Commun. Math. Phys. 291, 1–30 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0881-6
Communications in
Mathematical Physics
Generalized Kähler Potentials from Supergravity Nick Halmagyi1 , Alessandro Tomasiello2 1 Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA 2 ITP, Stanford University, Stanford, CA 94305-4060, USA.
E-mail:
[email protected] Received: 29 September 2007 / Accepted: 17 October 2008 Published online: 17 July 2009 – © Springer-Verlag 2009
Abstract: We consider supersymmetric N = 2 solutions with non–vanishing NS three– form. Building on worldsheet results, we reduce the problem to a single generalized Monge–Ampère equation on the generalized Kähler potential K recently interpreted geometrically by Lindström, Roˇcek, Von Unge and Zabzine. One input in the procedure is a holomorphic function w that can be thought of as the effective superpotential for a D3 brane probe. The procedure is hence likely to be useful for finding gravity duals to field theories with non–vanishing abelian superpotential, such as Leigh–Strassler theories. We indeed show that a purely NS precursor of the Lunin–Maldacena dual to the β–deformed N = 4 super–Yang–Mills falls in our class. Contents 1. 2. 3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . Pure Spinor Pairs and Vacua . . . . . . . . . . . . Dielectric Pure Spinors . . . . . . . . . . . . . . 3.1 From ordinary spinors . . . . . . . . . . . . . 3.2 From the definition . . . . . . . . . . . . . . 3.3 NS backgrounds . . . . . . . . . . . . . . . . 3.4 Branes . . . . . . . . . . . . . . . . . . . . . 3.5 A four–dimensional analogue . . . . . . . . . Geometrical Interpretation . . . . . . . . . . . . . 4.1 Pure spinors and Poisson geometry . . . . . . 4.2 Sigma–model interpretation . . . . . . . . . . 4.3 Restricting to four–dimensional leaves . . . . 4.4 Noncommutativity . . . . . . . . . . . . . . N = 2 NS Solutions from Superspace . . . . . . . 5.1 Generalized Kähler manifolds from a potential 5.2 Pure spinors from a potential . . . . . . . . .
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N. Halmagyi, A. Tomasiello
5.3 Checking the generalized Kähler condition . . . 5.4 Imposing the generalized Calabi–Yau condition A Special Case: Lunin–Maldacena NS Solution . . 6.1 Leigh–Strassler theories . . . . . . . . . . . . . 6.2 Moduli and Poisson bivectors . . . . . . . . . . 6.3 T–duality and bivector action . . . . . . . . . . 6.4 Lunin–Maldacena from a potential . . . . . . .
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1. Introduction Supersymmetric supergravity solutions with flux have recently started revealing their mathematical underpinnings. The concept of generalized complex geometry [1,2] has clarified for example the structure of N = 1 solutions with RR and NS field–strengths [3] and of N = 2 solutions with NS flux [2,4].1 The reason for these mathematical structures is likely to have its origin in the worldsheet action of the string. This is actually already manifest [2] in the case with only NS flux, the only one for which we currently have worldsheet control. In that case, generalized complex geometry has recently helped show [5] that the most general model with (2, 2) supersymmetry has an off–shell supersymmetric action. A by–product of the proof is that there exists locally a “generalized Kähler potential” K for any such model. This function K generalizes the familiar Kähler potential for the case without flux. A (2, 2) model need not have an N = 2 supergravity vacuum as a target, for the same reason that a Kähler manifold need not be Calabi–Yau. In the case without flux, K has to satisfy det(∂i ∂¯ j¯ K ) = const for the target to be Calabi–Yau. This is sometimes called the Monge–Ampère equation. We will see here something similar for the most general N = 2 background in which NS flux is also present. Namely, the generalized Kähler potential K has to satisfy a single differential equation, presented below in (5.21), in order for the manifold to be an N = 2 supergravity vacuum with NS three–form.2 We obtain this from supergravity, using the methods of generalized complex geometry, and this reproduces the one–loop computation in [8]. It also gives some new geometrical insight for the potential K ; for example, we see that K still appears in an expression ∂ ∂¯ K , see (5.23). On the way to showing this, we obtain some results of more general use. Generalized complex geometry approaches the supersymmetry problem via a compatible pure pair of two differential forms ± , satisfying certain algebraic constraints reviewed in Sect. 2. The ± also determine a metric and a B–field, so that in this paper we never write down the metric explicitly. Supersymmetry is then equivalent to simple–looking differential equations on the forms ± (see (2.8) and (2.11) below). We obtain in (3.1) a simple expression for the generic solution to the algebraic constraints for ± . This is a massage of an earlier computation [9]; we feel that the simplicity of (3.1) will allow to find RR solutions more easily, and to recognize more promptly their geometrical features. (For example, in the NS case (3.1) lets us recognize some similarities with four–dimensional studies in [1,10].) The specialization of the result (3.1) to NS N = 2 vacua reads (5.23) and leads to the generalized Monge–Ampère we mentioned earlier. We stress again that these ± , 1 In both cases, the results concern type II supergravity, which is the one of interest in this paper. 2 That a single equation should be sufficient was first suggested to us by M. Zabzine.
Generalized Kähler Potentials from Supergravity
3
for a K that satisfies (5.21), lead to a metric and B–field that satisfy the condition for an N = 2 vacuum. The other input in this construction is a holomorphic function w. One can see using [11] that w is the effective four–dimensional superpotential for a D3 brane sitting at the point in the internal manifold. This suggests an interpretation for the N = 2 vacua described in (5.23). Namely, by adding a large number of D3 branes to an NS background characterized by a function w, one would expect to obtain the gravity dual for a theory with a nonabelian version of w. One such solution already exists in the literature [12]. Although the main point of that paper is the gravity dual to the so–called β–deformation of N = 4 super–Yang–Mills, it also presents a purely NS analogue of that solution. We show in Sect. 6.4 that this NS solution is indeed in the form (5.23) for an appropriate choice of coordinates and of K , (6.24). For backgrounds with RR flux, in addition, there has also been recent progress relating the spacetime structure of generalized complex geometry and a worldsheet formalism [6]. The authors study SU(3)-structure deformations of Calabi-Yau backgrounds to first order in RR and NS flux using Berkovits’ hybrid formalism [7]. They find that the physical states must be modified and the constraints imposed by worldsheet (2, 2) supersymmetry are equivalent to the spacetime supersymmetry equations in the form derived in [3]. These calculations support the notion that generalized complex geometry is the natural framework in which to make contact with perturbative string theory. 2. Pure Spinor Pairs and Vacua In this section we will quickly review the definition of a compatible pure spinor pair and their uses in reformulating geometrically the supersymmetry conditions for vacua. For more thorough introductions to generalized complex geometry and its applications see [1,2,13]. First of all we will need an internal product in the space of forms (due to Chevalley): k
(A, B) vol ≡ (A ∧ λ(B))top , λ(Ck ) = (−)[ 2 ] Ck ,
(2.1)
where k denotes the degree of the form. This pairing is antisymmetric in six dimensions, and it is invariant under the action of O(6, 6) on forms (for more details on this action see for example [14, Sect. 2.1]). A pure spinor is a differential form (of mixed degree; thus a section of ⊕k k T ∗ ) 1. whose annihilator in (T ⊕ T ∗ ) ⊗ C has complex dimension 6, ¯ ) = 0 everywhere on the manifold. 2. such that (, Its type is the smallest degree occurring in the form. Thus ei J (which is pure, if J 3 is a volume form, because annihilated by ι∂m + i Jmn dxn ∧, ∀m) has type 0. A decomposable ¯ is never zero, because it is three–form = ξ1 ∧ ξ2 ∧ ξ3 (which is pure, if ∧ annihilated by ξi ∧ and their dual vector fields) has type 3. In fact, every pure spinor of type k [2, Prop. 2.24] can be written locally as = ξ1 ∧ · · · ∧ ξk ∧ e α for some one–forms ξi and two–form α. A pair of pure spinors ± is said to be compatible if
(2.2)
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N. Halmagyi, A. Tomasiello
1. the condition ¯ − , X + ) ∀X ∈ T ⊕ T ∗ (− , X + ) = 0 = (
(2.3)
is satisfied.3 2. The two pure spinors ± have equal norm, ¯ + , + ) = ( ¯ − , − ). (
(2.4)
3. If the pair ± satisfies the two previous conditions, it defines a metric (see [2,13] and below for details). Then we also impose that the metric defined by ± be positive definite. As an example of compatible pure pair, consider + = e−i J and − = , the two pure spinors of types 0 and 3 we have considered above. Condition 1 above then reduces to J ∧ = 0 (or, in other words, that J is of type (1, 1) in the almost complex structure ¯ = 4 J 3 . Now, one can determine a metric I defined by ). Condition 2 says that i ∧ 3 from the almost complex structure I and from J via gmn = Jmp I p n . Condition 3 then says that this metric should be positive definite. These conditions, together, make (J, ) an SU(3) structure on the manifold. More generally, the conditions for a compatible pure pair determine an SU(3)×SU(3) structure on T ⊕ T ∗ . By projecting the two SU(3) factors on T , one obtains two SU(3) structures on T . In the particular case we just saw, these two SU(3) structures coincide. Actually, two compatible pure spinors determine not just a metric g but also a B– field, a normalization function (which is going to be a combination of the dilaton and the warping), and two spinors η+1,2 of positive chirality. (One can think of these two spinors as the two SU(3) structures of the previous paragraph.) Concretely, this means that ± can be written as 2† , ± = (e B ∧)η+1 ⊗ η±
(2.5)
where the tensor product on the right-hand side is to be understood as a differential form using the Clifford map γ m 1 ...m k → dxm 1 ∧ · · · ∧ dxm k (see [13, Sect. 3.4] for a proof of (2.5) and for more explanations on the Clifford map). The spinors η+1,2 are in the spinor bundle associated to the metric g. This metric also determines a volume form vol, which we use from now on in the definition of the internal product (2.1). Finally, (2.5) can also be read backwards: namely, it is also true that given any B and η+1,2 , the right-hand side defines a compatible pure spinor pair. In particular, given such a pair ± and a two–form B, the new pair e B ∧ ± is also a compatible pure spinor pair; this is called a B-transform. The main reason to define a compatible pair is that the conditions for supersymmetric backgrounds of the form Minkowski4 × M6 can be reformulated completely in terms of a compatible pair on M6 . Namely, for an NS N = 2 solution [2,4], the metric has to be a product g10 = gMink4 + g6 , and there has to be a compatible pair on M6 with norm ¯ ± , ± )1/2 = e−φ (
(2.6)
(d + H0 ∧)± = 0.
(2.7)
such that
3 For the equivalence of this condition with the one defined in [2], see [14].
Generalized Kähler Potentials from Supergravity
5
Remembering that the pair ± might already define a non-zero B (see (2.5)), the total H curvature is H = H0 + d B. One is free to change the pair to one that has B = 0, which would then be closed under (d + H ∧);4 one cannot in general include all of H0 in the pair itself, unless one promotes the global behavior of ± from ordinary differential forms to something more ‘gerby’. This paper will be mainly concerned with local properties (we will work on C3 ); it will be convenient, then, to have all of the B field in the pure spinor pair, and the relevant condition will simply be d± = 0.
(2.8)
The condition for world–sheet (2, 2) supersymmetry is weaker than the condition (2.8) for target space N = 2 supersymmetry. Namely, since we have considered so far the case with no RR fluxes, we can consider the sigma model with target space described by a compatible pure pair ± and ask under what conditions it has (2, 2) worldsheet supersymmetry. The answer is known as bihermitian [15] or generalized Kähler [2] geometry, and it consists of the differential equations d± = (ιv± + ξ± ∧)±
(2.9)
for some vectors v± and one–forms ξ± .5 This condition is weaker than (2.8). The reason is that the generalized Kähler condition (2.9) guarantees (2, 2) worldsheet supersymmetry, but not necessarily conformal invariance, and hence (2.9) need not give rise to an N = 2 vacuum. This is very familiar for models with ordinary Kähler target spaces, that need not be Calabi–Yau. Finally, a similar result exists for backgrounds with non-vanishing RR fields [3,14]. For N = 1 supersymmetry, the metric can now be relaxed to be a warped product g10 = e2 A gMink4 + g6 , for A some function on the internal M6 ; the norm of the compatible pure spinor pair now has to be ¯ ± , ± ) = e3A−φ , (
(2.10)
and the differential equations are now (in IIB) d− = 0, d(e−A Re+ ) = 0, δ = −8 dd J− (e−3A Im+ ),
(2.11)
where δ is the given magnetic source, and d J− is a differential defined from − . (We have eliminated the RR field F from this equation, using the Bianchi identities. For more details, see [3,14].) Similarly to (2.7), if one wants a non–trivial NS curvature H0 , one can simply change d → (d + H0 ∧) in (2.11). We will see in the discussion after (4.3) that the first equation in (2.11) says that the manifold should be generalized complex. Both (2.8) and (2.11) are reformulations of the supersymmetry conditions. If one also satisfies the Bianchi identities and equations of motion for the fluxes, the remaining equations of motion follow [16,17]. For the NS flux H , we have assumed the Bianchi identity d H = 0 throughout; the equation of motion d ∗ H = . . . has recently been shown in [18] to follow from (2.11). By taking a limit in which the RR fluxes go to zero, (2.11) reproduces (2.8) (with the amount of supersymmetry doubling in the process), 4 H is always assumed to be closed, so (d + H ∧) is a differential. 5 The generalized Kähler condition is usually expressed in terms of generalized complex structures J , ±
tensors that we review succinctly in Sect. 4. There is a slight loss in generality here, in that for global reasons J± might exist without ± ; we are assuming that c1 of two line bundles are zero. This will not be important for our paper, that focuses on local solutions anyway.
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N. Halmagyi, A. Tomasiello
so [18] also shows that the equation of motion for H follows from (2.8). Turning to the RR fields, which are non–zero only in (2.11), their Bianchi identity can be easily shown to follow from (2.11) [13]; as for their equations of motion, they have been used to eliminate the RR flux from (2.11). 3. Dielectric Pure Spinors In this section we will show that a generic compatible pair of pure spinors can be written, up to B–transform and common overall rescalings, as ∧ z¯ + i(ω˜ 1 − ω˜ 2 ) . − = tan(2ψ)z ∧ exp i(ω˜ 1 + ω˜ 2 ) + = i exp
1
2z
(3.1)
Here, z is a one–form, ω˜ 1,2 are two complex two–forms that satisfy (ω˜ 1 )2 = 0 = (ω˜ 2 )2 ,
Imω˜ 1 = Imω˜ 2 ≡ Imω, ˜
(Imω) ˜ 2 = 0 everywhere, (3.2)
and ω˜ 1 ∧ ω˜ 2 + 2 sin2 (2ψ)(Imω) ˜ 2=0
(3.3)
for ψ a function (which is real, as follows from (3.2)). With purely algebraic manipulations, (3.1) can be demonstrated up to a B-transform where B is not necessarily closed. This will first be done in Sect. 3.1 from ordinary Cliff(6) spinors and then in Sect. 3.2 from the general definition of a compatible pair of pure spinors presented in Sect. 2. The B–field associated to the pair (3.1) is B = 2 sin2 (2ψ) Im ω, ˜
(3.4)
and the norm is ¯ ± , ± )1/2 = (
1 . cos(2ψ)
(3.5)
However, before we impose the differential constraints of supersymmetry (2.9), this is just a particularly nice choice. In Sect. 3.3 we will impose the constraints (2.9) and derive (3.1) up to a B-transform with d B = 0. 3.1. From ordinary spinors. We will now show how to obtain (3.1) if one defines the pure spinors as bilinears of two internal spinors. Let η+1,2 be two six–dimensional spinors of positive chirality such that ||η+1,2 ||2 = 1.
(3.6)
By multiplying one of them by a phase if necessary, we can arrange for the scalar η+1 † η+2
(3.7)
Generalized Kähler Potentials from Supergravity
7
to be purely imaginary. Now define η˜ + =
1 1 (η − iη+2 ), 2 +
χ+ =
1 1 (η + iη+2 ). 2 +
(3.8)
In general, given two spinors of the same chirality, we can always expand one in terms of the other; applying this to χ+ and η˜ + we get6 χ+ = a η˜ + + v · η˜ −
(3.9)
for some complex function a and vector v. However, using (3.6) and (3.7) we find i i χ+† η˜ + = − (η+1 † η+2 + η+2 † η+1 ) = − Re(η+1 † η+2 ) = 0. 4 2
(3.10)
Comparing this with (3.9) we see that a = 0, or in other words χ+ = v · η˜ − .
(3.11)
Going back to (3.8), we obtain by sum and difference η+1 = η˜ + + v · η˜ − ,
η+2 = i(η˜ + − v · η˜ − ).
(3.12)
1† 2 We can now define η+ = η˜ + /||η˜ + ||. Since η− vη+ = 0, we have ||η+1,2 ||2 = ||η˜ + ||2 + ||v · 2 η˜ − || ; recalling (3.6), it follows that ||η˜ + || cannot be larger than 1. Hence we can define
||η˜ + || = cos(ψ);
(3.13)
from (3.6) then it also follows that |v| = tan(ψ). Defining then z = v/ sin(ψ), we have obtained η1+ = cos(ψ)η+ + sin(ψ)z · η− , = i(cos(ψ)η+ − sin(ψ)z · η− ),
η2+
(3.14)
where now ||η+ || = 1 = |z|. We have shown that (3.14) is the most general pair of spinors one can write, up to a phase rotation for η+2 (that we fixed in (3.7)). The spinors in (3.14) are called dielectric spinors. In [19,20] it was realized that certain holographic RG flows are in fact neat realizations of the dielectric or Myers effect [21]. In those solutions, the ten–dimensional spinors have the schematic form = 1 + i2 → exp(iψ xy xy ∗),
(3.15)
where ∗ = ∗ . These spinors satisfy the projection conditions = (cos ψ + i sin ψ xy xy ∗)0123 .
(3.16)
Once we decompose the ten–dimensional 1,2 in terms of Minkowski and six–dimensional internal spinors as 1,2 = ζ4 ⊗ η+1,2 + c.c. as usual, (3.15) gives rise to (3.14). Physically, one interprets the projector (3.16) to be the rotation of a D3 brane projector into a D5 brane projector; thus it is called a dielectric projector. We will now show that one can derive from the “dielectric spinors” in (3.14) the expression for the “dielectric pure spinors” in (3.1). 6 We define η˜ = (η˜ )∗ ; a similar convention will be used for all chiral spinors. − +
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N. Halmagyi, A. Tomasiello
2† From η1,2 , one can define a compatible pair ± = η+1 ⊗ η± just like in (2.5). For (3.14), ± were computed in [9]. We can repackage them as follows: i j + 21 z ∧ z¯ + tan(2ψ) Im ω , + = i cos(2ψ) exp − cos(2ψ) (3.17) i Re ω − cos(2ψ) Im ω . − = sin(2ψ)z ∧ exp sin(2ψ) sin(2ψ)
Here, ω and j describe, together with z, an SU(2) structure on M6 . It is inside the SU(3) structure (J = j + 2i z ∧ z¯ , = ω ∧ z) defined by η+ through 8η+ ⊗ η+† ≡ e−i J and † ≡ . 8iη+ ⊗ η− i Each of the ηi alone also defines an SU(3) structure via 8η+i ⊗ η+i † ≡ e−i J and i† ≡ i ; and again each of the two SU(3) structures defines, together with z, 8iη+i ⊗ η− an SU(2) structure: i = ωi ∧ z,
i J i = j i + z ∧ z¯ . 2
(3.18)
One can compute
ω1,2 = cos(2ψ)Reω ∓ sin(2ψ) j + iImω = Re e±2iψ (Reω + i j) + iImω, (3.19) j 1,2 = cos(2ψ) j ± sin(2ψ)Reω = Im e±2iψ (Reω + i j) .
Now we want to try and reexpress the pure spinors (3.17) in terms of ω1,2 in (3.19). We also have the freedom of taking a B–transform ± → e−B ∧ ± ;
(3.20) 2† η± ,
notice that so far (3.17) have been defined by ± = η+1 ⊗ and hence, comparing with (2.5), they define a zero B. It so happens that the best choice for B is such that the exponent of + is purely imaginary: B = tan(2ψ)Imω.
(3.21)
In Sect. 3.3 we will see that the differential constraints of NS N = 2 backgrounds impose this choice (up to a closed B); it actually also happens to make the expression for ± more pleasant–looking: i (ω1 − ω2 ) , + = i cos(2ψ) exp 21 z ∧ z¯ + sin(4ψ) (3.22) i (ω1 + ω2 ) . − = sin(2ψ)z ∧ exp sin(4ψ) ¯ ± , ± ) = 1, since we have taken ||η|| = 1 in (3.14). We So far (3.22) have norm ( can also rescale (3.22), to obtain, after defining ω˜ a =
1 ωa , sin(4ψ)
(3.23)
the compatible pure spinor pair in (3.1), with norm given by (3.5); the B field in (3.21) turns into (3.4). In Sect. 3.3 we will see that this choice of normalization is forced on us in the case of NS N = 2 backgrounds, just as it was the case for the choice of B–field (as remarked after (3.21)). One can check using (3.19) that the ω˜ a defined in (3.23) satisfy (3.2) and (3.3).
Generalized Kähler Potentials from Supergravity
9
3.2. From the definition. We will now also show how to obtain (3.1) from the definition of a compatible pure spinor pair. First of all, a generic pure spinor pair in six dimensions has types 0 and 1; namely, the form of lowest degree in + is a zero–form, and in − , a one–form. We also know from (2.2) that a pure spinor of type 1 can always be written as − = φ 1 ∧ e α
(3.24)
for some one–form φ1 and two–form α. Similarly, a pure spinor + of type 0 can be written as the exponential of a two–form. Without loss of generality we can take ¯
+ = eβ+ f φ1 ∧φ1 ,
(3.25)
for some function f . One could also allow for another function in front of the exponential; however, given a compatible pair, even after taking into account (2.4), one has the freedom to rescale both ± by a function, and we will fix this ambiguity by taking the zero–form in + to be just 1. For v = vm dxm a one-form, we denote the contraction vm (E −1 )mn ι∂n by v. Here as usual E = g + B. Now we use this contraction operation to decompose an arbitrary two-form ω as ω = ω +
φ1 φ¯ 1 ω φ1 ∧ (φ¯ 1 ω) φ¯ 1 ∧ (φ1 ω) + − φ1 ∧ φ¯ 1 φ¯ 1 φ1 φ1 φ¯ 1 (φ¯ 1 φ1 )(φ1 φ¯ 1 )
(3.26)
so that φ1 ω = 0, φ¯ 1 ω = 0.
(3.27)
To find out about the properties of φ1 and its conjugate, one has to compute g and B from (3.25) and (3.24). Actually, one can take a shortcut by using (2.5) to translate the annihilators of ± in terms of gamma matrices; for more details see for example [13, Sect. 3.4]. This gives the conditions ¯ + = 0, (φ1 ∧ −φ1 ) (φ1 ∧ −φ1 )− = 0,
(φ1 ∧ +φ1 )± = 0,
(3.28)
and their conjugates. Imposing this and applying the decomposition (3.26) to the two– forms α, β, after some algebra we find that we can rewrite the pure spinors (3.25) and (3.24) again as
− = φ 1 ∧ e α , + = exp β + f φ1 ∧ φ¯ 1 ,
(3.29)
but this time in terms of new α , β , f such that φ1 and φ¯ 1 annihilate α and β , and the function f is real. From now on we will drop the primes. We now apply (2.3) to ± . If X is a one–form ζ , we get ¯ 2 φ1 ∧ ζ ∧ (β − α)2 = 0 = φ¯ 1 ∧ ζ ∧ (β − α)
(3.30)
¯ 2. (β − α)2 = 0 = (β − α)
(3.31)
which implies
The case in which X in (2.3) is a vector does not give any extra condition.
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We can already see from (3.31) that β − α and β − α¯ have the properties required by (3.2). However, the forms that appear in ± are α and β. To make contact between the two, recall once again that, given a pure spinor compatible pair, one can always produce another by ± → e B ∧ ± for B any real two–form. Using this, we can choose to make β purely imaginary. As in the previous subsection, this is just a choice at this point, but it will be pointed out in the next subsection that it is actually necessary for NS N = 2 backgrounds. Having made β purely imaginary, we can define α − β = 2i ω˜ 2 ,
α + β = 2i ω˜ 1 ,
(3.32)
and by (3.31) we conclude (3.2). To summarize, so far we have obtained that the pure spinors can be written, up to a B–transform, as + = exp[i(ω˜ 1 − ω˜ 2 ) + f φ1 ∧ φ¯ 1 ],
− = φ1 ∧ exp[i(ω˜ 1 + ω˜ 2 )].
(3.33)
This takes care of the first condition for compatibility that we saw in Sect. 2, namely (2.3). We now turn to (2.4). For that, notice first that (3.2) (which we just derived from (3.31)) implies (Reω˜ 1 )2 = (Reω˜ 2 )2 = (Imω) ˜ 2 and Reω˜ a ∧ Imω˜ = 0. The wedge prod1 2 uct Reω˜ ∧ Reω˜ , however, is not determined by this. Hence we can define the function ψ by Reω˜ 1 ∧ Reω˜ 2 + (1 − 2 cos2 (2ψ))(Imω) ˜ 2 = 0,
(3.34)
and then apply (2.4) to (3.33). We get that f = By taking now z =
φ1 tan(2ψ)
1 2 tan2 (2ψ)
.
(3.35)
one finally finds (3.1). (Equation (3.34) reduces then to (3.3).)
3.3. NS backgrounds. In deriving (3.1), we had to fix two ambiguities: under the B–transform (± → e B ∧ ± ) and rescaling (± → f ± ). We will now show that the choices we made are actually necessary in the case of NS backgrounds with N = 2 supersymmetry. The differential equations are quite simple: they say that ± are closed, (2.8). Let us focus on + , and let us go back to the expression for it given in (3.17) i 1 + = cos(2ψ) exp − j + z z¯ + tan(2ψ)Imω (3.36) cos(2ψ) 2 which has norm 1 and B = 0. (Both the norm and B field do depend on what − is; they would be different if we changed − in (3.17).) Now let us rescale and B–transform this + (supposing we also do the same to − ), and impose d( f e−B ∧ + ) = 0. First of all we see that f = 1/ cos(2ψ). This explains the rescaling made at the end of Sect. 3.1, to go from (3.22) to (3.1), which gives (3.5). In particular, remembering (2.6), we have eφ = cos(2ψ).
(3.37)
Generalized Kähler Potentials from Supergravity
11
Then we also see that the exponent of e B ∧ + should be closed, which means
i 1 d B+ j − z z¯ − tan(2ψ)Imω = 0. (3.38) cos(2ψ) 2 The real part of this equation implies that B = tan(2ψ)Imω + B0 , where B0 is closed. In (3.4) we took B0 = 0, since the focus of this paper is on local properties. Now that we have justified the choices made to fix the ambiguities in (3.1), we can also impose that they be closed. For completeness, we write them here:
i 1 2 1 2 d ω˜ − ω˜ − z ∧ z¯ = 0. (3.39) d(tan(2ψ)z) = 0, z ∧ d(ω˜ + ω˜ ) = 0, 2 3.4. Branes. The only reason we gave so far for being interested in the compatible pair (3.1) is that it is the most general pair of types 0 and 1, and hence the generic pure spinor pair. A more compelling and physical reason to be interested in backgrounds of this type is that the moduli space of D3 brane probes is partially lifted. This point was originally made in [11] but we repeat it here for the reader’s convenience. The fact that the moduli space of D3 branes is lifted can be seen in various ways. The most straightforward is to use the conditions in [11,22,23]: ι∗ [Re+ ]|top = 0,
ι∗ [(ι∂m + gmn dxn ∧)− ]|top = 0,
(3.40)
where ι : B → M6 is the inclusion, and as usual ι∗ is the pull–back. The symbol |top means that one should keep the form of highest degree on B. These conditions generalize, and are derived in the same way as the ones for backgrounds without fluxes: see for example [24,25]. Also, they reproduce physically the mathematical definition of generalized complex submanifolds given in [2]. So, consider D3 brane–probes extended along Minkowski4 and located at a point in M6 , in a background described by (3.1). The first condition in (3.40) is automatically satisfied, thanks to the i in front of + in (3.1). The second condition is satisfied only at points where tan(2ψ)z = 0.
(3.41)
Also, recall that, if (3.1) describe an N = 2 solution, ± must be closed. This implies (as we have seen already in (3.39)) that d(tan(2ψ)z) = 0. Locally, this means that tan(2ψ)z = dw
(3.42)
for some function w. In fact, one can go further and argue [11] that w is nothing else than the four–dimensional superpotential (for a single brane probe). As a check, supersymmetric vacua for the effective four–dimensional theory are critical points for w. At these points dw = 0, which, remembering (3.42) and (3.41), is precisely the condition for the D3 brane to be supersymmetric. This result is important for us: it tells us that the class of metrics we are considering can be trivially adjusted so as to produce an arbitrary assigned superpotential w on the four–dimensional effective theory on D3 branes. Another interesting case to consider is the case of a D7 wrapping the submanifold {w = w0 }. (We will see why this is interesting geometrically in Sect. 4.) In [26] it has
12
N. Halmagyi, A. Tomasiello
been shown for a few examples of solutions (albeit with nonvanishing RR fields) that this is a supersymmetric cycle. In general, wrapping the submanifold {w = w0 } with the B–field we gave in (3.4) (and hence with the pure spinors in (3.1)) does not satisfy the conditions (3.40) for a supersymmetric brane. However, in Sect. 6 we will consider a solution for which a different choice B = B + dλ exists7 (originally considered in [12]) so that the supersymmetry conditions with that B are satisfied. We will have more to say about this case in Sect. 4.4. 3.5. A four–dimensional analogue. It might be useful to notice how the results obtained so far extend those obtained in [10,27]. We will also see in Sect. 4.3 that this situation is relevant for us because of a certain foliation we will define in Sect. 4. The papers [10,27] consider the generalized Kähler condition (a weaker version of the N = 2 condition, which we reviewed around Eq. (2.9)). They point out that a 1,2 generalized Kähler manifold in four dimensions can be obtained by two two–forms ω˜ (4) , 1 )2 = 0 = (ω 2 )2 , ˜ (4) (ω˜ (4)
1 = Im ω 2 ≡ Im ω Imω˜ (4) ˜ (4) ˜ (4) ,
(Imω˜ (4) )2 = 0 everywhere,
1,2 d ω˜ (4) = 0.
(3.43)
The algebraic equations in (3.43) are exactly like those for ω˜ 1,2 in (3.2). The requirement 1,2 that ω˜ (4) should be closed is less exactly paralleled by (3.39). In fact, we can explain (3.43) from pure spinors, and explain the difference with (3.39) (relevant to the six–dimensional case) in the process. One can consider the pair (4)
1 = cos(2ψ)ei(ω˜ (4) −ω˜ (4) ) , 1
2
(4)
2 = sin(2ψ)ei(ω˜ (4) +ω˜ (4) ) , 1
2
(3.44)
which can be derived, as in Sect. 3.1, from four–dimensional spinors 1 = cos(φ)η ∗ η(4) (4) + sin(ψ)η(4) , 2 = i(cos(φ)η ∗ η(4) (4) − sin(ψ)η(4) ).
(3.45)
The condition for a pair ± to define a generalized Kähler structure is (2.9); we will show in Sect. 5.3 that applying (2.9) to a pure spinor of type 0, = f eα , implies that 1,2 the exponent is closed, dα = 0. Using this in (3.44), we obtain that d(ω(4) ) = 0, which is the differential equation in (3.43). 4. Geometrical Interpretation In this section, we will make some mathematical comments about the compatible pure spinor pair (3.1), many of which will be needed in Sect. 5. In Sect. 4.1 we will spell out some relations between pure spinors and Poisson geometry, and review a splitting of the tangent bundle T introduced in [5]. This splitting will then be given a sigma–model interpretation in Sect. 4.2, again following [5]. In Sect. 4.3 we will focus on one particular summand in this decomposition of T . Section 4.4 explains a possible spinoff in noncommutative geometry. 7 This different choice is not gauge equivalent to B, because we are not transforming A = A − λ at the same time.
Generalized Kähler Potentials from Supergravity
13
4.1. Pure spinors and Poisson geometry. The aim of this subsection is to show and explain the decomposition (4.6) of the tangent bundle T . We will start by recalling some mathematical definitions. A Poisson tensor P mn is a bivector (a section of 2 T ) such that P [m|q ∂q P |np] = 0.
(4.1)
In index–free notation, (4.1) can be written as [P, P]NS = 0, where [ , ]NS is the Nijenhuis–Schoutens bracket on k T . A foliation is a partition of a manifold M6 in disjoint connected sets (called leaves), such that in every chart of the atlas of M6 each leaf is homeomorphic to a vector subspace of R6 . A distribution D is a choice at every point x ∈ M6 of a subspace Dx of the tangent space Tx at that point. A distribution is said to be integrable if there exists a foliation of M6 such that, at every point x, Dx is the tangent space to a leaf of the foliation. According to Frobenius’ theorem, a distribution is integrable if it is closed under a Lie bracket. A less used, dual form of this theorem can be given by looking at the orthogonal of D with respect to the natural pairing between T and T ∗ , D ⊥ ⊂ T ∗ : namely, D is integrable if D ⊥ is closed under the action of d. Consider now the distribution defined on M6 by the image of P, im(P) ⊂ T . A classic result is that if P is a Poisson tensor, im(P) is an integrable distribution. We will see later how to show this using pure spinors. Given a pure spinor , one can associate to it a matrix J acting on sections of T ⊕ T ∗ , such that J 2 = −1. Concretely, in dimension 6 this is a 12×12 matrix that can be decomposed in 6×6 blocks:
I P . (4.2) J = J −I t (The lower–right block is not independent because of a certain hermiticity property of J that follows automatically from the non–degeneracy property of .) The way one associates such a matrix to is that the annihilator in T ⊕ T ∗ of (which has dimension 6, by ¯ is the −i–eigenspace of J . For definition) is the i–eigenspace of J ; the annihilator of example, we saw in Sect. 2 that a decomposable non–degenerate complex three–form defines an almost complex structure I . Then one can see that J I = 0I −I0t . One can also show that if d = 0
(4.3)
then J satisfies a certain integrability condition. In this case J is called a generalized complex structure.8 We recognize here the first condition in (2.11). As we anticipated there, it follows that a RR solution is also a generalized complex manifold. As an example, we can consider an ordinary complex structure I . We saw in Sect. 2 that a non–degenerate complex three–form defines an almost complex structure I . Then if one has (4.3), namely d = 0, the matrix J I = 0I −I0t defined above should be a generalized complex structure. One can see that this implies that I is a complex structure. It was indeed already known well before the definition of generalized complex geometry that d = 0 implies that the associated I is a complex structure. Going back to the general case, if J is generalized complex, it also follows that the upper–right block P in (4.2) is a Poisson tensor [28]. In fact, we can see how the 8 Because of (4.3), one can say that this generalized complex structure has a holomorphically trivial canonical bundle, in analogy with the particular case of an ordinary complex structure that we will consider shortly. [1] calls this case “generalized Calabi–Yau”.
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N. Halmagyi, A. Tomasiello
integrability of im(P) follows from d = 0. Here is how. We know from (2.2) the general expression of a pure spinor of type k. It is easy to see that the ξi in that formula generate ker(P), which is a subspace of T ∗ . Indeed, ξi ∧ is in the annihilator of , and hence (0, ξ ) ∈ T ⊕ T ∗ must be an eigenvector of J . Looking at (4.2), we see that this is the case only if Pξi = 0, ∀i = 1, . . . , k. As we mentioned earlier, a dual form of Frobenius’ theorem says that D is integrable if D ⊥ is closed under the action of d, where the orthogonality is intended with respect to the natural pairing between T and T ∗ . In our case, D = im(P), and D ⊥ = ker(P). From d = 0 it follows that d(ξ1 ∧ · · · ∧ ξk ) = 0, which means that {ξi } is closed under d; but {ξi } = ker(P). By the dual Frobenius theorem above, then, we have that im(P) is an integrable distribution, as we wanted to show. With a pair of pure spinors, there are two Poisson tensors P± that one might want to consider. In our case, ± in (3.1) have types 0 and 1 (in fact, we argued in Sect. 3.2 that they are the most general pair of those types). Hence P+ , the Poisson tensor associated to + , has no kernel. P− , associated to − , has kernel of dimension 1. It is easy to recognize for example what P+ is: since it is invertible, we can define the two–form P+−1 . When a Poisson tensor is invertible, a consequence of (4.1) is that its inverse two–form is closed. Hence d(P+−1 ) = 0. Then it is easy to guess that P+−1 is nothing but the exponent of + . (We assumed in this section that all pure spinors are closed; this condition implies that the upper–right block of (4.2) is Poisson. The generalized Kähler condition (2.9) would also be sufficient.) Actually, we can also associate P± to the pure spinors B–transformed in such a way as to have B = 0; this means for us the pair in (3.17). Remember (from (2.5)) that 2† . Each of the two spinors η+1,2 in that case the pair can be written as ± = η+1 ⊗ η± defines an almost complex structure I1,2 (by looking at the gamma matrices that annihilate them, for example: (1 + i I1,2 )n m γn η+1,2 = 0). An expression for the Poisson tensors P± associated to ± is then [2]9 P± = (I1 ± I2 )g −1 .
(4.4)
There is, however, a third Poisson tensor that one can consider [30]: σ ≡ [I1 , I2 ]g −1 ,
(4.5)
and this is the one that will be important for us in what follows. As noted in [5], since [I1 , I2 ] = (I1 − I2 )(I1 + I2 ), one has that ker[I1 , I2 ] = ker(I1 + I2 ) ⊕ ker(I1 − I2 ). One can then decompose the tangent space T as ker(I1 − I2 ) ⊕ ker(I1 + I2 ) ⊕ im(σ )
(4.6)
and the last distribution is integrable, being the image of a Poisson tensor. 4.2. Sigma–model interpretation. The decomposition (4.6) has a physical interpretation that will be important for us. Remember that, if the two pure spinors ± are closed (as in (2.8)), we are describing an N = 2 vacuum. Since there are no RR fields around, we can write down the sigma model with this background as a target; this sigma model turns out to have (2, 2) supersymmetry on the worldsheet. This condition is weaker than the one for an N = 2 vacuum, as we saw around Eq. (2.9). Suppose this supersymmetric model 9 These Poisson tensors were previously considered in [29].
Generalized Kähler Potentials from Supergravity
15
can be written in terms of (2, 2) superfields. There are three types of known superfields that include scalars: chiral multiplets (the usual ones, that describe for example a sigma model with a Kähler target space), twisted chiral multiplets, and semichiral multiplets (see for example [15]). Suppose there are in our model respectively dc , dt and ds of each type. Then one can see that 2dc = dim(ker(I1 − I2 )),
2dt = dim(ker(I1 + I2 )),
4ds = dim(im(σ )). (4.7)
In our case, since the pure spinors (3.1) have types 0 and 1 respectively, we know already that P± have kernels of dimensions 0 and 1.10 Looking at (4.6), we see that σ must have an image of dimension 4. Looking at (4.7), we see that this corresponds to having one semichiral multiplet. Whether the remaining multiplet is a chiral or twisted chiral multiplet is a matter of convention, and we choose it to be chiral. In conclusion, (3.1) corresponds on the worldsheet to having one chiral and one semichiral multiplet. The paper [5] gives the form of the action in terms of (2,2) superfields for a model with an arbitrary number of chirals, twisted chirals and semichirals. This gives a local construction of generalized Kähler manifolds. In Sect. 5 we will see how to promote this to a full–blown N = 2 supergravity solution. 4.3. Restricting to four–dimensional leaves. Before we do that, we need some more information on the Poisson tensor σ defined in (4.5). We have learned that any Poisson tensor P defines a foliation. Since the tangent space to that foliation is given by the distribution im(P), it follows that P is invertible on each leaf. In the case of σ , we know that its leaves are four–dimensional, and that they are given by {w = w0 } (remembering the discussion in Sect. 3.4, especially (3.41) and (3.42)).11 . Using local coordinates, one can define a restriction σ(4) of σ to the leaf {w = w0 }. In [30], it is shown that σ is of type (2, 0) + (0, 2); the (2, 0) part with respect to I1 (I2 ) is holomorphic with respect to I1 (I2 ). These properties remain true for σ(4) , which is also −1 invertible. The two–form σ(4) is holomorphic and of types (2, 0) + (0, 2). If we now take the (2, 0) part with respect to I1 , we have a holomorphic (2, 0)–form. A holomorphic version of the Darboux theorem implies that there exist coordinates q, p on {w = w0 } −1 is equal to dq ∧ dp (up to a constant). In particular, q such that the (2, 0) part of σ(4) and p are holomorphic coordinates with respect to I1 . Similar coordinates exist for I2 , which we call Q, P. The preceding paragraph condenses some arguments in [5], to which we refer the reader for more details; but the upshot for us is that one can define two complex structures (4) I1,2 on the four–dimensional leaves, just by using the holomorphic coordinates above. Now we can use that, in four dimensions [27], (4)
(4)
{I1 , I2 } = p14
(4.8)
for some function p. From this it also follows [I1(4) , I2(4) ]2 = ( p 2 − 4)14 . 10 If we call z the one–form part of , which is then in the kernel of I − I , it also follows that (I + I )z − 1 2 1 2 is in ker(I1 − I2 ); since by assumption this kernel is 1–dimensional, it follows that I1 z = I2 z = i z. 11 Strictly speaking, the generic leaves are four–dimensional; in the locus in which the one–form part of − vanishes, which is for us given by critical points of w, the two complex structures coincide, and σ vanishes.
16
N. Halmagyi, A. Tomasiello (4)
Each of the complex structures I1,2 on the leaf define, together with the pullback of the metric g(4) , an SU(2) structure. The two–forms for these two SU(2) structures satisfy the same relations as in (3.19), only now with every form replaced by its four–dimen−1 (4) (4) sional counterpart. Also, since σ(4) is invertible, we can write g(4) = σ(4) [I1 , I2 ]. We can now compute −1 (4) (4) (4) −1 j2(4) = g(4) I2(4) = σ(4) [I1 , I2 ]I2 = −σ(4) (2I1(4) + p I2(4) ),
(4.9)
(4)
and similarly for j1 ; by using then the four–dimensional analogues of (3.19) we can compute Imω(4) . One gets an expression containing a symmetric part; setting it to zero one obtains tan2 (2ψ) =
2+ p , 2− p
−1 Imω˜ (4) = 2σ(4) .
(4.10)
Recall that the tilde means division by sin(4ψ), as in (3.23). Equation (4.10) now gives 1 1 ¯ ¯ = − (d Q ∧ d P + d Q¯ ∧ d P) Imω˜ (4) = − (dq ∧ dp + d q¯ ∧ d p) 2 2
(4.11)
and 1 i ω˜ (4) = dq ∧ dp,
2 i ω˜ (4) = d Q ∧ d P.
(4.12)
We will see in Sect. 5.2 how these equations are extended to six dimensions. 4.4. Noncommutativity. Poisson tensors are used in classical mechanics to define Poisson brackets of functions on phase space. A natural question to ask is whether one can define naturally a noncommutative product among those functions, or in other words whether it is possible to quantize the Poisson bracket. It has been argued for example in [31] that this noncommutativity should be related to the one defined by the F–terms in the field theory duals, which we will review in Sect. 6. An immediate question is why string theory should quantize the Poisson tensor implicit in the geometry. One mechanism known in string theory to produce noncommutativity is via a B–field on the world–volume of a brane. In the case of a constant B–field along a flat–space brane, the noncommutativity parameter is given by [32,33] θ=
1 [(g + B)−1 − (g − B)−1 ] ≡ [(g + B)−1 ]A . 2
(4.13)
What we will show now is that if one computes (perhaps naively) this tensor θ for the D7 branes extended along a four–dimensional leaf {w = w0 }, one obtains exactly the Poisson tensor σ(4) we were dealing with in Sect. 4.3. We find this very suggestive, and one might consider it as a generalization of the “canonical coisotropic” brane in [34];12 it also seems to give an alternative view of [35]. On the cautionary side, the role of θ as “noncommutativity parameter” is far from clear in the non–flat case, and in any case the computation below does not show why the noncommutativity should be transferred to the field theory dual (a matter on which we will return in Sect. 6). Another problem is 12 We thank E. Witten for this remark.
Generalized Kähler Potentials from Supergravity
17
that a D7 wrapping the leaf {w = w0 } appears to be supersymmetric for a B = B + dλ different from the B we are using in this paper (as we remarked in Sect. 3.4). Be that as it may, pulling back (3.4) and using (4.10) gives −1 B(4) = 4 sin2 (2ψ)σ(4) .
(4.14)
Recall again that, on {w = w0 }, σ(4) is invertible, and hence we can rewrite (4.5) as −1 (4) (4) [I1 , I2 ]. We can then compute, with some manipulation, θ as defined by g(4) = σ(4) (4.13), with g(4) and B(4) as inputs: −1 θ = (2σ(4) (1 + I1(4) I2(4) ))−1 A =
1 8 sin2 (2ψ)
((1 + I2(4) I1(4) )σ(4) )A =
1 σ(4) , 4
(4.15)
which is what we claimed. 5. N = 2 NS Solutions from Superspace We will use the worldsheet construction of generalized Kähler structures in [5], and find that a single equation, (5.21), is enough to promote them to N = 2 supergravity solutions. We will do this in Sect. 5.4 by computing the pure spinors (5.23) associated to their “generalized Kähler potential” (which we will review in Sect. 5.1). After checking in Sect. 5.3 that the pure spinors indeed satisfy the conditions for the (2, 2) model, we will impose the stronger condition for an N = 2 supergravity vacuum in Sect. 5.4. 5.1. Generalized Kähler manifolds from a potential. We will first review some more results we need from the paper [5] (some part of it was already reviewed in Sect. 4.3). It is well-known that the off–shell supersymmetric action for a (2, 2) model without flux can be written as an integral over superspace of a single function real function K : S = d 2 σ d 2 θ d 2 θ¯ K (5.1) defined by J = i∂ ∂¯ K . The function K depends on chiral (2, 2) multiplets. There exist however more general (2, 2) models, those whose target space is a generalized Kähler manifold (as briefly reviewed by us around (2.9)). For these more general target spaces, it has been known for some time [15] that, to write the action in an off– shell supersymmetric fashion, one also needs new multiplets called twisted chiral and semichiral (as we mentioned already in Sect. 4, see for example (4.7)). The action can still be written as (5.1), but now K is a function of all three types of multiplets, and not of chirals only. The paper [5] found a geometrical interpretation of the “generalized Kähler potential”K , showing in the process that it, and an off–shell action, exist locally for any generalized Kähler manifold and hence for any (2, 2) model. Before we explain that interpretation, let us specialize our discussion to the number and types of multiplets we need. We reviewed in the previous section (see the discussion below (4.7)) that a compatible pure spinor pair of types 0 and 1 (for which (3.1) is the most general expression, as argued in Sect. 3) corresponds to having one semichiral and one ordinary chiral (2, 2) multiplets. There are two different complex structures I1,2 in a generalized Kähler geometry, as we saw in Sect. 4. In Sect. 4.3 we chose holomorphic coordinates for both I1 and I2 .
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N. Halmagyi, A. Tomasiello
Since ker(I1 − I2 ) = 1, or in other words, there is one chiral multiplet, I1 and I2 share an eigenform (see footnote 10); hence we took one of the holomorphic coordinates for I1 and one of the holomorphic coordinates for I2 to coincide. We called it w, since, as we reviewed in Sect. 3.4, it is the superpotential for a single D3 probe. Following [5], we called q, p the other two holomorphic coordinates for I1 ; and Q, P the other two for I2 . Obviously these four are redundant, and we will take q and P to be independent. Now for the geometrical interpretation of K in (5.1). On each leaf, the transformation 1 and ω2 preserves the form Imω . This form is closed because of (4.11), between ω(4) (4) (4) and non–degenerate because (3.2) is preserved by pull–back. In other words, on each leaf the change of coordinates between q, p and Q, P is a canonical transformation with respect to Imω(4) . Now, K in (5.1) is shown in [5] to be the “generating function” of this canonical transformation, in the sense that p = ∂q K ,
Q = ∂P K ,
(5.2)
just like in classical mechanics. We will actually see in the next subsection that an alternative definition exists: rather than being a function such that J = i∂ ∂¯ K as in the Kähler case, it is a function such that + = exp[−∂ ∂¯ K ], as we will see in (5.23). We should notice that, although our focus is on N = 2 supergravity solutions, this particular result is valid locally for all generalized Kähler manifolds, as we will explain after (5.17).
5.2. Pure spinors from a potential. We will now compute the pure spinors corresponding to the construction of generalized Kähler manifolds in [5] that we reviewed in Sect. 5.1. First some preliminary definition. Given that there are two different complex structures, there are two different Dolbeault operators ∂1 , ∂2 that it would be natural to work with. In what follows, somewhat surprisingly, it will be useful to consider the “mixed” Dolbeault operator ∂ ≡ dq∂q + d P∂ P + dw∂w ,
(5.3)
which utilizes a holomorphic coordinate with respect to the complex structure I1 and one with respect to I2 . Given this ∂, we can compute the two–form ∂ ∂¯ K . We will also need the hermitian matrix of its coefficients in the coordinates q, P, w: ⎞ ⎛ K q q¯ K q P¯ K q w¯ ⎟ ⎜ (5.4) K H = ⎝ K P q¯ K ¯ K P w¯ ⎠ , PP K wq¯ K w P¯ K ww¯ where for example K q q¯ = ∂q ∂q¯ K . It will also be useful to define its matrix of minors R. This is the same as −1 . R = det(K H )K H
(5.5)
Rww¯ = K q q¯ K P P¯ − K q¯ P K q P¯ .
(5.6)
So for example
Generalized Kähler Potentials from Supergravity
19
Now for the computation of the pure spinors. First of all we can compute the trigonometric functions present in the pure spinor Ansatz of the previous section. This is done by taking the anticommutator {I1 , I2 } (the complex structures I1,2 are explicitly given in [5]) and restricting it to the four semichiral directions. In those directions, it has to be proportional to the identity [27], as we saw in (4.8); from (4.10) we find that the proportionality factor p = 2(sin2 (2ψ) − cos2 (2ψ)). This gives sin2 (2ψ) =
K q q¯ K P P¯ − K q¯ P K q P¯ K q P K q¯ P¯ − K q¯ P K q P¯
,
cos2 (2ψ) =
K q P K q¯ P¯ − K q q¯ K P P¯ K q P K q¯ P¯ − K q¯ P K q P¯
.
(5.7)
Now a few remarks about the metric, which is given by [5] in their (3.33), which we can easily specialize to the case with one semichiral and one chiral multiplet. We saw in Sect. 4 that the Poisson tensor σ defines a foliation whose leaves are {w = w0 }. If we stay away from the critical loci of w, where the leaves change dimension, this foliation is just a fibration, and we can write the metric in a way adapted to it: g = gi j dy i dy j + 2gia dy i dxa + gab dxa dxb = gi j (dy i + Ai a dxa ) · (dy j + A j b dxb ) + z · z¯ .
(5.8)
Here · denotes the symmetric tensor product, and we have denoted the coordinates q, P, q, ¯ P¯ collectively by yi , and w, w¯ by xa . The right-hand side of (5.8) defines z. Any metric can be rewritten in the form (5.8); the presence of the foliation guarantees that this is globally well–defined (away from the critical loci of w), although this is not going to be too important for us, since we are eventually going to apply these metric to R6 . From the explicit form of the metric given in [5], one can compute A and z · z¯ in (5.8): Aq = −
Rwq¯ dw, Rww¯
Rw P¯ dw, Rww¯
(5.9)
det(K H ) dw · d w¯ ≡ ρ 2 dw · d w. ¯ Rww¯
(5.10)
AP = −
¯
along with Aq¯ = Aq and A P = A P , and z · z¯ = 4
gi j is not particularly interesting and we do not need its explicit expression. (A can also be extracted from {I1 , I2 }.) We now want to write down the pure spinors for the generalized Kähler metric in [5], using the complex structures I1,2 that they give explicitly in their formulas (6.72, 6.74). To do this, one can proceed in several ways. One can for example compute J1 = g I1 and J2 = g I2 , and then use (3.18) and sums and differences of (3.19) to compute ωr , ωi and j, and then ω1 and ω2 . Without giving more details, we will describe here the result. Define the following “push–forward” way to extend forms from the four dimensions spanned by q, P to the whole six–dimensional manifold: ι∗ (d4 q) ≡ Dq ≡ dq + Aq ,
ι∗ (d4 P) ≡ D P ≡ d P + A P ,
(5.11)
where d4 is the exterior differential along the leaves {w = w0 }. One has, then, i ω1 = ι∗ (d4 q ∧ d4 p), sin(4ψ)
i ω2 = ι∗ (d4 Q ∧ d4 P). sin(4ψ)
(5.12)
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N. Halmagyi, A. Tomasiello
To evaluate these expressions concretely, one should first express Q = ∂ P K and p = ∂q K , and then use (5.11), which results in i 1 sin(4ψ) ω i 2 sin(4ψ) ω
¯ = 8Dq ∧ (K q q¯ D q¯ + K q P D P + K q P¯ D P), ¯ ∧ DP ; = 8(K Pq Dq + K P q¯ D q¯ + K P P¯ D P)
(5.13)
one can see that with this definition ω˜ a = ωa / sin(4ψ) satisfy the conditions (3.2) and (3.3). One can also see that (5.13) imply, for the 1,2 defined in (3.18), i 1 2 cos2 (2ψ) i
2 cos2 (2ψ)
= 8dw ∧ dq ∧ dp,
2 = 8dw ∧ d Q ∧ d P,
(5.14)
after some simplification recalling that the coefficients in Aq , A P come from the inverse matrix of K H (see (5.5)). 1,2 are the (3, 0)–forms with respect to the two complex structures I1,2 ; since the latter are integrable, 1,2 had to be indeed conformally closed. Even more precisely, we see from (3.37) that d(e−2φ 1,2 ) = 0. This agrees with the computation in [36] in the case of N = 1 backgrounds, applied to η1,2 separately. Let us stress again that in expressions such as (5.14), p and Q are not independent variables, and are given by (5.2). So more explicitly dw ∧ dq ∧ dp = dw ∧ dq ∧ (K q q¯ d q¯ + K q P d P + K q P¯ d P¯ + K q w¯ d w). ¯ (5.15) Another combination of ω1,2 that can be simplified considerably is their difference: 1 (ω1 − ω2 ) + z z¯ = 8∂ ∂¯ K . s4ψ 2 i
(5.16)
Putting now together (3.1), (5.14) and (5.16) we get + = exp[8 ∂ ∂¯ K ], − = ρ tan(2ψ)dw ∧ exp 8 dq ∧ d(∂q K ) + 8 d(∂ P K ) ∧ d P ,
(5.17) (5.18)
where we have left the prefactor in − unspecified. This prefactor will be our focus in Sect. 5.4. For now, however, notice that the pair we have just obtained is applicable locally to any generalized Kähler manifold, since we have not yet derived nor imposed the extra conditions for it to be an N = 2 supergravity vacuum. In particular, this gives a possible definition of K which is formally identical to the more usual Kähler definition. A curious parallel to this situation exists for topological theories. If one defines the A model on a Kähler manifold, it is well–known [37] that the action can be written as S = {Q, V } + W , with W = J a “topological” action. The A model on a generalized Kähler manifold has, as topological action, precisely the two–form in the exponent of an even pure spinor [38,39]. The use of a potential to describe pure spinors also advocated in [9] for RR solutions; the reason a potential can still play a role in the RR case is that the real part of + is still (conformally) closed (see the second equation in (2.11)).
Generalized Kähler Potentials from Supergravity
21
5.3. Checking the generalized Kähler condition. Since the pure spinors (5.17), (5.18) have been derived from [5], they should define a generalized Kähler structure by construction. We will now check that they indeed satisfy the conditions (2.9). A pure spinor of type 0 can be written as + = eα for some two–form α. It is easy to see, then, that (2.9) can only be satisfied if dα = 0. Indeed, (2.9) says that we should have deα = (ιv + ξ ∧)eα
(5.19) α + ξ ) ∧ eα .
for some v and ξ . The right hand side of (5.19) can be rewritten as (ιv Then the zero–form part of (5.19) says that 0 = ιv α + ξ , hence dα = 0. In other words, the exponent in + in (5.17) has to be closed. Checking this is immediate because ¯ . ∂ ∂¯ K = d(∂ − ∂)K We now come to − . Again we take a step back and we ask ourselves what (2.9) says about a general pure spinor of type 1, namely 1 ∧ eβ with 1 a one–form and β a two–form. Proceeding in a way similar to the type 0 case, one can reduce (2.9) to the conditions 1 ∧ dσ = 0,
d( f 1 ) = 0 for some f.
(5.20)
For (5.18), the first condition in (5.20) is satisfied because the exponent is closed by itself. The second condition is satisfied by taking f = (ρ tan(2ψ))−1 . 5.4. Imposing the generalized Calabi–Yau condition. We have explained in Sect. 2 how the condition for an N = 2 supergravity vacuum (2.8) is in general stronger than the condition for (2, 2) worldsheet supersymmetry (namely, that the target space should be a generalized Kähler manifold). In the previous subsection we checked the generalized Kähler condition, which is equivalent to worldsheet (2, 2) supersymmetry. We now want to see what remains to be imposed for N = 2 supersymmetry in the target space, which is condition (2.8). We have noted in the previous subsection that actually this is already satisfied for + in (5.17). As for − in (5.18), we noted in the previous subsection that the exponent is already closed; the one–form in front, however, is only conformally closed (that is, up to a function, see (5.20)). This is good enough for the generalized Kähler condition to be true, but not quite for (2.8). So we have to impose that ρ tan(2ψ) in (5.18) be a constant; remembering the definition of ρ in (5.10) and the expressions for the trigonometric functions in (5.7), we get det(K H ) = const. K q P K q¯ P¯ − K q q¯ K P P¯
(5.21)
Recall that K H is a matrix given in (5.4). This equation is a generalization of the usual Monge-Ampère equation and reproduces the expression derived from worldsheet techniques in [8, Eq.(18)].13 The situation is very similar to the usual Kähler case. There, one can easily define a Kähler metric on C3 from a real function K , via J = i∂ ∂¯ K and trivial complex structure. One can define two pure spinors 13 A generalization of the Monge–Ampère equation was also derived in [40], which considers the case with chiral and twisted chiral multiplets.
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N. Halmagyi, A. Tomasiello
+ = ei J , − = det(∂ ∂¯ K ) dz 1 ∧ dz 2 ∧ dz 3 ,
(5.22)
which satisfy the algebraic constraints for two compatible pure spinors (in particular they have equal norm). The conditions (2.9) are indeed satisfied, as should be the case since a Kähler metric is also generalized Kähler; to obtain an N = 2 supergravity solution one also needs to impose that both pure spinors in (5.22) are closed. This imposes that det(∂ ∂¯ K ) = const, which is the Kähler analogue of (5.21). ¯ w) To summarize: for any function K (q, P, w, q, ¯ P, ¯ that satisfies (5.21), the pure spinors + = exp[−8∂ ∂¯ K ], − = dw ∧ exp −8dq ∧ d(∂q K ) − 8d(∂ P K ) ∧ d P
(5.23)
are compatible and satisfy (2.8). Adding some technical requirements, namely that the metric should be actually positive definite and nonsingular, and that the volume form defined by the pure spinors should have no zeros (in our case, this requires that the function det(∂ ∂¯ K ) have no zeros), (5.23) define an N = 2 solution in type II supergravity. At first sight, it looks like (5.23) gives now a very easy way of producing solutions: ¯ w¯ and any K one could think for example that any set of coordinates q, P, w, q, ¯ P, quadratic in those coordinates will do the job, since a quadratic K will surely satisfy (5.21).14 In fact, however, such solutions will correspond to flat metrics. To give a nontrivial example to this construction, we will now resort to a known solution. 6. A Special Case: Lunin–Maldacena NS Solution In this section we show that a certain solution given in [12, p. 23] is described by the pure spinors in (5.23) for an appropriate choice of coordinates and K . The solution is purely NS (we call it an LM–NS solution from now on) and is very similar in form to another solution in [12]; this one has RR fields too, and we will call it simply an LM solution. The LM solution is the gravity dual to the so–called β–deformation of N = 4 SYM, one of the Leigh–Strassler theories [42]. We hence start with a quick review of those theories. We then will show how one can derive the LM–NS solution in the pure spinor formalism. We finally show how it is a particular case of (5.23). 6.1. Leigh–Strassler theories. N = 4 super Yang–Mills is a conformal theory and as such it is interesting to study the space of its exactly marginal deformations. The arguments of Leigh and Strassler [42] rely on preserving a discrete Z3 symmetry which permutes the three N = 1 chiral superfields X i and as a result there are exactly two candidate deformations: 3 3 W = Tr [X 1 , X 2 ]X 3 + γ {X 1 , X 2 }X 3 + γ Xi . (6.1) i=1 14 Another Ansatz one could try to solve (5.21) is to take K = f (w, w) ¯ In this case, the ¯ + K 0 (q, P, q, ¯ P). determinant in the numerator of (5.21) factorizes, and one recovers [41, Eq.(5.1)], that describes a hyper–Kähler manifold. We thank M. Zabzine for this comment.
Generalized Kähler Potentials from Supergravity
23
The LM solution is the gravity dual to the family of theories with γ = 0 and is sometimes also referred to as the beta deformation. One of the original motivations for this paper was to find the gravity dual for the most general theory with γ = 0. Some approximate results have already been found, for example [44,45,43] in perturbation theory; these papers however also serve as an illustration of how complicated such a task can get at higher orders. The strategy we want to promote in this paper is that one might make progress by considering an “auxiliary” purely NS solution. The idea is that the gravity dual will then arise by placing D3 branes on this NS solution, much like the relation between flat space and AdS5 × S 5 . Thanks to the work of Lunin and Maldacena [12] we know this is the case for γ real and γ = 0. (The solution for γ imaginary is then obtained by S–duality and as such is obviously an RR background.) As described below, the reformulation of this problem in terms of generalized complex geometry suggests that there should be in addition an NS solution for γ = 0. We believe that some features of the full RR gravity dual can already be captured by the NS solution. For example, we saw in Sect. 3.4 that the moduli space of D3 branes is now partially lifted: a single, spacetime–filling D3 brane can only be placed at a critical point of the function we have called w. If we go back to (6.1), we see a similar feature. Unlike the usual N = 4 theory, this superpotential is non–zero even when we set the rank of the X i to 1. W becomes then a function w on C3 , and the supersymmetric vacua become its critical points. By using this w in the construction (5.23), one would find (if one were able to solve (5.21)) a geometry such that a D3 brane probe would have exactly w as an effective potential.15 Hence, at least an NS precursor of the full gravity dual should be in the class (5.23). In Sect. 6.4 we will see explicitly that this is indeed the case for the only known gravity dual. Probe D3 branes should experience a superpotential w in the full RR solution too [47]. Recall that a warped product Ad S5 × M5 can also be considered as a warped product R1,3 × M6 , with M6 the cone over M5 . It was shown in [17] that for any AdS5 solution of IIB supergravity, M6 must be either Calabi–Yau or have an SU(2) structure. Since by definition the gravity duals of Leigh–Strassler deformations have an AdS5 factor and are not Calabi–Yau, we see that they must be SU(2) structure solutions. This means that there are two complex structures I1,2 . As we stressed in Sect. 4, this is the case when the pure spinor − is of type 1, namely it has a non–zero one–form dw; in that case, w is then the superpotential for D3 brane probes that we claimed to be a common feature to the NS and RR solutions. Another possible check of this idea comes to mind [31] after noticing that the F–term equations for (6.22) look like [X 1 , X 2 ] = γ {X 1 , X 2 } + 3γ X 32 , [X 2 , X 3 ] = γ {X 2 , X 3 } + 3γ X 12 ,
(6.2)
[X 3 , X 1 ] = γ {X 3 , X 1 } + 3γ X 22 . If one interprets this as a noncommutativity on C3 , one might try to relate it to the computation in Sect. 4.4, although see the various caveats there.
15 Another way of finding a metric with this feature has been pointed out in [46]: it should be obtainable by dualities from their Eq. (20). We thank J. Maldacena for pointing this out.
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6.2. Moduli and Poisson bivectors. We just proposed that the class of solutions considered in Sect. 5 should be related to gravity duals of Leigh–Strassler theories. In this subsection we supplement these arguments, based on observations about the number of moduli in the field theory and in geometry. As a by–product, we will also see a possible generalization to more general conformal field theories. One of the achievements of generalized complex geometry [2] was the geometrical interpretation of Witten’s extended moduli space [37]. If one starts from a complex manifold M6 , its infinitesimal complex deformations are given by H 1 (M6 , T1,0 ). As we saw in Sect. 4, in the discussion after (4.3), to a complex structure I one can associate a generalized complex structure J I ; the associated pure spinor (which exists if c1 = 0) is the holomorphic three–form for I . It follows that, quite reasonably, a complex manifold can be considered as a generalized complex manifold. The infinitesimal deformations of J I are given by ⊕ p+q=2 H p (M6 , q T1,0 );
(6.3)
the obstructions to these moduli live in ⊕ p+q=3 H p (M6 , q T1,0 ).
(6.4)
Equation (6.3) are deformations of a complex structure as a generalized complex structure; indeed, ordinary complex structure deformations, that live in H 1 (M6 , T1,0 ), are a subset of (6.3). The other two summands, H 0 (M6 , 2 T1,0 ) and H 2 (M6 , O), vanish if M6 is Calabi–Yau; in that case, the only deformations are the more familiar complex structure deformations in H 1 (M6 , T1,0 ). However, an interesting observation was made in [31] by considering non–normalizable modes on M6 . Consider a Calabi–Yau threefold M6 which is a cone over a regular Sasaki–Einstein manifold M5 . This five dimensional manifold M5 is by definition a U(1) bundle over a Kähler–Einstein space B4 . The observation in [31] was that one could consider elements β i j ∈ H 0 (B4 , 2 T1,0 ) and then holomorphically extend these over the entire Calabi–Yau cone M6 to obtain a non-commutative deformation. Physically this is entirely reasonable since according to the AdS/CFT dictionary, non–normalizable modes in the bulk are dual to superpotential deformations (normalizable modes are dual to vev’s). These bivector deformations should be dual to the exactly marginal deformations of the field theory on a stack of D3 branes at the tip of this cone. It has been checked [31] in a few examples that the number of bivector deformations matches the total number of exactly marginal deformations, although one needs to restrict to regular Sasaki-Einstein spaces. The paper [48] gives a general reason for the match, and defines a map from bivectors to field theory deformations. On the conifold, the gravity dual of one particular deformation with SU(2) global symmetry was found in [49], and in [50] the space of exactly marginal deformations for the conifold was written down in the field theory following ideas in [51]. In the case of AdS5 × S 5 , the bivector deformations are the ones discussed by Leigh and Strassler. For more general field theories, we can observe that a bivector β i j deforms the pure spinor of a Calabi–Yau as δ− = β i j ι∂i ∧ ι∂i ,
(6.5)
at first order. This means that the action of β i j generates a one–form in − . Since for a vacuum this has to be closed (see (2.11)), it can be written locally as β i j ι∂i ∧ ι∂ j = dw. As we argued in 3.4 and again in 6.1, this means that D3 brane probes feel an effective superpotential w.
Generalized Kähler Potentials from Supergravity
25
This would seem to confirm further the intuition we promoted in Sect. 6.1, namely that the gravity duals to Leigh–Strassler theories should be obtained by placing D3 branes on an NS solution of the type considered in Sect. 5. In fact, it also seems to indicate that such solutions should be relevant to deformations of CFT’s other than N = 4. There are several obstacles, however, to using the observation in [31] directly. The first concerns obstructions, the second concerns introducing a second pure spinor + in the story. Although the two are connected, at first we will consider them separately. If one was ultimately concerned with the deformation of just B4 , then one would need only to calculate the obstruction group (6.4) to determine whether the deformation can be made for finite values of the parameter. This is indeed done in [2] for B4 = CP 2 . However, we are concerned not just with the generalized complex structure on B4 but with the one on M6 . The latter also has a holomorphically trivial canonical bundle, in that there exists a closed pure spinor for it (in the terminology of [1], it is a generalized Calabi–Yau structure). We may then appeal to a theorem in [1] which tells us the moduli of a generalized Calabi-Yau manifold are integrable. This issue is complicated, however, by the entrance in the scene of a second pure spinor + . Remember from Sect. 2 that this is needed both for NS vacua (2.8) and for RR vacua (2.11). One has to find a way to promote the bivector deformations (6.5) to deformations of a compatible pure pair ± . One way to deform a compatible pair 0± is to act on it with the same element O ∈ O(6, 6); since such an element keeps the internal product (·, ·) in (2.1) invariant, Conditions 1 and 2 in the definition of compatibility in Sect. 2 are kept satisfied (Condition 3 is an open condition, so it is not affected by small enough deformations). Next, since we are interested in deformations of − , we might further restrict deformations so that + is left unchanged. Remarkably, we will see in Sect. 6.3, and in particular in (6.20), that this is exactly what happens for the LM–NS solution. At first order, one can show [2, Sect. 6.5.1] that this leads to 1 1 ¯ ¯ J ¯ d z¯ k ) ∧ (ι∂ j − J j l¯d z¯ l )]± ≡ Re() · 0± ; (6.6) 2 ik 2 the operator Re() is indeed in the Lie algebra o(6, 6), and one can see that Re() ·+ = 0. The operator Re() is a linear combination of a bivector, of a tensor with one index up and one down16 , and of a two–form. At first order, the deformation (6.6) yields a generalized Kähler structure (defined by (2.9)), if one chooses these three components to be respectively in H 0 (M6 , 2 T1,0 ), H 1 (M6 , T1,0 ) and H 2 (M6 , O). If one now tries to extend (6.6) to all orders, the most natural possibility is just to write 1 1 ¯ ¯ ± = exp[Re(β i j (ι∂i + Ji k¯ d z¯ k ) ∧ (ι∂ j − J j l¯d z¯ l ))]0± . (6.7) 2 2 Since the exponential is now in O(6, 6), these pure spinors are now a compatible pair, as we noted above. On the other hand, for simple β i j it can happen that in (6.6) squares to zero, even if its real part Re() (which is in o(6, 6)) does not. In this case one has another option. One can check that the first order deformation (6.6) is also δ± = Re[β i j (ι∂i +
δ± = · 0± ;
(6.8)
in other words, one can drop the Re in (6.6). One can then simply define ± = e 0± = (1 + )0± . For this tentative pair, Condition 1 for compatibility is still satisfied, because 16 This part is present only if β i j has a symmetric part too.
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it only needs to be checked at first order, where it is still true (because it was true for (6.6), in which case we can use that Re() ∈ o(6, 6)). Condition 2 is not guaranteed this time; but one can simply rescale each ± by their new norm. Summarizing, ± =
(1 + ·)0± ||(1 + ·)0± ||
(if 2 = 0).
(6.9)
The finite–order deformation (6.7) applies more generally. It also avoids the step of dividing by the norm as in (6.9), so it is more appropriate for looking for supergravity vacua, namely solutions to (2.8). On the other hand, the second method, (6.9), can be applied to generalized Kähler manifolds, namely solutions to (2.9). In that case, a rescaling ± → e f ± can be reabsorbed by changing ξ± → ξ± − d log( f ) in (2.9). (In fact, for this very reason, for generalized Kähler manifolds part 2 of the compatibility condition can be omitted.) Both these ways of making (6.6) finite were utilized in [52,53] to produce examples of generalized Kähler structures for their general theory of generalized Kähler reduction. In this application, it is crucial that + is invariant under (6.6); morally, one can still use the two–form exponent of + for the symplectic quotient. This reduction procedure should be inherent to the gravity duals of the Leigh–Strassler deformations, because it would give a way to count the deformations more directly on the base B4 , as in [31]. The deformations considered on C3 in [52,53], however, satisfy (2.9) but do not satisfy (2.8), and hence do not give rise to N = 2 vacua. The pure spinors they consider either fail to satisfy Condition 2 for compatibility (as given in Sect. 2), or, if they are divided by their norms as in (6.9), are not closed. In Sect. 6.4, we will present the LM–NS solution as an example of (6.6) which does, on the contrary, satisfy (2.8). 6.3. T–duality and bivector action. From now on we will consider the case of γ = 0 in Eq. (6.1). The method used in [12] to generate their solutions was to act with an element of the symmetry group of supergravity O(6, 6) that does not belong to the stabilizer (isomorphic to O(6) × O(6)) of the initial solution. The action of T–duality on pure spinors was considered in [9] by using ordinary spinors. Here we want to present an alternative way of computing that action, which is more in line with other geometrical ideas present in this paper. The method is similar to the one explained in detail in [13, Sect. 6.1]. One exploits the fact that T–duality acts on T ⊕ T ∗ . One first computes the annihilator of the pure spinor we want to transform, Ann() ⊂ T ⊕ T ∗ ; then one acts on this annihilator with the ˜ such that Ann() ˜ = O(Ann()). T–duality element O ∈ O(6, 6); then one finds The result of this procedure has a subtle part, the mixing of the metric and B–field, and an easy part, the actual action on the pure spinor. The subtle part is that in general the manifold M6 is changed into some new manifold M˜ 6 . This is source of much of the agony in [13, Sect. 6.1], where care is needed because the connection on the S 1 –fibration on the original M6 is exchanged by T–duality with some component of the B–field. Fortunately, this kind of subtlety is not relevant for us. Indeed, let us consider a general T 3 –fibration with coordinates r i on the base and φ α on the fibre: β
g = gi j dr i · dr j + h αβ (dφ α + λiα dr i ) · (dφ β + λ j dr j ), 1 1 β B = bi j dr i ∧ dr j + bαi dr i ∧ (dφ α + λαj dr j ) + Bαβ (dφ α + λiα dr i ) ∧ (dφ β + λ j dr j ); 2 2
(6.10)
Generalized Kähler Potentials from Supergravity
27
the λiα are connections for the T 3 –fibration. Consider the action for example by an element of O(3, 3) (since the fibre is 3–dimensional) of the form 1 0 (6.11) βT 1 αβ
for βT some real bivector on T 3 . Then one can show that (taking the initial Bαβ = 0 for simplicity) the Buscher rules for this element can be summarized by acting on (6.10) by the simple rules
1 αβ α α ; (6.12) λi → λi + βT bβi , bαi → bαi , (h + B)αβ → h 1 + βh αβ this result is very similar to the one in [54] for the simpler inversion along the three direction of the fibre. The result of the previous paragraph can be applied to C3 , viewed as a T 3 fibration on R3+ . We see that, since the bαi are vanishing to begin with, the λiα do not change. Since the λiα are the connections for the T 3 –fibration, the manifold stays topologically the same. This leaves us with the easier part of the T–duality action. Since vectors and one– forms are mixed by an element of O ∈ O(6, 6), the pure spinor is also acted on17 . In general, an endomorphism of T ⊕T ∗ also acts naturally on differential forms, as detailed for example in [14, Sect. 2.1]. For the case of interest here, (6.11), this is nothing but the action of the bivector βT by contraction: ± = eβT 0± .
(6.13)
The bivector proposed in [12] is one that exhibits Z3 symmetry, as (6.1) does: βT = 4γ (ι∂ϕ1 ∧ ι∂ϕ2 + cycl. perm.).
(6.14)
We emphasize once again that βT is real, as opposed to β in Sect. 6.2, which was complex. (At the first order level, in (6.5), one can freely add to β its complex conjugate.) If one acts with (6.14) on the flat space pure spinors (0+ , 0− ) = (e−i J0 , 0 ) as in (6.13), one obtains the LM–NS solution we promised. One can easily check, for example, that βT (dz 1 ∧ dz 2 ∧ dz 3 ) = d(γ z 1 z 2 z 3 );
(6.15)
comparing with (6.1), we see that the right-hand side is exactly dw for the case we restricted ourselves to, γ = 0. In spite of this initial success, however, it turns out that the rest of the pure spinor pair (6.13) is not of the form (3.1). Since we advertised those as the most general pair of types 0 and 1, this would appear to be a problem. Fortunately, the difference is explained by a simple change of gauge in the B–field, as follows. In the language of [9], the B–field obtained after the action (6.13) is B0 = − cos(2ψ) sin(2ψ)y1 ∧ y2 ;
(6.16)
17 O also acts on the corresponding generalized complex structure J as (O t )−1 J O t , see [13, Eq.(6.9)]. Applying this to (6.11), we obtain what is called the β–transform in [2, Ex.2.2].
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this differs from the one in (3.4) by δb = tan(2ψ)x1 ∧ x2 =
1 2 dr ∧ dr22 + cycl. perm. 4 1
(6.17)
which is exact, and hence a gauge transformation. A curious fact is that if one now adds this to the action of β in (6.13), the flat–space + = e−i J0 is invariant in form: + = eβT +δb∧ e−i J0 = e−i J0 .
(6.18)
(This does not mean that there is no B–field, since B has to be read off from the pair. B is actually of the form (3.4).) Indeed, one can see that the LM–NS solution is a particular case of the procedure (6.7). We suspect this feature is general for duals to Leigh–Strassler theories, and some of the perturbative results in [45] seem to point in this direction. We were not able, however, to use this to find the solution. Summarizing, the pure spinors of the LM–NS solution are given by 1 ± = exp[γ (4ι∂ϕ1 ∧ ι∂ϕ2 + dr12 ∧ dr22 ) + cycl. perm.] ∧ ±0 , 4
(6.19)
where (+0 , −0 ) = (e−i J0 , ) is the flat solution and z i = ri eiϕi . Explicitly we have that + = +0 = e−i J0 , dz 1 ∧ dz 2 γ − = dw ∧ exp[ + (dr12 ∧ dr22 + cycl. perm.)]. γ z1 z2 4
(6.20) (6.21)
One prominent feature of this pair of pure spinors is that they are exact at second order18 in γ : the perturbation theory truncates. This is not true of the metric, which receives corrections at all orders. This interesting situation is due to the fact that passing from the pure spinors to the metric is non–linear; we regard this as encouragement that generalized complex geometry is the right framework to find more general solutions of this sort. 6.4. Lunin–Maldacena from a potential. We saw how the LM–NS solution is defined by the action of (6.14) as in (6.13), and that we can gauge–transform it in such a way as to fall in the class (3.1). Now we want to put this solution in the form (5.23). We have already noticed how the coordinate w has to be taken w = γ z1 z2 z3.
(6.22)
The semi-chiral co-ordinates are q = log(z 1 ) − γ2 (|z 2 |2 − |z 3 |2 ),
p = log(z 2 ) − γ2 (|z 3 |2 − |z 1 |2 ),
Q = log(z 1 ) + γ2 (|z 2 |2 − |z 3 |2 ),
P = log(z 2 ) + γ2 (|z 3 |2 − |z 1 |2 ),
(6.23)
which agrees with Eqs. (5.14), (5.23) and (6.21). 18 The deformations in [52,53] are exact at first order; however, as mentioned in Sect. 6.2, the pure spinors there are not closed.
Generalized Kähler Potentials from Supergravity
29
Moreover, if one chooses, as usual, q, P, w and their complex coordinates as independent variables, the generating function K is K = q P + q¯ P¯ + γ (|z 1 |2 + |z 2 |2 + |z 3 |2 ) + γ 2 (|z 2 |2 − |z 3 |2 )(|z 3 |2 − |z 1 |2 ), (6.24) ¯ w) ¯ P, ¯ given by inverting (6.22) and where one has to understand z i = z i (q, P, w, q, (6.23). (It is not necessary to invert them explicitly. To check that K in (6.24) satisfies ∂(q,P,w) (5.2), one can compute the Jacobian ∂(z 1 ,z 2 ,z 3 ) ) and invert it.) One can check that Eq. (5.21) is satisfied for this choice, and that K is not just quadratic in q, P, w. Acknowledgements We would like to thank S. Benvenuti, M. Headrick, O. Lunin, J. Maldacena, D. Martelli, I. Melnikov, M. Mulligan, M. Petrini, M. Roˇcek, E. Witten and M. Zabzine for discussions. A. T. is supported by the DOE under contract DEAC03-76SF00515 and by the NSF under contract 0244728. N. H. is supported in part by NSF Grants PHY-0094328 and PHY-0401814 and a Fermi-McCormick Fellowship. Both authors would like to thank the Aspen Center for Physics for hospitality during the completion of this project.
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22. Koerber, P.: Stable D–branes, calibrations and generalized Calabi–Yau geometry. JHEP 08, 099 (2005) 23. Martucci, L., Smyth, P.: Supersymmetric D–branes and calibrations on general n = 1 backgrounds. JHEP 11, 048 (2005) 24. Becker, K., Becker, M., Strominger, A.: Five-branes, membranes and nonperturbative string theory. Nucl. Phys. B456, 130–152 (1995) 25. Mariño, M., Minasian, R., Moore, G.W., Strominger, A.: Nonlinear instantons from supersymmetric p-branes. JHEP 01, 005 (2000) 26. Mariotti, A.: Supersymmetric D–branes on SU(2) structure manifolds. JHEP 0709, 123 (2007) 27. Hitchin, N.J.: Bihermitian metrics on Del Pezzo surfaces. http://arxiv.org/abs/math/0608213v1[math. DG], 2006 28. Lindström, U., Minasian, R., Tomasiello, A., Zabzine, M.: Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235–256 (2005) 29. Lyakhovich, S., Zabzine, M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B548, 243–251 (2002) 30. Hitchin, N.: Instantons, Poisson structures and generalized kaehler geometry. Commun. Math. Phys. 265, 131–164 (2006) 31. Wijnholt, M.: Parameter space of quiver gauge theories. http://arxiv.org/abs/hep-th/0512122v2, 2005 32. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 09, 032 (1999) 33. Schomerus, V.: D–branes and deformation quantization. JHEP 06, 030 (1999) 34. Kapustin, A., Witten, E.: Electric–magnetic duality and the geometric langlands program. http://arxiv. org/abs/hep-th/0604151v3, 2006 35. Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49– 81 (2004) 36. Gauntlett, J.P., Martelli, D., Waldram, D.: Superstrings with intrinsic torsion. Phys. Rev. D 69, 086002 (2004) 37. Witten, E.: Mirror manifolds and topological field theory. http://arxiv.org/abs/hep-th/9112056v1, 1991 38. Zucchini, R.: The bihermitian topological sigma model. JHEP 12, 039 (2006) 39. Chuang, W.-y.: Topological twisted sigma model with H –flux revisited. J. Phys. A 41, 115402 (2008) 40. Roˇcek, M.: Modified Calabi–Yau manifolds with torsion. In: Essays on Mirror Manifolds, S.-T. Yau (ed.), Hong Kong: International Press, 1992 41. Bogaerts, J., Sevrin, A., van der Loo, S., Van Gils, S.: Properties of semi-chiral superfields. Nucl. Phys. B562, 277–290 (1999) 42. Leigh, R.G., Strassler, M.J.: Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory. Nucl. Phys. B447, 95–136 (1995) 43. Kulaxizi, M.: Marginal deformations of N = 4 SYM and open vs. closed string parameters. http://arxiv. org/abs/hep-th/0612160v2, 2006 44. Graña, M., Polchinski, J.: Supersymmetric three–form flux perturbations on AdS5 . Phys. Rev. D 63, 026001 (2001) 45. Aharony, O., Kol, B., Yankielowicz, S.: On exactly marginal deformations of N = 1 SYM and type IIB supergravity on AdS5 × S 5. JHEP 06, 039 (2002) 46. Maldacena, J.M., Sheikh-Jabbari, M.M., Van Raamsdonk, M.: Transverse fivebranes in matrix theory. JHEP 01, 038 (2003) 47. Corrado, R., Halmagyi, N.: N = 1 field theories and fluxes in iib string theory. Phys. Rev. D71, 046001 (2005) 48. Bergman, A.: Deformations and D–branes. Adv. Theor. Math. Phys. 12, 781–815 (2008) 49. Halmagyi, N., Pilch, K., Romelsberger, C., Warner, N.P.: Holographic duals of a family of n = 1 fixed points. JHEP 0608, 083 (2006) 50. Benvenuti, S., Hanany, A.: Conformal manifolds for the conifold and other toric field theories. JHEP 08, 024 (2005) 51. Kol, B.: On conformal deformations. JHEP 09, 046 (2002) 52. Lin, H., Tolman, S.: Symmetries in generalized Kähler geometry. http://arxiv.org/abs/math/ 0509069v1[math.DG], 2005 53. Bursztyn, H., Cavalcanti, G., Gualtieri, M.: Reduction of Courant algebroids and generalized complex structures. Adv. Math. 211, 726–765 (2007) 54. Fidanza, S., Minasian, R., Tomasiello, A.: Mirror symmetric SU(3)–structure manifolds with NS fluxes. Commun. Math. Phys. 254, 401–423 (2005) Communicated by G. W. Gibbons
Commun. Math. Phys. 291, 31–61 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0867-4
Communications in
Mathematical Physics
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics Igor Rodnianski1, , Benjamin Schlein2, 1 Department of Mathematics, Princeton University, Princeton, NJ, USA 2 Institute of Mathematics, University of Munich, Theresienstr. 39,
D-80333 Munich, Germany. E-mail:
[email protected] Received: 20 November 2007 / Accepted: 22 August 2008 Published online: 9 July 2009 – © Springer-Verlag 2009
Abstract: The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N -boson states, in the limit of large N . In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N -body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.
1. Introduction We consider an N boson system described on the Hilbert space L 2s (R3N ) (the subspace of L 2 (R3N ) consisting of all functions symmetric with respect to arbitrary permutations of the N particles) by a mean field Hamiltonian of the form HN =
N
−x j +
j=1
N 1 V (xi − x j ). N
(1.1)
i< j
We will specify later assumptions on the interaction potential V . Note the coupling constant 1/N in front of the potential energy which characterizes mean-field models; it makes sure that in the limit of large N the potential and the kinetic energy are typically of the same order, and thus can compete to generate the nontrivial effective equation for the macroscopic dynamics of the system. Partially supported by NSF grant DMS-0702270.
On leave from the University of Cambridge. Supported by a Sofja Kovalevskaja Award of the Hum-
boldt Foundation. Current address: University of Cambridge, Centre for Mathematical Sciences, DPMMS, Wilberforce Rd, Cambridge CB3 0WB, UK.
32
I. Rodnianski, B. Schlein
We consider a factorized initial wave function L 2s (R3N ) ψ N (x) =
N
ϕ(x j ) for some ϕ ∈ H 1 (R3 )
(1.2)
j=1
with normalization ϕ L 2 (R3 ) = 1 (so that ψ N L 2 (R3N ) = 1) and we study its time-evolution ψ N ,t , given by the solution of the N body Schrödinger equation i∂t ψ N ,t = H N ψ N ,t
with initial data ψ N ,0 = ψ N .
(1.3) R3N .
In (1.2) and in what follows we use the notation x = (x1 , . . . , x N ) ∈ Clearly, because of the interaction among the particles, the factorization of the wave function is not preserved by the time evolution. However, due to the presence of the small constant 1/N in front of the potential energy in (1.1), we may expect the total potential experienced by each particle to be approximated, for large N , by an effective mean field potential, and thus that, in the limit N → ∞, the solution ψ N ,t of (1.3) is still approximately (and in an appropriate sense) factorized. We may expect, in other words, that in an appropriate sense ψ N ,t (x)
N
ϕt (x j )
for large N .
(1.4)
j=1
If (1.4) is indeed correct, it is easy to derive a self-consistent equation for the evolution of the one-particle wave function ϕt . In fact, it follows from (1.4) that the total potential experienced by a particle at x can be approximated by the convolution (V ∗|ϕt |2 )(x), and thus that the evolution of the one-particle wave function ϕt is described by the nonlinear Hartree equation i∂t ϕt = −ϕt + (V ∗ |ϕt |2 )ϕt .
(1.5)
To understand in which sense (1.4) holds true, we need to introduce marginal densities. The density matrix γ N ,t = |ψ N ,t ψ N ,t | associated with ψ N ,t is defined as the orthogonal projection onto ψ N ,t (we use here Dirac’s bracket notation; for f, g, h ∈ L 2 (Rd ), | f g| : L 2 (Rd ) → L 2 (Rd ) is the operator defined by | f g|(h) = g, h L 2 f ). The kernel of γ N ,t is thus given by γ N ,t (x; x ) = ψ N ,t (x)ψ N ,t (x ). (k)
For k = 1, . . . , N , we define then the k-particle marginal density γ N ,t associated with ψ N ,t by taking the partial trace of γ N ,t over the last N − k particles. In other words, we (k) define γ N ,t as a positive trace class operator on L 2 (R3k ) with kernel (k) (1.6) γ N ,t (xk ; xk ) = dx N −k γ N ,t (xk , x N −k ; xk , x N −k ). (k)
Since ψ N ,t L 2 (R3N ) = 1, we immediately obtain Tr γ N ,t = 1 for all N ≥ 1, k = 1, . . . , N , and t ∈ R. (k) By the choice of the initial wave function (1.2), at time t = 0 we have γ N ,0 = |ϕ ϕ|⊗k . It turns out that (1.4) should be understood in terms of convergence of marginal densities. For a large class of interaction potentials V , for every fixed k ≥ 1, and t ∈ R, one can in fact show that
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics (k)
γ N ,t → |ϕt ϕt |⊗k
as N → ∞,
33
(1.7)
where ϕt is a solution of the nonlinear Hartree equation (1.5). The convergence (1.7) holds in the trace norm topology. In particular, (1.7) implies that for arbitrary k and for an arbitrary bounded operator J (k) on L 2 (R3k ), ψ N ,t , J (k) ⊗ 1(N −k) ψ N ,t → ϕt⊗k , J (k) ϕt⊗k as N → ∞. The approximate identity (1.4) can thus be interpreted as follows: as long as we are interested in the expectation of observables depending non-trivially only on a fixed number of particles, the N -body wave function ψ N ,t can be approximated by the N -fold tensor product of the solution φt to the nonlinear Hartree equation (1.5). The first rigorous proof of (1.7) was obtained by Spohn in [11], under the assumption of a bounded interaction potential V . The problem of proving (1.7) becomes substantially more involved for singular potentials. In [7], Erd˝os and Yau extended Spohn’s approach to obtain a rigorous derivation of the Hartree equation (1.5) for a Coulomb interaction V (x) = const/|x| (partial results for the Coulomb interaction were also obtained by Bardos, Golse, and Mauser in [2]). In [4], the Hartree equation with Coulomb interaction was derived for semirelativistic bosons; in the semirelativistic setting, the dispersion of the bosons only grows linearly in the momentum (for large momenta), and thus the control of the Coulomb singularity is more delicate. In [3,5,6], models described by the Hamiltonian HN =
N
−x j +
j=1
N 1 3β N V (N β (xi − x j )) N
with β ∈ (0, 1]
i< j
with an N -dependent potential were considered (in the one-dimensional case, N -dependent potentials were considered by Adami, Golse and Teta in [1]). These models are used to describe systems of physical interest, such as Bose-Einstein condensates. Assuming the interaction to be positive (V (x) ≥ 0 for all x ∈ R3 ) and sufficiently small, the main result was again a proof of the convergence (1.7); this time, however, ϕt is a solution of the cubic nonlinear Schrödinger equation (with local nonlinearity) b0 if 0 < β < 1 2 i∂t ϕt = −ϕt + σ |ϕt | ϕt with σ = . (1.8) 8πa0 if β = 1 Here b0 = dx V (x) and a0 is the scattering length of V . The emergence of the scattering length a0 for β = 1 (for all other choices of 0 < β < 1 the coupling constant is given by b0 , which is the first Born approximation to 8πa0 ) is a consequence of the short scale correlation structure developed in solutions of the Schrödinger equation, which, in the case β = 1, is characterized by the same length scale O(1/N ) as the scale of the interaction potential. The results described above have been obtained by extensions of the approach introduced by Spohn in [11], which was based on the study of the BBGKY hierarchy (k)
i∂t γ N ,t =
k
(k)
[−x j , γ N ,t ] +
j=1
+
k 1 (k) [V (xi − x j ), γ N ,t ] N i< j
N −k N
k j=1
Trk+1 [V (x j − xk+1 ), γ N(k+1) ,t ]
(1.9)
34
I. Rodnianski, B. Schlein (k)
for the evolution of the marginal densities γ N ,t , k = 1, . . . , N (here Trk+1 denotes the partial trace over the (k + 1)th particle; this hierarchy is equivalent to the Schrödinger (k)
equation (1.3) for ψ N ,t ). Because of the compactness of the sequence γ N ,t , N ≥ k, the proof of (1.7) reduces to two main steps. The first step consists in proving that an (k) arbitrary family of limit points {γ∞,t }k≥1 satisfies the infinite hierarchy (k)
i∂t γ∞,t =
k k (k) (k+1) [−x j , γ∞,t ] + Trk+1 [V (x j − xk+1 ), γ∞,t ]. j=1
(1.10)
j=1
The second step is a proof of the uniqueness of the solution of (1.10). Since the factor(k) ized family γ∞,t = |ϕt ϕt |⊗k , with ϕt determined by (1.5), is a solution of the infinite hierarchy (1.10), these two steps are sufficient to obtain (1.7). Despite its many successes, this method has some limitations. The main one, from our point of view, is that, because of the use of abstract arguments related to the com(k) pactness of the sequence γ N ,t , this technique does not provide any information on the (k)
rate of convergence of γ N ,t to |ϕt ϕt |⊗k . In some cases, instead of comparing the solution of (1.9) with the solution of the infinite hierarchy (1.10), it is also possible to expand it in a Duhamel series and to compare it directly with the corresponding expansion for the factorized densities |ϕt ϕt |⊗k . This approach (see [11]) leads to bounds of the form Ck (1.11) Tr γ N(k),t − |ϕt ϕt |⊗k ≤ N for all sufficiently small times |t| ≤ t0 . The restriction to small times is needed to guarantee the convergence of the Duhamel expansion of the solution to (1.9). Iterating the arguments used to obtain (1.11), one can derive bounds of the form Ck (k) Tr γ N ,t − |ϕt ϕt |⊗k ≤ 1 N 2t which hold for all t ∈ R, but deteriorate very fast in time and are therefore not effective and not very useful. The next theorem, which is the main result of this paper, provides much stronger bounds on the difference between the true quantum mechanical evolution of the marginal densities and their Hartree evolution; in particular it shows that for every fixed time t ∈ R, the error is at most of the order O(N −1/2 ). Theorem 1.1. Suppose that there exists D > 0 such that the operator inequality V 2 (x) ≤ D (1 − x )
(1.12)
holds true. Let ψ N (x) =
N
ϕ(x j ),
(1.13)
j=1
for some ϕ ∈ H 1 (R3 ) with1 ϕ = 1. Denote by ψ N ,t = e−i HN t ψ N the solution to the (1) Schrödinger equation (1.3) with initial data ψ N ,0 = ψ N , and let γ N ,t be the one-particle 1 In what follows, for a function f we will always denote by f its L 2 norm, while, for an operator A, A will mean its L 2 operator norm.
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
35
density associated with ψ N ,t . Then there exist constants C, K , depending only on the H 1 norm of ϕ and on the constant D on the r.h.s. of (1.12) such that C (1) Tr γ N ,t − |ϕt ϕt | ≤ 1/2 e K t . (1.14) N Here ϕt is the solution to the nonlinear Hartree equation i∂t ϕt = −ϕt + (V ∗ |ϕt |2 )ϕt
(1.15)
with initial data ϕt=0 = ϕ. Remark 1.2. The assumption on the potential V means that the most singular potential we can handle is the Coulomb potential V (x) = κ/|x|. Note that our theorem applies both to the attractive (κ < 0) and the repulsive case (κ > 0). In particular Theorem 1.1 implies the result obtained by Erdös and Yau in [7]. Remark 1.3. Note that under the assumption (1.12) on the interaction potential V, the nonlinear equation (1.15) is known to be globally well-posed in H 1 (R3 ). This follows from the conservation of the mass ϕ and of the energy 1 E(ϕ) = dx |∇ϕ(x)|2 + dxdy V (x − y)|ϕ(x)|2 |ϕ(y)|2 2 and from the observation that there exist constants c1 , c2 such that E(ϕ) ≤ c1 ϕ2H 1 (1 + ϕ2 ) and ϕ2H 1 ≤ c2 E(ϕ) + ϕ4 + ϕ2 .
(1.16)
Both bounds can be proven using that, by (1.12), dy V (x − y)|ϕ(y)|2 ≤ ε∇ϕ2 + ε−1 ϕ2 for all ε > 0, uniformly in x ∈ R3 . Remark 1.4. Instead of (1.14) we will prove that γ N(1),t − |ϕt ϕt |HS ≤
C eK t , N 1/2
(1.17)
where .HS denotes the Hilbert-Schmidt norm. Although in general the trace norm is bigger than the Hilbert-Schmidt norm, in this case they differ at most by a factor of two.2 (1) In fact, since |ϕt ϕt | is a rank one projection, the operator A = γ N ,t − |ϕt ϕt | can only have one negative eigenvalue λneg < 0. Since moreover (1) Tr γ N ,t − |ϕt ϕt | = 0 it follows that the negative eigenvalue of A is equal, in absolute value, to the sum of all positive eigenvalues. The trace norm of A is equal, therefore, to 2|λneg | = 2A, where A denotes the operator norm of A. Since A ≤ AHS , we immediately obtain that Tr |A| ≤ 2AHS . 2 We would like to thank Robert Seiringer for pointing out this argument to us.
36
I. Rodnianski, B. Schlein
Remark 1.5. The bound (1.14) is not optimal. As mentioned above, for short times and bounded potentials, the quantity on the l.h.s. of (1.14) is known to be of the order 1/N . Nevertheless Theorem 1.1 is the first estimate on the rate of convergence towards the mean-field limit which holds for all times and remains of the same order N −1/2 for all fixed times. Remark 1.6. Although, in order to simplify the analysis, we only consider the rate of (1) convergence of the one-particle density γ N ,t to |ϕt ϕt |, our method can also be used to prove bounds of the form C( j) ( j) Tr γ N ,t − |ϕt ϕt |⊗ j ≤ 1/2 e K ( j) t N for all j, t, N and for j-dependent constants C( j), K ( j). In this paper we avoid the use of the BBGKY hierarchy and instead revive an approach, introduced by Hepp in [9] and extended by Ginibre and Velo in [8], to the study of a semiclassical limit of quantum many-boson systems3 . This approach is based on embedding the N -body Schrödinger system into the second quantized Fock-space representation and on the use of coherent states as initial data. The use of the Fock-space representation is in particular dictated by the fact that coherent states do not have a fixed number of particles. The Hartree dynamics emerges as the main component of the evolution of coherent states in the mean field limit (or, in the language of [8,9], in the semiclassical limit). The problem then reduces to the study of quantum fluctuations, described by an N -dependent two-parameter unitary group U N (t; s), around the Hartree dynamics. In [8,9], Hepp (for smooth interaction potentials) and Ginibre and Velo (for singular potentials) proved that, in the limit N → ∞, the fluctuation dynamics U N (t; s) approaches a limiting evolution U(t; s). This important result shows the relevance of the Hartree dynamics in the mean field limit (at least in the case of coherent initial states). It does not prove, however, the convergence (1.7) of the one-particle marginal density to the orthogonal projection onto the solution of the Hartree equation, nor does it imply convergence results for the evolution of factorized initial sates. The problem of convergence of marginals requires additional control on the growth of the number4 of fluctuations generated by the evolution U N (t; s). This analysis, which, technically, is the most difficult part of the present paper (see Proposition 3.3), is new.5 Another novel part of our work is the derivation of convergence towards Hartree dynamics for factorized initial states from the corresponding statements for the evolution of coherent states. Although we are mainly concerned with the dynamics of factorized initial data, the result we obtain for coherent states (see Theorem 3.1) is of independent interest, especially because, in this case, our bound is optimal in its N -dependence (for coherent states, we show that the error is at most of the order 1/N for every fixed time). The paper is organized as follows. In Sect. 2, we define the Fock space representation of the mean field system, introduce coherent states and review their main properties. In Sect. 3, we consider the evolution of a coherent state and we prove that, in this case, 3 Mathematically, the semiclassical limit considered in [8,9] is equivalent to the mean field limit considered in the present manuscript. 4 Fluctuations around the Hartree dynamics will be considered as particle excitations and thus it will be possible to compute their number. 5 A more precise discussion of the results of [8,9], and of their relation with our work can be found at the end of Sect. 3.
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
37
the rate of convergence to the mean field solution remains of the order 1/N for all fixed times. Finally, in Sect. 4, we show how to use coherent states to obtain information on the dynamics of factorized states, and we prove Theorem 1.1. 2. Fock Space Representation We define the bosonic Fock space over L 2 (R3 , dx) as the Hilbert space
F= L 2 (R3 , dx)⊗s n = C ⊕ L 2s (R3n , dx1 . . . dxn ), n≥0
n≥1
with the convention L 2 (R3 )⊗s 0 = C. Vectors in F are sequences ψ = {ψ (n) }n≥0 of n-particle wave functions ψ (n) ∈ L 2s (R3n ). The scalar product on F is defined by (n) (n) ψ1 , ψ2 L 2 (R3n ) ψ1 , ψ2 = n≥0
=
(0) (0) ψ1 ψ2
+
(n)
(n)
dx1 . . . dxn ψ1 (x1 , . . . , xn )ψ2 (x1 , . . . , xn ).
n≥1
An N particle state with wave function ψ N is described on F by the sequence {ψ (n) }n≥0 where ψ (n) = 0 for all n = N and ψ (N ) = ψ N . The vector {1, 0, 0, . . . } ∈ F is called the vacuum, and will be denoted by . On F, we define the number of particles operator N , by (N ψ)(n) = nψ (n) . Eigenvectors of N are vectors of the form {0, . . . , 0, ψ (m) , 0, . . . } with a fixed number of particles. For f ∈ L 2 (R3 ) we also define the creation operator a ∗ ( f ) and the annihilation operator a( f ) on F by n ∗ (n) 1 a ( f )ψ (x1 , . . . , xn ) = √ f (x j )ψ (n−1) (x1 , . . . , x j−1 , x j+1 , . . . , xn ), n j=1 √ (2.1) (a( f )ψ)(n) (x1 , . . . , xn ) = n + 1 dx f (x) ψ (n+1) (x, x1 , . . . , xn ).
The operators a ∗ ( f ) and a( f ) are unbounded, densely defined, closed operators. The creation operator a ∗ ( f ) is the adjoint of the annihilation operator a( f ) (note that by definition a( f ) is anti-linear in f ), and they satisfy the canonical commutation relations [a( f ), a ∗ (g)] = f, g L 2 (R3 ) ,
[a( f ), a(g)] = [a ∗ ( f ), a ∗ (g)] = 0.
(2.2)
For every f ∈ L 2 (R3 ), we introduce the self adjoint operator φ( f ) = a ∗ ( f ) + a( f ). We will also make use of operator valued distributions ax∗ and ax (x ∈ R3 ), defined so that ∗ a ( f ) = dx f (x) ax∗ , (2.3) a( f ) = dx f (x) ax
38
I. Rodnianski, B. Schlein
for every f ∈ L 2 (R3 ). The canonical commutation relations assume the form [ax , a ∗y ] = δ(x − y)
[ax , a y ] = [ax∗ , a ∗y ] = 0.
The number of particle operator, expressed through the distributions ax , ax∗ , is given by N =
dx ax∗ ax .
The following lemma provides some useful bounds to control creation and annihilation operators in terms of the number of particle operator N . Lemma 2.1. Let f ∈ L 2 (R3 ). Then a( f )ψ ≤ f N 1/2 ψ, a ∗ ( f )ψ ≤ f (N + 1)1/2 ψ, φ( f )ψ ≤ 2 f (N + 1)
1/2
(2.4)
ψ.
Proof. The last inequality clearly follows from the first two. To prove the first bound we note that 1/2 1/2
2 2 a( f )ψ ≤ dx | f (x)| ax ψ ≤ dx ax ψ dx | f (x)| = f N 1/2 ψ.
(2.5)
The second estimate follows by the canonical commutation relations (2.2) because a ∗ ( f )ψ2 = ψ, a( f )a ∗ ( f )ψ = ψ, a ∗ ( f )a( f )ψ + f 2 ψ2 = a( f )ψ2 + f 2 ψ2 ≤ f 2 N 1/2 ψ2 + ψ2 = f 2 (N + 1)1/2 ψ2 .
(2.6) (1)
Given ψ ∈ F, we define the one-particle density γψ associated with ψ as the positive trace class operator on L 2 (R3 ) with kernel given by (1)
γψ (x; y) =
1 ψ, a ∗y ax ψ. ψ, N ψ
(2.7)
By definition, γψ(1) is a positive trace class operator on L 2 (R3 ) with Tr γψ(1) = 1. For every N -particle state with wave function ψ N ∈ L 2s (R3N ) (described on F by the sequence {0, 0, . . . , ψ N , 0, 0, . . . }) it is simple to see that this definition is equivalent to the definition (1.6). (n) We define the Hamiltonian H N on F by (H N ψ)(n) = H N ψ (n) , with H(n) N =−
n j=1
j +
n 1 V (xi − x j ). N i< j
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
Using the distributions ax , ax∗ , H N can be rewritten as 1 ∗ H N = dx∇x ax ∇x ax + dxdy V (x − y)ax∗ a ∗y a y ax . 2N
39
(2.8)
By definition the Hamiltonian H N leaves sectors of F with a fixed number of particles invariant. Moreover, it is clear that on the N -particle sector, H N agrees with the Hamiltonian H N (the subscript N in H N is a reminder of the scaling factor 1/N in front of the potential energy). We will study the dynamics generated by the operator H N . In particular we will consider the time evolution of coherent states, which we introduce next. For f ∈ L 2 (R3 ), we define the Weyl-operator
(2.9) W ( f ) = exp a ∗ ( f ) − a( f ) = exp dx ( f (x)ax∗ − f (x)ax ) . Then the coherent state ψ( f ) ∈ F with one-particle wave function f is defined by ψ( f ) = W ( f ). Notice that ψ( f ) = W ( f ) = e− f
2 /2
(a ∗ ( f ))n n!
n≥0
= e− f
2 /2
1 √ f ⊗n , (2.10) n! n≥0
where f ⊗n indicates the Fock-vector {0, . . . , 0, f ⊗n , 0, . . . }. This follows from exp(a ∗ ( f ) − a( f )) = e− f
2 /2
exp(a ∗ ( f )) exp(−a( f ))
which is a consequence of the fact that the commutator [a( f ), a ∗ ( f )] = f 2 commutes with a( f ) and a ∗ ( f ). From Eq. (2.10) we see that coherent states are superpositions of states with a different number of particles (the probability of having n particles in ψ( f ) 2 is given by e− f f 2n /n!). In the following lemma we collect some important and well known properties of Weyl operators and coherent states. Lemma 2.2. Let f, g ∈ L 2 (R3 ). i) The Weyl operator satisfies the relations W ( f )W (g) = W (g)W ( f )e−2i Im f,g = W ( f + g)e−i Im f,g . ii) W ( f ) is a unitary operator and W ( f )∗ = W ( f )−1 = W (− f ). iii) We have W ∗ ( f )ax W ( f ) = ax + f (x),
and W ∗ ( f )ax∗ W ( f ) = ax∗ + f (x).
iv) From iii) we see that coherent states are eigenvectors of annihilation operators, ax ψ( f ) = f (x)ψ( f )
⇒
a(g)ψ( f ) = g, f L 2 ψ( f ).
