0 SU(2) (pm D ) , where the inducing representation pm Dj of A ×0 SU (2) is (a, h) 7→ eihpm , ai Dj (h) .
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Moreover, the set {U m,j : m ∈ R, m > 0, 2j ∈ N} is a family of physically inequivalent irreducible representations of G. Hence, by Theorem 3, the inequivalent irreducible symmetry actions of G are classified by two parameters m > 0 and 2j ∈ N. This result was obtained by Bargmann [6]. We end with some comments. 1. The characters of G∗ are of the form G∗ 3 (a, b, v, h) 7→ eiEb ∈ T , where E ∈ R. When restricted to K, any character is trivial. Hence, by b ' R × Z2 . The elements Theorem 2, the group H 2 (G, T) is isomorphic to K b of K are the maps R × Z2 3 (c, ξ) 7→ eimc (ξ) ∈ T where m ∈ R and is a character of Z2 . If s is a section for δ : SU (2) → SO(3), we have that any T-multiplier of G is equivalent to one of the form 0
1 0
((a, b, v, R), (a0 , b0 , v0 , R0 )) 7→ eim(v·Ra + 2 b v ) (s(R)s(R0 )s(RR0 )−1 ) . 2
2. Let U be an admissible representation and m ∈ R, ∈ Z2 be the corresponding algebraic and topological charges. Then the algebraic charge m parametrises the orbits in the dual group (as in the Poincar´e case) and it has the physical meaning of a mass. On the other hand, the topological charge is connected with the spin of the particles: the case = 1 characterises the bosonic representations, while = −1 corresponds to the fermionic ones. 3. If we consider a direct sum of irreducible representations, the admissibility condition (7) gives rise to two superselection rules. Namely, it does not allow superposition among particles with different masses (Bargmann superselection rule) and superposition among bosons and fermions. 4.3. The Galilei group in 2 + 1 dimensions From the physical point of view, the interest in the Galilei group in 2 + 1 dimensions arises in solid state physics where some genuine examples of two dimensional systems can be found. The analysis of the multipliers of this group has been done by Bose [18]. The classification of the representations of the corresponding central extensions has been done in [19]. In the latter paper no discussion of the physical equivalence is given and this leads to misleading conclusions regarding the spin of elementary particles. For these reasons we consider anew this case here as a nontrivial application of our theory. The Galilei group in 2 + 1 dimensions is G = T ×0 (V ×0 SO(2)) ,
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where T = Ts × Tt , Ts = R2 , Tt = R, and V = R2 . The semidirect product structure is the analogous of the 3 + 1 dimensional case. The covering group is G∗ = T ×0 (V ×0 R) and we denote its elements as (a, b, v, r), where a, v ∈ R2 , b ∈ R and r ∈ R. The kernel of the covering homomorphism δ is {(0, 0, 0, 2πk) : k ∈ Z} . The Lie algebra of G∗ is, as a vector space, Lie (G∗ ) = Lie (T ) ⊕ Lie (V) ⊕ Lie (R) = R3 ⊕ R2 ⊕ R , and we denote its elements by (a, b, v, r), with b, r ∈ R, a, v ∈ R2 . A result of Bose [18], shows that H 2 (Lie (G∗ ), R) is a three dimensional vector space and a basis is given by the equivalence classes of the following closed R-forms: F1 ((a1 , b1 , v1 , r1 ), (a2 , b2 , v2 , r2 )) = r1 b2 − r2 b1 , F2 ((a1 , b1 , v1 , r1 ), (a2 , b2 , v2 , r2 )) = v1 · a2 − v2 · a1 , F3 ((a1 , b1 , v1 , r1 ), (a2 , b2 , v2 , r2 )) = v1 ∧ v2 , where v1 ∧ v2 is a shorthand notation for v1x v2y − v2x v1y . Define F as the closed R3 -form F = (F1 , F2 , F3 ). To compute the corresponding R3 -multiplier τF of G∗ , we have to determine the simply connected Lie group G∗F with Lie algebra Lie (G∗F ) = R3 ⊕F Lie (G∗ ) . The algebra Lie (G∗F ) is, in fact, a semidirect sum. This can be shown as follows. Write Lie (G∗F ) = R2 ⊕Lie (G∗ )⊕R and its elements as (c1 , c2 , X, x) with c1 , c2 , x ∈ R and X ∈ Lie (G∗ ) in such a way that [(c1 , c2 , X, x), (c01 , c02 , X 0 , x0 )] = (F1 (X, X 0 ), F2 (X, X 0 ), [X, X 0 ], F3 (X, X 0 )) . By direct computation, the set {(v, r, x) ≡ (0, 0, 0, 0, v, r, x) : (v, r, x) ∈ Lie (V) ⊕ Lie (R) ⊕ R} is a subalgebra of Lie (G∗F ) with Lie brackets 0 ˙ ˙ 0 )v, 0, v ∧ v0 ) − δ(r [(v, r, x), (v0 , r0 , x0 )] = (δ(r)v
where (v, r, x), (v0 , r0 , x0 ) ∈ Lie (V)⊕Lie (R)⊕R. By this equation, Lie (V)⊕Lie (R)⊕ R is isomorphic to the Lie algebra of the covering group H of the diamond group, i.e. the Lie group H = V × R × R with product (v, r, x)(v0 , r0 , x0 ) = (v + δ(r)v0 , r + r0 , x + x0 + v ∧ δ(r)v0 ) with (v, r, x), (v0 , r0 , x0 ) ∈ H.
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Moreover, the set {(c1 , c2 , a, b) ≡ (c1 , c2 , a, b, 0, 0, 0) : (c1 , c2 , a, b) ∈ R2 ⊕ Lie (T )} is an abelian ideal of Lie (G∗F ) isomorphic to Lie (R2 × T ). Taking into account the previous results and the fact that, as a vector space, Lie (G∗F ) = R2 ⊕ Lie (T ) ⊕ (Lie (V) ⊕ Lie (R) ⊕ R) , then Lie (G∗F ) is isomorphic to the semidirect sum of Lie (R2 × T ) and Lie (H). Explicitly, if (v, r, x) ∈ Lie (H) and (c1 , c2 , a, b) ∈ Lie (R2 × T ) one has ˙ + bv, 0) [(v, r, x), (c1 , c2 , a, b)] = (rb, v · a, δ(r)a =: ρ(v, ˙ r, x)(c1 , c2 , a, b) , where ρ(v, ˙ r, x) is the 5 × 5 matrix
0 0
0 0 ρ(v, ˙ r, x) = 0 0 0 0 0 0
0
r
0 , v
v 0 −r r
0 0
0
which acts on the column vector (c1 , c2 , a, b) ∈ Lie (R2 × T ) ' R2 × T . If ρ is the representation of H such that its differential at the identity is ρ, ˙ ∗ 2 then GF is the semidirect product of R × T and H with respect to ρ. A simple calculation shows that 1 0 0 0 1 2 0 1 v v 2 ρ(v, 0, 0) = 0 0 1 0 v 0 0 0 1 0 0
1 0 ρ(0, r, 0) = 0 0
0
0
0
1
0
0 0
δ(r)
r
1
0 0
0 0
0
1
1 0
0
0
0 1 ρ(0, 0, a) = 0 0 0 0 0 0
0 1 0 0 1 0
0 . 0 1
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Hence the action of H on R2 × T is given by bv2 , δ(r)a + bv, b . (v, r, x)[(c1 , c2 , a, b)] = c1 + br, c2 + v · δ(r)a + 2 If g = (c1 , c2 , a, b, v, r, x) and g 0 = (c01 , c02 , a0 , b0 , v0 , r0 , x0 ) are in G∗F , then b0 v 2 , a + δ(r)a0 + b0 v, b + b0 , gg 0 = c1 + c01 + b0 r, c2 + c02 + v · δ(r)a0 + 2 v + δ(r)v0 , r + r0 , x + x0 + v ∧ δ(r)v0 , so that the explicit form of τF = (τ1 , τ2 , τ3 ) is τ1 (g, g 0 ) = b0 r τ2 (g, g 0 ) = v · δ(r)a0 + b0 v2 /2 τ3 (g, g 0 ) = v ∧ δ(r)v0 . By Theorem 1, the equivalence classes [τ1 ], [τ2 ], [τ3 ] form a basis of H 2 (G∗ , R). Moreover τ2 and τ3 satisfy the condition τi (k, g ∗ ) = τi (g ∗ , k) ,
k ∈ Ker δ, g ∗ ∈ G∗ ,
while τ1 does not. It follows that dim H 2 (G∗ , R)δ = 2 and we can choose τ = (τ2 , τ3 ) (notice that τ (k1 , k2 ) = 0 if k1 , k2 ∈ Ker δ) and the universal central extension G of G can be identified with the semidirect product of the vector group A = R × T and the Lie group H with respect to the action of H on A given by bv2 , δ(r)a + bv, b , (v, r, x)[(c, a, b)] = c + v · δ(r)a + 2 where the elements of A are denoted by (c, a, b), with c ∈ R, a ∈ Ts and b ∈ Tt , and the ones of H by (v, r, x), with x, r ∈ R and v ∈ V. As usual, we denote the elements of G as (c, a, b, v, r, x). Finally, one has that K = {(c, 0, 0, 0, 2πn, x) : c, x ∈ R, n ∈ Z} ' R2 × Z . Since G is a semidirect product we apply the results of Sec. 3.2 to classify the irreducible physically inequivalent representations of G. Let Aˆ be the dual group of A. We identify Aˆ with R4 using the pairing h(m, p, p0 ), (c, a, b)i = mc − p · a + p0 b . The dual action of G on Aˆ is 1 2 g[(m, p, p0 )] = m, δ(r)p + mv, p0 + δ(r)p · v + mv , 2
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where g = (c, a, b, v, r, x) ∈ G. We have the following orbits for the dual action. 1. For each l ∈ R, l > 0, G [(0, pl , 0)] = {(0, p, p0 ) : p2 = l2 } , where pl = (0, l). 2. For each E ∈ R, G [(0, 0, E)] = {(0, 0, E)} . 3. For each m ∈ R, E ∈ R, m 6= 0, p2 =E . G [(m, 0, E)] = (m, p, p0 ) : p0 − 2m ˆ hence the semidirect product is regular and Theorem 4 All the orbits are closed in A, holds. The set of singleton orbits is Aˆs = {(0, 0, E) : E ∈ R} , and the orbit classes of G are the following: 1. for each l ∈ R, l > 0,
e(0,p ,0) = G [(0, pl , 0)] ; O l e(0,0,0) = O
2.
[
G [(0, 0, E)] ;
E∈R
3. for any m > 0, e(m,0,0) = O
[
G [(m, 0, E)] ∪ G[(−m, 0, E)] .
E∈R
In the sequel we will exploit in detail only the third case, which presents some interesting physical features. e(m,0,0) . We have that Let m > 0 and pm = (m, 0, 0) ∈ O Gpm ∩ H = {(v, r, x) ∈ H : v = 0} is isomorphic to R2 and its irreducible representations are its characters. Explicitly, λ, µ ∈ R define the character of Gpm ∩ H (0, r, x) 7→ eiλx eiµr . Now we observe that 1. if y ∈ Aˆs , y 6= 0, then yG [pm ] 6= G [pm ]; 2. G [pm ] 6= G [pm ]−1 ; 3. the characters of H are of the form (v, r, x) 7→ eiµr .
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According to Theorem 4, every irreducible representation of G living on an orbit epm is equivalent to one of the form U m,λ = IndG (Dm,λ ) where Dm,λ is class O G pm
the representation of Gpm (c, a, b, 0, r, x) 7→ ei(mc+λx) . Moreover, the set {U m,λ : m, λ ∈ R, m > 0} is a family of physically inequivalent representations of G. To compute explicitly U m,λ , we observe that the orbit p2 =0 G[pm ] = (m, p, p0 ) : p0 − 2m can be identified with R2 using the map p2 2 ∈ G[pm ] . R 3 p ←→ m, p, 2m With respect to this identification the action of G on the orbit becomes (c, a, b, v, r, x)[p] = δ(r)p + mv so that the Lebesgue measure dp on R2 is G-invariant. We consider the section β : R2 → G p p 7→ 0, 0, 0, , 0, 0 m for the action of G on R2 . The representation U m,λ of G acts in L2 (R2 , dp) as 2 b 1 m,λ f (p) = ei( 2m p −p·a+mc) eiλ(x+ m v∧p) f (δ(−r)(p − mv)) . U(c,a,b,v,r,x) From the explicit form of U m,λ one readily gets that the angular momentum, i.e. the selfadjoint operator that generates the 1-parameter subgroup of rotations, has only the orbital part, so that the elementary particles in 2 + 1 dimensions have no spin. However, they acquire a new charge λ which is not of a space-time origin, but arises from the structure of the multipliers. If λ 6= 0, the two generators of velocity transformations do not commute. We add some final comments. 1. The characters of G∗ are G∗ 3 (a, b, v, r) 7→ eiEb eiµr ∈ T , where E, µ ∈ R. The set V of characters of K that extend to G∗ is V = {(c, 0, 0, 0, 2πn, x) 7→ z n : z ∈ T} ' T .
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b = R2 × T and K0 = R2 . Applying The group V is a closed subgroup of K 2 Corollary 1, H (G, T) is isomorphic to R2 . In particular, any T-multiplier of G is equivalent to one of the form 0
1 0
0
((a, b, v, R), (a0 , b0 , v0 , R0 )) 7→ eim(v·Ra + 2 b v ) eiλ(v∧Rv ) 2
where (m, λ) ∈ R2 . 2. From the explicit form of the characters of G∗ one has that, for all E, µ ∈ R, the representation m,λ (c, a, b, v, r, x) 7→ ei(Eb+µr) U(c,a,b,v,r,x)
is physically equivalent to U m,λ . Hence the angular momentum and the energy are both defined up to an additive constant. For the energy this phenomenon is well known in 3 + 1 dimensions, while it does not occur for the angular momentum. 3. The admissibility condition (7) gives rise to two superselection rules that do not allow superposition among states with different mass m and among states with different charge λ. However, there is no superselection rule connected with the spin. References [1] E. P. Wigner, Group Theory and Its Application to the Quantum Theory of Atomic Spectra, Academic Press Inc., New York, 1959, pp. 233-236. [2] I. Schur, J. Reine Angew. Math. 127 (1904) 20–50. [3] I. Schur, J. Reine Angew. Math. 132 (1906) 85–137. ´ ements de la th´eorie des repr´esentations, Editions ´ [4] A. Kirillov, El´ MIR, Moscou, 1974. [5] E. P. Wigner, Ann. Math. 40 (1939) 149–204. [6] V. Bargmann, Ann. Math. 59 (1954) 1–46. [7] G. W. Mackey, Acta Math. 99 (1958) 265–311. [8] C. C. Moore, Trans. Amer. Math. Soc. 113 (1964) 40–63. [9] C. C. Moore, Trans. Amer. Math. Soc. 113 (1964) 64–86. [10] C. C. Moore, Trans. Amer. Math. Soc. 221 (1976) 1–33. [11] C. C. Moore, Trans. Amer. Math. Soc. 221 (1976) 35–58. [12] V. S. Varadarajan, Geometry of Quantum Theory, Second ed., Springer-Verlag, Berlin, 1985. [13] P. T. Divakaran, Rev. Math. Phys. 6 (1994) 167–205. [14] G. Cassinelli, E. De Vito, P. Lahti and A. Levrero, Rev. Math. Phys. 9 (1997) 921. [15] J. Braconnier, J. Math. Pures Appl. 27 (1948) 1–85. [16] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995. [17] G. W. Mackey, Ann. Math. 55 (1952) 101–139. [18] S. K. Bose, Commun. Math. Phys. 169 (1995) 385–395. [19] S. K. Bose, J. Math. Phys. 36 (1995) 875–890.
STABILITY AND INSTABILITY OF THE WAVE EQUATION SOLUTIONS IN A PULSATING DOMAIN J. DITTRICH∗ Nuclear Physics Institute, Academy of Sciences of the Czech Republic, CZ-250 68 Rez, Czech Republic E-mail: [email protected]
P. DUCLOS and N. GONZALEZ† Centre de Physique Th´ eorique‡ CNRS Luminy, Case 907, F-13288 Marseille - Cedex 9, France and PhyMat, Universit´ e de Toulon et du Var, La Garde, France E-mail: [email protected] Received 17 June 1997 Revised 8 December 1997 The behavior of energy is studied for the real scalar field satisfying d’Alembert equation in a finite space interval 0 < x < a(t); the endpoint a(t) is assumed to move slower than the light and periodically in most parts of the paper. The boundary conditions are of Dirichlet and Neumann type. We give sufficient conditions for the unlimited growth, the boundedness and the periodicity of the energy E. The case of unbounded energy without infinite limit (0 < lim inf t→+∞ E(t) < lim supt→+∞ E(t) = +∞) is also possible. For the Neumann boundary condition, E may decay to zero as the time tends to infinity. If a is periodic, the solution is determined by a homeomorphism F¯ of the circle related to a. The behavior of E depends essentially on the number theoretical characteristics of the rotation number of F¯ .
Contents 1. Introduction 2. Preliminaries 2.1. Notations, definitions and preliminary results 2.2. Existence and unicity of the solution 2.3. Some useful lemmas 3. Dirichlet Problem 3.1. Stability 3.1.1. A universal lower bound 3.1.2. Periodicity 3.1.3. Absence of strong instability 3.1.4. A sufficient condition of stability ∗ Also
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member of the Doppler Institute of Mathematical Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague, Czech Republic. † Also Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Prague, Czech Republic. ‡ Unite Propre de Recherche 7061.
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3.2. Instability 3.2.1. A universal upper bound 3.2.2. A sufficient condition of instability 3.2.3. A sufficient condition of strong instability 3.2.4. Asymptotics 3.2.5. Instability: strong instability is not the rule 3.2.6. Perturbation of the boundary 4. Neumann Problem 4.1. Stability 4.1.1. An upper bound 4.1.2. Periodicity 4.1.3. Asymptotics 4.1.4. Absence of strong instability 4.1.5. A sufficient condition of stability 4.2. Instability 4.2.1. Universal lower and upper bounds 4.2.2. Singular initial data 4.2.3. Sufficient conditions for the decay of the energy A. Appendix B. Glossary
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1. Introduction The study of the so-called Fermi accelerators becomes more and more extensive. The name comes from Fermi’s considerations on the possible mechanism of cosmic rays acceleration [1]. In the later studies up to contemporary ones, they serve as simple prototypes of the externally driven dynamical systems, mainly in the connection with the deterministic and chaotic behavior of the classical and quantum systems. The first mechanical models were proposed by Ulam [2], the rigorous results in Newtonian mechanics (Pustyl’nikov [3, 4]) and in special-relativistic classical mechanics (Pustyl’nikov [5, 6, 4]) were obtained much later. Only as a sample ˇ of papers in nonrelativistic quantum mechanics let us mention Karner [7], Seba [8], Dembi´ nski, Makowski and Peplowski [9], Dodonov, Klimov and Nikonov [10], ˇˇtov´ıˇcek [11]. Similar problems for classical wave equation (Balazs [12], Duclos and S ˇ Cooper [13], Dittrich, Duclos and Seba [14], Cooper and Koch [15], M´eplan and Gignoux [16]) and Maxwell equations (Cooper [17]) were also considered. An analogous model in quantum field theory was treated, for example, by Moore [18], Calucci [19], Dodonov, Klimov and Nikonov [20], Johnston and Sarkar [21]. In the present paper, we continue and extend the study for the classical d’Alembert equation. Let us consider the one-dimensional wave equation in a domain with one spatial boundary fixed and the second one moving slower than the wave velocity. Let us assume that the boundary motion is described by a Lipschitz continuous function a and assume that the field satisfies either Dirichlet or Neumann boundary conditions. We describe the behavior of the energy E of the field in more details and for a wider class of functions a than the papers [13, 14, 15, 16] which treat only the case a ∈ C k (R) (k ≥ 2) and a periodic. The key to the results is that the orbits of the characteristics of the wave equation are given by a Lipschitz homeomorphism F of R which depends only on a and becomes the lift of a homeomorphism of the circle when a is periodic. According to the arithmetic properties of the rotation number of F and to the regularity of a,
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E will behave differently. It is worth remarking that our results are nonperturbative (in other words there are no assumptions on the smallness of time variations of a). Section 2 is devoted to the formulation of the problems (Dirichlet and Neumann), the link between a and F , the existence and unicity of a weak solution in the space of finite energy fields. Section 3 is divided into two subsections: Stability (E is bounded from below and above) and Instability (E is unbounded), both for the Dirichlet problem. In Subsec. 3.1, we give a universal lower bound (a needs just be bounded), a necessary and sufficient condition of periodicity and sufficient conditions for the existence of an upper bound. In Subsec. 3.2, we distinguish two types of instability: a weaker one (the limit superior is infinite) and a stronger one (the limit inferior is infinite). We first give a sufficient condition of instability and we state a sufficient condition of strong instability. The proof introduces the condition for occurence of “wandering characteristics” and therefore generalizes the condition of “periodic characteristics” given by Cooper [13]. We compute an asymptotics which shows the exponential increase of E due to the presence of periodic characteristics; in this case, under any small periodic perturbation of the boundary motion a, E keeps increasing exponentially. We also show on examples that E may have an infinite limit superior but a finite limit inferior in the case where there are no periodic characteristics and the boundary a is smooth; this answers negatively to a conjecture of Cooper [17]. Cooper and Koch [15] already used diffeomorphisms of the circle in the Dirichlet problem and they show that the spectrum of the evolution operator on one period depends on the rotation number. Section 4 is devoted to the Neumann problem. To our knowledge, this is the first treatment in the literature. As we shall see, E behaves in a completely different manner. In Subsec. 4.1, we prove that E is universally bounded from above except if the initial conditions are singular (in which case we show that E may diverge exponentially). The periodicity and lower bounds are proven like for the Dirichlet problem. We also propose conditions for an asymptotically periodic energy and give explicit asymptotics for this case. In Subsec. 4.2, we show that for appropriate initial conditions E decays (and even exponentially fast) to 0. Let a be a strictly positive real function to be precised later. The problems we consider are the Dirichlet problem: ϕtt − ϕxx = 0 ,
t ∈ R,
0 < x < a(t) ,
(1)
ϕ(x, 0) = ϕ0 (x) ,
0 < x < a(0) ,
(2)
ϕt (x, 0) = ϕ1 (x) ,
0 < x < a(0) ,
(3)
ϕ(0, t) = 0 ,
t ∈ R,
(4)
ϕ(a(t), t) = 0 ,
t ∈ R,
(5)
and the Neumann problem for which ϕx (0, t) = 0 , ϕx (a(t), t) = 0 ,
t ∈ R,
(6)
t∈R
(7)
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are required instead of (4)–(5). Since it will not play a role in the mathematical analysis, the wave velocity of the field ϕ is normalized to 1. In addition, if a is periodic, then by a rescaling in the parameters, one can also take the period equal to 1; this will simplify our notations. The energy of the field ϕ is given by the standard expression E(t) :=
1 2
Z
a(t)
(|ϕt (x, t)|2 + |ϕx (x, t)|2 ) dx .
(8)
0
2. Preliminaries 2.1. Notations, definitions and preliminary results We start with some notations and known results. Not to be very formal in trivialities, some equivalent spaces of functions (like functions on a circle and their liftings on a line) are indentified in our notation explained below. Let X be either the set Z (the integers) or N (the nonnegative integers) or Q (the rational numbers) or R (the real numbers). Then X ∗ := X \ {0}, X+ := {x ∈ ∗ := X ∗ ∩ X+ . For any set X, we denote by X˙ or Int X its X; x ≥ 0} and X+ interior. Denote by T the 1-dimensional torus (the circle of unit length) and by X either T or R. Let C 0 (X) be the space of the continuous functions and let C k (X), k ∈ N∗ , be the space of the k-times continuously differentiable functions; the kth derivative of a function F is denoted by F (k) or Dk F . Denote by C k (T) the set of 1-periodic and k-times continuously differentiable functions on R. One defines the norms kF k0 := supx∈T |F (x)| on C 0 (T) and kF kk := max0≤i≤k kDi F k0 on C k (T) for finite k ∈ N. By C ∞ (X) (resp. C ω (X)) one denotes the space of infinitely differentiable (resp. R-analytic) functions on X. For a measurable function F : X → R, we shall denote by Fmin and Fmax its essential infimum and its essential older continuous functions supremum respectively. Let Lipβ (X) be the space of H¨ with exponent β ∈ (0, 1]. By definition, if β ∈ (0, 1), C β (X) := Lipβ (X). If β = 1, Lip(X) := Lip1 (X) is the set of Lipschitz functions. We shall denote (y) the Lipschitz constant of a function F by L(F ) := supx,y∈X,x6=y F (x)−F . Let x−y π : R → T, x 7→ x + Z be the canonical projection. For any continuous map F¯ : T → T, the function F satisfying F¯ ◦ π = π ◦ F is called a lift of F¯ to R. Denote by Diffk (R) the C k -diffeomorphisms on R. Let [x] be the integer part of a real number x. One calls Dk (T), k ∈ R+ ∪ {+∞, ω}, the set of lifts of the orientationpreserving C k -diffeomorphisms of T, i.e. Dk (T) := {F ∈ Diff[k] (R); F − Id ∈ C ◦ (T), D[k] (F −Id) ∈ C k−[k] (T), D[k] (F −1 −Id) ∈ C k−[k] (T)}. One calls F ∈ D0 (T) a Lipschitz homeomorphism if F and F −1 are Lipschitz continuous. Recall that a 1periodic function F : R → R is of bounded variation on T if F is of bounded variation on [0, 1] and one denotes by BV(T) the set of functions of bounded variation on T. For any F ∈ D0 (T), the rotation number ρ(F ) is defined by F n (x) − x , n→+∞ n
ρ(F ) := lim
x ∈ R,
(9)
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where F n := F ◦ F ◦ . . . ◦ F is the nth iterate of F . In Herman [22, Prop. II.2.3, p. 20], the limit (9) is proven to exist (it is a real number independent of x) and to be uniform w.r.t. x. If in the sequel ρ(F ) = pq for p ∈ Z, q ∈ N∗ , it is always assumed that p and q are relatively primes. Definition 2.1. A point x0 ∈ R is said to be a periodic point of period q ∈ N∗ of F ∈ D0 (T) if there exists p ∈ N such that F q (x0 ) = x0 + p. The point x0 is said to be attracting if there exists a neighborhood U of x0 such that for all x ∈ U , F nq (x) − np tends to x0 as n tends to +∞. If x0 is an attracting periodic point of F −1 , then x0 is called a repelling periodic point of F . F ∈ D1 (T) is said to be a Morse–Smale diffeomorphism if F has a finite nonzero number of periodic points ak (with period q ∈ N∗ ), all of them hyperbolic (i.e. either DF q (ak ) < 1 and in this case the point is attracting or DF q (ak ) > 1 and in this case the point is repelling). One can show (e.g. Herman [22, Prop. II.5.3, p. 24]) that the existence of a periodic point x0 for F ∈ D0 (T), F q (x0 ) = x0 + p, is equivalent to ρ(F ) = pq ∈ Q. It means that if the rotation number is irrational then there are no periodic points. The minimal assumptions that we shall make on the moving boundary described by a are the following: Assumption 2.2. The function a is strictly positive, Lipschitz continuous on R with L(a) ∈ [0, 1). Other assumptions will be always given explicitly. In particular if a is 1-periodic, then a ∈ Lip(T). Let Id be the identity on R. Define h := Id − a, k := Id + a on R. Under Assumption 2.2, it is easy to see that h, k, h−1 , k −1 are Lipschitz homeomorphisms 1 . Hence k ◦ h−1 , h ◦ k −1 on R and L(k), L(h) ≤ 1 + L(a), L(h−1 ), L(k −1 ) ≤ 1−L(a) are also Lipschitz homeomorphisms on R and L(k ◦ h−1 ), L(h ◦ k −1 ) ≤ 1+L(a) 1−L(a) . If moreover a is 1-periodic, then h, k, k ◦ h−1 , h ◦ k −1 ∈ D0 (T). The proof of the following lemma is left to the reader. L(a) ∈ [0, 1)} Lemma 2.3. The sets An := {a ∈ C n (T); a > 0, a ∈ Lip(T), and Fn := F ∈ Dn (T); F > Id, F ∈ Lip(R), F −1 ∈ Lip(R) equipped with C n topologies are homeomorphic for n ∈ N ∪ {+∞, ω}. If n ≥ 1, An and Fn are open subsets of C n (T) and Dn (T) respectively. More generally there exists a bijection between the set of functions a satisfying Assumption 2.2 and the set of Lipschitz homeomorphisms F on R with F > Id. We will need the relations: F := (Id + a) ◦ (Id − a)−1 = Id + 2a ◦ (Id − a)−1 , −1 F + Id F − Id . ◦ a = 2 2
(10) (11)
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In the sequel there is a deep interplay between the invariant measures of diffeomorphisms of the circle and the physics of the problems. Let us remind some definitions. Let X be a compact metric space (for instance, X = T) and F : X → X be a continuous map. The measure µ is said to be an invariantmeasure of F iff µ 0 belongs to the set of probability measures on X (i.e. µ ∈ C 0 (X) the dual space of C 0 (X), µ ≥ 0 and µ(X) = 1) and for every µ-measurable set A, µ(F −1 (A)) = µ(A). According to Krylov and Bogolyubov’s theorem (e.g. Katok and Hasselblatt [23, Theorem. 4.1.1, p. 135]), for any continuous map of X, there exists at least one invariant measure. For the particular case of F ∈ D0 (T): if ρ(F ) ∈ R \ Q, the invariant measure µ is unique (F is said uniquely ergodic, see Herman [22, Prop. II.8.5, p. 28]). If ρ(F ) ∈ Q, the invariant measure is in general not unique and it may be atomic. As an illustration we prove the following lemma: Lemma 2.4. Let a ∈ A0 and F defined by (10). Let p, q ∈ N∗ , x0 ∈ R. Then F q (x0 ) = x0 + p ⇔
q−1 X
a ◦ h−1 ◦ F k (x0 ) =
k=0
p . 2
(12)
Proof. Assume that F q (x0 ) = x0 + p. Thus ρ(F ) = pq . Clearly 1X δF¯ k (x0 ) q q−1
µ :=
k=0
is an invariant probability measure for F¯ (δF¯ k (x0 ) is the Dirac measure at F¯ k (x0 )). Let ψ := F − Id ∈ C 0 (T). By formula (10), ψ = 2a ◦ h−1 . According to Herman [22, Prop. II.2.3, p. 20], Z ψ dµ ,
ρ(F ) = T
Z p = 2 a ◦ h−1 dµ . (13) q T Pq−1 Inserting µ into (13), the equality k=0 a ◦ h−1 ◦ F k (x0 ) = p2 is proven. Conversely, assume the formula on the right-hand side of the equivalence holds Pn−1 and let ψ := F − Id ∈ C 0 (T). Then F n = Id + k=0 ψ ◦ F k , so so that
F q (x0 ) = x0 + 2
q−1 X
a ◦ h−1 ◦ F k (x0 ) = x0 + p .
k=0
Remark 2.5. Let x0 ∈ I0 := [−a(0), a(0)). Then {(a ◦ h−1 ◦ F n (x0 ), h−1 ◦ F n (x0 )); n ∈ N} is the set of intersections between the characteristic t + x = x0 , reflecting against the boundaries, and the moving boundary in the (x, t) plane. The physical explanation of Lemma 2.4 is the following: the relation (12) expresses the existence of a periodic characteristic of period p after q bounces.
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In order to apply Denjoy’s theorem we will use the functions of class P (Herman [22, Def. VI.4.1, p. 74]). Definition 2.6. Let F ∈ D0 (T). Then F is of class P if F is differentiable up to a countable set (u.c.s.) and if its derivative is equal u.c.s. to a (1-periodic) function which is bounded from below by a positive number and is of bounded variation on T. Let us give a slight generalisation of the variation of a function and define a suitable class of functions a. Definition 2.7. Let f be a function defined u.c.s. in T. One denotes d ) := inf {Var(b); b : T → R, b(x) = f (x) u.c.s. in T} . Var(f A function a is said to be of class Q if a ∈ A0 , a is differentiable u.c.s. in T and d 0 ) < +∞. Var(a A link between the classes P and Q is established in the following lemma. Lemma 2.8. If a is of class Q, then F := (Id + a) ◦ (Id − a)−1 is of class P and 2 d 0) . d 0) ≤ Var(a Var(F (1 − L(a))2 Proof. For any > 0 there exists b1 : T → R such that b1 = a0 u.c.s. in T and d 0 ) + . Defining Var(b1 ) < Var(a L(a) if b1 (x) > L(a) ∀x ∈ T, b(x) := b1 (x) if −L(a) ≤ b1 (x) ≤ L(a) −L(a) if b1 (x) < −L(a) , d 0 ) + . We have F 0 = 1+a00 ◦h−1 then b = a0 u.c.s. in T and Var(b) ≤ Var(b1 ) < Var(a 1−a ◦h−1 exists u.c.s. and is bounded in the L∞ -norm. Since h−1 ∈ D0 (T), b ◦ h−1 ∈ BV(T) 1+b◦h−1 0 u.c.s. and and Var(b ◦ h−1 ) = Var(b). Let G := 1−b◦h −1 . Then G = F n X 1 + b ◦ h−1 (tk ) 1 + b ◦ h−1 (tk−1 ) Var(G) = sup 1 − b ◦ h−1 (tk ) − 1 − b ◦ h−1 (tk−1 ) k=1
= sup
n X k=1
2|b ◦ h−1 (tk ) − b ◦ h−1 (tk−1 )| |1 − b ◦ h−1 (tk )| · |1 − b ◦ h−1 (tk−1 )|
≤
2 Var(b ◦ h−1 ) (1 − L(a))2
=
2 2 d 0) + Var(a Var(b) < (1 − L(a))2 (1 − L(a))2
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where the supremum is taken over all 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn ≤ 1. Since this inequality holds for any > 0, d 0 ) ≤ Var(G) ≤ Var(F
2 d 0) . Var(a (1 − L(a))2
We recall that F := k ◦ h−1 and we define ∀n ∈ Z,
In := [F n (−a(0)), F n (a(0))) .
(14)
For convenience, denote xn := F n (a(0)), for all n ∈ Z. Since F (−a(0)) = a(0), In = [xn−1 , xn ). If a is bounded from below and from above, then ∀n ∈ Z,
2amin ≤ xn+1 − xn ≤ 2amax .
It can be easily seen that [ In = R
and ∀n 6= m, Im ∩ In = ∅ .
(15)
(16)
n∈Z
Clearly {In }n∈Z is a partition of R. The aim of the present paper is to study the energy E of the field: 1 2 2 kϕx (·, t)k(0,a(t)) + kϕt (·, t)k(0,a(t)) , E(t) = 2 where we denote by k · kX the L2 -norm on the measurable set X with respect to the Lebesgue measure m. It will appear useful to give the following definition. Definition 2.9. We shall say that the model is stable if E is bounded from below and from above by strictly positive constants. Equivalently the model is unstable if the limit superior (resp. limit inferior) of E is infinite (resp. zero). Moreover if the limit of E exists and is infinite or zero, then the model is said strongly unstable. Let Σ ⊂ Rn , n ≥ 1, be an open set. Let D(Σ) := C0∞ (Σ) and D0 (Σ) be its dual. 1 1 (Σ) and H0,loc (Σ) We will use also the standard Sobolev spaces H 1 (Σ), H01 (Σ), Hloc n 1 defined as the space of distributions φ such that for all ψ ∈ D(R ), ψφ ∈ H0 (Σ). 1 (Ω) is Definition 2.10. Let Ω := {(x, t) ∈ R2 ; 0 < x < a(t)}. Then ϕ ∈ Hloc called a weak solution of (1), (4)–(5) or of (1), (6)–(7) if
∀ψ ∈ D(Ω),
(ϕt , ψt )Ω − (ϕx , ψx )Ω = 0
(17)
(or simply ϕtt − ϕxx = 0 in D0 (Ω)) and the boundary conditions (4)–(5) or (6)–(7) are satisfied. Remark 2.11. The traces needed for the initial and boundary conditions do 1 (Ω). However they exist if ϕ satisfies (17); see not exist for general ϕ in Hloc Theorems 2.12 and 2.13 below.
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2.2. Existence and unicity of the solution We were not able to find in the literature a proof of the existence and unicity of a solution with finite energy to our problems when the boundary a is not smooth. Indeed the standard technique consists in transforming the problems into new ones with fixed boundaries and then to apply the general theory of hyperbolic partial differential equations with variable coefficients (cf. Ladyzhenskaya [25], Lions and Magenes [26], etc.). But this requires roughly that a is C 2 . That is why we give in details the proof for more general a. Theorem 2.12 (Dirichlet problem). If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0 and (ϕ0 , ϕ1 ) ∈ H01 ((0, a(0))) × L2 ((0, a(0))) , then there exists a unique weak solution ϕ of the Dirichlet problem satisfying the 1 (R) ∩ L∞ (R) such that initial conditions (2)–(3). Moreover there exists f ∈ Hloc ϕ(x, t) = f (t + x) − f (t − x)
a.e. in Ω
(18)
1 (Ω). and ϕ ∈ L∞ (Ω) ∩ H0,loc
Proof. First step: the form of general solution and the regularity of traces. By 1 (R) the assumptions on Ω, if a weak solution ϕ exists, then there exist f , g ∈ Hloc such that ϕ(x, t) = f (t + x) + g(t − x) a.e. in Ω. Moreover ϕ can be extended ¯ (so that the last relation holds in Ω). ¯ continuously to Ω The proof is given in appendix (Lemma B). Functions of such a form have traces on the boundaries of Ω and on the intersections of Ω with lines t = constant which are given by continuous 1 (∂Ω) and in H 1 ((0, a(t))) respectively. extension; moreover these traces are in Hloc Second step: the fixed boundary. Since ϕ has a trace ϕ(0, t) = f (t) + g(−t) on x = 0, ϕ(0, t) = 0 implies that g = −f . Then ϕ(x, t) = f (t + x) − f (t − x) a.e. in Ω. Third step: the initial conditions. Since ϕ has a trace on t = 0 given by ϕ(x, 0) = f (x) − f (−x) and ϕt (x, 0) = f 0 (x) − f 0 (−x) a.e. in (0, a(0)), the initial conditions (2)–(3) give Z x 1 ϕ1 (y) dy , (19) ϕ0 (x) + ∀x ∈ [0, a(0)], f (x) := f (0) + 2 0 Z x 1 ϕ1 (y) dy . (20) −ϕ0 (x) + f (−x) := f (0) + 2 0 The constant f (0) is arbitrary. Clearly f ∈ H 1 (I0 ), since ϕ0 (0) = 0 by assumption. Fourth step: the moving boundary. The function ϕ has a trace on x = a(t) and ϕ(a(t), t) = 0; this implies that f ◦F = f
on R
with F := (Id + a) ◦ (Id − a)−1 .
(21)
Fifth step: construction of a solution. By iteration of the formula (21): ∀n ∈ Z, f ◦ F n = f . Thus the function f is known on In for all n ∈ Z, and f ∈ C 0 (I˙n ) for
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all n ∈ Z. Since ϕ0 (a(0)) = 0 and F (−a(0)) = a(0), f is continuous at ±a(0), hence f ∈ C 0 (R) by iterations. Since F is Lipschitz continuous, f ∈ H 1 (In ) for all n ∈ Z, 1 1 (R) and ϕ ∈ H0,loc (Ω). Note that kf kL∞(R) = kf kL∞ (I0 ) < +∞. thus f ∈ Hloc Obviously ϕ ∈ L∞ (Ω), since kϕkL∞ (Ω) ≤ 2kf kL∞(I0 ) < +∞. Sixth step: unicity. Up to a constant, f is uniquely determined on I0 and by (21), f is then unique on R. Thus ϕ is unique, since the difference f (t+ x)− f (t− x) is independent of the constant f (0). Theorem 2.13 (Neumann problem). If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0 and (ϕ0 , ϕ1 ) ∈ H 1 ((0, a(0))) × L2 ((0, a(0))) , then there exists a unique weak solution ϕ of the Neumann problem satisfying the 1 (R) such that initial conditions (2)–(3). Moreover there exists f ∈ Hloc ϕ(x, t) = f (t + x) + f (t − x) a.e. in Ω .
(22)
The proof, very similar to that for the Dirichlet problem, is given in appendix. One just needs to notice that the moving boundary condition yields the essential functional equation (23) f 0 ◦ F = f 0 a.e. in R . In the sequel the initial conditions are assumed to satisfy the requirements for the existence and unicity of the weak solution if other assumptions are not mentioned. Let us give also conditions for the existence of classical C 2 -solutions. The proof is left to the reader. Theorem 2.14. Let a ∈ C 2 (R), |a0 | < 1, a > 0 and (ϕ0 , ϕ1 ) ∈ C 2 ([0, a(0)]) × C ([0, a(0)]). If 1
ϕ0 (0) = 0, ϕ1 (0) = 0,
ϕ0 (a(0)) = 0,
ϕ000 (0) = 0 ,
ϕ1 (a(0)) + a0 (0)ϕ00 (a(0)) = 0 ,
(1 + a02 (0))ϕ000 (a(0)) + 2a0 (0)ϕ01 (a(0)) + a00 (0)ϕ00 (a(0)) = 0 , then there exists a unique classical C 2 -solution of the Eqs. (1)–(5). If ϕ00 (0) = 0 ,
ϕ00 (a(0)) = 0 ,
ϕ01 (0) = 0 ,
ϕ01 (a(0)) + a0 (0)ϕ000 (a(0)) = 0 ,
then there exists a unique classical C 2 -solution of the Eqs. (1)–(3), (6)–(7). 2.3. Some useful lemmas We start by some considerations on the energy E. Lemma 2.15. Under the assumptions of Theorem 2.12 (Dirichlet problem) or Theorem 2.13 (Neumann problem) the following assertions hold:
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∀t ∈ R, E(t) = kf 0 k2(h(t),k(t)) < +∞. If E(0) = 0, then for all t ∈ R, E(t) = 0. E is absolutely continuous on R. Finally, E 0 = −2a0
k0 0 |f ◦ k|2 h0
E 0 = 2a0 |f 0 ◦ k|2
a.e. in R
a.e. in R
(Dirichlet problem),
(Neumann problem).
(24) (25)
Proof. Let us show 1) for the Dirichlet problem (the proof is similar for the Neumann problem). Since ϕ(x, t) = f (t + x) − f (t − x) a.e. in Ω, 1 kϕx (·, t)k2(0,a(t)) + kϕt (·, t)k2(0,a(t)) 2 1 0 2 2 kf (t + ·) + f 0 (t − ·)k(0,a(t)) + kf 0 (t + ·) − f 0 (t − ·)k(0,a(t)) = 2
E(t) =
= kf 0 k(h(t),k(t)) . 2
For all t ∈ R, k(t) − h(t) = 2a(t) is finite, since a is continuous. The function f 0 is in L2loc(R), therefore E(t) < +∞ for all t ∈ R. This proves 1). If E(0) = 0, then f 0 (x) = 0 a.e. on I0 . By the relations (21) or (23), f 0 (x) = 0 a.e. in R and 2) is proved. Part 3) follows from 1), the Lipschitz property of h and k and the fact that 0 f ∈ L2loc (R). Then the differentiation of E (the derivatives exist a.e. by 3)) and the use of the relations (21) (Dirichlet problem) or (23) (Neumann problem) give 4). Remark 2.16. Statement 4) of Lemma 2.15 says that E and a vary in opposite sense in the Dirichlet problem and in the same sense in the Neumann problem. This is a manifestation of the Doppler effect. In Lemma 2.17 we give a useful formula for E. We recall that the intervals {In }n∈Z are defined by (14). Lemma 2.17. Let a ∈ Lip(R), L(a) ∈ [0, 1), a > 0. Then for all t in R, there exists a unique n(t) in Z such that h(t) ∈ In(t) . This defines a function n : R → Z with n(0) = 0. For the Dirichlet problem there exists a function αD such that ∀t ∈ R,
2
f0
E(t) = αD (t) √
n(t) DF I0
(26)
and for the Neumann problem there exists a function αN such that ∀t ∈ R,
p
2
E(t) = αN (t) f 0 DF n(t) . I0
(27)
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Moreover αD (0) = αN (0) = 1 and 1 1 ≤ αD (t) ≤ 0 , 0 Fmax Fmin
∀t ∈ R,
0 0 Fmin ≤ αN (t) ≤ Fmax .
If in addition a is 1-periodic, then ∀t ∈ R∗ ,
a(t) 1 + a(0) n(t) 1 a(t) 1 + a(0) 1 − − < < − + . ρ(F ) ρ(F )t ρ(F )|t| t ρ(F ) ρ(F )t ρ(F )|t|
(28)
Proof. The first statement about the existence of the function n is true since {In }n∈Z is a partition of R. Assume E(0) > 0 (the opposite case is trivial), then f 0 6= 0 (as a function in
0 2
f 2 , since In is mapped by F −n onto L2 (I0 )) and for all n ∈ Z, kf 0 kIn = √DF n I0
I0 and DF n · f 0 ◦ F n = f 0 a.e. on R. Therefore by Lemma 2.15, kf 0 kIn > 0. It is well defined. To find the bounds on αD we use follows that αD (t) := kf E(t) 0 k2 In(t)
again Lemma 2.15 and the relation: ∀t ∈ R, kf 0 k(h(t),k(t)) = kf 0 k(h(t),xn(t) ) + kf 0 k(xn(t) ,k(t)) 2
2
=
2
2 kf 0 k(h(t),xn(t) )
0 2
f
+
√F 0
.
(xn(t)−1 ,h(t))
Similar considerations hold for the Neumann case. If a is 1-periodic, according to Herman [22, Prop. II.2.3, p. 20], ∀n ∈ Z∗ , ∀x ∈ R,
−
F n (x) − x 1 1 < − ρ(F ) < . |n| n |n|
Then, choosing x = ±a(0) and taking into account that F n (−a(0)) ≤ h(t) < F n (a(0)), we have ∀n ∈ Z,
∀t ∈ h−1 (In ),
nρ(F ) − 1 − a(0) < h(t) < nρ(F ) + 1 + a(0) .
Since ρ(F ) > 0 under our assumptions, (28) is proven.
3. Dirichlet Problem 3.1. Stability 3.1.1. A universal lower bound We prove that the energy cannot become arbitrarily small. Note that a is not necessarily periodic. We recall that f 0 (x) = 12 (ϕ00 (x) + ϕ1 (x)) a.e. in (0, a(0)) and f 0 (x) = 12 (ϕ00 (−x) − ϕ1 (−x)) a.e. in (−a(0), 0). Let us denote Mc := {x ∈ 4 f0 ˆ0 := supc∈R∗ k kM2c . It is easy to see that I0 ; 0 < |f 0 (x)| ≤ c} for c ∈ R∗+ and E 2a(0)c + ˆ0 ≤ E(0) if E(0) > 0. 0<E
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Theorem 3.1. Assume that E(0) > 0, a ∈ Lip(R), L(a) ∈ [0, 1), a > 0 and amax < +∞. Then a(0) ˆ0 . E (29) ∀t ∈ R, E(t) ≥ 0 amax Fmax If in addition ϕ00 , ϕ1 ∈ L∞ ((0, a(0))), then ∀t ∈ R,
E(t) ≥
E(0)2 . 0 Fmax 2amax kf 0 k2L∞ (I0 ) 1
(30)
Proof. By Cauchy–Schwarz inequality, for any c ∈ R∗+ :
√
f0
f0 p
2 0 n
√ √ DF ≤ 2amax c2 , ∀n ∈ Z, kf 0 kMc ≤
DF n f
DF n Mc Mc Mc
√
2
2
√
since DF n ≤ DF n = F n (a(0)) − F n (−a(0)) ≤ 2amax (see the formula Mc
I0
(15)). By Lemma 2.17, ∀t ∈ R,
E(t) ≥
1 0 Fmax
2 0
kf 0 k4Mc
√ f
≥ 1 .
0 Fmax 2amax c2 DF n(t) Mc
This proves (29). If moreover ϕ00 , ϕ1 ∈ L∞ ((0, a(0))), then f 0 ∈ L∞ (I0 ) and choosing c := kf 0 kL∞ (I0 ) , we have kf 0 k2Mc = E(0) and (30) follows. Remark 3.2. The condition amax < +∞ is necessary for Theorem 3.1, since the energy may vanish asymptotically otherwise. To see that, assume that a ∈ C 1 (R) is stricly increasing and a0 ≥ γ > 0 on R+ . Then amax = +∞ and limt→+∞ E(t) = 0. In fact, F 0 ≥ 1+γ 1−γ on [−a(0), +∞). Then, by Lemma 2.17, n(t) R n(t) 1−γ 0 2 |f (x)| dx = E(0) for t ≥ 0. Since h(t) tends to E(t) ≤ 1−γ 1+γ I0 1+γ infinity as t tends to infinity and h(t) ≤ F n(t) (a(0)), n(t) tends to infinity as t tends to infinity. Thus limt→+∞ E(t) = 0. 3.1.2. Periodicity Special case of bounded energy is the energy periodic in time. We find the necessary and sufficient condition for the energy to be periodic for all initial data. Theorem 3.3. 1) Let a ∈ A0 . If F q = Id + p for some q ∈ N∗ and p ∈ N∗ , then ∀(x, t) ∈ Ω,
ϕ(x, t + p) = ϕ(x, t)
and
∀t ∈ R,
E(t + p) = E(t) .
2) Let a ∈ Lip(R) such that L(a) ∈ [0, 1), a > 0. If there exists p ∈ R∗+ such that for all (ϕ0 , ϕ1 ) ∈ H01 ((0, a(0))) × L2 ((0, a(0))), the relation E(t + p) = E(t)
(31)
holds for all t ∈ R, then a is periodic of period p and if a is not constant in R, there exists q ∈ N, q ≥ 2, such that F q = Id + p.
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Proof. 1) Let a ∈ A0 , F q = Id + p with q ∈ N∗ , p ∈ N∗ . Then the relation f ◦ F = f implies f ◦ (Id + p) = f and the first statement follows from Lemma 2.15 1). 2) Let the assumptions of the second part hold now. First step. Using the relations (19), (20) and (21), it is not difficult R to see that for any t ∈ R and any ψ ∈ L2 (I(t)), I(t) := (h(t), k(t)), such that I(t) ψ dm = 0, there exist initial conditions (ϕ0 , ϕ1 ) ∈ H01 ((0, a(0))) × L2 ((0,R a(0))) such that the restriction of f 0 to I(t) is ψ. It is also easy to prove that if I(t) |ψ|2 g dm = 0 for some g ∈ L∞ (I(t)) and all ψ of the above properties, then g = 0 a.e. in I(t). To see that, choose e.g. ψ(x) := ±1 at the points where g(x) ≥ 0 and 0 elsewhere; the R signs can be chosen in such a way that I(t) ψ dm = 0. Then we see that g(x) > 0 only on a set of zero measure. The same holds for g(x) < 0. Second step. We recall the relation equivalent to (24): E 0 (t) = −2
h0 (t) 0 |f (h(t))|2 a0 (t) k 0 (t)
a.e. on R .
Substituting this expression into the derivative of Eq. (31) we obtain h0 (t) 0 h0 (t + p) 0 2 0 |f |f (h(t))|2 a0 (t) (h(t + p))| a (t + p) = k 0 (t + p) k 0 (t)
a.e. on R .
(32)
For any t ∈ R there exist t0 ∈ R and q ∈ N such that h(t) ∈ I(t0 ) and h(t + p) ∈ F q (I(t0 )). Here the integer q depends on t and t0 but can be chosen as constant in a neighborhood of a given t. Using the relation f 0 ◦ F −q · DF −q = f 0 a.e. on R, Eq. (32) reads h0 (t + p) DF −q (h(t + p))2 |f 0 (F −q (h(t + p)))|2 a0 (t + p) k 0 (t + p) =
h0 (t) 0 |f (h(t))|2 a0 (t) k 0 (t)
a.e. on R.
(33)
Here all the factors are nonzero except of a0 values and f 0 values corresponding to the arguments in the same interval I(t0 ). Let M0 := {x ∈ R; a0 (x) = 0}, M1 := {x ∈ R; a0 (x) exists, a0 (x) 6= 0 and (33) holds}. The relation (33) is valid by assumption for all initial conditions (ϕ0 , ϕ1 ) from the required set and consequently a0 (t) = a0 (t + p) = 0 for almost every t ∈ M0 and F −q (h(t + p)) = h(t) for t ∈ M1 ,
(34)
since otherwise we could simply construct (ϕ0 , ϕ1 ) such that one side of the equation (33) is zero while the other is nonzero. At almost every points of M1 which are accumulation points of M1 we can differentiate Eq. (34) and substitute the result into Eq. (33). Choosing the initial conditions such that f 0 (h(t)) 6= 0 and realizing that |a0 | < 1, a simple calculation gives: a0 (t + p) = a0 (t) at almost every considered accumulation points. Since the isolated points of M1 form a countable and therefore
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zero-measure set, the last relation holds a.e. in M1 and M0 , i.e. a.e. in R. So a(t + p) = a(t) + c with a constant c ∈ R. However, a is positive by assumption. Therefore c = 0 and p is a period of a, and also the relations h(t + p) = h(t) + p, k(t + p) = k(t) + p and F (t + p) = F (t) + p hold. If a is not constant in R, then m(M1 ) > 0 and there exists a point t such that (34) holds. The relation E(t + p) = E(t) expressed with the help of Lemma 2.15 1) leads to Z 1 2 1− |f 0 | dm = 0 . DF q I(t) According to the first step, the relation DF q (x) = 1 follows for almost every x ∈ I(t) and then F q (x) = x + p , taking into account (34). By application of F n we now see that the last relation holds in every interval F n (I(t)) with n ∈ Z and therefore in R. Example 3.4. When q = 1 the condition of the part 1) of the above theorem is satisfied iff a is constant and equal to a = p2 . For q = 2 we are able to prove that p p = . F 2 = Id + p ⇔ ∀t ∈ R, a(t) + a t + 2 2 p An example of such function is a(t) := 4 + α sin(2πt) for some p ∈ N∗ , p odd, and 1 . It would be interesting to find such an explicit characterization for any q. |α| < 2π 3.1.3. Absence of strong instability Proposition 3.5. Assume that a is of class Q and ρ(F ) ∈ R \ Q. Let V := d F 0 ). Then there exists a subsequence {I±qn }n∈N of the sequence {In }n∈Z deVar(ln fined by (14) such that qn tends to infinity as n tends to infinity and ∀n ∈ N,
∀t ∈ h−1 (I±qn ),
e−V E(0) eV E(0) ≤ E(t) ≤ . 0 0 Fmax Fmin
(35)
Proof. By the conditions assumed on a, F is of class P (see Definition 2.6 and Lemma 2.8). Then the Denjoy’s inequality holds (Herman [22, Prop. VI.4.4, p. 75]): ∀n ∈ N, e−V ≤ DF ±qn ≤ eV u.c.s. The rationals pqnn ∈ Q are the convergents of ρ(F ). The integers qn ∈ N satisfy the following properties: qn+1 > qn (n ≥ 1), qn ≥ n 2 2 (n ≥ 2). For further details see Herman [22, Sec. V, p. 57] or Lang [27]. It follows from (26) that ∀n ∈ N, ∀t ∈ h−1 (I±qn ), e−V E(0)αD (t) ≤ E(t) ≤ eV E(0)αD (t). Remark 3.6. The behavior described by Proposition 3.5 is in fact more general. Let m, n ∈ N and define the sequence of intervals Jn,m := ∪m k=−m Iqn +k . Then −1 we have for all t ∈ h (Jn,m ), m+1 1 0 , 0 e−V E(0) ≤ E(t) min Fmin Fmax m+1 1 0 eV E(0) . ≤ max Fmax , 0 Fmin
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Notice that under the condition of Proposition 3.5, the case of strong instability is excluded. The sequence {qn }n∈N used in the proof diverges at least exponentially. 3.1.4. A sufficient condition of stability Let us first give some definitions. If F ∈ Dl (T), 0 ≤ l ≤ ω, then F is said to be C -conjugate to Rρ(F ) := Id + ρ(F ), 0 ≤ k ≤ l, if there exists g ∈ Dk (T) such that g ◦ F = Rρ(F ) ◦ g. The function g is called the conjugacy of F . We start with a rather abstract but general theorem. k
Theorem 3.7. Assume a ∈ A0 , ρ(F ) ∈ R \ Q and denote by µ the unique invariant measure of F¯ . Assume that there exist λ1 , λ2 ∈ R such that the Radon– dµ ≤ λ2 < +∞. Then Nikodym derivative of µ w.r.t. m satisfies: 0 < λ1 ≤ dm ∀t ∈ R,
λ2 E(0) λ1 E(0) ≤ E(t) ≤ . 0 0 λ2 Fmax λ1 Fmin
(36)
Conversely, if a ∈ A0 , E(0) > 0 and E is bounded from above by a strictly positive constant, then there exists an invariant measure of F¯ which is not singular w.r.t. m. Rx dµ be the Radon–Nikodym derivative and set g(x) := 0 dµ. Proof. Let φ := dm By Herman [22, Sec. II, p. 19], g◦F = g+ρ(F ). Obviously g is absolutely continuous ≤ Dg −1 ≤ λ−1 and λ1 ≤ g 0 = φ ≤ λ2 a.e. It follows that g ∈ D0 (T) and λ−1 2 1 −1 n −1 a.e. Now F = g ◦ Rρ(F ) ◦ g, hence DF = Dg ◦ Rnρ(F ) ◦ g · g 0 a.e. Using Lemma 2.17, we have Z Z 2 2 1 |f 0 (x)| |f 0 (x)| 1 dx ≤ E(t) ≤ dx ∀t ∈ R, 0 0 n(t) (x) n(t) (x) Fmax Fmin I0 DF I0 DF and (36) follows immediately. The statement of the second part of the theorem is equivalent to: if all the invariant measures of F¯ are singular w.r.t. m, then supt∈R E(t) = +∞. Since by Corollary 3.15 (which is proven independently), the supremum of E is infinite under this condition, the proof is complete. Remark 3.8. From the proof of Theorem 3.7, it can be seen that if F ∈ F0 is Lipschitz conjugate to the translation Rρ(F ) (i.e. the conjugacy and its inverse are Lipschitz continuous), then the model is stable. Let us give an explicit example of stability. Example 3.9. Assume that a ∈ A0 is defined on one period by b b (λ + 1)t0 + b λ−1 t+ if ≤ t ≤ λ+1 λ+1 2 2 a(t) := −β −β b (λ + 1)t0 + b λ − 1t + 1 + b − λ ≤t≤ +1 if λ−β + 1 1 + λ−β 2 2
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−β
1−λ −1 for b, β ∈ R∗+ , λ > 1 and t0 := λ−λ , −β ∈ (0, 1). Computing F := (Id+a)◦(Id−a) it can be shown that F satisfies assumptions of Herman [22, Prop. VI.7.7.1, p. 82] β ∈ Zρ(F ) ( mod 1), and we have: if ρ(F ) ∈ R \ Q (by a suitable choice of b) and 1+β then this model is stable.
By adding some conditions, we can prove a general result of stability. First we recall the definition: Definition 3.10. Let β ∈ R+ . Then α ∈ R is a β-Diophantine number if there exists C > 0 such that p C p ∀ ∈ Q , α − ≥ 2+β . q q q The set of Diophantine numbers is of full measure and meager in the real numbers. For definitions and details see Herman [22, Sec. I, p. 11 and Sec. V, p. 57] or Lang [27]. Theorem 3.11. Assume (i) a ∈ C k (T), k ∈ R, k > 2, |a0 | < 1 and a > 0 and (ii) ρ(F ) is β-Diophantine with β < k − 2. Then (36) holds. Proof. The assumption (i) implies that F ∈ Dk (T). A result of Katznelson and Ornstein [28] gives: let F ∈ Dk (T) with ρ(F ) a β-Diophantine number and k > 2 + β. Then F is C 1 -conjugate to Id + ρ(F ). By Remark 3.8, the proof is complete. 3.2. Instability 3.2.1. A universal upper bound It is not assumed here that a is periodic. We recall that n(t) is defined in Lemma 2.17. Proposition 3.12. If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0, then ∀t ∈ R+ ,
E(t) ≤ E(0)
1 0 Fmin
n(t)+1 .
(37)
If moreover amin > 0 and E(0) > 0, then the maximal rate of exponential increase satisfies of E defined by r := lim supt→+∞ ln E(t) t r≤
1 1 ln 0 . 2amin Fmin
Proof. The first statement follows from Lemma 2.17,
2 n(t)+1
1 f0 2
≤ √ kf 0 kI0 . ∀t ∈ R+ , E(t) = αD (t) 0
n(t) F DF min I0
(38)
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Since amin ≤ a ≤ amax , then Id + 2amin ≤ F ≤ Id + 2amax (since F = Id + 2a ◦ h−1 ), hence Id + 2namin ≤ F n ≤ Id + 2namax. Combining these inequalities and the fact that by definition F n(t) (−a(0)) ≤ t − a(t) < F n(t) (a(0)), it is easy to prove that for all t ∈ R+ , h(t) + a(0) h(t) − a(0) ≤ n(t) ≤ 2amax 2amin (with the convention that if amax = +∞, then a−1 max = 0). Using (37), (38) follows. Remark 3.13. We will see in Remark 3.26 f) that this bound is almost optimal. Notice that in the case where a is periodic, one can deduce from the spectral analysis in Cooper and Koch [15] an other estimate, namely r ≤ 2 ln rU with rU the spectral radius of the evolution operator on one period. This estimate is more accurate if there exists no attracting hyperbolic periodic point, since in this case rU = 1 and therefore r = 0. It then follows that the rate of increase of E is subexponential. 3.2.2. A sufficient condition of instability We recall that if F ∈ D0 (T), then F¯ denotes its projection on the circle. Let ˜ := us identify [−a(0), −a(0) + 1] =: J ⊂ R with T and denote for any X ⊂ J, X S n ( m,n∈Z (F (X) + m)) ∩ I0 . Theorem 3.14. Let a ∈ A0 . Assume that for every invariant measure µ of F¯ , there exists a Borel set B in T such that µ(B) = 0 and >0 kϕ00 + ϕ1 k(B∩(0,a(0))) ˜
or
kϕ00 − ϕ1 k((−B)∩(0,a(0))) > 0, ˜
(39)
then m(B) > 0 (i.e. m is not absolutely continuous w.r.t. µ) and lim sup E(t) = +∞ . t→+∞
Proof. The relation m(B) > 0 follows from (39) and from the fact that F is a Lipschitz homeomorphism. It is also clear that there exist two integers m0 , n0 ∈ Z such that kf 0 kL1 (M) > 0 for the set M := (F n0 (B) + m0 ) ∩ I0 . Let us introduce a sequence of probability measures on T defined for all m-measurable set X ⊂ T by Z mn (X) := X
n−1 1 X ¯k DF dm . n
(40)
k=0
Since the probability measures form a weak* compact subset of C 0 (T)0 , there exists a subsequence {mni }i∈N which converges in the weak* topology to a Borel measure µ. It is easily seen that µ is an invariant measure of F¯ . By assumption, there exists a Borel set B in T such that µ(B) = 0 and m(B) > 0. Due to the regularity of µ and m, for every > 0, there exists an open set V in J such that V contains B and µ(V ) < ; there also exists a compact set K in J such that B contains K and
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m(B \ K ) < . According to Urysohn’s lemma, there exists a function g ∈ C 0 (T) such that the support of g is contained in V , the range of g is included in [0, 1] and g is equal to 1 on K . With the help of the function g it is easily seen that there exists i ∈ N such that for all i ≥ i , Z mni (K ) = K
ni −1 1 X DF¯ k dm < . ni k=0
Let us denote N := F n0 (K ) + m0 and M := N ∩ I0 . We will extend the measure mn to the Borel sets in R by formula (40) with F¯ replaced by F . A straightforward calculation shows that ! Z nX nX n0 +n i −1 0 −1 Xi −1 1 k k k DF¯ − DF¯ + DF¯ dm . mni (N ) = ni K k=0
k=0
k=ni
The integral of the last two sums is bounded by n0 . So there exists an integer i0 n0 such that for all i ≥ i0 , mni (N ) < 2. Since m(M \ M ) ≤ DFmax , we have: 1 0 0 kf kL1 (M ) > 2 kf kL1 (M) > 0 for sufficiently small . Now we will establish a useful inequality using Cauchy–Schwarz inequality. Let A ⊂ I0 be an m-measurable arbitrary set. Then kf 0 kL1 (A) =
n−1 1X 0 kf kL1 (A) n k=0
≤
n−1
0
√ 1 X
√ f
DF k
n A DF k A k=0
≤
2 ! 12 n−1 0
f 1 X
√
n DF k A k=0
n−1 √
2 1 X
DF k n A
! 12 .
k=0
Let {tk }k∈N be a sequence of real numbers such that tk ∈ h−1 (Ik ) for all k ∈ N and choose A := M ,
n−1 n−1
f 0 2 1X 1X
E(tk ) = αD (tk ) √
n n DF k I0 k=0
k=0
2 n−1 f0 1 X
≥ 0
√ k Fmax n DF M k=0 1
≥
1 0 Fmax
kf 0 k2L1 (M )
√
2 1 Pn−1 k k=0 DF n
M
≥
1
kf 0 k2L1 (M)
0 Fmax 4mn (N )
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and then for all i ≥ i0 ,
ni −1 kf 0 k2L1 (M) 1 1 X E(tk ) > 0 ni 8Fmax k=0
and since can take arbitrary small values, ni −1 1 X E(tk ) = +∞ . i→+∞ ni
lim
k=0
It follows that the limit superior of E is infinite as t tends to infinity.
Corollary 3.15. Let a ∈ A0 and assume that E(0) > 0. 1) If every invariant measure of F¯ is singular w.r.t. m, then lim sup E(t) = +∞ . t→+∞
2) If ρ(F ) ∈ R \ Q and the (unique) invariant measure of F¯ is not absolutely continuous w.r.t. m, then lim sup E(t) = +∞ . t→+∞
Proof. Assume that the assumptions of 1) hold: then the Borel set B of ˜ = 0. Theorem 3.14 can be chosen such that m(B) = 1 and µ(B) = 0, so m(I0 \ B) 0 Since E(0) > 0, kf kB˜ > 0 and the conclusion of Theorem 3.14 holds. Assume that the assumptions of 2) hold: then there exists a Borel set B in T such that m(B) = 0 and µ(B) > 0. Denote C := ∪n∈Z F¯ n (B). Since F ∈ F0 , F¯ (C) = C (i.e. C is F¯ -invariant) and m(C) = 0. Now we will prove that µ(C) = 1 which will show that µ and m are unusually singular. Then the first part of this corollary ends the proof. Let g be the conjugacy of F ; g is continuous and nondecreasing (see Herman [22, Prop. II.7.1, p. 25]) and g¯ ◦ F¯ (C) = g¯(C) + ρ(F ), hence g¯(C) is invariant by the irrational rotation Id + ρ(F ) and since m is its unique invariant measure, it follows that m(¯ g(C)) = 0 or m(¯ g(C)) = 1. But m(¯ g (B)) = µ(B) > 0, thus µ(C) = m(¯ g (C)) = 1. Example 3.16. If in Example 3.9, β = 1 and ρ(F ) ∈ R \ Q, then the limit superior of the energy as the time tends to infinity is infinite, whenever E(0) > 0 (see Herman [22, Theorem VI.7.4, p. 79] and Corollary 3.15 2)). Remark 3.17. In view of Theorems 3.7, 3.14 and Corollary 3.15, we put forward the question: whether there exists an example of F ∈ F0 with an invariant measure equivalent to m and leading to an infinite limit superior of the energy. Presently we do not know the answer.
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3.2.3. A sufficient condition of strong instability We give sufficient condition for the unlimited growth of energy in this section. The main results are contained in Theorem 3.20 and its corollaries. Definition 3.18. A point x ∈ R is called a wandering point for F ∈ F0 if there exists an open neighborhood U of x in R such that ∀n ∈ N∗ ,
∀m ∈ Z,
(F n (U ) + m) ∩ U = ∅ .
(41)
The set of wandering points in I˙0 will be denoted by W (F ) := {x ∈ I˙0 ; x is a wandering point for F } . Clearly, W (F ) is an open subset of I˙0 and W (F ) = W (F −1 ). Lemma 3.19. Let a ∈ A0 . Then for all x ∈ W (F ) there exists an open set U such that x ∈ U ⊂ W (F ) and lim kDF n kL1 (U) = 0,
n→±∞
+∞ X
kDF n kL1 (U) ≤ 1 .
(42)
n=−∞
Proof. Since x ∈ W (F ) is a wandering point, there exists an open set U 3 x in I˙0 such that U ⊂ W (F ) and (F n (U ) + m) ∩ U = ∅ for all n ∈ N∗ and m ∈ Z. It P+∞ P n n follows easily that +∞ n=−∞ kDF kL1 (U) = n=−∞ m(F (U )) ≤ 1. Theorem 3.20. If a ∈ A0 and kϕ00 + ϕ1 k(W (F )∩(0,a(0))) > 0
or
kϕ00 − ϕ1 k(−W (F )∩(0,a(0))) > 0 ,
then lim E(t) = +∞ .
t→±∞
(43)
Moreover, for all 0 < c < +∞ there exists an increasing sequence of integer numbers {nk }+∞ k=−∞ such that ∀t ∈ h−1 (Ink ),
E(t) ≥ c
|t| |t| ln . ρ(F ) ρ(F )
(44)
Proof. Since kf 0 kW (F ) > 0 by assumption, W (F ) is nonempty, so by Lemma 3.19, there exists a subset U ⊂ W (F ) such that lim kDF n kL1 (U) = 0
n→+∞
(45)
and kf 0 kU > 0. Since f 0 ∈ L2loc (R) and U is a bounded set, 0 < kf 0 kL1 (U) < +∞ .
(46)
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By Cauchy–Schwarz inequality, 0
kf kL1 (U)
√
f0 n
≤ √ DF
. U DF n
(47)
U
Then by Lemma 2.17, ∀t ∈ R,
E(t) ≥
1 0 Fmax
2 0 kf 0 k2L1 (U)
√ f
≥ 1
0 Fmax
DF n(t) 2 1 DF n(t) U L (U)
according to (47). Now, (45) and (46) imply (43). Due to Lemma 3.19, for any d > 0 there exists an increasing sequence of integer d nk (otherwise the series kL1 (U) ≤ numbers {nk }+∞ k=−∞ such that kDF |nk | ln |nk | −1 nk of nintegrals in (42) would diverge). If t ∈ h (Ink ), then h(t) ≤ F (a(0)) and F k (a(0)) − a(0) 1 as in proof of (28). Assume that t ≥ h−1 (a(0)), − ρ(F ) < nk |nk | < nk . Now for all t ∈ h−1 (Ink ), t > h−1 (a(0)) + 1, then nk > 0 and h(t)−a(0)−1 ρ(F ) E(t) ≥
kf 0 kL1 (U) |t − a(t) − a(0) − 1| |t − a(t) − a(0) − 1| ln . 0 Fmax d ρ(F ) ρ(F ) 1
2
The same arguments can be repeated for t < 0. The required relation (44) follows. Corollary 3.21. Assume a ∈ A0 , ρ(F ) = pq ∈ Q∗ and F q 6= Id + p. Denote by A := {x ∈ I0 ; F q (x) 6= x + p} the set of nonperiodic points. If kϕ00 + ϕ1 k(A∩(0,a(0))) 6= 0 or kϕ00 − ϕ1 k(−A∩(0,a(0))) 6= 0, then limt→±∞ E(t) = +∞. Proof. We shall verify that W (F ) consists of the complement of the set of periodic points (thus W (F ) = A) and m(W (F )) > 0. Since the set of wandering points is open it is sufficient to prove that at least one wandering point exists. By assumption, there exists a point x1 ∈ I0 such that F q (x1 ) 6= x1 + p. Evidently, F q (x1 ) 6= x1 + n for any n ∈ Z (otherwise ρ(F ) would have a value different from p p q q ). There also exists a point x0 ∈ I0 such that F (x0 ) = x0 + p as ρ(F ) = q . Let us pass from I0 and F to T and the projected map F¯ : T → T. Let us call ¯0 := π(x0 ). There exists a maximal (connected) open interval x ¯1 := π(x1 ) and x U in T containing x ¯1 such that F¯ q (x) 6= x for all x ∈ U . The interval U does not / U ). The endpoints of U are fixed points of F¯ q , one cover the whole T (e.g. x ¯0 ∈ is attracting and the second is repelling. Since F¯ q is orientation-preserving and ¯1 , F¯ (¯ x1 ), . . . , F¯ q (¯ x1 ) continuous, F¯ q (U ) = U . It is easily seen that all the points x are different. So there exists an open interval V such that x ¯1 ∈ V ⊂ U and the sets V, F¯ (V ), . . . , F¯ q (V ) are pairwise disjoint. Looking for the ordering of endpoints of the intervals F¯ nq (V ) in U it is seen that V ∩ F¯ nq (V ) = ∅ for n ∈ Z∗ (since F¯ q is monotone in U ). Now we prove that V ∩ F¯ nq+l (V ) = ∅ for n ∈ Z∗ , l ∈ {0, 1, . . . , q − 1}. Let us assume the opposite: there exist y ∈ V and z ∈ V such that y = F¯ nq+l (z) for some
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n ∈ Z∗ and l ∈ {0, 1, . . . , q − 1}. Then F¯ mq (y) = F¯ l (F¯ (n+m)q (z)) for any m ∈ Z. In the limit m → +∞, we have F¯ mq (y) → b, F¯ (n+m)q (z) → b where b is the attracting endpoint of U . So F¯ l (b) = b which contradicts the value of ρ(F ) = pq . As a result, we have proved that V ∩ F¯ m (V ) = ∅ for all m ∈ Z∗ and x1 is a wandering point. Since x1 is an arbitrary nonperiodic point, then A ⊂ W (F ) (and even A = W (F )). Thus assumptions of Theorem 3.20 are valid. Corollary 3.22. If a ∈ A0 , F has a finite nonzero number of periodic points and E(0) is strictly positive, then limt→±∞ E(t) = +∞. Proof: By assumption, there exist p, q ∈ N∗ such that ρ(F ) = pq . Moreover [−a(0), a(0)] \ W (F ) consists of a finite number of points; this can be seen in the proof of Corollary 3.21. Thus m(W (F )) = m(I0 ) = 2a(0) > 0, which implies that kf 0 kW (F ) > 0. Since the assumptions of Theorem 3.20 are satisfied, E tends to infinity as t tends to infinity. Corollary 3.23. Assume a ∈ A0 , ρ(F ) ∈ R \ Q, F is not C 0 -conjugate to Id+ρ(F ) and ϕ00 +ϕ1 6= 0 a.e. in W (F )∩(0, a(0)) or ϕ00 −ϕ1 6= 0 a.e. in (−W (F ))∩ (0, a(0)). Then limt→±∞ E(t) = +∞. Proof. To apply Theorem 3.20, let us prove that m(W (F )) > 0. As ρ(F ) ∈ R \ Q, there exists a unique invariant measure µ of F¯ in T and the support of µ satisfies supp µ = Ω(F¯ ) (Ω(F¯ ) is the set of nonwandering points on T). Since F is not C 0 -conjugate to Id + ρ(F ), supp µ 6= T. For these two facts see Herman [22, Sec. II.7.2, p. 25]. Since Ω(F¯ ) is closed, its nonempty complement in T is open and therefore of positive Lebesgue measure. The lift to R of this complement can be mapped by F into I0 (see Lemma 2.17). Thus m(W (F )) > 0. Remark 3.24. Corollary 3.23 is nonempty as show for instance Katok and Hasselblatt [23, Prop. 12.2.2., p. 405]: for all l ∈ [0, 1) and all β ∈ (0, 1), there exists F ∈ D1+β (T) such that ρ(F ) ∈ R \ Q and m(Ω(F¯ )) = l. Theorem 3.20 gives a new proof of divergence of the energy (for the previous ˇ results of that type see Cooper [13], Dittrich, Duclos and Seba [14], Cooper and Koch [15], M´eplan and Gignoux [16]). Assumptions based on the wandering points (instead of the periodic points like in [13, 14, 16]) give simplifications in the proof and more general results. Corollary 3.22 was already obtained by Cooper [13]. Corollary 3.21 is a slight generalization of Corollary 3.22: it allows to treat an infinite number of periodic points.a Corollary 3.23 was unexpected since ρ(F ) ∈ R \ Q: although there are no periodic points, E is increasing without bound. But classical solutions are excluded: if a ∈ A2 , then F is C 0 -conjugate to Id + ρ(F ) by Denjoy’s theorem (see Herman [22, Theorem VI.5.5, p. 76]). Note that Theorem 3.20 works with a boundary a differentiable a.e. In particular this includes the piecewise linear a Consider for instance the function a ∈ A equal to p (p ∈ N∗ ) on a half period and nonconstant 0 2 on the other half period. Then F has full intervals of periodic points.
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model. In Corollaries 3.21 and 3.22, there are no assumptions made on the nature of the periodic points (hyperbolic or not): this is the reason why we cannot have precise estimates on the rate of increase of E (see Subsec. 3.2.4 for a particular case). Compared to Theorem 3.14 the improvement is the following: if there exists a wandering set of strictly positive Lebesgue measure, then the model is strongly unstable. Finally we can extend the condition of Cooper for the growth of E without bound: “there exists a finite number of periodic characteristics” by “there exists a set of wandering characteristics which has an intersection of positive Lebesgue measure with {x ∈ (0, a(0)); ϕ00 (x) + ϕ1 (x) 6= 0} ∪ {x ∈ (−a(0), 0); ϕ00 (−x) − ϕ1 (−x) 6= 0}”. 3.2.4. Asymptotics In this section we will improve results given by Cooper [13], Dittrich, Duclos ˇ and Seba [14], Cooper and Koch [15]; see also Remark 3.26. For the asymptotics of the solution, see Cooper [13, Theorem 2, p. 79]. We recall that f is related to the initial conditions on I0 by the formulas (19)– (20). Although we have also worked out the case ρ(F ) := pq ∈ Q∗ , we present here only the case q = 1. Assume that there are only two periodic points of F in I0 : a1 := −a(0) which is attracting and hyperbolic and a2 which is repelling. Then a(0) is also an attracting hyperbolic periodic point with F 0 (a(0)) = F 0 (a1 ). For 2 simplicity take a(0) := p2 . Then In = I0 + np and E(np) = kf 0 kIn . We define for every x ∈ I0 , +∞ Y F 0 (a1 ) . l(x) := F 0 ◦ F k (x) k=0
Theorem 3.25. Assume a ∈ A1 , a00 ∈ L∞ (R), ρ(F ) := p ∈ N∗ and E(0) > 0. Assume also that F has only two periodic points in I0 : a1 := −a(0) and a2 ∈ I˙0 such that F 0 (a1 ) < 1 and F 0 (a2 ) ≥ 1. Take a(0) := p2 . Then l ∈ C 0 (I0 \ {a2 }), l > 0 on I0 \ {a2 } (l(a2 ) = 0) and
√ 2
lf 0 E(np) =
I0
F 0 (a1 )n
(1 + o(1))
as n → +∞ .
(48)
Proof. We use the following notations: J := [a1 , a2 ], for all n ∈ N, Kn :=
0 2 0
√ f ˆ n := 0 1 n , rn := Kn , ln (x) := Qn−1 F0 (ak1 ) > 0. Thus rn =
DF n , K ˆn k=0 F ◦F (x) F (a1 ) K J
√ 0 2
ln f . We first prove that ln (x) converges pointwise to l(x) (first step) and J that it is uniformly bounded in order to apply the Dominated Convergence Theorem and conclude that limn→+∞ rn =: L1 ∈ R+ (second step). It follows that Kn = ˆ n (L1 + o(1)) as n → +∞. Notice that by symmetry the proof needs just to be K done on one side, for instance [a1 , a2 ], and similar results hold for [a2 , a(0)].
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◦F (x) First step: pointwise convergence. Let vk (x) := F (aF1 )−F > −1. Then for 0 ◦F k (x) Qn−1 0 all x ∈ [a1 , a2 ], ln (x) = k=0 (1 + vk (x)). By continuity of F , there exists a set 0 J− := [a1 , a1 + δ− ], δ− > 0, such that for all x ∈ J− , F 0 (x) ≤ F (a21 )+1 < 1. By assumption, for all x ∈ [a1 , a2 ), there exists n0 (x) ∈ N such that for all n ≥ n0 (x), F n (x) ∈ J− . By the Mean Value Theorem, for any x in [a1 , a2 ),
1 k 00
and |vk (x)| ≤ |a1 − F (x)| · kF kL∞ (J) ·
F0 ∞ L (J) k
∀k ≥ n0 (x), |a1 − F k (x)| = F k−n0 (x) ◦ F n0 (x) (a1 ) − F k−n0 (x) ◦ F n0 (x) (x) ≤ |a1 − x| · kF 0 kL0∞ (J) · kF 0 kL∞ (J0 − ) . n (x)
k−n (x)
Q+∞ P It follows that +∞ k=0 |vk (x)| is convergent, therefore k=0 (1 + vk (x)) is absolutely convergent. The pointwise limit of ln (x) is l(x) (l > 0 on [a1 , a2 )). Since ln is clearly uniformly convergent on any compact included in [a1 , a2 ), l ∈ C 0 ([a1 , a2 )). Second step: uniform bound. Let J+ := [a2 − δ+ , a2 ] with δ+ small enough such that minJ+ F 0 ≥ F 0 (a1 ). For x ∈ J+ \ {a2 }, denote by nx ∈ N the greatest integer such that n < nx implies that F n (x) ∈ J+ . We know that for all k < nx , F 0 (a1 ) ≤ 1: it follows that for all x ∈ [a2 − δ+ , a2 ) and for all n ≤ nx , ln (x) ≤ 1. F 0 ◦F k (x) If J+ = J (this is the case if F 0 (a1 ) = minJ F 0 ) then nx = +∞ and immediately for all x ∈ J and for all n ∈ N, 0 ≤ ln (x) ≤ 1. If J+ 6= J, then nx < +∞ and for all x ∈ J+ \ {a2 }, for all n > nx , ln (x) = ln−nx (F nx (x))lnx (x) ≤ ln−nx (F nx (x)) ≤ sup sup ln (y) =: M < +∞ , n∈N y6∈J+
since the convergence of ln is uniform on [a1 , a2 − δ+ ]. We have proved that ln (x) ≤ max{1, M } on J and for all x ∈ J \ {a2 }, limn→+∞ ln (x) = l(x) > 0. By the Dominated Convergence Theorem we have that
√ 2
lim rn = lf 0 =: L1 ≥ 0 . n→+∞
J
Summing the √ results from both intervals [−a(0), a2 ] and [a2 , a(0)], the relation (48) follows with k lf 0 kI0 > 0. Remark 3.26. a) This proof can be generalized to an arbitrary finite number −1 of periodic points {ai }1≤i≤N by making a partition ∪N i=1 [ai , ai+1 ) of I0 by the periodic points, then considering E as a sum of terms over every interval [ai , ai+1 ) and finally applying the results proved in Theorem 3.25. b) Let us compare our result with previous ones (for p = 1): in Cooper [13], ˇ ≥ A + nc for Dittrich, Duclos and Seba [14], it is proven that for all n ∈ N∗ , ln E(n) n 0 ∗ = some constants 0 < A < − ln F (a1 ) and c ∈ R ; in Cooper and Koch [15], ln E(n) n
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− ln F 0 (a1 ) + o(1) as n → +∞ and now we have
√ 2
as n → +∞, with c := ln lf 0 .
ln E(n) n
= − ln F 0 (a1 ) +
c n
+o
1 n
I0
c) If F 0 (a2 ) > 1 and f 0 is L∞ in aneighborhood of a2 , we are even able to prove 1 1 that the remainder is not only o n but O n2 . We do not give the proof here [29]. d) The reader can deduce the behavior of E(t) from E(np) by Lemma 2.17. e) Even for nonsmooth boundary a, there might be an exponential increase of E. The following result can be proven: assume that a ∈ A0 , ρ(F ) = pq ∈ Q∗ and there exists a periodic point of F in I0 called a1 and satisfying lim sup x→a+ 1
F q (x) − F q (a1 ) =: γ < 1 . x − a1
Let a2 > a1 be the nearest periodic point of F and assume that kf 0 k(a1 ,a2 ) > 0. Then for all > 0, there exists a constant c > 0 such that ∀t ∈ R+ ,
E(t) ≥ c
1 γ+
pt .
f) By Proposition 3.12, we know that the maximal rate of exponential increase of 0 =: r0 . Assume p = 1, then in Theorem 3.25 we obtain E satisfies r ≤ − 2a1min ln Fmin 0 r = − ln F (a1 ). Since p = q = 1, then 12 ∈ Ran a by Lemma 2.4. Now assume |r 0 −r| r between the estimate of is e = 2a1min − 1 , which can be
0 , then the relative error e := that F 0 (a1 ) := Fmin
Proposition 3.12 and the result of Theorem 3.25 made arbitrarily small if amin is close to 12 . 3.2.5. Instability: strong instability is not the rule
Proposition 3.27. 1) Let G ∞ := D∞ (T) \ Int ρ−1 (Q). Then G ∞ is a closed set with empty interior and there exists A ⊂ G ∞ ∩ ρ−1 (R \ Q) such that A is a dense Gδ set in G ∞ . Moreover, for all F ∈ A ∩ F∞ , lim sup E(t) = +∞ , t→+∞
whenever E(0) > 0. 1 , β > |α| and α is fixed. Let 2) Let Fβ (x) := x + α sin(2πx) + β, 0 < |α| < 2π Kα := [|α|, |α| + 1] \ Int {β; ρ(Fβ ) ∈ Q}. Then Kα is a Cantor set and there exists B ⊂ Kα ∩ ρ−1 (R \ Q) such that B contains a dense Gδ set in Kα . Moreover, for all F ∈ B ∩ F∞ , lim supt→+∞ E(t) = +∞, whenever E(0) > 0. Proof. A direct consequence of Herman [22, Theorem XII.1.11, p. 168 and Prop. XII.1.13, p. 169] and Corollary 3.15. Remark 3.28. This is an application of Corollary 3.15 which shows that instability can be even generic. The example given in part 2) is the well known map
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of Arnol’d [30]. We see that the instability is possible even if there are no periodic points for F and if a is smooth. This answers negatively to a conjecture of Cooper [17]. By Proposition 3.5, one knows that the limit inferior of energy E is finite under the assumptions of Proposition 3.27. 3.2.6. Perturbation of the boundary In the following proposition we will prove that the instability of our model is in general the rule after perturbation. Let us denote Ea := E the energy of the model with a moving boundary a. Proposition 3.29. Let a ∈ Ak , k ≥ 1, and Ea (0) > 0. The following two assertions are true: 1) There exists a ˜ ∈ Ak arbitrarily near a in the C k -topology such that Ea˜ is increasing exponentially fast as t tends to infinity, whenever Ea˜ (0) > 0. 2) Assume that F := (Id + a) ◦ (Id − a)−1 is a Morse–Smale diffeomorphism. ˜kk < , Ea˜ Then there exists > 0 such that for all a ˜ ∈ Ak satisfying ka − a is still increasing exponentially fast, whenever Ea˜ (0) > 0. Proof. Let a ∈ Ak , k ≥ 1; then F := (Id +a)◦ (Id− a)−1 ∈ Fk . By a theorem of Peixoto (e.g. Nitecki [31, Sec. 1.2, p. 50]), one knows that there exists a dense, open subset of Dk (T) which consists of diffeomorphisms G with the following properties: the nonwandering set Ω(G) is finiteb and all periodic points are hyperbolic.c Consequently, there exists F˜ ∈ Dk (T) arbitrarily near F in the C k -topology. Since Fk is open for k ≥ 1, we can take F˜ ∈ Fk . By Lemma 2.3, Ak and Fk are homeomorphic and ! !−1 F˜ + Id F˜ − Id ◦ ∈ Ak (49) a ˜ := 2 2 (see formula (11)). Then Ea˜ is increasing exponentially fast by Theorem 3.25 or Remark 3.26 e). This proves 1). If we assume that F ∈ Fk , k ≥ 1, is a Morse–Smale diffeomorphism, then if F˜ is a sufficiently small perturbation of F in the C k -topology, F˜ is still a Morse–Smale ˜ by the formula (49), we can apply diffeomorphism and F˜ ∈ Fk . Thus, defining a Theorem 3.25 or Remark 3.26 e), which say that Ea˜ is increasing exponentially if Ea˜ (0) > 0. 4. Neumann Problem What will be remarkable is the opposite behavior of the Neumann problem w.r.t. the Dirichlet problem. Technicalities in the proofs are often the same, so we omit them. b Hence Ω(G) consists only of periodic points. c G is a Morse–Smale diffeomorphism.
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4.1. Stability 4.1.1. An upper bound Note that here a is not necessarily periodic. We recall that f 0 (x) =
1 0 (ϕ (x) + ϕ1 (x)) 2 0
f 0 (x) =
1 (−ϕ00 (−x) + ϕ1 (−x)) 2
a.e. in (0, a(0)) , a.e. in (−a(0), 0) .
(50) (51)
Proposition 4.1. If a ∈ Lip(R), L(a) ∈ [0, 1), a > 0, amax < +∞ and ϕ1 ∈ L∞ ((0, a(0))) ,
ϕ00 ∈ L∞ ((0, a(0))) ,
then ∀t ∈ R,
0 kf 0 kL∞ (I0 ) . E(t) ≤ 2amax Fmax 2
Moreover E ∈ Lip(R) and L(E) ≤ 2L(a) kf 0 kL∞ (I0 ) . 2
Proof. By the conditions on ϕ0 and ϕ1 , f 0 ∈ L∞ (I0 ). Then by Lemma 2.17, ∀t ∈ R,
p
2
0
0 E(t) ≤ Fmax
f DF n(t)
I0
2 0 ≤ Fmax kf 0 kL∞ (I0 ) F n(t) (a(0)) − F n(t) (−a(0)) 0 kf 0 kL∞ (I0 ) 2amax . ≤ Fmax 2
By the relation (23), we have f 0 ◦F = f 0 a.e. So by successive iterations we can show that f 0 ∈ L∞ (R) and kf 0 kL∞ (R) = kf 0 kL∞ (I0 ) . Using the formula E 0 = 2a0 |f 0 ◦ k|2 a.e. (see Lemma 2.15 4)), we get that E is Lipschitz continuous on R and L(E) ≤ 2L(a)kf 0 k2L∞ (I0 ) . Remark 4.2. The assumption that ϕ00 , ϕ1 ∈ L∞ ((0, a(0))) is crucial in the Neumann problem. There are examples for which the energy diverges if it is not the case as is shown in Sec. 4.2.2. Note also that if a is strictly increasing on R+ with a0 ≥ γ > 0, then amax = +∞ and limt→+∞ E(t) = +∞. The proof is similar as the one in Remark 3.2. 4.1.2. Periodicity With respect to the Dirichlet case (cf. Theorem 3.3) only minor changes are necessary to establish the following theorem: Theorem 4.3. 1) Let a ∈ A0 . If F q = Id + p for some q ∈ N∗ and p ∈ N∗ , then ∀t ∈ R, E(t + p) = E(t) .
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2) Let a ∈ Lip(R) such that L(a) ∈ [0, 1), a > 0. If there exists p ∈ R∗+ such that for all (ϕ0 , ϕ1 ) ∈ H 1 ((0, a(0))) × L2 ((0, a(0))), the relation E(t + p) = E(t) holds for all t ∈ R, then a is periodic of period p and if a is not constant in R, there exists q ∈ N, q ≥ 2, such that F q = Id + p. 4.1.3. Asymptotics In the next theorem we give an asymptotics in a simplified case: instead of a general rational rotation number ρ(F ) = pq , we consider ρ(F ) = p ∈ N∗ ; moreover we restrict the number of periodic points to two. This simplification has the advantage to propose a readable formula for the asymptotics. The general formula can be also worked out. Remember that f 0 is given on I0 by the formulas (50)–(51). Theorem 4.4. Let p ∈ N∗ . Assume that (i) a ∈ A1 , (ii) p2 ∈ Ran a, (iii) F has just two fixed points a1 , a2 in I0 such that a1 is attracting and a2 is repelling; take a1 := −a(0), (iv) (ϕ0 , ϕ1 ) ∈ C 1 ([0, a(0)]) × C 0 ([0, a(0)]) and ϕ00 (0) = ϕ00 (a(0)) = 0. Then E is asymptotically 1-periodic and more precisely lim E(t) − 2|f 0 (a2 )|2 a(t) = 0 . t→+∞
p Proof. p pBy assumption0 (ii), we can choose a(0)0 = 2 and then for all n ∈ Z, In = − 2 , 2 + np. Let fn be the restriction of f to In . Then Eq. (23) implies that fn0 = f00 ◦ F −n . By (iv), f 0 ∈ C 0 (I0 ), thus f 0 ∈ C 0 (R). Obviously, kf 0 kL∞ (R) = kf 0 kL∞ (I0 ) < +∞. Let φ : I0 → R be the function equal to f 0 (a1 ) if x = a1 and equal to f 0 (a2 ) otherwise. Assumption (iii) implies that for all x ∈ I0 , f 0 (x + np) converges pointwise to φ(x) as n → +∞. Let τ ∈ I0 and define the sequence tn := tn (τ ) := h−1 (τ ) + np, n ∈ N. Then h(tn ) = τ + np ∈ In and by obvious changes of variables:
√ 2 2 2
+ kf 0 k(τ +np,xn ) E(tn (τ )) = kf 0 k(h(tn (τ )),k(tn (τ ))) = f 0 F 0 (xn−1 ,τ +np)
√
2 = f 0 (· + np) F 0
(−a(0),τ )
+ kf 0 (· + np)k(τ,a(0)) . 2
Since f 0 ∈ L∞ (R) and limn→+∞ f 0 (x + np) = φ(x) for all x ∈ I0 , we can apply the Dominated Convergence Theorem, lim E(tn (τ )) = |f 0 (a2 )| (F (τ ) − F (−a(0))) + |f 0 (a2 )|2 (a(0) − τ ) 2
n→+∞
= |f 0 (a2 )| (F (τ ) − τ ) =: E∞ (τ ) . 2
It is easy to see that
|E(tn (τ )) − E∞ (τ )| ≤ kF 0 kL∞ (I0 ) |f 0 (· + np)|2 − |f 0 (a2 )|2 L1 (I
0)
.
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The right-hand side tends to zero when n tends to infinity and is independent of τ , thus E(tn ) converges uniformly w.r.t. τ to E∞ (τ ). Since F (τ ) − τ = 2a(h−1 (τ )) = 2a(tn (τ )), the proof is complete. 4.1.4. Absence of strong instability Like for the Dirichlet case (cf. Proposition 3.5) we have, by similar arguments: Proposition 4.5. Assume that a is of class Q and ρ(F ) ∈ R \ Q. Let V := d F 0 ). Then there exists a subsequence {I±qn }n∈N of the sequence {In }n∈Z deVar(ln fined by (14) such that qn tends to infinity as n tends to infinity and ∀n ∈ N,
∀t ∈ h−1 (I±qn ),
0 0 e−V E(0)Fmin ≤ E(t) ≤ eV E(0)Fmax .
4.1.5. A sufficient condition of stability The proofs of the following theorems are similar to Theorems 3.7 and 3.11. Theorem 4.6. Assume a ∈ A0 , ρ(F ) ∈ R \ Q and denote by µ the unique invariant measure of F¯ . Assume that there exist λ1 , λ2 ∈ R such that the Radon– dµ ≤ λ2 < +∞. Then Nikodym derivative of µ w.r.t. m satisfies: 0 < λ1 ≤ dm ∀t ∈ R,
λ2 λ1 0 0 E(0)Fmin ≤ E(t) ≤ E(0)Fmax . λ2 λ1
(52)
Theorem 4.7. If a ∈ C k (T), k ∈ R, k > 2, |a0 | < 1, a > 0 and ρ(F ) is β-Diophantine with β < k − 2, then (52) holds. Remark 4.8. Like for the Dirichlet problem, if F ∈ F0 is Lipschitz conjugate to Id + ρ(F ), then the Neumann model is stable. 4.2. Instability 4.2.1. Universal lower and upper bounds Note that here a is not necessarily periodic and n(t) is defined in Lemma 2.17. The more general bounds of the Neumann problem are given by the following proposition. The proof is similar to the one of Theorem 3.12. Proposition 4.9. If a ∈ Lip(R), L(a) ∈ [0, 1) and a > 0, then ∀t ∈ R+ ,
0 E(0) (Fmin )
n(t)+1
0 ≤ E(t) ≤ E(0) (Fmax )
n(t)+1
.
If moreover E(0) > 0, then the following two assertions are true: 1) If amin > 0, then the maximal rate of exponential increase of E defined by satisfies r+ := lim supt→+∞ ln E(t) t r+ ≤
1 0 ln Fmax . 2amin
(53)
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2) If limt→+∞ a(t) = 0, then the minimal rate of exponential increase of E t satisfies defined by r− := lim inf t→+∞ ln E(t) t r− ≥
1 0 ln Fmin , 2amax
(54)
with a−1 max = 0 if amax = +∞.
One could think that these estimates are unrealistic by the already existing bound given in Proposition 4.1. The results given in Proposition 4.10 and Theorem 4.13 show that this is not the case. 4.2.2. Singular initial data Proposition 4.10. There exist a ∈ Ak (1 ≤ k ≤ +∞ or k = ω) and initial conditions (ϕ0 , ϕ1 ) ∈ H 1 ((0, a(0))) × L2 ((0, a(0))) with ϕ00 6∈ L∞ ((0, a(0))) (or ϕ1 6∈ L∞ ((0, a(0)))) such that lim E(t) = +∞ t→+∞
and the rate of increase is exponential. Proof. Let F ∈ Fk , k ≥ 1, such that F (0) = p for some p ∈ N∗ , F 0 (0) > 1 and there exists x0 ∈ (0, a(0)] such that for all x ∈ [0, x0 ], F 0 (x) = F 0 (0). It is obvious that ρ(F ) = p and that we can build such F . To this F there corresponds a unique a ∈ Ak as stated in Lemma 2.3. For simplification we introduce F˜ := F − p. Then F˜ (0) = 0 and DF n = DF˜ n . For any n ∈ N∗ , let us look for x ∈ (0, x0 ] such that F˜ n (x) ≤ x0 . We define now the sequence y0 := x, yn := F˜ (yn−1 ), n ≥ 1. Immediately yn = F 0 (0)n x. Then for all n ∈ N, n ≥ 1, Fn (x) ≤ x0 ⇔ x ≤
x0 0 F (0)n
=: n → 0 as n → +∞ .
Assume for instance that ϕ0 ∈ H 1 ((0, a(0))) and ϕ1 ∈ L2 ((0, a(0))) satisfy ϕ0 (x) := 8 34 0 − 14 3 x and ϕ1 (x) := 0 for x ∈ (0, x0 ] and are arbitrary otherwise. Then f (x) = x on (0, x0 ]. Thus f 0 ∈ L2 ((0, x0 )) \ L∞ ((0, x0 )). Hence, by Lemma 2.17, ∀t ∈ R∗+ ,
p
2
0
0 E(t) ≥ Fmin
f DF˜ n(t)
(0,n(t) )
0 = Fmin F 0 (0)n(t) kf 0 k(0,n(t) ) 2
√ n(t) x0 0 0 √ = 2Fmin F 0 (0)n(t) p = 2Fmin x0 F 0 (0) 2 . F 0 (0)n(t) This construction is valid for k ≤ +∞. We can also easily build examples of F ∈ Dω (T). This lower bound and Proposition 4.9 show that E grows without bound exponentially fast.
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Remark 4.11. If in the above proof we assume more generally that ϕ0 (x) behaves on (0, x0 ] like xα , 12 < α < 1, E(t) grows faster than cF 0 (0)2(1−α)n(t) . The more singular is ϕ0 (i.e. α smaller and smaller) the faster the energy increases. In particular, the bound (53) on the maximal rate of exponential increase of E is almost reached for α close to 12 . The physical explanation of this proposition is simple: singularities of the initial data are trapped by an attractor and amplified by the Doppler effect. 4.2.3. Sufficient conditions for the decay of the energy As an immediate consequence of Theorem 4.4, we have: Corollary 4.12. If all the conditions of Theorem 4.4 hold and f 0 (a2 ) = 0, then limt→+∞ E(t) = 0. In the next theorem we give conditions to have an exponential decay of the energy. We recall that for every x ∈ I0 , we defined in Theorem 3.25, l(x) :=
+∞ Y k=0
F 0 (a1 ) . F 0 ◦ F k (x)
Theorem 4.13. Assume that (i) a ∈ A1 , a00 ∈ L∞ (R). (ii) ρ(F ) = pq ∈ Q∗ and F has just two periodic points a1 , a2 on I0 such that DF q (a1 ) < 1 and DF q (a2 ) ≥ 1. Take a1 := −a(0). (iii) (ϕ0 , ϕ1 ) ∈ C 1 ([0, a(0)]) × C 0 ([0, a(0)]), ϕ00 (0) = ϕ00 (a(0)) = 0 and E(0) > 0. (iv) There exists an open neighborhood of a2 denoted by V := (a2 − , a2 + ) for some > 0, such that for all x ∈ V , f 0 (x) = 0. Then E decreases exponentially to 0 and more precisely:
0 2
f q m
E(mp) =
√l DF (a1 ) (1 + o(1)) as m → +∞ , I0 where l is continuous and strictly positive on [−a(0), a(0)] \ V . Proof. Since the proof is very similar to the one of Theorem 3.25, we only sketch it. By (ii), a(0) is an attracting periodic point, DF q (−a(0)) = DF q (a(0)). Since for all t ∈ R+ , there exist a unique m(t) ∈ N and a unique r(t) ∈ {0, 1, . . . , q − 1} such that n(t) = m(t)q + r(t), it is possible to rewrite the energy as follows: p
p
2
2
0
0
n(t) n(t) + f DF E(t) = αN (t) f DF
(a1 ,a2 −)
p
2
= β(t) f 0 DF qm(t)
(a1 ,a2 −)
= β(t)E(m(t)p)
(a2 +,a(0))
p
2
+ f 0 DF qm(t)
(a2 +,a(0))
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for some strictly positive and bounded function β (which is obtained similarly as = p1 . Now similar arguments αN in Lemma 2.17). By Lemma 2.17, limt→+∞ m(t) t as in the proof of Theorem 3.25 show that l is continuous and strictly positive on [a1 , a2 − ] ∪ [a2 + , a(0)] and
0 2
2
√
f
0 mq
= DF q (a1 )m (1 + o(1)) as m → +∞ ,
f DF
√l (a1 ,a2 −) (a1 ,a2 −)
0 2
2
√
f
0 mq
= DF q (a1 )m (1 + o(1)) as m → +∞ .
f DF
√l (a2 +,a(0)) (a2 +,a(0))
The end of the proof is now obvious.
Remark 4.14. Like for Theorem 3.25 the assumptions of Theorem 4.13 are stronger than needed in order to obtain simpler results. A. Appendix Lemma A. Let Σ ⊂ R2 be a domain (i.e. nonempty, open and connected) such that for each y ∈ R, J y := {x ∈ R; (x, y) ∈ Σ} and for each x ∈ R, Jx := {y ∈ R; (x, y) ∈ Σ} are intervals of R. Let K1 := {x ∈ R; Σ ∩ ({x} × R) 6= ∅}, K2 := {y ∈ R; Σ ∩ (R × {y}) 6= ∅}. Then K1 and K2 are nonempty open intervals of R. 1 (Σ) and assume that ϕxy = 0 in D0 (Σ). Then there exist f ∈ Let ϕ ∈ Hloc 1 1 Hloc (K1 ), g ∈ Hloc (K2 ) such that ϕ(x, y) = f (x) + g(y)
a.e. in Σ .
Proof. Since Σ is nonempty, open and connected, the same holds for K1 and K2 ; so K1 and K2 are nonempty open intervals. 1 (Σ), ϕx ∈ L2loc (Σ) ⊂ L1loc (Σ). Further ϕxy = 0 in D0 (Σ) implies Since ϕ ∈ Hloc 1 that ∂y (ϕx ) ∈ L (Σ). Consequently (e.g. Kufner, John and Fuˇc´ık [24, Theorem 5.6.3, p. 274]), there exists a function u ∈ L1loc(Σ) such that: (i) u = ϕx ˆ 1 ⊂ K1 with m(K1 \ K ˆ 1 ) = 0 such that for all x ∈ K ˆ 1, a.e. in Σ, (ii) there exists K Jx 3 y 7→ u(x, y) is absolutely continuous and (iii) uy = ϕxy = 0 a.e. in Σ. ˜ 1) = 0 ˜ 1 ⊂ K1 with m(K1 \ K By Fubini’s theorem, (iii) ⇒ (iv): there exists K 0 ˆ1 ∩ K ˜ 1; ˜ 1 , uy (x, y) = 0 for almost every y ∈ Jx . Let K1 := K such that for all x ∈ K 0 0 m(K1 \ K1 ) = 0. The statements (ii) and (iv) imply that for all x ∈ K1 , Jx 3 y 7→ u(x, y) is constant. Let φ : K1 → R be the function defined by φ(x) := u(x, y) if x ∈ K10 and y ∈ Jx (φ is arbitrary on K1 \ K10 which is of zero measure). Let us show that φ ∈ L2loc(K1 ). It is sufficient to prove that for all x ∈ K1 , there exists r > 0 such that φ ∈ L2 ((x − r, x + r)). Let x0 ∈ K1 . Thus Jx0 6= ∅ and there exists r > 0 such that B ∞ ((x0 , y0 ), r) ⊂ Σ for some y0 ∈ Jx with B ∞ ((x0 , y0 ), r) := {(x, y) ∈ R2 ; max{|x − x0 |, |y − y0 |} < r}. Then for all x ∈ (x0 − r, x0 + r) ∩ K10 and for all y ∈ (y0 − r, y0 + r), one has u(x, y) = φ(x). The statement (i) implies
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that for almost every y ∈ (y0 − r, y0 + r), one has u(·, y) ∈ L2 ((x0 − r, x0 + r)). Let y1 ∈ (y0 − r, y0 + r) satisfying this assertion. Then for all x ∈ (x0 − r, x0 + r) ∩ K10 , one has φ(x) = u(x, y1 ); thus φ ∈ L2 ((x0 − r, x0 + r)). Since ϕ, ϕx ∈ L1loc(Σ), we use again the theorem in Kufner, John and Fuˇc´ık [24, Theorem 5.6.3, p. 274]: there exists a function v ∈ L1loc(Σ) such that: (i’) v = ϕ ˆ 2 ) = 0 such that for all y ∈ K ˆ 2, ˆ 2 ⊂ K2 with m(K2 \ K a.e. in Σ, (ii’) there exists K y J 3 x 7→ v(x, y) is absolutely continuous and (iii’) vx = ϕx = u a.e. in Σ. ˜ 2 ⊂ K2 with m(K2 \ Again by Fubini’s theorem, (iii’) ⇒ (iv’): there exists K ˜ ˜ K2 ) = 0 such that for all y ∈ K2 , vx (x, y) = u(x, y) for almost every x ∈ J y . By (ii’), with ay ∈ J y arbitrary: Z x ˆ 2 , ∀x ∈ J y , v(x, y) = v(ay , y) + vx (z, y) dz . (55) ∀y ∈ K ay
Using (iv’), (55) implies ˜ 2 =: K 0 , ˆ2 ∩ K ∀y ∈ K 2
Z ∀x ∈ J y ,
x
v(x, y) = v(ay , y) +
u(z, y) dz ay
Z
x
= v(ay , y) +
φ(z) dz . ay
Let a0 ∈ K1 and define: Z ∀x ∈ K1 ,
x
f (x) :=
φ(z) dz , a0
∀y ∈ K2 ,
Z
a0
g(y) := v(ay , y) +
φ(z) dz . ay
Since φ ∈ L1loc(K1 ), f is absolutely continuous on K1 and f 0 = φ a.e. on K1 : 1 (K1 ). We know that g : K2 → R is a function. Thus it follows that f ∈ Hloc ϕ(x, y) = v(x, y) = f (x) + g(y) a.e. on Σ. By changing the role of f and g, there 1 (K2 ) and a function f1 : K1 → R such that ϕ(x, y) = f1 (x) + g1 (y) exist g1 ∈ Hloc a.e. in Σ. Hence f1 = f − c a.e. and g1 = g + c a.e. for some c ∈ R. This yields ϕ(x, y) = f (x) + g(y) a.e. in Σ , 1 1 (K1 ), g ∈ Hloc (K2 ). with f ∈ Hloc
1 (Ω) satisfies Lemma B. Let a ∈ Lip(R), L(a) ∈ [0, 1) and a > 0. If ϕ ∈ Hloc 0 1 ϕtt − ϕxx = 0 in D (Ω), then there exist f, g ∈ Hloc (R) such that
ϕ(x, t) = f (t + x) + g(t − x)
a.e. in Ω .
¯ the traces of ϕ on each line t = Moreover ϕ can be continuously extended to Ω, 1 constant are in H ((0, a(constant))) and the traces of ϕ on the boundary ∂Ω of Ω 1 (∂Ω). are in Hloc
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Proof. The first statement is obvious by a change of variables and by using Lemma A. Then the second statement follows from the fact that f and g are in 1 (R) (and in particular f and g are continuous). Hloc Proof of Theorem 2.13. First step: by Lemma B, if a weak solution ϕ exists, then ϕ(x, t) = f (t + x) + 1 (R). g(t − x) a.e. in Ω with f , g ∈ Hloc Second step: the function ϕx has a trace on x = 0 by Lemma B and (6) implies that f 0 = g 0 a.e. Then g = f + c, where c ∈ R. Denote again by f the function f + 2c . Then ϕ(x, t) = f (t + x) + f (t − x) a.e. in Ω. Third step: the functions ϕ and ϕt have a trace on t = 0. Then f is given on I0 by the initial conditions ϕ0 and ϕ1 and by formulas similar to (19)–(20); thus f ∈ H 1 (I0 ). Fourth step: the function ϕx has a trace on x = a(t), thus (7) implies that f0 ◦ F = f0
a.e. in R .
Fifth step: by successive iterations, for all n ∈ Z, f 0 ◦ F n = f 0 a.e., thus f 0 is known a.e. on R. Up to a constant (depending on n) f is known on In by integration of f 0 . Using the third step the constants are determined by continuity 1 1 (R) and ϕ ∈ Hloc (Ω). of f . Obviously f ∈ Hloc Sixth step: by the third and fifth steps, f is unique on R. Thus ϕ is unique. B. Glossary a : R → R?+ , the law of the moving boundary, An := a ∈ C n (T); a > 0, a ∈ Lip(T), L(a) ∈ [0, 1)} a.e. almost everywhere BV(X), the space of functions on X of bounded variations, C k (X), the space of k-times continuously differentiable functions on X, C ω (X), the space of real analytic functions on X, C β (X) := Lipβ (X), whenever β ∈ (0, 1), Diffk (R), the C k -diffeomorphisms of R, Dk (T) the set of lifts to R of orientation-preserving C k diffeomorphisms of T, D0 (X) := (C0∞ (X))0 , the space of bounded linear functionals on the space of smooth functions with compact support in X, Diophantine number, a real number in a sense badly approximated by rationals, see Definition 3.10, E : R → R?+ , the energy of the field ϕ, F := k ◦ h−1 , F q , F composed by itself q times, F (k) or Dk F , the kth derivative of the function F , Fn , the set of functions F related to a ∈ An , i.e. Lipschitz homeomorphisms F ∈ Dn (T) such that F > Id, see Lemma 3.2.
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H 1 (X), Sobolev space of square integrable functions with square integrable generalized derivatives H01 (X) the closure of the space of smooth functions with compact support in H 1 (X); the fuctions satisfying Dirichlet boundary condition 1 1 (X), H0,loc (X), the spaces of functions the multiple of which by any smooth Hloc function of compact support are in H 1 (X) or H01 (X) h := Id − a, k := Id + a Id, the identity on R, In := [F n (−a(0)), F n (a(0))), n ∈ Z, ˙ the interior of a set X, Int X := X, (y) |, the Lipschitz constant of F ∈ Lip(X), L(F ) := supx,y∈X,x6=y | F (x)−F x−y Lipβ (X), the space of H¨older continuous functions with exponent β ∈ (0, 1], Lip(X) := Lip1 (X), the set of Lipschitz continuous functions, Morse–Smale diffeomorphism, a function from D1 (T) having nonzero finite number of periodic points which are all hyperbolic, m(B), the Lebesgue measure of a set B, P , function of class P , see Definition 2.6, periodic point of F ∈ D0 (T), a point in R with the projection in T invariant under F¯ q for some q ∈ N? , hyperbolic periodic point, a periodic point x with Dq F (x) 6= 1 ϕ, the real scalar field satistying the d’Alembert equation, ϕ0 , ϕ1 the initial value and initial velocity (the time derivative) of the field, π : R → T, x 7→ x + Z, the canonical projection, Q, function of class Q, see Definition 2.7, Rρ(F ) := Id + ρ(F ), the translation by ρ(F ) in R, i.e. the lift of rotation by the angle ρ(F ) in T, ρ(F ), the rotation number of F , whenever F ∈ D0 (T); a mean rotation angle of points in T under the projection F¯ ; see Eq. (9), T, the one dimensional torus (the circle of unit length), u.c.s., up to a countable set, d ) := inf{Var(b); b : T → R, b(x) = f (x) u.c.s. in T}, Var(f W (F ), the set of wandering point of F , see Definition 3.18, w.r.t., with respect to ? := {x ∈ X; x > 0} whenever X is a X ? := X \ {0}, X+ := {x ∈ X; x ≥ 0}, X+ subset of the real numbers, [x] the integer part of a real number x, kF k0 := kF kL∞ (X) := supx∈X |F (x)|, for a function F defined on X, kF kk := max1≤i≤k kF (i) k0 , kF kX , the L2 -norm of F on the set X, Fmin , Fmax , the essentiel infimum, resp. supremum of a real function F , F¯ , the projection of F on T whenever F is one-periodic, ¯ the closure of a set Ω, Ω, ∂Ω, the boundary of a set Ω
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Acknowledgments We wish to thank S. Vaienti for fruitful discussions. This work is partly supported by GACR grant No. 202/96/0218 (J. D.) and by a grant of the French Ministry of Foreign Affairs and the Czech Ministry of Education (N. G.). J. D. thanks the Centre de Physique Th´eorique of Marseille and the PhyMat and P. D., N. G. thank the Nuclear Physics Institute for hospitality. References [1] E. Fermi, “On the origin of the cosmic radiation”, Phys. Rev. 75 (1949) 1169–1174. [2] S. Ulam, “On some statistical properties of dynamical systems”, in Proceedings of the Fourth Berkeley Symposium on Mathematical and Statistical Problems, Vol. 3, Univ. of California Press, Berkeley, 1961, pp. 315–320. [3] L. D. Pustyl’nikov, “On Ulam’s problem”, Theoret. Math. Phys. 57 (1983) 1035–1038. [4] L. D. Pustyl’nikov, “Poincar´e models, rigorous justification of the second element of thermodynamics on the basis of mechanics, and the Fermi acceleration mechanism”, Russian Math. Surveys 50 (1995) 145–189. [5] L. D. Pustyl’nikov, “A new mechanism for particle acceleration and a relativistic analogue of the Fermi–Ulam model”, Theoret. Math. Phys. 77 (1988) 1110–1115. [6] L. D. Pustyl’nikov, “A new mechanism of particle acceleration and rotation numbers”, Theoret. Math. Phys. 82 (1990) 180–187. [7] G. Karner, “On the quantum Fermi accelerator and its relevance to “quantum chaos””, Lett. Math. Phys. 17 (1989) 329–339. ˇ [8] P. Seba, “Quantum chaos in the Fermi-accelerator model”, Phys. Rev. A41 (1990) 2306–2310. [9] S. T. Dembi´ nski, A. J. Makowski and P. Peplowski, “Quantum bouncer with chaos”, Phys. Rev. Lett. 70 (1993) 1093–1096. [10] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, “Quantum particle in a box with moving walls”, J. Math. Phys. 34 (1993) 3391–3404. ˇˇtov´ıˇcek, “Floquet hamiltonians with pure point spectrum”, Com[11] P. Duclos and P. S mun. Math. Phys. 177 (1996) 327–347. [12] N. Balazs, “On the solution of the wave equation with moving boundaries”, J. Math. Anal. Appl. 3 (1961) 472–484. [13] J. Cooper, “Asymptotic behavior for the vibrating string with a moving boundary”, J. Math. Anal. Appl. 174 (1993) 67–87. ˇ [14] J. Dittrich, P. Duclos and P. Seba, “Instability in a classical periodically driven string”, Phys. Rev. E49 (1994) 3535–3538. [15] J. Cooper and H. Koch, “The spectrum of a hyperbolic evolution operator”, J. Funct. Anal. 133 (1995) 301-328. [16] O. M´eplan and C. Gignoux, “Exponential growth of the energy of a wave in a 1D vibrating cavity: Application to the quantum vacuum”, Phys. Rev. Lett. 76 (1996) 408–410. [17] J. Cooper, “Long time behavior and energy growth for electromagnetic waves reflected by a moving boundary”, IEEE Transac. Ant. Prop. 41 (1993) 1365–1370. [18] G. T. Moore, “Quantum field theory of the electromagnetic field in a variable-length one-dimensional cavity”, J. Math. Phys. 11 (1970) 2679–2691. [19] G. Calucci, “Casimir effect for moving bodies”, J. Phys. A: Math. Gen. 25 (1992) 3873–3882 and corrigenda: J. Phys. A: Math. Gen. 26 (1993) 5636. [20] V. V. Dodonov, A. B. Klimov and D. E. Nikonov, “Quantum phenomena in resonators with moving walls”, J. Math. Phys. 34 (1993) 2742–2756. [21] H. Johnston and S. Sarkar, “A re-examination of the quantum field theory of optical cavities with moving mirrors”, J. Phys. A: Math. Gen. 29 (1996) 1741–1746.
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[22] M. Herman, “Sur la conjugaison diff´ erentiable des diff´eomorphismes du cercle ` a des rotations”, Publ. Math. I.H.E.S. 49 (1979) 5–234 (in French). [23] A. Katok and B. Hasselblatt, “Introduction to the Modern Theory of Dynamical Systems”, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995. [24] A. Kufner, O. John and S. Fuˇc´ık, Function Spaces, Noordhoff Int. Publ., Leyden; Academia, Prague, 1977. [25] O. A. Ladyzhenskaya, “The Boundary Value Problems of Mathematical Physics”, Appl. Math. Sci. 49, Springer-Verlag, New York-Berlin, 1985. [26] J. L. Lions and E. Magenes, “Probl`emes aux Limites Non Homog`enes et Applications. Vol. I”, Travaux et Recherches Math´ematiques 17, Dunod, Paris, 1968 (in French). [27] S. Lang, Introduction to Diophantine Approximations, Second Ed., Springer-Verlag, New York, 1995. [28] Y. Katznelson and D. Ornstein, “The differentiability of the conjugation of certain diffeomorphisms of the circle”, Erg. Theory Dynam. Systems 9 (1989) 643–680. [29] N. Gonzalez, “L’equation des ondes dans un domaine d´ependant du temps”, Doctoral Thesis, UTV Toulon and CTU Prague, 1997 (in French). [30] V. I. Arnol’d, “Small denominators I. Mappings of the circumference onto itself”, Translations A.M.S. Series 2 46 (1965) 213–284. [31] Z. Nitecki, Differentiable Dynamics, M.I.T. Press, Cambridge-Massachusetts-London, 1971.
THE UNIQUENESS OF THE SOLUTION OF ¨ THE SCHRODINGER EQUATION WITH DISCONTINUOUS COEFFICIENTS ¨ WILLI JAGER University of Heidelberg Institute for Applied Mathematics D-69120 Heidelberg Germany
¯ YOSHIMI SAITO Department of Mathematics University of Alabama at Birmingham Birmingham, Alabama 35294 USA Received 15 March 1997 Revised 20 October 1997
1. Introduction Let us consider the reduced wave equation −∆u(x) + q(x)u(x) = 0
(1.1)
Ω ⊃ ER0 = {x ∈ RN : |x| > R0 } ,
(1.2)
on the domain Ω such that
where R0 > 0 and N ≥ 2. Suppose that q(x) has the form q(x) = −`(x) + s(x) ,
(1.3)
where `(x) a positive function, and |s(x)| is supposed to be dominated by `(x). The Eq. (1.1) has been studied extensively especially in the relation to the operator H1 = −∆ + V (x)
(1.4)
in L2 (Ω), or 1 ∆ (1.5) µ(x) in the weighted Hilbert space L2 (Ω; µ(x)dx) with boundary conditions on the boundary ∂Ω and at infinity. In this work we are concerned with the asymptotic behavior of the solution u of Eq. (1.1) at infinity. One of the important conclusions of the study is that we can establish the nonexistence of a class of (nontrivial) solutions of (1.1) which includes the L2 -solutions. And this result plays an important role in the attempt (the limiting absorption method, see, e.g. [1, 4]) to prove the H2 = −
963 Reviews in Mathematical Physics, Vol. 10, No. 7 (1998) 963–987 c World Scientific Publishing Company
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existence of the boundary value of the resolvent (H1 − z)−1 or (H2 − z)−1 of the operator H1 or H2 when the complex parameter z approaches the real axis. Consider the equation −∆u(x) + (−k 2 + s(x))u(x) = 0 ,
(1.6)
with k > 0, i.e., `(x) = k 2 in (1.3). In the celebrated work Kato [9] he showed, among others, that, under the condition τ ≡ (2k)−1 lim |x||s(x)| < 1 , |x|→∞
a nontrivial solution u of Eq. (1.1) satisfies Z (|∇u(x)|2 + |u(x)|2 ) dS = ∞ lim r2τ + r→∞
(1.7)
(1.8)
|x|=r
for any > 0. One of the important features of the work [9] is that the coefficient s(x) does not need to be spherically symmetric which makes the scope of application much wider than the preceding works (cf., e.g., M¨ uller [10], Rellich [11]). Another important feature of [9] is that the method is based on differential inequalities satisfied by several functionals of the solution u so that the problem was successfully treated as a local problem at infinity. As a result we do not need to use any boundary conditions at the boundary ∂Ω of Ω or at infinity such as radiation condition (cf., e.g., Wienholtz [15]). As is well-known this result has many applications. In Ikebe [3], in which the spectral theory and scattering theory for the Schr¨ odinger operator −∆ + V (x) in R3 was developed under the condition that V (x) = O(|x|−γ ) (|x| → ∞, γ > 2) ,
(1.9)
the result of Kato [9] was used to prove the existence of the boundary value of the Green function on the positive real axis as well as the nonexistence of the positive eigenvalues. After the work [9] various extensions and modifications were presented as many efforts were made to treat more general operators in a similar method. See e.g., Ikebe–Uchiyama [5] for Schr¨ odinger operators with magnetic potentials, J¨ ager [6] for the second order elliptic operators, Weidmann [14] for the many body Schr¨ odinger operators, and Ikebe–Sait¯ o [4], Sait¯ o [13] for Schr¨ odinger operators with long-range potentials. Now let us consider the case that `(x) is a positive function which may be discontinuous. One of the motivations to consider such `(x) comes from the study of the reduced wave equations in layered media. Consider the equation −µ(x)−1 ∆u − λu = 0 (x ∈ RN )
(1.10)
in layered media, where µ(x) is a positive function on RN . Suppose that the function µ(x) is a simple function with surfaces of discontinuity (separating surfaces) which may extend to infinity. Roach and Zhang [12] proved the nonexistence of the solution of Eq. (1.10) under a geometric condition (“cone-like” discontinuity on the
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separating surface, see also [2]). Then J¨ager and Sait¯ o [7] proved a similar results under another geometric condition (“cylindrical” discontinuity) on the separating surfaces. In these works the method is not local at infinity, but some global integral identity of the solution u are used. And the method seems to need some modifications in the case where µ is a perturbation such as µ(x) = µ0 (x) + µ` (x) + µs (x) ,
(1.11)
µ0 being a simple function, and µ` (x) and µs (x) behaving like a long-range and short-range potentials at infinity, respectively (cf. [8]). In this work we are going to obtain an extension of the result (1.8) by Kato [9] which can be applied the reduced wave Eq. (1.10) with µ(x) satisfying (1.11) as well as Eq. (1.6) where s(x) is the sum of a short-range potential and a long-range potential. Under the several assumptions (Assumptions 2.1, 4.2, 5.5 and 5.8) on the coefficient q(x) the following (Theorem 5.10 in Sec. 5) will be proved: Suppose that a solution u of Eq. (1.1) satisfies Z ∂u 2 2 lim − Re (q(x))|u| dS = 0 . ∂r r→∞ |x|=r
(1.12)
Then u has a compact support. Our method is a local method at infinity which is similar to the method of Kato [9]. As in [9], some type of differential inequalities on functionals of the solution u will play important roles. However, we shall first establish the differential inequalities not in the ordinary sense but in the sense of distributions, and then they will be interpreted in the ordinary sense. In Sec. 2 we define our reduced wave equation and give the main assumption (Assumption 2.1) on the coefficients. In Sec. 3 we introduce and evaluate the first functional M + (v, r). In order to complete the evaluation of M + (v, r), another functional N (v, m, r) is introduced and evaluated in Sec. 4. Section 5 is devoted for proving the main theorem (Theorem 5.9). Some examples are discussed in Sec. 6. In Sec. 7 we shall discuss how our result can be applied to some reduced wave operators which were studied in [8]. A lemma on distributional derivative is given in Appendix. 2. Schr¨ odinger-Type Homogeneous Equation Consider the homogeneous Schr¨ odinger equation −∆u(x) + q(x)u(x) = 0 (x ∈ ER0 ) ,
(2.1)
ER = {x ∈ RN : |x| > R} .
(2.2)
where R0 > 0, and Let S N −1 be the unit sphere of RN . We set X = L2 (S N −1 ) and the inner product and norm of X is denoted by ( , ) and | |, respectively.
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Assumption 2.1. (i) Let N be an integer such that N ≥ 2. Let u ∈ H 2 (ER0 )loc , R0 > 0, be a solution of Eq. (2.1), where q(x) is a complex-valued, measurable, locally bounded function on ER0 . (ii) Set (N − 1)(N − 3) . (2.3) Q(x) = q(x) + 4r2 (a) Then Q(x) is decomposed as Q(x) = Q0 (x) + Q1 (x) ,
(2.4)
where Q0 (x) is a real-valued, measurable, locally bounded function on ER0 such that (2.5) Q0 (x) ≤ 0 , and Q1 (x) is a complex-valued, measurable, locally bounded function on ER0 . (b) For any x ∈ X = L2 (S N −1 ), (Q0 (r·)x, x) has the right limit for all r > R0 as a function of r = |x|. (c) There exist h0 > 0 and, for 0 < h < h0 , a real-valued, measurable function Q0r (x; h) on ER0 such that sup {|Q0r (x; h)|/x ∈ G ,
0 < h < h0 } < ∞
(2.6)
for any compact set G ⊂ ER0 , 1 ({Q0 ((r + h)·) − Q0 (r·)}φ, φ) h ≤ (Q0r (r·; h)φ, φ) (φ ∈ X, r > R0 , 0 < h ≤ h0 ) ,
(2.7)
and the limit lim(Q0r (r·; h)φ, φ) = (Q0r (r·)φ, φ) h↓0
(φ ∈ X)
(2.8)
exists with a real-valued, measurable, locally bounded function Q0r (x) on ER0 . (iii) There exists a positive, measurable function h(r) defined on (R0 , ∞) such that 2 (a) (r > R0 ) , (2.9) h(r) ≤ r (b) and, setting a(r) = h−1 (r) sup |Q1 (x)| , |x|=r (2.10) b(r) = inf [−(Q0 (x) + h−1 (r)Q0r (x))] , |x|=r
where h−1 (r) = 1/h(r), we have a(r)2 ≤ b(r)
(r > R0 ) .
(2.11)
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In order to transform Eq. (2.1) into a differential equation on (R0 , ∞) with operator-valued coefficients, we give the following: Definition 2.2. (i) For r > R0 define a selfadjoint operator B(r) in X by ( D(B(r)) = D(ΛN ) , (2.12) B(r) = −r−2 ΛN , where D(T ) is the domain of T , and ΛN is the (selfadjoint realization of) Laplace– Beltrami operator on S N −1 . (ii) tit For r > R0 define a bounded operators C0 (r), C0r (r; h), C0r (r) and C1 (r) on X by C0 (r) = Q0 (r·)× , C0r (r; h) = Q0r (r·; h)× , (2.13) C0r (r) = Q0r (r·)× , C1 (r) = Q1 (r·) × . Proposition 2.3. Let u be a solution of Eq. (2.1) and let v be as in Assumption 2.1, (ii). Let J = (R0 , ∞). Then, (i) v ∈ C 1 (J, X). (ii) v(r) ∈ D((−ΛN )1/2 ) for r ∈ J. (iii) We have Z s {|v 0 (r)|2 + |B 1/2 (r)v(r)|2 } dr < ∞
(R0 < r < s < ∞) ,
(2.14)
r
(iv) (v)
(vi) (vii)
where v 0 (r) = dv(r)/dr and B 1/2 (r) = B(r)1/2 . v(r) ∈ D(ΛN ) for almost all r ∈ J, and Bv ∈ L2 ((r, s), X) for R0 < r < s < ∞. v 0 (r) ∈ Cac ([r, s], X) for R0 < r < s < ∞, where Cac ([r, s], X) is all X-valued absolutely continuous functions on [r, s]. There exists the weak derivative v 00 (r) of v 0 (r) for r ∈ J. v 0 (r) ∈ D((−ΛN )1/2 ) for almost all r ∈ J, and B 1/2 v 0 ∈ L2 ((r, s), X) for R0 < r < s < ∞. B 1/2 v ∈ Cac ([r, s], X) for R0 < r < s < ∞, and we have 2 d (B 1/2 (r)v(r), B 1/2 (r)v(r)) = − (B 1/2 (r)v(r), B 1/2 (r)v(r)) dr r +2 Re (B 1/2 (r)v 0 (r), B 1/2 (r)v(r))
for almost all r ∈ J. (viii) We have − v 00 (r) + B(r)v(r) + C0 (r)v(r) + C1 (r)v(r) = 0 in X for almost all r ∈ J.
(2.15)
(2.16)
968
¨ ¯ W. JAGER and Y. SAITO
Proof. See [13], Proposition 1.3.
Proposition 2.4. Suppose that Q0r (x) satisfies Assumption 2.1, (ii-b) and (ii-c). Let η ∈ C 1 (J, X). Let g(r) =
d (C0 (r)η(r), η(r)) dr
(2.17)
be the derivative of f (r) = (C0 (r)η(r), η(r)) in the sense of distributions on (R0 , ∞). Then we have g(r) ≤ (C0r (r)η(r), η(r)) + 2 Re (C0 (r)η(r), η 0 (r)) ,
(2.18)
where Ineq. (2.18) should be taken in the sense of distributions on (R0 , ∞) again. Proof. (I) Let ϕ be a nonnegative C0∞ ((R0 , ∞)) function. Then, by definition hg, ϕi = −hf, ϕ0 i Z ∞ ϕ(r + h) − ϕ(r) dr f (r) = − lim h↑0 R0 h Z 1 ∞ (f (r + h) − f (r))ϕ(r) dr, = lim h↓0 h R0
(2.19)
where h , i denotes the dual pair bracketing. (II) Here we have f (r + h) − f (r) = (C0 (r + h)η(r + h), η(r + h)) − (C0 (r)η(r), η(r)) = ({C0 (r + h) − C0 (r)}η(r + h), η(r + h)) +(C0 (r)η(r + h), η(r + h) − η(r)) +(η(r + h) − η(r), C0 (r)η(r)) ,
(2.20)
and hence, using (2.7) in (ii-b) of Assumption 2.1, we obtain 1 (f (r + h) − f (r)) ≤ (C0r (r; h)η(r + h), η(r + h)) h 1 + C0 (r)η(r + h), (η(r + h) − η(r)) h 1 (η(r + h) − η(r)), C0 (r)η(r) . + h
(2.21)
(III) It is easy to see from (2.8) in (ii-c) of Assumption 2.1 and (i) of Proposition 2.3 that the right-hand side of (2.21) converges to g0 (r) ≡ (C0r (r)η(r), η(r)) + 2 Re (C0 (r)η(r), η 0 (r))
(2.22)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
969
boundedly on any compact interval in (R0 , ∞) as h ↓ 0. Therefore, noting that ϕ ≥ 0, we have Z ∞ g0 (r)ϕ(r) dr = hg0 , ϕi , (2.23) hg, ϕi ≤ R0
which completes the proof. 3. The Evaluation of the Functional M + (v, r)
Let v = v(r·) be as in (2.5). Then we are going to define the functional M + (v, r) by Definition 3.1. Let v be as in (ii-b) of Assumption 2.1. Then set M + (v, r) = |v 0 (r)|2 − (C0 (r)v(r), v(r)) − |B 1/2 (r)v(r)|2
(3.1)
for r > R0 . Proposition 3.2. Suppose that Assumption 2.1 is satisfied. Let M + (v, r) be as in Definition 3.1. (i) Then M + (v, r) is a real-valued, locally bounded function on J = (R0 , ∞). Further M + (v, r) is right continuous with its left limit for r ∈ J. (ii) We have d + M (v, r) ≥ −h(r)M + (v, r) (r > R0 ) , (3.2) dr where Ineq. (3.2) should be taken in the sense of distributions on (R0 , ∞). Proof. (i) follows from Assumption 2.1, (ii) and Proposition 2.3. From Propositions 2.3 and 2.4 we see that d + M (v, r) ≥ 2 Re (v 00 (r), v 0 (r)) dr −(C0r (r)v(r), v(r)) − 2 Re (C0 (r)v(r), v 0 (r)) 2 + (B(r)v(r), v(r)) − 2 Re (B(r)v(r), v 0 (r)) r
(3.3)
in the sense of distributions on (R0 , ∞). Using (2.16), we have from (3.3) d + M (v, r) ≥ −(C0r (r)v(r), v(r)) + 2 Re (C1 (r)v(r), v 0 (r)) dr 2 + (B(r)v(r), v(r)) r = −h(r)M + (v, r) +h(r) |v 0 (r)|2 − (C0 (r)v(r), v(r)) − (B(r)v(r), v(r)) −(C0r (r)v(r), v(r)) + 2 Re (C1 (r)v(r), v 0 (r)) 2 + (B(r)v(r), v(r)) r
(3.4)
¨ ¯ W. JAGER and Y. SAITO
970
Thus, using a(r) and b(r) defined by (2.10), and taking note of (2.9) in Assumption 2.1, we have d + M (v, r) ≥ −h(r)M + (v, r) dr +h(r)[|v 0 (r)|2 − 2a(r)|v 0 (r)||v(r)| + b(r)|v(r)|2 ] .
(3.5)
It follows from (2.11) in Assumption 2.1, that is, a(r)2 ≤ b(r), that (3.5) implies (3.2), which completes the proof. Proposition 3.3. Suppose that Assumption 2.1 is satisfied. For R1 > R0 we have Z r + h(t) dt M + (v, R1 ) (r ≥ R1 ) . (3.6) M (v, r) ≥ exp − R1
Proof. It follows from Proposition 3.2 that Z r d + M (v, r) h(t) dt exp dr R1 Z r h(t) dt M + (v, r) ≥ 0 (r > R1 ) , +h(r) exp
(3.7)
R1
and hence
Z r d + h(t) dt M (v, r) ≥ 0 exp dr R1
(r > R1 )
(3.8)
in the sense of distributions on (R1 , ∞). The Ineq. (3.6) follows from (3.8) and Lemma A of Appendix. 4. The Evaluation of the Functional N (v, m, r) Using M + (v, r), we are going to define another functional which will be used to evaluate M + (v, r) in Sec. 5. Definition 4.1. (i) Set N (v, m, r) = M + (w, r) + (m(m + 1) − F (r))r−2 |w|2
(w = rm v) ,
(4.1)
where m is a positive number and F (r) is a positive C 1 function on (R0 , ∞). (ii) For r > R0 define a bounded operators CR (r) on X by CR (r) = Re (Q(r·))× = (Q0 (r·) + Re (Q1 (r·)) × .
(4.2)
M (v, r) = |v 0 (r)|2 − (CR (r)v(r), v(r)) .
(4.3)
Set (iii) For r > R0 we set p(r) = inf [−(2Q0 (x) + rQ0r (x))] . |x|=r
(4.4)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
971
Assumption 4.2. The function F (r) introduced in Definition 4.1 satisfies the following (i)–(iii): (i) There exists a positive constant c0 such that F 2 (r) ≤ c0 r4 h2 (r)b(r)
(r > R0 ) ,
(4.5)
where b(r) is given in (2.10), F 2 (r) = F (r)2 , and h2 (r) = h(r)2 . (ii) We have F (r) → ∞ as r → ∞. (iii) There exists a positive constant c1 such that Fr (r) ≡
d F (r) ≤ c1 r−1 dr
(r > R0 ) .
(4.6)
Proposition 4.3. (i) Let b(r) be given by (2.10) and assume that h(r) satisfies Ineq. (2.9) and that Q0 (x) is nonpositive. Then r2 h2 (r)b(r) ≤ 2p(r)
(r > R0 ) .
(4.7)
(ii) Assume that Ineq. (2.11) holds. Then !2 r sup |Q1 (x)|
≤ 2p(r)
|x|=r
(r > R0 ) .
(4.8)
(iii) Suppose that Ineq. (4.5) holds. Then r−2 F 2 (r) ≤ 2c0 p(r)
(r > R0 ) .
(4.9)
Proof. Since 0 < rh(r) ≤ 2 and −Q0 (x) ≥ 0, we have r2 h2 (r)[−(Q0 (x) + h−1 (r)Q0r (x))] = rh(r)[rh(r)(−Q0 (x)) + r(−Q0r (x))] ≤ 2[2(−Q0 (x)) + r(−Q0r (x))] = 2[−(2Q0 (x) + rQ0r (x))] ,
(4.10)
which implies (4.7). It follows from (2.11) and (4.7) that !2 r sup |Q1 (x)|
!2 2 2
= r h (r) h
−1
|x|=r
(r) sup |Q1 (x)| |x|=r
≤ r h (r)b(r) 2 2
≤ 2p(r) .
(4.11)
From (4.5) and (4.7) we obtain F 2 (r) ≤ c0 r4 h2 (r)b(r) ≤ c0 r2 (2p(r)) = r2 (2c0 p(r)) for r > R0 , which implies (4.9).
(4.12)
¨ ¯ W. JAGER and Y. SAITO
972
Proposition 4.4. Suppose that Assumptions 2.1 and 4.2 hold. Then there exist m0 > 0 and r0 > R0 such that d 2 (r N (v, m, r)) ≥ 0 dr in the sense of distributions on (r0 , ∞).
(m ≥ m0 )
Proof. (I) By definition w = rm v satisfies 0 w = rm v 0 + mrm−1 v = rm v 0 + mr−1 w , 00 m 00 m−1 0 v + m(m − 1)rm−2 v w = r v + 2mr m 00 −1 w0 − mr−1 w + m(m − 1)r−2 w = r v + 2mr = rm v00 + 2mr−1 w0 − m(m + 1)r−2 w = rm (Bv + C0 v + C1 v) + 2mr−1 w0 − m(m + 1)r−2 w ,
(4.13)
(4.14)
and hence we have −w00 + 2mr−1 w0 + (B + C0 + C1 − m(m + 1)r−2 )w = 0 .
(4.15)
g(r, m) = (m(m + 1) − F (r))r−2 .
(4.16)
(II) Set Then, using (4.15) and Proposition 2.4, we have r−2
d 2 (r N (v, m, r)) ≥ 2r−1 (|w0 |2 − (C0 w, w) − (Bw, w) + (gw, w)) dr +2 Re (w00 − C0 w − Bw + gw, w0 ) 2 + |B 1/2 w|2 − (C0r w, w) + (gr w, w) r = 2r−1 (|w0 |2 − (C0 w, w) − (Bw, w) + (gw, w)) +2 Re (2mr−1 w0 + C1 w − m(m + 1)r−2 w + gw, w0 ) 2 + |B 1/2 w|2 − (C0r w, w) + (gr w, w) r = 2(1 + 2m)r−1 |w0 |2 + (2r−1 g + gr )|w|2 +r−1 ([− 2C0 − rC0r ]w, w) +2 Re (C1 w − m(m + 1)r−2 w + gw, w0 ) ,
where gr = dg/dr. Note that ( −1 2r g(r, m) + gr (r, m) = −Fr (r)r−2 , g(r, m) − m(m + 1)r−2 = −F (r)r−2 .
(4.17)
(4.18)
Then the above inequality (4.17) can be rewritten as r−2
d 2 (r N (v, m, r)) ≥ 2(1 + 2m)r−1 |w0 |2 + r−1 (p(r) − r−1 Fr (r))|w|2 dr −2 Re ((r−2 F (r) − C1 )w, w0 ) ,
where p(r) is as in (4.4).
(4.19)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
973
(III) It follows from Assumption 4.2, (iii) and Proposition 4.3, (ii), (iii) that
p(r) − r−1 Fr (r) ≥ p(r) − c1 r−2 = r−2 (r2 p(r) − c1 ) , |r−2 F (r) − Q1 (x)| ≤ r−1 (2c0 p(r))1/2 + r−1 (2p(r))1/2 √ √ = c2 r−1 p1/2 (r) (c2 = 2c0 + 2) .
(4.20)
Further, from (iii) of Proposition 4.3 and (ii) of Assumption 4.2 we see that r2 p(r) ≥ (2c0 )−1 F 2 (r) → ∞ (r → ∞) , and hence there exists r0 > R0 such that −2 2 r (r p(r) − c1 ) = p(r) 1 −
c1 r2 p(r)
≥ 2−1 p(r)
(4.21)
(4.22)
for r ≥ r0 . Thus, we obtain from (5.19) r−2
d 2 (r N (v, m, r)) dr ≥ 2(1 + 2m)r−1 |w0 |2 + 2−1 r−1 p(r)|w|2 − 2c2 r−1 p1/2 (r)|w0 ||w| = r−1 [2(1 + 2m)|w0 |2 + 2−1 p(r)|w|2 − 2c2 p1/2 (r)|w0 ||w|]
(4.23)
for r ≥ r0 . Therefore, there exists a sufficiently large m0 > 0 such that r2
d 2 (r N (v, m, r)) ≥ 0 dr
for r ≥ r0 and m ≥ m0 , which completes the proof.
(4.24)
5. Uniqueness Theorem We are going to prove our main theorem (Theorem 5.10) which shows, under Assumptions 2.1 and 4.1, and some additional conditions (Assumptions 5.5 and 5.8), that the solution u has compact support if u satisfies ∂u 2 − Re (q(x))|u|2 dS = 0 . ∂r |x|=r
Z lim r→∞
(5.1)
Proposition 5.1. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Let r0 and m0 be as in Proposition 4.4. Then there exist m1 ≥ m0 and r1 ≥ r0 such that N (v, m1 , r) > 0
(r ≥ r1 ) .
(5.2)
¨ ¯ W. JAGER and Y. SAITO
974
Proof. Since the support of u is assumed to be unbounded, there exists r1 ≥ r0 such that |v(r1 )| > 0. Since r1−2m N (v, m, r1 ) = r1−2m {|w0 (r1 )|2 − (C0 (r1 )w(r1 ), w(r1 )) −|B 1/2 (r1 )w(r1 )|2 + (m(m + 1) − F (r1 ))|w(r1 )|2 } ≥ −(C0 (r1 )v(r1 ), v(r1 )) − |B 1/2 (r1 )v(r1 )|2 +(m(m + 1) − F (r1 ))|v(r1 )|2 ,
(5.3)
we can choose a sufficiently large m1 ≥ m0 so that r1−2m1 N (v, m1 , r1 ) > 0 ,
or r12 N (v, m1 , r1 ) > 0 .
(5.4)
Note that, by (ii)-2 of Assumption 2.1, N (r, m, v) is right-continuous. Then Ineq. (5.4) is combined with (4.13) and Lemma A in Appendix to see that r2 N (r, m1 , v) > 0 on [r1 , ∞), which completes the proof. Definition 5.2. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Let F (r) and m1 be given in Definition 4.1 and Proposition 5.1, respectively. Then we introduce the following two alternative cases: Case I : There exists an infinite sequence {r`0 } such that R0 < r`0 , r`0 → ∞ as ` → ∞, and (5.5) 2 Re (v 0 (r`0 ), v(r`0 )) ≤ (2m1 r`0 )−1 F (r`0 )|v(r`0 )|2 for all ` = 1, 2, . . . . Case II : There exists r2 > r1 such that 2 Re (v 0 (r), v(r)) > (2m1 r)−1 F (r)|v(r)|2
(r ≥ r2 ) ,
(5.6)
where r1 is as in Proposition 5.1. Proposition 5.3. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Suppose that Case I in Definition 5.2 holds. Then there exists an infinite sequence {r`00 } such that R0 < r`00 , r`00 → ∞ as ` → ∞, and M + (v, r`00 ) > 0
(` = 1, 2, . . .) .
(5.7)
Proof. Let {r`0 } be as in Case I of Definition 5.2. Let w = rm1 v, where m1 is as in Proposition 5.1. Then we have for r = r`0 r−2m1 |w0 |2 = |v 0 + m1 r−1 v|2 = |v 0 |2 + 2m1 r−1 Re (v 0 , v) + m21 r−2 |v|2 ≤ |v 0 |2 + m1 r−1 (2m1 r)−1 F (r)|v|2 + m21 r−2 |v|2 = |v 0 |2 + (2−1 F (r) + m21 )r−2 |v|2 .
(5.8)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
975
Let r1 be as in Proposition 5.1. For r = r`0 such that r`0 ≥ r1 , it follows that 0 < N (v, m1 , r) = M + (w, r) + (m1 (m1 + 1) − F (r))r−2 |w|2 ≤ r2m1 {|v 0 |2 + (2−1 F (r) + m21 )r−2 |v|2 } −r2m1 {(C0 v, v) + (Bv, v)} +r2m1 (m1 (m1 + 1) − F (r))r−2 |v|2 = r2m1 {M + (v, r) + (m1 (2m1 + 1) − 2−1 F (r))r−2 |v|2 } .
(5.9)
Since F (r) → ∞ as r → ∞, there exists a positive integer `0 such that m1 (2m1 + 1) − 2−1 F (r`0 )) < 0 (` ≥ `0 ) .
(5.10)
Therefore we have only to define r`00 by r`00 = r`0 0 +`
(` = 1, 2, . . .) ,
(5.11)
which completes the proof.
Proposition 5.4. Suppose that Assumptions 2.1 and 4.2 hold. Suppose that the support of u is unbounded. Suppose that Case II in Definition 5.2 holds. Suppose, in addition, that (5.12) Re Q(x) ≤ 0 (x ∈ ER0 ) . Then there exist r3 > R0 and a positive constant c2 such that M (v, r) ≥ c2
(r ≥ r3 ) ,
(5.13)
where M (v, r) is given by (4.3). Proof. Since F (r) → ∞ as r → ∞, there exists r4 > R0 such that F (r) ≥2 2m1
(r ≥ r4 ) .
(5.14)
Then it follows from (5.6) that d |v(r)|2 ≥ 2r−1 |v(r)|2 dr
(r ≥ r4 ) .
Let r3 be such that r3 ≥ r4 and |v(r3 )| > 0. Then, since d d −2 2 −2 2 −1 2 (r |v(r)| ) = r |v(r)| − 2r |v(r)| ≥ 0 dr dr we have
r−2 |v(r)|2 ≥ r3−2 |v(r3 )|2 > 0
(5.15)
(r ≥ r4 ) ,
(r ≥ r3 ) .
(5.16)
(5.17)
Also, using (5.6) and (5.14) again, we see that 2r−1 |v(r)|2 ≤ (2m1 r)−1 F (r)|v(r)|2 ≤ 2|v(r)||v 0 (r)| ,
(5.18)
¨ ¯ W. JAGER and Y. SAITO
976
or r−1 |v(r)| ≤ |v 0 (r)|
(5.19)
for r ≥ r4 . Thus, it follows from (5.17) and (5.19) that |v 0 (r)|2 ≥ r−2 |v(r)|2 ≥ r2−2 |v(r3 )|2 > 0 for r ≥ r3 , which is combined with (5.12) to obtain (5.13).
(5.20)
Assumption 5.5. (i) Let h(r) be as above. Then h ∈ L1 ((R0 , ∞)). (ii) There exists a constant β ∈ (0, 1) such that 0 ≥ βQ0 (x) ≥ Re (Q(x))
(x ∈ ER0 ) .
(5.21)
Theorem 5.6. Suppose that Assumptions 2.1, 4.2 and 5.5 hold. Suppose that the support of u is unbounded. Then there exist a positive constant c3 and R2 > R0 such that (5.22) M (v, r) ≥ c3 (r ≥ R2 ) . Proof. Note that all the assumptions that are necessary for the conclusions of Propositions 3.2, 3.3, 4.3, 4.4, 5.1, 5.3 and 5.4 are satisfied. Suppose that Case I of Definition 5.2 is satisfied. Then, by Proposition 5.3 there exists R20 > R0 such that M + (v, R20 ) > 0. Therefore, setting R1 = R20 in Proposition 3.3, we have for r ≥ R20 , ! Z r
M + (v, r) ≥ exp −
R03
Z ≥ exp −
h(t) dt M + (v, R20 )
∞
R03
! h(t) dt M + (v, R20 ) .
(5.23)
Since we have from (5.21) M (v, r) = |v 0 (r)|2 − (CR (r)v(r), v(r)) ≥ |v 0 (r)|2 − β(C0 (r)v(r), v(r)) ≥ β|v 0 (r)|2 − β(C0 (r)v(r), v(r)) − β|B 1/2 (r)v(r)|2 = βM + (v, r) ,
(5.24)
it follows from (5.23) that M (v, r) ≥ c03 with c03
Z = β exp −
∞
R03
(r ≥ R20 )
(5.25)
! h(t) dt M + (v, R20 ) .
(5.26)
¨ THE UNIQUENESS OF THE SOLUTION OF THE SCHRODINGER EQUATION WITH
...
977
Suppose that Case II of Definition 5.2 is satisfied. Then from Proposition 5.4 we have (5.27) M (v, r) ≥ c2 (r ≥ r3 ) , where c2 and r3 are as in Proposition 5.4. Now set ( c3 = min{c03 , c2 } ,
(5.28)
R2 = max{R20 , r3 } .
Then (5.22) follows, which completes the proof.
Corollary 5.7. Suppose that Assumptions 2.1, 4.2 and 5.8 hold. Suppose that lim M (v, r) = 0 .
(5.29)
r→∞
Then u has compact support. In order to show our main theorem (Theorem 5.10) we need one more assumption. Assumption 5.8. We have 2 lim r inf Re (−q(x)) = ∞ . r→∞
|x|=r
(5.30)
Before we state and prove Theorem 5.10, we are going to unify Assumptions 2.1, 4.2, 5.5 and 5.8 in more organized form: Assumption 5.9. (1) Let N be an integer such that N ≥ 2. Let u ∈ H 2 (ER0 )loc , R0 > 0, be a solution of the homogeneous Schr¨ odinger Eq. (2.1), where ER0 is given by (2.2) (with R = R0 ). Here q(x) is a complex-valued, measurable, locally bounded function on ER0 which satisfies (5.30). (2) Set (N − 1)(N − 3) . (5.31) Q(x) = q(x) + 4r2 (a) Then Q(x) is decomposed as Q(x) = Q0 (x) + Q1 (x) ,
(5.32)
where Q0 (x) is a non-positive, measurable, locally bounded function on ER0 and Q1 (x) is a complex-valued, measurable, locally bounded function on ER0 such that 0 ≥ βQ0 (x) ≥ Re (Q(x)) with a constant β ∈ (0, 1).