40
I. Rodnianski, B. Schlein
v) The expectation of the number of particles in the coherent state ψ( f ) is given by f 2 , that is ψ( f ), N ψ( f ) = f 2 . Also the variance of the number of particles in ψ( f ) is given by f 2 (the distribution of N is Poisson), that is ψ( f ), N 2 ψ( f ) − ψ( f ), N ψ( f )2 = f 2 . vi) Coherent states are normalized but not orthogonal to each other. In fact 1
ψ( f ), ψ(g) = e− 2
f 2 +g2 −2( f,g)
⇒
1
| ψ( f ), ψ(g)| = e− 2 f −g . 2
3. Time Evolution of Coherent States Next we study the dynamics of coherent states with expected number of particles N in the limit N → ∞. We choose the initial data √ √ ψ( N ϕ) = W ( N ϕ)
for ϕ ∈ H 1 (R3 ) with ϕ = 1
(3.1)
√ and we study its time evolution ψ(N , t) = e−i H N t ψ( N ϕ) with the Hamiltonian H N defined in (2.8). Theorem 3.1. Suppose that there exists D > 0 such that the operator inequality V 2 (x) ≤ D(1 − x )
(3.2)
(1)
holds true. Let N ,t be the one-particle marginal associated with ψ(N , t) = e−i H N t √ W ( N ϕ) (as defined in (2.7)). Then there exist constants C, K > 0 (only depending on the H 1 -norm of ϕ and on the constant D appearing in (3.2)) such that C (1) eK t Tr N ,t − |ϕt ϕt | ≤ N
(3.3)
for all t ∈ R. Remark 3.2. The use of coherent states as initial data allows us to obtain the optimal rate of convergence 1/N for all fixed √ times (while for the evolution of factorized N -particle states we only get the rate 1/ N ; see (1.14)). Proof. The proof of Theorem 3.1 will occupy the remaining subsections of Sect. 3.
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
41 (1)
3.1. Dynamics U N of quantum fluctuations. By (2.7), the kernel of N ,t is given by
√ √ 1 , W ∗ ( N ϕ)ei H N t a ∗y ax e−i H N t W ( N ϕ) N √ √ √ ϕ t (y) , W ∗ ( N ϕ)ei H N t (ax − N ϕt (x))e−i H N t W ( N ϕ) = ϕt (x)ϕ t (y) + √ N √ √ √ ϕt (x) , W ∗ ( N ϕ)ei H N t (a ∗y − N ϕ t (y))e−i H N t W ( N ϕ) + √ N √ √ √ √ 1 , W ∗ ( N ϕ)ei H N t (a ∗y − N ϕ t (y))(ax − N ϕt (x))e−i H N t W ( N ϕ) . + N (3.4)
(1)
N ,t (x; y) =
It was observed by Hepp in [9] (see also Eqs. (1.17)-(1.28) in [8]) that √ √ √ W ∗ ( N ϕs ) ei H N (t−s) (ax − N ϕt (x))e−i H N (t−s) W ( N ϕs ) = U N (t; s)∗ ax U N (t; s) = U N (s; t) ax U N (t; s),
(3.5)
where the unitary evolution U N (t; s) is determined by the equation6 i∂t U N (t; s) = L N (t)U N (t; s) with the generator L N (t) =
and U N (s; s) = 1
dx ∇x ax∗ ∇x ax + dx V ∗ |ϕt |2 (x) ax∗ ax + dxdy V (x − y) ϕt (x)ϕt (y)a ∗y ax 1 + dxdy V (x − y) ϕt (x)ϕt (y)ax∗ a ∗y + ϕt (x)ϕt (y)ax a y 2 1 +√ dxdy V (x − y) ax∗ ϕt (y)a ∗y + ϕt (y)a y ax N 1 + dxdy V (x − y) ax∗ a ∗y a y ax . 2N
(3.6)
(3.7)
It follows from (3.4) that 1 , U N (t; 0)∗ a ∗y ax U N (t; 0) N ϕt (x) , U N (t; 0)∗ a ∗y U N (t; 0) +√ N ϕ t (y) (3.8) +√ , U N (t; 0)∗ ax U N (t; 0) . N √ In order to produce another decaying factor 1/ N in the last two terms on the r.h.s. of N (t; 0) the last equation, we compare the evolution U N (t; 0) with another evolution U defined through the equation (1)
N ,t (x, y) − ϕt (x)ϕ t (y) =
N (t; s) = L N (t) U N (t; s) i∂t U
N (s; s) = 1 with U
√ √ 6 Note that, explicitly, U (t, s) = W ∗ ( N φ )e−i H N (t−s) W ( N φ ). t s N
(3.9)
42
I. Rodnianski, B. Schlein
with the time-dependent generator N (t) = dx ∇x ax∗ ∇x ax + dx V ∗ |ϕt |2 (x) ax∗ ax L + dxdy V (x − y)ϕt (x)ϕt (y)a ∗y ax 1 + dxdy V (x − y) ϕt (x)ϕt (y)ax∗ a ∗y + ϕt (x)ϕt (y)ax a y 2 1 + dxdy V (x − y) ax∗ a ∗y a y ax . 2N
(3.10)
From (3.8) we find (1)
N ,t (x; y) − ϕt (x)ϕ t (y) 1 = , U N (t; 0)∗ a ∗y ax U N (t; 0) N ϕt (x) N (t; 0) , U N (t; 0)∗ a ∗y U N (t; 0) − U +√ N N (t; 0) N (t; 0)∗ a ∗y U + , U N (t; 0)∗ − U ϕ t (y) N (t; 0) , U N (t; 0)∗ ax U N (t; 0) − U +√ N N (t; 0) . N (t; 0)∗ ax U + , U N (t; 0)∗ − U
(3.11)
Here we used the fact that N (t; 0) = , U N (t; 0) = 0. N (t; 0)∗ ax∗ U N (t; 0)∗ a y U , U N (t) does not preserve This follows from the observation that, although the evolution U the number of particles, it preserves the parity (it commutes with (−1)N ). Multiplying (3.11) with the kernel J (x, y) of a Hilbert-Schmidt operator J over L 2 (R3 ) and taking the trace, we obtain (1) Tr J N ,t − |ϕt ϕt | 1 = dxdy J (x, y) a y U N (t; 0), ax U N (t; 0) N 1 N (t; 0) +√ dxdy J (x, y)ϕt (x) a y U N (t; 0), U N (t; 0) − U N 1 N (t; 0) N (t; 0) , a ∗y U +√ dxdy J (x, y)ϕt (x) U N (t; 0) − U N 1 N (t; 0) +√ dxdy J (x, y) ϕ t (y) ax∗ U N (t; 0), U N (t; 0) − U N 1 N (t; 0). N (t; 0) , ax U +√ dxdy J (x, y) ϕ t (y) U N (t; 0) − U N
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
43
Hence (1) Tr J N ,t − |ϕt ϕt | 1/2
1 2 ≤ dxax U N (t; 0)2 dxdy |J (x, y)| N 1 N (t; 0) dx |ϕt (x)|a(J (x, .))U N (t; 0) + √ U N (t; 0) − U N 1 N (t; 0) dx|ϕt (x)|a ∗ (J (x, .))U N (t; 0) + √ U N (t; 0) − U N 1 N (t; 0) dy |ϕt (y)|a ∗ (J (., y))U N (t; 0) + √ U N (t; 0) − U N 1 N (t; 0), dy |ϕt (y)|a(J (., y))U U N (t; 0) − U N (t; 0) +√ N and therefore J HS (1) U N (t; 0), N U N (t; 0) Tr J N ,t − |ϕt ϕt | ≤ N 2J HS N (t; 0)) (N + 1)1/2 U N (t; 0) + √ (U N (t; 0) − U N 2J HS N (t; 0). N (t; 0)) (N + 1)1/2 U + √ (U N (t; 0) − U N The proof of Theorem 3.1 now follows from Proposition 3.3, Lemma 3.8, Lemma 3.9, and from the remark that the trace norm can be controlled, in this case, by twice the Hilbert-Schmidt norm (see Remark 3 after Theorem 1.1). Proposition 3.3. Let U N (t; s) be the unitary evolution defined in (3.6). Then there exists a constant K , and, for every j ∈ N, constants C( j), K ( j) (depending only on ϕ H 1 and on the constant D appearing in (3.2)) such that U N (t; s)ψ, N j U N (t; s)ψ ≤ C( j) ψ, (N + 1)2 j+2 ψ e K ( j)|t−s|
(3.12)
for all ψ ∈ F, and for all t, s ∈ R. Remark 3.4. Proposition 3.3 states that the number of particles produced by the dynamics U N of quantum fluctuations is independent of N and grows in time with at most exponential rate. This N -independence plays an important role in our analysis. Its proof (M) requires the introduction of yet another dynamics U N , whose generator looks very similar to L N (t) but contains a cutoff, in the cubic term, guaranteeing that the number of particles is smaller than a given M. Proof. We start by introducing a new unitary dynamics with time-dependent generator (M) L N (t) similar to L N (t) but with a cutoff in the number of particles in the cubic term.
44
I. Rodnianski, B. Schlein (M)
3.2. Truncated dynamics U N . For a fixed M > 0 (at the end we will choose M = N ), we consider the time-dependent generator (M)
L N (t) =
dx ∇x ax∗ ∇x ax + dx V ∗ |ϕt |2 (x) ax∗ ax + dxdy V (x − y) ϕt (x)ϕt (y)a ∗y ax 1 + dxdy V (x − y) ϕt (x)ϕt (y)ax∗ a ∗y + ϕt (x)ϕt (y)ax a y 2 1 +√ dxdy V (x − y) ax∗ ϕ t (y)a y χ (N ≤ M) + ϕt (y)χ (N ≤ M)a ∗y ax N 1 + (3.13) dxdy V (x − y) ax∗ a ∗y a y ax 2N (M)
and the corresponding time-evolution U N (t; s), defined by (M)
(M)
(M)
i∂t U N (t; s) = L N (t)U N (t; s)
with
(M)
U N (s; s) = 1.
Step 1 in the proof of Proposition 3.3. Lemma 3.5. There exists a constant K (only depending on ϕ H 1 and on the constant D in (3.2)), such that, for all N , M ∈ N, ψ ∈ F, and t, s ∈ R, (M) (M) U N (t; s)ψ, N j U N (t; s)ψ ≤ ψ, (N + 1) j ψ exp 4 j K |t − s|(1 + M/N ) . (3.14) Proof (Proof of Lemma 3.5.). To prove (3.14) we compute the time-derivative of the expectation of (N + 1) j . It suffices to consider the case s = 0. We find d (M) (M) U (t; 0)ψ, (N + 1) j U N (t; 0)ψ dt N (M) (M) (M) = U N (t; 0)ψ, [iL N (t), (N + 1) j ]U N (t; 0)ψ = Im dxdyV (x − y) ϕt (x)ϕt (y) (M)
(M)
× U N (t; 0)ψ, [ax∗ a ∗y , (N + 1) j ]U N (t; 0)ψ 2 + √ Im dxdyV (x − y) ϕ t (y) N (M)
(M)
× U N (t; 0)ψ, [ax∗ a y χ (N ≤ M)ax , (N + 1) j ] U N (t; 0)ψ Using the pull-through formulae ax N = (N + 1)ax , ax∗ N = (N − 1)ax∗ , we find [ax∗ , (N + 1) j ] =
j−1 j k=0
k
(−1)k (N + 1)k ax∗ , [ax , (N + 1) j ] =
j−1 j k=0
k
(N + 1)k ax .
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
45
As a consequence, j−1 j ax∗ a ∗y , (N + 1) j = (−1)k ax∗ (N + 1)k a ∗y + (N + 1)k ax∗ a ∗y k k=0
j−1 k k k k j = (−1)k N 2 ax∗ a ∗y (N + 2) 2 + (N + 1) 2 ax∗ a ∗y (N + 3) 2 , k k=0
j−1 j−1 k k j j j k ax , (N + 1) = (N + 1) ax = (N + 1) 2 ax N 2 . k k k=0
k=0
Therefore d (M) (M) U (t; 0)ψ, (N + 1) j U N (t; 0)ψ dt N j−1 j k (−1) Im dxdy V (x − y) ϕt (x)ϕt (y) = k k=0 k k k k (M) (M) × U N (t; 0)ψ, N 2 ax∗ a ∗y (N + 2) 2 + (N + 1) 2 ax∗ a ∗y (N + 3) 2 U N (t; 0)ψ j−1 2 j Im dx +√ N k=0 k (M)
k
k
(M)
× U N (t; 0)ψ, ax∗ a(V (x − .)ϕt )χ (N ≤ M)(N + 1) 2 ax N 2 U N (t; 0)ψ. (3.15) To control contributions from the first term we use bounds of the form dxdyV (x − y) ϕt (x)ϕt (y) U (M) (t; 0)ψ, (N + 1) k2 a ∗ a ∗ (N + 3) k2 U (M) (t; 0)ψ x y N N k k (M) (M) ≤ dx|ϕt (x)|ax (N + 1) 2 U N (t; 0)ψ a ∗ (V (x − .)ϕt )(N + 3) 2 U N (t; 0)ψ
≤ const sup
1/2 dy V (x − y)2 |ϕt (y)|2
x
≤ K (N + 3)
k+1 2
(N + 3)
k+1 2
U N(M) (t; 0)ψ2
(M)
U N (t; 0)ψ2 .
Here we used that, by (3.2), sup x
dyV 2 (x − y)|ϕt (y)|2 ≤ Dϕt 2H 1 ≤ const Dϕ2H 1 ≤ K
(3.16)
is bounded uniformly in t (as follows from (1.16)). Similar estimates are applied to the k k term containing N 2 ax∗ a ∗y (N + 2) 2 .
46
I. Rodnianski, B. Schlein
On the other hand, to control contributions arising from the second integral on the r.h.s. of (3.15), we use estimates of the form dx U (M) (t; 0)ψ, a ∗ a(V (x − .)ϕt )χ (N ≤ M)(N + 1) k2 ax N 2k U (M) (t; 0)ψ x N N k k ≤ dx ax (N + 1) 2 U N(M) (t; 0)ψ a(V (x − .)ϕt )χ (N ≤ M) ax N 2 U N(M) (t; 0)ψ ≤ M 1/2 sup V (x − .)ϕt N
k+1 2
x
≤ K M 1/2 (N + 1)
k+1 2
(M)
k
(M)
U N (t; 0)ψ N 1/2 (N + 1) 2 U N (t; 0)ψ
(M)
U N (t; 0)ψ2 .
This implies d (M) U (t; 0)ψ, (N + 1) j U (M) (t; 0)ψ N dt N j j (M) (M) ≤ K (1 + M/N ) U N (t; 0)ψ, (N + 3)k U N (t; 0)ψ k k=0 (M) (M) ≤ 4 j K (1 + M/N ) U N (t; 0)ψ, (N + 1) j U N (t; 0)ψ. From the Gronwall Lemma, we find (3.14). Step 2 of the proof of Proposition 3.3 3.3. Weak bounds on the U N dynamics. To compare the evolution U N (t; s) with the cut(M) off evolution U N (t; s), we first need some (very weak) a-priori bound on the growth of the number of particle with respect to U N (t; s). Lemma 3.6. For arbitrary t, s ∈ R and ψ ∈ F, we have ψ, U N (t; s)N U N (t; s)∗ ψ ≤ 6 ψ, (N + N + 1)ψ.
(3.17)
Moreover, for every ∈ N, there exists a constant C() such that ψ, U N (t; s)N 2 U N (t; s)∗ ψ ≤ C() ψ, (N + N )2 ψ, ψ, U N (t; s)N 2+1 U N (t; s)∗ ψ ≤ C() ψ, (N + N )2+1 (N + 1)ψ
(3.18) (3.19)
for all t, s ∈ R and ψ ∈ F. Proof (Proof of Lemma 3.6.). Equation (3.19) follows from (3.18). In fact, assuming (3.18) to hold true, we have ψ, U N (t; s)N 2+1 U N (t; s)∗ ψ N 1 ψ, U N (t; s)N 2+2 U N (t; s)∗ ψ + ψ, U N (t; s)N 2 U N (t; s)∗ ψ ≤ 2N 2 C()N C( + 1) 2+2 ψ + ≤ ψ, (N + N ) ψ, (N + N )2 ψ 2N 2 ≤ D() ψ, (N + N )2+1 (N + 1)ψ (3.20) for an appropriate constant D().
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
47
To prove (3.17) and (3.18) we observe that, by (3.5), U N∗ (t; s)N U N (t; s) = dx U N∗ (t; s)ax∗ ax U N (t; s) √ √ = dx W ∗ ( N ϕs )ei H N (t−s) (ax∗ − N ϕ t (x)) √ √ ×(ax − N ϕt (x))e−i H N (t−s) W ( N ϕs ) √ √ √ = W ∗ ( N ϕs ) N − N ei H N (t−s) φ(ϕt )e−i H N (t−s) + N W ( N ϕs ). (3.21) (Recall that φ(ϕ) = a ∗ (ϕ) + a(ϕ) = dx(ϕ(x)ax∗ + ϕ(x)ax ).) From Lemma 2.1 and Lemma 2.2, we get √ √ ψ, U N∗ (t; s)N U N (t; s)ψ ≤ 2 ψ, W ∗ ( N ϕs )(N + N + 1)W ( N ϕs )ψ √ = 2 ψ, (N + N φ(ϕs ) + 2N + 1)ψ ≤ 6 ψ, (N + N + 1)ψ, (3.22) which shows (3.17). To complete the proof of (3.18), we define √ X t,s = (N − N ei H N (t−s) φ(ϕt )e−i H N (t−s) + N ). Then, using the notation ad A (B) = [B, A], it is simple to prove that there exists a constant C > 0 such that 2 for all m ∈ N. (3.23) ≤ C(N + N )2 and admXt,s (N ) ≤ C(N + N ) X t,s By induction it follows that, for every ∈ N, there exist constants D(), C() with −1 −1 X t,s (N + N )2 X t,s ≤ D()(N + N )2
and
2 X t,s ≤ C()(N + N )2 .
(3.24)
In fact, for = 1 (3.24) reduces to (3.23). Assuming (3.24) to hold for all < k, we can prove it for = k by noticing that k−1 k−1 2k−2 k−1 X t,s (N + N )2 X t,s ≤ 2(N + N )X t,s (N + N ) + 2|[X t,s , N ]|2 2k−2 ≤ 2(N + N )X t,s (N + N ) + 4k
k−2
2 m m k−1−m X t,s ad X t,s (N ) X t,s
m=0 2k−2 ≤ 2(N + N )X t,s (N + N ) + 4k C
k−2
m m X t,s (N + N )2 X t,s
m=0
≤ D(k) (N + N )
2k
for an appropriate constant D(k), and that, by (3.23) and (3.25), 2k k−1 k−1 X t,s ≤ C X t,s (N + N )2 X t,s ≤ C D(k)(N + N )2k = C(k)(N + N )2k .
In (3.25), we used the commutator expansion n−1 n [An , B] = Am adn−m A (B) m m=0
(3.25)
48
I. Rodnianski, B. Schlein
in the second line, the bound (3.23) in the third line, and the induction assumption in the last line. From (3.21) and (3.24), we obtain that √ √ 2 W ( N ϕs )ψ ψ, U N (t; s)N 2 U N (t; s)∗ ψ = W ( N ϕs )ψ, X t,s √ √ ≤ C() W ( N ϕs )ψ, (N + N )2 W ( N ϕs )ψ √ = C() ψ, (N + N φ(ϕs ) + 2N )2 ψ. (3.26) Analogously to (3.24), it is possible to prove that, for every ∈ N, there exists a constant C() with (N +
√
N φ(ϕs ) + 2N )2 ≤ C()(N + N )2 .
Equation (3.18) follows therefore from (3.26).
Step 3 of the proof of Proposition 3.3.
(M)
3.4. Comparison of the U N and U N
dynamics.
Lemma 3.7. For every j ∈ N there exist constants C( j), K ( j) (depending only on j, on ϕ H 1 and on the constant D in (3.2)) such that (M) U N (t; s)ψ, N j U N (t; s) − U N (t; s) ψ (N /M) j (N + 1) j+1 ψ2 ≤ C( j) exp K ( j)(1 + M/N )|t − s| (3.27) √ (1 + M/N ) and (M) (M) U N (t; s)ψ, N j U N (t; s) − U N (t; s) ψ (N + 1) j ψ2 ≤C j exp K ( j)(1 + M/N )|t − s| , √ M (1 + M/N )
(3.28)
for all ψ ∈ F and for all t, s ∈ R. Proof (Proof of Lemma 3.7.). To simplify the notation we consider the case s = 0 and t > 0 (but the other cases can be treated identically). To prove (3.27), we expand the
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
49
difference of the two evolutions: (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ = U N (t; 0)ψ, N j U N (t; 0) 1 − U N (t; 0)∗ U N(M) (t; 0) ψ t = −i ds U N (t; 0)ψ, N j U N (t; s) L N (s) − L(M) (s) U N(M) (s; 0)ψ N 0 t i = −√ ds dxdyV (x − y) × U N (t; 0)ψ, N 0 (M) N j U N (t; s)ax∗ ϕ t (y)a y χ (N > M) + ϕt (y)χ (N > M)a ∗y ax U N (s; 0)ψ t i = −√ ds dx ax U N (t; s)∗ N j U N (t; 0)ψ, N 0 (M)
a(V (x − .)ϕt )χ (N > M)ax U N (s; 0)ψ t i −√ ds dx ax U N (t; s)∗ N j U N (t; 0)ψ, N 0 (M)
χ (N > M)a ∗ (V (x − .)ϕt )ax U N (s; 0)ψ.
(3.29)
Hence (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ t 1 ≤ √ ds dxax U N (t; s)∗ N j U N (t; 0)ψ N 0 (M)
×a(V (x − .)ϕt )ax χ (N > M + 1)U N (s; 0)ψ t 1 +√ ds dxax U N (t; s)∗ N j U N (t; 0)ψ N 0 (M)
×a ∗ (V (x − .)ϕt )ax χ (N > M)U N (s; 0)ψ t 1 ≤ √ sup V (x − .)ϕt ds dxax U N (t; s)∗ N j U N (t; 0)ψ N x 0 (M)
× ax N 1/2 χ (N > M + 1)U N (s; 0)ψ t 1 + √ sup V (x − .)ϕt ds dxax U N (t; s)∗ N j U N (t; 0)ψ N x 0 (M)
× ax N 1/2 χ (N > M)U N (s; 0)ψ t C (M) ≤√ dsN 1/2 U N (t; s)∗ N j U N (t; 0)ψ N χ (N > M)U N (s; 0)ψ, N 0 where we used (3.16) once again. From Lemma 3.6, we obtain N 1/2 U N (t; s)∗ N j U N (t; 0)ψ2 = N j U N (t; 0)ψ, U(t; s)N U N (t; s)∗ N j U N (t; 0)ψ ≤ 6 N j U N (t; 0)ψ, (N + N + 1)N j U N (t; 0)ψ ≤ C( j) ψ, (N + N )2 j+1 (N + 1)ψ ≤ C( j)N 2 j+1 ψ, (N + 1)2 j+2 ψ. (3.30)
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I. Rodnianski, B. Schlein
Therefore, using the inequality χ (N > M) ≤ (N /M)2 j , we obtain (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ t (M) (M) ≤ C( j)N j (N + 1) j+1 ψ ds U N (s; 0)ψ, N 2 χ (N > M)U N (s; 0)ψ1/2 ≤ C( j)N (N + 1) j
j+1
0
t
ψ 0
(M)
ds U N (s; 0)ψ,
N 2 j+2 (M) U (s; 0)ψ1/2 . M2 j N
Finally, from (3.14), we conclude that (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ t ≤ C( j)(N /M) j (N + 1) j+1 ψ2 ds exp (K ( j) s (1 + M/N )) 0
≤ C( j)
(N /M) j (N 1+
√
+ 1) j+1 ψ2 M/N
exp (K ( j) t (1 +
M/N )).
To prove (3.28), we proceed similarly; analogously to (3.29) we find (M) (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ t i (M) ds dx ax U N (t; s)∗ N j U N (t; 0)ψ, = −√ N 0 (M)
a(V (x − .)ϕt )χ × (N > M)ax U N (s; 0)ψ t i (M) −√ ds dx ax U N (t; s)∗ N j U N (t; 0)ψ, N 0 (M)
χ (N > M)a ∗ (V (x − .)ϕt )ax U N (s; 0)ψ, and thus (M) (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ t C (M) (M) ≤√ dsN 1/2 U N (t; s)∗ N j U N (t; 0)ψ N χ (N > M)U N (s; 0)ψ. N 0 (3.31) Again, applying (3.18) and (3.14) we find (M) (M) U N (t; 0)ψ, N j U N (t; 0) − U N (t; 0) ψ ≤C
(N + 1) j+1 ψ2 exp (K ( j) t (1 + M/N )). √ j M (1 + M/N )
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
51
Step 4 Conclusion of the proof of Proposition 3.3. From (3.27), (3.28) and (3.14) we obtain, choosing M = N , (M)
U N (t; s)ψ, N j U N (t; s)ψ = U N (t; s)ψ, N j (U N (t; s) − U N (t; s))ψ (M)
(M)
+ (U N (t; s) − U N (t; s))ψ, N j U N (t; s)ψ (M)
(M)
+ U N (t; s)ψ, N j U N (t; s)ψ ≤ C( j)(N + 1) j+1 ψ2 e K ( j)|t−s| . N (t; s), defined N (t; s). We now consider the dynamics U 3.5. Approximate dynamics U in (3.9) by N (t; s) = L N (t) U N (t; s) N (s; s) = 1 with U i∂t U with the time-dependent generator ∗ L N (t) = dx ∇x ax ∇x ax + dx V ∗ |ϕt |2 (x) ax∗ ax + dxdy V (x − y)ϕt (x)ϕt (y)a ∗y ax 1 + dxdy V (x − y) ϕt (x)ϕt (y)ax∗ a ∗y + ϕt (x)ϕt (y)ax a y 2 1 + (3.32) dxdy V (x − y) ax∗ a ∗y a y ax . 2N Lemma 3.8. There exists a constant K > 0, only depending on ϕ H 1 and on the constant D appearing in (3.2), such that N (t; 0) ≤ e K t . N (t; 0), N 3 U U
(3.33)
Proof. We compute the derivative d N (t; 0) U N (t; 0), (N + 1)3 U dt N (t; 0), [i L N (t), (N + 1)3 ]U N (t; 0) = U N (t; 0), [ax∗ a ∗y , (N + 1)3 ]U N (t; 0) = −Im dxdyV (x − y)ϕt (x)ϕt (y) U N (t; 0), = 2Im dxdyV (x − y)ϕt (x)ϕt (y) U N (t; 0) ax∗ a ∗y (N + 1)2 + (N + 1)ax∗ a ∗y (N + 1) + (N + 1)2 ax∗ a ∗y U N (t; 0), = 2Im dxdyV (x − y)ϕt (x)ϕt (y) U N (t; 0) (N − 1)ax∗ a ∗y (N + 1)+(N + 1)ax∗ a ∗y (N + 1)+(N + 1)ax∗ a ∗y (N + 3) U N (t; 0), = 2Im dxdyV (x − y)ϕt (x)ϕt (y) U N (t; 0). 3(N + 1)ax∗ a ∗y (N + 1) + 4ax∗ a ∗y U
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I. Rodnianski, B. Schlein
Therefore d N (t; 0) U N (t; 0), (N + 1)3 U dt
N (t; 0), a ∗ (V (x − .)ϕt )(N + 1)U N (t; 0) dxϕt (x) ax (N + 1)U N (t; 0), a ∗ (V (x − .)ϕt )U N (t; 0). −8Im dxϕt (x) ax U
= 6Im
Taking the absolute value, we find d U N (t; 0) N (t; 0), (N + 1)3 U dt N (t; 0) a ∗ (V (x − .)ϕt )(N + 1)U N (t; 0) ≤ 6 dx|ϕt (x)|ax (N + 1)U N (t; 0) a ∗ (V (x − .)ϕt )U N (t; 0) + 8 dx|ϕt (x)|ax U N (t; 0)2 ≤ 14 sup V (x − .)ϕt (N + 1)3/2 U x
N (t; 0). N (t; 0), (N + 1)3 U ≤ C U Applying the Gronwall Lemma, we obtain (3.33). N dynamics. The final step in the proof of Theorem 3.6. Comparison of the U N and U N . 3.1 is the comparison of evolutions generated by U N and U N (t; s) be defined as in (3.6) and (3.9), Lemma 3.9. Let the evolutions U N (t; s) and U respectively. Then there exist constants C, K > 0, only depending on ϕ H 1 and on the constant D in (3.2), such that U N (t; 0) − U N (t; 0) ≤ √C e K t . N
(3.34)
Proof. We write N (t; 0) U N (t; 0) − U N (t; 0) = U N (t; 0) 1 − U N (t; 0)∗ U t N (s) U N (s; 0) ds U N (t; s) L N (s) − L = −i 0 t i N (s; 0) = −√ ds dxdy V (x − y) U N (t; s)ax∗ ϕt (y)a ∗y + ϕ t (y)a y ax U N 0 t i N (s; 0). = −√ ds dx U N (t; s)ax∗ φ(V (x − .)ϕt )ax U N 0 Hence t ∗ . U N (t; 0) − U N (t; 0) ≤ √1 dx a U ds φ(V (x − .)ϕ )a (s; 0) t x N x N 0 (3.35)
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
Next, we observe that 2 dx a ∗ φ(V (x − .)ϕt )ax U N (s; 0) x N (s; 0), φ(V (y − .)ϕt )a y ax∗ φ(V (x − .)ϕt )ax U N (s; 0) = dydx a y U N (s; 0), φ(V (y − .)ϕt )ax∗ a y φ(V (x − .)ϕt )ax U N (s; 0) = dydx a y U N (s; 0), φ(V (x − .)ϕt )φ(V (x − .)ϕt )ax U N (s; 0) + dx ax U N (s; 0), ax∗ φ(V (y − .)ϕt ) + V (y − x)ϕ t (x) = dydx a y U N (s; 0) × φ(V (x − .)ϕt )a y + V (x − y)ϕt (y) ax U N (s; 0), φ(V (x − .)ϕt )φ(V (x − .)ϕt )ax U N (s; 0). + dx ax U Therefore, we have 2 dx a ∗ φ(V (x − .)ϕt )ax U N (s; 0) x N (s; 0), φ(V (y − .)ϕt )φ(V (x − .)ϕt )a y ax U N (s; 0) = dydx ax a y U N (s; 0), φ(V (x − .)ϕt )a y ax U N (s; 0) + dydx V (x − y)ϕ t (x) a y U N (s; 0), φ(V (y − .)ϕt )ax U N (s; 0) + dydx V (x − y)ϕt (y) ax a y U N (s; 0), ax U N (s; 0) + dydx V (x − y)2 ϕ t (x)ϕt (y) a y U N (s; 0), φ(V (x − .)ϕt )φ(V (x − .)ϕt )ax U N (s; 0). + dx ax U It follows that 2 dx a ∗ φ(V (x − .)ϕt )ax U N (s; 0) x 2 N (s; 0)2 dydx(N + 2)1/2 ax a y U ≤ 4 sup V (x − .)ϕt x
+2 sup V (x − .)ϕt ×
x
N (s; 0)a y ax U N (s; 0) dydx|V (x − y)||ϕt (x)|(N + 1)1/2 a y U +2 sup V (y − .)ϕt
×
y
N (s; 0)(N + 1)1/2 ax U N (s; 0) dydx|V (x − y)||ϕt (y)|ax a y U
53
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I. Rodnianski, B. Schlein
N (s; 0)ax U N (s; 0) dydx V (x − y)2 |ϕt (x)||ϕt (y)|a y U 2 N (s; 0)2 . +4 sup V (x − .)ϕt dx(N + 1)1/2 ax U
+
x
Using (3.16), we obtain 2 dx a ∗ φ(V (x − .)ϕt )ax U N (s; 0) x N (s; 0)2 ≤ C dydxax a y N 1/2 U
dydx|V (x − y)| |ϕt (x)| 2
+C
×
+C
N (s; 0)2 dxdya y ax U
2
N (s; 0)2 a y N 1/2 U
1/2
1/2
N (s; 0)2 dydxax a y U
1/2
1/2
N (s; 0)2 × dydx|V (x − y)|2 |ϕt (y)|2 ax N 1/2 U N (s; 0)2 + dydx V (x − y)2 |ϕt (x)|2 a y U N (s; 0)2 . +C dxax N 1/2 U From
2 2 2 2 2 dydx|V (x − y)| |ϕt (y)| ax ψ ≤ sup dyV (x − y) |ϕt (y)| N 1/2 ψ2 x
≤ CN 1/2 ψ2 we thus find 2 ≤ C(N + 1)3/2 U dx a ∗ φ(V (x − .)ϕt )ax U N (t; 0)2 . (s; 0) N x Inserting the last bound in (3.35) and using the result of Lemma 3.8 we obtain (3.34). This concludes the proof of Theorem 3.1.
3.7. Discussion. As mentioned in the Introduction, our approach to the study of the mean field limit of the N -body Schrödinger dynamics mirrors that used by Hepp and GinibreVelo in [8,9] in the study of the semi-classical limit of quantum many-boson systems. In the language of the mean field limit, the main result obtained by Hepp (for smooth potentials) and by Ginibre and Velo (for singular potentials) was the convergence of the
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
55
fluctuation dynamics U N (t; s) (defined in (3.6)) to a limiting N -independent dynamics U(t; s) in the sense that s − lim U N (t; s) = U(t; s) N →∞
(3.36)
for all fixed t and s. Here the limiting dynamics U(t; s) is defined by i∂t U(t; s) = L(t)U(t; s)
with U(s; s) = 1
and with generator ∗ L(t) = dx ∇x ax ∇x ax + dx V ∗ |ϕt |2 (x) ax∗ ax + dxdy V (x − y)ϕt (x)ϕt (y)a ∗y ax 1 + dxdy V (x − y) ϕt (x)ϕt (y)ax∗ a ∗y + ϕt (x)ϕt (y)ax a y . (3.37) 2 The convergence (3.36) does not give any information about the convergence of the (1) one-particle marginal N ,t , associated with the evolution of the coherent initial state, to
the orthogonal projection |ϕt ϕt |. The definition of the marginal density (1) N ,t involves unbounded creation and annihilation operators. This also explains why the derivation of the bound (3.3) in Theorem 3.1 is in general more complicated than the proof of the convergence (3.36). The proof of (3.36) requires control of the growth of the expectation of powers of the number of particle operator N only with respect to the limiting dynamics. To prove (3.3), on the other hand, we need to control the growth of the expectation of N with respect to the N -dependent fluctuation dynamics U N (t; s). 4. Time Evolution of Factorized States This section is devoted to the proof of Theorem 1.1. The main idea in the proof is that we can write the factorized N -particle state ψ N = ϕ ⊗N (whose evolution is considered in Theorem 1.1) as a linear combination of coherent states, whose dynamics can be studied using the results of Sect. 3. Proof (Proof of Theorem 1.1). We start by writing ψ N = ϕ ⊗N or, more precisely, the sequence {0, 0, . . . , 0, ψ N , 0, 0, . . . } =
(a ∗ (ϕ)) N ∈F √ N!
as a linear combination of coherent states. While it is always possible in principle, our goal is to represent ψ N with the least number of coherent states. Lemma 4.1. We have the following representation: 2π √ (a ∗ (ϕ)) N dθ iθ N e W (e−iθ N ϕ) = dN √ 2π N! 0 with the constant
(4.1)
√
dN =
N! N 1/4 . N N /2 e−N /2
(4.2)
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I. Rodnianski, B. Schlein
Proof. To prove the representation (4.1) observe that, from (2.10) and since ϕ = 1,
2π 0
∞
√ dθ iθ N e W (e−iθ N ϕ) = e−N /2 N j/2 2π j=1
dθ iθ(N − j) e 2π
(a ∗ (ϕ)) j j!
e−N /2 N N /2 (a ∗ (ϕ)) N = . √ √ N! N!
(4.3)
(1)
The kernel of the one-particle density γ N ,t associated with the solution of the Schrödinger equation e−it H N
(a ∗ (ϕ)) N √ N!
is given by (see (2.7)) (1)
(a ∗ (ϕ)) N 1 (a ∗ (ϕ)) N , ei H N t a ∗y ax e−i H N t √ √ N N! N! 2π 2π 2 d dθ1 dθ2 −iθ1 N iθ2 N e e = N N 0 2π 0 2π √ √ × W (e−iθ1 N ϕ), a ∗y (t)ax (t)W (e−iθ2 N ϕ),
γ N ,t (x; y) =
(4.4)
where we introduced the notation ax (t) = ei H N t ax e−i H N t . It follows that (1)
γ N ,t (x; y) =
2π √ d N2 dθ1 2π dθ2 −iθ1 N iθ2 N e W (e−iθ1 N ϕ), e N 0 2π 0 2π √ √ √ a ∗y (t) − eiθ1 N ϕ t (y) ax (t) − e−iθ2 N ϕt (x) W (e−iθ2 N ϕ) d 2 ϕ t (y) 2π dθ1 2π dθ2 −iθ1 (N −1) iθ2 N e e + N√ 2π 0 2π N 0 √ √ √ × W (e−iθ1 N ϕ), ax (t) − e−iθ2 N ϕt (x) W (e−iθ2 N ϕ) d 2 ϕt (x) 2π dθ1 2π dθ2 −iθ1 N iθ2 (N −1) e e + N√ 2π 0 2π N 0 √ √ √ × W (e−iθ1 N ϕ), a ∗y (t) − eiθ1 N ϕ t (y) W (e−iθ2 N ϕ) 2π dθ1 2π dθ2 −iθ1 (N −1) iθ2 (N −1) 2 e e +d N ϕt (x)ϕ t (y) 2π 0 2π 0 √ √ × W (e−iθ1 N ϕ), W (e−iθ2 N ϕ) . (4.5)
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
57
Since
2π
dN 0
∞
√ dθ iθ(N −1) e W (e−iθ N ϕ) = d N e−N /2 2π j=0
×N j/2
2π
0
(a ∗ (ϕ)) j j!
dθ iθ(N −1− j) e 2π
e−N /2 N (N −1)/2 (a ∗ (ϕ)) N −1 √ N − 1! N − 1! = ϕ ⊗N −1 ,
= dN
(4.6)
we obtain, from (4.5), that (1)
γ N ,t (x; y) − ϕ t (x)ϕt (y) 2π √ √ d N2 dθ1 2π dθ2 −iθ1 N iθ2 N e W (e−iθ1 N ϕ), a ∗y (t) − eiθ1 N ϕ t (y) e = N 0 2π 0 2π √ √ × ax (t) − e−iθ2 N ϕt (x) W (e−iθ2 N ϕ) √ √ d N ϕt (x) 2π dθ1 W (e−iθ1 N ϕ), a ∗y (t) − eiθ1 N ϕ t (y) ϕ ⊗(N −1) + √ 2π N 0 2π √ √ dθ1 d N ϕ (y) a y (t) − eiθ1 N ϕt (y) W (e−iθ1 N ϕ), ϕ ⊗(N −1) , (4.7) + √t 2π N 0 and thus that d 2 2π dθ 2π dθ 1 2 (1) γ N ,t (x; y) − ϕt (x)ϕ t (y) ≤ N N 0 2π 0 2π √ √ × a y (t) − e−iθ1 N ϕt (y) W (e−iθ1 N ϕ) √ √ × ax (t) − e−iθ2 N ϕt (x) W (e−iθ2 N ϕ) |ϕt (x)|| f N (y)| |ϕt (y)|| f N (x)| + √ √ N N 2π 2π 2 d dθ1 dθ2 a y U Nθ1 (t; 0) ax U Nθ2 (t; 0) ≤ N N 0 2π 0 2π |ϕt (x)|| f N (y)| |ϕt (y)|| f N (x)| + + , (4.8) √ √ N N +
where the unitary evolutions U Nθ (t; s) are defined as in (3.6), but with ϕt replaced7 by e−iθ ϕt in the generator (3.7). Taking the square of (4.8) and integrating over x, y, we 7 We are making use here of the important fact that if ϕ solves the nonlinear equation (1.15), then eiθ ϕ t t is also a solution of the same equation, for any fixed real θ .
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I. Rodnianski, B. Schlein
obtain
(1) dxdy |γ N ,t (x; y) − ϕt (x)ϕ t (y) 2
2π d N4 dθ1 2π dθ2 N 1/2 U Nθ1 (t; 0)2 N 1/2 U Nθ2 (t; 0)2 N 2 0 2π 0 2π 4 + (4.9) dx| f N (x)|2 . N
≤2
Using Proposition 3.3 and the fact that d N N 1/4 to control the first term, and using Lemma 4.2 to control the second term on the r.h.s. of the last equation, we find constants C, K , only depending on ϕ H 1 and on the constant D in (1.12) such that γ N(1),t − |ϕt ϕt |HS ≤
C exp(K t). N 1/2
(4.10)
This proves (1.17) and thus concludes the proof of Theorem 1.1.
Lemma 4.2. Let ϕt be a solution to the Hartree equation (1.5) with initial data ϕ ∈ H 1 (R3 ) with ϕ = 1. Let
f N (x) =
2π
dθ1 2π
2π
dθ2 −iθ1 (N −1) iθ2 N e e 2π 0 0 √ √ √ × W (e−iθ1 N ϕ), ax (t) − e−iθ2 N ϕt (x) W (e−iθ2 N ϕ) .
d N2
Then there exist constants C, K (only depending on ϕ H 1 and on the constant D in (1.12) such that dx | f N (x)|2 ≤ Ce K t . Proof. Using that √ √ √ ax (t) − e−iθ2 N ϕt (x) W (e−iθ2 N ϕ) = W (e−iθ2 N ϕ)U Nθ2 (0; t)ax U Nθ2 (t; 0), where the unitary evolution U Nθ (t; s) is defined as in (3.6), but with ϕt replaced by e−iθ ϕt in the generator (3.7), we can rewrite f N (x) as
2π
f N (x) = 0
dθ2 ψ(θ2 ), U Nθ2 (0; t)ax U Nθ2 (t; 0) 2π
(4.11)
with ψ(θ2 ) =
d N2
2π 0
√ dθ1 iθ1 (N −1) −iθ2 N ∗ −iθ2 √ e e W (e N ϕ)W (e−iθ1 N ϕ). (4.12) 2π
Performing the integration over θ1 , we immediately obtain √ ψ(θ2 ) = d N e−iθ2 N W ∗ (e−iθ2 N ϕ)ϕ ⊗(N −1) .
(4.13)
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
59
It is also possible to expand ψ(θ2 ) in a sum of factors living in the different sectors of the Fock space. From Eq. (2.10) and Lemma 2.2, we compute √ √ √ √ W ∗ (e−iθ2 N ϕ)W (e−iθ1 N ϕ) = W (−e−iθ2 N ϕ)W (e−iθ1 N ϕ) √ i(θ2 −θ1 ) = ei N Ime W ((e−iθ1 − e−iθ2 ) N ϕ) = e−N e N e
i(θ2 −θ1 )
N m/2 (e−iθ1 − e−iθ2 )m ϕ ⊗m , √ m! m≥0
(4.14) which implies (using the periodicity in the variable θ1 ) ψ(θ2 ) = d N2 e−N
∞ N m/2 2π dθ1 iθ1 (N −1) −iθ2 (m+1) N e−iθ1 −iθ1 e e (e − 1)m ϕ ⊗m . e √ 2π m! 0 m=0
Switching to the complex variable z = e−iθ1 we obtain ψ(θ2 ) =
−d N2 e−N
N m/2 dz −N N z −iθ2 (m+1) z e (z − 1)m ϕ ⊗m , √ e 2πi m! m≥0
where the z integral is over the circle of radius one around the origin (in the clock-wise √ sense). Changing variables z → N z, and using that d N = e N /2 N !/N N /2 , we obtain ψ(θ2 ) = −(N − 1)! =
m ∞ N − 2 −iθ2 (m+1) dz −N z z e (z − N )m ϕ ⊗m √ e 2πi m! m=0
m ∞ N− 2 √ Rm e−iθ2 (m+1) ϕ ⊗m , m! m=0
(4.15)
where we defined Rm =
d N −1 z e (z − N )m |z=0 . N −1 dz
(4.16)
Comparing (4.15) with (4.13), we obtain the identity ∞ R2m = d N2 . N m m!
(4.17)
m=0
It is also possible to obtain pointwise bounds on the coefficients Rm . From (4.16) we deduce that for m ≤ (N − 1), Rm =
m (−1)m−k k=0
(N − 1)!m!N m−k k!(N − 1 − k)!(m − k)!
m = (−1)m−k N m−k (N − 1)...(N − k) k=0
m! . k!(m − k)!
(4.18)
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I. Rodnianski, B. Schlein
The coefficients Rm turn out to be intimately connected with the classical system of orthogonal Laguerre polynomials. Recall that the associated Laguerre polynomial L (α) n (x) admits the following representation: L (α) n (x) =
n (−1)k k=0
(n + α)! xk. k!(n − k)!(α + k)!
Therefore (N −m−1) (N ), Rm = (−1)m m! L m (α)
which, for N > m + 1, involves the value of the Laguerre polynomial L n (N ) with a positive index α. Asymptotic expansions and estimates for the Laguerre polynomials is a classical subject, see [12] and references therein. However for the indices α = N −m −1, n = m with N m, the value of x = N belongs to the oscillatory regime of the behav(α) ior of L n (x) and the sharp estimates for those values of parameters have been only obtained recently in [10], where it is proven that, for α > −1, n ≥ 2 and the values of (α) x ∈ (q 2 , s 2 ), the function L n (x) obeys the bound (n + α)! x(s 2 − q 2 ) x − α+1 (α) e2 x 2 , |L n (x)| < n! r (x) where 1
1
1
1
s = (n + α + 1) 2 + n 2 , q = (n + α + 1) 2 − n 2 , r (x) = (x − q 2 )(s 2 − x). As a consequence, we obtain that (N −m−1) (N )| |L m
<
(N − 1)! m!
√ 4N N m N − N −m e2 N 2 . 4N m − m 2
Assuming that m ≤ N and using the asymptotics (N −1)! ∼ N N −1/2 e−N , we obtain 1
1
m
(N −m−1) |L m (N )| m − 4 (m!)− 2 N 2 ,
and therefore Rm 1 2
(m!) N
1
m 2
m− 4 .
Summarizing, the coefficients Am = Rm /(m!1/2 N m/2 ) appearing in the expansion (4.15) of ψ(θ2 ) satisfy the bounds for all m ≤ N and |Am | ≤ Cm −1/4 ∞ A2m = d N2 ≤ C N 1/2 .
(4.19)
m=0
Inserting (4.15) into (4.11) we obtain 2π ∞ dθ iθ(m+1) ⊗m θ f N (x) = e ϕ , U N (0; t)ax U Nθ (t; 0) , Am 2π 0 m=0
(4.20)
Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics
and therefore
61
∞ dθ |Am | ⊗m √ ϕ , (N + 1)1/2 U Nθ (0; t)ax U Nθ (t; 0) 2π m + 1 0 m=0 ∞ 1/2 2π dθ |Am |2 ≤ (N + 1)1/2 U Nθ (0; t)ax U Nθ (t; 0) .(4.21) m+1 2π 0
| f N (x)| =
2π
m=0
From (4.19), we obtain ∞ N −1 |Am |2 1 1 + |Am |2 ≤ const. ≤C 3/2 m+1 (m + 1) N
m=0
m=0
(4.22)
m≥N
On the other hand, from Proposition 3.3, we have 2 2 (N + 1)1/2 U Nθ (0; t)ax U Nθ (t; 0) ≤ Ce K t (N + 1)2 ax U Nθ (t; 0) 2 ≤ Ce K t ax N 2 U Nθ (t; 0) . Thus, applying once more Proposition 3.3, we find 2π dθ θ t K 2 dx | f N (x)| ≤ Ce U N (t; 0), N 5 U Nθ (t; 0) ≤ Ce K t . 2π 0 Acknowledgements. B. Schlein is grateful to L. Erd˝os and H.-T. Yau for many stimulating discussions concerning the dynamics of mean field systems.
References 1. Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: Univ. Texas Math. Physics Archive, http://www.ma.utexas.edu, No. 05-211, 2005 2. Bardos, C., Golse, F., Mauser, N.: Weak coupling limit of the N -particle Schrödinger equation. Meth. Appl. Anal. 7, 275–293 (2000) 3. Elgart, A., Erd˝os, L., Schlein, B., Yau, H.-T.: Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal. 179(2), 265–283 (2006) 4. Elgart, A., Schlein, B.: Mean field dynamics of Boson stars. Commun. Pure Appl. Math. 60(4), 500–545 (2007) 5. Erd˝os, L., Schlein, B., Yau, H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167(3), 515–614 (2007) 6. Erd˝os, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of BoseEinstein condensate. To appear in Ann. of Math. http://arxiv.org/abs/math-ph/0606017v3, 2006 7. Erd˝os, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001) 8. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. I and II. Commun. Math. Phys. 66, 37–76 (1979), and 68, 45–68 (1979) 9. Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974) 10. Krasikov, I.: Inequalities for Laguerre polynomials. East J. Approx. 11, 257–268 (2005) 11. Spohn, H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980) 12. Szegö, G.: Orthogonal Polynomials. Colloq. pub. AMS. V. 23, New York: Amer. Math. Soc., 1959 Communicated by H.-T. Yau
Commun. Math. Phys. 291, 63–110 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0865-6
Communications in
Mathematical Physics
A Topos for Algebraic Quantum Theory Chris Heunen1 , Nicolaas P. Landsman1 , Bas Spitters2, 1 Radboud Universiteit Nijmegen, Institute for Mathematics, Astrophysics,
and Particle Physics, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands. E-mail:
[email protected]
2 Eindhoven University of Technology, Department of Mathematics and Computer Science,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Received: 7 April 2008 / Accepted: 17 April 2009 Published online: 21 July 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Dedicated to Ieke Moerdijk, at his 50th birthday Abstract: The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum (A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from to Scott’s interval domain. Noting that open subsets of (A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Supported by N. W. O.
64
C. Heunen, N. P. Landsman, B. Spitters
Motto: ‘Ces “nuages probabilistes”, remplaçant les rassurantes particules matérielles d’antan, me rappellent étrangement les élusifs “voisinages ouverts” qui peuplent les topos, tels des fantômes évanescents, pour entourer des “points” imaginaires.’ (A. Grothendieck [43])1 1. Introduction This introduction is intended for both mathematical physicists and topos theorists. We apologize in advance for stating the obvious for one or the other of these groups at various points, but we hope that most of it is interesting to both communities.
1.1. The logic of classical physics. In classical mechanics, the logical structure of a physical system is encoded in its phase space M. Indeed, elementary propositions (also called ‘yes-no questions’ in physics) correspond to suitable subsets of M (such as all sets, all measurable sets or all regular open sets), and the logical connectives are given by the standard set-theoretic operations. This makes the logic of the system ‘spatial’; its realization as a Boolean algebra (i.e. a distributive lattice with 0 and 1 in which every element has a complement) confirms the ‘classical’ nature of the situation [76]. Physicists do not usually describe a system in the above way. Instead, they work with observables a : M → R, like position or energy. (Such functions might be arbitrary, or else required to be measurable or continuous as appropriate.) From that perspective, the elementary propositions are of the form a ∈ , where ⊆ R is a (arbitrary, measurable, or regular open) subset of the reals. Either way, a state of the system may be construed as a catalogue of answers to all yes-no questions about the systems. We concentrate on pure states ρ ∈ M, which provide sharp (as opposed to probabilistic) answers. In the first description, a proposition U ⊆ M is true (equivalently, the answer to the corresponding question is ‘yes’) iff ρ ∈ U . In the second description, a ∈ is true for a state ρ iff a(ρ) ∈ , i.e. iff ρ ∈ a −1 (). Thus propositions of the second type fall into equivalence classes [a ∈ ] = a −1 (). As these are subsets of M, this leads us back to the purely spatial picture of the first description. This truth assignment has a very simple categorical description. We regard M as an object in the category Sets of all sets as objects and all functions as arrows, and interpret ρ / M , where 1 is any singleton. A subset ρ as an arrow (in fact, a monomorphism) 1 U ⊆ M may alternatively be described by its characteristic function χU : M → {0, 1}; anticipating the convention in topos theory, we relabel {0, 1} as and regard this as an object in Sets. Composition of 1 1
ρ,a∈
/
ρ
/ M and M =
1
ρ
χa −1 ()
/M
/ then yields an arrow
χa −1 ()
/ ,
(1)
i.e. we have defined ρ, a ∈ = χa −1 () ◦ ρ.
(2)
1 ‘These “probability clouds”, replacing the reassuring material particles of before, remind me strangely of the elusive “open neighborhoods” that populate the topoi, like evanescent phantoms, to surround the imaginary “points”.’
Topos for Algebraic Quantum Theory
65
The image of 1 under this map is a point of , which is precisely the above truth value of the proposition a ∈ in the state ρ (provided we identify {0, 1} with {false, true}). It is important for what follows to reformulate this description in terms of the topology O(M) of M (i.e. its collection of opens) instead of M itself. This makes sense if the subsets U ⊆ M above are open, which in our second description is the case if the observables a are continuous and the value sets ⊆ R are open as well. Hence a −1 () −1
a () / O(M) in Sets, but ρ ∈ M is now represented by the ‘state is an arrow 1 subobject’ [ρ] ⊆ O(M) given by
[ρ] = {V ∈ O(M) | ρ ∈ V } = {V ∈ O(M) | δρ (V ) = 1},
(3)
where δρ is the Dirac measure on M concentrated at ρ. We describe this object by its characteristic function χ[ρ] : O(M) → . The pairing map then becomes 1
a∈,ρ
/
=
1
a −1 ()
/ O(M)
χ[ρ]
/ ,
(4)
or, in other words, a ∈ , ρ = χ[ρ] ◦ a −1 ().
(5)
The reader may verify that a ∈ , ρ = ρ, a ∈ , so that our second categorical description of the state-proposition pairing is equivalent to the first. More generally, if µ is a probability measure on M, we might define a state object [µ] by replacing the Dirac measure δρ in (3) by µ, i.e. [µ] = {V ∈ O(M) | µ(V ) = 1}.
(6)
In physics, µ plays the role of a mixed state (unless it is a point measure, in which case it happens to be pure). Like the pure state ρ (or rather its associated probability measure δρ ), the mixed state µ defines a characteristic function χ[µ] : O(M) → . The latter, however, turns out not to share the attractive logical properties of χ[ρ] ≡ χδρ (unless µ is pure); see Subsect. 1.5. 1.2. Spatial quantum logic. The goal of this paper is to generalize this situation to quantum mechanics. In particular, we wish to find a spatial notion of quantum logic. This objective will be accomplished by: 1. Identifying an appropriate notion of a quantum phase ‘space’ . 2. Defining suitable ‘subsets’ of that act as elementary logical propositions of quantum mechanics. 3. Describing observables and states in terms of . 4. Associating a proposition a ∈ (and hence a ‘subset’ [a ∈ ] of ) to an observable a and an open subset ⊆ R. 5. Finding a pairing map between pure states and ‘subsets’ of (and hence between states and propositions of the type a ∈ ). In the last step, a state assigns a particular truth value to a given proposition; this is supposed to give empirical content to the formalism. The codomain of the pairing map in item 5, which may be called the ‘truth object’ of the theory, is by no means obvious and identifying it is explicitly part of the question. Certainly, 20th century physics shows that the ‘classical’ choice = {0, 1} is out of the question. The formulation of these objectives and the associated program goes back to von Neumann, who also famously proposed the following extremely elegant solution:
66
1. 2. 3. 4.
C. Heunen, N. P. Landsman, B. Spitters
A quantum phase space is a Hilbert space H . Elementary propositions correspond to closed linear subspaces of H . Observables are selfadjoint operators on H and pure states are unit vectors in H . The closed linear subspace [a ∈ ] is the image E()H of the spectral projection E() defined by a and (provided the latter is measurable). 5. The pairing map takes values in [0, 1] and is given by the “Born rule” , a ∈ = (, E()). Thus subsets of phase space became closed linear subspaces of Hilbert space, which, as Birkhoff and von Neumann [10] noticed, form a lattice L(H ) under inclusion as partial order. However, this lattice fails to be Boolean, basically because it is nondistributive. Nonetheless, Birkhoff and von Neumann interpreted the lattice operations ∧ and ∨ as ‘and’ and ‘or’, as in the classical case, and argued that the departure from the Boolean structure (and hence from classical logic) meant that one had to deal with a new kind of logic, which they aptly called quantum logic. This looked highly innovative, but on the other hand it conservatively preserved the spatial nature of the logic of classical physics, in that the logical structure of propositions is still associated with the spatial (i.e. Hilbert space) structure of the theory. Attractive and revolutionary as this spatial quantum ‘logic’ may appear [54,55,70, 76], it faces severe problems. The main logical drawbacks are: • Due to its lack of distributivity, quantum ‘logic’ is difficult to interpret as a logical structure. • In particular, despite various proposals no satisfactory implication operator has been found (so that there is no deductive system in quantum logic). • Quantum ‘logic’ is a propositional language; no satisfactory generalization to predicate logic has been found. Quantum logic is also problematic from a physical perspective. Since (by various theorems [14] and wide agreement) quantum probabilities do not admit an ignorance interpretation, [0, 1]-valued truth values attributed to propositions by pure states via the Born rule cannot be regarded as sharp (i.e. {0, 1}-valued) truth values muddled by human ignorance. This implies that, if x = [a ∈ ] represents a quantum-mechanical proposition, it is wrong to say that either x or its negation holds, but we just do not know which of these alternatives applies. However, in quantum logic one has the law of the excluded middle in the form x ∨ x ⊥ = 1 for all x. Thus the formalism of quantum logic does not match the probabilistic structure of quantum theory responsible for its empirical content. In fact, the above argument suggests that it is intuitionistic logic rather than quantum logic that is relevant in quantum mechanics (cf. [4]). More generally, as argued in particular by Butterfield and Isham [17,20], the fact that pure states fail to define truth assignments in the usual binary sense (i.e. true or false) renders the entire notion of truth in quantum mechanics obscure and calls for a complete reanalysis thereof [34–38]. As also probably first recognized by the same authors, such an analysis can fruitfully be attempted using topos theory, whose internal logic is indeed intuitionistic. From our perspective, another reason why topos theory offers itself on a silver tray in our search for a spatial quantum logic lies in the interplay between spatial and logical structures inherent in topos theory, as exemplified by the opening words of the renowned textbook by Mac Lane and Moerdijk: A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry and on the other hand, logic and set theory.
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We refer to [42,50,51,63] for accounts of topos theory; see also [9,57,65] for historical details. Briefly, a topos is a category in which one can essentially reason as in the category Sets of all sets (with functions as arrows), except for the fact that the logic is intuitionistic and the axiom of choice is generally not available. Briefly, the mathematics underlying topos theory is constructive. Specifically, a topos is a category with the following ingredients: 1. Terminal object. This is an object called 1 (unique up to isomorphism) such that for each object A there is a unique arrow A → 1, generalizing the singleton set in the category Sets. 2. Pullbacks. These generalize the fibered product B × A C = {(b, c) ∈ B ×C | f (b) = f / A and C g / A in Sets into a pullback square with appropriate g(c)} of B universality property. Cartesian products are a special case. 3. Exponentials. These generalize the idea that the class B A of functions from a set A to a set B is itself a set, and hence an object in Sets, equipped with the evaluation map ev : A × B A → B. 4. Subobject classifier. This generalizes the idea that one may characterize a subset A ⊆ B by its characteristic function χ A : B → {0, 1}. Subsets generalize to subobjects, i.e. monic (“injective”) arrows A B, and in a topos there exists / an object (the subobject classifier) with associated arrow 1 (“truth”)
such that for any subobject A B there is a unique arrow B Bo
f
/ 1 is a pullback of B
A χ
χA
/ and 1
χA
/ for which
/ . Conversely, given any
/ there exists a subobject A B of B (unique up to isomorphism) arrow B whose classifying arrow χ B equals χ . The subobject classifier in a topos plays the role of a “multi-valued truth object”, generalizing the simple situation in Sets, where = {0, 1} = {false, true}; see (2) and subsequent text. We assume that our topoi are cocomplete and have a natural numbers object. 1.3. Generalized notions of space. Our first objective in the list at the beginning of Subsect. 1.2, i.e. the identification of an appropriate notion of a quantum phase ‘space’, will be met by a combination of two profound notions of generalized space that have been around for some time. 1. First, let us recall the strategy of noncommutative geometry [25,26]. One starts with the replacement of a compact topological space X by the associated algebra of complex-valued continuous functions C(X, C). If X fails to be Hausdorff, this step loses information, but if it is, one may recover X from the commutative C*-algebra C(X, C) as its Gelfand spectrum. This yields a duality between the category of compact Hausdorff spaces and the category of unital commutative C*-algebras: nothing is lost, but nothing is gained either by abstracting spaces as commutative C*-algebras. The thrust of noncommutative geometry, then, is to allow C*-algebras to be noncommutative without losing the spatial perspective. That this can be done is impressive enough, but as the logical situation is obscured by moving from commutative to noncommutative C*-algebras, further ideas are needed (at least if one is interested in quantum logic). 2. A second approach to generalizing topological spaces would be to replace X by its topology O(X ). This has a natural lattice structure under inclusion, and in fact
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defines a highly structured kindof latticeknown as a frame. This is a complete distributive lattice such that x ∧ λ yλ = λ x ∧ yλ for arbitrary families {yλ } (and not just for finite ones, in which case the said property follows from the definition of a distributive lattice). For example, if X is a topological space, then the topology O(X ) of X is a frame with U V if U ⊆ V . A frame homomorphism preserves finite meets and arbitrary joins; this leads to the category Frm of frames and frame homomorphisms. Abstracting frames O(X ) coming from a topological space to general frames is a genuine generalization of the concept of a space, as plenty of frames exist that are not of the form O(X ). A simple example is the frame Oreg (R) of regular open subsets of R, i.e. of open subsets U with the property ¬¬U = U , where ¬U is the interior of the complement of U . This may be contrasted with the situation for unital commutative C*-algebras, which, as just recalled, are all of the form C(X ). Moreover, far from obscuring the logical structure of space, the generalization of spaces by frames rather explains and deepens this structure. Indeed, a frame is a complete Heyting algebra, with its intrinsic structure of an intuitionistic propositional logic. Here a Heyting algebra is a distributive lattice L with a map →: L × L → L satisfying x (y → z) iff x ∧ y z, called implication [42,63,80]. Every Boolean algebra is a Heyting algebra, but not vice versa; in fact, a Heyting algebra is Boolean iff ¬¬x = x for all x, which is the case iff ¬x ∨ x = for all x. Here negation is a derived notion, defined by ¬x = (x →⊥). For example, Oreg (R) is Boolean, but O(R) is not. In general, the elements of a Heyting algebra form an intuitionistic propositional logic under the usual logical interpretation of the lattice operations. A Heyting algebra is complete when arbitrary joins (i.e. sups) and meets (i.e. infs) exist. A complete Heyting algebra is essentially the same thing as a frame, for in a frame one may define y → z = {x | x ∧ y z}. Conversely, the infinite distributivity law in a frame is automatically satisfied in a Heyting algebra. The set of subobjects of a given object in a topos forms a complete Heyting algebra (as long as the topos in question is defined “internal to Sets”), generalizing the fact that the set of subsets of a given set is a Boolean algebra. The subobject classifier of such a topos is a complete Heyting algebra as well; in fact, these two statements are equivalent. (Note, however, frame maps do not necessarily preserve the implication → defining the Heyting algebra structure, as can already be seen in examples of the type f −1 : O(Y ) → O(X ), where f : X → Y is continuous [63]. Consequently, negation may not be preserved by frame maps either.) The category Loc of locales is the opposite category to Frm, i.e. it has the same objects but all arrows go in the opposite direction. Some topos theorists write X for a locale and O X or O(X ) for the same object seen as a frame [48,63,80]. Apart from the already unfortunate fact that this notation is applied also when O(X ) does not stand for the opens of a space X but denotes a general frame, it fails to distinguish between a topological space X and the associated locale (i.e. the frame O(X ) seen as a locale). Nonetheless, this notation often leads to elegant expressions and we will heavily use it. If X and Y are spaces, a continuous map f : X → Y induces a frame map f −1 : O(Y ) → O(X ) and hence an arrow O(X ) → O(Y ) in Loc, simply defined as f −1 read in the opposite direction. We write the latter arrow in Loc simply as f : X → Y . In general, an arrow in Frm is written as f −1 : O(Y ) → O(X ) (whether or not the frames in question come from topological spaces and if so, whether or not f −1 is indeed the pullback of a continuous function between these spaces), and the corresponding arrow in Loc is denoted by f : X → Y . Similarly, we will write C(X, Y ) for
Topos for Algebraic Quantum Theory
69
HomLoc (X, Y ) = HomFrm (O(Y ), O(X )). In particular, for a locale X , C(X, C) will denote the set of frame maps O(C) → O(X ). 1.4. Points and opens of locales. An element of a set X (and hence a fortiori also a point of a topological space X ) may be identified with an arrow ∗ → X , where ∗ is a given singleton (for simplicity we write ∗ instead of the more usual {∗}). The same goes for locales X , so that by definition a point of a locale X is a locale map p : ∗ → X , hence a frame map p −1 : O(X ) → O(∗) ∼ = {0, 1} = ; recall that the subobject classifier in Sets, seen as a topos, is = {0, 1} and note that ∗ is precisely the locale associated to , as our notation = O(∗) has indicated. A point of a locale X being defined as a locale map ∗ → X or as the corresponding frame map O(X ) → , an open of X is defined as a locale map X → S, where S is the locale defined by the so-called Sierpinski space, i.e. {0, 1} with {1} as the only open point. The corresponding frame map O(S) → O(X ) is determined by its value at 1 (since ∅ → ∅ and {0, 1} → X ), so that we may simply look at opens in X as arrows 1 → O(X ) (where the singleton 1 is seen as the terminal object in Sets). Clearly, if X is a genuine topological space with associated frame O(X ) of opens, then each such map 1 → O(X ) corresponds to an open subset of X in the usual sense. Using this concept, the set Pt(X ) of points of a locale X may be topologized in a natural way, by declaring its opens to be the sets of the form Pt(U ) = { p ∈ Pt(X ) | p −1 (U ) = 1},
(7)
where U ∈ O(X ) is some open. We say that a locale X is spatial if it is isomorphic (in the category of locales) to Pt(X ) (more precisely, to the locale associated to the frame O(Pt(X )) in the above topology). Conversely, a topological space X is called sober if it is homeomorphic to Pt(X ) (which, with the notation introduced above, really stands for the space of points of the locale associated to the frame O(X )). It is useful to know that X is sober when it is Hausdorff. If X is sober, any frame map ϕ : O(Y ) → O(X ) is induced by a continuous map f : X → Y as ϕ = f −1 . This provides additional justification for the notation f −1 : O(Y ) → O(X ) for a general frame map, and f : X → Y for the associated locale map. See [63, Sect. IX.3] for a very clear exposition of all this. For example, referring to Subsect. 1.1, the characteristic function χ[ρ] : O(M) → introduced below (3) is easily checked to define a frame map. Renaming this map as χ[ρ] ≡ ρ −1 , the associated locale map ρ : ∗ → M is therefore a point of the locale M in the above sense. In this special case, such a point may also be described by an arrow 1 → M, where 1 is the terminal object in Sets and M denotes M as a set rather than as a locale. This notion of points as elements of sets will be avoided in what follows. Thus frames and locales are two sides of the same coin: the elements 1 → O(X ) of the Heyting algebra O(X ) are the opens of the associated locale X , to be thought of as propositions, whereas the points of the locale correspond to models of the logical theory defined by these propositions. See [51,63] and especially [80] for a very clear explanation of this perspective. More precisely, recall that geometric propositional logic stands for the following fragment of intuitionistic propositional logic [51,63,80]. A formula ϕ in propositional geometric logic must be built from atomic propositions using the symbols (for “truth”), ∧ (for “and”), and ∨ (for “or”), where ∨ but not ∧ is allowed to carry an infinite index set. (This may be motivated by the remark that to verify a proposition ∨λ∈ pλ , one only needs to find a single pλ , whereas to verify ∧λ∈ pλ the truth of each pλ needs to be established, an impossible task in practice when is infinite.) Sequents or axioms must take the form ϕ → ψ, where ϕ and ψ are formulae.
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A frame O(X ), then, defines a geometric propositional theory whose propositions correspond to opens in X , combined by logical connectives given by the lattice operations in O(X ) à la Boole. This quite literally holds in the case of classical physics discussed in Subsect. 1.1, where the opens of the locale M are just the opens U of M as a topological space in the naive sense, construed as propositions “the system is in a state located within U ”. Conversely, a propositional geometric theory T has an associated Lindenbaum algebra O([T]), defined as the poset of formulae of T modulo provable equivalence, ordered by entailment. This poset turns out to be a frame, and the (standard) models of T (that by definition assign one of the two standard truth values 0 or 1 to the propositions of T in a consistent way) bijectively correspond to frame maps O([T]) → {0, 1}. Identifying {0, 1} with = O(∗) as explained above, we see that a model of the theory T is the same thing as a point ∗ → [T] of the locale [T]. More generally, one may consider a model of T in a frame O(Y ) (generalizing the standard models where Y = ∗) to be a locale map Y → [T].
1.5. Locales in topoi. The generalization from topological spaces to frames is an important step towards our goal, but it is not enough. Seeking further generality pertinent to quantum theory, one may proceed in at least two different ways. First, one may generalize locales to quantales [67]. This step leads to recognizable logical structures, but it does not relate well to the Copenhagen Interpretation of quantum mechanics we favour. Instead, we pass from frames as special objects in the category of sets (as defined above) to frames in more general topoi. This is indeed possible, as all of the above concepts can be defined in any topos by using its internal language [63]; see [12] for details. In particular, in a topos T one may consider the category FrmT of internal frames and its opposite category LocT of internal locales. The terminal object of the latter is the locale ∗ whose associated frame O(∗) is the subobject classifier of T . Opens, points and models are then defined in exactly the same way as in Sets, as long as one realizes that the identification of with {0, 1} and of ∗ with the singleton is peculiar to Sets. In particular, a point of a locale X in T is a frame map O(X ) → , whereas an open in X may be defined as an arrow 1 → O(X ). The collection Pt(X ) of a locale is still defined as the subobject of O(X ) corresponding to frame maps, its opens being given by interpreting (7) in the internal language of T , where U ∈ O(X ) is interpreted U
as an arrow 1 → O(X ) and p −1 (U ) = 1 means that p −1 ◦ U = , i.e. the truth arrow : 1 → in T . In any case, it is reassuring that topos theorists simply refer to ‘internal’ locales as ‘spaces’ [52,53,66]: returning to the opening words from Mac Lane and Moerdijk quoted earlier, one might say that the unification in question is exemplified by the idea of an internal locale with its associated Heyting algebra structure. Our quantum phase spaces , then, will be examples of locales in topoi. Their opens 1 → O() will correspond to the elementary propositions or yes-no questions about the system, and each physical state on the system will define a map O() → , where is the subobject classifier in the particular topos in which is defined. It is important to note that such maps generally fail to be frame maps, i.e. they do not define models in the above sense. This phenomenon already arises in classical physics if one considers mixed rather than pure states; indeed, the map χ[µ] : O(M) → introduced below 6 fails to be a frame map (except when µ happens to be pure). However, a fundamental difference between classical and quantum physics in this respect lies in the Kochen–Specker Theorem, which in its topos-theoretic incarnation
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(given in different versions in [20] and in Theorem 6 below) states that (generically) the quantum phase space has no points at all, although the quantum system has pure states (see Subsect. 4.1). Hence whereas pure states in classical physics - as defined in the usual sense through convexity - are also ‘pure’ in the logical sense, this is no longer the case in quantum physics. Nonetheless, pairing states and propositions into an internal truth value, i.e. taking the subobject classifier to be the codomain of the pairing map, is a central goal of this work, which we share with (and adopted from) the work of Isham et al. [20,34]. Unlike real-number valued pairings (which from a logical perspective might be preferable), an -valued pairing avoids both the problems with the ignorance interpretation of the Born probabilities (see Subsect. 1.2) and the bizarre ontology of the so-called Many-Worlds interpretation of quantum mechanics (cf. [14,17]). A philosophical defence of this goal may also be found in [18]. However, the final verdict about its validity, or rather its relevance to physics, can only be given once the Born rule has been derived from our -valued pairing, along with an appropriate interpretation of the Born probabilities. This derivation will be given in future work, in which the results of Sect. 6 of this paper will be combined with those in [61]. 1.6. Basic construction. The two notions of generalized space just described, i.e. noncommutative C*-algebra s and locales in arbitrary topoi, will be related by one of the main constructions in this paper, which we summarize in this subsection. This construction associates a certain internal locale to a noncommutative C*-algebra (assumed unital), and hinges on three ideas: 1. Algebraic quantum theory [39,44,58]; 2. Constructive Gelfand duality [5–7,27,29]; 3. Bohr’s doctrine of classical concepts [11,60,73]. From the first, we just adopt the methodology of describing a quantum system by a noncommutative C*-algebra A (defined in the usual topos Sets). This move generalizes the usual Hilbert space framework of quantum theory and has the advantage of being able to incorporate superselection rules in infinite systems, as well as other limiting situations like the transition from quantum to classical mechanics (and back). As to the second, it turns out that the notion of a C*-algebra makes sense in an arbitrary topos, so that one may, in particular, internalize commutative C*-algebras. Examples of such internal commutative C*-algebras arise from compact completely regular locales ([7,48], see also footnotes 2 and 3 below): if X is such a locale in some topos T , and if C is the locale defined by the complex numbers object in T (as in [7]), then the object C(X, C) of all locale maps from X to C is a commutative C*-algebra in T under natural operations. The Gelfand duality theorem of Banaschewski and Mulvey [7] states that, like in the case of the topos Sets, up to isomorphism these are the only examples of unital commutative C*-algebras: if A is a unital commutative C*-algebra in a topos T , there exists a compact completely regular locale such that A ∼ = C(, C). Here ∼ = denotes isomorphism in the category T and the arrows implementing this isomorphism are C*-algebra maps. Moreover, this isomorphism extends to a categorical duality between compact completely regular locales and unital commutative C*-algebras in T . We call the locale or (A) the Gelfand spectrum of A. It is defined up to isomorphism of locales. Third, Niels Bohr’s “doctrine of classical concepts” states that we can only look at the quantum world through classical glasses, measurement merely providing a “classical
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snapshot of reality”. The combination of all such snapshots should then provide a complete picture. In Bohr’s own words ([11], p. 209): However far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. (…) The argument is simply that by the word experiment we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics. This doctrine has a transparent formulation in algebraic quantum theory, to the effect that the empirical content of a quantum theory described by a certain noncommutative C*-algebra A is contained in suitable commutative C*-algebras associated to A. In the simplest case, which we study in this paper, these are simply the (unital) commutative C ∗ -subalgebras of A. (To understand classical behaviour in general, the pertinent commutative C*-algebras have to be extracted from A using limiting procedures like → 0 or N → ∞ [60].) The following construction weaves these three threads together. Let A be a unital C*-algebra (in the usual sense, i.e. in Sets) and let C(A) be the collection of its unital commutative C ∗ -subalgebras, partially ordered by inclusion. We regard the poset C(A) as a category, whose objects are the unital C ∗ -subalgebras C ⊆ A, and whose Hom-sets HomC (A) (C, D) consist of a single arrow if C ⊆ D and are empty otherwise. The category C(A) is a catalogue of all ‘classical snapshots of reality’ one may take of the quantum system described by A. Recall that for any category C, the topos SetsC has functors C → Sets as objects and natural transformations as arrows [63]. Put T (A) = SetsC (A).
(8)
The philosophical idea is that as observers we are confined to the topos T (A), whereas the physical system itself divinely exists in the ambient topos Sets. According to Bohr and Heisenberg, the system might seem to behave probabilistically from our limited classical perspective, but this behaviour is just a consequence of our confinement to T (A) (cf. Theorem 14 below). We will underline entities internal to T (A). It turns out that the tautological functor A : C → C, which (with some abuse of notation) maps a unital commutative C ∗ -subalgebra C of A (seen as an object of the category C(A)) into itself (seen as a set), is a unital commutative C*-algebra in T (A). We call A the Bohrification of A. It has an associated Gelfand spectrum (A), which is a locale in T (A). The map A → (A) associates a ‘space’ (A) in the sense of topos theory to a ‘space’ A in the sense of noncommutative geometry. In principle, this construction leads to the solution of all five problems listed at the beginning of Subsect. 1.2: 1. The quantum phase space of the system described by A is the locale ≡ (A) in the topos T (A). 2. The “subsets” of the locale acting as elementary propositions about A are simply the ‘opens’ in , defined as arrows 1 → O() in T (A). Thus the quantum logic of A is given by the Heyting algebra underlying (A). 3. Observables a ∈ A define locale maps δ(a) : → IR, where IR is the so-called interval domain. States ρ on A yield probability measures (more precisely, valuations) µρ on .
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4. An open interval ⊆ R defines an arrow 1
/ O(IR) of T (A) (where 1 is
the terminal object in T (A)), which, composed with the map O(IR) underlying δ(a), yields the desired proposition 1
[a∈]
/ O()
=
1
δ(a)−1
/ O()
−1
/ O(IR) δ(a) / O().
5. State-proposition pairing is defined exactly as in (4), i.e. by 1
a∈,ρ
/
=
1
[a∈]
/ O()
χ[ρ]
/ ,
(9)
where is the subobject classifier of T (A) and χ[ρ] is the characteristic map of the subobject [ρ] of O() consisting of all opens U of with µρ (U ) = 1 (defined through the internal language of T (A)). The construction of δ(a) is inspired by, and partly generalizes, the Daseinisation map of Döring and Isham ([35,36], cf. also Appendix B). The subobject classifier is the functor C(A) → Sets given by (C) = {S ⊆↑ C | S is an upper set},
(10)
where for any poset P an upper set in P is a subset U ⊆ P for which x ∈ U and x y implies y ∈ U , and one writes ↑ x = {y ∈ P | x y} for the so-called principal upper set on x. Note that (C) is a poset (and even a frame) under inclusion as partial ordering, with ∅ as bottom element, and ↑ C as top element. (One might think of the principal upper set ↑ C on the “classical snapshot of reality” C as the collection of all finer versions of the knowledge present in C.) The subobject classifier is a (covariant) functor by stipulating that if C ⊆ D, then the induced map (C) → (D) is given by S → S ∩ ↑ D. In this setup, we have taken Sets as the ambient topos. There are several reasons, however, one might want to consider other ambient topoi. Leaving the matter to future investigation, let us briefly indicate an important application. An algebraic quantum field theory (AQFT) [44] may be defined as a functor (O(M), ⊆) → CStar satisfying certain separability constraints, where M is Minkowski space-time and O(M) is its set of opens [13]. Analogous to Theorem 5 below, an AQFT may then be shown to be given by a single C*-algebra in the presheaf topos SetsO(M) (though not a commutative one). 1.7. Internal and external language. We have repeatedly used the word ‘internal’ for a construction intrinsic to a certain topos T ; for example, A and its Gelfand spectrum are internal to T (A), as is the Heyting algebra structure of . At this point, confusion may arise, for on the one hand the propositional logic carried by is intuitionistic, while on the other hand all constructions (ranging from the initial C*-algebra A to the locale (A) as an object in the associated functor topos SetsC (A) ) eventually arise from the topos Sets, whose underlying logic is classical. To clarify this, we remark that it is a very important aspect of topos theory that one may, indeed, usually adopt two points of view: an external one and an internal one. External constructions are carried out using classical mathematics, which (at least for the topoi used in this paper) takes place in the familiar topos Sets (even if the constructions in question are concerned with some other topos). Internal constructions, on the other
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Fig. 1. Illustration of universes of discourse
hand, only use concepts intrinsic to the topos one is studying. This idea is formalized by the internal or Mitchell-Bénabou language associated with each topos [12,51,63]. This is a logical language that for many instances and purposes allows one to reason within a given topos as if it were the topos Sets. For example, one may employ the usual logical and set-theoretic symbols (the latter even if an object X has no or few elements in the sense of arrows 1 → X ), whose meaning is determined by the so-called Kripke-Joyal semantics associated with the Mitchell-Bénabou language. We will actually use this semantics in our theory of state-proposition pairing. However, using the internal language one may (in general) neither appeal to the law of excluded middle x ∨ ¬x = , nor to the Axiom of Choice (although restricted versions thereof are sometimes valid). These limitations are a consequence of the fact that the internal language of a topos happens to be based on intuitionistic predicate logic (see [12,51] for the precise rules). Thus a topos can be seen as a universe of discourse, to which a mathematician or observer may wish to confine himself. On the other hand, even the internal language and associated logic can alternatively be studied externally with classical meta-logic. The various entities at play in our application of topos theory to quantum physics are illustrated in Fig. 1. This illustrates, in particular, that our quantum logic is meant to be the logic of an ‘internal’ observer, with all the restrictions this brings with it (whereas the quantum ‘logic’ of Birkhoff and von Neumann, to the extent it is a logic at all, rather pertains to a fictitious entity like Laplace’s demon, to whose intellect ‘nothing would be uncertain and the future just like the past would be present before its eyes.’) Let us give three closely related examples of internal versus external descriptions, each relevant to our logical approach to quantum theory. First, a fundamental fact of topos theory is that the subobjects SubT (A) of a given object A in a topos T (with subobject classifier T ) form a (complete) Heyting algebra. • Externally, one simply looks at SubT (A) as a set, equipped with the structure of a Heyting algebra in the category Sets. • Internally, SubT (A) is described as the exponential TA (or power ‘set’ P(A)), which is a Heyting algebra object in T . See [63, p. 201]. Second, as these Heyting algebras are complete, they are frames. The explicit internal description of a frame or locale is rather complicated as far as the completeness property of the underlying lattice is concerned [12]. However, if the topos T = Sh(X ) is that of op sheaves on a locale X (which, we recall, consists of those functors F in SetsO(X ) that satisfy a gluing condition stating that F(U ) can be computed from the F(Ui ) under any open covering U = ∪i Ui [63, Ch. II]), a simple external description is available [49,53]
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(also cf. [51, Sect. C1.6]): a locale L in Sh(X ) is externally described by a locale map f : L → X in Sets, with L = L(X ).
(11)
Furthermore, if L1 and L2 are locales in Sh(X ) with external descriptions f i : Li → X , then an internal locale map g : L1 → L2 in Sh(X ) is externally given by a locale map g : L1 → L2 in Sets such that f 2 ◦ g = f 1 . To see that this situation is relevant to our construction, first recall the Alexandrov topology on a poset P. Its opens are simply the upper sets, and the special upper sets of the form U = ↑x form a basis of the Alexandrov topology. Equipping P with the Alexandrov topology, one has an isomorphism of categories Sets P ∼ = Sh(P).
(12)
To understand this, just note that a sheaf F on P is determined by its values on the basis opens ↑x; a functor F : P → Sets then corresponds to F by F(x) = F(↑x).
(13)
It is, then, immediate from (8) and (12) that T (A) ∼ = Sh(C(A)),
(14)
so that we have the above-mentioned external description of locales to our avail, with X = C(A). Explicitly, to describe an internal locale L in Sh(X ) externally, i.e. in terms of the topos Sets, consider the set O(L) = HomSh(X ) (1, O(L)) of global sections of the associated frame O(L); this set coincides with O(L)(X ) (since a natural transformation in HomSh(X ) (1, O(L)) is determined by its value at X ) and defines a frame O(L)(X ) in Sets under the lattice structure borrowed from O(L). For V ⊆ U , let op LU V : O(L)(U ) → O(L)(V ) be the arrow part of the functor O(L) : O(X ) → Sets, X with special case LV ≡ LV . The completeness of O(L) implies that LV has a left adjoint L∗V : O(L)(V ) → O(L)(X ), which in turn defines a map f ∗ : O(X ) → O(L)(X ) by f ∗ : V → L∗V ( L(V ) ), where L(V ) is the top element of the lattice O(L)(V ). This is a frame map, and if we write O(Y ) = O(L)(X ), the corresponding locale map f : Y → X is the external description of L. Conversely, a locale L in Sets along with a locale map f : L → X (i.e. a frame map f ∗ : O(X ) → O(L)) induces a locale L in Sh(X ), defined as the sheaf L(U ) = {V ∈ L | V f ∗ (U )}. These constructions are adjoint to each other, yielding an equivalence of the category Loc(Sh(X )) of locales in Sh(X ) and the slice category Loc/ X relative to the category Loc of locales in Sets. For us, the external description of locales is useful for two reasons: 1. Returning to (9), we are clearly interested in the set
O() ∼ = HomSetsC(A) (1, O()) of opens in , as it encodes the quantum logic of our C*-algebra A. Let O() be the sheaf on C(A) that corresponds to O() by (13), so that O() ∼ = O(), with O() = HomSh(C (A)) (1, O()). Clearly, if a poset P has a bottom element
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⊥ and Z is any object in Sets P , then Z ∼ = Z (⊥). This applies to P = C(A) with ⊥= C · 1, so that
O() ∼ = O()(C · 1) ∼ = O()(C(A)) = O(),
(15)
where we have used (11). Hence the external description of the quantum logic of the C*-algebra A is entirely given in terms of the locale in Sets. 2. Important internal number systems in T (A) that are defined by geometric propositional theories T (see Subsect. 1.4) may be computed from (13) and their description in Sh(C(A)), which in turn is based on their external description in Sets. Specifically, if [T] is the locale defined by T in Sets, then the locale [T] ≡ [T]Sh(X ) giving the interpretation of T in Sh(X ) has external description π1 : X × [T] → X , where π1 is projection on the first component. It follows that the frame O([T]) in Sh(X ) corresponding to [T] is given by the sheaf U → O(U × [T]). Applying this to the case at hand, we see that the frame O([T]) corresponding to the interpretation [T] ≡ [T]C (A) of T in T (A) is given by the functor O([T]) : C → O(↑ C × [T]).
(16)
See Subsects. 2.3 and 5.1 for examples of this procedure. Our third example applies the second one to points of locales [63], and continues the discussion in Subsect. 1.4: • Internally, a point of a locale Y in a general topos T (internal to Sets for simplicity) is a locale map ∗ → Y , which is the same thing as an internal frame map O(Y ) → (where is the subobject classifier in T ). • Externally, we look at as the frame SubT (1) in Sets of subobjects of the terminal object 1 in T . The locale in Sets with frame SubT (1) is called the localic reflection Loc(T ) of T , i.e. O(Loc(T )) = SubT (1). For example, in T = Sh(X ) one has SubSh(X ) (1) ∼ = O(X ) and hence Loc(Sh(X )) ∼ = X . Applying the second example above, we find that the external description of the locale ∗ in Sh(X ) is just id : X → X , so that points in a locale L in Sh(X ) with external description f : Y → X are given by locale maps ϕ : X → Y that satisfy f ◦ ϕ = id, i.e. cross-sections of f . The fourth example continues both the previous one and the discussion of models in Subsect. 1.4. We initially defined a standard model of a geometric propositional theory T as a locale map ∗ → [T], and subsequently mentioned more general models Y → [T], still in Sets. We now consider even more general models of T in a topos T . • Externally, these are given by locale maps Loc(T ) → [T] in Sets. This is because the classifying topos of T is Sh([T]), and one has an equivalence between geometric morphisms T → Sh([T]) (which classify T-models in T , cf. [63, Thm. X.6.1]) and locale maps Loc(T ) → [T] (see [63, Sect. IX.5]). • Internally, one may interpret the theory T in T and thus define a locale [T]T internal to T . The points of this locale, i.e. the locale maps ∗ → [T]T or frame maps O([T]T ) → T , describe the models of T in T internally. One may explicitly verify the equivalence between the internal and the external descriptions for T = Sh(X ), for in that case the external description of [T]T is the map π1 : X × [T] → X . Hence locale maps ϕ : X → X × [T] that satisfy π1 ◦ ϕ = id are just (unconstrained) locale maps X → [T].
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1.8. Observation and approximation. Our construction of the locale map δ(a) : → IR in Sect. 1.6 involves the so-called interval domain IR [74]. To motivate its definition, consider the approximation of real numbers by nested intervals with endpoints in Q. For example, the real number π can be described by specifying the sequence [3, 4], [3.1, 3.2], [3.14, 3.15], [3.141, 3.142], . . . Each individual interval may be interpreted as finitary information about the real number under scrutiny, involving the single observation that the real number is contained in the interval. This description of the reals, which goes back to L.E.J. Brouwer, is formalized by the notion of the interval domain. Consider the poset IR whose elements are compact intervals [a, b] in R (including singletons [a, a] = {a}), ordered by reverse inclusion (for a smaller interval means that we have more information about the real number that the ever smaller intervals converge to). This poset is a so-called dcpo (directed complete partial order); directed suprema are simply intersections. As such, it carries the Scott topology [2,74], whose closed sets are lower sets that are closed under suprema of directed subsets. Here a lower set in a poset P is a subset L ⊆ P such that x ∈ L and y x implies y ∈ L; equivalently, ↓ L ⊆ L, where ↓ L = {y ∈ P | ∃ x ∈ L : y x}. (Lower sets are sometimes called down sets or downward closed sets.) Consequently, Scott opens must be upper sets U (defined in the obvious way) with the additional property that for every directed set D with D ∈ U the intersection D ∩ U is nonempty. In the case of IR, this means that each open interval ( p, q) in R (with p = −∞ and q = +∞ allowed) corresponds to a Scott open {[a, b] | p < a, b < q} in IR, and these opens form a basis of the Scott topology. The collection OScott (IR) is, of course, a frame, initially defined in Sets. The basis opens (r, s) may be reinterpreted as a collection of generators for this frame, which from the point of view of generators and relations differs from the frame O(R) of Dedekind reals in that the relation ( p, q) = ( p, q1 )∨( p1 , q) for p p1 q1 q holds for the reals, but not for the interval domain (see [51, D4.7.4] or Subsect. 2.3 below for the other relations for O(R)). The interval domain admits an internal definition in any topos. Its realization in T (A) will play an important role in this paper; see Subsect. 5.1. A related notion of approximation appears when considering an observable a ∈ Asa of a quantum system described by a C*-algebra A, as seen from inside its associated topos T (A). Specifically, we should approximate a within each classical snapshot C of A, where C ∈ C(A) is some commutative subalgebra. The difficulty is, of course, that a need not lie in C, but neither is there a single element of C that forms the ‘best approximation’ of a in C. The best one can do is approximate a by a family of elements of C, as follows. The self-adjoint part Asa of a C*-algebra A has a natural partial order , defined by a b iff b − a = c∗ c for some c ∈ A. (Equivalently, a b iff b − a = f 2 for some f ∈ Asa .) This partial order is linear—in the sense that a + c b + c whenever a b. For the C*-algebra A = C(X, C) one just recovers the pointwise order on (real-valued) functions, since Asa = C(X, R). For A = B(H ), the bounded operator on Hilbert space H , one has a ≥ 0 for a ∈ Asa iff (, a) ≥ 0 for all ∈ H . (Indeed, this is really a pointwise order as well, if one regards operators a on H as functions aˆ on H by a() ˆ = (, a). See e.g.[58].) Thus one may approximate a in C by collections of intervals of the type {[ f, g] | f, g ∈ Csa , f a g} (note that this is inhabited, since f = −a · 1 and g = a · 1 occur). Since the intervals [ f, g] in Csa fail to form a dcpo, however, a slight adaptation of this idea is needed, for which we refer to Subsect. 5.1 below.
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1.9. Structure of this article. Section 2 reviews the Gelfand duality theory in a topos due to Banaschewski and Mulvey. Our original results are as follows: 1. The construction of the ‘quantum phase space’ (A) from a C*-algebra A in Sets as the Gelfand spectrum of the Bohrification A of A in the topos T (A) is explained in Sect. 3. This section also contains our version of the Kochen–Specker Theorem. 2. Section 4 first describes the construction of states on A as probability integrals on the self-adjoint part of A. These, in turn, are equivalent to probability valuations on its Gelfand spectrum (A). On this basis, we eventually show that states define subobjects of the quantum phase space (A), as in classical physics (see (3)). 3. The interpretation of observables in A in terms of the Bohrification A is the subject of Sect. 5. In particular, we give our analogue of the Daseinisation map of Döring and Isham in Subsect. 5.2 (and more fully in Appendix A.2). 4. The pairing of states and propositions is elucidated in Sect. 6, yielding an element of the subobject classifier of T (A) that we explicitly compute. This pairing connects the mathematical constructions to quantum physics and completes Steps 1 to 5 of our general program mentioned at the beginning of Subsect. 1.2. Appendix A contains a number of technical results that somewhat distract from the main development of the paper. Finally, Appendix B discusses related work by Döring and Isham, which partly inspired the present article. 2. C*-Algebras and Gelfand Duality in a Topos This section recapitulates a constructive version of Gelfand duality, which is valid in every topos [5–7]. Recall that the usual version of Gelfand duality characterises unital commutative C*-algebras as algebras of complex-valued continuous functions on a compact Hausdorff space. More generally, the category cCStar of unital commutative C*-algebras and unital ∗ -homomorphisms is dual to the category KHausSp of compact Hausdorff spaces and continuous maps (see [48] for a proof aimed at algebraists and [59] for a proof in the spirit of C*-algebras). From a topos-theoretic point of view, this formulation is internal to the topos Sets, since both categories are defined relative to it. To understand the generalization of Gelfand duality to arbitrary topoi, a slight reformulation of the situation in Sets is appropriate: we replace topological spaces X by the associated locales, and hence replace KHausSp by the equivalent category KRegLoc of compact regular locales [48].2 Consequently, the duality cCStar KHausSp may be replaced by cCStar KRegLoc: the contravariant functor cCStar → KRegLoc is still given by A → (A), where (A) is the locale defined by the usual Gelfand spectrum of A (i.e. its pure state space), and in the opposite direction one has the familiar expression X → C(X, C), in which the right-hand side now stands for the locale maps from X to C. For technical reasons, in general topoi regular compact locales have to be replaced by completely regular compact locales,3 but otherwise one has a direct generalization of 2 A locale L is compact if every subset S ⊆ L with S = has a finite subset F with F = . It is
regular if every element of L is the join of the elements well inside itself, where a is well inside b (denoted a b) if there exists c with c ∧ a =⊥ and c ∨ b = . The (internal) categories KRegLoc and KHausSp in a topos T are equivalent when the full axiom of choice is available in T [48]. 3 See [48 or 7] for the definition of complete regularity. If the axiom of dependent choice (stating that for any nonempty set X and any relation R ⊆ X × X such that for all x there is an y with (x, y) ∈ R, there is a sequence (xn ) such that (xn , xn+1 ) ∈ R for each n ∈ N) is valid in a topos, then compact regular locales are automatically completely regular. This is the case in Sets, for example (where, of course, the full axiom of choice holds), and also in topoi like T (A) consisting of functors whose codomain validates dependent choice [41].
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the above reformulation of Gelfand duality in Sets. The following theorem is predicated on an internal definition of the category cCStar, which we shall give in Subsect. 2.1. Here and in what follows, all mathematical symbols are to be interpreted in the internal language of the topos T at hand. Theorem 1 (Gelfand duality in a topos) [5–7]. In any topos T , there is a categorical duality (i.e. contravariant equivalence) cCStar o
/
KRegLoc,
C(−,CT )
where the categories in question are defined internally to T . For A ∈ cCStar, the locale (A) is called the Gelfand spectrum of A. Here the symbol CT stands for the locale of Dedekind complex numbers in T . 2.1. C*-algebras in a topos. In any topos T (with natural numbers object), the rationals Q can be interpreted [63, Sect. VI.8], as can the Gaussian integers CQ = { p +qi : p, q ∈ Q}. For example, the interpretation of CQ in a functor topos SetsC (where, in our case, C is a poset) is the constant functor that assigns the set CQ to every C ∈ C. A *-algebra in T is a vector space A over CQ that carries an associative bilinear map · : A × A → A, and is furthermore equipped with a map (−)∗ : A → A satisfying (a + b)∗ = a ∗ + b∗ , (z · a)∗ = z · a ∗ , (a · b)∗ = b∗ · a ∗ , a ∗∗ = a, for all a, b ∈ A and z ∈ CQ . A is called commutative if a · b = b · a for all a, b ∈ A, and unital if there is a neutral element 1 for the multiplication. To define an internal C*-algebra, we define a seminorm on such an algebra; in general, a norm may not actually be definable in the internal language of a topos. This is a relation N ⊆ A × Q+ , which in Sets would have the meaning that (a, q) ∈ N iff a < q. In general, N must satisfy
(a, p) ∈ N (a, q) ∈ N (a, p) ∈ N ∧ (b, q) ∈ N (a, p) ∈ N ∧ (b, q) ∈ N (a, p) ∈ N for all a, b ∈ A, p, q ∈ Q+ , and z
→ ↔ → → → ∈
(0, p) ∈ N , ∃q∈Q+ [(a, q) ∈ N ], (a ∗ , p) ∈ N , ∃ p
1).
If the seminorm relation furthermore satisfies (a ∗ · a, q 2 ) ∈ N ↔ (a, q) ∈ N for all a ∈ A and q ∈ Q+ , then A is said to be a pre-semi-C*-algebra.
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To proceed to a C*-algebra, one requires a = 0 whenever (a, q) ∈ N for all q in Q+ , making the seminorm into a norm, and subsequently one requires this normed space to be complete in a suitable sense (see [7] for details). As a consequence of its completeness, a C*-algebra is automatically an algebra over a suitable completion of CQ (and not just over CQ itself, as baked into the definition). Note that in general topoi one has to distinguish certain real and complex number objects that coincide in Sets. From Q, one may construct the locale Rd ≡ R of Dedekind real numbers [63, Sect. VI.8] (see also Subsect. 2.3 below); we will drop the suffix d for simplicity. The object Pt(C) (which is the completion of CQ meant above) comprises the points of the complexified locale C = R + iR; see also [7] for a direct description that avoids R. In Sets, C is the locale with frame O(C), where (abusing notation) C are the usual complex numbers. In any topos, the one-dimensional C*-algebra C(∗, C) is nothing but Pt(C) and has Gelfand spectrum ∗ (i.e. the locale with frame ). A unital *-homomorphism between C*-algebras A and B is, as usual, a linear map f : A → B satisfying f (ab) = f (a) f (b), f (a ∗ ) = f (a)∗ and f (1 A ) = 1 B . Unital C*-algebras with unital *-homomorphisms form a category CStar (internal to T ); commutative unital C*-algebras form a full subcategory cCStar thereof.
2.2. Spectrum. The definition of the category KRegLoc of completely regular compact locales can be internalized without difficulty. The next step is to explicitly describe the Gelfand spectrum (A) ≡ of a given commutative C*-algebra A. We will do so following the reformulation in [27,29] of the pioneering work of Banaschewski and Mulvey [7]. To motivate the description, note that even in Sets the spectrum is now described (with the usual notational ambiguity explained in Subsect. 1.3) as the locale defined by the frame O() of open subsets of the usual Gelfand spectrum of A (defined as the subset of the dual A∗ consisting of space of nonzero multiplicative functionals on A in the relative weak∗ topology). The topology on the space can be described by giving a sub-base, for which one often takes U(a,ρ0 ,ε) = {ρ | |ρ(a) − ρ0 (a)| < ε} for a ∈ A, ρ0 ∈ , ε > 0. However, a much simpler choice of sub-base would be Da = {ρ ∈ | ρ(a) > 0},
(17)
where a ∈ Asa . Both the property that the ρ are multiplicative and the fact that the Da form a sub-base of the Gelfand topology may then be expressed lattice-theoretically by saying that O() is the frame FAsa freely generated by the formal symbols Da , a ∈ Asa , subject to the relations D1 = , Da ∧ D−a = ⊥, D−b2 = ⊥, Da+b Da ∨ Db , Dab = (Da ∧ Db ) ∨ (D−a ∧ D−b ),
(18) (19) (20) (21) (22)
supplemented with the ‘regularity rule’ Da
r ∈Q+
Da−r .
(23)
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This turns out to be a correct description of the spectrum of A also in an arbitrary topos T , in which case (18)–(23) have to be interpreted in T , of course.4 2.3. Gelfand transform. Classically, for a commutative unital C*-algebra A the Gelf∼ =
and transform A → C(, C) is given by a → aˆ with a(ρ) ˆ = ρ(a). In our setting it is convenient to restrict the Gelfand transform to Asa , yielding an isomorphism Asa ∼ = C(, R).
(24)
In a topos T , the Gelfand transform of an internal commutative unital C*-algebra A in T associates a locale map aˆ : → RT ,
(25)
to each a ∈ Asa , where is the spectrum of A and RT is the locale of internal Dedekind real numbers in T ; see below. Recalling from Subsect. 1.3 that aˆ is by definition a frame map aˆ −1 : O(RT ) → O(),
(26)
→Z and using the “λ-conversion rule” YY×X →Z [63, Sect. I.6], we note that the Gelfand transform may alternatively be regarded as a map X
ˆ· : Asa × O(RT ) → O().
(27)
Thus the use of the symbol a ∈ Asa in the internal language of T may be avoided in principle. In practice, however, we will often use the notation (25) or (26), and hence the formal symbols Da . For example, in the description (18)–(23) of the spectrum in terms of generators and relations, it is sufficient to define the frame map (26) on basic opens (−∞, r ) and (s, ∞) in RT . In the classical case (i.e. in Sets) discussed above, one has aˆ −1 (0, ∞) = Da from (17), and this remains true in general if aˆ −1 has the meaning (26). Using (18)–(21), one then finds aˆ −1 : (−∞, s) → Ds−a ; (r, ∞) → Da−r .
(28) (29)
As aˆ −1 is a frame map, for bounded open intervals (r, s) we therefore obtain5 aˆ −1 : (r, s) → Ds−a ∧ Da−r .
(30)
We now recall an explicit construction of the Dedekind reals [40,51, D4.7.4 & D4.7.5]. Define the propositional geometric theory TR generated by formal symbols ( p, q) ∈ Q × Q with p < q, ordered as ( p, q) ( p , q ) iff p p and q q , subject to the following axioms (or relations): 4 See [77] and the Appendix to this paper for the procedure of constructing a frame from generators and relations. Equivalently, in the spirit of [7] one could rephrase the above definition by saying that is the locale [T] corresponding to the propositional geometric theory T (in the sense explained in Subsect. 1.4) determined by the collection of propositions Da , a ∈ Asa , subject to the axioms (18)–(23), with replaced by . 5 Banaschewski and Mulvey [7] work with such intervals (r, s) as basic opens, in terms of which they write the Gelfand transform as aˆ −1 : (r, s) → a ∈ (r, s). Here the role of generators of the locale is played by elementary propositions of the logical theory generating as its Lindenbaum algebra, our generator Da corresponding to their proposition a ∈ (0, ∞). Classically, the proposition a ∈ (r, s) may be identified with the open a −1 (r, s) in the spectrum ; cf. Subsect. 1.1.
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1. ( p1 , q1 ) ∧ ( p2 , q2 ) = (max{ p1 , p2 }, min{q1 , q2 }) if max{ p1 , p2 } < min{q1 , q2 }, and ( p1 , q 1 ) ∧ ( p2 , q2 ) = ⊥ otherwise; 2. ( p, q)= {( p , q ) | p < p < q < q}; 3. = {( p, q) | p < q}; 4. ( p, q) = ( p, q1 ) ∨ ( p1 , q) if p p1 q1 q. This theory may be interpreted in any topos T , defining an internal locale (TR )T ≡ RT with associated frame O(RT ). Points m of RT , i.e. frame maps m −1 : O(RT ) → T , correspond bijectively to Dedekind cuts (L , U ) of Q (cf. [63, p. 321]) in the following way: a model m determines a Dedekind cut by L = { p ∈ Q | m | ( p, ∞)},
(31)
U = {q ∈ Q | m | (−∞, q)},
(32)
where ( p, ∞) and (−∞, formal generators of the frame q) are defined in terms of the O(Q) by ( p, ∞) = {( p, r ) | p < r } and (−∞, q) = {(r, q) | r < q}. The notation m | ( p, q) used here means that m −1 ( p, q) = , where : 1 → T is the truth element of T and ( p, q) is seen as an arrow ( p, q) : 1 → Q × Q → O(RT ). Conversely, a Dedekind cut (L , U ) uniquely determines a point m that maps a generator I = ( p, q) to m(I ) = iff I ∩ U = ∅ and I ∩ L = ∅. The Dedekind reals Pt(RT ), then, are defined in any topos T as the subobject of P(QT ) × P(QT ) consisting of those (L , U ) that are points of RT [63]. We mention four examples: 1. In T = Sets, a point m of R ≡ RSets corresponds to a real x described in the usual calculus way, so that L = { p ∈ Q | p < x} and U = {q ∈ Q | x > q}. Hence Pt(R) may be identified with R in the usual sense, and R is spatial as a locale; its frame O(R) is just the usual topology of R [51, D4.7.4]. From this perspective, the first condition in the definition of TR enforces that L and U are lower and upper sections of Q, respectively, the second implies that they are open, and the third means that L and U are both inhabited. The fourth – Dedekind – relation says that L and U ‘kiss’ each other.6 2. If X is a topological space (or, more generally, a locale), the structure of the locale RSh(X ) and its associated sheaf of Dedekind reals Pt(RSh(X ) ) in the topos Sh(X ) of sheaves on X follows from the argument above (16) in Subsect. 1.7. First, the frame of Dedekind reals is given by the sheaf O(R)Sh(X ) : U → O(U × R),
(33)
whereas the Dedekind real numbers object is the sheaf (see also [63]) Pt(R)Sh(X ) : U → C(U, R).
(34)
3. Consequently, using (12) and (13) we infer that in our functor topos T (A) = SetsC (A) , the frame of Dedekind reals is the functor O(R) : C → O((↑ C) × R);
(35)
6 The collection of L satisfying only the first three relations forms the locale of lower reals, which we denote by Rl . Locale maps to Rl are, classically, lower-semicontinuous real-valued functions. Analogously, there is a locale Ru of upper reals. See [51].
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the set on the right-hand side may be identified with the set of monotone functions from ↑ C to O(R).7 Perhaps surprisingly, the associated functor of points Pt(R) may be identified with the constant functor Pt(R) : C → R;
(36)
this follows from (34) and the fact that Alexandrov-continuous functions U → R (or, indeed, into any Hausdorff space) must be locally constant on any open U ⊆ C(A).8 4. If is the Gelfand spectrum of a commutative C*-algebra A in T , in the sheaf topos Sh() internal to T we similarly have Pt(R)Sh() : U → C(U, R).
(37)
Here we identify the open U of with its associated sublocale {V ∈ | V U } of . This locale, as well as R, is to be interpreted in the ambient topos T as explained in the above items. Example 4 leads to an elegant reformulation of the isomorphism (24) given by the Gelfand theory: since C(, R) = (Pt(R)Sh() ),
(38)
where is the global sections functor, one infers from (24) that Asa ∼ = (Pt(R)Sh() ).
(39)
In other words, the self-adjoint part of a unital commutative C*-algebra A in a topos is isomorphic to the global sections of the Dedekind reals in the internal topos of sheaves on its spectrum (and A itself “is” the complex numbers in the same sense).
3. The Internal C*-Algebra and its Spectrum In this section we explain the association of a particular commutative C*-algebra A, which is internal to a certain functor topos T (A), to a (generally) noncommutative C*-algebra A. As mentioned in the Introduction, this construction is motivated by Bohr’s doctrine of classical concepts, so that we call A the Bohrification of A. 7 This identification proceeds in two steps. First, for any topological space X one has a bijection O(X ) ∼ = C(X, S), where S = {0, 1} carries the Sierpinski topology, see Subsect. 1.4; explicitly, U ∈ O(X ) is mapped to χU , whereas in the opposite direction g ∈ C(X, S) is sent to g −1 ({1}). Hence O(↑ C ×R) ∼ = C(↑ C ×R, S) (with apologies for the double use of C, first for ‘continuous’ and second for C ∈ C(A)). Second, in general λ-conversion or ‘currying’ gives a bijection between functions Y × R → S and functions Y → S R ; with Y =↑ C equipped with the Alexandrov topology and C(R, S) ∼ = O(R), continuity then translates into monotonicity. 8 We take X = C(A), equipped with the Alexandrov topology, and prove that in this topology any f ∈ C(U, R) must be locally constant. Suppose C D in U , take and V ⊆ R open with f (C) ∈ V . Then tautologically C ∈ f −1 (V ) and f −1 (V ) is open by continuity of f . But the smallest open set containing C is ↑ C, which contains D, so that f (D) ∈ V . Taking V = ( f (C) − , ∞) gives f (D) > f (C) − for all > 0, whence f (D) f (C), whereas V = (−∞, f (C) + ) yields f (D) f (C). Hence f (C) = f (D).
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3.1. The topos associated to a C*-algebra. We first construct the topos T (A) in which A resides and draw attention to the functoriality of the map A → T (A). We denote the category of partially ordered sets and monotone functions by Poset. Proposition 2. There is a functor C : CStar → Poset, defined on objects as C(A) = {C ⊆ A | C ∈ cCStar}, ordered by inclusion. On a morphism f : A → B of CStar, it acts as C( f ) : C(A) → C(B) by the direct image C → f (C). As announced in (8) in the Introduction, the collection of functors C(A) → Sets forms a topos T (A) = SetsC (A) . This is the topos associated to A. We recall our convention to underline entities internal to T (A). The subobject classifier in T (A) has already been given in (10). Recall that a geometric morphism f : S → T between topoi is a pair of adjoint functors, consisting of a direct image part f ∗ : S → T and an inverse image part f ∗ : T → S, of which f ∗ is required to preserve finite limits. Denote the category of elementary topoi and geometric morphisms by Topos.9 Proposition 3. There is a functor T : CStar → Topos, defined on objects by T (A) = SetsC (A) , the category of functors from C(A) to the ambient topos. This immediately follows from Theorem VII.2.2 in [63] (p. 359) and Proposition 2. To close this subsection, note that instead of initially regarding C(A) as a poset as in the main text, we could have considered it as a category from the start, having the same objects, but with (equivalence classes of) monomorphisms as arrows (instead of inclusions). The functor in Proposition 2 would then have the category Cat of categories as its codomain. This would still have allowed us to define the associated topos, and also the internal C*-algebra we will define below. From then on, most constructions will be within the associated topos, and hence go through as well.
3.2. Bohrification. Whereas the previous subsection considered the topos T (A) associated to a C*-algebra A, this one is devoted to a particular object A in this topos. In fact, the definition of A is ‘tautological’ in a literal sense. Definition 4. Let A be a C*-algebra in Sets. The functor A : C(A) → Sets is given on objects by A(C) = C, and on morphisms D ⊆ C of C(A) as the inclusion A(D) → A(C). Note that the functor A : C(A) → Sets factors through CStar or cCstar via the forgetful embedding of CStar or cCstar in the ambient topos Sets. 9 We will not worry about the fact that Topos, like Poset and CStar, is a large category; when pressed one can limit these categories to a chosen universe to make them small.
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Theorem 5. A is a commutative C*-algebra in T (A) under the operations inherited from A. More precisely, A is a vector space over the internal complex numbers Pt(C) (given simply by the constant functor Pt(C) : C → C) by 0 : 1→ A + : A× A→ A · : Pt(C) × A → A
0C (∗) = 0, a +C b = a + b, z ·C a = z · a,
and an involutive algebra through · : A× A→ A (−)∗ : A → A
a ·C b = a · b, (a ∗ )C = a ∗ .
The norm relation is given by N : A × Q+ →
NC (a, q) iff a < q.
Proof. One easily checks that the arrows are natural transformations (and hence morphisms in T (A)) and that this structure satisfies the requirements for A to be a presemi-C*-algebra in T (A). Since each A(C) is a commutative C*-algebra in the ambient topos, A is commutative as well. (Alternatively, since the definition of a commutative pre-semi-C*-algebra consists only of geometrically definable objects (e.g. CQ ) and geometric formulae (see Appendix A and Sect. 2), it follows from Lemma 21 that A is a commutative pre-semi-C*-algebra in T (A), because every A(C) is a commutative C*-algebra in the ambient topos.) In fact, A is a pre-C*-algebra, i.e. internally the semi-norm is a norm: if for all q > 0 we have (a, q) ∈ N , then a = 0. To prove this, we need to show that C ∀a∈Asa ∀q∈Q+ .(a, q) ∈ N → a = 0, where we are using the internal language of T (A). In other words: for all C ⊇ C and a ∈ C , if C ∀q∈Q+ .(a, q) ∈ N , then C a = 0, i.e. for all C ⊇ C and a ∈ C , if for all C ⊇ C and q ∈ Q+ we have C (a, q) ∈ N , then C a = 0, i.e. for all C ⊇ C and a ∈ C , if a = 0, then a = 0. But this holds, since every C is a C*-algebra. Finally, A is in fact a C*-algebra, i.e. internally we have Cauchy completeness. By the axiom of dependent choice (which holds because T (A) is a functor topos whose codomain validates dependent choice [41]) it suffices to prove that every regular Cauchy sequence (i.e. a sequence (xn ) such that ||xn − xm || 2−n + 2−m for all n, m) converges. Thus we need to prove C ∀n,m .||xn − xm || 2−n + 2−m → ∃x∈A .∀n .||x − xn || 2−n , i.e. for all C ⊇ C, if C (∀n,m .xn − xm 2−n + 2−m ), then C ∃x∈A .∀n .||x − xn || 2−n , i.e. for all C ⊇ C, if C “x is regular”, then C ∃x∈A .∀n .x − xn 2−n . Once again, this holds because every C is a C*-algebra.
# "
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The functor A is our internal C*-algebra. By changing the universe of discourse from the ambient topos Sets to T (A), the (generally) noncommutative C*-algebra A has become a commutative C*-algebra A. Multiplication of two non-commuting operators is no longer defined, since they live in different commutative subalgebras.10 3.3. The Kochen–Specker theorem. Combining the material in Sects. 2 and 3 so far, we obtain a mapping A → (A), which associates a certain internal locale to a (generally) noncommutative C*-algebra. As argued in the Introduction, (A) describes the quantum logic of the physical system whose algebra of observables is A. An important property of the internal spectrum is that it may typically be highly non-spatial from an external point of view. First, recall (see Subsect. 1.4) that a point of a locale X in a topos T is a frame map O(X ) → , where is the subobject classifier in T . Theorem 6. Let H be a Hilbert space with dim(H ) > 2 and let A be the C*-algebra of bounded operators on H . Then the locale (A) has no points. Proof. We reason internally. A point ρ : ∗ → of the locale (see Subsect. 1.5) may be combined with a ∈ Asa with Gelfand transform aˆ : → R (see (25)), so as to produce a point aˆ ◦ ρ : ∗ → R of the locale R. This yields a map V ρ : Asa → Pt(R), which can be shown to be an internal multiplicative functional; see [6,7,27].11 Being an arrow in T (A), the map V ρ is a natural transformation, with components V ρ (C) : Asa (C) → Pt(R)(C); by Definition 4 and 36, this is just V ρ (C) : Csa → R. Hence one has a multiplicative functional V ρ (C) for each C ∈ C(A) in the usual sense, with the property (which follows from naturality) that if C ⊆ D, then the restriction of V ρ (D) to Csa coincides with V ρ (C). But this is precisely a valuation12 on B(H ), whose nonexistence was proved by Kochen and Specker [56]. " # This is a localic reformulation of the original topos-theoretic version of the KochenSpecker theorem due to Butterfield and Isham [20]. As in their work, the proof relies on the original version, but in being a statement about the nonexistence of models of a certain theory, our reformulation has a logical thrust that both the original version by Kochen and Specker and the reformulation by Butterfield and Isham lack. The theorem certainly holds for more general C*-algebras than just the collection of all bounded operators on a Hilbert space; see [32 and 47] for results on von Neumann algebras. For C*-algebras, one has the result that a simple infinite unital C*-algebra does not admit a dispersion-free quasi-state [45]. Evidently, Theorem 6 holds for such extensions as well. One way of looking at such results is to see them as illustrations of the failure of the Krein-Milman theorem in a constructive context [68]. Indeed, recall that the classical Krein-Milman theorem states that a compact convex set is the closed convex hull of its extreme points. The state space of A is still a compact convex set in an appropriate 10 Kochen and Specker refer to such a structure as a partial algebra [56] and stress its relevance for the foundations of quantum theory; in a partial algebra both addition and multiplication need only be defined for commuting operators. 11 This map may explicitly be given in the internal language of T (A), by noting that for each a ∈ A the sa expression ρ(a) ˜ = (L ρ,a , Uρ,a ) = ({r ∈ Q | ρ | Da−r }, {s ∈ Q | ρ | Ds−a }) is a Dedekind cut in T (A). 12 This terminology is to be distinguished from the one used in Subsect. 4.2 below. The naturality property just mentioned is often called noncontextuality in the philosophy of physics literature.
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localic sense (see Sect. 4), and the pure states on A would be its extreme boundary. These points, however, fail to exist, as we have just seen. 4. (Quasi-)States as Integrals This section about states, and the next one about observables, are both concerned with connections between the two levels we have developed (see Fig. 1): 1. the ambient topos Sets, containing the C*-algebra A; 2. the associated topos T (A), containing the internal commutative C*-algebra A and its spectrum . The main result of this section is Theorem 14, which gives an isomorphism between quasi-states on A at level 1 and, at level 2, either probability integrals on Asa , or, equivalently, probability valuations on the Gelfand spectrum . Subsequently, we show that probability valuations define subobjects of , as in classical physics. All this requires some preparation, firstly in the theory of quasi-states on C*-algebras (Subsect. 4.1) and secondly in abstract constructive integration theory (Subsect. 4.2). 4.1. States and quasi-states. A linear functional ρ : A → C on a C*-algebra A is called positive when ρ(a ∗ a) ≥ 0 for all a ∈ A. It is a state when it is positive and satisfies ρ(1) = 1. A state ρ is pure when ρ = tσ +(1−t)ω for some t ∈ (0, 1) and some states σ and ω implies ω = σ . Otherwise, it is called mixed. For example, if A ⊆ B(H ) for some Hilbert space H (which we may always assume by the Gelfand–Naimark Theorem), then each unit vector ∈ H defines a state ψ on A by ψ(a) = (, a). If A = B(H ), such states are pure. (If H is infinite-dimensional, not all pure states arise in this way, though.) Mixed states ρ on B(H ) arise from countable sequences ( pi ), 0 pi 1, i pi = 1, coupled with an orthonormal family of vectors (i ), through ρ(a) = i pi ψi (a). (By the spectral theorem, one may equivalently say that such states are given by positive operators ρˆ on H with unit trace, through ρ(a) = Tr (ρa).) ˆ A state ρ : A → C is called faithful when ρ(a ∗ a) = 0 implies a = 0. For example, if, in the situation just described, the i comprise an orthonormal basis of H and each pi > 0, then the associated state ρ is faithful. The states of a C*-algebra form a compact convex set, the extremal points of which are by definition the pure states. States are automatically hermitian, in the sense that ρ(a ∗ ) = ρ(a), or equivalently, ρ(a) ∈ R for self-adjoint a. In algebraic quantum physics, mathematical states as defined above are often used to model the physical states of the quantum system. However, when taking Bohr’s doctrine of classical concepts seriously, one should take into account that two observables can only be added in a physically meaningful way when they are jointly measurable, i.e. when the corresponding operators commute. Thus one may relax the definition of a quantum state, which ought to be linear only on commutative parts. This leads to the notion of a quasi-state [1]:13 13 Axiom VII of Mackey’s foundation of quantum mechanics [64] states that a measure on the projections of a von Neumann algebra extends to a state on the von Neumann algebra. Mackey stresses that, in contrast to his other axioms, Axiom VII does not have a physical justification. One can prove that a measure extends to a quasi-state, so one is led to ask whether every quasi-state is a state. This is not the case when the von Neumann algebra has a summand of type I2 , but it holds for all other von Neumann algebras [15]. For C*-algebras the question is more difficult. The main result seems to be the following [16]. Consider a C*-algebra with no quotient isomorphic to M2 (C) and let ρ be a quasi-linear functional. Then ρ is linear iff ρ restricted to the unit ball is uniformly weakly continuous.
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Definition 7. A quasi-linear functional on a C*-algebra A is a map ρ : A → C that is linear on all commutative subalgebras and satisfies ρ(a + ib) = ρ(a) + iρ(b) for all self-adjoint a, b ∈ A (possibly non-commuting). It is called positive when ρ(a ∗ a) ≥ 0 for all a ∈ A. When A is unital, a positive quasi-linear functional is called a quasi-state when ρ(1) = 1. This kind of quasi-linearity also determines when some property P of the C*-algebra A descends to a corresponding property P of the internal C*-algebra A, as the following lemma shows. To be precise, for P ⊆ A, define a subfunctor of A by P(C) = P ∩ C. Let us call a property P ⊆ A quasi-linear when a ∈ P and b ∈ P imply µa + iλb ∈ P for all µ, λ ∈ R and a, b ∈ Asa . Lemma 8. Let A be a C*-algebra, and let P ⊆ A be a quasi-linear property. Then P = A if and only if P = A. Proof. One implication is trivial; for the other, suppose that P = A. For a ∈ A, denote by C ∗ (a) the sub-C*-algebra generated by a. When a is self-adjoint, C ∗ (a) is commutative. So Asa ⊆ P, whence by quasi-linearity of P and the unique decomposition of elements in a real and imaginary part, we have A ⊆ P. " #
4.2. Algebraic integration theory. The well-known correspondence between states on commutative C*-algebras A and probability measures on the underlying Gelfand spectrum is an immediate consequence of the Gelfand isomorphism A ∼ = C(, C) and the Riesz-Markov representation theorem in measure theory. In the present topos-theoretical setting, it turns out to be more natural to work with integrals and valuations rather than measures. Recall the a priori difference between these three concepts: • measures are defined on Borel subsets of some space X ; • valuations are defined only on the open subsets of X ; • integrals are positive linear functionals on the (ordered) vector space Cc (X, R). Classically, if X is locally compact Hausdorff and the measures in question are suitably regular, there are isomorphisms between these notions. From a constructive point of view, however, there is a subtle difference between valuations and integrals.14 In any case, the fundamental role locales play in this paper as the Gelfand spectra of the internal C*-algebras A makes it quite natural to assign probabilities to opens (rather than Borel subsets) of the spectrum. The following string of definitions gives an abstract (and constructive) version of integration theory based on ordered vector spaces, abstracting from the Riemann, Lebesgue and Daniell integrals [30,31,75]. Several axiomatizations are possible, of which the one in terms of so-called f-algebras is the most convenient for our purposes. 14 The integral I ( f ) of a function f ∈ C(X ) is a Dedekind real, so that it can be approximated by rationals. This may not be the case for the valuation µ(U ) of an open U , as the ‘kissing’ property (if r < s then µ(U ) < s or r < µ(U )) may fail. Accordingly, µ(U ) is only a lower real, and can be thought of as a predicate r < µ(U ) on the rationals. This predicate is downward closed: if r < µ(U ) and s r , then s < µ(U ). But in general, given ε > 0 one cannot approximate µ(U ) up to ε with rationals. Given an integral I , we can define a corresponding valuation µ I (U ) by taking the sup of I ( f ) over all 0 f 1 with support in U . It is remarkable that for any valuation µ one can conversely find a (unique) integral I such that µ = µ I . So despite the fact that one may not be able to compute µ(U ), it is still possible to compute f dµ as a Dedekind real, which a priori is only a lower real.
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Definition 9. A Riesz space or vector lattice is a partially ordered vector space (R, ) over R (i.e. a real vector space R with partial ordering such that f g implies f + h g + h for all h and f ≥ 0 implies r f ≥ 0 for all r ∈ R+ ) that is a distributive lattice with respect to its partial order [62, Definition 11.1]. An f-algebra is a commutative, unital, real algebra R whose underlying vector space is a Riesz space in which f, g ≥ 0 implies f g ≥ 0, and f ∧ g = 0 implies h f ∧ g = 0 for all h ≥ 0. Moreover, the multiplicative unit 1 has to be strong in the sense that for each f ∈ R there exists a natural number n such that −n1 f n1 [82, Def. 140.8]. Note that although f-algebras are a priori defined over the real numbers, they can a fortiori be defined over Q as well. The self-adjoint part of any commutative C*-algebra A is an example of an f-algebra by defining a b in the usual way (i.e. iff ∃c∈A [b − a = c∗ c]); one has f ∨ g = max{ f, g} and f ∧ g = min{ f, g}. Conversely, by the Stone-Yosida representation theorem every f-algebra can be densely embedded in a space of real continuous functions on a compact space. Definition 10. An integral on an ordered vector space R is a linear functional I : R → R that is positive, i.e. if f ≥ 0 then also I ( f ) ≥ 0. If R has a strong unit 1 (e.g., the multiplicative unit in the case of f-algebras), then an integral I satisfying I (1) = 1 is called a probability integral. An integral is faithful when its kernel is {0}, i.e., when I ( f ) = 0 and f ≥ 0 imply f = 0. Except in the degenerate case I (1) = 0, any integral can obviously be normalised to a probability integral. The prime example of an integral is the Riemann or Lebesgue integral on the ordered vector space C[0, 1]. More generally, any positive linear functional on a commutative C*-algebra provides an example, states yielding probability integrals. We wish to use a certain generalization of the Riesz-Markov theorem that can be proved constructively [30] and hence can be used within our topos T (A). This requires a localic reformulation of Definition 9, as well as a similar approach to valuations. Let R be an f-algebra (in Sets, for the moment). In defining the following frame it is technically convenient to define R as a vector space over Q. Define Integral(R) as the distributive lattice freely generated by P f , f ∈ R, subject to the relations P1 = , P f ∧ P− f = ⊥, P f +g P f ∨ Pg , Pf = ⊥
(for f 0).
This lattice generates a frame O(I(R)) by adding the regularity condition P( f ) = P( f − q)
(40)
Q%q>0
to the relations above, just like (23) in the case of the spectrum. It can be shown (cf. (17)) that P f = {ρ : R → R | ρ( f ) > 0},
(41)
where each ρ is understood to be a positive linear functional. Models of this theory, i.e. points of the associated locale, precisely correspond to probability integrals on R; if I
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is such an integral, the associated model m I is given by m I (P f ) = 1 iff I ( f ) > 0. Conversely, a model m defines an integral Im by (compare with the proof of Theorem 6) Im ( f ) := ({ p | m | P f − p }, {q | m | Pq− f }), where the right-hand side is seen to be a Dedekind real from the relations on P• . All this may be internalized to any topos, where, of course, there is no a priori guarantee that points of the locale with frame O(I(R)) exist (and hence that expressions like (41) make good sense). The final ingredient of the constructive Riesz-Markov theorem is the definition of a locale of valuations. These were studied in [46 and 81]. Definition 11. A probability valuation on a locale X is a monotone map µ : O(X ) → [0, 1]l that satisfies the usual additivity and regularity conditions for measures, i.e. µ(U ) + µ(V ) = µ(U ∧ V ) + µ(U ∨ V ) and µ( λ Uλ ) = λ µ(Uλ ) for any directed family. (Here, [0, 1]l is the collection of lower reals between 0 and 1.) Like integrals, probability valuations on X organize themselves in a locale V(X ). The generalized Riesz-Markov Theorem, then, is as follows. Theorem 12 [30]. Let R be an f-algebra and let be its spectrum.15 Then the locales I(R) and V() are isomorphic. To obtain an integral from a valuation we define: Iµ f := (sup(si ) si µ(si < f < si+1 ), inf (si ) si+1 (1 − µ(si > f ) − µ( f > si+1 )), where (s < f ) is a notation for D f −s and (s < f < t) denotes D f −s ∧ Dt− f and si is a partition of [a, b] such that a f b. Conversely, to obtain a valuation from an integral I we define: µ I (Da ) := sup I (na + ∧ 1)|n ∈ N . Note that both locales in question are compact regular [30]. Logically speaking, the theorem follows from the existence of a bi-interpretation between the geometric theories Integral(R) and Valuation() (i.e. there are interpretation maps in two directions which are each other’s inverses) and the equivalence of the category of propositional geometric theories with interpretations to the category of frames. 4.3. From states on A to subobjects of O(). We return to our main topic. Since everything in this section so far may be interpreted in the internal language of a topos and the proof of Theorem 12 is constructive, we have: Corollary 13. Let A be a C*-algebra with Bohrification A and associated Gelfand spectrum in the topos T (A). Then the locale I(A) of probability integrals on Asa is homeomorphic to the locale V() of probability valuations on . As announced at the beginning of this section, the next theorem crosses two levels of Fig. 1. 15 See [27] for the notion of the spectrum of an f-algebra, which is described exactly as in Subsect. 2.2. If the f-algebra is the self-adjoint part of a commutative C*-algebra, then its spectrum as an f-algebra coincides with its spectrum as a C*-algebra.
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Theorem 14. There is a bijective correspondence between quasi-states on A and either probability integrals on Asa , or, equivalently, probability valuations on its Gelfand spectrum . This theorem may actually be extended to a correspondence between (faithful) positive quasi-linear functionals on A and (faithful) integrals on A, etc. Proof. Every positive quasi-linear functional ρ gives a natural transformation Iρ : Asa → R if we define its components (Iρ )C : Csa → R to be ρ|Csa (i.e. the restriction of ρ to Csa ⊆ Asa ). Conversely, let I : Asa → R be an integral. Define ρ : Asa → R by ρ(a) = IC ∗ (a) (a). For commuting a, b ∈ Asa , ρ(a + b) = IC ∗ (a+b) (a + b) = IC ∗ (a,b) (a + b) = IC ∗ (a,b) (a) + IC ∗ (a,b) (b) = IC ∗ (a) (a) + IC ∗ (b) (b) = ρ(a) + ρ(b), because I is a natural transformation, C ∗ (a) ∪ C ∗ (b) ⊆ C ∗ (a, b) ⊇ C ∗ (a + b), and I is locally linear. Moreover, ρ is positive because I is locally positive (see Lemma 8). Hence we have defined ρ on Asa and may extend it to A by complex linearity. It is clear that the two maps I → ρ and ρ → I are inverses of each other and that if I is a probability integral, then ρ is a quasi-state, and vice versa. " # In the Introduction, we have seen that in the classical case a (pure) state ρ defines a subobject [ρ] of the frame of opens of the classical phase space; see (3). As we shall now show, this remains true, mutatis mutandis, in the quantum case. The main technical difficulty is to adapt the condition δρ (V ) = 1 in (3). Theorem 14 yields a bijective correspondence between quasi-states ρ on A and probability valuations µρ on . Fix a state, or quasi-state, ρ on A. The logical formula µρ (−) = 1 (of the Mitchell-Bénabou language of T (A)) is a predicate on O() and hence defines a subobject [ρ] of O() with characteristic arrow χ[ρ] : O() → . This arrow is just the interpretation of µρ (−) = 1, i.e. χ[ρ] = µρ (−) = 1.
(42)
Compare with (3); beyond mimicking the notation, we see that we have been able to transfer the classical description of states to the quantum situation in every respect. 5. Observables and Propositions In this section and the next we give the details of Steps 2 to 5 of our five-step program for spatial quantum logic, cf. Subsects. 1.2 and 1.6. We start with the locale map δ(a) : → IR, then turn to the description of elementary propositions a ∈ as opens in the spectrum , and finally consider the pairing of states and propositions to arrive at a suitable notion of (multi-valued) truth in quantum theory.
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5.1. Interval domain. For a commutative unital C*-algebra A with Gelfand spectrum in Sets, the Gelfand transform of a ∈ Asa is a continuous function aˆ : → R. Equivalently, it is a locale map (25). As we have seen in Subsect. 2.3, mutatis mutandis the description (25) still applies when A is a commutative unital C*-algebra A with Gelfand spectrum in a topos T . In particular, one has the Gelfand transform aˆ : → R
(a ∈ Asa ).
(43)
Our problem, however, is to express an element a ∈ Asa of a noncommutative C*-algebra A in Sets in terms of some locale map δ(a) defined on the spectrum of the Bohrification A of A in T (A). As we shall see, this problem can be solved if we introduce some fuzziness, in that δ(a) no longer takes values in the internal Dedekind reals R in T (A), like a, ˆ but in the so-called interval domain IR, internalized in T (A) as IR. Thus, apart from (43) we are dealing with a second locale map δ(a) : → IR
(a ∈ Asa ).
(44)
In honour of Döring and Isham, we refer to δ(a) as the Daseinisation of a (although our map differs from theirs, cf. Appendix B). We have already encountered Scott’s interval domain IR in Subsect. 1.8 as the poset of compact intervals in R, ordered by inverse inclusion. Like the Dedekind real numbers, the interval domain is easily internalized and hence definable in any topos. In fact, the construction of the Dedekind real numbers in Subsect. 2.3 only requires a single modification so as to obtain the interval domain: the corresponding frame O(IR) is defined by the very same generators ( p, q) and relations as O(R), except that the fourth relation (i.e. ( p, q) = ( p, q1 ) ∨ ( p1 , q) if p p1 q1 q) is dropped. The models of O(IR) or points of the associated locale IR again correspond to pairs (L , U ) given by (31) and (32), but this time such a pair may fail to define a Dedekind cut; axiomatically, only the ‘kissing’ requirement no longer holds. In any topos T , we denote the locale defined by the geometric propositional theory given by the first three axioms in the list following (30) in Subsect. 2.3—interpreted in T —by O(IR)T , with the usual special case IR ≡ IRT (A) . Similarly, the subobject of P(Q) × P(Q) consisting of models of O(IR)T is denoted by Pt(IR)T , with Pt(IR)T (A) ≡ Pt(IR). The examples in Subsect. 2.3 now read as follows: 1. In Sets (or, more generally, when classical logic applies in T ), a cut (L , U ) defines a compact interval [sup L , inf U ] (where sup and inf are taken in R), so that Pt(IR) may be identified with the classical Scott interval domain IR. In that case, a generator ( p, q) ∈ O(IR) may be identified with the Scott open in IR that contains all intervals [a, b] such that p < a b < q. 2. In a topos Sh(X ) of sheaves one has O(R)Sh(X ) : U → O(U × IR),
(45)
but its points are not as easily described as (34); instead, one has Pt(IR) : U → {( f, g) | f, g : U → R | f g, f lower-semicont., g upper-semicont.}. This follows by carefully adapting the proof of [63, Theorem VI.8.2] for R.
(46)
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3. In particular, for T (A) = SetsC (A) , one has O(IR) : C → O((↑ C) × IR),
(47)
which may be identified with the set of monotone functions from ↑ C to O(IR).16 The object Pt(IR) will not be used in this paper.17
5.2. Daseinisation. After this preparation, we turn to the Daseinisation (44), or rather to the corresponding frame map δ(a)−1 : O(IR) → O().
(48)
A complete description of this map, based on the technique of generating (semi)lattices for frames, may be found in Appendix A.2. Here, we just look at the special case (1
δ(a)−1 (r,s)
(r,s)
−1
/ O()) = (1 −→ O(IR) δ(a) −→ O()),
(49)
where the arrow (r, s) : 1 → O(IR) maps into the monotone function with constant value ↓ (r, s).18 We may even simplify (49) even further by localizing it at C · 1; this, however, entails no loss of generality, for O()(C · 1) is the frame in Sets that (together with the frame map (88)) provides the external description of the internal locale in T (A) (see Subsect. 1.7 and Appendix A.2). The quantity δ(a)−1 (r, s)(C · 1) is a global element U of O()(C · 1) as described by Theorem 29 in Appendix A. Briefly, this theorem states that O()(C · 1) may be seen as the set of all subfunctors U of the functor C → L C that satisfy a certain regularity condition, where L C is the distributive lattice freely generated by the formal symbols Dc , c ∈ Csa subject to the relations (18)–(22) (simply interpreted in Sets).19 Abbreviating δ(a)−1 (r, s) = δ(a)−1 (r, s)(C · 1),
(50)
the ensuing element δ(a)−1 (r, s) of O()(C · 1) turns out to be the functor δ(a)−1 (r, s) : C → {D f −r ∧ Ds−g | f, g ∈ Csa , f a g}.
(51)
This follows from (106) and the definition of δ(a)−1 in Appendix A.2, combined with the equality {D f − p ∧ Dq−g | f, g ∈ Csa , f a g} { p
= {D f −r ∧ Ds−g | f, g ∈ Csa , f a g}. 16 The argument is the same as for R, see footnote 7. 17 For completeness, we mention that Pt(IR)(C) is the set of all pairs (L , U ), where L and U are subfunctors
of the constant functor Q, truncated to ↑ C ⊂ C(A), such that for all D ⊇ C, (L(D), U (D)) is a pair of the form (31)–(32). 18 Here (r, s) is seen as an element of the generating semilattice Q × Q, whereas ↓ (r, s) is its image in < the frame O(IR) through the canonical map (67); see Appendix A. 19 See Appendix A.1 for a detailed description of the functor C → L . C
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An alternative description of δ(a)−1 is as follows. For fixed a ∈ Asa , define functors L a ∈ T (A) and Ua ∈ T (A) by L a (C) = { f ∈ Csa | f a}; Ua (C) = {g ∈ Csa | a g}.
(52)
Each of these defines a subobject of Asa . In fact, the pair (L a , Ua ) is a directed subobject op of Asa × Asa . Now take a generator (r, s) of O(IR), and write (r, s) = (−∞, s)∧(r, ∞). The Gelfand transform (27) defines subobjects ˆ·(L a , (r, ∞)) and ˆ·(Ua , (−∞, s)) of O(). We then put δ(a)−1 : (r, s) →
ˆ·(L a , (r, ∞)) ∧ ˆ·(Ua , (−∞, s)).
(53)
Using (28) and (29), this gives20
δ(a)−1 (r, s) =
D f −r ∧ Ds−g .
(54)
f ∈L a ,g∈Ua
To illustrate what is going on, it is helpful to compute the right-hand side of (51) or (54) in Sets for A = C = C(, C). In that case the meaning of Da is given by (17), so that with ρ( f ) = f (ρ) one finds D f −r = {ρ ∈ | f (ρ) > r } and Ds−g = {ρ ∈ | g(ρ) < s}. One then obtains (with ∧ for ‘and’) δ(a)−1 A (r, s) =
{ρ ∈ | f (ρ) > r ∧ g(ρ) < s}
f,g∈Csa , f a g
= {ρ ∈ | ∃ f a [ f (ρ) > r ∧ f (ρ) < s] ∧ ∃g≥a [g(ρ) > r ∧ g(ρ) < s]} = {ρ ∈ | r < a(ρ) < s} = a −1 (r, s). (55) To close this subsection, we note the following: Proposition 15. The map δ : Asa → C(, IR) is injective, and a b iff δ(a) δ(b). Proof. Suppose that δ(a) = δ(b). Then for all C ∈ C(A), the sets L a (C) = { f ∈ Csa | f a} and Ua (C) = {g ∈ Csa | a g} must coincide with L b (C) and Ub (C), respectively. Imposing these equalities at C = C ∗ (a) and at C = C ∗ (b) yields a = b. The order in Asa is clearly preserved by δ, whereas the converse implication can be shown by the same method as the first claim of the proposition. " # 20 Using a generic point σ , we may even define
δ(a)(σ ) := (sup σ (L a ), inf σ (Ua )). Analoguously, one can view δ(a) as an interpretation of the geometric theory in the geometric theory of the intervals, see [30].
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5.3. Propositions. It immediately follows from the existence of the Daseinisation map (48) (see Subsect. 5.2 and Appendix A.2) that, as in the classical case, elementary propositions a ∈ define opens in phase space. For an open in the ‘quantum phase space’ is simply defined as a global element 1 → O() (cf. Subsect. 1.4)), so that given an observable a ∈ Asa and a Scott open ∈ O(IR), we may combine the corresponding arrows δ(a)−1 : O(IR) → O() and : 1 → O(IR) into (1
[a∈]
/ O() ) = ( 1
/ O(IR)
δ(a)−1
/ O()).
(56)
This generalises (49); in particular, : 1 → O(IR) is defined at C as the monotone function ↑ C → O(IR) taking constant value . In other words, [a ∈ ] = δ(a)−1 ◦ .
(57)
6. State-Proposition Pairing In Subsect. 4.3 we have shown how a state ρ on A gives rise to a subobject [ρ] of O() χ[ρ] / related to defined by the predicate µρ (−) = 1, and hence to an arrow O() the predicate in question by (42). Also, we have just seen the description (56) of propositions a ∈ as opens in . Hence we can pair a physical state ρ and a physical proposition a ∈ by composition, to end up with a ‘truth value’ a ∈ , ρ in the subobject classifier of T (A). Explicitly, one has (1
a∈,ρ
/ ) = (1
[a∈]
/ O()
χ[ρ]
/ ),
(58)
or a ∈ , ρ = χ[ρ] ◦ δ(a)−1 ◦ .
(59)
In what follows, we need the basic definitions of Kripke-Joyal semantics. If ϕ is some formula interpreted in a topos T as an arrow ϕ : F → , and α : B → F is any arrow in T (defining a ‘generalized element’ of F), then the notation B ϕ(α), or, less preα
ϕ
cisely, B ϕ (for ‘B forces ϕ’) means that the composite arrow B → F −→ factors through : 1 → . In a functor topos SetsC , where C is some category, the notation C ϕ for some C ∈ C is shorthand for y(C) ϕ, where y(C) : D → HomC (D, C) is the Yoneda functor. In our case T = T (A), the interpretation ϕ is a natural transformation F → , given by its components ϕ(C) : F(C) → (C), where C ∈ C(A). In that case the forcing condition C ϕ turns out to be equivalent to ϕ(C)(F(C)) = C , where C is the maximal upper set on C. Using the Kripke-Joyal semantics of T (A), we now explicitly compute the stateproposition pairing in case that = (r, s) is a rational interval. The computation is straightforward when using generating lattices (see Appendix A). From here on, ρ is a fixed state on A and we abbreviate µρ by µ. For D ∈ C(A), (a ∈ (r, s), ρ) D (∗)
(58)
=
(57),(42)
=
(χ[ρ] ◦ [a ∈ (r, s)]) D (∗) µ(δ(a)−1 (r, s)) = 1(D).
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Being a global element 1 → of the subobject classifier of T (A), the right-hand side is an element of the set (D), and hence an upper set on D. With slight abuse of notation, we simply call the latter a ∈ (r, s), ρ(D). It follows that a ∈ (r, s), ρ(D) = {C ∈ C(A) | C ⊇ D, C µ(δ(a)−1 (r, s)) = 1},
(60)
where µ ◦ δ(a)−1 (−) = 1 is the obvious predicate on O(IR) defined by µ(−) = 1 on O() and the Daseinisation map (48). Since a ∈ (r, s), ρ(D) is the truncation to ↑ D of the corresponding upper set at C · 1, we may use (51) or (54), from which we see that the forcing condition C µ(δ(a)−1 (r, s)) = 1 is equivalent to ⎛ ⎞ D f −r ∧ Ds−g ⎠ = 1. µC ⎝ f a g, f,g∈Csa
Here µC is the valuation defined as µρ , but with ρ restricted to C. Similarly, D f −r refers to an open in the spectrum of C (cf. Theorem 20, according to which the Da with a ∈ Csa may be seen as generators of the spectrum of C). Since the measure of the intersection of two opens equals one if the measures of both opens do, this means (for f, g ∈ Csa ) ⎛⎛ ⎞
⎞ µC ⎝⎝ D f −r ⎠ ∧ Ds−g ⎠ = 1, f a
g≥a
which happens if and only if ⎛ ⎞
µC ⎝ D f −r ⎠ = 1 and µC Ds−g = 1. f a
g≥a
The left conjunct means ∀n∈N ∃ f ∈Csa , f a [µC (D f −r ) > 1 −
1 ], n
(61)
since T (A) is a functor topos and hence the quantifiers above are interpreted locally. The construction of µρ from ρ (see Sect. 4) implies µC (Dh ) = lim ρ((mh + ) ∧ 1), m→∞
where the limit is a lower real. In other words, µC (Dh ) > q iff there exists m in N such that ρ((mh + ) ∧ 1) > q. So C µρ ( f a D f −r ) = 1 means that for each n ∈ N there exists f ∈ C with f a and µC (D f −r ) > 1 − n1 . Hence at the end of the day the state-proposition pairing a ∈ (r, s), ρ explicitly yields the upper set at D given by a ∈ (r, s), ρ(D) ⎧ ⎛ ⎨ = C ∈ C(A) | C ⊇ D, µC ⎝ ⎩
f a, f ∈Csa
⎞
⎛
D f −r⎠ = 1 and µC ⎝
a g,g∈Csa
⎫ ⎬ Ds−g ⎠ = 1 . ⎭ ⎞
(62)
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This formula can be put in a slightly more palpable form when A and each C ∈ C(A) are von Neumann algebras (in the ambient topos Sets). In that case, it can be shown [47] that the open D f −r in the spectrum gives rise to a projection operator [D f −r ], to which we can directly apply the state ρ. Moreover, unlike for general C*-algebras, the supremum P = {[D f −r ] | f a, f ∈ Csa } exists. One then simply has µC (P) = 1 when ρ(P) = 1. Similarly, the projection Q = {[Ds−g ] | a g, g ∈ Csa } exists and µC (Q) = 1 when ρ(Q) = 1. To close, we remark that one might consider a proposition µρ (−) > p, for some rational number p, instead of the proposition µρ (−) = 1 as in this paper. This would simplify the computations above slightly. For instance, (61) would become ∃ f ∈Csa , f a [µC (D f −r ) > p]. This eliminates a universal quantification, but otherwise the computations would continue mutatis mutandis as before. A. Generating Lattices for Frames At various places in this article we refer to a presentation of a frame (or locale) by a generating lattice with a covering relation. This technique has been developed in the context of formal topology [71,72], and extends an analogous construction due to Johnstone [48]. Note that formal topology may be developed in the framework of constructive set theory [3], and hence may be internalized in topos theory. Let (L , ) be a meet semilattice (i.e. a poset in which any pair of elements has a meet = g.l.b. = infimum; in most of our applications (L , ) is actually a distributive lattice). Definition 16. A covering relation on L is a relation ⊆ L × P(L) - equivalently, a function L → P(P(L)) - written x U when (x, U ) ∈ , such that: 1. If x ∈ U then x U ; 2. If x U and U V (i.e. y V for all y ∈ U ) then x V ; 3. If x U then x ∧ y U ; 4. If x ∈ U and x ∈ V , then x U ∧ V (where U ∧ V = {x ∧ y | x ∈ U, y ∈ V }). For example, if (L , ) = (O(X ), ⊆) one may take x U iff x U , i.e. iff U covers x. Let DL be the poset of all lower sets in L, ordered by inclusion; this is a frame [48, Sect. 1.2]. The structure gives rise to a closure operation21 A : DL → DL, given by AU = {x ∈ L | x U },
(63)
which has the following properties: ↓U ⊆ AU , U ⊆ AV ⇒ AU ⊆ AV , AU ∩ AV ⊆ A(↓U ∩ ↓V ). The frame F(L , ) generated by such a structure is then defined by F(L , ) = {U ∈ DL | AU = U } = {U ∈ P(L) | x U ⇒ x ∈ U };
(64)
the second equality follows because firstly the property AU = U guarantees that U ∈ DL, and secondly one has AU = U iff x U implies x ∈ U . An equivalent description of F(L , ) is F(L , ) ∼ (65) = P(L)/ ∼, 21 As a map, A is also defined on P(L). Let χ : L × P(L) → be the characteristic function of the subset ⊆ L × P(L). Then A = χˆ is just the ‘curry’ or ‘λ-conversion’ of χ .
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where U ∼ V iff U V and V U . Indeed, the map U → [U ] from F(L , ) (as defined in (64)) to P(L)/ ∼ is a frame map with inverse [U ] → AU ; hence the idea behind the isomorphism (65) is that the map A picks a unique representative in the equivalence class [U ], namely AU . The frame F(L , ) comes equipped with a canonical map f : L → F(L , ); x → A(↓x),
(66) (67)
which satisfies f (x) f (U ) if x U . In fact, f is universal with this property, in that any homomorphism g : L → G of meet semilattices into a frame G such that g(x) g(U ) whenever x U has a factorisation g = ϕ ◦ f for some unique frame map ϕ : F(L , C) → G. This suggests that the point of the construction is that F(L , ) is (isomorphic to) a frame defined by generators and relations, provided the covering relation is suitably defined in terms of the relations. More precisely [3, Thm. 12]: Proposition 17. Suppose one has a frame F and a meet semilattice22 L with a map f : L → F of meet semilattices that generates F in the sense that for each U ∈ F one has U = { f (x) | x ∈ L , f (x) ≤ U }. Define a cover relation on L by x U iff f (x) f (U ). (68) Then one has a frame isomorphism F ∼ = F(L , ). We now turn to maps between frames. Definition 18. Let (L , ) and (M, ) be meet semilattices with covering relation as above, and let f ∗ : L → P(M) be such that: 1. f ∗ (L) = M;23 2. f ∗ (x) ∧ f ∗ (y) f ∗ (x ∧ y); 3. x U ⇒ f ∗ (x) f ∗ (U ) (where f ∗ (U ) = u∈U f (U )). Define two such maps f 1∗ , f 2∗ to be equivalent if f 1∗ (x) ∼ f 2∗ (x) (i.e. f 1∗ (x) f 2∗ (x) and f 2∗ (x) f 1∗ (x)) for all x ∈ L. A continuous map f : (M, ) → (L , ) is an equivalence class of such maps f ∗ : L → P(M).24 Our main interest in continuous maps lies in the following result.25 Proposition 19. Each continuous map f : (M, ) → (L , ) is equivalent to a frame map F( f ) : F(L , ) → F(M, ), given by F( f ) : U → A f ∗ (U ).
(69)
All results in this subsection may be internalized in any topos; for example, a covering relation on an internal meet semilattice L in a topos T is simply a subobject of L × L , where is the subobject classifier in T . The defining properties of a covering relation are then interpreted in the internal language of T . Proposition 19 holds in this generality, since its proof is constructive; see especially [3]. 22 This even works in case that L is just a set preordered by x y when f (x) f (y). 23 If L and M have top elements and , respectively, then this condition may be replaced by f ∗ ( ) = L M L M . 24 Instead of taking equivalence classes, one could demand as a fourth condition that f ∗ (x) = A f ∗ (x) for
all x ∈ L. 25 In fact, one may extend this into an equivalence F between the category of formal topologies and the category of frames. A formal topology is a generalization of the above triples (L , , ), where is merely required to be a preorder. In this more general case, the axioms on the cover relation take a slightly different form. See [8,69].
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A.1. Localization of the spectrum. We now consider some applications pertinent to the main body of the paper. First, we return to the Gelfand spectrum in Subsect. 2.2. In its presentation by means of generators and relations, Eqs. (18)–(22) play a different role from the regularity rule (23), and we will treat the latter separately. First, for an arbitrary unital commutative C*-algebra A in some topos, consider the distributive lattice L A freely generated by the formal symbols Da , a ∈ Asa (i.e. a is a variable of type Asa ), subject to the relations (18)–(22). As shown in [27,31], L A can be described more explicitly, as follows. Let A+ := {a ∈ Asa | a ≥ 0}. Define p q iff there exists n ∈ N such that p nq. Define p ≈ q iff p q and q p. The lattice operations on A respect ≈ and hence A+ / ≈ is a lattice. We then have LA ∼ = A+ / ≈ .
(70) A+ /
The image of the generator Da in L A , seen as an element of ≈, may also be described explicitly: decomposing a ∈ Asa as a = a + − a − with a ± ∈ A+ in the usual way, under the isomorphism (70) this image coincides with the equivalence class [a + ] in A+ / ≈. In explicit computations [23,47], one may therefore simply identify L A with A+ / ≈ and Da (seen as an element of L A ) with [a + ], respectively. Such computations are also greatly facilitated by the following ‘locality’ theorem. Theorem 20. For each C ∈ C(A) one has L A (C) = L C ,
(71)
where the right-hand side is simply defined in Sets (where it may be computed through (70)). Furthermore, if C ⊆ D, then the map L A (C) → L A (D) given by the functoriality of L A simply maps each generator Dc for c ∈ Csa to the same generator for the spectrum of D (this is well defined because c ∈ Dsa , and this inclusion preserves the relations (18)–(22)); we write this as L C → L D . A proof of this theorem by explicit computation may be found in [22, Thm. 5.2.3]. Here, we give an alternative proof, which requires some familiarity with geometric logic [51,63,80].26 It relies on the following lemmas. Lemma 21. Let T be a geometric theory. For any category C, there is an isomorphism of categories Mod(T, SetsC ) ∼ = Mod(T, Sets)C . Here Mod(T, T ) is the category of T-models in T .27 This lemma may be found in [51, Cor. D.1.2.14]. Lemma 22. The lattice L A generating the spectrum of an internal commutative C*-algebra A is preserved under inverse images of geometric morphisms. 26 Further to our remarks in Subsect. 1.4 on geometric propositional logic, we recall that a geometric predicate logic is a theory whose formulae are as described there (where the atomic formulae may now involve relations and equalities and all the usual structures allowed in first-order logic as well), now also involving finitely many free variables x = (x1 , . . . , xn ), and the existential quantifier ∃, with axioms taking the form ∀x : ϕ(x) → ψ(x). Geometric formulae form an important class of logical formulae, because they are precisely the ones whose truth value is preserved by inverse images of geometric morphisms between topoi. From their syntactic form alone, it follows that their interpretation in the external language is determined locally. 27 This lemma is, in fact, valid for any topos E replacing Sets; Johnstone’s proof just relies on the fact that the functor evC : E C → E that evaluates at C ∈ C is (the inverse image part of) a geometric morphism. The stated generalization follows because the functor (evC )∗ : E → E C given by (evC )∗ (S) = S C(−,C) determines the direct image part [63, Exercise VII.10.1].
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To prove the second lemma, we first use the characterization of the real part Asa of a commutative C*-algebra A as an f-algebra over the rationals (see Definition 9). Moreover, the spectrum of a C*-algebra coincides with the spectrum of the f-algebra of its self-adjoint elements [29]. We claim that the theory of f-algebras is geometric. First, we observe that an f-algebra is precisely a uniquely divisible lattice ordered ring [27, p. 151], since unique divisibility turns a ring into a Q-algebra. The definition of a lattice ordered ring is algebraic: it can be written using equations only. The theory of torsionfree rings, i.e. (nx = 0 x x = 0) for all n > 0, is also algebraic. The theory of divisible rings is obtained by adding infinitely many geometric axioms x ∃ y ny = x, one for each n > 0, to the algebraic theory of rings. A torsion-free divisible ring is the same as a uniquely divisible ring: Suppose that ny = x and nz = x, then n(y − z) = 0, and so y − z = 0. We conclude that the theory of uniquely divisible lattice ordered rings, i.e. f-algebras, is geometric. In particular, Asa and hence A+ are definable by a geometric theory. Secondly, the relation ≈ in (70) is defined by an existential quantification, so that the generating lattice A+ / ≈—and hence by (70) also L A —is preserved under inverse images of geometric morphisms. This proves Lemma 22. Combining Lemma 22 with Lemma 21, we obtain (71) and hence Theorem 20. For later we use, we put an important property of L A on record. Definition 23. A distributive lattice is normal if for all b1 , b2 such that b1 ∨ b2 = there are c1 , c2 such that c1 ∧ c2 = ⊥ and c1 ∨ b1 = and c2 ∨ b2 = . A distributive lattice is called strongly normal if for all a, b there exist x, y such that a b ∨ x and b a ∨ y and x ∧ y = ⊥. Lemma 24. The lattice L A is strongly normal, and hence normal. This lemma is due to Coquand [27, Thm. 1.11], but we give a proof. Proof. First, every strongly normal lattice is normal. To prove this, let b1 ∨ b2 = and choose x, y such that b1 b2 ∨ x, b2 b1 ∨ y, and x ∧ y = ⊥. Then b1 ∨ b2 (b2 ∨ x) ∨ b2 = b2 ∨ x. Similarly, = b1 ∨ y. Second, to check that L A is strongly normal, it is enough to verify the defining property on the generators Da . So we pick a, b in Asa . Then one has Da Da−b ∨ Db , Db Db−a ∨ Da , and Da−b ∧ Db−a = ⊥. " # We now turn to the relation (23), which is to be imposed on L A . It turns out that 23 is a special case of a relation that can be defined on any distributive lattice L by x y iff there exists z such that x ∧ z = ⊥ and y ∨ z = .28 Lemma 25. For all Da , Db ∈ L A , the following are equivalent:29 1. There exists Dc ∈ L A such that Dc ∨ Da = and Dc ∧ Db = ⊥; 2. There exists q > 0 such that Db Da−q . Proof. 1 ⇒ 2: By [27, Cor 1.7] there exists q > 0 such that Dc−q ∨ Da−q = . Hence Dc ∨ Da−q = , so Db = Db ∧ (Dc ∨ Da−q ) = Db ∧ Da−q Da−q . 2 ⇒ 1: Choose Dc := Dq−a . " # 28 Banaschewski and Mulvey write that x is ‘rather below’ y [7], whereas Johnstone [48] says that x is ‘well inside’ y. The notation is usually reserved for the so-called ‘way below’ relation, but this relation coincides with the ‘well inside’ relation on compact regular locales (see [48, p.303] and Theorem 27), so we feel entitled to identify them notationally. 29 In what follows, one may take q > 0 either in Q or in R.
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Hence in what follows we write Db Da iff ∃q>0 Db Da−q ,
(72)
and note with Coquand [27] that in view of the above lemma the relation (23) just states that the frame O() is regular.30 This leads to the following description. For any distributive lattice L, an ideal I ∈ Idl(L) is called regular if I ⊇ ↓ ↓x implies x ∈ I , where ↓ ↓x = {y ∈ L | y x}.
(73)
Expressed in logical language, I is therefore a regular ideal if ∀ y∈L (y x ⇒ y ∈ I ) ⇒ x ∈ I,
(74)
and hence one has the frame RIdl(L) of regular ideals of L, defined by RIdl(L) = {U ∈ Idl(L) | (∀ y∈L y x ⇒ y ∈ U ) ⇒ x ∈ U };
(75)
for the sake of completeness, U ∈ Idl(L) as a predicate on P(L) stands for ⊥ ≡ 0 ∈ U and x ∈ U, y x ⇒ y ∈ U ; x, y ∈ U ⇒ x ∨ y ∈ U.
(76) (77)
Any ideal U ∈ Idl(L) can be turned into a regular ideal AU by means of the closure operation A : DL → DL defined by [24] AU = {x ∈ L | ∀ y∈L y x ⇒ y ∈ U },
(78)
and the canonical map f : L → RIdl(L) is given in terms of (78) by (67). Combining Theorem 27 in [24] (which states that the regular ideals in a normal distributive lattice form a compact regular frame) with Theorem 1.11 in [27] (which applies this to the case at hand), we finally obtain: Theorem 26. The Gelfand spectrum O() of a commutative unital C*-algebra A is isomorphic (as a frame) to the frame of all regular ideals of L A , i.e. O() ∼ = {U ∈ Idl(L A ) | (∀Db ∈L A Db Da ⇒ Db ∈ U ) ⇒ Da ∈ U }.
(79)
In this realization, the canonical map f : L A → O() is given by f (Da ) = {Dc ∈ L A | ∀Db ∈L A Db Dc ⇒ Db Da }.
(80)
By construction, we then have f (Da )
{ f (Da−q ) | q > 0}.
(81)
For later use, also note that (80) implies f (Da ) = ⇔ Da = .
(82)
30 See [48, III.1.1] for this notion. Recall that by the general theory of Banaschewski and Mulvey [7], the spectrum has to be a compact regular frame.
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We may now equip L A with the covering relation defined by (68), given (79) and the ensuing map (80).31 Consequently, by Proposition 17 one has O() ∼ = F(L A , ).
(83)
This description becomes computable by the following two results. Theorem 27. In any topos, the covering relation on L A defined by (68) with (79) and (80) is given by D a U iff for all q > 0 there exists a (Kuratowski) finite U0 ⊆ U such that Da−q U0 . (If U is directed, this means that there exists Db ∈ U such that Da−q Db .) Proof. The easy part is the “⇐” direction: from (81) and the assumption we have f (Da ) f (U ) and hence Da U by definition of the covering relation. In the opposite direction, assume Da U and take some q > 0. From (the proof of) Lemma 25, Da ∨ Dq−a = , hence f (U ) ∨ f (Dq−a ) = . Since O() is compact, there is a finite U0 ⊂ U for which f (U0 ) ∨ f (Dq−a ) = , so that by (82) we have Db ∨ Dq−a = , with Db = U0 . By (19) we have Da−q ∧ Dq−a = ⊥, and hence Da−q = Da−q ∧ = Da−q ∧ (Db ∨ Dq−a ) = Da−q ∧ Db Db =
U0 . # "
Thus we have two alternative expressions for the spectrum: O() ∼ = {U ∈ Idl(L A ) | ∀q>0 Da−q ∈ U ⇒ Da ∈ U }, ∼ = {U ∈ P(L A ) | Da U ⇒ Da ∈ U }.
(84) (85)
The first follows from (79), the second from (64) and (83). To apply this to our functor topos T (A), we apply the Kripke–Joyal semantics for the internal language of the topos T (A) (see [63, Sect. VI.7], whose notation we will use, and [12, Sect. 6.6]) to the statement Da U . This is a formula φ with two free variables, namely Da of type L A , and U of type P(L A ) ≡ L A . Hence in the forcing statement C φ(α) in T (A), we have to insert α ∈ (L A × L A )(C) ∼ = L C × Sub(L A|↑C ), where L A|↑C is the restriction of the functor L A : C(A) → Sets to ↑ C ⊂ C(A). Here we have used (71), as well as the isomorphism [63, Sect. II.8] L A (C) ∼ = Sub(L A|↑C ).
(86)
Consequently, we have α = (Dc , U ), where Dc ∈ L C for some c ∈ Csa (note the change of typefont between the formal variable Da and the actual element Dc ) and U : ↑ C → Sets is a subfunctor of L A|↑C . In particular, U (D) ⊆ L D is defined whenever D ⊇ C, and the subfunctor condition on U simply boils down to U (D) ⊆ U (E) whenever C ⊆ D ⊆ E. 31 Alternatively, writing D U iff U ⊇ ↓D , the covering relation is inductively generated by , as a 0 0 ↓ a explained in [28,79]. The triple (L A , , 0 ) is a flat site as defined in [79].
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Corollary 28. In the topos T (A) the cover of Theorem 27 may be computed locally, in the sense that for any C ∈ C(A), Dc ∈ L C and U ∈ Sub(L A|↑C ), one has C Da U (Dc , U ) iff Dc C U (C), in that for all q > 0 there exists a finite U0 ⊆ U (C) such that Dc−q U0 . Proof. For simplicity, assume that U0 ∈ U , so that we may replace U0 by Db = U0 ; the general case is analogous. We then have to inductively analyze the formula Da U , which, under the stated assumption, in view of Theorem 27 may be taken to mean ∀q>0 ∃Db ∈L A (Db ∈ U ∧ Da−q Db ).
(87)
We now infer from the rules for Kripke–Joyal semantics in a functor topos that:32 1. C (Da ∈ U )(Dc , U ) iff for all D ⊇ C one has Dc ∈ U (D); since U (C) ⊆ U (D), this happens to be the case iff Dc ∈ U (C). 2. C (Db Da )(Dc , Dc ) iff Dc Dc in L C . 3. C (∃Db ∈L A Db ∈ U ∧ Da−q Db )(Dc , U ) iff there is Dc ∈ U (C) such that Dc−q Dc . 4. C (∀q>0 ∃Db ∈L A Db ∈ U ∧ Da−q Db )(Dc , U ) iff for all D ⊇ C and all q > 0 there is Dd ∈ U (D) such that Dc−q Dd , where Dc ∈ L C is seen as an element of L D through the injection L C → L D of Theorem 20, and U ∈ Sub(L A|↑C ) is seen as an element of Sub(L A|↑D ) by restriction. This, however, is true at all D ⊇ C iff it is true at C, because U (C) ⊆ U (D) and hence one can take Dd = Dc for the Dc ∈ L C that makes the condition true at C. " # This brings us to our recipe for computing the spectrum in T (A) locally: Theorem 29. The spectrum O() of A in T (A) can be computed as follows: 1. At C ∈ C(A), the set O()(C) consists of those subfunctors U ∈ Sub(L A|↑C ) such that for all D ⊇ C and all Dd ∈ L D one has Dd D U (D) ⇒ Dd ∈ U (D). 2. In particular, at C·1, the set O()(C·1) consists of those subfunctors U ∈ Sub(L A ) such that for all C ∈ C(A) and all Dc ∈ L C one has Dc C U (C) ⇒ Dc ∈ U (C). 3. The condition that U = {U (C) ⊆ L C }C∈C (A) be a subfunctor of L A comes down to the requirement that U (C) ⊆ U (D) whenever C ⊆ D. 4. The map O()(C) → O()(D) given by the functoriality of O() whenever C ⊆ D is given by truncating an element U :↑ C → Sets of O()(C) to ↑ D. 5. The external description of O() is the frame map π∗ : O(C(A)) → O()(C · 1),
(88)
given on the basic opens ↑ D ∈ O(C(A)) by π∗ (↑ D) = χ↑D : E → (E ⊇ D); E → ⊥ (E D),
(89)
where the top and bottom elements , ⊥ at E are given by {L E } and ∅, respectively. 32 The first one follows from [12, Prop. 6.6.10] and a routine computation. The others are obvious from either [63, Sect. VI.7] or [12, Sect. 6.6].
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Proof. By (85), O() is the subobject of L A defined by the formula φ given by ∀Da ∈L A Da U ⇒ Da ∈ U,
(90)
whose interpretation in T (A) is an arrow from L A to . In view of (86), we may identify an element U ∈ O()(C) with a subfunctor of L A|↑C , and by (90) and Kripke–Joyal semantics in functor topoi (see, in particular, [63, Sect. IV.7]), we have U ∈ O()(C) iff C φ(U ), with φ given by (90). Unfolding this using the rules for Kripke–Joyal semantics and using Corollary 28 (including part 1 of its proof), we find that U ∈ O()(C) iff ∀ D⊇C ∀ Dd ∈L D ∀ E⊇D Dd E U (E) ⇒ Dd ∈ U (E),
(91)
where Dd is regarded as an element of L E . This condition, however, is equivalent to the apparently weaker condition ∀ D⊇C ∀ Dd ∈L D Dd D U (D) ⇒ Dd ∈ U (D);
(92)
condition (91) clearly implies (92), but the latter applied at D = E actually implies the first, since Dd ∈ L D also lies in L E . Items 2 to 4 are now obvious, and the last follows by the explicit prescription for the external description of frames recalled in Subsect. 1.7. Note that each O()(C) is a frame in Sets, inheriting the frame structure of the ambient frame Sub(L A|↑C ). " # An equivalent way to compute the spectrum, which derives from (84) rather than (85), is as follows: O()(C · 1) (and similarly all the other O()(C)) consists of those subfunctors U ∈ Sub(L A ) such that for all C ∈ C(A), U (C) is a regular ideal in L C . To prove this, according to (84) the formula expressing that U ∈ P(L A ) be a regular ideal is U ∈ Idl(L) ∧ ∀Da ∈L A ∀q>0 Da−q ∈ U ⇒ Da ∈ U,
(93)
where the condition U ∈ Idl(L) is spelled out in (76) and (77). The locality of this first condition and of the conjunction in (93) being almost trivial, we concentrate on the second term, calling it φ as usual. We then find that C · 1 φ(U ) iff for all C ∈ C(A), all Dc ∈ U (C), and all D ⊇ C one has: if Dc−q ∈ U (E) for all q > 0 at all E ⊇ D, then Dc ∈ U (D). Now the antecedent automatically holds at all E ⊇ D iff it holds at D, and similarly the if …then statement holds at all D ⊇ C if it holds at C. A.2. Daseinisation map. Our next aim is to construct the Daseinisation map (44), which, read as a frame map, for fixed a ∈ Asa is δ(a)−1 : O(IR) → O().
(94)
We will use the realization (83) of the spectrum O() of A as the frame F(L A , ) defined in the preceding subsection. The second frame we deal with is that of the interval domain O(IR), cf. Subsect. 5.1. Following [69], we construct the interval domain as a frame F(Q ×< Q, ) defined by a covering relation. Here the pertinent meet semilattice Q ×< Q consists of pairs ( p, q) ∈ Q × Q with p < q, ordered by inclusion (i.e. ( p, q) ( p , q ) iff p p and q q ), with a bottom element ⊥ added. The covering relation is defined by ⊥ U for all U and ( p, q) U iff for all rational p , q with
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p < p < q < q there exists ( p , q ) ∈ U with ( p , q ) ( p , q ). In Sets one easily verifies the frame isomorphism F(Q ×< Q, ) ∼ = O(IR),
(95)
so that, in particular, we may regard O(IR) as a subset of the power set P(Q ×< Q). Proposition 30. The functor O(IR) internalizing the interval domain in T (A) is given by O(IR) ∼ = F(Q ×< Q, ).
(96)
O(IR)(C · 1) ∼ = {S ∈ Sub(Q ×< Q) | S(C) ∈ O(IR) for all C ∈ C(A)},
(97)
Explicitly, we have
where O(IR) ⊂ P(Q ×< Q) through (95), as just explained. Furthermore, O(IR)(C) is the truncation of (97) to ↑ C (cf. Theorem 29), and the functorial map O(IR)(C) → O(IR)(D) whenever C ⊆ D is given by truncation. Finally, the external description of O(IR) is given by the frame map ∗ πIR : O(C(A)) → O(IR)(C · 1),
(98)
∗ is given by a formula similar to (89). where πIR
Proof. This follows from a computation analogous to but simpler than the proof of Theorem 29, combined with the remark following (47) and the observation that the condition that S : C(A) → P(Q ×< Q) in the right-hand side of (97) be a subfunctor of Q ×< Q means that we may identify S with a monotone function from C(A) to O(IR). # " We now give the external description δ(a)−1 : O(IR)(C · 1) → O()(C · 1)
(99)
of our Daseinisation map (94). In view of (83) and (96), we will define (99) as a frame map δ(a)−1 : F(Q ×< Q, )(C · 1) → F(L A , )(C · 1).
(100)
Internalizing Proposition 19 to T (A), we proceed by constructing a continuous map d(a) : (L A , ) → (Q ×< Q, ),
(101)
for in that case we may put δ(a)−1 = F(d(a))(C · 1).
(102)
By definition, as a map in the functor topos T (A) the continuous map d(a) is a natural transformation d(a)∗ : Q ×< Q → P(L A ) = L A
(103)
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∗ : Q × Q(C) → L A (C). One has Q × Q(C) ∼ Q × Q, with components d(a)C = < < < ∗ so by (86) the d(a)C are maps ∗ : Q ×< Q → Sub(L A|↑C ). d(a)C
(104)
∗ is determined by d(a)∗ : Q × Q → Sub(L ) as By naturality, d(a)C < A C·1 ∗ d(a)C (r, s)(D) = d(a)∗C·1 (r, s)(D),
d(a)∗
d(a)∗
is determined by for all D ⊇ C, so L A (C) by Theorem 20, we may now define
C·1 .
(105)
Using the description of the lattice
d(a)∗C·1 (r, s) : C → {D f −r ∧ Ds−g | f, g ∈ Csa , f a g},
(106)
which is indeed a subset of L A (C) = L C , as required. Lemma 31. The map (101) defined by (103), (105) and (106) is continuous (in the sense of Definition 18, internalized to T (A)). Proof. First, we claim that d is continuous iff each d(a)C is. Indeed, with regard to the first condition in Definition 18 this is obvious; for the second cf. [63, Prop. I.8.5], and for the third this is true because both covering relations are described locally in C (cf. Corollary 28). By Proposition 19, continuity of d, in turn, would mean that (100) is well defined as a frame map. Thus what remains is to verify that each map d(a)C is continuous in the sense of Definition 18. This is indeed the case; we spare the readers the details.33 " # We now compute the associated frame map (100). The map (103) induces a map Sub(Q ×< Q) → Sub( L A ) as the left adjoint of the pullback in the opposite direction (see, e.g., [63, Exercise I.10]), which by composition with ∪ yields a map Sub(Q ×< Q) → Sub(L A ). The latter restricts to a map F(Q ×< Q, )(C · 1) → F(L A , )(C · 1), which by definition is the map (100) and hence gives the external description (99) of our Daseinisation map.34 This is a frame map by construction; see Lemma 31 and Proposition 19. The associated locale map δ(a) : → IR is our version of the Döring–Isham Daseinisation map. It is unenlightening to write it down explicitly, but we give an appealing special case in Subsect. 5.2. A.3. Localization of integrals. Finally, to compute the interpretation of the locale of integrals we may proceed analogously to the case of the spectrum. The free distributive lattice satisfying the relations in Sect. 4.2 may alternatively be defined by an entailment relation [24]. Consequently, it suffices to describe when ∧A ∨B in the lattice. As proved in [27,30], this holds if a positive combination of elements in A is below a positive combination of elementsin B - in symbols, if there are ri , s j > 0 and ai in A and b j in B such that ri ai s j b j . This is an existential quantification over finite subsets of an f-algebra. The construction of taking the (Kuratowski) finite powerset is geometric, see e.g. [80]. So existential quantification over it is preserved by geometric morphisms. Applying this to the internal C*-algebra and applying Lemma 21 we obtain (cf. Theorem 20): 33 These will appear in the PhD Thesis of the first author. 34 According to (69), this map is just the component of F (d(a)) at C · 1. This component, however, deter-
mines F (d(a)) as a whole, since F (d(a))(C) is just the restriction of F (d(a))(C · 1) to the truncation of each subfunctor S in (97) to ↑ C.
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Proposition 32. The interpretation of the lattice generating the locale of integrals of the internal C*-algebra is given by the functor assigning to each commutative subalgebra C the lattice generating the integrals on C. If C ⊆ D, then the inclusion maps generators of the lattice for C to generators of the lattice for D and preserves relations. The covering relation for the space of integrals is also interpreted locally. A similar statement holds for valuations; see [30,81]. Vickers [78] uses a presentation of locales which is similar to formal topology, but which is tailored for geometric reasoning. B. Related Work The present article was to a considerable extent motivated by the fundamental work of Butterfield and Isham [20,21] and Döring and Isham [34–38]. We refrain from a full comparison, but restrict ourselves to what we see as the key points. As to Butterfield and Isham, our reformulation of the Kochen–Specker Theorem is in their spirit, but we feel our version is more powerful, especially from a logical perspective: our statement that a certain locale has no points has a logical interpretation in terms of (the lack of) models of a certain geometric theory, whereas the original reformulation [20] merely claims that some presheaf lacks global sections (i.e. points). Compared with Döring and Isham, our overall programme and philosophy, as explained in the Introduction, are quite different from theirs: our ambitions are limited to finding a spatial notion of quantum logic (although we do hope that locales in topoi might provide a generalized concept of space that will be useful in quantum gravity). The principal technical differences between the two approaches lie in our use of: 1. covariant functors (instead of contravariant ones); 2. C*-algebras (instead of von Neumann algebras); 3. locales (instead of Stone spaces); 4. internal reasoning and the associated use of Kripke–Joyal semantics; 5. states as internal integrals and the correspondence between integrals and valuations (i.e. measures defined on open sets).35 This has many technical advantages, which has made it possible to obtain our main results (see Subsect. 1.9). Conceptually, the two programs in question overlap to the effect that the Gelfand spectrum O() of the Bohrification A of A provides a pointfree realization of Döring and Isham’s notion of a state object in a topos, whereas the interval domain O(IR) realizes their quantity object, again in the sense of pointfree topology internalized to a suitable topos.36 These objects are linked by observables, which define arrows from the state object to the quantity object. Thus for each a ∈ Asa , our Daseinisation δ(a) : → IR is an observable in the sense of Döring and Isham. Restricted to the special case A = B(H ), our construction resembles the single example of such a topos that both Butterfield, Hamilton and Isham [19] and Döring and Isham [35,36] give, namely that of presheaves over the preorder category of commutative von Neumann subalgebras of B(H ) (ordered by inclusion). Acknowledgement. The authors are indebted to Andreas Döring, Ieke Moerdijk, Chris Mulvey, Isar Stubbe, and Steve Vickers for guidance and useful feedback on talks and earlier drafts of this article. We are exceptionally grateful to the referee of this paper for unusually detailed and helpful comments. 35 An analogous external result has meanwhile been found by Döring [33]. 36 In fact, our use of pointfree techniques leads to topoi of covariant functors just as inevitably as the more
conventional methods in [19–21,34–37] lead to topoi of presheaves.
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Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Commun. Math. Phys. 291, 111–150 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0870-9
Communications in
Mathematical Physics
On Transonic Shocks in a Nozzle with Variable End Pressures Jun Li1, Zhouping Xin2, Huicheng Yin1, 1 Department of Mathematics & IMS, Nanjing University, Nanjing 210093, P.R. China 2 Department of Mathematics and IMS, CUHK, Shatin, N.T., Hong Kong.
E-mail: [email protected] Received: 29 April 2008 / Accepted: 2 June 2009 Published online: 23 July 2009 – © Springer-Verlag 2009
Abstract: In the book, Courant and Friedrichs (Supersonic Flow and Shock Waves. New York: Interscience Publishers, 1948) described the following transonic shock phenomena in a de Laval nozzle: Given the appropriately large receiver pressure pr , if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed. The position and the strength of the shock front are automatically adjusted so that the end pressure at the exit becomes pr . When the end pressure pr varies and lies in an appropriate scope, in general, it is expected that a curved transonic shock is still formed in a nozzle. In this paper, we solve this problem for the two-dimensional steady Euler system with a variable exit pressure in a nozzle whose divergent part is an angular sector. Both existence and uniqueness are established. 1. Introduction and Main Results This paper is concerned with the transonic shock problem in a nozzle when the given variable end pressure at the exit of the nozzle lies in an appropriate scope. In [22–25], the authors have studied the well-posedness or ill-posedness of a transonic shock for the supersonic flow through a general 2-D or 3-D slowly-varying nozzle with an appropriately large exit pressure. However, the end pressure or the position of the shock in [22–25] are either induced by the appropriate boundary conditions on the exit or determined by the ordinary differential equations which result from the assumptions on symmetric properties of the supersonic incoming flow, the nozzle walls and the end pressure. In this paper, under the more natural and physical boundary condition (i.e. the variable exit pressure in a suitable scope), we will study the transonic shock problem Supported by the National Natural Science Foundation of China (No.10571082) and the National Basic Research Programm of China (No.2006CB805902). Supported in part by Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK4028/04P, CUHK4040/06P and RGC Central Allocation Grant CA05/06.SC01.
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when a supersonic flow goes through a 2-D curved nozzle with a straight diverging part. In particular, we will verify the following transonic shock phenomena for the steady Euler flow as illustrated in [8]: Given the appropriately large receiver pressure pe (x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed; moreover, the position and the strength of the shock front are automatically adjusted so that the end pressure at the exit becomes pe (x). To simplify the presentation, we only consider the isentropic gases. In fact, by a slight modification, our discussions are also available to the non-isentropic case. The steady isentropic Euler system in two dimensional space is ⎧ ⎨∂1 (ρu 1 ) + ∂2 (ρu 2 ) = 0, ∂1 (P + ρu 21 ) + ∂2 (ρu 1 u 2 ) = 0, ⎩ ∂1 (ρu 1 u 2 ) + ∂2 (P + ρu 22 ) = 0,
(1.1)
where u = (u 1 , u 2 ), ρ and P are the velocity, the density and the pressure respectively. Moreover, pressure function P = P(ρ) is smooth with P (ρ) > 0 for ρ > 0, and √ the c(ρ) = P (ρ) being the sound speed. For the ideal polytropic gas, the equation of state is given by P = Aρ γ , here A and γ are positive constants, and 1 < γ < 3 (especially γ ≈ 1.4 with respect to the air). 4,α Assume that the nozzle walls 1 and 2 are C −regular for X 0 − 1 ≤ r = x12 + x22 ≤ X 0 + 1 (X 0 > 1 is a fixed constant and the constant α ∈ (0, 1)) and i
consists of two curves i1 and i2 with 11 and 21 including the walls for the converging part of the nozzle, while 12 and 22 are straight line segments so that the divergent part of the nozzle is part of a symmetric angular sector. Assume that i2 is represented by x2 = (−1)i tgθ0 x1 with x1 > 0 and X 0 < r < X 0 + 1, where 0 < θ0 < π2 is sufficiently small. Furthermore, it is assumed that the C 4,α −smooth supersonic incoming − − − flow (ρ0− (x), u − 1,0 (x), u 2,0 (x)) is symmetric near r = X 0 so that ρ0 (x) = ρ0 (r ) and U − (r )x − u i,0 (x) = 0 r i (i = 1, 2) near r = X 0 (this assumption can be easily realized by the hyperbolicity of the supersonic incoming flow and the symmetric property of the nozzle walls for X 0 < r < X 0 + 1, see [13]):
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Suppose that the possible shock and the flow field state behind are denoted by x1 = η(x2 ) and (ρ + (x), u +1 (x), u +2 (x)) respectively. Then the Rankine-Hugoniot conditions on become ⎧ ⎨[ρu 1 ] − η (x2 )[ρu 2 ] = 0, [P + ρu 21 ] − η (x2 )[ρu 1 u 2 ] = 0, (1.2) ⎩ [ρu 1 u 2 ] − η (x2 )[P + ρu 22 ] = 0. In addition, P + (x) satisfies the physical entropy condition (see [8]): P + (x) > P − (x)
x1 = η(x2 ).
on
(1.3)
On the exit of the nozzle, the end pressure is prescribed by P + (x) = Pe + ε P0 (θ )
on
r = X 0 + 1;
(1.4)
here ε > 0 is sufficiently small, θ = ar ctan xx2 , P0 (θ ) ∈ C 3,α [−θ0 , θ0 ] with 1
(3)
P0 (±θ0 ) = P0 (±θ0 ) = 0,
x2 P0 (ar ctan ) 3,α ≤ C, x1 C {(x1 ,x2 ): x12 +x22=X 0 +1,|x2 |≤x1 tanθ0 }
the constant Pe denotes the end pressure for which a symmetric shock lies at the position r = r0 with r0 ∈ (X 0 , X 0 + 1), and the supersonic incoming flow is given by (ρ0− (r ), U0− (r )). For more details, one can see Proposition 2.1 in Sect. 2. Since the flow is tangent to the nozzle walls x2 = (−1)i tgθ0 x1 (i = 1, 2), then u +2 = (−1)i tgθ0 u +1
on
x2 = (−1)i tgθ0 x1 .
(1.5)
Finally, X 0 and θ0 are assumed to be suitably large and small respectively so that X 0 θ0 = 1
and
η0 < θ0 < η0 2
(1.6)
with η0 > 0 suitably small constant. It is noted here that the assumption (1.6) implies that the nozzle wall i2 : x2 = (−1)i tgθ0 x1 is close to the line segment x2 = (−1)i for X 0 ≤ r ≤ X 0 + 1. As will be shown in Sect. 2, under the above assumptions on the nozzle and the symmetric supersonic incoming flow near the throat of the nozzle, there exists a unique symmetric transonic shock solution for the given constant end pressure. Furthermore, the position of the shock location, r = r0 , depends monotonically on the given end pressure. This solution will be the background solution. Let (P0+ (r ), U0+ (r )) be the subsonic part of the background solution for r0 < r < X 0 + 1, which can be extended into the domain {r : X 0 ≤ r ≤ X 0 + 1} and the extension will be denoted by ( Pˆ0+ (r ), Uˆ 0+ (r )). For more details, one can see Proposition 2.1 and Remark 2.2 in Sect. 2. The first main result in this paper is Theorem 1.1 (Uniqueness). Let the assumptions above hold and M0− (X 0 ) ≡ γ +1 U0− (X 0 ) > 2 γ − 2 . Then there exists a constant ε0 = X13 such that for all c(ρ0− (X 0 )) 0 ε ∈ (0, ε0 ], the problem (1.1)–(1.5) has no more than one solution (P + (x), u +1 (x), u +2 (x); η(x2 )) with the following properties:
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(i) η(x2 ) ∈ C 4,α [x21 , x22 ], and η(x2 )− r02 − x22 L ∞ [x 1 ,x 2 ] ≤ C0 X 0 ε, (η(x2 )− r02 − x22)C 3,α [x 1 ,x 2 ] ≤ C0 ε, 2
2
2
2
where (x1i , x2i )(i = 1, 2) stands for the intersection points of x1 = η(x2 ) with x2 = (−1)i tgθ0 x1 for i = 1, 2, and C0 is a positive constant depending on α and the supersonic incoming flow. ¯ + ), and (ii) (P + (x), u +1 (x), u +2 (x)) ∈ C 3,α ( (P + , u +1 , u +2 ) − ( Pˆ0+ , uˆ +1,0 , uˆ +2,0 )C 3,α ( ¯ + ) ≤ C0 ε, where + is the subsonic region given by + = {(x1 , x2 ) : η(x2 ) < x1 < (X 0 + 1)2 − x22 , |x2 | < tgθ0 x1 }, and ( Pˆ0+ , uˆ +1,0 , uˆ +2,0 ) = ( Pˆ0+ (r ), Uˆ 0+ (r ) rx ). Remark 1.1. Besides the uniqueness result described by Theorem 1.1, it will also be shown that the position of the shock depends on the given end pressure monotonically. This will be stated more precisely in Proposition 3.2 in Sect. 3. In addition, the order
X 0 ε in the assumption of η(x2 ) − r02 − x22 L ∞ [x 1 ,x 2 ] comes essentially from the rela2 2 tion between the shock position and the end pressure (one can see Proposition 5.1 and Remark 5.2 in Sect. 5). This actually means that the shock position will move with the order X 0 O(ε) when the end pressure adds or decreases a quantity O(ε) in (1.4). Remark 1.2. Due to the corners in the subsonic region, the requirement of C 3,α regularity for the uniqueness in Theorem 1.1 seems stringent. However, based on such a uniqueness, we can obtain the existence of a transonic solution in the same regularity class, see Theorem 1.2 below and the Appendix. Remark 1.3. The previous uniqueness result in [22,24,25] requires that either the transonic shock curve goes through a fixed point or one solution has special symmetry. Remark 1.4. The assumption that r0 ∈ (X 0 , X 0 + 1) is made to ensure a positive distance between the shock position and the exit of the nozzle; this will be used in the analysis in Sect. 4. Based on Theorem 1.1, we can establish the following existence theorem. Theorem 1.2 (Existence of a transonic shock solution). For the nozzle and the supersonic incoming flow defined as above, the problem (1.1) with (1.2)–(1.5) has a unique transonic shock solution such that (ρ + , u +1 , u +2 ; η(x2 )) satisfies the estimates in Theorem 1.1. Remark 1.5. It should be noted that similar results as in Theorem 1.1 and Theorem 1.2 hold for the 2-D Euler flows with the pressure depending on both the density and the entropy. However, neither the uniqueness nor the existence as in Theorem 1.1 and Theorem 1.2 holds for irrotational flows, see [22,24]. Remark 1.6. Theorem 1.1 can be extended into the 3-D nozzle case, which will be given in our forthcoming paper [15].
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Remark 1.7. For nozzles with part of the symmetric angular sector as the converging part, one can also establish a corresponding theory of uniqueness and existence of a transonic shock lying in the converging part of the nozzle as Theorem 1.1 and 1.2. However, it is shown in [25] that such a transonic shock is dynamically unstable. It is noted that there have been many works on the steady transonic problems (see [3–9,12,14,18–28] and references therein). In particular, in [25], it is shown that under the same assumptions as in this paper on the nozzle and the supersonic incoming flows, there exist two constant pressures P1 and P2 with P1 < P2 , such that if the exit constant pressure Pe ∈ (P1 , P2 ), then the symmetric transonic shock exists uniquely in the diverging part of the nozzle, and the position and the strength of the shock is completely determined by Pe . More importantly, we establish the dynamically asymptotic stability of such a transonic shock for the unsteady Euler system in both 2-D and 3-D cases. Various related results, such as uniqueness with additional assumptions, non-existence, compatibility conditions, and many useful analytical tools on transonic shocks in slightly curved nozzles with given appropriate end pressure at the exit of the nozzle for either steady irrotational flows or steady Euler flows have been established in [22–25], see also [5–7,26]. Next we would like to comment on the proofs of the main results in this paper. In should be noted that in almost all the previous works mentioned above except [15,22], the uniqueness is obtained under the additional assumption that the shock curve goes through a fixed point in advance [7,22–26]. This condition is crucial in the proofs. However, for non-flat nozzles, this additional condition may lead the transonic shock problem to be over-determined [22]. In the present paper, we have found a new way to determine the position of the transonic shock and remove the undesired assumption that the shock curve goes through a fixed point so that the transonic shock problem as described by Courant-Friedrichs is well-posed. Besides the analytical tools developed in [22–25], the new key ingredients in this paper are to establish the monotonic property of the pressure along the nozzle walls and to estimate the gradients of the solution instead of the solution itself so that one can avoid the difficulties induced by the unknown position of the shock; for more detailed explanations, see Sect. 5. It follows from this that the position of the shock can be uniquely determined in Theorem 1.1 when the end pressure is given, and the continuous dependence and monotonic property of the end pressure on the position of the shock are also derived. With these crucial results, we can complete the proofs on Theorem 1.1 and Theorem 1.2. The rest of the paper is organized as follows. In Sect. 2, for the reader’s convenience, the description of the background solution is given although it has been done in [25]. In Sect. 3, we reformulate the 2-D problem (1.1) with the boundary conditions (1.2)–(1.5) so that one can obtain a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system for the angular velocity U2 and an algebraic equation on (ρ, u 1 , u 2 ) along a streamline. In Sect. 4, using the decomposition techniques in Sect. 3, we establish some a priori estimates on the derivatives of difference between two possible solutions. In Sect. 5, based on the estimates given in Sect. 4, and through looking for an ordinary differential inequality along a nozzle wall, we can derive that the end pressure on the wall is monotonic with respect to the position of the shock; thus the position of the shock can be uniquely determined by the end pressure, and the proof of the uniqueness result in Theorem 1.1 is then completed. In Sect. 6, along the nozzle walls, by establishing the continuous dependence of the shock position on the end pressure, we can determine the position of the shock and complete the proof on Theorem 1.2. Finally, in Appendix, we will give a proof on the existence
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of a transonic shock C 3,α solution when the transonic shock is assumed to go through a fixed point and the end pressure is suitably adjusted by a constant. In what follows, we will use the following convention: O(ε) means that there exists a generic constant C1 > 0 independent of X 0 and ε such that C 2,α ≤ C 1 ε. O(ε)
(m > 0) means that there exists a generic constant C2 > 0 independent of X 0 and ε such that O X1m C 1,α ≤ XCm2 . O
1 X 0m
0
0
2. The Background Solution In this section, we will describe the transonic solution to the problem (1.1) with (1.2)– (1.5) when the end pressure is a suitable constant Pe under the assumptions on the nozzle walls and the supersonic incoming flow in Sect. 1. Such a solution is called the background solution and can be obtained by solving the related ordinary differential equations. In fact, the related analysis has been given in Sect. 147 of [8] and the details can be seen in [22,25]. But for the reader’s convenience and later computations in this paper, we still outline it here. Proposition 2.1 (Existence of a transonic shock for the constant end pressure). For the 2-D nozzle and the supersonic incoming flow given in Sect. 1, there exist two constant pressures P1 and P2 with P1 < P2 , which are determined by the incoming flow and the nozzle, such that if the end pressure Pe ∈ (P1 , P2 ), then system (1.1) has a symmetric transonic shock solution − (P0− (r ), u − 1,0 (x), u 2,0 (x)), for r < r0 , (P, u 1 , u 2 ) = (P0+ (r ), u +1,0 (x), u +2,0 (x)), for r > r0 , ± (x) = U0± (r ) xri (i = 1, 2), and (P0± (r ), U0± (r )) are C 4,α − smooth. Moreover, here u i,0 the position r = r0 with X 0 < r0 < X 0 + 1 and the strength of the shock are determined by Pe .
Proof. Let r = r0 be the location of the shock which is to be found. It follows from (1.1) that the supersonic incoming flow (ρ0− (r ), U0− (r ))(X 0 < r < r0 ) and the subsonic flow (ρ0+ (r ), U0+ (r ))(r0 < r < X 0 + 1) satisfy respectively
± ± d dr (rρ0 U0 ) = 0, ± 2 ± ± ± 1 1 2 2 (U0 ) + h(ρ0 ) = 2 (U0 (r0 )) + h(ρ0 (r0 ));
here h(ρ0± ) is the enthalpy given by h (ρ0± ) =
(2.1)
c2 (ρ0± ) . ρ0±
The corresponding Rankine-Hugoniot conditions across the shock r = r0 are [ρ0 U0 ] = 0, (2.2) [ρ0 U02 + P0 ] = 0.
As in [22,25], the proof of the proposition can be divided into four steps.
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Step 1. For the supersonic state (ρ0− (r0 ), U0− (r0 )), there exists a unique subsonic state (ρ0+ (r0 ), U0+ (r0 )) satisfying (2.2). The proof can be found in Sect. 147 of [8], so it is omitted here. Step 2. For a given supersonic state (ρ0− (X 0 ), U0− (X 0 )), (2.1) has a unique supersonic solution (ρ0− (r ), U0− (r )) for r ∈ [X 0 , X 0 + 1]. In fact, it follows from (2.1) that f 1 (ρ0− , U0− , r ) ≡ rρ0− (r )U0− (r ) − C0 = 0, f 2 (ρ0− , U0− , r ) ≡ 21 (U0− (r ))2 + h(ρ0− (r )) − C1− = 0 with C0 = X 0 ρ0− (X 0 )U0− (X 0 ) and C1− = 21 (U0− (X 0 ))2 + h(ρ0− (X 0 )). Since ⎧ − ρ0− (U0− )2 ⎪ ⎪ dρ0 = − , ⎨ dr r ((U0− )2 − c2 (ρ0− )) − ⎪ U0− c2 (ρ0− ) ⎪ dU0 = ⎩ dr r ((U0− )2 − c2 (ρ0− )) and
− 2P (ρ0 ) + ρ0− P (ρ0− ) (U0− )2 d((U0− )2 − c2 (ρ0− )) , = dr r ((U0− )2 − c2 (ρ0− ))
then one has 1 − (U0 (X 0 ))2 − c2 (ρ0− (X 0 )) > 0 for X 0 ≤ r ≤ X 0 + 1. 2 (2.3)
− ∂( f 1 , f 2 ) ∂( f 1 , f 2 ) 2 2 − − In addition >
− − − =r (U0 (r )) −c (ρ0 (r )) and ∂(ρ0 , U0 ) ∂(ρ0− , U0− ) ρ0 (X 0 ),U0 (X 0 ),X 0 0. This, together with the implicit function theorem and (2.3), yields that (2.1) has a unique supersonic solution (ρ0− (r ), U0− (r )) for r ∈ [X 0 , X 0 + 1]. (U0− (r ))2 −c2 (ρ0− (r )) ≥
Step 3. For a given subsonic state (ρ0+ (r0 ), U0+ (r0 )), (2.1) has a unique subsonic solution (ρ0+ (r ), U0+ (r )) for r ∈ [r0 , X 0 + 1]. The proof is similar to that in Step 2, so is omitted. Step 4. The shock position r0 is a continuously decreasing function of Pe when the end pressure Pe lies in an appropriate scope. In fact, it follows from (2.1) and (2.2) that for r ≷ r0 , ± rρ0 (r )U0± (r ) ≡ C0 , (2.4) ± ± ± 1 2 2 (U0 (r )) + h(ρ0 (r ))) ≡ C 1 , here C1± are Bernoulli’s constants. Note that C1− and C1+ are different in general; moreover, C1+ depends on the end pressure P0+ (X 0 + 1) = Pe . Especially, ± r0 ρ0 (r0 )U0± (r0 ) ≡ C0 , (2.5) ± ± ± 1 2 2 (U0 (r0 )) + h(ρ0 (r0 )) ≡ C 1 . Next we derive the dependence of r0 on the end pressure P0+ (X 0 + 1) = Pe .
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It follows from the first equation in (2.4) and the second equation in (2.5) that ⎧ d(ρ0± (r0 )U0± (r0 )) ⎪ dr0 ⎪ ⎨ = −ρ0± (r0 )U0± (r0 ) , dρ0+ (X 0 + 1) r0 dρ0+ (X 0 + 1) (2.6) ⎪ dU + (r ) c2 (ρ0+ (r0 )) dρ0+ (r0 ) dC1+ ⎪ ⎩U0+ (r0 ) + 0 0 + = . dρ0 (X 0 + 1) ρ0+ (r0 ) dρ0+ (X 0 + 1) dρ0+ (X 0 + 1) In addition, by C1+ = and (2.6), one has [ρ0 U02 ]
C02 2(X 0 +1)2 (ρ0+ (X 0 +1))2
+ h(ρ0+ (X 0 + 1)), the second equation in (2.2)
dC1+ dr0 + = ρ (r ) 0 0 r0 dρ0+ (X 0 + 1) dρ0+ (X 0 + 1) =
ρ0+ (r0 )(c02 (X 0 + 1) − (U0+ (X 0 + 1))2 ) . ρ0+ (X 0 + 1)
(2.7)
Since [ρ0 U02 ] < 0 due to [ρ0 U02 + P0 ] = 0 and [P0 ] > 0, then (2.7) implies that r0 is a continuous and strictly decreasing function of the end pressure P0+ (X 0 + 1). Next, we complete the proof on Proposition 2.1. For r0 ∈ [X 0 , X 0 + 1), it follows from Step 2 that there exists a unique supersonic flow in [X 0 , r0 ]. Moreover, due to Step 1 and Step 3, there exists a unique shock at r0 and a unique subsonic flow in [r0 , X 0 + 1]. Thus one can define a function F(r0 ) = P0+ (X 0 + 1) for r0 ∈ (X 0 , X 0 + 1). By Step 4, F(r0 ) is a strictly decreasing and continuous differentiable function on P0+ (X 0 + 1). When r0 = X 0 or r0 = X 0 + 1, one can obtain two different end pressures P1 and P2 with P1 < P2 . Therefore, by the monotonicity of F(r0 ), one can obtain a symmetric transonic shock for P0+ (X 0 + 1) ≡ Pe ∈ (P1 , P2 ). Remark 2.1. By the assumption (1.6) and Eq. (2.4), one can easily conclude that there exists a constant C > 0 independent of X 0 such that for r0 ≤ r ≤ X 0 + 1,
d k U + (r ) d k P + (r ) C
0 0 k = 1, 2, 3, 4. (2.8)
+
≤ k,
k k
dr
dr X0 Remark 2.2. It follows from the analysis in Step 2 and Remark 2.1 that for suitably large X 0 one can extend the subsonic flow (ρ0+ (r ), U0+ (r )) into (ρˆ0+ (r ), Uˆ 0+ (r )) defined for r ∈ (X 0 , X 0 + 1) and satisfies (2.1) and (2.8) on (X 0 , X 0 + 1). This extension will be often used later on. 3. The Reformulation on Problem (1.1) with (1.2)–(1.5) In this section, the nonlinear problem (1.1) with (1.2)–(1.5) will be reformulated so that one can obtain a second order elliptic equation for ρ + (x) and two 2×2 first order systems for the radial speed U1+ and the angular speed U2+ . To this end, as in [22,25], we first derive the relations between (P + , U1+ ) and U2+ on the shock . It is more convenient to use the polar coordinates x1 = r cosθ, (3.1) x2 = rsinθ
Transonic Shocks in a Nozzle with Variable End Pressures
and decompose (u +1 , u +2 ) as
119
+ u 1 = U1+ cosθ − U2+ sinθ, u +2 = U1+ sinθ + U2+ cosθ.
Then, (1.1) and (1.2) become respectively ⎧ ρ+U + 1 ⎪ ⎨∂r (ρ + U1+ ) + r ∂θ (ρ + U2+ ) + r 1 = 0,
ρ + ((U + )2 −(U + )2 )
1 2 ∂ (ρ + (U + )2 + P + ) + r1 ∂θ (ρ + U1+ U2+ ) + = 0, ⎪ r ⎩ r + +1 + 1 2 + + + + + + 2 ∂r (ρ U1 U2 ) + r ∂θ (P + ρ (U2 ) ) + r ρ U1 U2 = 0,
and
⎧ ⎪ ⎪ [ρU1 ] − r˜ (θ ) [ρU2 ] = 0, ⎪ ⎪ r˜ (θ ) ⎨ [ρU12 + P] − r˜ (θ ) [ρU1 U2 ] = 0, r˜ (θ ) ⎪ ⎪ ⎪ ⎪ ⎩[ρU1 U2 ] − r˜ (θ ) [P + ρU 2 ] = 0, 2 r˜ (θ )
where r = r˜ (θ ) stands for the shock in the coordinate (r, θ ). In addition, for any C 1 solution, (3.3) is equivalent to ⎧ ρ + U1+ 1 + + + + ⎪ 0, ⎪ ⎨∂r (ρ U1 ) + r+ ∂θ (ρ U2 ) ++ r = U2 (U2+ )2 ∂r P + + + U1 ∂r U1 + r ∂θ U1 + ρ + − r = 0, ⎪ ⎪ ⎩U + ∂ U + + U2+ ∂ U + + 1 ∂θ P + + U1+ U2+ = 0. 1 r
2
r
θ
2
r ρ+
(3.2)
(3.3)
(3.4)
(3.5)
r
It follows from (3.4) that on r = r˜ (θ ), + (ρ − ρˆ0+ (r0 ))Uˆ 0+ (r0 ) + ρˆ0+ (r0 )(U1+ − Uˆ 0+ (r0 )) = g1 , (P + − Pˆ0+ (r0 )) + (ρ + − ρˆ0+ (r0 ))(U1+ )2 + 2ρˆ0+ (r0 )Uˆ 0+ (r0 )(U1+ − Uˆ 0+ (r0 )) = g2 ; here g1 = g2 =
(ρ + )2 U1+ (U2+ )2 − (ρ0− (r0 )U0− (r0 ) − ρ0− U0− ) − (ρ + − ρˆ0+ (r0 ))(U1+ − Uˆ 0+ (r0 )), [P] + ρ + (U2+ )2 (ρ + U1+ U2+ )2 − (P0− (r0 ) − P − ) − (ρ0− (r0 )(U0− (r0 ))2 [P] + ρ + (U2+ )2 −ρ − (U − )2 ) − ρˆ0+ (r0 )(U1+ − Uˆ 0+ (r0 ))2 . 0
0
Thus, a direct computation yields that on r = r˜ (θ ), + U1 − Uˆ 0+ (r0 ) = g˜ 1 ((U2+ )2 , P0− − P0− (r0 ), U0− − U0− (r0 )), P + − Pˆ0+ (r0 ) = g˜ 2 ((U2+ )2 , P0− − P0− (r0 ), U0− − U0− (r0 ))
(3.6)
with g˜i (0, 0, 0) = 0 for i = 1, 2; here the quantities on the right hand sides of the above will be suitably small. For convenience, the following transformation y1 = r, (3.7) y2 = X 0 θ, will be used to change the fixed walls into y2 = ±1 respectively.
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In addition, the superscripts “+” will be neglected for convenience in case of no confusion. Then, (3.5) and (3.4) can be rewritten respectively as ⎧ ρU1 X0 ⎪ ⎪ ⎪∂ y1 (ρU1 ) + y1 ∂ y2 (ρU2 ) + y1 = 0, ⎨ 2 X 0 U2 ∂ U + ∂ y1 P − U2 = 0, (3.8) U ∂ U + 1 y 1 y 1 1 2 y ρ y ⎪ 1 1 ⎪ ⎪ ⎩U ∂ U + X 0 U2 ∂ U + X 0 ∂ y2 P + U1 U2 = 0, 1 y1 2 y1 y2 2 y1 ρ y1 and
⎧ X ξ (y2 ) ⎪ ⎪ [ρU1 ] − 0 [ρU2 ] = 0, ⎪ ξ(y2 ) ⎪ ⎨ X ξ (y2 ) (3.9) [ρU1 U2 ] = 0, [ρU12 + P] − 0 ξ(y2 ) ⎪ ⎪ ⎪ ⎪ ⎩[ρU1 U2 ] − X 0 ξ (y2 ) [P + ρU 2 ] = 0, 2 ξ(y2 ) where ξ(y2 ) = r˜ Xy20 . Note that the transformation between the coordinate systems (x1 , x2 ) and (y1 , y2 ) keeps the equivalence of C 4,α norm independent of X 0 . So from now on, we will use the coordinate system (y1 , y2 ) instead of (x1 , x2 ). Let y2 = y2 (y1 , β) be the characteristics starting from the point (ξ(β), β) for the 2 first order differential operator U1 ∂ y1 + X 0yU ∂ y2 . Then 1 dy2 (y1 , β) X 0 U2 (y , y (y , β)), = 1 2 1 y1 U1 dy1 (3.10) y2 (ξ(β), β) = β, β ∈ [−1, 1]. It follows from the second and the third equations in (3.8) that the following Bernoulli’s law holds: 1 1 2 2 (U1 ) + (U2 ) + h(ρ) (y1 , y2 (y1 , β)) = G 0 (β) (3.11) 2 2 with G 0 (β) =
1 1 (U1 (ξ(β), β))2 + (U2 (ξ(β), β))2 + h(ρ)(ξ(β), β). 2 2
Next, we derive the governing problems for U1 and U2 . It follows from (3.11) that ⎧ ∂y P d G (β)∂ β(y , y ), ⎪ ⎨ U1 ∂ y2 U1 + U2 ∂ y2 U2 + ρ2 = dβ y2 0 1 2 (3.12) ⎪ ∂ P y X 0 U2 d ⎩ U ∂y U + U ∂y U + 1 (y1 , y2 (y1 , β)) dβ G 0 (β)∂ y2 β(y1 , y2 ); (3.13) =−y 1 1 1 2 1 2 ρ 1 U1
here β(y1 , y2 ) denotes the inverse function of y2 = y2 (y1 , β). Combining (3.12)–(3.13) with the first equation and the third equation in (3.8) yields ∂ y1 U1 = h 1 (P, U1 , U2 , ∂ y1 P, ∂ y2 P), (3.14) ∂ y2 U1 = h 2 (P, U1 , U2 , ∂ y1 P, ∂ y2 P),
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and ⎧ ⎨∂ y1 U2 = h 3 (P, U1 , U2 , ∂ y1 P, ∂ y2 P), ∂ y U2 = h 4 (P, U1 , U2 , ∂ y1 P, ∂ y2 P), ⎩U 2(y , ±1) = 0, 2 1 here h i =
i 0 (1
(3.15)
≤ i ≤ 4) with
(U1 )2 + (U2 )2 , y 1 U2 X 0 ∂ y2 P U1 U2 + = y1 y1 ρ y1 U1 ∂ y1 P X 0 U2 d − + G 0 (β(y1 , y2 ))∂ y2 β(y1 , y2 ) y1 ρ y1 U1 dβ (U2 )2 U1 1 X 0 U2 U1 ∂ y1 P + − + 2 ∂ y2 P , y1 y1 ρc (ρ) y1 ∂y P d U1 (U2 )2 G 0 (β(y1 , y2 ))∂ y2 β(y1 , y2 ) − 2 = + y1 y1 U1 dβ ρ U1 U2 X 0 U2 U1 ∂ y1 P + + ∂ y2 P X 0 ρc2 (ρ) y1 X 0 U2 d G 0 (β(y1 , y)))∂ y2 β(y1 , y2 ) − y1 U1 dβ ∂ y1 P U2 (U1 )2 (U2 )2 X 0 ∂ y2 P U1 U2 + + + − , X 0 U1 y1 ρ y1 y1 ρ X0 U1 U2 X 0 U2 X 0 U1 U1 ∂ y1 P + = ∂ y2 P − 2 ∂ y2 P y1 ρc2 (ρ) y1 y1 ρ U2 ∂ y1 P X 0 U2 d + G 0 (β(y1 , y2 ))∂ y2 β(y1 , y2 ) , − y1 ρ y1 U1 dβ 2 ∂ y2 P U1 U2 U1 (U1 ) U2 + =− 2 ∂ y1 P + ∂ y2 P − U2 ρc (ρ) X 0 y1 y1 ρ X 0 y1 2 ∂ y1 P U2 d (U1 ) + G 0 (β(y1 , y2 ))∂ y2 β(y1 , y2 ) − U1 . − y1 dβ X 0 y1 X 0ρ
0 = 1
2
3
4
Next, we derive the governing problem for the density ρ. It follows from U2 = 0 on y2 = ±1 and the third equation in (3.8) that ∂ y2 ρ = 0
on
y2 = ±1.
(3.16)
Furthermore, applying the first order operator U1 ∂ y1 + Xy10 U2 ∂ y2 to the first equation in (3.8) and subsequently subtracting ∂ y1 (ρ × {the second equation in (3.8)}) and ∂ y2 (ρ × {the third equation in (3.8)}) yield the following boundary value problem for the density:
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⎧ ⎪ ∂ y1 (U12 − c2 (ρ))∂ y1 ρ + X 0 Uy11 U2 ∂ y2 ρ + Xy10 ∂ y2 U1 U2 ∂ y1 ρ + Xy10 ((U2 )2 − (c(ρ))2 )∂ y2 ρ = ⎪ ⎪ ⎪ ⎪ ⎪ ρU 2 ρU 2 ⎪ 2 ⎪ ∂ y2 U1 + y12 − Xy10 ∂ y2 ρU2 ∂ y1 U1 − ρU1 ∂ y1 U2 −∂ y1 Xy10 ρU1 ∂ y2 U2 + y11 − X 0yρU ⎪ ⎪ 1 ⎪ ⎨ 1 , + ∂ y1 U1 + Xy10 ∂ y2 U2 )(U1 ∂ y1 ρ + Xy10 U2 ∂ y2 ρ + ρ∂ y1 U1 + Xy10 ρ∂ y2 U2 + ρU (3.17) y1 ⎪ ⎪ ⎪ P(ρ) − Pˆ + (r0 ) = g˜ 2 ((U2 )2 , P − − P − (r0 ), U − − U − (r0 )) on y = ξ(y ), ⎪ 1 2 0 0 0 ⎪ ⎪ ⎪ ⎪ on y2 = ±1, ⎪ ∂ y2 ρ = 0 ⎪ ⎪ ⎩ P(ρ) = Pe + ε P0 y2 on y1 = X 0 + 1. X0
In addition, due to the third equation of (3.9), it holds that ξ (y2 ) =
ξ(y2 )[ρU1 U2 ] . X 0 [ρU22 + P]
(3.18)
Therefore, in order to prove Theorem 1.1, it suffices to show Theorem 3.1. Let the assumptions of Theorem 1.1 hold. Then for ε <
1 , X 03
the free
boundary value problem (3.14)–(3.15) and (3.17) with (3.9) has no more than one solution (P(y), U1 (y), U2 (y); ξ(y2 )) satisfying the following estimates with a uniform constant C > 0 (depending on α and the supersonic incoming flow): (i) ξ(y2 ) ∈ C 4,α [−1, 1], and ξ(y2 ) − r0 L ∞ [−1,1] ≤ C X 0 X 0 ε,
ξ (y2 )C 3,α [−1,1] ≤ Cε.
(ii) (P(y), U1 (y), U2 (y)) ∈ C 3,α (ω¯ + ) satisfies C ∂ yk1 (P, U1 )(y) − ( Pˆ0+ , Uˆ 0+ )(y1 ) C k,α (ω¯ + ) ≤ 2 , k = 0, 1, 2, 3 X0 and ∂ y2 (P, U1 )C 2,α (ω¯ + ) + U2 C 3,α (ω¯ + ) ≤ Cε, where ω+ = {(y1 , y2 ) : ξ(y2 ) < y1 < X 0 + 1, −1 < y2 < 1}. Remark 3.1. It is noted that (i) and (ii) in Theorem 3.1 relax somewhat the conditions (i) and (ii) in Theorem 1.1. To show Theorem 3.1, as in [25], one may reduce the free boundary problem (3.14)– (3.15) and (3.17) with (3.9) into a fixed boundary value problem. Indeed, set y1 − ξ(y2 ) , z1 = X 0 + 1 − ξ(y2 ) (3.19) z 2 = y2 . Then the domain ω+ becomes E + = {(z 1 , z 2 ) : 0 < z 1 < 1, −1 < z 2 < 1}.
(3.20)
Transonic Shocks in a Nozzle with Variable End Pressures
For convenience, one sets ⎧ 1 ⎪ D ≡ y1 = , ⎪ 1 ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) ⎪ 0 ⎪ ⎪ 1 ⎪ ⎨D1 ≡ ∂ y1 = ∂ , X 0 + 1 − ξ(z 2 ) z 1 X 0 (z 1 − 1)ξ (z 2 ) X0 ∂ = ⎪ ∂ ≡ D ⎪ 2 y 2 ⎪ y1 (ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )))(X 0 + 1 − ξ(z 2 )) z 1 ⎪ ⎪ ⎪ X 0 ⎩ ∂ . + ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) z 2
123
(3.21)
Then, (3.9) and (3.14)–(3.15) can be rewritten respectively as ⎧ X ξ (z 2 ) ⎪ ⎪ [ρU1 ](ξ(z 2 ), z 2 ) = 0 [ρU2 ](ξ(z 2 ), z 2 ), ⎪ ⎪ ξ(z 2 ) ⎨ X ξ (z 2 ) [ρU1 U2 ](ξ(z 2 ), z 2 ), [ρU12 + P](ξ(z 2 ), z 2 ) = 0 ξ(z 2 ) ⎪ ⎪ ⎪ ⎪ ⎩[ρU1 U2 ](ξ(z 2 ), z 2 ) = X 0 ξ (z 2 ) [P + ρU 2 ](ξ(z 2 ), z 2 ), 2 ξ(z 2 ) h 1 (P, U1 , U2 , D1 P, D2 P), D1 U 1 = D2 U 1 = h 2 (P, U1 , U2 , D1 P, D2 P)
(3.22)
(3.23)
and ⎧ ⎪ h 3 (P, U1 , U2 , D1 P, D2 P), ⎨ D1 U 2 = h 4 (P, U1 , U2 , D1 P, D2 P), D2 U 2 = ⎪ ⎩U (z , ±1) = 0, 2 1
(3.24)
here hi (i = 1, 2, 3, 4) has the same property as h i . In addition, (3.17) becomes ⎧ ⎪ D1 (U12 − c2 (ρ))D1 ρ + U1 U2 D2 ρ + D2 U1 U2 D1 ρ + (U22 − c2 (ρ))D2 ρ = ⎪ ⎪ ⎪ ⎪ −D1 (D0 ρU12 + D0 ρU22 ) + D0 ρU1 D2 U2 − D1 (ρU1 )D2 U2 + D2 (ρU1 )D1 U2 ⎪ ⎪ ⎪ −D0 ρU2 D2 U1 + D1 (ρU2 )D2 U1 − D2 (ρU2 )D1 U1 ⎪ ⎨ +(D1 U1 + D2 U2 )(U1 D1 ρ + U2 D2 ρ + ρ D1 U1 + ρ D2 U2 + D0 ρU1 ), ⎪ on z 1 = 0, ⎪ P(ρ) − Pˆ0+ (r0 ) = g˜ 2 ((U2 )2 , P − − P0− (r0 ), U − − U0− (r0 )) ⎪ ⎪ ⎪ ∂z 2 ρ = 0 on z 2 = ±1, ⎪ ⎪ ⎪ ⎪ ⎩ P(ρ) = Pe + ε P0 z 2 on z 1 = 1. X0 (3.25) We conclude this section by noting that as a by-product of the analysis for Theorem 3.1 and Theorem 1.1, one can further obtain more estimates on the location of the shock and its monotonic dependence on the end pressure. Indeed, suppose that the problem (1.1) with (1.2)–(1.3) and (1.5) has two C 3,α solutions (ρ, U1 , U2 ; ξ1 ) and (q, V1 , V2 ; ξ2 ) which satisfy the exit pressure conditions Pe + ε P01 (θ ) and Pe + ε P02 (θ ) at r = X 0 + 1 respectively and admit the estimates in Theorem 1.1. In terms of the transformations (3.1), (3.7) and the end conditions of P(ρ) and P(q) can be (3.19), z2 written as P(ρ) = Pe + ε P01 X 0 and P(q) = Pe + ε P02 Xz 20 . Then we can arrive at
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Proposition 3.2. Under the assumptions of Theorem 1.1, if (ρ, U1 , U2 ; ξ1 ) and (q, V1 , V2 ; ξ2 ) are defined as above, then the following estimates hold: 1 1 ξ1 (z 2 ) − ξ2 (z 2 )C 2,α [−1,1] ≤ C |ξ1 (1) − ξ2 (1)| + ε X0 X 02 z2 z2 − P02 C 1,α [−1,1] (3.26) × P01 X0 X0 and ρ − qC 2,α (E + ) + (U1 , U2 ) − (V1 , V2 )C 2,α (E + ) z2 z2 1 1 P01 − P02 C 1,α [−1,1] , (3.27) ≤C |ξ1 (1) − ξ2 (1)| + ε X0 X0 X0 X0 where E + is given in (3.20). Furthermore, if P01 Xz 20 = P02 Xz 20 + C1 with the constant C1 > 0, then ξ1 (z 2 ) < ξ2 (z 2 ).