(x ∈ ER0 )
(5.33)
¨ ¯ W. JAGER and Y. SAITO
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(b) For any φ ∈ X = L2 (S N −1 ), (Q0 (r·)φ, φ) has the right limit for all r > R0 as a function of r = |x|, where ( , ) is the inner product of X. (c) There exist h0 > 0 and, for 0 < h < h0 , a real-valued, measurable function Q0r (x; h) on ER0 such that (2.6), (2.7) and (2.8) hold. (d) There exists h(r) ∈ L1 ((R0 , ∞)) such that 0 < h(r) ≤
2 r
(r > R0 ) ,
(5.34)
and, setting a(r) = h−1 (r) sup |Q1 (x)| , |x|=r b(r) = inf [−(Q0 (x) + h−1 (r)Q0r (x))] , |x|=r (h−1 (r) = 1/h(r)) ,
(5.35)
we have a(r)2 ≤ b(r)
(r > R0 ) .
(5.36)
(3) The function F (r) introduced in Definition 4.1 satisfies Assumption 4.2. Theorem 5.10. Suppose that Assumptions 5.9 hold. Suppose that the solution u satisfies (5.1). Then u has compact support. Proof. Note that 0 ≤ |v 0 (r)|2 − (CR (r)v(r), v(r)) = |(r(N −1)/2 u(r·))0 |2 − rN −1 (Re (Q(r·))u(r·), u(r·)) = rN −1 |∂r u(r·) + 2−1 (N − 1)r−1 u(r·)|2 −rN −1 (Re (Q(r·))u(r·), u(r·)) ≤ 2rN −1 |∂r u(r·)|2 + 2−1 (N − 1)2 rN −3 |u(r·)|2 (N − 1)(N − 3) −rN −1 Re (q(r·)) + u(r·), u(r·) , 4r2 ≤ 2rN −1 |∂r u(r·)|2 +
N 2 − 1 N −3 r |u(r·)|2 4
−rN −1 (Re (q(r·))u(r·), u(r·)) ,
(5.37)
where ∂r = ∂/∂r and we have used
Q(x) = q(x) +
(N − 1)(N − 3) , 4r2
2 2 (N − 1) − (N − 1)(N − 3) = N − 1 . 2 4 4
(5.38)
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Therefore we have 0 ≤ |v 0 (r)|2 − (CR (r)v(r), v(r)) ≤ 2rN −1 {|∂r u(r·)|2 − (Re (q(r·))u(r·), u(r·))} −rN −3 ([−r2 Re (q(r·)) − 4−1 (N 2 − 1)]u(r·), u(r·)) .
(5.39)
It follows from (1) of Assumption 5.9 (5.30) that there exists R3 > R0 such that −r2 Re (q(x)) − 4−1 (N 2 − 1) > 0
(|x| = r, r ≥ R3 ) ,
(5.40)
and hence, for |x| ≥ R3 , 0 ≤ |v 0 (r)|2 − (CR (r)v(r), v(r)) ≤ 2
∂u 2 − Re (q(x))|u|2 dS , ∂r |x|=r
Z
(5.41)
which, together with (5.1), implies (5.29). Thus Corollary 5.7 can be applied to see that u has compact support, which completes the proof. 6. Examples In this section we are going to give some applications of Theorem 5.10. Example 6.1. Let R > 0 and let u ∈ H 2 (ER )loc be a solution of the equation (−∆ + V` (x) + Vs (x) − λ(x))u = 0
(x ∈ ER ) .
(6.1)
Here λ(x) is a real-valued, measurable, locally bounded function on ER satisfying the following (i) and (ii): (i) There exists m0 > 0 such that λ(x) ≥ m0
(x ∈ ER ) .
(6.2)
(ii) For any φ ∈ X the function fφ (r) = (λ(r·)φ, φ)
(6.3)
is a right continuous, nondecreasing function on (R, ∞). The functions V` (x) and Vs (x) are real-valued and complex-valued functions, respectively, satisfying the following (iii) and (iv): (iii) The long-range potential V` (x) is assumed to be C 1 function on ER such that lim sup |V` (x)| = 0 , r→∞ |x|=r (6.4) 1+ ∂V` < ∞ |x| sup ∂|x| r>R,|x|=r with ∈ (0, 2).
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(iv) The short-range potential Vs (x) is assumed to be measurable such that {r1+ |Vs (x)|} < ∞ ,
sup
(6.5)
r>R,|x|=r
where is as above. Suppose, in addition, that ∂u 2 + λ(x)|u|2 dS = 0 . ∂r |x|=r
Z lim r→∞
(6.6)
Then u is identically zero in ER . In fact, set Q0 (x) = −λ(x) + V` (x) , Z r+h ∂V` −1 (sω) ds Q0r (rω; h) = h ∂r r
Q0r (x) =
(ω ∈ S N −1 , h > 0) ,
∂V` , ∂r
Q1 (x) = Vs (x) +
(6.7)
(N − 1)(N − 3) . 4r2
Then, since Re (−q(x)) = λ(x) − V` (x) − Re (Vs (x)) ≥ m0 − V` (x) − Re (Vs (x))
(6.8)
(1) of Assumption 5.9 is satisfied for sufficiently large r. For φ ∈ X, we have 1 1 ([Q0 ((r + h)·) − Q0 (r·)]φ, φ) = − ([λ((r + h)·) − λ(r·)]φ, φ) h h 1 + ([V` ((r + h)·) − V` (r·)]φ, φ) h ≤ (Q0r (r·; h)φ, φ) → (Q0r (r·)φ, φ)
(6.9)
as h → 0 with h > 0. Thus 2(c) of Assumption 5.9 is satisfied. Set h(r) = r−1−/2
(r > R) .
(6.10)
Then h(r) ∈ L1 ((R, ∞)) and Ineq. (5.34) is satisfied for sufficiently large r. Also we have a(r) → 0 as r → ∞ and b(r) ≥ m0 /2 for sufficiently large r, and hence 2(d) of Assumption 5.9 is now satisfied. Noting that βQ0 (x) − Re (Q(x)) = (1 − β)λ(x) + (β − 1)V` (x) − Re (Vs (x)) ,
(6.11)
and that λ(x) ≥ m0 (6.2), we see that 2(a) of Assumption 5.9 holds for sufficiently large r with any β ∈ (0, 1). The condition 2(b) of Assumption 5.9 is verified by (ii) of Example 6.1 and the smoothness of V` (x). Define F (r) by F (r) = log r. Obviously (ii) and (iii) of Assumption 4.2 are satisfied by definition. Since r4 h2 (r)b(r) = r2− (λ(x) + o(1))
(6.12)
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as r → ∞, (4.5) in Assumption 4.2 holds for sufficiently large r. Therefore, by setting R0 sufficiently large, all the conditions of Assumption 5.9 are satisfied, which implies that the solution u has compact support in ER . Therefore it follows from the unique continuation theorem that u is identically zero in ER . We remark here that, if λ(x) is assumed to be bounded from above, too, then the condition (6.6) is equivalent to Z ∂u 2 2 (6.13) lim + |u| dS = 0 . ∂r r→∞ |x|=r Another remark is that, if Vs (x) is real-valued, then the condition (6.6) is implied by the generalized radiation condition p ∂u − i λ(x)u ∈ L2 (ER ) (6.14) rδ−1 ∂r with δ > 1/2 and R > R. Example 6.2. Let R > 0 and let u ∈ H 2 (ER )loc be a solution of the equation 1 ∆ − λ u = 0 (x ∈ ER ) (6.15) − µ(x) Here λ > 0 and the real-valued function µ(x) on ER is decomposed as µ(x) = µ0 (x) + µ` (x) + µs (x)
(x ∈ ER ) ,
(6.16)
where µ0 (x), µ` (x) and µs (x) satisfy the following (i)–(iv): f0 > 0 such that (i) µ0 (x) is real-valued and measurable and there exists m f0 µ0 (x) ≥ m
(x ∈ ER ) .
(6.17)
(ii) For any φ ∈ X the function gφ (r) = (µ0 (r·)φ, φ)
(6.18)
is a right continuous, nondecreasing function on (R, ∞). (iii) The real-valued function µ` satisfies Example 6.1(iii) with V` (x) replaced by µ` . (iv) The complex-valued function µs (x) satisfies Example 6.1(iv) with Vs (x) replaced by µs . Set
λ(x) = λµ0 (x) , V` (x) = λµ` (x) , Vs (x) = λµs (x) .
(6.19)
Then u satisfy Eq. (6.1) in Example 6.1, where λ(x), V` (x) and Vs (x) satisfy (i)–(iv) in Example 6.1. Thus the condition Z ∂u 2 (6.20) lim + λµ0 (x)|u|2 dS = 0 ∂r r→∞ |x|=r implies that u is identically zero.
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7. Reduced Wave Operator in Layered Media In [8] we considered the reduced wave operator H=−
1 ∆ in H = L2 (RN , µ(x) dx) , µ(x)
(7.1)
where µ(x) is a real-valued function such that 0 < inf µ(x) ≤ sup µ(x) < ∞ . x
(7.2)
x
By defining the domain D(H) of H by D(H) = H 2 (RN ), where H 2 (RN ) is the second order Sobolev space on RN , H becomes a self-adjoint operator on H. In this section we shall show that the nonexistence of the eigenvalues of H can be proved in some cases discussed in [8] by using the result of Sec. 6 (Example 6.2). Suppose that µ(x) has the decomposition (6.16) with a position function µ0 , a long-range perturbation µ` and a short-range perturbation µs . The functions µ0 , µ` , µs are assumed to satisfy (i)–(iv) of Example 6.2. In [8], for the sake of simplicity, we assumed that only one of a long-range perturbation or short-range perturbation appeared with the main term µ0 (x), but we can easily modify the arguments in [8] so that we can treat µ(x) of the form (6.16). Let K− be a nonpositive integer or K− = −∞ and let K+ be a nonnegative integer or K+ = ∞. Let K be a set of integers given by (7.3) K = {k/K− ≤ k ≤ K+ } . Let {Ωk }k∈K be a sequence of open sets of RN such that Ωk ∩ Ω` = ∅ (k 6= `) , [ Ωk = R N ,
(7.4)
k∈K
where A is the closure of A. Further we assume that the boundary ∂Ωk of Ωk has the form (−) (+) (7.5) ∂Ωk = Sk ∪ Sk , (−)
(+)
(−)
(+)
where Sk ∩ Sk = ∅, and each of Sk and Sk is a continuous surface which is a finite union of smooth surfaces. We also assume that (+) (−) (k ∈ K) , Sk = Sk+1 (+) (−) SK+ = SK+ +1 = ∅ (if K+ 6= ∞) , (7.6) S (−) = ∅ (if K− 6= −∞) . K− Now the function µ0 (x) is assumed to be a simple function which takes a constant value νk on each Ωk such that {νk }k∈K ia a bounded, positive sequence. We assume that the origin 0 of the coordinates is in Ω0 , and µ0 (x) satisfies the condition (+) (−) (7.7) (νk+1 − νk )(n(k) (x) · x) ≥ 0 x ∈ Sk = Sk+1 , k ∈ K ,
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where n(k) (x) is the unit outward normal of Ωk at x ∈ ∂Ωk and n(k) (x)·x is the inner product of n(k) (x) and x in RN . Then the following theorem has been obtained in [8] ([8, Theorem 4.6]): Theorem 7.1. Let H be as above. Suppose, in addition, that µ takes the form of either µ = µ0 + µs or µ = µ0 + µ` . Let σp (H) be the set of the point spectrum of H. Then the multiplicity of each λ ∈ σp (H) is finite, σp (H) does not have any accumulation points except at 0 and ∞. It is not difficult to extend this result to the general case that µ = µ0 + µs + µ` . Using the Example 6.2, we can show a sufficient condition for the nonexistence of the point spectrum of the operator H. Theorem 7.2. Let H be as above. Suppose that, for almost all ω ∈ S N −1 , µ0 (rω) is a nondecreasing function of r ∈ [0, ∞). Then σp (H) = 0. Proof. The condition (ii) of Example 6.2 is now satisfied since µ0 (rω) is nondecreasing. Here we are going to give some examples. N such that Example 7.3. Let {Uk }∞ k=0 be a sequence of open sets of R
Uk ⊂ Uk+1 (k ≥ 0) , ∞ [ Uk = RN ,
(7.8)
k=0
where the boundary ∂Uk of Uk is a continuous surface which is a finite union of smooth surfaces. Suppose that n ˜ (k) (x) · x ≥ 0 (k = 0, 1, 2, . . .) ,
(7.9)
where n ˜ (k) (x) is the unit outward normal of Uk at x ∈ ∂Uk . Set Ω0 = U 0 , Ωk = Uk \Uk−1 (k ≥ 1) , (+) Sk = ∂Uk (−) Sk = ∂Uk−1
(k ≥ 0) ,
(7.10)
(k ≥ 1) .
This is the case that K− = 0 and K+ = ∞. Let µ0 (x) be given by µ0 (x) = νk
(x ∈ Ωk ) ,
(7.11)
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where {νk }∞ k=0 ia a bounded, positive, increasing sequence. Then we see that not only the condition (7.7) is satisfied but also µ0 (rω) is a nondecreasing function of r ∈ [0, ∞) for almost all ω ∈ S N −1 . Thus Theorem 7.2 can be applied to see that there is no point spectrum of H. Therefore the limiting absorption principle holds on the whole positive interval (0, ∞) (see Sec. 5 of [8]). Example 7.4. Let {ck /k = ±1, ±2, . . .} be an increasing sequence of real numbers such that c−1 < 0 < c1 , (7.12) lim ck = ±∞ . k→±∞
Let xN be the N th coordinate of x = (x1 , x2 , . . . , xN ), and set {x ∈ RN /c−1 < xN < c1 } (k = 0) , N Ωk = {x ∈ R /ck−1 < xN < ck } (k = −1, −2, . . .) , {x ∈ RN /c < x < c } (k = 1, 2, . . .) . k
N
(7.13)
k+1
We also set (±)
S0
= {x ∈ RN /xN = c±1 } , (
(+) Sk
=
(−) Sk
{x ∈ RN /xN = ck }
(k = −1, −2, . . .) ,
{x ∈ R /xN = ck+1 }
(k = 1, 2, . . .) ,
{x ∈ RN /xN = ck−1 }
(k = −1, −2, . . .) ,
{x ∈ RN /xN = ck }
(k = 1, 2, . . .) ,
N
(
and =
(7.14) (7.15)
(7.16)
(+)
Note that, for x ∈ Sk ,
( n(k) (x) · x
≥0
(k ≥ 0) ,
≤0
(k < 0) .
(7.17)
Define a simple function µ0 (x) by (7.11), where the sequence {νk }∞ k=−∞ is assumed ∞ is decreasing and {ν to be bounded and positive such that {νk }−1 k }k=1 is increask=−∞ ing. Then, as in Example 7.3, Theorem 7.2 can be applied to show that σp (H) = ∅. The planes {x ∈ RN /xN = ck } can be perturbed as far as the condition (7.17) is satisfied. Appendix Here we are going to prove a lemma on distributions on a half interval (a, ∞) which was used when we evaluate the functionals M + (v, r) and N (v, m, r). Lemma A. Let f (r) be a real-valued function on I = (a, ∞) such that f is locally L1 and right continuous on I. Suppose that f 0 ≥ 0, where f 0 is the distributional derivative of f and the inequality should be taken in the sense of distributions. Then f is nondecreasing on I.
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Proof. Here we are giving a rather elementary proof. (I) Let r ∈ I and h > 0. Then, for φ ∈ C0∞ (I), we have Z Z [f (r + h) − f (r)]φ(r) dr = − f (r)[φ(r) − φ(r − h)] dr I
I
Z
Z =−
r
f (r) I
φ0 (s) ds dr ,
(A.1)
r−h
where φ is supposed to be extended on the whole line (−∞, ∞) by setting φ(r) = 0 for r ≤ a. Since Z h Z r 0 φ (s) ds = φ0 (t + r − h) dt , (A.2) r−h
0
it follows that Z Z [f (r + h) − f (r)]φ(r) dr = I
0
h
Z − f (r)φ0 (t + r − h) dr dt.
(A.3)
I
(II) Let φ ∈ C0∞ (I) and φ ≥ 0. Then, since Z − f (r)φ0 (t + r − h) dr = hf 0 , φ(· + t − h)i ≥ 0
(A.4)
I
for h > 0 and 0 ≤ t ≤ h, where hF, Gi denotes the value of the distribution F for the test function G, it follows from (A.3) that Z [f (r + h) − f (r)]φ(r) dr ≥ 0 (A.5) I
for any φ ∈ C0∞ (I) with φ ≥ 0. (III) Suppose that there exist r0 ∈ I, h0 > 0 and η0 > 0 such that f (r0 + h0 ) − f (r0 ) = −η0 . Since f is right continuous, there exists r1 > r0 such that ( |f (r0 ) − f (r)| < η0 /3 , |f (r0 + h0 ) − f (r + h0 )| < η0 /3
(A.6)
(A.7)
for r0 ≤ r ≤ r1 . Then, for r0 ≤ r ≤ r1 , we have f (r + h0 ) − f (r) = f (r0 + h0 ) − f (r0 ) + {f (r + h0 ) − f (r0 + h0 )} + {f (r0 ) − f (r)} ≤ f (r0 + h0 ) − f (r0 ) + |f (r + h0 ) − f (r0 + h0 )| + |f (r0 ) − f (r)| < −η0 /3 .
(A.8)
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Let φ ∈ C0∞ (I) such that
supp φ ⊂ [r0 , r1 ] , φ ≥ 0, Z r1 φ(r) dr = 1 .
(A.9)
r0
Then, it follows that Z Z [f (r + h) − f (r)]φ(r) dr = I
r1
r0
≤− =−
η0 3
[f (r + h) − f (r)]φ(r) dr Z
r1
φ(r) dr r0
η0 3
< 0, which contradicts (A.5). This completes the proof.
(A.10)
Acknowledgements This work was finished when the second author was visiting the University of Heidelberg for February 1997. Here he would like to thank Deutsche Forschungs Gemeinschaft for its support through SFB 359. Also the second author is thankful to Professor Willi J¨ ager for his kind hospitality during this period. References [1] D. Eidus, “The principle of limiting absorption”, Amer. Math. Soc. Translations 47 (1965) 157–191 (Mat. Sb. 57 (1962)). [2] D. Eidus, “The limiting absorption and amplitude problems for the diffraction problem with two unbounded media”, Commun Math. Phys. 107 (1986) 29–38. [3] T. Ikebe, “Eigenfunction expansions associated with the Schr¨ odinger operators and their applications to scattering theory”, Arch. Rational Mech. Anal. 5 (1960) 1–34. [4] T. Ikebe and Y. Sait¯ o, “Limiting absorption method and absolute continuity for the Schr¨ odinger operator”, J. Math. Kyoto Univ. 12 (1972) 513–612. [5] T. Ikebe and J. Uchiyama, “On the asymptotic behavior of eigenfunctions of secondorder elliptic differential operators”, J. Math. Kyoto Univ. 11 (1971) 425–448. [6] W. J¨ ager, “Zur Theorie der Schwingugsgleichung mit variablen Koeffizienten in Aussengebieten”, Math. Z. 102 (1969) 62–88. [7] W. J¨ ager and Y. Sait¯ o, “On the Spectrum of the Reduced Wave Operator with Cylindrical Discontinuity”, Forum Mathematicum 9 (1997) 29–60. [8] W. J¨ ager and Y. Sait¯ o, “The reduced wave equation in layered materials”, to appear in Osaka J. Math. [9] T. Kato, “Growth properties of solutions of the reduced wave equation with a variable coefficients”, Commun. Pure Appl. Math. 12 (1959) 403–425. [10] C. M¨ uller, Grundprobleme der Mathematischen Theorie Elektromagnetischer Schwingungen, Springer, Berlin 1957. ¨ [11] F. Rellich, “Uber das asymptotische Verhalten des L¨ osungen von ∆u + λu = 0”, Jber. Deutsche. Math. Verein. 53 (1943) 57–65.
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[12] G. Roach and B. Zhang, “On Sommerfeld radiation conditions for the diffraction problem with two unbounded media”, Proc. Royal Soc. Edinburgh 121A (1992) 149– 161. [13] Y. Sait¯ o, Spectral Representations for Schr¨ odinger Operators with Long-Range Potentials, Lecture Notes in Mathematics, Vol. 727, Springer, Berlin, 1979. [14] J. Weidmann, “On the Continuous spectrum of Schr¨ odinger operators”, Commun. Pure and Appl. Math. 19 (1966) 107–110. [15] E. Wienholtz, “Halbbeschr¨ anke partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus”, Math. Ann. 135 (1958) 50–80.
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD ERIK SKIBSTED Institut for Matematiske Fag Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark E-mail : [email protected] Received 16 April 1997 We develop an extension of the abstract Mourre theory which consecutively is used to prove spectral properties of various systems coupled to a massless bosonic field. Our models include the spin-boson model and the standard model of quantum electrodynamics for a non-relativistic atom considered recently in [8] and [3] respectively.
1. Introduction It is well known that Mourre’s commutator method [9] is a powerful machinery for proving basic spectral properties of N -body Schr¨ odinger operators H: Letting F (H), σpp (H) and σsc (H) denote the set of thresholds (i.e. eigenvalues of sub-Hamiltonians), the set of eigenvalues and the continuous singular spectrum, respectively, these properties include: (1) F (H) is closed and countable. (2) The eigenvalues (counted with multiplicity) can only a‘ccumulate at F (H). (3) There is a limiting absorption principle away from F (H) ∪ σpp (H). In particular σsc (H) = ∅. The purpose of the paper is to prove these or similar properties for various systems coupled weakly to a massless bosonic field. We discuss N -body as well as finite number of states systems motivated by the recent papers [3] and [8], respectively. For the spin-boson Hamiltonians of [8] the above properties should be replaced by: (2)0 σpp (H) is finite. (3)0 There is a limiting absorption principle away from σpp (H). In particular σsc (H) = ∅. To explain a basic idea of [8] let us consider bosons with positive mass m which p 2 means that the 1-boson energy function is given by ω(k) = |k| + m2 . We introduce B = 12 (F · p + p · F ); F (k) = −ω(k)|k|−2 k, p = −i∇k , as an operator on the 1-boson space L2 (Rνk ), and compute i[ω, B] = I. Upon lifting to the symmetric 989 Reviews in Mathematical Physics, Vol. 10, No. 7 (1998) 989–1026 c World Scientific Publishing Company
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E. SKIBSTED
Fock space F (by second quantization) this gives i[dΓ(ω), dΓ(B)] = N , where N denotes the “number operator”. Of course it is non-negative which in the applications of [8] is used to establish a Mourre estimate for spin-boson Hamiltonians. To get the consequences (2)0 and (3)0 listed above the authors developed an extended abstract Mourre theory since their “conjugate operator” is not selfadjoint and consequently does not meet the requirements of [9]. (Notice that the above conjugate operators B and A = dΓ(B) do not have self-adjoint realizations.) It is noticed in [8] that the massless case m = 0 is worse. As a matter of fact the authors did not obtain results in this case (except for a certain compressed model in which N is bounded). The basic obstacle for m = 0 is that N is not bounded relatively to H = dΓ(ω). More general for all nonzero real-valued f ∈ C0∞ (σ(H)), f (H)i[H, A]f (H) is unbounded.
(1.1)
In this paper we develop a Mourre theory that overcomes the above difficulty. We notice that (1.1) does not fit into the existing refinements of Mourre’s paper (see for example [10, 4, 1, 12]). Of course our refinement of the Mourre theory involves new conditions. One of these is that the H-unbounded piece M of a decomposition i[H, A] = M + G has that property that i[H, M ] is H-bounded. (Notice that for the above example the latter condition is trivial: i[H, N ] = 0.) Another complication is the one mentioned above: A might not be self-adjoint. The procedure of [8] essentially consists in mimicking [9] using the semigroup generated by −iA (assumed to exist). One ingredient of our approach is the introduction of a sequence of self-adjoint operators (An ) which (by assumption) in some sense converges to A. Also each An is assumed to have the usual good commutator property i[H, An ] being H-bounded. Hence our procedure is facilitated by the basic technical results of [9] that are directly applicable for the approximating operators. For the example discussed above (1.1) and the lack of self-adjointness are due to the singularity at k = 0 of the vector field F , and therefore the approximating self-adjoint operators may naturally be constructed by suitably smoothing our the vector field. Our examples concern Hamiltonians defined on a Hilbert space of the form L2 (X) ⊗ F . The spin-boson model corresponds to X being finite. For N -body systems X is a finite dimensional vector space. In [3] the authors propose the conjugate operator A = Ael ⊗ I − I ⊗ Af , where Ael is the generator of the dilations and Af is the second quantized generator of the dilations lifted from the 1-photon space. The basic idea is to invoke the well-known Mourre estimate for “the N -body part” Hel of the Hamiltonian (given in terms of Ael ) and then show local positivity away from the thresholds of Hel for small perturbations of Hel ⊗ I + I ⊗ dΓ(ω) .
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This procedure does not yield to the properties (1)–(3) listed above even for perturbations for which there is a natural notion of thresholds for the perturbed Hamiltonians. Explicitly the authors do not prove spectral properties in neighbourhoods of thresholds of Hel . Moreover to get spectral information in neighbourhoods of eigenvalues of Hel they need implicit conditions (cf. Fermi’s Golden Rule) and an analyticity assumption on the perturbation. In this paper we shall consider perturbations of the above tensor sum for which there is a natural notion of thresholds for the full Hamiltonian. We shall obtain all of the properties (1)–(3) for some classes of perturbations. Our basic idea is to consider the (non-self-adjoint) conjugate operators A = CBR ⊗ I + I ⊗ dΓ(B) ,
(1.2)
where C and R are adjustable positive parameters, B is given as in the beginning of this section, and BR is somewhat similar to Ael . More precisely BR is given in terms of a scaled version of the vector field invented by Graf [7]. We notice that by a previous result of the author [14, Theorem B2] this operator provides another Mourre estimate for “the N -body part” Hel . In [14] the latter estimate had obvious applications in studying propagation properties with locally singular potentials. In the present context the use of BR is convenient not only for dealing with local singularities but for another reason too. (As a matter of fact we don’t see how to obtain (1)–(3) for non-trivial perturbations say by using (1.2) with BR replaced by Ael .) We emphasize that for all our examples we only have results for weak coupling. It is an open problem to go beyond this restriction for the classes of perturbations we consider. For the spin-boson Hamiltonians and our first model of an N -body system coupled to the bosonic field, the latter phrased as the electron-boson model, we give a very explicit bound on the perturbation (in fact similar to the one of [8] obtained for positive mass). For the more complicated model, the so-called standard model of quantum electrodynamics for a non-relativistic atom, we do not derive an explicit bound. (In that case our methods would probably not yield realistic bounds.) Although we shall not discuss it in this paper our limiting absorption principle may be used to show absence of embedded eigenvalues (in a weak coupling regime) in accordance to Fermi’s Golden Rule and to treat related issues. This does not require the notion of resonance as defined in [3] or any other such notion. We notice that in the theory of N -body Schr¨ odinger operators the analyticity assumption of [13] was similarly removed in the treatment of instability of embedded eigenvalues of [2]. In Sec. 2 we formulate and derive the extended abstract Mourre theory. In Sec. 3 we discuss in detail the applications to the spin-boson model. We prove (2)0 and (3)0 . In the next section we introduce “the electron-boson model” and do some preliminaries. As an easy corollary of the main result Proposition 4.4 we recover the Mourre estimate [14, Theorem B2]. In Sec. 5 we use the proposition to complete “the electron-boson model”, that is to establish (1)–(3). Finally in Sec. 6 we discuss the standard model of quantum electrodynamics for a non-relativistic atom. For a
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certain class of perturbations we obtain again the above properties. In that section we skip many details referring the reader to various procedures and preliminary results for the previously discussed models. 2. Abstract Theory For a (densely defined) operator T on a Hilbert space we denote by D(T ) and ρ(T ) its domain and resolvent set, respectively. Assumptions 2.1. Let H, M, An , n ∈ N, be self-adjoint operators on a Hilbert space H. Suppose M ≥ δI for some positive number δ, and that there exists a closed symmetric operator A on H with the properties −i ∈ ρ(A) and for some core C of A with C ⊆ D(An ) the identity limn→∞ An φ = Aφ holds for all φ ∈ C. Suppose (1) (Compatibility) The set D(H)∩D(M ) is dense in D(H) as well as in D(M ). The form i[H, M ] defined on this set extends to an H-bounded operator i[H, M ]0 , and sup kHeitM (H − i)−1 k < ∞ . |t|<1
(2) (First commutators) For all n ∈ N the set D(H) ∩ D(An ) is dense in D(H). The form i[H, An ] defined on this set extends to an H-bounded operator Hn , and sup kHeitAn (H − i)−1 k < ∞ . |t|<1
Moreover there exists an H-bounded (symmetric) operator G such that for all φ ∈ D(H) ∩ D(M ), lim Hn φ = (M + G)φ .
n→∞
(3) (Second commutators) For all n ∈ N the set D(M ) ∩ D(An ) is dense in D(M ). The form i[M, An ] defined on this set extends to an M -bounded operator Mn , and sup kM eitAn M −1 k < ∞ . |t|<1
Moreover there exists an M -bounded operator M∞ such that for all φ ∈ D(M ), lim Mn φ = M∞ φ . n→∞
The form i[G, An ] defined on D(H)∩D(An ) extends to an H-bounded operator Gn . There exists an H-bounded operator G∞ such that for all φ ∈ D(H), lim Gn φ = G∞ φ .
n→∞
(4) (Positivity at E) For a given E ∈ R there exist α > 0, an open neighbourhood U of E and a compact operator K such that for all real-valued f ∈ C0∞ (U) the form inequality M + f (H)Gf (H) ≥ αf (H)2 − f (H)Kf (H) − (I − f (H))L(I − f (H)) 1
holds on D(H)∩D(M 2 ) for some symmetric H-bounded operator L = L(f ).
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
993
The following result may be derived as in [9]. Lemma 2.2. Suppose Assumption 2.1 (2) and (3). Let for n ∈ N and λ ∈ R \ {0}An (λ) = λ2 (An + iλ)−1 . Then for |λ| sufficiently large (possibly depending on n) An (λ) preserves D(H) as well as D(M ). Moreover, (1) For all φ ∈ D(H), Hn φ = lim i[H, An (λ)]φ , λ→∞
Gn φ = lim i[G, An (λ)]φ . λ→∞
(2) For all φ ∈ D(M ), Mn φ = lim i[M, An (λ)]φ . λ→∞
Corollary 2.3. (The virial theorem) Suppose Assumptions 2.1 (2) and (4) and (possible replacing the second part of Assumption 2.1 (2)) that for all φ ∈ 1 D(H) ∩ D(M 2 ), 1
lim hHn iφ = hM + Giφ = kM 2 φk2 + hGiφ .
n→∞
(Here hT iφ = hφ, T φi.) Suppose {φm } ⊆ D(H) is a sequence of eigenstates, (H − Em )φm = 0, kφm k = 1, such that φm → 0 weakly and Em → E. Then 1 / D(M 2 ) for m ≥ m0 . there exists m0 ∈ N such that φm ∈ Proof. We pick f ∈ C0∞ (U) equal to one on a neighbourhood of E and compute 1 for φm ∈ D(M 2 ) the expectation hM + Giφm = lim lim hi[H, An (λ)])φm = 0 . n→∞ λ→∞
On the other hand, if m is large enough hM + Giφm ≥ α − hKiφm ≥ yielding a contradiction.
α 2
Our main result is Theorem 2.4. (The limiting absorption principle) Suppose Assumptions 2.1 with E not being an eigenvalue of H. Then there exists a neighbourhood V of E such that (1) supIm z6=0, Re z∈V k(A + i)−1? (H − z)−1 (A + i)−1 k < ∞. 1 (2) supRe z1 , Re z2 ∈V, Im z1 Im z2 >0,z1 6=z2 {|z1 − z2 |− 3 k(A + i)−1? {(H − z1 )−1 − (H − z2 )−1 }(A + i)−1 k} < ∞.
994
E. SKIBSTED
Remark 2.5. The results of [9] are obtained from ours by letting M = I and An = A. (Notice that Assumption 2.1 (4) follows from the estimate for M = 0, cf. (3.16).) For simplicity of presentation we have not attempted to impose the weakest possible conditions under which our results may be obtained. Below we list some modified conditions under which Corollary 2.3 and Theorem 2.4 remain valid as can be justified actually by minor modifications of the proofs. Two of these conditions are reminiscences of [9] and [10] to which we refer for further details. (1) Introducing the dual space H−2 = H2? , H2 = D(H), the conditions of Assumptions 2.1 (3) and (4), Gn and L being H-bounded, may be replaced by Gn and L being bounded as maps from H2 to H−2 . The condition on the existence of G∞ may be replaced by supn k(H + i)−1 Gn (H − i)−1 k < ∞. (2) The condition of Assumption 2.1 (3) on the existence of M∞ may be replaced by supn kM −s Mn M −s k < ∞ for some s < 1. (3) The weight (A + i)−1 of Theorem (2.4.1) may be replaced by (A + i)−s = R sπ ∞ e− 2 i Γ(s)−1 0 eitA e−t ts−1 dt, s > 12 , where we use the notation eitA for the semigroup generated by −iA. In order to prove Theorem 2.4 we need some preliminary results. Lemma 2.6. Suppose Assumption 2.1 (1). Consider for ∈ R \ {0}H() = H − iM with the domain D = D(H) ∩ D(M ). The adjoint operator is given by H()? = H(−) . In particular z ∈ ρ(H()) for either Im z + δ and both positive or both negative. (Here δ refers to the delta of Assumptions 2.1.) Moreover in these cases the resolvent Rz0 () = (H() − z)−1 obeys the bound kRz0 ()k ≤ |Im z + δ|−1 .
(2.1)
Proof. Obviously H()? ⊇ H(−). Therefore for given ψ ∈ D(H()? ) we just need to show that ψ ∈ D. To that end we shall use that with C = kH()? ψk |hH()φ, ψi| ≤ Ckφk for all φ ∈ D .
(2.2)
For any non-real complex η we can compute (cf. [9]) M (H − η)−1 M −1 = (H − η)−1 − i(H − η)−1 i[H, M ]0 (H − η)−1 M −1 .
(2.3)
Since obviously the right-hand side is bounded we get that D is preserved by (H − η)−1 . Therefore we may replace φ in (2.2) by (H − η¯)−1 φ. Commuting the resolvent by using (2.3) we thus obtain |hM φ, (H − η)−1 ψi| ≤ Cψ,η kφk for all φ ∈ D .
(2.4)
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
995
Since (by assumption) D is dense in D(M ) we conclude that (H − η)−1 ψ ∈ D(M ). By repeating the argument we obtain that φκ = Dκ ψ ∈ D where for κ > 0, Dκ = H(κH − i)−1 (κH + i)−1 = κ−1 (I + i(κH − i)−1 )(κH + i)−1 . Inserting in (2.2) yields |Re hH()φκ , ψi| ≤ Ckφκ k ≤ CkH(κH + i)−1 ψk .
(2.5)
On the other hand, |Re hH()φκ , ψi| ≥ kH(κH + i)−1 ψk2 − | Im hM φκ , ψi| .
(2.6)
To estimate the last term we notice that (by using a technique of [9] cf. Lemma 2.2) Im hM φκ , ψi = lim
λ→∞
=
i h[M iλ(M + iλ)−1 , Dκ ]iψ 2
1 + 0 + hR H R + iR− H 0 R− R+ + iR− R+ H 0 R+ iψ ; 2
R+ = (κH + i)−1 ,
R− = (κH − i)−1 ,
H 0 = i[H, M ]0 .
In particular, 3 | Im hM φκ ψi| ≤ || kψk kH 0 (H − i)−1 k k(H − i)(κH + i)−1 ψk 2 ≤
1 kH(κH + i)−1 ψk2 + C1 kψk2 , 2
(2.7)
where C1 is independent of κ (and ψ). Combining (2.5), (2.6) and (2.7) yields (by subtraction) 1 kH(κH + i)−1 ψk2 − C1 kψk2 ≤ kH()? ψk kH(κH + i)−1 ψk , 2
(2.8)
from which we obtain the bound kH(κH + i)−1 ψk ≤ Cψ ,
(2.9)
the constant Cψ being independent of κ. Letting κ → 0 we obtain that ψ ∈ D(H). This fact combined with (2.2) implies that also ψ ∈ D(M ). The second part of the lemma follows from the first part just proven and the fact that the numerical range of H() is a subset of {η|Im η ≤ −δ}. Lemma 2.7. Suppose the assumption of Lemma 2.6 and that f ∈ C0∞ (R) is equal to one on a neighbourhood of E. Then there exist constants C, 0 > 0 and neighbourhood V of E such that k(H − i)(I − f (H))Rz0 ()k ≤ C provided || ≤ 0 , Im z > 0 and Re z ∈ V.
996
E. SKIBSTED
Proof. We pick an almost analytic extension f˜ ∈ C0∞ (C) of f so that we can represent Z 1 (∂¯f˜)(η)(H − η)−1 dudv , η = u + iv . (2.10) f (H) = π C In conjunction with (2.3) we obtain that M f (H)M −1 and [f (H), M ] are bounded.
(2.11)
In particular, for any φ ∈ H, ψ = (I − f (H))Rz0 ()φ ∈ D = D(H) ∩ D(M ) . We commute (H() − z)ψ = (I − f (H))φ − i[f (H), M ]Rz0 ()φ , yielding together with (2.11) and Lemma 2.6 k(H() − z)ψk ≤ k(I − f (H))φk + ||C1 kRz0 ()φk ≤ C2 kφk
(2.12)
for constants C1 and C2 independent of and z. We can also compute k(H() − z)ψk2 = k(H − Re z)ψk2 + k(M + Im z)ψk2 − hi[H, M ]iψ .
(2.13)
Clearly the last term on the right-hand may be estimated by |hi[H, M ]iψ | ≤ ||C3 k(H − i)ψk2 .
(2.14)
Combining (2.12), (2.13) and (2.14) gives the bound k(H − Re z)ψk2 + k(M + Im z)ψk2 ≤ C22 kφk2 + ||C3 k(H − i)ψk2 .
(2.15)
Clearly we can find a neighbourhood V of E and a positive number κ such that for all t ∈ R and s ∈ V, {(t − s)2 − 2κ|t − i|2 }|1 − f (t)|2 ≥ 0 .
(2.16)
For ||C3 ≤ κ and Re z(= s) ∈ V, (2.15) and (2.16) give the bound (by subtraction) (2.17) k(H − i)ψk2 ≤ κ−1 C22 kφk2 . In the rest of this section we impose Assumptions 2.1 with E not being an eigenvalue of H. We pick a real-valued f ∈ C0∞ (R) equal to one on a neighbourhood of E such that the form inequality M + f (H)Gf (H) ≥
α f (H)2 − (I − f (H))L(I − f (H)) 2
(2.18)
997
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD 1
holds on D(H) ∩ D(M 2 ). We would like to prove Lemmas 2.6 and 2.7 for the perturbed operator H() − if (H)Gf (H). Introducing the notation Rz () = (H − i(M + f (H)Gf (H)) − z)−1 for its resolvent, we have: Lemma 2.8. There exist constants C, 0 > 0 and a neighbourhood V of E such that kRz ()k ≤
C ||
,
(2.19)
k(H − i)(I − f (H))Rz ()k ≤ C
(2.20)
provided || ≤ 0 , Im z > 0 and Re z ∈ V. Proof. It is not obvious that the resolvent exists for all z in question. Clearly a perturbation argument based on (2.1) gives the existence for large vales of |Im z|. Below we shall prove (2.19) in a domain of the desired form assuming that the resolvent exists. Then by a simple connectedness argument it follows that it exists in the whole domain. So suppose z is given such that Rz () exists. Then from Rz () = Rz0 ()(I + if (H)Gf (H)Rz ()) and Lemma 2.7 we obtain (with 0 and V given as in Lemma 2.7) k(H − i)(I − f (H))Rz ()k ≤ C1 (1 + || kRz ()k) .
(2.21)
By (2.18) and Lemma 2.6, 2 Im Rz () 2 + Rz¯(−)(I − f (H))L(I − f (H))Rz () , Rz¯(−)f (H) Rz () ≤ α (2.22) yielding
kf (H)Rz ()k2 ≤ C2
kRz ()k + k(H − i)(I − f (H))Rz ()k2 ||
.
(2.23)
Combining (2.21) and (2.23) we obtain kRz ()k2 ≤ 2(k(I − f (H))Rz ()k2 + kf (H)Rz ()k2 ) kRz ()k 2 2 2 2 + C1 (1 + || kRz ()k) ≤ 2C1 (1 + || kRz ()k) + 2C2 .(2.24) || We may assume that 2C12 (1 + C2 )20 < 1. Then (by subtraction) (2.24) implies the bound (2.19). Upon combining with (2.21) we get (2.20). With Rz () given as in Lemma 2.8, we introduce the notation Fz () = (A + i)−1? Rz ()(A + i)−1 .
998
E. SKIBSTED
Lemma 2.9. In addition to the bounds of Lemma 2.8 we have (with a possibly larger constant C) kRz ()(A + i)−1 k ≤ ||− 2 C(1 + kFz ()k) 2 ,
(2.25)
kM Rz ()(A + i)−1 k ≤ C(1 + kFz ()k) 2
(2.26)
1
1
1
for all and z given as in the lemma. Proof. For any φ ∈ H, we put ψ = Rz ()(A + i)−1 φ. The expectation of (2.22) in the state (A + i)−1 φ gives 2 kFz ()k kφk2 + C1 k(H − i)(I − f (H))ψk2 . kf (H)ψk2 ≤ α || Upon combining with Lemma 2.8 we thus obtain 4 kFz ()k 2 2 2 + C2 kφk2 , kψk ≤ 2kf (H)ψk + 2k(I − f (H))ψk ≤ α ||
(2.27)
which clearly gives (2.25). As for (2.26) we compute (A + i)−1 φ + (H() − z)ψ − if (H)Gf (H)ψ , which in conjunction with (2.13), (2.14) and Lemma 2.8 leads to k(A + i)−1 φk2 ≥
1 k(H() − z)ψk2 − 2 C3 kψk2 2
≥
1 k(M + Im z)ψk2 − (||C4 + 2 C3 )k(H − i)ψk2 2
≥
1 kM ψk2 − 2(||C4 + 2 C3 )k(H − i)f (H)k2 kψk2 − C5 kφk2 . 2
So by (2.27), kM ψk2 ≤ C6 (|| kψk2 + kφk2 ≤ C6
4 kFz ()k + ||C2 + 1 kφk2 . α
which clearly gives (2.26).
Proof of Theorem 2.4. We shall use Lemmas 2.8 and 2.9 to prove the differential inequality
d
Fz () ≤ ||− 12 C(1 + kFz ()k) . (2.28)
d In conjunction with (2.19) this will give Theorem 2.4 (1) by three integrations (iterations) with respect to using the fact that for any φ ∈ H, lim Rz ()φ = (H − z)−1 φ ,
→0
which in turn is readily obtained from Assumption 2.1 (1) and (2.1).
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
999
To prove (2.28) we compute (for 6= 0), d Fz () = (A + i)−1? Rz ()i(M + f (H)Gf (H))Rz ()(A + i)−1 . d The middle term is rewritten as M + f (H)Gf (H) = (M + G) − (I − f )Gf − G(I − f ) .
(2.29)
(2.30)
Upon substituting (2.30) into the right-hand side of (2.29) we obtain three terms. The second and third terms are bounded by ||− 2 C(1 + kFz ()k) 2 (≤ ||− 2 c(1 + kFz ()k)) 1
1
1
(2.31)
by (2.20) and (2.25). It remains to bound the operator Bz () = (A + i)−1? Rz ()i(M + G)Rz ()(A + i)−1 .
(2.32)
For that we rewrite hφ1 , Bz ()φ2 i for φ1 , φ2 ∈ C1 = (A + i)C as follows. We insert (cf. Assumption 2.1 (2) and Lemma 2.2 (1)) M + G = limn→∞ limλ→∞ i[H, An (λ)] = T1 + · · · + T5 ; T1 = limn→∞ limλ→∞ i[H − i(M + f Gf ), An (λ)] , T2 = − limn→∞ limλ→∞ [M, An (λ)] , T3 = − limn→∞ limλ→∞ [f, An (λ)]Gf ,
(2.33)
T4 = − limn→∞ limλ→∞ f [G, An (λ)]f , T5 = − limn→∞ limλ→∞ f G[f, An (λ)] . The contribution from T1 is given by lim lim hφ1 , {(A + i)−1? i[Rz (), λAn (An + iλ)−1 ](A + i)−1 }φ2 i
n→∞ λ→∞
= hRz¯(−)(A + i)−1 φ1 , A(A + i)−1 φ2 i − hA(A + i)−1 φ1 , Rz ()(A + i)−1 φ2 i . (2.34) Here we used that for φ ∈ C1 , lim lim iλAn (An + iλ)−1 (A + i)−1 φ = lim An (A + i)−1 φ = A(A + i)−1 φ .
n→∞ λ→∞
n→∞
Since C1 is dense in H we obtain from (2.34), the Cauchy Schwarz inequality and (2.25) that the contribution from T1 to (2.32) is bounded by the constant (2.31). We are left with bounding the contributions from T2 , . . . , T5 to (2.32). For that we compute as forms on D(M ) using Assumptions 2.1, Lemmas 2.2 and 2.6, (2.10) and (2.3) T2 = iM∞ , Z 1 (∂¯f˜)(η)(H − η)−1 (M + G)(H − η)−1 dudv Gf , T3 = −i π C T4 = if G∞ f , Z 1 (∂¯f˜)(η)(H − η)−1 (M + G)(H − η)−1 dudv . T5 = −if G π C
(2.35)
1000
E. SKIBSTED
By (2.3) the terms T3 and T5 are of the form: T3 = M B and T5 = −B ? M ;
B bounded.
Hence by (2.25), (2.26) and the Cauchy Schwarz inequality all terms of (2.35) contribute to (2.32) by operators with the bounding constant ||− 2 C(1 + kFz ()k) . 1
We have proved (2.28). The second statement Theorem 2.4 (2) may be proved using (2.25) and (2.28) (without the last factor) exactly as in [10]. 3. The Spin-Boson Model Let F denoteR the symmetric Fock space over L2 (Rν ), ν ∈ N. Let m ∈ N. On H = Cm ⊗ F = X ⊗Fdx, X = {1, . . . , m}, we consider a Hamiltonian of the form: Z ⊕{a(λx ) + a? (λx )}dx , (3.1) H = S ⊗ I + I ⊗ Hf + V, V = X
where S is self-adjoint on C , Hf is the second quantization dΓ(w) on F of the operator of multiplication by the function ω(k) = |k| on L2 (Rν ), and a and a? are the operator of annihilation and creation, respectively, of a function λx (·) ∈ L2ω := L2 (Rν , (1 + ω(k)−1 )dk). We use the notation kλx kω for the corresponding weighted L2 -norm. It is well known (cf. [3] and [8]) that for any φ in the form domain of I ⊗ Hf ,
Z
Z
1
⊕a(λx )dxφ , ⊕a? (λx )dxφ ≤ sup kλx kω hI ⊗ Hf + Ii 2 . (3.2) φ
m
X
X
x∈X
Hence by the Kato–Rellich theorem [11, Theorem X.12] the Hamiltonian (3.1) with domain D(H) = Cm ⊗ D(Hf ) is self-adjoint. We shall impose the following stronger condition. Let for i, j ∈ {0, 1, 2}, i + j ≤ 2, x ∈ X i ∂ (i,j) −j λx (k) λx (k) = |k| ∂|k| considered as distributions on C0∞ (Rν \ {0}). We demand λx(i,j) (·) ∈ L2ω .
(3.3)
We can now verify Assumptions 2.1 (1)–(3). Up to this point we have only specified H. Consider the semigroup U (t) acting on L2 (Rν ) given by ν−1 2 k t φ (|k| − t) ,t ≥ 0. U (t)φ(k) = F (|k| > t) 1 − |k| |k| It is generated by −iB (that is U (t) = eitB ), where B is the closure on C0∞ (Rν \{0}) of k ∂ ∂ 1 k ·p+p· ,..., , p = −i . B=− 2 |k| |k| ∂k1 ∂kν
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1001
The A of Assumptions 2.1 is given by A = I ⊗ dΓ(B), where −idΓ(B) is the generator of the second quantized semigroup Γ(U (t)). Generating a contraction semigroup −1 ∈ ρ(−iA). Consequently, −i ∈ ρ(A). We specify (the last product consisting of all finite symmetric linear combinations of products of C0∞ -functions) C=
p [M
Cm ⊗ (C0∞ (Rν \ {0})⊗sq )(⊆ H) .
(3.4)
p∈N q=0
It follows immediately from [11, Theorem X.49] that C is a core of A. By the formal computations on F [dΓ(B), a? (λx )] = a? (Bλx ) , [dΓ(B), a(λx )] = −a(Bλx ) ,
(3.5)
we can (on a formal level) understand (3.3) as conditions guaranteeing H-boundedness of i[V, A] and i[i[V, A, A]. As for the commutator with I ⊗ Hf we have (still formally) i[I ⊗ Hf , A] = I ⊗ i[dΓ(ω), dΓ(B)] = I ⊗ dΓ(i[ω, B]) = I ⊗ dΓ(I) = I ⊗N,
(3.6)
where N is the “number operator”. Motivated by the above computations we define with PΩ given as the projection in F onto the vacuum M = I ⊗ (N + PΩ )(≥ I) , An = I ⊗ dΓ(Bn ) ; Bn =
(3.7)
1 −k (Fn · p + p · Fn ), Fn (k) = p . 2 |k|2 + n−2
The flow generated by the smooth vector field Fn is given by d θn (k, t) = Fn (θn (k, t)), θn (k, 0) = k , dt and satisfies the bound |θn (k, t)| ≤ en . |k| |t|<1, k∈Rν \{0} sup
sup
(3.8)
To verify Assumption 2.1 (2) we notice that the formula (eitBn φ)(k) = (det θn0 (k, t)) 2 φ(θn (k, t)) 1
(3.9)
1002
E. SKIBSTED
combined with (3.8) gives the bound sup kHf eitdΓ(Bn ) (Hf − i)−1 k
|t|<1
!−1 p p X X ≤ en . sup ω(θn (kq , −t)) ω(kq ) − i ≤ sup sup ν |t|<1, p∈N, k1 ,...,kp ∈R q=1 q=1
Clearly the same bound holds upon extending from F to H = Cm ⊗ F . Hence (cf. (3.2)) the bound of Assumption 2.1 (2) holds. To verify that i[H, An ] extends to a H-bounded operator we notice that C given by (3.4) is invariant under eitAn (cf. (3.8) and (3.9)). By a result of Mourre [9, Proposition II.1] we therefore only need to show that i[H, An ] defined as form on C extends to a H-bounded operator. But in this sense we obtain from (3.5) and (3.6) that i[H, An ] = I ⊗ dΓ(ωn ) + Vn(1) ; |k|
, ωn (k) = p |k|2 + n−2 λ(1) n,x
Vn(1)
Z ? (1) ⊕{a(λ(1) n,x ) + a (λn,x )}dx ,
=
(3.10)
X
|k|
= −iBn λx = p |k|2 + n−2
+ λ(1,0) x
+ (ν − 1)|k|2 (0,1) λ 2(|k|2 + n−2 ) x
ν n2
.
(1)
Since ωn ≤ nω and λn,x ∈ L2ω both terms are H-bounded. We readily obtain the last part of Assumption 2.1 (2) by using (3.2), (3.10) and the Lebesgue dominated convergence theorem. Here we put Z ? (1) ⊕{a(λ(1) G= x ) + a (λx )}dx − I ⊗ PΩ ; X
ν − 1 (0,1) λx . 2 Clearly for all φ ∈ C limn→∞ An φ = Aφ. As for Assumption 2.1 (1) we notice that C is also a core of M and that it is invariant under eitM . The bound of Assumption 2.1 (1) is trivial. So by the same argument as above we only need to compute i[H, M ] as a form on C: Z ⊕{P¯Ω a(iλx ) + a? (iλx )P¯Ω }dx ; i[H, M ] = − (3.11) X ¯ PΩ = I − PΩ , λ(1) x = λx (1, 0) +
which obviously is H-bounded. As for Assumption 2.1 (3) we compute by techniques familiar by now Mn = M∞ = 0 , Z (2) ? (2) ⊕{a(λ(2) Gn = Vn = n,x ) + a (λn,x )}dx ; X (1) λ(2) n,x = −iBn λx =
2 X i=0
cn,i λ(i,2−i) , x
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1003
Z ? (2) ⊕{a(λ(2) x ) + a (λx )}dx ;
G∞ = V (2) = X
λ(2) x =
2 X
ci λ(i,2−i) , x
i=0
for smooth coefficient functions obeying sup |cn,i (k)| < ∞ .
ci (k) = lim Cn,i (k), n→∞
i=0,1,2
n∈N,k∈Rν
Finally we shall verify Assumption 2.1 (4) for all E ∈ R assuming Z 2 |λ(1) β := sup x (k)| dk < 1 .
(3.12)
Rν
x∈X
For that we introduce the unitary operator (cf. [8]) Z (1) ? (1) ⊕{a(λx ) − a (λx )}dx . T = exp
(3.13)
X
By the proof of [11, Theorem X.41] C constitutes a set of analytic vectors for the exponent. This fact justifies a small computation yielding to: Z ? (1) ⊕{a(λ(1) I ⊗N + x ) + a (λx )}dx X
Z Z ⊕dx =T I ⊗N −
Rν
X
2 |λ(1) (k)| dk T −1 x
≥ T (I ⊗ N − β)T −1 ,
(3.14)
as a form on C. 1 By a simple density argument (3.14) extends to D(M 2 ) and therefore in parti1 cular to D(H) ∩ D(M 2 ). For any real-valued f ∈ C0∞ (R), we decompose the form: M + f (H)Gf (H) = P − L + f (H)Lf (H) − f (H)Kf (H) ; Z ? (1) −1 ⊕{a(λ(1) , P = I ⊗N + x ) + a (λx )}dx + T I ⊗ PΩ T Z
X
⊕{a(λ(1) x )
L=
(3.15) +a
?
(λ(1) x )}dx
+ T I ⊗ PΩ T
−1
− I ⊗ PΩ ,
X
K = T I ⊗ PΩ T −1 . Clearly K is compact. By (3.12) and (3.14) P ≥ (1 − β) =: 2α > 0. We decompose 2αI = 2αf (H)2 + 2α(I − f (H)2 ) and write 2α − L + f (H)(L − 2α)f (H) = −(I − f (H))(L − 2α)(I − f (H)) + (I − f (H))B ? f (H) + f (H)B(I − f (H)) ,
1004
E. SKIBSTED
where B = B(f ) is bounded. Since the last two terms on the right-hand side may be estimated for any > 0 as (I − f (H))B ? f (H) + f (H)B(I − f (H)) ≥ −2 f (H)BB ? f (H) − −2 (I − f (H))2 (3.16) we obtain the inequality of Assumption 2.1 (4) by choosing such that 2 kBk2 ≤ α. We have verified Assumptions 2.1. The additional assumption of Corollary 2.3 can be verified by (3.10) and the argument after (3.10). In fact we can prove the following stronger result. Theorem 3.1. Suppose (3.3) and (3.12). Let Hpp be the space of eigenstates of 1 H. Then Hpp ⊆ D(M 2 ), (3.17) dim Hpp ≤ m(1 − β)−1 , and the bounds of Theorem 2.4 hold for any E not being an eigenvalue of H. In particular, Hsc , the continuous singular subspace of H, is empty. Proof. For any eigenstate φ of H the expectation hHn iφ vanishes, cf. the proof of Corollary 2.3. Hence we obtain from (3.10) and the Lebesgue monotone and dominated convergence theorems that hM + Giφ vanishes (by letting n → ∞). In 1 particular, φ ∈ D(M 2 ). Moreover, cf. the above verification of Assumption 2.1 (4), Z Z (1) 2 ⊕dx |λx (k)| dk = 0. I ⊗N − X
Rν
T −1 φ
Estimating yields hI(1 − β) − I ⊗ PΩ iT −1 φ ≤ 0 , which in turn (by summing over an orthonormal basis in Hpp ) gives (3.17).
4. The Electron-Boson Model, Preliminaries WeR shall now study Hamiltonian given by (3.1) with H replaced by H = L2 (X)⊗ F = X ⊕Fdx, where X is a finite dimensional real vector space with an inner product. The operator S is now replaced by the (minus) Laplace operator on L2 (X), that is S = −∆ and Hf is given exactly as before while finally V is replaced by X Vb , V = b∈B
Z Vb =
Xb
(4.1) ⊕{a(λbxb )
+a
?
(λbxb )
b
b
b
+ v (x )I}dx .
Here B is a finite index set for a collection of subspaces X b ⊂ X on each of which the corresponding “pair potential”, Vb , has the specified form. If λb = 0 for all b then Hel := −∆ + V acts on the factor L2 (X) and is usually called a generalized Schr¨ odinger operator.
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1005
In this section we shall require λb = λb(·) (·) ∈ L∞ (X b , L2ω ) , k∂yα (|y|α λby (·))kω → 0 for |y| → ∞; |α| ≤ 1 ,
(4.2)
and v b (−∆b + 1)−1 compact, v b real-valued, |∂yα (|y||α| v b (y))| → 0 for |y| → ∞, |α| ≤ 1 .
(4.3)
The above derivatives are only required to exist outside a compact subset of X b and may possibly there be defined as distributional derivatives. (See the beginning of Sec. 5 for more details.) By (3.2), (4.2), (4.3) (the latter conditions only used with |α| = 0) and the Kato–Rellich theorem we infer that H is essentially self-adjoint on D(∆) ⊗ D(Hf ). We now introduce various notation well known for generalizes Schr¨odinger operators. We may assume that the family {Xb |b ∈ B}, defined by writing Xb for the orthogonal complement of X b in X, is closed with respect to intersection and that there exist bmin , bmax ∈ B such that X bmin = 0 and X bmax = X. The set B is ordered by writing b1 ⊂ b2 if X b1 ⊂ X b2 . We introduce projections on X by putting Πb x = xb and Πb x = xb for x = xb ⊕xb ∈ X = X b ⊕Xb . Similar notation p = pb ⊕pb will be used for components of the momentum operator p = −i∇ = (p1 , . . . , pdim X ). R We define on Hb = L2 (X b ) ⊗ F = X b ⊕F dx the sub-Hamiltonian X Vc . (4.4) H b = (pb )2 ⊗ I + I ⊗ Hf + V b ; V b = c⊂b
For b = bmin , the first term on the right-hand side of (4.4) is interpreted as zero, that is H bmin = Hf + V bmin on F . We also define (on H = L2 (Xb ) ⊗ Hb ) Hb = (pb )2 ⊗ I + I ⊗ H b ,
(4.5)
and more general for b ⊂ c (on Hc = L2 (Xbc ) ⊗ Hb ), Hbc = (pcb )2 ⊗ I + I ⊗ H b . As for the usual generalized Schr¨odinger operators the latter Hamiltonians are useful for conducting induction arguments. We define for b 6= bmin the set of thresholds as the set of eigenvalues of subHamiltonians, that is [ σpp (H c ) . (4.6) Fb = c$b
Let d : R → R be given by (E − τ ), F b (E) := F b ∩ (−∞, E] 6= ∅ , inf τ ∈F b (E) b d (E) = 1, F b (E) = ∅ . b
(4.7)
We abbreviate dbmax (E) = d(E) and F bmax = F (H) (assuming here bmax 6= bmin ).
1006
E. SKIBSTED
Similarly we define for c ⊂ b (b = bmin not excluded) with Pcb = ∪c⊂d⊂b σpp (H d ) and some constant C ≥ 0, (E − τ ), Pcb (E) := Pcb ∩ (−∞, E] 6= ∅ , inf τ ∈Pcb (E) b (4.8) dc (E) = b C, Pc (E) = ∅ . Clearly for any > 0, dbc (E + ) ≤ dbc (E) + ,
(4.9)
and similarly for the function defined in (4.7). For R ∈ R, the notation F (· < R) stands for the characteristic function of the interval (−∞, R). Let F (· ≥ R) = 1 − F (· < R). For δ > 0 the notation ηδ stands for any function η ∈ C0∞ (R) obeying 0 ≤ η ≤ 1, η(t) = 1 for |t| ≤ δ and η(t) = 0 for |t| > 2δ. We shall prove the following result expressed in terms of the definition (4.8). Lemma 4.1. Let b, c ∈ B, c ⊂ b, E0 ∈ R, > 0, Kf be compact on F and finally hc ∈ L∞ (X c ) be compactly supported. Then sup kBcb (δ, E, )k → 0 f or δ → 0 ;
E≤E0
Bcb (δ, E, ) = ηδ (H b − E)(B b ⊗ Kf ) , B b = hc (xc )F ((pbc )2 < dbc (E + ) − 2) . (For b = bmin B b is the number hc (0)F (0 < dbc (E + ) − 2).) Proof. We consider the statement of the lemma as a collection of statements labelled by b ∈ B and holding for all other quantities (including the constant C of (4.8)). Then we shall proceed by induction using the ordering of B. Thus the first step will be to verify the statement for b = bmin. For that we may proceed more general by showing it for c = b in which case we may actually assume that b = bmax (since bmin = bmax is not excluded at this point). To do that we notice that for δ < 12 and E ≤ E0 , Bc (δ, E, ) = ηδ (H − E)B1 KB2 ; B1 = F (H < E0 + 1)((p2 + 1) ⊗ I) , K = ((p2 + 1)−1 hc ) ⊗ Kf ,
(4.10)
B2 = B2 (E, ) = F ((pc )2 < dc (E + ) − 2) ⊗ I . Here we have omitted the notation b = bmax at various places while we have kept the notation c for a latter purpose. (If b = bmin we may interpret B1 = I and K = hc (0)Kf .) Clearly B1 is bounded and K is compact. If E ∈ σpp (H) then B2 = 0. On the other hand if E ∈ / σpp (H) then by the compactness kBc (δ, E, )k → 0 for δ → 0. To obtain this convergence uniformly with respect to E ≤ E0 we proceed
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1007
by the way of contradiction. So suppose for some κ > 0 that kBc (δn , En , )k > κ
(4.11)
for sequences δn → 0 and En → E. We notice that (4.9) implies the bound − ; |E − En | ≤ , dc (En + ) − 2 ≤ dc E + 2 2 yielding for large enough n that
B2 (En , ) . Bc (δn , En , ) = ηδn (H − En )Bc δ, E, 2
(4.12)
Here δ > 0 may be chosen arbitrarily. Choosing it so small that kBc (δ, E, 2 )k ≤ κ we obviously have a contradiction to (4.11). Next we conduct the induction step assuming again b = bmax , so that we can now use the statement of the lemma for all b 6= bmax . Suppose c 6= bmax . Since the last argument above involving (4.12) did not use the property c = bmax , it suffices to show convergence for fixed E. For that purpose we pick a family {jb }, b 6= bmax , of functions on X each one being smooth and homogeneous of degree 0 outside a compact set. We assume that X jb = 1 0 ≤ jb ≤ 1, b 0
/b and for any b ⊂ 0
|x|jb (x) ≤ C|xb |jb (x) ; We insert this partition ηδ (H − E)B ⊗ Kf =
X
C = C(b, b0 ) < ∞ .
(4.13)
ηδ (H − E)jb B ⊗ Kf .
(4.14)
b
Clearly we may assume that the constant C entering in the definition of B through max it is readily seen that (4.8) is large. Assuming C ≥ E0 + − inf Pbbmin 00
0
dbc00 (E 0 ) ≤ dbc0 (E 0 ); c00 ⊂ c0 ⊂ b0 ⊂ b00 ,
E 0 ≤ E0 + .
(4.15)
In particular, Fc = Fbmax Fc ;
Fc0 = Fc0 ((p0c )2 < dc0 (E + ) − 2) ,
and thus we can write for b with c ⊂ /b ηδ (H − E)jb B ⊗ Kf = ηδ (H − E)B1 KB2 B3 ; B1 = F (H < E0 + 1)((p2 + 1) ⊗ I) , K = ((p2 + 1)−1 jb hc ) ⊗ Kf , B2 = Fbmax ⊗ I ,
B3 = Fc ⊗ I .
By (4.13) K is compact and hence we recognize the form (4.10) treated above.
1008
E. SKIBSTED
Thus we only need to bound those terms with c ⊂ b. To do that we split ηδ (H − E)jb = ηδ (H − E)jb ηδ0 (Hb − E) + ηδ (H − E)T (δ 0 ) ; T (δ 0 ) = ηδ0 (H − E)jb − jb ηδ0 (Hb − E), δ 0 ≥ 2δ .
(4.16)
We may write ηδ (H − E)T (δ 0 )B ⊗ Kf = ηδ (H − E)KB2 B3 ; K = T (δ 0 )(hc ⊗ Kf ) , B2 = Fbmax ⊗ I, B3 = Fc ⊗ I . By a computation using (2.10) and (4.13) we see that kT (δ 0 )(F (|x| ≥ R) ⊗ I)k → 0 for R → ∞ , implying that K is compact. Hence for fixed δ 0 > 0 we can again proceed as above to treat the contribution from the second term on the right-hand side of (4.16). It remains to show that the contribution from the first term on the right-hand side of (4.16) can be estimated arbitrarily small by choosing δ 0 > 0 small enough. For that purpose we use (4.5) to reduce to a statement for the sub-Hamiltonian H b . We decompose (pc )2 = (pbc )2 + (pb )2 and estimate (cf. (4.9) and 4.15)) dc (E + ) − (pb )2 ≤ dc (E − (pb )2 + ) ≤ dbc (E − (pb )2 + ) ,
(4.17)
yielding kηδ0 (Hb − e)B ⊗ Kf k ≤ kηδ0 (H b − (E − (pb )2 )) · · · (hc (xc )F ((pbc )2 < dbc (E − (pb )2 + ) − 2)) ⊗ Kf k ≤ sup kηδ0 (H b − E 0 )(hc (xc )F ((pbc )2 E 0 <E0
< dbc (E 0 + ) − 2)) ⊗ Kf k .
(4.18)
Clearly by the induction hypothesis the right-hand side of (4.18) → 0 for δ 0 → 0 completing the proof. In the rest of this section bmin 6= bmax . We can now prove a similar result expressed in terms of the definition (4.7). Corollary 4.2. Let c ∈ B, E ∈ R, > 0, Kf be compact on F and finally hc ∈ L∞ (X c ) be compactly supported. Then for any given ε > 0, there exists for all small enough δ > 0 a compact operator K = K(ε, δ) on H such that kBc − Kk ≤ ε ; Bc = ηδ (H − E)(B ⊗ Kf ) ,
B = hc (xc )F ((pc )2 < d(E + ) − 2) .
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1009
Proof. We use the partition of unity in the proof of Lemma 4.1 to obtain (4.14) (with the present definition of B). The terms with c ⊂ / b are compact. As for the terms with c ⊂ b we use (4.16) and the discussion following (4.16) to conclude that we only need to find a δ 0 > 0 such that X kηδ0 (Hb − E)B ⊗ Kf k ≤ ε . (4.19) c⊂b6=bmax
For that we pick C = max(1, E + − inf F (H)) to be used as input in the definition (4.8) for any b and c such that c ⊂ b 6= bmax . Then it is readily seen (cf. (4.17)) that (4.20) d(E + ) − (pb )2 ≤ d(E − (pb )2 + ) ≤ dbc (E − (pb )2 + ) , which implies the bound (4.18). Using Lemma 4.1 to the right-hand side of (4.18) we clearly obtain (4.19) for some small δ 0 > 0. In order to apply Corollary 4.2 we introduce a certain vector field on X invented by Graf [7]. We have enlisted a collection of its characteristics in the following lemma. These may be derived by mimicking [7] or a simplified procedure due to Derezinski [5]. The property (5) was derived in [14, Appendix A]. Lemma 4.3. There exist on X a smooth vector field ω with symmetric derivative qb } indexed by b ∈ B and consisting of smooth functions, ω∗ and a partition of unity {˜ 0 ≤ q˜b ≤ 1, such that for some positive constants r1 and r2 P (1) ω∗ (x) ≥ b Πb q˜b (x). (2) ω b (x) = 0 if |xb | < r1 . qb ) if c ⊂ / b. (3) |xc | > r1 on supp (˜ b qb ). (4) |x | < r2 on supp (˜ (5) For all multi-indicies α and n ∈ N ∪ {0} there exists C ∈ R: |∂xα q˜b (x)| + |∂xα (x · ∆)n (ω(x) − x)| ≤ C ;
x∈X.
We state some consequences. Combining (3) and (4) yields to qb (kx) = 0 if c ⊂ / b; q˜c (x)˜
k=
r1 . r2
(4.21)
We define for any b ∈ B, X
qb (x) = q˜b (kx)
!− 12 q˜c (kx)2
.
c
Then by (1) and (4.21) ω? (x) ≥
X
Πb qb (x)2 .
(4.22)
b
Finally using (2) and the fact that ω is a gradient field, we obtain ω(x) − ω(xb ) if |xb | < r1 .
(4.23)
1010
E. SKIBSTED
We shall now consider the operator BR on L2 (X) given by BR =
x x 1 Rω ·p+p·ω R ; 2 R R
R > 0.
More precisely, we shall consider the formal commutator x x X 1 (∆(∇ · ω)) Vb0 ; p− ⊗ I + i[H, BR ⊗ I] := 2pω? R 2R2 R b∈B x b Z · ⊕{a(∇b λbxb ) + a? (∇b λbxb ) + (∇b v b )(xb )I}dxb . Vb0 = −Rω R Xb
(4.24)
(4.25)
Here R should be chosen so large that the right-hand side makes sense as a symmetric operator on D(H). Notice that this is possible due to our conditions (4.2) and (4.3), (3.2) and Lemma 4.3 (2) and (5). In fact we observe that kVb0 (H − i)−1 k → 0 for R → ∞ .
(4.26)
To state the main result of this section we notice (cf. [11, Theorem X.41]) that P ˜ b obeying (4.2) is any potential V˜ = b∈B V˜b of the form (4.1) with v˜b = 0 and λ essentially self-adjoint on C=
p [O
C0∞ (X) ⊗ (C0∞ (Rν \ {0})⊗s q )(⊆ H) .
(4.27)
p∈N q=0
In the next section C will play a similar role as the space defined by (3.4) did in Sec. 3. Proposition 4.4. Let V˜ be given as above, E ∈ R and > 0. Then there exists R0 > 0 so that for any R ≥ R0 there exist an open neighbourhood U = U(R) of E and a compact operator K = K(R) on H such that with T = exp(iV˜ ) f (H)i[H, BR ⊗ I]f (H) ≥ f (H){2(d(E + ) − 3)T I ⊗ PΩ T −1 − K − I}f (H)
(4.28)
for all real-valued f ∈ C0∞ (U). Proof. Let in the following the notation o(1) stands for an R-depending bounded operator obeying ko(1)k → 0 for R → ∞. Operators or forms on the form o(1)(H − i) or (H + i)o(1)(H − i) are denoted by oH (1) and oHH (1), respectively. By (4.22), Lemma 4.3 (5) and (4.26) x x X (pb )2 qb , (4.29) Qb (R) ⊗ I + oH (1) ; Qb (R) = qb i[H, BR ⊗ I] ≥ 2 R R b
as a form inequality on D(H).
1011
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
Pdim Xb 2 For each of the terms Qb (R) ⊗ I we may write (pb )2 = pj , (p1 , . . . , j=1 pdim Xb ) being the moment operator with respect to some orthonormal basis in Xb , and then estimate Qb (R) ⊗ I ≥
dim Xb X
Pj? T I ⊗ PΩ T −1 Pj = T
j=1
dim XXb
Pj? I ⊗ PΩ Pj T −1 + oHH (1)
j=1
= T Qb (R) ⊗ PΩ T −1 + oHH (1) ; x ⊗I. Pj = pj qb R
(4.30)
The commutations of the middle step require justification: We introduce T (t) = exp(itV˜ ); −1 ≤ t ≤ 1, and obtain by a differentiation on the core C ⊂ D(H) XZ 1 T (t)i[V˜c , Pj ]T (−t)dt . (4.31) T Pj T −1 − Pj = c
0
Here by assumption Z V˜c = Xc
˜ c c ) + a? (λ ˜ c c )}dxc , ⊕{a(λ x x
˜ b satisfies (4.2). We compute where λ Z ˜c c ) + a? (∂j λ ˜ c c )}qb x dxc . ˜ ⊕{a(∂j λ i[Vc , Pj ] = − x x R Xc
(4.32)
(4.33)
Clearly this operator is zero for c ⊂ b and in general of the form oH (1) (cf. Lemma 4.3 (3)). By iterating the formula (4.31), we obtain X XXZ 1 Z t i[V˜c , Pj ] = dt dsT (s)i2 [V˜d , [V˜c , Pj ]]T (−s) T Pj T −1 − Pj − d
c⊂ /b
=
c⊂ /b
0
XX 1 d
c⊂ /b
2
0
fc,d ⊗ I ,
(4.34)
where by the canonical commutation relations and (4.33) we have computed Z x 2 ˜ d c ˜ ˜ ˜ Im λxd (k)∂j λxc (k)dk ⊗ I =: fc,d (x) ⊗ I . i [Vd , [Vc , Pj ]] = −2qb R Rν (4.38) Since fc,d ⊗ I = o(1) we can write (4.34) on the form T −1 Pj = Pj T −1 + oH (1)
(4.35)
as an operator on C and thus (by density) on D(H). Clearly we can justify the middle step of (4.30) by using (4.35) and the adjoint formula.
1012
E. SKIBSTED
Using Qb (R) ≥ (d(E + ) − 3)qb
x R
F ((pb )2 ≥ d(E + ) − 3)qb
x R
to the right-hand side of (4.30) and substituting in (4.20) yields to i[H, BR ⊗ I] ≥ 2(d(E + ) − 3)T I ⊗ PΩ T −1 X − 2d(E + ) T Pb (R) ⊗ PΩ T −1 + oHH (1) ;
(4.36)
b
Pb (R) = qb
x R
F ((pb )2 < d(E + ) − 3)qb
x R
.
For each of the terms T Pb (R) ⊗ PΩ T −1 , we shall now use the following analogue of (4.31). XZ 1 −1 T (t1 )iad1c1 (Pb (R) ⊗ PΩ )T (−t1 )dt1 , T Pb (R) ⊗ PΩ T − Pb (R) ⊗ PΩ = c1
0
(4.37) utilizing here and below the notation m−1 ˜ adm c1 ,...,cm (B) = [Vcm , adc1 ,...,cm−1 (B)] ,
ad0 (B) = B; c1 , . . . , cm ∈ B . The commutator in (4.37) is given by (1)
ad1c1 (Pb (R) ⊗ PΩ ) = Bc1 + Pb (R) ⊗ Iad1c1 (I ⊗ PΩ ) ; (m) (I ⊗ PΩ ) . Bc1 ,...,cm = [V˜cm , Pb (R) ⊗ I]adcm−1 1 ,...,cm−1 (m)
(4.38)
(m)
For cm ⊂ bBc1 ,...,cm = 0. In general Bc1 ,...,cm = o(1) (cf. (4.42) below and Lemma 4.3 (3)). To treat the contribution from the second term on the right-hand side of (4.38) to (4.37), we iterate by inserting T (t1 )Pb (R) ⊗ Ii ad1c1 (I ⊗ PΩ )T (−t1 ) − Pb (R) ⊗ Ii ad1c1 (I ⊗ PΩ ) X Z t1 T (t2 )(Pb (R) ⊗ I)i2 ad2c1 ,c2 (I ⊗ PΩ )T (−t2 )dt2 + o(1) , =
(4.39)
0
c2
(2)
where we used that Bc1 ,c2 = o(1). By iterating the above procedure n(∈ N) times we obtain the formula T Pb (R) ⊗ PΩ T −1 = Tn + Rn + o(1) ; Tn =
n−1 X
X
(m!)−1 Pb (R) ⊗ Iim adm c1 ,...,cm (I ⊗ PΩ ) ,
(4.40)
m=0 c1 ,...,cm
Rn =
X Z c1 ,...,cn
0
Z
1
dt1 · · · 0
tn−1
dtn T (tn )Pb (R) ⊗ Iin adnc1 ,...,cn (I ⊗ PΩ )T (−tn ) .