(3.28)
4. A Priori Estimates on the Solutions of (3.22)–(3.25) In this section, we establish some key a priori estimates on the gradients of the difference between two solutions (ρ, U1 , U2 ; ξ1 (z 2 )) and (q, (3.22)– 1 , V2 ; ξ2 (z 2 )) to the V problem z z 2 2 (3.25) with exit pressure conditions Pe + ε P01 X and Pe + ε P02 X respectively. 0 0 These estimates will play crucial roles in proving Theorem 1.1 and Theorem 1.2. For convenience, let Q = P(q) denote the pressure for the density q, and (D0 , D1 , D2 ) 0 , D 1 , D 2 ) will denote the expressions as in (3.21) corresponding to (ρ, U1 , U2 ; and ( D ξ1 (z 2 )) and (q, V1 , V2 ; ξ2 (z 2 )) respectively. Set ⎧ (Wi , W3 )(z) = (Ui , ρ) (ξ1 (z 2 ) + z 1 (X 0 + 1 − ξ1 (z 2 )), z 2 ) ⎪ ⎪ ⎪ −(Vi , q) (ξ2 (z 2 ) + z 1 (X 0 + 1 − ξ2 (z 2 )), z 2 ) , i = 1, 2, ⎨ W4 = ξ1 (z 2 ) − ξ2 (z 2 ), ⎪ ⎪ j = 1, 2, 3, ⎪ ⎩ M j = ∂z 1 W j , N k = ∂ z 2 Wk , k = 1, 2, 3, 4. A key technical point here is that we will focus on the estimates on M j ( j = 1, 2, 3) and Nk (k = 1, 2, 3, 4) directly, which will be established in a series of Lemmas (Lemma 4.1–Lemma 4.5). In the following lemmas, we always assume that the assumptions in Theorem 3.1 hold. Lemma 4.1.
⎧ ⎪ 1 ⎪ D W4 , − D = O ⎪ 0 0 ⎨ X 02 1 = O(1)W4 ∂z 1 , D1 − D ⎪ ⎪ ⎪ ⎩ D2 − D 2 = O(ε)W4 ∂z 1 + O(1)N4 ∂z 1 + O 1 W4 ∂z 2 . X0
(4.1)
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1 since the other terms can be treated Proof. We only check the estimate on D1 − D similarly. Indeed, ξ1 (z 2 ) − ξ2 (z 2 ) ∂z (X 0 + 1 − ξ1 (z 2 ))(X 0 + 1 − ξ2 (z 2 )) 1 W4 ∂z , = (X 0 + 1 − ξ1 (z 2 ))(X 0 + 1 − ξ2 (z 2 )) 1
1 = D1 − D
1 so the second inequality in (4.1) holds since (X 0 + 1 − ξ1 (z 2 ))(X 0 + 1 − ξ2 (z 2 )) C 1,α ≤ C and ξi (z 2 ) is a small perturbation of r0 due to the assumptions.
Lemma 4.2 (Estimate of N4 ). It holds that N4 C 2,α ≤ Cε (W1 , W2 , W3 )C 1,α + W4 L ∞ + (N1 , ε−1 N2 , N3 )C 1,α . (4.2) Proof. It follows from (3.18) that X 0 ξ1 (z 2 )[ρU22 + P](ξ1 (z 2 ), z 2 ) = ξ1 (z 2 )[ρU1 U2 ](ξ1 (z 2 ), z 2 ), X 0 ξ2 (z 2 )[q V22 + Q](ξ2 (z 2 ), z 2 ) = ξ2 (z 2 )[q V1 V2 ](ξ2 (z 2 ), z 2 ). The difference of these two equations yields
N4 (z 2 ) = O(ε) · (W1 , W2 , W3 , W4 ) + O(ε) · (N1 , ε−1 N2 , N3 , X 0−1 N4 ); N4 (±1) = 0,
here and in what follows, for notational convenience, O(ε) · A denotes the inner product of vectors A and O(ε) whose component is of order ε. Equation (4.2) then follows from this initial value problem for an ODE.
Lemma 4.3 (Estimates of Mi ). C (W1 , ε X 0 W2 , W3 , W4 )C 1,α + Cε(N1 , ε−1 N2 , N3 , N4 )C 1,α , X0 (4.3) −1 ≤ Cε (W1 , (ε X 0 ) W2 , W3 , W4 )C 1,α (4.4) +(εN1 , N2 , ε−1 N3 , (ε X 0 )−1 N4 )C 1,α .
(M1 , M3 )C 1,α ≤ M2 C 1,α
Proof. Rewrite (3.8) as ⎧ ⎪ 1 (ρU1 ) + D 2 (ρU2 ) + D 0 (ρU1 ) = O 1 W4 + O(ε)N4 D ⎪ ⎪ X0 ⎨ 0 (ρU 2 ) = O 1 W4 + O ε N4 (4.5) 1 P + ρU1 D 1 U1 + ρU2 D 2 U1 − D D 2 X 0 ⎪ X0 ⎪ ⎪ ⎩ρU1 D 2 P + D 0 (ρU1 U2 ) = O(ε)W4 + O 1 N4 . 1 U2 + ρU2 D 2 U2 + D X0
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Then tedious but elementary computations using the assumptions in Theorem 3.1 show that ⎧ 1 W1 = O(X −1 ) · (W1 , ε X 0 W2 , W3 , W4 ) + O(ε) · (M2 , εM3 ) + O(ε) · (ε−1 N2 , N3 , N4 ), W + ρD ⎪ U D ⎪ 0 ⎨ 1 1 3 2 1 W1 = O(X −1 ) · (W1 , ε X 0 W2 , W3 , W4 ) + O(ε2 )M1 + O(ε)N1 + O ε N4 , c (ρ) D1 W3 + ρU1 D 0 X0 ⎪ ⎪ ⎩ ρU D W = O(ε) · (W , (ε X )−1 W , W , W ) + O(ε) · (εM , M ) + O(ε)N + O(1)N + O 1 N . 1
1
2
1
0
2
3
4
2
3
2
3
X0
4
(4.6) 1 W1 , D 1 W2 , D 1 W3 ), the determinant of the coefWith respect to the variables ( D ficient matrix in (4.6) is ρ 2 U1 (c2 (ρ) − U12 ) > 0 for subsonic states, then a direct computation yields Lemma 4.3.
Lemma 4.4 (Estimates of N1 and N2 ). For i = 1, 2, Ni C 1,α ≤ Cε(W1 , W2 , W3 , W4 )C 1,α + Proof. First, we deal with the term
d dβ G 0
C N4 C 2,α + CN3 C 1,α . X0
(4.7)
(β(z 1 , z 2 ))D2 β(z 1 , z 2 ) contained in (3.23)
and (3.24). Let z 21 (s; z) and z 22 (s; z) be the characteristics of (3.10) going through (z 1 , z 2 ) with z 21 (0; z) = β and z 22 (0; z) = β˜ corresponding to (U1 , U2 ) and (V1 , V2 ) respectively, i.e., ⎧ ⎨ dz 21 (s; z) X 0 (X 0 + 1 − ξ1 (z 21 )) U2 ξ1 (z 21 ) + s(X 0 + 1 − ξ1 (z 21 )), z 21 , = ds A1 ⎩z 1 (z ; z) = z , z 21 (0; z) = β, 2 2 1 where A1 = (ξ1 (z 21 ) + s(X 0 + 1 − ξ1 (z 21 )))U1 + U2 X 0 (s − 1)ξ1 (z 21 ). ˜ Set l(s; z) = Similarly, one can define z 22 (s; z) corresponding to (V1 , V2 ), ξ2 and β. Then one has dl 1 1 1 1 2 ds = O(ε)l + O(ε)W1 (s, z 2 ) + O(1)W2 (s, z 2 ) + O(ε)W4 (z 2 ) + O(ε )N4 (z 2 ) , l(0; z) = β − β l(z 1 ; z) = 0.
z 21 (s; z) − z 22 (s; z).
(4.8) Since the solution has the C 3,α −regularities, then all the coefficients in (4.8) are in ˜ C 2,α ; this will lead to the C 2,α estimate of β − β. By (4.8), one has L ∞ ≤ C(εW1 , W2 , εW4 , ε2 N4 ) L ∞ , β − β and
= z1 O(ε)W1 (t, z 1 ) + O(1)W2 (t, z 1 ) + O(ε)W4 (z 1 ) + O(ε 2 )N4 (z 1 ) + O(ε)l(t; z) dt, β −β 2 2 2 2 0s l(s; z) = z1 O(ε)W1 (t, z 21 ) + O(1)W2 (t, z 21 ) + O(ε)W4 (z 21 ) + O(ε 2 )N4 (z 21 ) + O(ε)l(t; z) dt,
(4.9)
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and ˜ C 2,α ≤ Cε, ∂z 1 (β, β)
˜ C 2,α ≤ C. ∂z 2 (β, β)
This yields C 2,α ≤ C(εW1 , W2 , εW4 , ε2 N4 )C 2,α . β − β In addition, set
(4.10)
⎧ ⎨B1 = d G 0 (β(z 1 , z 2 ))D2 β(z 1 , z 2 ), dβ ⎩B2 = d G 2 β 0 (β (z 1 , z 2 ). (z 1 , z 2 )) D dβ
Then, direct computations using (3.11) and (3.21) show that ε ) + O(ε)∂z 2 (β − β ) W4 + O(ε2 )N4 + O(ε2 )∂z 1 (β − β B1 − B2 = O X0 + O(ε) · (W1 , W2 , W3 )(0, β) ). + O(1) · (N1 , εN2 , N3 )(0, β) + O(ε)(β − β Thus, by (4.10), the following estimate holds B1 − B2 C 1,α C 2,α ) ≤ Cε((W1 , W2 , W3 , X 0−1 W4 , εN4 )C 1,α + β − β + C(N1 , εN2 , N3 )(0, z 2 )C 1,α [−1,1] ≤ Cε(W1 , W2 , W3 , X 0−1 W4 , εN4 )C 2,α + C(N1 , εN2 , N3 )(0, z 2 )C 1,α [−1,1] .
(4.11)
Note that the third equation in R-H conditions (3.9) implies that X 0 ξ1 (z 2 ) X 0 ξ1 (z 2 )∂τ [ρU22 + P] X 0 (ξ1 (z 2 ))2 1 = ∂τ (ρU1 U2 ) − − ; ξ1 (z 2 ) (ξ1 (z 2 ))2 ξ1 (z 2 ) [ρU22 + P] here ∂τ denotes the tangent derivative of z 1 = ξ1 (z 2 ) and a similar expression holds for ξ2 (z 2 ). This, together with the first two equations in (3.9), yields for i = 1, 3, 1 −1 N4 . (4.12) Ni (0, z 2 ) = O(ε) · (W1 , W2 , W3 , X 0 W4 ) + O(ε)N2 + O X0 It follows from (4.11), (4.12) and (4.3)–(4.5) that B1 − B2 C 1,α C ≤ Cε(W1 , W2 , W3 , X 0−1 W4 )C 2,α + N4 C 1,α + CεN2 C 1,α X0 ≤ Cε (W1 , W2 , W3 , X 0−1 W4 )C 1,α + (N1 , N2 , N3 )C 1,α +
C N4 C 2,α . X0
(4.13)
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By (3.23) and (3.24), one has
⎧ ⎨D 1 W1 = O 1 · (W1 , ε X 0 W2 , W3 , W4 ) + O(ε) · N3 , X −1 N4 , ε −1 M3 , B1 − B2 , 0 X 0 ⎩D 2 W1 = O 1 · (εW1 , W2 , ε X 0 W3 , W4 ) + O(1) · N3 , X −1 N4 , εM3 , B1 − B2 , 0 X0
(4.14) and
⎧ ⎨D 1 W2 = O(ε) · W1 , (ε X 0 )−1 W2 , W3 , W4 + O(1) · N3 , X −1 N4 , εM3 , ε 2 (B1 − B2 ) , 0 ⎩D 2 W2 = O 1 · (W1 , ε X 0 W2 , W3 , W4 ) + O(ε) · N3 , N4 , ε −1 M3 , B1 − B2 . X0
(4.15) We can now estimate N2 . Since W2 (z 1 , ±1) = 0, then there exists z 2 = z 2 (z 1 ) such that N2 (z 1 , z 2 (z 1 )) = 0 holds. So, (4.15) implies that ⎧ ∂z 1 N2 = O(ε) · (W N1 , (ε X 0 )−1 N2 , (ε X 0 )−1 N3 , N4 , M3 ⎪ 1 , W2 , W3 , W4 ) + O(ε) · ⎪ ⎪ ⎪ ⎪ +O(1) · ε∂z 1 N3 , ∂z 2 N3 , X 0−1 ∂z 2 N4 + O(ε2 ) · B1 − B2 , ∂z 2 (B1 − B2 ) , ⎪ ⎨ ∂z 2 N2 = O(ε) · (W1 , W2 , W3 , W4 ) + O(ε)M3 + O X10 · (N1 , N2 , N3 , N4 ) ⎪ ⎪ ⎪ ⎪ ⎪ +O(ε) · ε−1 ∂z 1 N3 , ∂z 2 N3 , ∂z 2 N4 , (B1 − B2 ), ∂z 2 (B1 − B2 ) , ⎪ ⎩ N2 (z 1 , z 2 (z 1 )) = 0. It should be emphasized here that instead of estimating W2 in (4.15) directly, one differentiates (4.15) with respect to z 2 and uses the structure of the background solution to derive the desired system for N2 (with order O(ε) coefficients for Wi (i = 1, 2, 3, 4) and M3 ). This will make it possible to obtain a control on N2 C 1,α in terms of ε(W1 , W2 , W3 , W4 )C 1,α , X 0−1 (N1 C 1,α + N4 C 2,α ), and N3 C 1,α . Indeed, it follows from this system for N2 , (4.3)–(4.4), (4.13), and a direct computation that N2 C 1,α ≤ C∇z N2 C α ≤ Cε(W1 , W2 , W3 , W4 )C 1,α C + (N1 C 1,α + N4 C 2,α ) + CN3 C 1,α . X0
(4.16)
Next, note that (4.14) shows that
⎧ 1 ⎪ + O N = O(ε) · , W , W , W ∂ ) (W ⎪ z 1 1 2 3 4 ⎪ 1 X 0 · (ε X 0 M3 , N1 , ε X 0 N2 , N3 , N4 ) ⎪ ⎪ ⎪ +O(1)∂z 1 N3 + O(ε) ∂z 2 N3 , X 0 −1 ⎪ 2 N4 , B1 − B2 , ∂z 2 (B1 − B2 ) , ⎨ ∂z ∂z 2 N1 = O(ε) · (W1 , W2 , W3 , W4 ) + O X10 · (ε X 0 M3 , N1 , N2 , ε X 0 N3 , N4 ) ⎪ ⎪ ⎪ ⎪ +O(ε)∂z 1 N3 + O(1)∂z 2 N3 + O X10 ∂z 2 N4 + O(ε)(B1 − B2 ) + O(1)∂z 2 (B1 − B2 ), ⎪ ⎪ ⎪ ⎩ N1 (z 1 , ±1) = 0,
where N1 (0, ±1) = 0 follows from the compatibility condition derived in [22]. Then one can estimate N1 as above to obtain N1 C 1,α ≤ Cε(W1 , W2 , W3 , W4 )C 1,α +
C X 0 (N2 C 1,α
Combining this with (4.16) shows Lemma 4.4. Finally, we estimate N3 .
+ N4 C 2,α ) + CN3 C 1,α .
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Lemma 4.5 (Estimate of N3 ). N3 satisfies C N3 C 1,α ≤ Cε(W1 , W2 , W3 , W4 )C 1,α + (N1 , N2 )C 1,α + N4 C 2,α X 0 z z2 1 2 − P02 )C 1,α [−1,1] . + Cε (P01 (4.17) X0 X0 X0 Proof. Due to (3.25), one has by a direct computation that
⎧ 2 2 U1 U2 D 2 W3 1 W3 + (U 2 − c2 (ρ)) D ⎪ 2 ⎪ D1 (U 1 −c (ρ) D1 W3 + U1 U2 D2 W3 + D2 ⎪ ⎪ ⎪ ⎪ ⎪= O X12 · (W1 , ε X 0 W2 , W3 , W4 ) + O X10 · (M1 , ε X 0 M2 , M3 ) + O(ε)N1 + O X10 N2 ⎪ ⎨ 0 ε 1 + O(ε)N 3 + O X 0 N4 + O X 0 ∂z 2 N4 , ⎪ ⎪ ⎪ ⎪ z ⎪ W3 (1, z 2 ) = P −1 (Pe + ε P01 2 ) − P −1 (Pe + ε P02 z 2 ), ⎪ ⎪ X X0 0 ⎪ ⎩ N3 (z 1 , ±1) = 0.
It follows from this and (4.12) that
⎧ 1 (U 2 − c2 (ρ)) D 2 U1 U2 D 1 N3 + U1 U2 D 2 N3 2 N3 + D 1 N3 + (u 2 − c2 (ρ)) D ⎪ D ⎪ 1 2 ⎪ ⎪ ⎪ 1 −1 N , N , X −1 N ) + O ⎪ = ∂ · (M , N )) + ∂ , (ε X ) N O(ε) · (N ∂ (O(ε) z 3 3 z 1 0 2 3 4 z 4 ⎪ 1 2 2 0 X0 ⎪ ⎪ ⎪ ⎪ +O(ε) · (W1 , W2 , W3 , W4 ) + O(ε) · (M1 , M2 , εM3 ) + O(X 0−2 ) · (N1 , ε X 0 N2 , N3 , N4 ) ⎪ ⎪ ⎨ +O X10 ∂z 1 N1 + O(ε)∂z 1 N2 + O X10 ∂z 1 N3 , ⎪ ⎪ ⎪ ⎪ N3 (0, z 2 ) = O(ε) · W1 , W2 , W3 , X 0−1 W4 + O(ε)N2 + O X1 N4 , ⎪ ⎪ 0 ⎪ ⎪ z2 z2 ⎪ ⎪ N3 (1, z 2 ) = O(ε)W3 (1, z 2 ) + O Xε0 (P01 − P ⎪ 02 X 0 ), X0 ⎪ ⎪ ⎩ N (z , ±1) = 0. 3 1
(4.18) As in [22], it can be verified that the compatible condition holds at the corner points (0, ±1) and (1, ±1). Furthermore, these compatible conditions guarantee the C 1,α regularity of solution. So by the regularity estimates of second order elliptic equations with divergence forms in [1,2], one can arrive at (4.18) and from (4.3)–(4.4) that C N3 C 1,α ≤ Cε(W1 , W2 , W3 , W4 )C 1,α + (N1 , N2 )C 1,α + N4 C 2,α X 0 z z 1 2 2 − P02 )C 1,α [−1,1] , +Cε (P01 X0 X0 X0 which shows Lemma 4.5.
Finally, we point out that all the estimates above can be improved. Remark 4.1. Let (ρ, U1 , U2 ; ξ1 (z 2 )) = (ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 )), Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 )), 0; r0 ) be the background solution and (q, V1 , V2 ; ξ2 (z 2 )) be any solution to the problem (3.22)-(3.25) as before. Then the corresponding estimates in Lemma 4.2 and Lemma 4.3–Lemma 4.5 can be improved C 3,α and C 2,α respectively. This fact is used to get the high regularity estimates in Theorem 1.2. Indeed, by Proposition 2.1, in this case, (ρ, U1 , U2 ; ξ1 (z 2 )) is C 4,α -smooth. It follows that the coefficients of l(s; z) in (4.8) are C 3,α -smooth. Hence, one may obtain the desired higher regularity estimates just following the proofs of Lemma 4.2–Lemma 4.5.
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Based on Lemma 4.1–Lemma 4.5, the uniqueness result of Theorem 3.1 (and thus Theorem 1.1) will be proved in next section. 5. Proof of Theorem 1.1. Due to the equivalence between Theorem 1.1 and Theorem 3.1, it suffices to prove Theorem 3.1 only. In Sect. 4, we have established a priori estimates for the gradients of the solution instead of the solution itself. If trying to derive a priori estimates on the solution itself, one then can obtain from (4.6) and (4.15) that M3 C 1,α ≤ C1 N2 C 1,α + some positive terms with small coefficients and N2 C 1,α ≤ C2 M3 C 1,α + some positive terms with small coefficients, with C1 and C2 being some order one positive constants. Thus, it seems difficult to get any useful information on M3 and N2 . To overcome this difficulty, we derive the gradient estimates on the solution instead of the solution itself. Furthermore, we also estimate N3 instead of M3 from the corresponding second order elliptic equation (4.18) to avoid the difficulties caused by the constant Pe in the variable end pressure. Combining these estimates with properties of the background solution, we can derive the monotonic and continuous dependence between the shock position and the exit pressure along the nozzle wall, which will be crucial in proving Theorem 1.1 and Theorem 1.2. Assume that there exist two solutions (ρ, U1 , U2 ; ξ1 ) and (q, V1 , V2 ; ξ2 ) to the problem (3.22)–(3.25). First, we intend to show ξ1 (1) = ξ2 (1) holds by contradiction. Otherwise, without loss of generality, one may assume ξ1 (1) < ξ2 (1).
(5.1)
Under this assumption, it will be shown that the corresponding end pressures are different, which is contradictory with (1.4). Indeed, we have first − 2γ +1 − 2 U (X ) Lemma 5.1. Under the assumption (5.1) and M0− (X 0 ) ≡ 0 − 0 > , it c(ρ0 (X 0 ) γ holds that ρ(ξ1 (1), 1) > q(ξ2 (1), 1).
(5.2)
Proof. Note that the background supersonic solution (ρ0− (y1 ), U0− (y1 )) satisfies ⎧ − dρ0 ρ0− (M0− )2 ⎪ ⎪ = , ⎪ ⎪ dy1 ⎪ y1 (1 − (M0− )2 ) ⎪ ⎪ ⎪ dU − ⎨ U0− 0 (5.3) dy1 = − y1 (1 − (M − )2 ) , 0 ⎪ ⎪ ⎪ γ −1 − 2 ⎪ ⎪ (M0 ) ) M0− (1 + ⎪ d M0− ⎪ 2 ⎪ =− , ⎩ dy 1 y1 (1 − (M0− )2 )
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U0− (y1 ) denotes the Mach number of the supersonic incoming flow. c(ρ0− (y1 )) This yields that for large X 0 and y ∈ (X 0 − 1, X 0 + 1), 1 − − − − − − . (ρ0 , U0 , M0 )(y1 ) = (ρ0 , U0 , M0 )(X 0 ) + O X0
here M0− (y1 ) =
In addition,
⎧ − − 2 − − − 2 ⎪ ⎨ d(ρ0 (U0 ) + P0 ) = − ρ0 (U0 ) < 0, y1 dy1 − − − − d(ρ ρ U ) U ⎪ ⎩ 0 0 = − 0 y 0 < 0. dy1 1
(5.4)
Next, we analyze the relation between the density ρ(y1 , 1) and the shock position (y1 , 1). Since U2 = 0 for y2 = ±1, then the Rankine-Hugoniot conditions (3.9) imply that [ρU1 ](y1 , 1) = 0,
[ρU12 + P](y1 , 1) = 0.
This yields for the polytropic gas, A(ρ(y1 , 1))γ +1 − B(y1 )ρ(y1 , 1) + C(y1 ) = 0, P0− (y1 ) + ρ0− (y1 )(U0− (y1 ))2
with B(y1 ) = It follows from (5.5) that
and C(y1 ) =
(5.5)
(ρ0− (y1 )U0− (y1 ))2 .
dρ(y1 , 1) ρ(y1 , 1)B (y1 ) − C (y1 ) = . dy1 ρ(y1 , 1)(c2 (ρ(y1 , 1)) − U12 ) In addition, (5.4) shows that ρ(y1 , 1)B (y1 ) − C (y1 ) =
ρ0− (y1 )(U0− (y1 ))2 (2ρ0− (y1 ) − ρ(y1 , 1)). y1
Next, elementary calculations show that 2ρ0− (y1 ) < ρ(y1 , 1) for y1 ∈ (X 0 , X 0 + 1). Indeed, set f (x) = Ax γ +1 − B(y1 )x + C(y1 ),
(5.6)
then by the expressions of B(y1 ) and C(y1 ), it holds that f (ρ0− (y1 )) = 0,
f (ρ0− (y1 )) < 0,
f (x) > 0 for x > 0,
f (+∞) = +∞,
so there exists a unique point ρ(y1 , 1) > ρ0− (y1 ) such that f (ρ(y1 , 1)) = 0, namely, (5.5) holds. On the other hand, noting that M0− (y1 ) > M0− (X 0 ) > 1 due to (5.3), one has for large X 0 , γ +1 −2 2 − (M0− (y1 ))2 < 0, f (2ρ0− (y1 )) = ρ0− (y1 )P0− (y1 ) γ γ +1 − where one has used the assumption M0 (X 0 ) > 2 γ − 2 .
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Thus, one concludes that ρ(y1 , 1) > 2ρ0− (y1 ). This implies that
dρ(y1 ,1) dy1
< 0, and consequently ρ(ξ1 (1), 1) > q(ξ2 (1), 1).
Remark 5.1. It follows from the assumption (5.1) and Lemma 5.1 that W3 (0, 1) > 0 holds in Sect. 4. This property will play an important role in proving Theorem 3.1. Next, we establish some estimates which will be used to derive the monotonic property of the shock position on the end pressure. Lemma 5.2. For ε0 ≤
1 X 03
in Proposition 3.2, the following estimates hold
⎧ z2 z2 ⎪ (W1 , W3 , X 0−1 W4 , M1 , M3 )C 1,α ≤ C|W3 (0, 1)| + Cε X1 P01 − P02 C 1,α [−1,1] , ⎪ X X0 ⎪ 0 0 ⎪ ⎪ ⎨ (W , M ) 1,α ≤ C |W (0, 1)| + Cε 1 P z2 − P z2 C 1,α [−1,1] , 2 2 C 3 01 X 0 02 X 0 X0 X0 4 ⎪ ⎪ z2 z2 C 1 ⎪ ⎪ P01 − P02 C 1,α [−1,1] . Ni C 1,α ≤ |W3 (0, 1)| + Cε ⎪ ⎩ X X X X 0
i=1
0
0
0
(5.7) Proof. As in the derivation of (5.5), one may obtain that A(ρ(ξ1 (1), 1))γ +1 − B(ξ1 (1))ρ(ξ1 (1), 1) + C(ξ1 (1)) = 0.
(5.8)
Same expression holds for (q, V1 , ξ2 ). Then, A(ρ(ξ1 (1), 1))γ +1 − ρ(ξ1 (1), 1)A(q(ξ2 (1), 1))γ −q(ξ2 (1), 1)V12 (ξ2 (1), 1)(ρ(ξ1 (1), 1) − q(ξ2 (1), 1)) = ρ(ξ1 (1), 1)(B(ξ1 (1)) − B(ξ2 (1))) − (C(ξ1 (1)) − C(ξ2 (1))). This, together with (5.4) and the definitions of B(y1 ) and C(y1 ), yields a0 W3 (0, 1) =
ρ0− (ξ˜ )(U0− )2 (ξ˜ ) (2ρ0− (ξ˜ ) − ρ(ξ1 (1), 1))W4 (1), ξ˜
(5.9)
for some ξ˜ between ξ1 (1) and ξ2 (1), and ρ(ξ1 (1), 1)(P(ρ(ξ1 (1), 1)) − P(q(ξ2 (1), 1))) − q(ξ2 (1), 1)V12 (ξ2 (1), 1) ρ(ξ1 (1), 1) − q(ξ2 (1), 1) q(ξ2 (1), 1)(P(ρ(ξ1 (1), 1)) − P(q(ξ2 (1), 1))) − q(ξ2 (1), 1)V12 (ξ2 (1), 1) ≥ ρ(ξ1 (1), 1) − q(ξ2 (1), 1) > 0.
a0 =
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Then under the assumptions of Lemma 5.1, it follows from (5.9) and the R-H conditions for (ρ, U1 ) and (q, V1 ) respectively that |W1 (0, 1)| ≤ C|W3 (0, 1)|, C |W3 (0, 1)| ≤ |W4 (1)|. X0
|W4 (1)| ≤ C X 0 |W3 (0, 1)|, (5.10)
Since W1 C 1,α W2 C 1,α W3 C 1,α W4 C 1,α then one has by (5.10) that ⎧ ⎪ W1 C 1,α ⎪ ⎪ ⎨W 2 C 1,α ⎪ W 3 C 1,α ⎪ ⎪ ⎩W 4 C 1,α
≤ ≤ ≤ ≤
≤ ≤ ≤ ≤
|W1 (0, 1)| + M1 C 1,α + N1 C 1,α , M2 C 1,α + N2 C 1,α , |W3 (0, 1)| + M3 C 1,α + N3 C 1,α , |W4 (1)| + N4 C 1,α , C|W3 (0, 1)| + M1 C 1,α + N1 C 1,α , M2 C 1,α + N2 C 1,α , |W3 (0, 1)| + M3 C 1,α + N3 C 1,α , C X 0 |W3 (0, 1)| + N4 C 1,α .
(5.11)
On the other hand, it follows from Lemma 4.2–Lemma 4.5 that (N1 , N2 , N3 )C 1,α + N4 C 2,α ≤ Cε((W1 , W2 , W3 , W4 )C 1,α + (M1 , M2 , M3 )C 1,α ) z2 z2 Cε − P02 C 1,α [−1,1] , P01 + X0 X0 X0 C (W1 , ε X 0 W2 , W3 , W4 )C 1,α + CεM2 C 1,α (M1 , M3 )C 1,α ≤ X0 C (N1 , ε X 0 N2 , N3 )C 1,α + N4 C 2,α , + X0 C M2 C 1,α ≤ Cε(W1 , W3 , W4 )C 1,α + W2 C 1,α X0 + Cε((M1 , M3 , N1 , N2 )C 1,α C +N4 C 2,α ) + N2 C 1,α . X0 This implies that (N1 , N2 , N3 )C 1,α + N4 C 2,α ≤ Cε(W1 , W2 , W3 , W4 )C 1,α z2 z2 1 P01 − P02 C 1,α [−1,1] , + Cε X0 X0 X0 C (W1 , W3 , W4 )C 1,α + CεW2 C 1,α (M1 , M3 )C 1,α ≤ X0 z2 z2 1 P01 − P02 C 1,α [−1,1] , + Cε X0 X0 X0 C W2 C 1,α M2 C 1,α ≤ Cε(W1 , W3 , W4 )C 1,α + X0 z2 z2 1 P01 − P02 C 1,α [−1,1] . + Cε X0 X0 X0
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Consequently, combining this with (5.11) yields (5.7), which completes the proof of Lemma 5.2.
Now we are ready to prove Theorem 3.1. Proof of Theorem 3.1. Under the assumption of P01 prove
z2 X0
= P02
z2 X0
, it suffices to
W1 = W2 = W3 = W4 = 0. It can be verified directly that (4.6) may be rewritten as ⎧ ⎪ 1 W1 = a1 W4 + O 1 · (W1 , ε X 0 W2 , W3 ) + O(ε) · (M2 , εM3 ) 1 W3 + ρ D U1 D ⎪ ⎪ X0 ⎪ ⎪ ⎨ +O(1)N2 + O(ε) · (N3 ,N4 ), 1 W3 + ρU1 D 1 W1 = a2 W4 + O 1 · (W1 , ε X 0 W2 , W3 ) + O(ε2 )M1 (5.12) c2 (ρ) D ⎪ X0 ⎪ ⎪ ⎪ ⎪ ⎩ +O(ε)N1 + O Xε0 N4 , where 1 ∂z (ρU1 ) (X 0 + 1 − ξ1 (z 2 ))(X 0 + 1 − ξ2 (z 2 )) 1 (1 − z 1 )ρU12 ε + , +O (ξ1 (z 2 ) + z 1 (X 0 + 1 − ξ1 (z 2 )))(ξ2 (z 2 ) + z 1 (X 0 + 1 − ξ2 (z 2 ))) X0 1 ε . (c2 (ρ)∂z 1 ρ + ρU1 ∂z 1 U1 ) + O a2 = − (X 0 + 1 − ξ1 (z 2 ))(X 0 + 1 − ξ2 (z 2 )) X0 a1 = −
Then, it follows from (3.21) and (5.12) that for every z 2 ∈ [−1, 1], 1 · (W1 , ε X 0 W2 , W3 ) + O(ε) · (εM1 , M2 , εM3 ) ∂z 1 W3 = a(z)W4 + O X0 +O(1)N2 + O(ε) · (N1 , N3 , N4 ), (5.13) where (X 0 + 1 − ξ2 (1)) (a2 − U1 a1 ) c2 (ρ) − U12 ∂z 1 ρ =− X 0 + 1 − ξ1 (z 2 ) (X 0 + 1 − ξ2 (1))(1 − z 1 )ρU13 − 2 (c (ρ) − U12 )(ξ1 (z 2 ) + z 1 (X 0 + 1 − ξ1 (z 2 )))(ξ2 (z 2 ) + z 1 (X 0 + 1 − ξ2 (z 2 ))) ε . +O X0
a(z) =
Under the assumptions of Theorem 3.1, we have 1 , U1 > 0, ∂z 1 ρ > 0, ∂z 1 ρ = O X0 c2 (ρ) − U12 > 0,
c2 (ρ) − U12 = O(1).
U1 = O(1),
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Hence, a(z) is a negative function in the subsonic domain. Then it follows from Remark 5.1, Lemma 5.2 and (5.13) that for every z 2 ∈ [−1, 1], ∂z 1 W3 = a(z)W4 (z 2 ) + b(z)W3 (0, 1) (5.14) with b(z) L ∞ ≤ O X10 . In addition, W4 (1) < 0 due to assumption (5.1). This means that the term a(z)W4 (1) is always non-negative. Therefore, along the line z 2 = 1, (5.14) yields ∂z 1 W3 ≥ b(z 1 , 1)W3 (0, 1), W3 (0, 1) > 0.
Thus, for suitably large X 0 , W3 (z 1 , 1) > C1 W3 (0, 1) > 0 for some constant C1 > 0. However, this contradicts W3 (1, 1) = 0 due to the end pressure condition (1.4). This implies that the assumption (5.1) is not right. Thus, ξ1 (1) = ξ2 (1) holds, namely, W4 (1) = 0. As a consequence of this and (5.10), W3 (0, 1) = 0. This, together with Lemma 5.2, yields W1 = W2 = W3 = W4 ≡ 0, which completes the proof of Theorem 3.1.
Proof of Proposition 3.2. Estimates (3.26) and (3.27) in Proposition 3.2 follow immediately from 5.2.So it suffices to show (3.28). Lemma z2 z2 z2 − P By P01 X 0 = P02 Xz 20 + C1 , we get P01 02 X 0 C 1,α [−1,1] = 0 in X0 (5.7). Thus, it follows from the third inequality in (5.7) and (5.10) that there exists a generic constant C > 1 such that 1 |W3 (0, 1)| ≤ |W3 (0, z 2 )| ≤ C|W3 (0, 1)|. C First, we claim that ξ1 (1) < ξ2 (1) holds. Otherwise, it follows from the proof of Lemma 5.1 that W4 (1) ≥ 0 and W3 (0, 1) ≤ 0. This, together with (5.14), shows that ∂z 1 W3 ≤ b(z 1 , 1)W3 (0, 1) on z 2 = 1, (5.15) W3 (0, 1) ≤ 0 with b(z 1 , 1) L ∞ = O X10 . Hence W3 (1, 1) ≤ 0 for suitably large X 0 , which contradicts P01 Xz 20 > P02 Xz 20 . Similarly, one can obtain ξ1 (−1) < ξ2 (−1). Next, we show that ξ1 (z 2 ) < ξ2 (z 2 ) for z 2 ∈ [−1, 1]. Note that W4 (z 2 ) = W4 (1) + N4 (˜z 2 )(z 2 −1) for some z˜ 2 ∈ [z 2 , 1]. By (5.7) and (5.10), N4 L ∞ ≤ C X 0−1 |W3 (0, 1)| ≤ C X 0−2 |W4 (1)|. Hence W4 (z 2 ) < 0 holds for all z 2 ∈ [−1, 1] for suitably large X 0 since W4 (1) < 0. So we complete the proof of Proposition 3.2.
In the end of this section, based on Lemma 5.1 and Remark 2.1–Remark 2.2 in Sect. 2, we can give the interesting estimates on differences of two shock positions and the related subsonic flows in the domain {(r, θ ) : X 0 ≤ r ≤ X 0 + 1, −θ0 ≤ θ ≤ θ0 } corresponding to two different background transonic shock solutions.
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+ (r), U ˆ + (r) and (ρˆ + (r ), Uˆ + (r )) Proposition 5.1. For two extended subsonic flows (ρˆ0,1 0,1 0,2 0,2 with r ∈ [X 0 , X 0 + 1] given in Remark 2.2, which correspond to the shock positions r0,1 and r0,2 with r0,i ∈ (X 0 , X 0 + 1), and the constant end pressures P1,e and P2,e respectively, then we have for large X 0 , + (r ), U ˆ + (r )) − ( Pˆ + (r ), Uˆ + (r ))C 4,α [X ,X +1] ≤ C|P2,e − P1,e |, ( Pˆ0,2 0,2 0,1 0,1 0 0 (5.16) |r0,2 − r0,1 | ≤ C X 0 |P2,e − P1,e |.
Remark 5.2. It follows from Proposition 5.1 that if the difference of two end pressures is of order O(ε), then the differences of related shock positions and extended subsonic flows will be of order X 0 O(ε) and O(ε) respectively. In addition, it also implies that the assumptions in Theorem 1.1 are plausible although the real shock position and further the related background transonic shock are not known in advance for such an end pressure condition Pe + O(ε). Proof. Without loss of generality, we assume that X 0 < r0,2 < r0,1 < X 0 + 1 holds true. Then it holds trivially that X 0 − r0,1 X 0 − r0,2 < < 0. X 0 + 1 − r0,1 X 0 + 1 − r0,2 X 0 − r0,2 , 1 . As in Sect. 4, we set Denoted by the interval L = X 0 + 1 − r0,2 ⎧ + (r ˆ + (r0,1 + z 1 (X 0 + 1 − r0,1 )), ⎨W1 (z 1 ) = Uˆ 0,2 0,2 + z 1 (X 0 + 1 − r0,2 )) − U 0,1 + + (r W (z ) = ρˆ0,2 (r0,2 + z 1 (X 0 + 1 − r0,2 )) − ρˆ0,1 0,1 + z 1 (X 0 + 1 − r0,1 )), ⎩ 3 1 W4 = r0,2 − r0,1 .
z1 ∈ L , z1 ∈ L ,
+ (r ), U ˆ + (r ))(i = 1, 2) satisfy Since (ρˆ0,i 0,i
⎧ + + (r )(U ˆ + (r ))2 d ρˆ0,i (r ) ρˆ0,i ⎪ 0,i ⎪ ⎪ = − , ⎪ ⎨ dr + (r ))2 − c2 (ρˆ + (r ))) r ((Uˆ 0,i 0,i + (r ) + (r )c2 (ρˆ + (r )) ⎪ d Uˆ 0,i Uˆ 0,i ⎪ 0,i ⎪ ⎪ = , ⎩ dr + + (r ))) 2 ˆ r ((U0,i (r )) − c2 (ρˆ0,i
r ∈ [X 0 , X 0 + 1], r ∈ [X 0 , X 0 + 1],
then it follows from Remark 2.2 and a direct computation that ⎧ dW1 ⎪ ⎨ = O(X 0−1 ) · (W1 , W3 ) + b1 (z 1 )W4 , z1 ∈ L , dz 1 dW 3 ⎪ ⎩ = O(X 0−1 ) · (W1 , W3 ) + b3 (z 1 )W4 , z1 ∈ L , dz 1
(5.17)
where ⎧ + c2 (ρˆ + ) Uˆ 0,1 ⎪ X0 + 1 0,1 ⎪ ⎪ b (z ) = − ⎪ 1 1 ⎨ + )2 − c2 (ρˆ + ) (r0,2 + z 1 (X 0 + 1 − r0,2 ))(r0,1 + z 1 (X 0 + 1 − r0,1 )) (Uˆ 0,1 0,1 + (U + )2 ˆ ⎪ ρ ˆ + 1 X ⎪ 0 0,1 0,1 ⎪ . ⎪ ⎩b3 (z 1 ) = (r + z (X + 1 − r ))(r + z (X + 1 − r )) ˆ + 2 + ) 0,2 1 0 0,2 0,1 1 0 0,1 (U0,1 ) − c2 (ρˆ0,1
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Obviously, b3 (z 1 ) < 0 and bi (z 1 ) = O(X 0−1 )(i = 1, 3) hold true for large X 0 . By the assumption W4 < 0, Lemma 5.1 and (5.10), one has W3 (0) > 0,
|W1 (0)| ≤ C|W3 (0)|, C |W3 (0)| ≤ |W4 |. X0
|W4 | ≤ C X 0 |W3 (0)|, (5.18)
Thus, it follows from (5.17)–(5.18) and the positivity of b3 (z 1 )W4 that ⎧ −1 z 1 ∈ [0, 1], ⎨W3 (z 1 ) ≥ C(W3 (0) − X 0 W1 L ∞ [0,1] ), −1 W1 L ∞ [0,1] ≤ C(X 0 W3 L ∞ [0,1] + W3 (0)), ⎩ W3 L ∞ [0,1] ≤ C(X 0−1 W1 L ∞ [0,1] + W3 (0)). This derives that W3 (z 1 ) ≥ C(W3 (0) − X 0−2 W3 L ∞ ), W3 L ∞ [0,1] ≤ C W3 (0).
z 1 ∈ [0, 1],
Therefore, W3 (z 1 ) ≥ C W3 (0) for z 1 ≥ 0 and W3 (0) ≤ C W3 (1). On the other hand, by (5.17), one has W1 C 3,α (L) ≤ C(|W1 (0)| + X 0−1 |W4 | + X 0−1 W3 C 2,α (L) ), W3 C 3,α (L) ≤ C(|W3 (0)| + X 0−1 |W4 | + X 0−1 W1 C 2,α (L) ).
(5.19)
Combining (5.19) with (5.18) and 0 < W3 (0) ≤ C W3 (1) yields (W1 , W3 )C 3,α (L) ≤ C|W3 (1)|.
(5.20)
This, together with |W4 | ≤ C X 0 |W3 (0)| in (5.18), yields |W4 | ≤ C X 0 |W3 (1)|. Namely, the second inequality in (5.16) is shown. Next, we prove the first estimate in (5.16). In fact, by Remark 2.1 in Sect. 2 we have + + (( Pˆ0,1 ) , (Uˆ 0,1 ) )C 3,α [X 0 ,X 0 +1] ≤ C X 0−1 .
Thus, + + (r ) − Pˆ0,1 (r )C 3,α [X 0 ,X 0 +1] Pˆ0,2 r − r0,2 + r0,2 + ≤ Pˆ0,2 (X 0 + 1 − r0,2 ) X 0 + 1 − r0,2 r − r0,2 + r0,1 + (X 0 + 1 − r0,1 ) C 3,α [X 0 ,X 0 +1] − Pˆ0,1 X 0 + 1 − r0,2 r − r0,2 + + Pˆ0,1 (r0,1 + (X 0 + 1 − r0,1 )) X 0 + 1 − r0,2 r − r0,1 + − Pˆ0,1 (r0,1 + (X 0 + 1 − r0,1 ))C 3,α [X 0 ,X 0 +1] X 0 + 1 − r0,1 + ≤ C(W3 C 3,α (L) + ( Pˆ0,1 ) C 3,α [X ,X +1] |W4 |)
≤ C|W3 (1)|.
0
0
(5.21)
+ −U ˆ + can be estimated. Then we complete the proof of ProposiAnalogously, Uˆ 0,2 0,1 tion 5.1.
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6. Proof of Theorem 1.2 In this section, based on Theorem 1.1 and the related estimates given in Sect. 4–Sect. 5, we will show the existence result in Theorem 1.2. First, note that if the transonic shock is required to go through some fixed point on the wall, then as in [26], one can prove that problem (1.1) with (1.2)–(1.3) and (1.5) has a unique transonic shock solution when the end pressure Pe + ε P0 (θ ) in (1.4) is adjusted by an appropriate constant. It follows from this that if one can show that there exists a point at the wall such that the shock goes through this point and the corresponding adjustment constant on the end pressure is zero, then Theorem 1.2 will be proved. Next we state an existence result when the shock is assumed to go through a fixed point on the wall, whose proof will be given in the Appendix. Consider the 2-D nozzle and the supersonic incoming flow as givenin Sect. 1. Let (x10 , x10 tanθ0 ) be a given point on the wall of the nozzle with r0 = x10 1 + tan 2 θ0 ∈ (X 0 , X 0 + 1). Denote by Pe ∈ (P1 , P2 ) the constant exit pressure when the shock position is given by r = r0 with P1 and P2 being given in Proposition 2.1 of Sect. 2. Then one has Theorem 6.1. Under the assumptions as in Theorem 1.1, there exists a constant C0 such that the transonic shock problem (1.1) with (1.2), (1.3) and (1.5) has a solution with the following properties: (ρ + , u +1 , u +2 ; η(x2 )) ∈ C 3,α , η(x10 tanθ0 ) P = Pe + ε P0 (θ ) + C0 +
x10 ,
= on
r = X 0 + 1.
(6.1) (6.2) (6.3)
Moreover,the solution satisfies the analogous estimates in Theorem 1.1. In particular, |η(x2 ) − r02 − x22 | ≤ Cε holds true. In terms of the coordinates (y1 , y2 ) in (3.7), Theorem 6.1 can be restated equivalently as follows Theorem 6.1 . Under the assumptions in Theorem 6.1, there exists an appropriate constant C0 such that the free boundary value problem (3.14)–(3.15), (3.9), and (3.14) has a C 3,α solution (ρ, U1 , U2 ; ξ ) satisfying P(ρ) = Pe + ε P0
ξ(1) = r0 , y2 + C0 on X0
(6.4) y1 = X 0 + 1.
(6.5)
Moreover, the solution admits the same estimates as in Theorem 3.1. In particular, |η(y2 ) − r0 | ≤ Cε. It follows from Proposition 3.2 that the adjustment constant C0 in Theorem 6.1 (Theorem 6.1 ) depends continuously on the position where the shock intersects with the wall of the nozzle. More precisely, one has Lemma 6.1 (Continuity and uniqueness). (i) Assume that two variable exit pressures P˜1 and P˜2 have the form (1.4) and satisfy P˜1 = P˜2 + C0 with a constant C0 . Let (ρ, U1 , U2 ; ξ1 ) and (q, V1 , V2 ; ξ2 ) be solutions to the free boundary value problems (3.14)–(3.15), (3.9) and (3.17)
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corresponding to the exit pressure P˜1 and P˜2 respectively and satisfy the corresponding estimates in Theorem 3.1. Then |C0 | ≤
C |ξ1 (1) − ξ2 (1)|, X0
(6.6)
with a uniform constant C. (ii) If the transonic shock goes through a fixed point on the wall of the nozzle, then the corresponding exit pressure is uniquely determined. Namely, if there exist two constants C1 and C2 such that the end pressures of two solutions are P˜1 = Pe + ε P0 (θ ) + C1 and P˜2 = Pe + ε P0 (θ ) + C2 , then C1 ≡ C2 . Remark 6.1. By Theorem 3.1, the solutions corresponding to the variable exit pressure P˜1 and P˜2 are unique respectively. Based on Lemma 6.1, we now prove Theorem 1.2. √ √ Proof of Theorem 1.2. Denote by P¯1 = Pe − X 0 ε and P¯2 = Pe + X 0 ε the exit pressures of the symmetric transonic shock solutions with corresponding shocks at y1 = r1 and y1 = r2 respectively. Then it follows from Remark 5.1 and the uniqueness result in Theorem 1.1 that r1 > r2 . For each fixed point (y1 ∗, 1) with y1 ∗ ∈ [r2 , r1 ], it follows from Theorem 6.1 that there exists a constant C0 such that problem (1.1) with (1.2)–(1.3), (1.5) and the end pressure P = Pe + ε P0 (θ ) + C0 has a unique solution (ρ, U1 , U2 ; ξ(y2 )) which satisfies ξ(1) = y1 ∗ and the estimates in Theorem 3.1. If y1 ∗ = r2 , then this corresponding adjustment constant, C0 , must be positive. Indeed, if not, then C0 ≤ 0. Applying the estimate (3.27) to (ρ, U1 , U2 ; ξ(y2 )) and the background solution (q, V1 , V2 ; r2 ) which corresponds to the constant end pressure P¯2 , and noting that W4 (1) = 0, one has 1 z2 C 1,α < Cε. P0 (6.7) W3 L ∞ ≤ Cε X0 X0 √ On the other hand, W3 (1, 1) = O(1)( P¯2 −(Pe +ε P0 ( X10 )+C0 )) ≥ C X 0 ε > Cε, which contradicts (6.7) for large X 0 . Hence, C0 > 0. Similarly, for y1 ∗ = r2 , the corresponding adjustment constant, C0 , must be negative. It follows from Theorem 6.1 and Lemma 6.1 that C0 is a Lipschitz continuous function of y1 ∗, i.e. C0 = C0 (y1 ∗). We have shown that C0 (r2 ) > 0 and C0 (r1 ) < 0, thus there exists a y10 ∈ (r2 , r1 ) such that C0 (y10 ) = 0. Consequently, it follows from Theorem 6.1 that the problem (1.1)–(1.5) has a transonic shock solution (P(y), U1 (y), U2 (y); ξ(y)) and the transonic shock passes through (y10 , 1). By Theorem 1.1 such a solution is unique. Thus Theorem 1.2 is proved.
Appendix In this section, we will focus on the proof on Theorem 6.1 . In [26], for almost parallel nozzle walls and a special exit pressure boundary condition, when the shock is required to go through a fixed point, it is proved that problem (1.1) with the related boundary conditions has a solution in some weighted Hölder space if the exit pressure is adjusted by an appropriate constant. It should be noted that the exit boundary in [26] is straight; this makes it possible to straighten out both the solid walls and the exit of the nozzle
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simultaneously by a Langrange transformation. This ingredient plays an important role in the proof of the main result in [26]. However, in our case, the exit of the nozzle is curved, so it is related to the solution itself under a Langrange transformation. Thus, in order to overcome this difficulty and also obtain higher regularities (than weighted Hölder regularity) of the solution, we will use a different method. In particular the reformulation of the system (1.1) in Sect. 3 will be used. Before starting to prove Theorem 6.1 , we now state a regularity result for the Laplacian equation with mixed boundary conditions satisfying suitable compatibility conditions at the corners. Lemma A.1. Let ⎧ ⎨u(x1 , x2 ) = f (x1 , x2 ) in = (−1, 1) × (−1, 1), u(x1 , x2 ) = g(x1 , x2 ) on x2 = ±1, ⎩∂ u(x , x ) = 0 on x1 = ±1, x1 1 2
(A.1)
¯ g ∈ C 4,α ( ) ¯ and ∂x1 g(±1, ±1) = ∂x3 g(±1, ±1) = 0, ∂x1 where f ∈ C 2,α ( ), 1 ¯ f (±1, x2 ) = 0, then Eq. (A.1) has a unique solution u(x1 , x2 ) ∈ C 4,α ( ). Proof. First it is noted that the Dirichlet boundary is not empty, so it follows from [16] that (A.1) has a unique solution, ¯ u ∈ C 4,α ( ) ∩ C 4,α ({±1} × (−1, 1)) ∩ C 4,α ((−1, 1) × {±1}) ∩ C 0 ( ). To obtain the higher regularities of the solution at the corners, one can use the standard reflection method such as in [1,2 or 6]. Without loss of generality, we deal only with the corner (−1, −1) as an example since the treatments on other corners are the same. Set −1 ≤ x1 < 1, u(x1 , x2 ), U (x1 , x2 ) = u(−2 − x1 , x2 ), −3 < x1 ≤ −1, f (x1 , x2 ), −1 ≤ x1 < 1, F(x1 , x2 ) = f (−2 − x1 , x2 ), −3 < x1 ≤ −1, −1 ≤ x1 < 1, g(x1 , x2 ), G(x1 , x2 ) = g(−2 − x1 , x2 ), −3 < x1 ≤ −1. Then it follows from the compatibility conditions of f and g that F(x) ∈ C 2,α , G(x) ∈ C 4,α and U (x), F(x), G(x) satisfy in = (−3, 1) × (−1, 1), U (x1 , x2 ) = F(x1 , x2 ) (A.2) U (x1 , x2 ) = G(x1 , x2 ) on x2 = ±1. So it follows from the local regularity estimates in [11], that U (x) ∈ C 4,α in a small ¯ admits the following neighborhood of the point (−1, −1). Hence, u(x) ∈ C 4,α ( ) estimate: uC 4,α ( ) ¯ ≤ C(gC 4,α ( ) ¯ + f C 2,α ( ) ¯ ). Thus, Lemma A.1. is proved.
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Lemma A.2. If the system (3.8), with (3.9) and (1.4)–(1.5), has a solution (ρ(y), U1 (y), U2 (y); ξ(y2 )) with (ρ(y), U1 (y), U2 (y)) ∈ C 3,α and ξ(y2 ) ∈ C 4,α , then the following compatible conditions at the corners hold ⎧ ∂ y ρ(y1 , ±1) = 0, ∂ y32 ρ(y1 , ±1) = 0, ⎪ ⎪ ⎨ 2 ∂ y2 U1 (y1 , ±1) = 0, (A.3) 2 U ⎪ 2 (y1 , ±1) = 0, ∂ y2 U2 (y1 , ±1) = 0, ⎪ ⎩ ξ (±1) = 0, ξ (3) (±1) = 0. Proof. It follows from boundary condition (1.5), the jumping condition, and (3.8) that U2 (y1 , ±1) = 0, ∂ y2 ρ(y1 , ±1) = 0, ξ (±1) = 0. Applying ξ (y2 )∂ y1 + ∂ y2 to the first and the second equations in (3.9) yields ∂ y2 U1 (ξ(±1), ±1) = 0, ∂ y2 ρ(ξ(±1), ±1) = 0. Thus, in terms of the second equation in (3.8), ∂ y2 U1 (y1 , ±1) satisfies U1 ∂ y1 (∂ y2 U1 ) + (∂ y1 U1 + ∂ y2 U1 (ξ(±1), ±1) = 0,
X0 y1 ∂ y2 U2 )∂ y2 U1
=0
on
y2 = ±1,
which implies ∂ y2 U1 (y1 , ±1) = 0. In addition, differentiating the first equation of (3.8) with respect to y2 , one can get ∂ y22 U2 (y1 , ±1) = 0. And taking ξ (y2 )∂ y1 + ∂ y2 on the third equation of (3.9) twice yields ∂ y32 ξ(±1) = 0. The other equalities can be obtained similarly, and then the proof of Lemma A.2 is completed.
Next, we reformulate the problem in Theorem 6.1 for easy presentation. Let (ρ, U1 , U2 ; ξ ) be a solution to (3.8)–(3.9) such that ξ(1) = r0 . In terms of the transformation (3.19), the domain ω+ = {(y1 , y2 ) : ξ(y2 ) < y1 < X 0 + 1, −1 < y2 < 1} is transformed into E + = {(z 1 , z 2 ) : 0 < z 1 < 1, −1 < z 2 < 1}.
(A.4)
2 Set w = U U1 . Then it follows from a direct computation that the system (3.8) with (1.4), (3.6), (3.16) and (3.18) is equivalent to the following problem ⎧ ξ(z 2 )(ρU1 U2 )(ξ(z 2 ), z 2 ) ⎨ξ (z ) = , 2 2 X 0 ((ρU2 )(ξ(z 2 ), z 2 ) + P(ξ(z 2 ), z 2 ) − P0− (ξ(z 2 ))) (A.5) ⎩ξ(1) = r ,
0
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and ⎧ ∂1 w + λ1 ∂2 ρ = F1 (ρ, U1 , U2 , w; ξ ), ⎪ ⎪∂ w − λ ∂ ρ = F (ρ, U , U , w; ξ ), ⎪ ⎪ 2 1 2 1 2 ⎨ 2 2 , P − − P − (r ), U − − U − (r )) P(ρ) = Pˆ0+ (r0 ) + g˜2 ((U ) 0 0 0 0 2 ⎪ z2 ⎪ on z + ε P = 1, P(ρ) = P ⎪ e 0 1 ⎪ X0 ⎩ w=0 on z 2 = ±1,
on
z 1 = 0, (A.6)
where X 0 (X 0 + 1 − r0 ) c2 (ρˆ + (r0 )) , r0 ρˆ + (r0 ) (Uˆ 0+ (r0 ))2 c2 (ρˆ + (r0 )) 1 r0 −1 , λ2 = X 0 ρˆ + (r0 )(X 0 + 1 − r0 ) (Uˆ + (r0 ))2 λ1 =
(A.7)
1
and c2 (ρ) w X 0 (1 − z 1 )ξ (z 2 ) F1 (ρ, U1 , U2 , w; ξ ) = ∂z 1 ρ − − w 2 ∂z 1 ρ ρ ρ(ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 ))) U12 X 0 (X 0 + 1 − ξ(z 2 )) c2 (ρ) 2 − − w −λ1 ∂z 2 ρ, (A.8) ρ(ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 ))) U12 (ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) c2 (ρ) F2 (ρ, U1 , U2 , w; ξ ) = − 1 − λ2 ∂ z 1 ρ X 0 (X 0 + 1 − ξ(z 2 ))ρ U12 (z 1 − 1)ξ (z 2 ) w (z 1 − 1)ξ (z 2 ) ∂z 1 + ∂z 2 ρ − ∂z w − ρ X 0 + 1 − ξ(z 2 ) X 0 + 1 − ξ(z 2 ) 1 −
1 + w2 . X0
(A.9)
After determination of w and ρ, U1 can be obtained through the Bernoulli’s law ⎧ X 0 (z 1 − 1)ξ (z 2 )U2 X 0 (X 0 + 1 − ξ(z 2 ))U2 ⎪ ⎪ ⎪ { U ∂z 1 + ∂z } + 1 ⎪ ⎪ ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ × 21 U12 (1 + w 2 ) + h(ρ) = 0, (A.10) ⎪ ⎪ 1 1 1 ⎪ 2 2 + 2 2 + 2 ⎪ U (1 + w ) + h(ρ) (ξ(z 2 ), z 2 ) = (Uˆ + g1 ) + w (Uˆ 0 + g1 ) ⎪ ⎪ ⎪ 2 1 2 0 2 ⎪ ⎪ ⎪ ⎩ + + h(ρˆ0 (r0 ) + g2 (ξ(z 2 ), z 2 ); here g˜i (i = 1, 2) in (A.6) and (A.10) have the analogous expressions as in (3.6). With a slight abuse of notations, we still set ρ(z) = ρ(ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )), z 2 ), Ui (z) = Ui (ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )), z 2 ), i = 1, 2.
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Now we begin to prove Theorem 6.1 . This will be achieved by using the contractible mapping theorem. To this end, we introduce the iteration spaces as Sσ = {ξ(z 2 ) ∈ C 4,α [−1, 1] : ξ − r0 C 4,α [−1,1] ≤ σ, ξ(1) = r0 , ξ (±1) = ξ (3) (±1) = 0}
(A.11)
and δ = {(ρ(z), U1 (z), U2 (z)) : (ρ1 U1 ) − (ρˆ0+ , Uˆ 0+ )(r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) +U2 C 3,α (E + ) ≤ δ, ∂z 2 U1 (z 1 , ±1) = 0, U2 (z 1 , ±1) = ∂z22 U2 (z 1 , ±1) = 0, ∂z 2 ρ(z 1 , ±1) = ∂z32 ρ(z 1 , ±1) = 0},
(A.12)
where σ > 0 and δ > 0 will be determined later on. The proof of Theorem 6.1 will be divided into five steps.
Step 1. Approximate Shock. For every (q, V1 , V2 ) ∈ δ , the approximate shock is defined as follows: ⎧ (q V1 V2 )(0, z 2 ) ⎨ξ (z ) = ξ(z 2 ) 2 X 0 P(q(0, z 2 )) − P − (ξ(z 2 )) + (q V 2 )(0, z 2 ) , (A.13) 2 0 ⎩ξ(1) = r . 0
Obviously, (A.13) has a unique solution ξ = ξ(z 2 ) ∈ C 4,α ([−1, 1]); moreover, one has ξ (±1) = 0,
ξ (3) (±1) = 0,
(A.14)
and ξ(z 2 ) − r0 C k,α [−1,1] ≤ CV2 C k−1,α (E + ) ,
k = 2, 3, 4.
(A.15)
If δ > 0 is chosen such that Cδ ≤ σ,
(A.16)
ξ(z 2 ) − r0 C 4,α [−1,1] ≤ σ,
(A.17)
then (A.15) yields
namely, ξ(z 2 ) ∈ Sσ . Step 2. Approximate ρ and w. In this Step, we will look for the solution (ρ(z), w(z)) to the following problem: ⎧ 2 ⎪ ∂1 w + λ1 ∂2 ρ = F1 (q, V1 , V2 , V ⎪ V1 ; ξ ), ⎪ ⎪ ⎪ ⎪ 2 ⎪ ∂ w − λ2 ∂1 ρ = F2 (q, V1 , V2 , V ⎪ V1 ; ξ ), ⎪ ⎨2 ρ = ρˆ0+ (r0 ) + g˜2 (V22 (0, z 2 ), P0− (ξ(z 2 )) − P0− (r0 ), U0− (ξ(z 2 )) − U0− (r0 )) ⎪ ⎪ ⎪ ⎪ ⎪P(ρ) = P + ε P z 2 + C ⎪ on z 1 = 1, e 0 0 ⎪ X0 ⎪ ⎪ ⎩ w=0 on z 2 = ±1,
on
z 1 = 0,
(A.18)
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where F1 and F2 are defined by (A.8) and (A.9), and C0 is a constant to be adjusted so that (A.18) has a solution. Let (ρ1 , w1 ) and (ρ2 , w2 ) solve ⎧ ∂1 w1 + λ1 ∂2 ρ1 = F1 (q, V1 , V2 , VV21 ; ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨∂2 w1 − λ2 ∂1 ρ1 = 0, ρ1 = 0 on z 1 = 0, (A.19) ⎪ ρ = 0 on z 1 = 1, 1 ⎪ ⎪ ⎪ on z 2 = −1, ⎪ ⎩w1 = 0 w1 = 0 on z 2 = 1, and ⎧ ⎪∂1 w2 + λ1 ∂2 ρ2 = 0, ⎪ ⎪ ⎪ ∂2 w2 − λ2 ∂1 ρ2 = F2 (q, V1 , V2 , VV21 ; ξ ), ⎪ ⎪ ⎪ ⎨ρ = ρˆ + (r ) + g˜ (V 2 (0, z ), P − (ξ(z ))− P − (r ), U − (ξ(z )) − U − (r )) 2 2 2 0 2 0 0 2 0 0 0 0 0 2 ⎪P(ρ2 ) = Pe + ε P0 Xz 2 + C0 on z = 1, 1 ⎪ 0 ⎪ ⎪ ⎪ ⎪ on z 2 = −1, ⎪w2 = 0 ⎩ on z 2 = 1, w2 = 0
on
z 1 = 0,
(A.20) respectively. Set w = w1 + w2 and ρ = ρ1 + ρ2 . Then (ρ, w) solves (A.18). To solve the elliptic systems (A.19) and (A.20), one may introduce potential functions φ1 (z) and φ2 (z) as follows: ∂1 φ 1 = w 1 ,
∂2 φ1 = λ2 ρ1 ,
φ1 (0, 0) = 0
(A.21)
φ2 (0, 0) = 0.
(A.22)
and ∂1 φ2 = −λ1 ρ2 ,
∂2 φ2 = w2 ,
Then (A.19) becomes 2 ∂1 φ1 + λλ21 ∂22 φ1 = F1 (q, V1 , V2 , φ1 = 0 on ∂ E+,
V2 V1 ; ξ )
in
E+,
(A.23)
while (A.20) is changed into ⎧λ V2 2 2 2 ⎪ in E+, ⎪ λ1 ∂1 φ2 + ∂2 φ2 = F2 (q, V1 , V2 , V1 ; ξ ) ⎪ ⎪ − + 2 ⎪ ∂ φ = −λ ( ρ ˆ (r ) + g ˜ (V (0, z ), P (ξ(z )) − P0− (r0 ), U0− (ξ(z 2 )) ⎪ 1 2 1 0 2 2 2 0 0 2 ⎪ ⎪ − ⎪ −U0 (r0 ))) on z 1= 0, ⎨ z2 −1 on z 1 = 1, ⎪ ∂1 φ2 = −λ1 P (Pe + ε P0 X 0 + C0 ) ⎪ ⎪ ⎪ φ = 0 on z = −1, ∂ 2 2 2 ⎪ ⎪ ⎪ ⎪ on z 2 = 1, ⎪ ∂2 φ2 = 0 ⎩ φ2 (0, 0) = 0. (A.24) First, due to (A.8), one can check that F1 (q, V1 , V2 , VV21 ; ξ ) ∈ C 2,α (E +) and F1 (z 1 , ±1) = ∂22 F1 (z 1 , ±1) = 0, so the compatible conditions for (A.23) are satisfied. Then, similar
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to the proof of Lemma A.1, (A.23) has a unique solution φ1 (z) ∈ C 4,α (E + ) and admits the following estimate: V2 w1 C 3,α (E + ) + ρ1 C 3,α (E + ) ≤ φ1 C 4,α (E + ) ≤ CF1 (q, V1 , V2 , ; ξ )C 2,α (E + ) V1 1 ≤O + δ + σ V2 C 3,α (E + ) + O(δ)q X0 − ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + O(δ + σ )V1 − Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) +O(δ)ξ − r0 C 4,α [−1,1] 1 ≤O + δ + σ δ. (A.25) X0 Furthermore, the following compatible conditions hold ∂22 φ1 (z 1 , ±1) = ∂1 ∂22 φ1 (z 1 , ±1) = ∂24 φ1 (z 1 , ±1) = 0.
(A.26)
Next we solve problem (A.24). It follows from (A.9) and (q, V1 , V2 ) ∈ σ that V2 ; ξ ) ∈ C 2,α (E + ), V1 ∂1 φ2 (1, z 2 ) ∈ C 3,α [−1, 1] F2 (q, V1 , V2 ,
∂1 φ2 (0, z 2 ) ∈ C 3,α [−1, 1],
and V2 ; ξ )(z 1 , ±1) = 0, V1 ∂2k (∂1 φ2 )(0, ±1)) = ∂2k (∂1 φ2 )(1, ±1)) = 0, k = 1, 3. In addition, it can be verified directly that the background solution ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 )), Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 )), 0; r0 satisfies ⎧ λ2 2 ˆ 2ˆ + ˆ+ ⎪ ⎪ λ1 ∂1 φ2 + ∂2 φ2 = F2 (ρˆ0 (r0 + z 1 (X 0 + 1 − r0 )), U0 (r0 + z 1 (X 0 + 1 − r0 )), ⎪ ⎪ ⎪ 0, 0, r0 ) in E+, ⎪ ⎪ ⎪ + (r ) ⎪ ˆ φ ∂ = −λ ρ ˆ on z = 0, ⎨ 1 2 1 0 0 1 (A.27) ∂1 φˆ 2 = −λ1 P −1 (Pe ) on z 1 = 1, ⎪ ⎪ ˆ ⎪ ∂2 φ2 = 0 on z 2 = −1, ⎪ ⎪ ⎪ ˆ2 = 0 ⎪ φ ∂ on z 2 = 1, ⎪ 2 ⎪ ⎩ ˆ φ2 (0, 0) = 0, ∂2 F2 (q, V1 , V2 ,
where ∂1 φˆ 2 = −λ1 ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 )), ∂2 φˆ 2 = 0. So as in [11], the solvability condition for (A.24) is V2 F2 (q, V1 , V2 , ; ξ ) V1 E+ −F2 ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 )), Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 )), 0, 0, r0 dz
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g˜2 V2 (0, z 2 ), P0− (ξ(z 2 )) − P0− (r0 ), U0− (ξ(z 2 )) − U0− (r0 ) + P −1 (Pe ) −1 z2 −1 Pe + ε P0 + C0 dz 2 . −P (A.28) X0
= λ2
1
It is easy to check that (A.28) has a unique solution C0 . Hence, (A.24) has a unique solution φ2 ∈ C 4,α (E + ) with the following estimate: w2 C 3,α (E + ) + ρ2 − ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + |C0 | ≤ Cφ2 − φˆ 2 C 4,α (E + ) ≤ C(F2 (q, V1 , V2 ,
V2 ; ξ ) − F2 (ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 )), V1
Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 )), 0, 0, r0 )C 2,α (E + ) +g˜2 (V22 (0, z 2 ), P0− (ξ(z 2 )) − P0− (r0 ), U0− (ξ(z 2 )) z2 C 3,α [−1,1] ) −U0− (r0 ))C 3,α (E + ) + εP0 X0 1 1 ≤O + δ ξ − r0 C 4,α [−1,1] + O(δ)V2 C 3,α (E + ) + O + δ V1 X0 X0 1 −Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + O +δ+σ X0 z2 C 3,α (E + ) × q − ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + CεP0 X0 1 ≤O + δ + σ (σ + δ) + Cε. (A.29) X0 Meanwhile, it follows from (A.20)–(A.24) that the following compatible conditions hold: ∂2 φ2 (z 1 , ±1) = ∂23 φ2 (z 1 , ±1) = ∂1 ∂2 φ2 (z 1 , ±1) = ∂1 ∂23 φ2 (z 1 , ±1) = 0. (A.30) Due to (A.21), (A.22), (A.25)–(A.26) and (A.29)–(A.30), it holds that ρ − ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + wC 3,α (E + ) + |C0 | 1 1 ≤O +δ+σ δ+O + δ σ + Cε X0 X0
(A.31)
and ∂2 ρ(z 1 , ±1) = ∂23 ρ(z 1 , ±1) = 0,
w(z 1 , ±1) = ∂22 w(z 1 , ±1) = 0.
(A.32)
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Step 3. Approximate U1 . By (A.10), U1 is obtained by solving ⎧ X 0 (z 1 − 1)ξ (z 2 )V2 X 0 (X 0 + 1 − ξ(z 2 ))V2 ⎪ ⎪ ∂z 1 + ∂z } + { V ⎪ 1 ⎪ ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) ξ(z 2 ) + z 1 (X 0 + 1 − ξ(z 2 )) 2 ⎪ ⎪ ⎪ ⎪ 1 2 ⎪ ⎪ × U (1 + w 2 ) + h(ρ) = 0, ⎪ ⎪ 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 U12 (1 + w 2 ) + h(ρ) (0, z 2 ) = 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ 1 ⎪ ⎪ 1 + w 2 (0, z 2 ) Uˆ 0+ (r0 ) + g1 V22 (0, z 2 ), P0− (ξ(z 2 )) − P0− (r0 ), U0− (ξ(z 2 )) − U0− (r0 ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ +h ρˆ + (r ) + g (V 2 (0, z ), P − (ξ(z )) − P − (r ), U − (ξ(z )) − U − (r ) . 0
0
2
2
2
0
2
0
0
0
2
0
0
(A.33) It follows from the characteristics method and the analysis in Sect. 4 that (A.33) has a unique solution U1 = U1 (z) ∈ C 3,α (E + ) such that U1 − Uˆ 0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) ≤ O(1)(ρ − ρˆ0+ (r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + O
1 + δ ξ − r0 C 4,α [−1,1] X0
+O(δ + σ )wC 3,α (E + ) 1 1 ≤O +δ+σ δ+O + δ σ + Cε. X0 X0
(A.34)
Due to (A.32) and the definition of δ , one can check from (A.33) by following the proof of Lemma A.2 that ∂2 U1 (z 1 , ±1) = 0.
(A.35)
Step 4. A mapping on δ . Note that the coefficients of ε in (A.31) and (A.34) depend only on the background solution and then are uniformly bounded. Hence, one can select proper constants σ = O(1)ε > 0 and δ = O(1)ε > 0 such that the solution (ρ, U1 , U2 ; ξ ) obtained in Step 1-Step 3 satisfies ξ − r0 C 4,α [−1,1] ≤ σ, and (ρ1 U1 ) − (ρˆ0+ , Uˆ 0+ )(r0 + z 1 (X 0 + 1 − r0 ))C 3,α (E + ) + |U2 C 3,α (E + ) ≤ δ. This, together with (A.14), (A.32) and (A.35), shows that ξ ∈ Sσ ,
(ρ(z 1 , z 2 ), U1 (z 1 , z 2 ), U2 (z 1 , z 2 )) ∈ δ .
Therefore, for each (q, V1 , V2 ) ∈ δ , by use of Step 1-Step 3, we can define a mapping T from δ into itself by T (q, V1 , V2 ) = (ρ, U1 , U2 ).
(A.36)
In order to prove Theorem 6.1 , it suffices to show that the mapping T is contractible in C 2,α (E + ).
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Step 5. Contractible estimate on the mapping T . For any given (ρ, ˜ U˜ 1 , U˜ 2 ) and (q, ˜ V˜1 , V˜2 ) in δ , set T (ρ, ˜ U˜ 1 , U˜ 2 ) = (ρ, U1 , U2 ),
T (q, ˜ V˜1 , V˜2 ) = (q, V1 , V2 ).
The corresponding approximate shocks ξ1 (z 2 ) and ξ2 (z 2 ) can be obtained from (A.12). As in Sect. 4, define Wi (i = 1, 2, 3), M j ( j = 1, 2, 3), Nk (k = 1, 2, 3) corresponding to (ρ, U1 , U2 ) and (q, V1 , V2 ), and define W˜ i (i = 1, 2, 3, 4), M˜ j ( j = 1, 2, 3), N˜ k (k = V2 2 1, 2, 3, 4) in terms of (ρ, ˜ U˜ 1 , U˜ 2 ; ξ1 ) and (q, ˜ V˜1 , V˜2 ; ξ2 ). In addition, set W5 = U U1 − V1 ˜
˜
and W˜ 5 = U˜ 2 − V˜1 . U1 V2 We first establish some estimates on T . By (A.13), one has in N˜ 4 (z 2 ) = O(δ)W˜ 1 + O(1)W˜ 2 + O(δ)W˜ 3 + O Xδ0 W˜ 4 W˜ 4 (1) = 0.
(−1, 1),
(A.37)
This implies that W˜ 4 C 3,α [−1,1] ≤ CδW˜ 1 C 2,α ( E¯ + ) + CW˜ 2 C 2,α ( E¯ + ) + CδW˜ 3 C 2,α ( E¯ + ) .
(A.38)
By (A.18) and (A.28), one has ⎧ ε+σ 1 ⎪ ∂1 W5 + λ1 ∂2 W3 = O Xσ W˜ 1 + O( + δ 2 )W˜ 3 + O(δ 2 )W˜ 4 + O W˜ 5 ⎪ ⎪ ⎪ 0 X X 0 0 ⎪ ⎪ ⎪ ⎪ +O(δ) M˜ 3 + O(δ) N˜ 3 + O X10 N˜ 4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪∂2 W5 − λ2 ∂1 W3 = O 1 + 1 W˜ 1 + O 1 + δ 2 W˜ 3 + O δ + σ δ + σ δ W˜ 4 ⎪ 2 ⎪ X X 0 0 X X X ⎨ 0 0 0 σ + δ W˜ + O(δ) M˜ + O(δ) N˜ + O(σ )∂ W˜ + O δ N˜ , +O 3 3 z 4 5 5 1 ⎪ X X0 ⎪ 0 ⎪ ⎪ 1 ⎪ W3 = O(σ )W˜ 1 + O(σ )W˜ 2 + O(δσ )W˜ 3 + O X 0 + σ δ W˜ 4 + O(δ) N˜ 4 on z 1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ on z 1 = 1, W3 = O(σ )W˜ 1 + O(σ )W˜ 2 + O(δσ )W˜ 3 + O X 0 + σ δ W˜ 4 + O(δ) N˜ 4 ⎪ ⎪ ⎪ ⎪ ⎪ on z 2 = −1, ⎪ ⎩ W5 = 0 W5 = 0 on z 2 = 1.
(A.39) So following the arguments in Step 2, one can arrive at 1 W3 C 2,α ( E¯ + ) + W5 C 2,α ( E¯ + ) ≤ C + σ W˜ 1 C 2,α ( E¯ + ) + Cσ W˜ 2 C 2,α ( E¯ + ) X0 1 1 ˜ +C + δ W3 C 2,α ( E¯ + ) + C + δ W˜ 4 C 2,α [−1,1] X0 X0 1 +C + σ + δ W˜ 5 C 2,α ( E¯ + ) . X0 It follows from (A.38) and the expression of W5 that 1 + σ W˜ 1 C 2,α ( E¯ + ) W3 C 2,α ( E¯ + ) + W5 C 2,α ( E¯ + ) ≤ C X0 1 1 +C + σ + δ W˜ 2 C 2,α ( E¯ + ) + C + δ W˜ 3 C 2,α ( E¯ + ) . X0 X0
(A.40)
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149
In addition, due to (A.33), one can calculate to obtain W1 = O(δ)W2 + O(1)W3 + O(σ δ)W˜ 1 + O(σ )W˜ 2 + O(σ δ)W˜ 3 1 ˜ +O + σ δ W˜ 4 + O(δ) N˜ 4 + O(δ)(β − β), X0
(A.41)
where β and β˜ stand for the starting points from the transonic shock of two characteristics respectively, whose definitions are given in (4.8). As in Lemma 4.4, one can obtain the following estimate: ˜ C 2,α ( E¯ ) ≤ C(δW˜ 1 C 2,α ( E¯ ) + W˜ 2 C 2,α ( E¯ ) + δW˜ 4 C 2,α [−1,1] ). β − β + + +
(A.42)
Then, it follows from (A.41) and (A.42) that W1 C 2,α ( E¯ + ) ≤ CδW2 C 2,α ( E¯ + ) + CW3 C 2,α ( E¯ + ) + C(σ δ + δ 2 )W˜ 1 C 2,α ( E¯ + ) + C(σ + δ)W˜ 2 C 2,α ( E¯ + ) + Cσ δW˜ 3 C 2,α ( E¯ + ) + C
1 + δ W˜ 4 C 3,α [−1,1] . X0
This, together with (A.38) and (A.40), yields 1 W1 C 2,α ( E¯ + ) ≤ C + σ (W˜ 1 , W˜ 3 )C 2,α ( E¯ + ) X0 1 +C + σ + δ W˜ 2 C 2,α ( E¯ + ) . X0 Combining (A.38) with (A.43) yields
(A.43)
1 (W1 , W2 , W3 )C 2,α ( ) + σ (W˜ 1 , W˜ 3 )C 2,α ( ) ¯ ≤C ¯ X0 1 C ˜ +C + σ + δ W˜ 2 C 2,α ( ) W4 C 2,α [−1,1] . ¯ + X0 X0
Thus, combining this with the estimate (A.38), we arrive at 1 (W1 , W2 , W3 )C 2,α ( ) ≤ C + σ (W˜ 1 , W˜ 3 )C 2,α ( ) ¯ ¯ X0 1 +C + σ + δ W˜ 2 C 2,α ( ) ¯ . X0
(A.44)
This shows that the mapping T is contractible in C 2,α (E + ) for suitably small δ, σ and X 0−1 . In fact, as stated in Step 4, we can choose σ = O(1)ε > 0 and δ = O(1)ε > 0. Therefore, the system (A.5), (A.6) and (A.10) has a unique solution (ρ(z), U1 (z), U2 (z); ξ(z 2 )) when the exit pressure condition in (A.6) is adjusted by a unique constant C0 (determined by the integral equality (A.28)). Since the coordinate transformation (A.4) is reversible and keeps the equivalence of C 4,α norms between the two coordinates (z 1 , z 2 ) and (y1 , y2 ) for r0 ∈ (X 0 , X 0 + 1) and suitably small σ = O(ε), then we finish the proof of Theorem 6.1 and Theorem 6.1.
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References 1. Azzam, A.: On Dirichlet’s problem for elliptic equations in sectionally smooth n-dimensional domains. SIAM. J. Math. Anal. 11(2), 248–253 (1980) 2. Azzam, A.: Smoothness properties of mixed boundary value problems for elliptic equations in sectionally smooth n-dimensional domains. Ann. Polon. Math. 40, 81–93 (1981) 3. Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. New York/London: John Wiley & Sons, Inc. Chapman & Hall, Ltd. 1958. 4. Canic, S., Keyfitz, B.L., Lieberman, G.M.: A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. LIII, 484–511 (2000) 5. Chen, G.-Q., Chen, J., Song, K.: Transonic nozzle flows and free boundary problems for the full Euler equations. J. Differ. Eq. 229(1), 92–120 (2006) 6. Chen, G., Feldman, M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J.A.M.S. 16(3), 461–494 (2003) 7. Chen, S.: Stability on transonic shock fronts in two-dimensional Euler systems. Trans. Amer. Math. Soc. 357(1), 287–308 (2005) 8. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. New York: Interscience Publishers Inc., 1948 9. Embid, P., Goodman, J., Majda, A.: Multiple steady states for 1-D transonic flow. SIAM J. Sci. Statist. Comput. 5(1), 21–41 (1984) 10. Gilbarg, D., H¨ormander, L.: Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74(4), 297– 318 (1980) 11. Gilbarg, D., Tudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224, Berlin-New York: Springer, 1983 12. Glaz, H.M., Liu, T.-P.: The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow. Adv. in Appl. Math. 5(2), 111–146 (1984) 13. John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Comm. Pure Appl. Math. 27, 377–405 (1974) 14. Kuz’min, A.G.: Boundary-Value Problems for Transonic Flow. New York: John Wiley & Sons, LTD, 2002 15. Li, J., Xin, Z., Yin, H.: The uniqueness of multidimensional transonic shock in a 3-D curved nozzle with the variable end pressures. Preprint, 2007 16. Lieberman, G.M.: Mixed boundary value problems for elliptic and parabolic differential equation of second order. J. Math. Anal. Appl. 113(2), 422–440 (1986) 17. Lieberman, G.M.: Oblique derivative problems in Lipschitz domains II. J. Reine Angew. Math. 389, 1–21 (1988) 18. Liu, T.-P.: Nonlinear stability and instability of transonic flows through a nozzle. Comm. Math. Phys. 83(2), 243–260 (1982) 19. Liu, T.-P.: Transonic gas flow in a duct of varying area. Arch. Rational Mech. Anal. 80(1), 1–18 (1982) 20. Morawetz, C.S.: Potential theory for regular and Mach reflection of a shock at a wedge. Comm. Pure Appl. Math. 47, 593–624 (1994) 21. Morawetz, C.S.: On the nonexistence of continuous transonic flows past profiles, I, II, III. Comm. Pure Appl. Math. 9, 45–68 (1956); 10, 107–131 (1957); 11, 129–144 (1958) 22. Xin, Z., Yan, W., Yin, H.: Transonic shock problem for the Euler system in a nozzle. Arch. Rational Mech. Anal. (2009). doi:10.1007/s00205-009-0251-8 23. Xin, Z., Yin, H.: Transonic shock in a nozzle I, 2-D case. Comm. Pure Appl. Math. LVIII, 999–1050 (2005) 24. Xin, Z., Yin, H.: Three-dimensional transonic shock in a nozzle, Pacific J. Math. 236(1), 139–193 (2008) 25. Xin, Z., Yin, H.: Transonic shock in a curved nozzle, 2-D and 3-D complete Euler systems. J. D. E. 245(4), 1014–1085 (2008) 26. Yuan, H.: On transonic shocks in two-dimensional variable-area ducts for steady Euler system. SIAM J. Math. Anal. 38(4), 1343–1370 (2006) 27. Zheng, Y.: A global solution to a two-dimensional Riemann problem involving shocks as free boundaries. Acta Math. Appl. Sin. Engl. Ser. 19(4), 559–572 (2003) 28. Zheng, Y.: Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. Engl. Ser. 22(2), 177–210 (2006) Communicated by P. Constantin
Commun. Math. Phys. 291, 151–176 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0871-8
Communications in
Mathematical Physics
Colliding Solitons for the Nonlinear Schrödinger Equation W. K. Abou Salem1, , J. Fröhlich2,3 , I. M. Sigal1, 1 Department of Mathematics, University of Toronto, Toronto, M5S 2E4 Ontario,
Canada. E-mail: [email protected]; [email protected]
2 Institute for Theoretical Physics, ETH Zürich, Zürich CH 8093, Switzerland.
E-mail: [email protected]
3 Institut des Hautes Études Scientifique, F-91440 Bures-sur-Yvette, France.
E-mail: [email protected] Received: 19 May 2008 / Accepted: 1 June 2009 Published online: 24 July 2009 – © Springer-Verlag 2009
Abstract: We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed v of the solitons, we show that, up to times of order log v after the collision, the solitons preserve their shape (in L 2 -norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential. 1. Introduction 1.1. Heuristic discussion and overview of earlier results. In this paper, we study the collision of two fast solitons in the presence of a (time-dependent) external potential that varies slowly in space as compared to the size of the solitons. For a class of typical local and nonlocal nonlinearities, we show that if the initial relative speed of the solitons is v 1 and the spatial variation of the external potential is sufficiently slow, then the solitons pass through each other almost “blindly”: The L 2 -norm of the difference between the true solution and the one corresponding to a configuration of two solitons moving in the external potential decays algebraically with v, up to times of order log v, after the collision. This is an example where the solitary waves for NLS display both their “wave” and “particle” nature. They pass through each other almost blindly because they are localized waves with high relative propagation speed and relative phase, while their center of mass dynamics is approximately that of a classical particle in a slowly varying external potential. The problem of asymptotic stability of multi-soliton configurations (scattering theory) for the nonlinear Schrödinger equation without an external potential has been
Supported in part by NSERC grant NA 7901.
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W. K. Abou Salem, J. Fröhlich, I. M. Sigal
addressed in [1 and 2]; see also [3]. In these papers, the authors prove, under rather strong spectral assumptions on the linearized equation, the asymptotic stability of multisoliton solutions (in three or higher dimensions). The main ingredient of their analysis is asymptotic stability of single solitons and dispersive estimates (which are related to the “charge-transfer model”). Our results and approach are different: We study the long-time dynamics of fast colliding solitons in the presence of an external potential, rather than the asymptotics, and we use softer, yet more robust techniques that allow us to analyze a wide class of systems under weak assumptions. Furthermore, our analysis holds in any dimension N ≥ 1. There has been considerable progress in understanding the long-time dynamics of a single soliton in slowly varying external potentials and in the presence of nonlinear perturbations, [4–9]. In [4], the long-time dynamics of a single soliton in a slowly varying external potential has been studied using the symplectic structure of the soliton manifold and the Hamiltonian flow generated by the nonlinear Schrödinger equation. This geometric and Hamiltonian structure has been further elucidated and considerably developed in [6 and 7]. Furthermore, the analysis of [4] has been generalized in [8] to study the long-time dynamics of solitons in time-dependent external potentials (over much longer time scales), and to study the effective dynamics of solitons in the presence of nonlinear perturbations, [9]. The analysis below, supplemented by some additional spectral assumptions, can be extended to study the effective dynamics of multiple solitons with low velocities in slowly varying external potentials (and in the presence of nonlinear perturbations) as long as the soliton centers of mass are well separated. We note that a similar time-scale appears in studying the splitting of a fast soliton that scatters from a delta-potential for the cubic NLS in 1-dimension, see [10]. Very recently, there appeared new work on the collision of solitons for a different class of nonintegrable equations, the gKdV in 1-dimension, in the absence of external perturbations.1 In [12], the collision of a slow soliton with a fast one for the quartic gKdV in 1-dimension is studied, where explicit expressions for the shifts of the solitons are given. Moreover, it is shown that both solitons survive the collision, up to a small perturbation. The analysis in the interacting (overlap) regime relies on an algebraic approach which is based on expanding in a nonlinear basis, while the analysis in the noninteracting regime relies on asymptotic techniques that were developed to study asymptotic stability of solitons for the gKdV equation, see for example [13]. The analysis of [12] is extended in [14] to more general power nonlinearities that admit (nonlinearly) stable solitary wave solutions. Here, our problem and analysis are different. We study the fast collision of solitons for the generalized NLS, with power (local) and/or Hartree (nonlocal) nonlinearities, in arbitrary dimensions and in the presence of an external potential. As for our approach, we use the fact that the relative velocity and phase of the solitons is high, together with semiclassical techniques, in order to control the overlap region, while the effective dynamics of the center of mass of the solitons is studied using the symplectic structure of the multi-soliton manifold and the Hamiltonian nature of the dynamics generated by the nonlinear Schrödinger equation. 1.2. Description of the problem. By x = (x1 , . . . , x N ) ∈ R N we denote points in physical space, and by t ∈ R time. We consider the nonlinear Schrödinger equation i∂t ψ(x, t) = (− + Vh (x, t))ψ(x, t) − f (ψ(x, t)), 1 See [11] for a recent review about problems related to the stability of solitons.
(1)
Colliding Solitons for the Nonlinear Schrödinger Equation
where =
N
∂2 i=1 ∂ x 2 i
153
is the N -dimensional Laplacian, with N ≥ 1, Vh denotes the
external potential, with
Vh (x, t) ≡ V (hx, t),
and f is a nonlinearity f : H 1 (R N ; C) → H −1 (R N ; C), such that f (ψ) = f (ψ) and f (0) = 0. Next, we discuss the various assumptions, which are satisfied by typical local and Hartree nonlinearities that appear in physical applications of (1), see Remark 1 below. (A1) Global well-posedness. The nonlinear Schrödinger equation (1) is globally wellposed in H 1 . We refer the reader to [15], Chap. 6, for well-posedness of (1) in energy space for timeindependent potentials, and [8] for the case of time-dependent external potentials and nonlinearities. We make the following assumption on the regularity and symmetries of the nonlinearity. (A2) Nonlinearity. Let F : H 1 → R be the functional with the property that its Fréchet derivative F = f. We assume that F ∈ C 3 (H 1 ; R) and that F(T ·) = F(·), where T is a translation Tatr : u(x) → u(x − a), a ∈ R N , or a rotation TRr : u(x) → u(R −1 x), R ∈ S O(N ), or a gauge transformation Tγg : u(x) → eiγ u(x), γ ∈ [0, 2π ), or a boost i
Tvb : u(x) → e 2 v·x u(x), v ∈ R N . We are interested in the dynamics of multi-soliton configurations, so we assume the existence of solitary wave solutions of (1) when V = 0; see for example [15], Chap. 8, for a discussion of solitary waves for NLS. (A3) Solitary waves. When V = 0, there exists an interval I ⊂ R such that, for all µ ∈ I, (1) admits solitary wave solutions of the form i
u σ = eiµt+iγ + 2 v·(x−a−vt) ηµ (x − a − vt), where σ = (a, v, γ , µ) ∈ R N × R N × [0, 2π ) × I. Here, ηµ is a positive, spherically symmetric function solving the nonlinear eigenvalue problem (− + µ)ηµ − f (ηµ ) = 0,
(2)
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with ηµ ∈ L 2 (R N ) ∩ C 2 (R N ), |x|3 ηµ L 2 + |x|2 |∇ηµ | L 2 + |x|2 ∂µ ηµ L 2 < ∞, ∀µ ∈ I,
(3)
and √
ηµ ∝ e −
µx
, as x → ∞.
Let m(µ) =
1 2
2 , d x ηµ
the “charge” of the soliton. We assume that ∂µ m(µ) > 0, which implies orbital stability, see [16–18]. We require some local properties of the nonlinearity (satisfied for classes of local and nonlocal nonlinearities; see Remark 1). (A4) Localization. We assume that, for ξ > 0 and a constant C < ∞ (both independent of a2 and v2 ), ( f (ηµ1 + u σ2 ) − f (ηµ1 ))X ηµ1 L 2 ≤ Ce−ξ a2 , and ( f (ηµ1 + u σ2 ) − f (ηµ1 ))X ηµ1 L 1 ≤ Ce−ξ a2 , √ √ where ηµ1 ≡ u (0,0,0,µ1 ) , X = 1, x or ∂x . We note that ξ ≤ min( µ1 , µ2 ). n (A5) For g ∈ L p (R N ), p ≥ 1, u = i=1 u σi , where u σ has been introduced in 1 Assumption (A3), and w ∈ H with w L 2 ≤ 1, we have that |g, f (u + w) − f (u) − f (u)w| ≤ Cw2L 2 , where C is a constant that depends on g and µi , i = 1, . . . , n. We make the following assumption on the external potential, which, among other things, guarantees well-posedness of (1) in H 1 (in spite of the fact that the energy is no longer conserved, see [8]). (A6) The external potential V ∈ W 1,∞ (R; C 2 (R N )). Next, we specify the class of the initial conditions for Eq. (1) that we will consider. We are interested in the collision of solitons with high relative speed. A 2-soliton configuration, perturbed by a small-amplitude wave, is given by i
i
v1 ·x v2 ·x ψ(t = 0) = φ(x) = e 2 η a1 ) + e 2 η a2 ) + w , µ1 (x − µ2 (x −
(4)
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155
with a1 , a2 , v1 , v2 ∈ R N , w ∈ H 1 , and µ1 , µ2 ∈ I0 , where I0 ⊂ I \∂ I is a bounded interval and its closure I 0 ⊂ I \∂ I. We assume that v1 − v2 ( inf m (µ))−1 µ∈I0
a1 − a2 with v1 − v2 = O(1). We assume that the wave w (hereafter sometimes called “fluctuation” ) is small. More specifically, w ∈ H 1 , with
w 2L 2 < C v1 − v2 −1 . We are not imposing any condition on the directions of the relative speed and position of the solitons. In particular, we can have that ( a1 − a2 ) · ( v1 − v2 ) < 0, which is the case corresponding to colliding solitons. We will remark later how one may improve our estimates in case the solitons are escaping from each other. In what follows, we denote by v0 := v1 − v2 , the initial relative velocity of the solitons.
1.3. Main result. We are now in a position to state our main result on the collision of two solitons. (Generalizations of this result for fast n-solitons, n ≥ 2, are straightforward.) Theorem 1. Consider the nonlinear Schrödinger equation (1) with initial condition as in (4), and suppose Assumptions (A1)-(A6) hold. Then, for any fixed α ∈ (0, 1), v0 2 1 (inf µ∈I0 m (µ))− 1−α and h (inf µ∈I0 m (µ)) 1−α , the solution of the initial value problem can be written as i
i
ψ(x, t) = eiγ1 t+ 2 v1 ·(x−a1 ) ηµ1 (x − a1 ) + eiγ2 t+ 2 v2 ·(x−a2 ) ηµ2 (x − a2 ) + w(x, t), for all t ∈ [0, τα ), with τα := Cα min(log v0 , 2| log h|), and sup w L 2 ≤ C (v0 −
t∈[0,τα )
1−α 2
+ h 1−α ),
(5)
where the constants C, C > 0 are independent of v0 , h and α. Furthermore, for t ∈ [0, τα ), the parameters ai , vi , γi , µi , i = 1, 2, satisfy the following equations: ∂t ai = vi + O(v0 −(1−α) + h 2(1−α) + e−ξ a1 −a2 ), ∂t vi = −2∇Vh (ai , t) + O(v0 −(1−α) + h 2(1−α) + e−ξ a1 −a2 ), v2 ∂t γi = µi + i − Vh (ai , t) + O(v0 −(1−α) + h 2(1−α) + e−ξ a1 −a2 ), 4 ∂t µi = O(v0 −(1−α) + h 2(1−α) + e−ξ a1 −a2 ), √ √ for some ξ ∈ (0, min( µ1 , µ2 )) independent of v0 and h.
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In particular, for v0 1 sufficiently large, and h = O(v0 − 2 ), the solitons preserve their shape, in L 2 -norm, up to times of order log v0 after the collision, and the dynamics of the centers of mass of the solitons is approximately determined by the Hamilton equations for two classical particles in an external potential Vh . Our analysis relies on three main ingredients. First, using a skew-orthogonal (or Lyapunov-Schmidt) decomposition (Prop. 1, Sect. 4), we decompose the solution of (1) with initial condition close to a 2-soliton configuration, as described by (4), into a trajectory belonging to a symplectic submanifold of 2-soliton states, and a part describing a small-amplitude wave skew-orthogonal to the manifold. The dynamics of the trajectory contained in the 2-soliton manifold is obtained by the skew-orthogonal projection of the Hamiltonian flow generated by the nonlinear Schrödinger equation in a small tubular neighbourhood of the 2-soliton manifold onto the latter (Prop. 2, Sect. 5). As for the small-amplitude dispersive wave, we control its L 2 -norm using charge conservation and skew-orthogonal decomposition (Prop. 3, Sect. 6). The main difference between our approach and a previous approach designed to study the effective dynamics of a single soliton in an external potential, see, for example, [4], is that we control the L 2 -norm of the small-amplitude wave (fluctuation) using charge conservation, rather than controlling its H 1 -norm by using an approximate Lyapunov functional and proving the constraint positivity of the Hessian, Eq. (13) below, under additional assumptions that are verified in the case of special local nonlinearities. Unlike the L 2 -norm, the H 1 -norm of ψ grows like v0 , and we lose control over w H 1 , as v0 → ∞. Remark 1. Let us consider some concrete examples for which all Assumptions, (A1)-(A5), are satisfied. An example where Assumptions (A1)-(A3) are satisfied is when f is a Hartree nonlinearity, f (ψ) = (W |ψ|2 )ψ, where W is a positive, spherically symmetric function on R N belonging to L p + L ∞ , with p > N2 , p ≥ 1, and decaying at infinity, W → 0, as x → ∞; see [15,18]. The localization property, Assumption (A4), is satisfied if in addition W decays exponentially fast. We next verify that (A5) holds for p ≥ 2. From the form of the nonlinearity, we have that |g, f (u + w) − f (u) − f (u)w| ≤ C(|g, (W |w|2 )u| + |g, (W |uw|)w|). Applying Hölder’s and Young’s inequalities, we find |g, (W |w|2 )u| ≤ gu L q W |w|2 L q ≤ g L q u L ∞ W L q |w|2 L 1 n ≤ g L q u i L ∞ W L q w2L 2 i=1
≤ Cw2L 2 ,
Colliding Solitons for the Nonlinear Schrödinger Equation
157
where q = p or ∞ (W ∈ L p + L ∞ ) and 1 = 1/q + 1/q . Similarly, |g, (W |uw|)w| ≤ gW |uw| L 2 w L 2 ≤ g 2q W (|u||w|) L q w L 2 ≤ g ≤ g ≤ g
L q−2 2q
L q−2 2q
L q−2 2q
L q−2
W L q uw L 1 w L 2 W L q u L 2 w2L 2 n W L q ηµi L 2 w2L 2 i=1
≤ Cw2L 2 . Thus, Assumption (A5) is satisfied. Another example where our assumptions are satisfied is to choose f to describe a local nonlinearity. For example, (A1) and (A2) are satisfied if f is of the form f (ψ)(x) = h(|ψ(x)|2 )ψ(x), where h ∈ C 2 (R+ , R) with ∂rk h(r ) ≤ C(1 + r α−k ), k = 0, 1, 2, α ∈ (0, N 2−2 ), N ≥ 3, and α ∈ (0, ∞) if N = 1, 2 (see, for example, [15,18] for a discussion of well-posedness in H 1 ). Solitary wave solutions appearing in (A3) exist, if, in addition, −∞ < lim h(r ) < µ, r →∞
−∞ ≤ lim r −α h(r ) ≤ C, r →∞
and there exists r0 > 0 with
r0
h(r )dr > µr0 ,
0
see [19,20]. The condition of orbital stability can be checked for all these nonlinearities; see [16–18]. Assumption (A4) follows directly from (A3) and the form of the local nonlinearity. Furthermore, Assumption (A5) is satisfied if sup r
r ∈R+
2k−1 2
∂rk h(r ) < ∞, k = 1, 2.
An explicit example of a local nonlinearity that satisfies all our hypotheses is 4 , f (ψ) = |ψ|s ψ χθ,s (|ψ|), s ∈ 0, N where χθ,s , θ 1, is a smooth regularization which is chosen such that (A5) is satisfied. For example, 1, if|y|sgn(s−1) < θ/2 χθ,s (y) = . 1−s |y| , if|y|sgn(s−1) > θ More generally, f can be a sum of local and nonlocal (Hartree) nonlinearities.
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Remark 2. We wish to describe special initial conditions where one can obtain control of the fluctuation w over longer time scales. Assume (A1)-(A6) hold, and suppose, for the sake of simplicity, that h = 0, which corresponds to a spatially flat potential. (1) Large separation. If the soliton centers of mass are initially separated by a distance d max( √1µ1 , √1µ2 , | log inf µ∈I0 m (µ)|) and v1 , v2 = O(1), with w L 2 = O(e−χ d ) for some √ χ > 0, then one obtains a result similar to Theorem 1 with supt∈[0,d ) w L 2 < C/ d, for any ∈ (0, 1) and, for t ∈ [0, d ), ai (t) = ai + t vi + O(d −(1−) ), vi (t) = vi + O(d −(1−) ), µi (t) = µi + O(d −(1−) ), v2 γi (t) = γi + t ( µi + i − V (0, t)) + O(d −(1−) ), 4 with i = 1, 2, see Sect. 8. (2) Escaping solitons. Suppose that the solitons escape from each other with a high relative speed ( a1 − a2 ) · ( v1 − v2 ) ≥ 0, and v0 (inf µ∈I0 m (µ))−2 , w L 2 = O(e−χ v0 ), for some χ > 0, then, for any fixed ∈ (0, 1), there exists a contant C independent of v0 and such that sup
t∈[0,v0 )
1
w(t) L 2 ≤ Cv0 − 2 ,
and, for t ∈ [0, v0 ), ai (t) = ai + t vi + O(v0 −1+ ), vi (t) = vi + O(v0 −1+ ), µi (t) = µi + O(v0 −1+ ), vi2 γi (t) = − V (0, t) + O(v0 −1+ ), γi + t µi + 4 with i = 1, 2, see Sect. 8. The organization of this paper is as follows. In Sect. 2, we recall some basic properties of the nonlinear Schrödinger equation. In Sect. 3, we recall the definition of the one-soliton manifold and we introduce the 2-soliton (or, more generally, n-soliton) manifold. In Sect. 4, we prove the skew-orthogonal decomposition property for elements in some neighbourhood of H 1 that are close in (L 2 -norm) to a two-soliton manifold, which is a central tool in our analysis. In Sect. 5, we use the skew-orthogonal property and the nonlinear Schrödinger equation (1) to find the reparametrized equations of motion corresponding to the parameters on the two-soliton manifold, and in Sect. 6, we control the L 2 -norm of the fluctuation, using charge conservation and the skew-orthogonal decomposition. In Sect. 7 we prove Theorem 1 by combining the results of Propositions 1, 2 and 3. We finally end with some remarks on solitons escaping from each other in Sect. 8.
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1.4. Notation. • In the following, L p (I ) denotes the standard Lebesgue space, 1 ≤ p ≤ ∞, with norm 1 p d x | f (x)| p , f ∈ L p (I ), p < ∞, f L p = I
f L ∞ = ess sup(| f |), f ∈ L ∞ (I ). • We denote by ·, · the real inner product on L 2 (R N ), u, v = Re uv, u, v ∈ L 2 (R N ). RN
N αi . Further• Given the multi-index α = (α1 , . . . , α N ) ∈ N N , we denote |α| = i=1 more, ∂xα := ∂xα11 · · · ∂xαNN . • For 1 ≤ p ≤ ∞ and s ∈ N, the (complex) Sobolev space is given by W s, p (R N ) := {u ∈ S (R N ), ∂xα u ∈ L p (R N ), |α| ≤ s}, where S (R N ) is the space of tempered distributions. We equip W s, p with the norm uW s, p = ∂xα u L p , α,|α|≤s
which makes it a Banach space. We use the shorthand W s,2 = H s . • Given f and g real functions on R N , we denote their convolution by , f g(x) := dy f (y)g(x − y).
2. Hamiltonian Structure of the Nonlinear Schrödinger Equation In this section, we recall some basic properties of the nonlinear Schrödinger equation (1), see for example [4,18]. We will use these properties in the following sections. The phase space for the nonlinear Schrödinger equation (1) is chosen to be H 1 (R N ; C). The space H 1 (R N ; C) has a real inner product (Riemannian metric) u, v := Re d x uv (6) for u, v ∈ H 1 (R N , C). 2 On H 1 (R N ; C) one can define with a symplectic 2-form ω(u, v) := Im d x uv = u, iv. (7) The Hamiltonian functional corresponding to the nonlinear Schrödinger equation (1) is 1 1 2 HV (ψ) := (8) |∇ψ| d x + V |ψ|2 − F(ψ). 2 2 2 The tangent space at an element ψ ∈ H 1 is T H 1 = H 1 . ψ
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Using the correspondence H 1 (R N ; C) ←→ H 1 (R N ; R) ⊕ H 1 (R N ; R) ψ ←→ (Reψ Imψ) i −1 ←→ J,
0 1 is the complex structure on H 1 (R N ; R2 ), the nonlinear Schröding−1 0 er equation can be written as
where J :=
∂t ψ = J HV (ψ). Furthermore,
Rev , Imv Rev 0 −1 . ω(u, v) = d x (Reu Imu) Imv 1 0
u, v =
d x (Reu Imu)
We note that since the Hamiltonian functional HV defined in (8) is nonautonomous, energy is not conserved. For ψ ∈ H 1 satisfying (1), 1 ∂t HV (ψ) = d x (∂t V )|ψ|2 , 2 see [8] for a proof of this statement. Yet, HV is invariant under global gauge transformations, HV (eiγ ψ) = HV (ψ), and the associated conserved quantity is the “charge” 1 N (ψ) := d x |ψ|2 . 2 The assumption ∂µ m(µ) > 0 implies that ηµ appearing in Assumption (A3) is a local minimizer of HV =0 (ψ) restricted to the balls Bm := {ψ ∈ H 1 : N (ψ) = m}, for m > 0; see [16,17]. They are critical points of the functional 1 Eµ (ψ) := (9) d x (|∇ψ|2 + µ|ψ|2 ) − F(ψ), 2 where µ = µ(m) is a Lagrange multiplier. 3. Soliton Manifolds In this section, we recall the definition and properties of a single soliton manifold (see [4–9]), and we introduce the multi-soliton manifold.