1013
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
We claim that kRn k → 0 for n → ∞ ,
(4.41)
uniformly with respect to R. To show (4.41) we notice the bound (cf. the proof of [11, Theorem X.41]) kadnc1 ,...,cn (I
⊗ PΩ )k ≤ 4
n
√
Z ˜ c c (k)|2 dk |λ x
n! sup c,xc
Rν
n2 ,
(4.42)
which upon computing the integrals implies kRn k ≤ C n (n!)− 2 ; 1
Z C = 4(#B) sup c,xc
Rν
˜ c c (k)|2 dk |λ x
12 ,
and therefore clearly (4.41). 1 We fix n such that 2d(E + )(#B)C n (n!)− 2 ≤ 4 bounding the contribution from all Rn (one for each b) to the second term on the right-hand side of (4.36) by 4 . Next we look at the term Tn . Clearly Z (I ⊗ P ) = ⊕Kf (xc )dxc =: Dc adm Ω c1 ,...,cm Xc
with X c = X c1 + · · · + X cm and Kf (·) being a bounded weakly measurable compact operator-valued function with kKf (xc )k → 0 for |xc | → ∞. Up to a small bounded operator on Hc = L2 (X c ) ⊗ F Dc ((pc )2 + 1)−1 ⊗ I ≈ Dc K c ⊗ I , where K c is on the form K c = g c (xc )((pc )2 + 1)−1 , g c ∈ L∞ (X c ) being compactly supported. Again up to a small bounded operator we can write Z ⊕QKf (xc )Qdxc K c ⊗ I , Dc K c ⊗ I ≈ Xc
where Q is a finite dimensional projection on F (by the Lebesgue dominated convergence theorem). Finally using that (pc )2 is H-bounded we conclude that − 2d(E + )(m!)−1 Pb (R) ⊗ Iim adm c1 ,...,cm (I ⊗ PΩ )η1 (H − E) X (Pb (R)hc (xc )) ⊗ Kf η1 (H − E) , ≈
(4.43)
hc ,Kf
where the summation is finite and ranges over compactly supported hc ∈ L∞ (X c ) and compact operators Kf on F . Of course the approximation in (4.43) is uniform with respect to R. We fix for each term an approximation such that the total contribution from the errors to the second term on the right-hand side of (4.36) is bounded by 4 (uniformly in R).
1014
E. SKIBSTED
x x Since we can write qb ( R )hc (xc ) = qb ( R )hdR (xd ) with X d = X b + X c and hdR ∈ d L (X ) being compactly support (cf. Lemma 4.3 (4)), and for this d ∞
Pb (R) = Pb (R)F ((pd )2 < d(E + ) − 2) + o(1) (cf. (2.10), (4.16) and Lemma 4.3 (5)), we conclude (Pb (R)hc (xc )) ⊗ Kf = (Pb (R)F ((pd )2 < d(E + ) − 2)hdR (xd )) ⊗ Kf + o(1) .
(4.44)
Now we fix R0 such that the contributions from the terms o(1) on the right-hand side of (4.44) and (4.40) plus the one from η1 (H −E)oHH (1)η1 (H −E) (with oHH (1) given in (4.36)) altogether are bounded by 4 for R ≥ R0 . Finally by Corollary 4.2 we can find a compact operator K = K(R) on H such that for a small δ > 0, X Pb,d,Kf ⊗ Kf ηb (H − E)k ≤ ; kK − 4 b,d,Kf
Pb,d,Kf = Pb (R)F ((pd )2 < d(E + ) − 2)hdR (xd ) . Upon symmetrizing all previously encountered terms we readily obtain the result using that the four 4 -bounds appeared up to this point adds to . The proof of Proposition 4.4 was involved due to the appearance of the operator T . Partly as a warm up for the next section we close Sec. 4 by specializing to the case T = I and λb = 0 (for all b). We shall derive spectral properties for the generalized Schr¨ odinger operator Hel := −∆ + V acting (as a factor) on L2 (X). Notice that in this case H = Hel ⊗ I + I ⊗ Hf and that the set of thresholds of H as defined by (4.6) coincides with the set Fel of thresholds of Hel (defined similarly in terms of sub-Hamiltonians on the Hilbert space L2 (X)). Corollary 4.5 (A Mourre estimate). Under the condition (4.3) there exists for any given E ∈ R and > 0, a positive number R0 so that for any R ≥ R0 we can find an open neighbourhood U = U(R) of E and a compact operator K = K(R) on L2 (X) such that f (Hel i[Hel , BR ]f (Hel ) ≥ f (Hel ){(2d(E) − )I − K}f (Hel )
(4.45)
for all real-valued f ∈ C0∞ (U). Moreover the eigenvalues of Hel (counted with multiplicity) can only accumulate at Fel , the latter set being closed and countable. Proof. We proceed by induction with respect to the ordering of B (cf. the proof of Lemma 4.1). So suppose we know the statements of the corollary for subHamiltonians (leaving the start of induction to the reader), then we need to verify them for Hel . Since (by assumption) the eigenvalues of any sub-Hamiltonian can
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1015
only accumulate at the set of thresholds of this operator we obtain that Fel is closed and countable. Now we can verify the other statements without further reference to the induction hypothesis. We apply (4.28) to the invariant subspace L2 (X) = (I ⊗ PΩ )H yielding f (Hel )i[Hel , BR ]f (Hel ) ≥ f (Hel {(2d(E + ) − 7)I − K}f (Hel ) .
(4.46)
If E ∈ Fel then clearly (4.45) follows from (4.46). Using that Fel is closed then also for E ∈ / Fel (4.46) implies (4.45). The remaining statement that the eigenvalues of Hel can only accumulate at Fel follows readily from (4.45) and a virial theorem (cf. Corollary 2.3). 5. The Electron-Boson Model, Continued We proceed somewhat similar to Sec. 3 skipping some arguments given there. The basic set-up is the same as the one in Sec. 4, but we shall need the following stronger conditions compared to (4.2) and (4.3). We require b(i,j)
λ(·) b(i,j) λxb (k)
−j
= |k|
∂ ∂|k|
(·) ∈ L∞ (X b , L2ω ) ;
i
b(i,j)
k∂yα (|y|α λy
λbxb (k) ,
i, j ∈ {0, 1, 2}, i + j ≤ 2 ,
(5.1)
(·))kω → 0 for |y| → ∞ ;
i + j + |α| ≤ 2 , and v b (−∆b + 1)−1 compact, v b real-valued, |∂yα (|y||α| v b (y))| → 0 for |y| → ∞; |α| ≤ 2 .
(5.2)
As in Sec. 3 the function λbxb (·) in (5.1) (for fixed xb ) is considered as a distribution on C0∞ (Rν \ {0}). Moreover all the above derivatives with respect to xb are only required to exist outside a compact subset Kb ⊂ X b in the sense of distribution. More precisely we consider in the case of (5.1) λb(i,j) as a distribution on C0∞ (X b \ Kb ) ⊗ C0∞ (Rν \ {0}) and in the case of (5.2) v b as a distribution on C0∞ (X b \ Kb ). We shall need the following analogue of (3.12). Assume that for all b ∈ B, 2 Z X c(1) λxc (k) dk < 1 ; β b := sup b b ν x ∈X R c⊂b (5.3) ν − 1 c(0,1) c(1) c(1,0) λxc . + λxc = λxc 2 For later convenience, we abbreviate X X b(1) λbxb , λ(1) λxb , β = β bmax . (5.4) λx = x = b⊂B
b∈B
1016
E. SKIBSTED
We shall verify Assumptions 2.1 for some fixed energy E ∈ / F(H) in Assumption 2.1 (4) assuming that F (H) is closed and countable. The latter condition can be verified inductively, cf. the proof of Corollary 4.5. For that let BR be defined as the closure in L2 (X) of the expression (4.24) on ∞ C0 (X) and Bf = dΓ(B) as in Sec. 3. We shall consider A = CBR ⊗ I + I ⊗ Bf
(5.5)
defined as the closure on C, the latter given by (4.27). Here C is an arbitrary constant chosen such that (with d(E) given by (4.7)) C2d(E) ≥ 1 .
(5.6)
It is readily proved (cf. [11, Theorem X.49]) that −iA generates a contraction semigroup. With Bn given by (3.7) we define An = CBR ⊗ I + I ⊗ dΓ(Bn ) as the closure on C, which constitutes a family of self-adjoint operators approaching A as n → ∞. Let (5.7) M = I ⊗ (N + PΩ )(≥ I) . We can now verify Assumptions 2.1 for R large enough. We shall mainly focus on Assumption 2.1 (4) referring to Sec. 3 for a more detailed account. (Many arguments there may readily be modified.) We notice that the invariance of D(H) and D(M ) under eitAn follows from the formula eitAn = (eitCBR ⊗ I)(I ⊗ eitdΓ(Bn ) )
(5.8)
since by explicit computation these properties hold for both factors on the righthand side. Moreover (5.8) implies invariance of C, and therefore it suffices to calculate commutators as forms on C (cf. Sec. 3). We obtain Z ? (1) ⊕{a(λ(1) (5.9) G= x ) + a (λx )}dx − I ⊗ PΩ + Ci[H, BR ⊗ I] . X
Clearly by (5.1) and (4.25) G is bounded relative to H. (1) To verify Assumption 2.1 (4) we let again T be given by (3.13) (with λx given in (5.4)). Then using (5.7) and (5.9) we split for any real-valued f ∈ C0∞ (R) (cf. (3.15)) M + f (H)Gf (H) = P − L + f (H)(L + Q)f (H) ; Z ? (1) −1 ⊕{a(λ(1) , P = I ⊗N + x ) + a (λx )}dx + T I ⊗ PΩ T Z L=
X ? (1) −1 ⊕{a(λ(1) , x ) + a (λx )}dx − I ⊗ PΩ + T I ⊗ PΩ T
X
Q = Ci[H, BR ⊗ I] − T I ⊗ PΩ T −1 .
(5.10)
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1017
Due to (5.3) and (3.14) P ≥ (1 − β) =: 2α > 0. By (4.28) and (5.6) f (H)Qf (H) ≥ −Cf (H){K + I}f (H) ,
(5.11)
for > 0 small, K = K(, R), R large, and finally for all f supported sufficiently close to E. We fix = α2 C −1 and a corresponding large R. Then we may proceed exactly as in Sec. 3 after the formula (3.15). Using (3.16) with the there chosen such that 2 kBk2 ≤ α2 we obtain now from (5.10) and (5.11) the inequality of Assumption 2.1 (4) (with the above α and K replaced by CK). We have now outlined a verification of Assumptions 2.1 for our electron-boson Hamiltonian. Leaving it to the reader to use Corollary 2.3 (see the first part of the proof of Theorem 3.1) and to perform an induction argument (cf. the proof of Corollary 4.5, the start of induction using Theorem 3.1 with m = 1) we summarize our results as follows. Theorem 5.1. Suppose (5.1), (2.5) and (5.3). Then any eigenstate of H belongs 1 to D(M 2 ), with M given by (5.7), and the eigenvalues of H (counted with multiplicity) can only accumulate at the set of thresholds F (H), the latter being closed and countable. Moreover the bounds of Theorem 2.4 hold with A given by (5.5) and E not being an eigenvalue nor threshold energy of H. In particular Hsc , the continuous singular subspace of H, is empty. 6. Extended Models We shall now consider an extension of the model of the previous section. Up to inclusion of spin, polarization and various cut-offs the standard (Dirac) model of quantum electrodynamics (see [3]) can be put on this form as to be discussed at the end of this section. We consider again H = p2 ⊗ I + I ⊗ Hf + V˜ on H = L2 (X) ⊗ F , P where the “potential” V˜ = b∈B V˜b now is more general. We demand
(6.1)
¯ b · pb + pb · U ¯b + (W ¯ b )2 ; V˜b = Vb + U Z ⊕{a(λVxbb + a? (λVxbb ) + v Vb (xb )}dxb , Vb = Xb
Z ¯b )l = (U
Xb
bl bl ⊕{a(λU ) + a? (λU ) + v Ubl (xb )}dxb , xb xb
(6.2)
Z ¯ b )l = (W
Xb
bl bl ⊕{a(λW ) + a? (λW ) + v Wbl (xb )}dxb , xb xb
where we write (with respect to an orthonormal basis in X b ) ¯ b = (Wb1 , . . . , Wb dim X b ) , ¯b = (Ub1 , . . . , Ub dim X b ), W U
pb = (pb1 , . . . , pbdim X b ) .
We aim at an analogue of Theorem 5.1 for H on this form by verifying Assumptions 2.1 with the same inputs for A, An and M .
1018
E. SKIBSTED
We impose the conditions (5.1) and (5.2) for the first term Vb on the right-hand ¯ b , which ¯b and W side of (6.2) and need to specify conditions on the “vectors” U again will allow us to apply the Kato–Rellich theorem. A related problem is to find conditions such that our new G is bounded relatively to H. For the latter purpose we (formally) compute the following analogue of (5.9): G = T1 + T2 + T3 − I ⊗ PΩ + Ci[H, BR ⊗ I] ; Z ⊕{a(λVx (1) ) + a? (λVx (1) )}dx , T1 = X
T2 =
X
X
(6.3)
i{[Ubl , Bf ]pbl + pbl [Ubl , Bf ]} ,
b∈B 1≤l≤dim X b
T3 =
X
X
i{Wbl [Wbl , Bf ] + [Wbl , Bf ]Wbl } .
b∈B 1≤l≤dim X b
P V (1) V (1) = b∈B λxbb is given as in (5.3) and (5.4). With a similar notation Here λx the commutators are given by Z U (1) U (1) ⊕{a(λxbbl ) + a? (λxbbl )}dxb , i[Ubl , Bf ] = Xb
Z i[Wbl , Bf ] =
Xb
(6.4) W (1) ⊕{a(λxbbl )
+a
?
W (1) (λxbbl )}dxb
.
Moreover we may write ∂ Ubl (1) ∂ Ubl (1) ? + ⊕ a λ b λ b +a dxb . ∂xl x ∂xl x Xb Z
ipbl [Ubl , Bf ]
=
i[Ubl , Bf ]pbl
(6.5) Motivated by the above computations we can now specify conditions on the new terms. Clearly (6.4), (6.5) and (3.2) suggest the condition U (1)
∂xβb λxbbl
(·) ∈ L∞ (X b , L2ω ) ;
|β| ≤ 1 ,
(6.6)
since it assures that the term T1 on the right-hand side of (6.3) is H-bounded. 1 1 (Notice that (−∆b + 1) 2 ⊗ (Hf + 1) 2 is H-bounded.) Similarly the condition bl (·) ∈ L∞ (X b , L2ω ) ; ∂xβb λU xb
|β| ≤ 1 ,
assures relative boundedness of the contribution to H from the terms involving λUbl in the definition of V˜b . To obtain relative boundedness of the term T3 , we use the bounds
Z
− 32
(Hf + 1) ⊕a(λx )dx(Hf + 1) ≤ 2 sup k(1 + ω)λx kω ,
X
x∈X
Z
? − 32
(Hf + 1) ⊕a (λx )dx(Hf + 1) ≤ 2 sup k(1 + ω)λx kω ,
X
x∈X
(6.7)
1019
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
obtained from (3.2), (3.5) and commutations. By interpolating (3.2) and (6.7) we obtain
Z
1 −1
(Hf + 1) 21 2 sup k(1 + ω)λ k , ⊕a(λ )dx(H + 1) x f x ω
≤ 2 x∈X
X (6.8)
Z
1 1 ? −1
(Hf + 1) 2 2 ⊕a (λx )dx(Hf + 1)
≤ 2 sup k(1 + ω)λx kω ,
x∈X
X
We expand each term of T3 into a sum of products using (6.4). Those terms that are products of two factors each one either being of the form a(λx ) or a∗ (λx ) are handled by inserting 1 1 I = (Hf + 1)− 2 (Hf + 1) 2 between the factors and then invoke (3.2) and (6.8). This procedure leads to the conditions (6.9) (1 + ω)λWbl , (1 + ω)λWbl (1) ∈ L∞ (X b , L2ω ) . W (1)
To handle the contribution from the remaining terms a(λxbbl vxWbbl , we impose (v Wbl )2 (−∆b + 1)−1 bounded, Wbl
b
W (1)
)vxWbbl and a? (λxbbl
)
(6.10) − 12
which by interpolation implies boundedness of v (−∆ + 1) . Combining this property with (6.9) we readily treat the above terms. Moreover we remark that (6.9) and (6.10) assure the relative boundeness of the ¯ b in the definition of V˜b . contribution from the term involving W It remains to elaborate on the last term on the right-hand side of (6.3). Formally (cf. (4.25)) x x X 1 p− V˜b0 ; (∆(∇ · ω)) ⊗ I + i[H, BR ⊗ I] := 2pω∗ R 2R2 R b∈B
V˜b0
=
0 T1b
+
0 T2b
+
0 T3b
,
0 = i[Vb , BR ⊗ I] , T1b
(6.11)
0 ¯ b · pb + pb · U ¯b , BR ⊗ I] , = i[U T2b 0 ¯ b )2 , BR ⊗ I] , = i[(W T3b
where we may compute x b Z 0 = Vb0 = −Rω · ⊕{a(∇b λVxbb ) + a? (∇b λVxbb ) + (∇b v Vb )(xb )}dxb , T1b R Xb X 0 0 0 = {Ubl0 pbl + pbl Ubl0 + Ubl BRl + BRl Ubl } ; T2b 1≤l≤dim X b
x 1 0x ∂ ωl · p + p · ωl0 , ωl0 = ω, 2 R R ∂xl X = {Wbl Wbl0 + Wbl0 Wbl } .
0 = BRl 0 T3b
1≤l≤dim X b
1020
E. SKIBSTED
Here Ubl0 and Wbl0 are given as Vb0 by replacing Vb by Ubl and Wbl , respectively. The conditions bl (·)) ∈ L∞ (X b , L2ω ) , ∂xαb (|xb ||α| ∂xβb λU xb
∂xαb (|xb ||α| ∂xβb v Ubl (·)) ∈ L∞ (X b ) , bl (·)) ∈ L∞ (X b , L2ω ) , ∂xαb (|xb ||α| (1 + w)λW xb
∂xαb (|xb ||α| v Wbl (·)) ∈ L∞ (X b ) ; |β|, |α| ≤ 1 , 0 and and Lemma 4.3 assure H-boundedness of the contribution from the terms T2b 0 T3b . Actually due to Lemma 4.3 and (4.23) we here only need boundedness outside a compact set in X b . Since we shall verify more than just relative boundeness (6.6), (6.9), (6.10) and (6.12) would not suffice. For example, we shall need decay at infinity in (6.9) and (6.12). Also we shall need assumptions for |α| = 2. Explicitly (and as a conclusion of the previous discussion) we impose in addition to (5.1) and (5.2) for Vb the following conditions. Let for i, j ∈ {0, 1, 2}, i + j ≤ 2, i ∂ Ubl (i,j) −j bl (k) = |k| λU (k) . λxb xb ∂|k| W (i,j)
(k) be defined similarly. Let λxbbl ¯b , we demand As for U U (i,j)
∂xβb λxbbl
(·) ∈ L∞ (X b , L2ω ) ; U (i,j)
k∂yα (|y|α ∂yβ λy bl
i + j ≤ 2, |β| ≤ 1 ,
(·))kω → 0 for |y| → ∞ ;
i + j + |α| ≤ 2 ,
(6.12)
|β| ≤ 1 ,
and (∂xβb v Ubl (xb ))(−∆b + 1)−
1+β 2
compact;
|β| ≤ 1 ,
|∂yα (|y||α| ∂yβ v Ubl (y))| → 0 for |y| → ∞ ;
v Ubl real-valued,
|α| ≤ 2 ,
|β| ≤ 1 .
(6.13)
¯ b , we demand As for W W (i,j)
(1 + ω)λxbbl
∈ L∞ (X b , L2ω ) ;
i+j ≤ 2,
k∂yα (|y|α (1 + ω)λWbl (i,j) )kω → 0 for |y| → ∞ ;
(6.14)
i + j + |α| ≤ 2 , and (v Wbl )2 (−∆b + 1)−1 compact,
v Wbl real-valued,
|∂yα (|y||α| v Wbl (y))| → 0 for |y| → ∞; |α| ≤ 2 .
(6.15)
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1021
To verify the Kato–Rellich criterion, kV˜ b φk ≤ (1 − b )k((pb )2 ⊗ I + I ⊗ Hf )φk + Cb kφk
(6.16)
(for some b > 0; b ∈ B) one would need an additional smallness assumption. It may readily be shown that the following bound implies (6.17): X X 5 Ubl Wbl 2 2 sup kλxb kω + sup k(1 + ω)λxb kω < 1 . 2 b∈B 1≤l≤dim X b
xb ∈X b
xb ∈X b
With the above assumptions we may justify the previous computations. In particular G is given by the H-bounded expression(6.3). To verify Assumption 2.1 (4) we notice that the decay conditions of (5.1), (5.2) and (6.13)–(6.16), and the property (4.23), imply that the terms V˜b0 in (6.11) in the form sense vanish for R → ∞. Explicitly using the notation of the proof of Proposition 4.4 V˜b0 = oHH (1). The other terms on the right-hand side of (6.11) may be treated as in the proof of Proposition 4.4 (with T = I), and since we may generalize the previous results of Sec. 4 to the present more general class of Hamiltonians we conclude (4.28) with T = I under the present assumptions (5.1), (5.2) and (6.13)–(6.17). We shall need a lower bound for the form T1 + T2 + T3 on the right-hand side of (6.3) (substituting (5.3)). Due to the appearance of momentum operators in the definition of T2 we need at this point to restrict our attention to a bounded energy regime. So let I0 := (−∞, E0 ] for some fixed E0 . Similarly to Sec. 5 we assume that F (H) ∩ I0 is closed and countable. We claim that the latter property may be proved inductively under the following additional smallness condition replacing (5.3). We demand that for all b ∈ B, F0b (T1b + T2b + T3b )F0b ≥ −β b F0b ; F0b = F (H b < E0 + 1) ,
β b ∈ [0, 1) ,
(6.17)
where for b = bmax (T1b + T2b + T3b ) is given by T1 + T2 + T3 on the right-hand side of (6.3) and for b 6= bmax by similarly expressions in terms of the potentials V˜c with c ⊂ b (cf. (5.3)). We notice that (6.17) and (6.18) are smallness conditions in the sense that they are satisfied upon replacing the potential V˜ subjected to the previous conditions by eV˜ for any sufficiently small constant e > 0. Assuming (6.18) and that E ∈ I0 \ F (H) we verify Assumption 2.1 (4) by choosing the constant C (defining A and An ) such that (5.6) holds (with d(E) defined similarly). Then we decompose (with F0 = F0bmax ) M + f (H)Gf (H) = P − L + f (H)(L + Q)f (H) ; P = I ⊗ N + F0 (T1 + T2 + T3 )F0 + I ⊗ PΩ , L = F0 (T1 + T2 + T3 )F0 , Q = Ci[H, BR ⊗ I] − I ⊗ PΩ .
(6.18)
1022
E. SKIBSTED
Again P ≥ (1 − β) =: 2α (with β = β bmax ). Since (due to arguments given above) we can use (5.11) again we can from this point proceed as in Sec. 5. We conclude: Theorem 6.1. Let E0 ∈ R and I0 := (−∞, E0 ]. Suppose (5.1) and (5.2) for the ¯b and W ¯ b , respectively, and the smallness terms Vb , (6.13)–(6.16) for the “vectors” U 1 conditions (6.17) and (6.18). Then any eigenstate of H belongs to D(M 2 ), with M given by (5.7), and the eigenvalues of H in I0 (counted with multiplicity) can only accumulate at the set of thresholds in I0 , F (H) ∩ I0 , the latter being closed and countable. Moreover the bounds of Theorem 2.4 hold with A given by (5.5) and E ∈ I0 not being an eigenvalue nor threshold energy of H. In particular, the restriction of H to the spectral subspace F (H < E0 )H does not have continuous singular spectrum. For high energies we can obtain a similar result under some other assumptions on the potential. The idea is to verify Assumptions 2.1 with M = I and A = An =
1 (x · p + p · x) ⊗ I + I ⊗ dΓ(−(k · pk + pk · k)) 2
(6.19)
(cf. [3]). Formally i[H, A] = 2(H − V˜ ) + i[V˜ , A] .
(6.20)
If V˜ and i[V˜ , A] are “small” then the right-hand side of (6.21) is large for high energies and hence in particular positive. To implement this idea we need the following conditions on V˜ : (xb · ∇xb )m1 (k · ∇k )m2 λVxbb (k) ∈ L∞ (X b , L2ω ) ; m1 , m2 ∈ {0, 1, 2} ,
m1 + m2 ≤ 2 .
(6.21)
((xb · ∇xb )m v Vb (xb ))(−∆b + 1)−1 bounded; m ∈ {0, 1, 2} .
(6.22)
bl (k) ∈ L∞ (X b , L2ω ) ; (xb · ∇xb )m1 ∂xβb (k · ∇k )m2 λU xb
m1 , m2 ∈ {0, 1, 2} ,
m1 + m2 ≤ 2, |β| ≤ 1 .
((xb · ∇xb )m ∂xβb v Ubl (xb ))(−∆b + 1)− m ∈ {0, 1, 2} ,
1+β 2
(6.23)
bounded;
|β| ≤ 1 .
(6.24)
bl ∈ L∞ (X b , L2ω ) ; (xb · ∇xb )m1 (1 + ω)(k · ∇k )m2 λW xb
m1 , m2 ∈ {0, 1, 2} ,
m1 + m2 ≤ 2 .
(6.25)
((xb · ∇xb )m (v Wbl (xb ))j )(−∆b + 1)− 2 bounded; 1
m ∈ {0, 1, 2} ,
j ∈ {1, 2} .
(6.26)
Also we need (6.17) and the form inequality to be valid for some (large) E0 ∈ R, any b ∈ B and some αb ∈ (0, 2),
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
F¯0b (i[V˜ b , Ab ] − 2V˜ b )F¯0b ≥ −(2 − αb )H b F¯0b ; F¯ b = F (H b ≥ E0 − 2) .
1023
(6.27)
0
Here Ab is given by (6.20) for b = bmax and similarly for b 6= bmax . Similar to a previous discussion of the conditions (6.17) and (6.18) the conditions (6.22)–(6.27) imply (6.17) and (6.28) upon introducing a small enough coupling constant. With the above conditions it is now straightforward to justify the form identity (6.21) (on the domain D(H)) and the inequality (with α = αbmax ), η1 (H − E)i[H, A]η1 (H − E) ≥ α(E0 − 2)η1 (H − E)2 ;
E ≥ E0 .
(6.28)
Hence we have verified Assumption 2.1 (4). Moreover the strong form of (6.29) implies absence of eigenvalues ≥ E0 (by a virial theorem). We may argue similarly for sub-Hamiltonians. Therefore we have outlined a proof of: Theorem 6.2. Suppose (6.22)–(6.27) and the smallness conditions (6.17) and (6.28), the latter for some E0 > 2. Let I¯0 := [E0 , ∞). Then (σpp (H)∩F(H))∩I¯0 = ∅. Moreover the bounds of Theorem 2.4 hold for any E ∈ I¯0 and with A given by (6.20). In particular the restriction of H to the spectral subspace F (H ≥ E0 )H is purely absolutely continuous. Under all of the previous conditions (5.1), (5.2), (6.13)–(6.16), (6.22)–(6.27) we may combine Theorems 6.1 and 6.2 upon introducing a small coupling constant. This is done by using the E0 of Theorem 6.2 as input in Theorem 6.1 and would imply absence of continuous singular spectrum and also the same conclusion on the set of eigenvalues and the set of thresholds as in Theorem 5.1. We end this section essentially by proceeding in this way for an example in a slightly more general framework, namely the standard model of quantum electrodynamics (see [3] for a more detailed account). We consider the following model for an atom coupled to a photon field: Including spin and polarization the Hilbert space is given by H = Hel ⊗Hf , where Hel is the N times anti-symmetric tensor product of the 1-electron space L2 (R3x ) ⊗ C2 while Hf denotes the symmetric Fock space built from the 1-photon space L2 (R3k ) ⊗ C2 . (The first factor C2 accounts for spin while the latter factor accounts for polarization.) Let m and e denotes the electron mass and charge, respectively, and σj the triple consisting of the standard Pauli matrices (each one considered as acting on the spin space C2 of the jth electron). We shall use units in which h(bar), c = 1, and √ therefore e = α where the feinstructure constant α ' (137)−1 . We shall consider e as a small parameter. The Hamiltonian reads N X 1 [σj · (−i∇j − eA(xj ))]2 + I ⊗ Hf + e2 V (x) ⊗ I , (6.29) H= 2m j=1 where A(y) is a quantized vector potential to be elaborated on below and with Z being the charge of the nucleus
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E. SKIBSTED
V (x) =
N X j=1
−
Z + |xj |
X 1≤i≤j≤N
1 . |xi − xj |
(6.30)
The theory of Dirac amounts to putting X Z 1 p [µ (k)eiky aµ (k) + µ (k)e−iky a?µ (k)]dk , A(y) = 2π ω(k) µ=1,2
(6.31)
where the vectors 1 (k), 1 (k) and k|k|−1 constitute an orthonormal basis in R3 , and the operators a1 (k), a?1 (k), a2 (k) and a?2 (k) obey [a1 (k), a2 (k 0 )] = [a1 (k), a?2 (k 0 )] = 0 , [a1 (k), a?( k 0 )] = [a2 (k), a?2 (k 0 )] = δ(k − k 0 ) . Since the latter operators are only formally defined (cf. [11, Sec. X.7]) the expression (6.32) is a formal object. In order to have a well-defined operator one makes the so-called ultraviolet cut-off. Then the form is X ¯ yµ )+a? (λ ¯ ¯yµ (k) = (λyµ1 (k), λyµ2 (k), λyµ3 (k)) , [aµ (λ λ (6.32) A(y) = µ yµ )] ; µ=1,2
where ˜µl (k) λyµl = e−iky λ ˜ ∈ L2 (R3 ). for a suitable real-valued function λµl k The methods discussed previously in this section fail to handle λyµl of this form. The problem is the lack of decay as y → ∞ of the y-dependence which may be overcome by cutting off suitably at y = ∞. There is another problem at k = 0 ˜ µl indicated above which may be (the infrared region) for the concrete value of λ overcome by cutting off suitably in this region. In any case imposing similar conditions as before on each component λyµl (k) the methods may be generalized to the present extended model. Notice that we may rewrite the first term on the right-hand side of (6.30) as N N X X 1 [σj · (−i∇j − eA(xj ))]2 = −∆ ⊗ I + V˜j ; 2m j=1 j=1
∆=
N X 1 ∆j , 2m j=1
Vj = −
¯ j · pj + pj · U ¯j + (W ¯ j )2 , V˜j = Vj + U
e σj · curl A(xj ) , 2m
¯j = − e A(xj ) , U 2m
pj = −i∇j ,
(6.33)
¯ j = √ e A(xj ) . W 2m
Except for the additional spin and polarization structure the form of (6.30) and (6.34) fits into the framework given by (6.1) and (6.2). The conditions we need on λyµl (k) may be derived from (6.34) by comparing with (5.1), (5.2), (6.13)–(6.16), (6.22)–(6.27). Notice that the last term on the right side of (6.30) is in agreement
SPECTRAL ANALYSIS OF N-BODY SYSTEMS COUPLED TO A BOSONIC FIELD
1025
with (5.2) and (6.23). We consider the following simplified condition given in terms of some > 0: m+i |α|+ ∂ λyµl (k) ∈ L∞ (R3y , L2ω ) ; (1 + y 2 ) 2 ∂yα+β (1 + ω(k))1−|β| |k|m−j ∂|k| (6.34) |α| + m + i + j ≤ 2 , |β| ≤ 1 . For the above model our methods imply: Theorem 6.3. Consider the Hamiltonian H (6.30) with V (x) given by (6.31) and A(y) by (6.33), each component λyµl obeying (6.35). Then for small enough e > 0 the following statements hold: The eigenvalues of H (counted with multiplicity) can only accumulate at the set of thresholds F (H), the latter being closed and countable. Moreover there is a limiting absorption principle away from the eigenvalues and thresholds. In particular, the continuous singular subspace of H is empty. Remarks 6.4. (1) We have not attempted to prove bounds on e > 0 comparably 1 with the physically relevant value e ' (137)− 2 . It is an open problem whether the smallness of e is strictly needed for the conclusion of Theorem 6.3. (2) For molecules consisting of static nuclei there is a similar model (see [3]). In this case the Coulomb potentials describing the interaction between the electrons and the nuclei do not fulfil (6.23). On the other hand, the weaker analogue condition (5.2) of Theorem 6.1 is fulfilled. Consequently the conclusion of that theorem holds for the molecule-photon model too. References [1] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, “C0 -groups, commutator methods and spectral theory of N -body Hamiltonians”, Progress in Math. Series, 135 Birkh¨ auser, Basel, 1996. [2] S. Agmon, I. Herbst and E. Skibsted, “Perturbation of embedded eigenvalues in the generalized N -body problem”, Commun. Math. Phys. 122 (1989), 411–438. [3] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, preprint July 1996, to appear in Adv. Math. [4] A. M. Boutet de Monvel-Bertheir, D. Manda and R. Purice, “The commutator method for form-relatively compact perturbations”, Lett. Math. Phys. 22 (1991) 211–223. [5] J. Derezi´ nski, “Asymptotic completeness for N -particle long-range quantum systems”, Ann. Math. 38 (1993) 427–476. [6] R. Froese and I. Herbst, “A new proof of the Mourre estimate”, Duke Math. J. 49 (4) (1982) 1075–1085. [7] G. M. Graf, “Asymptotic completeness for N -body short-range quantum system: a new proof”, Commun. Math. Phys. 132 (1990) 73–101. [8] M. H¨ ubner and H. Spohn, “Spectral properties of the spin-boson Hamiltonian”, Ann. Inst. Henri Poincar´e 62 (3) (1995) 289–323. [9] E. Mourre, “Absence of singular continuous spectrum for certain self-adjoint operators”, Commun. Math. Phys. 91 (1981) 391–408.
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E. SKIBSTED
[10] P. Perry, I. M. Sigal and B. Simon, “Spectral analysis of N -body Schr¨ odinger operators”, Ann. Math. 114 (1981) 519–567. [11] M. Reed and B. Simon, Fourier Analysis, Self-Adjointness. Methods of Modern Mathematical Physics II, New York, Academic Press, 1975. [12] J. Sahbani, “The conjugate operator method for locally regular Hamiltonians”, preprint June 1996, J. Operator Theory 38 (2) (1997) 297–322. [13] B. Simon, “Resonances in N -body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory”, Ann. Math. 97 (1973) 247–274. [14] E. Skibsted, “Propagation estimates for N -body Schr¨ odinger operators”, Commun. Math. Phys. 142 (1991) 67–98.
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION C. FRØNSDAL Physics Department, University of California Los Angeles CA 90024, USA
A. GALINDO Departamento de F´ısica Te´ orica Universidad Complutense 28040 Madrid, Spain Received 10 February 1998 1991 Mathematics Subject Classification: 81R50, 70G50 We study the vertex operators Φ(z) associated with standard quantum groups. The element Z = RRt is a “Casimir operator” for quantized Kac–Moody algebras and the quantum Knizhnik–Zamolodchikov (q-KZ) equation is interpreted as the statement :ZΦ(z) := Φ(z). We study the covariance of the q-KZ equation under twisting, first within the category of Hopf algebras, and then in the wider context of quasi Hopf algebras. We obtain the intertwining operators associated with the elliptic R-matrix and calculate the two-point correlation function for the eight-vertex model.
1. Introduction In this paper we study the quantum Knizhnik–Zamolodchikov equation [12] for quasi Hopf algebras, with its covariance properties with respect to twisting, and its relation to matrix elements of intertwining operators. The conclusions bear on the interpretation of the solutions of similar equations with exotic R-matrices. We calculate the correlation functions for the eight-vertex model. Correlation Functions for the Eight-Vertex Model d Baxter [2] introduced the trigonometric and elliptic quantum R-matrix for sl(2); this paper is mostly about the elliptic case, and about the generalization [4] to \). The trigonometric R-matrices found their interpretation in elliptic quantum sl(N terms of quantized Kac–Moody algebras, viewed as Hopf algebras; that is, quantum groups [7]. The elliptic R-matrices had, until recently, not found their place in an algebraic framework. Surprisingly the elliptic R-matrices also turned out to be related to quantized Kac–Moody algebras, but with a quasi Hopf structure [14, 15]. More precisely, the algebraic structure is the same as in the trigonometric case, while the coproduct ∆ of the trigonometric quantum group is replaced by a new, deformed coproduct ∆ (“elliptic coproduct”) that depends on a deformation parameter . It can be expressed as ∆ = (Ft )−1 ∆Ft ; the twistor F must satisfy a cocycle 1027 Reviews in Mathematical Physics, Vol. 10, No. 8 (1998) 1027–1059 c World Scientific Publishing Company
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C. FRØNSDAL and A. GALINDO
condition that has been solved to give an explicit expression for F as a power series in . The quotient of the elliptic quantum group, by the ideal generated by the center, is a Hopf algebra; it is the quantization, in the sense of Drinfel’d, of the classical, affine Lie bialgebra with elliptic r-matrix in the classification of Belavin and Drinfeld [5]. To understand the role of these elliptic quantum groups in the context of integrable models and conformal field theory, we calculate the correlation functions of the eight-vertex model. The premise is that Baxter’s vertex operators can be interpreted mathematically as intertwining operators for representations of quantized Kac–Moody algebras [17]; this is the interpretation that affords the most direct link between statistical models and conformal field theory. Here we define new intertwining operators in terms of the elliptic coproduct and calculate the correlation functions that are associated with them; that is, matrix elements of products of intertwining operators. We find that these functions satisfy equations similar to the quantum Knizhnik–Zamolodchikov equations of Frenkel and Reshetikhin [12], but that they can be described much more easily in terms of the familiar correlation functions that govern the six-vertex model. Twist Covariance The larger issue is the question of the covariance of the q-KZ equation under twisting in the category of quasi Hopf algebras. To begin with, we point out that the q-KZ of Frenkel and Reshetikhin [12] can be easily generalized to all simple, affine quantum groups endowed with what we call a “standard” R-matrix: a universal R-matrix (expressed as a series in Chevalley–Drinfeld generators, see Definition 2.1.) that commutes with the Cartan subalgebra. Reshetikhin [22] has described a highly specialized form of twisting under which a standard R-matrix remains of standard type. From now on, by the term “twisting” we always have in mind a more radical twist that transforms a standard R-matrix to a nonstandard or esoteric R-matrix. A quantum group in the sense of this paper is a quantized, affine Kac–Moody algebra ˆ g based on a simple Lie algebra g. The structure of coboundary Hopf algebra is given by a coproduct, an antipode and a counit, but only the coproduct plays a direct role in this paper. A coboundary Hopf algebra is a Hopf algebra ˆg with an invertible element R ∈ ˆ g ⊗ gˆ that satisfies the Yang–Baxter relation and that intertwines the coproduct ∆ with its opposite ∆0 : R∆0 = ∆R .
(1.1)
The q-KZ equation is a holonomic system of difference equations that are satisfied by certain intertwining operators, Φ, Ψ : Vµ,k → V (z) ⊗ Vν,k ,
(1.2)
where Vµ,k and Vν,k are irreducible, highest weight gˆ-modules of level k and V (z) is an evaluation module. The intertwining property of Φ and of Ψ is expressed as Φx = ∆(x)Φ ,
Ψx = ∆0 (x)Ψ ,
(1.3)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1029
for x ∈ ˆ g. When R is of standard type (Definition 2.1), then the q-KZ equation for Ψ takes the form (Z 0 − 1)Ψ = 1 , (1.4) where Z 0 is a Casimir operator (acting in V (z) ⊗ Vν,k ) for ˆg. To define this operator let us express R as R = Ri ⊗ Ri , where we use the summation convention for the index i; then formally, Z 0 = Rt R ,
Rt := Ri ⊗ Ri .
(1.5)
However, to make sense of an operator product such as Z 0 Ψ it is necessary to renormalize it. The correct form of the q-KZ equation is indeed (1.4), but with Z 0 Ψ replaced by the normal-ordered product ˆ
:Z 0 Ψ: = Rt (Ri q H ⊗ 1)ΨRi ,
(1.6)
ˆ ˆ. where the factor q H belongs to the Cartan subalgebra of g We study a deformation of the initial, standard quantum group, implemented by twisting with an invertible element F ∈ ˆg ⊗ ˆg that is a formal power series in a deformation parameter . The twisted quantities are:
R = (Ft )−1 RF , Ψ = F−1 Ψ ,
∆0 = F ∆0 F−1 , Z0 = F−1 Z 0 F ,
and the twisted KZ equation is :Z0 Ψ : = Ψ ; it has the same form as in the standard case. However, Eq. (1.6) is not covariant; we mean by that it cannot be generalized by simply replacing R by R , since the expression Rt (Ri ⊗ 1)Ψ Ri is not well defined. Instead, the correct expression for the normal-ordered product is ˆ :Z0 Ψ : = F−1 :Z 0 Ψ: = F−1 Rt (Ri q H ⊗ 1)ΨRi . Therefore, although there is a clear sense in which “the q-KZ equation” is covariant, the normal-ordered product (1.6) is not. This observation has analogous implications for correlation function. To illustrate this, consider the two-point correlation function g(z1 , z2 ) = hΨ(z1 )Ψ(z2 )i. In the standard case the q-KZ equation reduces to g(q −k−g z1 , z2 ) = q A1 R−1 (z1 , z2 )g(z1 , z2 ) .
(1.7)
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The twisted correlation function obeys g (q −k−g z1 , z2 ) = (F−1 (z2 , q −k−g z1 )q A1 R−1 (z1 , z2 )F (z2 , z1 ))g (z1 , z2 ) , and this is not the same as Eq. (1.7) with R replaced by R . This conclusion casts some light on the proposed generalization of of the q-KZ equations for correlation functions. Integrability is assured by the Yang–Baxter relation for the R-matrix. It is natural to study the equations that result from replacing the trigonometric R-matrix in (1.7) and the rest, by more exotic R-matrices. Since this requires a knowledge of such R-matrices in finite dimensional representations only, it is possible, in particular, to use the elliptic R-matrix of Baxter in this connection. As long as the elliptic quasi Hopf algebra was not known, it was possible to speculate that the solutions of such “elliptic q-KZ equations” relate in some way to (unknown) elliptic intertwiners. Our conclusion is that this interpretation is not the correct one. Outline of the Paper Section 2 summarizes some facts about standard, universal R-matrices and sets our notation. Section 3 examines certain intertwining operators and draws some conclusions (Proposition 3.1) that are used later to determine the correct approach to regularizing operator products. Sections 4 and 5 present a view of the KZ and q-KZ equations. Both can be interpreted very simply as eigenvalue equations, ζΦ = 0 or (Z − 1)Φ = 0, for the Casimir operators ζ or Z of affine Kac–Moody or quantized, affine Kac– Moody algebras. Section 4 deals with the classical KZ equation ζΦ = 0; the effect of different polarizations is discussed, as well as the invariance of the operator ζ (Propositions 4.1 and 4.2). The quantum case is taken up in Sec. 5; the correct normal-ordered action of the Casimir elements Z and Z 0 on the intertwiners Φ and Ψ is established (Proposition 5.1), and the q-KZ equations are presented in Eqs. (5.6) and (5.8). Sections 6 and 7 explore the effect on intertwiners of twisting in the categories of Hopf and quasi Hopf algebras. In Sec. 6 we stress the distinction between “finite” and “elliptic” twisting. The twisted q-KZ equation is presented (Definition 6.3). In Sec. 7 quasi Hopf twisting is discussed and a recursion relation to actually calculate the elliptic twistor is given. Sections 8 and 9 apply the results to correlation functions. In Sec. 8 the classical and quantum q-KZ equations for correlation functions are given; the effect of twisting is exhibited and a certain lack of covariance is emphasized. In Sec. 9 the two-point correlation function for the eight-vertex model is calculated, as well as explicit expressions for the twisting matrix in the fundamental representation of d sl(2). Finally, some auxiliary material is relegated to an Appendix. Relation to Other Work (1) Our original goal was to discover the enigmatic “elliptic quantum groups” and to use it to define and calculate the correlation functions for the eight-vertex
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1031
model. This is precisely the problematics of a series of paper by Jimbo, Miwa and others; see especially the review [17] and the papers [11, 18]. These authors did not have available the universal, elliptic R-matrix and did not anticipate the fact that the algebraic structure of the elliptic quantum group would turn out to be the same as in the trigonometric case. (Only the coproduct is changed.) They postulated a new algebraic structure, but in the absence of a coproduct they could not define intertwiners. In spite of this they did succeed in calculating correlation functions that stand up to analysis and that reproduce some of Baxter’s results on the eightvertex model. Nevertheless, the correlation functions that we here propose for the eight-vertex model are quite different. (2) One of the most interesting aspects of the elliptic quantum group is its quasi Hopf nature. Quasi Hopf algebras, characterized by a modified quantum Yang– Baxter relation, are basic to the Knizhnik–Zamolodchikov–Bernard generalization of the KZ equation that was discovered by Bernard [6]. This equation also arises in connection with Felder’s elliptic quantum groups [10]. However, these developments are not concerned with highest weight matrix elements of intertwiner operators, and the quasi Hopf algebras of Felder et al. are not related to the elliptic R-matrices of Baxter and Belavin. The new r-matrices discovered by Enriquez and Rubtsov [9] and by Frenkel, Reshetikhin and Semenov–Tian–Shansky [13] are of a different sort. These interesting developments go beyond the classification of classical r-matrices by Belavin and Drinfel’d [5] and are outside the scope of this paper. 2. Standard, Affine, Universal, Quantum R-Matrices This section contains basic definitions and notation. Let M , N be two finite sets, ϕ, ψ two maps, ϕ : M × M → C , a, b 7→ ϕab , ψ : M × N → C,
a, β 7→ Ha (β) ,
and q a complex parameter. Let A or A(ϕ, ψ) be the universal, associative, unital algebra over C with generators {Ha }a∈M , {e±α }α∈N , and relations [Ha , Hb ] = 0 ,
[Ha , e±β ] = ±Ha (β)e±β ,
[eα , e−β ] = δαβ (q ϕ(α,·) − q −ϕ(·,α) ) , with ϕ(α, ·) = ϕab Ha (α)Hb , ϕ(·, α) = ϕab Ha Hb (α) and q ϕ(α,·)+ϕ(·,α) 6= 1, α ∈ N . The algebra of actual interest is a quotient A0 = A/I, where I is a certain ideal; in this paper we suppose that I is generated by a complete set of (quantized) Serre relations among the eα ’s and among the e−α ’s; then A0 is a quantized (generalized) Kac–Moody algebra. In the case when A0 is a quantized Kac–Moody algebra of affine type, based on a simple Lie algebra g, we sometimes write ˆg for A0 . The “Cartan subalgebra” A00 is generated by {Ha }a∈M , extended by the inclusion of exponentials.
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C. FRØNSDAL and A. GALINDO
Definition 2.1. The standard, universal R-matrix has the form R = qϕ T = qϕ where t0 = 1 ⊗ 1, t1 = and tn has the form
P
∞ X
tn ,
ϕ=
X
ϕab Ha ⊗ Hb ,
(2.1)
n=0
e−α ⊗ eα (the sum is over the Serre generators, α ∈ N ) (α0 )
tn = t(α) e−α1 . . . e−αn ⊗ eα01 . . . eα0n .
(2.2)
Sums over repeated indices are implied; the multi-index (α0 ) runs over the permutations of (α). (α0 )
The coefficients t(α) ∈ C are essentially determined (the elements tn are determined uniquely) by the imposition of the Yang–Baxter relation, R12 R13 R23 = R23 R13 R12 .
(2.3)
It has been shown that, for a universal R-matrix of the type (2.1), this relation is equivalent to the recursion relation [14] [eγ ⊗ 1, tn ] = tn−1 (q ϕ(γ,.) ⊗ eγ ) − (q −ϕ(.,γ) ⊗ eγ )tn−1 ,
(2.4)
with the initial condition t0 = 1. There is exactly one solution in A0 ⊗ A0 . ˆ is a quantized, affine Kac–Moody algebra based on We suppose now that A0 = g a simple Lie algebra g. The coproduct is then generated by the following formulas: ∆(eα ) = 1 ⊗ eα + eα ⊗ q ϕ(α,.) ,
∆(e−α ) = q −ϕ(.,α) ⊗ e−α + e−α ⊗ 1 ,
(2.5)
and ∆Ha = Ha ⊗ 1 + 1 ⊗ Ha . Let π1 , π2 be finite dimensional representations of g, and πi (zi ) the associated evaluation representations of gˆ with spectral parameters zi . Let (2.6) R(z1 , z2 ) := π1 (z1 ) ⊗ π2 (z2 )R . The spectral parameters are regarded as formal variables; R(z1 , z2 ) is a formal power series R12 (z2 /z1 ) in z2 /z1 . The effectiveness of the recursion relation (2.4) is illustrated in the Appendix. Finally, given A = ai ⊗ai ∈ A0 ⊗A0 , we shall write At := ai ⊗ai and mA := ai ai . 3. Highest Weight Modules and Intertwining Operators Let Vµ be an irreducible, finite dimensional, highest weight ˆg-module, and Vµ,k = ⊕n≥0 Vµ,k [−n] the associated level k, highest weight, irreducible, graded ˆg-module. The intertwining operators of greatest interest are imbeddings Φ = Φ(z) : Vµ,k → V (z) ⊗ Vν,k , where V (z) is an evaluation module over ˆg. The defining property of Φ is Φx = ∆(x)Φ , for all x in ˆ g.
(3.1)
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8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
We shall obtain some very essential information about the structure of the intertwining operators. Proposition 3.1. Let v be a homogeneous element of Vµ,k . Then Φv = bn , with bn ∈ Vν,k [−n], where the sum is not, in general, finite.
P n
an ⊗
Proof. It will be enough to verify that the sum is effectively infinite in one d with V the typical case. Thus consider the quantized Kac–Moody algebra sl(2) fundamental representation. In this case, for any v ∈ Vµ,k , Φv takes the form: ! A(z)v A Φv = v. (3.2) = B B(z)v Necessary conditions to be satisfied by the operators A, B : Vµ,k → Vν,k are A A eα v . v= ∆(eα ) (3.3) B B The first space is two-dimensional, 0 e1 ⊗ 1 = κ 0
with 1 , 0
e0 ⊗ 1 = κ
0 z
0 0
.
(3.4)
The parameter κ is related to q, κ2 = q − q −1 . In full detail, [e0 , A] = 0 , [e1 , A] = −κq ϕ(1,.)B , [e1 , B] = 0 , [e0 , B] = −κzq ϕ(0,.)A .
(3.5)
On the highest weight vector v0 in Vµ,k , we have e0 Av0 = 0 ,
e1 Bv0 = 0 ,
e0 Bv0 = −κzq ϕ(0,.)Av0 ,
e1 Av0 = −κq ϕ(1,.) Bv0 , (3.6)
with a unique solution of the form: Av0 =
∞ X
0 z n v2n ,
Bv0 =
n=0
∞ X
0 z n+1 v2n+1 ,
(3.7)
n=0
with v00 a highest weight vector in Vν,k and vectors vn0 ∈ Vν,k determined recursively by 0 0 = e1 v2n+1 = 0, e0 v2n
0 0 e1 v2n = −κq ϕ(1,.) v2n−1 ,
The solutions have the form: X 0 n 0 v2n = A2n σ σ(e−1 e−0 ) v0 , σ∈S2n
0 v2n+1 =
0 0 e0 v2n+1 = −κq ϕ(0,.) v2n . (3.8)
X
Bσ2n+1 σ(e−1 e−0 )n e−0 v00 ,
σ∈S2n+1
where the sum is over all permutations of the generators. It is clear that vn0 6= 0 for all n and the proposition is proved.
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ˆ is a quantized Kac–Moody algebra of We return to the general case, A0 = g affine type, based on a simple Lie algebra g, Vµ,k is a highest weight module over ˆ g, Vi (zi ) finite dimensional evaluation modules. Remark 3.2. The product Φ2 Φ1 is a compound map Φ
Φ
1 2 Φ2 Φ1 : Vµ,k −→V 1 ⊗ Vν,k −→V1 ⊗ V2 ⊗ Vλ,k .
(3.9)
It has the property Φ2 Φ1 x = Φ2 ∆(x)Φ1 = (id ⊗ ∆)∆(x)Φ2 Φ1 .
(3.10)
By coassociativity of ∆, Φ2 Φ1 is an intertwiner of the same type as Φ1 and Φ2 : Φ2 Φ1 : Vµ,k → (V1 (z1 ) ⊗ V2 (z2 )) ⊗ Vλ,k .
(3.11)
Consequently, universal statements about intertwiners apply to products of intertwiners as well. This observation will be of use in Sec. 8. Of course, it does not apply in the quasi Hopf case (Sec. 9). 4. The Classical KZ Equation The object Z = RRt ∈ A0 ⊗ A0 ,
(4.1) 0
if it exists, is invariant in the sense that it commutes with ∆(x), ∀x ∈ A . It plays the role of a Casimir element for the quantized Kac–Moody algebra. Since the intertwiner Φ projects on an irreducible representation, one expects that there is hZi ∈ C such that (Z − hZi)Φ = 0 . (4.2) We shall begin our study of this equation by considering its classical limit. The result is Propositions 4.2 and 4.3. The important concepts are normal ordering and “polarization”. Then we shall return to the quantum case to show that (4.2) is the q-KZ equation of Frenkel and Reshetikhin [12, Sec. 5]. The classical limit is defined by setting q = eη , expanding in powers of η, and retaining the first nonvanishing term. When A0 = ˆg is a quantized Kac–Moody algebra of finite type, one finds that X E−α ⊗ Eα , (4.3) R = 1 + ηr + O(η 2 ) , r = ϕ + α∈∆+
where the sum runs over the positive roots of g. For simple roots one has eα = √ η(Eα + O(η)); the others are normalized so that the Casimir element in g ⊗ g takes the form: C = r + rt . In the case of an untwisted affine loop algebra one gets X X E−α ⊗ Eα + (z2 /z1 )n C , R = 1 + ηr + O(η 2 ) , r = ϕ + α∈∆+
n≥1
(4.4)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1035
where ∆+ is the set of positive roots of the underlying Lie algebra and where z1 , z2 are the spectral parameters in the first, resp. second space. It is important to keep in mind that this expression is, until further development, nothing more than a formal power series in z2 /z1 . In terms of the basis n = z n E±α , E±α
Han = z n Ha ,
(4.5)
the expression for r becomes r = ϕ + E−α ⊗ Eα +
X
Cn ,
(4.6)
n≥1
with −n n ⊗ Eαn + Eα−n ⊗ E−α , C n = K ab Ha−n ⊗ Hbn + E−α
K ab = (ϕ + ϕt )ab .
(4.7)
Summation over a, b and α ∈ ∆+ will henceforth be taken for granted. Note that Eqs. (4.6) and (4.7) are valid in the case of twisted loop algebras as well. Returning to affine Kac–Moody algebras, it will be convenient to change our conventions just a little. Retain the above notation for the loop algebra, so that, in particular, (4.8) ϕ = ϕab Ha ⊗ Hb , where the sum runs over the basis of the Cartan subalgebra of a simple Lie algebra g. The form that characterizes the full, quantized affine Kac–Moody algebra ˆg is ϕˆ = ϕ + uc ⊗ d + (1 − u)d ⊗ c ,
(4.9)
where d is the degree operator, c is a basis for the central extension and u is a parameter. For the full quantized Kac–Moody algebra the limit is R = 1 + ηˆ r + O(η 2 ) ,
rˆ = r + uc ⊗ d + (1 − u)d ⊗ c .
(4.10)
The classical limit of Z is Z = 1 + ηζ + O(η 2 ) , Formally, ζ = rˆ + rˆt =
∞ X
ζ = rˆ + rˆt .
(4.11)
Cn + c ⊗ d + d ⊗ c .
−∞
When both spaces are evaluation modules, where c 7→ 0, ζ=
+∞ X
(z2 /z1 )n C .
(4.12)
n=−∞
This sum becomes zero when projected on a quotient algebra of meromorphic functions.
1036
C. FRØNSDAL and A. GALINDO
We try to make sense out of the classical limit of (4.2), namely (ζ − hζi)Φ = 0 . By abuse of notation we retain the notation Φ for the classical limit of the intertwiner. Now the first space is an evaluation module, where c vanishes, and if c 7→ k (k is the level) on the second space, then formally ζΦ(z) = kz
X X d Φ(z) + C −n Φ(z) + C n Φ(z) . dz n>0
(4.13)
n≥0
Let us introduce a uniform basis {La } for g, so that the Casimir element takes the form C = La ⊗ La (summation implied). Then (in the untwisted case) (4.13) takes the form X X d + La ⊗ z n L−n z −n Lna Φ(z) . ζΦ(z) = kz (4.14) a + La ⊗ dz n>0 n≥0
However, the significance of this formula is doubtful, as we shall see. This is the reason for the introduction of normal-ordered products in [12]. Polarization It is usual, at this point of the development, to replace the operator products by normal-ordered products. It is a step that merits comment. Normal-ordered operator products are introduced in field theory when ordinary operator products fail to make sense. The typical example is this product of destruction and creation operators: ! ! X X inω −imω ∗ e an e am . n
m
When it is applied to the vacuum one gets ! ! +∞ X X X inω −imω ∗ e an e am |0i = |0i , n
m
n=−∞
which is without meaning. The last term in (4.14) is of this kind; the degreedecreasing operators in Φ correspond to the creation operators, and the degreeincreasing operators Lna correspond to the destruction operators. Collecting all terms of the same degree in the product one gets a divergent series. We want to avoid having to interpret such infinite series, if it is possible. Using the (classical) intertwining property of the intertwiner, ! N N N X X X z −n (La ⊗ Lna )Φ = z −n (La ⊗ 1)ΦLna − La La ⊗ 1 Φ . (4.15) n=1
n=1
n=1
P Passing with N to infinity we encounter the meaningless expression n>0 La La , an exact analogue of the divergent sum that is thrown away when a field operator
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1037
product is replaced by the normal-ordered product. It is tempting to redefine the operator ζ, by dropping this offensive term, thus ζΦ = kz
d Φ + (La ⊗ 1) :Ja (z)Φ: (?) , dz
with Ja = Ja+ + Ja− =
X n≥0
z n L−n a +
X
z −n Lna ,
(4.16)
(4.17)
n>0
and :Ja Φ: := Ja+ Φ + ΦJa− .
(4.18)
This new operator is well defined on highest weight modules and it will serve if it has the property that the formal expression (4.1) was intended to assure; that is, if it is invariant. Actually it is, almost. Proposition 4.1. The covariant definition of the operator product ζΦ(z) is ζΦ(z) = (k + g)z
d Φ(z) + (La ⊗ 1):Ja (z)Φ(z): , dz
(4.19)
where g is the dual Coxeter number of g. This result of [12] is an analogue of Proposition 4.2 that we prove below. The replacement of the factor k by k + g, at first sight somewhat mysterious, is thus required by covariance. For sl(N ), g = N We calculate the value hζi. The operator J− annihilates the highest weight vector v0 ; therefore d + + La ⊗ Ja Φ(z)v0 . ζΦ(z)v0 = (k + g)z dz In terms of the contravariant bilinear form (.,.), with v00 the highest weight vector of Vν,k , one gets a V (z)-valued function: (v00 , Φ(z)v0 ) =: Φv00 v0 (z) ∈ V (z) , and (v00 , ζΦ(z)v0 ) = (k + g)z
d Φv0 v (z) + (v00 , La ⊗ Ja+ Φ(z)v0 ) . dz 0 0
In Ji+ only the zero mode contributes, and the second term reduces to const. × Φv00 v0 (z). The constant has the value 1 (C(µ) − C(ν) − C(π)) , 2
(4.20)
where C(µ) is the value of the Casimir operator C = m C in Vµ,k [0]. (Recall that if A = a ⊗ b ∈ A0 ⊗ A0 , then mA = ab ∈ A0 .)
1038
C. FRØNSDAL and A. GALINDO
We can reduce the value hζi of ζ to zero by choosing the grading of Φ according to Φ(z) =
X
Φ[n]z −n−(µ|vπ) ,
(µ|ν, π) :=
n∈Z
1 (C(µ) − C(ν) − C(π)) ; 2(k + g)
then for any weight vector w in V , (w ⊗ v00 , Φ(z)v0 ) = z −(µ|vπ) (w ⊗ v00 , Φ[0]v0 ) ,
(4.21)
and
d Φ(z) + :La ⊗ Ja Φ(z): = 0 . dz This is the “classical” Knizhnik–Zamolodchikov equation [20]. ζΦ(z) = (k + g)z
(4.22)
Alternative Polarizations The polarization defined by (4.17) and (4.18) is ad hoc. We have the freedom of shifting any finite set of summands from J + to J − , as in X X z n L−n z −n Lna ; Ja = Ja+ + Ja− = a + n>0
n≥0
the effect in this particular case is merely to change the sign of C(π) in (4.20). The result now agrees with [12]. Another polarization is suggested by (4.11), ζ = La ⊗ Ja = La ⊗ Ja+ + La ⊗ Ja− = rˆt + rˆ . Here we are dealing directly with the full Kac–Moody algebra, including the c, dterms in rˆ. Formally, the intertwining property gives ri rˆi ⊗ 1)Φ + (ˆ ri ⊗ 1)Φˆ ri , rˆΦ = (ˆ ri ⊗ rˆi )Φ = −(ˆ and
1 1 X n ri , rˆi ] . (4.23) C + c ⊗ d + d ⊗ c + [ˆ 2 2 The first term on the right-hand side of this last equation, although meaningless, looks like it may be a scalar, and thus ignorable. Proceeding heuristically up to ˆ Proposition 4.2, we begin by dropping this term. The other term is an element H of the Cartan subalgebra of ˆ g, rˆi rˆi =
ˆ = 1 [ˆ ri , rˆi ] ; H 2
(4.24)
it is determined up to an additive central element by ˆ eα ] = [eα , rˆi rˆi ] = m[eα ⊗ 1 + 1 ⊗ eα, rˆ] = m(ϕ(α, .) ∧ eα ) = ϕ(α, α)eα . [H,
(4.25)
If we restrict the relation (4.25) to the real simple roots, then it determines a unique element in the Cartan subalgebra of g, namely 1X [Eα , E−α ] . H= 2 α>0
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1039
Therefore, there is a unique element in the extended Cartan subalgebra, of the form: ˆ = H + gd , H
(4.26)
such that (4.25) holds for the affine root e0 as well. The integer g is the dual Coxeter number of g. The redefined operator is ˆ ⊗ 1)Φ(z) + :(ˆ ζΦ(z) := (H rt + rˆ)Φ(z): ˆ ⊗ 1)Φ(z) + rˆt (z)Φ(z) + (ˆ = (H ri (z) ⊗ 1)Φ(z)ˆ ri .
(4.27)
ˆ as the element of Notice that we did not actually use (4.24); instead we defined H ˆ g that has the same commutator with eα as (4.23). This makes it plausible that the term that was dropped is a scalar and that covariance is preserved. Indeed we have: Proposition 4.2. The operator product ζΦ(z) defined in (4.27) is covariant; that is, if Φ is an intertwiner then so is ζΦ. Proof. The coproduct is that of the classical limit, ∆(x) = x ⊗ 1 + 1 ⊗ x for x∈ˆ g. ˆ ⊗ 1)Φ(z) + [∆(x), rˆt (z)]Φ(z) ∆(x)ζΦ(z) − (ζΦ(z))x = ([x, H] ri + (ˆ ri (z) ⊗ 1)Φ[x, rˆi ] . + ([x, rˆi (z)] ⊗ 1)Φ(z)ˆ
(4.28)
Suppose first that x ∈ g; then in terms 2, 3, 4 only the zero modes contribute. The sums over the degree are now finite and ∆(x)ζΦ(z) − ζΦ(z)x = ([x, H] ⊗ 1)Φ(z) + ([x, ri (0)ri (0)] ⊗ 1)Φ(z) ,
(4.29)
which vanishes in view of (4.26). We shall verify that (4.28) holds for x = e0 . Besides (4.29) there are additional terms that arise from the extension term in the commutation relations, others that arise from the fact that e0 does not commute with the degree operator, and finally the more subtle contributions that come from the fact that the degree of [e0 , y] is shifted by 1 from that of y: ˆ ⊗ 1)Φ = (−ge0 ⊗ 1)Φ + ([e0 , H] ⊗ 1)Φ , ([e0 , H] X [∆(e0 ), rˆt (z)]Φ = [∆(e0 ), rˆt (0)]Φ + [∆(e0 ), Lna ⊗ L−n a ]Φ n>0
= [∆(e0 ), rˆ (0)]Φ + (L1a ⊗ [e0 , L−1 a ])Φ , X n ([e0 , rˆi (z)] ⊗ 1)Φˆ ri = ([e0 , rˆi (0)] ⊗ 1)Φˆ ri + ([e0 , L−n a ] ⊗ 1)ΦLa , t
n>0
(ˆ ri (z) ⊗ 1)Φ[e0 , rˆi ] = (ˆ ri (0) ⊗ 1)Φ[e0 , rˆi ] +
X
n (L−n a ⊗ 1)Φ[e0 , La ] .
n>0
1040
C. FRØNSDAL and A. GALINDO
The two infinite sums almost cancel, leaving only the first term of the second one. The sum of the last two expressions is 1 [∆(e0 ), rˆ(0)]Φ + ([e0 , ri (0)ri (0)] ⊗ 1)Φ + ([e0 , L−1 a ] ⊗ 1)ΦLa .
Adding the second expression we obtain 1 1 −1 [∆(e0 ), rˆ(0) + rˆt (0)]Φ + ([e0 , L−1 a ] ⊗ La )Φ + (La ⊗ [e0 , La ])Φ 1 + ([e0 , ri (0)ri (0)] ⊗ 1)Φ + ([e0 , L−1 a ]La ⊗ 1)Φ .
The first three terms cancel exactly and we have ∆(e0 )ζΦ(z) − ζΦ(z)e0 = −g(e0 ⊗ 1)Φ + ([e0 , H] ⊗ 1)Φ 1 + ([e0 , ri (0)ri (0)] ⊗ 1)Φ + ([e0 , L−1 a ]La ⊗ 1)Φ .
Terms two and three cancel as in (4.29) and the proposition is proved when we 1 verify that, in the evaluation module, [e0 , L−1 a ]La = [e0 , La ]La = ge0 , and repeat the calculation with e0 replaced by e−0 . Normalization Returning to (4.27) we put the degree operator into evidence: ζΦ(z) = (k + g)z
d Φ + (H ⊗ 1)Φ(z) + rt (z)Φ(z) + (ri (z) ⊗ 1)Φ(z)ri . dz
(4.30)
Again we fix the grading of the intertwiner as in (4.21), but now with (µ|ν, π) replaced by (µ|ν) =
A(w) , k+g
A := ϕ(., v0 ) + ϕ(v00 , .) + H =
1 (C(µ) − C(ν)) , 2
(4.31)
so that the operator form of the Knizhnik–Zamolodchikov equation takes the form: ζΦ(z) = 0 .
(4.32)
Note that this makes the grading of Φ independent of the choice of evaluation module; this grading/normalization is thus “universal”. Remark 4.3. In view of the interpretation of the quantum field Φ(z) as an intertwiner for highest weight affine Kac–Moody modules, the appearance of the rational r-matrix in the original KZ equation (4.22) has always seemed somewhat mysterious. The mystery is deepened by the discovery [19] that the monodromy associated with the solutions yields a representation of Uq (g). The alternative, to use the polarization based on the decompostion ζ = rˆ+ˆ rt , was first suggested in [12]; it seems to be more natural. However, Φ is defined as an intertwiner of Kac–Moody modules, with the classical coproduct; it knows nothing about r-matrices. Normal ordering is an example of additive renormalization, or “subtraction”, necessary
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1041
only if the ordinary product is ill defined. Any two polarizations that eliminate the divergent term by subtracting a scalar (that is; without compromising covariance) are equivalent, and one is not more natural than the other, in the present context at least. The fact that renormalization is required is revealed by the fact that the subtracted term, the last term in (4.15), is divergent. It is related to the fact that the classical r-matrix has a pole at z1 /z2 = 1. Remark 4.4. The appearance of the factor k + g as a coefficient of the degree operator in both versions is justified by covariance, as is indeed implied by the proof of Proposition 4.1 in [12]. The term (H ⊗ 1)Φ(z) has exactly the same origin. Perhaps it should be pointed out that the concept of “covariance” that is evoked in this section is quite distinct from the covariance under twisting that is alluded to in the title of the paper and in from Sec. 6 onward. 5. The Quantum KZ Equation Here we shall make sense of Eq. (4.2), (Z − hZi)Φ(z) = 0 ,
Z = RRt ,
in the quantized Kac–Moody algebra, to recover the q-KZ equation of Frenkel and Reshetikhin [12]. The action of Rt Φ on Vµ,k is well defined (in terms of formal series), since both Φ and Rt act by degree-decreasing operators in the second space (Proposition 3.1 and Eq. (3.9)), but the subsequent action of R is not. We therefore investigate the effect of normal ordering. Thus if R = Ri ⊗ Ri , we set (tentatively) :ZΦ: = (Ri ⊗ 1)ΨRi ,
Ψ := Rt Φ
(?) ,
and try to prove that the operator Z : Φ 7→ :ZΦ: is invariant; that is, that it commutes with the coproduct. In view of the intertwining property of Φ this is the same as ∆(x):ZΦ: = :ZΦ: x (?) . Attempts to verify this equation leads to: ˆ be the element in the Cartan subalgebra A0 of A0 with Proposition 5.1. Let H 0 the property ˆ ˆ (5.1) q H eα q −H = q ϕ(α,α) eα , and define the normal-ordered product :ZΦ: by ˆ i ⊗ 1)ΨRi , :ZΦ: := (R
Ψ := Rt Φ ,
ˆ i := Ri q Hˆ . R
(5.2)
Then ∆(x):ZΦ: = :ZΦ: x ,
∀x ∈ A0 .
(5.3)
1042
C. FRØNSDAL and A. GALINDO
Proof. We begin with R ∆0 (eα ) = ∆(eα )R; that is (Ri ⊗ Ri )(eα ⊗ 1 + q ϕ(α,.) ⊗ eα ) = (1 ⊗ eα + eα ⊗ q ϕ(α,.) )(Ri ⊗ Ri ) , and thus Ri ⊗ Ri eα = −Ri eα q −ϕ(α,.) ⊗ Ri + Ri q −ϕ(α,.) ⊗ eα Ri + eα Ri q −ϕ(α,.) ⊗ q ϕ(α,.) Ri , which gives us ˆ
:ZΦ: eα = (Ri q H ⊗ 1)ΨRi eα ˆ
= −(Ri eα q −ϕ(α,.) q H ⊗ 1)ΨRi ˆ
ˆ
+ (Ri q −ϕ(α,.) q H ⊗ 1)Ψeα Ri + (eα Ri q −ϕ(α,.) q H ⊗ 1)Ψq ϕ(α,.) Ri . Using the intertwining property of Ψ we convert the last two terms to (Ri q H ⊗ eα )ΨRi + (Ri q −ϕ(α,.) q H eα ⊗ 1)ΨRi + (eα Ri q H ⊗ q ϕ(α,.) )ΨRi . ˆ
ˆ
ˆ
As for the first term, we shift the operator eα to the right; since eα commutes with ˆ − ϕ(α, .) we get the required cancellation and the result is H ˆ i ⊗ eα )ΨRi + (eα R ˆ i ⊗ 1)ΨRi . ˆ i ⊗ q ϕ(α,.) )ΨRi = ∆(eα )(R :ZΦ: eα = (R ˆ In the classical limit the q-factor in (5.2) produces the H-term in Eq. (4.27). A similar calculation with e−α completes the proof of Proposition 5.1, and we have an independent confirmation of the covariance of (4.27). Normalization Our next task is to pull out the degree operator. Since the first space is an evaluation module, on which the central element c is zero, the degree operator d . We define L∓ by appears only in the first factors of R and Rt , as z dz ˆ i (z) ⊗ Ri = q (1−u)kd+gd (Ad(q −gd ) ⊗ 1)L− (z) , R
Rt (z) = q ukd L+ (z) ,
(5.4)
d acts in the evaluation module and Ad(x)y = xyx−1 . Objects denoted where d = z dz by the letter L (with ornamentation) do not contain d. We also need the expansions
L− (z) = L−i (z) ⊗ L− i ,
L+ (z) = L+i (z) ⊗ L+ i .
Now :ZΦ(z): = q (1−u)kd+gd (Ad(q −gd )L−i (z) ⊗ 1)(q ukd ⊗ 1)L+ (z)Φ(z)L− i = q (k+g)d (L−i (q −g−uk z) ⊗ 1)L+ (z)Φ(z)L− i =: q (k+g)d :L(z)Φ(z): .
(5.5)
Thus, the q-KZ equation for Φ: Φ(q −k−g z) = :L(z)Φ(z): = (L−i (q −g−uk z) ⊗ 1)L+ (z)Φ(z)L− i .
(5.6)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1043
Here we have fixed hZi = 1. It means that the grading of Φ is so chosen that, on any weight vector w ∈ V and the highest weight vectors v0 ∈ Vµ,k , v00 ∈ Vν,k , we have (w ⊗ v00 , Φ(q −k−g z)v0 ) = q (µ|ν) , (w ⊗ v00 , Φ(z)v0 ) with (µ|ν) as in (4.31). The other intertwiner, Ψ ∝ Rt Φ, satisfies Ψx = ∆0 (x)Ψ and :Z 0 Ψ: = Ψ where 0 Z := Rt R. We find ˆ i ⊗ 1)Ψ(z)Ri = q ukd L+ (z)q (k−uk+g)d (L−i (q −g z) ⊗ 1)Ψ(z)L− , :Z 0 Ψ(z): = Rt (R i (5.7) and thus, the q-KZ equation for Ψ: Ψ(q −k−g z) = :L0 (z)Ψ(z): = L+ (q −g−k+uk z)(L−i (q −g z) ⊗ 1)Ψ(z)L− i .
(5.8)
6. Hopf Twisting It is remarkable that the elliptic quantum group can be viewed as deformation of the trigonometric quantum group. The deformation does not affect the algebraic structure, which remains that of a quantized, affine Kac–Moody algebra. Only the coproduct distinguishes the elliptic case from the trigonometric one. The deformation is implemented by a twist in the category of Hopf algebras (this section) or quasi Hopf algebras (next section). The full elliptic quantum group is quasi Hopf; it becomes Hopf on the quotient by the ideal generated by the center. In this section we investigate the effect of twisting on the intertwiners and on the KZ equation, in the quantum case where the relationship between the intertwiner and the R-matrix is clearer. Definition 6.1. A formal Hopf deformation of a standard R-matrix R is a formal power series R = R + R1 + · · · , that satisfies the Yang–Baxter relation to each order in . It turns out [14] that the deformations of greatest interest have the form of a twist. Theorem 6.2. Let R be the R-matrix, ∆ the coproduct, of a coboundary Hopf algebra A0 , and F ∈ A0 ⊗ A0 , invertible, such that ((1 ⊗ ∆21 )F )F12 = ((∆13 ⊗ 1)F )F31 . Then ˜ := (F t )−1 RF R
(6.1)
1044
C. FRØNSDAL and A. GALINDO
(a) satisfies the Yang–Baxter relation and (b) defines a Hopf algebra A˜ with the same product and with coproduct ˜ = (F t )−1 ∆F t . ∆ This is a result of Drinfel’d [8]; a detailed proof was given in [14]. We say that a deformation R of a standard R-matrix R is implemented by a twistor F if there is a formal power series F = 1 + F1 + · · · that satisfies (6.1) to each order in and R = (Ft )−1 RF .
(6.2)
In this case the deformed R-matrix intertwines a deformed coproduct, R ∆0 = ∆ R ,
∆0 := F−1 ∆0 F .
(6.3)
Known solutions of (6.1) have the following structure [14]. We need a pair of ˆ i ⊂ {eα }α∈N , and a diagram subalgebras Γ1 , Γ2 of A0 = gˆ, generated by sets Γ ˆ ˆ isomorphism τ : Γ1 → Γ2 . A deformation exists when the parameters of A0 satisfy the following condition: ϕ(σ, .) + ϕ(., τ σ) = 0 ,
ˆ1 . σ∈Γ
ˆ 1 . Then there is a cocycle F of the form: Note that eτ σ is defined only if eσ ∈ Γ Y Fm := F1 F2 . . . Fm . . . , F = m≥1
Fm =
X
m(ρ)
mn F(σ) fσ1 . . . fσn ⊗ f−ρ1 . . . f−ρn ,
(6.4)
(σ)
fσ := q −ϕ(σ,.) eσ ,
f−ρ := e−ρ q ϕ(.,ρ) ,
where the sum is over all (σ) = σ1 , . . . , σn , and all permutations (σ 0 ) of (σ), such m(ρ) that ρi = τ m σi0 is defined. We take F(σ) = 1 when the set (σ) is empty. Note that the family of deformation of this type is large enough to contain the quantization of all the classical Lie bialgebras classified by Belavin and Drinfel’d, with r-matrices of constant, trigonometric and elliptic type. Two cases need to be distinguished. (a) Finite twisting is by definition the case when there is k such that for all σ, ˆ 1 ; then Γ ˆ1, Γ ˆ 2 are distinct and the product over m is finite. /Γ τ kσ ∈ (b) Elliptic twisting. The only other possibility (see [14, Sec. 16]) is that A0 = \) and Γ1 = Γ2 is generated by all the simple roots. This section deals sl(N with twisting in the category of Hopf algebras; elliptic twisting within the context of Hopf algebras implies [15] that we drop the central extension and descend to loop algebras. The full elliptic Kac–Moody algebra is quasi Hopf and will be discussed in the next section.
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1045
The deformed R-matrix and coproduct are R = (Ft )−1 RF ,
∆ = (Ft )−1 ∆Ft .
(6.5)
Here are some products that seem ill defined; thus R has degree-increasing operators in the second space, where F has degree-decreasing operators. This problem can be handled in a general way by adopting an interpretation that is quite natural in deformation theory. One notes that F is a formal power series in the deformation parameter . One interprets all the operators this way; then the problem reduces to making sure that the coefficients are well defined. Indeed, to any fixed order in , the product RF is, in the second space, a power series in the operators eα multiplied by a polynomial in the other generators. It is, nevertheless, of some interest to determine whether singularities arise as one assigns a value to and attempts to sum up the deformation series. In this respect cases (a) and (b) are quite different. (a) Finite twisting. The sum in (6.4) becomes finite when projected on a finite dimensional representation in either one of the two spaces. Infinite sums will appear if both representations are infinite, but there is a finite number of terms with fixed weight; therefore no infinite, purely numerical series will appear. Infinite sums with operator coefficients are beyond (our power of) analysis in the general case, and of no immediate concern to us. The value of is basis dependent; the only distinct possibilities are = 0, 1. (b) Elliptic twisting. We note that the range of is in this case || < 1. Here the situation is more delicate, and of some interest. Under twisting, the Casimir element Z suffers an equivalence transformation Z = (Ft )−1 ZFt ,
(6.6)
and one expects that an intertwiner Φ , satisfying ∆ (x)Φ = Φ x ,
x ∈ A0 ,
(6.7)
may be expressed as Φ = (Ft )−1 Φ. However, Ft has a structure similar to that of R, with degree-increasing operators in the second space, and we must consider the possibility that normal ordering may be required. In fact probably not, but since we have not proved this, we shall switch our attention to the other intertwiner. We consider instead the alternative intertwiner Ψ, and the alternative Casimir operator Z 0 that commutes with ∆0 (x), namely Z 0 = Rt R ,
Z 0 ∆0 (x) = ∆0 (x)Z 0 ,
(Z 0 − 1)Ψ = 0 .
(6.8)
We have Z0 = F−1 Z 0 F , F−1 Ψ
The operator product property of Ψ , namely
is therefore in the clear.
Ψ = F−1 Ψ .
(6.9)
is well defined as an operator on Vµ,k . The intertwining ∆0 (x)Ψ = Ψ x ,
(6.10)
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C. FRØNSDAL and A. GALINDO
We define the operator product Z0 Ψ (z). Formally, Z0 Ψ = Rt R Ψ = F−1 Z 0 Ψ , and this too is well defined, provided we define the untwisted product as in (5.7); that is ˆ Z 0 Ψ → :Z 0 Ψ: = Rt (Ri q H ⊗ 1)ΨRi . The equation satisfied by the twisted correlation function is (Z0 − 1)Ψ = 0 or more precisely: Definition 6.3. The twisted q-KZ equation is the following equation for the twisted intertwiner operator: Ψ (q −k−g z) = F−1 (q −k−g z)L+ (zq −g−k+uk )(L−i (zq −g ) ⊗ 1)F (z)Ψ (z)L− i . (6.11) It should be noted that the polarization used is the same as before deformation. To justify this we repeat that the definition of the intertwining operators is independent of normal ordering conventions, normal ordering is relevant only when the ordinary product does not exist, it is required to be well defined and covariant, nothing more. Of course, it is also true that, if Ψ is defined as in (6.9), then (6.11) is equivalent to (5.8). The top matrix element of Ψ is (v00 , Ψ v0 ) = (v00 , F−1 Ψv0 ) = (v00 , Ψv0 ) .
(6.12)
This shows that, in a complete description of, say, the eight-vertex model, both periodic and non-periodic functions appear. We had naively expected to encounter nothing but elliptic functions, that “the eight-vertex model lives on the torus”. Having thus discarded a prejudice, we are comfortable with the continued use, in the twisted case, of the original polarization based on the standard trigonometric R-matrix. The alternative of defining a normal-ordered product such that R Φ = (Ri ⊗ 1)Φ Ri is entirely redundant. Another idea is to replace matrix elements by traces, as suggested by Bernard [6] and in [12]. However, since we know that the elliptic quantum group, as an algebra, is the same as the standard quantum group (that is, a Kac–Moody algebra), there seems to be no reason to take less interest in the highest weight matrix elements in the elliptic case. Continuity of physics also suggests that we continue to work with the usual module structure, as was argued in [18, Sec. 4]. Trace functionals are interesting in themselves, but there seems to be no reason to neglect the matrix elements. The intertwiners of Kac–Moody modules, and the solutions of the KZ equation, know nothing about r-matrices. For all that we may derive different versions of the
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1047
equation, the solutions remain the same. To base the polarization on the R-matrix is not an imperative; more important is to adopt a workable definition that gives a meaning to the objects of interest; to wit, matrix elements of intertwiners. In the setting of conformal field theory twisting does not affect the quantization paradigm, but it does change the quantum fields (the intertwiners) and their operator product expansions. We shall need to know the twistor F . It is determined, uniquely, by the recursion relationsa [1 ⊗ fρ , Fm ] = m (Fm (fτ −m ρ ⊗ q −ϕ(ρ,.) ) − (fτ −m ρ ⊗ q ϕ(.,ρ) )Fm ) ,
(6.13)
with the initial conditions Fm = 1 − m
X
fτ −m ρ ⊗ f−ρ + · · ·
(6.14)
(ρ)
These equations were solved in a special case, and used to calculate the elliptic d in the fundamental representation [14]. Later, we shall exploit the R-matrix of sl(2) similarity between this relation and the recursion relation (2.4) for the universal R-matrix. 7. Quasi Hopf Twisting We are interested in the elliptic quantum groups, in the sense of Baxter [2] and Belavin [4]. This takes us out of the framework of Hopf algebras, but just barely so. The special nature of these quasi Hopf algebras is that they become Hopf algebras at level zero; that is, on the quotient by the ideal generated by the center. Quasi Hopf deformations are constructed in the same way as Hopf deformations, except that the element F need not satisfy the cocycle condition (6.1). The deformed R-matrix and coproduct are given by (6.5), but the former no longer satifies the Yang–Baxter relation and the latter is not coassociative, in general. m(ρ) If F(σ) are the coefficients of the elliptic Hopf twistor in (6.4), then the elliptic quasi Hopf twistor has the form [15] Y X m(ρ) F = Fm , Fm = nm F(σ) fσ1 . . . fσn ⊗ f−ρ1 . . . f−ρn Q(m, ρ) , m=1,2,...
(σ)
(7.1) where Q(m, ρ) ∈ A00 ⊗ A00 and A00 is the Cartan subalgebra of the quantized Kac– Moody algebra A0 . This factor is equal to unity in the Hopf case, and (7.1) then reduces to (6.4). The F -twisted algebra is a Hopf algebra when the parameters satisfy the condition ˆ1 , ϕ(σ, .) + ϕ(., τ σ) = 0 , σ ∈ Γ where now τ is the cyclic diagram automorphism that takes each simple root of sl(N ) to its neighbour. This condition can be satisfied on the loop algebra (when a The fact that the recursion relation (6.13) is necessary was shown in [14, 15]; sufficiency was proved quite recently, by Jimbo, Konno, Odake and Shiraishi [16].
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C. FRØNSDAL and A. GALINDO
c 7→ 0). We are interested in the full Kac–Moody algebra (c 6= 0); in that case the best that can be done is to choose parameters such that ϕ(σ, .) + ϕ(., τ σ) = [(1 − u)δσ0 + uδτ0σ ]c .
(7.2)
This algebra is what we mean by “elliptic quantum group in the sense of Baxter and Belavin”; it is a quasi Hopf algebra of a particularly benevolent type, where the deviation from coassociativity is confined to the center. Instead of the cocycle condition (6.1) we now have ((id ⊗ ∆21 )F )F12 = ((∆13 ⊗ id)F )F31,2 ,
(7.3)
where Fij,k is an extension of Fij , supported on the center, to the k’th space. In the case of interest, when we are dealing with modules with fixed level c 7→ k, this amounts to a modification of the coefficients in Fij . From (7.3) one gets the Cartan factors Q(m, ρ) [15] and the recursion relation [1 ⊗ fρ , Fm ] = m (Fm (fτ −m ρ ⊗ q −ϕ(ρ,.) ) − (fτ −m ρ ⊗ q ϕ(.,ρ) )Fm )Q(m, ρ) ,
(7.4)
with the initial conditions Fm = 1 − m
X
(fτ −m ρ ⊗ f−ρ )Q(m, ρ) + · · ·
(7.5)
ρ m m is known, F12,3 is obtained by means The solutions will be given later. Once F12 of the substitution 1 ⊗ c 7→ 1 ⊗ ∆(c) . (7.6)
8. Correlation Functions The main objects of interest, in conformal field theory as well as in the study of statistical models, are the correlation functions. In their simplest form they are matrix elements of products of intertwiners, fv0 v (z1 , . . . , zN ) = hv 0 , Φ(z1 ) . . . Φ(zN ) vi ∈ V1 (z1 ) ⊗ · · · ⊗ VN (zN ) , gv0 v (z1 , . . . , zN ) = hv 0 , Ψ(z1 ) . . . Ψ(zN ) vi ∈ V1 (z1 ) ⊗ · · · ⊗ VN (zN ) .
(8.1)
Here Φ(zp ) and Ψ(zp ) are intertwiners between highest weight modules, Φ(zp ), Ψ(zp ) : Vµp ,k → Vp (zp ) ⊗ Vµp−1 ,k ,
p = 1, . . . , N ,
with {Vp (zp )} a set of evaluation modules, and v ∈ VµN ,k , v 0 ∈ Vµ0 ,k . These “functions” are formal, V1 ⊗ · · · ⊗ VN -valued series in N distinct variables. Classical Correlation Functions We begin with the classical case and the polarization (4.18), X X z n L−n z −n Lna , Ja = Ja+ + Ja− = a + n>0
n≥0
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1049
and the normalization that leads to (4.22): (k + g)z
d Φ(z) + La ⊗ Ja+ Φ(z) + (La ⊗ 1)Φ(z)Ja− = 0 . dz
Then for any p ∈ {1, . . . , N }, (k + g)zp
d fv0 v (z1 , . . . , zN ) = −La(p) hv 0 , . . . Φ(zp−1 )Ja+ (zp )Φ(zp ) . . . vi dzp − La(p) hv 0 , . . . Φ(zp−1 )Φ(zp )Ja− (zp ) . . . vi .
(8.2)
Here La denotes the action of La in Vp . Suppose now that the vectors v0 and v00 are highest weight vectors of the respective highest weight modules. The intertwiners (p) satisfy [Lna , Φ(zp )] = −La zpn Φ(zp ); this allows us to permute J + through to the left, where it dies on the highest weight vector, and to permute J − towards the right, where only the zero modes survive, to contribute the last term in (p)
d fv0 v (z1 , . . . , zN ) dzp 0 0 X X zp n (p) 0 L(q) =− a La hv0 , . . . Φ(zp−1 )Φ(zp ) . . . v0 i z q n>0
(k + g)zp
1≤q
+
X
X zq n
N ≥q>p n≥0
zp
0 La(p) L(q) a hv0 , . . . Φ(zp−1 )Φ(zp ) . . . v0 i
+ La(p) hv 0 , . . . Φ(zp−1 )Φ(zp ) . . . La v0 i . Hence (k + g)
X d 1 fv00 v0 (z1 , . . . , zN ) = L(p) L(q) fv00 v0 (z1 , . . . , zN ) , p = 1, . . . , N , dzp zp − zq a a q6=p
(8.3) (N +1)
acts on v0 . The last where q takes the values 1, . . . , N + 1, zN +1 = 0, and La expression must be supplemented by the instruction X (1/z ) (zq /zp )n , q > p, p n≥0 1 (8.4) := X zp − zq n (−1/z ) (z /z ) , q < p . q p q n≥0
The domain of convergence is thus |z1 | > |z2 | > · · · > |zN +1 | = 0. In the simplest, nontrivial case N = 1. Projecting on a vector w ∈ V we get (k + g)
d w c fv0 v0 (z1 ) = fvw0 v0 (z1 ) , 0 dz1 z 0
c=
1 hw ⊗ v 0 , CΦ vi = (C(ν) − C(µ) − C(π)) , 0 hw ⊗ v , Φvi 2
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C. FRØNSDAL and A. GALINDO
which simply reflects the choice of grading of Φ. The case N = 2 is not much more complicated; the equations are (k + g)
df c12 f c13 f = + , dz1 z1 − z2 z1
(k + g)
df c12 f c23 f = + , dz2 z2 − z1 z2
(8.5)
d and funwhere cij = La La and “3” refers to the source space. In the case of sl(2) damental evaluation modules it is a simple matter to work out the hypergeometric solutions. The general structure of the solution was exploited by Khono [19] and Drinfel’d [8] to construct representations of the braid group and examples of quasi Hopf algebras. If instead we use the polarization of (4.27) we obtain from (4.30) and (4.31), on the vectors of highest weight, ! p−1 N X X d fv0 v (z1 , . . . , zN ) + Ap + rqp − rpq fv00 v0 (z1 , . . . , zN ) = 0 , (k + g)zp dzp 0 0 q=1 q=p+1 (i)
(j)
(8.6) Ap := (H +
ϕ(v00 , .)
+ ϕ(., v0 ))p ,
for p = 1, . . . , N . When N = 2, (k + g)z1
d fv0 v (z1 , z2 ) + A1 fv00 v0 (z1 , z2 ) − r12 fv00 v0 (z1 , z2 ) = 0 , dz1 0 0
(k + g)z2
d fv0 v (z1 , z2 ) + A2 fv00 v0 (z1 , z2 ) + r12 fv00 v0 (z1 , z2 ) = 0 . dz2 0 0
The solutions are, of course, the same, up to normalization. q-Deformed Correlation Functions We turn to the q-KZ equation (5.6), Φ(q −k−g z) = :L(z)Φ(z): . For functions of the type (8.1) the implication is Tp fv00 v0 (z1 , . . . , zN ) := fv00 v0 (. . . , zp−1 , q −k−g zp , zp+1 , . . .) = hv00 , . . . Φ(zp−1 )L−i (zp0 )L+ (zp )Φ(zp )L− i Φ(zp+1 ) . . . v0 i , with z 0 = q −g−uk z. More transparently, − Tp fv00 v0 (z1 , . . . , zN ) = [L−i (zp0 )L+j (zp )]hv00 , . . . Φ(zp−1 )L+ j Φ(zp )Li Φ(zp+1 ) . . . v0 i .
(8.7) For N = 2, 0
fv00 v0 (q −k−g z1 , z2 ) = [L−i (z10 )q −ϕ(v0 ,.) ]1 hv00 , Φ(z1 )L− i Φ(z2 )v0 i , fv00 v0 (z1 , q −k−g z2 ) = [q ϕ(.,v0 )+H L+i (z2 )]2 hv00 , Φ(z1 )L+ i Φ(z2 )v0 i .
(8.8)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1051
We reduce this using the quasi triangularity conditions in the Appendix. The final result is z2 g+k A1 −1 q fv00 v0 (z1 , z2 ) , q T1 fv00 v0 (z1 , z2 ) = R12 z1 z2 fv00 v0 (z1 , z2 ) , T2 fv00 v0 (z1 , z2 ) = q A2 R12 (8.9) z1 Ai := (ϕ(v00 , .) + ϕ(., v0 ) + H)i ,
i = 1, . . . , N .
These two equations can be combined in two ways. The result is the same in either case, in consequence of the fact that the operator A1 + A2 (the subscripts refer to the two evaluation modules) commutes with R12 . From the fact that the correlation function is invariant for the action of the Cartan subalgebra in the four spaces it follows in fact that we can replace 1 A1 + A2 → (C(µ) − C(ν)) . 2 The result is that T1 T2 fv00 v0 (z1 , z2 ) = q A1 +A2 fv00 v0 (z1 , z2 ) .
(8.10)
The two equations (8.9) are thus mutually consistent. For correlators with more than two intertwiners one obtains similar equations ([12] and below), and for them consistency depends on the fact that R satisfies the Yang–Baxter relation. For the other two-point function we have from (5.8), with z 00 = q −g−k+uk z, Tp gv00 ,v0 (z1 , . . . , zN ) − = L+i (zp00 )L−j (q −g zp )hv00 , . . . Ψ(zp−1 )L+ i Ψ(zp )Lj Ψ(zp+1 ) . . . v0 i ,
(8.11)
and, in particular, − T1 gv00 ,v0 (z1 , z2 ) = L+i (z100 )L−j (q −g z1 )hv00 , L+ i Ψ(z1 )Lj Ψ(z2 )v0 i ,
(8.12)
− T2 gv00 ,v0 (z1 , z2 ) = L+i (z200 )L−j (q −g z2 )hv00 , Ψ(z1 )L+ i Ψ(z2 )Lj v0 i ,
(8.13)
and with the help of the Appendix, −1 T1 gv00 ,v0 (z1 , z2 ) = q A1 R12
z2 z1
gv00 ,v0 (z1 , z2 ) ,
z2 −k−g A2 q gv00 ,v0 (z1 , z2 ) . q T2 gv00 ,v0 (z1 , z2 ) = R12 z1 The q-KZ equations for the 3-point functions are z2 k+g z3 k+g A1 −1 −1 T1 f (z1 , z2 , z3 ) = R12 R13 q f (z1 , z2 , z3 ) , q q z1 z1 z3 k+g A2 z2 −1 q R12 f (z1 , z2 , z3 ) , T2 f (z1 , z2 , z3 ) = R23 q z2 z1 z3 z3 A3 R23 f (z1 , z2 , z3 ) , T3 f (z1 , z2 , z3 ) = q R13 z1 z2
(8.14)
(8.15)
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C. FRØNSDAL and A. GALINDO
with A as before. Integrability is expressed as a cocycle condition that is precisely ˜ 12 is the Yang–Baxter relation for R, Eq. (A.6) with c2 = 0. (The tilde on R redundant.) Remarks 8. (1) It is interesting to note that 1
T1 T2 T3 f (z1 , z2 , z3 ) = q A1 +A2 +A3 f (z1 , z2 , z3 ) = q 2 (C(µ)−C(ν)) f (z1 , z2 , z3 ) . (8.16) This is what one expects, since the product of any number of intertwiners should have the universal property; see Remark 3.2, also (4.31) and (8.10). (2) The first and the last equations in (8.15) can be written as follows: −1 (k+g)d1 A1 q q f (z1 , z2 , z3 ) , T1 f (z1 , z2 , z3 ) = q −(k+g)d1 R1,32
T3 f (z1 , z2 , z3 ) = q A3 R21,3 f (z1 , z2 , z3 ) .
(8.17)
Here Ri,jk is the action of the universal R-matrix in the evaluation module via the opposite coproduct, R1,32 = (id ⊗ ∆0 )R, R21,3 = (∆0 ⊗ id)R. This too is an expression of universality; compare the first of (8.17) with the first of (8.9). Similarly one finds directly, using the formulas in the Appendix that, if g(z1 , z2 , z3) is the alternative 3-point function in (8.1), then z3 −k−g z3 −k−g A3 R13 q g(z1 , z2 , z3 ) q q T3 g(z1 , z2 , z3 ) = R23 z2 z1 = q −(k+g)d3 R12,3 q (k+g)d3 q A3 g(z1 , z2 , z3 ) .
(8.18)
The other two formulas cannot be obtained so directly, but the principle of universality encountered in Remarks 8 tells us that −1 g(z1 , z2 , z3 ) . T1 g(z1 , z2 , z3 ) = q A1 R1,23
(8.19)
Finally, from (8.16), T2 g(z1 , z2 , z3 ) = T1−1 q A1 +A2 +A3 T3−1 g(z1 , z2 , z3 ) −1 −(k+d)d1 q g(z1 , z2 , z3 ) . = q (k+g)d1 R12 q A2 R23
(8.20)
Summing up, we have −1 g(z1 , z2 , z3 ) , T1 g(z1 , z2 , z3 ) = q A1 R1,23 −1 T2 g(z1 , z2 , z3 ) = q −(k+g)d2 R12 q (k+g)d2 q A2 R23 g(z1 , z2 , z3 ) ,
(8.21)
T3 g(z1 , z2 , z3 ) = q −(k+g)d3 R12,3 q (k+g)d3 q A3 g(z1 , z2 , z3 ) . Twisting and Covariance Let us evaluate one of the two-point functions of the twisted model, g (z1 , z2 ) = hv 0 , Ψ (z1 )Ψ (z2 ) vi = hv 0 , F−1 Ψ(z1 )F−1 Ψ(z2 ) vi .
(8.22)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1053
On highest weight vectors, g (z1 , z2 ) = hv00 , Ψ(z1 )F−1 (z2 )Ψ(z2 ) v0 i = F−1,i (z2 )hv00 , ∆0 (Fi−1 )Ψ(z1 )Ψ(z2 ) v0 i . (8.23) Now ∆0 (f−ρ ) = f−ρ ⊗ 1 + q ϕ(.,ρ) ⊗ f−ρ and so, for Hopf deformations, when F is a series of the type (6.4), g (z1 , z2 ) = F−1 (z2 , z1 )g(z1 , z2 ) .
(8.24)
An alternative derivation of this result makes direct use of the cocycle condition. It can be written as follows −1 −1 ((id ⊗ ∆31 )F−1 ) = F21 ((∆12 ⊗ id)F−1 ) . F13
(8.25)
Applying v00 we get, because this vector is a highest weight vector, −1 ... , hv00 , (id ⊗ ∆31 )F−1 ) . . . = hv00 , F21
(8.26)
which is just what we need to reduce (8.23) to (8.24). The transformation formula (8.24) shows that the result (8.14) is not covariant with respect to twisting, in the following sense. The equation satisfied by g is z2 −1 F (z2 , z1 )g (z1 , z2 ) ; T1 g (z1 , z2 ) = F−1 (z2 , q −k−g z1 )q A1 R12 z1 the right-hand side is very different from the naive analogue of (8.14), z2 A1 −1 q R12 g (z1 , z2 ) . z1 Thus twisting does not preserve the form of the equations satisfied by matrix elements of intertwining operators; one cannot simply replace R in these equations by a twisted R-matrix. In fact, it is clear that our calculations made use of the specific form of the standard R-matrix. The factors q A , in particular, are characteristic of the standard R-matrix. Of course, we do not deny the existence of holonomic difference equations that involve R-matrices of a more general type. The claim is that the solutions to such equations are not, in general, matrix elements of intertwining operators for highest weight, quantized Kac–Moody modules. The elliptic correlation functions can be found by solving a “modified” q-KZ equation, but much more simply by the intermediary of the solutions of the standard q-KZ equations for the 6-vertex model, as in Eq. (8.24). For three-point functions the effect of twisting is g (z1 , z2 , z3 ) = hv00 , F−1 Ψ(z1 )F−1 Ψ(z2 )F−1 Ψ(z3 )v0 i −1 Ψ(z3 )v0 i = F−1,i (z2 )F−1,j (z3 )hv00 , Ψ(z1 )Fi−1 Ψ(z2 )Fj −1 )Ψ(z2 )Ψ(z3 )v0 i = F−1,i (z2 )F−1,j (z3 )hv00 , ∆41 (Fi−1 )Ψ(z1 )∆42 (Fj −1 −1 i )i (z2 )F−1,j (z3 )hv00 , Ψ(z1 )(Fj ) Ψ(z2 )Ψ(z3 )v0 i = F−1 (z2 , z1 )(Fj
1054
C. FRØNSDAL and A. GALINDO
−1 −1 i = F−1 (z2 , z1 )(Fj )i (z2 )F−1,j (z3 )hv00 , ∆41 ((Fj ) )Ψ(z1 )Ψ(z2 )Ψ(z3 )v0 i −1 −1 i )i (z2 )F−1,j (z3 )hv00 , ((Fj ) )(z1 )Ψ(z1 )Ψ(z2 )Ψ(z3 )v0 i = F−1 (z2 , z1 )(Fj −1 )(z1 , z2 )F−1,j (z3 )g(z1 , z2 , z3 ) = F−1 (z2 , z1 )∆12 (Fj
= F−1 (z2 , z1 )((id ⊗ ∆)F−1 )(z3 , z1 , z2 )g(z1 , z2 , z3 ) .
(8.27)
Thus we conclude that the twisted correlation functions can be obtained from the untwisted ones. The latter are found by solving equations that are known to be integrable by virtue of the fact that the standard R-matrix satisfies the Yang– Baxter relation. It is possible, but redundant and unrewarding, to write down the equations satisfied by the twisted correlation functions; they are complicated and uninstructive whether expressed in terms of R or R . 9. Correlation Function for the 8-Vertex Model Here we try to understand what, if any, are the qualitative new features that result from the fact that the elliptic quantum group is not a Hopf algebra. Technically, the difference is that the reduction of (8.23) to (8.24) is no longer valid, because the twistor is no longer of the type (6.4). Instead of the cocycle condition (6.1) that gave us (8.25) we now have the modified cocycle condition (7.3), which yields −1 −1 ((id ⊗ ∆31 )F−1 ) = F21,3 ((∆12 ⊗ id)F−1 ) (9.1) F13 and, instead of (8.24), −1 (z2 , z1 )g(z1 , z2 ) . g (z1 , z2 ) = F21,3
(9.2)
To calculate F21,3 see (7.6). This is the two-point function for the eight-vertex model. The quasi Hopf nature of the elliptic quantum group is parameterized by the level k of the highest weight module and the effect on the two-point function is in the numerical modification of the matrix F that is indicated by the third index. Equation (8.27) gets modified in the same manner. To get an idea of the importance of this effect it is enough to calculate the modified matrix in the case N = 2 with V the fundamental sl(2)-module. The result is as follows. The trigonometric R-matrix is given for comparison, with the two spaces interchanged: Rt = 2m F12,3 =
2m−1 = F12,3
A(q, x) ϕ q ((1 − q −2 x)H+ + (1 − x)H− + e1 ⊗ e−1 + e0 ⊗ e−0 ) , 1 − q −2 x
(9.3)
A2m (q, x, ) ((1 − q 2 αα0 x)H+ + (1 − αα0 x)H− − αf1 ⊗ f−1 − α0 f0 ⊗ f−0 ) , 1 − q 2 αα0 x (9.4) A2m−1 (q, x, ) ((1 − β 2 x)H+ + (1 − q 2 β 2 x)H− − βf1 ⊗ f−0 − βf0 ⊗ f−1 ) . 1 − q2 β 2 x (9.5)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
1055
Here x = z1 /z2 , H± =
1 [(1 + H) ⊗ (1 ± H) + (1 − H) ⊗ (1 ∓ H)] 4
(9.6)
(in another notation, H+ = e11 ⊗ e11 + e22 ⊗ e22 , H− = e11 ⊗ e22 + e22 ⊗ e11 ), and α = q uk (2 q −k )m , α0 = q (1−u)k (2 q −k )m , β 2 = q k (2 q −k )2m−1 . d Remember that k denotes the level of the highest weight sl(2)-module. It enters here because it appears in the extension F21,3 of the twistor in Eq. (9.2). This operator acts in three spaces, but its action on the highest weight module is limited to the center. In the level zero case we recover the Hopf twistor. The calculation that leads to (9.3) is given in detail in the Appendix, with an explicit formula for the normalizing factor A(q, x). The matrix factors in (9.4) and (9.5) are obtained in the same way, and the scalar factors as follows: Proposition 9.1. (a) The normalizing factor in (9.4) is A2m (q, x, ) = A(1/q, αα0 x) .
(9.7)
(b) The normalizing factor in (9.5) is A2m−1 (q, x, ) = A(1/q, β 2 x) .
(9.8)
Proof of (a). Consider the universal R-matrix, and the algebra map generated by e1 → α−1 f1 ,
e−1 → −αf−1 ,
−1
e0 → α0 f0 ,
e−0 → −α0 f−0
(9.9)
in the second space, but ei → fi in the first space. This maps the original algebra to another algebra with the q replaced by q −1 . Now consider the factorization (2.1) of the universal R-matrix, R = q ϕ T . After (9.9), the first two terms in T t agree with the first two terms of F2m . The recursion relations (2.4) and (7.4) also agree, after replacing q by 1/q, and so do the solutions. Then we pass to the evaluation representation, setting f0 = zˆ1 f−1 . In the R-matrix (more precisely, in T t ) we have set e0 = z1 e−1 , which after the substitution (9.9) becomes α0 f0 = (z1 /α)f−1 , so that z1 = αα0 zˆ1 . Under these transformations, including transposition of the two spaces, the polynomial factor in (9.3) is transformed into that of (9.4), and the normalizing factor also agree. Proof of (b). In this case, in the second space let e1 → β −1 f0 , e−1 → −βf−0 , e0 → β −1 f1 , e−0 → −βf−1 , and in the first space ei → fi . Putting it all together, we have after a simple change of basis a dˆ b cˆ , x = z1 /z2 , F12,3 (z1 , z2 ) = A(F ) cˆ b ˆ d a
(9.10)
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C. FRØNSDAL and A. GALINDO
with A(F ) =
Y
A(q −1 , q k ¯4m+2 x) =
m≥0
=
Y (1 − xq k ¯4m+2 q 4n )(1 − xq k ¯4m+2 q 4n+4 ) (1 − xq k ¯4m+2 q 4n+2 )2
m,n≥0
(q k ¯2 x; q 4 , ¯4 )∞ (q k+4 ¯2 x; q 4 , ¯4 )∞ . (q k+2 ¯2 x; q 4 , ¯4 )2∞
and a ± dˆ =
Y (1 ± q −1+k/2 √x ¯2m−1 ) √ , (1 ± q 1+k/2 x ¯2m−1 ) m≥1
(9.11)
b ± cˆ =
Y (1 ± q −1+k/2 √x ¯2m ) √ , (1 ± q 1+k/2 x ¯2m ) m≥1
with ¯2 = 2 q −k . Finally, we give the result of projecting the universal elliptic d on the evaluation representation (k = 0), R-matrix of sl(2) α δ β γ R (z1 , z2 ) = ((Ft )−1 RF )(z1 , z2 ) = A (q, x−1 ) , γ β δ α where α+δ = q β+γ =
θ3 (u − ρ, τ ) θ2 (u − ρ, τ ) , α−δ =q , θ3 (u + ρ, τ ) θ2 (u + ρ, τ )
θ1 (u − ρ, τ ) , θ1 (u + ρ, τ )
β−γ =
θ(u − ρ, τ ) , θ(u + ρ, τ )
with x = z1 /z2 = e4πiu ,
q = e2πiρ ,
= eπiτ ,
A (q, x−1 ) = A(q, x−1 )A(F )/A(Ft ) .
In terms of the Jacobian elliptic functions one has α+δ :α−δ :β+γ :β−γ =
1 cn(2K(u − ρ), k) 1 sn(2K(u − ρ), k) dn(2K(u − ρ), k) :1: : , dn(2K(u + ρ), k) q cn(2K(u + ρ), k) q sn(2K(u + ρ), k)
where K, k are the real quarter-period and modulus, respectively, for the nome : 2 4 √ Y π Y 1 + 2n−1 1 − 2n 1 + 2n · , k = 4 . K= 2 1 − 2n−1 1 + 2n 1 + 2n−1 n≥1
n≥1
Acknowledgements We thank Olivier Babelon, Benjamin Enriquez, Moshe Flato, Tetsuji Miwa and Nikolai Reshetikhin for advice. We thank Moshe Flato for an incisive and constructive criticism of the original manuscript. A. G. thanks the Fundaci´on Del Amo for financial support and the Department of Physics of UCLA for hospitality.
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
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Appendix Solving the Recursion Relations We shall solve the recursion relation (2.4) in the fundamental evaluation repred Here we set sentation of sl(2). 1 1 0 0 1 0 0 . e1 = κ , e−1 = κ , ϕ= H ⊗H, H = 0 −1 0 0 1 0 2 (A.1) The commutation relations hold with κ2 = q − q −1 . The factor T in R = q ϕ T has the form a b cx T = , x = z1 /z2 , c b a and (2.4) is equivalent to [T, 1 ⊗ e−γ ] = (e−γ ⊗ q ϕ(γ,.) )T − T (e−γ ⊗ q −ϕ(.,γ)) ,
γ = 1, 0 ,
with ϕ(1, .) = ϕ(., 1) = H, ϕ(0, .) = ϕ(., 0) = −H. Taking γ = 1 we get two relations, q(a − b) = c = (aq − b/q)/x , (A.2) and taking γ = 0 the same two relations. Hence R(q, x−1 ) =
A(q, x−1 ) ϕ q ((1 − q −2 x−1 )H+ + (1 − x−1 )H− + e−1 ⊗ e1 + e−0 ⊗ e0 ) . 1 − q −2 x−1
The matrices in (9.4) and (9.5) are found in the same way. In the special case of d the Cartan factors in (7.5) are, for m ≥ 1, sl(2) Q(2m, 1) = q (u−m)c ,
Q(2m, 0) = q (1−u−m)c ,
Q(2m − 1, 1) = Q(2m − 1, 0) = q (1−m)c . In the structure, R is determined uniquely by the recursion relations and the initial conditions, but in the evaluation representation the normalizing factor A(q, x−1 ) remains undetermined. Fortunately Levendorskii, Soibelman and Stukopin [21], starting from an equivalent expression for the standard, universal R-matrix for d obtain the following result, sl(2) X 1 q k − q −k xk . A(q, x) = exp (A.3) k q k + q −k k≥1
The sum converges for |q| 6= 1, |x| < 1 and the formula can be manipulated to yield (xq 2 ; q 4 )2∞ |q| < 1 , (x; q 4 )∞ (xq 4 ; q 4 )∞ , (A.4) A(q, x) = (x; q −4 )∞ (xq −4 ; q −4 )∞ , |q| > 1 . (xq −2 ; q −4 )2∞
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C. FRØNSDAL and A. GALINDO
Hence A(q, x)A(q −1 , x) = 1 ,
|q| 6= 1 .
(A.5)
This is also clear from (A.3). The inverse of R can also be represented as a series, similar to (2.1), X eˆ−α ⊗ eˆα + · · · , R−1 = q −ϕ Tˆ , Tˆ = 1 + α
with eˆα := q −ϕ(α,.) eα ,
eˆ−α = −e−α q ϕ(.,α) .
The commutation relations for the eˆ’s agree with those of the e’s, and the recursion relations for Tˆ agrees with that of T , all up to the sign of ϕ. (We get a recursion relation for Tˆ from the fact that R−1 also satisfies the Yang–Baxter relation.) Consequently, in the structure, R(ϕ, e)−1 = R(−ϕ, eˆ) , and in any evaluation representation, R(q, x)−1 = R(q −1 , x) . These results are quite general and imply, in particular, Eq. (A.5). Reduced Formulas We list here the formulas that are obtained from the Yang–Baxter relation R12 R13 R23 = R23 R13 R12 and the quasi triangular conditions (id ⊗ ∆)R = R12 R13 ,
(∆ ⊗ id)R = R23 R13
when the c, d factors are removed as in ˜, R = q uc ⊗ d+(1−u)d ⊗ c R namely ˜ 23 = R ˜ 23 (q −(1−u)d1 c2 R ˜ 12 ˜ 13 q uc2 d3 )R ˜ 13 q (1−u)d1 c2 )R ˜ 12 (q −uc2 d3 R R
(A.6)
and ˜ = (q −(1−u)d1 c3 R ˜ 13 , (∆ ⊗ id)R ˜ = (q −uc1 d3 R ˜ 13 . ˜ 12 q (1−u)d1 c3 )R ˜ 23 q uc1 d3 )R (id ⊗ ∆)R (A.7) These last two relations give us what we need to reduce (8.8), namely −1 z2 k−uk − ˜ L (z1 )13 Φ(z2 ) = R12 q (L−i (z1 ) ⊗ 1)Φ(z2 )L− i , z1 L
+i
(z2 )Φ(z1 )L+ i
˜ 12 = L (z2 )R +
z2 z1
(A.8)
Φ(z1 ) .
(A.9)
8-VERTEX CORRELATION FUNCTIONS AND TWIST COVARIANCE OF q-KZ EQUATION
For the other intertwiner, there is an analogue of (A.9), ˜ 12 z2 q −uk L+ (z2 )Ψ(z1 ) , = R L+i (z2 )Ψ(z1 )L+ i z1
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(A.10)
but we could not find an analogue of (A.8). To obtain (8.14) we used the method that was explained for the derivation of (8.21). References [1] O. Babelon and D. Bernard, “A Quasi Hopf interpretation of quantum 3-j and 6-j symbols and difference equations”, q-alg/9511019. [2] R. J. Baxter, “Partition Function of the Eight-Vertex Model”, Ann. Phys. 70 (1972) 193–228. [3] R. J. Baxter and S. B. Kelland, J. Phys. C: Solid State Phys. 7 (1974) L403–6. [4] A. A. Belavin, “Dynamical symmetry of integrable systems”, Nucl. Phys. 180 (1981) 198–200. [5] A. A. Belavin and V. G. Drinfeld, “Triangle equation and simple Lie algebras”, Sov. Sci. Rev. Math. 4 (1984) 93–165. [6] D. Bernard, “On the WZW model on the torus”, Nucl. Phys. B303 (1988) 77–174. [7] V. G. Drinfeld, “Quantum groups”, in Proc. Int. Congress Math. Berkeley, ed. A. M. Gleason, A. M. S., Providence, 1987. [8] V. G. Drinfeld, “Quasi Hopf Algebras”, Leningrad Math. J. 1 (1990) 1419–1457. [9] B. Enriquez and I. V. Rubtsov, “Quasi Hopf algebras associated with sl(2) and complex curves”, q-alg/9608005. [10] G. Felder, “Elliptic quantum groups”, hep-th/9412207. [11] O. Foda, K. Iohara, M. Jimbo, T. Miwa and H. Yan, “An elliptic algebra for sl(2)”, RIMS preprint 974. [12] I. B. Frenkel and N. Yu. Reshetikhin, “Quantum affine algebras and holonomic difference equations”, Commun. Math. Phys. 146 (1992) 1–60. [13] I. B. Frenkel, N. Yu. Reshetikhin and M. Semenov-Tian-Shansky, “Drinfeld–Sokolov reduction for difference operators and deformations of W -algebras I. The case of Virasoro algebra”, q-alg/9704011. [14] C. Frønsdal, “Generalization and deformations of quantum groups”, RIMS Publ. 33 (1997) 91–149 (q-alg/9606020). [15] C. Frønsdal, “Quasi Hopf deformation of quantum groups”, Lett. Math. Phys. 40 (1997) 117–134 (q-alg/9611028). [16] M. Jimbo, H. Konno, S. Odake and J. Shiraishi, “Quasi Hopf twistors for elliptic quantum groups”, q-alg/9712029. [17] M. Jimbo and T. Miwa, “Algebraic analysis of solvable lattice models”, Regional Conference Series in Math. (1995) Number 85. [18] M. Jimbo, T. Miwa and A. Nakayashiki, “Difference equations for the correlation functions of the eight-vertex model”, J. Phys. A: Math. Gen. 206 (1993) 2199–2209. [19] T. Kohno, “Monodromy representations of braid groups and Yang–Baxter equations”, Ann. Inst. Fourier (Grenoble) 37 (4) (1987) 139–160. [20] V. G. Knizhnik and A. B. Zamolodchikov, “Current algebra and Wess-Zumino model in two dimensions”, Nucl. Phys. B247 (1984) 83–103. [21] S. Levendorskii, Y. Soibelman and V. Stukopin, “The Quantum Weyl Group and (1) the Universal Quantum R-Matrix for Affine Lie Algebra A1 ”, Lett. Math. Phys. 27 (1993) 253–264. [22] N. Yu. Reshetikhin, “Multiparameter quantum groups and twisted quasitriangular Hopf algebras”, Lett. Math. Phys. 20 (1990) 331–336.
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS ` MICHELE IRAC-ASTAUD and GUY RIDEAU Laboratoire de Physique Th´ eorique de la mati` ere condens´ ee Universit´ e Paris VII 2 place Jussieu F-75251 Paris Cedex 05 France Received 15 July 1997 Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators a, a† , N and the unity 1 such as [a, N ] = a, [a† , N ] = −a† , a† a = ψ(N ) and aa† = ψ(N + 1). We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the eigenstates of a (or a† ). We give various examples, in particular we consider functions ψ that are linear combinations of q N , q −N and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.
1. Introduction The harmonic oscillator Lie-algebra is defined by four operators: the annihilation operator a, the creation operator a† , the energy operator N and the unity 1 satisfying the following commutation relations: [a, N ] = a ,
[a† , N ] = −a†
(1)
and [a, a† ] = 1 ,
(2)
†
where a is the adjoint of a and N is self-adjoint. This algebra has been deformed in many different ways (see in particular [1–6]) and the representations of the deformed algebras were widely studied. In this paper, the deformed harmonic oscillator is defined by the relations (1) and by the following relations between the three operators a, a† and N : a† a = ψ(N ) ,
aa† = ψ(N + 1) ,
(3)
where ψ is a real analytical function. In the other formulations encountered in the literature, (3) is replaced by [a, a† ] = f (N, q) ,
(4)
[a, a† ]q ≡ aa† − qa† a = fq (N, q) .
(5)
or
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M. IRAC-ASTAUD and G. RIDEAU
In these approach, the function ψ is not given but results of solving the equations: f (N, q) = ψ(N + 1) − ψ(N ) or fq (N, q) = ψ(N + 1) − qψ(N ) .
(6)
The resolution of these equations leads to some arbitrariness that is eliminated in our formulation, f and fq being uniquely determined in terms of ψ. Let us give some examples: — the usual harmonic oscillator defined by f (N ) = 1 corresponds to ψusual (N ) = N + σ. — the q-oscillator [1, 2] defined by fq (N, q) = q −N corresponds to ψqosc (N ) = −q −N /(q − q −1 ) + σq N /(q − q −1 ) ,
∀σ.
(7)
— the q-oscillator defined by fq (N, q) = 1 corresponds to 0 (N ) = (1 − q)−1 + σq N , ψqosc
— with the usual notation: [x] =
∀σ.
q x − q −x q − q −1
(8)
(9)
the function ψsuq (2) (N ) = σ − [N − 1/2]2 , ∀ σ, corresponds to f (N ) = −[2N ]; that is to the deformation suq (2) of the Lie-algebra su(2) after the identification a = L− , a† = L+ and Lz = N . — suq (1, 1) is obtained for the ψsuq (1,1) (N ) = −ψsuq (2) (N ). Generalizing the pioneer work of Bargmann [7] for the usual harmonic oscillator, the purpose of this paper is to study if the deformed harmonic oscillator defined by (1) and (3) admits representations on one space of complex variable functions. In [8], we restricted to the case where the function ψ does not vanish. The scalar product of the representations we are looking for, is written with a true integral as in [7–12] and contrarily to the works of [13–19] where a q-integration occurs. In Sec. 2, we recall how to build the irreducible representations on the basis of the eigenvectors of N . They are determined by the spectrum of N which is depending on the zeros of ψ. Then, in Sec. 3, we discuss the existence of the coherent states that are defined as the eigenstates of the operators a (or a† ). In Sec. 4, we study the possibility of Bargmann representations. The formulation of the problem is done in a general framework. We show on various examples how the construction works: Sec. 5 is devoted to strictly positive function ψ , other cases are considered in Sec. 6. 2. Representations Let |0i be the eigenvector of N with eigenvalue µ. We built the normalized vectors |ni ( λn a†n |0i , n ∈ N + (10) |ni = λn a−n |0i , n ∈ N −
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
with
λ−2 n = ψ(µ + n)! =
n Y ψ(µ + i) ,
1063
n ∈ N+
i=1
n+1 Y ψ(µ + i) , n ∈ N −
(11)
i=0
N + and N − are the set of integers ≥ 0 and < 0. The vectors |ni are the eigenvectors of N with eigenvalue µ + n and span the Hilbert space H. As hn|aa† |ni is necessarily positive or zero, the construction of the increasing states stops if it exists an integer ν+ such that ψ(µ + ν+ + 1) = 0
(12)
in which case the representation labelled by µ and ν+ admits a highest weight state |ν+ i. We have an analogous situation for the decreasing states built with a, when it exists an integer ν− such as ψ(µ + ν− ) = 0. The representation labelled by µ and ν− then admits a lowest weight state |ν− i. We get different types of representations [3, 4, 5, 6]: 1) ψ has no zero. The inequivalent representations are labelled by the decimal part of µ and are defined by: † 1/2 a |ni = (ψ(µ + n + 1)) |n + 1i a|ni = (ψ(µ + n))1/2 |n − 1i , N |ni = (µ + n) |ni
n∈Z
(13)
The spectrum of N , SpN , is µ + Z. The operator N has no lowest and no highest eigenstates. These representations, thus, are non equivalent to Fock-representations and are called non-Fock-representations [20, 21]. It is the case when ψ is equal to ψqosc with q ∈ [0, 1] and σ ≤ 0. An interpretation of this case [12] is obtained by identifying the states |ni to the functions on a circle. 2) ψ has zeros. We are interested in the intervals where ψ is positive: a) finite intervals We can associate a representation to the intervals that have a length equal to an T integer. The spectrum of N is [µ + ν− , µ + ν+ ] Z + µ. For example, in the case ψsuq (2) , when σ = [l + 1/2]2 , l being a positive integer, the dimension of the representation is 2l + 1 and verifies: 1 a† |l, mi= l + 1 2 − m + 1 − 1 2 2 |l, m + 1i 2 2 1 (14) 1 2 1 2 2 − m − |l, m − 1i a|l, mi = l + 2 2 N |l, mi= m|l, mi
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b) infinite intervals The representations are similar to the Fock-representation of the usual harmonic oscillator. The spectrum of N is µ + ν− + N + or µ + ν+ + N − . Let us give an example: when ψ is equal to ψqosc with σ = 1, we recover the usual q-oscillator case such as a† |ni=[n + 1]1/2 |n + 1i a|ni =[n]1/2 |n − 1i , n ∈ Z+ (15) N |ni= n |ni The first step to build a Bargmann representation requires to study the coherent vectors. 3. Coherent States We call coherent states [22], the eigenvectors of the operator a or a† . The state P |zi = p cp |pi is an eigenvector of a if the coefficients cp verify the recursive relation zcp = ψ(µ + p + 1)1/2 cp+1 .
(16)
— When the spectrum of N is upper bounded, (16) implies that all the cp vanish and then that a has no eigenvectors. If we look for the eigenvectors of a† , the situation is analogous: a† has no eigenvectors if the spectrum of N is lower bounded. Therefore, in the case (2.a) of the previous section as the spectrum of N is finite, a and a† have no eigenvectors, hence no Bargmann representation exists. — When the spectrum of N is no upper bounded, the eigenvectors |zi of a take the form: when SpN = Z + µ , −∞ ∞ X X zn n 1/2 |zi = z (ψ(µ + n)!) |ni + |ni , ψ(µ + n)!1/2 n=−1 n=0 when SpN = ν− + µ + N + , (17) ν− ∞ X X zn n 1/2 z (ψ(µ + n)!) |ni + |ni , ν− < 0 |zi = ψ(µ + n)!1/2 n=−1 n=0 ∞ X z n (ψ(µ + n)!)−1/2 |ni , ν− ≥ 0 |zi = n=ν−
with the convention ψ(µ)! = 1. The domain D of existence of the coherent states depends on the function ψ. Indeed, |zi belongs to the Hilbert space spanned by the basis |ni only if the series in the right-hand side of (17) are convergent in norm. — When SpN = Z + µ, this implies that: |z| < lim ψ(p)1/2 = r2 , p→∞
(18)
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
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and |z| > lim ψ(p)1/2 = r1 . p→−∞
(19)
Thus when r2 = 0, the annihilation operator has no eigenvectors. When r1 is smaller than r2 , the eigenstates of a exist and their domain is r1 < |z| < r2 . When r1 is larger than r2 , the annihilation operator a has no eigenstates, but then we can establish by analogous reasoning that the creation operator a† has eigenstates if r1 6= 0. — When the spectrum of N is lower bounded, SpN = µ + ν− + N + , the second condition (19) does not exist and the eigenstates of a always exist provided r2 6= 0; their domain is defined by |z| < r2 . When the spectrum of N is upper bounded, SpN = µ + ν+ + N − , the eigenvectors of a† exist only if |z| < r1 . To summarize, the eigenvectors of a exist if: — SpN = µ + Z and r22 ≡ ψ(+∞) > r12 ≡ ψ(−∞), the domain of existence D of the coherent states is D = {z; r1 < |z| < r2 } or — SpN = µ + ν− + N + , then D = {z; |z| < r2 }. The eigenvectors of a† exist if: — SpN = µ + Z and r2 < r1 , then D = {z; r2 < |z| < r1 } or — SpN = µ + ν+ + N − , then D = {z; |z| < r1 }. The part played by a and a† is analogous, in the following we restrict to the case where the eigenstates of a exist, that is: — ψ is a strictly positive function with r1 < r2 — ∃x0 such that ψ(x0 ) = 0 and ψ(x) > 0 when x > x0 . We do not study here the case where r1 = r2 . Although µ is a significant quantity of labelling inequivalent representations, it does not play a part in the present problem. So we simplify the notation in assuming µ = 0 from now on. Indeed, this is equivalent to substituting N − µ to N and ψµ (N ) = ψ(µ + N ) to ψ(N ). 4. Bargmann Representation 4.1. Representation space Following the construction [7], in the Bargmann representation any state |f i of H: |f i =
X n∈SpN
fn |ni ,
X n∈SpN
|fn |2 < ∞
(20)
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M. IRAC-ASTAUD and G. RIDEAU
is represented as the function of a complex variable z, f (z) = hz|f i, with a Laurent expansion: X z n fn X + z n fn (ψ(n)!)1/2 , SpN = Z , 1/2 ψ(n)! n<0 n≥0 ν− X znf X n + z n fn (ψ(n)!)1/2 , SpN = ν− + N + , ν− < 0 , (21) f (z) = 1/2 ψ(n)! n≥0 n<0 X z n fn (ψ(n)!)−1/2 , SpN = ν− + N + , ν− ≥ 0 , n≥ν− on the domain D of definition of the eigenvectors of a, r1 < |z| < r2 . The space of the representation S is constituted with holomorphic functions in D, strongly depending on ψ (cf. Subsec. 4.3). In particular, to the basis vectors |ni, n ∈ SpN , correspond the monomials: ( hz|ni =
z n (ψ(n)!)−1/2 , n ≥ 0 z n (ψ(n)!)1/2 ,
n<0
(22)
4.2. Resolution of unity A Bargmann representation exists if we can exhibit a positive real function F (x) such that Z (23) F (zz)|zihz|dzdz = 1 , where the integration is extended to the whole complex plane and where F (|z|2 ) contains the characteristic function of the domain D of existence of the coherent states. Denoting z = ρeiθ , Eq. (23) reads Z 0
2π
dθ 2
Z
F (ρ2 )dρ2 |ρe−iθ ihρeiθ | = 1 .
(24)
D
From the resolution of unity (23), we obtain the scalar product in S: Z (g|f ) =
F (zz)f (z)g(z)dzdz
(25)
and the representation of any linear operator K of H by the kernel K(ζ, z) = hζ|K|zi such that Z (26) (Kf )(ζ) = F (zz)K(ζ, z)f (z)dzdz .
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4.3. Reproducing kernel Let G(x) be the function: X X xn (ψ(n)!)−1 + xn ψ(n)! SpN = Z , n<0 n≥0 ν− X X n −1 x (ψ(n)!) + xn ψ(n)! , SpN = ν− + N + , ν− < 0 G(x) = n<0 n≥0 X xn (ψ(n)!)−1 , SpN = ν− + N + , ν− ≥ 0 n≥ν
(27)
−
We easily verify that G(ζz) = hz|ζi, so that G(ζz) is the function of S representing the coherent state |ζi. The function G(x) plays a prominent part for, if (23) should be true, we will have Z F (zz)hζ|zihz|f idzdz = hζ|f i
(28)
and G(ζz) appears to be one reproducing kernel. Moreover, we get from the Schwarz inequality: |f (z)| ≤ |f |1/2 |G(zz)|1/2 .
(29)
Thus, S is the set of holomorphic functions the growth of which on the edge of D, is controlled by the growth of |G(x)|1/2 . We easily prove that, in the Bargmann representation, a† is the multiplication by z and N is the operator zd/dz, as in the usual case, while a is the operator z −1 ψ(zd/dz). As G(ζz) corresponds to the coherent state |ζi, we obtain for G the following equation: d G(x) , (30) xG(x) = ψ x dx which could be obtained directly from the expansion (27). 4.4. Weight function Let us introduce the Mellin transform Fˆ (ρ) of the weight function F (x): Z Fˆ (ρ) =
∞
Z F (x)xρ−1 dx =
0
r22
F (x)xρ−1 dx .
(31)
r12
From (22) and (23), we deduce that F (x) must verify the following condition: ( ψ(n)! , n≥0 Fˆ (n + 1) = (32) −1 (ψ(n)!) , n < 0 , n ∈ SpN Let us remark that Fˆ (ρ) ≤ Fˆ (n) + Fˆ (n + 1) ,
n ≤ Re ρ < n + 1
(33)
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M. IRAC-ASTAUD and G. RIDEAU
as F (x) is a positive function. As the Mellin transform of F exists for all the integers belonging to the spectrum of N , then, due to (33), it exists for any ρ such as Re ρ is greater than the lowest bound ν− of SpN , ν− can be finite or −∞. Formula (32) is equivalent to Fˆ (n + 1) = ψ(n)Fˆ (n) ,
with Fˆ (1) = 1
(34)
which ensures that the operators a† = z, a = z −1 ψ(zd/dz) be adjoint on the basis |ni. The function ψ being given, Fˆ verifying (34) must be interpolated in order to get F as the Mellin inverse of (31). Equation (34) can be obviously interpolated by Fˆ (ρ + 1) = ψ(ρ)Fˆ (ρ) .
(35)
Any other interpolation of Fˆ (n) is obtained by ˆ , Fˆh (ρ) = Fˆ (ρ)h(ρ)
(36)
where Fˆ (ρ) is some solution of (35) and where ˆh(ρ) is equal to 1 on the spectrum of N . In order to tackle the discussion on the existence of a Bargmann representation, we first state the following lemmas: Lemma 1. If there exists one particular solution of (35) which does not admit a Mellin inverse, the same holds for every solution of (35). ˆ Proof. Two solutions of (35) differ by a multiplicative factor k(ρ) which is a periodic function of period 1: ˆ + 1) = k(ρ) ˆ . k(ρ
(37)
Let Fˆ (ρ) be one particular solution of (35), without Mellin inverse. We cannot ˆ ˆ Fˆ (ρ) has a Mellin inverse. Indeed, the behavior of k(ρ) ˆ find k(ρ) such as k(ρ) at infinities on a parallel to the imaginary axis must compensate the corresponding ˆ bad behavior of Fˆ (ρ). This implies that k(ρ) has a Mellin inverse k(x). Then from Eq. (37), we obtain that k(x) must verify xk(x) = k(x) ,
(38)
ˆ and then k(x) is equal to kδ(x − 1), so that k(ρ) is a constant in contradiction to ˆ ˆ the assumption that k(ρ)F (ρ) have a Mellin inverse. ˆ ˆ Lemma 2. If h(ρ) is equal to 1 on SpN and admits a Mellin inverse h(x), h(ρ) is the identity. Proof. Let us assume that SpN contains N + . Then we have Z ∞ ˆ + 1) = h(n h(x)xn dx = 1 . 0
(39)
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
From this equation, we easily deduce Z ∞ h(x)eρx dx = eρ ,
1069
(40)
0
ˆ so that h(x) = δ(x − 1) and h(ρ) = 1. The same method works when SpN = ν− + N + , ν− > 0. Discussion 1) If there exists a solution of (35) with a positive Mellin inverse, our problem is solved. 2) If we exhibit a solution of (35) without Mellin inverse, we know from Lemma 1 that no solution of (35) with Mellin inverse exists and from Lemma 2 that these solutions cannot be corrected by a suitable multiplicative factor. 3) If there exists a solution of (35) with a Mellin inverse not strictly positive on D, we cannot conclude, for we cannot ensure this solution cannot be improved by a suitable multiplicative factor leading to positivity. Therefore, a Bargmann representation can be obtained only if there exist solution of (35) with Mellin inverse, so that the interpolation problem (34) is completely solved by (35) precisely. Concerning the remaining positivity condition, let us point out that it will be satisfied if ψ(ρ) is such that ψ(ρ + iσ) is a definite positive function of σ for any ρ ∈ SpN . This can be proved by an adaptation to the Mellin transform of the well-known Bochner theorem for Fourier transform. But this condition on ψ(ρ) is only a sufficient condition as it is readily seen from the counter-example of the usual oscillator: ψ(ρ) = ρ. Therefore, the only interpolation of (34) to be considered in the following is precisely the simplest one, namely (35). Remains one case where we cannot conclude, namely when the Mellin inverse of the solution of (35) exists but is not positive on D, we cannot ensure that the result can be improved by a suitable factor leading to the positivity. To summarize, a Bargmann representation can be defined on a deformed harmonic oscillator algebra if the coherent states exist and if it exists at least one solution of (35) with a positive Mellin inverse on D. Furthermore, let us remark that (35) can be written: Z Z F (x)xρ dx = F (x)ψ(x∂x + 1)xρ−1 dx .
(41)
R The right member is equal to (ψ(−x∂x )F (x))xρ−1 dx if the integration by parts (or the change of variables) can be done without extra terms and this gives d F (x) . (42) xF (x) = ψ −x dx This equation, when it holds, can be used to study the positivity of F (x), when we cannot obtain an explicit expression for this function.
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M. IRAC-ASTAUD and G. RIDEAU
In the next section [8], we restrict to strictly positive ψ and to F solution of (42), fast decreasing at infinity and at the origin. In Sec. 6, we illustrate the construction of Bargmann representation with specific vanishing functions ψ, in particular we discuss the cases of the q-oscillators, looking for a resolution of the identity involving a true integral. 5. ψ Strictly Positive In this section, the eigenvalues of N are all the integers. The coherent states lie in the whole complex plane, except the origin, in the two first subsections and in a ring in the last one. 5.1. The q-oscillator The function ψ associated to the q-oscillator (7) is strictly positive if q < 1 and σ ≤ 0. We easily see that when σ 6= 0, the coherent states do not exist. Let ψλ,q (N ) be equal to λq −N , q ≤ 1, λ > 0. When λ = (q −1 − q), we get the function ψqosc for σ = 0 previously introduced and corresponding to the q-oscillator algebra (7) with non-Fock representation and with coherent states. Let us remark that this function also corresponds to aa† = q −1 a† a. In this case, the functional equations (35): Fˆλ,q (ρ + 1) = λq −ρ Fˆλ,q (ρ)
(43)
xFλ,q (x) = λFλ,q (qx)
(44)
and are equivalent for Fλ,q (x) has to be fast decreasing at infinity and at the origin. Introducing ln Fˆλ,q (ρ), we solve the resulting equation in terms of first and second Bernouilli polynomials so that we get the particular solution: 1 2 0 ˆ Fλ,q (ρ) = exp ρ ln λ − (ρ − ρ) ln q . (45) 2 Taking the inverse Mellin transform, we obtain the following particular solution of (44) [12, 8]: 2 ln (xλ−1 ) ln(xλ−1 ) 0 − . (46) Fλ,q (x) = exp 2 ln(q) 2 The general solution of (44) is given by 0 Fλ,q (x) = Fλ,q (x)hλ,q (x) ,
(47)
where hλ,q (x) is a function satisfying hλ,q (x) = hλ,q (qx)
(48)
that is a periodic function of ln x/ ln q of period 1. We verify directly that up to 0 (x) are well equal to λ−n q −n(n+1)/2 for a constant the momentums M (n) of Fλ,q
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1071
n ∈ Z, as wanted, and this is yet true for Fλ,q (x) given in (47) as implied by the definition of hλ,q (x). As the absolute value of a solution of Eq. (48) is also a solution, the function hλ,q (x) can always be taken strictly positive. Nevertheless, there exist an infinite number of hλ,q (x), in particular all the powers of any solution, so that we get a lot of candidates to define the norm we are looking for. But mutatis mutandis, the situation is analogous to that initially studied by Bargmann and we can extend the largest part of its proof. Particularly we can prove that the set S of holomorphic functions under consideration is the set of functions f (z) such that ln2 (xλ−1 ) ln(xλ−1 ) − |f (z)| ≤ C exp − 4 ln q 4 2 ! +∞ X 1 ln q ln(xλ−1 ) +n+ . (49) exp × 2 ln q 2 −∞ The representation Hilbert space is the Hilbert space defined on S by Z (f, g)F = Fλ,q (zz)g(z)f (z)dzdz ,
(50)
where Fλ,q (zz) is a positive function of the form given in Eq. (47). The various norms obtained are proportional as verified on the basis elements. Furthermore, following Bargmann’s procedure, we can prove the closedness of d . Moreover we have the operators z and z −1 q z dz |zf (z)| = 2
∞ X
|fn |2 ψ(n + 1) ,
(51)
−∞
and
2 X ∞ −1 z ψ z d f (z) = |fn |2 ψ(n) . dz
(52)
−∞
Using the two previous equations, we obtain |zf (z)|2 = q|z −1 q z dz f (z)|2 . d
(53)
†
This proves that the operators a and a have the same domain of definition and are mutually adjoint [7]. We have proved the existence of Bargmann representations for the deformed oscillator defined by (1) and (3) when ψq−1 −q,q (N ) = (q −1 − q)q −N . The resolution of identity is not unique and is given explicitly by (23), (46) and (47). 5.2. Generalization of the previous example In this section, we point out some directions to extend the results of the previous section when ψ(x) is of the form: ! 2p+1 X n , a2p+1 > 0 . an x (54) ψ(x) = exp 0
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M. IRAC-ASTAUD and G. RIDEAU
A solution of Eq. (35) is Fˆ (ρ) = exp
2p+1 X 0
! an Bn+1 (ρ) , n+1
(55)
where Bn+1 (ρ) are the Bernouilli polynomials [26] defined by the difference equation: Bn (x + 1) − Bn (x) = nxn−1 .
(56)
The term of highest degree in (55) is a2p+1 ρ2p+2 /(2p + 1). When ρ is pure imaginary ρ = iσ, σ ∈ R, this term is a2p+1 (−1)p+1 σ 2p+2 /(2p + 1), therefore the inverse Mellin transform of Fˆ (ρ) exists only if p is an even number. The function F (x) thus obtained is always real, but not necessarily positive. Nevertheless, in specific cases, for example when the exponent in (55) contains only the term of highest degree, F (x) is actually strictly positive and the Bargmann procedure works as before. Since ψ(n + 1)/ψ(n) grows indefinitely as n → ±∞, Eqs. (51) and (52) imply d ) is included in the domain of z but cannot be identical. that the domain of z −1 ψ(z dz Nevertheless, the mutual adjointness can be proved on the basis, using Eq. (34). It is worthwhile to underline that this generalization gives an example where the Bargmann representation does not exist, namely when p is odd. 5.3. The ring case Let us consider the simple case ψ(x) = a + q x , q > 1, a > 0, that is mainly involved in the study of the q-oscillator (8) with non-Fock representations. The domain of existence of the coherent states is a ≤ |z|2 . The momentum Fˆ (n) reads Z +∞ ˆ F (n) = F (x)xn−1 dx . (57) a
We first prove that Eqs. (34) and (42) are not equivalent if F (x) is positive on the whole positive axis. Indeed, let us start with a solution of (42), Eq. (34) reads Z a F (x)xn−1 dx = 0 , (58) q−1 a
that is obviously impossible if F (x) is positive on [q −1 a, a]. Therefore in this case, the momentums deduced from the weight function solution of (42) are not the expected ones (solutions of (34)). Moreover, in [8], we proved that the solution of (42) is identically zero. Let us look for a solution of (35) that cannot have poles, due to (33): Fˆ (ρ + 1) = (q ρ + a)Fˆ (ρ) . We have as a convenient particular solution the following entire function: Y Fˆ (ρ) = aρ (1 + a−1 q ρ−p−1 ) , p≥0
(59)
(60)
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1073
but it is not a Mellin transform of a true function F (x). Indeed if it has an inverse Mellin transform, it can be calculated on any parallel to the imaginary, for instance Q on Re ρ = ln a/ ln q. On this axis, |Fˆ (iy)| ≥ p≥0 (1 − q −p−1 ), so that (60) is not the Mellin transform of a true function. Nevertheless, we can write (60) in the form [24]: −n nρ X a q . Fˆ (ρ) = aρ 1 + (61) (q − 1) · · · (q n − 1) n≥1
The series is absolutely convergent as q > 1. It is easily verified that this expression can be seen as the Mellin transform of the following measure: F (x) =
X n≥0
a−n δ(ln a + ln q − ln x) . (q − 1) · · · (q n − 1)
(62)
Therefore, in this case we obtain a Bargmann representation if we accept the weight function to be a true measure. The same is true when we consider ψ(x) = 1/(q x + a), q < 1. Equation (35) reads 1 ˆ Fˆ (ρ + 1) = ρ F (ρ) . (63) q +a The domain of existence of the coherent states is the disc of radius 1/a. We then obtain F (x) =
X n≥0
(q −1
q −n a−n δ(− ln a + n ln q − ln x) . − 1) · · · (q −n − 1)
(64)
In this subsection, we gave examples where the Bargmann representations exist only if we admit that the scalar product be expressed by means of true measures. 6. ψ Vanishes In this section, we consider two cases where the spectrum of N is the set N + of the positive integers and where the coherent states are defined in the whole complex plane. 6.1. q-oscillators The first example corresponds to (7) with σ = 1 and the second one to (8) with σ = (q − 1)−1 . a) ψ(x) = [x] ≡ (q x − q −x )/(q − q −1 ) A resolution of the identity was shown to be obtained with a q-integration [13]. The Rx x∂x −q−x∂x q-integration 0 dq x is the inverse operator of the q-derivative Dq = x1 q q−q −1 that vanishes at the origin: Z x X q − q −1 −1 dq x = x∂x x = (q − q) q (2n+1)x∂x x , when q < 1 . (65) q − q −x∂x 0 n≥0
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M. IRAC-ASTAUD and G. RIDEAU
The q-exponential is defined by Expq (qx) − Expq (q −1 x) x(q − q −1 )
Expq (x) = Dq Expq (x) =
(66)
with the condition that it is equal to one when x is zero. This function reads Expq (x) =
X xn , [n]!
(67)
n≥0
and vanishes on the negative axis [13]. Denoting by −ζ the first zero at the left of the origin the resolution of identity then reads Z
dθ 2
Z
ζ2
dq ρ2 Expq (−ρ2 )|ρe−iθ ihρeiθ |2ρdρ = 1 .
(68)
0
Here we look for a Bargmann representation where the scalar product involves a true integral. First, it is easy to verify that in this case as in the following, if F verifies (42), its Mellin transform is solution of (35) and the moments are the expected ones. In both cases, we choose to define the weight function, not through its Mellin transform but directly as solution of (42). Equation (42) for this particular case reads q −x∂x − q x∂x F (x) . q − q −1
xF (x) =
(69)
The obvious solution of this equation: F (x) = Expq (−x)
(70)
is not positive for all positive values of x and the Bargmann representation as defined in Sec. 4 does not exist. Following the trick used to get (68), we can try to limit the integration to the domain where Expq (x) is positive. Let us see if Z Fˆ (n) =
ζ
Expq (−x)xn−1 dx
(71)
0
could work. Equation (34) gives Z
ζ
Expq (−q −1 x)xn−1 dx −
qζ
Z
ζ
q−1 ζ
Expq (−qx)xn−1 dx = 0 .
(72)
The problem is symmetric under the change q into q −1 . Let us choose q > 1, (72) takes the form: Z ζ (Expq (−x)q n + Expq (−qx))xn−1 dx = 0 . (73) q−1 ζ
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1075
The integrand of (73) reads (Expq (−x)(q n − x(q − q −1 )) + Expq (−q −1 x))xn−1 and is positive for n enough large; this leads to Expq (−x)(q n − x(q − q −1 ) + Expq (−q −1 x) = 0, that is impossible. Therefore in order to obtain a Bargmann representation, we must look for another solution of (42) that will be positive. As already noticed, the problem being symetric under the change q into q −1 , we assume q > 1. Let us start with (35) that reads qρ q ρ − q −ρ ˆ (1 − q −2ρ )Fˆ (ρ) . (74) F (ρ) = Fˆ (ρ + 1) = q − q −1 q − q −1 Let us write Fˆ on the form: ρ Fˆ (ρ) = φq 2 (ρ−1) (q − q −1 )−ρ fˆ(ρ) .
(75)
The function fˆ(ρ) must verify fˆ(ρ + 1) = (1 − q −2ρ )fˆ(ρ) , and is given by fˆ(ρ) =
X n≥0
(76)
q −2nρ . (1 − q −2 ) · · · (1 − q −2n )
(77)
The condition Fˆ (1) = 1, furnishes the normalization factor: φ = (q − q
−1
X
q −2n ) 1+ −2 (1 − q ) · · · (1 − q −2n ) n>0
!−1 .
(78)
Putting (77) and (78) in (75), we obtain Fˆ (ρ), and then we can calculate its inverse Mellin transform: 2 ! 1 1 ln x + ln(q − q −1 ) + ln q exp − 2 ln q 2 F (x) = X q −2n 1+ −2 (1 − q ) · · · (1 − q −2n ) n>0 ×
X q −n(2n+1) ((q − q −1 )x)−2n q − q −1 ×√ . −2 −2n (1 − q ) · · · (1 − q ) 2π ln q n≥0
(79)
This function being positive, we have obtained a Bargmann representation where the scalar product is written with a true integral. Let us stress that F (−x) is solution of (66) and is thus a possible candidate to write the resolution of identity with a q-integration and a positive function on the whole positive axis. The same is true in the next example where two resolutions of the identity coexist. b) ψ(x) = (x) ≡ (q x − 1)/(q − 1), with q > 1 First we show that the resolution of the identity can be obtained with a q-integral as in [13].
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M. IRAC-ASTAUD and G. RIDEAU
The q-integration [23–25] is defined to be the inverse of the q-derivative Dq ≡ 1 qx∂x −1 x q−1 : Z x X q−1 x = (q − 1) dq x ≡ x∂x q −(n+1)x∂x x . (80) q − 1 0 n≥0
The q-exponential, solution of the equation: Expq (x) = Dq Expq (x) = is given by Expq (x) =
Y
Expq (qx) − Expq (x) x(q − 1)
(81)
X xn , (n)!
(82)
(1 + x(1 − q −1 )q −p ) =
p≥0
n≥0
and vanishes for x = −q p (1 − q −1 )−1 . The nearest zero on the left of the origin is of the identity takes the same form −ζ = −(1 − q −1 )−1 . Therefore the resolution Rx as in (68) with the new expressions for 0 dq x, Expq and ζ. Let us now look for a Bargmann representation as defined in Sec. 4. We see that the Eq. (42) can be written: F (q −1 x) = (x(q − 1) + 1)F (x) .
(83)
We easily prove that the weight function is given by F (x) =
1 . Expq (qx)
(84)
It is a positive function when x > 0 and its Mellin transform fulfills qρ − 1 ˆ F (ρ) . Fˆ (ρ + 1) = q−1
(85)
This ensures that the momentum Fˆ (n) are the expected one (32). Thus, in this case, coexist two resolutions of the identity ,one involving a true integral and a weight function F (x) = (Expq (qx))−1 and one with a q-integral, the weight function being Expq (−x). 6.2. ψ(x) = xn , n > 0 The Mellin transform of the weight function is solution of the equation deduced from (35): (86) Fˆ (ρ + 1) = ρn Fˆ (ρ) , R ∞ −t z−1 and can be expressed with the gamma-function Γ(z) = 0 e t dt: Fˆ (ρ) = (Γ(ρ))n . When n is an integer, the inverse Mellin transform gives F (x): Z ∞ Z ∞ F (x) = ··· e−(t1 +···+tn ) dt1 · · · dtn δ(x − t1 × · · · × tn ) . 0
0
(87)
(88)
BARGMANN REPRESENTATIONS FOR DEFORMED HARMONIC OSCILLATORS
1077
On this expression, we see that F (x) is a positive function so that the Bargmann representation exists. In the case n = 1, we recover the usual harmonic oscillator where F (x) = e−x . 7. Conclusion We have studied the possibility of Bargmann representations for any deformed oscillator algebra characterized by a function ψ. We gave the conditions to be verified by this function for admitting representations with coherent states. We get the unique functional equation to be satisfied by the Mellin transform of the weight function defining the scalar product. We were able to get definite and positive answer in many cases including in particular some types of q-oscillators. Although we did not succeed in obtaining a general characterization of the function ψ leading to Bargmann representations, we underline two points: — We exhibited cases where the Bargmann representations do not exist even when coherent states do (Subsec. 5.2); — The analysis of Subsec. 5.3 showed that the scope of our study have to be extended up to include true measures for writing the scalar product. Finally let us remark that we have obtained scalar products for the Bargmann representations of the usual q-oscillators, involving true integrals instead of q-integrations as previously proposed in literature. References [1] L. C. Biedenharn, J. Phys. A: Math. Gene. 22 (1989) L873. [2] A. J. Mac Farlane, J. Phys. A: Math. Gene. 22 (1989) 4581. [3] M. Irac-Astaud and G. Rideau, “Deformed quantum harmonic oscillator”, Proc. Third Int. Wigner Symposium, Oxford, 1993, to appear. [4] M. Irac-Astaud and G. Rideau, “On the existence of quantum bihamiltonian systems: the harmonic oscillator case”, preprint PAR-LPTM 92, Lett. Math. Phys. 29 (1993) 197, Theor. Math. Phys. 99 (1994) 658. [5] C. Quesne and N. Vansteenkiste, “Representation theory of deformed oscillator”, Helv. Phys. Acta 69 (1996) 141, and many references therein. [6] P. Kosinski, M. Mazewski and P. Maslanka, “Representations of generalized oscillator algebra”, Czech. J. Phys. 47 (1997) 41. Fifth Colloquium on quantum groups and integrable systems, Prague, 20–22 June, 1996. [7] V. Bargmann, “On a Hilbert space of analytic functions and an associated integral transform”, Commun. Pure and Appl. Math. 14 (1961) 187. [8] M. Irac-Astaud and G. Rideau, “Bargmann representation for some deformed harmonic oscillators with non-Fock representation”, Proc. Symposium in honor of Jiri Patera and Pavel Winternitz for their 60th birthday, Algebraic Methods and Theoretical Physics, January 9–11, 1997, Centre de recherches math´ e matiques, Universit´e de Montr´eal. [9] A. D. Janussis, P. Filippakis and J. C. Papaloucas, “Commutation relations and coherent states”, Lettere al Nuovo Cimento 29(15) (1980) 481. [10] J. A. de Azcarraga and D. Ellinas, “Complex analytic realizations for quantum algebras”, J. Math. Phys. 35(3) (1994) 1322. [11] V. Spiridonov, “Coherent states of the q-Weyl algebra”, Lett. Math. Phys. 35 (1995) 179.