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3.1. Soliton Manifold. We define the transformation Tavγ by 1
ψavγ := Tavγ ψ = ei( 2 v·(x−a)+γ ) ψ(x − a), where v, a ∈ R N and γ ∈ [0, 2π ). We then define the soliton manifold as Ms := {ησ := Tavγ ηµ , σ = (a, v, γ , µ) ∈ R N × R N × [0, 2π ) × I }, where I appears in Assumption (A3). If f (0) = 0, where f appears in (1), then I ⊂ R+ . The tangent space to the soliton manifold Ms at ηµ ∈ Ms is given by Tηµ Ms = span{E t , E g , E b , E s }, where E t := ∇a Tatr ηµ |a=0 = −∇ηµ , E g := ∂γ Tγg ηµ |γ =0 = iηµ , E b := 2∇v Tvb ηµ |v=0 = i xηµ , E s := ∂µ ηµ . In the following, we denote by e j := −∂x j , j = 1, · · · , N , e j+N := i x j , j = 1, · · · , N , e2N +1 := i, e2N +2 := ∂µ ,
(10)
+2 which, when acting on ησ ∈ Ms , generate the basis vectors {eα ησ }2N α=1 of Tησ Ms . The soliton manifold Ms inherits a symplectic structure from (H 1 , ω). For σ = (a, v, γ , µ) ∈ R N × R N × [0, 2π ) × I, the matrices
σ := Pσ J −1 Pσ ∈ Tη∗σ Ms ∧ Tη∗σ Ms , where Pσ is the L 2 -orthogonal projection onto Tησ Ms , define the induced symplectic structure on Ms . We have the following easy lemma, which we prove in the Appendix. Lemma 1. If ∂µ m(µ) > 0, then σ is invertible. Explicitly, we find that σ |Tησ Ms : = {eα ησ , ieβ ησ }1≤α,β≤2N +2 ⎞ ⎛ 0 − 21 vm (µ) 0 −m(µ)1 N ×N ⎜m(µ)1 N ×N 0 0 am (µ) ⎟ ⎟, =⎜ ⎝ 0 0 0 m (µ) ⎠ 1 T T 0 2 v m (µ) −a m (µ) −m (µ)
(11)
where 1 N ×N is the N × N identity matrix, and (·)T stands for the transpose of a vector in R N ; see the proof of Lemma 1 in the Appendix.
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3.2. Group structure. The anti-selfadjoint operators {eα }α=1,...,2N +1 defined in (11) form the generators of the Lie algebra g corresponding to the Heisenberg group H2N +1 , where the latter is given by (a, v, γ ) · (a , v , γ ) = (a , v , γ ), with a = a + a , v = v + v , and γ = γ + γ + 21 v · a .3 Elements of g satisfy the commutation relations [ei , e j+N ] = −e2N +1 δi j , i, j = 1, . . . , N ,
(12)
and the rest of the commutators are zero. 3.3. Zero modes. The solitary wave solutions transform covariantly under translations and gauge transformations, i.e., Eµ (Tatr Tγg ηµ ) = 0 for all a ∈ R and γ ∈ [0, 2π ). Here, the prime stands for the Fréchet derivative. There are zero modes of the Hessian, Lµ := − + µ − f (ηµ ),
(13)
associated to these symmetries. We have the following lemma about the action of the Hessian on elements of the tangent space of the soliton manifold. Lemma 2. iLµ : Tηµ Ms → Tηµ Ms with (iLµ )2 X = 0, for any vector X ∈ Tηµ Ms . Proof. Differentiating Eµ (Tatr ηµ ) = 0 with respect to a and setting a to zero gives E (ηµ )∇a ηµ (x − a)|a=0 = Lµ E t = 0.
(14)
Similarly, differentiating Eµ (Tγ ηµ ) = 0 with respect to γ and setting γ to zero gives g
Lµ E g = 0.
(15)
Lµ E b = (− + µ − f (ηµ ))i xηµ = −i∇x ηµ = i E t .
(16)
Using (2), we have
Furthermore, differentiating (2) with respect to µ gives (− + µ − f (ηµ ))E s + ηµ = 0, and hence Lµ E s = i(iηµ ) = i E g .
(17)
3 This structure was noted for the case N = 1 in [6].
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3.4. Two-soliton manifold. Next, we discuss a manifold corresponding to two solitons. It is given by 2s := {(ησ1 , ησ2 ), σi = (ai , vi , γi , µi ) ∈ R N × R N × [0, 2π ) × I, i = 1, 2}. M 2s is The tangent space to M 2s = {(X 1 , X 2 ), X i ∈ Tησ Ms , i = 1, 2}. T(ησ1 ,ησ2 ) M i We introduce the embedding map 2s → H 1 , E: M 2s and T M 2s is given, respectively, by whose action on M E(ησ1 , ησ2 ) = ησ1 + ησ2 ∈ H 1 , E(X 1 , X 2 ) = X 1 + X 2 ∈ T H 1 + T H 1 . 2s ) and E(T M 2s ) respectively. In what follows, M2s and T M2s are identified with E(M 4. Skew-Orthogonal Decomposition Let I be the same as in Assumption (A3). We define := {σ = (a, v, γ , µ) ∈ R N × R N × [0, 2π ) × I }, and let 0 := {σ = (a, v, γ , µ) ∈ R N × R N × [0, 2π ) × I0 , with I 0 ⊂ I \∂ I bounded}. We define 2 := {(σ1 , σ2 ) ∈ 0 × 0 , a1 − a2 > d or v1 − v2 > κ}. d,κ 2 , the centers of mass of η In other words, for (σ1 , σ2 ) ∈ d,κ σ1 and ησ2 are either separated by a distance larger than d or their relative speed is larger than κ. We consider the neighbourhood Uδ,d,κ ⊂ H 1 defined by
Uδ,d,κ := {ψ ∈ H 1 ,
sup 2 (σ1 ,σ2 )∈d,κ
ψ − ησ1 − ησ2 L 2 < δ}.
We have the following proposition. Proposition 1. Suppose (A2) and (A3) hold. Then, for δ inf µ∈I0 m (µ) and κ inf µ∈I 1 m (µ) (or d max( √1µ1 , √1µ2 , | log inf µ∈I0 m (µ)|)), there exist unique 0
σ1 (·), σ2 (·) : Uδ,d,κ → such that ψ = ησ1 (ψ) + ησ2 (ψ) + w,
(18)
ω(w, X i ) = 0, i = 1, 2,
(19)
and for all X i ∈ Tησi (ψ) Ms , i = 1, 2.
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Remark 3. The group element Tavγ ∈ H2N +1 is given by Tavγ = e−a·∂x ei
v·x 2
eiγ .
−1 Y T It follows from (12) that Tavγ avγ ∈ g if Y ∈ g. Furthermore, it follows from translational invariance that ω(Tavγ u, Tavγ v) = ω(u, v), for u, v ∈ L 2 . Therefore, we have from Proposition 1 that
ω(w, Y (ησ1 + ησ2 )) = ω(w , Y (ησ1 + ησ2 )) = 0, −1 Y T −1 −1 ∀Y ∈ g, where Y Tavγ avγ ∈ g, w = Tavγ w, and ησ = Tavγ ησ .
Proof. . Without loss of generality, we choose a reference frame in which max(v1 , v2 ) ≤ 2κ, for example, the rest frame of one of the solitons. Using Remark 3, the result of our analysis holds in any reference frame obtained by Galilean boosts and spatial translations. We define the mapping 2 G : Uδ,d,κ × d,κ → R4N +4
by ⎞ ω(ψ − ησ1 − ησ2 , e1 ησ1 ) · ⎟ ⎜ ⎟ ⎜ · ⎟ ⎜ ⎟ ⎜ · ⎟ ⎜ ⎜ω(ψ − ησ − ησ , e2N +2 ησ )⎟ 1 2 1 ⎟ ⎜ G(ψ, (σ1 , σ2 )) := ⎜ ⎟. ⎜ ω(ψ − ησ1 − ησ2 , e1 ησ2 ) ⎟ ⎟ ⎜ · ⎟ ⎜ ⎟ ⎜ · ⎠ ⎝ · ω(ψ − ησ1 − ησ2 , e2N +2 ησ2 ) ⎛
Then (18) and (19) are equivalent to (σ1 (ψ), σ2 (ψ)) satisfying, for a given ψ ∈ Uδ,d,κ , the equation G(ψ, (σ1 (ψ), σ2 (ψ))) = 0. We use the implicit function theorem to show that there exist unique σ1 (ψ), σ2 (ψ) ∈ such that G(ψ, (σ1 (ψ), σ2 (ψ))) = 0. First, note that, by construction, G(ησ1 + ησ2 , (σ1 , σ2 )) = 0.
(20)
2 G ∈ C 1 (Uδ,d,κ × d,κ ; R4N +4 ),
(21)
Furthermore,
since it is linear in ψ and ησi , i = 1, 2, and it is differentiable in σi ∈ 0 , i = 1, 2. We still need to show that ∂(σ σ1 , σ2))| σ1 =σ1 , σ2 =σ2 is invertible for 1 ,σ2 ) G(ησ1 + ησ2 , ( κ
1 inf µ∈I0 m (µ)
(or d max
1 √1 , √1 , µ1 µ2 inf µ∈I0 m (µ)
).
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We know that the matrix +2 {ω(eα ησ , eβ ησ )}2N α,β=1 ,
is invertible, see (11), Lemma 1. We write i
eα ησ1 ieβ ησ2 =: e 2 (v1 −v2 )·x h αβ (x), α, β = 1, . . . , 2N + 2,
(22)
which corresponds to a decomposition where the fast oscillating term (in space) i e 2 (v1 −v2 )·x is separated from the slowly oscillating term (in space) h αβ . Let vm := max(v1 , v2 , 1). It follows from the fact that f ∈ C 2 (Assumption (A2)) and the exponential localization √ √ in space of the solitons (Assumption (A3)), that there exists ξ ∈ (0, min( µ1 , µ2 )), which is independent of v1 − v2 , and a constant C that depends only on µ1 and µ2 , such that h αβ W 3,1 (R N ) < Cvm 2 e−ξ a1 −a2 ,
(23)
for α, β = 1, . . . , 2N + 2.4 Suppose that κ 1. Let v := v1 − v2 . Using that i
i
Le 2 v·x = e 2 v·x , where L := −2i
v · ∇x , v2
and integrating by parts three times, we obtain i ω(eα ησ1 , eβ ησ2 ) = (L 3 e 2 v·x )h αβ (x) d x i = e 2 v·x (L ∗ )3 h αβ (x) d x.
(24)
Moreover, (L ∗ )3 h αβ L 1 ≤ v−3 h αβ W 3,1 .
(25)
Equations. (23)–(25) yield |ω(eα ησ1 , eβ ησ2 )| ≤ Cv2 v−3 ≤ Cv−1 ≤ Cκ −1 . (26) (Suppose alternatively that d max √1µ1 , √1µ2 with v0 = O(1) fixed. Then it follows from (23) that |ω(eα ησ1 , eβ ησ2 )| < Ce−ξ d ,
√ √ for some positive constant C that depends on µ1 and µ2 and ξ ∈ (0, min( µ1 , µ2 )).) 4 More generally, if f ∈ C r (H 1 , H −1 ),
h αβ W r +1,1 (R N ) < Cvm 2 e−ξ a1 −a2 . For example, in the case of local nonlinearities, the above estimate holds for any r ≥ 1, in which case we obtain better estimates.
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Hence, for δ inf m (µ) µ∈I0
and κ ( inf m (µ))−1 µ∈I0
(or d max
√1 , √1 , | log inf µ∈I0 µ2 µ1
m (µ)| ), the (4N + 4) × (4N + 4) matrix {ω(eα ησ1 , eµ ησ1 )} {ω(eα ησ1 , eν ησ2 )} {ω(eα ησ2 , eµ ησ1 )} {ω(eα ησ2 , eν ησ2 )}
σ1 , σ2 ))| ∂(σ1 ,σ2 ) G(ησ1 + ησ2 , ( σ1 =σ1 , σ2 =σ2 =
is invertible. Invertibility of ∂(σ1 ,σ2 ) G, together with (20), (21) and the implicit function theorem,5 imply that there exist unique C 1 maps σ1 (ψ) and σ2 (ψ) such that G(ψ, (σ1 (ψ), σ2 (ψ))) = 0. 5. Reparametrized Equations of Motion In this section, we apply the skew-orthogonal property to obtain reparametrized equations of motion for the parameters that characterize the projection of the true solution of (1) with initial condition φ onto M2s . We assume that the hypotheses for the skew-orthogonal decomposition, Sect. 4, hold. We will verify in the proof of the main theorem that for large enough v0 and small h, this is indeed the case over a certain time interval. Proposition 2. Consider (1) with initial condition (4), and suppose that (A1)-(A6) hold. Assume further that there exists τ > 0 such that, for t ∈ [0, τ ), ψ(t), the solution of (1) with initial condition φ, is in Uδ,d,κ , where δ is given in Proposition 1. Then, for v0 1, there exists a positive constant C0 < 1 independent of v0 and h, such that, for w L 2 < C0 , the parameters σi = (ai , vi , γi , µi ), i = 1, 2, satisfy the equations ∂t ai = vi + O(w2L 2 + h 2 + e−ξ a1 −a2 ), ∂t vi = −2∇ai Vh (ai , t) + vi2
O(w2L 2
2
+h +e
(27) −ξ a1 −a2
),
− Vh (ai , t) + O(w2L 2 + h 2 + e−ξ a1 −a2 ), 4 ∂t µi = O(w2L 2 + h 2 + e−ξ a1 −a2 ), √ √ for some constant ξ ∈ (0, min( µ1 , µ2 )) that is independent of v0 and h. ∂t γi = µi +
(28) (29) (30)
In what follows, we denote by C a positive constant that is independent of v and h, but that may change from one line to another. 5 See for example [21].
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Proof. We first find the equation of motion for i
ψ = e− 2 v1 ·x−iγ1 ψ(x + a1 ). u 1 = Ta−1 1 v1 γ1 Using Proposition 1, we have u 1 = ηµ1 + ησ2 + w ,
(31)
where ησ2 = Ta−1 η and w = Ta−1 w. Here, a2 = a2 − a1 , v2 = v2 − v1 , 1 v1 γ1 σ2 1 v1 γ1 γ2 = γ2 − γ1 , and µ2 = µ2 . It follows from Remark 3 that ω(w , X 1 + X 2 ) = 0,
(32)
for all X 1 ∈ Tηµ1 Ms and X 2 ∈ Tησ Ms . 2 We define the coefficients c j := ∂t a1, j − v1, j , j = 1, . . . , N , 1 c N + j := − ∂t v1, j − ∇a1 Vh (a1 , t), j = 1, . . . , N , 2 1 1 c2N +1 := µ1 − v12 + ∂t a1 · v1 − Vh (a1 , t) − ∂t γ , 4 2 c2N +2 := −∂t µ.
(33)
v12 , 4
(34)
Note that i
e− 2 (v1 ·x+2γ1 ) ψ(x + a1 ) = u 1 + iv1 · ∇u 1 − i
e− 2 (v1 ·x+2γ1 ) f (ψ(x + a1 )) = f (u 1 ).
(35)
Differentiating u 1 with respect to t and using (1), (34)–(35), we get ∂t u 1 = −i((− + µ1 )u 1 − f (u 1 )) +
2N +1
cα eα u 1 − iRV u 1 ,
(36)
α=1
where RV = Vh (x + a1 , t) − Vh (a1 , t) − x · ∇Vh (a1 , t). In other words, ∂t u 1 = −iEµ 1 (u 1 ) +
2N +1
cα eα u 1 − iRV u 1 ,
(37)
α=1
where Eµ is defined in (9). Recall that Eµ 1 (ηµ1 ) = 0, which implies Eµ 1 (u 1 ) = Lµ1 (ησ2 + w ) + Nµ1 (ησ2 + w ),
(38)
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where Lµ1 = (− + µ1 − f (ηµ1 )) ≡ Eµ1 (ηµ1 ) and Nµ1 (ησ2 + w ) = f (ηµ1 + ησ2 + w ) − f (ηµ1 ) − f (ηµ1 )(ησ2 + w ). Substituting (31) and (38) into (37), we obtain
∂t w = −iLµ1 +
2N +1
cα eα − iRV
α=1
−∂t ησ2 + −iLµ1 +
2N +1
w + Nµ1 (ησ2 + w ) +
2N +2
cα eα − iRV
α=1
cα eα ηµ1 − iRV ηµ1
α=1
ησ2 .
(39)
To obtain the equations of motion for a1 , v1 , γ1 and µ1 , we use the skew-orthogonal property to project (39) onto Tηµ1 Ms . It follows from (32) that iw , X = 0 for all X ∈ Tηµ1 Ms . Therefore, ∂t iw , X = ∂t µ1 iw , ∂µ1 X + i∂t w , X = 0.
(40)
Substituting the expression for ∂t w given by (39) in (40), and using eα∗ = −eα , α = 1, . . . , 2N + 2,
(41)
we have Lµ1 w , X + i + i
2N +2 α=1
2N +2 α=1
cα eα w X + RV w , X + i Nµ1 (ησ2 + w ), X
cα eα ηµ1 , X + RV ηµ1 , X − i∂t ησ2 , X
+ (Lµ1 + i
2N +1 α=1
cα eα + RV )ησ2 , X = 0.
(42)
Some of the terms in the above equation drop-out due to the zero modes of the Hessian. It follows from (14)–(17), Lemma 2, that X = iLµ1 X ∈ Tηµ1 Ms if X ∈ Tηµ1 Ms , and hence Lµ1 w , X = w , Lµ1 X = −ω(w, X ) = 0.
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Together with (41) and (42), this yields 2N +2
2N +2 cα ω(eα ηµ1 , X ) = RV ηµ1 , X + cα ieα w , X
α=1
α=1
+ RV w , X + i Nµ1 (ησ2 + w ), X + RV ησ2 , X + i∂t ησ2 , X 2N +1 cα eα )X . + ησ2 , (Lµ1 + i
(43)
α=1
We now estimate each term appearing in the right-hand-side of (43) with X = eβ ηµ1 , β = 1, . . . , 2N + 2. Note that it follows from Assumptions (A3) and (A6) that RV eβ ηµ1 L 2 = O(h 2 ), β = 1, . . . , 2N + 2, and from (A3) that X L 2 = eβ ηµ1 L 2 = O(1), β = 1, . . . , 2N + 2, eα X L 2 = eα eβ ηµ1 L 2 = O(1), α, β = 1, . . . , 2N + 2. Hence, Hölder’s inequality, (A3), (A6) and the fact that V is real yield the estimates |RV ηµ1 , X | = |ηµ1 , RV X | ≤ ηµ1 L 2 RV eβ ηµ1 L 2 ≤ Ch 2 ,
(44)
|RV w , X | = |w , RV X | ≤ RV eβ ηµ1 L 2 w L 2 ≤ Ch w L 2 , 2
|RV ησ2 , X | = |ησ2 , RV X | ≤ ησ2 L 2 RV eβ ηµ1 L 2 ≤ Ch . 2
(45) (46)
We also have from (A3) and Hölder’s inequality that 2N +2 2N +2 cα ieα w , X = cα iw , eα X α=1
α=1
≤ Ccw L 2 ,
(47)
where c := maxα=1,...,2N +2 |cα |. We now use Assumptions (A3)-(A4) to evaluate |i Nµ1 (ησ2 + w ), X |. It follows from (A3) that i X = ieβ ηµ1 ∈ L p , p ≥ 1, which, together with (A5), yield |i( f (ηµ1 + ησ2 + w ) − f (ηµ1 + ησ2 ) − f (ηµ1 + ησ2 )w ), X | ≤ Cw 2L 2 . It follows from the boundedness and the exponential localization of the solitons in space, (A3), and the fact that f ∈ C 2 , (A2), that −i f (ηµ1 )ησ2 , X | ≤ f (ηµ1 ) L ∞ ησ2 X L 1 ≤ Ce−ξ a1 −a2 ,
√ √ for some ξ ∈ (0, min( µ1 , µ2 )) which is independent of v0 and h. Moreover, it follows directly from (A4) that |i( f (ηµ1 + ησ2 ) − f (ηµ1 )), X | ≤ Ce−ξ a1 −a2
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and |i( f (ηµ1 + ησ2 ) − f (ηµ1 ))w , X | ≤ w L 2 ( f (ηµ1 + ησ2 ) − f (ηµ1 ))X L 2 ≤ Ce−ξ a1 −a2 w L 2 . Therefore, |i Nµ1 (ησ2 + w ), X | ≤ C(w 2L 2 + e−ξ a1 −a2 ).
(48)
To evaluate the remaining terms, we use the fact that ηµ1 and ησ2 are exponentially localized in space, while their relative fast oscillating phase is v = v1 − v2 ≥ vm , where vm := max(1, v1 , v2 ). When estimating an upper bound for |i∂t ησ2 , eβ ηµ1 |, the partial derivative with time contributes vm 2 , since ⎡ ⎤ N 2N 1 v · a ˙ ∂t ησ = ⎣ e2N +1 + ∂t µe2N +2 ⎦ ησ . ∂t a j e j + ∂t v j e j + ∂ t γ + 2 2 j=1
j=N +1
(49) However, using (22) and (23), and integrating by parts twice in space, we can pull a i factor of v1 − v2 −2 from the fast oscillating term e 2 (v2 −v1 )·x , see the discussion below (23) in the proof of Proposition 1. Hence |∂t ησ2 , eβ ηµ1 | ≤ Ce−ξ a1 −a2 .
(50)
Furthermore, (A2) and (A3) yield |ησ2 , Lµ1 eβ ηµ1 | ≤ Ce−ξ a1 −a2 .
(51)
Again, using (22) and (23) and integrating by parts twice in space to pull a factor of i v1 − v2 −2 from the fast oscillating factor e 2 (v2 −v1 )·x , we have 2N +1 cα eα eβ ηµ1 ≤ Ccv−2 e−ξ a1 −a2 . (52) ησ2 , i α=1
From (43)–(52), we have 2N +2 cα ω(eα ηµ1 , eβ ηµ1 ) α=1 ≤ C w2L 2 + c(w L 2 + v−2 ) + h 2 + e−ξ a1 −a2 , for β = 1, . . . , 2N + 2, where we used w L 2 = Ta−1 w L 2 = w L 2 1 v1 γ1 due to translational invariance.
(53)
Colliding Solitons for the Nonlinear Schrödinger Equation
Using Lemma 1, (11) and (53), and assuming w L 2 and v−2 ≤ obtain the estimate
171 1 4C µ1 ,
we
c ≤ C(w2L 2 + h 2 + e−ξ a1 −a2 ). Recalling now the definition of cα , α = 1, . . . , 2N +2 (see (34)), we conclude (27)–(30), with i = 1. To get the equations of motion for a2 , v2 , γ2 and µ2 , we consider u 2 = Ta−1 ψ, 2 v2 γ2 and we repeat the above analysis with 1 ↔ 2. 6. Control of the Fluctuation We now control the L 2 -norm of the fluctuation w using conservation of charge, the skew-orthogonal property, Sect. 4, and the reparametrized equations of motion, Sect. 5. Proposition 3. Consider (1) with initial condition (4), and suppose that (A1)-(A6) hold. Assume further that there exists τ > 0 such that, for t ∈ [0, τ ), ψ(t) ∈ Uδ,d,κ , where δ is given in Proposition 1. Then, for v0 1 and h 1, sup
t∈[0,Cα min(log v0 ,2| log h|))
w(t)2L 2 ≤ C (v0 −1+α + h 2(1−α) ),
for some positive constants C and C that are independent of v0 , h, and α ∈ (0, 1). Proof. From conservation of charge (L 2 -norm) of the solution of (1), ψ(t) L 2 = φ L 2 , and skew-orthogonal decomposition (Proposition 1), we have ψ2L 2 = w2L 2 + ηµ1 2L 2 + ηµ2 2L 2 + 2Reησ1 , ησ2 = φ2L 2 ,
(54)
where we used w, ησ j = −ω(w, iησ j ) = 0, and ησ j L 2 = ηµ j L 2 , for j = 1, 2. Differentiating (54) with respect to t, and recalling that m(µ) = 21 ηµ 2L 2 , we get ∂t w2L 2 = −2∂t µ1 ∂µ1 m(µ1 ) − 2∂t µ2 ∂µ2 m(µ2 ) − 2∂t Reησ1 , ησ2 .
(55)
First, using the exponential localization of solitons in space and the fast relative phase of the solitons, we estimate an upper bound for |∂t Reησ1 , ησ2 | = |∂t ω(ησ1 , e2N +1 ησ2 )|. From (27)–(30) and (49), it seems a priori that |∂t Reησ1 , ησ2 | is of order v2 . Howi ever, we can pull a factor of v1 − v2 −2 from the fast oscillating phase e 2 (v1 −v2 )·x by integrating by parts twice, as in (23)–(26) in Sect. 4. Therefore, |∂t Reησ1 , ησ2 | ≤ Ce−ξ a1 −a2 .
(56)
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Furthermore, (30) implies that |∂t µ1 ∂µ1 m(µ1 ) + ∂t µ2 ∂µ2 m(µ2 )| ≤ C(h 2 + e−ξ a1 −a2 + w2L 2 ).
(57)
Now, (55)–(57) yield |∂t w2L 2 | ≤ C(h 2 + e−ξ a1 −a2 + w2L 2 ),
(58)
for some positive constant C independent of v0 and h. It follows from (58) and the Duhamel formula that w 2L 2 ) + w2L 2 ≤ C(ect (h 2 +
t
ds ec(t−s) e−ξ a1 −a2 ).
(59)
0
For times t < Cv0 , ∈ (0, 1), we know from (28) that v1 (t) − v2 (t) ≥ c0 v0 ,
(60)
for some constant c0 > 0. Making the change of variables s → a(s), where a(s) := a1 (s) − a2 (s), and using that e−ξ a(s) = −
1 ∂s e−ξ a(s) , ξ ∂s a(s)
(27), (28) and (60), we have t ct ds ec(t−s) e−ξ a1 −a2 ≤ C e . v 0
0
Together with (59), we get the estimate w2L 2 ≤ C h 2 ect +
1 ct e , v0
(61)
for some positive constants C and c that are independent of v0 and h. Let τ := αc min(log v0 , 2| log h|) for some α ∈ (0, 1). For t < τ, (61) implies sup w2L 2 < C(v0 −1+α + h 2(1−α) ).
t∈[0,τ )
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7. Proof of Theorem 1 We now show that, for v0 1 large enough and h 1 small enough, the hypotheses of Propositions 1, 2 and 3 can be simultaneously satisfied. Let T := sup{t ≥ 0, ψ(t) ∈ Ud,κ,δ with δ as in Proposition 1}. By continuity of w(t) L 2 , T > 0. If T ≤ Cα min(log v0 , 2| log h|), then by Proposition 3, sup w(t) L 2 ≤ C (v0
−1+α 2
t∈[0,T )
+ h 1−α ).
(62)
Here, C, C appear in Proposition 3. We need δ inf m (µ), µ∈I0
where δ appears in Proposition 1. Consider v0 and h satisfying C (v0
−1+α 2
+ h 1−α ) ≤
2 δ ( inf m (µ)) 1−α inf m (µ). µ∈I0 µ∈I0 2
Then w(T ) L 2 ≤ 2δ , and T is not the maximal time unless T = Cα min(log v0 , 2| log h|). Then (62) yields (5). Furthermore, the hypotheses of Proposition 2 are satisfied. Using (5) in (27)–(30) gives the estimates on the evolution of the parameters in Theorem 1. 8. Comments on Separating Solitons We now discuss Remark 2 in Subsect. 1.3, whose hypotheses we assume. (1) Suppose that the soliton centers of mass are initially separated by a distance d max √1µ1 , √1µ2 , | log inf µ∈I0 m (µ)| , such that w L 2 = O(e−χ d ) for some χ > 0, and that v1 , v2 = O(1). Then the analysis above (Propositions 1, 2 and 3) holds, except that (58) implies
w2L 2 ≤ Ce−C d+C
d
< Cd −1 ,
for t < d , ∈ (0, 1), from which follows the claim of (1) of Remark 2. (2) In the case of escaping solitons, (58) in the proof of Proposition 3 implies that
w2L 2 ≤ Ce−C v0 +C
v
0
< C/v0
for t ≤ v0 , ∈ (0, 1). Hence claim (2) of Remark 2 also holds.
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9. Appendix Proof of Lemma, Sect. 2. Explicitly, σ |Tησ Ms := {eα ησ , ieβ ησ }1≤α,β≤2N +2 , where eα ησ , α = 1, . . . , 2N + 2, are basis vectors of Tησ Ms . For α = 1, . . . , N and β = N + 1, . . . , 2N , i
i
eα ησ , ieβ ησ = −∂xα (e 2 v·(x−a)+iγ ηµ (x − a)), −xβ e 2 v·(x−a)+iγ ηµ (x − a) vα iηµ (x − a), xβ ηµ (x − a) + ∂xα ηµ (x − a), xβ ηµ (x − a). = 2 It follows from translational invariance of the integral, and positivity and spherical symmetry of ηµ (x) that iηµ (x − a), xβ ηµ (x − a) = iηµ (x), xβ ηµ (x) + iηµ (x), aβ ηµ (x) = 0, and, by integration by parts, ∂xα ηµ (x − a), xβ ηµ (x − a) = −δαβ ηµ (x), ηµ (x) − xβ ηµ (x − a), ∂xα ηµ (x − a) = −2δαβ m(µ) − ∂xα ηµ (x − a), xβ ηµ (x − a), where m(µ) = 21 ηµ L 2 and δαβ stands for the Kroenecker delta. Therefore, eα ηµ , ieβ ηµ = −eβ ησ , ieα ησ = −δαβ m(µ), α = 1, . . . , N , β = N + 1, . . . , 2N . Furthermore, ∂µ ησ , i∂γ ησ = −∂µ ηµ (x − a), ηµ (x − a) = −∂µ m(µ), and hence e2N +2 ησ , ie2N +1 ησ = −e2N +1 ησ , ie2N +2 ησ = −m (µ), where m (µ) = ∂µ m(µ). For α, β = 1, . . . , N , i
i
eα ησ , ieβ ησ = ∂xα (e 2 v·(x−a)+iγ ηµ (x − a)), i∂xβ (e 2 v·(x−a)+iγ ηµ (x − a)) i i = vα + ∂xα ηµ (x), i( vβ + ∂xβ )ηµ (x) 2 2 vβ 1 vα = vα vβ ηµ , iηµ +∂xα ηµ , i∂xβ ηµ + ηµ , ∂xβ ηµ + ∂xα ηµ , ηµ , 4 2 2 where we used translational invariance in the second line. It follows from spherical symmetry of ηµ (x) that ηµ , ∂xβ ηµ = 0, ∂xα ηµ , i∂xβ ηµ = 0, α = β. Furthermore, since ηµ is real, ηµ , iηµ = ∂xα ηµ , i∂xα ηµ = 0.
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Therefore, eα ησ , ieβ ησ = 0, α, β = 1, . . . , N . For α = 1, . . . , N ,
i eα ησ , ie2N +1 ησ = vα + ∂xα ηµ (x − a), ηµ (x − a) = 0, 2
and
i 1 eα ησ , ie2N +2 ησ = − vα + ∂xα ηµ (x − a), i∂µ ηµ (x − a) = − vα m (µ). 2 2
For α = N + 1, . . . , 2N , eα ησ , ie2N +1 ησ = −i xα ηµ (x − a), ηµ (x − a) = 0, and eα ησ , ie2N +2 ησ = i xα ηµ (x − a), i∂µ ηµ (x − a) = aα m (µ). Explicitly, we have ⎞ 0 − 21 vm (µ) 0 −m(µ)1 N ×N ⎜m(µ)1 N ×N 0 0 am (µ) ⎟ ⎟, σ |Tησ Ms = ⎜ ⎝ 0 0 0 m (µ) ⎠ 1 T T 0 2 v m (µ) −a m (µ) −m (µ) ⎛
where 1 N ×N is the N × N identity matrix, and (·)T stands for the transpose of a vector in R N . One may easily verify that the (2N + 2) × (2N + 2) skew-symmetric matrix σ given in (11) is invertible if ∂µ m(µ) > 0. Acknowledgements. W.A.S. thanks Catherine Sulem for pointing out references [11–14].
References 1. Perelman, G.: Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations. Commun. Part. Diff. Eq. 29, 1051–1095 (2004) 2. Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. Commun. Pure Appl. Math. 58, 149–216 (2005) 3. Martel, Y., Merle, F., Tsai, T.-P.: Stability in H 1 for the sum of K solitary waves to some nonlinear Schrödinger equations. Duke Math. J. 133, 405–466 (2006) 4. Fröhlich, J., Gustafson, S., Jonsson, B.L.G., Sigal, I.M.: Solitary wave dynamics in an external potential. Commun. Math. Phys. 250(3), 613–642 (2004) 5. Fröhlich, J., Jonsson, B.L.G., Gustafson, S., Sigal, I.M.: Long time motion of NLS solitary waves in a confining potential. Ann. Henri Poincare 7, 621–660 (2006) 6. Holmer, J., Zworski, M.: Slow soliton interaction with delta impurities. J. Mod. Dyn. 1, 689–718 (2007) 7. Holmer, J., Zworski, M.: Soliton interaction with slowly varying potentials. IMRN ArtID 026, 36pp (2008) 8. Abou Salem, W.K.: Solitary wave dynamics in time-dependent potentials. J. Math. Phys. 49, 032101 (2008) 9. Abou Salem, W.K.: Effective dynamics of solitons in the presence of rough nonlinear perturbations. Nonlinearity 22, 747–763 (2009)
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10. Holmer, J., Marzuola, J., Zworski, M.: Fast soliton scattering by delta impurities. Commun. Math. Phys. 274, 187–216 (2007) 11. Tao, T.: Why are solitons stable?. Bull. Amer. Math. Soc. 46, 1–33 (2009) 12. Martel, Y., Merle, F.: Description of two soliton collision for the quartic gKdV equation. http://arxiv.org/ abs/0709.267vL[math.AP], 2007 13. Martel, Y., Merle, F.: Refined asymptotics around solitons for gKdV equations. Discrete Contin. Dyn. Syst. 20, 177–218 (2008) 14. Martel, Y., Merle, F.: Stability of two soliton collision for nonintegrable gKdV equations. Commun. Math. Phys. 286, 39–79 (2009) 15. Cazenave, T.: An Introduction to Nonlinear Schrödinger Equations. Textos de Métodos Matemáticos 26. Rio de Janeiro: Instituto de Matemática, 1996 16. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987) 17. Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94(2), 308–348 (1990) 18. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Number 130 in Applied Mathematical Sciences. Springer, New York, 1999 19. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch Rat. Mech. Anal. 82, 313–345 (1983) 20. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch Rat. Mech. Anal. 82, 347–375 (1983) 21. Munkres, J.R.: Analysis on Manifolds. Reading, MA. Addison-Wesley, 1991 Communicated by P. Constantin
Commun. Math. Phys. 291, 177–224 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0874-5
Communications in
Mathematical Physics
The Ginibre Ensemble of Real Random Matrices and its Scaling Limits A. Borodin1 , C. D. Sinclair2 1 Mathematics 253-37, California Institute of Technology, Pasadena,
California 91125, USA. E-mail: [email protected]
2 Department of Mathematics, University of Colorado, Boulder,
Colorado 80309, USA. E-mail: [email protected] Received: 27 May 2008 / Accepted: 2 April 2009 Published online: 18 July 2009 – © Springer-Verlag 2009
Abstract: We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2×2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large. Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Point Processes on the Space of Eigenvalues of R N ×N . . . . . Point Processes on X N Associated to Weights . . . . . . . . . Examples of Point Processes Associated to Weights . . . . . . 4.1 The real Ginibre ensemble . . . . . . . . . . . . . . . . . 4.2 The range of Mahler measure . . . . . . . . . . . . . . . . 4.3 The range of other multiplicative functions on polynomials 5. Correlation Measures and Functions . . . . . . . . . . . . . . . 6. A Matrix Kernel for Point Processes on X N . . . . . . . . . . . 7. Correlation Functions in Terms of the Matrix Kernel . . . . . . 8. Limiting Correlation Functions for the Real Ginibre Ensemble . 8.1 In the bulk . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 At the edge . . . . . . . . . . . . . . . . . . . . . . . . . 9. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The proofs of Proposition 5 and Proposition A.3 . . . . . . 9.2 The proofs of Proposition 6 and Proposition A.4 . . . . . . 9.3 The proofs of Theorem 8 and Corollary 9 . . . . . . . . . 9.4 The proofs of Theorem 10 and Theorem 11 . . . . . . . . 9.5 The proofs of Theorem 12 and Theorem 13 . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Correlation Functions for β = 1 and β = 4 Hermitian Ensembles
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B. The Pfaffian Cauchy-Binet Formula . . . . . . . . . . . . . . . . . C. Limiting Correlation Functions for the Complex Ginibre Ensemble D. Plots of Correlation Functions for the Real Ginibre Ensemble . . . D.1 The real bulk . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 The complex bulk . . . . . . . . . . . . . . . . . . . . . . . . D.3 The real edge . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 The complex edge . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The principal subject of this paper is the Ginibre ensemble of real random matrices — square real valued matrices with i.i.d. normal entries. Random matrix models have been very successful in describing various physical phenomena. (See e.g. [17] and references therein.) Physical applications of the Ginibre ensembles are described in [14,18] and [13]. Mathematically, the ability to analyze spectra of random matrices is largely based on determinantal or Pfaffian formulas for spectral correlations that have been derived for a variety of models. The real Ginibre ensemble is one of the few models for which such formulas remained unavailable for over 40 years (the model was introduced by Ginibre in 1965 [15]). The goal of this paper is to prove Pfaffian formulas for all the correlation functions of the real Ginibre ensemble and to evaluate their bulk and edge scaling limits. The algebraic techniques that we use were developed in a recent work of the second author [22]. Starting from that paper, Forrester and Nagao [13] have independently obtained similar Pfaffian formulas for correlations and constructed certain skeworthogonal polynomials necessary for the asymptotic analysis. We take the next step and obtain the asymptotics of the correlation functions using these polynomials. Some of the results presented here have been independently developed by Sommers [24]—in particular he produces a Pfaffian formulation for the 2-point correlation functions and some of their scaling limits in the bulk. Some further results of Sommers and Wieczorek on the correlations of the real Ginibre ensemble appear in [25]. In the algebraic part of this paper we consider a general class of probability measures that includes the real Ginibre ensembles and ensembles arising in the study of Mahler measure of polynomials; the latter are of interest in number theory, see [23]. We show that the correlation functions for an ensemble from this class can be expressed as the Pfaffian of a block matrix whose entries are expressed in terms of a 2 × 2 matrix kernel associated to the ensemble. We find much inspiration from Tracy and Widom’s paper on correlation and cluster functions of Hermitian and related ensembles [29]. However, instead of using properties of the Fredholm determinant to calculate the correlation functions via the cluster functions, we use the notion of the Fredholm Pfaffian to determine the correlation functions directly. A Pfaffian analog of the Cauchy-Binet formula introduced by Rains [21] lies at the heart of our proof. For completeness we will include Rains’ proof here. In place of the identities of de Bruijn [7] used by Tracy and Widom we will use an identity of the second author [22] to compute the correlation functions for ensembles of asymmetric matrices. Rains’ Cauchy-Binet formula has applicability in a wider context than just the determination of the correlation function for ensembles of asymmetric matrices, and we will use it to give a simplified proof of the correlation function of Hermitian ensembles of random matrices when β = 1 and β = 4. We then apply our theory to the real Ginibre ensemble. It is known (see [2,8,9,16]) that the density functions of real and complex eigenvalues is approximately constant on
Ginibre Ensemble of Random Matrices and its Scaling Limits
179
√ √ √ (− N , N ) and the disc of radius N respectively (here N is the size of the matrices). We√study the local correlations of eigenvalues in √four different regions: near a real point u N with√−1 < u < 1 (real bulk), near ± N (real edge), √ √ near a (non-real) √ complex point u N with |u| < N (complex bulk), and near u N with |u| = N , Im(u) = 0 (complex edge). Four different limit processes arise, and we compute their correlation functions explicitly. The complex bulk and edge limits turn out to be the same as in the case of the much simpler complex Ginibre ensembles. The correlations of real eigenvalues in the real bulk region were obtained in [13] and the density functions were computed earlier using different techniques [8,9]. All other results appear to be new. The paper is organized as follows. Sect. 2 introduces necessary notation. In Sect. 3, we introduce a class of ensembles relevant for our study. In Sect. 4 we show that the real Ginibre ensemble and the ensemble related to Mahler measure fall into this class. In Sect. 5, we introduce the correlation functions. In Sect. 6 we construct the correlation kernel, and in Sect. 7 we state how the correlation functions are expressed through that kernel. Sect. 8 contains statements of asymptotic results as well as the limiting correlation kernels. Sect. 9 contains the proofs. Appendix A shows how to apply Rains’ Cauchy-Binet formula to the β = 1, 4 Hermitian random matrix ensembles. Appendix B contains the proof of Rains’ formula. In Appendix C we compute the bulk and edge limits of the complex Ginibre ensemble. (We thank the referee for providing a reference, [11], for the results in Appendix C—we include them here for completeness.) Appendix D provides plots of the first and second correlation functions in various limit regimes. 2. Point Processes on the Space of Eigenvalues of R N×N We start rather generally since the results in this manuscript can be used to describe not only the statistics of eigenvalues of ensembles of real matrices but also the statistics of roots of certain ensembles of real polynomials. We begin with random point processes on two-component systems. Let X be the set of finite multisets of the closed upper half plane H ⊂ C. An element ξ ∈ X is called a configuration, and X is called the configuration space of H . Given a Borel set A of H , we define the function N A : X → Z≥0 by specifying that N A (ξ ) be the cardinality (as a multiset) of (ξ ∩ A). We define to be the sigma-algebra on X generated by {N A : A ⊆ H Borel}. A probability measure P defined on is called a random point process on X . For each pair of non-negative integers (L , M) we define X L ,M to be the subset of X consisting of those configurations ξ which consist of exactly L real points and M points in the open upper half plane H . That is, X L ,M := {ξ ∈ X : NR (ξ ) = L and N H (ξ ) = M} . Clearly, X L ,M is measurable and X can be written as the disjoint union X L ,M . X= L≥0,M≥0
Given a point process P on X , we may define the measure P L ,M on X by P L ,M (B) := P(B ∩ X L ,M ) for each B ∈ . The measure P L ,M induces a measure on X L ,M (and we will also use the symbol P L ,M for this measure).
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Given a matrix Y ∈ R N ×N there must be a pair of non-negative integers (L , M) with L + 2M = N such that, counting multiplicities, Y has L real eigenvalues and M non-real complex conjugate pairs of eigenvalues. By representing each pair of complex conjugate eigenvalues by its representative in H , we may identify all possible multisets of eigenvalues of matrices in R N ×N with the disjoint union X N := X L ,M . (2.1) (L ,M) L+2M=N
Similarly, we may identify all possible multisets of roots of degree N real polynomials with this disjoint union. Thus, when studying the statistics of eigenvalues of ensembles of asymmetric real matrices (respectively of the roots of degree N real polynomials) we may restrict ourselves to random point processes on H which are supported on the disjoint union given in (2.1). That is, the eigenvalue statistics of an ensemble of real matrices is determined by a set of finite measures P L ,M on X L ,M for each pair of non-negative integers (L , M) with L + 2M = N . In this situation we will say that P is a random point process on X N associated to the family of finite measures {P L ,M : L + 2M = N }. From here forward we will assume that L , M and N are non-negative integers such that L + 2M = N , and a sum indexed over (L , M) will be taken to be a sum over all pairs satisfying this condition. Moreover, when we refer to a point process we will always mean a point process on X N . 3. Point Processes on X N Associated to Weights In this section we will introduce an important class of point processes associated to Borel measures on C. We will be particularly interested in measures which are a sum of two mutually singular measures: one of which is absolutely continuous with respect to Lebesgue measure on R and one which is absolutely continuous with respect to Lebesgue measure on C. The corresponding densities with respect to Lebesgue measure will allow us to construct a weight function which uniquely determines the associated point process on X N . Point processes of this sort arise in the study of asymmetric random matrices and the range of multiplicative functions on polynomials with real coefficients. It will be useful to distinguish non-real complex numbers and we set C∗ = C\R. We start rather generally by constructing measures on X L ,M from measures on R L × M C∗ . The benefit in doing this is that it allows us to express important quantities associated to point processes (averages, correlation functions, etc.) as rather pedestrian integrals over R L × C∗M . To each (α, β) ∈ R L × C∗M we associate a configuration in X L ,M given by 1 , . . . , β M }, where β {α, β} := {α1 , . . . , α L , β m = {βm , βm } ∩ H. A given configuration ξ ∈ X L ,M may correspond to several vectors in R L × C∗M and we will call {(α, β) : {α, β} = ξ } the set of configuration vectors of ξ . A function F on X L ,M induces a function on R L × C∗M specified by (α, β) → F{α, β}, and given a measure ν L ,M on R L × C∗M there exists a unique measure P L ,M on X L ,M specified by demanding that 1 F(ξ ) dP L ,M (ξ ) = F{α, β} dν L ,M (α, β), (3.1) L!M!2 M R L ×C∗M X L ,M
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for every -measurable function F on X . The normalization constant L!M!2 M arises since a generic element ξ ∈ X L ,M corresponds to L!M!2 M configuration vectors. By specifying measures ν L ,M on R L × C∗M for all pairs (L , M) and normalizing so that the total measure of X N is 1, we define a point process on X N . A very important class of point processes arises when we demand that the various ν L ,M are all related to a single measure on C. Given a measure ν1 on R and a measure ν2 on C∗ , we set ν to be the measure ν1 + ν2 on C. We will write ν L for the product measure of ν1 on R L and ν2M will be the product measure of ν2 on C M . By combining ν L and ν2M with a certain Vandermonde determinant we will arrive at the desired measures on R L × C∗M . Given a vector γ ∈ C N we define V (γ ) to be the N × N matrix whose n, n entry is given by γnn −1 . We will denote the determinant of V (γ ) by (γ ), and define the function : R L × C M → C by (α, β) := (α1 , . . . , α L , β1 , β1 , . . . , β M , β M ). Using these definitions we set ν L ,M to be the measure on R L × C∗M given by dν L ,M (α, β) = 2 M |(α, β)|, d(ν L × ν2M )
(3.2)
and we will write P Lν ,M for the measure on X L ,M given specified by ν L ,M as in (3.1). If P Lν ,M is finite for each pair (L , M), then we set P ν :=
1 ν P L ,M Zν
where
Z ν :=
(L ,M)
(L ,M)
P Lν ,M (X L ,M ).
We will call P ν the point process on X N associated to the weight measure ν and Z ν will be referred to as the partition function of P ν . We set λ1 and λ2 to be Lebesgue measures on R and C respectively and let λ := λ1 +λ2 . If there exists a function w : C → [0, ∞) such that ν = wλ (by which we mean dν/dλ = w), then we will call w the weight function of P ν . In this situation we set P Lw,M := P Lν ,M and define P w and Z w analogously. If P ν has weight function w then L M dν L ,M M (3.3) (α, β) = 2 w(α ) w(βm ) |(α, β)| , d(λ L × λ2M ) =1
m=1
and we define L ,M : R L × C M to be the function given on the right-hand side of (3.3). The collection { L ,M : L + 2M = N } plays the role of the joint eigenvalue probability density function, and we will call L ,M the L , M-partial joint eigenvalue probability density function of P w . 4. Examples of Point Processes Associated to Weights Point processes on X N associated to weights arise in a variety of contexts. It is often the case that the weight function w is invariant under complex conjugation. In this situation, there necessarily exists some function ρ : C → [0, ∞) such that ⎧ if γ ∈ R; ⎨ ρ(γ ) w(γ ) := ⎩ ρ(γ )ρ(γ ) if γ ∈ C . ∗
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When it exists, we will call ρ the root or eigenvalue function of P w (depending on whether P w models the roots of random polynomials or eigenvalues of random matrices). The root/eigenvalue function is often a more natural descriptor of P w than w. In this situation we will write P [ρ] and Z [ρ] for P w and Z w . 4.1. The real Ginibre ensemble. In 1965 J. Ginibre introduced three ensembles of random matrices whose entries were respectively chosen with Gaussian density from R, C and Hamilton’s quaternions [15]. The real Ginibre ensemble is given by R N ×N together with the probability measure η given by dη(Y) = (2π )−N
2 /2
e− Tr(Y
T Y)/2
dλ N ×N (Y),
where λ N ×N is Lebesgue measure on R N ×N . This ensemble has since been named GinOE due to certain similarities with the Gaussian Orthogonal Ensemble. Among Ginibre’s original goals was to produce a formula for the partial joint eigenvalue probability density functions of GinOE. He was only able to do this for the subset of matrices with all real eigenvalues. In the 1990s Lehmann and Sommers [19] and later Edelman [8] proved that the L , M-partial eigenvalue probability density function of GinOE is given by L ,M for the eigenvalue function √ Gin(γ ) := exp(−γ 2 /2) erfc( 2|Im(γ )|). Consequently, the investigation of the eigenvalue statistics of GinOE reduces to the study of P [Gin] . 4.2. The range of Mahler measure. The Mahler measure of a polynomial f (x) =
N n=0
an x N −n = a0
N
(x − γn )
(4.1)
n=1
is given by
1
µ( f ) = exp 0
N log f (e2πiθ ) dθ = |a0 | max{1, |γn |}. n=1
The second equality comes from Jensen’s formula. Mahler measure arises in a number of contexts, i.e. ergodic theory, potential theory and Diophantine geometry and approximation (a good reference covering the many aspects of Mahler measure is [10]). One problem in the context of the geometry of numbers is to estimate the number of degree N integer polynomials with Mahler measure bounded by T > 0 as T → ∞. Chern and Vaaler produced such an estimate in [6] using the general principal that the number of lattice points in a ‘reasonable’ domain in R N +1 is roughly equal to the volume (Lebesgue measure) of the domain. That is, if we identify degree N polynomials with their vector of coefficients in R N +1 and use the approximation, #{g(x) ∈ Z[x] : deg g = N , µ(g) ≤ T } ≈ λ N +1 {g(x) ∈ R[x] : deg g = N , µ(g) ≤ T } =T N +1 λ N +1 {g(x)∈R[x] : deg g = N , µ(g) ≤ 1},
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then the main term in the asymptotic estimate Chern and Vaaler were interested in can be expressed in terms of the volume of the degree N star body of Mahler measure, U N = {g(x) ∈ R[x] : deg g = N , µ(g) ≤ 1}.
(4.2)
The volume of this set is given by 2 Z [ψ] , N +1
(4.3)
where ψ(γ ) = max{1, |γ |}−N −1 [6]. Consequently, the volume of the star body which leads to an asymptotic estimate of interest in Diophantine geometry essentially equals the partition function for the random point process P [ψ] . 4.3. The range of other multiplicative functions on polynomials. We can generalize the Mahler measure by replacing the function γ → max{1, |γ |} with other functions of γ . Given a continuous function φ : C → (0, ∞) which satisfies the asymptotic formula, φ(γ ) ∼ |γ |
|γ | → ∞,
as
(4.4)
we define the function : C[x] → [0, ∞) by : a0
N
(x − γn ) → |a0 |
n=1
N
φ(γn ).
n=1
The function is known as a multiplicative distance function (so named because it is a distance function in the sense of the geometry of numbers on finite dimensional subspaces of C[x]). The asymptotic condition in (4.4) ensures that is continuous on finite dimensional subspaces of C[x] (one of the axioms of a distance functions). We define the degree N starbody of in analogy with (4.2): U N () = {g(x) ∈ R[x] : deg g = N , (g) ≤ 1}. In this situation the volume of U N () equals λ N +1 (U N ()) =
2 Z [ψ] , N +1
where
ψ(γ ) = φ(γ )−N −1 .
We may discover further information about the range of values takes on degree N real polynomials by considering the point process on X N corresponding to the root function ψ(γ ) = φ(γ )−σ for σ > N . The partition function of P [ψ] is therefore a function of σ , and Z [ψ] (σ ) is known as the degree N moment function of . In fact, we may extend the domain of real moment functions to a function of a complex variable s on the half-plane Re(s) > N . Moreover, in this domain Z [ψ] (s) is analytic. Any analytic continuation beyond this half-plane gives information about the range of values of which may not be realizable by other methods. For instance, when ψ(γ ) = max{1, |γ |}−s , Chern and Vaaler discovered that Z [ψ] (s) has an analytic continuation to a rational function of s with rational coefficients and poles at positive integers ≤ N [6]. Similar results have been found for moment functions of other multiplicative distance functions; see [23].
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5. Correlation Measures and Functions Suppose and m are non-negative integers, not both equal to 0. Then, given a function f : R × Cm ∗ → C, we define the function F f : X N → C by F f (ξ ) = f (x, z), {x,z}⊆ξ
where the sum is over all (x, z) ∈ R × Cm ∗ such that {x, z} ⊆ ξ . We take an empty sum to equal 0, and thus if ξ ∈ X L ,M with L ≤ or M ≤ m, then F f (ξ ) = 0. Given a point process P on X N , if there exists a measure ρ ,m on R × Cm ∗ such that for all Borel measurable functions f , f (x, z) dρ ,m (x, z) = F f (ξ ) dP(ξ ), (5.1) R ×C m ∗
XN
then we call ρ ,m the ( , m)–correlation measure of P. Furthermore, if ρ ,m has a density with respect to λ × λ2m , then we will call this density the , m–correlation function of P and denote it by R ,m . By convention we will take R0,0 to be the constant 1. Proposition 1. 2m Zν
(L ,M) L≥ ,M≥m
dρ ,m d(ν ×ν2m ) (x, z)
equals
1 (L − )!(M − m)!
|(x∨α, z∨β)| dν L− (α) dν2(M−m) (β),
R L− C M−m
where x ∨ α ∈ R L is the vector formed by concatenating the vectors x ∈ R and α ∈ R L− (and similarly for z∨β ∈ C M ). Corollary 2. If ν = wλ, then R ,m (x, z) equals
1 Zw
(L ,M) L≥ ,M≥m
1 (L − )!(M − m)!2 M−m
R L−
C M−m
×
L ,M (x∨α, z∨β) dλ L− (α) dλ2(M−m) (β).
Proof of Proposition 1. Assume that L ≥ and M ≥ m, from (3.1) and (3.2), 1 ν F f (ξ ) dP L ,M (ξ )= F f {α, β} |(α, β)| dν L (α) dν2M (β). L!M! R L C∗M X L ,M
(5.2)
The function (α, β) → |(α, β)| is invariant under any permutation of the coordinates of α and β, since such a permutation merely permutes the columns of the Vandermonde matrix. Similarly, replacing any of the coordinates of β with their complex conjugates merely transposes pairs of columns of the Vandermonde matrix. That is, if (α, β) and (α , β ) are elements in R L × C∗M such that {α, β} = {α , β }, then |(α, β)| = |(α , β )|. Moreover, if any of the β are real, then the Vandermonde matrix has two identical columns and is therefore zero. We may thus replace the domain of integration on the right-hand side of (5.2) with R L × C M . In fact, we may assume
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that the domain of integration on the right-hand side is over the subset of R L × C M consisting of those vectors with distinct coordinates. Now, F f {α, β} =
f (x, z).
{x,z}⊆{α ,β }
Assuming that the coordinates of (α, β) are distinct, and if (x, z) ∈ R × Cm is such that {x, z} ⊆ {α, β} we may find a vector (a, b) ∈ R L− × C M−m such that (x ∨ a, z ∨ b) is given by permuting the coordinates of (α, β). Clearly |(α, β)| = |(x ∨ a, z ∨ b)| and dν L (α) dν2M (β) = dν (x) dν L− (a) dνm (z) dν M−m (b). These observations together with an application of Fubini’s Theorem imply that (5.2) can be written as 1 f (x, z) L!M! R Cm {x,z} ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ × |(x∨a, z∨b)| dν L− (a) dν M−m (b) dν (x) dνm (z). ⎪ ⎪ ⎩ L− M−m ⎭ C
R
Now, it is easily seen that there are 2m
L M ! m! m
vectors corresponding to each {x, z}, and thus we find X L ,M
×
⎧ ⎪ ⎨
F f (ξ ) dP Lν ,M (ξ )
2m
⎪ ⎩ (L− )!(M−m)!
=
R Cm
R L− C M−m
f (x, z)
⎫ ⎪ ⎬ |(x∨a, z∨b)| dν L− (a) dν M−m (b) dν (x) dνm (z). ⎪ ⎭
The proposition follows since
F f (ξ ) dP ν (ξ ) = X
1 F f (ξ ) dP Lν ,M (ξ ). Zν X L ,M (L ,M)
From here forward, and unless otherwise stated, ( , m) will represent an ordered pair of non-negative integers such that + 2m ≤ N .
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6. A Matrix Kernel for Point Processes on X N From here forward we will assume that N is even. Let P ν be the point process on X N associated to the weight w, and as before let ν = ν1 + ν2 . We define the operator ν on the Hilbert space L 2 (ν) by ⎧ 1 ⎪ ⎨ g(y) sgn(y − γ ) dν1 (y) if γ ∈ R, ν g(γ ) := 2 R ⎪ ⎩ ig(γ ) sgn(Im(γ )) if γ ∈ C∗ , and we use this to define the skew-symmetric bilinear form on L 2 (ν) given by ·|·ν g|hν := g(γ )ν h(γ ) − ν g(γ )h(γ ) dν(γ ). C
If ν = wλ then g|hν = g | hλ where, for instance, h(γ ) := h(γ )w(γ ). Theorem 3. Let q = {q0 (γ ), q2 (γ ), . . . , q N −1 (γ )} ⊆ R[γ ] be such that each qn is monic and deg qn = n. Then, Z ν = Pf Uqν ,
(6.1)
the Pfaffian of Uqν , where Uqν := [qn |qn ν ];
n, n = 0, 1, . . . , N − 1.
We will call q a complete family of monic polynomials. Remark. It is at this point that it is necessary that N be even, since the Pfaffian is only defined for antisymmetric matrices with an even number of rows and columns. A similar formula to (6.1) exists for Z ν in the case when N is odd; see [22]. However, we have not pursued the subsequent analysis necessary to recover the correlation functions in this case. Theorem 3 follows from results proved in [22]. In fact, [22] gives a formula for the average of a multiplicative class function over the point process on X N determined by the weight function of the real Ginibre ensemble. However, the combinatorics necessary to arrive at such averages is independent of any specific feature of Gin(γ ) and the measure on C specified by Gin(γ )dλ(γ ) can formally be replaced by any measure ν. In order to express the correlation functions for the point process P w associated to the weight w we will define η to be a measure on C given by a linear combination of point masses, and then use the definition of the partition function and properties of Pfaffians to expand both sides of the equation Z w(λ+η) 1 w(λ+η) = w Pf Uq . w Z Z
(6.2)
The coefficients in the linear combination defining η appear again in terms on both sides of the expanded equation, and after identifying like coefficients on both sides of the expanded equation we will be able to read off a closed form for the correlation functions in terms of the Pfaffian of a matrix whose entries depend on a 2 × 2 matrix kernel.
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In order to define the 2 × 2 matrix kernel for P w , we let q be a complete family of monic polynomials, and we define qn (γ ) := qn (γ )w(γ )
n = 0, 1, . . . , N − 1.
(6.3)
We then define S N (γ , γ ) := 2
N −1 N −1 n=0
µn,n qn (γ ) λ qn (γ ),
n =0
where we define µn,n to be the n, n entry of (Uqw )−T . Similarly we define, I S N (γ , γ ) := 2
−1 N −1 N
µn,n λ qn (γ ) λ qn (γ ),
n=0 n =0
and
DS N (γ , γ ) := 2
−1 N −1 N n=0
µn,n qn (γ ) qn (γ ).
n =0
Remark. The functions S N , I S N and DS N can be shown to be independent of the family q. By setting q to be skew-orthogonal with respect to the bilinear form ·|·λ we arrive at particularly simple representations for these expressions. Finally, in order to define the matrix kernel P w we define the function E : C2 → {− 21 , 0, 21 } by ⎧ 1 ⎪ ⎨ sgn(γ − γ ) if γ , γ ∈ R; 2 E(γ , γ ) := ⎪ ⎩ 0 otherwise. The matrix kernel of P w is then given by DS N (γ , γ ) S N (γ , γ ) . K N (γ , γ ) := −S N (γ , γ ) I S N (γ , γ ) + E(γ , γ )
(6.4)
Remark. The explicit N -dependence of K N and its constituents is traditional, since one is often interested in the N → ∞ asymptotics of K N . 7. Correlation Functions in Terms of the Matrix Kernel We may state one of the main results of this manuscript. Theorem 4. The , m–correlation function of P w is given by j, j = 1, 2, . . . , ; K N (x j , x j ) K N (x j , z k ) ; R ,m (x, z) = Pf K N (z k , x j ) K N (z k , z k ) k, k = 1, 2, . . . , m.
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Remark. The matrix on the right-hand side of this expression is composed of 2 × 2 blocks, so that, for instance, the first row of 2 × 2 blocks is given by [K N (x1 , x1 ) . . .
K N (x1 , x ) K N (x1 , z 1 ) . . .
K N (x1 , z m )] .
We define the measure δ on C to be the measure with unit point mass at 0. Given U real numbers x1 , x2 , . . . , xU and V non-real complex numbers, z 1 , z 2 , . . . , z V , we define the measure η by dη(γ ) :=
U
au dδ(γ − xu ) +
V
bv (dδ(γ − z v ) + dδ(γ − z v )) ,
v=1
u=1
where a1 , a2 , . . . , aU and b1 , b2 , . . . , bV are indeterminants. It does no harm to assume that U and V are both greater than N . By reordering and renaming the x, z, a and b we will also write dη(γ ) :=
T
ct d δ (γ − yt ),
t=1
where we define d δ (γ − y) =
⎧ ⎨ dδ(γ − y)
if y ∈ R;
⎩ dδ(γ − y) + dδ(γ − y)
if y ∈ C∗ .
Clearly T = U + V . As we alluded to previously, the proof of Theorem 4 relies on expanding Z w(λ+η) /Z w in two different ways and then equating the coefficients of certain products of c1 , c2 , . . . , cT . One of the expansions of Z w(λ+η) /Z w comes from Theorem 3, while the other comes directly from the definition of the partition function. Proposition 5. √ Z w(λ+η) = Pf J + ct ct K N (yt , yt ) ; w Z
t, t = 1, 2, . . . , T,
(7.1)
where J is defined to be the 2T × 2T matrix consisting of 2 × 2 blocks given by 0 1 ; t, t = 1, 2, . . . , T. J := δt,t −1 0 Remark. The Pfaffian which appears in (7.1) is an example of a Fredholm Pfaffian. This is the Pfaffian formulation of the notion of a Fredholm determinant and is discussed in [21]. We defer the proof of Proposition 5 and Proposition 6 until Sect. 9. For each ( , m) and (L , M) we define the , m, L , M-partial correlation function of P w to be R ,m,L ,M : R × Cm → [0, ∞), where R ,m,L ,M (x, z) := 1 (L − )!(M −m)!2 M−m
R L− C M−m
L ,M (x ∨ α, z ∨ β) dλ L− (α) dλ2(M−m) (β).
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When = m = 0, we take x ∨ α = α and z ∨ β = β, so that R0,0,L ,M is a constant equal to P L ,M (X L ,M ). The partial correlation functions are related to the correlations functions by the formula R ,m (x, z) =
1 Zw
R ,m,L ,M (x, z),
(7.2)
(L ,M) L≥ ,M≥m
and thus the partial correlation functions are one path to the correlation functions. Equation (7.2) is still valid when = m = 0, though the constituent correlation ‘functions’ are actually constants. The partial correlation functions of P [Gin] of the special forms R L ,M,0,M and R0,N ,0,n have been studied by Akemann and Kanzieper in [1 and 18]. For general weight function w, the partial correlation functions of the form R N ,0,n,0 are (up to normalization) equal to the correlation functions of the β = 1 Hermitian ensemble with weight w. This connection will be exploited in A. Before stating the next proposition we need a bit of notation. Given non-negative integers n and W , we define InW to be the set of increasing functions from {1, 2, . . . , n} into {1, 2, . . . , W }. Clearly if W < n then InW is empty. Given a vector a ∈ CW and an element u ∈ InW , we define the vector au ∈ Cn by au = {au(1) , au(2) , . . . , au(n) }. Proposition 6. For each pair (L , M), 1
L ,M (α, β) d(λ + η) L (α) d(λ + η)2M (β) L!M!2 M R L C M ⎧ ⎫ L M ⎨ m ⎬ = au( j) bv(k) R ,m,L ,M (xu, zv). ⎩ ⎭ U V =0 m=0 u∈I v∈Im
j=1
k=1
Remark. We will use the convention that 0 j=1 u∈IU 0
0
au( j) =
bv(k) = 1.
v∈I0V k=1
This will allow us to keep from having to deal with the pathological correlation ‘functions’ R0,0,L ,M and R0,0 separately. Proposition 7. Suppose K : {1, 2, . . . , T } × {1, 2, . . . , T } → R2×2 is such that K (t, t ) = −K (t , t)T , and define K to be the 2T × 2T block antisymmetric matrix whose t, t entry is given by K (t, t ). Then, Pf[J + K] = 1 +
T
Pf Kt,
S=1 t∈IT S
where for each t ∈ ITS , Kt is the 2S × 2S antisymmetric matrix given by Kt = [K (t(s), t(s ))];
s, s = 1, 2, . . . , S.
(7.3)
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Proof. This is a special case of the formula for the Pfaffian of the sum of two antisymmetric matrices. See [23 or 26] for a proof. Using these lemmas we may complete the proof of Theorem 4. First notice that Proposition 6 and (7.2) imply that ⎧ ⎫ m ⎬ ⎨ Z w(λ+η) = a b (7.4) R (x , z ). u ( j) v (k) ⎩ ⎭ ,m u v Zw U V (l,m) u∈I v∈Im
j=1
k=1
This follows by summing both sides of the expression in Proposition 7 over all (L , M) and then reorganizing the sums over (L , M), and m. U +V V Given u ∈ IU and v ∈ Im we define (u : v) to be the element in I +m given by
u(n) if n ≤ (u ∨ v)(n) := + v(n) if n > . V Notice that each t ∈ ITS is equal to (u ∨ v) for some u ∈ IU and v ∈ Im with + m = S. (This does not preclude the possibility that either U or V equals 0, in which case t = v or t = u.) It follows that we can rewrite (7.3) as Pf(J + K) = Pf K(u∨v) . ( ,m) +m≤T
V v∈Im u∈IU
√ If we set K (t, t ) = ct ct K N (yt , yt ), then ⎧ ⎫ m ⎨ ⎬ K N (xu(k) , xu(k ) ) K N (xu(k) , z v(n ) ) , au( j) bv(k) Pf Pf K(u∨v) = K N (z v(n) , xu(k ) ) K N (z v(n) , z v(n ) ) ⎩ ⎭ j=1
k=1
where k and k are indices that run from 1 to and n and n are indices which run from 1 to m. Thus, Pf(J + K) equals ⎧ ⎫ m ⎬ ⎨ K N (xu(k) , xu(k ) ) K N (xu(k) , z v(n ) ) au( j) bv(k) Pf , (7.5) K N (z v(n) , xu(k ) ) K N (z v(n) , z v(n ) ) ⎩ ⎭ V ( ,m) u∈IU v∈Im +m≤T
j=1
k=1
and Theorem 4 follows by equating the coefficients of (7.5).
j=1 a j
m
k=1 bk
in (7.4) and
8. Limiting Correlation Functions for the Real Ginibre Ensemble We now turn to the large N asymptotics of the matrix kernel for the Ginibre ensemble of real matrices. In fact, we maintain our restriction to the case where N = 2M is even, and consider the asymptotics of K 2M as M → ∞. Throughout this section we will take φ to be the function given by √ 2 2 φ(γ ) = exp(−γ /4 − γ /4) erfc 2|Im(γ )| . Notice that, since erfc(0) = 1, when γ ∈ R this reduces to exp(−γ 2 /2).
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The skew-orthogonal polynomials for this weight are reported in [13] to be π2m (γ ) = γ 2m
π2m+1 (γ ) = γ 2m+1 − 2mγ 2m−1 √ with normalization π2m |π2m+1 ν = 2 2π (2m)!. A detailed account of the derivation of these skew-orthogonal polynomials will appear in [12]. The skew-orthogonality of these polynomials and formulas from Sect. 6 imply that M−1 π2m (γ )λ π2m+1 (γ ) − π2m+1 (γ )λ π2m (γ ) 1 S2M (γ , γ ) = √ , (2m)! 2π m=0 M−1 π2m (γ ) π2m+1 (γ ) − π2m+1 (γ ) π2m (γ ) 1 , DS2M (γ , γ ) = √ (2m)! 2π m=0
(8.1)
(8.2)
M−1 π2m (γ )λ π2m+1 (γ ) − λ π2m+1 (γ )λ π2m (γ ) 1 λ I S2M (γ , γ ) = √ . (8.3) (2m)! 2π m=0
The correlation functions are all of the form Pf K for an appropriate matrix K whose entries are given in terms of (8.1), (8.2) and (8.3). If D is a square matrix such that the product DKDT makes sense, then Pf(DKDT ) = Pf K · det D. And thus, if det D = 1 we have Pf K = Pf(DKDT ). That is, we may alter (and potentially simplify) the presentation of the Pfaffian representation of the correlation functions by modifying K in this manner by a matrix with determinant 1. When D is diagonal, the process of modifying K by D preserves the block structure of K. That is, the effect of modifying K by D affects changes at the kernel level and the correlation functions can be represented as N the Pfaffian of a block matrix (cf. Theorem 4) with respect to a new matrix kernel K dependent on K N and D. This will allow us to write the correlation functions of the real Ginibre ensemble in the simplest manner possible by ‘factoring’ unnecessary terms out of the kernel. It will be convenient to define c M , s M and e M to be the degree 2M − 2 Taylor polynomials for cosh, sinh and exp respectively. Explicitly, c M (γ ) :=
M−1 m=0
γ 2m , (2m)!
s M (γ ) :=
M−1 m=1
γ 2m−1 (2m − 1)!
and e M (γ ) := c M (γ ) + s M (γ ) =
2M−2 m=0
γm . m!
Theorem 8. The , m–correlation function of the real Ginibre ensemble of 2M × 2M matrices is given by 2M (x j , z k ) 2M (x j , x j ) K j, j = 1, 2, . . . , ; K ; R ,m (x, z) = Pf 2M (z k , z k ) k, k = 1, 2, . . . , m, K 2M (z k , x j ) K where
! S2M (γ , γ ) 2M (γ , γ ) = DS 2M (γ , γ ) K − S2M (γ , γ ) " I S 2M (γ , γ ) + E(γ , γ )
is given as follows. Let x, x ∈ R and z, z ∈ C∗ , then:
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2M (x, x ), are given by 1. The entries in the real/real kernel, K • S2M (x, x ) = r M (z, x) e−z /2 =√ 2π 2
1
e− 2√(x−x 2π
)2
e−x x e M (x x ) + r M (x, x ), where
√ 2 M−3/2 1 x2 2M−1 , (8.4) erfc 2Im(z) · γ M− , sgn(x)z (2M−2)! 2 2
and γ is the lower incomplete gamma function; • ! DS 2M (x, x ) = • " I S 2M (x, x ) =
1
e− 2√(x−x 2π
)2
(x − x)e−x x e M (x x );
x 2 /2 −t x 2 /2 −t 2 2 √ √ e e e−x /2 e−x /2 √ sgn(x ) √ c M (x 2t) dt − √ sgn(x) √ c M (x 2t) dt. 2 π 2 π t t 0 0 2M (z, z ), are given by 2. The entries in the complex/complex kernel, K √ 1 2 √ − 2 (z−z ) • S2M (z, z ) = ie √ (z − z) erfc 2Im(z) erfc 2Im(z ) e−zz e M (zz ); 2π √ 1 2 √ e− 2√(z−z ) ! • DS 2M (z, z ) = (z − z) erfc 2Im(z) erfc 2Im(z ) e−zz e M (zz ); 2π √ √ − 1 (z−z )2 • " I S 2M (z, z ) = − e 2√ (z − z) erfc 2Im(z) erfc 2Im(z ) e−zz e M (zz ). 2π
2M (x, z), are given by 3. The entries in the real/complex kernel, K √ 1 2 − (x−z) • S2M (x, z) = ie √2 (z − x) erfc 2Im(z) e−x z e M (x z); 2π √ 1 2 2 (x−z) e− √ • S2M (z, x) = erfc 2Im(z) e−x z e M (x z) + r M (z, x). 2π √ 1 2 2 (x−z) e− √ ! • DS 2M (x, z) = (z − x) erfc 2Im(z) e−x z e M (x z); 2π √ − 1 (x−z)2 • " I S 2M (x, z) = − ie √2 erfc 2Im(z) e−x z e M (x z) − ir M (z, x). 2π
2M . Theorem 8 allows us to derive the M → ∞ limit of K Corollary 9. (Limit at the origin) Let x and x be real numbers, and suppose z and z 2M . Then, are complex numbers in the open upper half plane. We define K = lim K M→∞
the limit exists, and 1. The limiting real/real kernel, K (x, x ), is given by ⎡ K (x, x ) =
1 2 √1 (x − x)e− 2 (x−x ) 2π ⎣ 1 2 1 − √1 e− 2 (x−x ) 2 2π
1 2 √1 e− 2 (x−x ) 2π √ | sgn(x − x ) erfc |x−x 2
⎤ ⎦.
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2. The limiting complex/complex kernel, K (z, z ), is given by √ √ 1 K (z, z ) = √ erfc 2Im(z) erfc 2Im(z ) 2π ' ( 1 1 2 2 (z − z)e− 2 (z−z ) i(z − z)e− 2 (z−z ) × 1 1 2 . 2 i(z − z)e− 2 (z−z ) −(z − z)e− 2 (z−z ) 3. The limiting real/complex kernel, K (x, z), is given by ( ' 1 1 2 2 √ 1 (z − x)e− 2 (x−z) i(z − x)e− 2 (x−z) K (x, z) = √ erfc 2Im(z) . 1 1 2 2 2π −e− 2 (x−z) −ie− 2 (x−z) Remark. Observe that all blocks of the kernel are invariant with respect to real shifts. That is, if c ∈ R and γ , γ are in C then K (γ + c, γ + c) = K (γ , γ ). 8.1. In the bulk. The circular law for N × N matrices with i.i.d Gaussian entries says that, when normalized by N −1/2 the density of eigenvalues becomes uniform on the unit disk as N → ∞ (See [16] for a proof of this fact when the entries are i.i.d. Gaussian, and [2 and 27] for more general results.) This gives us the appropriate scaling when considering the matrix kernel in the in this section we will be interested √ bulk. Specifically, √ 2M (u 2M + s, u 2M + s ), where u is a point in the open unit in the large M limit of K disk, and s and s are complex numbers. When u is real we expect that the limiting kernel under this scaling should yield (s, s ); indeed this is the case. When u is nonreal a different kernel arises. K Theorem 10. Let −1 < u < 1 be a real number, let r1 , r2 , . . . , r ∈ R and s1 , s2 , . . . , sm be in the open upper half plane. Set, √ √ j = 1, 2, . . . , ; and z k = u 2M + sk k = 1, 2, . . . , m. x j = u 2M + r j Then,
lim R ,m (x, z) = Pf
M→∞
K (r j , r j ) K (r j , sk ) ; K (sk , r j ) K (sk , sk )
j, j = 1, 2, . . . , ; k, k = 1, 2, . . . , m,
where K is given as in Corollary 9. Theorem 11. Let u be in the open upper half plane such that |u| < 1 and suppose s1 , s2 , . . . , sm ∈ C. Set, √ z k = u 2M + sk k = 1, 2, . . . , m. Then, lim R0,m (−, z) = det
M→∞
m |sk |2 |sk |2 1 exp − − + sk s k . π 2 2 k,k =1
Remark. The limiting correlation functions in the complex bulk are invariant with respect to any complex shift.
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Remark. The function 1 |s|2 |s |2 (s, s ) → exp − − + ss π 2 2
is, up to a factor of 1/2, the limiting (scalar) kernel of the complex Ginibre ensemble. Thus, the limiting correlation functions in the bulk of the real Ginibre ensemble off the real line is almost identical to the limiting correlation functions in the bulk of the complex Ginibre ensemble. See Ginibre’s original paper [15], or [20, Sect. 15.1], for the derivation of the finite N correlation functions for the complex Ginibre complex ensemble. We derive the large N asymptotics of the correlation functions of Ginibre’s complex ensemble in Appendix C. 8.2. At the edge. At the edge of the spectrum new kernels emerge. Theorem 12. Let u = ±1, let r1 , r2 , . . . , r ∈ R and s1 , s2 , . . . , sm be in the open upper half plane. Set √ √ x j = u 2M + r j j = 1, 2, . . . , ; and z k = u 2M + sk k = 1, 2, . . . , m. Then, lim R ,m (x, z) = Pf
M→∞
K edge (r j , r j ) K edge (r j , sk ) ; K edge (sk , r j ) K edge (sk , sk )
where K edge (γ , γ ) =
j, j = 1, 2, . . . , ; k, k = 1, 2, . . . , m,
DSedge (γ , γ ) Sedge (γ , γ ) , −Sedge (γ , γ ) I Sedge (γ , γ )
and: 1. The real/real kernel at the real edge, K r ), is given by edge (r, 1 2 2 • Sedge (r, r ) = √1 e− 2 (r −r ) erfc u (r√+r ) + 4√1 π e−r erfc(−ur ); 2 2π 2 1 2 • DSedge (r, r ) = √1 (r − r )e− 2 (r −r ) erfc u (r√+r ) ; 2 2π 2 −r | • I Sedge (r, r ) = 21 sgn(r − r ) erfc |r√ . 2
2. The complex/complex kernel at the real edge, K edge (s, s ), is given by √ √ • Sedge (s, s ) = √i erfc 2Im(s) erfc 2Im(s ) 2 2π
× (s − s)e
− 21 (s−s )2
• DSedge (s, s ) =
√1 2 2π
erfc
× (s − s)e
(s + s ) ; erfc u √ 2
√ √ 2Im(s) erfc 2Im(s )
− 21 (s−s )2
(s + s ) ; erfc u √ 2
Ginibre Ensemble of Random Matrices and its Scaling Limits
• I Sedge (s, s ) = −
√1 2 2π
195
√ √ erfc 2Im(s) erfc 2Im(s )
1 2 (s + s ) . × (s − s)e− 2 (s−s ) erfc u √ 2 3. The real/complex kernel at the real edge, K edge (r, s), is given by √ 1 2 • Sedge (r, s) = √i e− 2 (r −s) erfc 2Im(s) (s − r ) erfc u (r√+s) ; 2 2π 2 √ 1 2 • Sedge (s, r ) = √1 e− 2 (r −s) erfc 2Im(s) erfc u (r√+s) 2 2π
2
1 2 + √ e−s erfc(−ur ); 4 π √ 1 − 21 (r −s)2 √ • DSedge (r, s) = e erfc 2Im(s) (s − r ) erfc u (r√+s) ; 2 2π 2 √ 1 2 • I Sedge (r, s) = √−i e− 2 (r −s) erfc 2Im(s) erfc u (r√+s) 2 2π
2
2 i − √ e−s erfc(−ur ). 4 π
Remark. The kernel when u = −1 is the image of the kernel at u = 1 under the involution on the closed upper half plane given by z → −z. At the complex edge we have the following: Theorem 13. Let u be in the open upper half plane such that |u| = 1 and suppose s1 , s2 , . . . , sm ∈ C. Set, √ z k = u 2M + sk k = 1, 2, . . . , m. Then, lim R0,m (−, z) m sk u + s k u 1 |sk |2 |sk |2 = det exp − − + sk s k erfc . √ π 2 2 2 k,k =1
M→∞
Remark. The kernel at the complex edge are identical to that of the kernel at the edge of the complex Ginibre complex ensemble. 9. Proofs 9.1. The proofs of Proposition 5 and Proposition A.3. In the case of the real asymmetric ensembles Y = C and b = 1. In the case of the Hermitian ensembles Y = R and √ b = β. We start with qn |qn w(λ+η) = qn (γ ) λ+η qn | qn λ+η = qn (γ ) Y − λ+η qn (γ ) qn (γ ) d(λ + η)(γ ).
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An easy calculation reveals that this is equal to 2 ct ( qn (yt ) − qn (yt )λ qn (yt )) qn (yt )λ b T
qn |qn wλ +
t=1
−
2 b
T T
ct ct qn (yt ) qn (yt ) E(yt , yt ).
t=1 t =1
Next we define A to be the N b × 2T matrix given by ' A :=
2ct qn (yt ) b
( 2ct qn (yt ) ; λ b
n = 0, 1, . . . , N b − 1
and t = 1, 2, . . . T, and the 2T × 2T matrix B, B := −J +
√
ct ct E(yt , yt ) 0 ; 0 0
t, t = 1, 2, . . . , T.
We define C to be the N b × N b matrix given by C = (Uqwλ )−T ; the n, n entry of C is µn,n . A bit of matrix algebra reveals that Z w(λ+η) Pf(B−T − AT CA) Pf(C−T − ABAT ) = , = Zw Pf(C−T ) Pf(B−T )
(9.1)
where the second equality comes from the Pfaffian Cauchy-Binet formula (see B). Notice that B−T = −J −
0√ 0 ; 0 ct ct E(yt , yt )
t, t = 1, 2, . . . , T.
And from Proposition 7, Pf(B)T = Pf(−J) = (−1)T . A bit more matrix algebra reveals, √ √ c√t ct DS N (yt , yt ) √ ct ct S N (yt , yt ) ; A CA = − ct ct S N (yt , yt ) ct ct I S N (yt , yt ) T
t, t = 1, 2, . . . T.
Using these facts and simplifying (9.1) we find Z w(λ+η) √ = (−1)T Pf −J − [ ct ct K N (yt , yt )] ; w Z
t, t = 1, 2, . . . , T,
and the lemma follows by using the fact that if E is an antisymmetric 2T × 2T matrix, then Pf(−E) = (−1)T Pf(E).
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9.2. The proofs of Proposition 6 and Proposition A.4. We start with 1
L ,M (α, β) d(λ + η) L (α) d(λ + η)2M (β). L!M!2 M R L C M
(9.2)
Notice that in the case of the Hermitian ensembles that this is equal to Z w(λ+η) when L = N and M = 0, and the proof of Proposition A.4 follows from the proof recorded here by setting every instance of M to 0. First we write d(λ + η) L (α) =
L L dλ1 (α j ) + dη1 (α j ) = dη (α u)dλ L− (α u ), (9.3) =0 u∈IL
j=1
where given t ∈ InW , we define t to be the unique element in IW W −n whose range is disjoint from t. Notice that since u appears in the summand on the right-hand side of (9.3), the inner sum is not actually empty when = 0; in this situation the summand is equal to dλ L (α). Similarly, d(λ + η)2M (β) =
M
dη2m (β v)dλ2(M−m) (β v ).
M m=0 v∈Im
Thus, (9.2) equals L M M =0 m=0 u∈IL v∈Im
1 L!M!2 M
×
L ,M (α, β)dη (α u)dλ L− (α u ) dη2m (β v)dλ2(M−m) (β v ).
RL CM
We can relabel the α and β in the integrand in any manner we wish, and in particular we may make the integrand independent of u and v. In particular, if we set i ∈ InW to be the identity function on {1, 2, . . . , n}, and since the cardinality of InW is Wn , we find that (9.2) is equal to L M
1 !(L − )!m!(M − m)!2 M =0 m=0 ×
L ,M (α, β) dη (α i) dη2m (β i) dλ L− (α i ) dλ2(M−m) (β i ). R L− C M−m
R Cm
(9.4) Now, dη (α i) =
j=1
η1 (α j ) =
U j=1 u=1
au dδ(α j − xu ).
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We may exchange the sum and the integral on the right-hand side of this expressions by using the set, FU of all functions from {1, 2, . . . , } into {1, 2, . . . , U }. Specifically, ⎫ ⎧ ⎬ ⎨ dη (α i) = au( j) dδ(α j − xu( j) ) , ⎭ ⎩ U u∈F
j=1
and similarly, dη2m (β i) =
V v∈Fm
m
bv(k) d δ (βk − z v(k) ) .
k=1
Thus,
L ,M (α, β) dη (α i) dη2m (β i) =
L ,M (α, β)
R Cm
V v∈Fm u∈FU
×
=
⎧ ⎨ ⎩
R Cm
au( j) dδ(α j − xu( j) )
j=1
⎧ ⎨ u∈FU
V v∈Fm
⎩
au( j)
j=1
m k=1
⎫ m ⎬ ⎭
bv(k)
bv(k) d δ (βk − z v(k) )
k=1
⎫ ⎬ ⎭
2m L ,M (xu ∨ α i , zv ∨ β i ).
(9.5)
Notice that if u or v is not one-to-one then |(xu∨α i, zv∨β i)| = 0. We may consequently V replace the sums over FU and Fk with their respective subsets of one-to-one functions. Moreover, since L ,M is symmetric in the coordinates of each of its arguments, we may replace each one-to-one function in these sums with the increasing function with the same range so long as we compensate by multiplying by ! and m!. Proposition 6 follows from the definition of R ,m,L ,M by substituting (9.5) into (9.4). Proposition A.4 follows from the fact that Rn = Rn,0,N ,0 /Z w . 9.3. The proofs of Theorem 8 and Corollary 9. It shall be convenient to introduce the following variants of S2M and DS2M : S2M (γ , γ ) :=
M−1 m=0
π2m (γ )λ π2m+1 (γ ) − π2m+1 (γ )λ π2m (γ ) , (2m)!
and ) DS 2M (γ , γ ) :=
M−1 m=0
π2m (γ )π2m+1 (γ ) − π2m+1 (γ )π2m (γ ) . (2m)!
The following lemma gives a closed form for these functions.
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Lemma 9.1. Let x be a real number, and suppose z and z are complex numbers. 2 2 M−3/2 1. S2M (z, x) = φ(x)e M (zx) + (2M−2)! sgn(x)z 2M−1 · γ M − 21 , x2 . 2. ) DS 2M (z, z ) = (z − z)e M (zz ). π2m and λ π2m+1 . We start by noticing, Proof. First we compute λ ∞ 1 1 x 1 ∞ g (x) = g(y) sgn(y − x) dy = − g(y) dy + g(y) dy. λ 2 ∞ 2 −∞ 2 x When g(y) = e−y /2 y n , we may evaluate the latter two integrals in terms of the incomplete gamma functions, ⎧ (n−1)/2 sgn(x) · γ n+1 , x 2 ⎪ if n is even; ⎪ ⎨ −2 2 2 λ g (x) = ⎪ ⎪ ⎩ 2(n−1)/2 n+1 , x 2 if n is odd. 2 2 2
We immediately conclude that
and
1 x2 , λ π2m (x) = −2m−1/2 sgn(x) · γ m + , 2 2
(9.6)
x2 x2 2 − m m, = x 2m e−x /2 , λ π2m+1 (x) = 2m m + 1, 2 2
(9.7)
where in the second equality we used the fact that (a + 1, x) = a(a, x) + x a e−x . Using (9.6) and (9.7), we may write S2M (z, x) = φ(x)c M (zx) M−1 M−1 2m−1/2 2m−1/2 1 x2 1 x2 2m+1 2m−1 z − z + sgn(x) γ m+ , γ m+ , . (2m)! 2 2 (2m −1)! 2 2 m=0
m=1
Next, we use the fact that γ (a + 1, x) = aγ (a, x) − x a e−x ,
(9.8)
so that the second sum in this expression becomes −
M−1 m=1
M−1 z 2m−1 |x|2m−1 2(m−1)−1/2 2(m−1)+1 1 x2 z + φ(x) . γ (m − 1) + , (2(m − 1))! 2 2 (2m − 1)! m=1
Consequently, S2M (z, x) = φ(x) (c M (zx) + sgn(x)s M (z|x|)) 1 x2 2 M−3/2 2M−1 γ M− , + sgn(x)z (2M − 2)! 2 2 2 M−3/2 1 x2 sgn(x)z 2M−1 γ M − , . = φ(x)e M (zx) + (2M − 2)! 2 2
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Turning to ) DS 2M , ) DS 2M (z, z ) =
M−1 z 2m (z 2m+1 m=0
− 2mz 2m−1 ) − (z 2m+1 − 2mz 2m−1 )z 2m (2m)!
M−1 − z 2m+1 z 2m z 2m−1 z 2m − z 2m z 2m−1 + (2m)! (2m − 1)! m=0 m=1 M−1 z 2m z 2m M−1 z 2m−1 z 2m−1 + = (z − z) (2m)! (2m − 1)!
=
M−1 z 2m z 2m+1
= (z −
m=0 z)e M (z z).
m=1
With a closed form for S2M and ) DS 2M in hand, we are ready to prove Theorem 8. Proof of Theorem 8. From Lemma 9.1 we have that φ(x) φ(x)φ(x ) S2M (x, x ) = √ e M (x x ) + r M (x, x ) S2M (x, x ) = √ 2π 2π and DS2m (x, x ) =
φ(x)φ(x ) ) φ(x)φ(x ) DS 2M (x, x ) = √ (x − x)e M (x x ). √ 2π 2π
Now, 1
φ(x)φ(x ) = e− 2 (x
2 +x 2 )
1
2
= e− 2 (x−x ) e−x x ,
and therefore, 2
1
e− 2 (x−x ) −x x S2M (x, x ) = √ e M (x x ) + r M (x, x ), e 2π
and 1
2
e− 2 (x−x ) (x − x)e−x x e M (x x ). DS2M (x, x ) = √ 2π
The computation of I S2M (x, x ) is a bit more involved. From (8.3), (9.6) and (9.7), we see M−1 1 x 2 1 2m 2 I S2M (x, x ) = √ ·γ m+ , sgn(x )x 2m e−x /2 (2m)! 2 2 2 π m=0
M−1 1 x2 1 2m 2 ·γ m+ , sgn(x)x 2m e−x /2 . − √ (2m)! 2 2 2 π m=0
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201
I S2M (x, x ) is clearly skew-symmetric in its arguments; looking at the first sum in this expression we thus find, M−1
1 √
2 π
m=0
2m 2 sgn(x )x 2m e−x /2 (2m)!
1 2 = √ e−x /2 sgn(x ) 2 π
x 2 /2 0
x 2 /2
0
e−t √ t
t m−1/2 e−t dt
M−1 2m m 2m dt, t x (2m)! m=0
where on the left-hand side we have replaced the lower incomplete gamma function with its integral definition, and on the right-hand side we have exploited the linearity of the √ integral. The sum on the right-hand side of this equation is equal to c M (x 2t), and thus I S2M (x, x ) = e−x /2 √ sgn(x ) 2 π 2
x 2 /2 0
2
√ e−t e−x /2 √ c M (x 2t) dt − √ sgn(x) 2 π t
x 2 /2 0
√ e−t √ c M (x 2t) dt. t
Turning to the complex/complex entries of K 2M , if z is assumed to be in the open upper half plane then λ πn (z) = iπn (z). From this we see that S2M (z, z ) = iφ(z)φ(z )) DS 2M (z, z ), DS2M (z, z ) = φ(z)φ(z )) DS 2M (z, z ) and I S2M (z, z ) = −φ(z)φ(z )) DS 2M (z, z ). Next, we define 1
ψ(z) = e 4 (z
2 −z 2 )
.
Notice that 1
e− 4 (z
2 +z 2 )
1
e− 4 (z
2 +z 2 )
and thus
φ(z)φ(z ) = ψ(z)ψ(z )e
− 21 (z−z )2
1
2
= ψ(z)ψ(z )e− 2 (z−z ) e−zz , erfc
√
2 Im(z) erfc
√
2 Im(z) e−zz .
Using this and Lemma 9.1, we conclude that S2M (z, z )
1 2 √ √ ie− 2 (z−z ) = ψ(z)ψ(z ) √ 2Im(z) erfc 2Im(z) e−zz e M (zz ), (z − z) erfc 2π DS2M (z, z ) 1 2 √ √ e− 2 (z−z ) = ψ(z)ψ(z ) √ 2Im(z) e−zz e M (zz ), 2Im(z) erfc (z − z) erfc 2π
and S2M (z, z )
1 2 √ √ −e− 2 (z−z ) = ψ(z)ψ(z ) 2Im(z) erfc 2Im(z) e−zz e M (zz ). (z − z) erfc √ 2π
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Lastly we look at the real/complex entries of K 2M . As in all other cases, DS2M (x, z) =
φ(x)φ(z) ) DS 2M (x, z), √ 2π
and it is easily verified that
1
φ(x)φ(z) = ψ(z)e− 2 (x−z)
2
erfc
√
2 Im(z) e−x z .
Thus, 1 2 √ e− 2 (x−z) DS2M (x, z) = ψ(z) √ 2 Im(z) e−x z e M (x z). (z − x) erfc 2π Since S2M (x, z) = i
φ(x)φ(z) ) DS 2M (x, z), √ 2π
φ(z) S2M (z, x) = √ S2M (z, x), 2π
and φ(z) S2M (z, x). I S2M (x, z) = −i √ 2π It follows that
1 2 √ ie− 2 (x−z) 2Im(z) e−x z e M (x z), (z − x) erfc √ 2π 1 2 √ e− 2 (x−z) −x z erfc 2Im(z) e e M (x z) + r M (z, x) , S2M (z, x) = ψ(z) √ 2π
S2M (x, z) = ψ(z)
and
1
e− 2 (x−z) I S 2M (x, z) = −iψ(z) √ 2π
2
√ −x z erfc 2Im(z) e e M (x z) + r M (z, x) .
Clearly, ψ(x) = ψ(x ) = 1, and thus we find that ψ(γ ) 0 N (γ , γ ) ψ(γ ) 0 . K N (γ , γ ) = K 0 ψ(γ ) 0 ψ(γ ) It follows that, if we define K to be the matrix N (x j , z k ) N (x j , x j ) K K K= N (z k , z k ) ; K N (z k , x j ) K
j, j = 1, 2, . . . , ; k, k = 1, 2, . . . , m,
and D to be the diagonal matrix D = diag (ψ(x1 ), ψ(x1 ), . . . , ψ(x ), ψ(x ), ψ(z 1 ), ψ(z 1 ) . . . , ψ(z m ), ψ(z m )) , then R ,m (x, z) = Pf(DKD). But, since ψ(z) = ψ(z)−1 , we have that det D = 1, and R ,m (x, z) = Pf K as claimed.
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Proof of Corollary 9. We first make use of the fact that lim e M (z) = e z
M→∞
pointwise on C. This simplifies all terms in the kernel except the I S2M term. It remains to show that r M (z, x) → 0 as M → ∞, and that 1 |x − x | 1 . (9.9) sgn(x − x ) + lim I S2M (x, x ) = sgn(x − x ) erfc √ M→∞ 2 2 2 The first of these facts is easily seen by noting that γ (M − 1/2, x 2 /2) < (M − 1/2), and by Legendre’s duplication formula, √ |z|2M−1 2−M+3/2 π (2M − 2) |z|2M−1 2 M−3/2 (M − 1/2) = . (2M − 1) (2M − 1)(M − 1) Thus, |r M (z, x)| < e
− 12 Re(z 2 )
erfc
√
2Im(z)
|z|2M−1 , 2 M (M)
and it is easy to see that this goes to 0 as M → ∞, independent of the value of z. To establish (9.9) we start with
x 2 /2
I M (x, x ) := 0
√ e−t √ c M (x 2t) dt. t
(9.10)
Since the terms in c M are all positive, from the Monotone Convergence Theorem, x 2 /2 −t √ e I (x, x ) := lim I M (x, x ) = √ cosh(x 2t) dt M→∞ t 0 √ |x | − x |x | + x π x 2 /2 − erf . erf e = √ √ 2 2 2
(9.11)
The latter equality follows from the fact that √ x −t a √ + √ e π a 2 /4 * a √ e + x − erf − x , erf √ cosh(a t) dt = 2 2 2 t 0 which can be verified via differentiation. Now, 1 , −x 2 /2 2 lim I S2M (x, x ) = √ e sgn(x )I (x, x ) − e−x /2 sgn(x)I (x , x) M→∞ 2 π |x | + x |x | − x 1 − sgn(x ) erf sgn(x ) erf = √ √ 4 2 2 x − |x| x + |x| + sgn(x) erf − sgn(x) erf √ √ 2 2 x − x 1 . = − erf √ 2 2
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It follows that (9.9) can be written as 1 1 1 1 |x − x | x − x = sgn(x − x ) − sgn(x − x ) erf , sgn(x − x ) − erf √ √ 2 2 2 2 2 2 where we have exploited the fact that erf is an odd function. We arrive at the form for I S2M stated in the corollary using the fact that erfc = 1 − erf. 9.4. The proofs of Theorem 10 and Theorem 11. In order to prove Theorems 10 and 11, it is necessary to investigate the asymptotics of the partial sums and the exponential function. Lemma 9.2. Let u = ±1 be a complex number, and let (vm )∞ m=1 be a sequence of complex numbers satisfying v M = u 2 + O M −1/2 as M → ∞. Then, as M → ∞, e2M(1−u ) u 4M e−2(1−u ) · . e M (2Mv M ) ∼ 1 − √ 2π u 2 (1 − u 2 ) M 2
e
−2Mv M
2
In particular, when u is real and 0 < |u| < 1, lim e−2Mv M e M (2Mv M ) = 1.
M→∞
Proof. Set v = v M . We start by writing 2Mv = v(2M − 2) + 2v. Thus, v v −2Mv e M (2M − 2) v+ . e M (2Mv) = exp −(2M − 2) v+ e M −1 M −1 We write w = wM = v +
v . M −1
Clearly w M = u 2 + O(M −1/2 ). Under this hypothesis, and since u 2 = 1 , Eqs. (2.9), (2.15) and (1.7) of [5] imply that e2M(1−u ) u 4M e−2(1−u ) · . √ 2 2 2π u (1 − u ) M 2
e−(2M−2)w e M ((2M − 2)w) ∼ 1 −
2
The second statement of the lemma follows from the fact that if u is real and 0 < |u| < 1, then e2M(1−u ) u 4M = e2M(1−u 2
2 +2 log |u|)
and 1 − u 2 + 2 log |u| is negative when u is in (−1, 1). We are ready to prove Theorem 11.
,
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Proof of Theorem 11. Let u be a point in the open upper half plane with modulus less than 1, and√suppose s and s are√complex numbers. For all but finitely many values of M, z = u 2M + s and z = u 2M + s are in C∗ . Thus, in this case, we need only consider the asymptotics of the complex/complex kernel under these substitutions. We will make use of the fact that if x is a real number, 2 1 2 1 e−x erfc(x) = √ . ,x ∼ √ 2 π π|x| Consequently, erfc
√
√ exp −2MIm(u)2 − 2 2MIm(u)Im(s) − Im(s)2 2Im(u 2M + s) ∼ . √ √ 2Im(u) 4 Mπ (9.12) √
Now, by Theorem 8, √ √ (s − s) − 1 (s−s )2 ! e 2 DS 2M (u 2M + s, u 2M + s ) = √ 2π √ √ √ √ × erfc 2Im(u 2M + s) erfc 2Im(u 2M + s ) √ √ × exp −2Mu 2 − u 2M(s + s ) − ss e M 2Mu 2 + u 2M(s + s ) + ss . Therefore, by Lemma 9.2 and (9.12), √ √ (s − s) − 1 (s−s )2 ! DS 2M (u 2M + s, u 2M + s ) ∼ √ e 2 2π e−4MIm(u) e−2 2
×
√
2MIm(u)(Im(s)+Im(s )) e−Im(s)2 −Im(s )2
√ 2Im(u) Mπ
.
e2M(1−u ) u 4M e−2(1−u ) · × 1− √ 2 2 2π u (1 − u ) M 2
2
/ .
It is easily seen that e−4MIm(u) e−2 2
lim
M→∞
and,
√
2MIm(u)(Im(s)+Im(s ))
√ 2Im(u) Mπ
= 0,
2 −4MIm(u)2 2M(1−u 2 ) 4M e u = e2M(1−|u| +2 log |u|) . e
Since |u| < 1, we have 1 − |u|2 + 2 log |u| < 0, and therefore √ 2 2 lim e−2 2MIm(u)(Im(s)+Im(s )) e−4MIm(u) e2M(1−u ) u 4M = 0. M→∞
We conclude that
√ √ lim ! DS 2M (u 2M + s, u 2M + s ) = 0.
M→∞
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And, since " I S 2M (z, z ) = −! DS 2M (z, z ), √ √ I S 2M (u 2M + s, u 2M + s ) = 0. lim " M→∞
√ √ Turning to S2M (u 2M + s, u 2M + s ), we set √ η M (s) = exp −2i 2MIm(u)Re(s) . From Theorem 8, we have + √ √ −i * √ η M (s)η M (−s ) S2M (u 2M + s, u 2M + s ) = √ 2i 2MIm(u) + (s − s ) 2π √ √ × exp 2i 2MIm(u)Re(s) exp −2i 2MIm(u)Re(s ) √ 1 × exp 4M(Im(u))2 − 2i 2MIm(u)(s − s ) − (s − s )2 2 √ √ √ √ × erfc 2 MIm(u) + 2Im(s) erfc 2 MIm(u) + 2Im(s ) √ √ × exp −2M|u|2 − (su + s u) 2M −ss e M 2M|u|2 + (su + s u) 2M + ss . Using (9.12), we see
√ √ η M (s)η M (−s ) S2M (u 2M + s, u 2M + s ) 1 1 ∼ exp − (s − s )2 − Im(s)2 − Im(s )2 π 2 √ √ ×exp −2M|u|2 − (su + s u) 2M −ss e M 2M|u|2 + (su +s u) 2M + ss .
And thus, by Lemma 9.2,
√ √ η M (s)η M (−s ) S2M (u 2M + s, u 2M + s ) 1 1 ∼ exp − (s − s )2 − Im(s)2 − Im(s )2 π 2 1 |s|2 |s |2 = exp − − + ss . π 2 2
Next, we set D M to be the 2m × 2m diagonal matrix given by D M = diag (η M (s1 ), η M (−s1 ), . . . , η M (sm ), η(−sm )), noting that det D M = 1. It follows that 2M (sk , sk ) m lim R0,m (−, z) = lim Pf D M K M→∞
⎡ = Pf ⎣
k,k =1
M→∞
0
1 π
D−1 M
2 exp − |sk2| −
2 |s |2 − π1 exp − |sk2| − k2 + sk s k m |sk |2 |sk |2 1 exp − − + sk s k = det , π 2 2 k,k =1 where the last equation follows from Sect. 4.6 of [22].
0
|sk |2 2
⎤m + sk s k
⎦ k,k =1
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In√ order to prove √ Theorem 10 we also need to analyze the large M asymptotics of r M (u 2M + s, u 2M + r ), where r and u are real numbers with 0 < |u| < 1 and s is a complex number. Lemma 9.3. Let r and u be real numbers with 0 < |u| < 1 and let s be a complex number in the closed upper half plane. Then, √ √ lim r M (u 2M + s, u 2M + r ) = 0. M→∞
Proof. 2 2 √ √ √ e−s /2 2 M−3/2 e−Mu −us √2M r M (u 2M + s, u 2M + r ) = sgn(u 2M + r ) √ e · (2M − 1) 2π √ √ 1 r2 . ×(u 2M + s)2M−1 · γ M − , Mu 2 + u 2Mr + 2 2
We simplify this using Legendre’s duplication formula for (2M − 1) and by setting P(a, x) = γ (a, x)/ (a), √
√
2M−1 s 1+ √ e 2 u 2M 2 √ 1 r · P M − , Mu 2 + u 2Mr + . 2 2 √
e−s r M (u 2M + s, u 2M + r ) = sgn(u 2M + r ) √ M M+1/2 e−Mu u 2M−1 × M! 2
2 /2
√ −u 2Ms
Next we use Stirling’s approximation for M! in the denominator to show that 2M−1 2 √ √ √ s e−s /2 r M (u 2M + s, u 2M + r ) ∼ sgn(u) √ e−u 2Ms 1 + √ 2u π u 2M √ 1 r2 2 2 . × exp M(1 − u + 2 log |u|) · P M − , Mu + u 2Mr + 2 2 Using the fact that 2M−1 √ s 2 2 1+ √ ∼ es 2M/u e−s /2u , u 2M we find
√ √ r M (u 2M + s, u 2M + r ) 2 √ 1 + u2 sgn(u) s 1 − u2 exp s ∼ 2M √ exp − 2 u2 u 2u π √ 1 r2 2 2 . (9.13) × exp M(1 − u + 2 log |u|) · P M − , Mu + u 2Mr + 2 2
Finally, we notice that if 0 < |u| < 1 then 1 − u 2 + 2 log |u| < 0. It follows that √ lim exp s 2M(1/u − u) exp M(1 − u 2 + 2 log |u|) = 0, M→∞
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and the lemma follows since √ 1 r2 < 1. 0 < P M − , Mu 2 + u 2Mr + 2 2 Proof of Theorem 10. First we consider the case where s and s are both in the open upper half plane. From Theorem 8, √ √ − 12 (s−s )2 √ √ ie S2M (u 2M +s, u 2M +s )= √ 2Im(s) erfc 2Im(s ) (s − s) erfc 2π √ √ × exp −2Mu 2 − u 2M(s + s ) − ss e M 2Mu 2 + u 2M(s + s ) + ss , 1 2 √ √ √ √ e− 2 (s−s ) ! DS 2M (u 2M +s, u 2M +s )= √ 2Im(s) erfc 2Im(s ) (s −s) erfc 2π √ √ × exp −2Mu 2 − u 2M(s + s ) − ss e M 2Mu 2 + u 2M(s + s ) + ss , and 1 2 √ √ √ √ −e− 2 (s−s ) " 2Im(s) erfc 2Im(s ) I S 2M (u 2M +s, u 2M +s )= √ (s −s) erfc 2π √ √ × exp −2Mu 2 − u 2M(s + s ) − ss e M 2Mu 2 + u 2M(s + s ) + ss . By Lemma 9.2, these converge to the appropriate entries of the complex/complex kernel K (s, s ) as M → ∞. Next we turn to the case where u and r are real, and s is in the open upper half plane. In this case, Theorem 8 yields, 1 2 √ √ √ ie− 2 (r −s) S2M (u 2M + r, u 2M + s) = √ 2Im(s) (s − r ) erfc 2π √ √ × exp −2Mu 2 − u 2M(r + s) − r s e M 2Mu 2 + u 2M(r + s) + r s , 1 2 √ √ √ e− 2 (r −s) S2M (u 2M + s, u 2M + r ) = √ erfc 2Im(s) 2π √ √ × exp −2Mu 2 − u 2M(r + s) − r s e M 2Mu 2 + u 2M(r + s) + r s , √ √ + r M (u 2M + r, u 2M + s), and thus, by Lemmas 9.2 and 9.3, √ − 12 (r −s)2 ie 2Im(s) , lim S2M (u 2M + r, u 2M + s) = √ (s − r ) erfc M→∞ 2π √
√
and 2 √ √ e− 2 (r −s) lim S2M (u 2M + s, u 2M + r ) = √ M→∞ 2π 1
erfc
√ 2Im(s) .