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[12] K. Kowalski, J. Rembielinski and L. C. Papaloucas, “Coherent states for a particle on a circle”, J. Phys. A: Math. Gene. 29 (1996) 4149. [13] R. W. Gray and C. A. Nelson, “A completeness relation for the q-analogue coherent states by q-integration”, J. Phys. A: Math. Gen. 23 (1990) L945. [14] M. Chaichian, D. Ellinas and P. Kulish, “Quantum algebra as the dynamical symmetry of deformed Jaynes–Cummings model”, Phys. Rev. Lett. 65 (1990) 980. [15] A. J. Bracken, D. S. McAnally, R. B. Zhang and M. D. Gould, “A q-analogue of Bargmann space and its scalar product”, J. Phys. A: Math. Gen. 24 (1991) 1379. [16] B. Jurco, “On coherent states for the simplest quantum groups”, Lett. Math. Phys. 21 (1991) 51. [17] C. Quesne, “Coherent states, K-matrix theory and q-boson realizations of the quantum algebra suq (2)”, Phys. Lett. A153 (1991) 303. [18] A. Odzijewicz, “Quantum algebras and q-special functions related to coherent states maps of the disc”, preprint IFT 18/95. [19] A. M. Perelomov, Helv. Phys. Acta 68 (1996) 554; A. M. Perelomov,“On the completeness of some subsystems of q-deformed coherent states”, preprint FTUV 96-38, IFIC 96-46. [20] P. P. Kulish, “Contraction of quantum algebra and q-oscillators”, Theor. Math. Phys. 86 (1991) 108. [21] G. Rideau, “On the representations of quantum oscillator algebra” Lett. Math. Phys. 24 (1992) 147. [22] J. R. Klauder and B. S. Skagerstam, Coherent States, World Scientific, 1985. [23] E. H. Jackson, “On q-definite integrals”, Q. J. Pure Appl. Math. 41 (1910) 193. [24] H. Exton, q-Hypergeometric Functions and Applications, Ellis Horwood Series, New York, 1983. [25] D. S. McAnally, “q-exponential and q-gamma functions”, J. Math. Phys. 36(1) (1995) 546. [26] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, Academic Press, 1965.
SOME PROPERTIES OF MASSLESS PARTICLES IN ARBITRARY DIMENSIONS MOURAD LAOUES Laboratoire Gevrey de Math´ ematique Physique Universit´ e de Bourgogne 9, avenue Alain Savary B.P. 400, F-21011 Dijon Cedex, France E-mail : [email protected] [email protected] Received 26 January 1998 Various properties of two kinds of massless representations of the n-conformal (or (n+1)˜n = g SO0 (2, n) are investigated for n ≥ 2. It is found that, for spaceDe Sitter) group G time dimensions n ≥ 3, the situation is quite similar to the one of the n = 4 case for SO0 (2, n − 1). These representations Sn -massless representations of the n-De Sitter group g ˜ n . The main difference is that they are not are the restrictions of the singletons of G contained in the tensor product of two UIRs with the same sign of energy when n > 4, whereas it is the case for another kind of massless representations. Finally some examples of Gupta–Bleuler triplets are given for arbitrary spin and n ≥ 3.
1. Introduction The (ladder) representations D(s + 1, s, s), 2s ∈ N and || = 1 of the universal g0 (2, 4) of the conformal group remain irreducible when restricted covering C˜4 = SO g0 (1, 3) n T4 of the Poincar´e group and each nonto the universal covering P˜4 = SO trivial positive energy representation of the conformal group with that property is equivalent to one of them. However the restriction to the universal covering S˜4 of the De Sitter group is irreducible only if s > 0; indeed one has D(s + 1, s) if s > 0; D(s + 1, s, s)| ˜ = S4 D(1, 0) ⊕ D(2, 0) if s = 0. These representations are called massless (relatively to the De Sitter group) for a variety of reasons [2]. In the present paper we call them S4 -massless representations g0 (2, 3) because, as indicated in [2, 11, 13] they satisfy of the De Sitter group S˜4 = SO the following masslessness conditions: (a) They contract smoothly to a massles discrete helicity representation of the g0 (1, 3) n R4 ; Poincar´e group P˜4 = SO (b) Any massless discrete helicity representation U P of the Poincar´e group has ˆ (called C4 -massless representation in this a unique extension to a UIR U ˆ to the SO0 (2, 4). The restriction of U paper) of the conformal group C˜4 = g De Sitter group is precisely one of the massless representations of S˜4 recalled above;
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M. LAOUES
(c) For spin s ≥ 1 one may construct a gauge theory on the Anti-de Sitter space for massles particles, quantizable only by the use of an indefinite metric and a Gupta Bleuler triplet; (d) The massless representations in question distinguish themselves by the fact that the physical signals propagate on the Anti-de Sitter light cone. Other interesting representations of S˜4 are the Dirac singletons Di = D(1, 12 ) and Rac = D( 12 , 0) (which are also C3 -massless representations in the sense defined below). Some of their properties are: 1. Dirac singletons are, up to equivalence, the only unitary irreducible positive energy representations of S˜4 which remain irreducible when restricted to the ˜ 4 of the Lorentz group; universal covering L ˜ 4 ) they contract 2. In the limit of zero curvature (of the De Sitter space S˜4 /L ˜ to unitary irreducible representations (UIR) of P4 that are trivial on the translation part T4 ; 3. Let χ(µ1 ) ⊗ π(µ2 ) denote the IR (up to equivalence), with highest weight ˜ 4 of the maximal compact subgroup of (µ1 , µ2 ) of the universal covering K ˜ 4 of the Dirac singletons UIRs the De Sitter group. Then the restriction to K is given by M 1 1 1 + s, s | ˜ = +s+l ⊗ π(s + l), s = 0 or . D χ − K 2 2 2 4 l∈N
4. Finally the Dirac singletons satisfy the following [10]: Rac ⊗ Rac =
M
D(s + 1, s);
s∈N
Rac ⊗ Di =
M
D(s + 1, s);
s− 12 ∈N
Di ⊗ Di =
M
D(s + 1, s) ⊕ D(2, 0).
s−1∈N
Note [2] that the Dirac singletons are not massless representations of the De Sitter group. But if one considers S4 as the conformal group of the 3-dimensional Minkowski space then the Dirac singletons are massless, i.e. their restriction to the corresponding Poincar´e group P3 is irreducible [2, 3, 13]. In this case it is clear from the context what kind of masslessness is considered. However, for general n, some confusion may arise. To avoid it we shall introduce a prefix to the word “massless” (see Definition (1)), to distinguish between “conformal masslessness” and “De Sitter masslessness” in any dimension, to precise which group we are representing. A common property to both types of massless representations is the existence of Gupta–Bleuler (GB) quantization; see for example [2, 6, 14, 15]. The purpose of this work is to continue the study performed in [3] and more specifically to look for properties of maslessness (both types) which persist when
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
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the space-time dimension becomes an arbitrary integer n ≥ 2. In Sec. 2 we fix the notations and recall some results. In Sec. 3 we discuss the irreducibility of a massless representation of the n-conformal group when restricted to the (n + 1)Lorentz group and its contractibility to UIRs of the n-Poincar´e group. Reduction to the maximal compact subgroup of the conformal group is studied in Sec. 4. Finally Dirac singletons and Gupta–Bleuler triplets are treated in (respectively) Secs. 5 and 6. It is found that almost all the properties of massless representations in dimension n = 4 are conserved when n ≥ 3; however the property that massless representations are, when n = 4, contained in the tensor product of two positive energy UIRs (of the De Sitter group) fails for general n. After a first version of this paper was written appeared a preprint [9] with somewhat different conclusions, based on a less-demanding notion of masslessness in higher dimensions. Since we need the definitions and results of this paper to compare both notions, we shall discuss this point at the end of the paper. 2. Generalities We suppose n ≥ 2. Let R1,n−1 be the n-dimensional Minkowski space-time, Tn its group of translations, Ln = SO0 (1, n − 1) the n-Lorentz group, Pn = Ln n Tn the n-Poincar´e group and Sn = SO0 (2, n − 1) the n-De Sitter group. We write Tn , Ln , Pn and Sn the corresponding Lie algebras. Let Gn = SO0 (2,n). The preceding groups may be considered as subgroups of Gn . Indeed let Mab −1≤a
(1)
[Mab , Mcd] = ηad Mbc + ηbc Mad − ηac Mbd − ηbd Mac ,
(2)
and
where η=
12 −1n
.
We now imbed the above-mentioned Lie algebras in Gn in the following way Tn = hM−1,α + Mα,n ,
0 ≤ α ≤ n − 1i
Ln = hMαβ ,
0 ≤ α, β ≤ n − 1i
Sn = hMab ,
−1 ≤ a, b ≤ n − 1i.
Let T n = hM−1,α − Mα,n ,
0 ≤ α ≤ n − 1i
Dn = hM−1,n i and T n (resp. Dn ) the connected subgroup of Gn , the Lie algebra of which is T n (resp. Dn ). Then we define the n-conformal group of R1,n−1 as the closed subgroup
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Cn of Gn generated by Tn , Ln , Dn and T n . Cn , Gn and Sn+1 are locally isomorphic; one has, if Cn denotes the Lie algebra of Cn : Cn = Gn = Sn+1 . Note that with our definition by “conformal group of R1,1 ” we mean here the group SO0 (2, 2)/Z2 . ˜ be the spinorial covering of G when G is Let G a Lie group. We denote by G isomorphic to L3 or P3 and the universal covering otherwise. Let U be a non trivial highest weight unitary representation of C˜n . Definition 1. We say that U is Cn -masslessa whenever U and U| ˜ are irrePn ducible. We say that U is Sn+1 -masslessb whenever U is a restriction to S˜n+1 ' C˜n of a Cn+1 -massless representation of C˜n+1 . Note that the notions of Cn -massless and Sn -massless refer to n-dimensional ˜ n (which we call the n-De Sitter space-times R1,n−1 (n-Minkowski space) and S˜n /L space, though n-Anti De Sitter space might be a more appropriate expression). The g0 (2, n) while the conformal group of both of them is C˜n , locally isomorphic to SO ˜ g invariance groups are respectively Pn and Sn = SOn (2, n − 1). For example, usual massless particles in 4-Minkowski space, for the Poincar´e group, are in fact C4 massless (under extension to the conformal group SO0 (2, 4)/Z2 ) and S4 -massless (under deformation to the De Sitter group SO0 (2, 3)); usual Dirac singletons are C3 -massless, and their restrictions to SO0 (2, 2) (reducible in a sum of two) are S3 -massless. For simplicity we identify the group representation U , the Lie algebra representation it defines dU and the extension of the latter to U(Cn ) and denote I = kerU (Cn ) (U ). Then one has [3] Theorem 1. U is Cn -massless ⇐⇒ η cd Mac Mbd −
n 2 Mab + ηab C2 = 0 2 n+2
(mod I)
(3)
∀a, b ∈ {−1, . . . , n}, where C2 is the Casimir operator and where we have used the Einstein summation conventionc . Definition 2. We call the right-hand side of the preceding equivalence (3) the fundamental relation (FR). a Massless relatively to the n-conformal group. b Massless relatively to the (n + 1)-De Sitter group. c Summation on repeated indices.
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˜ n+1 and 3. Irreducibility of U When Restricted to L ˜ its Contractibility to UIRs of Pn+1 ˜ n+1 3.1. Irreducibility of U when restricted to L The following proposition is a characterization of UIRs U which remain irre˜ n+1 . ducible when restricted to L Proposition 1. Let (U, H) be a highest weight UIR of C˜n . Then U| ˜ is irreducible Ln+1
⇐⇒
U satisfies the FR.
(4)
is irreducible. Then both the Proof. Assume that the restriction U| ˜ Ln+1 Casimir operators of Gn and Ln+1 are sent to the scalars by U and U| ˜ , reLn+1 spectively. It follows that the difference of these operators is also sent to the scalars and thanks to the adjoint action of Gn one obtains the FR. The converse is proved in [3]. It easily follows: is irreducible. Corollary 1. If U is Cn -massless then U| ˜ Ln+1 3.2. A contraction of Cn -massless representations Consider a family (Sρ )0<ρ≤1 of operators defined on the underlying vector space Vn of Sn+1 = Gn by: Sρ (Mαβ ) = Mαβ √ Sρ (M−1α ) = ρ M−1α
if
0 ≤ α, β ≤ n
if
0 ≤ α ≤ n.
(5)
It defines a contraction of Sn+1 to Pn+1 . We are using here the notion of contractions of representations (on Hilbert spaces) given in [2] (see also [3]). Let Gnρ be the Lie algebra isomorphic to Gn defined by the bracket: [x, y]ρ = S−1 ρ [Sρ x, Sρ y],
x, y ∈ Vn
(6)
and let Uρ the representation of Gnρ defined on the corresponding space H by Uρ (x) = Z−1 ρ ◦ U (Sρ x) ◦ Zρ ,
x ∈ Vn ,
(7)
where (Zρ )0<ρ≤1 is a continuous family of closed invertible operators of H, Z1 being the identity. We choose them here such that Z−1 ρ U (Mαβ )Zρ = U (Mαβ ),
0 ≤ α, β ≤ n.
(8)
Thus one has Uρ (Mαβ ) = U (Mαβ ),
0 ≤ α, β ≤ n
(9)
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M. LAOUES
and, using the FR Uρ (M−1α )Uρ (M−10 ) = Z−1 ρ U (Sρ M−1α )U (Sρ M−10 )Zρ = ρZ−1 ρ U (M−1α M−10 )Zρ = ρU
n X k=1
n 2 η0α C2 Mαk M0k − M0α − 2 n+2
(10) ! .
2 Thus the operator Uρ (M−10 ) has limit zero when ρ → 0, in the sense that it sends a dense subspace to {0} when ρ → 0. It follows that the limit of Uρ (M−1α ), 0 ≤ α ≤ n, is zero too. If one chooses ρ to be the curvature of the space Sn+1 /Ln+1 then one can write, from what precedes:
Proposition 2. In the limit of zero curvature the contracted Cn -massless representation is trivial on Tn+1 , the translation part of P˜n+1 . ˜n 4. Reduction of U on the Maximal Compact Subgroup of C The following results are proved in [3]: n+2 Theorem 2. Let λ = (λ1 , . . . , λr ), r = [[ n+2 2 ]] (the integer part of 2 ), the highest weight (HW) of the Cn -massless representation U . Then there exists a real number s such that
λ1 = −s − where
n−2 and λ2 = · · · = λr−1 = |λr | = s, 2
s>0 2s ∈ N λr ≥ 0 and s = 0 or 1/2
(11)
if n = 2; if n is even and n ≥ 4; if n is odd.
Proposition 3. Let kn = so(2) ⊕ so(n) the maximal compact subalgebra of Cn and let χ(µ1 ) ⊗ π(µ2 , . . . , µr ) be an IR of kn with HW µ = (µ1 , . . . , µr ). Then one has ∞ M n−2 − l ⊗ π(s + l, s, . . . , s, s), (12) χ −s − U| = kn 2 l=o
where || = 1 (resp. = 1) if n is even (resp. odd). Thus Cn -massless representations are very degenerate and are, in some sense, “singleton” representations.
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5. Dirac Singletons Definition 3. A positive (resp. negative) energy representation of S˜n is a lowest (resp. highest) weight representation. We say that U and U 0 are Dirac singletons (DS) if a Sn -massless representation of S˜n occurs in the reduction of the product U ⊗ U 0 and if U and U 0 have the same sign of energy. It has been proved by M. Flato and C. Frønsdal in [10] for the n = 4 case that the (irreducible and unitary) representations Di = D(1, 1/2) and Rac = D(1/2, 0) are Dirac singletons and that the product (Di ⊕ Rac) ⊗ (Di ⊕ Rac) reduces to a SO0 (2, 3). Recall that Di and Rac are direct sum of S4 -massless representations of g SO0 (2, 3). Unfortunately when n ≥ 5 things behave C3 -massless representations of g differently; the next proposition treats this case. Proposition 4. Assume n ≥ 5. Let U and U 0 be DS. Then only a finite number of Sn -massless representations of S˜n can occur in the reduction of the product U ⊗ U 0 . Moreover U and U 0 cannot be simultaneously unitary. Proof. Since U and U 0 have the same sign of energy we can assume they are g0 (2, n − 1). Let λ and λ0 their respective HW. Let HW representations of S˜n ' SO ν = 0 (resp. 1) if n is even (resp. odd). Then n + 2 = 2r + ν and the rank r0 of SnC is given by r0 = r − (1 − ν), thus SnC and CnC have the same rank if and only if n is 0 odd. Let (ei )1≤i≤r0 be the canonical basis of Cr and let ( ∆+ n+1
=
{ei ± ej , 1 ≤ i < j ≤ r0 }
if n + 1 is even;
{ei ± ej , 1 ≤ i < j ≤ r0 } ∪ {ej , 1 ≤ j ≤ r0 } if n + 1 is odd.
(13)
C Then ∆+ n+1 defines a set of positive roots for Sn . Thus if U is a HW irreducible ˜ representation of Sn with HW λ = (λ1 , . . . , λr0 ) then a weight µ of U has the form
X
X
α∈∆+ n+1
pα ∈N
µ = λ−
pα α
r0 h i X =− E+ (qj + pj ) + (1 − ν)m1 e1 j=2 r0 h r0 i−1 i X X X λi + qi − pi − + (qij + pij ) − (1 − ν)mi + (qji − pji ) ei , (14) i=2
j=i+1
j=2
where E = −λ1 and (qj )j , (pj )j , (mj )j (qij )i<j and (pij )i<j are families of natural P1 Pr0 integers, such that j=2 (qj2 − pj2 ) = j=r0 +1 (qir0 + pir0 ) = 0. Let σ = (−s − r0 + 2 + ν/2, s, . . . , s) = (−s − n−2 2 , s, . . . , s) where 2s ∈ N (resp. s = 0 or 1/2) if n is even (resp. odd) and let Λ be the set of such σ’s. Then each
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Sn -massless representation of S˜n has at least one element of Λ as a HW. Indeed if U0 is an Sn -massless representation of S˜n then one has n−2 U0 ∼ D(s + 2 , s, . . . , s), 2s ∈ N and s 6= 0 (15) or n even =⇒ n−2 n U0 ∼ D( 2 , 0, . . . , 0) ⊕ D( 2 , 0, . . . , 0), n−1 1 1 n−1 1 1 1 U0 ∼ D( 2 , 2 , . . . , 2 ) ⊕ D( 2 , 2 , . . . , 2 , − 2 ) n odd =⇒ or n U0 ∼ D( n−2 2 , 0, . . . , 0) ⊕ D( 2 , 0, . . . , 0).
(16)
Now assume that there exists, for an Sn -massless representation U0 , for which σ ∈ Λ is a HW, two DS U and U 0 with HW λ and λ0 respectively. Then it is well known that there exists a weight µ of U such that σ = µ + λ0 . Since µ is given by (14) one has, for some s 0
0
E+E +
r X
(qj + pj ) + (1 − ν)m1 = s + r0 − 2 − ν/2
(17)
j=2
and, for each i, 2 ≤ i ≤ r0 , 0
λi +
λ0i
+ qi − pi −
r X
(qij + pij ) − (1 − ν)mi +
j=i+1
i−1 X
(qji − pji ) = s.
(18)
j=2
Now assume n ≥ 5. Then r0 ≥ 3 and (18) becomes, for i = 2: 0
λ2 +
λ02
+ q2 = p2 +
r X
(q2j + p2j ) + (1 − ν)m2 + s.
(19)
j=3
Adding λ2 + λ02 − E − E 0 to both sides of (17) and using (19), one gets 0
0
r r X X (qj + pj ) + 2p2 + (1 − ν)m1 + (1 − ν)m2 + (q2j + p2j ) j=3
j=3
= λ2 − E +
λ02
0
0
− E + r − 1 − ν/2.
(20)
Thus one has 0
r X
(qj + pj ) + (1 − ν)m1 + p2 ≤ λ2 − E + λ02 − E 0 + r0 − 1 − ν/2.
(21)
j=3
Now adding λ3 + λ03 + q23 − E − E 0 to both sides of (17) and using (18) for i = 3 one finds 0
q2 + p2 + 2p3 +
r X
0
(qj + pj ) + (1 − ν)m1 + (1 − ν)m3 +
j=4
= λ3 − E + λ03 − E 0 + r0 − 1 − ν/2 + q23 ,
r X
(q3j + p3j ) + p23
j=4
(22)
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1087
thus q2 ≤ λ3 − E + λ03 − E 0 + r0 − 1 − ν/2 + q23 .
(23)
But from (20) one gets q23 ≤ λ2 − E + λ02 − E 0 + r0 − 1 − ν/2,
(24)
q2 ≤ λ2 + λ3 − 2E + λ02 + λ03 − 2E 0 + 2r0 − 2 − ν.
(25)
so that Finally one has, thanks to (21), (25) and (17): 0
s=
r X
(qj + pj ) + (1 − ν)m1 + E + E 0 − r0 + 1 + ν/2
j=2
≤ 2λ2 − 2E + λ3 + 2λ02 − 2E 0 + λ03 + 2r0 − 2 − ν.
(26)
The right-hand side being finite for fixed U and U 0 , only Sn -massless representations whose parameter s satisfies (26), thus a finite number, may occur in the reduction of U ⊗ U 0 . Now unitarity of U and U 0 implies [1, 7]: E ≥ λ2 + r0 − 3/2 − ν/2
(27)
E 0 ≥ λ02 + r0 − 3/2 − ν/2,
(28)
and but that is not compatible with (20). Indeed the left-hand side of (20) is a natural integer whereas the right-hand one satisfies: λ2 − E + λ02 − E 0 + r0 − 1 − ν/2 ≤ −r0 + 2 + ν/2 ≤ −1 + ν/2 < 0.
(29)
Remark 1. Unitarity of U or U 0 is however possible for n ≥ 5. Indeed, the Sn n massless representation D( n−2 2 , 0, . . . , 0) ⊕ D( 2 , 0, . . . , 0) is contained in the tensor product of the Cn−1 -massless representation U = D( n−3 2 , 0, . . . , 0), which is unitary, by the representation U 0 = D( 12 , 0, . . . , 0) ⊕ D( 32 , 0, . . . , 0), which is not unitary. 1 1 Another example is given by the Sn -massless representation D( n−1 2 , 2 , . . . , 2 ), the n−2 1 1 unitary Cn−1 -massless representation U = D( 2 , 2 , . . . , 2 ) and the non-unitary representation U 0 = D( 12 , 0, . . . , 0). Now let us look to the other values of n. As seen above the case n = 4 is treated in [10], thus we examine only the cases n = 3 and n = 2. Let n = 3. Then the De Sitter algebra S3 ' so(2, 2) is isomorphic to so(2, 1) ⊕ so(2, 1). The C3 -massless representations of the conformal algebra C3 ' so(2, 3) are the Rac = D(1/2, 0) and the Di = D(1, 1/2) or, more shortly, D(1/2 + s, s), s being 0 or 1/2. The S3 -massless representations of S3 are thus D(1/2 + s, s)S3 , s = 0 or
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M. LAOUES
1/2. Having in mind that an irreducible HW representation of so(2, 2) is equivalent to a tensor product (which we write ) of two irreducible representations of so(2, 1) one gets D(1/2, 0)| ∼ D0 (1/2, 0) ⊕ D0 (3/2, 0) S3 ∼ D(1/4) D(1/4) ⊕ D(3/4) D(3/4)
(30)
and D(1, 1/2)| ∼ D0 (1, 1/2) ⊕ D0 (1, −1/2) S3 ∼ D(1/4) D(3/4) ⊕ D(3/4) D(1/4).
(31)
Here we have denoted by D0 (E, j) (resp. D(α)) the irreducible representation with HW (−E, j) (resp. (−α)) of so(2, 2) (resp. so(2, 1)). Now a large number of UIRs π and π 0 of S˜3 may have the property that S3 -massless representations are contained in π ⊗ π 0 . To reduce that number we shall suppose that π and π 0 are C2 -massless, in analogy with the 4-dimensional case where the Dirac singletons Di and Rac are C3 -massless. Then, if one assumes that π and π 0 are HW representations, each one has the form D0 (α, ±α) where α > 0. Thus one must consider the products D0 (α, ±α) ⊗ D0 (β, ±β) D0 (α, ±α) ⊗ D0 (β, ∓β),
α, β > 0
or, equivalently, the products h i h i D(0) D(α) ⊗ D(0) D(β) h i h i D(0) D(α) ⊗ D(β) D(0) h i h i D(α) D(0) ⊗ D(β) D(0) h i h i D(α) D(0) ⊗ D(0) D(β) . Now using D(α) ⊗ D(β) ∼
∞ M
D(α + β + l),
(32)
(33)
(34)
l=0
one finds D0 (α, ±α) ⊗ D0 (β, ±β) =
∞ M
D0 (α + β + l, ±[α + β + l]),
(35)
l=0
D0 (α, ±α) ⊗ D0 (β, ∓β) = D0 (α + β, ±[α − β]).
(36)
Finally it is easily seen that D0 (1/4, 1/4) ⊗ D0 (1/4, −1/4) = D0 (1/2, 0),
(37)
D0 (3/4, 3/4) ⊗ D0 (3/4, −3/4) = D0 (3/2, 0),
(38)
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
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thus h i h i D0 (1/4, 1/4) ⊕ D0 (3/4, 3/4) ⊗ D0 (1/4, −1/4) ⊕ D0 (3/4, −3/4) i h i h = D0 (1/2, 0) ⊕ D0 (3/2, 0) ⊕ D0 (1, 1/2) ⊕ D0 (1, −1/2) .
(39)
The right-hand side is a sum of S3 -massless representations. Using (35) and (36), one can see that this is a unique solution (up to equivalence) of the problem Singleton⊗Singleton = ⊕S3 -massless for unitary Dirac singletons, which are, here, D0 (1/4, ±1/4) ⊕ D0 (3/4, ±3/4). They are not irreducible, but each component is irreducible on both S3 and L3 . Let n = 2. Then C2 ' so(2, 2), S2 ' so(2, 1) and L2 ' so(1, 1). A (HW) C2 massless representation of C2 has the form D0 (α, ±α), α > 0. Thus the S2 -massless ∼ D(α). Now C1 -masslessness representations of S2 have the form D0 (α, ±α)| S2 on D(β) (or irreducibility on the 2-Lorentz group L2 ) implies β = 0, so that, instead of the n = 4 and n = 3 cases, Dirac singletons are not compatible with C1 -masslessness. But one has D(α/2) ⊗ D(α/2) ∼ D(α) ⊕
∞ M
D(α + 1 + l).
(40)
l=0
Thus S2 -massless representations occur in the tensor product of two S2 -massless ones. 6. Indecomposability. Gupta Bleuler Triplets Gupta–Bleuler triplets are used to quantize gauge theories, in a way similar to the quantization of (4-dimensional flat) QED. This kind of quantization is done on an indefinite metric space which carries indecomposable representations, as in the Gupta–Bleuler quantization of the electromagnetic field. Let us see how it works in the case of our massless representations. If U2 is a massless representation of Gn then it can be obtained as a component of an indecomposable representation. Indeed one can find UIRs Uε , ε > 0, and U3 such that limε→0 Uε is a non trivial extension U2 → U3 (i.e. we have an exact sequence 0 → H3 → H → H2 → 0 where Hi is the carrying space of Ui , i = 2 or 3). The elements of H3 , the gauge states, are obtained from those of H by applying a constraint similar to the Lorentz condition in QED. The elements of H2 , the physical states, are realized on the quotient H/H3 . Now the indecomposable representation (U2 + U3 , H) has no invariant nondegenerate metric, thus covariant quantization is not possible. But if one extends the representation U3 by U2 + U3 in a non trivial way (U3 → U2 → U3 ) to a bigger space endowed with an invariant nondegenerate (but indefinite) Hermitian form then quantization of the gauge theory under construction becomes possible. In the following we construct some examples of Gupta–Bleuler triplets for the massless representations when n ≥ 3.
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M. LAOUES
6.1. Massless representations and indecomposability Let us recall that the massless representations for Gn = so(2, n) are the Cn -massless and the Sn+1 -massless ones. Below we write them again, according to the parity of n. In analogy with 4-dimensional physics we call the parameter s the spin of the representation. Case 1: n is even Cn -massless representations are n−2 , s, . . . , ±s , D s+ 2
2s ∈ N.
(41)
Sn+1 -massless representations are 1 1 n 1 ⊕D , ,..., ,− D 2 2 2 2 n−1 n+1 D , 0, . . . , 0 ⊕ D , 0, . . . , 0 2 2
1 n 1 , ,..., 2 2 2
1 2
(42)
for spin 0.
(43)
for spin
Case 2: n is odd Cn -massless representations are n−2 , s, . . . , s , D s+ 2
s∈
1 0, . 2
(44)
Sn -massless representations are n+1 n−1 , 0, . . . , 0 ⊕ D , 0, . . . , 0 for spin 0 2 2 n−1 D s+ , s, . . . , s , 2s ∈ N and s ≥ 12 . 2
D
(45) (46)
Some of the above irreducible representations correspond to the limit of unitarity [1, 7]. It is the case of the Cn -massless ones and, when n is odd, of the Sn+1 massless representations for which s ≥ 1. Then one can look for indecomposability and Gupta–Bleuler (GB) triplets. That is what we do in the next subsections. In the next two subsections the cases of the representations D( n−2 2 , 0, . . . , 0) and 1 n−2 1 1 D( 2 + 2 , 2 , . . . , 2 ) are treated without separating the n even and n odd cases, since those representations are Cn -massless for both n even and n odd. Finally, when s ≥ 1 the Cn -massless D(s + n−2 2 , s, . . . , s) for n even and the Sn+1 -massless , s, . . . , s) for n odd are investigated successively. D(s + n−1 2
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
6.2. Cn -masslessness, spin 0 6.2.1. Reduction of D(E0 , 0, . . . , 0) to kn and its indecomposability Recall that kn ' so(2) ⊕ so(n) is the maximal compact subalgebra of Gn . Let Gn = kn + pn the Cartan decomposition of Gn . Let Xjk −r≤j,k≤r a basisd of GnC such that (47) Xjk = −Xkj , and [Xjk , Xj 0 k0 ] = δj,−j0 Xkk0 + δk,−k0 Xjj0 − δj,−k0 Xkj 0 − δk,−j0 Xjk0 ,
(48)
n± = hX±j,±k , (1 − ν) ≤ j, k ≤ ri + hX±j,∓k , 1 ≤ j < k ≤ ri,
(49)
and let
h = hX−j,j = Hj , 1 ≤ j ≤ ri.
(50)
Then GnC = n+ + h + n− is a triangular decomposition of GnC . ± C + C − Let p± = pC n ∩ n . Then Gn = p + kn + p . The basis (Xjk )jk is chosen such that p± = hX±1,j , −r ≤ j ≤ r
and |j| 6= 1i,
(51)
kC n = hXjk , −r ≤ j, k ≤ r
and |j|, |k| 6= 1i.
(52)
The root system ∆n+2 is defined by the set of positive roots ∆+ n+2 which is given ]]) instead of n + 1 (resp. r0 = [[ n+1 by (13), but with n + 2 (resp. r = [[ n+2 2 2 ]]). P (±α) ± the The new basis is also chosen such that in the decomposition n = α>0 Gn (e ±e )
(e )
subspace Gn j k is, for 1 ≤ j < k ≤ r, generated by Xj,±k and, if n is odd, Gn j is, for 1 ≤ j ≤ r, generated by X0j . The roots which correspond to kC n are the compact roots and the others the noncompact ones. The set of positive compact +n (resp. noncompact) roots is denoted by ∆+c n+2 (resp. ∆n+2 ). +c Let λ = (λ1 , . . . , λr ) a ∆n+2 -dominant integer weight and let K(λ) denote the irreducible (finite dimensional) HW kn -module. We write N (λ) for the induced HW Gn -module, with HW λ, and L(λ) for the irreducible quotient. The HW vectors for both N (λ) and L(λ) are, for simplicity, identified and denoted by vλ . Proposition 5. Let E0 > 0, λ = (−E0 , 0, . . . , 0), Oλ = D(E0 , 0, . . . , 0), P Z = |h|6=1 X−1,h X−1,−h ∈ U(GnC ) and, for l, k ∈ N, vlk = (X−1,2 )l Zk vλ ∈ N (λ). Then ∞ M U(kC (53) N (λ) = n )vlk l,k=0
and / N (λ) is irreducible ⇐⇒ E0 ∈
n n − 1, . . . , − 2 2
n−1 2
d This basis is more appropriate to the triangular decomposition of G C than the n
. Mab
(54) a,b
basis.
1092
M. LAOUES
− j for some j ∈ {1, . . . , [[ n−1 2 ]]}, then n N (λ) L(λ) = L − + j, 0, . . . , 0 ' ∞ M 2 U(kC n )vl,j+k
Moreover if E0 =
n 2
l,k=0
'
j−1 ∞ M M
U(kC n )vlk .
(55)
l=0 k=0
Corollary 2. Let us write χ(µ1 ) ⊗ π(µ2 , . . . , µr ) for the irreducible representation, with HW µ, of kn on K(µ). Then D(E0 , . . . , 0)| = kn
∞ M ∞ M
if E0 ∈ / D(E0 , . . . , 0)| = kn
χ(−[E0 + l + 2k]) ⊗ π(l, 0, . . . , 0)
l=0 k=0
j−1 ∞ M M
n−1 − 1, . . . , 2
(56)
,
χ(−[E0 + l + 2k]) ⊗ π(l, 0, . . . , 0)
l=0 k=0
if E0 =
n 2
n 2
n−1 . − j for some j ∈ 1, . . . , 2
(57)
Remark 2. 1. The value j = 1 corresponds to the Cn -massless case: ∞ M n−2 , 0, . . . , 0 | = χ(−[E0 + l]) ⊗ π(l, 0, . . . , 0) D kn 2 l=0
which is a particular case of Proposition 3. 2. Thanks to the preceding results one can see that indecomposability arises when E0 reaches the value n2 − j (we use the same notations): n n − j, 0, . . . , 0 + D + j, 0, . . . , 0 . (58) D D(E0 , 0, . . . , 0) −→ n E0 → 2 −j 2 2 Proof of the Proposition. The vlk ’s are maximal vectors for D(E0 , 0, . . . , 0)| ; kn C + C indeed one has n+ ∩ kC n , X−1,2 = 0 and kn , Z = 0, thus n ∩ kn vlk = 0. N (λ) is r Q qj vλ , where (qj )|j|6=1 is a family of natural generated by the monomials j=−r X−1,j |j|6=1
integers and, if |j| = 6 1: 1 X−2,j vl+1,k l+1 X−1,j vlk = vl+1,k X l 1 v − X−2,−h X−2,h vl+1,k l−1,k+1 l+1 (l + 1)(l + n) |h|6=1,2
if |j| 6= 2, if j = 2, if j = −2,
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1093
− C C where v−1,k = 0; thus one has p− vlk ⊂ U(kC n )vl−1,k+1 +U(kn )vl+1,k . Since p , kn ⊂ p− one can conclude that N (λ) = U(p− )vλ ⊂
∞ M
U(kC n )vlk .
l,k=0
Now p+ = h X1j , −r ≤ j ≤ r and |j| 6= 1 i and |j| 6= 1 implies n X1j vlk = δj,−2 l(E0 + 2k + l − 1)vl−1,k + 2k E0 − + k X−1,j vl,k−1 , 2 with v−1,k = vl,−1 = 0; thus for a maximal vector for which the weight is strictly less than λ, necessarily proportional to some vlk , one must have k(E0 − n2 + k) = 0 and l = 0, i.e. l = 0, k 6= 0 and E0 − n2 + k = 0. E0 being strictly positive one has 1 ≤ k ≤ [ n−1 2 ]. L∞ L∞ n C Finally let j ∈ {1, . . . , [ n−1 l=0 k=j U(kn )vlk . Then 2 ]}, E0 = 2 − j and Kj = the relation C pC n vlk ⊂ U(kn )h vl−1,k ; vl+1,k ; vl−1,k+1 ; vl+1,k+1 i implies U(GnC )Kj ⊂ Kj , so that
h i N − n − j , 0, . . . , 0 i hn 2 − j , 0, . . . , 0 = L − . 2 Kj
6.2.2. A Gupta–Bleuler triplet for the Cn -massless D( n−2 2 , 0, . . . , 0) Using the preceding notations and results one can see that D( n−2 2 + ε, 0, . . . , 0) sends the operator Z to zero if ε = 0 but it does not if ε 6= 0. It is precisely this fact which gives us the desired indecomposable representations. Indeed, let ε > 0 and E0 = n−2 2 + ε. Then D(E0 , 0, . . . , 0) is irreducible, but when ε → 0 one obtains, from Remark 2 and for j = 1, an indecomposable representation: n+2 n−2 n−2 + ε, 0, . . . , 0 −→ D , 0, . . . , 0 + D , 0, . . . , 0 . (59) D ε→0 2 2 2 In order to construct explicitly a Gupta–Bleuler (GB) triplet [4], let ρ > 0 and let ( ) n X 2,n a 2,n 2 Hρ = y, y = y ea ∈ R such that y = 1/ρ , a=−1 2 − y02 − y2 . The De Sitter space-time is the universal covering where y 2 = y a ya = y−1 of Hρ2,n . The action of Gn on C ∞ -functions defined on Hρ2,n is well known:
Uλ (Mab ) = Lab = ya ∂b − yb ∂a , where ∂c =
∂ ∂y c .
(60)
Let ∂ 2 = ∂ a ∂a and δ = y a ∂a . Then one has 1 Uλ (C2 ) = − Lab Lab = −y 2 ∂ 2 + δ(δ + n). 2
(61)
1094
M. LAOUES
Now the resolution of the Laplace–Beltrami equation on Hρ2,n is standard [16]. One finds that the following solutions form a Hilbertian basis for L2µ (Hρ2,n ), with dµ(y) = ρ−11+y2 dtdn y:
1/2
n−2 E0 ψklm (t, y) = ρ−(2k+E0 − 2 )
Γ(k + E0 + l)Γ(k + 1) n − 2 Γ(k + n2 + l)Γ k + E0 − 2 E0 +l
×e−i(E0 +l+2k)t (ρ−1 + y2 )− 2 (y2 ) 2 ! −1 2 n − y y ρ (l+ n−2 ,E − ) 0 l 2 Ym p ×Pk 2 , ρ−1 + y2 (y2 ) l
(62)
(α,β)
are the Jacobi polynomials, l = (l2 , . . . , l[ n+1 ] ) and m = (m1 , . . . , m[ n2 ] ) where Pk 2 are vectors, in Nr−1+ν and Nr−1 respectively, subject to certain conditions, l = l2 , −1 0 1/2 l Ym are the spherical harmonics on S n−1 and eit = yy−1 +iy . The scalar +iy 0 product we use to normalize these functions is given by Z ←→ dn y 0 (ψ, ψ ) = ψ(y) : i∂t : ψ 0 (y) −1 , (63) ρ + y2 Rn ↔
E0 where ψ(y)Aψ 0 (y) = Aψ(y)ψ 0 (y) + ψ(y)Aψ 0 (y). We extend the functions ψklm to 2,n 2,n H+ = ∪ρ>0 Hρ by fixing the degree of homogeneity: δψ = −E0 ψ. Then, ψ being in the kernel of ∂ 2 , one has
Uλ (C2 )ψ = E0 (E0 − n)ψ. Let
x±j
i √ (y −1 ± iy 0 ) 2 1 = √ (y 2j−1 ± iy 2j ) 2 n y ∂j =
∂ ∂x−j
(64)
if j = 1, if 2 ≤ j ≤ r, if n is odd and j = 0, and
∂−j = ∂j .
P P P Then one has y 2 = − rj=−r x−j xj , ∂ 2 = − rj=−r ∂−j ∂j , δ = rj=−r x−j ∂j and one can choosee Xjk such that Uλ (Xjk ) = xk ∂j − xj ∂k .
(65)
0 . Then ϕ2 is, up to a multiplicative constant, the maximal vector Let ϕ2 (y) = x−E 1 E0 of Uλ thus ψklm ∈ U(Gn )ϕ2 . Moreover one finds that 0 −2 0 −1 − 2εE0 x−1 x−E , (Zϕ2 )(y) = −E0 (E0 + 1)y 2 x−E 1 1
e We use the notations of the preceding subsubsection.
(66)
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
thus lim (Zϕ2 )(y) = −
ε→0
n − 2 n 2 − n+2 y x1 2 . 2 2 −n 2
Now assume ε = 0 and let ϕ1 (y) = x−1 x1
− n+2 2
and ϕ3 (y) = y 2 x1
2 −n
1 nZ
(67) . Then
2 n−2 Z
−−→ ϕ2 p−−−−−−−− −−→ ϕ3 ϕ1 p−−−−−−−− where Z =
P |j|6=1
(68)
X1j X1,−j and one has ∂ 2 ϕ2 = ∂ 2 ϕ3 = 0,
whereas ∂ 2 ϕ1 =
1 ϕ 6= 0, y2 3
but (∂ 2 )2 ϕ1 = 0.
H
Let cl(V ) denotes the closure of any topological space V and let i = cl(U(GnC )ϕi ), i taking the value 1, 2 or 3. Then it is not difficult to prove the following
H
(0)
Proposition 6. 1. 1 (0) invariant subspace of i−1 ;
H and H 2. H /H (0) 1
(0) 2
⊃
H
(0) 2
⊃
H
(0) 3
and
H
(0) i
(0)
, i = 2 or 3, is a closed
H /H
(0) 3
carry the IR D( n+2 2 , 0, . . . , 0), while , 0, . . . , 0). ries the Cn -massless representation D( n−2 2 3. (n − 2)(n + 2) ϕi = 0 if i = 2 or 3, Uλ (C2 ) + 4 (n − 2)(n + 2) ϕ1 = nϕ3 6= 0, Uλ (C2 ) + 4 2 (n − 2)(n + 2) Uλ (C2 ) + ϕ1 = 0. 4
H
(0) 2
(0) 3
car-
(69)
(0)
4. limy2 →0 ϕ(y) = 0 ∀ϕ ∈ 3 . Thus the Cn -massless D( n−2 2 , 0, . . . , 0) may be realized irreducibly on the cone Q2,n = {y, y ∈ R2,n such that y 2 = 0}. Definition 4. In analogy with QED on 4-dimensional Minkowski space we (0) (0) (0) (0) (0) (0) (0) call the elements of S = 1 / 2 (resp. P = 2 / 3 , resp. G = (0) 3 ) scalar (resp. physical, resp. gauge) states.
H
H
K
H H
H
H H
H
− n+2
Remark 3. Let (0) the closure of the GnC -module generated by y 7−→ x1 2 ; 4 2 2 2 it carries the IR D( n+2 2 , 0, . . . , 0). Let ∂ = (∂ ) and let us identify y to the corresponding operator. Then the GB triplet n−2 n+2 n+2 , 0, . . . , 0 −→ D , 0, . . . , 0 −→ D , 0, . . . , 0 D 2 2 2
1096
M. LAOUES
defined by ϕ1 , ϕ2 and ϕ3 may be defined by (0) = positive energy solutions f of ∂ 4 f = 0 1
H H H
(0) 2 (0) 3
= f∈ = f∈
H H
(0) 1
such that ∂ 2 f = 0 ,
(0) 2
such that
f ∈ y2
K
(0)
n−2 f and δf = − 2
,
.
H
←→ R n (0) y Now, for ϕ and ϕ0 in 1 , define (ϕ, ϕ0 )1 = S 1 ×Rn ϕ(y)y 2 ∂ 2 ϕ0 (y) ρdtd −1 +y 2 and R n−2 (ϕ, ϕ0 )2 = S 1 ×S n−1 (y2 ) 2 ϕ(y)ϕ0 (y)dtdΩ, where y belongs to some Hρ2,n (resp. Q2,n ) in the first (resp. second) integral. Then it is not difficult to choose the constant c such that the form defined by hϕ, ϕ0 i = (ϕ, ϕ0 )1 + c(ϕ, ϕ0 )2 is an invariant non degenerate indefinite metric such that hϕi , ϕj i 6= 0 if and only if (i, j) ∈ {(1, 3), (3, 1), (2, 2)}.
Definition 5. Again in analogy with 4-dimensional Minkowskian QED, the (0) (0) condition ∂ 2 f = 0, on f ∈ 1 , which fixes the space 2 will be called Lorentz condition; the equation ∂ 4 f = 0 will be called the dipole equation.
H
H
6.3. Cn -masslessness, spin 1/2 6.3.1. Reduction on kn and indecomposability of D(E0 , 12 , . . . , 12 ) The following result is known; see for example [1, 7]. Proposition 7. D(E0 , 12 , . . . , 12 ) is unitarizable if and only if E0 ≥ Here we consider only the unitary case, i.e. E0 ≥
n−1 2 .
n−1 2 .
Proposition 8. Let λ = (−E0 , 12 , . . . , 12 ) and recall that ν = 0 (resp. 1) if n is even (resp. odd). 1. If E0 >
n−1 2
then D(E0 , 12 , . . . , 12 ) is irreducible and one has
∞ M 1 1 1 = χ (−[E + l + 2k]) ⊗ π D E0 , , . . . , 0 2 2 |kn 2 l,k=0 ∞ M 1 1 1 + l, , . . . , , ν − χ (−[E0 + l + 2k + 1]) ⊗ π ⊕ 2 2 2 l,k=0
1 1 + l, , . . . , 2 2 1 . 2
(70) then N (λ) is not simple; it contains a maximal submodule 2. If E0 = n−1 2 1 1 1 n+1 1 isomorphic to L(− n+1 2 , 2 , . . . , 2 , ν − 2 ) which carries the UIR D( 2 , 2 , . . . , 1 1 n−1 1 1 2 , ν − 2 ). The irreducible one D( 2 , 2 , . . . , 2 ) is carried by the quotient.
1097
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1 1 1 1 Proof. 1. If E0 > n−1 2 , then D(E0 + 2 , 0, . . . , 0) ⊗ D(− 2 , 2 , . . . , 2 ) = 1 1 1 1 1 D(E0 , 2 , . . . , 2 ) ⊕ D(E0 + 1, 2 , . . . , 2 , ν − 2 ). If we denote by vσ the maximal vector of D(− 12 , 12 , . . . , 12 ) one finds that, for l, k ∈ N, the vectors vlk ⊗ vσ and vlk ⊗ (X−1,[ν−1]r vσ ) generate a submodule (of the tensor product) isomorphic to L(λ). Pr 1 ν 2. Now assume E0 = n−1 j=2 X−1,j X−j,[ν−1]r . 2 and let Y = ν+1 X−1,[ν−1]r − Then one can see that Yν (v00 ⊗vσ ) generates an irreducible submodule of U(GnC )(v00 ⊗ 1 1 n−1 1 1 vσ ) isomorphic to L(− n+1 2 , 2 , . . . , ν − 2 ) while D( 2 , 2 , . . . , 2 ) is carried by the quotient U(GnC )(v00 ⊗ vσ )/U(GnC )Yν (v00 ⊗ vσ ). 1 1 6.3.2. A Gupta–Bleuler triplet for D( n−1 2 , 2, . . . , 2) ν Let ε ≥ 0 such that E0 = n−1 2 + ε. Proposition 8 says that if ε = 0 then Y is 1 1 sent to 0 by Uλ = D(E0 , 2 , . . . , 2 ). Now assume ε > 0, then Uλ is irreducible; but when ε → 0 one obtains an indecomposable representation: 1 1 1 1 1 n+1 1 n−1 1 n−1 + ε, , . . . , −→ D , ,..., +D , ,..., ,ν − . D 2 2 2 ε→0 2 2 2 2 2 2 2 (71) To construct a Gupta–Bleuler triplet we need explicit realizations of the representations concerned. Let σ = ( 12 , . . . , 12 ) and let, if n is even, σ − = ( 12 , . . . , 12 , − 21 ). We denote by Sσ the irreducible spinor representation D(− 12 , 12 , . . . , 12 ) and, when n is even, by Sσ− the irreducible one D(− 12 , 12 , . . . , 12 , − 21 ). Let + be the carrier space of Sσ and − the carrier one of Sσ− when n is even (resp. {0} when n is = + ⊕ − be the spinor module of Gn . odd). Finally let Let γ−1 , . . . , γ2r−2 be 2r matrices in gl( ) such that [γa , γb ]+ = 2ηab ,f where 2 = −1. Then [A, B]+ = AB + BA, and let γ2r−1 ∈ Cγ−1 · · · γ2r−2 such that γ2r−1
S
S S S S
S
[γa , γb ]+ = 2ηab The following realization of Sσ on
∀a, b ∈ {−1, . . . , n}.
S is well known:
−−→ Sab = Mab p−−−−−−−−
1 1 [γa , γb ] = (γa γb − ηab ). 4 2
Later we shall also need the generators ωj defined by i √ (γ−1 ± iγ0 ) if j = 1, 2 1 ω±j = √ (γ2j−1 ± iγ2j ) if 2 ≤ j ≤ r, 2 if n is odd and j = 0. γn Thus one has [ωj , ωk ]+ = −2δj,−k f We identify the identity of
gl(S ) with 1.
∀j, k ∈ {−r, . . . , r},
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M. LAOUES
and the preceding realization of Sσ may be written: Xjk p−−−−−−−− −−→
1 1 [ωj , ωk ] = (ωj ωk + δj,−k ). 4 2
S
2,n −→ + such that We realize D(E0 , 12 , . . . , 12 ) on spinor fields Ψ : H+ 1 Ψ. ∂2Ψ = 0 and δΨ = − E0 + 2
The action of Gn on spinor fields is given by Uλ (Mab ) = Lab + Sab . Let y= /
n X a=−1
Theng
r X
y a γa =
x−j ωj
and ∂/ =
n X
∂ a γa = −
a=−1
j=−r
r X
∂−j ωj .
j=−r
(n + 1)(n + 2) y − /∂/ Ψ Uλ (C2 )Ψ = −y 2 ∂ 2 + δ(δ + n + 1) + 8 (n + 1)(n + 2) y 1 1 = E0 + E0 − − n + − /∂/ Ψ. 2 2 8
(72)
It is easy to prove the following Lemma. y and ∂/ commute with the action of Gn ; Lemma 1. 1./ y 2. [/ , ∂/ ]+ = 2δ + n + 2; 3. −y 2 ∂ 2 = y/ ∂/ (y/ ∂/ − 2δ − n); 4. if ε > 0, then (−2ε)−1 (∂//y − 2) and(−2ε)−1 /y∂/ are projectors on the irreducible subspaces of the tensor product L(−[E0 + 12 ], 0, . . . , 0) ⊗ L( 12 , . . . , 12 ), namely the spaces L(−E0 , 12 , . . . , 12 ) and L(−[E0 + 1], 12 , . . . , 12 , ν − 12 ) respectively. Let us consider the spinor fields Ψ2 and Ψ3 defined by −E0 − 12
Ψ2 (y) = x1
vσ
and
−E0 − 32
Ψ3 (y) = /y x1
ω[ν−1]r vσ .
Then one has 1 1 1 , 0, . . . , 0 ⊗ L ,..., ' U(GnC )Ψ2 ⊕ U(GnC )Ψ3 . L − E0 + 2 2 2 −E0 − 12
Moreover, let Ψ1 (y) = x1
ω−1 ω[ν−1]r . 1 ν E0 + Y Ψ2 = 2
thus
Then 1 1 Ψ3 − ε Ψ1 , 2 2
1 1 E0 + Ψ3 . lim Y Ψ2 = ε→0 2 2 ν
g We identify y 2 with the function y 7−→ y 2 , / y with y 7−→ y /, and so on.
(73)
(74)
1099
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
From now on we assume ε = 0, i.e. E0 =
n−1 2 .
1 − 2−ν X1,−[ν−1]r
Then
ν 4 nY
−−→ Ψ2 p−−−−−−−− −−→ Ψ3 . Ψ1 p−−−−−−−−
H
(1/2)
Let i to prove.
= cl(U(GnC )Ψi ), i being 1, 2 or 3. The next proposition is not difficult
H
(1/2)
Proposition 9. 1. 1 closed invariant subspace of
H
(1/2) 1
2.
H
(1/2) 2
3.
(75)
/H
/H
(1/2) 3
(1/2) 2
H
⊃
H
(1/2) 2
⊃
(1/2) i−1 ;
and
H
(1/2) 3
H
(1/2) 3
and
H
(1/2)
i
, i = 2 or 3, is a
1 1 carry the IR D( n+1 2 , 2 , . . . , ν − 2 ), while
1 1 carries the Cn -massless representation D( n−1 2 , 2 , . . . , 2 );
∂/ Ψi = 0
if
y/∂/ Ψ1 = nΨ3 6= 0
i = 2 or i = 3, but
(y/∂/ )2 Ψ1 = 0.
(76)
H
(1/2) y Ψ)(y) = 0 ∀Ψ ∈ 4. limy2 →0 (/ and limy2 →0 (/y Ψ2 )(y) 6= 0. Thus the Cn 3 n−1 1 1 massless representation D( 2 , 2 , . . . , 2 ) may be realized irreducibly on the cone Q2,n .
H
H
(1/2)
(1/2)
H
(1/2)
= / 2 (resp. Definition 6. The elements of the space S 1 (1/2) (1/2) (1/2) (1/2) (1/2) = 2 / 3 , resp. G = 3 ) are called scalar (resp. physical, P resp. gauge) states.
H
H
H
Remark 4. Let − n+1 2
H
K
(1/2)
H
be the closure of the GnC -module generated by the field
1 1 1 ω[ν−1]r vσ ; it carries the IR D( n+1 y 7−→ Φ(y) = x1 2 , 2 , . . . , 2 , − 2 ). Then the Gupta–Bleuler triplet
D
n+1 1 1 1 , ,..., ,ν − 2 2 2 2
−→ D
n−1 1 1 , ,..., 2 2 2
−→ D
n+1 1 1 1 , ,..., ,ν − 2 2 2 2
defined by Ψ1 , Ψ2 and Ψ3 may be redefined by
H H H
(1/2) 1 (1/2) 2 (1/2) 3
n = positive energy solutions of ∂ 2 Ψ = 0, δΨ = − Ψ and (/y∂/ )2 Ψ = 0 , 2 (1/2) = Ψ∈ 1 such that ∂/Ψ = 0 , (77) (1/2) = Ψ∈ 2 such that Ψ ∈ /y (1/2) .
H H
K
H
←→ R n (1/2) y , define (Ψ, Ψ0 )1 = ρ−1 S 1 ×Rn Ψ∗ (y) /y∂/ Ψ0 (y) ρdtd Now, for Ψ and Ψ0 in 1 −1 +y 2 R n and (Ψ, Ψ0 )2 = S 1 ×S n−1 (y2 ) 2 Ψ∗ (y)Ψ0 (y)dtdΩ, y being in some Hρ2,n (resp. Q2,n )
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M. LAOUES
in the first (resp. second) integral. Again it is not difficult to choose the constant c such that the form defined by hΨ, Ψ0 i = (Ψ, Ψ0 )1 + c(Ψ, Ψ0 )2 is an invariant non degenerate indefinite metric such that hΨi , Ψj i 6= 0 if and only if (i, j) ∈ {(1, 3), (3, 1), (2, 2)}. Definition 7. The equation ∂/Ψ = 0, which fixes the space called the Lorentz condition.
H
(1/2) , 2
will be
6.4. Indecomposability and GB triplets for spin s ≥ 1 We assume in this subsection that s ≥ 1 and 2s ∈ N. 6.4.1. Indecomposability of D(E0 , s, . . . , s, sν ) Let λ = (−E0 , s, . . . , s, sν ), where |sν | = s and, if n is odd, sν ≥ 0. Proposition 10. 1. D(E0 , s, . . . , s, sν ) is unitarizable ⇐⇒ E0 ≥ n−2+ν + s; 2 n−2+ν 2. if E0 > 2 + s, then N (λ) is simple; + s, then N (λ) contains, up to a multiplicative constant, a 3. if E0 = n−2+ν 2 ν unique maximal vector of weight (−E0 −1, s, . . . , s, sν − ssν ); it is given by Y−1,− sν vλ , s r where 0 Y−1,±r = 2sX−1,±r −
r X
X−1,j X−j,±r ,
j=2 1 = 2sX−1,−r + 2X−1,0 X−r,0 Y−1,−r
2(s − 1) X 2 X − X−1,j X−j,−r − X−1,j X−j,0 X−r,0 . 2s − 1 j=2 2s − 1 j=2 r
r
Since, for n even, the treatment of Uλ is similar for both signs of sν we shall consider from now on that sν = s. Proof of the Proposition. For the first two items see [1, 7]. For the last one, a maximal vector of weight (−E0 − 1, s, . . . , s, s − 1) for n even has the general form: v 0 = aX−1,−r +
r X
bj X−1,j X−j,−r vλ ,
j=2 a and n+ v 0 = 0 implies bj = − 2s for each j. The same technique works for odd n.
Remark 5. The situation for s ≥ 1, for both n even and n odd, is more complicated than for the spin 0 and spin 1/2 cases. Indeed more than one submodule for N (λ) exists when E0 = n−2+ν + s, thus it is a priori possible to construct very 2 different examples of Gupta–Bleuler triplets U 0 −→ Uλ −→ U 0 with U 0 unitary.
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
6.4.2. A GB triplet for D( n−2+ν + s + i, s, . . . , s, s − i), i = 1 or 2 2 + s + ε, ε ≥ 0. To realize our Gupta–Bleuler triplet we need Let E0 = n−2+ν 2 explicitly the representations D(E0 , s, . . . , s) and D(E0 + 1, s, . . . , s, s − 1), especially for ε = 0. Both of them are contained in the reduction of the tensor product D(E0 + s, 0, . . . , 0) ⊗ D(−s, s, . . . , s). The representation S[2sσ] = D(−s, s, . . . , s) itself is contained in the tensor power Sσ⊗2s of the irreducible spinorial representation. We define the action of Mab ∈ Gn on a tensor v1 ⊗ · · · ⊗ v2s ∈ +⊗2s by
S
Sab (v1 ⊗ · · · ⊗ v2s ) =
=
2s X
1 v1 ⊗ · · · ⊗ [γa , γb ]vt ⊗ · · · ⊗ v2s 4 t=1
2s X
(78)
(t)
Sab (v1 ⊗ · · · ⊗ vt ⊗ · · · ⊗ v2s ).
t=1
S
Let γa (t) be defined on the tensorsh of
⊗2s
S ⊕S
=(
⊗2s −)
+
by
γa (t) (v1 ⊗ · · · ⊗ v2s ) = v1 ⊗ · · · ⊗ γa vt ⊗ · · · ⊗ v2s . Then the action defined in (78) may be written more simply: −−→ Sab = Mab p−−−−−−−−
2s X t=1
S
Let Sym( 0
γ a (t) γa (t ) = −
⊗2s + ) r X
(t)
Sab =
2s X 1 (t) (t) γa , γb . 4 t=1
be the space of symmetric tensors in (t)
S
⊗2s +
(79) 0
and let γ (t) ·γ (t ) =
(t0 )
ω−j ωj .
j=−r
Proposition 11. S[2sσ] = D(−s, s, . . . , s) is realized irreducibly on the space:
S
V S = Sym(
⊗2s + )
∩
h\
0
ker γ (t) · γ (t ) − ν
i .
t,t0
t6=t0
2,n We realize the unitary representations of interest on tensor-spinors Ψ : H+ −→ V S such that ∂2Ψ = 0 and δΨ = −(E0 + s)Ψ.
To this effect, we define the action of Gn on them by Mab 7−→ Lab + Sab . Let y(t) = y a γa (t) = /
r X j=−r
h Recall that
(t)
x−j ωj
S− = {0} for n odd.
and ∂/(t) = ∂ a γa (t) = −
r X j=−r
(t)
∂−j ωj ,
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then one has 2s i h X y/(t)∂/(t) Ψ Uλ (C2 )Ψ = −y 2 ∂ 2 + δ(δ + n + 2s) + rs(s + r − 1 + ν) − t=1 2s i h X y/(t)∂/(t) Ψ. (80) = (E0 + s)(E0 − s − n) + rs(s + r − 1 + ν) − t=1
Lemma 2. 1. For fixed t, /y (t) and ∂/(t) satisfy the three first items of Lemma 1; y (t) , / y (t0 ) ] = 0 and [∂/(t) , ∂/(t0 ) ] = 0; 2. if t 6= t0 , then [/ 0 0 3. if t 6= t0 , then [∂/(t) , y/(t ) ] = γ (t) · γ (t ) (= ν on V S ). Let us define, for non-negative integers k, l and spinors v1 , . . . , vk , symmetric tensors in ⊗2s by
S
v1 · · · vk =
1 X τ (v1 ) ⊗ · · · ⊗ τ (vk ), k! τ ∈Sk
v1l = v1 · · · v1 | {z }
(81)
l terms
and let Ψ1 , Ψ2 and Ψ3 be defined by h i 0 −s Ψ1 (y) = x−E ω−1 ω−r vσ vσ − ν ω−1 vσ ω−r vσ vσ2s−2 , 1 0 −s 2s Ψ2 (y) = x−E vσ , 1 h i Ψ3 (y) = x1−E0 −s−1 /y ω−r vσ vσ − ν /yvσ ω−r vσ vσ2s−2 .
Then one has U(GnC )Ψ2 ⊕ U(GnC )Ψ3 ⊂ L(−[E0 + s], 0, . . . , 0) ⊗ L(s, . . . , s) and one finds that ν Ψ2 = s(E0 + s)Ψ3 − εsΨ1 , Y−1,−r
(82)
ν lim Y−1,−r Ψ2 = s(E0 + s)Ψ3 .
(83)
thus ε→0
From now on we assume E0 = − 12 X1,r
n−2+ν 2
+ s. Then ν 2 Y−1,−r s(n−2+ν+4s)
Ψ1 p−−−−−−−− −−→ Ψ2 p−−−−−−−− −−→ Ψ3 .
H
(84)
= cl(U(GnC )Ψi ), i being equal to 1, 2 or 3. The next proposition is Let i straightforward: (s)
1103
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
H
(s)
Proposition 12. 1. 1 (s) invariant subspace of i−1 .
H
⊃
H
(s) 2
⊃
H
(s) 3
H /H and H carry the H /H carries the representation D( (s) 1 (s) 3
2.
(s) 2
(s) 2
(s) 3
and
H
(s)
i
, i = 2 or 3, is a closed
IR D( n+ν 2 + s, s, . . . , s, s − 1), while + s, s, . . . , s);
n−2+ν 2
3. ∂/(t) Ψi = 0 ∀t ∈ {1, . . . , 2s} 2s X
if
i = 2 or i = 3;
y (t)∂/ (t) Ψ1 = (n − 2 + ν + 4s)Ψ3 6= 0 /
but
t=1
2s X
(85) !2
/y (t)∂/(t)
Ψ1 = 0;
t=1
H
(s) y (1) · · · / y (2s) Ψ)(y) = 0 ∀Ψ ∈ and limy2 →0 (/y (1) · · · /y(2s) Ψ2 )(y) 6= 4. limy2 →0 (/ 3 n−2+ν 0. Thus the representation D( 2 + s, s, . . . , s) may be realized irreducibly on the cone Q2,n .
H
(s)
H H (s)
(s)
H
(s)
Definition 8. The elements of the space S = 1 / 2 (resp. P = (s) (s) (s) = 3 ) are called scalar (resp. physical, resp. gauge) 3 , resp. G states.
H H
H
(s) 2 /
H
Let, for t ∈ N, v t ⊗ (v ∧ v 0 ) = v t+1 ⊗ v 0 − v t ⊗ v 0 ⊗ v, and let τ(t,t0 ) t≤t0 be the system of generators (permutations t ↔ t0 if t 6= t0 and identity if t = t0 ) of the group-algebra of S2s . Let
Y
i 1 hP τ(t,2s) /y(2s) 1≤t≤2s 2s hP 1 = 1≤t≤2s−1 τ(t,2s−1) 2s(2s − 1) i P + y/(2s−1) − y/(2s) 0 ,2s−1) τ τ 0 (t,2s) (t 1≤t
if n is even;
if n is odd.
Finally, let ( Φ(y) =
− n+ν −2s x1 2
×
vσ2s−1 ⊗ ω−r vσ
if n is even;
vσ2s−2 ⊗ (vσ ∧ ω−r vσ )
if n is odd.
As in the cases s = 0 and s = 1/2 we have here:
K
Remark 6. Let (s) be the closure of the simple GnC -module generated by the n+ν s, s, . . . , s, s − 1). Then the Gupta–Bleuler triplet field Φ; it carries the IR D( 2 +n−2+ν n+ν + s, s, . . . , s → D n+ν D 2 + s, s, . . . , s, s − 1 → D 2 2 + s, s, . . . , s, s − 1
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M. LAOUES
defined by Ψ1 , Ψ2 and Ψ3 may be redefined by (
H
(s) 1
=
positive energy solutions of ∂ 2 Ψ = 0, such that
n−2+ν − 2s Ψ and δΨ = − 2
H H
(s) 2 (s) 3
= Ψ∈ = Ψ∈
H H
2s X
!2 /y (t) /(t)
t=1
(s) 1
such that ∂/(t) Ψ = 0
(s) 2
such that Ψ ∈
YK
∂
) Ψ=0
,
(86)
∀t ∈ {1, . . . , 2s} , (s) .
Now, as in spin 0 and spin 1/2 cases, one can find an invariant non-degenerate form ←−− − − − − − − −→ P −−−−−−−−−− n ν R (s) y y/(j)∂/(j) Ψ0 (y) dtd on 1 . Let (Ψ, Ψ0 )1 = (ρ−1 )2s+ 2 S 1 ×Rn Ψ∗ (y) 2s j=1 ρ−1 +y2 and R n−2+ν (s) (Ψ, Ψ0 )2 = S 1 ×S n−1 (y2 ) 2 +2s Ψ∗ (y)Ψ0 (y)dtdΩ, Ψ and Ψ0 being in 1 and 2,n 2,n y belongs to some Hρ (resp. Q ) in the first (resp. second) integral. Again it is not difficult to choose the constant c such that the form defined by hΨ, Ψ0 i = (Ψ, Ψ0 )1 + c(Ψ, Ψ0 )2 is an invariant non degenerate indefinite metric for which 6 0 if and only if (i, j) ∈ {(1, 3), (3, 1), (2, 2)}. hΨi , Ψj i =
H
H
Definition 9. The system of equations ∂/ (t) Ψ = 0, t ∈ {1, . . . , 2s}, which fixes (s) the space 2 , will be called the Lorentz condition.
H
6.5. Further remarks on GB triplets The above considerations show that the true generalization of Dirac singletons from 4-dimensional De Sitter space to space-time in dimension n ≥ 5 are in fact the Cn−1 -massless representations. Indeed, though (Cn−1 -massless) ⊗(Cn−1 -massless) does not contain Sn -massless representations in general (this is true only for n = 3 or 4), their restriction to the n-Lorentz group g SO0 (1, n− 1) is irreducible and they are, together with their conjugates, the only representations with that property. Thus they contract to UIRs of the n-Poincar´e group which are trivial on the translations and their weight diagram is very degenerate, in some sense 1-dimensional. Let us look at indecomposability and construction of gauge theories with GB quantization. The most interesting case is when n ≥ 5. GB triplets are easily constructed for Cn−1 -massless representations and for arbitrary spin s (2s ∈ N or {0, 1}, depending on the parity of n). But for Sn -massless representations, which represent, for us, massless particles on n-De Sitter space-time, the situation is different. Indeed, if n is even, Sn -massless representations exist for arbitrary spin s, but one can construct a GB triplet, with our method, only for s ≥ 1, because for s = 0 or 1/2 no indecomposability arises around the corresponding highest (or lowest) weight. Nevertheless these representations occur (once) in the tensor product namely D( n−3 of 2 , 0, . . . , 0) ⊗ by a non-unitary one, a 1Cn−1 -massless representation 3 n−2 1 D( 2 , 0, . . . , 0) ⊕ D( 2 , 0, . . . , 0) for spin 0 and D( 2 , 2 , . . . , 12 ) ⊗ D( 12 , 0, . . . , 0)
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1105
for spin 1/2, for which it seems that construction of GB triplets is possible. Thus one can hope to construct, for even n ≥ 6, a gauge theory analogous to that of D(1, 0) ⊕ D(2, 0) and D( 32 , 12 ) with the usual Dirac singletons when n = 4. Now assume n is odd. Then Sn -massless representations exist only for spin 0 or 1/2 and, again, one cannot construct a GB triplet, but they are contained in the reduction of the same tensor products as in the even case (here the last term in the spin 12 unitary factor is ± 21 ). However, unlike the latter, the representations D( 12 , 0, . . . , 0) and D( 32 , 0, . . . , 0), which are also below the unitary limit, cannot naturally be considered as quotients of extensions, while this is the case for the n−2 1 1 1 Cn−1 -massless representations D( n−3 2 , 0, . . . , 0) and D( 2 , 2 , . . . , 2 , ± 2 ). Thus for odd n only one factor in the tensor product has naturally GB triplets. One can ask the question of what would be the analogue of a gauge theory in this context. 7. Discussion First we recall that in the n = 2 case the situation is drastically different from n ≥ 3. Indeed though C2 -masslessness is easily defined, there is no good notion of spin and there exist infinitely many C2 -massless non-equivalent representations with equivalent restrictions to the 2-Poincar´e group; but this is not the case of the restrictions to the 2-De Sitter group, locally isomorphic to SO0 (2, 1). Note however that in this case (as is well known) the full conformal group is infinite-dimensional. We have shown here that most properties of massless representations are, in some sense, independent of the space-time dimension n. But if n ≥ 3, the property of occuring in the tensor product of two UIRs of the same energy sign is true only for n = 3 and n = 4. An interpretation is that compositeness of massless particles (in the sense of the present paper) is not possible in De Sitter space-time with dimension n ≥ 5. Concerning the Gupta–Bleuler quantization, it can be seen that for general n ≥ 3 the construction of triplets works with no problem, but for a given massless representation there is no unique solution to the construction of a Gupta–Bleuler triplet. There is however some ambiguity when one tries to generalize to the g0 (1, n−1), n ≥ 5, the notion of g0 (2, n−1)/SO n-dimensional Anti-De Sitter space SO masslessness. For n ≥ 5, there is no canonical definition of a massless representation SO0 (2, n), especially for “spin” s ≥ 1. Indeed the of the n-De Sitter group S˜n = g rank of the compact subalgebra so(n − 1) of the De Sitter algebra is ≥ 2, instead of 1 in the n = 4 case. Thus there are several slightly different alternatives to describe massless particles in De Sitter world in higher dimensions, which coincide for n = 4. The two extreme are, for spin s ≥ 1 (2s ∈ N), U (s) = D(s + n−2 2 , s, . . . , s, s), where || = 1 if n is odd and = 1 if n is even, and U 0 (s) = D(s + n − 3, s, 0, . . . , 0) if s is an integer or U 0 (s) = D(s + n − 3, s, 1/2, . . . , 1/2, /2) if s − 1/2 is an integer. The former are what we call here Sn -massless representations for n even (for n odd, s ≥ 1 there are no Sn -massless representations). The latter have very recently been called, when n = 5, massless (in the bulk) in [8, 9].
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In what follows we shall compare somewhat in detail various properties of both alternatives. In order to do this we need first to look more closely at the notion of masslessness in the n-dimensional Minkowski space R1,n−1 . On this basis we then compare the notions of Anti-De Sitter masslessness in n ≥ 5 dimensions, also in both alternatives. As in the 4-dimensional Minkowski space one needs, for massless representations of the n-Poincar´e group P˜n , the mass operator to be zero (and the representation non trivial). Thus the massless representation, say U P , of interest must be induced by a UIR of a subgroup which is a semi-direct product of a subgroup of the n˜ − 2) = SO(n g − 2) n Rn−2 by Lorentz group isomorphic to the Euclidean group E(n the group of space-time translations R1,n−1 . Moreover, for physical reasons, it seems reasonable to eliminate the “continuous spin” in the inducing representation, i.e. we assume that the Euclidean group part of the inducing representation is trivial on the translations subgroup Rn−2 . It is thus finite dimensional and essentially g0 (n − 2) with HW λ. determined by a UIR πλ of SO A first problem (not appearing in the comparison between our approach and that of [8, 9]) is that the choice of λ, to define a spin s ≥ 1, is not unique for n ≥ 6; to make this choice easier one can use the physically sensible fact that the wave equations for massless particles are invariant under the action of the n-conformal group C˜n . Thus we may add the following extension condition, always satisfied for ˆ of the n-conformal massless representations when n = 4: U P extends to a UIR U group. An interesting consequence of this (strong) condition is that λ depends now on a unique parameter s such that [3]: λ=
(s, . . . , s, s), 2s ∈ N and || = 1 if n is even, (s, . . . , s), s ∈ {0, 1/2} if n is odd.