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√ √ √ √ The limiting values for ! DS 2M (u 2M +r, u 2M + s) and " I S 2M (u 2M +r, u 2M + s) follow from this as well, since ! DS 2M (z, z ) = −i S2M (z, z )
and
" I S 2M (z, z ) = i S2M (z , z).
Finally, we turn to the case where u, r and r are all real. Here, Theorem 8 implies that √ √ 1 1 2 S2M (u 2M + r, u 2M + r ) = √ e− 2 (r −r ) 2π √ √ 2 × exp −2Mu − u 2M(r + r ) − rr e M 2Mu 2 + u 2M(r + r ) + rr √ √ +r M (u 2M + r, u 2M + r ), and √ √ (r − r ) − 1 (r −r )2 ! e 2 DS 2M (u 2M + r, u 2M + r ) = √ 2π √ √ × exp −2Mu 2 − u 2M(r + r ) − rr e M 2Mu 2 + u 2M(r + r ) + rr . From Lemmas 9.2 and 9.3, we see that √ √ 1 1 2 lim S2M (u 2M + r, u 2M + r ) = √ e− 2 (r −r ) , M→∞ 2π and √ √ (r − r ) − 1 (r −r )2 lim ! . e 2 DS 2M (u 2M + r, u 2M + r ) = √ M→∞ 2π All that remains to show is √ √ 1 |r − r | " lim I S 2M (u 2M + r, u 2M + r ) = sgn(r − r ) erfc . √ M→∞ 2 2 First we write
x 2 /2 −t 2 √ e e−x /2 " I S 2M (x, x ) = √ sgn(x ) I (x, x ) − √ C M (x 2t) dt 2 π t 0 2 2 x /2 e−t √ e−x /2 − √ sgn(x) I (x , x) − √ C M (x 2t) dt , 2 π t 0
where I (x, x ) is given as in (9.11) and C M = cosh −c M . That is, 2
e−x /2 e−x /2 " I S 2M (x, x ) = √ sgn(x )I (x, x ) − √ sgn(x)I (x , x) 2 π 2 π 2 2 x /2 e−t √ e−x /2 + √ sgn(x) √ C M (x 2t) dt 2 π t 0 x 2 /2 −t 2 /2 −x √ e e − √ sgn(x ) √ C M (x 2t) dt. 2 π t 0 2
(9.14)
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√ √ Making the substitutions x = u 2M + r and x = u 2M + r , and assuming that M is sufficiently large, (9.11) yields ⎧ 2 x /2 ⎪ r − r ⎪ ⎪e if u > 0; erfc √ ⎪ ⎪ ⎨ 4 2 sgn(x ) √ I (x, x ) = ⎪ 2 2 π ⎪ ⎪ e x /2 r −r ⎪ ⎪ erfc √ if u < 0. ⎩− 4 2 Consequently,
2
e−x /2 e−x /2 lim √ sgn(x )I (x, x ) − √ sgn(x)I (x , x) M→∞ 2 π 2 π r − r r − r 1 1 = erfc √ − erfc √ 4 4 2 2 r −r 1 . = − erf √ 2 2 2
Thus, if we can show that the second line of (9.14) goes to 0 as M → ∞, we will have
r − r 1 1 1 lim sgn(x − x ) + I S2M (x, x ) = sgn(r − r ) − erf √ M→∞ 2 2 2 2 1 |r − r | = sgn(r − r ) erfc √ 2 2 as desired. We thus consider 2
e−x /2 √ sgn(x) 2 π
x 2 /2 0
√ e−t √ C M (x 2t) dt. t
For any v > 0, C M (v) ≤
∞ vm = P(2M, v)ev , m!
m=2M
where P(2M, v) = γ (2M, v)/ (2M). Among other things, this implies that C M (v) ≤ ev and 2 (x−x )/√2 2 √ √ e−x /2 x /2 e−t 1 2 e−t P 2M, x (x + 2t) dt. √ √ C M (x 2t) dt ≤ √ 2 2 π 0 π −x /2 t √ √ Under the substitutions x = u 2M + r and x = u 2M + r , the right-hand side of this is less than or equal to 1 √ π
√ (r −r )/ 2 −∞
√ u 1 2 √ + 2 r + 2tr ) dt. (2r + 2t + e−t P 2M, 2M u 2 + √ 2M 2M (9.15)
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In [28], Temme gives the uniform asymptotic expansion for P(a, x) when a > 0 and x ∈ R: P(a, x) ∼
0 1 erfc sgn(1 − λ) a(λ − 1 − log λ) ; 2
λ=
x . a
(9.16)
In our situation, √ u 1 2 √ r + 2tr ), (2r + 2t + λ = u2 + √ 2M 2M which implies that λ − 1 − log λ = u 2 − 1 − 2 log |u| + O(M −1/2 ) and sgn(1 − λ) = sgn 1 − u 2 + O(M −1/2 ) = 1 as M → ∞. Since, for |u| < 1, u 2 − 1 − 2 log |u| > 0, we conclude that 0 sgn(1 − λ) 2M(λ − 1 − log λ) → ∞ as M → ∞. Temme’s asymptotic for P(a, x) gives that, √ u 1 2 √ + r + 2tr ) = 0. (2r + 2t + lim P 2M, 2M u 2 + √ M→∞ 2M 2M Thus, by applying the Dominated Convergence Theorem to (9.15) (which we may do since 0 < P(a, x) < 1), we must have that 2
e−x /2 √ sgn(x) M→∞ 2 π
x 2 /2
lim
and the lemma is proved.
0
√ e−t √ C M (x 2t) dt = 0, t
9.5. The proofs of Theorem 12 and Theorem 13. In order to prove Theorem 12 we need analogs of Lemmas 9.2 and 9.3 for the case where u 2 = 1. Lemma 9.4. Suppose a is a complex number and vM = 1 + √
a 2M
+ O(M −1 ).
Then, lim e
M→∞
−2Mv M
a 1 e M (2Mv M ) = erfc √ . 2 2
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Proof. As in the proof of Lemma 9.2 we set v = v M and write v w = wM = v + , M −1 so that e−2Mv e M (2Mv) = e−(2M−2)w e M ((2M − 2)w). We are now in position to use a result of Bleher and Mallison [3, Theorem B.1], which shows that 0 1 e−(2M−2)w e M ((2M − 2)w) ∼ erfc ξ(w) (2M − 2) , (9.17) 2 where ξ(w) =
(w − 1) (w − 1)2 (w − 1)3 − + + ··· . √ √ √ 2 6 2 36 2
In our case, a ξ(w) = √ + O(M −1 ), 2 M and the lemma now follows from (9.17) and (9.18).
(9.18)
Lemma 9.5. Let u = ±1, r ∈ R and let s be in the closed upper half plane. Then, √ √ 1 2 lim r M (u 2M + s, u 2M + r ) = √ e−s erfc(−ur ). M→∞ 4 π Proof. From (9.13) we have that
√ 1 −s 2 1 r2 . r M (u 2M + s, u 2M + r ) = √ e P M − , M + u 2Mr + 2 2 2 π √
√
(9.19)
Using Temme’s asymptotic for P(a, x) given in (9.16), √
λ=
r2 2M M 1 1 − 2M
1 + ur √ 2 +
√ ur 2 + O(M −1 ). =1+ √ M
It follows that, as M → ∞, sgn(1 − λ) → −u sgn(r ), and λ − 1 − log λ = Thus,
r2 + O(M −3/2 ). M
√ 1 r2 1 ∼ erfc (−ur ), P M − , M + u 2Mr + 2 2 2
and the lemma follows from (9.19).
Proof of Theorem 12. The proof of Theorem 12 is the same, mutatis mutandis, as that of Theorem 10 replacing the asymptotics in Lemmas 9.2 and 9.3 with those in Lemmas 9.4 and 9.5. Proof of Theorem 13. The proof of Theorem 13 is the same, mutatis mutandis, as that of Theorem 11 replacing the asymptotics in Lemmas 9.2 with those in Lemmas 9.4.
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Appendices A. Correlation Functions for β = 1 and β = 4 Hermitian Ensembles In this appendix we will use the Pfaffian Cauchy-Binet Formula (see B) in order to derive the correlation functions of the β = 1 and β = 4 Hermitian ensembles. We will keep the exposition brief, but will introduce all notation necessary for this appendix to be read independently from the main body of the paper. We reuse much of the notation from the main body of the paper so that we may also reuse the same proofs. For convenience, N will be a fixed even integer; similar results are true for odd integers. Given a Borel measure ν on R we define the associated partition function to be 1 Z ν := |(γ )| dν N (γ ), N ! RN where (γ ) is the Vandermonde determinant in the variables γ1 , γ2 , . . . , γ N and ν N is the product measure of ν on R N . When β = 1 we define the function E : R2 → {− 21 , 0, 21 } and the operator ν on L 2 (ν) by 1 and ν g(γ ) := g(y)E(y, γ ) dν(y). E(γ , γ ) := sgn(γ − γ ) 2 R When β = 4 we define E(γ , γ ) := 0 and ν g(y) := g (y). We use ν to define the skew-symmetric bilinear form ·|·ν on L 2 (ν) given by g|hν := (g(γ )ν h(γ ) − ν g(γ )h(γ )) dν(γ ). Theorem A.1. Let b := deg qn = n. Then,
√
R
β, and let q be a family of N b monic polynomials such that Z ν = b N Pf Uqν ,
where Uqν = [qn |qn ν ]; n, n = 0, 1, . . . , N b − 1. This theorem follows from de Bruijn’s identities [7]. We set λ to be Lebesgue measure on R. If there is some Borel measurable function w : R → [0, ∞) so that ν = wλ (that is, dν/dλ = w) then we define Z w := Z ν . Clearly, N 1 w Z =
N (γ ) dλ N (γ ) where
N (γ ) := w(γn ) |(γ )|β . N ! RN n=1
We may specify an ensemble of Hermitian matrices by demanding that its joint probability density function is given by N . The n th correlation function of this ensemble is then defined to be Rn : Rn → [0, ∞), where 1 1 Rn (y) := w ·
N (y ∨ γ ) dλ N −n (γ ), (A.1) Z (N − n)! R N −n where y ∨ γ ∈ R N is the vector formed by concatenating the vectors y ∈ Rn and γ ∈ R N −n . By definition, R0 = 1. Here we take (A.1) as the definition of the n th correlation function; one can use the point process formalism to show that this definition is consistent with the definition derived in that manner. See [4] for details.
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We set µn,n to be the n, n entry of (Uqwλ )−T , and we define qn := wqn and ⎧ ⎪ ⎨ w(y)qn (y)E(y, γ ) dλ(y) if β = 1; R λ qn (γ ) := ⎪ ⎩ w(γ )qn (γ ) if β = 4. (In the case of β = 4 this contradicts the earlier definition of ν , but it has the benefit of allowing us to treat the β = 1 and β = 4 cases simultaneously.) Using this notation we define the functions S N , I S N and DS N : R2 → R by S N (γ , γ ) :=
N b−1 2 µn,n qn (γ ) λ qn (γ ), b n,n =0
I S N (γ , γ ) :=
N b−1 2 µn,n λ qn (γ ) λ qn (γ ) b n,n =0
and DS N (γ , γ ) :=
N b−1 2 µn,n qn (γ ) qn (γ ). b n,n =0
The matrix kernel of our ensemble is then defined to be DS N (γ , γ ) S N (γ , γ ) K N (γ , γ ) := −S N (γ , γ ) I S N (γ , γ ) + E(γ , γ ). Theorem A.2.
Rn (y) = Pf K N (y j , yj ) ;
j, j = 1, 2, . . . , n.
Our proof of this theorem begins by setting η to be the measure on R given by dη(γ ) =
T
ct dδ(γ − yt ),
t=1
where y1 , y2 , . . . , yT are real numbers and c1 , c2 , . . . , cT are indeterminants and δ is the probability measure on R with point mass at 0. We will assume that T ≥ N . As with Theorem 4 in the main body of this paper, we will prove Theorem A.2 by expanding Z w(λ+η) /Z w in two different ways and then equating the coefficients of certain products of c1 , c2 , . . . , cT . Proposition A.3. √ Z w(λ+η) K N (yt , yt ) ; = Pf J + c c t, t = 1, 2, . . . , T, t t Zw where J is defined to be the 2T × 2T matrix consisting of 2 × 2 blocks given by 0 1 ; t, t = 1, 2, . . . , T. J := δt,t −1 0 Proposition A.3 is proved in Sect. 9.1.
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For each N ≥ 0 we define InT to be the set of increasing functions from {1, 2, . . . , n} into {1, 2, . . . , T }. Given a vector y ∈ RT and an element t ∈ InT , we define the vector yt ∈ Rn by yt = {yt(1) , yt(2) , . . . , yt(n) }. Proposition A.4. Z w(λ+η) Zw
=1+
⎧ N ⎨ n n=1 t∈InT
⎩
⎫ ⎬ ct( j)
j=1
⎭
Rn (yt).
(A.2)
The proof of Proposition A.4 is given in Sect. 9.2. Finally, we set K to be the 2T × 2T block matrix given by √ K := ct ct K N (yt , yt ) ; t, t = 1, 2, . . . , T. From the formula for the Pfaffian of the sum of two antisymmetric matrices (see Proposition 7) and Proposition A.3 we have that T Z w(λ+η) = Pf[J + K] = 1 + Pf Kt, Zw T
(A.3)
n=1 t∈In
where for each t ∈ InT , Kt is the 2n × 2n antisymmetric matrix given by √ Kt = [ ct( j) ct( j ) K N (yt( j) , yt( j ) )]; j, j = 1, 2, . . . , S. Finally, Pf Kt =
⎧ n ⎨ ⎩
j=1
⎫ ⎬ ct( j)
⎭
Pf[K N (yt( j) , yt( j ) )];
j, j = 1, 2, . . . , n,
and Theorem A.2 follows from Proposition A.4 by comparing coefficients of c1 c2 · · · cn in (A.2) and (A.3). B. The Pfaffian Cauchy-Binet Formula Theorem B.1. (Rains) Suppose B and C are respectively 2J × 2J and 2K × 2K antisymmetric matrices with non-zero Pfaffians. Then, given any 2J × 2K matrix A, Pf(C−T − AT BA) Pf(B−T − ACAT ) = . Pf(C−T ) Pf(B−T ) Proof. Let I2K and I2J be respectively the 2K × 2K and 2J × 2J identity matrices, and let O be the 2J × 2K matrix whose entries are all 0. Then, an easy calculation shows that −T −T I2J O B B −A I2J −BA O = , AT B I2K AT C−T OT I2K OT C−T − AT BA and similarly −T I2J −AC B−T −A I2J O B − ACAT O = . OT I2K AT C−T CAT I2K C−T OT
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Now, if D and E are 2N × 2N matrices and D is antisymmetric, then it is well known that Pf(EDET ) = Pf D · det E, from which we conclude that −T −T B O B − ACAT O . = Pf Pf OT C−T − AT BA C−T OT The theorem follows since the Pfaffian of the direct sum of two even rank antisymmetric matrices is the product of the Pfaffians of the two matrices. That is Pf(B−T ) Pf(C−T − AT BA) = Pf(B−T − ACAT ) Pf(C−T ). C. Limiting Correlation Functions for the Complex Ginibre Ensemble The complex Ginibre ensemble consists of N × N complex matrices with i.i.d. normal entries. In this section we derive the scaling limits of the correlation functions of the Ginibre complex ensemble in the bulk and at the edge. As the quantities of interest are similar to those in the main body of the paper we will reuse much of our previous notation for the analogous quantities. In his original paper on the subject, [15], Ginibre showed that the joint density of eigenvalues is given by N 1
N (γ ) = w(γn ) |(γ )|2 , Z n=1
where w(γ ) = e−|γ | , (γ ) is the Vandermonde determinant whose columns are given in terms of γ1 , γ2 , . . . , γ N , and 1
N (γ ) dλ2N (γ ). Z= N ! CN 2
We may take the n th correlation function of this ensemble to be the function Rn : Cn → [0, ∞) given by 1 1 Rn (z) := ·
N (z ∨ γ ) dλ2(N −n) (γ ). Z (N − n)! C N −n The correlation functions can also be defined as densities with respect to Lebesgue measure which satisfy an identity analogous to (5.1). Ginibre gave a closed form for Rn in terms of a scalar kernel. Specifically, he showed that Rn (z) = det [K N (z k , z k )]nk,k =1 , where K N (z, z ) =
|z|2 |z |2 1 exp − − e M (zz ). 2π 2 2
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Clearly then, lim Rn (z) = det
N →∞
n |z k |2 |z k |2 1 exp − − + zk zk . 2π 2 2 k,k =1
This is the limiting correlation function of the complex Ginibre ensemble at the origin. Notice that an almost identical expression appears in Theorem 11. Like the real Ginibre ensemble, the complex Ginibre ensemble satisfies the circular law. We therefore expect that √ limiting correlation functions will emerge after scaling eigenvalues by a factor of N . Theorem C.1. Let u be in the closed unit disk, and suppose s1 , s2 , . . . , sn are complex numbers. Set √ z k = u N + sk k = 1, 2, . . . , n. Then: 1. Limiting correlation functions in the bulk. If |u| < 1, n |sk |2 |sk |2 1 exp − − + sk s k lim Rn (z) = det . N →∞ 2π 2 2 k,k =1
2. Limiting correlation functions at the edge. If |u| = 1, lim Rn (z) = det
N →∞
n sk u + s k u |sk |2 |sk |2 1 exp − − + sk s k erfc . √ 2π 2 2 2 k,k =1
Remark. These results appear in [11]. We present them here for completeness, and since these scaling limits follow easily from some of the asymptotics employed in the derivation of the scaling limits for the real Ginibre ensemble. Proof. Let N = 2M, ψ M (s) = exp
(su − su) √ M √ 2
and define D to be the n × n matrix given by D = diag (ψ M (s1 ), ψ M (s2 ), . . . , ψ(sn )). It is easily seen that |ψ M (s)| = 1 and det D = 1. We also define K to be the n × n matrix given by K=
n 1 sk u + s k u |sk |2 |sk |2 exp − − + sk s k erfc . √ 2π 2 2 2 k,k =1
Then, lim Rn (z) = lim det DKD−1 .
M→∞
M→∞
(C.1)
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The k, k entry of K can be computed to be 1 |sk |2 |sk |2 (sk u + s k u) √ (s k u + sk u) √ exp −2M|u|2 − − M− M− √ √ 2π 2 2 2 2 √ ×e M 2M|u|2 + (sk u + s k u) 2M + sk s k √ |sk |2 |sk |2 1 exp −2M|u|2 − (sk u + s k u) 2M − sk s k exp − − + sk s k = 2π 2 2 (sk u − s k u) √ −(sk u − s k u) √ × exp M exp M √ √ 2 2 √ ×e M 2M|u|2 + (sk u + s k u) 2M + sk s k . It follows that the k, k entry of DKD−1 is given by √ 1 |sk |2 |sk |2 exp − − + sk s k exp −2M|u|2 − (sk u + s k u) 2M − sk s k 2π 2 2 √ ×e M 2M|u|2 + (sk u + s k u) 2M + sk s k . Statement 1 of the theorem now follows from (C.1) and Lemma 9.2. Statement 2 follows from (C.1) and Lemma 9.4. D. Plots of Correlation Functions for the Real Ginibre Ensemble This appendix contains various visualizations of the limiting correlation functions of the real Ginibre ensemble. As usual, H (respectively H ) is the open (closed) upper half plane. Given a point u ∈ H with |u| ≤ 1, r1 , r2 , . . . , r ∈ R and s1 , s2 , . . . , sm ∈ H m , we set √ √ j = 1, 2, . . . , ; and z k = u 2M + sk k = 1, 2, . . . , m. x j = u 2M + r j We will use the notation u R ,m (r1 , . . . , r , s1 , . . . , sm ) = lim R ,m (x, z), M→∞
where R ,m (x, z) is the , m correlation function of the real Ginibre ensemble of 2M × 2M matrices. D.1. The real bulk. Let u be a√point in the real bulk. The local density of real eigenu (r , r ) is values is constant (equal to 1/ 2π ). The limiting correlation function, R2,0 1 2 u u invariant under real shifts, and hence R2,0 (r1 , r2 ) = R2,0 (r1 − r2 , 0). We may therefore plot this correlation function as a function of a r1 − r2 . As |r1 − r2 | → ∞, this quantity approaches (2π )−1 —the square of the density of real eigenvalues. See Fig. 1. u (s). Due to The local density of complex eigenvalues in the real bulk is given by R0,1 the invariance of the correlations functions with respect to real shifts, this is a function
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Ru2,0 r1 ,r2 1 2
.16
2
2
r1 r2
u (r , r ) in the real bulk as a function of r − r Fig. 1. R2,0 1 2 1 2
Ru0,1 s 1 p
.32
Im s
2
Fig. 2. The density of complex eigenvalues in the real bulk as a function of Im(s)
of Im(s) only. As Im(s) → ∞ this density tends toward the density of eigenvalues in the complex bulk. Specifically, lim
Im(s)→∞
u R0,1 (s) =
1 . π
See Fig. 2. u (r, s) is invariant under real shifts, and thus can be The correlation function R1,1 u (r, s) approaches plotted as a function of r − Re(s) and Im(s). As Im(s) → ∞, R1,1 2−1/2 π −3/2 —the product of the density of real eigenvalues in the real bulk and the density of eigenvalues in the complex bulk. See Fig. 3. D.2. The complex bulk. When u is in the complex bulk; that is u ∈ H and |u| < 1, the density of eigenvalues is constant (equal to 1/π ). The only non-trivial correlation
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1
.13
2 p3 2
Ru1,1 r,s
4
0
2
2
Im s
0
r
Re s 0
2
u (r, s) in the real bulk as a function of r − Re(s) and Im(s) Fig. 3. R1,1
Ru0,2 s,s' 1 p2
2
.10
2
s
s
u (s, s ) in the complex bulk as a function of s − s Fig. 4. A plot of R0,2
u (s, s ). This function is invariant under both real and function we can visualize is R0,2 u (s, s ) as a function of s −s . As |s −s | → ∞, imaginary shifts. That is, we may plot R0,2 u (s, s ) approaches 1/π 2 —the square of the density of eigenvalues in the complex R0,2 bulk. See Fig. 4
D.3. The real edge. For concreteness we will concentrate on the real edge corresponding to u = 1. At the real edge the local density of real eigenvalues is no longer constant. 1 (r ). As r → −∞ we expect the local density of eigenHere the density is given by R1,0 √ values to approach the density of real eigenvalues in the real bulk, 1/ 2π . Indeed, this is the case. As r → ∞ the local density of eigenvalues decreases to 0. See Fig. 5.
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R11,0 r 1
.40
2p
2
r
2
Fig. 5. The density of real eigenvalues at the real edge as a function of r
1 p
R10,1 s
.32
4
0
2
2
Im s
0
Re s 2
0
Fig. 6. The density of complex eigenvalues at the real edge as a function of Re(s) and Im(s)
1 (s). This The local density of complex eigenvalues at the real edge is given by R0,1 can be plotted as a function of Re(s) and Im(s). If s ‘moves’ in the direction of the complex bulk (loosely speaking, Re(s) → −∞ while simultaneously, Im(s) → ∞) 1 (s) approaches 1/π —the density of eigenvalues in the complex bulk. Since then R0,1 1 (s) approaches 0 with Im(s). See Fig. 6. the real axis repels complex roots, R0,1 1 We may also plot R2 (r, r ). Here we expect that if r → −∞ and r → ∞ and |r − r | → ∞, then R21 (r, r ) should approach the square of the density of real eigenvalues in the real bulk, (2π )−1 . If r − r approaches 0, then the repulsion of eigenvalues implies that R21 (r, r ) → 0. Similarly, if either r or r is large and positive, then we are
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1 2p
.16
R12,0 r,r'
3
0
0
3
r
0
r 3
3
1 (r, r ) at the real edge as a function of r and r Fig. 7. A plot of R2,0
Ri0,1 s 2 p
2
.64
2
Im s
i (s)) as a function of the radius (given here Fig. 8. The radial density of eigenvalues (represented here as R0,1 by Im(s)) at the complex edge
looking at the local correlation of involving a real eigenvalue away from the bulk, and therefore R21 (r, r ) is small. See Fig. 7 D.4. The complex edge. The limiting kernel at the complex edge, when u is on the unit circle, is invariant under shifts in the direction of the tangent line of the unit circle at u. For simplicity we take u = i, so that, for instance, the local density of complex i (s) which is a eigenvalues is invariant under real shifts. This density is given by R0,1 function of Im(s) only. As Im(s) → ∞ we are moving away from the bulk and thus i (s) → 0. If Im(s) → −∞ then R i (s) approaches 2/π , the density of complex R0,1 0,1 eigenvalues in the bulk. See Fig. 8.
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Acknowledgements. We thank Percy Deift for helpful early discussions and for introducing the authors to each other. The second author would also like to thank Brian Rider for many helpful discussions regarding the real Ginibre ensemble. We are grateful to Eric Rains for allowing us to include his proof of the Pfaffian Cauchy-Binet formula (Appendix B). We also thank the referee for helpful suggestions and for providing references previously unknown to us. Finally we would like to thank Peter Forrester for keeping us updated about his work. The first named author (A.B.) was partially supported by the NSF grant DMS-0707163.
References 1. Akemann, G., Kanzieper, E.: Integrable structure of Ginibre’s ensemble of real random matrices and a pfaffian integration theorem. J. Stat. Phys. 129, 1159–1231 (2007) 2. Bai, Z.D.: Circular law. Ann. Probab. 25(1), 494–529 (1997) 3. Bleher, P., Mallison, R. Jr.: Zeros of sections of exponential sums. Int. Math. Res. Not., Art. ID 38937, 49, pp. (2006) 4. Borodin, A., Olshanski, G.: Representation theory and random point processes. In: European Congress of Mathematics, Zürich: Eur. Math. Soc., 2005, pp. 73–94 5. Carpenter, A.J., Varga, R.S., Waldvogel, J.: Asymptotics for the zeros of the partial sums of e z . I. In Proceedings of the U.S.-Western Europe Regional Conference on Padé Approximants and Related Topics (Boulder, CO, 1988), Vol. 21, 1991, pp. 99–120 6. Chern, S.-J., Vaaler, J.D.: The distribution of values of Mahler’s measure. J. Reine Angew. Math. 540, 1–47 (2001) 7. de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. (N.S.) 19, 133–151 (1956) 8. Edelman, A.: The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60(2), 203–232 (1997) 9. Edelman, A., Kostlan, E., Shub, M.: How many eigenvalues of a random matrix are real? J. Amer. Math. Soc. 7(1), 247–267 (1994) 10. Everest, G., Ward, T.: Heights of Polynomials and Entropy in Algebraic Dynamics. Universitext. London: Springer-Verlag London Ltd., 1999 11. Forrester, P.J., Honner, G.: Exact statistical properties of the zeros of complex random polynomials. J. Phys. A 32(1), 2961–2981 (1999) 12. Forrester, P., Nagao, T.: Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A 41, 375003 (2008) 13. Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007) 14. Fyodorov, Y.V., Sommers, H.-J.: Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A 36(12), 3303–3347 (2003) 15. Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965) 16. Girko, V.L.: The circular law. Teor. Veroyatnost. i Primenen. 29(4), 669–679 (1984) 17. Guhr, T., Mueller-Groeling, A., Weidenmueller, H.A.: Random matrix theories in quantum physics: Common concepts. Phys. Rep. 299, 189 (1998) 18. Kanzieper, E., Akemann, G.: Statistics of real eigenvalues in Ginibre’s ensemble of random real matrices. Phys. Rev. Lett. 95, 230201 (2005) 19. Lehmann, N., Sommers, H.-J.: Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67, 941–944 (1991) 20. Mehta, M.L.: Random Matrices. Third ed., Volume 142 of Pure and Applied Mathematics (Amsterdam). Amsterdam: Elsevier/Academic Press, 2004 21. Rains, E.M.: Correlation functions for symmetrized increasing subsequences. http://arxiv.org/abs/math/ 0006097v1[math.Co], 2000 22. Sinclair, C.D.: Averages over Ginibre’s ensemble of random real matrices. Int. Math. Res. Not. 2007, 1–15 (2007) 23. Sinclair, C.D.: The range of multiplicative functions on C[x], R[x] and Z[x]. Proc. London Math. Soc. 96(3), 697–737 (2008) 24. Sommers, H.-J.: Symplectic structure of the real Ginibre ensemble. J. Phys. A 40, F671–676 (2007) 25. Sommers, H.-J., Wieczorek, W.: General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A 41, 40 (2008) 26. Stembridge, J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83(1), 96–131 (1990) 27. Tao, T., Vu, V.: Random matrices: The circular Law. Commun. Cont. Math. 10, 261–307 (2007)
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28. Temme, N.M.: Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function. Math. Comp. 29(132), 1109–1114 (1975) 29. Tracy, C.A., Widom, H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92(5–6), 809–835 (1998) Communicated by S. Zelditch
Commun. Math. Phys. 291, 225–255 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0872-7
Communications in
Mathematical Physics
Flat Galaxies with Dark Matter Halos—Existence and Stability Roman Fiˇrt, Gerhard Rein, Martin Seehafer Mathematisches Institut der Universität Bayreuth, D 95440 Bayreuth, Germany. E-mail: [email protected] Received: 16 June 2008 / Accepted: 4 May 2009 Published online: 15 July 2009 – © Springer-Verlag 2009
Abstract: We consider a model for a flat, disk-like galaxy surrounded by a halo of dark matter, namely a Vlasov-Poisson type system with two particle species, the stars which are restricted to the galactic plane and the dark matter particles. These constituents interact only through the gravitational potential which stars and dark matter create collectively. Using a variational approach we prove the existence of steady state solutions and their nonlinear stability under suitably restricted perturbations.
1. Introduction Around 1970 astrophysicists noticed that in typical spiral galaxies the rotation velocities of the stars, when computed in the gravitational potential of the visible matter, do not fit with their observed rotation velocities. It was then conjectured that such galaxies are surrounded by a halo of so far not directly observable dark matter in such a way that the rotation velocities of the stars are consistent with the resulting gravitational potential [11]. For an introduction to dark matter we refer to [3, Chap. 10] and the references there. The distribution of the stars in a galaxy is usually modeled by a density function on phase space, and it is assumed that collisions are sufficiently rare to be neglected and that the stars interact only by the gravitational potential which they create collectively. In a non-relativistic setting this results in a system of partial differential equations which in the mathematics literature is known as the Vlasov-Poisson system, cf. [27]. While the true physical nature (and existence) of dark matter are still conjectural, we are aware of at least one astrophysics investigation where it is also modeled as Vlasov-type matter, cf. [28]. Given the fact that the only role which galactic dark matter has to play is to provide the mass and hence the gravitational potential needed to resolve the discrepancy concerning the rotation velocities of the stars, such a description of dark matter seems natural.
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In the present paper we investigate a model for a flat, disk-like galaxy with a halo of dark matter where both the distribution of the stars in the galactic plane and the distribution of the dark matter particles in the halo obey a Vlasov equation, and the interaction among stars, dark matter, and between these two constituents is through the gravitational potential which all the particles (stars and dark matter) create collectively. Following the practice in astrophysics we assume that the stars are restricted to a plane which we take to be the x1 , x2 plane. Their distribution on phase space is given by f˜ = f˜(t, x, ˜ v) ˜ ≥ 0, where t ≥ 0 denotes time and x, ˜ v˜ ∈ R2 denote position and velocity in the galactic plane. The distribution of the dark matter particles is given by f = f (t, x,v) ≥ 0, where x,v ∈ R3 denote position and velocity in three dimensional space. The evolution of the galaxy and its halo is then governed by the following Vlasov-Poisson type system of equations: ∂t f + v · ∇x f − ∇x Ue · ∇v f = 0, ∂t f˜ + v˜ · ∇x˜ f˜ − ∇x˜ Ue (·,0) · ∇v˜ f˜ = 0, ρ(t, ˜ y˜ ) Ue (t, x) = U (t, x) + U˜ (t, x) = − R3 ρ(t,y) |x−y| dy − R2 |x−( y˜ ,0)| d y˜ , ρ(t, x) = R3 f (t, x,v)dv, ρ(t, ˜ x) ˜ = R2 f˜(t, x, ˜ v)d ˜ v. ˜
(1.1) (1.2) (1.3) (1.4)
Here ρ and ρ˜ are the spatial mass densities of dark matter respectively stars, U and U˜ are the induced Newtonian potentials, and Ue denotes the potential of the system as a whole, i.e., the effective potential which determines the particle orbits. In order that the stars remain in their plane it is sufficient to require that f (t, x, ˜ x3 , v,v ˜ 3 ) = f (t, x,−x ˜ ˜ 3 , v,−v 3 ), a condition which at least formally is preserved by solutions of the system and which implies that ∇U (t, x,0) ˜ is parallel to the plane; for ∇ U˜ (t, x) ˜ this is true automatically. Throughout this paper we use the convention that variables with (without) tilde denote flat (non-flat) quantities. To our knowledge a fully non-linear model where the gravitational interaction within both types of matter and between the two types is taken into account has so far not been investigated. Our aim is to prove the existence and non-linear stability of steady state configurations to this system. We obtain such stable steady states as minimizers of the total energy 1 1 2 |v| f d x dv + |v| ˜ 2 f˜ d x˜ d v˜ 2 2 1 1 ˜ ρ( ˜ x)d ˜ x, ˜ + Ue (x)ρ(x)d x + Ue (x,0) 2 2 satisfying suitable constraints. This so-called energy-Casimir approach was developed for the usual, three dimensional Vlasov-Poisson system, i.e., f˜ = 0 in the above, in [12– 16,25], see also [7,19,27,29]. The approach has also been used to prove the existence of stable steady states for flat galaxies without a halo, i.e., with f = 0 in the above, cf. [9,10,24]. The fact that in the present situation the energy is a functional acting on two functions together with the potential interaction terms between the flat and the non-flat component requires substantial new ingredients in the basic scheme. One pitfall to avoid is that for a minimizer of the above energy functional one of the two components might vanish. Besides the above stability results it is known that global classical solutions to the initial value problem for the usual three dimensional Vlasov-Poisson system exist,
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cf. [21,22], while local classical and global weak solutions exist in the flat case without halo, cf. [6]. For the situation at hand nothing is known about the initial value problem, but we conjecture that the analogue of [6] for weak solutions remains true. Our stability result is conditional in the sense that it holds for solutions as long as they exist and preserve the required conserved quantities. For more information on the Vlasov-Poisson system in general we refer to the review article [27]. The paper proceeds as follows. In the next section we formulate our variational problem and our main result on the existence of minimizers. In Sect. 3 we establish properties of the potentials which allow us to define and control the potential energies, in particular the interaction terms. Next we collect some relevant results about the decoupled variational problems where one of the two components is missing; these facts are established in an Appendix. In Sect. 5 we show that the energy functional is bounded from below, that not all the mass can escape to infinity along a minimizing sequence, and we investigate the splitting properties of the functional. With these prerequisites we can then prove the existence of minimizers in Sect. 6. The fact that such minimizers are steady states together with some of their properties are established in Sect. 7. In Sect. 8 we finally investigate the stability estimate resulting from their minimizing property. 2. Variational Setup We denote the set of non-negative, Lebesgue integrable functions by L 1+ (Rn ). For f ∈ L 1+ (R6 ) and ρ ∈ L 1+ (R3 ) we denote by ρ f (x) :=
R3
f (t, x,v)dv, Uρ (x) := −
ρ(y) dy R3 |x − y|
the induced spatial density and gravitational potential; we write U f = Uρ f . Similarly, for f˜ ∈ L 1+ (R4 ) and ρ˜ ∈ L 1+ (R2 ), ˜ := ρ f˜ (x)
R2
f˜(t, x, ˜ v)d ˜ v, ˜ Uρ˜ (x) := −
ρ( ˜ y˜ ) d y˜ , R2 |x − ( y˜ ,0)|
and to abbreviate we sometimes write ρ˜ and U˜ instead of ρ f˜ and Uρ˜ ; notice that the latter is defined on R3 . In what follows we do not explicitly denote the domain of integration—R3 or R2 —unless in cases of ambiguity. The integrability properties of these potentials are investigated in Sect. 3. Next we define the various parts of the energy functional. For f ∈ L 1+ (R6 ) and f˜ ∈ L 1+ (R4 ), 1 1 ˜ v)d ˜ v˜ d x, ˜ |v|2 f (x,v)dv d x, E kin ( f˜) := |v| ˜ 2 f˜(x, 2 2 ρ˜ (x) ρ f (x)ρ f (y) 1 1 f˜ ˜ ρ˜ f˜ ( y˜ ) d x d y, E pot ( f˜) := − d x˜ d y˜ , E pot ( f ) := − 2 |x − y| 2 |x˜ − y˜ | E kin ( f ) :=
denote the kinetic and potential energies of the non-flat and flat components. The total energy of each component is then defined by H( f ) := E kin ( f ) + E pot ( f ), H( f˜) := E kin ( f˜) + E pot ( f˜).
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Finally,
1 1 ˜ ˜ x, ˜ H( f, f˜)=H( f ) + H( f˜) + U f˜ (x)ρ f (x)d x + U f (x,0)ρ f˜ ( x)d 2 2 =H( f ) + H( f˜) + U f˜ (x)ρ f (x)d x
is the total energy of the state ( f, f˜). In Sect. 3, where we investigate the existence of all these integrals on the constraint set defined below, we will also see that the two interaction terms are equal. We wish to minimize this functional over the constraint set FM := ( f, f˜)| f ∈ L 1+ (R6 ), f˜ ∈ L 1+ (R4 ), || f ||1 ≤ M, || f ||1+1/k ≤ N , ˜ || f˜|| ˜ ˜ || f˜||1 ≤ M, 1+1/k˜ ≤ N , E kin ( f ) + E kin ( f ) < ∞, f (x, ˜ x3 , v,v ˜ 3 ) = f (x,−x ˜ ˜ 3 , v,−v 3) , ˜ N˜ ) denotes the constraint vector whose components are all where M := (M, N , M, strictly positive, || · || p denotes the usual L p norm, and 0 < k < 7/2, 0 < k˜ < 2. In Sect. 3 and 5 we will see that the total energy functional is well defined and bounded from below on this set. The following theorem is our main result. Theorem 2.1. Let ( f j , f˜j ) ⊂ FM be a minimizing sequence of H. Then there exists ( f 0 , f˜0 ) ∈ FM , a subsequence again denoted by ( f j , f˜j ) and a sequence of shift vectors ˜ v) ˜ := f˜j (x˜ + a˜ j , v), ˜ (a˜ j ) ⊂ R2 such that with T j f j (x,v) := f j (x + (a˜ j ,0),v), T j f˜j (x, ˜ T j f j f 0 , T j f˜j f˜0 weakly in L 1+1/k (R6 ) or L 1+1/k (R4 ) respectively, E pot (T j f j − f 0 ) → 0, E pot (T j f˜j − f˜0 ) → 0,
and
(ρT j f j − ρ f0 )UT j f˜j − f˜0 d x → 0.
Moreover ( f 0 , f˜0 ) is a minimizer of H over FM . The spatial shifts parallel to the (x1 , x2 ) plane are necessary due to the invariance of the total energy and the constraint set under such shifts. If ( f 0 , f˜0 ) is a minimizer of H, then (T j f 0 , T j f˜0 ) is a minimizing sequence for any choice of shift vectors a˜ j ∈ R2 which is weakly convergent to a minimizer only if we shift our frame of reference accordingly. The constraints on || f ||1+1/k and || f˜||1+1/k˜ in the definition of the set FM play the role of the Casimir constraints, and it does not seem to be possible to include these Casimirs into the functional to be minimized, as was done for example in [14] for the purely three dimensional and in [24] for the purely flat problem. In the latter cases these Casimir functionals can be replaced by more general ones of the form ( f (x,v))dv d x with some suitable prescribed function . The Casimir constraint determines the microscopic equation of state of the resulting steady states, and the choice in the present paper
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restricts these steady states to the so-called polytropic case, cf. Thm. 7.2. In astrophysics polytropic states have been and are studied extensively, also in the context of dark matter, cf. the discussion in [5,8,18,23,31] and the references there. However, from the applications point of view it is desirable to extend the present analysis to non-polytropic and possibly non-isotropic states along the lines in [14–16]. Such an extension does not seem straightforward to the authors since the form of the constraints and in particular their scaling properties play an important role. 3. Preliminaries We start by collecting some well known estimates for the spatial densities and potential energies induced by elements from the constraint set FM . Lemma 3.1. Let ( f, f˜) ∈ FM and define n := k + 3/2, L 1+1/n (R3 ), ρ f˜ ∈ L 1+1/n˜ (R2 ) with
n˜ := k˜ + 1. Then ρ f ∈
||ρ f ||1+1/n ≤C N (k+1)/(n+1) E kin ( f )3/(2k+5) , ˜
˜
˜ ||ρ f˜ ||1+1/n˜ ≤C N˜ (k+1)/(n+1) E kin ( f˜)1/(k+2) ,
and −E pot ( f )≤C||ρ f ||26/5 ≤ CM E kin ( f )1/2 , −E pot ( f˜)≤C||ρ˜ f˜ ||24/3 ≤ CM E kin ( f˜)1/2 , where the constant C > 0 is universal and CM > 0 depends on the constraint vector M. ˜ 1 + 1/n > 6/5 and 1 + 1/n˜ > 4/3 so that ρ f ∈ L 6/5 (R3 ), By the restrictions on k and k, 4/3 2 ρ f˜ ∈ L (R ). Proof. Given R > 0 we split the v-integral and use Hölder’s inequality and the definition of the kinetic energy to find that ρ f (x)= f (x,v)dv + f (x,v)dv |v|≤R
≤
4π 3 R 3
1/(k+1)
|v|>R
k/(k+1)
f 1+1/k (x,v)dv
+
1 R2
|v|2 f (x,v)dv.
We optimize this estimate in R, take the resulting estimate for ρ f (x) to the power 1 + 1/n and integrate with respect to x to obtain the estimate for ρ f . The estimate for ρ f˜ follows the same lines. The last two inequalities follow by interpolation and the Hardy-Littlewood-Sobolev inequality. In order to analyze the mixed term in H( f, f˜) we need some information on the integrability of the flat potential in R3 . Lemma 3.2. Let ρ˜ ∈ L 4/3 (R2 ). Then Uρ˜ ∈ L 6 (R3 ) and ˜ L 4/3 (R2 ) . ||Uρ˜ || L 6 (R3 ) ≤ C||ρ||
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Proof. We use the general form of the Minkowski inequality, cf. [20, 2.4], and the weak Young inequality to obtain 6 ρ( ˜ y˜ ) 6 ||Uρ˜ ||6 = d y˜ d x3 d x˜ R2 R R2 |x − ( y˜ ,0)| ⎡
1/6 ⎤6 6 ( y˜ ) ρ ˜ ≤ ⎣ d x3 d y˜ ⎦ d x˜ (|x˜ − y˜ |2 + x32 )3 6 ρ( ˜ y˜ ) d y˜ d x˜ = C||ρ˜ ∗ | · |−5/6 ||6L 6 (R2 ) =C |x˜ − y˜ |5/6 ≤C||ρ|| ˜ 6L 4/3 (R2 ) ||| · |−5/6 ||6 12/5 Lw
(R 2 )
;
the function | · |−λ is an element of the weak L p space L w (Rn ), cf. [20, 4.3]. n/λ
We also need to investigate the integrability of Uρ , restricted to the (x1 , x2 ) plane. Lemma 3.3. There exists a bounded linear operator S : L 6/5 (R3 ) → L 4 (R2 ) such that for ρ ∈ Cc (R3 ) compactly supported and continuous, Sρ = Uρ (·,0); notice that for such ρ the induced potential Uρ is a continuous, pointwise defined function. We write Uρ (·,0) = Sρ ∈ L 4 (R2 ) also for ρ ∈ L 6/5 (R3 ) so that ||U (·,0)|| L 4 (R2 ) ≤ C||ρ|| L 6/5 (R3 ) . If in addition ρ˜ ∈ L 4/3 (R2 ), then the following mixed potential energies exist and are equal: ˜ ρ( ˜ x)d ˜ x. ˜ Uρ˜ (x)ρ(x)d x = Uρ (x,0) Proof. Fubini’s theorem together with the Hölder inequality and Lemma 3.2 imply that for ρ ∈ Cc (R3 ), ρ˜ ∈ Cc (R2 ), |ρ(y)ρ( ˜ x)| ˜ |Uρ (x,0) d x˜ dy = |U|ρ| ˜ ρ( ˜ x)|d ˜ x˜ ≤ ˜ (y)ρ(y)|dy |(x,0) ˜ − y| ≤C||ρ|| L 6/5 (R3 ) ||ρ|| ˜ L 4/3 (R2 ) . The estimate for ||Uρ (·,0)|| L 4 (R2 ) follows by taking the supremum over all ρ˜ ∈ Cc (R2 ) with ||ρ|| ˜ L 4/3 (R2 ) = 1. This shows that the operator S is bounded with respect to the indicated norms on the dense subset Cc (R3 ) of L 6/5 (R3 ) so that it can be extended as stated. Since the mixed potential energies now exist they are equal again by Fubini’s theorem. It will be useful to view the potential energy as a bilinear form which induces a scalar product. More precisely we define for ρ,σ ∈ L 6/5 (R3 ), 1 ρ(x)σ (y) ρ,σ pot := dy dx 2 |x − y|
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with the analogous definition for ρ, ˜ σ˜ pot , ρ, ˜ σ˜ ∈ L 4/3 (R2 ), and 1 ρ(x)ρ( ˜ y˜ ) ρ, ρ ˜ pot := d y˜ d x. 2 |x − ( y˜ ,0)|
(3.1)
It is well known that ·,· pot is a scalar product on L 6/5 (R3 ), cf. [20, 9.8], and the same is true on L 4/3 (R2 ). The induced norms are denoted by 1/2
1/2
||ρ||pot := ρ,ρ pot , ||ρ|| ˜ pot := ρ, ˜ ρ ˜ pot . Finally, f, g pot := ρ f ,ρg pot etc., provided that the induced spatial densities belong to the proper L p space, so that with this notation, E pot ( f ) = − f, f pot = −|| f ||2pot
(3.2)
etc. The Cauchy-Schwarz inequality corresponding to the mixed case (3.1) is established next. It tells us how strong the mixed potential energy term is in comparison to the potential energies of its individual components. 6/5
4/3
Lemma 3.4. Let ρ ∈ L + (R3 ), ρ˜ ∈ L + (R2 ). Then ρ, ρ ˜ pot ≤ ||ρ||pot ||ρ|| ˜ pot . Proof. We first show the assertion under the additional assumption that ρ, ρ˜ ∈ Cc∞ are compactly supported and smooth. In that case Uρ is smooth and bounded. Let d ∈ Cc∞ (R) be such that d ≥ 0 and d = 1, and let δ (x) := −1 d(x/ ) denote the induced δ-sequence; > 0. Then ˜ x3 )δ (x3 )d x3 = Uρ (x,0) ˜ lim Uρ (x, →0
pointwise for x˜ ∈ R2 , while the latter integral is bounded in modulus by ||Uρ ||∞ . Using Lebesgue’s theorem and the fact that ·,· pot is a scalar product on L 6/5 (R3 ) we can now argue as follows: 1 1 ρ, ρ ˜ pot = Uρ (x, ˜ ρ( ˜ x)d ˜ x˜ = lim ˜ x3 )δ (x3 )ρ( ˜ x)d ˜ x3 d x˜ Uρ (x,0) 2 2 →0 = lim ρ, ρ˜ ⊗ δ pot →0
1/2 1 ρ( ˜ x) ˜ ρ( ˜ y˜ )δ (x3 )δ (y3 ) ≤ ||ρ||pot lim dx dy →0 2 |x − y| 1/2 1 ρ( ˜ x) ˜ ρ( ˜ y˜ )δ (x3 )δ (y3 ) dx dy ≤ ||ρ||pot lim →0 2 |x˜ − y˜ | =||ρ||pot ||ρ|| ˜ pot ; in the last step we used that δ = 1 for > 0. The general case follows by approximating ρ and ρ˜ by compactly supported, smooth functions, observing the fact that both sides of the inequality are continuous with respect to the L 6/5 (R3 )-norm for ρ and the L 4/3 (R2 )-norm for ρ. ˜
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4. The Decoupled Minimizers In the next sections the existence and properties of the minimizers of the decoupled problems where one of the components is missing will become important. Here we briefly collect the relevant facts. A function g on Rd × Rd is called spherically symmetric if for every A ∈ SO(d), g(Ax, Av) = g(x,v). For each M, N > 0 the energy H( f ) has a minimizer f 03D in the set 3D F M,N := f ∈ L 1+ (R6 ) | || f ||1 ≤ M, || f ||1+1/k ≤ N , E kin ( f ) < ∞ . The minimizer is unique up to spatial shifts, spherically symmetric, has negative energy, i.e., H( f 03D ) < 0, saturates the constraints, i.e., || f 03D ||1 = M, || f 03D ||1+1/k = N , and has compact spatial support. There exists a constant R ∗ > 0 which is independent of M and N such that the radius of this spatial support is R = R ∗ M (2k−1)/3 N −(2k+2)/3 .
(4.1)
By spherical symmetry, f 03D (x, ˜ x3 , v,v ˜ 3 ) = f 03D (x,−x ˜ ˜ 3 , v,−v 3 ). Similarly, the energy H( f˜) has a minimizer f 0FL in the set FL ˜ || f˜|| ˜ , E kin ( f˜) < ∞ . F M,N := f˜ ∈ L 1+ (R4 ) | || f˜||1 ≤ M, ≤ N ˜ 1+1/k A slight complication arises from the fact that we do at the moment not know whether this minimizer is again unique up to spatial shifts. However, there does exist a twoFL ) FL FL parameter family ( f M,N M,N >0 such that f M,N is a minimizer of H( f˜) over F M,N which saturates the constraints, has negative energy, is axially symmetric with respect to the x3 axis, i.e., spherically symmetric as a function of x, ˜ v, ˜ and has compact spatial support. There exists a constant R˜ ∗ independent of M and N such that the radius of this spatial support is ˜ ˜ R˜ = R˜ ∗ M k N −(k+1) .
(4.2)
In what follows f 0FL always denotes the corresponding member of the above family. In particular, if ˜ N˜ ) =: M3D ,MFL , M = (M, N , M, 3D and f FL denotes f FL . then f 03D denotes the minimizer of H over F M,N 0 ˜ N˜ M, Since the above facts are known or follow by arguments already available in the literature we defer their discussion to the Appendix.
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5. Properties of H First we establish a lower bound for H on FM and certain a-priori bounds along minimizing sequences. Lemma 5.1. (a) The functional H is bounded from below on FM , i.e., −∞ < inf H =: h M < 0. FM
(b) Along every minimizing sequence ( f j , f˜j ) ⊂ FM of H both the kinetic and the potential energies are bounded, more precisely, for j sufficiently large, E kin ( f j ) + E kin ( f˜j ) + |E pot ( f j )| + |E pot ( f˜j )| ≤ CM , where the constant CM > 0 depends only on M. Proof. Lemma 3.1 and Lemma 3.4 imply that for ( f, f˜) ∈ FM , f, f˜ pot ≤|| f ||pot || f˜||pot ≤ CM E kin ( f )1/4 E kin ( f˜)1/4 ≤CM E kin ( f )1/2 + CM E kin ( f˜)1/2 ; the value of the constant CM may change from line to line. Using Lemma 3.1 again this yields the estimate H( f, f˜) ≥ E kin ( f ) − CM E kin ( f )1/2 + E kin ( f˜) − CM E kin ( f˜)1/2 . Hence h M > −∞. Moreover,
h M ≤ H( f 03D , f 0FL ) = H( f 03D ) + H( f 0FL ) +
(5.1)
U˜ 0 ρ0 d x < 0.
Hence along a minimizing sequence H( f j , f˜j ) ≤ 0 for j sufficiently large, and by (5.1),
E kin ( f j )1/2 − CM /2
2 2 2 + E kin ( f˜j )1/2 − CM /2 ≤ CM /2.
Another reference to Lemma 3.1 completes the proof. In order to pass to the limit along a minimizing sequence we need the following compactness properties of the potential energies; by 1 S we denote the indicator function of the set S, and we recall (3.2) and the corresponding notation. Lemma 5.2. Let (ρ j ) ⊂ L 1+1/n (R3 ) and (ρ˜ j ) ⊂ L 1+1/n˜ (R2 ) be such that ρ j ρ0 weakly in L 1+1/n (R3 ), ρ˜ j ρ˜0 weakly in L 1+1/n˜ (R2 ). Then for each R > 0, ||1 B R (ρ j − ρ0 )||pot → 0, ||1 B˜ R (ρ˜ j − ρ˜0 )||pot → 0 as j → ∞. Proof. The convergence of the non-flat potential energy is proved for example in [27, Lemma 2.5]. For the flat case we refer to [10, Lemma 3.6].
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A crucial step in the analysis is to show that minimizing sequences do not spread out in space and that up to spatial shifts not all the mass can leak out to infinity. This is the content of the next result. Proposition 5.3. Let ( f j , f˜j ) ⊂ FM be a minimizing sequence of H. Then there exists a sequence (a˜ j ) ⊂ R2 of shift vectors, 0 > 0, and R0 > 0 such that for all sufficiently large j ∈ N, f j dv d x ≥ 0 , f˜j d v˜ d x˜ ≥ 0 . (a˜ j ,0)+B R0
a˜ j + B˜ R0
Here B R0 and B˜ R0 denote the closed ball of radius R0 about the origin in R3 or R2 respectively. Remark. It is important that the same shift vectors work for both the non-flat and the flat component. Proof. Let U j := U f j , ρ˜ j := ρ f˜j , and let R 3D and R FL denote the radii of the decoupled minimizers f 03D and f 0FL subject to constraints M3D and MFL , cf. Sect. 4. Since lim H( f j , f˜j ) ≤ H( f 03D , f 0FL ) = H( f 03D ) + H( f 0FL ) + U03D ρ0FL d x, ˜ j→∞
we get that for j sufficiently large, 1 ˜ H( f j ) + H( f˜j ) + U j ρ˜ j d x˜ < H( f 03D ) + H( f 0FL ) + U03D ρ0FL d x. 2 Since H( f j ) ≥ H( f 03D ) and H( f˜j ) ≥ H( f 0FL ) this implies that 3D ˜ ρ0 (y)ρ0FL (x) 1 1 d x˜ dy U j ρ˜ j d x˜ ≤ U03D ρ0FL d x˜ = − 2 2 |(x,0) ˜ − y| M M˜ <− 3D 2(R + R FL ) for all sufficiently large j ∈ N. For R > 1 we write 1 1 1 1 = 1{|x|≤1/R} (x) + 1{1/R<|x|
(5.2)
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For J1 we first apply the Hölder inequality and then the general form of the Minkowski inequality as in the proof of Lemma 3.2 to obtain the estimate ρ˜ j (x) ˜ d x˜ J1 ≤ρ j 1+1/n |( x,0) ˜ − ·| |x−˜ ˜ ·|<1/R n+1 ≤C ρ j 1+1/n ρ˜ j ∗ ( K˜ R1 )n/(n+1) n+1 ≤C |ρ j 1+1/n ρ˜ j 1+1/n˜ ( K˜ R1 )n/(n+1) γ ≤C ρ j 1+1/n ρ˜ j 1+1/n˜ R −σ ≤ CM R −σ , where
1 1 + γ := n + 1 n˜ + 1
−1
> 1, σ :=
n 2 − > 0; γ n +1
recall that 3/2 < n < 5 and 1 < n˜ < 3. With (5.2) we find that M M˜ − > 3D 2(R + R FL )
U j ρ˜ j d x˜ = −J1 − J2 − J3 ,
and hence J2 ≥
M M˜ 2(R 3D + R FL )
−
M M˜ − CM R −σ . R
For R sufficiently large the right hand side is positive, so that
M M˜ M M˜ −1 −σ − − CM R ρ j (x)ρ˜ j ( y˜ )d x d y˜ . 0< R ≤ 2(R 3D + R FL ) R |x−( y˜ ,0)|
R2
and such that for some δ0 , r0 > 0, |x−( y˜ ,0)|
ρ(x)σ ( y˜ )d x d y˜ > δ0 .
Then there exist 0 , R0 > 0 depending only on δ0 , r0 , and M such that 0 < ρ(x)d x and 0 < σ ( y˜ )d y˜ |x−(a,0)|
for some a˜ ∈ R2 .
| y˜ −a|
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Proof. Let z ∈ R3 be given. Note first that (x, y˜ ) ∈ R5 | |z − ( y˜ ,0)| < r0 , |x − ( y˜ ,0)| < r0 ⊂ (x, y˜ ) ∈ R5 | |z − x| < 2r0 , |x − ( y˜ ,0)| < r0 , and hence
|z−( y˜ ,0)|
≤
|z−x|<2r0
σ ( y˜ )d x d y˜ |x−( y˜ ,0)|
Multiplying with ρ(z) and integrating with respect to z ∈ R3 we obtain ρ(z) σ ( y˜ )d x d y˜ dz |z−( y˜ ,0)|
|x−( y˜ ,0)|
(5.4)
Changing the order of integration, the right hand side of (5.4) can be rewritten as ρ(z) σ ( y˜ )d y˜ dz d x x∈R3 |z−x|<2r0 |x−( y˜ ,0)|
where the functions R and S are defined by R(x) := ρ(z)dz, S(x) := |x−z|<2r0
|x−( y˜ ,0)|
σ ( y˜ )d y˜ .
(5.5)
From our hypothesis, ρ(z) σ ( y˜ )d y˜ dz δ0 < z∈R3 |z−( y˜ ,0)|
(5.6)
As a direct consequence of our definitions (5.5) we find that R∞ ≤ M, S∞ ≤ M.
(5.7)
Flat Galaxies with Dark Matter Halos—Existence and Stability
Furthermore,
R1 =
and
S1 =
x∈R3 |z−x|<2r0
≤ 8M
ρ(z)dz d x =
z∈R3 |z−x|<2r0
ρ(z )d x dz
4π 3 r , 3 0
x∈R3 |x−( y˜ ,0)|
≤M
237
σ ( y˜ )d y˜ d x =
y˜ ∈R2 |x−( y˜ ,0)|
σ ( y˜ )d x d y˜
4π 3 r . 3 0
We may thus continue with (5.6) as follows: 3 R(x) S(x)d x δ0 < 4πr03 R3 3 1/2 ≤ (R S) (R(x) S(x))1/2 d x ∞ 3 4πr03 R 1/2 1/2 3 1/2 ≤ R S R(x)d x S(x)d x ∞ 4πr03 √ 1/2 ≤ 2 2MR S∞ . So there exists a ∈ R3 such that
R(a) S(a) >
δ0 √ 2 2M
2 .
In view of (5.7) this implies that R(a) >
δ02 δ02 and S(a) > . 8M 3 8M 3
Finally we write a = (a,a ˜ 3 ) with a˜ ∈ R2 , a3 ∈ R and observe that δ2 σ ( y˜ )d y˜ ≥ σ ( y˜ )d y˜ = S(a) > 0 3 . 8M |a− ˜ y˜ | 0 clearly implies |a3 | < r0 , so that δ2 ρ(z)dz ≥ ρ(z)dz = R(a) > 0 3 , 8M |z−(a,0)|<3r ˜ |a−z|<2r0 0 which is exactly our claim with 0 :=
δ02 , R0 := 3r0 . 8M 3
(5.8)
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In what follows it is important to control the parameters 0 and R0 in Proposition 5.3 if the constraint vector M varies. This is the content of the following corollary. Corollary 5.5. Let the constraint vector M satisfy the bounds 0 < Ml ≤ M ≤ Mu , 0 < M˜ l ≤ M˜ ≤ M˜ u , 0 < Nl ≤ N ≤ Nu , 0 < N˜ l ≤ N˜ ≤ N˜ u . Then the parameters 0 and R0 in Proposition 5.3 can be chosen independently of M and ( f j , f˜j ), depending only on the bounds Ml and Mu . Proof. Under the given bounds on M we can choose R > 0 depending only on these bounds such that the left hand side in (5.3) is bounded from below by a parameter δ0 > 0 also depending only on these bounds. To this end, observe that M and M˜ are bounded both from below and above, CM is bounded from above, and the radii R 3D and R FL are bounded from above in view of (4.1) and (4.2). Given (5.8) this completes the proof. The last tool needed for the proof of Theorem 2.1 is the fact that the energy infimum h M is sub-additive in M. While up to now all components of the constraint vector M were strictly positive, this sub-additivity is for technical reasons needed also in situations where the flat or the non-flat component of a constraint vector vanishes, i.e., M = (M3D ,MFL ) and MFL = 0 or M3D = 0. In such a case h M is obviously taken to 3D denote H( f 03D ) or H( f 0FL ) respectively, where f 03D is the minimizer of H over FM 3D FL , cf. Sect. 4. We say that the constraint vector M ∈ [0,∞[4 and f 0FL is the one over FM FL is nontrivial, if (M > 0 and N > 0) or ( M˜ > 0 and N˜ > 0). Proposition 5.6. For all M1 , M2 ∈ [0,∞[4 , h M1 +M2 ≤ h M1 + h M2 . If both M1 and M2 are nontrivial, then this inequality is strict. If M1 satisfies uniform bounds from above and below as in Corollary 5.5, and if either this is true also for M2 or one component of M2 vanishes and the other one satisfies such uniform bounds, then there exists > 0 depending only on these bounds such that h M1 +M2 ≤ h M1 + h M2 − . Proof. Consider two minimizing sequences ( f j1 , f˜j1 ) ⊂ FM1 and ( f j2 , f˜j2 ) ⊂ FM2 with H( f j1 , f˜j1 ) → h M1 , H( f j2 , f˜j2 ) → h M2 . We can assume that the minimizing sequences are shifted in such a way that the assertions of Proposition 5.3 hold with 01 , 02 , R01 , and R02 , and without spatial shifts. If one component of M1 or M2 vanishes, say the flat one of M2 , the corresponding trivial minimizing sequence ( f 03D ,0) need of course not be shifted, and we take for R02 the radius of the minimizer f 03D and for 02 its mass.
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By the Minkowski inequality, ( f j1 + f j2 , f˜j1 + f˜j2 ) ∈ FM1 +M2 , and hence h M1 +M2 ≤ H( f j1 + f j2 , f˜j1 + f˜j2 ) =H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) − 2 f j1 , f j2 pot − 2 f˜j1 , f˜j2 pot − 2 f j1 , f˜j2 pot − 2 f j2 , f˜j1 pot ≤ H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) → h M1 + h M2 . If M1 and M2 have both at least one nontrivial component, say, in both cases the non-flat one, then the corresponding potential interaction energy is bounded away from zero so that the estimate above is strict: ρ 1j (x)ρ 2j (y) 1 dx dy f j1 , f j2 pot = ρ 1j ,ρ 2j pot ≥ 2 B 1 ×B 2 |x − y| R0
≥
1 2
R0
01 02 . R01 + R02
Assume now that we have positive uniform lower and upper bounds for M1 and M2 . Then the above lower bound for the interaction term is bounded from below by some > 0 depending only on the uniform bounds on the constraint vectors, where we use Corollary 5.5 and in addition (4.1) if the flat component of M2 vanishes. If the non-flat component of M2 vanishes we use (4.2) instead. Remark. The uniform sub-additivity is also valid if both M1 and M2 have exactly one nontrivial component which is uniformly bounded from below and above, but this case is not needed in what follows. 6. Proof of Theorem 2.1 Let ( f j , f˜j ) ∈ FM be a minimizing sequence for H. We choose shift vectors a˜ j ∈ R2 such that the assertion of Proposition 5.3 holds. To keep the notation simple we redefine ( f j , f˜j ) as the minimizing sequence shifted by these vectors as in the statement of Theorem 2.1. Hence according to Proposition 5.3, ˜ 0 ≤ (6.1) f j dv d x ≤ M, 0 ≤ f˜j d v˜ d x˜ ≤ M. |x|≤R0
|x|≤R ˜ 0
This new sequence is of course minimizing as well. The definition of FM implies the a-priori bounds || f j ||1+1/k ≤ N , || f˜j ||1+1/k˜ ≤ N˜ . Hence after extracting a subsequence which we denote by the same symbol, f j f 0 weakly in L 1+1/k (R6 ),
˜ f˜j f˜0 weakly in L 1+1/k (R4 ).
From this weak convergence it follows that ˜ || f 0 ||1+1/k ≤ N , || f˜0 || ˜ || f 0 ||1 ≤ M, || f˜0 ||1 ≤ M, 1+1/k˜ ≤ N ,
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and E kin ( f 0 ) ≤ lim sup E kin ( f j ) < ∞, E kin ( f˜0 ) ≤ lim sup E kin ( f˜j ) < ∞. j→∞
j→∞
By Lemma 3.1 the corresponding spatial densities ρ j := ρ f j and ρ˜ j := ρ f˜j are bounded in L 1+1/n (R3 ) or L 1+1/n˜ (R2 ) respectively. After extracting a subsequence again, ρ j ρ0 weakly in L 1+1/n (R3 ), ρ˜ j ρ˜0 weakly in L 1+1/n˜ (R2 ). It is easy to see that in fact ρ0 = ρ f0 and ρ˜0 = ρ f˜0 . The essential step is to prove that up to extracting yet another subsequence the potential energy terms converge, i.e., || f j − f 0 ||pot + || f˜j − f˜0 ||pot → 0 as j → ∞; by Lemma 3.4 it then follows that also f j − f 0 , f˜j − f˜0 pot → 0. For R > R1 ≥ R0 we define B R1 ,R := {x ∈ R3 | R1 ≤ |x| < R} with the obvious definition of B˜ R1 ,R , and we split the functions f j and f˜j as follows: f j =1 B R
1 ×R
3
f j + 1 BR
1 ,R ×R
f˜j =1 B˜ R
1 ×R
2
f˜j + 1 B˜ R
1 ,R ×R
3 2
f j + 1 B R,∞ ×R3 f j =: f j1 + f j2 + f j3 , f˜j + 1 B˜ R,∞ ×R2 f˜j =: f˜j1 + f˜j2 + f˜j3 .
Lemma 5.2 implies that for R > R1 ≥ R0 fixed, || f j1 + f j2 − f 01 − f 02 ||pot + || f˜j1 + f˜j2 − f˜01 − f˜02 ||pot → 0 as j → ∞. So we only need to show that for any > 0 and R sufficiently large, lim inf || f j3 ||pot + || f˜j3 ||pot < . j→∞
(6.2)
(6.3)
Once this is established we use the triangle inequality for || · ||pot to conclude that || f j − f 0 ||pot ≤ || f j1 + f j2 − f 01 − f 02 ||pot + || f j3 ||pot + || f 03 ||pot . We can surely find R > 1 such that the right hand side is as small as we want for j sufficiently large. Hence for j → ∞, E pot ( f j ) → E pot ( f 0 ) and with the same argument, E pot ( f˜j ) → E pot ( f˜0 ). Finally by Lemma 3.4,
U˜ j ρ j d x →
U˜ 0 ρ0 d x,
and all together this implies that H( f 0 , f˜0 ) ≤ lim H( f j , f˜j ) = h M . j→∞
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This is the desired minimizing property of ( f 0 , f˜0 ). We prove (6.3) by contradiction, so assume that (6.3) is false, i.e. ∃ 1 > 0∀R > 1∃ j0 ∈ N∀ j ≥ j0 : || f j3 ||pot + || f˜j3 ||pot ≥ 1 . Then we can choose a subsequence such that without change of labeling it satisfies either ∀R > 1∃ j0 ∈ N∀ j ≥ j0 : || f j3 ||pot ≥ 1 /2
(6.4)
∀R > 1∃ j0 ∈ N∀ j ≥ j0 : || f˜j3 ||pot ≥ 1 /2.
(6.5)
or
In the following we consider the first case; the second one can be treated analogously. The contradiction is arrived at by splitting f j and f˜j as above and then using the uniform sub-additivity from Proposition 5.6. Let us denote f j0 := 1 B R
0 ×R
3
f j , f˜j0 := 1 B˜ R
0 ×R
2
f˜j .
Since the splitting parameters satisfy the relation R > R1 ≥ R0 , (6.1) implies that ˜ 0 ≤ || f j0 ||1 ≤ || f j1 ||1 ≤ M, 0 ≤ || f˜j0 ||1 ≤ || f˜j1 ||1 ≤ M.
(6.6) ˜
We also need uniform lower bounds for the L 1+1/k -norm and L 1+1/k -norm. By Lemma 3.1, (k+1)/(n+1)
f j0 1 = ρ 0j 1 ≤ C(R0 )ρ 0j 1+1/n ≤ C(R0 ) f j0 1+1/k
,
with an analogous estimate for f˜j . Hence with (6.1), 0 < C( 0 ) ≤ || f j1 ||1+1/k ≤ N , 0 < C( 0 ) ≤ || f˜j1 ||1+1/k˜ ≤ N˜ .
(6.7)
From the assumption (6.4) we now derive such bounds also for f j3 . By Lemma 3.1 with θ ∈]0,1[ an interpolation parameter and σ := (1 − θ )(1 + k)/(1 + n), 2(1−θ)
2θ 2σ || f ||2pot = |E pot ( f )| ≤ C ||ρ||26/5 ≤ C ||ρ||2θ 1 ||ρ||1+1/n ≤ C || f ||1 || f ||1+1/k . (6.8)
With f = f j3 this implies that 0 < C( 1 ) ≤ || f j3 ||1 ≤ M, 0 < C( 1 ) ≤ || f j3 ||1+1/k ≤ N .
(6.9)
To arrive at a contradiction we insert the splitting of f j and f˜j into the energy functional: H( f j , f˜j ) = H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) + H( f j3 , f˜j3 ) −2 f j2 , f j1 + f j3 pot − 2 f j1 , f j3 pot − 2 f˜j2 , f˜j1 + f˜j3 pot − 2 f˜j1 , f˜j3 pot −2 f j2 , f˜j1 + f˜j3 pot − 2 f j1 , f˜j3 pot − 2 f j1 + f j3 , f˜j2 pot − 2 f j3 , f˜j1 pot =:H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) + H( f j3 , f˜j3 ) −I1 − I2 − I˜1 − I˜2 − J1 − J2 − J˜1 − J˜2 .
(6.10)
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Using the Cauchy-Schwarz inequality for ·,· pot , i.e., Lemma 3.4 for the mixed terms J1 and J˜1 , and the boundedness of potential energies along the minimizing sequence, cf. Lemma 5.1, we obtain the estimates I1 + J1 ≤C || f j2 ||pot ≤ C || f j2 − f 02 ||pot + || f 02 ||pot , I˜1 + J˜1 ≤C || f˜j2 ||pot ≤ C || f˜j2 − f˜02 ||pot + || f˜02 ||pot . For R > 2R1 and x ∈ B R1 , y ∈ B R,∞ we note that 1 1 1 2 ≤ ≤ = , |x − y| |y| − R1 |y| − |y|/2 |y| and we combine this with the Hölder inequality to estimate I2 , I˜2 , J2 , and J˜2 as follows:
I2 ≤2 I˜2 ≤2
ρ j (x)d x
B R1
B˜ R1
J2 ≤2 J˜2 ≤2
ρ˜ j (x)d ˜ x˜ ρ j (x)d x
B R1
B˜ R1
|y|
ρ j (y)dy ≤ C||ρ j ||26/5
| y˜ |
−1
ρ˜ j ( y˜ )d y˜ ≤ C||ρ˜ j ||24/3
B R,∞
B˜ R,∞
B˜ R,∞
−1
R1 R R1 R
1/2 , 1/2 ,
| y˜ |−1 ρ˜ j ( y˜ )d y˜ ≤ C||ρ j ||6/5 ||ρ˜ j ||4/3 |y|−1 ρ j (y)dy ≤ C||ρ j ||6/5 ||ρ˜ j ||4/3
ρ˜ j (x)d ˜ x˜
B R,∞
R1 R R1 R
1/2 , 1/2 .
We wish to apply the uniform sub-additivity from Proposition 5.6 to the constraint vectors induced by ( f j1 , f˜j1 ) and ( f j3 , f˜j3 ). To this end, let M j :=( f j 1 , f j 1+1/k , f˜j 1 , f˜j 1+1/k˜ ), Mij :=( f ji 1 , f ji 1+1/k , f˜ji 1 , f˜ji 1+1/k˜ ), i = 1,2,3. From (6.6), (6.7), (6.9) we have the required uniform bounds for M1j and (M 3j , N 3j ). With respect to f˜j3 we now distinguish two cases. Either this function also satisfies such non-zero uniform bounds or it is negligible, more precisely: Case 1. ∃ 2 > 0∀ R > 1∃ j0 ∈ N∀ j ≥ j0 : |E pot ( f˜j3 )| ≥ 2 . In this case the analogue of the potential energy estimate (6.8) for f˜3 implies that j
˜ 0 < C( 2 ) ≤ || f˜j3 || ˜ 0 < C( 2 ) ≤ || f˜j3 ||1 ≤ M, 1+1/k˜ ≤ N . So we have obtained uniform positive bounds for each entry of the quantities M1j and M3j from above and below. By Proposition 5.6, H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) + H( f j3 , f˜j3 ) ≥ h M1 + h M2 + h M3 ≥ h M + j
j
j
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with > 0 independent of the splitting parameters R > 2R1 and of j. Recalling (6.10) we find that h M − H( f j , f˜j ) + ≤ I1 + I2 + I˜1 + I˜2 + J1 + J2 + J˜1 + J˜2 ≤ C1 || f 02 ||pot + || f˜02 ||pot + || f j2 − f 02 ||pot + || f˜j2 − f˜02 ||pot + (R1 /R)1/2 . We choose R1 ≥ R0 such that C1 (|| f 02 ||pot + || f˜02 ||pot ) < /4. Next we choose R > 2R1 such that C1 (R1 /R)1/2 ≤ /4. For j large, 1 h M − H( f j , f˜j ) + ≤ + C1 || f j2 − f 02 ||pot + || f˜j2 − f˜02 ||pot , 2 and by (6.2) this contradicts the fact that ( f j , f˜j ) is minimizing. Case 2. ∀ > 0∃ R > 1∀ j0 ∈ N∃ j ≥ j0 : || f˜j3 ||pot < , provided R ≥ R . In this case we neglect f˜3 in the sub-additivity argument and recall that Proposition 5.6 yields 2 > 0 j
only depending on the bounds for M1j , M 3j , and N 3j such that h M1 + h M2 + h (M 3 ,N 3 ,0,0) ≥h M1 +(M 3 ,N 3 ,0,0) + 2 + h M2 j
j
j
j
j
j
j
j
≥h M1 +(M 3 ,N 3 ,0,0)+M2 + 2 j
j
j
j
≥h M1 +(M 3 ,N 3 ,0,0)+M2 + 2 + h (0,0, M˜ 3 , N˜ 3 ) ≥ h M j + 2 . j
j
j
j
j
j
By the assumption of the present case we can choose a subsequence which we keep on denoting as before such that || f˜j3 ||pot < for all j ∈ N, where will be determined in terms of 2 below; if necessary we increase R so that R ≥ R . By Lemma 3.4, | f˜j3 , f j3 pot | ≤ C . Together with the assumption of the present case, H( f j3 , f˜j3 ) = H( f j3 ) + H( f˜j3 ) − 2 f˜j3 , f j3 pot ≥ H( f j3 ) − 2 − C . We choose R > 2R1 ≥ 2R0 such that in (6.10), I1 + ··· + J˜2 < . Hence by (6.10), H( f j , f˜j )≥H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) + H( f j3 , f˜j3 ) − ≥H( f j1 , f˜j1 ) + H( f j2 , f˜j2 ) + H( f j3 ) − 2 − C ≥h M1 + h M2 + h (M 3 ,N 3 ,0,0) − 2 − C j
j
j
j
≥h M j + 2 − − C . 2
If is chosen properly in terms of 2 , H( f j , f˜j ) ≥ h M + 2 /2 as j → ∞. This contradicts the minimizing property of ( f j , f˜j ). If one considers the case (6.5) instead of (6.4), all the arguments remain the same with the roles of the flat and non-flat components interchanged. The proof of Theorem 2.1 is complete.
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7. Properties of the Minimizer First we exclude the possibility that for a minimizer f 0 = 0 or f˜0 = 0. Indeed the next result shows that the constraints are to some extent saturated by any minimizer. Proposition 7.1. Let ( f 0 , f˜0 ) ∈ FM be a minimizer of H over FM . Then ˜ || f 0 ||1 = M ∨ || f˜0 ||1 = M, ˜ || f 0 ||1+1/k = N , || f˜0 || ˜ = N. 1+1/k
Proof. We define for a,b,c,d,e > 0 a rescaled state ( f 0∗ , f˜0∗ ) as f 0∗ (x,v) := a f 0 (bx,cv),
f˜0∗ (x, ˜ v) ˜ := d f˜0 (b x,e ˜ v); ˜
because of the mixed potential energy term x and x˜ must be scaled in the same way. Then E kin ( f 0∗ ) = ab−3 c−5 E kin ( f 0 ), E kin ( f˜0∗ ) = db−2 e−4 E kin ( f˜0 ), E pot ( f 0∗ ) = a 2 b−5 c−6 E pot ( f 0 ), E pot ( f˜0∗ ) = d 2 b−3 e−4 E pot ( f˜0 ), ∗ ∗ −4 −3 −2 ˜ U˜ 0 ρ0 d x. U0 ρ0 d x = adb c e Assume that || f 0 ||1+1/k < N . Then we choose a = c3 , b = d = e = 1. For this choice of parameters f˜0∗ = f˜0 , || f 0∗ ||1 = || f 0 ||1 , || f 0∗ ||1+1/k = c3/(k+1) || f 0 ||1+1/k , and H( f 0∗ , f˜0∗ ) = c−2 E kin ( f 0 ) + E pot ( f 0 ) + H( f˜0 ) +
U˜ 0 ρ0 d x.
We can choose c > 1 so that the rescaled state still lies in FM and has lower energy which is a contradiction. The analogous argument shows that || f˜0 ||1+1/k˜ = N˜ . In order to prove that at least one of the two mass constraints is saturated we assume ˜ and we choose the scaling parameters that || f 0 ||1 < M ∧ || f˜0 ||1 < M, a = c−7 , b = d = c−4 , e = c. For this choice, || f 0∗ ||1 = c2 || f 0 ||1 , || f˜0∗ ||1 = c2 || f˜0 ||1 , ˜ ˜ || f 0∗ ||1+1/k = c(2k−7)/(k+1) || f 0 ||1+1/k , || f˜0∗ ||1+1/k˜ = c(2k−4)/(k+1) || f˜0 ||1+1/k˜ ,
and H( f 0∗ , f˜0∗ ) = H( f 0 , f˜0 ).
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Since 0 < k < 7/2 and 0 < k˜ < 2 we can choose c > 1 such that ( f 0∗ , f˜0∗ ) ∈ FM and both M∗ := (|| f 0∗ ||1 ,|| f 0∗ ||1+1/k ,|| f˜0∗ ||1 ,|| f˜0∗ ||1+1/k˜ ) and M − M∗ are non-trivial. The strict sub-additivity in Proposition 5.6 implies the desired contradiction: h M < h M∗ + h M−M∗ < H( f 0∗ , f˜0∗ ) = H( f 0 , f˜0 ) = h M . The main result of this section is the fact that the minimizers are functions of the particle or local energy. We use the Lagrange multiplier method presented for example in [4,14,15,27,29]. Theorem 7.2. Let ( f 0 , f˜0 ) be a minimizer as obtained in Theorem 2.1 with induced potentials (U0 , U˜ 0 ). Then f 0 (x,v) = ˜ v) ˜ = f˜0 (x,
k E 0 −E(x,v) λ +
a.e.,
k˜ E˜ 0 −E(x,0, ˜ v,0) ˜ λ˜ +
a.e.,
where E(x,v) := 21 |v|2 +U0 (x) + U˜ 0 (x) and (·)+ denotes the positive part. The Lagrange multipliers are defined as 2k + 5 1 E kin ( f 0 ) + 2E pot ( f 0 ) + U0 ρ˜0 d x , E 0 := || f 0 ||1 3 1 ˜ ˜ ˜ ˜ (k + 2)E kin ( f 0 ) + 2E pot ( f 0 ) + U0 ρ˜0 d x , E 0 := || f˜0 ||1 and λ :=
2(k + 1)E kin ( f 0 ) 1+1/k 3|| f 0 ||1+1/k
, λ˜ :=
(k˜ + 1)E kin ( f˜0 ) || f˜0 ||
1+1/k˜ 1+1/k˜
.
Proof. Let ( f 0 , f˜0 ) be a minimizer of H with corresponding potentials (U0 , U˜ 0 ). For f such that ( f, f˜0 ) ∈ FM we define G( f ) := H( f, f˜0 ). Then G( f ) − G( f 0 )= E kin ( f ) − E kin ( f 0 ) + E pot ( f ) − E pot ( f 0 ) + (ρ f − ρ0 )U˜ 0 d x. For each fixed > 0 we define the set S := (x, y) ∈ R6 | ≤ f 0 (x,v) ≤ −1 .
(7.1)
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Let η ∈ L ∞ (R6 ) be a real-valued function with compact support such that η ≥ 0 a.e. for (x,v) ∈ R6 \ supp f 0 and suppη ⊂ (R6 \ supp f 0 ) ∪ S . For t ∈ [0, T ] and T = (||η||1 + ||η||1+1/k + ||η||∞ )−1 /2 we define
f t (x,v) := α 3 (t)|| f 0 ||1
f 0 + tη (x,α(t)v), || f 0 + tη||1
where α(t) :=
|| f 0 ||1+1/k || f 0 + tη||1 || f 0 ||1 || f 0 + tη||1+1/k
(k+1)/3
.
For t ∈ [0, T ], || f t ||1 = || f 0 ||1 , || f t ||1+1/k = || f 0 ||1+1/k and f 0 + tη ≥ 0 a.e. For small enough, || f 0 ||1 2 || f 0 ||1+1/k1 2
≤ || f 0 + tη||1 ≤ || f 0 ||1 + 2 ,
≤ || f 0 + tη||1+1/k ≤ || f 0 ||1+1/k + 2 ,
which implies that α is a smooth function on [0, T ] and ⎡ α (t) =
||η||1 k +1 α(t) ⎣ − 3 || f 0 + tη||1
( f0
+ tη)1/k η d x dv 1+1/k
|| f 0 + tη||1+1/k
⎤ ⎦.
Moreover, α is bounded on [0, T ]. From (7.1) we conclude that for t ∈ [0, T ], G( f t ) − G( f 0 )=
|| f 0 ||1 || f 0 ||1 t − 1 E kin ( f 0 ) + 2 E kin (η) 2 α (t)|| f 0 + tη||1 α (t)|| f 0 + tη||1
|| f 0 ||21 || f 0 ||21 t + ρη U0 d x − 1 E ( f ) + pot 0 || f 0 + tη||21 || f 0 + tη||21 || f 0 ||21 t 2 || f 0 ||1 + E (η) + − 1 ρ0 U˜ 0 d x pot || f 0 + tη||1 || f 0 + tη||21 || f 0 ||1 t + ρη U˜ 0 d x. (7.2) || f 0 + tη||1
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By Taylor expansion at t = 0, ⎡ ⎞⎤ ⎛ 1/k η d x dv f ||η|| ||η|| k + 1 || f 0 ||1 1 1 0 ⎠⎦ + O(t 2 ), ⎝ − 1=−t ⎣ +2 − 1+1/k α 2 (t)|| f 0 + tη||1 || f 0 ||1 3 || f 0 ||1 || f 0 || 1+1/k
|| f 0 ||1 t =t + O(t 2 ), 2 α (t)|| f 0 + tη||1 || f 0 ||21 2||η||1 t − 1=− + O(t 2 ), || f 0 ||1 || f 0 + tη||21 || f 0 ||21 t
=t + O(t 2 ), || f 0 + tη||21 || f 0 ||1 ||η||1 t − 1=− + O(t 2 ), || f 0 + tη||1 || f 0 ||1 || f 0 ||1 t =t + O(t 2 ). || f 0 + tη||1 If we substitute these expansions into (7.2), we find that 1/k G( f t ) − G( f 0 ) = t (E − E 0 + λ f 0 )η dv d x + O(t 2 ) with E 0 and λ as given in the theorem. Since G( f t ) attains its minimum at t = 0, the choice of η and → 0 imply that E − E 0 ≥ 0 on R6 \ supp f 0 and E0 − E k f0 = a.e. on supp f 0 . λ If we repeat this argument with the roles of flat and non-flat states exchanged, i.e., for G( f˜) := H( f 0 , f˜), we obtain the assertion for f˜0 . The previous theorem states that for a minimizer ( f 0 , f˜0 ) both components are functions of the local or particle energy in the induced potential U0,e = U0 + U˜ 0 . Since the latter is time-independent, the particle energy is conserved along particle orbits, i.e., along the characteristics of the Vlasov equations (1.1) and (1.2) respectively. Hence f 0 and f˜0 satisfy these equations at least formally, and we are justified to refer to ( f 0 , f˜0 ) as a steady state of the system (1.1)–(1.4). We do not discuss the regularity of this steady state further. However, to conclude this section we want to address the question whether these states have spatially compact support. Proposition 7.3. Let ( f 0 , f˜0 ) be a minimizer as obtained in Theorem 2.1 and assume that 0 < k < 5/2 and 0 < k˜ < 1. Then U0 , U˜ 0 , ρ0 , ρ˜0 ∈ L ∞ (R3 ) with lim U0 (x) = 0, lim U˜ 0 (x) = 0,
|x|→∞
|x|→∞
E 0 , E˜ 0 < 0, and ρ0 and ρ˜0 have compact support.
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Proof. Consider a density ρ˜ ∈ L 1+ ∩ L p (R2 ). Then Uρ˜ ∈ L ∞ (R3 ) with lim|x|→∞ Uρ˜ (x) = 0, provided p > 2. If ρ is defined on R3 then the same is true provided p > 3/2. We prove this assertion for the flat case. Here ρ( ˜ y˜ ) d y˜ −Uρ˜ (x)= 2 |x − ( y ˜ ,0)| R ρ( ˜ y˜ ) ρ( ˜ y˜ ) = d y˜ + d y˜ . |x − ( y ˜ ,0)| |x − ( y˜ ,0)| |x−( y˜ ,0)|≤R |x−( y˜ ,0)|>R ||ρ|| ˜ 1 ρ( ˜ y˜ ) ≤ d y˜ + 2 R |x− ˜ y˜ |≤ R 2 −x3 |x − ( y˜ ,0)| ≤C R ( p−2)/ p ||ρ|| ˜
+ L p ({|x− ˜ y˜ |≤ R 2 −x32 })
||ρ|| ˜ 1 . R
Since this holds for any R > 0 the assertion follows; notice that for R > 0 fixed the first term goes to zero for |x| → ∞. By the weak Young inequality and Lemma 3.3, U0 ∈ L 6 (R3 ) and U0 (·,0) ∈ L 4 (R2 ), and again by the weak Young inequality and Lemma 3.2, U˜ 0 ∈ L 4 (R2 ) and U˜ 0 ∈ L 6 (R3 ). Hence U0,e ∈ L 4 (R2 ) ∩ L 6 (R3 ). If we integrate the relations between f 0 , f˜0 , and U0,e from Theorem 7.2 with respect to v or v˜ respectively we obtain the relations ˜ E˜ 0 −U0,e (·,0))n+˜ , ρ = c(E 0 −U0,e )n+ , ρ˜ = c(
(7.3)
where c and c˜ depend on λ and k or λ˜ and k˜ respectively. From the integrability assertions for the potential we conclude that the spatial densities have the required integrability provided 6/n > 3/2, i.e., n < 4 which means k < 5/2, and 4/n˜ > 2 i.e., n˜ < 2 which means k˜ < 1. It therefore remains to show that E 0 < 0 and E˜ 0 < 0 as claimed; the assertion on the support of the densities then follows. Assume that E 0 > 0. Then for |x| large, ρ0 (x) > c(E 0 /2)n which contradicts its integrability, and the same argument works for ρ˜0 . Now assume that E 0 = 0. Then ρ0 (x) = c(−U0,e (x))n , and this again contradicts the integrability of ρ0 since −U0,e ≥ C/|x| for large |x| and C > 0. We prove this for U˜ 0 , the argument for U0 being completely analogous. We choose R > 0 such that | y˜ |≤R
ρ˜0 ( y˜ )d y˜ =: m > 0.
Next we observe that for | y˜ | ≤ R and |x| ≥ 2R, R 1 1 |x − ( y˜ ,0)| − |x| ≤ (|x| − R)2 . If we restrict the convolution integral defining U˜ 0 to the set {|y| ≤ R} and expand the kernel as indicated the assertion on U˜ 0 follows. The same argument works for U0 so n˜ which ˜ = c(−U0,e (x,0)) ˜ that indeed −U0,e ≥ C/|x| as claimed. If E˜ 0 = 0 then ρ˜0 (x) contradicts the integrability of ρ. ˜ Notice that under the present assumptions on k and k˜ it follows that n < 3 and n˜ < 2.
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8. Stability In this section we show how the minimizing property of a minimizer ( f 0 , f˜0 ) ∈ FM leads to a stability estimate. Given a second state ( f, f˜) ∈ FM and denoting the effective potential of the minimizer by U0,e , a simple computation shows that 1 2 ˜ ˜ |v| +U0,e (x) ( f − f 0 )(x,v)dv d x H( f, f )=H( f 0 , f 0 ) + 2 1 2 |v| ˜ +U0,e (x,0) ˜ ( f˜ − f˜0 )(x, ˜ v)d ˜ v˜ d x˜ + 2 − || f − f 0 ||2pot − || f˜ − f˜0 ||2pot − 2 f − f 0 , f˜ − f˜0 pot . With E(x,v) = 21 |v|2 +U0,e (x) and d(( f, f˜),( f 0 , f˜0 )):= E(x,v)( f − f 0 )(x,v)dv d x + E(x,0, ˜ v,0)( ˜ f˜ − f˜0 )(x, ˜ v)d ˜ v˜ d x, ˜
(8.1)
we can rewrite this expansion as H( f, f˜)=H( f 0 , f˜0 ) + d(( f, f˜),( f 0 , f˜0 )) − || f − f 0 ||2pot − || f˜ − f˜0 ||2pot − 2 f − f 0 , f˜ − f˜0 pot .
(8.2)
We need to show that d(( f, f˜),( f 0 , f˜0 )) ≥ 0 with equality only if ( f, f˜) = ( f 0 , f˜0 ). To this end we restrict ourselves to states ( f, f˜) ∈ FM such that ˜ 1+1/k 1+1/k˜ f 1+1/k = f 0 , f˜ = f˜0 , f˜1+1/k = f˜0 . (8.3) f = f0 , Remark. From a physics point of view a galaxy and its halo are typically perturbed by the gravitational field of some distant exterior object. In particular, the perturbation will result in a measure preserving redistribution of the particles in phase space, and will hence preserve the constraints in (8.3). On the other hand, the fact that the perturbations lie in FM means that the stars are only shifted within the galactic plane and not perpendicularly to it. This is certainly an unphysical restriction. To remove it is a non-trivial problem for future research. Using (8.3) and the strict convexity of the function [0,∞[ ζ → ζ p for p > 1 we find that λ 1+1/k d(( f, f˜),( f 0 , f˜0 ))= (E − E 0 )( f − f 0 ) + ) ( f 1+1/k − f 0 1 + 1/k λ˜ ˜ 1+1/k˜ ˜ ˜ ˜ ) + (E − E 0 )( f − f 0 ) + ( f˜1+1/k − f˜0 ˜ 1 + 1/k 1/k ( f − f0 ) ≥ (E − E 0 ) + λ f 0 1/k˜ ( f˜ − f˜0 ) ≥ 0; + (E − E˜ 0 ) + λ˜ f˜0
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Theorem 7.2 implies that the last expressions are non-negative, and the strict convexity implies that equality holds only if ( f, f˜) = ( f 0 , f˜0 ). In order to establish a stability result, we now wish to apply the above estimates to the time evolution ( f (t), f˜(t)) of a perturbation of ( f 0 , f˜0 ). Clearly, we need to require that ( f (0), f˜(0)) ∈ FM satisfies the constraints (8.3). More importantly, in view of the fact that nothing is known on the initial value problem for the system (1.1)–(1.4), we have to assume that this system has solutions t → ( f (t), f˜(t)) which preserve the total energy, the constraints (8.3), and ( f (t), f˜(t)) ∈ FM . To keep the rest of the discussion simple we furthermore assume that the minimizer ( f 0 , f˜0 ) is unique in FM up to spatial shifts. If the minimizer is up to spatial shifts only isolated with respect to the distance measurement used in the stability estimate below, the result remains unchanged. If the minimizers are not even isolated one can prove the stability of the whole set of minimizers; we refer to [26] for the corresponding modifications of the arguments. Stability estimate. Assume the minimizer ( f 0 , f˜0 ) is unique in FM . Then for every > 0 there exists δ > 0 such that for any solution t → ( f (t), f˜(t)) of (1.1)–(1.4) satisfying the above assumptions the following is true: If d(( f (0), f˜(0)),( f 0 , f˜0 )) + || f (0) − f 0 ||pot + || f˜(0) − f˜0 ||pot < δ, then d(( f (t), f˜(t)),( f 0 , f˜0 )) + || f (t) − f 0 ||pot + || f˜(t) − f˜0 ||pot < up to spatial shifts parallel to the (x1 , x2 ) plane and as long as the solution exists. We do not call this a theorem because it is not clear that sufficiently regular solutions to the initial value problem do indeed exist. Assuming the latter the proof is by contradiction. If the assertion were false, there exists > 0, a sequence of solutions (t → ( f j (t), f˜j (t))) and a sequence of times (t j ) such that for all j ∈ N, d(( f j (0), f˜j (0)),( f 0 , f˜0 )) + || f j (0) − f 0 ||pot + || f˜j (0) − f˜0 ||pot < 1/j,
(8.4)
d(( f j (t j ), f˜j (t j )),( f 0 , f˜0 )) + || f j (t j ) − f 0 ||pot + || f˜j (t j ) − f˜0 ||pot ≥ ,
(8.5)
but regardless of how we shift ( f j (t j ), f˜j (t j )) in space. Now (8.4) and the fact that d is non-negative imply that all three terms in (8.4) converge to zero. By Lemma 3.4 this is true also for the mixed term f j (0) − f 0 , f˜j (0) − f˜0 pot , and since H is preserved, (8.2) implies that H( f j (t j ), f˜j (t j )) = H( f j (0), f˜j (0)) → H( f 0 , f˜0 ). Since ( f j (t j ), f˜j (t j )) ∈ FM this means that ( f j (t j ), f˜j (t j )) is a minimizing sequence for H in FM . By Theorem 2.1 there exists a subsequence such that up to spatial shifts, || f j (t j ) − f 0 ||pot + || f˜j (t j ) − f˜0 ||pot → 0; at this point the uniqueness assumption for ( f 0 , f˜0 ) enters. Again by Lemma 3.4 this is true also for the mixed term f j (t j ) − f 0 , f˜j (t j ) − f˜0 pot , and (8.2) with ( f, f˜) = ( f j (t j ), f˜j (t j )) implies that also d(( f j (t j ), f˜j (t j )),( f 0 , f˜0 )) → 0. Hence all three terms in (8.5) converge to zero, which is a contradiction.
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9. Appendix: Facts on the Decoupled Minimizers Here we establish the claims on the decoupled variational problems referred to in Sect. 4. Several of these claims, in particular for the non-flat case, can be found in the literature. Existence. In most of the previous investigations the existence of minimizers in the decoupled cases was not done exactly for the problems stated in Sect. 4: Either the Casimir functional, which in the case at hand corresponds to the L 1+1/k norm, was included into the functional to be minimized, i.e., an energy-Casimir functional instead of the energy was minimized, and only the constraint on the L 1 norm was posed, or the energy was minimized under the constraint that the sum of the mass and the Casimir functional was fixed. In the three dimensional case the problem with two constraints in the form stated in Sect. 4 has been dealt with in [2,29]. We briefly show how the method used in [2] can be adapted to the flat case. With the help of the Riesz rearrangement inequality [20, 3.7] and the fact that the kinetic energy as well as the constraints are invariant under spherically symmetric rearrangements, the problem is reduced to minimizing H( f˜), where the functions f˜ in the constraint set have the form f˜(x, ˜ v) ˜ = ϕ(|x|,| ˜ v|), ˜ with ϕ : [0,∞[2 → [0,∞[ non-increasing in each argument. This monotonicity implies ˜ that for 1 ≤ q ≤ 1 + 1/k, |x| ˜ |v| ˜ q q 2 2 ˜ ˜ v)| ˜ x| ˜ |v| ˜ ≤C ϕ q (r,s)r s ds dr ≤ C|| f˜||q , f (x, 0
f˜(x, ˜ v)| ˜ x| ˜ 2 |v| ˜ 4 ≤C
0
Hence
0 |x| ˜ |v| ˜
ϕ(r,s)r s 3 ds dr ≤ C E kin ( f˜).
0
−2/q −2/q |v| ˜ , for |v| ˜ ≤ V (|x|), ˜ |x| ˜ ˜ f (x, ˜ v) ˜ ≤ g(x, ˜ v) ˜ := C |x| ˜ −2 |v| ˜ −4 , for |v| ˜ > V (|x|), ˜
where V (|x|) ˜ > 0 is an arbitrary function and the constant C depends on E kin ( f˜), || f˜||1 , ˜ and || f ||1+1/k˜ , quantities which are bounded along minimizing sequences. The function g induces the spatial density V (|x|) ∞ ˜ −2/q 1−2/q −2 ρg (x)=C| ˜ x| ˜ |v| ˜ d|v| ˜ + C|x| ˜ |v| ˜ −3 d|v| ˜ =C|x| ˜
−2/q
0 2−2/q
V
V (|x|) ˜
(|x|) ˜ + C|x| ˜
−2
V
−2
(|x|). ˜
The choice V (|x|) ˜ = Vq (|x|) ˜ := |x| ˜ (1−q)/(2q−1) yields the estimate ˜ ≤ C |x| ˜ −2q/(2q−1) ρg (x) with the exponent s := −2q/(2q − 1) being such that s < −3/2 for 1 < q < 3/2, s > −3/2 for q > 3/2.
(9.1)
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We split the estimate for f˜ by choosing q = 1 + 1/k˜ > 3/2 for |x| ˜ ≤ 1 and q ∈]1,4/3[ for |x| ˜ > 1 so that ⎧ ˜ ˜ ⎪ |x| ˜ −2/(1+1/k) |v| ˜ −2/(1+1/k) for |x| ˜ ≤ 1 ∧|v| ˜ ≤ V1+1/k˜ (|x|), ˜ ⎪ ⎪ ⎨ −2 −4 | v| ˜ for | x| ˜ ≤ 1 ∧| v| ˜ > V (| x|), | x| ˜ 1+1/k˜ ˜ f˜(x, ˜ v) ˜ ≤ g(x, ˜ v) ˜ := C −2/q −2/q ⎪ |v| ˜ for |x| ˜ > 1∧|v| ˜ ≤ Vq (|x|), ˜ |x| ˜ ⎪ ⎪ ⎩ −2 −4 |x| ˜ |v| ˜ for |x| ˜ > 1∧|v| ˜ > Vq (|x|). ˜ By (9.1) we can obtain exponents s1 > −8/5 and s2 < −8/5 such that s ˜ ≤ 1, Cr 1 for |x| ρ f˜ (x) ˜ ≤ ρg (x) ˜ ≤ Cr s2 for |x| ˜ > 1. By the Hardy-Littlewood-Sobolev inequality this implies that g has finite potential energy. The crucial step in the existence proof for a minimizer is the convergence of the potential energy along a minimizing sequence ( f˜j ). Since 0 ≤ f˜j ≤ g, the finiteness of the potential energy for g allows us to pass to the limit using the dominated convergence theorem. Saturation of the constraints. Minimizers of the decoupled problems always saturate the constraints, i.e., || f 03D ||1 = M, || f 03D ||1+1/k = N , and similarly for the flat case, because if for a minimizer one (or both) equalities were replaced by strict inequalities, then this minimizer can be rescaled in such a way that the constraints become saturated but the energy strictly decreases, which is a contradiction. A similar argument was used in the proof of Proposition 7.1. The Euler-Lagrange relation and symmetry. For minimizers of the flat or non-flat problem the phase space distributions are functions of the local energy, more precisely, they satisfy relations as stated in Theorem 7.2, the only differences being that 1 1 2 E(x,v) := |v|2 +U03D (x), E(x, ˜ +U0FL (x), ˜ v) ˜ := |v| ˜ 2 2 and the interaction term U03D ρ0FL in the definitions of E 0 and E˜ 0 is dropped. The asserted symmetry of the minimizers follows from the fact that symmetric decreasing rearrangements in x or x˜ strictly decrease the energy except when ρ03D and ρ0FL and hence also the induced potentials are symmetric with respect to some point, cf. [20, 3.7, 3.9]. Virial identity and compact support. Both flat and non-flat minimizers satisfy the virial identity 2E kin ( f 0FL ) = −E pot ( f 0FL ), 2E kin ( f 03D ) = −E pot ( f 03D ). This follows from the fact that these minimizers are time-independent solutions of the corresponding Vlasov-Poisson systems. A direct proof based on their minimizing property and scaling is given in [2, 3.2]. The virial identities together with the restrictions on k and k˜ immediately imply that the cut-off energies E 0 and E˜ 0 in the Euler-Lagrange relations are strictly negative. In order to show that the minimizers have compact support it therefore suffices to show that their potentials converge to zero at spatial infinity. We indicate the corresponding argument for the 3D case, the flat one being completely analogous. Applying the Hölder inequality to the first term in the estimate ρ03D (y) M 3D dy + , R > 0 −U0 (x) ≤ R |x−y|≤R |x − y|
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implies that the potential is bounded and vanishes at spatial infinity, provided ρ03D ∈ L p (R3 ) with p > 3/2. While a-priori this need not be the case for 0 < k < 7/2 we can use the fact that similar to (7.3), ρ03D = c(E 0 −U03D )n+ , start with the known integrability U03D ∈ L 6 (R3 ) to conclude that ρ03D ∈ L 6/n (R3 ), and obtain a new integrability estimate for the potential through the weak Young inequality. After finitely many iterations the desired integrability of ρ03D follows, cf. [27, Prop. 2.7]. Uniqueness for the 3D problem. First we notice that by the virial relation the Lagrange multipliers are uniquely determined by the constraint parameters M and N . In the 3D case we can now continue as follows. A minimizer is completely determined by its potential. The latter satisfies the Emden-Fowler equation 1 2 1 (r U ) = c(E 0 −U )n+ , i.e., 2 (r 2 y ) = −cy+n , r2 r where y = E 0 −U , and the constant c depends only on k and M, N . The solutions of this equation which are regular at the origin are uniquely determined by their value at the origin. Moreover, the scaling yα (r ) = αy(α λr ) with λ = (k + 1/2)/2 turns solutions into solutions. But the mass constraint fixes this scaling, and uniqueness of the minimizer follows. For more details we refer to [27, p. 464]. Unfortunately, there is no analogue to the Emden-Fowler equation in the flat case—the flat potential does not satisfy the Poisson equation—and uniqueness in the flat case is not known. The radius relation (4.1) in the 3D case. Each minimizer is a spherically symmetric steady state ( f,ρ,U ) of the three dimensional Vlasov-Poisson system, and for each choice of M and N there is a unique such steady state. If ( f,ρ,U ) is a steady state and α,β > 0 are arbitrary, then f αβ (x,v) = α 2 β f (αx,βv), ραβ (x) = α 2 β −2 ρ(αx), Uαβ (x) = β −2 U (αx) defines another one. There is a unique steady state ( f ∗ ,ρ ∗ ,U ∗ ) with || f ∗ ||1 = 1 = || f ∗ ||1+1/k , and the minimizer with general M and N is obtained by rescaling f ∗ with the parameters α = M (1−2k)/3 N (2k+2)/3 , β = M (k−2)/3 N −(k+1)/3 . Since R = R ∗ /α this implies (4.1). The radius relation (4.2) in the flat case. We do not know whether for each choice of M and N there exists a unique minimizer f 0FL , and so we cannot use the argument above to prove (4.2). To obtain a two-parameter family of minimizers which obeys the radius relation (4.2) we proceed as follows. Since minimizers a-posteriori saturate the constraints we redefine FL F M,N := f˜ ∈ L 1+ (R4 ) | || f˜||1 = M, || f˜||1+1/k˜ = N , E kin ( f˜) < ∞ . FL we define the rescaled function For f˜ ∈ F1,1
f˜µ,ν (x, ˜ v) ˜ := µ f˜(µx,ν ˜ v). ˜ Then ˜
˜
˜
˜
|| f˜µ,ν ||1 = µ−1 ν −2 , || f˜µ,ν ||1+1/k˜ = µ(1−k)/(1+k) ν −2k/(k+1) , H( f˜µ,ν ) = µ−1 ν −4 H( f˜).
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FL , i.e., For M, N > 0 arbitrary we choose µ, ν such that f˜µ,ν ∈ F M,N ˜
˜
˜
˜
µ = M −k N k+1 , ν = M (k−1)/2 N −(k+1)/2 . FL f˜ → f˜ ∈ F FL FL denote an The mapping F1,1 µν M,N is one-to-one and onto. Let f FL . Then f FL is a minarbitrary, spherically symmetric minimizer of H over the set F1,1 µ,ν FL FL imizer of H over F M,N which we denote by f M,N . To see this we observe that any FL can be written as g˜ = f˜ FL function g˜ ∈ F M,N µ,ν where f˜ ∈ F1,1 , and hence FL H(g) ˜ = H( f˜µ,ν ) = µ−1 ν −4 H( f˜) ≥ µ−1 ν −4 H( f FL ) = H( f µ,ν ). FL If R˜ ∗ denotes the radius of the spatial support of f FL , then the spatial support of f M,N has radius ˜ ˜ R˜ = µ−1 R˜ ∗ = R˜ ∗ M k N −(k+1) ,
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Commun. Math. Phys. 291, 257–302 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0873-6
Communications in
Mathematical Physics
Random Quantum Circuits are Approximate 2-designs Aram W. Harrow, Richard A. Low Department of Computer Science, University of Bristol, Bristol, U.K. E-mail: [email protected]; [email protected] Received: 20 August 2008 / Accepted: 1 May 2009 Published online: 17 July 2009 – © Springer-Verlag 2009
Abstract: Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group. 1. Introduction: Pseudo-Random Quantum Circuits There are many examples of algorithms that make use of random states or unitary operators (e.g. [5,28]). However, exactly sampling from the uniform Haar distribution is inefficient. In many cases, though, only pseudo-random operators are required. To quantify the extent to which the pseudo-random operators behave like the uniform distribution, we use the notion of k-designs (often referred to as t-designs). A k-design has k th moments equal to those of the Haar distribution. For most uses of random states or unitaries, this is sufficient. Constructions of exact k-designs on states are known (see [3] and references therein) and some are efficient. Ambainis and Emerson [3] introduced the notion of approximate state k-designs, which can be implemented efficiently for any k. However, the known constructions of unitary k-designs are inefficient to implement. Approximate unitary 2-designs have been considered [10,14,18], although the approaches are specific to 2-designs. We consider a general class of random circuits where a series of two-qubit gates are chosen from a universal gate set. We give a framework for analysing the k th moments of these circuits. Our conjecture, based on an analogous classical result [23], is that a random circuit on n qubits of length poly(n, k) is an approximate k-design. While we do not prove this, we instead give a tight analysis of the k = 2 case. We find that in a
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Fig. 1. An example of a random circuit. Different lines indicate a different gate is applied at each step
broad class of natural random circuit models (described in Sect. 1.1), a circuit of length O(n(n + log 1/)) yields an -approximate 2-design. Our definition of an approximate k-design is in Sect. 2.2. Our results also apply to an alternative definition of an approximate 2-design from [10], for which we show random circuits of length O(n(n +log 1/)) yield -approximations, thus extending the results of that paper to a larger class of circuits. Moreover, our results also apply to random stabiliser circuits, meaning that a random stabiliser circuit of length O(n(n + log 1/)) will be an -approximate 2-design. This both simplifies the construction and tightens the efficiency of the approach of [14], which constructed -approximate 2-designs in time O(n 6 (n 2 + log 1/)) using O(n 3 ) elementary quantum gates.