(87)
The bad news, with this condition, is that in odd dimensional space-times (and already for n = 5), one can define naturally neither massless particles with spin s ≥ 1 nor helicity. Nevertheless we call here, for uniformity of presentation, a mass zero representation of the n-Poincar´e group which satisfies the extension condition a massless discrete helicity representation (MDHR). Remark 7. In the fact one may define helicity in a natural way for a large class of representations of P˜n , especially when n is even. Indeed, take n even and denote by ε = (εµ1 ...µn )0≤µ1 ,...,µn ≤n−1 the completely skew-symmetric tensor such that ε01...(n−1) = 1. Define the generalized Pauli–Lubanski vector by: (r−2)(r−3) 2
(i/2)r−2 (−1) Wµ = − (r − 2)! 0
X
εµν1 ...νn−1 M ν1 ν2 · · · M νn−3 νn−2 P νn−1
ν1 ...νn−1
where the M νν ’s and the P ν ’s stand for the generators of the n-Poincar´e algebra ˜ − 2) and where r − 2 is the rank of the maximal compact subgroup Kn−2 of E(n (in fact, we have denoted by r the rank of the Lie algebra of the n-conformal group
MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
1107
Gn ). Then one can show easily that in a massless (UI) representation P˜n one has, if λ = (λ3 , . . . , λr ) is the HW of the irreducible representation πλ of Kn−2 : Wµ = ε(λ3 + r − 3) · · · (λr−1 + 1)|λr |Pµ where ε is the sign of λr . Thus one may define naturally helicity thanks to this relation provided that λr 6= 0, in which case it could not. Let us look at two examples. The first one is when all the components of λ are equal to s modulo ε (case of U 0 (s)). In this case one has, in the same conditions as above: Wµ = ε(s + r − 3) · · · (s + 1)sPµ , This relation not only fixes the sign of the helicity but determines also the spin s. The second is when λ3 = s and the other components equal to σ modulo ε (case of U 0 (s)) where σ, being 0 or 1/2, is such that s − σ is an integer. Then one has: Wµ = ε(s + r − 3) · · · (σ + 1)σPµ , which equals 0 when s is an integer. Thus this relation, in this example, is not appropriate to define helicity for two kinds of particles (bosons and fermions) simultaneously. For these reasons and some others (for example the conformal invariance of equations) we prefer the first example to induce representations which describe massless particles in the Minkowski space-time Mn . If one drops the extension condition, for example for odd n, then πλ with λ = (s, . . . , s, s) or even λ = (s, 0, . . . , 0) or (s, 1/2, . . . , 1/2, 1/2) and s ≥ 1 may be used to induce a “massless” representation U P in order to represent a massless particle with spin s in the n-dimensional Minkowski space. Now consider the following masslessness conditions, analogous to the n = 4 ones: (a) Massless representations of the n-De Sitter group S˜n contracts smoothly to a MDHR of the n-Poincar´e group P˜n ; ˆ of the n-conformal group of any MDHR of (b) The unique extension to a UIR U ˆ the n-Poincar´e group is such that U |S˜n is precisely a massless representation of S˜n ; (c) For s ≥ 1, one may construct a gauge theory on the n-dimensional Anti-De Sitter space for massless particles, quantizable by the use of an indefinite metric and a GB triplet; (d) The massless representations are such that the physical signals propagate on the Anti-De Sitter light cone. We define also what we shall call here a singleton property (SP): Singleton ⊗ Singleton contains Massless representations. Then it was proved in [3] that representations of S˜n which satisfy conditions (a), (b), (c) above have the form U0 with U0 given by (15) and (16) in Sec. 5. Thus there
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M. LAOUES
is no Sn -massless representation for s ≥ 1 and n odd. This, of course, is related to what happens in flat n-dimensional space where (for n odd) the spin can be only 0 or 1/2 (for a MDHR). As a consequence, when s ≥ 1, conditions (a), (b), (c) are relevant for U (s) only if n is even and not at all for U 0 (s) (n ≥ 5). We conjecture that a representation of the n-de Sitter group that satisfies (a), (b), (c) must satisfy also condition (d) (this will be proved in a forthcoming paper). Unfortunately (for n ≥ 5) property (SP) is not satisfied by Sn -massless representations, i.e. by representations which satisfy (a), (b), (c), as shown in Proposition 4. Now if one drops one (or more) of the masslessness conditions then one can define other representations of S˜n to be massless ones, i.e. to represent massless particles on the n-dimensional Anti-de Sitter space-time. Between (a), (b) and (c) (and probably (d)) only dropping the stronger condition (b) has actually an effect because (b) implies both (a) and (c) (and probably (d)). Indeed if one drops (b) then things change radically. For example let D = D(E0 , λ2 , . . . , λr0 ) a UIR of S˜n . If the weight (−E0 , λ2 , . . . , λr0 ) reaches the limit of unitarity then usually one obtains an indecomposable representation from which one may construct a GB triplet and then a gauge theory. Thus condition (c) is still satisfied by a large number of representations. A contraction of D to a UIR U P of P˜n is usually possible but the contracted representation U P is not a MDHR in general. Let us look at the example of U 0 (s), s ≥ 1. From what precedes U 0 (s) does not satisfy (b). Moreover U 0 (s) contracts naturally to a representation UλP , of the n-Poincar´e group, for which πλ = π(s,0,...,0) or πλ = π(s,1/2,...,1/2,/2) , depending on whether s or s − 1/2 is integer. But U P is not a MDHR. Even more, U 0 (s) does not, in general, contract to a MDHR, for which the weight λ of πλ must satisfy the relation (87), especially when n is odd, case of which the allowed spin is 0 or 1/2. Among the 3 masslessness conditions we have studied so far, only (c) is totally satisfied by U 0 (s). Indeed let σ = 0 (resp. 1/2) if s (resp. s − 1/2) is integer (≥ 1). Then the representation D(s + n − 3 + ε, s, σ, . . . , σ, σ) becomes indecomposable if ε → 0 (see some examples in [9] for low values of n) and one may construct a GB triplet D(s + n − 2, s − 1, σ, . . . , σ, σ) → D(s + n − 3, s, σ, . . . , σ, σ) → D(s + n − 2, s − 1, σ, . . . , σ, σ). Moreover, if we consider the (Cn−1 -massless) representations D( n−3 2 , 0, . . . , 0) , 1/2, . . . , 1/2, /2) as singletons, because they have properties similar to and D( n−2 2 the singletons Rac and Di (see Secs. 1, 3 and 4 and Subsec. 6.5), though they are not Dirac singletons in the sense of Definition 3, then U 0 (s) satisfies property (SP) because the following is true: n−3 n−3 , 0, . . . , 0 ⊗ D , 0, . . . , 0 contains D 2 2 ∞ M s=0
D(n − 3 + s, s, 0, . . . , 0);
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MASSLESS PARTICLES IN ARBITRARY DIMENSIONS
D
n−3 , 0, . . . , 0 2
⊗D
n−2 , 1/2, . . . , 1/2, /2 2
∞ M
contains
D(n − 3 + s, s, 1/2, . . . , 1/2, /2.
s−1/2=0
Acknowledgements The author thanks Professors M. Flato and E. Angelopoulos for stimulating discussions and Professor D. Sternheimer for helpful criticism of the manuscript. References [1] E. Angelopoulos, “On the unitary dual of SO 0 (2r, 2)”, Lett. Math. Phys. 7 (1983) 121–127. [2] E. Angelopoulos, M. Flato, C. Fronsdal and D. Sternheimer, “Massless particles, conformal group and de Sitter universe”, Phys. Rev. D23 (1981) 1278–1289. [3] E. Angelopoulos and M. Laoues, “Masslessness in n dimensions”, to be published in Rev. Math. Phys. [4] H. Araki, “Indecomposable Representations with Invariant Inner Product”, Commun. Math. Phys. 97 (1985) 149–159. [5] B. Binegar, C. Fronsdal and W. Heidenreich, “Conformal QED”, J. Math. Phys. 24(12) (1983) 2828–2846. [6] B. Binegar, C. Fronsdal and W. Heidenreich, “de Sitter QED”, Ann. Phys. 149 (1983) 254–272. [7] T. Enright, R. Howe and N. Wallach, “A classification of unitary highest weight modules”, in Proc. Univ. Utah Conf., 1982, Progress in Mathematics 40 (1983) 97– 143, ed. P. C. Trombi, Birkh¨ auser. [8] S. Ferrara and C. Fronsdal, “Conformal Maxwell theory as a singleton field theory on ADS5 , IIB three branes and duality”, hep-th/9712239 (version 4). [9] S. Ferrara and C. Fronsdal, “Gauge fields as composite boundary excitations”, hepth/9802126. [10] M. Flato and C. Fronsdal, “One massless particle equals two dirac singletons”, Lett. Math. Phys. 2 (1978) 421–426. [11] M. Flato and C. Fronsdal, “Singletons: Fundamental gauge theory”, in Proc. Int. Symposium on Topological and Geometrical Methods in Field Theory, (Espoo, Finland, eds. J. Hietarinta and J. Westerholm), World Scientific Publ., 1983, pp. 273–290. [12] M. Flato and C. Fronsdal, “The singleton dipole”, Commun. Math. Phys. 108 (1987) 469–482. [13] M. Flato, C. Fronsdal and J.P. Gazeau, “Masslessness and light-cone propagation in 3 + 2 de Sitter and 2 + 1 Minkowski spaces”, Phys. Rev. D33 (1986) 415–420. [14] C. Fronsdal, “Semisimple gauge theories and conformal gravity”, AMS Lecture in Appl. Math. 21 (1985) 165–178. [15] W. Heidenreich, “Quantum theory of spin- 12 fields with gauge freedom”, Il Nuovo Cimento 80 (1984) 220–240. [16] R. Ra¸czka, N. Limi´c and J. Niederle, “Discrete degenerate representations of noncompact rotation groups”, J. Math. Phys. 7 (1966) 1861–1876.
QUANTUM AFFINE TODA SOLITONS N. J. MACKAY School of Mathematics, University of Sheffield Hicks Building, Hounsfield Rd Sheffield S3 7RH, UK E-mail : [email protected] Received 10 November 1997 We review some of the progress in affine Toda field theories in recent years, explain why known dualities cannot easily be extended, and make some suggestions for what should be sought instead.
1. A Brief Review The affine Toda field theories are models of real scalar fields, in one space dimension (here taken to be the real linea ), with exponential interactions. The Lagrangian is n m2 X 1 nj (eβαj ·φ − 1) , (1) L = (∂µ φ)(∂ µ φ) − 2 2 β j=0 where the field φ(x, t) is an n-dimensional vector, n being the rank of the finite Lie algebra g. The αj (j = 1, . . . , n) are the simple roots of g; α0 is the lowest root, so that {α0 , αj } are described by one of the extended Dynkin diagrams of an affine algebra gˆ. The fields can be rescaled so that β only appears in L through an overall factor 1/β 2 ; expanding in powers of β 2 is thus equivalent in the quantum theory to expanding in ~. A great deal has been discovered about these models in the last ten years, and the resulting spectra of perturbative and solitonic particles are surprisingly rich. It is clear, however, that vastly more remains to be learned about the setting for these results. This paper does not provide a pedagogic review: excellent introductions already exist [1, 2], and the reader is referred to these for full details of masses, S-matrices and so on. Rather I shall provide a brief qualitative summary, and then describe the current frontier. In particular I want to explain why there appears to be no duality among the full spectra of solitons and particles. 1.1. Real β, simply-laced g The Lagrangian (1), with β real, is amenable to simple techniques. The vacuum φ = 0 is unique; the exponentials may be expanded about it, and the masses (of order m) and the three-point couplings of the mass eigenstates may then be read a The
theory on the half-line is another story [45]. 1111
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off [3, 4]. The masses form an eigenvector of the Lie algebra’s Cartan matrix [5], and so may be put into correspondence with spots on its Dynkin diagram — we shall refer to this as the particle’s “species”. The theory is further known to be classically integrable, with local, commuting conserved charges of spins equal (for g simply-laced) to the exponents of the Lie algebra [6]: the first exponent is always one, corresponding to energy-momentum. Their conservation constrains the admissible three-point couplings very strongly, indeed to such an extent that it might be surprising that solutions exist at all. Nevertheless an admissible set does exist for each algebra, as is made beautifully clear in the construction due to Dorey [7, 8]. When two particles fuse to form a third, the conservation of each charge requires the existence of a triangle with sides of lengths equal to the charges and angles determined by the particles’ momenta. In Dorey’s construction, each three-point coupling corresponds to a triangle of roots, each being chosen from an orbit of a root under a Coxeter element of the Weyl group. The projection of this equation in the (higher-dimensional) root space onto various planes, on which the Coxeter element acts as a rotation, gives the triangles for the charges. These results at tree-level can be quantized perturbatively; and at one-loop order, for simply-laced g, the particle masses all renormalize in the same ratio. The expectation is that this will be true to all orders, and exact S-matrices (which, since the particles are scalar, will be scalar factors, in fact products of simple trigonometric functions) may be hypothesized for the particles [9, 3, 4], with rigid mass ratios and hence pole structure, described by Dorey’s rule. Further, the dependence on β is such that there is a strong ↔ weak coupling duality: the S-matrices are invariant under β 7→ 4π/β. 1.2. Imaginary β, simply-laced g If in the simplest case, g = a1 , we take β to be imaginary, we have moved from the sinh- to the sine-Gordon model. The vacuum is now degenerate, and there exist solitons, with masses proportional to m/β 2 , and “breathers”, consisting of an oscillating soliton-antisoliton pair (at imaginary rapidity difference), with a continuous mass spectrum. Exact S-matrices, now with matrix structure, may be proposed for the solitons, and the bound states include a discretized spectrum of breathers. The perturbative particle is still present, and its mass and S-matrix are equal to those of the lowest breather state [10]. It is therefore natural to suspect that in the exact quantum theory the breather and the particle are identical. For other g, however, L is complex for imaginary β. Nevertheless there exist solitonsb , constructed by finding Hirota τ -functions [13] or with vertex operators [14], of each species and with topological charges in (but not filling) the corresponding fundamental representation [16], interpolating the (now degenerate) vacua. They have real masses, of order 1/β 2 and proportional to those of the particles, and real higher conserved charges [15], and they “fuse” — double solitons truncate b Issues of reality and stability are still in doubt [11–13].
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and appear as single solitons — according to Dorey’s rule. They also obey an exclusion rule: static multisolitons exist, but only if no two of the constituent single solitons are of the same species. There are also breathers in every species, and excited or “breathing” solitons in some [17]; these have the expected continuous mass spectra. Any quantization of such an apparently non-unitary theory must be very suspect, but it is possible to plough ahead with, for example, semiclassical quantization, and obtain sensible results: that, as with the particles, the soliton mass ratios do not renormalize. The expectation is that, in a sense yet to be discovered, there is an embedded unitary theory, of solitons and their bound states, within the larger non-unitary one [18]. Whether such a unitary theory can be defined for all values of the coupling constant is more equivocal. Finally it is worth pointing out that, in contrast to the sine-Gordon ↔ massive Thirring model relationship, there is no known quantum field theory of which the affine Toda solitons are the elementary excitations, though the particle S-matrices have been expressed as exchange relations [19]. As with sine-Gordon solitons, we can try to construct exact soliton S-matrices. Traditionally the route used has been the implication from conservation of the local charges that multiparticle scattering must factorize into products of two-particle scattering processes, as described by the Yang–Baxter equation (YBE). However, it is now recognized that underlying the YBE is a set of conserved charges forming a quantum group [20]. In contrast to the charges described earlier, these are nonlocal: that is, if measured on two asymptotically separate physical states, instead of yielding simply the sum of the values of the charges on the individual states, there are further interaction terms. Further, they do not commute — in fact they form a quantum group — and they have indefinite or non-integer spin. The multiplets in the theory will form representations of them (scalar for the particles, non-scalar for the solitons), and their conservation can be used to determine the S-matrices up to a scalar factor. In the affine Toda theories the charges have non-integral spins of order 4π/β 2 −1, and form a quantized affine algebra [21] in the principal gradation, with deforma2 2 tion parameter q = e4iπ /β . There is a (topological charge-dependent) similarity transformation to the homogeneous gradation, in which the spectral parameter x 2 is e(4π/β −1)θ (where θ is the incoming particles’ rapidity difference). It is important to distinguish this realization of the quantum group from that obtained by quantizing the auxiliary algebra of the Lax pair, where the deformation parameiβ 2
ter is q = e 2 , and which thus has the conventional classical (β → 0) limit, the undeformed algebra. In contrast, in the classical limit of the S-matrices, whilst the quantum charge algebra itself becomes highly deformed, the ambiguities (multiples of 2iπ in the exponents) in θ at the poles (which occur at x equal to some power of q) become relatively small, and the discrete bound state spectrum becomes continuous, as expected. The two forms of q are very suggestive of some strong-weak coupling relationship between the auxiliary and quantum spaces. This is not a phenomenon only of affine
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Toda theories: it also occurs with Yangians [22], where there is a similar ~ ↔ 1/~ relationship. It is unclear how or whether to attach any significance to this, relating as it does a mathematical artefact (the auxiliary algebra) to physical charges (the quantum algebra). The physical states are expected to fall into fundamental representations of the quantized affine algebra. These are reducible representations of the (q-deformed) Lie subalgebra, containing its corresponding fundamental representation as a component. If we believe (as all the classical and semiclassical information leads us to) that these multiplets are quantum solitons, an oustanding issue is how they succeed in filling representations which are classically only part-filled. Pressing on, we encounter the outstanding open problem of the general construction of the relevant solutions of the YBE. The only simply laced case for which it is solved completely is an , and the results are precisely what one might hope for [2, 23, 24]. If we apply the bootstrap principle, that all poles are to be interpreted in terms of on-shell diagrams of physical particles, and that, when the diagram is tree-level, S-matrices for the intermediate state can be constructed by fusion, the spectrum needed to close the bootstrap consists of the solitons and discretized spectra of breathers and breathing solitons. The analysis of the particles for real β can still be carried out for imaginary β, and their masses and S-matrices are the same as for the lowest breathers, leading us to identify them. Even in the absence of explicit S-matrices for other g, something can still be said about their expected pole structure. Mathematically, the S-matrix fusings are the analogue for the quantum group of the Clebsch–Gordan rule for Lie algebras (to which Dorey’s rule is related [25], but not identical): they are its rule for tensoring representations. For the fundamental representations, this rule has recently been shown to be precisely Dorey’s rule [26]. The proof is exhaustive, however, so its truth is very suggestive of unseen mathematical structure beyond our current understanding of quantum groups. Mirroring these facts about the algebraic structure is a general observation that the scalar prefactors of the expected S-matrices are related to the “quantum dilogarithm” [27]. A relationship between two such different pieces of mathematics would come as no surprise: as we have seen, ATFTs have many ways of exhibiting Dorey’s rule. 1.3. g nonsimply-laced When g is nonsimply laced, the classical soliton masses are no longer proportional to those of the particles. In fact, they are proportional to those of the particles in the dual theory [14], based on g ∨ , with the root lengths inverted; we shall call this “Lie duality”. In addition to these untwisted algebras, whose affine extensions we denote g (1) and g ∨(1) , we must now also consider the twisted algebras g (1)∨ , obtained from a nonsimply laced g by inverting the lengths of both the αi and α0 . (Because of the involvement here of α0 , we shall call this “affine duality”.) Here, both the particles and the solitons are a subset of those for a simply-laced algebra h(1) , of which the twisted algebra is a subalgebra invariant under an automorphism of order k: in the usual notation, g (1)∨ = h(k) .
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For nonsimply laced algebras, the particle mass ratios [3, 32] (computed to one-loop order) are no longer β-independent. In fact the strong ↔ weak duality mentioned earlier extends to this case, using affine duality of algebras: the masses [28] and S-matrices [28, 29, 30] of the (g (1) , β) theory are those of the (twisted) (g (1)∨ , 4π/β) theory. Dorey’s rule has not yet been fully generalized to nonsimply laced cases. The rule for g predicts the correct (tree-level) three-point couplings, but careful examination of the particle S-matrices [29, 30] shows that not all of these are allowed fusings in the exact quantum theory. In fact Dorey’s rule can be extended to cover twisted algebras [26, 35], and it is the intersection (which we shall denote D(g)) of the sets of couplings allowed by the rules for g and for g (1)∨ which gives the correct set. However, a full generalization needs also to give the correct flexible mass and charge triangles, and, despite some progress, this has not yet been achieved. The soliton masses [31, 32, 33] (computed semiclassically) no longer renormalize in the same ratio either: in fact the quantum-to-classical soliton mass ratio is, up to a species-independent factor, that of the particle.c Because of the Lie duality mentioned above, however, there is no affine duality of soliton masses. This might have been expected: remember that whereas the particles have masses of order m, solitons have masses of order m/β 2 , heavy in the weak limit, which would be expected to become very light for strong coupling. Further, the S-matrices, as can be seen from x and q (whose form is similar for nonsimply-laced algebras), will look very different for strong and weak couplings. In calculating the soliton S-matrices, we encounter the subtlety that for the gˆ g ∨ ) [21]. Their spins now depend on the root theory, the non-local charges form Uq (ˆ length, with the algebra in what has been termed the “spin gradation”. It can still be transformed to the homogeneous gradation, however, though with a slightly different form for x. The first complete set of S-matrices for a nonsimply laced theory was found [37] (2) for the twisted algebra dn+1 , and contains the expected results: solitons, excited solitons and breathers, with the lowest breather state identifiable with the particle. The soliton fusing rule is D(cn ), as expected. The most subtle case is that of the untwisted nonsimply laced algebras, where we must construct YBE solutions corresponding to twisted affine algebras [38]. This (1) was done [39] for the bn theory, and the classical result mentioned in the first paragraph for the particle was found to apply to the lowest breather state as well: its mass is not always proportional to that of the soliton. Thus we now believe that in all cases the lowest breather is to be identified with the particle; in support of this, the exact masses and S-matrices turn out to be identical. The soliton fusing rule turns out to be D(cn ), suggesting a Lie duality (not yet proven in general): that g solitons have a D(g ∨ ) fusing rule. c Although almost certainly true [36], this matter is not settled: the semiclassical calculations have never been completed entirely satisfactorily.
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2. Towards an Overview We can discuss these facts within the following scheme [37]: g (1)∨ , 10 solitons g ∨(1) , solitons l Lie ↓? g (1) , 0 particles
Affine
←→
g (1)∨ , 10 particles
Here ≡ β 2 /4π, where β is the real coupling: thus in the top-left, classical solitons would be obtained when is negative. Everything in this diagram has the same (exact) mass ratios, extracted from the proposed S-matrices, and the same D(g) fusings. Solitons are in the top row, particles in the bottom; the left column is weakly coupled, the right strongly-coupled. 2.1. Lie duality The only fact present which has not already been described is the relationship of 0 to . Recall that Lie duality operated classically (at zeroth order in ) in the left-hand column. At first order (i.e. from semiclassical calculations) [32, 33] it also applied with 0 = − (with real-coupled particles and imaginary-coupled solitons). However, on examination of the exact mass ratios we find that it works to all orders only with 0 ⇒ 0 = − , (2) =− ∨ ˜∨ ˜∨ h∨ 0 h h +h 1+ 1+ 1+ h h h where h is the Coxeter number (of g and g ∨ ), h∨ is the dual Coxeter number of g, ˜ ∨ that of g ∨ . and h To see this we require that breather poles in S-matrices for general algebras work analogously to the bn case, the only case in which the S-matrices have been calculated and thoroughly examined. Our conjecture (which we do not believe to be too strong, since the form of the poles is highly constrained) is that for any Lie algebra and any species, 1 h∨ π|α|2 ; (3) , where λ = − − m = 2M sin 2hλ h m denotes particles, M solitons, and α is the relevant simple root, with |long root|2 = 2. A simple illustrative example √ is given by g2 , which is Lie self-dual. Hence there is only one classical mass ratio, 3. Duality exchanges the long and short roots, and hence also the heavy and light species of solitons compared with particles. Under the strong ↔ weak affine duality, the particles’ exact quantum mass ratio is π mh = 2 cos ml H() (h means “heavy”, l “light”) with 1 1 1 = − H() 6 18
2 1+ 3
,
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(3)
which runs to the affine dual (d4 ) ratio in its strong-coupling limit (checked [40] in perturbation theory to order β 4 ). Does this expression also give the exact soliton (1) (3) mass ratio? — that is, does the g2 ↔ d4 pair contain only one exact quantum mass ratio to describe both solitons and particles?d The answer is no. Whilst this (3) ratio’s strong limit gives the correct ratio for the d4 soliton masses, we can see from 0 6= in (2) that its weak value does not give the correct g2 soliton mass ratio. (1) (3) Unfortunately, whilst some of the soliton S-matrices in the g2 ↔ d4 pair have been calculated [34], they are not enough to prove (3) for g2 , which, with h∨ = 4, h = 6, yields π , ml = 2Mh sin 18λ π . mh = 2Ml sin 6λ √ The solitons then have a new mass ratio whose → 0 limit is 3, π Mh , = 2 cos Ml H(0 ) with 0 related to by (2). √ In the strong limit → ∞ this ratio runs not to the d4 ratio 2 cos(π/12) but to 2. Is there some overall duality scheme into which these results among the masses fit? Suppose we seek a duality between the (ˆ g, ) and (ˆ g 0 , 0 ) theories, in which, 0 for some functional dependence of on , solitons and particles are exchanged. Such a relationship would of course take no account of excited solitons or higher breathers, and so might seem unlikely. Consideration of the classical masses requires gˆ = g (1) , gˆ0 = g ∨(1) : that is, Lie duality. For g2 we seek a relationship: (3)
↔ 0 Mh /Ml = mh /ml mh /ml = Mh /Ml Mh /ml = mh /Ml mh /Ml = Mh /ml The first two are what we considered above: they require 2 cos
π π = 2 cos . 0 H( ) H()
For all to be satisfied we require π 1 π = sin sin 18λ0 6λ 4 π 1 π = sin sin 6λ0 18λ 4 d Thanks to Ed Corrigan for persistently asking me this question.
(4)
(5) (6)
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(We find that (4) is equivalent to (5)/(6).) But there is no 0 = f () which solves (5) and (6), and hence no exact quantum Lie duality in this form. This analysis generalizes to the bn ↔ cn case. The bn spectrum is extracted from the S-matrices [39], and gives (3) with h∨ = 2n − 1, h = 2n: aπ sin Ma H = nπ Mn sin H
(a = 1, 2, . . . , n − 1)
π 2nλ π mn = 2Mn sin 4nλ aπ ma . = 2 sin ⇒ mn H ma = 2Ma sin
For cn , the known particle mass ratios aπ sin 0 ma H = nπ mn sin 0 H (again for a = 1, 2, . . . , n − 1) will lead via (3), now with h∨ = n + 1, h = 2n, to soliton masses satisfying π , 4nλ0 π , mn = 2Mn sin 2nλ0 ma = 2Ma sin
giving in turn aπ Ma = 2 sin 0 Mn H (where our priming of quantities simply denotes their association with cn rather than bn ). We now seek 0 = f () such that (bn , ) ↔ (cn , 0 ) Mi /Mj = mi /mj mi /mj = Mi /Mj Mi /mj = mi /Mj mi /Mj = Mi /mj for all i, j = 1, 2, . . . , n. The first of these is what we considered earlier, and is satisfied if (2) holds, giving a weak ↔ weak relationship.
QUANTUM AFFINE TODA SOLITONS
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The full set is satisfied if π 1 π = sin 4nλ0 2nλ 4 π 1 π = sin sin 2nλ0 4nλ 4 sin
(7) (8)
holds. In the semiclassical limit (Eqs. (6.5) and (6.6) of [32]) it was pointed out that full bn ↔ cn Lie duality requires a strong ↔ weak relationship, β 2 β 02 = 8h2 (that paper contained a misprint), which is the same as the leading term obtained from either (7) or (8), 1 π2 = . 2h2 λλ0 4 However, at all-orders (7) and (8) again have no solution for arbitrary , and we find no Lie duality in this form. There will be solutions for certain fixed , 0 , but we have no indication that these would be significant. 2.2. Strongly-coupled solitons As we pointed out, such a duality would seem too naive given the presence in the spectrum of many other excited states. However, as the coupling becomes strong ( large and negative) such states disappear from the spectrum. Strongly coupled solitons are problematic classically and semiclassically, but there is no problem with examining their proposed exact S-matrices at such values of the coupling. How do solitons behave at strong coupling → ∞? Their S-matrices are of course non-diagonal, and we can never hope to identify soliton and particle S-matrices. However, in the infinite coupling limit the leading term in the S-matrix is just the identity: might the solitons behave like particles in this limit? If we were to use the discussion at the beginning of this section as a guide, we might expect to see a relation between (g ∨(1) , = −1/α) solitons and (g (1)∨ , ∨ ˜∨ (1) h ) particles (where α is small and positive). In the a2 case (to pick the α− h + h (1) (1) simplest example), the relation would be between (a2 , −1/α) solitons and (a2 , α − 2) particles, which does not occur: the particle S11 , for example, is certainly not unity at = −2, and indeed is so only in the weak- and strong-coupling limits. The unlikeliness of (2) in simply laced cases (where the rigid mass ratios give us no guidance on how 0 and are to be related) leads us to suspect that it is only an artefact. If the solitons are to behave like weakly coupled particles at all, their masses, of order m/, seem to make it more relevant to investigate mid-ranges of the coupling, ∼ 1. 2.3. An observation on soliton S-matrices In our initial scheme, the g ∨(1) quantum solitons carry representations of Uq (g ∨(1)∨ ) non-local charges (and thus the g (1)∨ solitons, Uq (g (1) ) charges). Both
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contain a Uq (g) subalgebra, although the multiplets form different representations of it [39]. (2) (1) Specializing in g = cn , we find that we are comparing a2n−1 with cn YBE solutions, and these share a very similar algebraic structure, only differing in certain algebraic parameters, and hence in the form of dependence on the quantum group spectral parameter x. That is, making the change (n + 1, 2n) 7→ (2n, 2n − 1) in (1) (2) the cn solutions yields those associated with a2n−1 . Unfortunately this cannot be achieved by a transformation on : if we effect the composite Lie-affine transformation on implicit in our scheme we also rescale the exponents in x and q. This is inevitable: the change in exchanges strong and weak couplings, whereas the (n + 1, 2n) 7→ (2n, 2n − 1) observation relates S-matrices which are both at weak coupling. 2.4. Folding non-local charges (1)
(2)
Whereas quantum bn soliton multiplets carry representations of Uq (a2n−1 ) (1) (1) charges, classically the bn solitons are “folded” from dn+1 : that is, they are dn+1 (1) solitons invariant under an automorphism of the dn+1 Dynkin diagram. The quan(1) (1) tum dn+1 soliton multiplets represent Uq (dn+1 ) charges, and linear combinations of (2) these, invariant under a twisting automorphism, form a Uq (dn+1 ) subalgebra. To (1) describe the quantum bn solitons, however, we would need to add roots and so (2) (1) multiply charges, and would expect to find a Uq (a2n−1 ) subalgebra of Uq (dn+1 ). It would be useful to understand how the two are related. 2.5. Further directions All we have done here is to set up a few easy targets to shoot down. Perhaps the overall point is that we should seek to understand strongly coupled solitons on their own terms rather than expect straightforward dualities with particles. Certainly, more investigation of the known soliton S-matrices at strong and, especially, at mid-range coupling is needed. However, we should also stand back a little and look for alternative ways forward. Of primary importance is the generalization of the fusing rule to the full spectrum. In simply laced cases, and for solitons only or particles only, mass ratios do not renormalize and Dorey’s rule is therefore rigid. Any extension of it, either to nonsimply laced cases or to a rule for the rich arrangement [2] of couplings between solitons and particles or breathers, requires a flexible, β-dependent and therefore intrinsically quantum construction. Steps have been made towards both former [41] and latter [42], but the full construction remains some way off. At a grander level, the goal is an algebraic scheme in which g (1) , g ∨(1) , g (1)∨ and ∨(1)∨ all make appearances. Such a scheme would have eventually to incorporate g all of our results, of course, but let us point out some of those likely to be more salient. First, the Coxeter number plays a central role in Dorey’s construction, and so the incorporation of the flexible Coxeter number H is a correspondingly crucial step. Second, an emerging fact [43] is that when (the non-commuting, non-additive,
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non-integer spin) quantum group charges exist, so do Wilson’s local charges (which are commuting, additive and have spins equal to the exponents). Unfortunately, the constructions are always rather model-specific; what we would like is a modelindependent construction incorporating both. Finally, in the background remains our lack of understanding of quantum group representations: why Dorey’s rule applies to their tensor products remains a central unexplained fact. Remarkably, such a construction may be emerging in the theory of deformed Walgebras [44]. Much remains conjectural, but it is becoming clear that different limits of these deformations yield the centres of different quantized affine algebras, dual in various ways, and that the W-algebras can be used to construct representations of these. Discovering the facts of ATFTs within this picture is likely to make an interesting quest. Acknowledgements I would like to thank G´erard Watts for helpful comments, and Pembroke College, Cambridge, for a Stokes Fellowship. References [1] E. Corrigan, “Recent developments in affine Toda quantum field theory”, preprint DTP-94/55, hep-th/9412213. [2] G. M. Gandenberger, “Exact S-matrices for quantum affine Toda solitons and their bound states”, Ph.D. thesis, Cambridge University 1996, available at http://www.damtp.cam.ac.uk/user/hep/publications.html [3] H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, “Affine Toda field theory and exact S-matrices”, Nucl. Phys. B338 (1990) 689; “Multiple poles and other features of affine Toda field theory”, Nucl. Phys. B356 (1991) 469. [4] P. Christe and G. Mussardo, “Elastic S-matrices in (1+1) dimensions and Toda field theories”, Int. J. Mod. Phys. A5 (1990) 4581. [5] M. Freeman, “On the mass spectrum of affine Toda field theory”, Phys. Lett. B261 (1991) 57. [6] G. Wilson, “The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras”, Ergod. Th. Dyn. Sys. 1 (1981) 1. [7] P. E. Dorey, “Root systems and purely elastic S-matrices (I)”, Nucl. Phys. B358 (1991) 654; “Root systems and purely elastic S-matrices (II)”, Nucl. Phys. B374 (1992) 741. [8] A. Fring, H.-C. Liao and D. I. Olive, “The mass spectrum and coupling in affine Toda field theories”, Phys. Lett. B66 (1991) 82; A. Fring and D. I. Olive, “The fusing rule and the scattering matrix of affine Toda theory”, Nucl. Phys. B379 (1992) 429. [9] A. E. Arinshtein, V. A. Fateev and A. B. Zamolodchikov, “Quantum S-matrix of the 1 + 1-dimensional Toda chain”, Phys. Lett. B87 (1979) 389. [10] A. B. Zamolodchikov and Al. B. Zamolodchikov, “Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models”, Ann. Phys. (NY) 120 (1979) 253. [11] J. M. Evans, “Complex Toda theories and twisted reality conditions”, Nucl. Phys. B390 (1993) 225. [12] S. P. Khastgir and R. Sasaki, “Instability of solitons in imaginary-coupling affine Toda theory”, Prog. Theor. Phys. 95 (1996) 503.
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[13] T. J. Hollowood, “Solitons in affine Toda field theory”, Nucl. Phys. B384 (1992) 523. [14] D. I. Olive, N. Turok and J. W. R. Underwood, “Solitons and the energy-momentum tensor for affine Toda theory”, Nucl. Phys. B401 (1993), 663; “Affine Toda solitons and vertex operators”, Nucl. Phys. B409 (1993) 509. [15] M. Freeman, “Conserved charges and soliton solutions in affine Toda theory”, Nucl. Phys. B433 (1995) 657; hep-th/9408092. (1) [16] W. A. McGhee, “The topological charges of the an affine Toda solitons”, Int. J. Mod. Phys. A9 (1994) 2648, hep-th/9307035; Durham Ph.D. thesis, 1994, unpublished. (1) [17] U. Harder, A. Iskandar and W. McGhee, “On the breathers of an affine Toda field theory”, Int. J. Mod. Phys. A10 (1995) 1879; hep-th/9409035. [18] J. Underwood and B. Spence, “Restricted quantum theory of affine Toda solitons”, Phys. Lett. B349 (1995) 83; hep-th/9501078. [19] E. Corrigan and P. Dorey, “A representation of the exchange relation for affine Toda field theory”, Phys. Lett. B273 (1991) 237; hep-th/9109056; A. Fring, “Braid relations in affine Toda field theory”, Int. J. Mod. Phys. A11 (1996) 1337; hep-th/9503019. [20] D. Bernard and A. LeClair, “Quantum group symmetries and non-local currents in 2-dimensional quantum field theories”, Commun. Math. Phys. 142 (1990) 99. [21] G. Felder and A. LeClair, “Restricted quantum affine symmetry of perturbed minimal conformal models”, Int. J. Mod. Phys. A7 (Proc. Suppl.) 1A (1992) 239. [22] N. J. MacKay, “Lattice quantization of Yangian charges”, Phys. Lett. B349 (1995) 94; hep-th/9505179. [23] T. J. Hollowood, “Quantizing sl(N) solitons and the Hecke algebra”, Int. J. Mod. Phys. A8 (5) (1993) 947. (1) [24] G. M. Gandenberger, “Exact S-matrices for bound states of a2 affine Toda solitons”, Nucl. Phys. B449 (1995) 375; hep-th/9501136. [25] H. W. Braden, “A note on affine Toda couplings”, J. Phys. A25 (1992) L15. [26] V. Chari and A. Pressley, “Yangians, integrable quantum systems and Dorey’s rule”, Commun. Math. Phys. 181 (1996) 265; hep-th/9505085. [27] P. R. Johnson, “Exact quantum S-matrices for solitons in simply-laced affine Toda field theories”, Nucl. Phys. B496 (1997) 505; hep-th/9611117. [28] G. W. Delius, M. T. Grisaru and D. Zanon, “Exact S-matrices for non simply-laced affine Toda theories”, Nucl. Phys. B382 (1992) 365. [29] P. E. Dorey, “A remark on the coupling-constant dependence in affine Toda field theories”, Phys. Lett. B312 (1993) 291; hep-th/9304149. [30] E. Corrigan, P. E. Dorey and R. Sasaki, “On a generalised bootstrap principle”, Nucl. Phys. B408 (1993) 579; hep-th/9304065. [31] T. J. Hollowood, “Quantum soliton mass corrections in sl(n) affine Toda field theory”, Phys. Lett. B300 (1993) 73; hep-th/9209024. [32] N. J. MacKay and G. M. T. Watts, “Quantum mass corrections for affine Toda solitons”, Nucl. Phys. B441 (1995) 277; hep-th/9411169; G. M. T. Watts, “Quan(1) tum mass corrections for c2 affine Toda solitons”, Phys. Lett. B338 (1994) 40; hep-th/9404065. [33] G. W. Delius and M. T. Grisaru, “Toda soliton mass corrections and the particlesoliton duality conjecture”, Nucl. Phys. B441 (1995) 259; hep-th/9411176. [34] G. Takacs, “Quantum affine symmetry and scattering amplitudes of the imaginary(3) coupled d4 affine Toda field theory”, Nucl. Phys. B502 (1997) 629; hep-th/9701118; (3) (1) “The R-matrix of the Uq (d4 ) algebra and g2 affine Toda field theory”, Nucl. Phys. B501 (1997) 711; hep-th/9702196. [35] P. E. Dorey, unpublished. [36] G. M. Gandenberger and N. J. MacKay, “Remarks on excited states of affine Toda solitons”, Phys. Lett. B390 (1997) 18; hep-th/9608055.
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[37] G. M. Gandenberger and N. J. MacKay, “Exact S-matrices for dn+1 affine Toda solitons and their bound states”, Nucl. Phys. B457 (1995) 240; hep-th/9506169. [38] G. W. Delius, M. D. Gould and Y.-Z. Zhang, “Twisted quantum affine algebras and solutions to the Yang–Baxter equation”, Int. J. Mod. Phys. A11 (1996) 3415; math.QA/9508012. [39] G. M. Gandenberger, N. J. MacKay and G. M. T. Watts, “Twisted algebra R-matrices (1) and exact S-matrices for bn affine Toda solitons and their bound states”, Nucl. Phys. B465 (1996) 329; hep-th/9509007. (3) [40] H. S. Cho, I.-G. Koh and J. D. Kim, “Duality in the d4 affine Toda theory”, Phys. Rev. D47 (1993) 2625. [41] T. Oota, “q-deformed Coxeter element in nonsimply-laced affine Toda field theories”, Yukawa Inst. (Kyoto) preprint; hep-th/9706054. [42] A. A. P. Iskandar, “On breathers in affine Toda theories”, Ph.D. thesis, Durham, 1995. [43] J. M. Evans, N. J. MacKay and M. Ul-Hassan, “Conserved charges and supersymmetry in principal chiral models”, DAMTP-97-120, hep-th/9711140. [44] E. Frenkel and N. Reshetikhin, “Deformations of W-algebras associated to simple Lie algebras”, math.QA/9708006; P. Bouwknegt and K. Pilch, “On deformed W-algebras and quantum affine algebras”, math.QA/9801112. [45] E. Corrigan, “Integrable field theory with boundary conditions”, hep-th/9612138.
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS ´ HEILES SYSTEM VIA DEGREE FOR OF HENON S1 -EQUIVARIANT ORTHOGONAL MAPS ANDRZEJ MACIEJEWSKI Toru´ n Centre for Astronomy Nicolaus Copernicus University PL-87-100 Toru´ n, ul. Gagarina 11, Poland E-mail : [email protected]
SLAWOMIR RYBICKI∗ Faculty of Mathematics and Informatics Nicolaus Copernicus University PL-87-100 Toru´ n, ul. Chopina 12/18, Poland E-mail : [email protected] Received 25 June 1998 Revised 19 January 1998 1991 Mathematics Subject Classification: Primary 34C23, 58E09; Secondary 34C25 In this article we investigate global bifurcations of non-stationary periodic solutions of autonomous Hamiltonian systems. As the main tool we use the degree theory for S1 equivariant orthogonal maps (see [24]–[26]). In order to illustrate our abstract results, we study global bifurcations of periodic solutions of the famous H´enon–Heiles Hamiltonian system.
1. Introduction In their seminal paper, [15] M. H´enon and C. Heiles introduced the celebrated polynomial Hamiltonian system which is now called their names. This system originated from the study of motion of a star in an axially symmetric galaxy. Later on, the name of H´enon–Heiles system was attached to any family of natural Hamiltonian systems with polynomial potential of degree three. The aim of H´enon and Heiles studies was to answer the question about the existence of “the third integral”. In fact, using the Poincar´e cross section, they showed the presence of the deterministic chaos in the system. Since the publication of paper [15] many aspects of H´enon–Heiles system have been studied. Two of them, complementary in some sense, connected with integrability and non-integrability, seem to be very important. On the one hand, in a multi-parameter family of H´enon–Heiles systems few integrable cases were found. Important studies of these integrable cases can be found, e.g. in [6, 8] and in references therein. On the other hand, many efforts ∗ Partially
supported by KBN grant 2 PO3A 047 12 and Nicholas Copernicus University grant
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were done to show that the H´enon–Heiles system is not integrable except for known cases (see e.g. [14, 16, 18]). Still less is known about families of particular solutions, or invariant sets of H´enon–Heiles system. To the best of the authors’ knowledge, till now, the most complete description of periodic solutions in this system can be found in [10, 23]. It is well known that solutions of Hamiltonian systems can be considered as critical points of functionals defined on suitably chosen functional Hilbert spaces (see [20, 22, 27]), and that symmetry has a strong impact on the number and shape of solutions of variational problems (see [7]). This can be observed when we look for periodic solutions of autonomous Hamiltonian systems and exploit the invariance under time shifts, as well as the symmetries of Hamiltonian. The aim of this paper is to show applications of a new tool — degree theory for S1 -equivariant orthogonal maps, defined in [24], for the study of global bifurcations of periodic solutions in a Hamiltonian system. In order to do this, we study the H´enon–Heiles Hamiltonian system from the global bifurcation theory point of view. The plan of this paper is following. In Sec. 2, we present without proof a finite-dimensional global bifurcation theorem for S1 -equivariant gradient maps (Theorem 2.2). In Subsecs. 3.1 and 3.2, we prove global bifurcation theorems for Hamiltonian systems (Theorems 3.1 and 3.2). As tools, we use the Amann–Zehnder reduction (see [2]–[5]) and the degree theory for S1 -equivariant orthogonal maps. Since any gradient map is orthogonal (see [24] for the definition of the class of orthogonal maps), we can apply degree for orthogonal maps in our situation. In Sec. 4, we study families of periodic solutions, of an arbitrary period, emanating from stationary solutions of H´enon–Heiles system (4.12). The main result of this section is Theorem 4.1. It is worth pointing out that in this theorem we use Z3 -symmetry of H´enon–Heiles Hamiltonian. Section 5 is devoted to the study of 2π-periodic solutions of H´enon–Heiles family (5.16). In Theorems 5.1, 5.2 and 5.3 we describe topological properties of the set of 2π-periodic, π-periodic and 2π/jperiodic solutions of family (5.16), respectively. 2. Mathematical Background The aim of this section is to set up the terminology and notation. We also formulate here a finite-dimensional version of the famous Rabinowitz alternative for the class of S1 -equivariant gradient maps. We consider compact Lie group S1 = {z ∈ C : |z| = 1} and its proper subgroups Zk = {0, 1, . . . , k − 1}. Let V be a representation of S1 and v ∈ V , then S1v = {g ∈ S1 : g · v = v}, is called the isotropy group of v. If K ∈ {S1 , {e}, Z2 , Z3 , . . .}, then V K = {v ∈ V : g · v = v ∀ g ∈ K}, is the set of fixed points of the action of group K. For j ∈ N = {1, 2, 3, . . .}, we define a map ρj : S1 → GL(2, R) as follows: " # cos j · θ − sin j · θ j i·θ ρ (e ) = , 0 ≤ θ ≤ 2·π. sin j · θ cos j · θ For k, j ∈ N, we denote by R[k, j] the direct sum of k copies of representation (R2 , ρj ); the trivial k-dimensional representation of S1 we denote by R[k, 0]. We
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say that two representations V and W are equivalent (V ≈ W ) if there exists an S1 -equivariant, linear isomorphism L : V → W . The following classical result gives a complete classification (up to the equivalence) of finite-dimensional representations of the group S1 . Theorem 2.1 (Classification theorem). If V is a finite-dimensional representation of S1 , then there exist finite sequences {ki }, {ji } satisfying conditions ji ∈ {0} ∪ N ,
ki ∈ N , 1 ≤ i ≤ r , j1 < j2 < · · · < jr , (∗) L such that V is equivalent to ri=1 R[ki , ji ]. Moreover, the equivalence class of V (V ≈ Lr i=1 R[ki , ji ]) is uniquely determined by {ji }, {ki } satisfying (∗). See [1] for a proof of this theorem. Let V be a finite-dimensional representation of the group S1 , and let f : V ×R → R be a S1 -equivariant C 2 -function, such that: (H.1) there exist v1 , . . . , vk ∈ V S such that ∇f −1 (0)∩(V S ×R) = {v1 , . . . , vk }× R, where ∇f denotes the gradient of f with respect to the first coordinate, (H.2) for any vi ∈ {v1 , . . . , vk }, and any real numbers α < β 1
1
#({(vi , λ) : det(∇2 f (vi , λ)) = 0} ∩ ({vi } × [α, β])) < ∞ , where #(A) denotes the number of elements of the set A, and ∇2 f denotes the Hessian of f . For any vi ∈ {v1 , . . . , vk }, we define A(vi ) = {λ ∈ R : det(∇2 f (vi , λ)) = 0}. Definition 2.1. A set {v1 , . . . , vk } × R is said to be the set of trivial solutions of the equation ∇f (v, λ) = 0. A trivial solution (vi , λ0 ) is called the bifurcation point of nontrivial solutions of the equation ∇f (v, λ) = 0, if (vi , λ0 ) ∈ Z(∇f ) := closure({(v, λ) ∈ V × R : ∇f (v, λ) = 0 and v ∈ / {v1 , . . . , vk }}) . The set of bifurcation points will be denoted by B(∇f ). The connected component of set Z(∇f ) containing (vi , λ0 ) is denoted by C(vi , λ0 ). Remark 2.1. By implicit function theorem we obtain the following: if (vi , λ0 ) is a bifurcation point of nontrivial solutions of the equation ∇f (v, λ) = 0, then Sk λ0 ∈ A(vi ). In other words, only the points of the set i=1 {vi }×A(vi ) are suspected to be the bifurcation points. In the rest of this section, for any (vi , λ0 ) ∈ bifurcation index
Sk
i=1 {vi }
× A(vi ), we define the
! ∞ M η(vi , λ0 ) = ηS1 (vi , λ0 ), η{e} (vi , λ0 ), ηZ2 (vi , λ0 ), . . . , ηZk (vi , λ0 ), . . . ∈ Z⊕ Z , i=1
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λ λο + ε
Dα ( v ) x { λ ο + ε } i
λο T
( V S1 )
λο − ε
vi
Dα ( v ) x { λ ο − ε } i
1
VS
Fig. 1. For definition of bifurcation index.
whose non-triviality implies (vi , λ0 ) ∈ B(∇f ). In fact, the non-triviality of our index will imply the global bifurcation in the sense of Rabinowitz. Sk Let us fix (vi , λ0 ) ∈ i=1 {vi } × A(vi ). By assumption (H.2) there is ε > 0 / A(vi ), ∇2 f (vi , λ0 ± ε) are such that [λ0 − ε, λ0 + ε] ∩ A(vi ) = {λ0 }. Since λ0 ± ε ∈ non-degenerate. Therefore, there is α > 0 such that ∇2 f (·, λ0 ± ε)−1 (0) ∩ (Dα (vi ) × {λ0 ± ε}) = {(vi , λ0 ± ε)}, where Dα (vi ) = {v ∈ V : |v − vi | < α}, see Fig. 1. Now we are in a position to define the bifurcation index η(vi , λ0 ) by the formula ! ∞ M Z , η(vi , λ0 ) = DEG(∇f (·, λ0 +ε) , Dα (vi ))−DEG(∇f (·, λ0 −ε), Dα (vi )) ∈ Z⊕ i=1
where DEG(·, ·) = DEGS1 (·, ·), DEG{e} (·, ·), DEGZ2 (·, ·), . . . , DEGZk (·, ·), . . . , denotes the degree of S1 -equivariant orthogonal maps defined by the second author in [24]. Index η(vi , λ0 ) can be effectively computed in the following way. By Theorem 2.1 one can assume that V = R[k, 0] ⊕ R[k1 , j1 ] ⊕ · · · ⊕ R[kr , jr ], where 0 ≤ k, 0 < ki , 0 < j1 < · · · < jr . It is known that ± A0 (vi ) 0 0 0 0 A± 0 1 (vi ) 0 2 ∇ f (vi , λ0 ± ε) = . .. 0 . 0 0 0 0 0 A± (v ) i r
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By Lemma 4.1 in [24] we get η(vi , λ0 ) = DEG(∇2 f (vi , λ0 + ε), Dα (0)) − DEG(∇2 f (vi , λ0 − ε), Dα (0)) . In other words, in order to compute the bifurcation index it is enough to compute the degrees of isomorfisms ∇2 f (vi , λ0 ± ε) on a disc centred at the origin and subtract them. Applying the formula for degree of isomorphism, see Corollary 4.3 in [24], we compute the coordinate η(vi , λ0 )Q of the bifurcation index η(vi , λ0 ), which corresponds to the isotropy group Q, as follows: η(vi , λ0 )Q =
−
(−1)m
(A+ (vi )) 0
−
− (−1)m
(A− (vi )) 0
0 + − − − + m (A (v )) − i 0 · m (Ak (vi )) − (−1)m (A0 (vi )) · m− (A− k (vi )) (−1) 2
Q = S1 , Q = Zjm , jm ∈ / J (V ) Q = Zjm , jm ∈ J (V )
where m− (A) denotes the Morse index of a symmetric matrix A and J (V ) = + − − − {j1 , . . . , jr }. It is understood that if k = 0, then (−1)m (A0 (vi )) = (−1)m (A0 (vi )) = 1. Throughout this paper Θ = (0, 0, 0, . . .) denotes the trivial element in the group L∞ Z ⊕ ( i=1 Z). We can now formulate the main theorem of this section. This theorem provides a sufficient condition for the global bifurcation of zeros of a family of S1 -equivariant gradient maps. Moreover, it yields information about global behavior of a continuum of nontrivial solutions which bifurcate from the set of trivial solutions. Theorem 2.2 (Finite-dimensional global bifurcation theorem). If η(vi , λ0 ) 6= Θ, then a) either continuum C(vi , λ0 ) is unbounded in V × R, Sk Sk m b) or C(vi , λ0 )∩({v1 , . . . , vk }×R) = m=1 {vm }×{λm 1 , . . . , λim } ⊂ m=1 {vm }× Avm ), and im k X X
η(vm , λm i ) = Θ.
m=1 i=1
This theorem can be proved in a standard way using the complementing map trick. In fact in order to prove this theorem it is enough to repeat the reasoning given by Ize in [21] for the Leray–Schauder degree. But in our proof we must replace the Leray–Schauder degree with the degree for S1 -equivariant orthogonal maps. Remark. Actually, we do not use the simple structure of the set of trivial solutions in the above theorem. That is why one can prove a generalization of Theorem 2.2 in a case when the set of trivial solutions consists of a finite number 1 of unbounded curves in V S × R.
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3. Global Bifurcations of Periodic Solutions of Hamiltonian Systems In this section, we consider a one parameter family of Hamiltonian systems defined by C 2 -Hamiltonian function H : R2N × R → R. We distinguish two different cases. In the first case, we assume that H(x, λ) = λH(x), and this kind of parametrisation is an effect of rescaling of time variable (see next section). The second case corresponds to a nonlinear dependence of H on λ. 3.1. Case H(x, λ) = λH(x) In this subsection, we are interested in finding sufficient conditions for the existence of bifurcation points of non-stationary periodic solutions of a fixed period 2π, for a parametrized family of Hamiltonian systems of the following form: x˙ = λJ∇H(x) ,
(3.1)
0 −Id ] is the standard symplectic matrix, and H : R2N → R is a where J = [ Id 0 2 C -map, such that
1. ∇H −1 (0) = {s1 , . . . , sk }, 2. ∇2 H(si ) is non-degenerate for all i = 1, . . . , k. Following [5], one can consider the space L2 ((0, 2π); R2N ) a representation of the group S1 with an action of S1 given as follows: ϕ · u(t) = u(t + ϕ) , for fixed u(t) ∈ L2 ((0, 2π); R2N ), and ϕ ∈ S1 (in this notation we identify a complex number of module 1 with its argument). The space L2 ((0, 2π); R2N ) ⊕ R will be considered a representation of the group S1 with action of S1 given by the formula: ϕ · (u(t), λ) = (ϕ · u(t), λ) . 1 Let H2π = {x ∈ H 1 ([0, 2π]; R2N ) : x(0) = x(2π)} be a Hilbert space with a scalar product defined by the formula Z 2π 1 hx, yiH2π = [(x(t), y(t)) + (x(t), ˙ y(t))] ˙ dt , 0
1 for x, y ∈ H2π , where the space H 1 ([0, 2π]; R2N ) is the Sobolev space of absolutely continuous functions whose first derivative is in L2 ((0, 2π); R2N ). It is known that 2π-periodic solutions of Hamiltonian system (3.1) can be considered critical points 1 × (with respect to the first coordinate) of an S1 -equivariant C 2 -functional f : H2π R → R, given by the formula: Z Z 2π 1 2π f (x, λ) = (−J x(t), ˙ x(t)) dt − λ H(x(t)) dt . (3.2) 2 0 0
Definition 3.1. The set {s1 , . . . , sk } × R is said to be the set of trivial solutions 1 × R is said to be a bifurcation point of of system (3.1). A point (si , λ0 ) ∈ H2π
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non-stationary 2π-periodic solutions of Hamiltonian system (3.1) if for its any open 1 × R there exists a non-stationary 2π-periodic solution of neighborhood O ⊂ H2π system (3.1) in O. Let us denote by C(si , λ0 ) a connected component of the set 1 closure {(x(t), λ) ∈ H2π × R : x(t) is a non-stationary solution of (3.1) on level λ} ,
containing (si , λ0 ). For j ∈ N, and a symmetric matrix A, we define matrices Q(j, A) as follows: # " −A jJ T , Q(j, A) = jJ −A and for (i, j) ∈ {1, . . . , k} × N, we define sets J (i, j) = {λ ∈ R : det Q(j, λ∇2 H(si )) = 0} . Remark 3.1. The following statement holds true ∀ a < b ∃ j0 ∈ N ∀ i ∈ {1, . . . , k} ∀ j > j0 1. J (i, j) ∩ [a, b] = ∅, 2. m− (Q(j, λ∇2 H(si ))) = 2N . Remark 3.2. Applying the Hartman–Grobman theorem we can easily prove a necessary condition for the existence of bifurcation point of non-stationary 2π1 × R is a periodic solutions of Hamiltonian system (3.1). Namely, if (si , λ0 ) ∈ H2π bifurcation point then there is j ∈ N such that λ0 ∈ J (i, j). Only the elements of S S the set ki=1 {si } × ∞ j=1 J (i, j) are suspected to be bifurcation points. Let us fix si ∈ {s1 , . . . , sk }, j ∈ N and λ0 ∈ J (i, j). By Remark 3.1 one can choose ε > 0 such that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 } . L The bifurcation index η(si , λ0 ) ∈ Z ⊕ ( ∞ i=1 Z) is defined as follows: η(si , λ0 )S1 = 0 ,
(3.3)
and η(si , λ0 )Zj = (−1)m
−
(−λ0 ∇2 H(si ))
m− (Q(j, (λ0 + ε)∇2 H(si ))) − m− (Q(j, (λ0 − ε)∇2 H(si ))) . (3.4) 2 From Remark 3.1 it follows that our bifurcation index is well defined, i.e. only a finite number of coordinates of our bifurcation index is different from 0. We can now formulate the main theorem of this subsection. ·
Theorem 3.1 (Global bifurcation theorem for Hamiltonian systems, I). Let us fix j ∈ N, i ∈ {1, . . . , k}, λ0 ∈ J (i, j) and choose ε > 0, such that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 }. If m− (Q(j, (λ0 + ε)∇2 H(si ))) 6= m− (Q(j, (λ0 − ε)∇2 H(si ))), then continuum C(si , λ0 ) is
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A. MACIEJEWSKI and S. RYBICKI
1 1. either unbounded in H2π × R, 1 2. or bounded in H2π × R, and Sk Sk m (a) C(si , λ0 ) ∩ m=1 {sm } × R = m=1 ({sm } × {λm 1 , . . . , λim }) , im k X X
(b)
η(sm , λm i ) = Θ.
(3.5)
m=1 i=1 1 × R is bounded. Let |x|∞ = Proof. Suppose that continuum C(si , λ0 ) ⊂ H2π maxt∈[0,2π] |x(t)|. It is known that there exists a positive constant c, such that for 1 , any x ∈ H2π 1 . |x|∞ ≤ c · |x|H2π
As an immediate consequence of the above inequality, we obtain that the corresponding set of solution curves is bounded in R2N × R. We suppose that this set is included in an open disk Dα = (x, λ) ∈ R2N × R : |x|2 + λ2 < α2 , of a sufficiently large radius α 1. Let us define a smooth function ψ : R2N × R → R, such that ψ(x, λ) = 1 in D2α , and ψ(x, α) = 0 in (R2N × R) − D4α . We define H1 (x, λ) = λψ(x, λ)H(x) and modify Hamiltonian system (3.1) as follows: x˙ = J∇H1 (x, λ) .
(3.6)
Since the vector fields in systems (3.1) and (3.6) coincide in D2α , the set C(si , λ0 ) is also a bounded continuum of solutions of system (3.6). Moreover, the vector field in system (3.6) vanishes outside of D4α , and that is why sup (x,λ)∈R2N ×R
|∇2 H1 (x, λ)| < ∞ .
(3.7)
The rest of this proof falls naturally into two steps. In the first step, we will apply the S1 -equivariant Amann–Zehnder reduction given in [5]. Applying this reduction, we obtain a parametrized family of S1 -equivariant functions, defined on a finitedimensional representation of the group S1 . Step 1. Following the Amann–Zehnder reduction, see [5], we define functional 1 × R → R as follows: f : H2π Z Z 2π 1 2π (−J x, ˙ x) dt − H1 (x, λ) dt . (3.8) f (x, λ) = 2 0 0 It is well known that f is a C 2 -functional. Since ∇f (x, λ) = −J x˙ − ∇H1 (x, λ), looking for the 2π-periodic solutions of system (3.6) is equivalent to looking for the critical points of f with respect to x. In order to find the critical points of the functional f , it is enough to find critical points of a finite-dimensional function a ∈ C 2 (Z ⊕ R). Namely, similarly to the proof of Theorem 5 in [5], and thanks to Remark 2.2 of [4], in order to prove this theorem, it is enough to establish critical
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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points of a function a ∈ C 2 (Z ⊕ R), where linear space Z is defined as follows. Let A(x) = J x. ˙ It is known, see Lemma 3 of [5], that operator A is self-adjoint, and that σ(A) = Z, where σ(A) denotes the spectrum of A. By (3.7) one can choose ( ) jinf = inf
j∈N:
sup (x,λ)∈R2N ×R
|∇2 H1 (x, λ)| < j
and define a positive number β in the following way: 1 β= 2
|∇ H1 (x, λ)| + jinf 2
sup (x,λ)∈R2N ×R
,
! .
It is evident that β 6∈ σ(A), and that, for any (x, λ) ∈ R2N ⊕ R, we have σ(∇2 H1 (x, λ)) ⊂ (−β, β) , where σ(B) denotes the spectrum of matrix B. Let us denote by E(µ) a linear 1 , which is an eigenspace of A corresponding to the eigenvalue µ, subspace of H2π spanned by the following vectors: cos(µt)ek + sin(µt)Jek
for k = 1, 2, . . . , 2N .
We put n = max{j ∈ N : j < β}, and Z = E(0) ⊕
n M
(E(−j) ⊕ E(j)) .
j=1
Using notation of the Classification Theorem it can be easily seen that Z≈
n M
R[2N, j].
j=0
It was shown that the map a is constant on the orbits of the action of the group S1 (see p. 177 in [5]). From now on, we will be interested in the orbits of zeros of an S1 -equivariant, gradient map ∇a : Z ⊕ R → Z , where ∇a denotes the gradient of a with respect to the first coordinate. Let us list the main properties of the map ∇a (see Lemma 4 of [5]). There exists an S1 1 defined equivariant, C 1 -map y : Z × R → Z ⊥ , such that a map x : Z × R → H2π by the formula x(z, λ) = z + y(z, λ), satisfying x(si , λ) = si for any λ ∈ R and i = 1, . . . , k, has the following properties: 1. orbits of zeros of ∇a (critical orbits of a) are in an one-to-one correspondence with the critical points of f given by (3.8), i.e. ∇a(z, λ) = 0 iff x(z, λ) is a critical point of functional f . Moreover, a is of the form: Z Z 2π 1 2π (−J w, ˙ w) dt − H1 (w(t), λ) dt , a(z, λ) = 2 0 0 where w(t) = x(z, λ)(t),
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A. MACIEJEWSKI and S. RYBICKI
2. ∇a(si , λ) = 0 for all λ ∈ R and i = 1, . . . , k, 3. ∇2 a(si , λ)|R[2N,0] = −∇2 H1 (si , λ) = −λ∇2 H(si ), for all λ ∈ R and i = 1, . . . , k, 4. ∇2 a(si , λ)|R[2N,j] = Q(j, ∇2 H1 (si , λ)) = Q(j, λ∇2 H(si )), for all (si , λ) ∈ Dα and i = 1, . . . , k. Hence, for any i = 1, . . . , k 1
ker(∇2 a(si , λ)) ∩ Z S = ker(∇2 a(si , λ)) ∩ R[2N, 0] ( {0} for any λ 6= 0 , = R[2N, 0] for λ = 0 . Step 2. Now it is not difficult to verify that the map ∇a : Z ⊕ R → Z satisfies all the assumptions of Theorem 2.2. Thus, applying Theorem 2.2, we complete our proof. A similar theorem has been announced by Dancer in [13]. However, Dancer used as a tool another version of degree for S1 -equivariant gradient maps. Remark 3.3. The period of periodic solution whose existence is guaranteed by the above theorem need not be minimal period. This is the usual feature of results obtained by means of variational methods. 3.2. Case ∇H(x, λ)6= 6 λ∇H(x) In this subsection, we are interested in finding sufficient conditions for the existence of bifurcation points of non-stationary periodic solutions of a fixed period 2π for a parametrized family of Hamiltonian systems of the following form: x˙ = J∇H(x, λ) ,
(3.9)
where H : R2N × R → R is a C 2 -map, such that ∇H −1 (0) = {ϕ1 (R1 ), . . . , ϕk (Rk )}, where for any i = 1, . . . , k 1. Ri is either an open half-line or R, 2. ϕi : Ri → R2N × R are continuous maps, and ϕi (λ) = (xi (λ), λ), where xi : Ri → R2N , is a continuous map. Definition 3.2. A set ϕ1 (R1 ) ∪ · · · ∪ ϕk (Rk ) is said to be the set of trivial 1 × R is said solutions of system (3.9). A point ϕi (λ0 ) ∈ ϕ1 (R1 ) ∪ · · · ∪ ϕk (Rk ) ⊂ H2π to be a bifurcation point of non-stationary 2π-periodic solutions of Hamiltonian 1 × R there exists a nonsystem (3.9) if for its any open neighborhood O ⊂ H2π stationary 2π-periodic solution of system (3.9) in O. Let us denote by C(ϕi (λ0 )) a connected component of the set 1 × R : x(t) is a non-stationary solution of (3.9) on level λ} , closure{(x(t), λ) ∈ H2π
containing ϕi (λ0 ).
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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As in the previous subsection, for (i, j) ∈ {1, . . . , k} × N, we define matrices Q(j, ∇2 H(ϕi (λ))) as follows: # " jJ T −∇2 H(ϕi (λ)) 2 , Q(j, ∇ H(ϕi (λ))) = jJ −∇2 H(ϕi (λ)) and sets J (i, j) = {λ ∈ Ri : det Q(j, ∇2 H(ϕi (λ))) = 0}. Assume that for any (i, j) ∈ {2, . . . , k} × N 1. #(J (i, j)) < ∞, 2. if λ0 ∈ J (i, j) then ∇2 H(ϕi (λ0 )) is non-degenerate. S∞ Sk Remark 3.4. Notice that only elements of the set i=1 q j=1 ϕi (J (i, j)) can be the bifurcation points of non-stationary 2π-periodic solutions of system (3.9). For fixed (i, j) ∈ {2, . . . , k} × N, λ0 ∈ J (i, j) we choose ε > 0, such that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 } , L∞ and define the bifurcation index η(ϕi (λ0 )) ∈ Z ⊕ ( i=1 Z) as follows η(ϕi (λ0 ))S1 = 0, and η(ϕi (λ0 ))Zj = (−1)m ·
−
(−∇2 H(ϕi (λ0 )))
m− (Q(j, ∇2 H(ϕi (λ0 + ε))) − m− (Q(j, ∇2 H(ϕi (λ0 − ε))) . (3.10) 2
The bifurcation index is well defined. Theorem 3.2 (Global bifurcation theorem for Hamiltonian systems, II). Let us assume that m− (Q(j, ∇2 H(ϕi (λ0 + ε)))) 6= m− (Q(j, ∇2 H(ϕi (λ0 − ε)))), for fixed i ∈ {2, . . . , k}, j ∈ N, λ0 ∈ J (i, j), and ε > 0 chosen is such a way that [λ0 − ε, λ0 + ε] ∩ J (i, j) = {λ0 }. Then either C(ϕi (λ0 )) ∩ ϕ1 (R1 ) 6= ∅ or continuum C(ϕi (λ0 )) is 1 1. either unbounded in H2π × R, 1 2. or bounded in H2π × R and Sk Sk m (a) C(ϕi (λ0 )) ∩ m=2 ϕm (Rm ) = m=2 {ϕm (λm 1 ), . . . , ϕm (λim )} ,
(b)
im k X X
η(ϕm (λm i )) = Θ .
(3.11)
m=2 i=1
Theorem 3.2 is a slight generalization of Theorem 3.1. The proof of this theorem is in fact the same as the proof of Theorem 3.1 (see Remark 2.2). Remark 3.5. Let us fix j ∈ N. Then it is easy to check that C(ϕi (λ0 ))Zj = 1 Zj 1 ) × R ⊂ H2π × R is a continuum of 2π C(ϕi (λ0 )) ∩ (H2π j -periodic solutions of system (3.9).
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A. MACIEJEWSKI and S. RYBICKI
4. Global Bifurcations of Periodic Solutions of H´ enon Heiles System. Solutions of an Arbitrary Period In this section, we consider H´enon–Heiles system and, using the theory developed in the previous sections, study the global behavior of branches of periodic solutions, which emanate from the stationary solutions. The H´enon–Heiles Hamiltonian system has the form: x˙ = J∇H(x) ,
(4.12)
where the Hamiltonian H : R4 → R is given by the formula: H(x1 , x2 , x3 , x4 ) =
1 2 1 (x + x22 + x23 + x24 ) + x33 − x3 x24 . 2 1 3
(4.13)
System (4.12), written explicitly has the following form: x˙1 = −x3 − x23 + x24 , x˙2 = −x4 + 2x3 x4 , x˙3 = x1 , x˙4 = x2 . Lemma 4.1. The H´enon–Heiles system has the following properties: 1. if
2πi cos 3 2πi sin 3 g(i) = 0 0
− sin cos
2πi 3
0
2πi 3
0
0
cos
2πi 3
0
sin
2πi 3
0
0 , 2πi − sin 3 2πi cos 3
then J∇H(g(i)x) = g(i)J∇H(x) , for i = 0, 1, 2, 2. there are four stationary solutions
s3 =
s1 = (0, 0, 0, 0) , √ ! 1 3 = g(1)s2 , 0, 0, , 2 2
s2 = (0, 0, −1, 0) , √ ! 1 3 = g(2)s2 , s4 = 0, 0, , − 2 2
3. σ(∇2 H(s1 )) = {1, 1, 1, 1}, σ(∇2 H(s2 )) = σ(∇2 H(s3 )) = σ(∇2 H(s4 )) = {−1, 1, 1, 3},
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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1137
4. (a) σ(Q(j, λ∇2 H(s1 ))) = {−λ + j, −λ − j}, multiplicity of any eigenvalue equals 4, p p p (b) p σ(Q(j, λ∇2 H(si ))) = { λ2 + j 2 , − λ2 + j 2 , −2λ + λ2 + j 2 , −2λ − λ2 + j 2 , multiplicity of any eigenvalue equals 2, for i = 2, 3, 4. The following lemma is a direct consequence of Lemma 4.1. Lemma 4.2. It is evident that 1. J (1, j) = {±j}, 2. for sufficiently small ε > 0 m− (Q(j, (j + ε)∇2 H(s1 )) = 8 , m− (Q(j, (j − ε)∇2 H(s1 )) = 4 , m− (Q(j, (−j + ε)∇2 H(s1 )) = 4 , m− (Q(j, (−j − ε)∇2 H(s1 )) = 0 , −
2
H(s1 )) = 1, 3. (−1)m (±j∇ √ 3 4. J (i, j) = {± 3 j} for i = 2, 3, 4, 5. for sufficiently small ε > 0
! !! √ 3 2 j + ε ∇ H(si ) Q j, = m 3 ! !! √ 3 j − ε ∇2 H(si ) = m− Q j, 3 ! !! √ 3 − 2 j + ε ∇ H(si ) Q j, − = m 3 ! ! √ 3 − 2 j − ε ∇ H(si ) = Q(j, − m 3 −
6,
4,
4,
2,
for i = 2, 3, 4, √ − 2 3 6. (−1)m (± 3 j∇ H(si )) = −1, for i = 2, 3, 4. Let us change the study of periodic solutions of an arbitrary period of system (4.12) into the bifurcation problem of periodic solutions with a fixed period 2π. Namely, applying the change of variables s(t) = λt , we obtain ( x˙ = λJ∇H(x), (4.14) x(0) = x(2π) . It is clear that 2π-periodic solutions of system (4.14) on level λ correspond to 2πλperiodic solutions of system (4.12). We are now in a position to formulate the main theorem of this section. This theorem ensures the existence of unbounded (in
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A. MACIEJEWSKI and S. RYBICKI
period or amplitude) components of non-stationary periodic solutions of H´enon– Heiles system which emanate from stationary solutions of this system. Theorem 4.1. Let us fix any i ∈ {1, 2, 3, 4}, j ∈ N and λ0 ∈ J (i, j). Then continuum C(si , λ0 ) of non-stationary 2π-periodic solutions of system (4.14) is 1 × R. unbounded in H2π Proof. Our proof starts with the observation that the H´enon–Heiles system has Z3 × S1 symmetry, where Z3 -symmetries come from the symmetry of Hamiltonian (4.13). In fact, the H´enon–Heiles Hamiltonian is fixed on the orbits of the action 1 × R, but we do not need such rich symmetries in our of the group D3 × S1 on H2π proof. Functional (3.2) constructed for system (4.14) is fixed on the orbits of the action of the group Z3 × S1 given by (g(i), ϕ) ? (x(t), λ) := (g(i)x(t + ϕ), λ) , 1 × R is a solution where g(i) is defined in Lemma 4.1. Hence, if (x(t), λ) ∈ H2π 1 of system (4.14), and (g(i), ϕ) ∈ Z3 × S , then (g(i), ϕ) ? (x(t), λ) is a solution of system (4.14). By (3.3), (3.4) and Lemma 4.2, we compute the bifurcation indices ( 2 Q = Zj , η(s1 , j)Q = 0 Q 6= Zj ,
and
√
3 j η si , 3
!
( = Q
−1
Q = Zj ,
0
Q 6= Zj ,
for i = 2, 3, 4 and j ∈ N. Let us suppose, contrary to our claim, that the continuum C(si , λ0 ) is bounded in 1 × R. From Lemma 4.1 it follows that system (4.14) satisfies all the assumptions H2π of Theorem 3.1. Notice that the bifurcation indices η(s1 , j) are nontrivial and their √ nonzero coordinates are positive. On the other hand, the bifurcation indices η(si , 33 j) are nontrivial but their nonzero coordinates are negative. By the above and formula (3.5) there exists j ∈ N such that (s1 , j) ∈ C(si , λ0 ). In other words C(s1 , j) = C(si , λ0 ). The proof will be completed if we show that C(s1 , j) is unbounded for any j ∈ N. Let us fix j ∈ N. Since (g(i), 0) ? (s1 , j) = (s1 , j), (g(i), 0) ? C(s1 , j) = C(s1 , j) for i = 0, 1, 2. √Therefore, by Lemma 4.1, √ we have: if s1 6= sm and √ there exists j0 such that (sm , 33 j0 ) ∈ C(s1 , j) then (s2 , 33 j0 ) ∈ C(s1 , j), (s3 , 33 j0 ) ∈ C(s1 , j) and √ (s4 , 33 j0 ) ∈ C(s1 , j). Thus, by Theorem 3.1, we obtain (√ √ ) 3 3 j1 , . . . , jp ∪ {s1 } C(s1 , j) ∩ ({s1 , s2 , s3 , s4 } × R) = {sm } × 3 3 m=2 4 [
×{j, l1 , . . . , lq } ,
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
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1139
and, consequently, by (3.5) √ ! q X 3 ji + η sm , η(s1 , li ) + η(s1 , j) = Θ ∈ Z ⊕ 3 i=1 i=1
p 4 X X
∞ M
m=2
i=1
It is clear that p X
√
3 ji η sm , 3· 3 i=1
! +
q X
η(s1 , li ) + η(s1 , j) = Θ ∈ Z ⊕
∞ M
i=1
! Z
.
! Z
.
(4.15)
i=1
Let us look at the coordinate of (4.15) which corresponds to the isotropy group Zj . We check at once that this coordinate is equal to 2 or −1, contrary to (4.15). Corollary 4.1. As an immediate consequence of Theorem 4.1, we obtain the following. For any stationary solution si ∈ {s1 , s2 , s3 , s4 } of H´enon–Heiles system there exists a connected set C(si ) of periodic solutions of this system, emanating from si and satisfying at least one of the following conditions: 1. for any arbitrary large period T there exists a T -periodic function in C(si ), 2. there exists a function in C(si ) of arbitrary large amplitude. The results of this section were obtained earlier in [11, 17], see the remark below. The aim of this section was to show how our general machinery works in the case of the well-known H´enon–Heiles Hamiltonian system. However, we would like to point out that our methods also work for more general class of Hamiltonian systems for which approach used in [11, 17] cannot be applied, see Sec. 5. In Sec. 5 we prove new results on periodic solutions of H´enon–Heiles family of Hamiltonian systems. Remark 4.1. The sets C(si ) have the following properties: 1. C(s1 ) has property 1. stated in Corollary 4.1 (see [17]), 2. C(s1 ) has property 2. stated in Corollary 4.1, because C(s1 ) ∩ H −1 (h) 6= ∅ for any h > 0 (see [11]), 3. C(si ) has property 2. stated in Corollary 4.1, because C(si ) ∩ H −1 (h) 6= ∅ for any h > 0 (see [11]), for i = 2, 3, 4. 5. Global Bifurcations of Periodic Solutions of H´ enon Heiles System. Solutions of Fixed Period T = 2π In this section, we consider an 1-parameter H´enon–Heiles family of Hamiltonian systems and, using the theory developed in previous sections, study the global behavior of branches of 2π-periodic solutions, which emanate from the curves of stationary solutions. Let us consider a family of Hamiltonian systems of the form: x˙ = J∇H(x, λ) ,
(5.16)
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A. MACIEJEWSKI and S. RYBICKI
where the Hamiltonian H : R4 × R → R is given by the formula: H(x1 , x2 , x3 , x4 , λ) =
1 2 λ (x1 + x22 + x23 + x24 ) + x33 − x3 x24 . 2 3
Family (5.16), written explicitly, has the form: x˙1 = −x3 − λx23 + x24 , x˙2 = −x4 + 2x3 x4 , x˙3 = x1 , x˙4 = x2 . It was considered in [9, 19]. Lemma 5.1. The H´enon–Heiles family (5.16) has the following properties: 1. there are five curves of stationary solutions of family (5.16), namely ϕ1 (λ) = ((0, 0, 0, 0), λ) 1 0, 0, − , 0 , λ ϕ2 (λ) = λ 1 0, 0, − , 0 , λ ϕ3 (λ) = λ 1 √λ+2 ,λ 0, 0, , 2 ϕ4 (λ) = 2 √ 1 λ+2 ,λ ϕ5 (λ) = 0, 0, , − 2 2
R1 = R , R2 = {λ ∈ R : λ > 0} , R3 = {λ ∈ R : λ < 0} , R4 = {λ ∈ R : λ ≥ −2} , R5 = {λ ∈ R : λ ≥ −2} ,
2. moreover, σ(Q(j, ∇2 H(ϕ1 (λ)))) = {±j − 1} ,
for λ ∈ R1 ,
σ(Q(j, ∇2 H(ϕ2 (λ)))) ) ( p 2 λ2 p 2λ + 2 ± 1 + j , = ± 1 + j2, − 2λ σ(Q(j, ∇2 H(ϕ3 (λ)))) ) ( p 2 λ2 p 2λ + 2 ± 1 + j , = ± 1 + j2, − 2λ
for λ ∈ R2 ,
for λ ∈ R3 ,
σ(Q(j, ∇2 H(ϕ4 (λ)))) ) ( p p 3 + λ ± (λ + 1)2 + 4j 2 2 , = ± 1 + j ,− 2
for λ ∈ R4 ,
GLOBAL BIFURCATIONS OF PERIODIC SOLUTIONS OF
...
σ(Q(j, ∇2 Hϕ5 (λ)))) ) ( p 2 + 4j 2 p 3 + λ ± (λ + 1) , = ± 1 + j2, − 2
1141
for λ ∈ R5 ,
and the multiplicity of any eigenvalue is equal to 4, 2, 2, 2, 2, respectively. For any (i, j) ∈ {1, 2, 3, 4, 5} × N, we define J (i, j) = {λ : det(Q(j, ∇2 H(ϕi (λ)))) = 0} . The following lemma is a direct consequence of Lemma 5.1. Lemma 5.2. It is easy to check that ( J (1, j) =
1.
for j = 1 ,
∅
for j > 1 ,
∅ ∅ J (2, j) = ( ) p 4 + 3j 2 + 4 λ2,j = j2 − 4
2.
for j = 1 , for j = 2 , for j > 2 ,
( ) √ √ 4+ 7 4− 7 λ3,0 = − , λ3,1 = − 3 3 3 λ3,2 = − J (3, j) = 8 ) ( p 4 − 4 + 3j 2 λ3,j = j2 − 4
3.
4. 5. 6. 7.
R
J (4, j) = {λ4,j = −2 + j 2 } for j ≥ 1, J (5, j) = {λ5,j = −2 + j 2 } for j ≥ 1, − 2 (−1)m (−∇ H(ϕ1 (λ))) = 1, − 2 (−1)m (−∇ H(ϕ2 (λ))) = −1, ( (−1)m
8. −
2
−
(−∇2 H(ϕ3 (λ)))
=
1 −1
9. (−1)m (−∇ H(ϕ4 (λ))) = −1 for λ > −2, − 2 10. (−1)m (−∇ H(ϕ5 (λ))) = −1 for λ > −2,
for j = 1 ,
for j = 2 ,
for j > 2 ,
for λ ∈ (−2, 0) , for λ ∈ (−∞, −2) ,
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A. MACIEJEWSKI and S. RYBICKI
11. for j > 2: (−1)m
−
(−∇2 H(ϕ2 (λ2,j )))
· (m− (Q(j, ∇2 H(ϕ2 (λ2,j + ε))))
−m− (Q(j, ∇2 H(ϕ2 (λ2,j − ε)))) = −1 · (4 − 6) = 2 , 12.
(−1)m
−
(−∇2 H(ϕ3 (λ3,0 )))
(5.17)
· (m− (Q(2, ∇2 H(ϕ3 (λ3,0 + ε))))
−m− (Q(2, ∇2 H(ϕ3 (λ3,0 − ε)))) = −1 · (4 − 6) = 2 , (−1)m
−
(−∇2 H(ϕ3 (λ3,1 )))
(5.18)
· (m− (Q(2, ∇2 H(ϕ3 (λ3,1 + ε))))
−m− (Q(2, ∇2 H(ϕ3 (λ3,1 − ε)))) = 1 · (2 − 4) = −2 , 13.
(−1)m
−
(−∇2 H(ϕ3 (λ3,2 )))
(5.19)
· (m− (Q(2, ∇2 H(ϕ3 (λ3,2 + ε))))
−m− (Q(2, ∇2 H(ϕ3 (λ3,2 − ε)))) = 1 · (2 − 4) = −2 ,
(5.20)
14. for j > 2: (−1)m
−
(−∇2 H(ϕ3 (λ3,j )))
· (m− (Q(j, ∇2 H(ϕ3 (λ3,j + ε))))
−m− (Q(j, ∇2 H(ϕ3 (λ3,j − ε)))) = 1 · (2 − 4) = −2 ,
(5.21)
15. for j ≥ 1: (−1)m
−
(−∇2 H(ϕ4 (λ4,j )))
· (m− (Q(j, ∇2 H(ϕ4 (λ4,j + ε))))
−m− (Q(j, ∇2 H(ϕ4 (λ4,j − ε)))) = −1 · (6 − 4) = −2 ,
(5.22)
16. for j ≥ 1: (−1)m
−
(−∇2 H(ϕ5 (λ5,j )))
· (m− (Q(j, ∇2 H(ϕ5 (λ5,j + ε))))
−m− (Q(j, ∇2 H(ϕ5 (λ5,j − ε)))) = −1 · (6 − 4) = −2 .
(5.23)
Now we are in a position to formulate the main theorem of this section. Namely, we describe topological properties of continua of non-stationary 2π-periodic solutions of H´enon–Heiles family (5.16) which bifurcate from the set of stationary solutions. Theorem 5.1. Continua of 2π-periodic solutions of family (5.16) have the following properties.
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1. If C(ϕ3 (λ3,0 )) ∩ ϕ1 (R1 ) = ∅, then 1 ×R (a) either C(ϕ3 (λ3,0 )) is unbounded is H2π (b) or C(ϕ3 (λ3,0 )) is bounded and there is i ∈ {3, 4, 5}, such that ϕi (λi,1 ) ∈ C(ϕ3 (λ3,0 )); moreover, either C(ϕk (λk,1 )) ∩ ϕ1 (R1 ) 6= ∅ or C(ϕk (λk,1 )) 1 × R, for k ∈ {3, 4, 5} − {i}, is unbounded in H2π 3. for any i ∈ {3, 4, 5}, if C(ϕi (λi,1 )) ∩ ϕ1 (R1 ) = ∅, then 1 × R, (a) either C(ϕi (λi,1 )) in unbounded is H2π (b) or C(ϕi (λi,1 )) is bounded, and ϕ3 (λ3,0 ) ∈ C(ϕi (λi,1 )); moreover, either 1 × R, for C(ϕk (λk,1 )) ∩ ϕ1 (R1 ) 6= ∅ or C(ϕk (λk,1 )) is unbounded in H2π k ∈ {3, 4, 5} − {i}, 3. for any i ∈ {3, 4, 5}; if C(ϕi (λi,2 )) ∩ ϕ1 (R1 ) = ∅, then C(ϕi (λi,2 )) is un1 × R, bounded in H2π 4. for any j ≥ 3, if C(ϕ2 (λ2,j )) ∩ ϕ1 (R1 ) = ∅, then 1 × R, (a) either C(ϕ2 (λ2,j )) is unbounded in H2π (b) or C(ϕ2 (λ2,j )) is bounded, and there exists i ∈ {3, 4, 5}, such that ϕi (λi,j ) ∈ C(ϕ2 (λ2,j )); moreover, either C(ϕk (λk,j )) ∩ ϕ1 (R1 ) 6= ∅ or 1 × R, for k ∈ {3, 4, 5} − {i}, C(ϕk (λk,j )) is unbounded in H2π 5. for any i ∈ {3, 4, 5} and j ≥ 3, if C(ϕi (λi,j )) ∩ ϕ1 (R1 ) = ∅, then 1 ×R (a) either C(ϕi (λi,j )) is unbounded in H2π (b) or C(ϕi (λi,j )) is bounded, and ϕ2 (λ2,j ) ∈ C(ϕi (λi,j )); moreover, either 1 × R, for C(ϕk (λk,j )) ∩ ϕ1 (R1 ) 6= ∅ or C(ϕk (λk,j )) is unbounded in H2π k ∈ {3, 4, 5} − {i}. Proof. 1. By (5.18), (5.19), (5.22), (5.23) and (3.10), we obtain ( ( 2 Q = Z1 , −2 Q = Z1 , η(ϕ3 (λ3,1 ))Q = η(ϕ3 (λ3,0 ))Q = 0 Q 6= Z1 , 0 Q 6= Z1 , ( η(ϕ4 (λ4,1 ))Q =
−2
Q = Z1 ,
0
Q 6= Z1 ,
( η(ϕ5 (λ5,1 ))Q =
−2
Q = Z1 ,
0
Q 6= Z1 .
(5.24)
(5.25)
Moreover, for any i = 2, 3, 4, 5, and j > 1 η(ϕi (λi,j ))Z1 = 0 .
(5.26)
Let us suppose that C(ϕ3 (λ3,0 )) ∩ ϕ1 (R1 ) = ∅ and that C(ϕ3 (λ3,0 )) is bounded 1 × R. in H2π By (3.11) C(ϕ3 (λ3,0 ) ∩ {ϕ3 (λ3,1 ), ϕ4 (λ4,1 ), ϕ5 (λ5,1 )} 6= ∅, because the sum of L∞ bifurcation indices equals the trivial element in Z ⊕ ( i=1 Z). We fix i ∈ {3, 4, 5} such that ϕi (λi,1 ) ∈ C(ϕ3 (λ3,0 ). By (5.24)–(5.26) and (3.11), we have / C(ϕ3 (λ3,0 ) ϕk (λk,1 ) ∈
for
k ∈ {3, 4, 5} − {i} .
We fix k ∈ {3, 4, 5} − {i} and suppose that C(ϕk (λk,1 )) ∩ ϕ1 (R1 ) = ∅. By (5.24)– (5.26) and (3.11), continuum C(ϕk (λk,1 )) is unbounded.
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A. MACIEJEWSKI and S. RYBICKI
2. The proof for 2. is similar to the proof of 1. 3. By (5.20), (5.22), (5.23) and (3.10), we obtain ( ( −2 Q = Z2 , −2 η(ϕ4 (λ4,2 ))Q = η(ϕ3 (λ3,2 ))Q = 0 Q 6= Z2 , 0 ( η(ϕ5 (λ5,2 ))Q =
−2
Q = Z2 ,
0
Q 6= Z2 .
Q = Z2 , Q 6= Z2 ,
(5.27)
(5.28)
Moreover, for any i = 2, 3, 4, 5, and j 6= 2 η(ϕi (λi,j ))Z2 = 0 .
(5.29)
We fix i ∈ {3, 4, 5} and suppose that C(ϕi (λi,2 )) ∩ ϕ1 (A1 ) = ∅. By (5.27)–(5.29) and (3.11) continuum C(ϕi (λi,2 )) is unbounded. 4. The proof of 4. is similar to the proof of 1. 5. The proof of 5. is similar to the proof of 1. Combining Theorem 3.2 and Remark 3.5 with the fact that ∇2 H(ϕ1 (λ)) = Id for any λ ∈ R1 , one can prove the following two theorems. These theorems yield information about 2π/j-periodic solutions of family (5.16). The proofs of these theorems are in fact similar to the proof of Theorem 5.1. The only difference is that we must restrict functional (3.2) to the set of fixed points of the action of the group Z2 in Theorem 5.2 and Zj , j > 2 in Theorem 5.3. Theorem 5.2. Continua of π-periodic solutions of family (5.16) have the following properties: 1 × 1. continua C(ϕ3 (λ3,2 ))Z2 , C(ϕ4 (λ4,2 ))Z2 , C(ϕ5 (λ5,2 ))Z2 are unbounded in H2π R, 2. for any even j > 2 (a) either C(ϕ2 (λ2,j ))Z2 , C(ϕ3 (λ3,j ))Z2 , C(ϕ4 (λ4,j ))Z2 , C(ϕ5 (λ5,j ))Z2 are 1 × R, unbounded in H2π Z2 (b) or C(ϕ2 (λ2,j )) is bounded and there is i ∈ {3, 4, 5} such that ϕi (λi,j ) ∈ 1 × R, for C(ϕ2 (λ2,j ))Z2 ; moreover, C(ϕk (λk,j ))Z2 is unbounded in H2π k ∈ {3, 4, 5} − {i},
3. for any i ∈ {3, 4, 5} and even j > 2 1 × R, (a) either C(ϕi (λi,j ))Z2 is unbounded in H2π Z2 (b) or C(ϕi (λi,j )) is bounded and ϕ2 (λ2,j ) ∈ C(ϕi (λi,j ))Z2 ; moreover, 1 × R, for k ∈ {3, 4, 5} − {i}. C(ϕk (λk,j ))Z2 is unbounded in H2π Theorem 5.3. Let j ≥ 3, then continua of have the following properties. For any k ∈ N:
2π j -periodic
solutions of family (5.16)
1. either continua C(ϕ2 (λ2,kj))Zj , C(ϕ3 (λ3,kj))Zj , C(ϕ4 (λ4,kj))Zj , C(ϕ5 (λ5,kj))Zj 1 are unbounded in H2π × R,
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2. or C(ϕ2 (λ2,kj ))Zj is bounded and there is i ∈ {3, 4, 5}, such that ϕi (λi,kj ) ∈ 1 × R for r ∈ C(ϕ2 (λ2,kj ))Zj ; moreover, C(ϕr (λr,kj ))Zj are unbounded in H2π {3, 4, 5} − {i}.
References [1] J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969. [2] H. Amann, “Multiple positive fixed points of asymptotically linear maps”, J. Funct. Anal. 17 (1974) 174–213. [3] H. Amann, “Saddle points and multiple solutions of differential equations”, Math. Z. 169 (1979) 127–166. [4] H. Amann and E. Zehnder, “Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations”, Ann. Sc. Norm. Super. Pisa, CC. Sci. IV, Ser 7, (1980) 539–603. [5] H. Amann and E. Zehnder, “Periodic solutions of asymptotically linear Hamiltonian systems”, Manuscr. Math. 32 (1980) 149–189. [6] M. Antonowicz and S. Rauch-Wojciechowski, “Bi-Hamiltonian formulation of the Henon–Heiles system and its multidimensional extension”, Phys. Lett. A163 (1992) 167. [7] T. Bartsch, “Topological methods for variational problems with symmetries”, Lect. Notes in Math. 1560, Springer-Verlag, Berlin-Heidelberg-New York, 1993. [8] M. Blaszak and S. Rauch-Wojciechowski, “A generalized Henon–Heiles system and related integrable Newton equations”, J. Math. Phys. 35 (1994) 1693. [9] M. Braun, “On the applicability of third integral of motion”, J. Diff. Eq. 13 (1973) 300–318. [10] R. C. Churchill, G. Pecelli and D. L. Rod, “Hyperbolic periodic orbits”, J. Diff. Eq. 24 (1977) 329–348. [11] R. C. Churchill, G. Pecelli and D. L. Rod, “A survey of the H´ enon–Heiles Hamiltonian with applications to related examples”, Lectures Notes in Phys. 93 (1979) 76–136. [12] R. C. Churchill and D. L. Rod, “Pathology in dynamical systems II: Applications”, J. Diff. Eq. 21 (1976) 66–112. [13] E. N. Dancer, “A new degree for S 1 -invariant mappings and applications”, Ann. Inst. H. Poincar´e, Analyse Nonlin´eaire 2 (1985) 473–486. [14] A. P. F. Fordy, “The Henon–Heiles system revisited”, Physica D52 (1991) 204. [15] M. H´enon and Heiles, “The applicability of the third integral of motion: Some numerical experiments”, Astronomical J. 69(1) (1964) 73–79. [16] H. Ito, “Non-integrability of H´ enon–Heiles system and a theorem of Ziglin”, Kodai Math. J. 8(1) (1985) 120–138. [17] R. H. G. Helleman, “Periodic solutions of arbitrary period, variational methods”, Lectures Notes in Phys. 93 (1979) 353–375. [18] S. Kasperczuk, “Homoclinic chaos in generalized H´enon–Heiles system”, Acta Phys. Polonica A88 (1995) 1073–1079. [19] M. Kummer, “On resonant nonlinearly coupled oscillators with two equal frequencies”, Commun. Math. Phys. 48 (1976) 53–79. [20] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Berlin Heidelberg New York, Springer, 1989. [21] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Inst. of Mathematical Sciences, New York, 1974. [22] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reg. Conf. Ser. Math. 35, Providence, R.I., Am. Math. Soc., 1986.
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[23] D. L. Rod and R. C. Churchill, “A guide to the H´ enon–Heiles Hamiltonian”, in Singularities and Dynamical Systems, ed. S. N. Pnevmatikos, Elsevier Sci. Publ., 1985, 385–395. [24] S. Rybicki, “S1 -degree for orthogonal maps and its applications to bifurcation theory”, Nonlinear Anal. TMA 23(1) (1994) 83–102. [25] S. Rybicki, “On periodic solutions of autonomous Hamiltonian systems via degree for S1 -equivariant gradient maps”, to appear in Nonlinear Anal. TMA, (1998). [26] S. Rybicki, “Applications of degree for S1 -equivariant gradient maps to variational nonlinear problems with S1 -symmetries”, Top. Meth. in Nonlin. Anal. 9(2) (1997) 383–417. [27] M. Struwe, “Variational methods; Applications to nonlinear partial differential equations and Hamiltonian systems”, A series of modern surveys in mathematics 34, Springer, (1996).
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES IN 1+1 DIMENSIONS ∗ ¨ MICHAEL MUGER
Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133 Roma Italy E-mail : [email protected] Received 6 May 1997 Revised 2 December 1997 We show that a large class of massive quantum field theories in 1 + 1 dimensions, characterized by Haag duality and the split property for wedges, does not admit locally generated superselection sectors in the sense of Doplicher, Haag and Roberts. Thereby the extension of DHR theory to 1 + 1 dimensions due to Fredenhagen, Rehren and Schroer is vacuous for such theories. Even charged representations which are localizable only in wedge regions are ruled out. Furthermore, Haag duality holds in all locally normal representations. These results are applied to the theory of soliton sectors. Furthermore, the extension of localized representations of a non-Haag dual net to the dual net is reconsidered. It must be emphasized that these statements do not apply to massless theories since they do not satisfy the above split property. In particular, it is known that positive energy representations of conformally invariant theories are DHR representations.
1. Introduction It is well known that the superselection structure, i.e. the structure of physically relevant representations or “charges”, of quantum field theories in low dimensional spacetimes gives rise to particle statistics governed by the braid group and is described by “quantum symmetries” which are still insufficiently understood. The meaning of “low dimensional” in this context depends on the localization properties of the charges under consideration. In the framework of algebraic quantum field theory [?, ?] several selection criteria for physical representations of the observable algebra have been investigated. During their study of physical observables obtained from a field theory by retaining only the operators invariant under the action of a gauge group (of the first kind), Doplicher, Haag and Roberts were led to singling out the class of locally generated superselection sectors. A representation is of this type if it becomes unitarily equivalent to the vacuum representation when restricted to the observables localized in the spacelike complement of an arbitrary double cone (intersection of future and past directed light cones): ∗ Supported
by the Studienstiftung des deutschen Volkes and the CEE. 1147
Reviews in Mathematical Physics, Vol. 10, No. 8 (1998) 1147–1170 c World Scientific Publishing Company
¨ M. MUGER
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π A(O0 ) ∼ = π0 A(O0 ) ∀ O ∈ K .
(1.1)
Denoting the set of all double cones by K we consider a quantum field theory to be defined by its net of observables K 3 O 7→ A(O). This is a map which assigns to each double cone a C ∗ -algebra A(O) satisfying isotony: O1 ⊂ O2 ⇒ A(O1 ) ⊂ A(O2 ) .
(1.2)
This net property allows the quasilocal algebra to be defined by A=
[
A(O)
k·k
.
(1.3)
O∈K
The net is local in the sense that [A(O1 ), A(O2 )] = {0}
(1.4)
if O1 , O2 are spacelike to each other. The algebra A(G) associated with an arbitrary subset of Minkowski space is understood to be the subalgebra of A generated (as a C ∗ -algebra) by all A(O) where G ⊃ O ∈ K. Furthermore, the Poincar´e group acts on A by automorphisms αΛ,x such that αΛ,x (A(O)) = A(ΛO + x)
∀ O.
(1.5)
This abstract approach is particularly useful if there is more than one vacuum. One requires of a physically reasonable representation that at least the translations (Lorentz invariance might be broken) are unitarily implemented: π ◦ αx (A) = Uπ (x)π(A)Uπ (x)∗ ,
(1.6)
the generators of the representation x 7→ U (x), i.e. the energy-momentum operators, satisfying the spectrum condition (positivity of the energy). Vacuum representations are characterized by the existence of a unique (up to a phase) Poincar´e invariant vector. Furthermore we assume them to be irreducible and to satisfy the Reeh–Schlieder property, the latter following from the other assumptions if weak additivity is assumed. In the analysis of superselection sectors satisfying (??) relative to a fixed vacuum representation one usually assumes the latter to satisfy Haag dualitya π0 (A(O))0 = π0 (A(O0 ))00
∀ O ∈ K,
(1.7)
which may be interpreted as a condition of maximality for the local algebras. In [?, ?], based on (??), a thorough analysis of the structure of representations satisfying (??) was given, showing that the category of these representations a M0 = {X ∈ B(H)|XY = Y X ∀ Y ∈ M} denotes the algebra of all bounded operators commuting with all operators in M. If M is a unital ∗-algebra then M00 is known to be the weak closure of a M.
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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together with their intertwiners is monoidal (i.e. there is a product or, according to current fashion, fusion structure), rigid (i.e. there are conjugates) and permutation symmetric. In particular, the Bose–Fermi alternative, possibly with parastatistics, came out automatically although the analysis started from observable, i.e. strictly local, quantities. A lot more is known in this situation (cf. [?]) but we will not need that. A substantial part of this analysis, in particular concerning permutation statistics and the Bose–Fermi alternative, is true only in at least 2 + 1 spacetime dimensions. The generalization to 1 + 1 dimensions, where in general only braid group statistics obtains, was given in [?] and applied to conformally invariant theories in [?]. Whereas for the latter theories all positive energy representations are of the DHR type [?], it has been clear from the beginning that the criterion (??) cannot hold for charged sectors in gauge theories due to Gauss’ law. Implementing a programme initiated by Borchers, Buchholz and Fredenhagen proved [?] for every massive one-particle representation (where there is a mass gap in the spectrum followed by an isolated one-particle hyperboloid) the existence of a vacuum representation π0 such that π A(C 0 ) ∼ = π0 A(C 0 ) ∀ C .
(1.8)
Here the C’s are spacelike cones which we do not need to define precisely. In ≥ 3 + 1 dimensional spacetime the subsequent analysis leads to essentially the same structural results as the original DHR theory. Due to the weaker localization properties, however, the transition to braid group statistics and the loss of group symmetry occur already in 2 + 1 dimensions, see [?]. In the 1 + 1 dimensional situation with which we are concerned here, spacelike cones reduce to wedges (i.e. translates of WR = {x ∈ R2 | x1 ≥ |x0 |} and the spacelike complement WL = WR0 ). Furthermore, the arguments in [?] allow us only to conclude the existence of two a priori different vacuum representations π0L , π0R such that the restriction of π to left handed wedges (translates of WL ) is equivalent to π0L and similarly for the right handed ones. As for such representations, of course long well-known as soliton sectors, an operation of composition can only be defined if the “vacua fit together” [?], there is in general no such thing as permutation or braid group statistics. For lack of a better name soliton representations with coinciding left and right vacuum, i.e. representations which are localizable in wedges, will be called “wedge representations (or sectors)”. There have long been indications that the DHR criterion might not be applicable to massive 2d-theories as it stands. The first of these was the fact, known for some time, that the fixpoint nets of Haag-dual field nets with respect to the action of a global gauge group do not satisfy duality even in simple sectors, whereas this is true in ≥ 2 + 1 dimensions. This phenomenon has been analyzed thoroughly in [?] under the additional assumption that the fields satisfy the split property for wedges. This property, which is expected to be satisfied in all massive quantum field theories, plays an important role also in the present work which we summarize briefly.
¨ M. MUGER
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In the next section we will prove some elementary consequences of Haag duality and the split property for wedges (SPW), in particular strong additivity and the time-slice property. The significance of our assumptions for superselection theory derives mainly from the fact that they preclude the existence of locally generated superselection sectors. More precisely, if the vacuum representation satisfies Haag duality and the SPW then every irreducible DHR representation is unitarily equivalent to the vacuum representation. This important and perhaps surprising result, to be proved in Sec. 3, indicates that the innocent-looking assumptions of the DHR framework are quite restrictive when they are combined with the split property for wedges. Although this may appear reasonable in view of the non-connectedness of O0 , our result also applies to the wedge representations which are only localizable in wedges provided left and right handed wedges are admitted. In Sec. 4 we will prove the minimality of the relative commutant for an inclusion of double cone algebras which, via a result of Driessler, implies Haag duality in all locally normal irreducible representations. In Sec. 5 the facts gathered in the preceding sections will be applied to the theory of quantum solitons thereby concluding our discussion of the representation theory of Haag-dual nets. Summing up the results obtained so far, the representation theory of such nets is essentially trivial. On the other hand, dispensing completely with a general theory of superselection sectors including composition of charges, braid statistics and quantum symmetry for massive theories is certainly not warranted in view of the host of more or less explicitly analyzed models exhibiting these phenomena. The only way to accommodate these models seems to be to relax the duality requirement by postulating only wedge duality. In Sec. 6 Roberts’ extension of localized representations to the dual net will be reconsidered and applied to the theories considered already in [?], namely fixpoint nets under an unbroken inner symmetry group. In this work we will not attempt to say anything concerning the quantum symmetry question. 2. Strong Additivity and the Time-Slice Axiom Until further notice we fix a vacuum representation π0 (which is always faithful) on a separable Hilbert space H0 and omit the symbol π0 (·), identifying A(O) ≡ π0 (A(O)). Whereas we may assume the algebras A(O), O ∈ K to be weakly closed, for more complicated regions X, in particular infinite ones like O0 , we carefully distinguish between the C ∗ -subalgebra A(X) ≡
[
A(O)
k·k
(2.1)
O∈K,O⊂X
of A ≡ π0 (A) and its ultraweak closure R(X) = A(X)00 . Definition 2.1. An inclusion A ⊂ B of von Neumann algebras is standard [?] if there is a vector Ω which is cyclic and separating for A, B, A0 ∧ B. Due to the Reeh–Schlieder property, the inclusion A(O1 ) ⊂ A(O2 ) (R(W1 ) ⊂ R(W2 )) is standard whenever O1 ⊂⊂ O2 (W1 ⊂⊂ W2 ), i.e. the closure of O1 is
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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contained in the interior of O2 . (W1 ⊂⊂ W2 is equivalent to the existence of a double cone O such that W1 ∪ W20 = O0 .) Definition 2.2. An inclusion A ⊂ B of von Neumann algebras is split [?], if there exists a type-I factor N such that A ⊂ N ⊂ B. A net of algebras satisfies the split property (for double cones) [?] if the inclusion A(O1 ) ⊂ A(O2 ) is split whenever O1 ⊂⊂ O2 . The importance of these definitions derives from the following result [?, ?]: Lemma 2.3. Let A ⊂ B be a standard inclusion. Then the following are equivalent: (i) The inclusion A ⊂ B is split. (ii) The is a unitary Y such that Y ab0 Y ∗ = a ⊗ b0 , a ∈ A, b0 ∈ B 0 . Remarks. 1. The implication (ii)⇒(i) is trivial, an interpolating type-I factor being given by N = Y ∗ (B(H0 ) ⊗ 1)Y . 2. The natural spatial isomorphism A(O1 ) ∨ A(O2 )0 ∼ = A(O1 ) ⊗ A(O2 )0 implied by the split property whenever O1 ⊂⊂ O2 clearly restricts to A(O1 ) ∨ R(O20 ) ∼ = A(O1 ) ⊗ R(O20 ) .
(2.2)
As an important consequence, every pair of normal states φ1 ∈ A(O1 )∗ , φ2 ∈ R(O20 )∗ extends to a normal state φ ∈ (A(O1 ) ∨ R(O20 ))∗ . Physically this amounts to a form of statistical independence between the regions O1 and O20 . 3. We emphasize that in the case where Haag duality fails (A(O) ( A(O0 )0 ), requiring (??) whenever O1 ⊂⊂ O2 defines a weaker notion of split property since one can conclude only the existence of a type-I factor N such that A(O1 ) ⊂ N ⊂ A(O20 )0 = Ad (O2 ). In 1 + 1 dimensions (and only there, cf. [?, p. 292]) the split property may be strengthened by extending it to wedge regions. In this paper we will examine the implications of the split property for wedges (SPW). The power of this assumption in combination with Haag duality derives from the fact that one obtains strong results on the relation between the algebras of double cones and of wedges. Some of these have already been explored in [?], where, e.g., it has been shown that the local algebras associated with double cones are factors. We recall some terminology introduced in [?]: the left and right spacelike complements of O are denoted by O O O 0 O 0 and WRR , respectively. Furthermore, defining WLO = WRR and WRO = WLL WLL we have O = WLO ∩ WRO . Before we turn to the main subject of this section, we remark on the relation between the two notions of Haag duality which are of relevance for this paper. In [?], as apparently in a large part of the literature, it was implicitly assumed that Haag duality for double cones implies duality for wedges, i.e. R(W )0 = R(W 0 ) ∀ W ∈ W ,
(2.3)
¨ M. MUGER
1152
where W is the set of all wedge regions. Whereas there seems to be no general proof of this claim, for theories in 1 + 1 dimensions satisfying the SPW we can give a straightforward argument, thereby also closing the gap in [?]. In view of Remark 3 after Lemma ?? the following definition of the split property for wedges is slightly weaker than the obvious modification of Definition ??, but seems more natural from a physical point of view (cf. Remark 2): Definition 2.4. A net of algebras satisfies the split property for wedges if the map x ⊗ y 7→ xy, x ∈ R(W1 ), y ∈ R(W2 ) extends to an isomorphism between R(W1 ) ⊗ R(W2 ) and R(W1 ) ∨ R(W2 ) whenever W1 ⊂⊂ W20 . By standardness this isomorphism is automatically spatial in the sense of Lemma ?? (ii). In the case O O , W2 = WRR the canonical implementer [?] will be denoted Y O . where W1 = WLL Proposition 2.5. Let A(O) be a net of local algebras in 1 + 1 dimensions, satisfying Haag duality (for double cones) and the SPW. Then A satisfies wedge duality and the inclusion R(W1 ) ⊂ R(W2 ) is split whenever W1 ⊂⊂ W2 . Proof. Appealing to the definition (??), duality for double cones is clearly equivalent to O O ) ∨ R(WRR ) ∀ O ∈ K. A(O)0 = R(WLL
(2.4)
Given a right wedge W , let Oi , i ∈ N be an increasing sequence of double cones all of which have the same left corner as W and satisfying ∪i Oi = W . Then we clearly W have R(W ) = i A(Oi ) and ^ ^ Oi R(W )0 = R(W 0 ) ∨ R(WRR A(Oi )0 = ) . (2.5) i
i
O1 O1 Using the unitary equivalence Y O1 R(W 0 ) ∨ R(WRR ) Y O1 ∗ = R(W 0 ) ⊗ R(WRR ), the right-hand side of (??) is equivalent to O^ ^ R(W 0 ) ⊗ R(WROi ) = R(W 0 ) R(WROi ) = R(W 0 ) ⊗ C1 , (2.6) i
i
∧i R(WROi )
where we have used the consequence = C1 of irreducibility. This clearly proves R(W )0 = R(W 0 ). The final claim follows from Lemma ??. Now we are prepared for the discussion of additivity properties, starting with the easy Lemma 2.6. O ) ∨ A(O) = R(WLO ) , R(WLL
(2.7)
O ) ∨ A(O) = R(WRO ) . R(WRR
(2.8)
O O Remark. Equivalently, the inclusions R(WLL ) ⊂ R(WLO ), R(WRR ) ⊂ R(WRO ) are normal.
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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O O O O Proof. Under the unitary equivalence R(WLL )∨R(WRR )∼ )⊗R(WRR ) = R(WLL O O O O ∼ O ∼ we have R(WLL ) = R(WLL ) ⊗ 1 and A(O) = R(WL ) ∩ R(WR ) = R(WR ) ⊗ O O ) ∨ A(O) ∼ ) ∨ R(WRO )) ⊗ R(WLO ). Due to wedge R(WLO ). Thus R(WLL = (R(WLL duality and factoriality of the wedge algebras this equals B(H0 )⊗R(WLO ) ∼ = R(WLO ). We emphasize that all above equivalences are established by the same unitary transformation. The second equation is proved in the same way.
Remark. The proof of factoriality of wedge algebras in [?] relies, besides the usual net properties, on the spectrum condition and on the Reeh–Schlieder theorem. This is the only place where positivity of the energy and weak additivity enter into our analysis. ˜ are spacelike Consider now the situation depicted in Fig. 1. In particular, O, O separated double cones the closures of which share one point. Such double cones will be called adjacent.
@@ @@ @@ @@W @@ @@ W @ O @ O˜ @ @@ @@ @@ @ @ @ O L
O LL
Fig. 1. Double cones sharing one point.
ˆ = sup(O, O) ˜ be the smallest double cone containing O, O. ˜ Lemma 2.7. Let O Then ˜ = A(O) ˆ . A(O) ∨ A(O)
(2.9) ˜
ˆ = W O ∩ W O . Under the unitary Proof. In the situation of Fig. 1 we have O L R ∼ ˜ ˜ since O ˜ ⊂ W O . Thus equivalence considered above we have A(O) = 1 ⊗ A(O) RR ˜ O O O O ∼ ˜ ˜ A(O)∨A(O) = R(WR )⊗(R(WL )∨A(O)). But now WL = WLL leads to R(WLO )∨ ˜ ˜ ∼ ˜ = R(W O˜ ) via the preceding lemma. Thus A(O) ∨ A(O) A(O) = R(WRO ) ⊗ R(WLO ) L ˜ O O ˆ which in turn is unitarily equivalent to R(W ) ∧ R(W ) = A(O). R
L
Remark. In analogy to chiral conformal field theory we denote this property strong additivity. With these lemmas it is clear that the quantum field theories under consideration are n-regular in the sense of the following definition for all n ≥ 2. Definition 2.8. A quantum field theory is n-regular if R(W1 ) ∨ A(O1 ) ∨ · · · ∨ A(On−2 ) ∨ R(W2 ) = B(H0 ) ,
(2.10)
¨ M. MUGER
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whenever Oi , i = 1, . . . , n − 2 are mutually spacelike double cones such that the sets Oi ∩ Oi+1 , i = 1, . . . , n − 3 each contain one point and where the wedges W1 , W2 are such that !0 n−2 [ Oi . (2.11) W1 ∪ W2 = i=1
Corollary 2.9. A quantum field theory in 1 + 1 dimensions satisfying Haag duality and the SPW fulfills the (von Neumann version of the) time-slice axiom, i.e. R(S) = B(H0 ) ,
(2.12)
whenever S = {x ∈ R2 | x · η ∈ (a, b)} where η ∈ R2 is timelike and a < b. Proof. The time-slice S contains an infinite string Oi , i ∈ Z of mutually spacelike double cones as above. Thus the von Neumann algebra generated by all these double cones contains each A(O), O ∈ K from which the claim follows by irreducibility. Remarks. 1. We wish to emphasize that this statement on von Neumann algebras is weaker than the C ∗ -version of the time-slice axiom, which postulates that the C ∗ -algebra A(S) generated by the algebras A(O), O ⊂ S equals the quasilocal algebra A. We follow the arguments in [?, Sec. III.3] to the effect that this stronger assumption should be avoided. 2. It is interesting to confront the above result with the investigations concerning the time-slice property [?] and the split property [?, Theorem 10.2] in the context of generalized free fields (in 3 + 1 dimensions). In the cited works it was proved that generalized free fields possess the time-slice property iff (roughly) the spectral measure vanishes sufficiently fast at infinity. On the other hand, the split property imposes strong restrictions on the spectral measure, in particular it must be atomic without an accumulation point at a finite mass. The split property (for double cones) is, however, neither necessary nor sufficient for the time-slice property. 3. Absence of Localized Charges Whereas the results obtained so far are intuitively plausible, we will now prove a no-go theorem which shows that the combination of Haag duality and the SPW is extremely strong. Theorem 3.1. Let O 7→ A(O) be a net of observables satisfying Haag duality and the split property for wedges. Let π be a representation of the quasilocal algebra A which satisfies π A(W ) ∼ = π0 A(W )
∀ W ∈W,
(3.1)
where W is the set of all wedges (left and right handed). Then π is equivalent to an at most countable direct sum of representations which are unitarily equivalent
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
to π0 : π=
M
πi ∼ = π0 .
πi ,
...
1155
(3.2)
i∈I
In particular, if π is irreducible it is unitarily equivalent to π0 . Remark. A fortiori, this applies to DHR representations (??).
@
@@
@ @@O @@W @ @W @ @@ W @@ @@ @ @ @ O2
@@
1
1
2
Fig. 2. A split inclusion of wedges.
Proof. Consider the geometry depicted in Fig. 2. If π is a representation satisfying (??) then there is a unitary V : Hπ → H0 such that, setting ρ = V π(·)V ∗ , we have ρ(A) = A if A ∈ A(W 0 ). Due to normality on wedges and wedge duality, ρ continues to normal endomorphisms of R(W ), R(W1 ). By the split property there are type-I factors M1 , M2 such that R(W ) ⊂ M1 ⊂ R(W1 ) ⊂ M2 ⊂ R(W2 ) .
(3.3)
Let x ∈ M1 ⊂ R(W1 ). Then ρ(x) ∈ R(W1 ) ⊂ M2 . Furthermore, ρ acts trivially on M10 ∩ R(W2 ) ⊂ R(W )0 ∩ R(W2 ) = A(O2 ), where we have used Haag duality. Thus ρ maps M1 into M2 ∩ (M10 ∩ R(W2 ))0 ⊂ M2 ∩ (M10 ∩ M2 )0 = M1 , the last identity following from M1 , M2 being type-I factors. By [?, Corollary 3.8] every endomorphism of a type-I factor is inner, i.e. there is a (possibly infinite) family of P isometries Vi ∈ M1 , i ∈ I with Vi∗ Vj = δi,j , i∈I Vi Vi∗ = 1 such that ρ(A) = η(A) where η(A) ≡
X
∀ A ∈ M1 ,
Vi A Vi∗ , A ∈ B(H0 ) .
(3.4)
(3.5)
i∈I
(The sum over I is understood in the strong sense.) Now, ρ and thus η act trivially on M1 ∩ R(W )0 ⊂ R(W1 ) ∩ R(W )0 = A(O1 ), which implies Vi ∈ M1 ∩ (M1 ∩ R(W )0 )0 = R(W ) .
(3.6)
ˆ ⊃⊃ W Thanks to Lemma ?? we know that for every wedge W ˆ ) = R(W ) ∨ A(O) , R(W
(3.7)
¨ M. MUGER
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ˆ ∩ W 0 . From the fact that ρ acts trivially on A(W 0 ) it follows that where O = W ˆ ) which (??) is true also for A ∈ A(O). By assumption, ρ is normal also on A(W ˆ ). As this holds for every W ˆ ⊃⊃ W , we conclude that leads to (??) on A(W X V ∗ Vi A Vi∗ V ∀ A ∈ A . (3.8) π(A) = i∈I
Remarks. 1. The main idea of the proof is taken from [?, Proposition 2.3]. 2. The above result may seem inconvenient as it trivializes the DHR/FRS superselection theory [?, ?, ?] for a large class of massive quantum field theories in 1 + 1 dimensions. It is not so clear what this means with respect to field theoretical models since little is known about Haag duality in nontrivial models. 3. Conformal quantum field theories possessing no representations besides the vacuum representation, or “holomorphic” theories, have been the starting point for an analysis of “orbifold” theories in [?]. In [?], which was motivated by the desire to obtain a rigorous understanding of orbifold theories in the framework of massive two-dimensional theories, the present author postulated the split property for wedges and claimed it to be weaker than the requirement of absence of nontrivial representations. Whereas this claim is disproved by Theorem ??, as far as localized (DHR or wedge) representations of Haag dual theories are concerned, none of the results of [?] is invalidated or rendered obsolete. 4. Haag Duality in Locally Normal Representations A further crucial consequence of the split property for wedges is observed in the following: Proposition 4.1. Let O 7→ A(O) be a net satisfying Haag duality (for double ˆ we have cones) and the split property for wedges. Then for every pair O ⊂⊂ O ˆ ∧ A(O)0 = A(OL ) ∨ A(OR ) , A(O)
(4.1)
where OL , OR are as in Fig. 3.
@@
@@ @@ @@ @W@ @@ @@ @W@ W @@O @@O @@O @@W @@ @@ @@ @@ ˆ O
O LL
ˆ O LL
O RR
L
R
Fig. 3. Relative commutant of double cones.
ˆ O RR
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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Proof. By the split property for wedges there is a unitary operator Y O : H0 → O O O O ) ∨ R(WRR ) = Y O∗ (R(WLL ) ⊗ R(WRR ))Y O . More H0 ⊗ H0 such that R(WLL specifically, Y O xy Y O∗ = x ⊗ y
O O ∀ x ∈ R(WLL ), y ∈ R(WRR ).
(4.2)
O O O O ˆ 0= By Haag duality A(O)0 = R(WLL ) ∨ R(WRR )∼ ) ⊗ R(WRR ) and A(O) = R(WLL ˆ ˆ ˆ ˆ O O O O O 0 ∼ ˆ R(WLL ) ∨ R(WRR ). Now R(WLL/RR ) ⊂ R(WLL/RR ) implies A(O) = R(WLL ) ⊗ ˆ R(W O ) under the same equivalence ∼ = provided by Y O , and thus RR
ˆ ˆ ˆ ˆ O O ˆ ∼ ) ⊗ R(WRR ))0 = R(WRO ) ⊗ R(WLO ) , A(O) = (R(WLL
(4.3)
where we have used wedge duality and the commutation theorem for tensor products. Now we can compute the relative commutant as follows: ˆ ˆ O O ˆ ∧ A(O)0 ∼ A(O) ) ⊗ R(WRR )) = (R(WRO ) ⊗ R(WLO )) ∧ (R(WLL ˆ
ˆ
O O )) ⊗ (R(WLO ) ∧ R(WRR )) = (R(WRO ) ∧ R(WLL
= A(OL ) ⊗ A(OR ) ∼ = A(OL ) ∨ A(OR ) .
(4.4) ˆ
O We have used Haag duality in the form R(WRO ) ∧ R(WLL ) = A(OL ) and similarly for A(OR ).
Remarks. 1. Readers having qualms about the above computation of the intersection of tensor products are referred to [?, Corollary 5.10], which also provides the justification for the arguments in Sec. 2. 2. Recalling that R(O) = A(O) and that the algebras of regions other than double cones are defined by additivity, (??) can be restated as follows: ˆ ∩ O0 ) . ˆ ∩ R(O)0 = R(O R(O)
(4.5)
In conjunction with the assumed properties of isotony, locality and Haag duality for double cones (??) entails that the map O 7→ R(O) is a homomorphism of orthocomplemented lattices as proposed in [?, Sec. III.4.2]. While the discussion in [?, Sec. III.4.2] can be criticized, the class of models considered in this paper provides examples where the above lattice homomorphism is in fact realized. The proposition should contribute to the understanding of Theorem ?? as far as DHR representations are concerned. In fact, it already implies the absence of DHR sectors as can be shown by an application of the triviality criterion for local 1-cohomologies [?] given in [?], see also [?]. Sketch of proof. Let z ∈ Z 1 (A) be the local 1-cocycle associated according to [?, ?] with a representation π satisfying the DHR criterion. Due to Proposition ?? it satisfies z(b) ∈ A(|∂0 b|) ∨ A(|∂1 b|) for every b ∈ Σ1 such that |∂0 b| ⊂⊂ |∂1 b|0 . Thus
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¨ M. MUGER
the arguments in the proof of [?, Theorem 3.5] are applicable despite the fact that we are working in 1+1 dimensions. We thereby see that there are unique Hilbert spaces H(O) ⊂ A(O), O ∈ Σ0 ≡ K of support 1 such that z(b)H(∂1 b) = H(∂0 b) ∀ b ∈ Σ1 . Each of these Hilbert spaces implements an endomorphism ρO of A such that ρO ∼ = π. This implies that ρ is either reducible or an inner automorphism. Remark. This argument needs the split property for double cones. It is not completely trivial that the latter follows from the split property for wedges. It is clear that the latter implies unitary equivalence of A(O1 ) ∨ A(O2 ) and A(O1 ) ⊗ A(O2 ) if O1 , O2 are double cones separated by a finite spacelike distance. The split ˆ 0 property for double cones requires more, namely unitary equivalence of A(O)∨A(O) 0 ˆ ˆ and A(O)⊗A(O) whenever O ⊂⊂ O, which is equivalent to the existence of a type-I ˆ factor N such that A(O) ⊂ N ⊂ A(O). Lemma 4.2. Let A be a local net satisfying Haag duality and the split property for wedges. Then the split property for double cones holds. Proof. Using the notation of the preceding proof we have A(O) ∼ = R(WRO ) ⊗ R(WLO ) ,
(4.6)
ˆ ˆ ˆ ∼ A(O) = R(WRO ) ⊗ R(WLO ) .
(4.7) ˆ
By the SPW there are type-I factors NL , NR such that R(WLO ) ⊂ NL ⊂ R(WLO ) ˆ and R(WRO ) ⊂ NR ⊂ R(WRO ). Thus Y O∗ (NR ⊗ NL )Y O is a type-I factor sitting ˆ between A(O) and A(O). Having disproved the existence of nontrivial representations localized in double cones or wedges, we will now prove a result which concerns a considerably larger class of representations. Theorem 4.3. Let O 7→ A(O) be a net of observables satisfying Haag duality and the SPW. Then every irreducible, locally normal representation of the quasilocal algebra A fulfills Haag duality. Proof. We will show that our assumptions imply those of [?, Theorem 1]. A satisfies the split property for double cones (called “funnel property” in [?, ?]) by Lemma ??, whereas we also assume condition (1) of [?, Theorem 1] (Haag duality and irreducibility). Condition (3), which concerns relative commutants A(O2 ) ∩ A(O1 )0 , O2 ⊃⊃ O1 in the vacuum representation, is an immediate consequence of Proposition ?? (we may even take O = O1 , O2 = O3 ). Finally, Lemma ?? implies ˆ 0 ∨ A(OL ) ∨ A(OR ) , A(O)0 = A(O)
(4.8)
where we again use the notation of Fig. 3. This is more than required by Driessler’s condition (2). Now [?, Theorem 1] applies and we are done.
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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Remarks. 1. In [?] a slightly simplified version of [?, Theorem 1] is given which dispenses with condition (2) at the price of a stronger form of condition (3). This condition is still (more than) fulfilled by our class of theories. 2. Observing that soliton representations are locally normal with respect to both asymptotic vacua [?, ?], we conclude at once that Haag duality holds for every irreducible soliton sector where at least one of the vacua satisfies Haag duality and the SPW. Consequences of this fact will be explored in the next section. We remark without going into details that our results are also of relevance for the construction of soliton sectors with prescribed asymptotic vacua in [?]. 5. Applications to the Theory of Quantum Solitons In [?] it has been shown that every factorial massive one-particle representation (massive one-particle representation) in ≥ 2 + 1 dimensions is a multiple of an irreducible representation which is localizable in every spacelike cone. (Here, massive one-particle representation means that the lower bound of the energy-momentum spectrum consists of a hyperboloid of mass m > 0 which is separated from the rest of the spectrum by a mass gap.) In 1 + 1 dimensions one is led to irreducible soliton sectors [?] which we will now reconsider in the light of Theorems ?? and ??. In this section, where we are concerned with inequivalent vacuum representations, we will consider a QFT to be defined by a net of abstract C ∗ -algebras instead of the algebras in a concrete representation. Given two vacuum representations π0L , π0R , a representation π is said to be a soliton representation of type (π0L , π0R ) if it is translation covariant and L/R A(WL/R ) , π A(WL/R ) ∼ = π0
(5.1)
where WL , WR are arbitrary left and right handed wedges, respectively. An obvious consequence of (??) is local normality of π0L , π0R with respect to each other. In order to formulate a useful theory of soliton representations [?] one must assume L/R to satisfy wedge duality. After giving a short review of the formalism in [?], π0 we will show in this section that considerably more can be said under the stronger assumption that one of the vacuum representations satisfies duality for double cones and the SPW. (Then the other vacuum is automatically Haag dual, too.) Let π0 be a vacuum representation and W ∈ W a wedge. Then by A(W )π0 we denote the W ∗ -completion of the C ∗ -algebra A(W ) with respect to the family of seminorms given by kAkT = |tr T π0 (A)| ,
(5.2)
where T runs through the set of all trace class operators in B(Hπ0 ). Furthermore, R we define extensions AL π0 , Aπ0 of the quasilocal algebra A by AL/R = π0
[ W ∈WL/R
A(W )π0
k·k
,
(5.3)
¨ M. MUGER
1160
where WL , WR are the sets of left and right wedges, respectively. Now, it has been demonstrated in [?] that, given a (π0L , π0R )-soliton representation π, there are to AR such that homomorphisms ρ from AR πR πL 0
0
π∼ = π0L ◦ ρ .
(5.4)
(Strictly speaking, π0L must be extended to AR , which is trivial since A(W )π0 is π0L 00 isomorphic to π0 (A(W )) .) The morphism ρ is localized in some right wedge W in the sense that ρ A(W 0 ) = id A(W 0 ) .
(5.5)
Provided that the vacua of two soliton representations π, π 0 “fit together” π0R ∼ = π00L 0 L 0R one can define a soliton representation π × π of type π0 , π0 via composition of the corresponding morphisms: π × π0 ∼ = π0L ◦ ρρ0 A .
(5.6)
Alternatively, the entire analysis may be done in terms of left localized morphisms to AL . As proved in [?], the unitary equivalence class of the composed η from AL π0L π0R representation depends neither on the use of left or right localization nor on the concrete choice of the morphisms. Whereas for soliton representations there is no analog to the theory of statistics [?, ?, ?], there is still a “dimension” ind(ρ) defined by ind(ρ) ≡ [A(W )π0L : ρ(A(W )π0R )] ,
(5.7)
where ρ is localized in the right wedge W and [M : N ] is the Jones index of the inclusion N ⊂ M . Proposition 5.1. Let π be an irreducible soliton representation such that at least one of the asymptotic vacua π0L , π0R satisfies Haag duality and the SPW. Then π and both vacua satisfy the SPW and duality for double cones and wedges. The associated soliton-morphism satisfies ind(ρ) = 1. Proof. By symmetry it suffices to consider the case where π0L satisfies HD + SPW. By Theorem ?? also the representations π and π0R satisfy Haag duality since they are locally normal w.r.t. to π0L . Let now W1 ⊂⊂ W2 be left wedges. By Proposition ??, wedge-duality holds for π0L and π0L (A(W1 ))00 ⊂ π0L (A(W2 ))00 is split. Since π0L (A(W2 ))00 is unitarily equivalent to π(A(W2 ))00 , also π(A(W1 ))00 ⊂ π(A(W2 ))00 splits. A fortiori, π satisfies the SPW in the sense of Definition ?? and thus wedge duality by Proposition ??. By a similar argument the SPW is carried over to π0R . Now, for a right wedge W we have π0L ◦ ρ(A(W ))− = π0L ◦ ρ(A(W 0 ))0 = π0L (A(W 0 ))0 = π0L (A(W ))− .
(5.8)
By ultraweak continuity on A(W ) of π0L and of ρ this implies ρ(A(W )π0R ) = A(W )π0L , whence the claim.
(5.9)
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This result rules out soliton sectors with infinite index so that [?, Theorem 3.2] applies and yields equivalence of the various possibilities of constructing antisoliton sectors considered in [?]. In particular the antisoliton sector is uniquely defined up to unitary equivalence. Now we can formulate our main result concerning soliton representations. Theorem 5.2. Let π0L , π0R be vacuum representations, at least one of which satisfies Haag duality and the SPW. Then all soliton representations of type (π0L , π0R ) are unitarily equivalent. Remark. Equivalently, up to unitary equivalence, a soliton representation is completely characterized by the pair of asymptotic vacua. Proof. Let π, π 0 be irreducible soliton representations of types (π0 , π00 ) and respectively. They may be composed, giving rise to a soliton representation of type (π0 , π0 ) (or (π00 , π00 )). This representation is irreducible since the morphisms ρ, ρ0 must be isomorphisms by the proposition. Now, π × π 0 is unitarily equivalent to π0 on left and right handed wedges, which by Theorem ?? and irreducibility implies π × π 0 ∼ = π0 . We conclude that every (π00 , π0 )-soliton is an antisoliton of 0 every (π0 , π0 )-soliton. This implies the statement of the theorem since for every soliton representation with finite index there is a corresponding antisoliton which is unique up to unitary equivalence.
(π00 , π0 ),
Remark. The above proof relies on the absence of nontrivial representations which are localizable in wedges. Knowing just that DHR sectors do not exist, as follows already from Proposition ??, is not enough. 6. Solitons and DHR Representations of Non-Haag Dual Nets 6.1. Introduction and an instructive example We have observed that the theory of localized representations of Haag-dual nets of observables which satisfy the SPW is trivial. There are, however, quantum field theories in 1 + 1 dimensions where the net of algebras which is most naturally considered as the net of observables does not fulfill Haag duality in the strong form (??). As mentioned in the introduction, this is the case if the observables are defined as the fixpoints under a global symmetry group of a field net which satisfies (twisted) duality and the SPW. The weaker property of wedge duality (??) remains, however. This property is also known to hold automatically whenever the local algebras arise from a Wightman field theory [?]. However, for the analysis in [?, ?, ?] as well as Sec. 4 above one needs full Haag duality. Therefore it is of relevance that, starting from a net of observables satisfying only (??), one can define a larger but still local net Ad (O) ≡ R(WLO ) ∧ R(WRO )
(6.1)
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which satisfies Haag duality, whence the name dual net. Here WLO , WRO are wedges such that WLO ∩ WRO = O and duality is seen to follow from the fact that the wedge algebras R(W ), W ∈ W are the same for the nets A, Ad . (For observables arising as group fixpoints the dual net has been computed explicitly in [?].) It is known [?, ?] that in ≥ 2+1 dimensions representations π satisfying the DHR criterion (??) extend uniquely to DHR representations π ˆ of the (appropriately defined) dual net. Furthermore, the categories of DHR representations of A and Ad , respectively, and their intertwiners are isomorphic. Thus, instead of A one may as well study Ad to which the usual methods are applicable. (The original net is needed only to satisfy essential duality, which is implied by wedge duality.) In 1 + 1 dimensions things are more complicated. As shown in [?] there are in general two different extensions ˆ R . They coincide iff one (thus both) of them is a DHR representation. Even π ˆL, π before defining precisely these extensions we can state the following consequence of Theorem ??. Proposition 6.1. Let A be a net of observables satisfying wedge duality and the SPW. Let π be an irreducible DHR or wedge representation of A which is not unitarily equivalent to the defining (vacuum) representation. Then there is no extension π ˆ to the dual net Ad which is still localized in the DHR or wedge sense. Proof. Assume π to be the restriction to A of a wedge-localized representation π ˆ of Ad . As the latter is known to be either reducible or unitarily equivalent to π0 , the same holds for π. This is a contradiction. The fact that the extension of a localized representation of A to the dual net Ad cannot be localized, too, partially undermines the original motivation for considering these extensions. Nevertheless, one may entertain the hope that there is something to be learnt which is useful for a model-independent analysis of the phenomena observed in models. Before we turn to the general examination of the extensions ˆ R we consider the most instructive example. π ˆL, π It is provided by the fixpoint net under an unbroken global symmetry group of a field net as studied in [?]. We briefly recall the framework. Let O 7→ F (O) be a (for simplicity) bosonic, i.e. local, net of von Neumann algebras acting on the Hilbert space H and satisfying Haag duality and the SPW. On H there are commuting strongly continuous representations of the Poincar´e group and of a group G of inner symmetries. Both groups leave the vacuum Ω invariant. Defining the fixpoint net A(O) = F (O)G = F (O) ∩ U (G)0
(6.2)
A(O) = A(O) H0
(6.3)
and its restriction
to the vacuum sector (= subspace of G-invariant vectors) we consider A(O) as the observables. It is well-known that the net A satisfies only wedge duality. Nevertheless, one very important result of [?] remains true, namely that the restrictions of
SUPERSELECTION STRUCTURE OF MASSIVE QUANTUM FIELD THEORIES
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ˆ interpreted A to the charged sectors Hχ which are labeled by the characters χ ∈ G, ∗ as representations of the abstract C -algebra A, satisfy the DHR criterion and are connected to the vacuum by charged fields, i.e. the representation of A in Hχ is of the form πχ (A) = A Hχ ∼ = πχO (A) = ψ A ψ ∗ H0 ,
(6.4)
where ψ ∈ F (O) and αg (ψ) = χ(g)ψ. It was shown in [?, Theorem 3.10] that the dual net in the vacuum sector is given by Ad (O) = AˆL (O) H0 = AˆR (O) H0 ,
(6.5)
AˆL/R (O) = FˆL/R (O)G = FˆL/R (O) ∩ U (G)0 .
(6.6)
where
Here the nonlocal nets FˆL/R (O) are obtained by adjoining to F (O) the disorder operators [?] ULO (G) or URO (G), respectively, which satisfy O O Ad ULO (g) F(WLL ) = αg = Ad URO (g) F (WRR ), O O Ad ULO (g) F(WRR ) = id = Ad URO (g) F (WLL )
(6.7)
and transform covariantly under the global symmetry: O O (h) U (g)∗ = UL/R (ghg −1 ) . U (g) UL/R
(6.8)
For the moment we restrict to the case of abelian groups G. The disorder O (G)00 . On the C ∗ operators commuting with G, AˆL/R (O) is simply A(O) ∨ UL/R ˆ which acts trivially on algebras AˆL and AˆR there is an action of the dual group G A and via O O (g)) = χ(g) UL/R (g) α ˆ χ (UL/R
∀O∈K
(6.9)
on the disorder operators. Since this action commutes with the Poincar´e group and ˆ χ 6= ω0 ∀ χ 6= eGˆ ) it gives rise to inequivalent since it is spontaneously broken (ω0 ◦ α vacuum states on Aˆ via ˆχ . ωχ = ω0 ◦ α
(6.10)
ˆχ,R of πχ to the dual net Ad can now defined using the The extensions π ˆχ,L , π right-hand side of (??) by allowing A to be in AˆL or AˆR . As is obvious from the commutation relation (??) between fields and disorder operators, the extenπχ,R ) is nothing but a soliton sector interpolating between the vacua ω0 sion π ˆχ,L (ˆ and ωχ−1 (ωχ and ω0 ). The moral is that the net Ad , while not having nontrivial localized representations by Theorem ??, admits soliton representations. Furthermore, with respect to Ad , the charged fields ψχ are creation operators for
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solitons since they intertwine the representations of Ad on H0 and Hχ . Due to ULO (g) URO (g) = U (g) and U (g) Hχ = χ(g)1 we have ULO (g) Hχ = χ(g) URO (g −1 ) Hχ ,
(6.11)
so that the algebras AˆL/R (O) Hχ are independent of whether we use the left or right localized disorder operators. In particular, in the vacuum sector ULO (g) and URO (g −1 ) coincide, but due to the different localization properties it is relevant whether ULO (g), considered as an element of Ad , is represented on Hχ by ULO (g) or by χ(g) URO (g −1 ). This reasoning shows that the two possibilities for extending a localized representation of a general non-dual net to a representation of the dual net correspond in the fixpoint situation at hand to the choice between the nets AˆL and AˆR arising from the field extensions FˆL and FˆR . 6.2. General Analysis We begin by first assuming only that π is localizable in wedges. Let O be a double cone and let WL , WR be left and right handed wedges, respectively, containing O. By assumption the restriction of π to A(WL ), A(WR ) is unitarily equivalent to π0 . Choose unitary implementers UL , UR such that Ad UL A(WL ) = π A(WL ) , Ad UR A(WR ) = π A(WR ) .
(6.12)
ˆ R are defined for A ∈ Ad (O) by Then π ˆL, π π ˆ L (A) = UL A UL∗ , π ˆ R (A) = UR A UR∗ .
(6.13)
Independence of these definitions of the choice of WL , WR and the implementers UL , UR follows straightforwardly from wedge duality. We state some immediate consequences of this definition. ˆ R are irreducible, locally normal representations of Proposition 6.2. π ˆL , π ˆL, π ˆ R are normal on left and right handed wedges, Ad and satisfy Haag duality. π respectively. Proof. Irreducibility is a trivial consequence of the assumed irreducibility of π whereas local normality is obvious from the definition (??). Thus, Theorem ?? applies and yields Haag duality in both representations. Normality of, say, πˆL on left handed wedges W follows from the fact that we may use the same auxiliary wedge WL ⊃ W and implementer UL for all double cones O ⊂ W . Clearly, the extensions π ˆL, π ˆ R cannot be normal w.r.t. π0 on right and left wedges, respectively, for otherwise Theorem ?? would imply unitary equivalence to π0 . In general, we can only conclude localizability in the following weak sense. Given
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an arbitrary left handed wedge W , π ˆ L is equivalent to a representation ρ on H0 such that ρ(A) = A ∀ A ∈ A(W ). Furthermore, by duality ρ is an isomorphism of A(W 0 ) onto a weakly dense subalgebra of R(W 0 ) which is only continuous in the norm. In favorable cases like the one considered above this is a local symmetry, acting as an automorphism of A(W 0 ). But we will see shortly that there are perfectly non-pathological situations where the extensions are not of this particularly nice type. In complete generality, the best one can hope for is normality with respect to another vacuum representation π00 . In particular, this is automatically the case if π is a massive one-particle representation [?] which we did not assume so far. If the representation π satisfies the DHR criterion, i.e. is localizable in double cones, we can obtain stronger results concerning the localization properties of the ˆR . By the criterion, there are unitary operators extended representations π ˆL , π O X : Hπ → H0 such that π O (A) ≡ X O π(A) X O∗ = A ∀ A ∈ A(O0 ) .
(6.14)
(By wedge duality, X O is unique up to left multiplication by operators in Ad (O).) Considering the representations O = X Oπ ˆL/R X O∗ π ˆL/R
(6.15)
on the vacuum Hilbert space H0 , it is easy to verify that O O O Ad (WLL ) = id Ad (WLL ), π ˆL
(6.16)
O O O Ad (WRR ) = id Ad (WRR ). π ˆR
(6.17)
O ˜ , the other extension behaving similarly. If A ∈ A(O) We restrict our attention to π ˆL ∗ ˜ Therefore then π ˆL (A) = X Or A X Or whenever Or > O. ∗
O (A) = X O X Or A X Or X O∗ , π ˆL
(6.18)
∗
where the unitary X O X Or intertwines π O and π Or . Associating with every pair (O1 , O2 ) two other double cones by ˆ = sup(O1 , O2 ) , O
(6.19)
ˆ ∩ O10 ∩ O20 O0 = O
(6.20)
(O0 may be empty) and defining ˆ ∩ A(O0 )0 , C(O1 , O2 ) = Ad (O)
(6.21)
we can conclude by wedge duality that ∗
X O X Or ∈ C(O, Or ) .
(6.22)
O ˜ Or )) which already (A) as given by (??) is contained in Ad (sup(O, O, Thus π ˆL O d shows that π ˆL maps the quasilocal algebra A into itself (this does not follow if
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π is only localizable in wedges). Since the double cone Or > O may be chosen arbitrarily small and appealing to outer regularity of the dual net Ad we even have ˜ and thus finally π ˆ O (A) ∈ Ad (sup(O, O)) L
O ˜ ⊂ C(O, O) ˜ . π ˆL (Ad (O))
(6.23)
This result has two important consequences. Firstly, it implies that the representaO tion π ˆL maps the quasilocal algebra into itself: O π ˆL (Ad ) ⊂ Ad .
(6.24)
O O This fact is of relevance since it allows the extensions π ˆ1,L , π ˆ2,L of two DHR representations π1 , π2 to be composed in much the same way as the endomorphisms of A derived from DHR representations in the Haag dual case. In this respect, the extensions π ˆL/R are better behaved than completely general soliton representations as studied in [?]. O O (and π ˆR ), while The second consequence of (??) is that the representations π ˆL still mapping local algebras into local algebras, may deteriorate the localization. We will see below that this phenomenon is not just a theoretical possibility but really occurs. Whereas one might hope that one could build a DHR theory for nondual nets upon the endomorphism property of the extended representations, their ˆR seem to constitute weak localization properties and the inequivalence of π ˆL and π serious obstacles. It should be emphasized that the above considerations owe a lot to Roberts’ local 1-cohomology [?, ?, ?], but (??) seems to be new.
6.3. Fixpoint nets: non-abelian case We now generalize our analysis of fixpoint nets to non-abelian (finite) groups G, P ˜ where the outcome is less obvious a priori. Let Aˆ = g∈G Fg ULO (g) ∈ AˆL (O˜1 ) (Fg must satisfy the condition given in [?, Theorem 3.16]) and let ψi ∈ F (O2 ), where O1 ) be a multiplet of field operators transforming according O2 < O1 (i.e. O2 ⊂ WLL to a finite dimensional representation of G. Then ! X X X X O ∗ ∗ 1 ψi Fg UL (g) ψi = ψi αg (ψi ) Fg ULO1 (g) . (6.25) i
g∈G
g∈G
i
In contrast to the abelian case where ψαg (ψ ∗ ) is just a phase, Og ≡ is a nontrivial unitary operator X Og−1 = Og∗ = αg (ψi )ψi∗
P i
ψi αg (ψi∗ ) (6.26)
i
satisfying αk (Og ) = Okgk−1 .
(6.27)
In particular (??) is not contained in Ad (O1 ) which implies that the map Aˆ 7→ P O2 ˆ ˆ ∗ i ψi A ψi does not reduce to a local symmetry on AL (WRR ). Rather, we obtain
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ˆ and O0 as above we clearly see that a monomorphism into AˆL (WRO2 ). Defining O d ˆ (??) is contained in A (O). Furthermore, due to the relative locality of the net A with respect to Ad and F, (??) commutes with A(O0 ). Thus we obtain precisely the localization properties which were predicted by our general analysis above. We close this section with a discussion of the duality properties in the extended representations π ˆ . In the case of abelian groups G Haag duality holds in all charged sectors since these are all simple. Our abstract result in Theorem ?? to the effect that duality obtains in all locally normal irreducible representations of the dual net applies, of course, to the situation at hand. We conclude that Haag duality also holds for the non-simple sectors which by necessity occur for non-abelian groups G. Since this result is somewhat counterintuitive (which explains why it was overlooked in [?]) we verify it by the following direct calculation. Lemma 6.3. The commutants of the algebras AˆL (O) are given by O O ) ∨ FˆL (WRR ) AˆL (O)0 = AˆL (WLL
∀ O ∈ K.
(6.28)
Proof. For simplicity we assume F to be a local net for a moment. Then AˆL (O)0 = (FˆL (O) ∧ U (G)0 )0 = FˆL (O)0 ∨ U (G)00 = (FL (O) ∨ ULO (G)00 )0 ∨ U (G)00 = (FL (O)0 ∧ ULO (G)0 ) ∨ U (G)00 O O = ((FL (WLL ) ∨ FL (WRR )) ∧ ULO (G)0 ) ∨ U (G)00 O O = (FL (WLL ) ∧ ULO (G)0 ) ∨ FL (WRR ) ∨ U (G)00 O O = AˆL (WLL ) ∨ FˆL (WRR ).
(6.29)
The fourth line follows from the third using the split property. In the last step we have used the identities AˆL (WL ) = AL (WL ) and FL (WR )∨U (G)00 = FˆL (WR ) which hold for all left (right) handed wedges WL (WR ), cf. [?, Proposition 3.5]. Now, if F satisfies twisted duality, (2.23) of [?] leads to F (O) ∨ ULO (G)00 ∼ = F (WRO ) ∨ U (G)00 ⊗ O O 00 0 ∼ O O t F(WL ) and (F(O)∨UL (G) ) = A(WR )⊗F (WRR ) . Using this it is easy to verify that (??) is still true. Proposition 6.4. The net AˆL satisfies Haag duality in restriction to every invariant subspace of H on which AˆL acts irreducibly. Proof. We recall that the representation π of AˆL/R on H is of the form π = ⊕ξ∈Gˆ dξ πξ . Let thus P be an orthogonal projection onto a subspace Hξ ⊂ H on which AˆL acts as the irreducible representation πξ . Since P commutes with AL (O) O ) we have and AL (WLL O O ) ∨ FˆL (WRR )P P AˆL (O)0 P = P AˆL (WLL O O = AˆL (WLL ) ∨ (P FˆL (WRR )P) O O = P AˆL (WLL ) ∨ AˆL (WRR )P ,
(6.30)
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which implies O O ) ∨ AˆL (WRR ) Hξ . (AˆL (O) Hξ )0 = AˆL (WLL
(6.31)
This provides a concrete verification of Theorem ?? in a special, albeit important situation. 7. Conclusions and Outlook We have seen that the combination of Haag duality with the split property for wedges has remarkable unifying power. It implies factoriality of the double cone algebras, n-regularity for all n and irreducibility of time-slice algebras. As a consequence of the minimality of relative commutants of double cone algebras we obtain Haag duality in all irreducible, locally normal representations. The strongest result concerns the absence not only of locally generated superselection (DHR) sectors but also of charges localized in wedges. This in turn implies the uniqueness up to unitary equivalence of soliton sectors with prescribed asymptotic vacua. In the following we briefly relate these results to what is known in concrete models in 1 + 1 dimensions. (a) The free massive scalar field . Since this model is known to satisfy Haag duality and the SPW, Theorem ?? constitutes a high-brow proof of the well-known absence of local charges. Furthermore, there are no non-trivial soliton sectors, since the vacuum representation is unique [?]. Thus, the irreducible representations constructed in [?], which are inequivalent to the vacuum, must be rather pathological. In fact, they are equivalent to the (unique) vacuum only on left wedges. (b) P(φ)2 -models. These models have been shown [?] to satisfy Haag duality in all pure phases, but there is no proof of the SPW. Yet, the split property for double cones, the minimality of relative commutants and strong additivity, thus also the time slice property, follow immediately from the corresponding properties for the free field via the local Fock property. These facts already imply the nonexistence of DHR sectors and Haag duality in all irreducible locally normal sectors. All these consequences are compatible with the conjecture that the SPW holds. There seems, however, not to be a proof of the absence of wedge sectors. (c) The sine-Gordon/Thirring model. For this model neither Haag duality nor the SPW are known. In the case β 2 = 4π, however, for which the SG model corresponds to the free massive Dirac field, there seems to be no doubt that the net Aˆ constructed like in Sec. 6 from the free Dirac field is exactly the local net of the SG model. As shown in [?], also Aˆ satisfies Haag duality and the SPW. Since from the point of view of constructive QFT there is nothing special about β 2 = 4π one may hope that both properties hold for all β ∈ [0, 8π). In view of the results of this paper as well as of [?] it is highly desirable to clarify the status of the SPW in interacting massive models like (b) and (c) as well as that of Haag duality in case (c). (Also the Gross–Neveu model might be expected
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to satisfy both assumptions.) The most promising approach to this problem should be identifying conditions on a set of Wightman (or Schwinger) distributions which imply Haag duality and the SPW, respectively, for the net of algebras generated by the fields. For a first step in this direction see [?, Sec. IIIB]. Acknowledgments I am greatly indebted to K.-H. Rehren for his interest and encouragement, many helpful discussions and several critical readings of the manuscript. Conversations with K. Fredenhagen, J. Roberts, B. Schroer, and H.-W. Wiesbrock are gratefully acknowledged. The work was completed at the Erwin Schr¨odinger Institute, Vienna which kindly provided hospitality and financial support. Last but not least, I thank P. Croome for a very thorough proofreading of a preliminary version. References [1] C. D’Antoni and R. Longo, “Interpolation by type-I factors and the flip automorphism”, J. Funct. Anal. 51 (1983) 361–371. [2] J. J. Bisognano and E. H. Wichmann, “On the duality condition for a Hermitian scalar field”, J. Math. Phys. 16 (1975) 985–1007. [3] D. Buchholz, “Product states for local algebras”, Commun. Math. Phys. 36 (1974) 287–304. [4] D. Buchholz and K. Fredenhagen, “Locality and the structure of particle states”, Commun. Math. Phys. 84 (1982) 1–54. [5] D. Buchholz, G. Mack and I. Todorov, “The current algebra on the circle as a germ of local field theories”, Nucl. Phys. B (Proc. Suppl.) 5B (1988) 20–56. [6] D. Buchholz, “On quantum fields that generate local algebras”, J. Math. Phys. 31 (1990) 1839–1846. [7] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, “The operator algebra of orbifold models”, Commun. Math. Phys. 123 (1989) 485–527. [8] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations I”, Commun. Math. Phys. 13 (1969) 1–23. [9] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics I”, Commun. Math. Phys. 23 (1971) 199–230. [10] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics II”, Commun. Math. Phys. 35 (1974) 49–85. [11] S. Doplicher, “Local aspects of superselection rules”, Commun. Math. Phys. 85 (1982) 73–86. [12] S. Doplicher and R. Longo, “Standard and split inclusions of von Neumann algebras”, Invent. Math. 75 (1984) 493–536. [13] S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics”, Commun. Math. Phys. 131 (1990) 51–107. [14] W. Driessler, “Comments on lightlike translations and applications in relativistic quantum field theory”, Commun. Math. Phys. 44 (1975) 133–141; “On the type of local algebras in quantum field theory”, Commun. Math. Phys. 53 (1977) 295–297. [15] W. Driessler, “Duality and absence of locally generated superselection sectors for CCR-type algebras”, Commun. Math. Phys. 70 (1979) 213–220. [16] K. Fredenhagen, “Generalizations of the theory of superselection sectors”, in [?]; “Superselection sectors in low dimensional quantum field theory”, J. Geom. Phys. 11 (1993) 337–348.
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