1.1. Random Circuits. The random circuit we will use is the following. Choose a 2-qubit gate set that is universal on U (4) (or on the stabiliser subgroup of U (4)). One example of this is the set of all one qubit gates together with the controlled-NOT gate. Another is simply the set of all of U (4). Then, at each step, choose a random pair of qubits and apply a gate from the universal set chosen uniformly at random. For the U (4) case, the distribution will be the Haar measure on U (4). One such circuit is shown in Fig. 1 for n = 4 qubits. This is based on the approach used in Refs. [9,26] but our analysis is both simpler and more general. Since the universal set can generate the whole of U (2n ) in this way, such random circuits can produce any unitary. Further, since this process converges to a unitarily invariant distribution and the Haar distribution is unique, the resulting unitary must be uniformly distributed amongst all unitaries [15]. Therefore this process will eventually converge to a Haar distributed unitary from U (2n ). This is proven rigourously in Lemma 3.2. However, a generic element of U (2n ) has 4n real parameters, and thus to even have (4−n ) fidelity with the Haar distribution requires (4n ) 2-qubit unitaries. We address this problem by considering only the lower-order moments of the distribution and showing these are nearly the same for random circuits as for Haar-distributed unitaries. This claim is formally described in Theorem 2.2. Our paper is organised as follows. In Sect. 2 we define unitary k-designs and explain how a random circuit could be used to construct a k-design. In Sect. 3 we work out how the state evolves after a single step of the random circuit. We then extend this to multiple steps in Sect. 4 and prove our general convergence results. A key simplification will be (following [26]) to map the evolution of the second moments of the quantum circuit onto a classical Markov chain. We then prove a tight convergence result for the case where the gates are chosen from U (4) in Sect. 5. This section contains most of the technical content of the paper. Using our bounds on mixing time we put together the proof that random circuits yield approximate unitary 2-designs in Sect. 6. Section 7 concludes with some discussion of applications.
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2. Preliminaries 2.1. Pauli expansion. Much of the following will be done in the Pauli basis. The Pauli operators will be taken as {σ0 , σ1 , σ2 , σ3 } and defined to be 1 0 0 1 0 −i 1 0 , σ1 = , σ2 = , σ3 = . σ0 = 0 1 1 0 i 0 0 −1 n
If |ψ ∈ C2 is a state on n qubits then we write ψ = |ψψ|. We can expand ψ in the Pauli basis as γ ( p)σ p , (2.1) ψ = 2−n/2 p
where σ p = σ p1 ⊗ . . . ⊗ σ pn for the string p = p1 · · · pn . Inverting, the coefficients γ ( p) are given by γ ( p) = 2−n/2 tr σ p ψ.
(2.2)
It is easy to show that the coefficients γ ( p) are real and, with the chosen normalisation, the squares sum to tr ψ 2 , which is 1 for pure ψ. In general γ 2 ( p) ≤ 1 p
with equality if and only if ψ is pure. Note also that tr ψ = 1 is equivalent to γ (0) = 2−n/2 . This notation is extended to states on nk qubits by treating γ as a function of k strings from {0, 1, 2, 3}n . Thus a state ρ on nk qubits is written as γ0 ( p1 , · · · , pk )σ p1 ⊗ . . . ⊗ σ pk . (2.3) ρ = 2−nk/2 p1 ,..., pk
2.2. k-designs. We will say that a k-design is efficient if the effort required to sample a state or unitary from the design is polynomial in n and k. Note that we do not require the number of states to be polynomial because, even for approximate unitary designs, an exponential number of unitaries is required. Rather, the number of random bits needed to specify an element of the design should be poly(n, k). 2.2.1. State designs A (state) k-design is an ensemble of states such that, when one state is chosen from the ensemble and copied k times, it is indistinguishable from a uniformly random state. This is a way of quantifying the pseudo-randomness of the state and is a quantum analogue of k-wise independence. Hayashi et al. [20] give an inefficient construction of k-designs for any n and k. The state k-design definition we use is due to Ref. [3]: Definition 2.1. An ensemble of quantum states { pi , |ψi } is a state k-design if pi (|ψi ψi |)⊗k = (|ψψ|)⊗k dψ, i
ψ
(2.4)
where the integration is taken over the left invariant Haar measure on the unit sphere in Cd , normalised so that ψ dψ = 1.
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+k , where +k is the projector onto (k+d−1 k ) the symmetric subspace of k d-dimensional spaces. For a rigourous proof, see Ref. [16] and for a less precise proof, but from a quantum information perspective, see Ref. [7].
It is well known that the above integral is equal to
2.2.2. Unitary designs A unitary k-design is, in a sense, a stronger version of a state design. Just as applying a Haar-random unitary to an arbitrary pure state results in a uniformly random pure state, applying a unitary chosen from a unitary k-design to an arbitrary pure state should result in a state k-design. Another way to say this is that the state obtained by acting U ⊗k , where U is drawn from a unitary k-design on U (d), on any d k -dimensional state should be indistinguishable from the case where U is drawn uniformly from U (d). Formally, we have: Definition 2.2. Let { pi , Ui } be an ensemble of unitary operators. Define GW (ρ) =
pi Ui⊗k ρ(Ui† )⊗k
(2.5)
U ⊗k ρ(U † )⊗k dU.
(2.6)
i
and G H (ρ) = U
Then the ensemble is a unitary k-design iff GW = G H . Unitary designs can also be defined in terms of polynomials, so that if p is a polynomial with degree k in the matrix elements of U and k in the matrix elements of U ∗ , then averaging p over a unitary k-design should give the same answer as averaging over the Haar measure. To see the equivalence with Definition 2.2 note that averaging a monomial over our ensemble can be expressed as i 1 , . . . , i k |GW (| j1 , . . . , jk j1 , . . . , jk |)|i 1 , . . . , i k , and so if GW = G H then any polynomial of degree k will have the same expectation over both distributions.
2.3. Approximate k-designs. 2.3.1. Approximate state designs Numerous examples of exact efficient state 2-design constructions are known (e.g. [8]) but general exact constructions are not efficient in n and k. Approximate state designs were first introduced by Ambainis and Emerson [3] and they constructed efficient approximate state k-designs for any k. Aaronson [1] also gives an efficient approximate construction. We define approximate state designs as follows. Definition 2.3. An ensemble of quantum states { pi , |ψi } is an -approximate state k-design if ⊗k ⊗k (1 − ) (|ψψ|) dψ ≤ pi (|ψi ψi |) ≤ (1 + ) (|ψψ|)⊗k dψ. ψ
i
ψ
(2.7)
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In [3], a similar definition was proposed but with the additional requirement that the ensemble also forms a 1-design (exactly), i.e. pi |ψi ψi | = |ψψ|dψ. i
ψ
This requirement was necessary there only so that a suitably normalised version of the ensemble would form a POVM. We will not use it. By taking the partial trace one can show that a k-design is a k -design for k ≤ k. Thus approximate k-designs are always at least approximate 1-designs. 2.3.2. Approximate unitary designs It was shown in Ref. [4] that a quantum analogue of a one time pad requires 2n bits to exactly randomise an n qubit state. However, in Ref. [5] it was shown that n + o(n) bits suffice to do this approximately. Translated into k-design language, this says an exact unitary 1-design requires 22n unitaries but can be done approximately with 2n+o(n) . So approximate designs can have fewer unitaries than exact designs. Here, we are interested in improving the efficiency of implementing the unitaries. There are no known efficient exact constructions of unitary k-designs; it is hoped that our approach will yield approximate unitary designs efficiently. We will require approximate unitary k-designs to be close in the diamond norm [24]: Definition 2.4. The diamond norm of a superoperator T , ||(T ⊗ idd )X ||1 ||T || = sup ||T ⊗ idd ||∞ = sup sup , ||X ||1 d d X =0 where idd is the identity channel on d dimensions. Operationally, the diamond norm of the difference between two quantum operations tells us the largest possible probability of distinguishing the two operations if we are allowed to have them act on part of an arbitrary, possibly entangled, state. In the supremum over ancilla dimension d, it can be shown that d never needs to be larger than the dimension of the system that T acts upon. The diamond norm is closely related to completely bounded norms (cb-norms), in that ||T || is the cb-norm of T † and can also be interpreted as the L 1 → L 1 cb-norm of T itself [11,27]. We can now define approximate unitary k-designs. Definition 2.5. GW is an -approximate unitary k-design if ||GW − G H || ≤ ,
(2.8)
where GW and G H are defined in Definition 2.2. In Ref. [10], they consider approximate twirling, which is implemented using an approximate 2-design. They give an alternative definition of closeness which is more convenient for this application: Definition 2.6 ([10]). Let { pi , Ui } be an ensemble of unitary operators. Then this ensemble is an -approximate twirl if max EW W ((W † ρW ))W † − EU U ((U † ρU ))U † ≤ 2 , (2.9) d where the first expectation is over W chosen from the ensemble and the second is the Haar average. The maximisation is over channels and d is the dimension (2n in our case). Our results work for both definitions with the same efficiency.
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2.4. Random Circuits as k-designs. If a random circuit is to be an approximate k-design then Eq. 2.8 must be satisfied where the Ui are the different possible random circuits. We can think of this as applying the random circuit not once but k times to k different systems. Suppose that applying t random gates yields the random circuit W . If W ⊗k acts on an nk-qubit state ρ, then following the notation of Eq. 2.8, the resulting state is γ0 ( p1 , . . . , pk )W σ p1 W † ⊗ · · · ⊗ W σ pk W † . ρW := W ⊗k ρ(W † )⊗k = 2−nk/2 p1 ,..., pk
(2.10) For this to be a k-design, the expectation over all choices of random circuit should match the expectation over Haar-distributed W ∈ U (2n ). We are now ready to state our main results. Our results apply to a large class of gate sets which we define below: Definition 2.7. Let E = { pi , Ui } be a discrete ensemble of elements from U (d). Define an operator G E by G E := pi Ui⊗k ⊗ (Ui∗ )⊗k . (2.11) i
More generally, we can consider continuous distributions. If µ is a probability measure on U (d) then we can define G µ by analogy as G µ := dµ(U )U ⊗k ⊗ (U ∗ )⊗k . (2.12) U (d)
Then E (or µ) is k-copy gapped if G E (or G µ ) has only k! eigenvalues with absolute value equal to 1. For any discrete ensemble E = { pi , Ui }, we can define a measure µ = i pi δUi . Thus, it suffices to state our theorems in terms of µ and G µ . The condition on G µ in the above definition may seem somewhat strange. We will see in Sect. 3 that when d ≥ k there is a k!-dimensional subspace of (Cd )⊗2k that is acted upon trivially by any G µ . Additionally, when µ is the Haar measure on U (d) then G µ is the projector onto this space. Thus, the k-copy gapped condition implies that vectors orthogonal to this space are shrunk by G µ . We will see that G µ is k-copy gapped in a number of important cases. First, we give a definition of universality that can apply not only to discrete gates sets, but to arbitrary measures on U (4). Definition 2.8. Let µ be a distribution on U (4). Suppose that for any open ball S ⊂ U (4) there exists a positive integer such that µ (S) > 0. Then we say µ is universal [for U (4)]. Here µ is the -fold convolution of µ with itself; i.e. µ = δU1 ···U dµ(U1 ) · · · dµ(U ). When µ is a discrete distribution over a set {Ui }, Definition 2.8 is equivalent to the usual definition of universality for a finite set of unitary gates.
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Theorem 2.1. The following distributions on U (4) are k-copy gapped: (i) Any universal gate set. Examples are U (4) itself, any entangling gate together with all single qubit gates, or the gate set considered in [26]. (ii) Any approximate (or exact) unitary k-design on 2 qubits, such as the uniform distribution over the 2-qubit Clifford group, which is an exact 2-design. Proof. (i) This is proven in Lemma 3.2. (ii) This follows straight from Definition 2.2. Theorem 2.2. Let µ be a 2-copy gapped distribution and W be a random circuit on n qubits obtained by drawing t random unitaries according to µ and applying each of them to a random pair of qubits. Then there exists C (depending only on µ) such that for any > 0 and any t ≥ C(n(n + log 1/)), GW is an -approximate unitary 2-design according to either Definition 2.5 or Definition 2.6. To prove Theorem 2.2, we show that the second moments of the random circuits converge quickly to those of a uniform Haar distributed unitary. For W a circuit as in Theorem ⊗2 2.2, write γW ( p1 , p2 ) for the Pauli coefficients of ρW = W ⊗2 ρ W † . Then write γt ( p1 , p2 ) = EW γW ( p1 , p2 ) where W is a circuit of length t. Then we have Lemma 2.1. Let µ and W be as in Theorem 2.2. Let the initial state be ρ with γ0 ( p, p) ≥ 0 and p γ0 ( p, p) = 1 (for example the state |ψψ| ⊗ |ψψ| for any pure state |ψ). Then there exists a constant C (possibly depending on µ) such that for any > 0, (i) γt ( p 1 , p 2 ) − δ p1 p2 p1 , p2 p1 p2 =00
1
2 ≤
2n (2n + 1)
(2.13)
for t ≥ Cn log 1/. (ii)
1 ≤ γ t ( p 1 , p 2 ) − δ p p 1 2 n n 2 (2 + 1)
(2.14)
p1 , p2 p1 p2 =00
for t ≥ Cn(n + log 1/) or, when µ is the uniform distribution on U (4) or its stabiliser subgroup, t ≥ Cn log n . We can then extend this to all states by a simple corollary: Corollary 2.1. Let µ, W and γW be as in Lemma 2.1. Then, for any initial state ρ = 1 p1 , p2 γ0 ( p1 , p2 )σ p1 ⊗ σ p2 , there exists a constant C (possibly depending on µ) 2n such that for any > 0, (i)
γt ( p 1 , p 2 ) − δ p1 p2
p1 , p2 p1 p2 =00
for t ≥ Cn(n + log 1/).
p =0 γ0 ( p, 4n − 1
p)
2 ≤
(2.15)
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(ii) p =0 γ0 ( p, p) γ t ( p 1 , p 2 ) − δ p 1 p 2 ≤ n 4 −1 p ,p
(2.16)
1 2 p1 p2 =00
for t ≥ Cn(n + log 1/). By the usual definition of an approximate design (Definition 2.5), we only need convergence in the 2-norm (Eq. 2.15), which is implied by 1-norm convergence (Eq. 2.16) but weaker. However, Definition 2.6, which requires the map to be close to the twirling operation, requires 1-norm convergence (i.e. Eq. 2.16). Thus, Theorem 2.2 for Definition 2.5 follows from Corollary 2.1(i) and Theorem 2.2 for Definition 2.6 follows from Corollary 2.1(ii). Theorem 2.2 is proved in Sect. 6 and Corollary 2.1 in Sect. 4. We note that, in the course of proving Lemma 2.1, we prove that the eigenvalue gap (defined in Sect. 4.3) of the Markov chain that gives the evolution of the γ ( p, p) terms is O(1/n). It is easy to show that this bound is tight for some gate sets. Related work. Here we summarise the other efficient constructions of approximate unitary 2-designs. – The uniform distribution over the Clifford group on n qubits is an exact 2-design [14]. Moreover, [14] described how to sample from the Clifford group using O(n 8 ) classical gates and O(n 3 ) quantum gates. Our results show that applying O(n(n +log 1/)) random two-qubit Clifford gates also achieve an -approximate 2-design (although not necessarily a distribution that is within of uniform on the Clifford group). – Dankert et al. [10] gave a specific circuit construction of an approximate 2-design. To achieve small error in the sense of Definition 2.5, their circuits require the same O(n(n + log 1/)) gates that our random circuits do. However, when we use Definition 2.6, the circuits from [10] only need O(n log 1/) gates while the random circuits analysed in this paper need to be length O(n(n + log 1/)). – The closest results to our own are in the papers by Oliveira et al. [9,26], which considered a specific gate set (random single qubit gates and a controlled-NOT) and proved that the second moments converge in time O(n 2 (n + log 1/)). Our strategy of analysing random quantum circuits in terms of classical Markov chains is also adapted from [9,26]. In Sect. 3, we generalise this approach to analyse the k th moments for arbitrary k. The main results of our paper extend the results of [9,26] to a larger class of gate sets and improve their convergence bounds. Some of these improvements have been conjectured by [30], which presented numerical evidence in support of them. 3. Analysis of the Moments In order to prove our results, we need to understand how the state evolves after each step of the random circuit. In this section we consider just one step and a fixed pair of qubits. Later on we will extend this to prove convergence results for multiple steps with random pairs of qubits drawn at every step. We consider first the Haar distribution over the full unitary group and then will discuss the more general case of any 2-copy gapped distribution. In this section, we work in general dimension d and with a general Hermitian orthogonal basis σ0 , . . . , σd 2 −1 . Later we will take d to be either 4 or 2n and the σi to be
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Pauli matrices. However, in this section we keep the discussion general to emphasise the potentially broader applications. Fix an orthonormal basis for d × d Hermitian matrices: σ0 , . . . , σd 2 −1 , normalised so that tr σ p σq = d δ p,q . Let σ0 be the identity. We need to evaluate the quantity
(3.1) EU U ⊗k σ p1 ⊗ · · · ⊗ σ pk (U † )⊗k =: T (p), where the expectation is over Haar distributed U ∈ U (d). We will need this quantity in two cases. Firstly, for d = 2n , these are the moments obtained after applying a uniformly distributed unitary so we know what the random circuit must converge to. Secondly, for d = 4, this tells us how a random U (4) gate acts on any chosen pair. Call the quantity in Eq. 3.1 T (p) (we use bold to indicate a k-tuple of coefficients; take p = ( p1 , . . . , pk )) and write it in the σ p basis as ˆ T (p) = (3.2) G(q; p)σq1 ⊗ · · · ⊗ σqk . q
ˆ Here, G(q; p) is the coefficient in the Pauli expansion of T (p) and we define Gˆ as the ˆ matrix with entries equal to G(q; p). We have left off the usual normalisation factor because, as we shall see, with this normalisation Gˆ is a projector. Inverting this, we have ˆ G(q; p) = d −k tr σq1 ⊗ · · · ⊗ σqk T (p)
= d −k EU tr (σq1 ⊗ · · · ⊗ σqk )U ⊗k (σ p1 ⊗ · · · ⊗ σ pk )(U † )⊗k . (3.3) Note that Gˆ is real since T and the basis are Hermitian. We can gain all the information we need about the Haar integral in Eq. 3.1 with the following observations: Lemma 3.1. T (p) commutes with U ⊗k for any unitary U . Proof. Follows from the invariance of the Haar measure on the unitary group. Corollary 3.1. T (p) is a linear combination of permutations from the symmetric group Sk . Proof. This follows from Schur-Weyl duality (see e.g. [16]). From this, we can prove that Gˆ is a projector and find its eigenvectors. ˆ ˆ Theorem 3.1. Gˆ is symmetric, i.e. G(q; p) = G(p; q). Proof. Follows from the invariance of the trace under cyclic permutations. Theorem 3.2. Pπ is an eigenvector of Gˆ with eigenvalue 1 for any permutation operator Pπ i.e. ˆ G(p; q)tr (σq1 ⊗ · · · ⊗ σqk Pπ ) = tr (σ p1 ⊗ · · · ⊗ σ pk Pπ ). q
Further, any vector orthogonal to this set has eigenvalue 0.
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Proof. For the first part, ˆ G(p; q)tr (σq1 ⊗ . . . ⊗ σqk Pπ ) q
= d −k
EU tr σq1 U σ p1 U † . . . tr σqk U σ pk U † tr σq1 ⊗ . . . ⊗ σqk Pπ
q
=d
−k
tr Pπ EU
tr σq1 U σ p1 U † σq1 ⊗ . . . ⊗ tr σqk U σ pk U † σqk
q1
(3.4)
qk
Writing U † σ p U in the σ p basis, we find
1 tr σq U σ p U † σq = U σ p U † . d q Therefore Eq. 3.4 becomes
tr Pπ EU U † σ p1 U ⊗ . . . ⊗ U † σ pk U = tr σ p1 ⊗ . . . ⊗ σ pk Pπ . For the second part, consider any vector v which is orthogonal to the permutation operators (we can neglect the complex conjugate because Pπ is real in this basis), i.e. (3.5) tr σq1 ⊗ · · · ⊗ σqk Pπ v(q) = 0 q
for any permutation π . Then ˆ tr σq1 ⊗ · · · ⊗ σqk T (p) v(q) G(p; q)v(q) = d −k q
q
which is zero since T (p) is a linear combination of permutations and v is orthogonal to this by Eq. 3.5. ˆ i.e. q G(p; ˆ ; q) = G(p; ˆ ˆ Theorem 3.3. Gˆ 2 = G, q). q )G(q Proof. Using Eq. 3.3,
ˆ ; q) = ˆ ˆ G(p; q )G(q G(p; q )d −k tr σq1 ⊗ · · · ⊗ σqk T (q) . q
q
From Corollary 3.1, T (q) is a linear combination of permutations. This implies, using Theorem 3.2 that
ˆ G(p; q )d −k tr σq1 ⊗ . . . ⊗ σqk T (q) = d −k tr σ p1 ⊗ · · · ⊗ σ pk T (q) q
ˆ = G(p; q) as required. Corollary 3.2. Gˆ is a projector so has eigenvalues 0 and 1. We now evaluate Gˆ and T for the cases of k = 1 and k = 2 since these are the cases we are interested in for the remainder of the paper.
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3.1. k = 1. The k = 1 case is clear: the random unitary completely randomises the state. Therefore all terms in the expansion are set to zero apart from the identity i.e. σ0 p = 0 (3.6) T ( p) = 0 p = 0. 3.2. k = 2. For k = 2, there are just two permutation operators, identity I and swap F. Therefore there are just two eigenvectors with non-zero eigenvalue (n > 1). In normalised form, taking them to be orthogonal, their components are f 1 (q1 , q2 ) = δq1 0 δq2 0 , 1 δq q (1 − δq1 0 ). f 2 (q1 , q2 ) = 2 d −1 1 2 We will now prove three properties of Gˆ that we need: ˆ p1 , p2 ; q1 , q2 ) = 0 if p1 = p2 or q1 = q2 . 1. G( Proof. Consider the function f (q1 , q2 ) = δq1 a δq2 b with a = b. This function has zero overlap with the eigenvectors f 1 and f 2 so it goes to zero when acted on by ˆ Therefore G( ˆ p1 , p2 ; a, b) = 0. The claim follows from the symmetry property G. (Theorem 3.1). ˆ p; q) ≡ G( ˆ p1 , p2 ; q1 , q2 ). With this we will write G( ˆ 2. G( p; 0) = δ p0 . Proof. Let Gˆ act on eigenvector f 1 . ˆ p; a) = 3. G(
1 d 2 −1
for a, p = 0.
Proof. Let Gˆ act on the input δqa . This has zero overlap with f 1 and overlap with f 2 .
1 d 2 −1
Therefore we have
⎧ ⎪ ⎨0 ˆ G( p1 , p2 ; q1 , q2 ) = 1 ⎪ ⎩
1 d 2 −1
p1 = p2 or q1 = q2 p1 = p2 = q 1 = q 2 = 0 . p1 = p2 = 0, q1 = q2 = 0
ˆ p1 , p2 ; q1 , q2 )σq1 ⊗ σq2 , we have G( ⎧ ⎪ p1 = p2 , ⎨0, p1 = p2 = 0, T ( p1 , p2 ) = σ0 ⊗ σ0 , ⎪ ⎩ 1 σ ⊗ σ p = p = 0. 1 2 p p 2 p =0 d −1
Since T ( p1 , p2 ) =
(3.7)
q1 ,q2
(3.8)
Therefore the terms σ p1 ⊗ σ p2 with p1 = p2 are set to zero. Further, the sum of the diagonal coefficients γ ( p, p) is conserved. This allows us to identify this with a probability distribution (after renormalising) and use Markov chain analysis. To see this, write again the starting state 1 ρ= γ0 (q1 , q2 )σq1 ⊗ σq2 d q ,q 1
2
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with the state after application of any unitary W , ρW =
1 γW (q1 , q2 )σq1 ⊗ σq2 = 2−n γ (q1 , q2 ) W σq1 W † ⊗ W σq2 W † . d q ,q q ,q 1
2
1
2
Then q
γW (q, q) =
1 tr σq ⊗ σq ρW d q
= tr (FρW )
1 γ (q1 , q2 )tr F W σq1 W † ⊗ W σq2 W † = d q ,q 1
2
1 = γ (q1 , q2 )tr σq1 σq2 d q ,q 1 2 = γ (q, q) q
as required, where F is the swap operator and we have used Lemmas A.2 and A.1. 3.3. Moments for general universal random circuits. We now consider universal distributions µ that in general may be different from the uniform (Haar) measure on U (d). Our main result in this section will be to show that a universal distribution on U (4) is also 2-copy gapped. In fact, we will phrase this result in slightly more general terms and show that a universal distribution on U (d) is also k-copy gapped for any k. Universality (Definition 2.8) generalises in the obvious way to U (d), whereas when we say that µ is k-copy gapped, we mean that G µ − G U (d) ∞ < 1,
(3.9)
where G ? = EU U ⊗k ⊗ (U ∗ )⊗k , with the expectation taken over µ for G µ or over the Haar measure for G U (d) . The reason Eq. 3.9 represents our condition for µ to be k-copy gapped is as follows: Observe that Gˆ and G are unitarily related, so the definition of k-copy gapped could ˆ We have shown above that Gˆ U (d) (and thus G U (d) ) equivalently be given in terms of G. has all eigenvalues equal to 0 or 1; i.e. is a projector. By contrast, G µ may not even be Hermitian. However, we will prove below that all eigenvectors of G U (d) with eigenvalue 1 are also eigenvectors of G µ with eigenvalue 1. Thus, Eq. 3.9 will imply that limt→∞ (Gˆ µ )t = Gˆ U (d) , just as we would expect for a gapped random walk. We would like to show that Eq. 3.9 holds whenever µ is universal. This result was proved in [6] (and was probably known even earlier) when µ had the form (δU1 + δU2 )/2. Here we show how to extend the argument to any universal µ. Lemma 3.2. Let µ be a distribution on U (d). Then all eigenvectors of G U (d) with eigenvalue 1 are eigenvectors of G µ with eigenvalue one. Additionally, if µ is universal then µ is k-copy gapped for any positive integer k (cf. Eq. 3.9). In particular, if k = 2 this lemma implies that µ is 2-copy gapped (cf. Theorem 2.1).
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Proof. Let V ∼ = Cd be the fundamental representation of U (d), where the action of U ∈ U (d) is simply U itself. Let V ∗ be its dual representation, where U acts as U ∗ . The operators G µ and G U (d) act on the space V ⊗k ⊗ (V ∗ )⊗k . We will see that G U (d) is completely determined by the decomposition of V ⊗k ⊗ (V ∗ )⊗k into irreducible representations (irreps). Suppose that the multiplicity of (rλ , Vλ ) in V ⊗k ⊗ (V ∗ )⊗k is m λ , where the Vλ ’s are the irrep spaces and rλ (U ) the corresponding representation matrices. In other words V ⊗k ⊗ (V ∗ )⊗k ∼ Vλ ⊗ Cm λ , (3.10) = U
⊗k
∗ ⊗k
⊗ (U )
∼
λ
|λλ| ⊗ rλ (U ) ⊗ Im λ .
(3.11)
λ
Here ∼ indicates that the two sides are related by conjugation by a fixed (U -independent) unitary. Let λ = 0 denote the trivial irrep: i.e. V0 = C and r0 (U ) = 1 for all U . We claim that EU rλ (U ) = 0 whenever λ = 0 and the expectation is taken over the Haar measure. To show this, note that EU rλ (U ) commutes with rλ (V ) for all V ∈ U (d) and thus, by Schur’s Lemma, we must have EU rλ (U ) = cI for some c ∈ C. However, by the translation-invariance of the Haar measure we have cI = EU rλ (U ) = EU rλ (U V ) = c rλ (V ) for all V ∈ U (d). Since λ = 0, we cannot have rλ (V ) = I for all V and so it must be that c = 0. Thus, if we write G U (d) and G µ using the basis on the RHS of Eq. 3.11, we have G U (d) = |00| ⊗ Im 0 ,
(3.12)
where |00| is a projector onto the trivial irrep. On the other hand, |λλ| ⊗ G µ = |00| ⊗ Im 0 + rλ (U )dµ(U ) ⊗ Im λ .
(3.13)
λ =0
Thus, every eigenvector of G U (d) with eigenvalue one is also fixed by G µ . For the remainder of the space, the direct sum structure means that . r G U (d) − G µ ∞ = max (U )dµ(U ) (3.14) λ λ =0 m λ =0
∞
Note that this maximisation only includes λ with dim Vλ > 1. This is because nontrivial one-dimensional irreps of U (d) have the form det U m for some non-zero integer m. Under the map U → eiφ U , such irreps pick up a phase of eimφ . However, U ⊗k ⊗ (U ∗ )⊗k is invariant under U → eiφ U . Thus V ⊗k ⊗ (V ∗ )⊗k cannot contain any non-trivial one-dimensional irreps. Now suppose by contradiction that there exists λ = 0 with m λ = 0 and rλ (U ) dµ(U )∞ = 1. (We do not need to consider the case rλ (U )dµ(U )∞ > 1, since rλ (U )∞ = 1 for all U and · ∞ obeys the triangle inequality.) Indeed, the triangle inequality further implies that there exists a unit vector |v ∈ Vλ such that dµ(U ) rλ (U )|v = ω|v, for some ω ∈ C with |ω| = 1.
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By the above argument we can assume that dim Vλ > 1. Since Vλ is irreducible, it cannot contain a one-dimensional invariant subspace, implying that there exists U0 ∈ U (d) such that |v|rλ (U0 )|v| = 1 − δ, for some δ > 0. Since U → |v|rλ (U )|v| is continuous, there exists an open ball S around U0 such that |v|rλ (U )|v| ≤ 1 − δ/2 for all U ∈ S. Define S¯ := U (d)\S. Now use the fact that µ is universal to find an such that µ (S) > 0. Next, observe we
that dµ (U ) v|rλ (U )|v = ω . Taking the absolute value of both sides yields dµ (U ) v|rλ (U )|v 1 = U (d) ≤ dµ (U ) |v|rλ (U )|v| U (d) = dµ (U ) |v|rλ (U )|v| + dµ (U ) |v|rλ (U )|v| S S¯
δ + 1 − µ (S) ≤ µ (S) 1 − 2 < 1, a contradiction. We conclude that G U (d) − G µ ∞ < 1. 4. Convergence In Sect. 3 we saw that iterating any universal gate set on U (d) eventually converges to the uniform distribution on U (d). Since the set of all two-qubit unitaries is universal on U (2n ), this implies that random circuits eventually converge to the Haar measure. In this section, we turn to proving upper bounds on this convergence rate, focusing on the first two moments. Let Gˆ (i j) be the matrix with Gˆ (with d = 4) acting on qubits i and j and the identity on the others. Then, if the pair (i, j) is chosen at step t, we can find the coefficients at step t + 1 by multiplying by Gˆ (i j) . In general, a random pair is chosen at each step. So 1 γt+1 (p) = (4.1) Gˆ (i j) (p; q)γt (q), n(n − 1) q i = j
where γt+1 are the expected coefficients at step t. We can think of this evolution as repeated application of the matrix 1 P= (4.2) Gˆ (i j) . n(n − 1) i = j
For k = 2, the key idea of Oliveira et al. [26] was to map the evolution of the γ ( p, p) coefficients to a Markov chain. The γ ( p1 , p2 ) coefficients with p1 = p2 just decay as each qubit is chosen and can be analysed directly. However, we can only map the γ ( p, p) coefficients to a probability distribution when they are non-negative, which is not the case for general states. Most of the rest of the paper is dedicated to proving Lemma 2.1, which only applies to states with γ ( p, p) ≥ 0 and normalised so their sum is 1. Corollary 2.1 then extends this to all states:
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Proof (of Corollary 2.1). Lemma 2.1 still applies to the γ ( p1 , p2 ) terms with p1 = p2 . Therefore we just need to show how to apply Lemma 2.1 to states that initially have some negative γ ( p, p) terms. For the γ ( p, p) terms, Lemma 2.1 says that the random walk starting with any initial probability distribution converges to uniform in some bounded time t. Let gt ( p, p; q, q) be the coefficients after t steps of the walk starting at a particular point q (i.e. g0 ( p, p; q, q) = δ p,q ). Now, for any starting state ρ, let the initial coefficients be γ0 ( p, p). Then, by linearity, we can write the expected coefficients after t steps γt ( p, p) := EγW ( p, p) as γt ( p, p) = γ0 (q, q)gt ( p, p; q, q) (4.3) q =0
for p = 0. We can now prove convergence rates for the expected coefficients γt ( p, p): (i) For the 2-norm, we have from Lemma 2.1 that for t ≥ Cn log 1/, 2 1 gt ( p, p; q, q) − n ≤ 4 −1
(4.4)
p =0
for any q. Note that the normalisation for the γ ( p, p) terms with p = 0 has changed from Lemma 2.1 since we are neglecting the γ (0, 0) term here. Now
2 q =0 γ0 (q, q) γt ( p, p) − 4n − 1 p =0 ⎛ ⎞ 2 1 ⎝ ⎠ = γ0 (q, q) gt ( p, p; q, q) − n 4 −1 p =0 q =0 2 1 gt ( p, p; q , q ) − n ≤ γ0 (q, q)2 4 −1 q =0 q =0 p =0 ≤ (4n − 1) γ0 (q, q)2 ≤4 n
q =0
γ0 (q1 , q2 )2
q1 ,q2
= 4 tr ρ 2 ≤ 4n , n
where the first inequality is the Cauchy-Schwarz inequality. Therefore for t ≥ Cn(n + log 4n /), the 2-norm distance from stationarity for the γ ( p, p) terms is at most . Choose C such that C n(n + log 1/) ≥ Cn(n + log 4n /) to obtain the result. (ii) For the 1-norm, Lemma 2.1 says that for t ≥ Cn(n + log 1/), gt (q; p, p) − 1 ≤ . (4.5) n 4 − 1 p =0
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We can then proceed much as for the 2-norm case: q =0 γ0 (q, q) γt ( p, p) − 4n − 1 p =0 1 = γ0 (q, q) gt ( p, p; q, q) − n 4 − 1 p =0 q =0 1 ≤ |γ0 (q, q)| gt ( p, p; q, q) − 4n − 1 q =0 p =0 ≤ |γ0 (q, q)| q =0 n
≤ 2 .
The last inequality follows from |σq ⊗ σq | = σ0 ⊗ σ0 . Therefore for t ≥ Cn(n + log 2n /), the 1-norm distance from stationarity for the γ ( p, p) terms is at most . We now proceed to prove Lemma 2.1. Firstly, we will consider the simple case of k = 1 to prove this process forms a 1-design as this will help us to understand the more complicated case of k = 2. 4.1. First moments convergence. Recall that ρ = 2−n/2 p γ ( p)σ p and we wish to evaluate the moments of the coefficients. So for the first moments to converge, we want to know Eγ ( p). For k = 1, the U (4) random circuit uniformly randomises each pair that is chosen. More precisely, a pair of sites i, j are chosen at random and all the coefficients with pi = 0 or p j = 0 are set to zero. Thus we get an exact 1-design when all sites have been hit. For other gate sets, the terms do not decay to zero but decay by a factor depending ˆ Call the gap ; for U (4) = 1 and for others 0 < ≤ 1 and is on the gap of G. independent of n. Therefore once each site has been hit m times the terms have decayed by a factor (1 − )m . For a bound like the mixing time (see Sect. 4.3 for definition), we want to bound the quantity p =0 |EW γW ( p)|, where γW ( p) is the Pauli coefficient after applying the random circuit W . We also want 2-norm bounds, so we bound p =0 (EW γW ( p))2 too. We will in fact find bounds on p =0 EW |γW ( p)| and p =0 (EW |γW ( p)|)2 , which are stronger. A standard problem in the theory of randomised algorithms is the ‘coupon collector’ problem. If a magazine comes with a free coupon, which is chosen uniformly randomly from n different types, how many magazines should you buy to have a high probability of getting all n coupons? It is not hard to show that n ln n samples (magazines) have at least a 1 − probability of including all n coupons. Using this, we expect all sites to be hit with probability at least 1 − after (n log n ) steps. This argument can be made precise in this context by bounding the non-identity coefficients. We find, as expected, that the sum is small after O(n log n) steps:
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Lemma 4.1. After O(n log 1/) steps
(EW |γW ( p)|)2 ≤ ,
p =0
and after O(n log n ) steps,
EW |γW ( p)| ≤ .
(4.6)
p =0
Proof. At each step, a pair of sites is chosen at random and any terms with non-identity ⊗(n−1) coefficients for this pair decay by a factor (1 − ). For example, the term σ1 ⊗ σ0 decays whenever the first site is chosen. Thus the probability of each term decaying depends on the number of zeroes. We start with the 1-norm bound. Suppose the circuit applied after t steps is Wt . Consider EWt |γWt ( p)| for any p with d non-zeroes. Since the state ρ is physical, tr ρ 2 ≤ 1, so p γ02 ( p) ≤ 1. Now, in each step, if any site is chosen where p is non-zero, this term decays by a factor (1 − ). This occurs with probability 1 − (d−n)(d−n−1) ≥ d/n, the probability of choosing a pair n(n−1) where at least one site is non-zero. Therefore E|γWt ( p)| ≤ ((1 − )d/n + (1 − d/n)) |γWt−1 ( p)|, where the expectation is over the circuit applied at step t. If we iterate this t times we find EW |γW ( p)| ≤ exp(−td/n)|γ0 ( p)|, where the expectation here is over all random circuits for the t steps. We now sum over all p:
EW |γW ( p)| ≤
p =0
n d=1
exp(−td/n)
|γ0 ( p)|,
d( p)=d
where d( p) is the number of non-zeroes in p. For the 1-norm bound, we can simply bound |γ0 ( p)| ≤ 1 to give d( p)=d |γ0 ( p)| ≤ dn 3d so
EW |γW ( p)| ≤ (1 + 3 exp(−t/n))n − 1,
p =0
where we have used the binomial theorem. Now let t = p =0
n
ln
3n .
EW |γW ( p)| ≤ (1 + /n)n − 1 = O().
This gives
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For the 2-norm bound,
(EW |γW ( p)|)2 ≤
p =0
exp(−2td/n)γ02 ( p)
p =0
=
n
exp(−2td/n)
d=1
≤
n
γ02 ( p)
d( p)=d
exp(−2td/n)
d=1
≤ where we have used
2 p γ0 ( p)
exp(−2t/n) , 1 − exp(−2t/n)
≤ 1. We find after
(EW |γW ( p)|)2 ≤
p =0
n 2
ln 1/ steps that
1−
4.2. Second moments convergence. Firstly, the σ p1 ⊗ σ p2 terms for p1 = p2 decay in a similar way to the non-identity terms in the 1-design analysis. In fact, the proof of Lemma 4.1 carries over almost identically to this case to give Lemma 4.2. After O(n log 1/) steps
(EW |γW ( p1 , p2 )|)2 ≤
p1 = p2
and after O(n(n + log 1/)) steps
EW |γW ( p1 , p2 )| ≤ .
p1 = p2
Proof. Instead of the number of zeroes governing the decay rate, we need to count the number of places where p1 and p2 differ. This gives E|γWt ( p1 , p2 )| ≤ ((1 − )d/n + (1 − d/n)) |γWt−1 ( p1 , p2 )|, where now d is the number of differing sites. There are d places so we find
n d n−d states that differ in d 12 4
EW |γW ( p1 , p2 )| ≤ 4n [(1 + 3 exp(−t/n))n − 1].
p1 = p2 n (n ln 4 + ln 1/) to make this O(). The 2-norm bound follows in the same Set t = way as for Lemma 4.1.
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We now need to prove the γ ( p, p) terms converge quickly. We have seen above that the sum of the terms γ ( p, p) is conserved and, for the purposes of proving Lemma 2.1, we assume the sum is 1 and γ ( p, p) ≥ 0 for all p. To illustrate the evolution, consider the simplest case when the gates are chosen from U (4). We have evaluated Gˆ in Sect. 3.2 for k = 2 for this case. Translated into coefficients this yields the following update rule, where we have written it for the case when qubits 1 and 2 are chosen: γt+1 (r1 , r2 , r3 , . . . , rn , s1 , s2 , s3 , . . . , sn ) ⎧ 0 (r1 , r2 ) = (s1 , s2 ) ⎪ ⎪ ⎨ 0, r , . . . , rn , 0, 0, s3 , . . . , sn ) (r1 , r2 ) = (s1 , s2 ) = (0, 0) = γ1t (0, 3 , r , r , . . . , r , r , r , s , . . . , s ) (r , r ) = (s , s ) = (0, 0). ⎪ γ (r ⎪ t 1 2 3 n 1 2 3 n 1 2 1 2 r1 ,r2 ⎩ 15 r1 r2 =0
(4.7) The key idea of Oliveira et al. [26] was to map the evolution of the γ ( p, p) coefficients to a Markov chain. We can apply this here to get, on state space {0, 1, 2, 3}n , the evolution: 1. Choose a pair of sites uniformly at random. 2. If the state is 00 it remains 00. 3. Otherwise, choose the state uniformly at random from {0, 1, 2, 3}2 \{00}. This is the correct evolution since, if the initial state is distributed according to γt (q, q), the final state is distributed according to γt+1 ( p, p). The evolution for other gate sets will be similar, but the states will not be chosen uniformly randomly in the third step. However, the state 00 will remain 00 and the stationary distribution on the other 15 states is the same. We will find the convergence times for general gate sets and then consider the U (4) gate set since we can perform a tight analysis for this case.
4.3. Markov chain analysis. Before finding the convergence rate for our problem, we will briefly introduce the basics of Markov chain mixing time analysis. All of these standard results can be found in Ref. [25] and references therein. A process is Markov if the evolution only depends on the current state rather than the full state history. Therefore the evolution of the state can be thought of as a matrix, the transition matrix, acting on a vector which represents the current distribution. We will only be interested in discrete time processes so the state after t steps is given by the t th power of the transition matrix acting on the initial distribution. We say a Markov chain is irreducible if it is possible to get from one state to any other state in some number of steps. Further, a chain is aperiodic if it does not return to a state at regular intervals. If a chain is both irreducible and aperiodic then it is said to be ergodic. A well known result of Markov chain theory is that all ergodic chains converge to a unique stationary distribution. In matrix language this says that the transition matrix P has eigenvalue 1 with no multiplicity and all other eigenvalues have absolute value strictly less than 1. We will also need the notion of reversibility. A Markov chain is reversible if the time reversed chain has the same transition matrix, with respect to some distribution. This condition is also known as detailed balance: π(x)P(x, y) = π(y)P(y, x).
(4.8)
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It can be shown that a reversible ergodic Markov chain is only reversible with respect to the stationary distribution. So above π(x) is the stationary distribution of P. An immediate consequence of this is that for a chain with uniform stationary distribution, it is reversible if and only if it is symmetric (i.e. P(x, y) = P(y, x)). Note also that reversible chains have real eigenvalues, since they are similar to the symmetric matrix π(x) π(y) P(x,
y). With these definitions and concepts, we can now ask how quickly the Markov chain converges to the stationary distribution. This is normally defined in terms of the 1-norm mixing time. We use (half the) 1-norm distance to measure distances between distributions: 1 1 ||s − t|| = ||s − t||1 = |si − ti |. (4.9) 2 2 i
We assume all distributions are normalised so then 0 ≤ ||s − t|| ≤ 1. We can now define the mixing time: Definition 4.1. Let π be the stationary distribution of P. Then if P is ergodic the mixing time τ is (4.10) τ () = max min{t ≥ 0 : P t s − π ≤ }. s
t
We will also use the (weaker) 2-norm mixing time (note this is not the same as τ2 in Ref. [25]): Definition 4.2. Let π be the stationary distribution of P. Then if P is ergodic the 2-norm mixing time τ2 is (4.11) τ2 () = max min{t ≥ 0 : P t s − π 2 ≤ }. s
t
Unless otherwise stated, when we say mixing time we are referring to the 1-norm mixing time. There are many techniques for bounding the mixing time, including finding the second largest eigenvalue of P. This gives a good measure of the mixing time because components parallel to the second largest eigenvector decay the slowest. We have (for reversible ergodic chains) Theorem 4.1 (see Ref. [25], Corollary 1.15). τ () ≤
1 1 ln , π∗
where π∗ = min π(x) and = min(1 − λ2 , 1 + λmin ), where λ2 is the second largest eigenvalue and λmin is the smallest. is known as the gap. If the chain is irreversible, it may not even have real eigenvalues. However, we can bound the mixing time in terms of the eigenvalues of the reversible matrix PP∗ , where P ∗ (x, y) = π(y) π(x) P(y, x). In this case we have ([25], Corollary 1.14) τ () ≤
2 1 , ln PP∗ π∗
(4.12)
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277
where now PP∗ is the gap of the chain PP∗ . Note that for a reversible chain P = P ∗ and PP∗ ≈ 2, so the bounds are approximately the same. This can also be converted into a 2-norm mixing time bound: τ2 () ≤
2 ln 1/. PP∗
(4.13)
To bound the gap, we will use the comparison theorem in Theorem 4.2 below. In this theorem, we are thinking of the Markov chain as a directed graph where the vertices are the states and there are edges for allowed transitions (i.e. transitions with non-zero probability). For irreducible chains, it is possible to make a path from any vertex to any other; we call the path length the number of transitions in such a path (which will in general depend on the choice of path). Theorem 4.2 (see Ref. [25], Theorem 2.14). Let P and Pˆ be two Markov chains on the same state space with the same stationary distribution π . Then, for every x = y ∈ ˆ with P(x, y) > 0 define a directed path γx y from x to y along edges in P and let its length be |γx y |. Let be the set of all such paths. Then ˆ ≥ /A ˆ where for the gaps and A = A() =
1 a =b,P(a,b) =0 π(a)P(a, b)
max
ˆ π(x) P(x, y)|γx y |.
x = y:(a,b)∈γx y
For example, when comparing 1-dimensional random walks there is no choice in the paths; they must pass through every point between x and y. Further, the walk can only progress one step at a time so (without loss of generality, for reversible chains) let b = a + 1 to give A = max a
= max a
1 ˆ π(x) P(x, y)(y − x) π(a)P(a, a + 1) x≤a y≥a+1
ˆ P(a, a + 1) . P(a, a + 1)
(4.14)
A generalisation of the comparison theorem involves constructing flows, which are weighted sets of paths between states. This can give a tighter bound since bottlenecks are averaged over. This gives a modified comparison theorem: Theorem 4.3 ([12], Theorem 2.3). Let P and Pˆ be two Markov chains on the same state space with the same stationary distribution π . Then, for every x = y ∈ with ˆ P(x, y) > 0, construct a set of directed paths Px y from x to y along edges in P. We define the flow function f which maps each path γx y ∈ Px y to a real number in the interval [0, 1] such that ˆ f (γx y ) = P(x, y). γx y ∈Px y
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A. W. Harrow, R. A. Low
Again, let the length of each path be |γx y |. Then ˆ ≥ /A ˆ where for the gaps and
1 a = b,P(a,b) = 0 π(a)P(a, b)
A = A( f ) =
max
π(x) f (γx y )|γx y |.
(4.15)
x = y,γx y ∈ Px y :(a,b) ∈ γx y
Note that we recover the comparison theorem when there is just one path between each x and y. 4.3.1. log-Sobolev constant We will need tighter, but more complicated, mixing time results to prove the tight result for the U (4) case. We use the log-Sobolev constant: Definition 4.3. The log-Sobolev constant ρ of a chain with transition matrix P and stationary distribution π is 2 x = y ( f (x) − f (y)) P(x, y)π(y) ρ = min . 2 f 2 f (x) 2 x π(x) f (x) log π(y) f (y) y
The mixing time result is: Lemma 4.3 (see Ref. [13], Theorem 3.7’). The mixing time of a finite, reversible, irreducible Markov chain is 1 d 1 1 log log , (4.16) τ () = O + log ρ π∗ where ρ is the Sobolev constant, π∗ is the smallest value of the stationary distribution, is the gap and d is the size of the state space. Further, the comparison theorem (Theorem 4.2) works just the same to give ρ ≥ ρ/A. ˆ We will need one more result, due to Diaconis and Saloff-Coste: Lemma 4.4 ([13], Lemma 3.2). Let Pi , i = 1, . . . , d, be Markov chains with gaps i and Sobolev constants ρi . Now construct the product chain P. This chain has state space equal to the product of the spaces for the chains Pi and at each step one of the chains is chosen at random and run for one step. Then P has spectral gap given by: =
1 min i d i
ρ=
1 min ρi . d i
and Sobolev constant:
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4.4. Convergence proof. We now prove the Markov chain convergence results to show that the γ ( p, p) terms converge quickly. We have already shown that the γ ( p1 , p2 ) terms with p1 = p2 converge quickly and that there is no mixing between these terms ˆ and the γ ( p, p) terms. Therefore, in this section, we remove such terms from G. We want to prove the Markov chain with transition matrix (Eq. 4.2) P=
1 Gˆ (i j) n(n − 1) i = j
converges quickly. Firstly, we know from Sect. 3.3 that P has two eigenvectors with eigenvalue 1. The first is the identity state (σ0 ⊗ σ0 ) and the second is the uniform sum of all non-identity terms ( 4n1−1 p =0 σ p ⊗ σ p ). From now on, we remove the identity state. This makes the chain irreducible. Since we know it converges, it must be aperiodic also so the chain is ergodic and all other eigenvalues are strictly between 1 and −1. We show here that the gap of this chain, up to constants, does not depend on the choice of 2-copy gapped gate set. In the second half of the paper we find a tight bound on the gap for the U (4) case which consequently gives a tight bound on the gap for all universal sets. Since the stationary distribution is uniform, the chain is reversible if and only if P is a symmetric matrix. A sufficient condition for P to be symmetric is for Gˆ (i j) to be symmetric. We saw in Theorem 3.1 that for the U (4) gate set case Gˆ (i j) is symmetric. In fact, the proof works identically to show that Gˆ (i j) is symmetric for any gate set, provided the set is invariant under Hermitian conjugation. However, 2-copy gapped gate sets do not necessarily have this property so the Markov chain is not necessarily reversible. We will find equal bounds (up to constants) for the gaps of both P (if Gˆ is symmetric) and PP∗ (if Gˆ is not symmetric) below: Theorem 4.4. Let µ be any 2-copy gapped distribution of gates. If µ is invariant under Hermitian conjugation then let P be the eigenvalue gap of the resulting Markov chain matrix P. Then P = (U (4) ),
(4.17)
where U (4) is the eigenvalue gap of the U (4) chain. If µ is not invariant under Hermitian conjugation, then let PP∗ be the eigenvalue gap of the resulting Markov chain matrix PP∗ . Then PP∗ = (U (4) ).
(4.18)
Proof. We will use the comparison method with flows (Theorem 4.3). Firstly consider the case where µ is closed under Hermitian conjugation, i.e. Gˆ is symmetric. We will compare P to the U (4) chain, which we call PU (4) . Recall that this chain chooses a pair at random and does nothing if the pair is 00 and chooses a random state from {0, 1, 2, 3}2 \{00} otherwise. To apply Theorem 4.3, we need to construct the flows between transitions in PU (4) . We will choose paths such that only one pair is modified throughout. For example (with n = 4), the transition 1000 → 2000 is allowed in PU (4) . To construct a path in P, we need to find allowed transitions between these two paths in P. Gˆ may not include the transition 10 → 20 directly, however, Gˆ is irreducible on this subspace of just two pairs. This means that a path exists and can be of maximum length 14 if it has to cycle through
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all intermediate states (in fact, since Gˆ is symmetric the maximum path length is 8; all that is important here is that it is constant). For example, the transitions 10 → 11 → 20 might be allowed. Then we could choose the full path to be 1000 → 1100 → 2000. In this case we have chosen the path to involve transition pairing sites 1 and 2. However, we could equally well have chosen any pairing; we could pair the first site with any of the others. We can choose 3 paths in this way. For this example, the flow we want to choose will be all 3 of these paths equally weighted. We now use this idea to construct flows between all transitions in PU (4) to prove the result. Let x = y ∈ and let d(x, y) be the Hamming distance between the states (d(x, y) gives the number of places at which x and y differ). There are two cases where PU (4) (x, y) = 0: 1. d(x, y) = 2. Here we must choose a unique pairing, specified by the two sites that differ. Make all transitions in P using this pair giving just one path. 2. d(x, y) = 1. For this case, choose all possible pairings of the changing site that give allowed transitions in PU (4) . For each pairing, construct a path in P modifying only this pair. If the differing site is initially non-zero then there are n − 1 such pairings; if the differing site is initially zero then there are n − z(x) pairings where z(x) is the number of zeroes in the state x. All the above paths are of constant length since we have to (at most) cycle through all states of a pair. We must now choose the weighting f (γx y ) for each path such that f (γx y ) = PU (4) (x, y), (4.19) Px y
where Px y is the set of all paths from x to y constructed above. We choose the weighting of each path to be uniform. We just need to calculate the number of paths in Px y to find f: 1. d(x, y) = 2. There is just one path so f (γx y ) = PU (4) (x, y) = (1/n 2 ). 2. d(x, y) = 1. If the differing site is initially non-zero then PU (4) (x, y) = (1/n) P (x,y) = (1/n 2 ). If the differing site and there are n − 1 paths so f (γx y ) = U (4) n−1 is initially zero then PU (4) (x, y) = n−z(x) and there are n − z(x) paths so n2 f (γx y ) =
PU (4) (x,y) n−z(x)
= (1/n 2 ).
So for all paths, f = (1/n 2 ). We now just need to know how many times each edge (a, b) in P is used to calculate A: A=
max
a =b,P(a,b) =0
A(a, b),
(4.20)
where A(a, b) =
1 P(a, b)
f (γx y ).
(4.21)
x = y,γx y ∈Px y :(a,b)∈γx y
We have cancelled the factors of π(x) because the stationary distribution is uniform. We have also ignored the lengths of the paths since they are all constant. To evaluate A(a, b), we need to know how many paths pass through each edge (a, b). We again consider the two possibilities separately:
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1. d(a, b) = 2. Suppose a and b differ at sites i and j. Firstly, we need to count how many transitions from x to y in PU (4) could use this edge, and then how many paths for each transition actually use the edge. To find which x and y could use the edge, note that x and y must differ at sites i, j or both. Furthermore, the values at the sites other than i and j must be the same as for a (and therefore b). There is a constant number of x, y pairs that satisfy this condition. Now, for each x, y pair satisfying this, paths that use this edge must use the pairing i, j for all transitions. Since in the paths we have chosen above there is a unique path from x to y for each pairing, there is at most one path for each x, y pair that uses edge a, b. For d(a, b) = 2, P(a, b) = (1/n 2 ) so A(a, b) is a constant for this case. 2. d(a, b) = 1. Let there be r pairings that give allowed transitions in P between a and b. As above, each pairing gives a constant number of paths. So the numerator is (r/n 2 ). Further, P(a, b) = (r/n 2 ). So again A(a, b) is constant. Combining, A is a constant so the result is proven for the case Gˆ is symmetric. We now turn to the irreversible case. We now need to bound the gap of PP∗ = PP T . This chain selects two (possibly overlapping) pairs at random and applies Gˆ to one of them and Gˆ T to the other. We can use the above exactly by choosing Gˆ to perform the transitions above and Gˆ T to just loop the states back to themselves. By aperiodicity (the greatest common divisor of loop lengths is 1), we can always find constant length paths that do this. Now we need to know the gap of the U (4) chain. We can, by a simple application of the comparison theorem, show it is (1/n 2 ). However, in the second half of this paper we show it is (1/n). This gives us (using Theorem 4.1): Corollary 4.1. The Markov chain P has mixing time O(n(n + log 1/)) and 2-norm mixing time O(n log 1/). We conjecture that the mixing time (as well as Lemma 4.2) can be tightened to (n log n ), which is asymptotically the same as for the U (4) case: Conjecture 4.1. The second moments for the case of general 2-copy gapped distributions have 1-norm mixing time (n log n ). It seems likely that an extension of our techniques in Sect. 5 could be used to prove this. Combining the convergence results we have proved our general result Lemma 2.1: Proof (of Lemma 2.1). Combining Corollary 4.1 (for the γ ( p, p) terms) and Lemma 4.2 (for the γ ( p1 , p2 ), p1 = p2 terms) proves the result. We have now shown that the first and second moments of random circuits converge quickly. For the remainder of the paper we prove the tight bound for the gap and mixing time of the U (4) case and show how mixing time bounds relate to the closeness of the 2-design to an exact design. Only for the U (4) case is the matrix Gˆ a projector so in this sense the U (4) random circuit is the most fundamental. While we expect the above mixing time bound is not tight, we can prove a tight mixing time result for the U (4) case. However, using our definition of an approximate k-design, the gap rather than the mixing time governs the degree of approximation.
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5. Tight Analysis for the U(4) Case We have already found tight bounds for the first moments in Lemma 4.1: just set = 1.
5.1. Second moments convergence. We need to prove a result analogous to Lemma 4.2 for the terms σ p1 ⊗ σ p2 , where p1 = p2 . We already have a tight bound for the 2-norm decay, by setting = 1 into Lemma 4.2. We tighten the 1-norm bound: Lemma 5.1. After O(n log n ) steps
EW |γW ( p1 , p2 )| ≤ .
(5.1)
p1 = p2
Proof. We will split the random circuits up into classes depending on how many qubits have been hit. Let H be the random variable giving the number of different qubits that have been hit. We can work out the distribution of H and bound the sum of |γW ( p1 , p2 )| for each outcome. Firstly we have, after t steps, P(H ≤ h) ≤
n h(h − 1) t n ≤ (h/n)t . n(n − 1) h h
Now, for each qubit hit, each coefficient which has p1 and p2 differing in this place is (n−h) terms in the sum in set to zero. So after h have been hit, there are only (at most) 16 2 Eq. 5.1. As before, the state is a physical state, tr ρ ≤ 1 so p1 p2 γ 2 ( p1 , p2 ) ≤ 1 so √ N if there are at most N non-zero terms in the sum. Therefore p1 p2 |γ ( p1 , p2 )| ≤ we have, after t steps,
EW |γW ( p1 , p2 )| ≤
p1 = p2
n−1
P(H = h)16(n−h)/2
h=1
≤
n−1
P(H ≤ h)4(n−h)
h=1
≤
n−1 n (h/n)t 4(n−h) h h=1
n−1 n (1 − h/n)t 4h = h h=1
≤
n−1 n h=1
h
exp(−ht/n)4h .
h →n−h
Random Quantum Circuits are Approximate 2-designs
Now, let t = n ln n :
EW |γW ( p1 , p2 )| ≤
p1 = p2
n−1 n 4 h h=1
h
n
283
n 4 n 4 = 1+ −1− = O(), n n
where the last line follows from the binomial theorem.
This, combined with the mixing time result we prove below, completes the proof that the second moments of the random circuit converge in time O(n log n ). 5.2. Markov chain of coefficients. The Markov chain acting on the coefficients is reducible because the state {0}n is isolated. However, if we remove it then the chain becomes irreducible. The presence of self loops implies aperiodicity, therefore the chain is ergodic. We have already seen that the chain converges to the Haar uniform distribution (in Sect. 1.1), therefore the stationary state is the uniform state π(x) = 1/(4n − 1). Further, since the chain is symmetric and has uniform stationary distribution, the chain satisfies detailed balance (Eq. 4.8) so is reversible. We now turn to obtaining bounds on the mixing time of this chain. We want to show that the full chain converges to stationarity in time (n log n ). This implies (see later) that the gap is (1/n). To prove this, we will construct another chain called the zero chain. This is the chain that counts the number of zeroes in the state. Since it is the zeroes that slow down the mixing, this chain will accurately describe the mixing time of the full chain. Lemma 5.2. The zero chain has transition matrix P on state space (we count non-zero positions) = {1, 2, . . . , n}. ⎧ 1 − 2x(3n−2x−1) y=x ⎪ 5n(n−1) ⎪ ⎪ ⎨ 2x(x−1) y = x −1 5n(n−1) P(x, y) = 6x(n−x) (5.2) ⎪ y = x +1 ⎪ 5n(n−1) ⎪ ⎩ 0 otherwise for 1 ≤ x, y ≤ n. Proof. Suppose there are n − x zeroes (so there are x non-zeroes). Then the only way the number of zeroes can decrease (i.e. for x to increase) is if a non-zero item is paired with a zero item and one of the 9 (out of 15) new states is chosen with no zeroes. The 9 2x(n−x) probability of choosing such a pair is 2x(n−x) n(n−1) so the overall probability is 15 n(n−1) . The number of zeroes can increase only if a pair of non-zero items is chosen and one 6 x(x−1) of the 6 states is chosen with one zero. The probability of this occurring is 15 n(n−1) . The probability of the number of zeroes remaining unchanged is simply calculated by requiring the probabilities to sum to 1. We see that the zero chain is a one-dimensional random walk on the line. It is a lazy random walk because the probability of moving at each step is < 1. However, as the number of zeroes decreases, the probability of moving increases monotonically: 1 − P(x, x) =
2x(3n − 2x − 1) ≥ 2x/5n < 1. 5n(n − 1)
(5.3)
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Lemma 5.3. The stationary distribution of the zero chain is 3x nx . π0 (x) = n 4 −1
(5.4)
Proof. This can be proven by multiplying the transition matrix in Lemma 5.2 by the state Eq. 5.4. Alternatively, it can be proven by counting the number of states with n − x zeroes. There are nx ways of choosing which sites to make non-zero and each non-zero site can be one of three possibilities: 1, 2 or 3. The total number of states is 4n − 1, which gives the result. Below we will prove the following theorem: Theorem 5.1. The zero chain mixes in time (n log n ). The 2-norm mixing time follows easily: Theorem 5.2. The zero chain has 2-norm mixing time O(n log 1/). Proof. We use a lower bound on the 1-norm mixing time to show that the gap of the zero chain is (1/n) and then use the 2-norm mixing bound Eq. 4.13. In [25], Theorem 4.9, they prove the lower bound: τ1 () ≥
1 1− ln , 2
where is the eigenvalue gap. In Theorem 5.1, we showed τ1 () ≤ Cn ln constant C. Combining, 1 n 1− ln ≤ Cn ln 2
(5.5) n
for some
(5.6)
for all > 0. Divide by ln 1/ and take the limit → 0 to find 1− ≤ Cn
(5.7)
which implies the gap is (1/n). The 2-norm bound now follows from Eq. 4.13. Before proving Theorem 5.1, we will show how the mixing time of the full chain follows from this. Corollary 5.1. The full chain mixes in time (n log n ). Proof. Once the zero chain has approximately mixed, the distribution of zeroes is almost correct. We need to prove that the distribution of non-zeroes is correct after O(n log n ) steps too. Once each site of the full chain has been hit, meaning it is chosen and paired with another site so not both equal zero, the chain has mixed. This is because, after each site has been hit, the probability distribution over the states is uniform. When the zero chain has approximately mixed, a constant fraction of sites are zero so the probability of hitting a site at each step is (1/n). By the coupon collector argument, each site will have been hit with probability at least 1 − in time O(n log n ). Once the zero chain has mixed to , we can run the full chain this extra number of steps to ensure each site has
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285
been hit with high probability. Since the mixing of the zero chain only increases with time, the distance to stationarity of the full chain is now 1 − − . We make this formal below. After t0 = O(n log n ) steps, the number of zeroes is -close to the stationary distribution π0 by Theorem 5.1 and only gets closer with more steps since the distance to stationarity decreases monotonically. The stationary distribution Eq. 5.4 is approximately a Gaussian peaked at 3n/4 with O(n) variance. This means that, with high probability, the number of non-zeroes is close to 3n/4. We will in fact only need that there is at least a constant fraction of non-zeroes; with probability at least 1 − − exp(−(n)) there will be at least n/2. To prove the mixing time, we run the chain for time t0 so the zero chain mixes to . Then run for t1 additional steps. Let Hi,t be the event that site i is hit at step t. Let n H . We want to show P(H ) is close to 1, or, in other 0 +t1 Hi = ∪tt=t H and H = ∩i=1 i 0 +1 i,t words, that all sites are hit with high probability. Further let X t be the random variable giving the number of non-zeroes at step t. If at step t − 1 site i is non-zero then the event Hi,t occurs if the qubit is chosen, which occurs with probability 2/n. If, however, it was zero then it must be paired with a non-zero thing for Hi,t to hold. Conditioned on any history with X t−1 ≥ n/2, this probability is ≥ 1/n. In particular, we can condition on not having previously hit i and the bound does not change. Combining we have ⎛ ⎞⎞ ⎛ t−1 c c ⎠⎠ ⎝ P ⎝ Hi,t Hi,t ≤ 1 − 1/n. X t−1 ≥ n/2 t =t0 +1
Then, after t1 extra steps,
t +t −1 1 0 c P Hi [X t ≥ n/2] ≤ (1 − 1/n)t1 , t=t0
which, using the union bound, gives
t +t −1 1 0 c P H [X t ≥ n/2] ≤ n(1 − 1/n)t1 . t=t0
Now, since the zero chain has mixed to ,
t +t −1 n−1 0 1 π0 (x) + ≤ t1 exp(−O(n)) + , P [X t ≥ n/2] ≤ t1 t=t0
x=n/2
so P(H c ) ≤ n(1 − 1/n)t1 + t1 exp(−O(n)) + . c Now, choose t1 = n ln 2n so that P(H ) ≤ δ, where δ = + t1 exp(−O(n)). Choose = 1/n so that δ is 1/ poly(n). Now, using the bound on P(H c ), we can write the state v after t1 = O(n log n) steps as
v = (1 − δ)π + δπ ,
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where π is the stationary distribution and π is any other distribution. Using this, ||v − π || ≤ δ. We now apply Lemma A.14 to show that after O(n log n ) steps the distance to stationarity of the full chain is .
5.3. Proof of Theorem 5.1. We will now proceed to prove Theorem 5.1. We present an outline of the proof here; the details are in Sect. A.2. Firstly, note that by the coupon collector argument, the lower bound on the time is (n log n). We need to prove an upper bound equal to this. Intuition says that the mixing time should take time O(n log n) because the walk has to move a distance (n) and the waiting time at each step is proportional to n, n/2, n/3, . . . which sums to O(n log n), provided each site is not hit too often. We will show that this intuition is correct using the Chernoff bound and log-Sobolev (see later) arguments. We will first work out concentration results of the position after some number of accelerated steps. The zero chain has some probability of staying still at each step. The accelerated chain is the zero chain conditioned on moving at each step. We define the accelerated chain by its transition matrix: Definition 5.1. The transition matrix for the accelerated chain is
Pa (x, y) =
⎧ ⎪ 0 ⎪ ⎪ ⎨
x−1 3n−2x−1 3(n−x) ⎪ ⎪ ⎪ 3n−2x−1
⎩
0
y=x y = x −1 . y = x +1 otherwise
(5.8)
We use the accelerated chain in the proof to firstly prove the accelerated chain mixes quickly, then to bound the waiting time at each step to obtain a mixing time bound for the zero chain. To prove the mixing time bound, we will split the walk up into three phases. We will split the state space into three (slightly overlapping) parts and the phase can begin at any point within that space. So each phase has a state space i ⊂ [1, n], an entry space E i ⊂ i and an exit condition Ti . We say that a phase completes successfully if the exit condition is satisfied in time O(n log n) for an initial state within the entry space. When the exit condition is satisfied, the walk moves onto the next phase. The phases are: 1. 1 = [1, n δ ] for some constant δ with 0 < δ < 1/2. E 1 = 1 (i.e. it can start anywhere) and T1 is satisfied when the walk reaches n δ . For this part, the probability of moving backwards (gaining zeroes) is O(n δ−1 ) so the walk progresses forwards at each step with high probability. This is proven in Lemma A.8. We show that the waiting time is O(n log n) in Lemma A.9. 2. 2 = [n δ /2, θ n] for some constant θ with 0 < θ < 3/4. E 2 = [n δ , θ n] and T2 is satisfied when the walk reaches θ n. Here the walk can move both ways with constant probability but there is a (1) forward bias. Here we use a monotonicity argument: the probability of moving forward at each step is
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287
3(n − x) 3n − 2x − 1 3(n − x) ≥ 3n − 2x 3(1 − θ ) . ≥ 3 − 2θ
p(x) =
If we model this random walk as a walk with constant bias equal to 3(1−θ) 3−2θ we will find an upper bound on the mixing time since mixing time increases monotonically with decreasing bias. Further, the waiting time at x = a stochastically dominates the waiting time at x = b for b ≥ a. The true bias decreases with position so the walk with constant bias spends more time at the early steps. Thus the position of this simplified walk is stochastically dominated by the position of the real walk while the waiting time stochastically dominates the waiting time of the real walk. 3. 3 = [ θ2 n, n] and E 3 = [θ n, n]. T3 is satisfied when this restricted part of the chain has mixed to distance . Here the bias decreases to zero as the walk approaches 3n/4 but the moving probability is a constant. We show that this walk mixes quickly by bounding the log-Sobolev constant of the chain. Showing these three phases complete successfully will give a mixing time bound for the whole chain. We now prove in the Appendix that the phases complete successfully with probability at least 1 − 1/ poly(n): Lemma 5.4. P(Phase 1 completes successfully) ≥ 1 − n 2δ−1 − 2n −δ . Lemma 5.5. 3 2 4 2µ P(Phase 2 completes successfully) ≥ 1 − exp − µθ n − 3 θn
−µn δ 2 exp 4 δ − (q/ p)n /2 , − 1 − exp(−µ/2) where µ =
6(1−θ) 3−2θ
− 1.
Lemma 5.6. P(Phase 3 completes successfully) ≥ 1 −
θ 3(2 − θ )
θn/2
.
We can now finally combine to prove our result: Proof (of Theorem 5.1). The stationary distribution has exponentially small weight in the tail with lots of zeroes. We show that, provided the number of zeroes is within phase 3, the walk mixes in time O(n log n ). We also show that if the number of zeroes is initially within phase 1 or 2, after O(n log n) steps the walk is in phase 3 with high probability. We can work out the distance to the stationary distribution as follows. Let p f be the probability of failure. This is the sum of the error probabilities in Lemmas 5.4, 5.5 and 5.6. The key point is that p f = 1/ poly(n). Then after O(n log n )
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steps (the sum of the number of steps in the 3 phases), the state is equal to (1 − p f )v3 + p f v , where v3 is the state in the phase 3 space and v is any other distribution, which occurs if any one of the phases fails. Since the distance to stationarity in phase 3 is , ||v3 − π3 || ≤ , where π3 is the stationary distribution on the state space of phase 3. In θn/2−1 Lemma A.12 we show that π3 (x) = π(x)/(1 − w), where w = x=1 π(x). Since π(x) is exponentially small in this range, w is exponentially small in n. Now use the triangle inequality to find ||v3 − π || ≤ ||v3 − π3 || + ||π3 − π ||.
(5.9)
Since the chain in phase 3 has mixed to , the first term is ≤ . We can evaluate ||π3 −π ||: 1 ||π3 (x) − π(x)|| 2 x=1 ⎞ ⎛ θn/2−1 n 1⎝ = π(x) + (π(x)/(1 − w) − π(x))⎠ 2 n
||π3 − π || =
x=1
x=θn/2
1 = (w + 1 − (1 − w)) = w. 2 So now, ||(1 − p f )v3 + p f v − π || = ||(1 − p f )(v3 − π ) + p f (v − π )|| ≤ (1 − p f )||v3 − π || + p f ||v − π || ≤ (1 − p f )( + w) + p f ≤ δ, where δ = +w + p f . We are free to choose : choose it to be 1/n so that δ is 1/ poly(n). So now the running time to get a distance δ is t = O(n log n). We then apply Lemma A.14 to obtain the result. This concludes the proof of Theorem 5.1 so Corollary 5.1 is proved. We have now proven Lemma 2.1 and consequently Corollary 2.1. We now show how Theorem 2.2 follows.
6. Main Result We will now show how the mixing time results imply that we have an approximate 2-design. Proof (Proof of Theorem 2.2). We will go via the 2-norm since this gives a tight bound when working with the Pauli operators. The supremum can be taken over just physical states ρ [29]. We write ρ in the Pauli basis as usual (as Eq. 2.3).
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289
||GW − G H ||2 = sup ||(GW ⊗ I )(ρ) − (G H ⊗ I )(ρ)||21 ρ
≤ 24n sup ||(GW ⊗ I )(ρ) − (G H ⊗ I )(ρ)||22 ρ = sup γ0 ( p1 , p2 , p3 , p4 )(GW (σ p1 ⊗ σ p2 ) ⊗ σ p3 ⊗ σ p4 ρ p1 , p2 , p3 , p4 p1 p2 =00
2 −G H (σ p1 ⊗ σ p2 ) ⊗ σ p3 ⊗ σ p4 ) . 2
Now, write (for p1 p2 = 00) σq2 . We get sup ρ
GW ( 21n σ p1
⊗ σ p2 ) =
p1 , p2 , p3 , p4 ,q1 ,q2 p1 p2 =00,q1 q2 =00
sup ρ
≤ 24n sup ρ
q1 ,q2 q1 q2 =00
gt (q1 , q2 ; p1 , p2 )σq1 ⊗
p1 , p2 , p3 , p4 ,q1 ,q2 p1 p2 =00,q1 q2 =00
p1 , p2 , p3 , p4 p1 p2 =00
2 ⊗ σ p4 2
=2
δq q δ p p γ0 ( p1 , p2 , p3 , p4 ) gt (q1 , q2 ; p1 , p2 ) − n1 2n 1 2 2 (2 + 1)
×σq1 ⊗ σq2 ⊗ σ p3 4n
1 2n
γ02 ( p1 ,
δq q δ p p p2 , p3 , p4 ) gt (q1 , q2 ; p1 , p2 ) − n1 2n 1 2 2 (2 + 1)
2
γ02 ( p1 , p2 , p3 , p4 ) 2
≤ 24n 2 , where the first equality comes from the orthogonality of the Pauli operators under the Hilbert-Schmidt inner product and the last inequality comes from the fact that ρ is a physical state so has tr ρ 2 ≤ 1. This proves the result for the diamond norm, Definition 2.5. For the distance measure defined in Definition 2.6, the argument in [10] can be used together with the 1-norm bound to prove the result. It is unfortunate that there is still a dimension factor remaining in the above proof. To get a distance we have to run the random circuit for O(n(n + log 1/)) steps. However, closeness in the diamond-norm may be too stringent a requirement. After O(n(n + log 1/)) steps, the random circuit gives a 2-design in the measure used by Dankert et al. (see [10] and Definition 2.6). This is in contrast to the O(n log 1/) steps required by the explicit circuit construction of Dankert et al. 7. Conclusions We have proved tight convergence results for the first two moments of a random circuit. We have used this to show that random circuits are efficient approximate 1- and 2-unitary designs. Our framework readily generalises to k-designs for any k and the next step in this research is to prove that random circuits give approximate k-designs for all k. We have shown that, provided the random circuit uses gates from a universal gate set that is also universal on U (4), the circuit is still an efficient 2-design. We also see that the
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random circuit with gates chosen uniformly from U (4) is the most natural model. We note that the gates from U (4) can be replaced by gates from any approximate 2-design on two qubits without any change to the asymptotic convergence properties. One application of this work is to give an efficient method of decoupling two quantum systems by applying a random unitary from a 2-design to one system and then discarding part of it. This technique is used in [2] to construct a variety of encoding circuits for tasks in quantum Shannon theory; thus, we (like [10]) reduce the encoding complexity in [2] (and related works, such as [21]) to O(n 2 ). Unfortunately, the decoding circuits still remain inefficient. An algorithmic application of random circuits was given in [19], where they were used to construct a new class of superpolynomial quantum speedups. In that paper, random circuits of length O(n 3 ) were used in order to guarantee that they were so-called “dispersing” circuits. Our results immediately imply that circuits of length O(n 2 ) would instead suffice. We believe that this could be further improved with a specialised argument, since [19] assumed that the input to the random circuit was always a computational basis state. Another potential application of random circuits is to model the evolution of black holes [22]. In Ref. [22], they conjecture that short random local quantum circuits are approximately 2-designs, and thus can be used for decoupling quantum systems (as in [2]). This, in turn, is used to make claims about the rate at which black holes leak information. While our model differs from that of Ref. [22] in that they consider nearest-neighbour interactions and we do not, our techniques and results could be readily extended to cover the case they consider. Finally, random circuits are interesting physical models in their own right. The original purpose of [26] was to answer the physical question of how quickly entanglement grows in a system with random two party interactions. Lemma 2.1(i) shows that O(n(n + log 1/)) steps suffice (in contrast to O(n 2 (n + log 1/)) which they prove) to give almost maximal entanglement in such a system. Acknowledgements We are grateful for funding from the Army Research Office under grant W9111NF-051-0294, the European Commission under Marie Curie grants ASTQIT (FP6-022194) and QAP (IST-200515848), and the U.K. Engineering and Physical Science Research Council through “QIP IRC.” We thank Raphaël Clifford, Ashley Montanaro and Dan Shepherd for helpful discussions.
A. Appendix A.1. Permutation operators. The following theorems about permutation operators will be used repeatedly. Lemma A.1. Let C be a cycle of length c in Sc . Then tr (C (A1 ⊗ A2 ⊗ . . . ⊗ Ac )) = tr AC(1) AC ◦2 (1) AC ◦3 (1) . . . A1 . Proof. We have tr (C (A1 ⊗ A2 ⊗ . . . ⊗ Ac )) =
i 1 i 2 . . . i c |C (A1 ⊗ A2 ⊗ . . . ⊗ Ac ) |i 1 i 2 . . . i c
i 1 ,i 2 ,...,i c
=
i 1 |AC(1) |i C(1) i 2 |AC(2) |i C(2)
i 1 ,i 2 ,...,i c
. . . i c |AC(c) |i C(c)
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=
i 1 ,i 2 ,...,i c
291
i 1 |AC(1) |i C(1) i C(1) |AC ◦2 (1) |i C ◦2 (1) . . . i C ◦c−1 (1) |A1 |i 1
C ◦c (1)
since result.
= 1. Evaluate the sum using the resolution of the identity to get the
With this we can work out the Pauli expansion of the swap operator: Lemma A.2. The swap operator F on two d dimensional systems can be written as 1 σp ⊗ σp, d p where {σ p } form a Hermitian orthogonal basis with tr σ p2 = d. Proof. Expand F in the basis and use Lemma A.1: tr σ p ⊗ σq F = tr σ p σq d p=q = 0 otherwise. The given sum has the correct coefficients in the basis, therefore
1 d
p
σ p ⊗σ p = F.
A.2. Zero chain mixing time proofs. A.2.1. Asymmetric simple random walk We will use some facts about asymmetric simple random walks, i.e. a random walk on a 1D line with probability p of moving right at each step and probability q = 1 − p of moving left. The position of the walk after k steps is tightly concentrated around k( p − q): Lemma A.3. Let X k be the random variable giving the position of a random walk after k steps starting at the origin with probability p of moving right and probability q = 1 − p of moving left. Let µ = p − q. Then for any η > 0, 2 η P(X k ≥ µk + η) ≤ exp − 2k and
2 η . P(X k ≤ µk − η) ≤ exp − 2k
Proof. The standard Chernoff bound for 0/1 variables Y˜i gives, with Y˜i equal to 1 with k probability p and for Yk = i=1 Y˜i , 2η2 , P(Yk ≥ kp + η) ≤ exp − k 2η2 P(Yk ≤ kp − η) ≤ exp − . k For our case, set Y˜i = 2 X˜ i − 1 to give the desired result.
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This result is for a walk with constant bias. We will need a result for a walk with varying (but bounded from below) bias: Lemma A.4. Let X k be the random variable giving the position of a random walk after k steps starting at the origin with probability pi ≥ p of moving right and probability qi ≤ p of moving left at step i. Let µ = p − (1 − p). Then for any η > 0, 2 η P(X k ≥ µk + η) ≤ exp − 2k and 2 η P(X k ≤ µk − η) ≤ exp − . 2k Proof. Let Y˜i be a random variable equal to 1 with probability p and 0 with probability 1 − p. Then let Z˜ i be a random variable equal to 1 with probability pi and 0 with k k probability 1 − pi . Let Yk = i=1 Y˜i and Z k = i=1 Z˜ i . Then following the standard Chernoff bound derivation (for λ > 0),
P(Z k ≥ kp + η) = P eλZ k ≥ eλ(kp+η) eλ(kp+η) EeλZ k eλ(kp+η) ≤ EeλYk 2η2 . ≤ exp − k ≤
We can then, as above, set Z˜i = 2 X˜ i − 1. The calculation is similar for the bound on P(X k ≤ µk − η). From Lemma A.3 we can prove a result about how often each site is visited. If the walk runs for t steps the walk is at position tµ with high probability so we might expect from symmetry that each site will have been visited about 1/µ times. Below is a weaker concentration result of this form but is strong enough for our purposes. It says that the amount of time spent ≤ x is about x/µ. Lemma A.5. For γ > 2 and integer x > 0,
∞ µx(γ − 2) P , I(X k ≤ x) ≥ γ x/µ ≤ 2 exp − 2 k=1
where I is the indicator function. Proof. Let Yk = I(X k ≤ x). From Lemma A.3, (kµ − x)2 P(Yk = 0) ≤ exp − 2k
Random Quantum Circuits are Approximate 2-designs
for k ≤ x/µ and
293
(kµ − x)2 P(Yk = 1) ≤ exp − 2k
for k ≥ x/µ. Then the quantity to evaluate is
P
∞
Yk ≥ γ x/µ .
k=1
We use a standard trick to split this into two mutually exclusive possibilities and then bound the probabilities separately. Write
∞ P Yk ≥ γ x/µ k=1
⎛
⎛ ⎞⎞ x/µ ∞ γ ⎝ = P⎝ Y j = 1 ⎠⎠ Yk ≥ γ x/µ k=1
j=1
⎛
⎛ ⎞⎞ x/µ ∞ γ ⎝ +P⎝ Yk ≥ γ x/µ Y j = 0 ⎠⎠ . k=1
(A.1)
j=1
We can bound the first term: ⎛
⎛ ⎞⎞ ⎛ ⎞ x/µ γ x/µ ∞ γ ⎝ P⎝ Yk ≥ γ x/µ Yk = 1⎠ Y j = 1 ⎠⎠ = P ⎝ k=1
j=1
k=1
≤ P Yγ x/µ = 1 µx(γ − 1)2 ≤ exp − 2γ µx(γ − 2) . ≤ exp − 2
The second term is done similarly: ⎛ ⎞ ⎛
⎛ ⎞⎞ x/µ ∞ ∞ γ ⎜ ⎟ ⎝ Y j = 0 ⎠⎠ ≤ P ⎝ P⎝ Yk ≥ γ x/µ [Yk = 1]⎠ k=1
k= γµx +1
j=1
≤
∞ k= γµx +1
≤
∞ k= γµx +1
P (Yk = 1)
(kµ − x)2 exp − 2k
µx(γ − 2) . ≤ exp − 2
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The last fact we need about asymmetric simple random walks is a bound on the probability of going backwards. If p > q then we expect the walk to go right in the majority of steps. The probability of going left a distance a is exponentially small in a. This is a well known result, often stated as part of the gambler’s ruin problem: Lemma A.6 (see e.g. [17]). Consider an asymmetric simple random walk that starts at a > 0 and has an absorbing barrier at the origin. The probability that the walk eventually absorbs at the origin is 1 if p ≤ q and (q/ p)a otherwise. This result is for infinitely many steps. If we only consider finitely many steps, the probability of absorption must be at most this. A.2.2. Waiting time From above we saw that the probability of moving is at least 2x/5n when at position x. The length of time spent waiting at each step is therefore stochastically dominated by a geometric distribution with parameter 2x/5n. The following concentration result will be used to bound the waiting time (in our case β = 2/5): Lemma A.7. Let the waiting time at each site be W (x) ∼ Geo (βx/n), the total waiting time W = tx=1 W (x) and t = n βln t . Then P(W ≥ Ct ) ≤ 2t (1−C)/2 . Proof. By Markov’s inequality for λ > 0, P(W ≥ Ct ) ≤
EeλW . eλCt
The W (x) are independent so EeλW =
t
EeλW (x) .
x=1
Summing the geometric series we find EeλW (x) = provided eλ <
1 1− βx n
βx n
e−λ − 1 +
βx n
,
for all 1 ≤ x ≤ t. Therefore eλ is of the form
0 < α < 1. With this, EeλW (x) =
1 1− αβ n
, where
x x −α
and EeλW =
t!(1 − α) . (t + 1 − α)
We are free to choose α within its range to optimise the bound. However, for simplicity, we will choose α = 1/2. From Lemma A.13, √ EeλW ≤ 2 t. The result follows, using the inequality 1 − x ≤ e−x .
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A.2.3. Phase 1 Here we prove that phase 1 completes successfully with high probability. The bias here is large so the walk moves right every time with high probability: Lemma A.8. The probability that the accelerated chain moves right at each step, starting from x = 1 for t steps, is at least 1 − t 2 /n. Proof. The probability of moving right at each step is t x=1
(n − 2)(n − 3) . . . (n − t) 3(n − x) = 3n − 2x − 1 (n − 5/3)(n − 7/3) . . . (n − (2t + 1)/3) ≥ (1 − 2/n)(1 − 3/n) . . . (1 − t/n) ≥ (1 − t/n)t ≥ 1 − t 2 /n.
Let t = n δ . Provided δ < 1/2 this probability is close to one. Therefore, with high probability, the walk moves to n δ in n δ steps. Using Lemma A.7 the waiting time can be bounded: Lemma A.9. Let W (1) be the waiting time during phase 1. Let H be the event that the walk moves right at each step. Then
P W (1) ≥ Ct |H ≤ 2n δ(1−C)/2 , (A.2) where t =
5δn ln n . 2
Proof. This follows directly from Lemma A.7, since each site is hit exactly once. We now combine these two lemmas to prove that phase 1 completes successfully with high probability: Proof (Proof of Lemma 5.4). In Lemma A.8, we show that in n δ accelerated steps, the walk moves right at each step with probability ≥ 1 − n 2δ−1 . Call this event H . Then P(H ) ≥ 1 − n 2δ−1 . Lemma A.9 shows that the waiting time W (1) is bounded with high probability (choosing C = 3): P(W (1) ≤ 15nδ ln n/2|H ) ≥ 1 − 2n −δ . Then we can bound the probability of phase 1 completing successfully: P(Phase 1 completes successfully) ≥ P(H ∩ W (1) ≤ 15nδ ln n/2) = P(H )P(W (1) ≤ 15nδ ln n/2|H ) ≥ (1 − n 2δ−1 )(1 − 2n −δ ) ≥ 1 − n 2δ−1 − 2n −δ .
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A.2.4. Phase 2 Phase 2 starts at n δ /2 and finishes when the walk has reached θ n for some constant 0 < θ < 3/4. We show that, with high probability, this also takes time O(n log n). The probability of moving right during this phase is at least p = 3(1−θ) 3−2θ . We first define some constants that we will derive bounds in terms of. Let γ be a constant > 2. Let µ = p − (1 − p) and µ˜ = µ/γ . Finally let s = µt ˜ for some t (which will be the number of accelerated steps). Then, with high probability, the walk will have passed s after t steps: Lemma A.10. Let X t be the position of the walk at accelerated step t, where X 0 = n δ . Then P(X t ≤ s) ≤ exp(−µ2 t (1 − 1/γ )2 /2). Proof. Let X t = X t − n δ . Then from Lemma A.4, 2 η . P(X t ≤ µt − η) ≤ exp − 2t Now let η = µt − s and use P(X t ≤ s) = P(X t ≤ s − n δ ) ≤ P(X t ≤ s) to complete the proof. We now prove a bound on the waiting time: Lemma A.11. Let W (2) be the waiting time in phase 2. Then, assuming the walk does not go back beyond n δ /2,
δ 2 exp −µn 4 15n ln s ≤ (4/s)3/2µ + P W (2) ≥ (A.3) . µ 1 − exp −µ 2
, where X k is the position of the walk at accelerated step k (X 0 = n δ ). We want to bound (w.h.p.) the waiting time W (2) = tk=1 Wk of t steps of the accelerated walk. Define the event H to be ⎧ "⎫ !∞ ⎨ ⎬ H= I(X k ≤ x) ≤ x/µ˜ . (A.4) ⎩ δ ⎭
Proof. Let Wk ∼ Geo
2X k 5n
x≥n /2
k=1
If H occurs, no sites have been hit too often and the walk has not gone back further than n δ /2. It is important that we also use the restriction that X k ≥ n δ /2 because the waiting time grows the longer the walk moves back. However, it is very unlikely that the walk will go backwards (even to n δ /2). We now define some more notation to bound the waiting time. Let X = (X 1 , X 2 , . . . , X t ) be a tuple of positions and let N x (X) be the number of times that x appears in X and let N(X) = (N1 (X), N2 (X), . . . , Nn (X)). Then we have x N x (X) = t.
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As we said above, the waiting time at x = a stochastically dominates the waiting time at x = b for b ≥ a. In other words, Wk Wk
if X k ≤ X k ,
(A.5)
where X Y means that X stochastically dominates Y . Now write the waiting time for all steps: W (2) (X) =
t
Wk
k=1
=
x (X) N
x
Wh (x),
(A.6)
h=1
where Wh (x) ∼ Geo 2x 5n . If event H occurs, we can put some bounds on N x . We find that, for all x ≥ n δ /2, x
N y (X) ≤ x/µ˜
(A.7)
y=n δ /2
and N x (X) = 0 for x < n δ /2. Now let Xm be such that Nn δ /2 (Xm ) = 1/µ˜ for x > n δ /2. Then x
N y (Xm ) = x/µ. ˜
nδ 2µ˜
and N x (Xm ) =
(A.8)
y=n δ /2
Now we introduce the relation : Definition A.1. Let x and y be n-tuples. Then x y if k
xi ≤
i=1
k
yi
(A.9)
i=1
for all 1 ≤ k ≤ n with equality for k = n. Note that this is like majorisation, except the the tuples are not sorted. Using elements of this, we find that N(X) N(Xm ). (Using y N y (X) = y N y (X ) = t for all X, X .) If we combine Eqs. A.5 and A.6 we find that W (2) (X) W (2) (X ) if N(X) N(X ). Roughly speaking, this is simply saying that the waiting time is larger if the earlier sites are hit more often. But since for all X that satisfy H , X Xm , we have W (2) (X) W (2) (Xm ) provided H occurs. We will simplify further by noting that Xm X0 , where N x (X0 ) = 1/µ˜ for 1 ≤ x ≤ µt ˜ = s and zero elsewhere. Therefore 5Cn ln s 5Cn ln s (2) (2) H ≤ P W (X0 ) ≥ . P W (X) ≥ 2µ˜ 2µ˜ We can bound this by applying Lemma A.7. Let Wh = sx=1 Wh (x). From Lemma A.7, P(Wh ≥ Ct ) ≤ 2s
1−C 2
,
(A.10)
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1/µ˜ ˜ . The same where t = 5n 2ln s . However, we want a bound on P h=1 Wh ≥ Ct /µ reasoning as in Lemma A.7 bounds this as ⎛ ⎞ 1/µ˜ 1−C 1/µ˜ P⎝ Wh ≥ Ct /µ˜ ⎠ ≤ 2s 2 . (A.11) h=1
Therefore
(1−C)/2 5Cn ln s ≤ 21/µ˜ s µ˜ . P W (2) (X0 ) ≥ 2µ˜
(A.12)
To complete the proof, we just need to find P(H c ). We can bound it using the union bound and Lemma A.5: ⎛ "⎞ !∞ n P(H c ) = P ⎝ I(X k ≤ x) > x/µ˜ ⎠ x=n δ /2 n
≤
P
x=n δ /2 n
≤
x=n δ /2 ∞
k=1 ∞
I(X k ≤ x) ≥ x/µ˜
k=1
−µx(γ − 2) 2 exp 2
−µx(γ − 2) ≤ 2 exp 2 x=n δ /2
δ 2 exp −µn 4(γ −2)
. = 1 − exp −µ(γ2 −2) Now, for any events A and B,
P(A) = P(A ∩ B) + P(A ∩ B c ) = P(A|B)P(B) + P(A ∩ B c ) ≤ P(A|B) + P(B c ), and set C = 2 and γ = 3 to obtain the result. We now combine these two lemmas to prove that phase 2 completes successfully with high probability: Proof (Proof of Lemma 5.5). Phase 2 can fail if: – The walk does not reach θ n. The probability of this is bounded by Lemma A.10: 2 P(X t ≤ θ n) ≤ exp − µθ n . 3 This follows from setting t =
3θn µ
and γ = 3.
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– The waiting time is too long. This probability is bounded by Lemma A.11:
−µn δ 3 2 exp 2µ 4 4 15n ln(θ n) δ ≤ P W (2) ≥ + + (q/ p)n /2 . µ θn 1 − exp(−µ/2) – The walk gets back to n δ /2. This is bounded by Lemma A.6: δ P Walk gets to n δ /2 ≤ (q/ p)n /2 . So, using the union bound we can bound the overall probability of failure:
δ 3 2 exp −µn 4 4 2µ 2 δ + (q/ p)n /2 . + P(Phase 2 fails) ≤ exp − µθ n + 3 θn 1 − exp(−µ/2) A.2.5. Phase 3 This phase starts at θ n. We show that this mixes quickly using log-Sobolev arguments. Lemma A.12. The zero chain on the restricted state space x ∈ [m, n], where m = θ n/2 for 0 ≤ θ ≤ 3/4, has mixing time O n log n . Proof. We restrict the Markov chain to only run from m by adjusting the holding probability at m, P(m, m). Construct the chain P with transition matrix ⎧ ⎪ x < m or y < m ⎨0 , (A.13) P (x, y) = 1 − P(m, m + 1) x = y = m ⎪ ⎩ P(x, y) otherwise where P is the transition matrix of the full zero chain. This chain then has stationary distribution π(x)/(1 − w) m ≤ x ≤ n π (x) = , (A.14) 0 otherwise where w = m−1 x=1 π(x). To see this, first note that the distribution is normalised. We want to show that n
P (x, y)π (x) = π (y).
(A.15)
x=m
When y = m we are required to prove that P (m, m)π (m) + P (m + 1, m)π (m + 1) = π (m). This follows from the reversibility of the unrestricted zero chain, using P (m, m) = 1 − P(m, m + 1). For y > m, Eq. A.15 is satisfied simply because π(x) is the stationary distribution of P and related by a constant factor to π (x). We can now prove this final mixing time result, making use of Lemma 4.4. Let Q i be the chain that uniformly mixes site i. This converges in one step and has a log-Sobolev constant independent of n; call it ρ1 . Let Q be the chain that chooses a site at random and then uniformly mixes that site. This is the product chain of the Q i so, by Lemma 4.4, has gap 1/n and Sobolev constant ρ Q = ρ1 /n. We can construct the zero chain for this and find its Sobolev constant.
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The Sobolev constant is defined (Definition 4.3) in terms of a minimisation over functions on the state space. For the chain Q we can write ρ Q = inf f (φ). φ
If we restrict the infimum to be over functions φ with φ(x) = φ(y) for x and y containing the same number of zeroes then we obtain the Sobolev constant for the zero-Q chain, ρ Q 0 , which is the chain which counts the number of zeroes in the full chain Q. Since taking the infimum over less functions cannot give a smaller value, ρ Q 0 ≥ ρ Q ≥ ρ1 /n. We can now compare this chain to the zero-P chain. The stationary distributions are the same. The transition matrix for the zero-Q chain is
Q 0 (x, y) =
⎧ n+2x ⎪ ⎪ 4n ⎪ ⎨x 4n
3(n−x) ⎪ ⎪ ⎪ ⎩ 4n 0
y=x y = x −1 . y = x +1 otherwise
Then construct Q 0 by restricting the space to only run from m in exactly the same way as P is constructed from P. Q 0 has the same stationary distribution as P . Now we can perform the comparison. From Eq. 4.14: Q 0 (a, a + 1) a≥m P (a, a + 1) 5 5(n − 1) ≤ . = max a≥m 8a 8θ
A = max
1 Therefore ρ P ≥ 8θρ 5n . Exactly the same argument applies to show the gap is (1/n), so the mixing time is (from Eq. 4.16) O(n log n ).
Now we can prove that phase 3 completes successfully with high probability: Proof (of Lemma 5.6). In Lemma A.12, we show that after O n log n steps the chain mixes to distance . We just need to show that the walk goes back to θ n/2 with small probability. This follows from Lemma A.6.
A.3. Moment generating function calculations. The following lemma is needed in the moment generating function calculations. Lemma A.13. For Integer s > 0, √ (s + 1)(1/2) ≤ 2 s. (s + 1/2)
(A.16)
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301
Proof. From expanding the functions, Eq. A.16 becomes s!2s 2 × 4 × 6 × · · · × 2(s − 1) × 2s = (2s − 1)!! 1 × 3 × 5 × · · · × (2s − 3) × (2s − 1) s
= x=1
We then proceed by induction. s+1 x=1
It is easy to show that
2(s+1) 2(s+1)−1
2x . 2x − 1
&1
2x x=1 2x−1
= 2 and by the inductive hypothesis
2(s + 1) √ 2x ≤ 2 s. 2x − 1 2(s + 1) − 1 ≤
s+1 s
and the result follows.
A.4. Mixing times. We find bounds for the mixing time above that are valid with high probability. Below we turn these into full mixing time bounds. Lemma A.14. If after O(n log n) steps the state v of a random walk satisfies ||v − π || ≤ δ, where π is the stationary distribution and δ is 1/ poly(n), then the number of steps required to be at most a distance from stationarity is n O n log . Proof. Let s be the slowest mixing initial state. Then, after t = O(n log n) steps we have at worst the state (1 − δ)π + δs, and if we repeat kt times δ becomes δ k . So to get a distance , k = Now we evaluate the mixing time: ) kt = O(n log n)
log log δ
*
'
log log δ
( .
)
* log 1/ log 1/δ = O(n max(log n, log 1/)) n . = O n log
= O(n log n)
References 1. Aaronson, S.: Quantum Copy-Protection. Talk at QIP, New Delhi, India, December 2007, available at http://www.scottaaronson.com/talks/copy.ppt, 2007
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2. Abeyesinghe, A., Devetak, I., Hayden, P., Winter, A.: The mother of all protocols: Restructuring quantum information’s family tree. http://arxiv.org/abs/:quant-ph/0606225v1, 2006 3. Ambainis, A., Emerson, E.: Quantum t-designs: t-wise independence in the quantum world. IEEE Conference on Computational Complexity 2007, http://arxiv.org/abs/:quant-ph/0701126v2, 2007 4. Ambainis, A., Mosca, M., Tapp, A., de Wolf, R.: Private Quantum Channels. FOCS 2000, Washington, DC: IEEE, 2000, pp. 547–553 5. Ambainis, A., Smith, A.: Small pseudo-random families of matrices: derandomizing approximate quantum encryption. Lecture Notes in Computer Science 3122, Berlin-Heidelberg-NewYork: Springer, 2004, pp. 249–260 6. Arnold, V.I., Krylov, A.L.: Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex domain. Sov. Math. Dokl. 4(1), 1962 7. Barenco, A., Berthiaume, A., Deutsch, D., Ekert, A., Jozsa, R., Macchiavello, C.: Stabilization of quantum computations by symmetrization. SIAM J. Comput. 26(5), 1541–1557 (1997) 8. Barnum, H.: Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases. http://arxiv.org/abs/:quant-ph/0205155v1, 2002 9. Dahlsten, O.C.O., Oliveira, R., Plenio, M.B.: The emergence of typical entanglement in two-party random processes. J. Phys. A Math. Gen. 40, 8081–8108 (2007) 10. Dankert, C., Cleve, R., Emerson, J., Livine, E.: Exact and approximate unitary 2-designs: constructions and applications. http://arxiv.org/abs/:quant-ph/0606161v1, 2006 11. Devetak, I., Junge, M., King, C., Ruskai, M.B.: Multiplicativity of completely bounded p-norms implies a new additivity result. Commun. Math. Phys. 266, 37–63 (2006) 12. Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible markov chains. Ann. Appl. Probab. 3(3), 696–730 (1993) 13. Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6(3), 695–750 (1996) 14. DiVincenzo, D., Leung, D., Terhal, B.: Quantum data hiding. Information Theory. IEEE Transactions 48(3), 580–598 (2002) 15. Emerson, J., Livine, E., Lloyd, S.: Convergence conditions for random quantum circuits. Phys. Rev. A 72, 060302 (2005) 16. Goodman, R., Wallach, N.: Representations and Invariants of the Classical Groups. Cambridge: Cambridge University Press, 1998 17. Grimmett, G., Welsh, D.: Probability: An Introduction. Oxford: Oxford University Press, 1986 18. Gross, D., Audenaert, K., Eisert, J.: Evenly distributed unitaries: On the structure of unitary designs. J. Math. Phys. 48, 052104 (2007) 19. Hallgren, S., Harrow, A.W.: Superpolynomial speedups based on almost any quantum circuit. In: Proc. 35th Intl. Colloq. on Automate Languages an Programming LCUS 5125, 2, pp. 782–795, 2008 20. Hayashi, A., Hashimoto, T., Horibe, M.: Reexamination of optimal quantum state estimation of pure states. Phys. Rev. A 72, 032325 (2006) 21. Hayden, P., Horodecki, M., Yard, J., Winter, A.: A decoupling approach to the quantum capacity. Open Syst. Inf. Dyn. 15, 7–19 (2008) 22. Hayden, P., Preskill, J.: Black holes as mirrors: quantum information in random subsystems. JHEP 09, 120 (2007) 23. Hoory, S., Brodsky, A.: Simple Permutations Mix Even Better. http://arxiv.org/abs/math/ 0411098v2[math.CO] 2004 24. Kitaev, A.Yu., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Providence, RI Amer. Math. Soc. (2002) 25. Montenegro, R., Tetali, P.: Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1(3), 237–354 (2006) 26. Oliveira, R., Dahlsten, O.C.O., Plenio, M.B.: Efficient generation of generic entanglement. Phys. Rev. Lett. 98, 130502, (2007) 27. Paulsen, V.I.: Completely Bounded Maps and Dilations. New York: John Wiley & Sons, Inc., 1987 28. Sen, P.: Random measurement bases, quantum state distinction and applications to the hidden subgroup problem. IEEE Conference on Computational Complexity 2006, 2005, pp. 274–287 29. Watrous, J.: Notes on super-operator norms induced by Schatten norms. Quantum Information and Computation 5(1), 58–68 (2005) 30. Znidaric, M.: Optimal two-qubit gate for generation of random bipartite entanglement. Phys. Rev. A 76, 012318 (2007) Communicated by A. Connes
Commun. Math. Phys. 291, 303–320 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0879-0
Communications in
Mathematical Physics
Spin (7)-Manifolds with Parallel Torsion Form Christof Puhle Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. E-mail: [email protected] URL: www.math.hu-berlin.de/~puhle Received: 31 July 2008 / Accepted: 30 May 2009 Published online: 25 July 2009 – © Springer-Verlag 2009
Abstract: Any Spin (7)-manifold admits a metric connection ∇ c with totally skewsymmetric torsion Tc preserving the underlying structure. We classify those with ∇ c -parallel Tc = 0 and non-Abelian isotropy algebra iso (Tc ) spin (7). These are isometric to either Riemannian products or homogeneous naturally reductive spaces, each admitting two ∇ c -parallel spinor fields. Contents 1. Introduction . . . . . . 2. Parallel Torsion . . . . 3. Spin (7)-Manifolds . . 4. Subalgebrae of spin (7) 5. Algebraic Classification 6. Geometric Results . . . References . . . . . . . . . .
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303 305 305 307 309 314 319
1. Introduction In the early ‘80’s physicists tried to incorporate torsion into superstring and supergravity theories in order to get a physically flexible model. Strominger described the mathematics of the underlying superstring theory of type II. It consists of a Riemannian spin manifold (M n , g) equipped, amongst other things, with a spinor field and a 3-form T that satisfy a certain set of field equations (see [19]), including g
∇X +
1 (X 4
T) · = 0 ∀ X ∈ T M n .
Supported by the SFB 647: ‘Space–Time–Matter’, German Research Foundation DFG.
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We denote by τ · the Clifford product of a differential form τ with a spinor field . In the theory T is seen as a field strength of sorts, whilst is the so-called supersymmetry. With the metric connection ∇ whose torsion is the 3-form T, g 1 g (∇ X Y, Z ) = g ∇ X Y, Z + · T (X, Y, Z ) ∀ X, Y, Z ∈ T M n , 2 the above equation transforms to ∇ = 0. In other words, the spinor field is parallel with respect to ∇, a fact imposing restrictions on the holonomy group Hol (∇). In the case T = 0, i.e. when ∇ is the Levi-Civita connection of (M n , g), the holonomy group is one of the following (see [20]): SU (n), Sp (n), G2 , Spin (7). In order to construct models with T = 0, it is therefore reasonable to study manifolds that admit a metric connection ∇ with totally skew-symmetric torsion whose Hol (∇) is contained in SU (n), Sp (n), G2 or Spin (7). Surprisingly, the existence of such a connection is unobstructed for Spin (7)-manifolds M 8 (see [17]). Furthermore this connection, denoted by ∇ c , is unique, preserves the underlying Spin (7)-structure and makes a nontrivial spinor field parallel. The more general point of view of [2,12,18] indicates that structures with parallel torsion form Tc , ∇ c Tc = 0, are of particular interest and provide a starting point to solve the entire system of Strominger’s equations. For example, δTc = 0 is automatically satisfied in this setup. Moreover, many geometric properties become algebraically tractable by assuming the parallelism of Tc , for this implies, for instance, that the holonomy algebra hol (∇ c ) becomes a subalgebra of the isotropy algebra iso (Tc ) spin (7). The aim of the paper is the classification of Spin (7)-manifolds with parallel torsion form Tc = 0 and non-Abelian iso (Tc ) spin (7). We show that the latter fall into eight types. For all of these algebrae we describe the admissible torsion forms Tc and Ricci tensors Ricc with respect to ∇ c . Finally we discuss the geometry of the space M 8 relative to its holonomy algebra hol (∇ c ) iso (Tc ) and to the Spin (7)-orbit of the torsion form Tc ∈ 3 = 38 ⊕ 348 . The main result is that these spaces are isometric to either a Riemannian product or a homogeneous naturally reductive space; some of them are uniquely determined (see Theorem 6.2 and Theorem 6.6). Moreover, every structure admits at least two ∇ c -parallel spinor fields. There are examples exhibiting 16 ∇ c -parallel spinor fields and satisfying the additional constraint g
Ricc = Rici j −
1 c T Tc = 0 4 imn jmn
for the energy-momentum tensor (see Examples 6.2 and 6.3). The paper is structured as follows: In Sect. 2 we state basic facts on metric connections with parallel, totally skew-symmetric torsion. We then specialize to the case of Spin (7)-structures in Sect. 3. Section 4 is devoted to the study of the non-Abelian subalgebrae of spin (7) used for the algebraic classification (see Sect. 5) in terms of the torsion form. In the last section we discuss the geometry of each of these classes.
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2. Parallel Torsion Fix a Riemannian spin manifold (M n , g), a 3-form T, and denote the Levi-Civita connection by ∇ g . The equation g 1 g (∇ X Y, Z ) = g ∇ X Y, Z + · T (X, Y, Z ) ∀ X, Y, Z ∈ T M n , 2 defines a metric connection ∇ with totally skew-symmetric torsion T. We will consider the case of parallel torsion, ∇T = 0. Then the 3-form T is coclosed (see [12]), δT = 0, its differential is given by dT = (ei T) ∧ (ei T) =: 2 σ T i
for a chosen orthonormal frame (e1 , . . . , en ), and the curvature tensor R∇ of ∇ is a field of symmetric endomorphisms of 2 . If there exists a ∇-parallel spinor field one can compute the Ricci tensor Ric∇ of ∇ algebraically (see for example [2]), 2 Ric∇ (X ) · = (X
dT) · .
Moreover, the following relation holds (see [1]): 4 T2 · := 4 T · (T · ) = 2 Scalg + T2 · . Consequently, T2 acts as a scalar on the space of ∇-parallel spinor fields, hence it gives an algebraic restriction on T. 3. Spin (7)-Manifolds Consider the space R8 , fix an orientation and denote a chosen oriented orthonormal basis by (e1 , . . . , e8 ). The compact simply connected Lie group Spin (7) can be described (see for example [16]) as the isotropy group of the 4-form , = φ + ∗φ,
()
where φ, Z and D denote the following forms: φ := (Z ∧ e7 + D) ∧ e8 ,
Z := e12 + e34 + e56 ,
D := e246 − e235 − e145 − e136 .
Here and henceforth we shall not distinguish between vectors and covectors and use the notation ei1 ...im for the exterior product ei1 ∧ . . . ∧ eim . The so-called fundamental form is self-dual with respect to the Hodge star operator, ∗ = , and the 8-form ∧ is a non-zero multiple of the volume form of R8 . A Spin (7)-structure/manifold is a triple M 8 , g, consisting of an 8-dimensional 8 Riemannian manifold M , g and a 4-form such that there exists an oriented orthonormal adapted frame (e1 , . . . , e8 ) realizing () at every point. Equivalently, these structures can be defined as a reduction of the structure group of orthonormal frames of the tangent bundle to Spin (7). The space of 3-forms decomposes into two irreducible Spin (7)modules, 3 = 38 ⊕ 348 , which can be characterized using the fundamental form as 38 := ∗ (β ∧ ) : β ∈ 1 , 348 := γ ∈ 3 : γ ∧ = 0 .
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The subscript specifies the dimension of the respective space. We will denote the projection of a 3-form T onto one of these spaces by T8 or T48 respectively. Any Spin (7)-manifold admits (see [17]) a unique metric connection ∇ c (the characteristic connection) with totally skew-symmetric torsion Tc (the characteristic torsion) preserving the Spin (7)-structure, ∇ c = 0, and Tc is given by Tc = −δ −
7 ∗ (θ ∧ ). 6
Here θ ∈ 1 denotes the so-called Lee form θ :=
1 1 6 ∗ (δ ∧ ) = ∗ ∧ Tc = − ∗ (∗d ∧ ). 7 7 7
The Riemannian scalar curvature Scalg and the scalar curvature Scalc of ∇ c are given by (see [17]) Scalg =
49 1 7 3 θ 2 − Tc 2 + δθ, Scalc = Scalg − Tc 2 . 18 2 2 2
( )
Analyzing the algebraic type of Tc we obtain Cabrera’s description [5] – by differential equations involving the Lee form – of the Fernández classification [10] of Spin (7)-structures. For example, a Spin (7)-structure is of class W1 , i.e. a balanced structure, if and only if the Lee form vanishes. Equivalently, these structures can be characterized by T8c = 0. Spin (7)-structures of class W0 – the so-called parallel structures – are defined by a closed fundamental form, d = 0. These are the structures with vanishing torsion, Tc = 0. In [5] Cabrera shows that the Lee form of a Spin (7)-structure of class W2 (for c = 0) is closed, and therefore such a manwhich d = θ ∧ holds or, equivalently, T48 ifold is locally conformally equivalent to a parallel Spin (7)-manifold. These are called locally conformal parallel. Finally, structures of class W = W1 + W2 are characterized c = 0. We summarize the previous facts in the following table: by T8c = 0 and T48 class W0 (parallel) W2 (locally conformal parallel) W1 (balanced) W = W1 + W2
characteristic torsion c =0 T8c = 0, T48 c =0 T48 T8c = 0 c = 0 T8c = 0, T48
differential equations d = 0, θ = 0 d = θ ∧ θ =0 —
We now restrict to parallel characteristic torsion, ∇ c Tc = 0. Lemma 3.1. The following formulae hold in the presence of parallel characteristic torsion: θ 2 =
36 c 2 T8 , 7
δθ = 0.
Proof. We prove the second equation, 6 6 ∗ d ∗ ∗ ∧ Tc = ∗ d ∧ Tc . 7 7 The 7-form ∧ Tc is ∇ c -parallel and the sum i (ei T) ∧ (ei ( ∧ T)) vanishes for arbitrary 3-forms T ∈ 3 (R8 ).
δθ = − ∗ d ∗ θ = −
This lemma and ( ) result in the following proposition:
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Proposition 3.1. Let M 8 , g, be a Spin (7)-manifold with ∇ c Tc = 0. Then the Riemannian scalar curvature Scalg and the scalar curvature Scalc of ∇ c are given in terms of the torsion form by Scalg =
27 c 2 1 c 2 c 2 T8 − T48 , Scalc = 12 T8c 2 − 2 T48 . 2 2
A direct computation shows that for arbitrary 3-forms T, vector fields X and spinor fields the following equation is satisfied: −4 X σ T · = T2 − 7 T8 2 · X · . The previous proposition together with this equation and the facts of Sect. 2 eventually prove the following: Proposition 3.2. Let M 8 , g, be a Spin (7)-manifold with ∇ c Tc = 0. Any ∇ c parallel spinor field on M 8 satisfies c 2 2 T · = 7 T8c 2 · , −4 Ricc (X ) · = Tc − 7 T8c 2 · X · . From now on we assume Spin (7)-structures to be non-parallel, Tc = 0, and to have parallel characteristic torsion, ∇ c Tc = 0. 4. Subalgebrae of spin (7)
It is known that the group Spin (7) ⊂ SO (8) acts on spinors. Let Cliff R8 denote the real Clifford algebra of the Euclidean space R8 . We will use the following real representation of this algebra on the space of real spinors 8 := R16 :
0 Mi 0 Id , for i = 1, . . . , 7 , e8 = ei = −Id 0 Mi 0 M1 M3 M5 M7
:= E 18 + E 27 − E 36 − E 45 , M2 := −E 17 + E 28 + E 35 − E 46 , := −E 16 + E 25 − E 38 + E 47 , M4 := −E 15 − E 26 − E 37 − E 48 , := −E 13 − E 24 + E 57 + E 68 , M6 := E 14 − E 23 − E 58 + E 67 , := E 12 − E 34 − E 56 + E 78 .
Here E ij denotes the standard basis of the Lie algebra so (8). We fix an orthonormal basis 1 := [1, 0, . . . , 0]T , . . . , 16 := [0, . . . , 0, 1]T of real spinors. The 4-form corresponds via the Clifford product to the real spinor 0 := 9 − 10 ∈ 8 , · 0 = −14 · 0 , and therefore Spin (7) can be seen as the isotropy group of 0 . Its Lie algebra spin (7) is the subalgebra of spin (8) containing all 2-forms ω= ωij · eij ∈ 2 R8 i< j
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such that the Clifford product ω · 0 = 0. This is satisfied if and only if ω18 = −ω27 + ω36 + ω45 , ω28 = ω17 + ω35 − ω46 , ω38 = −ω16 − ω25 − ω47 , ω48 = −ω15 + ω26 + ω37 , ω58 = ω14 + ω23 − ω67 , ω68 = ω13 − ω24 + ω57 , ω78 = −ω12 − ω34 − ω56 . We fix the following basis of spin (7): P1 := e35 + e46 , P5 := e13 + e24 ,
P2 := e36 − e45 , P6 := e14 − e23 ,
P3 := e15 + e26 , P4 := e16 − e25 , P7 := e12 − e34 , P8 := e34 − e56 ,
Q 1 := 2 · e17 − e35 + e46 , Q 2 := 2 · e27 + e36 + e45 , Q 3 := 2 · e37 + e15 − e26 , Q 4 := 2 · e47 − e16 − e25 , Q 5 := 2 · e57 − e13 + e24 , Q 6 := 2 · e67 + e14 + e23 , S1 := e18 − e27 , S2 := e28 + e17 , S3 := e38 − e47 , S4 := e48 + e37 , S5 := e58 − e67 , S6 := e68 + e57 , S7 := e78 − e56 . For a given Lie subalgebra g of spin (7), i.e. g spin (7), we denote by 3 R8 g and ( 8 )g the spaces of g-invariant 3-forms and spinors respectively. We assume 3 R8 g is non-trivial. The action of spin (7) coincides on the 8-dimensional vector spaces R8 = span (e1 , . . . , e8 ) , 38 R8 = span (∗ (e1 ∧ ) , . . . , ∗ (e8 ∧ )). Consequently g preserves a T ∈ 3 R8 g with T8 = 0, if and only if it preserves a vector. A long but elementary computation for the other case 38 R8 g = {0} proves that any non-Abelian g is conjugate to R ⊕ su (2) = span (P7 + 2 P8 − 4 S7 , P5 , P6 , P7 ) < u (3) < su (4) < spin (7). To conclude, a non-Abelian subalgebra of spin (7) that preserves a nontrivial 3-form is either a subalgebra of g2 or the algebra R ⊕ su (2) above. Dynkin’s results [8,9] on maximal subalgebrae of exceptional Lie algebrae like g2 allow to state the following: Theorem 4.1. Let g be a non-Abelian subalgebra of spin (7). If there exists a non-trivial g-invariant 3-form T, i.e. 0 = T ∈ 3 R8 g, then g is conjugate to one of the following algebrae: g2 = span (P1 , . . . , P8 , Q 1 , . . . , Q 6 ) < spin (7) , su (3) = span (P1 , . . . , P8 ) < g2 , su (2) ⊕ suc (2) = span (P5 , P6 , P7 , P7 + 2 P8 , Q 5 , Q 6 ) < g2 , u (2) = span (P7 + 2 P8 , P5 , P6 , P7 ) < su (3) , R ⊕ suc (2) = span (P7 , P7 + 2 P8 , Q 5 , Q 6 ) < su (2) ⊕ suc (2) , so (3) = span (P1 + P5 , P2 + P6 , P7 + P8 ) < su (3) , su (2) = span (P5 , P6 , P7 ) < u (2) , suc (2) = span (P7 + 2 P8 , Q 5 , Q 6 ) < R ⊕ suc (2) , √ √ soir (3) = span (P5 − 3/5 Q 2 , P6 + 3/5 Q 1 , P7 + 3 P8 ) < g2 , R ⊕ su (2) = span (P7 + 2 P8 − 4 S7 , P5 , P6 , P7 ) < su (4) < spin (7) .
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Here suc (2) denotes the centralizer of su (2) inside g2 which is isomorphic, but not conjugate, to su (2). soir (3) denotes the maximal subalgebra of g2 generating an irreducible 7-dimensional real representation. The Lie algebrae g2 and R ⊕ su (2) are of rank 2. Their maximal tori are given by t2 := k · P7 + l · (P7 + 2 P8 ) < g2 , ˜t2 := k˜ · P7 + l˜ · (P7 + 2 P8 − 4 S7 ) < R ⊕ su (2). 1-dimensional tori contained in these will be denoted by t1 or ˜t1 respectively. 5. Algebraic Classification Given a non-parallel Spin (7)-structure let iso (Tc ) be the isotropy algebra of the characteristic torsion Tc and hol (∇ c ) the holonomy algebra of the characteristic connection ∇ c . Obviously, these two are Lie subalgebrae of spin (7), and a non-Abelian iso (Tc ) is one of the algebrae in Theorem 4.1. But not all of those algebrae can occur as the isotropy algebra of a non-trivial 3-form. A direct computation proves the following: Proposition 5.1. If the isotropy algebra iso (T) < spin (7) of a non-trivial 3-form T contains suc (2) or su (2), then dim (iso (T)) ≥ 4. Since we restricted the consideration to parallel characteristic torsion the holonomy algebra hol (∇ c ) is a subalgebra of iso (Tc ), hol ∇ c iso Tc < spin (7). Conversely, fix h g < spin (7). Suppose there exists a Spin (7)-manifold with c c hol (∇ c ) = h and ∇ c -parallel torsion Tc = 0 satisfying iso (T ) = g. Then T is 3 8 necessarily contained in the space of g-invariant 3-forms R g satisfying
Tc
2
· = 7 T8c 2 · , −4 Ricc (X ) · =
2 Tc − 7 T8c 2 · X ·
(•)
for all h-invariant spinors ∈ ( 8 )h and all vectors X ∈ R8 (cf. Prop. 3.2). Furthermore, two torsion forms 0 = T1c , T2c ∈ 3 R8 g define equivalent geometric structures if they are equivalent under the action of the algebra . := x ∈ spin (7) : x 3 R8 ⊆ 3 R8 inv 3 R8 g
g
g
Define the space K (h) of algebraic curvature tensors with values in h by
K (h) := R ∈ 2 R8 ⊗ h : σ X,Y,Z {R (X, Y, Z , V )} = 0 ∀ X, Y, Z , V ∈ R8 . Here σ X,Y,Z denotes the cyclic sum over X, Y, Z . If the space K (hol (∇ c )) is trivial for a Spin (7)-manifold with parallel torsion, the curvature operator Rc : 2 R8 → hol (∇ c ) of the characteristic connection is ∇ c -parallel (see [7]) and thus hol (∇ c )invariant. A case-by-case study proves the following: Proposition 5.2. Let h g < spin (7) with g non-Abelian and suppose there exists a non-trivial g-invariant 3-form. Then K (h) is non-trivial if and only if su (2) h.
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The recipe to obtain necessary conditions on Tc and Ricc goes as follows: (1) (2) (3) (4) (5)
c ) = g with g < spin (7) non-Abelian. Fix h = hol (∇ c ) iso (T 3 Determine the spaces R8 g and ( 8 )h. Solve (•). Quotient out the action of inv 3 R8 g on Tc = 0. If su (2) h, analyze the h-invariance and the symmetry of Rc .
Applying this, we determine Tc and Ricc for all admissible combinations of hol (∇ c ) and non-Abelian iso (Tc ), iso Tc = g2 , su (3), su (2) ⊕ suc (2), u (2), R ⊕ suc (2) , so (3) , soir (3) , R ⊕ su (2) . The condition of (5), su (2) hol (∇ c ), is satisfied for hol (∇ c ) R ⊕ suc (2), so (3), soir (3) or ˜t2 . For clarity we define the following forms: Z 2 := e56 , Z 3 := e12 − e34 , D1 := e246 − e145 , Z 1 := e12 + e34 , D2 := −e235 − e136 , D3 := −e135 + e245 , D4 := e146 + e236 , D5 := e123 − e356 , so that we have Z = Z1 + Z2,
D = D1 + D2 ,
D¯ := D3 + D4 .
5.1. The cases iso (T c ) = g2 , su (2) ⊕ suc (2), R ⊕ suc (2), soir (3). The characteristic torsion is an element of the family Tc = a1 · (Z ∧ e7 + D) + b1 · ((Z 1 − 6 Z 2 ) ∧ e7 + D) + b2 · (Z 3 ∧ e8 ), where a1 , b1 , b2 ∈ R. The constraints on the torsion parameters relative to the considered isotropy algebrae are arranged in the following table: iso Tc constraints
g2 , soir (3) b1 = b2 = 0
su(2) ⊕ suc (2) b1 = 0, b2 = 0
R ⊕ suc (2) b2 = 0
.
The corresponding Spin (7)-structure is of type W1 or W2 if and only if a1 = 0 or b1 = b2 = 0 respectively. The characteristic Ricci tensor Ricc has the shape Ricc = diag (λ, λ, λ, λ, κ, κ, κ, 0) depending on the parameters of the torsion form, λ = 3 (a1 + b1 ) (4 a1 − 3 b1 ) − b22 , κ = 4 (a1 + b1 ) (3 a1 − 4 b1 ). We proceed with the holonomy classification. System (•) becomes inconsistent for su (2) h g = g2 . Moreover, we deduce κ = 0 in the case of hol (∇ c ) = u (2) or su (2) with iso (Tc ) = su (2) ⊕ suc (2).
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5.1.1. The subcases hol (∇ c ) R ⊕ suc (2), so (3). Applying step (5) we are able to compute the characteristic curvature tensor Rc = r1 · (P7 ⊗ P7 ) + r2 · ((P7 + 2 P8 ) ⊗ (P7 + 2 P8 ) + Q 5 ⊗ Q 5 + Q 6 ⊗ Q 6 ), which depends on the torsion parameters in the following way: r1 =
3 3 1 κ − λ = − (a1 + b1 ) (5 a1 − 2 b1 ) + b22 , r2 = − κ. 8 2 8
We arranged the necessary conditions on the parameters r1 and r2 for each holonomy algebra hol (∇ c ) R ⊕ suc (2) or so (3) in the following table: hol ∇ c constraints
suc (2) r1 = 0
t2 , t1 [l = 0] r2 = 0
so(3), t1 [l = 0], 0 r1 = r2 = 0
.
Here 0 denotes the zero algebra. c 5.1.2. The subcase hol (∇ ) = soir (3) There exists only one soir (3)-invariant curvature tensor Rc : 2 R8 → soir (3), namely the projection onto the algebra soir (3),
Rc = −a12 · (U1 ⊗ U1 + U2 ⊗ U2 + U3 ⊗ U3 ). Here (U1 , U2 , U3 ) denotes the following basis of soir (3): U1 := 5/2 P5 − 3/2 Q 2 , U2 := 5/2 P6 + 3/2 Q 1 , U3 := P7 + 3 P8 . 5.2. The cases iso (T c ) = so (3), su (3). There are two admissible families of characteristic torsions. The first one depends on a single positive parameter, TcI = a1 · Z ∧ e7 , a1 ∈ R, a1 > 0, whilst the second is a 3-parameter family TcII = a1 · D¯ + a2 · (2 D1 + 5 D2 + 3 D5 ) + b1 · (D1 − D2 − 2 D5 ) with a1, a2, b1 ∈ R, b1 > 0. The isotropy algebra of type I is the algebra su (3), i.e. iso TcI = su (3). The condition iso TcII = su (3) holds if and only if a1 = 0, b1 = 23 a2 , TcII ∼ D = D1 + D2 . The Spin (7)-structures of this subsection are not of type W2 . They are of type W1 if only if Tc is of type II and a1 = a2 = 0. The characteristic Ricci tensor is Ricc = diag (λ, λ, λ, λ, λ, λ, 0, 0), and depends on the torsion type λ I = 2 a12 , λ II = 4 a12 + 4 (2 a2 + b1 ) (5 a2 − b1 ) accordingly.
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5.2.1. The subcase hol (∇ c ) so (3). The curvature operator Rc : 2 R8 → so (3) is the projection onto the subalgebra so (3) < spin (7) scaled by the parameter λ above, Rc = −
λ · (V1 ⊗ V1 + V2 ⊗ V2 + V3 ⊗ V3 ) . 2
The basis (V1 , V2 , V3 ) of so (3) is V1 :=
1/2 (P1 + P5 ), V2 := 1/2 (P2 + P6 ), V3 := P7 + P8 .
For hol (∇ c ) = t1 < so (3) and in the case of trivial holonomy, hol (∇ c ) = 0, the parameter λ has to vanish necessarily. 5.2.2. The subcase hol (∇ c ) = t2 The characteristic curvature Rc : 2 R8 → t2 is Rc = −
λ · (3 (P7 ⊗ P7 ) + (P7 + 2 P8 ) ⊗ (P7 + 2 P8 )) 4
for the parameter λ above. 5.3. The case iso (T c ) = u (2). The characteristic torsion is an element of one of the following two 3-parameter families: TcI = a1 · (Z 1 + 5 Z 2 ) ∧ e8 + a2 · (Z 1 + 5 Z 2 ) ∧ e7 + b1 · (Z 1 − 2 Z 2 ) ∧ e7 ,
TcII
7 ¯ 7 = a1 · (Z 1 − 2 Z 2 ) ∧ e8 + D + a2 · (Z 1 − 2 Z 2 ) ∧ e7 + D 4 4 +b1 · (Z 1 − 2 Z 2 ) ∧ e7 .
Here a1 , a2 , b1 ∈ R, b1 > 0. Not all parameter configurations are admissible: the isotropy algebra iso TcI of type I contains the algebra su (2) ⊕ suc (2) if a1 = 0 and b1 = −a2 , and the condition su (2) ⊕ suc (2) iso TcII holds if a1 = 0 and b1 = 43 a2 . The isotropy algebra is iso (Tc ) = su (3) if and only if a1 = 0 and b1 = 43 a2 (for type I ) or a1 = 0 and b1 = −a2 (for type II ) respectively. These four cases have to be excluded. The considered Spin (7)-structures are not of type W2 . Those of type W1 satisfy a1 = a2 = 0. In this particular case both torsion families coincide, i.e. TcI = TcII . The Ricci tensor of ∇ c is Ricc = diag (λ, λ, λ, λ, κ, κ, 0, 0). The constants λ and κ depend on the torsion type, λ I = 6 a12 + (a2 + b1 ) (6 a2 − b1 ) , κ I = 10 a12 + 2 (a2 + b1 ) (5 a2 − 2 b1 ), 45 2 33 2 a1 + a22 − 2 a2 b1 − b12 , κ II = a1 + a22 − 8 a2 b1 − 4 b12 . λ II = 4 4
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5.3.1. The subcase hol (∇ c ) t2 . Proposition 5.2 allows to compute the curvature tensor of the characteristic connection, Rc = r1 · (P7 ⊗ P7 ) + r2 · (P7 + 2 P8 ) ⊗ (P7 + 2 P8 ) , where r1 and r2 are given in terms of the parameters λ and κ above, r1 =
1 1 κ − λ, r2 = − κ. 4 4
Conditions on these parameters, given a specific holonomy algebra hol (∇ c ) t2 , are the following: hol ∇ c constraints
t1 [k = 0] r1 = 0
t1 [l = 0] r2 = 0
t1 [k, l = 0], 0 r1 = r2 = 0
.
The condition r1 = r2 = 0 can only be realized for TcI with a1 = 0 and b1 = −a2 , one of the excluded possibilities. Consequently, there exists no Spin (7)-structure with parallel characteristic torsion, iso (Tc ) = u (2) and hol (∇ c ) = t1 [k, l = 0] or 0.
5.4. The case iso (T c ) = R ⊕ su (2). Here the characteristic torsion form Tc is an element of the 1-parameter family Tc = b1 · (D3 − D4 ) , b1 ∈ R, b1 > 0, and Ricc is given by Ricc = diag 0, 0, 0, 0, −4 b12 , −4 b12 , 0, 0 . The corresponding Spin (7)-structure is of type W1 . The Ricci tensor of a ˜t2 -invariant and symmetric curvature operator is an element of the 3-parameter family diag (α + β + γ , α + β + γ , α + β − γ , α + β − γ , 4 β, 4 β, 16 β, 16 β) , α, β, γ ∈ R. Thus R ⊕ su (2) is the only admissible (i.e. Tc = 0) characteristic holonomy algebra for iso (Tc ) = R ⊕ su (2).
5.5. The admissible isotropy and holonomy algebrae. Summarizing the previous subsections, the following table provides an overview of the isotropy and holonomy algebrae which comply with the requirements of Steps (1) to (5) and lead to non-vanishing characteristic torsion:
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iso Tc g2 su(3) su(2) ⊕ suc (2) u(2) R ⊕ suc (2) so(3) soir (3) R ⊕ su(2)
hol ∇ c c K hol ∇ K hol ∇ c = 0 (⇒ ∇ c Rc = 0) = 0 g2 , su(2) ⊕ suc (2) R ⊕ suc (2), soir (3) su(3), u(2) so(3), t2 su(2) ⊕ suc (2), u(2), su(2) R ⊕ suc (2), suc (2), so(3), t2 , t1 , 0 u(2), su(2) t2 , t1 — R ⊕ suc (2), suc (2), t2 , t1 , 0 — so(3), t1 , 0 — soir (3) R ⊕ su(2) —
6. Geometric Results In this section we discuss the geometries related to the algebraic cases of Sect. 5. The most important tool in these considerations is the splitting theorem of de Rham generalized to geometric structures with totally skew-symmetric torsion (see [6]): Theorem 6.1. Let (M n , g, T) be a complete, simply connected Riemannian manifold with 3-form T. Suppose the tangent bundle T M n = T M+ ⊕ T M− g
splits under the action of the holonomy group of ∇ X Y = ∇ X Y +
1 2
· T (X, Y, · ) so that
T (X + , X − , · ) = 0, T (X + , Y+ , · ) ∈ T M+ , T (X − , Y− , · ) ∈ T M−
(∗)
for all X + , Y+ ∈ T M+ and X − , Y− ∈ T M− . Let T = T+ + T− denote the corresponding decomposition of the 3-form T. Then (M, g, T) is isometric to a Riemannian product (M+ , g+ , T+ ) × (M− , g− , T− ). The condition ∇T = 0 results in ∇ + T+ = 0, ∇ − T− = 0 for g
∇ X+ Y := ∇ X+ Y +
1 1 g · T+ (X, Y, · ) , ∇ X− Y := ∇ X− Y + · T− (X, Y, · ). 2 2
We split the consideration in the same manner as in Sect. 5 and start with the most obvious cases. 6.1. The case iso (T c ) = soir (3). The characteristic holonomy hol (∇ c ) is equal to c c soir (3), the ∇8 -parallel torsion form T is proportional to (Z ∧ e7 + D) and the tangent bundle of M , g, splits into the following soir (3)-invariant components: T M 8 = E ⊕ R · e8 . There exist two spinor fields which are parallel with respect to ∇ c , namely 1 − 2 and 9 − 10 (cf. Sect. 4). The curvature tensor Rc is uniquely determined, soir (3)invariant, ∇ c -parallel and Rc = Rc | E ⊕ 0|R·e8 (see Sect. 5). Since the torsion form Tc does not depend on e8 and ∇ c e8 = 0, we conclude that e8 is ∇ g -parallel. Consequently, a complete and simply connected M 8 is the Riemannian product of a 7-dimensional manifold Y 7 with R. We furthermore conclude that the space Y 7 is isometric to a naturally reductive, nearly parallel G2 -structure with fundamental form (Z ∧ e7 + D) and
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characteristic holonomy algebra soir (3) (see [12]). Consider the embedding of SO (3) into SO (5) given by the 5-dimensional irreducible SO (3)-representation. This gives rise to the homogeneous naturally reductive space SO (5) /SOir (3). With [11,14] we obtain that Y 7 is isometric to SO (5) /SOir (3). Theorem 6.2. A complete, simply connected Spin (7)-manifold with parallel characteristic torsion, ∇ c Tc = 0, and iso (Tc ) = soir (3) is isometric to the Riemannian product of SO (5) /SOir (3) with R. 6.2. The cases iso (T c ) = so (3), su (3). The Spin (7)-structure M 8 , g, admits two ∇ c -parallel vector fields e7 , e8 and four ∇ c -parallel spinor fields 1 , 2 , 9 , 10 . Moreover, the differential forms Z and D are parallel with respect to ∇ c , and we can reconstruct a Spin (7)-structure by using (). There are two types of characteristic torsion. First we discuss type I : Tc ∼ Z ∧ e7 and iso (Tc ) = su (3). The torsion form vanishes along e8 , i.e. e8 Tc = 0, and the tangent bundle splits into two hol (∇ c )-invariant components, T M 8 = E ⊕ R · e8 . Consequently, a complete, simply connected M 8 is isometric to the Riemannian product of a7-dimensional Y 7 with R. The torsion Tc is contained in the space 7manifold 73-form 3 3 7 3 1 R ⊕ 27 R of the decomposition of R into irreducible G2 -components. Up to isometry the space Y 7 admits a cocalibrated G2 -structure with fundamental form (Z ∧ e7 + D) and a characteristic connection with totally skew-symmetric torsion equal to Tc . The holonomy algebra of this connection coincides with hol (∇ c ). Consequently, Y 7 is homothetic to an η-Einstein Sasakian 7-manifold with contact vector field e7 and fundamental form Z (see [11]). Theorem 6.3. Let M 8 , g, be a complete, simply connected Spin (7)-manifold with parallel characteristic torsion Tc and iso (Tc ) = su (3). Suppose that the torsion form is of type I , i.e. Tc ∼ Z ∧ e7 . Then M 8 is isometric to the Riemannian product of a 7-dimensional, simply connected, η-Einstein Ricg¯ = 10 · g¯ − 4 · e7 ⊗ e7 Sasakian manifold Y 7 , g, ¯ e7 , Z with R. Conversely, such a product admits a Spin (7)structure with parallel characteristic torsion and hol (∇ c ) contained in su (3). Remark 6.1. Simply connected Sasakian manifolds M 7 , g, ξ, ϕ which admit the Ricci tensor of the last theorem can be constructed via the Tanno deformation of 7 ˜ ˜ ξ˜ , ϕ˜ . This deformation is a 7-dimensional Einstein-Sasakian structure M , g, ϕ := ϕ, ˜ ξ := a 2 · ξ˜ , g := a −2 · g˜ + a −4 − a −2 · η˜ ⊗ η˜ with the deformation parameter a 2 = 23 (see [15]). We recommend the article [4] for further constructions of Sasakian structures of η-Einstein type.
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Example 6.1. The algebraic classification in Sect. 5 proves that for hol (∇ c ) = so (3) or t2 the corresponding Spin (7)-structure M 8 is a homogeneous naturally reductive space. Since the curvature tensor of the characteristic connection does not depend on e8 , we can conclude the same for the Sasakian manifold Y 7 and denote Y 7 = G/H. In [11] Y 7 was identified for characteristic holonomy h = hol (∇ c ) = so (3). Here the corresponding naturally reductive space is isometric to the Stiefel manifold Y 7 = SO (5) /SO (3). We finally discuss the case h = hol (∇ c ) = t2 . The Lie algebra g of the 9-dimensional automorphism group G is given by g = t2 ⊕ R7 with the bracket [A + X, B + Y ] = [A, B] − Rc (X, Y ) + A · Y − B · X − Tc (X, Y ) . It turns out that the corresponding Killing form is non-degenerate, and thus g is semisimple. Consequently, g is isomorphic to su (2) ⊕ c (su (2)), where c (su (2)) denotes the centralizer of su (2) inside spin (7). We proceed with torsion type II . The torsion form does not depend on e7 and e8 , and the tangent bundle splits into the following hol (∇ c )-invariant components: T M 8 = E1 ⊕ E2 .
Here E 2 is spanned by {e7 , e8 }. The torsion form Tc belongs to the 32 R6 ⊕ 312 R6 component of the decomposition of 3 R6 under the action of U (3) (see [3]). Consequently, if M 8 is simply connected and complete, then it is isometric to the Riemannian product of R2 with an almost Hermitian manifold X 6 of Gray-Hervella type W1 ⊕ W3 , with Kähler form Z and characteristic holonomy contained in iso (Tc ). The latter structures have been exhaustively studied in [3,21]. Theorem 6.4. A complete, simply connected Spin (7)-manifold with parallel characteristic torsion of type II in the class iso (Tc ) = so (3) or su (3) is isometric to the Riemannian product X 6 × R2 , where X 6 is an almost Hermitian manifold of GrayHervella type W1 ⊕ W3 with characteristic holonomy contained in iso (Tc ). Remark 6.2. Consider the special case hol (∇ c ) = su (3) = iso (Tc ) and torsion type II . Here Tc is proportional to D and X 6 is isometric to a strictly (i.e. non-Kähler) nearly Kähler manifold. Conversely, any Riemannian product X 6 × R2 with X 6 a strictly nearly Kähler manifold admits a Spin (7)-structure with parallel characteristic torsion and characteristic holonomy contained in su (3). 6.3. The cases iso (T c ) = g2 , su (2) ⊕ suc (2), R ⊕ suc (2). The vector field e8 , the spinor fields 1 − 2 , 9 − 10 and the globally defined 3-form (Z ∧ e7 + D) are parallel with respect to ∇ c . The tangent bundle T M 8 of M 8 , g, splits into two components preserved by ∇ c , T M 8 = E ⊕ R · e8 . The torsion form Tc satisfies e8 Tc = b2 (e12 − e34 ) and the real parameter b2 vanishes if and only if iso (Tc ) = g2 or su (2) ⊕ suc (2) (see Sect. 5). Consequently, we split the discussion into iso (Tc ) = g2 or su (2) ⊕ suc (2) and iso (Tc ) = R ⊕ suc (2).
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Suppose iso (Tc ) = g2 or su (2) ⊕ suc (2). Here the torsion form does not depend on e8 , and ∇ g e8 = 0. Therefore a complete and simply connected M 8 is isometric to the Riemannian product Y 7 with R. The 3-form Tc is contained 7 of a 37-manifold 7 3 in the component 1 R ⊕ 27 R of the decomposition of 3-forms on R7 into G2 -irreducible components (see [12]). Consequently, the space Y 7 is isometric to a cocalibrated G2 -manifold with fundamental form (Z ∧ e7 + D) and parallel characteristic torsion Tc . The corresponding characteristic holonomy of Y 7 is the subalgebra hol (∇ c ) of iso (Tc ). Finally, we can reconstruct the considered Spin (7)-structure using Eq. (). Theorem 6.5. Let M 8 , g, be a complete, simply connected Spin (7)-manifold with parallel characteristic torsion Tc and iso (Tc ) = g2 or su (2) ⊕ suc (2). Then M 8 is isometric to the Riemannian product of R with a cocalibrated G2 -manifold with parallel characteristic torsion and characteristic holonomy contained in iso (Tc ). Those Y 7 with non-Abelian characteristic holonomy were extensively studied in [11]. We provide an example for Y 7 with Abelian characteristic holonomy. This restricts necessarily to iso (Tc ) = su (2) ⊕ suc (2). Example 6.2. If hol (∇ c ) = 0 the torsion parameters satisfy b2 = 0 and b1 = −a1 . Moreover, the spinor fields 1 , . . . , 16 are ∇ c -parallel, the characteristic curvature tensor vanishes Rc = 0 and the torsion form Tc is proportional to Z 2 ∧ e7 = e567 . In particular, the latter implies that Tc does not depend on the ∇ c -parallel vector fields e1 , e2 , e3 , e4 and e8 . Computing the Lie bracket [ · , · ] of the Lie algebra corresponding to M 8 via Tc (X, Y, Z ) = −g ([X, Y ] , Z ) results in the conclusion that a complete and simply connected M 8 is isometric to the Riemannian product R5 × SU (2). We proceed with iso (Tc ) = R ⊕ suc (2). With the algebraic considerations of Sect. 5 we immediately obtain the following result: Proposition 6.1. Any Spin (7)-manifold with parallel characteristic torsion Tc and iso (Tc ) = R ⊕ suc (2) is isometric to a homogeneous naturally reductive space. Example 6.3. Let hol (∇ c ) = 0. Then the spinor fields 1 , . . . , 16 are ∇ c -parallel, Rc = 0 and the characteristic torsion is proportional to one of the following two 3-forms: α± = (Z 1 − 2 Z 2 ) ∧ e7 + D ±
√
3 · (Z 3 ∧ e8 ).
Each of these 3-forms can be reconstructed with α± = −g ([X, Y ] , Z ) using the bracket [ · , · ] of the respective Lie algebra √
su (3) = span (P4 , P3 , P1 , P2 , −P5 , −P6 , P7 , ±
1/3
(P7 + 2 P8 )).
We conclude that the considered Spin (7)-manifold M 8 is isometric to SU (3).
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6.4. The case iso (T c ) = R ⊕ su (2). The characteristic holonomy hol (∇ c ) is 8 necessarily equal to R ⊕ su (2). A Spin (7)-structure M , g, with non-trivial parallel characteristic torsion and hol (∇ c ) = R ⊕ su (2) = iso (Tc ) admits two ∇ c -parallel 2-forms Z 1 = e12 + e34 and Z 2 = e56 , two ∇ c -parallel spinor fields 9 and 10 , and the tangent bundle T M 8 splits into the sum of two R ⊕ su (2)-invariant subbundles, T M 8 = E1 ⊕ E2 . Here E 2 is spanned by {e7 , e8 }. The torsion form Tc does not depend on e7 and e8 , and therefore a complete, simply connected M 8 is isometric to the Riemannian product of a 2-dimensional manifold with a 6-dimensional manifold X 6 . The space X 6 is isometric to an almost Hermitian manifold with Kähler form Z = Z 1 + Z 2 and non-trivial parallel characteristic torsion Tc . This torsion form is contained in the 312 R6 -com ponent in the decomposition of 3 R6 under the action of U (3) (see [3]). Analyzing the representation of Hol (∇ c ) ⊂ U (1) × U (2) ⊂ U (3) on R6 ∼ = E 1 , we conclude with [14] that X 6 carries the structure of a twistor space and is homothetic to either CP3 or F (1, 2). The representation of Hol (∇ c ) on R2 ∼ = E 2 defines a non-trivial rotation. The 2-dimensional component is consequently isometric to S2 . We can reconstruct the considered Spin (7)-structure from the Hermitian structure of its components, =
1 · ω ∧ ω + Re (F) , 2
where ω := Z + e7 ∧ e8 ,
F := (e1 + i e2 ) ∧ (e3 + i e4 ) ∧ (e5 + i e6 ) ∧ (e7 + i e8 ).
Finally we obtain the following result. Theorem 6.6. A complete, simply connected Spin (7)-manifold with parallel characteristic torsion Tc and iso (Tc ) = R ⊕ su (2) is isometric to the Riemannian product of a 2-sphere with either the projective space CP3 or the flag manifold F (1, 2), both equipped with their standard nearly Kähler structure from the twistor construction. 6.5. The case iso (T c ) = u (2). The differential forms Z 1 , Z 2 , D, the vector fields e7 , c e8 and thespinor fields 1 , 2 , 9 , 10 are parallel with respect to ∇ and the tangent bundle of M 8 , g, splits into the following u (2)-invariant components: T M 8 = E 1 ⊕ E 2 ⊕ R · e7 ⊕ R · e8 . The subbundle E 2 is spanned by e5 and e6 . There are two types of characteristic torsion. We start to discuss the case of torsion type I . Setting 2 a2 − b1 · e7 + a1 · e8 T M+ = E 1 ⊕ ((a2 + b1 ) · e7 + a1 · e8 ), T M− = E 2 ⊕ 5 and T = T+ + T− = Tc satisfies system (∗) and T+ = Z 1 ∧ ((a2 + b1 ) · e7 + a1 · e8 ), T− = 5 · Z 2 ∧
2 a2 − b1 · e7 + a1 · e8 . 5
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The equation e7 Tc = 0 or e8 Tc = 0 holds, if a2 = b1 = 0 or a1 = 0 respectively. Consequently, a simply connected and complete M 8 is isometric to the Riemannian product of a 5-manifold with a 3-manifold each carrying a Sasakian structure. The respective fundamental forms are Z 1 and Z 2 . Theorem 6.7. Let M 8 , g, be a complete, simply connected Spin (7)-manifold with parallel characteristic torsion Tc and iso (Tc ) = u (2). Suppose that the torsion form is of type I . Then M 8 is isometric to the Riemannian product of a Sasakian 3-manifold with a 5-dimensional Sasakian structure. We proceed with the discussion of torsion type II . Solving the equation X Tc = 0 leads to X = a1 · e7 − (a2 + b1 ) · e8 = 0 (see Sect. 5). Consequently, a complete, simply connected M 8 is isometric to the Riemannian product of a 7-dimensional integrable G2 -manifold Y 7 with R (see [13]). Theorem 6.8. Let M 8 , g, be a complete, simply connected Spin (7)-manifold with parallel characteristic torsion Tc and iso (Tc ) = u (2). Suppose that the torsion form is of type II . Then M 8 is isometric to the Riemannian product of R with an integrable G2 -manifold with parallel characteristic torsion and characteristic holonomy contained in u (2). Thanks. We wish to thank Nils Schömann and Richard Cleyton for discussions and the SFB 647: Space–Time–Matter for financial support.
References 1. Agricola, I., Friedrich, T.: The Casimir operator of a metric connection with totally skew-symmetric torsion. J. Geom. Phys. 50, 188–204 (2004) 2. Agricola, I., Friedrich, T., Nagy, P.-A., Puhle, C.: On the Ricci tensor in the common sector of type II string theory. Class. Quant. Grav. 22, 2569–2577 (2005) 3. Alexandrov, B., Friedrich, T., Schoemann, N.: Almost Hermitian 6-manifolds revisited. J. Geom. Phys. 53, 1–30 (2005) 4. Boyer, C.P., Galicki, K., Matzeu, P.: On eta-Einstein sasakian geometry. Commun. Math. Phys. 262, 177–208 (2006) 5. Cabrera, F.M.: On Riemannian manifolds with Spin (7)-structure. Publ. Math. 46, 271–283 (1995) 6. Cleyton, R., Moroianu, A.: Connections with parallel torsion in Riemannian geometry. to appear, can be obtained from the authors. 7. Cleyton, R., Swann, A.: Einstein metrics via intrinsic or parallel torsion. Math. Z. 247, 513–528 (2004) 8. Dynkin, E.B.: Semisimple subalgebras of semisimple Lie algebras. Am. Math. Soc., Transl., II. Ser. 6, 111–243 (1957) 9. Dynkin, E.B.: Maximal subgroups of the classical groups. Am. Math. Soc., Transl., II. Ser. 6, 245–378 (1957) 10. Fernández, M.: A classification of Riemannian manifolds with structure group Spin (7). Ann. Mat. Pura Appl., IV. Ser. 143, 101–122 (1986) 11. Friedrich, T.: G2 -manifolds with parallel characteristic torsion. Diff. Geom. Appl. 25, 632–648 (2007) 12. Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math 6, 303–336 (2002) 13. Friedrich, T., Ivanov, S.: Killing spinor equations in dimension 7 and geometry of integrable G2 -manifolds. J. Geom. Phys. 48, 1–11 (2003) 14. Friedrich, T., Kath, I., Moroianu, A., Semmelmann, U.: On nearly parallel G2 -structures. J. Geom. Phys. 23, 256–286 (1997) 15. Friedrich, T., Kim, E.C.: The Einstein-Dirac equation on Riemannian spin manifolds. J. Geom. Phys. 33, 128–172 (2000) 16. Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982) 17. Ivanov, S.: Connections with torsion, parallel spinors and geometry of Spin (7)-manifolds. Math. Res. Lett. 11, 171–186 (2004)
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18. Puhle, C.: The Killing spinor equation with higher order potentials (2007). J. Geom. Phys. 58, 1355–1375 (2008) 19. Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253–284 (1986) 20. Wang, M.Y.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7, 59–68 (1989) 21. Schoemann, N.: Almost hermitian structures with parallel torsion. J. Geom. Phys. 57, 2187–2212 (2007) Communicated by G. W. Gibbons
Commun. Math. Phys. 291, 321–345 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0877-2
Communications in
Mathematical Physics
Making Almost Commuting Matrices Commute M. B. Hastings Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] Received: 20 August 2008 / Accepted: 1 May 2009 Published online: 18 July 2009 – © Springer-Verlag 2009
Abstract: Suppose two Hermitian matrices A, B almost commute ([A, B] ≤ δ). Are they close to a commuting pair of Hermitian matrices, A , B , with A− A , B − B ≤ ? A theorem of H. Lin [3] shows that this is uniformly true, in that for every > 0 there exists a δ > 0, independent of the size N of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how δ depends on . We give uniform bounds relating δ and . The proof is constructive, giving an explicit algorithm to construct A and B . We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements. The problem of when two almost commuting matrices are close to matrices which exactly commute, or, equivalently, when a matrix which is close to normal is close to a normal matrix, has a long history. See, for example [1,2], and other references in [3] where it is mentioned that the problem dates back to the 1950s or earlier. Finally in 1995, Lin [3] proved that for any > 0, there is a δ > 0 such that for all N , for any pair of Hermitian N -by-N matrices, A, B, with A, B ≤ 1, and [A, B] ≤ δ, there exists a pair A , B with [A , B ] = 0 and A − A ≤ and B − B ≤ This proof was later shortened and generalized by Friis and Rordam [4]. Interestingly, the same is not true for almost commuting unitary matrices [5] or for almost commuting triplets [6,7]. The importance of the above results is that the bound is uniform in N . That is, δ depends only on . Unfortunately, the proofs do not give any bounds on how δ depends on . Further, the proofs of Lin and Friis and Rordam are nonconstructive, so there is no known way to find the matrices A and B . In this paper, we present a construction of matrices A and B which enables us to give lower bounds on how small δ must be to obtain a given error .
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Specifically, we prove that Theorem 1. Let A and B be Hermitian, N -by-N matrices, with A, B ≤ 1. Suppose [A, B] ≤ δ. Then, there exist Hermitian, N -by-N matrices A and B such that 1 [A , B ] = 0. 2 A − A ≤ (δ) and B − B ≤ (δ), with (δ) = E(1/δ)δ 1/6 ,
(1)
where the function E(x) grows slower than any power of x. The function E(x) does not depend on N . Throughout this paper, we use . . . to denote the operator norm of a matrix, and | . . . .| to denote the l 2 -norm of a vector. The proof of Theorem 1 involves first constructing a related problem involving a block tridiagonal matrix, H , and a block identity matrix X (we use the term “block identity matrix” to refer to a block diagonal matrix that is proportional to the identity matrix in each block). For such matrices we prove the theorem Theorem 2. Let X be a block identity Hermitian matrix and let H be a block tridiagonal Hermitian matrix, with the j th block of X equal to c + j times the identity matrix, for some constants c and . Let H , X ≤ 1. Then, there exist Hermitian matrices A and B such that 1 [A , B ] = 0. 2 A − H ≤ () and B − X ≤ (), with () = E (1/)1/5 ,
(2)
where the function E (x) grows slower than any power of x. The function E (x) does not depend on the dimension of the matrices. After proving these results, we prove a tighter bound in the case where H is a tridiagonal matrix, rather than a block tridiagonal matrix: Theorem 3. Let X be a diagonal Hermitian matrix and let H be a tridiagonal Hermitian matrix, with the j th diagonal entry of X equal to c + j, for some constants c and . Let H , X ≤ 1. Then, there exist Hermitian matrices A and B such that 1 [A , B ] = 0. 2 A − H ≤ () and B − X ≤ (), with () = E (1/)1/2 ,
(3)
where the function E (x) grows slower than any power of x. The function E (x) does not depend on the dimension of the matrices. The proofs rely heavily on ideas relating to Lieb-Robinson bounds [8–11]. These bounds, combined with appropriately chosen filter functions, have been used in recent years in Hamiltonian complexity to study the dynamics and ground states of quantum systems, obtaining results such as a higher dimensional Lieb-Schultz-Mattis theorem [9], a proof of exponential decay of correlations [14], studies of dynamics out of equilibrium [15–17], new algorithms for simulation of quantum systems [18–22], an area law for entanglement entropy for general interacting systems [23], study of harmonic
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lattice systems [24], a Goldstone theorem with fewer assumption [25], and many others. The present paper represents a different application, to the study of almost commuting matrices. Before beginning the proof, we give some discussion of physics intuition behind the result. The next few paragraphs are purely to motivate the problems from a physics viewpoint. In the last section on quantum measurement and in the discussion at the end we give additional applications to quantum measurement and construction of Wannier functions. The section on quantum measurement is intended to be self-contained. As mentioned, we begin by relating this problem to the study of block tridiagonal matrices. We then interpret the matrix H as a Hamiltonian for a single particle moving in one dimension, and apply the Lieb-Robinson bounds. The result (2) implies that we can construct a complete orthonormal basis of states which are simultaneously localized in both position (X ) and energy (H ). It is certainly easy to construct an overcomplete basis of states which is localized in both position and energy, by considering, for example, Gaussian wavepackets. The interesting result is the ability to construct an orthonormal basis which satisfies this. Additional physics intuition can be obtained by considering the case where H is a tridiagonal matrix with 0 on the diagonal and elements just above and below the diagonal equal to 1, and where X is a diagonal matrix with entries 1/N , 2/N , . . .. We refer to this as a uniform chain. In the uniform chain case, if we define a new matrix H by randomly perturbing H , replacing each diagonal element of H with a small diagonal number chosen at random, the eigenvectors of H are localized with high probability [26,27]. Then, we can construct a matrix X which exactly commutes with H as follows: if v is an eigenvector of H , we choose it to have eigenvalue for X equal to (v, X v). Then, since the eigenvectors are localized, we find that X − X is small. The difference X − X depends on the localization length which depends inversely on the amount of disorder, while the difference H − H depends on the amount of disorder. Unfortunately, we do not have a good enough understanding of the effect of disorder for matrices H which are block tridiagonal, rather than just tridiagonal, to turn this approach into a proof for general H and X , and thus we rely on an alternative, constructive approach.
I. Proof of Main Theorem We now outline the proof of Theorem 1. The proof is constructive, and is described by the following algorithm: 1 Construct H from A as described in Sect. II A and Lemma 1. We will bound H − A. 2 Construct X from B as described in Sect. II B. We will bound X − B. In a basis of eigenvalues of X , the matrix H will be block tridiagonal. 3 Construct a new basis as described in Sect. III such that in this basis H is close to a block diagonal matrix. That is, we will bound the operator norm of the block offdiagonal part of H . The blocks will be different from the blocks considered in Step 2 above and will be larger. Further, we will show that X is close to a block identity matrix in this basis. 4 Set A to be the block diagonal part of H in the basis constructed in Step 3 and set B to the block identity matrix constructed in Step 3, so that [A , B ] = 0. This algorithm involves several choices of constants. In a final Section, V, we indicate how to pick the constants to obtain the error bound (1). The key step will be Step 3.
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II. Reduction to Block Tridiagonal Problem The first two steps of the proof above (1,2) reduce Theorem 1 to Theorem 2, while the last two steps (3,4) prove Theorem 2. In this section we present the first two steps. A. Construction of finite-range H . We begin by constructing matrix H as given in the following lemma, where the constant will be chosen later. Lemma 1. Given Hermitian matrices A and B, with [A, B] ≤ δ, for any there exists a Hermitian matrix H with the following properties: 1 [H, B] ≤ δ. 2 For any two vectors v1 , v2 which are eigenvectors of B with corresponding eigenvalues x1 , x2 , and with |x1 − x2 | ≥ , we have (v1 , H v2 ) = 0. 3 A − H ≤ 1 , with 1 = c0 δ/, where c0 is a numeric constant given below. Proof. We define
H =
dt exp(i Bt)A exp(−i Bt) f (t),
(4)
where the function f (t) is defined to have the Fourier transform f˜(ω) = (1 − ω2 )3 , |ω| ≤ 1, f˜(ω) = 0, |ω| ≥ 1,
(5)
and hence the Fourier transform of f (t) is supported on the interval [−, ]. Properties (1) and (2) follow immediately from Eq. (4). Property (3) follows from H − A = dt exp(i Bt)A exp(−i Bt) f (t) − A = dt (exp(i Bt)A exp(−i Bt) − A) f (t) ≤ dt (exp(i Bt)A exp(−i Bt) − A) | f (t)| ≤ dt |t|[A, B] | f (t)| ≤ δ dt |t f (t)| = c0 δ/,
(6)
where we define the constant c0 by c0 =
dt|t f (t)|.
(7)
The second line in Eq. (6) follows because f˜(0) = 1 so that dt A f (t) = A. Note that since the first and second derivatives of f˜(ω) vanish at ω = ±1, the function f (t) decays as 1/t 3 for large t and hence c0 is finite. Since f˜ is an even function, H is Hermitian.
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Note that the precise form of the function f (t) is unimportant: all we require is that f˜(0) = 1; that f˜ is supported on the interval [−1, 1]; that f˜ is sufficiently smooth that f (t) decays fast enough for the integral over t (7) to converge; and that f˜ is an even function. Remark. In a basis of eigenvectors of B, Property 3 in the above lemma implies that H is “finite-range”, in that the off-diagonal elements are vanishing for sufficiently large |x1 −x2 |. The next theorem is a Lieb-Robinson bound for such finite range Hamiltonians, similar to those proven for many-body Hamiltonians [8–11]. This result is also similar to results on the decay of entries of smooth functions of matrices proven in [12,13]. We now introduce some terminology. Given two sets of real numbers, S1 , S2 , we define dist(S1 , S2 ) =
min
x1 ∈S1 ,x2 ∈S2
|x1 − x2 |.
(8)
Remark. The reason for introducing this “distance function” is that we think of H as defining the Hamiltonian for a one-dimensional, finite-range quantum system, with different “sites” of the system corresponding to different eigenvectors of B, and then the distance function is the distance between different sets of sites. Further, we say that a vector w is “supported on set S for position operator B” if w is a linear combination of eigenvectors of B whose corresponding eigenvalues are in set S. Finally, for any set S we define the projector P(S, B) to be the projector onto eigenvectors of B whose corresponding eigenvalues lie in set S. We now give the Lieb-Robinson bound: Theorem 4. Let H have the properties 1 H ≤ 1. 2 For any two vectors v1 , v2 which are eigenvectors of B with corresponding eigenvalues x1 , x2 , and with |x1 − x2 | ≥ , we have (v1 , H v2 ) = 0. Define v L R = e2 .
(9)
Then, for any vector v supported on a set S1 for position operator B, and for any projector P(S2 , B), we have |P(S2 , B) exp(−i H t)v| ≤ e−dist(S1 ,S2 )/ |v|
(10)
|t| ≤ dist(S1 , S2 )/v L R .
(11)
for any Proof. Expand exp(−i H t)v in a power series as v − i H tv − (H 2 /2)t 2 v + . . .. Then, by assumption, P(S2 , B)(−it)n (H n /n!)v vanishes for n < dist(S1 , S2 )/. Let m = dist(S1 , S2 )/ . Then, |
(−it)n (H n /n!)v| ≤
n≥m
(|t|n /n!)|v|
n≥m
≤
1 1 1 |v|. (12) (e|t|/n)n |v| ≤ (e|t|/m)m e n≥m e 1 − e|t|/m
For the given v L R and t, the result follows.
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Remark. The proof of this Lieb-Robinson bound is significantly simpler than the proofs of the corresponding bounds for many-body systems considered elsewhere. The power series technique used here does not work for such systems.
B. Construction of X . In this subsection, we construct the matrix X from B. We define a function Q(x) by Q(x) = x/ + 1/2 .
(13)
X = Q(B).
(14)
Then, we set
Note that |Q(x) − x| ≤ /2 for all x, and Q(x)/ is always an integer. Then, X − B ≤ 2 ,
(15)
2 = /2.
(16)
with
By 2 in Lemma 1, the matrix H is a block tridiagonal matrix when written in a basis of eigenstates of X , with eigenvalues of X ordered in increasing order. III. Construction of New Basis In this section we construct the basis to make H close to a block diagonal matrix and X close to a block identity matrix. This completes Step 3 of the construction of A and B . We refer to the basis constructed in this step as the “new basis” and we refer to the basis in which X is diagonal as the “old basis”. There will be a total of n cut + 1 different blocks in the new basis, where n cut is chosen later. Before constructing the new basis, we give some definitions. We define an interval Ii by Ii = [−1 + 2(i − 1)/n cut , −1 + 2i/n cut ) for 1 ≤ i < n cut and Ii = [−1 + 2(i − 1)/n cut , −1 + 2i/n cut ] for i = n cut . Let Ji be the matrix given by projecting H onto the subspace of eigenvalues of X lying in this interval Ii , and call this subspace Bi . Then, in the old basis of eigenvalues of X , Ji is block tridiagonal with at least L different blocks, where L = (2/n cut ) − 1 (some of these blocks might have dimension zero if B happens to have fewer than L distinct eigenvalues in that interval). We will choose n cut later so that L >> 1 and so the new basis will have fewer blocks than the old basis. Before constructing the new basis we need the following lemma. We claim that: Lemma 2. Let J be a Hermitian block tridiagonal matrix, with J ≤ 1, acting on a space B. Let there be L blocks, so that the space B has L orthogonal subspaces, which we write V j for j = 1, . . . , L, with (v, J w) = 0 for v ∈ Vi and w ∈ V j with |i − j| > 1. Then, there exists a space W which is a subspace of B with the following properties: 1 The projection of any normalized vector v ∈ V1 onto the orthogonal complement of W has norm bounded by 3 where 3 is equal to 1/L 1/4 times a function growing slower than any power of L.
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2 For any normalized vector w ∈ W, the projection of J w onto the orthogonal complement of W has norm bounded by 4 , where 4 is equal to 1/L 1/4 times a function growing slower than any power of L. 3 The projection of any normalized vector v ∈ V L onto W has norm bounded by 5 , where 5 is equal to a function decaying faster than any power of L. Proof. This lemma is the key step in the proof of the main theorem, and the proof of this lemma is given in the next section. For each i, 1 ≤ i ≤ n cut , we apply Lemma 2 to the matrix J = Ji defined on the space B = Bi . For given i, we refer to the space W as constructed in Lemma 2 as Wi and we refer to the subspaces V j defined in Lemma 2 as V j (i). Let Bi have dimension D B (i) and let Wi have dimension DW (i). Let Wi⊥ denote the D B (i) − DW (i)-dimensional space which is the orthogonal complement of Wi . Let V j (i) have dimension d j (i). By Properties 1 ,2 in Lemma 2, D B (i) ≥ d1 (i) and D B (i) ≤ DW (i) − d L (i). The new basis has n cut + 1 blocks, which we label by i = 0, 1, . . . , n cut . For 1 ≤ i < n cut , we define the i th block of the new basis to be the space spanned by Wi+1 and Wi⊥ . For i = 0, the i th block is the space Wi+1 = W1 . For i = n cut , the i th block is the space Wi⊥ = Wn⊥cut . Then, the matrix H is block tridiagonal in this new basis. The block-off-diagonal terms arise from three sources. First, the matrix Ji contains non-vanishing matrix elements between the spaces Wi and Wi⊥ , and those spaces are now in different blocks. However, by Property 2 in Lemma 2, these matrix elements are bounded by 4 . Second, ⊥ and V i , and V i there are non-vanishing matrix elements between the subspace Wi−1 1 1 may not be completely contained in subspace Wi . However, by Property 1 in Lemma 2, these contribute only 3 to the norm of the block-off-diagonal terms of H in the new basis. Third, there are non-vanishing matrix elements between Wi and V Li , and V Li may not be completely contained in subspace Wi⊥ . However, by Property 1 in Lemma 2, these contribute only 5 to the norm of the block-off-diagonal terms of H in the new basis. Therefore, the block-off-diagonal terms in H are bounded in operator norm by 2(3 + 4 + 5 ).
(17)
Define B to be the block identity matrix (in the new basis) which is equal to −1 + 2i/n cut times the identity matrix in the i th block. Since each block i in the new basis lies within the space spanned by Bi and Bi+1 we have B − B ≤ 2/n cut .
(18)
Remark. Here is a sketch of the above procedure, in a case where H has 8 blocks and n cut = 2. The matrix originally looks like ⎞ ⎛ ... ... ⎟ ⎜. . . . . . . . . ⎟ ⎜ ⎟ ⎜ ... ... ... ⎟ ⎜ ... ... ... ⎟ ⎜ (19) ⎟ ⎜ ... ... ... ⎟ ⎜ ⎟ ⎜ ... ... ... ⎟ ⎜ ⎝ . . . . . . . . .⎠ ... ...
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where the . . . indicate non-vanishing entries. We combine the entries in the first 4 blocks into a matrix J1 and the entries in the last 4 into a matrix J2 so H looks like J1 . . . , (20) . . . J2 , where the . . . couples only the L th block of space B1 to the 1st block of space B2 . Then, we apply Lemma 2 to decompose B1 into spaces W1 and W1⊥ so that J1 looks like . . . O(4 ) (21) O(4 ) . . . and similarly for J2 . Inserting Eq. (21) into Eq. (20), H looks like (in the new basis) ⎛ ⎞ . . . O(4 ) O(5 ) . . . O(3 )⎟ ⎜ O(4 ) . . . (22) ⎝ O( ) . . . . . . O(4 )⎠ 5 O(3 ) O(4 ) . . . which is close to the block diagonal matrix, ⎛ ... ⎜ ... ... ⎝ ... ...
⎞ ⎟ ⎠
(23)
...
which has 3 = n cut + 1 blocks. IV. Proof of Lemma 2 Let the space V1 be d1 dimensional, with orthonormal basis vectors v1 , . . . , vd1 . Let S denote the D B -by-d1 matrix whose columns are these basis vectors, so that S is an isometry. Define a function F(ω0 , r, w, ω) as follows. Let F(0, 0, 1, ω) = 1 for ω = 0. Let F(0, 0, 1, ω) = 0 for |ω| ≥ 1. Let F(0, 0, 1, ω) = F(0, 0, 1, −ω). For 0 ≤ ω ≤ 1, choose F(0, 0, 1, ω) to be infinitely differentiable so that the Fourier transform of ˜ F(0, 0, 1, ω), which we write F(0, 0, 1, t), is bounded by a function which decays faster than any polynomial. Finally, we impose F(0, 0, 1, ω) + F(0, 0, 1, 1 − ω) = 1 for 0 ≤ ω ≤ 1. For general ω0 , r, w, define the function F(ω0 , r, w, ω) by F(ω0 , r, w, ω) = 1 for |ω − ω0 | ≤ r , and F(ω0 , r, w, ω) = F(0, 0, 1, (|ω − ω0 | − r )/w) for |ω − ω0 | ≥ r . Then F(ω0 , r, w, ω) = 0 for |ω − ω0 | ≥ r + w. For r ≥ 0 and w > 0, the function F(ω0 , r, w, ω) is infinitely differentiable with respect to ω everywhere. The functions F(0, 1, 1, ω) and F(0, 0, 1, ω) are sketched in Fig. 1a, b; the variable r denotes the width of the flat part at the center of the function, while w denotes the width of the changing part of the function. Since F(0, 0, 1, ω) is infinitely differentiable, there is a function T (x) which decays faster than any polynomial such that: ˜ 0 , w, w, t)| ≤ T (wt0 ), dt|F(ω |t|≥t0
|t|≥t0
(24) ˜ 0 , 0, w, t)| ≤ T (wt0 ). dt|F(ω
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(b)
(a)
Fig. 1. Sketch of (a) F (0, 1, 1, ω) = F (−1, 0, 1, ω) + F (0, 0, ω) + F (1, 0, 1, ω) and (b) F (0, 0, 1, ω)
The operator norm of J is bounded by 1. The idea of the proof is to divide the interval of eigenvalues of J , which is [−1, 1], into various small overlapping windows. Then, for each interval centered on a frequency ω, we will construct vectors given by approximately projecting vectors in V1 onto the space spanned by eigenvectors of J with eigenvalues lying in that interval; we call the spaces of these vectors Xi , where i labels the particular window. Then, each of these projected vectors x will have the property that J x is close to ωx. This will be the key step in ensuring Property 2 in the claims of the lemma. The idea of approximate projection is important here. In fact, we will use the smooth filter functions F(ω0 , r, w, ω) above. The smoothness will be essential to ensure that the vectors x have most of their amplitude in the first blocks rather than the last blocks. Since the vectors in the spaces Xi are approximate projections of vectors in V1 into different windows, we will be able to approximate any vector v1 ∈ V1 by a vector in the space spanned by the Xi simply by adding up the projections of v1 in each different window. Because the windows overlap, the vectors may not be orthogonal to each other; the overlap between vectors is something we will need to bound (see Eq. (43) below). To control the overlap, we choose W to be a subpace of the space spanned by the Xi as explained below; this will then require us to be careful to ensure that we are still able to approximate vectors in V1 by vectors in W. Let n win be some even integer chosen later. We will choose n win = L/F(L),
(25)
where the function F(L) is a function that grows slower than any power of L and is defined further below. The choice of function F(L) will depend only on the function T (x) defined above. For each i = 0, . . . , n win − 1, define ω(i) = −1 + 2i/(n win − 1).
(26)
κ = 2/(n win − 1),
(27)
Define
so ω(i) = −1 + iκ. n win −1 F(ω(i), 0, κ, ω) = 1 for When ω(i) and κ are chosen as above, we have i=0 −1 ≤ ω ≤ 1. See Fig. 2a to see a sketch of three functions F(ω(i − 1), 0, κ, ω), F(ω(i), 0, κ, ω), and F(ω(i + 1), 0, κ, ω); as F(ω(i), 0, κ, ω) decreases for ω(i) ≤ ω ≤ ω(i + 1), the function F(ω(i + 1), 0, κ, ω) is increasing to keep the sum constant. A. Construction of spaces Xi . To construct Xi , we define the matrix τi by τi = F(ω(i), 0, κ, J )S.
(28)
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(a)
(b)
Fig. 2. a) Sketch of overlapping windows. b) Re-arrangement of windows as discussed in section on tridiagonal matrices
Define λmin = 1/(n win L 2 ).
(29)
Compute the eigenvectors of the matrix τi† τi . For each eigenvector xa with eigenvalue greater than or equal to λmin compute ya = τi xa . Let Xi be the space spanned by all such vectors ya . Let Z i project onto the eigenvectors xa with eigenvalue less than λmin ; the projector Z i will be used later in computing the error estimates. Remark. To understand this construction, in Fig. 2a we sketch the functions F(ω(i − 1), 0, κ, ω), F(ω(i), 0, κ, ω), and F(ω(i +1), 0, κ, ω), which form partially overlapping windows. Note that the vectors F(ω(i), 0, κ, J )Sx1 and F(ω(i ± 1), 0, κ, J )Sx2 , for arbitrary x1 , x2 , need not be orthogonal. B. Properties of Xi . This subsection establishes certain properties of the Xi . It is primarily intended to motivate the construction thus far. We will show that the Xi have three properties which are closely related to the three properties we desire to show in Lemma 2. First, for any normalized vector v ∈ V1 , the projection√of v onto the orthogonal √ complement of the space spanned by the Xi is bounded by 2n win λmin = 2/L. To show this, for any v ∈ V1 , with |v| = 1, we write v = Sx with |x| = 1, and then |v −
n win −1
τi (1 − Z i )x| = | 2
n win −1
i=0
≤2
τi Z i x|2
i=0 n win −1
|τi Z i x|2
i=0
≤ 2n win λmin ≤ 2/L 2 .
(30)
The factor of 2 in the first inequality follows because (τi Z i x, τ j Z j x) = 0 for |i − j| > 1, but may be non-vanishing for i = j ±1. Similar factors of 2 occur in several other places. Second, each space Xi is an approximate eigenspace of J . That is, for any vi ∈ Xi , we have |(J − ω(i))vi | ≤ κ|vi |.
(31)
Third, for any vector y ∈ Xi , the norm of the projection of y onto V L is bounded by |y| times a function growing slower than any power of L. It is here that we will pick the
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˜ 0 , r, w, t) denote the Fourier function F(L) and use the Lieb-Robinson bounds. Let F(ω transform of F(ω0 , r, w, ω) with respect to the last variable ω. Then, for any x with (1 − Z i )x = x, we find that y = τi x = F(ω(i), 0, κ, J )Sx is equal to y=
˜ dt F(ω(i), 0, κ, t) exp(i J t)Sx.
(32)
We use the Lieb-Robinson bounds for matrix J , by defining a position matrix which is equal to i in the i th block. Using the Lieb-Robinson bounds, for time t ≤ L/v L R , with v L R = e2 , we find that the norm of the projection ofexp(i J t)Sx onto the space ˜ V L is bounded by exp(−L). At the same time, the integral |t|≥L/v L R dt F(ω(i), 0, κ, t) 2 is bounded by T (2L/v L R n win ) = T (2F(L)/e ). Since T (x) decays faster than any negative power of x, we can choose an F(x) which grows slower than any power of x such that T (2F(L)/e2 ) still decays faster than any negative power of L. Thus, since |y| ≥ λmin |x| by construction, for this choice of F(x) the norm of the projection of any vector y ∈ Wi onto V L is bounded by |y| times a function decaying faster than any negative power of L. The reason for picking λmin > 0 is to help establish the third property above. Let us give an example of a situation where we would encounter problems if we have taken λmin = 0. Consider a matrix of the form ⎞ 0 1/4 ⎟ ⎜1/4 0 1/4 ⎟ ⎜ 1/4 0 1/4 ⎟ ⎜ ⎟ ⎜ 1/4 0 1/4 ⎟ ⎜ ⎟ ⎜ ... ⎟ ⎜ ⎝ 1/4 0 1/4⎠ 1/4 1/2 ⎛
(33)
Here, each block has size one. If it weren’t for the “1/2” in the last line, this matrix would have operator norm slightly less than 1/2. However, because of the 1/2, this matrix has one eigenvalue greater than 1/2. For this particular choice of matrix, this eigenvalue is close to 5/8. The corresponding eigenvector is localized near the last block, and is exponentially small in the first block. If we project a vector in V1 into a narrow window centered on ω(i) = 5/8, the result will project onto this eigenvector, and thus the resulting state will have large amplitude on V L . However, for such a window, we would find that τi would be exponentially small, and so we would not include this vector in Xi . The properties we have established for spaces Xi are closely related to the properties in Lemma 2 that we are trying to establish. Unfortunately, the spaces Xi need not be orthogonal, and in fact may be very far from orthogonal. This can lead to problems like the following: suppose we have two vectors, v1 ∈ X1 and v2 ∈ X2 . We know that the projection of v1 onto V L is small compared to |v1 |, and we know the same thing for v2 ; however, we don’t know that the projection of v1 + v2 onto V L is small compared to |v1 + v2 | because we don’t know how |v1 + v2 | compares to |v1 | and |v2 |. We have two different ways of dealing with this: in the next subsection, we present a construction for block tridiagonal matrices that involves choosing a subspace of the space spanned by the Xi . In a later section on tridiagonal matrices, we present a much simpler construction that involves combining several windows into one; the reader may prefer to read that section first.
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C. Construction of W. We now construct the space W. Let each space Xi have dimension Di . In each space Xi we can find an orthonormal basis of vectors, vi,b , for b = 1, . . . , Di . We define a block tridiagonal matrix ρ of inner products of vectors vi,b as follows: the i th block (for 0 ≤ i < n win ) has dimension Di , and on the diagonal the matrix is equal to the identity matrix. Above the diagonal, the block in the i th row and i + 1st column is equal to the matrix of inner products (vi,b , vi+1,c ) for b = 1, . . . , Di and c = 1, . . . , Di+1 . Note that for |i − j| > 1, the spaces Xi and X j are orthogonal, so that the matrix ρ is block tridiagonal. We define a new vector space R to be a space of n win−1 dimension i=0 Di . The matrix ρ is Hermitian and positive semidefinite. It is equal to ρ = A† A, for some matrix A which has entries only on the block diagonal and on the diagonal above the block diagonal. The matrix A is a linear operator from R to B. Remark. The matrix ρ is block tridiagonal. To motivate what follows, consider the following circular reasoning: given that ρ is block tridiagonal, if we knew that Theorem 2 were true, we could find a basis in which ρ was approximately diagonal and in which a position operator, a block diagonal matrix equal to i/n win in the i th block, was also approximately diagonal. Then we choose W to be the space spanned by vectors of the form Awi , where wi are basis vectors in this basis for which the diagonal entry of ρ are not too close to zero (how close is something we would pick later). Then, we would know that the vectors Awi and Aw j are not degenerate for i = j, and the operator A would be an approximate isometry from the space spanned by the wi to W. Also, we would find that Awi was an approximate eigenvector of J . We would know that any vector v ∈ V1 had small projection orthogonal to W, since v could be written as Sx and, while x may have some projection onto vectors wi for which the diagonal entry of ρ is very close to zero, the error in v we make by dropping those vectors from x is small. This would give the space W the properties we are trying to construct. Unfortunately, of course, we are trying to prove Theorem 2, so this line of reasoning does not help. However, we do not need such a strong result in the present construction as will be seen below. If there is a vector w = w1 + w2 such that (w1 + w2 , ρ(w1 + w2 )) is small, then this means that Aw1 is close to −Aw2 . Suppose Aw1 ∈ X1 and Aw2 ∈ X2 . Then, we can take W to be the space spanned by X2 , X3 , . . . and spanned by the subspace of X1 orthogonal to Aw1 , and this leads to only a small error in our ability to approximate vectors v ∈ V1 by vectors in W. This is the basic idea behind the construction that follows. We define spaces Yi , for i = 0, . . . , n sb − 1, as follows, where n sb is the smallest even integer greater than or equal to n win /lb with the “block length” lb being an integer equal to 1/4
lb = n win .
(34)
Here, “sb” stands for “super-block” as we combine several blocks into one superblock. We pick Yi to be the subspace of R spanned by the vectors in blocks from the (i − 1)lbth block to the (i + 1)lb − 1th block. That is, it is the subspace spanned by vectors in blocks (i − 1)lb , (i − 1)lb + 1, (i − 1)lb + 2, . . . , (i + 1)lb − 1. Therefore, Yi is orthogonal to Y j for |i − j| > 1. The space spanned by the Xi is the same as the space spanned by the AYi for odd i; we will choose the space W to be a subspace of this space. Let Pi project onto the subspace of R spanned by the blocks from the ilbth block to the (i + 1)lb − 1th block. For notational convenience later (and to avoid various off-by-one errors), we define P−1 = 0, and we define Xi for i < 0 to be the empty set. In Fig. 3 we sketch the blocks used to define the spaces Yi for the case n sb = 6. The horizontal position in the figure indicates increasing block number, as marked in the top
Making Almost Commuting Matrices Commute
333
Fig. 3. Sketch of which blocks are in which subspaces Yi for n sb = 6, as well as which blocks are in the range of the Pi and also the subspaces NiL ,R and Yi which are defined below
row. Space Yi overlaps with space Yi±1 , as seen. We have also sketched the range of the operators Pi . We claim that Lemma 3. There exist spaces Ni , for i = 0, . . . , n sb − 1 with the properties that: 1 Ni is a subspace of Yi . 2 For any vector v ∈ Ni , the quantity (v, ρv) is bounded by |v|2 /lb2 times a function F0 (lb ), which is growing slower than any power of lb . 3 Let Ni project onto Ni . For any vector v which is in the space spanned by eigenvectors of ρ with eigenvalue less than 1/lb2 , the sum i |Ni v|2 is greater than or equal to (1 − F1 (lb ))|v|2 , where F1 (lb ) is a function decaying faster than any negative power of lb . Proof. Define a matrix M by
M=
Define a matrix O by O=
0 A . A† 0
˜ dt exp(i Mt)F(0, G(lb )/lb , G(lb )/lb , t),
(35)
(36)
where G(lb ) is a function growing slower than any power of lb to be chosen later. For each space X j , define X j to project onto that space. Define X j by Xj 0 . (37) Xj = 0 0 Compute the eigenvalues of X j O 2 X j . For each eigenvalue greater than λ0 , for some λ0 chosen later, compute the corresponding eigenvector x. The quantity λ0 will be chosen later to go to zero faster than any power of lb . Let Ni be the space spanned by (Pi−1 + Pi )O x, for all j with (i − 1/2)lb ≤ j < (i + 1/2)lb . Ni is a subspace of Yi , as claimed.
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M. B. Hastings
By construction, (O x, ρ O x) is bounded by (2G(lb )/lb )2 . We now use the Lieb-Robinson bounds for matrix M, by defining a position matrix which is equal to i in the i th block. Using the Lieb-Robinson bounds, for time t ≤ lb /2v L R , with v L R = e2 , we find that the norm of (1 − Pi−1 + Pi ) exp(i Mt)Sx, for x ∈ X j , is bounded by exp(−lb /2). ˜ At the same time, the integral |t|≥lb /2v L R dt F(0, G(lb )/lb , G(lb )/lb , t) is bounded by T (G(lb )/2v L R ). Since T (x) decays faster than any negative power of x, we can choose a G(x) which grows slower than any power of x such that T (G(lb )/2v L R ) still decays faster than any negative power of L. Thus, since |y| ≥ λ0 |x| by construction, we can find a λ0 going to zero faster than any power of lb such that |(Pi−1 + Pi )O x − O x| goes to zero faster than any power of lb . This verifies 2. 2 Also, for any vector v which is an eigenvector of ρ with eigenvalue less than 1/lb , we can write v = i vi , with vi in the space spanned by X j for (i −1/2)lb ≤ j < (i +1/2)lb . Then, i |(Ovi , v)|2 = |v|2 . Using the Lieb-Robinson bounds and the particular choice of λ0 , we arrive at 3. Remark. The definition of M and O as block matrices in the above lemma is simply a trick to make the claims 2,3 in the lemma depend on lb−2 rather than lb−1 as we would have found without this trick of introducing block matrices. In physics jargon, near the edge of the band (eigenvalues close to zero for ρ which is a positive semi-definite matrix), we have dynamic critical exponent 2 rather than 1. Consider any even i. By Jordan’s lemma for pairs of projectors1 , applied to Ni , which is the projector onto Ni , and to Pi−1 , we can find an orthonormal basis for Ni , with basis vectors n i,b such that Pi−1 n i,b is orthogonal to n i,c for b = c. Let NiL be the space spanned by vectors Pi−1 n i,b for the n i,b such that |Pi−1 n i,b |2 ≥ 1/2, and let NiR be the 2 space spanned by vectors Pi n i,b for the n i,b such that |Pi−1 n i,b | < 1/2. Any vector L v ∈ Ni can be written as v = b A(b)Pi−1 n i,b , where the A(b) are some coefficients and the n i,b are the basis vectors with |Pi−1 n i,b |2 ≥ 1/2. For any such vector v ∈ NiL , there exists a vector w, given by w = b A(b)Pi n i,b , such that the following hold. The vector w is in the subspace of R projected onto by Pi , with w ∈ NiR . Also, |w| ≤ |v|, and (v − w, ρ(v − w)) ≤ |v − w|2 F0 (lb )/lb2 ≤ 2|v|2 F0 (lb )/lb2 .
(38)
Similarly for any vector v ∈ NiR , there exists a vector w with |w| ≤ |v|, which is in the subspace of R projected onto by Pi−1 , with w ∈ NiL for which Eq. (38) again holds. Note that NiL is a subspace of the range of Pi−1 and NiR is a subspace of the range of Pi . For each odd i, define Yi to be the subspace of Yi which is orthogonal to the space R and N L (it is possible that this subspace Y is empty). By Jordan’s spanned by Ni−1 i+1 i lemma for pairs of projectors, applied to the projector onto Yi and the projector Ni , we can find an orthonormal basis for Yi , with basis vectors yi,b such that Ni yi,b is orthogonal to yi,c for b = c. We now define Wi , for odd i, 1 ≤ i ≤ n sb − 1, to be the space 1 Jordan’s lemma states that given any two projectors, M and N , we can find a basis in which both projectors are block diagonal with blocks of size 1 or 2. Equivalently (this is the formulation we use), if M projects onto a subspace M, then we can find an orthonormal basis of vectors m i for M with the property that (m i , N m j ) = 0 for i = j. The original reference is [28]
Making Almost Commuting Matrices Commute
335
spanned by the vectors Ayi,b for b such that (yi,b , Ni yi,b ) ≤ 1 − 1/lb2 .
(39)
Define Ri to be the space spanned by vectors yi,b for b such that (39) holds, so that Wi = ARi . The space W is the space spanned by the Wi for odd i. Let P be the projector onto W. D. Properties of W and Wi . In this section we establish certain properties for the spaces W and Wi . The main results are Eq. (40), showing that the spaces Wi are approximate eigenspaces; Eq. (43), controlling the overlap between different spaces Wi ; and Eq. (45), showing that for any vector v in the space spanned by Xi with v = Ax, the vector Pv ∈ W is close to v, where the maximum distance |Pv − v| between the vectors depends on |x|. First Property. For any odd i, for any vi ∈ Wi , we have |(1 − P)J vi | ≤ |(J − ω(ilb ))vi | (40) ≤ lb κ|vi |. Second Property. By construction, for any vector r = i=1,3,5,... ri , with ri ∈ Ri , r ∈ R, and |r | = 1, we have (r, ρr ) ≥ (1/lb2 )1/(8lb2 − F3 (lb )),
(41)
where F3 (lb ) is a function decreasing faster than lb−2 . To show Eq. (41), consider any vector w which is in the space spanned by eigenvectors of ρ with eigenvalue less than 1/lb2 , with |w| = 1. By 3 of Lemma 3, either i=1,3,... |Ni w|2 ≥ (1 − F1 (lb ) − 1/4l 2 ) or i=0,2,... |Ni w|2 ≥ 1/4lb2 . Suppose the second possibility is true. Then, b L R 2 2 L R L R 2 i=1,3,... (|Ni w| + |Ni | ) ≥ 1/8lb , where Ni and Ni project onto Ni and Ni , L R respectively. However, |Ni r | = |Ni r | = 0 for all even i by construction since R j is
orthogonal to the spaces NiR and NiL . Therefore, |(r, w)| ≤ 1 − 1/4lb2 ≤ 1 − 1/8lb2 . On the other hand, if the first possibility is true, since i=0,2,... |Ni r |2 ≤ 1 − 1/lb2 , we have
|(r, w)| ≤ (1 − 1/lb2 )(1 − F1 (lb ) − 1/4lb2 )
+ 1/lb2 F1 (lb ) + 1/4lb2 ≤ 1 − 1/8lb2 + F3 (lb ), (42) −2 where F3 (lb ) is a function decreasing faster than lb . Therefore, from Eq. (41), for any v = i=1,3,... vi with vi ∈ Wi , we have
|v| ≥ (1 − F2 (lb ))(1/8lb ) 2
4
n sb
|vi |2 ,
(43)
i=1,3,...
where F2 (lb ) is a function decaying to zero as lb → ∞, as may be seen by setting vi = Ari and v = Ar .
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M. B. Hastings
For any vi ∈ Wi , we can find x j ∈ X j such that vi =
(i+1)lb −1
|x j ≤ |vi with xi ∈ Xi and v = i xi with j=(i−1)lb
|2
|2 . Therefore, for any v
j=(i−1)lb
x j with
∈ W, we can find xi , i = 0, . . . , n win − 1
|v| ≥ (1 − F2 (lb ))(1/8lb ) 2
(i+1)lb −1
4
n win −1
|xi |2 .
(44)
i=0
Third Property. We also claim that for any vector v in the space spanned by the Xi , such that v = Ax that |Pv − v| ≤ (2 +
√ 2)( F0 (lb )/lb )|x|.
(45)
Let w1 be the projection of x orthogonal to the space spanned by the NiL and the NiR . Let w2 = x − w1 . Thus, we can write v = Aw1 + Aw2 , with w1 orthogonal to the space spanned by NiL and NiR for even i, and w2 in the space spanned by the NiL and the NiR for even i. Then, by Eq. (38), we can find a vector z orthogonal to the space spanned by ANiL and ANiR , with |z − Aw2 |2 ≤ 2(F0 (lb )/lb2 )|w2 |2 .
(46)
Then, consider the vector Aw1 + z. This is equal to a vector Az 1 in W plus a vector Az 2 such that z 2 = i z i with (z i , Ni z i ) ≥ 1 − 1/lb2 . Then, (z 2 , ρz 2 ) ≤ 4/lb2 . Therefore, √ since |Pv − v| ≤ |z − Aw2 | + (z 2 , ρz 2 ), Eq. (45) follows. n win −1 Any vector v = i=0 xi with xi ∈ Xi can be written as v = Ax with |x|2 ≤ n win −1 n win −1 xi , with xi ∈ Xi , 2 i=0 |xi |2 . Therefore, Eq. (45) implies that for any v = i=0 we have |Pv − v| ≤
√
2(2 +
√
n win −1 2)( F0 (lb )/lb ) |xi |2 .
(47)
i=0
E. Verification of claims. We now verify the claims regarding the subspace W. Proof of First Claim. To prove 1, note that for any vector v ∈ B we have v=
n win −1
F(ω(i), 0, 2n win , J )v.
(48)
i=0
For any v ∈ V1 , with |v| = 1, we can write v = Sx with |x| = 1, and then, from Eq. (30), |v −
n win −1 i=0
τi (1 − Z i )x|2 ≤ 2/L 2 .
(49)
Making Almost Commuting Matrices Commute
337
The vector τi (1 − Z i )x is in Xi . So, by Eq. (47), |(1 − P)
n win −1
τi (1 − Z i )x| ≤
i=0
√
n win −1 √ 2(2 + 2)( F0 (lb )/lb ) |τi (1 − Z i )x|2 i=0
n win −1 √ √ ≤ 2(2 + 2)( F0 (lb )/lb ) |τi x|2 ≤
√
√ 2(2 + 2)( F0 (lb )/lb ).
i=0
(50)
Combining Eqs. (49, 50) with a triangle inequality verifies the first claim, given that F(L) is chosen to grow slower than any power of L. Proof of Second Claim. To prove the second claim 2, consider any vector v ∈ W, with v = i vi for odd i with vi ∈ Wi . By Eqs. (40,43), and by the fact that Wi is orthogonal to W j for |i − j| > 2, we have |(1 − P)J v|2 ≤ | (J − ω(ilb ))vi |2 i
≤2
|(J − ω(ilb ))vi |2
i
≤2
(lb κ)2
i
|vi |2
i
1 ≤2 (lb κ))2 |v|2 (1 − F2 (lb ))(1/8lb )4 2 8 l 3 κ |v|2 , =2 1 − F2 (lb ) b
(51)
verifying the second claim. Remark. The factor of 2 on the second line of Eq. (51) arises because ((J − ω(ilb ))vi , (J − ω( jlb ))v j ) = 0 for |i − j| > 1 but may be non-zero for i = j ± 1. Proof of Third Claim. As we established before, using the Lieb-Robinson bound, for the given choice of F(x) the norm of the projection of any vector y ∈ Wi onto V L is bounded by |y| times a function decaying faster than any negative power of L. Let PV L project onto V L . Using Eq. (44), we find that the projection of any vector v ∈ W onto n win −1 V L is bounded by (writing v = i=0 wi with wi ∈ Xi ) |PV L v| ≤ n win 2
n win −1
|PV L wi |2
i=0
≤ n win maxi (|PV L wi |2 /|wi |2 )
1 8l 4 |v|2 . 1 − F2 (lb ) b
(52)
Since (|PV L w j |2 /|w j |2 ) is bounded by a function decaying faster than any negative power of L, this verifies the third claim. This completes the proof of Lemma 2. After giving the error bounds in the next section, we explain some of the motivation behind the above construction, and comment on the easier case in which J is a tridiagonal matrix, rather than a block tridiagonal matrix.
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M. B. Hastings
V. Error Bounds We finally give the error bounds to obtain Theorems 1, 2. To obtain Theorem 2, we pick n cut = −1/5 ,
(53)
so that L = (2/n cut )/) − 1 is of order 2/4/5 . Then, from Lemma 2 and Eq. (17), in the new basis the block-off-diagonal terms in H are bounded in operator norm by a constant times 1/5 times a function growing slower than any power of 1/. By Eq. (18), the difference between B and B is bounded in operator norm by a constant times 1/5 . Therefore, Theorem 2 follows. To obtain Theorem 1, we pick = δ 5/6
(54)
in Lemma 1. We omit the detailed analysis, but it is possible to choose E(x) to be a polylog as follows. We can pick T (x) to decay like exp(−x η ), for any η < 1 [29,30]. Then we can pick F(L) to equal log(L)θ , for θ > 1/η, so that T (F(L)) ∼ exp(−(log(L))θ/η ) decays faster than any power. VI. Tridiagonal Matrices In this section, we present tighter bounds for the case in which H is a tridiagonal matrix, rather than a block tridiagonal matrix. Remark. The difficulty we face is that the Xi are not orthogonal to each other. If they were orthogonal, then many of the estimates would be easier. Consider the case in which J is a block diagonal matrix, so that V1 is one dimensional. Let ρ(E) be a smoothed density of states at energy E: ρ(E) = tr(S † F(E, 1/L , 1/L , J )† F(E, 1/L , 1/L , J )S). Suppose ρ(E) is such that it has a peak in the crossing points of Fig. 2a (the points where one function F is decreasing and the other is increasing and they cross). Then, with the overlapping windows as shown, we find that most of the smoothed density of states lies in the overlap between the windows, rather than in the windows themselves. The overlap between the vectors in different windows is large. In the case of a tridiagonal matrix, we can combine two of the windows as shown in Fig. 2b to reduce the overlap of the normalized vectors; this general idea will motivate the construction in this section. We prove that Lemma 4. Let J be an L-by-L Hermitian tridiagonal matrix, with J ≤ 1 acting on a space B. Let v j denote the vector with a 1 in the j th entry and zeroes elsewhere. Then, there exists a space W which is a subspace of B with the following properties: 1 The projection of v1 onto the orthogonal complement of W has norm bounded by 3 , where 3 is equal to a constant times 1/L. 2 For any normalized vector w ∈ W , the projection of J w onto the orthogonal complement of W has norm bounded by 4 , where 4 is equal to 1/L times a function growing slower than any power of L. 3 The projection of v L onto W has norm bounded by 5 , where 5 is a function decaying faster than any power of L.
Making Almost Commuting Matrices Commute
339
This lemma implies Theorem 3: we construct A , B as before, following Steps 3 to construct the new basis, but because of the tighter bounds in Lemma 4 we can choose n cut = −1/2 when constructing the new basis. Now, in Step 4, we find that A , B are diagonal matrices, rather than just block diagonal matrices. For each i = 0, 1, . . . , n win − 1, define ω(i) = −1 + iκ,
(55)
as before. Define ρi = tr S † F(ω(i), 0, κ, J )† F(ω(i), 0, κ, J )S = |F(ω(i), 0, κ, J )v1 |2 .
(56)
Set λmin = 1/(n win L 2 ),
(57)
n win = L/F(L)
(58)
as before with
as before. To prove Lemma 4, we use the following algorithm. There are n win windows, labeled 0, . . . , n win − 1. We label various windows as either “unmarked” or “marked”; windows which are marked get marked by an integer label. 1 Set i = 0. Initialize a real variable x to 0. Initialize an integer counter a to 1. Initialize all windows to unmarked. 2 Set x to 0. If ρi < λmin , then 2a. Increment i by one. 2b. If i ≥ n win , terminate. Otherwise, go to Step 2. 3 Mark window i with label a. 4 Set x to x + ρi . If x < 9ρi , then 4a. Increment i by one. 4b. If i ≥ n win , terminate. Otherwise, go to Step 3. 5 Increment a by one. Increment i by one. If i ≥ n win , terminate. Otherwise, go to Step 2. After running this algorithm, there will be sequences of marked windows, with all windows in any given sequence marked with the same integer label a. There may be one or more unmarked windows separating the sequences of marked windows. In Step 2, we scan along to find an i with ρi ≥ λmin , and then in Step 4 we mark a sequence of windows. We claim that the length of a sequence of marked windows is at most 1 + log10/9 (2/λmin ) . This bound on the length of a sequence of marked windows holds because at the start of a sequence x is at least λmin , x grows exponentially along the sequence (otherwise in Step 4 we find that ρi+1 > (1/9)x for some i), and x can be n win −1 at most 2 since i=0 ρi ≤ 2. Let the total number of sequences be n seq . Note that n seq ≤ n win .
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M. B. Hastings
For each sequence of windows marked with a given integer a, from window i to j, construct the vector ya given by ya =
j
F(ω(k), 0, κ, J )v1
k=i
= F((ω(i) + ω( j))/2, (ω( j) − ω(i))/2, κ, J )v1 .
(59)
The inner product (ya , ya+1 ) is equal to (F(ω( j), 0, κ, J )v1 , ya+1 ). By Cauchy-Schwarz, this is bounded by |(F(ω( j), 0, κ, J )v1 ||ya+1 |. To estimate |(F(ω( j), 0, κ, J )v1 |, we j j j use |(F(ω( j), 0, κ, J )v1 |2 = ρ j ≤ k=i ρk /9 ≤ |ya |2 /9 = k=i k =i (F(ω(k), 0, κ, J )v1 , F(ω(k ), 0, κ, J )v1 )/9, where the first inequality is by construction and the second inequality follows from the√fact that (F(ω(k), 0, κ, J )v1 , F(ω(k ), 0, κ, J )v1 ) ≥ 0. Therefore, (ya , ya+1 ) ≤ (|ya |/ 9)|ya+1 | , so (ya , ya+1 ) ≤ (1/3)|ya ||ya+1 |.
(60)
We define W to be the space spanned by all such vectors ya , and we define P to project onto W. Consider any vector v ∈ W, with v=
n seq
va ,
(61)
a=1
with va parallel to ya . By Eq. (60), n seq
1 |v| ≥ |va |2 . 3 2
(62)
a=1
Remark. The function F((ω(i) + ω( j))/2, (ω( j) − ω(i))/2, κ, ω) is equal to unity for ω(i) ≤ ω ≤ ω( j). We now prove Lemma 4 as follows: to prove the first claim, note that by construction, |Pv1 − v1 | ≤ | 2
n seq
ya − v1 |2
a=1
≤ 2n win λmin ≤ 2/L 2 .
(63)
The second line of the above equation follows because the difference a ya − v1 is equal to − iunmarked F(ω(i), 0, κ, J )v1 , where the sum ranges over i such that the corresponding window is unmarked. To prove the second claim, consider the a th sequence of marked windows, from window i to window j. Let ωa = (ω− (i) + ω+ ( j))/2. Then 2 + log10/9 (2/λmin ) |ya |, |(J − ωa )ya | ≤ (64) n win which is bounded by 1/L times a function growing slower than any power of L. Therefore, 2 + log10/9 (2/λmin ) |ya |. |(1 − P)J ya | ≤ (65) n win
Making Almost Commuting Matrices Commute
Using the bound Eq. (62), for any vector v ∈ W, √ 2 + log10/9 (2/λmin ) |v|, |(1 − P)J v| ≤ 2 3 n win
341
(66)
which is bounded by 1/L times a function growing slower than any power of L, verifying the second claim. The proof of the third claim is identical to the previous case. VII. Quantum Measurement A. Construction and results. The constructions above can be applied to operators which arise in various physical quantum systems. For example, consider a quantum spin for a large spin S. Then, the operators Sx /S and S y /S have operator norm 1 and have a commutator that is of order 1/S. Thus, we can find a basis in which both operators are almost diagonal. While it is well known that one can use a POVM (positive operatorvalued measure) to approximately measure Sx and S y at the same time, the existence of the given basis implies that one can approximately measure Sx and S y simultaneously with a single projective measurement. Interestingly, while the operator Sz2 is also almost diagonal in this basis (since it equals S(S + 1) − Sx2 − S y2 ), it is not possible to find a basis in which Sx , S y , and Sz are all almost diagonal (this obstruction is similar to that in [6]). Therefore, to approximately measure Sx , S y , and Sz simultaneously will require a POVM, rather than a projective measurement. For completeness, we now briefly show how to construct a POVM to approximately measure several almost commuting operators simultaneously. Consider any number N of Hermitian matrices, labeled A1 , . . . , A N , with [Ai , A j ] ≤ δ for all i, j and with Ai ≤ 1 for all i. We now construct a POVM to approximately measure all N operators simultaneously. The physical idea is very simple: we first do a “soft” measurement of A N , then A N −1 , and so on, until all operators are measured. Let n win be some integer given by n win = δ −1/2 (N − 1)−1/2
(67)
(n win will typically be much larger than unity). For i = 1, . . . , N and n = 0, . . . , n win − 1, define ω(i) = −1 + 2i/(n win − 1) = −1 + iκ, where κ = 2/(n win − 1) as before, and define M(i, n) = F(ω(n), 0, κ, Ai ).
(68)
(69)
The definition of F is given at the start of Sect. IV; we will see later that in this section we do not actually need F to be infinitely differentiable as it is defined there, but we have only weaker requirements on F. Define O(n 1 , n 2 , . . . , n N ) = M(1, n 1 )† M(2, n 2 )† . . . M(N , n N )† (M(N , n N ) . . . M(2, n 2 )M(1, n 1 )).
(70)
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M. B. Hastings
Then, n win −1
O(n 1 , n 2 , . . . , n N ) = 1,
(71)
n 1 ,n 2 ,...=0
and all of the operators O(n 1 , n 2 , . . . , n N ) are positive semidefinite by construction. Therefore, the operators O(n 1 , n 2 , . . . , n N ) form a POVM. Note that M(i, n i ) = M(i, n i )† , but we continue to write daggers on the operators for clarity. We claim that this POVM approximately measures all operators simultaneously. That is, we will show that for any density matrix ρ, if the outcome of the measurement is n 1 , n 2 , . . . , n N , then if we perform a subsequent measurement of any operator Ai , the outcome will be close to ω(n i ) with high probability. We show this by computing the expectation value (Ai − ω(n i ))2 averaged over all measurement outcomes. For any density matrix ρ, for any i, the average over all outcomes of (Ai − ω(n i ))2 is equal to tr (Ai − ω(n i ))2 M(1, n 1 )M(2, n 2 ) . . . ρ . . . M(2, n 2 )† M(1, n 1 )† . (72)
n win −1 n 1 ,n 2 ,...=0
The main result in this section is that tr (Ai − ω(n i ))2 M(1, n 1 )M(2, n 2 ) . . . ρ . . . M(2, n 2 )† M(1, n 1 )†
n win −1 n 1 ,n 2 ,...=0
≤ const. × (N − 1)δ.
(73)
We show this in the next subsection. √ B. Bounds. Note that n i M(i, n i ) ≤ 2. To bound Eq. (72), we need three results, Eqs. (74, 75, 76) below. First, n win −1
(Ai − ω(n i ))M(i, n i ) ≤ const. × κ
n i =0
≤ const. × 1/n win .
(74)
Second, we need
n win −1
[M( j, n j ), (Ai − ω(n i ))]O M( j, n j )† ≤ const. × (δ/κ)O
n j =0
for any operator O.
(75)
Making Almost Commuting Matrices Commute
343
Third, we need
n win −1
[M( j, n j ), (Ai − ω(n i ))]O[M( j, n j )† , (Ai − ω(n i ))]
n j =0
≤ const. × (δ/κ)2 O
(76)
for any operator O. Equation (74) follows immediately from the support of F. To show Eq. (75), define (77) A0 = κ dt exp(i A j t)(Ai − ω(n i )) exp(−i A j t) f (κt), where the function f (t) is defined to have the Fourier transform as in Eq. (5). Then, A0 − (Ai − ω(n i )) ≤ const. × δ/κ as in Lemma 1. Also, if v1 , v2 are eigenvectors of A j with corresponding eigenvalues x1 , x2 with |x1 − x2 | ≥ κ, then (v1 , A0 v2 ) = 0, which implies that
n win −1
[M( j, n j ), A0 ]O M( j, n j )† ≤ 2 maxn j ([M( j, n j ), A0 ]O M( j, n j )† ). (78)
n j =0
Equation (78) is the reason for introducing the operator A0 . We can bound the commutator [M( j, n j ), A0 ] as follows. Note that [A j , A0 ] ≤ δ. Write
(79) M( j, n j ) = dt exp(i A j t) F(ω(n j ), 0, κ, t), √ ., ., t) denotes the Fourier transform of the square-root of F. Then since where F(., [exp(i A j t), A0 ] ≤ const. × |t|δ, we can use a triangle inequality to show that
(80) [M( j, n j ), A0 ] ≤ dt F(ω(n j ), 0, κ, t)|t|δ. √ Then, since F(., ., ., ω) is infinitely differentiable, the Fourier transform decays faster than any power of t and the integral over t converges, so we have [M( j, n j ), A0 ] ≤ const. × δ/κ. Using Eq. (78) gives Eq. (75). Equation (76) is derived similarly. Using Eqs. (74, 75, 76), we can bound the sum in Eq. (72) by writing (Ai −ω(n i ))2 = (Ai − ω(n i ))(Ai − ω(n i )), and commuting one of the terms (Ai − ω(n i )) to the right through M( j, n j ) for j < i until it hits the M(i, n i ) and commuting the other term (Ai − ω(n i )) to the left through M( j, n j )† for j < i until it hits M(i, n i )† . Therefore, n win −1
tr((Ai − ω(n i ))2 M(1, n 1 )M(2, n 2 ) . . . ρ . . . M(2, n 2 )† M(1, n 1 ))†
n 1 ,n 2 ,...=0
≤ const. × (i − 1)2 δ 2 n 2win + (i − 1)δn win /n win + 1/n 2win ≤ const. × (N − 1)2 δ 2 n 2win + (N − 1)δ + 1/n 2win .
(81)
The first term on the right-hand side of Eq. (81) arises from two non-vanishing commutators (if the non-vanishing commutators are with M( j, n j ) and M(k, n k )† for j = k
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then we use Eq. (75) twice, but if j = k we use Eq. (76) once). The second term arises from one non-vanishing commutator and one use of Eq. (74), and the last term arises from using Eq. (74) twice. Choosing n win = δ −1/2 (N − 1)−1/2 ,
(82)
we find that we measure all operators to within a mean-square error of order (N − 1)δ, as claimed. Note that we did not actually require that F(., ., ., ω) be infinitely differentiable in √ this section. We only required that the Fourer transform F(., ., ., t) decay sufficiently rapidly in t that the integral (80) converges. The other properties of F we used are that n F(ω(n), 0, κ, ω) = 1 for −1 ≤ ω ≤ 1 and that F(ω(n), 0, κ, ω) vanish for |ω − ω(n)| ≥ κ. VIII. Discussion The main result is an explicit construction of a pair of exactly commuting matrices which are close to a pair of almost commuting matrices. The construction of the matrix is explicit and can be handled easily on a computer for modest N . We have in fact implemented the construction in Lemma 4 for the uniform chain. In practical applications, we expect that, for many tridiagonal matrices, the lack of orthogonality of the Xi will not cause a problem, and choosing W to be the space spanned by the Xi will lead to satisfactory results, without having to follow the more complicated procedure above. If, for some particular J , the lack of orthogonality of the Xi does cause a problem, an alternative procedure that might be more useful in practice than the deterministic procedure above is to add small, randomly chosen matrices to each diagonal block of J . This may smooth out the spectrum of J and then allow one to choose W to be the space spanned by the Xi . We gave the above applications to quantum measurement. Another application of this result is to construct Wannier functions for any two dimensional quantum system for a spectral gap. In [31], it was pointed out that given a two dimensional quantum system with a gap between bands, one could define an operator G which projected onto the bands below the gap. Then, define the operator X and Y to measure the X and Y position of particles, and define G X G and GY G as projections of X and Y into the lowest band. Let X , Y = L, where L is the linear size of the system. Since the operator G was constructed in [31] as a short-range operator, the commutator [G X G, GY G] is small compared to L 2 , and thus we can use the results here to construct a basis of Wannier functions which is localized in both the x- and y-directions. Acknowledgement. I thank T. J. Osborne and J. Yard for useful conversations, and I thank T. J. Osborne for many comments on a draft of this paper. I thank D. Poulin for explaining Jordan’s lemma. This work was supported by U. S. DOE Contract No. DE-AC52-06NA25396.
References 1. Rosenthal, P.R.: Are almost commuting matrices near commuting pairs? AMS Monthly 76, 925 (1969) 2. Halmos, P.R.: Some unsolved problems of unknown depth about operators on Hilbert space. Proc. Roy. Soc. Edinburgh A 76, 67 (1976) 3. Lin, H.: Almost commuting self-adjoint matrices and applications. Fields. Inst. Commun. 13, 193 (1995)
Making Almost Commuting Matrices Commute
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4. Friis, P., Rordam, M.: Almost commuting self-adjoint matrices—a short proof of Huaxin Lin’s theorem. J. Reine Angew. Math. 479, 121 (1996) 5. Voiculescu, D.: Asymptotically commuting finite rank unitaries without commuting approximants. Acta Sci. Math. 451, 429 (1983) 6. Voiculescu, D.: Remarks on the singular extension in the C∗ -algebra of the Heisenberg group. J. Op. Thy. 5, 147 (1981) 7. Davidson, K.R.: Almost commuting Hermitian matrices. Math. Scand. 56, 222 (1985) 8. Lieb, E.H., Robinson, D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251 (1972) 9. Hastings, M.B.: Lieb-Schultz-Mattis in higher dimensions. Phys. Rev. B 69, 104431 (2004) 10. Hastings, M.B., Koma, T.: Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781 (2006) 11. Nachtergaele, B., Sims, R.: Lieb-Robinson bounds and the exponential clustering theorem. Commun. Math. Phys. 265, 119 (2006) 12. Demko, S., Moss, W.F., Smith, P.W.: Decay rates for inverses of band matrices. Math. Comp. 43, 491 (1984) 13. Benzi, M., Golub, G.H.: Bounds for the entries of matrix functions with applications to preconditions. BIT 39, 417 (1999) 14. Hastings, M.B.: Locality in quantum and Markov dynamics on lattices and networks. Phys. Rev. Lett. 93, 140402 (2004) 15. Cramer, M., Dawson, C.M., Eisert, J., Osborne, T.J.: Exact relaxation in a class of nonequilibrium lattice systems. Phys. Rev. Lett. 100, 030602 (2008) 16. Eisert, J., Osborne, T.J.: General entanglement scaling laws from time evolution. Phys. Rev. Lett. 97, 150404 (2006) 17. Bravyi, S., Hastings, M.B., Verstraete, F.: Lieb-Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006) 18. Osborne, T.J.: A renormalization-group algorithm for eigenvalue density functions of interacting quantum systems. http://arXiv.org/abs/cond-mat/0605194v1[cond-mat.str-el], 2006 19. Osborne, T.J.: Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev. Lett. 97, 157202 (2006) 20. Hastings, M.B.: Quantum belief propagation. Phys. Rev. B Rapids 76, 201102 (2007) 21. Osborne, T.J.: Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev. A 75, 042306 (2007) 22. Hastings, M.B.: Observations outside the light-cone: algorithms for non-equilibrium and thermal states. Phys. Rev. B 77, 144302 (2008) 23. Hastings, M.B.: An area law for one dimensional quantum systems. J. Stat. Mech., P08024 (2007) 24. Nachtergaele, B., Raz, H., Schlein, B., Sims, R.: Lieb-Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286, 1073–1098 (2009) 25. Hastings, M.B.: Quasi-Adiabatic continuation in gapped spin and fermion systems: goldstone’s theorem and flux periodicity. J. Stat. Mech. P05010 (2007) 26. Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108, 41 (1987) 27. von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124, 285 (1989); Shubin, C., Vakilian, R., Wolff, T.: Some harmonic analysis questions suggested by Anderson-Bernoulli models. Geom. Funct. Anal. 8, 932 (1998) 28. Jordan, C.: Essai sur la géométrie à n dimensions. Bulletin de la S. M. F. 3, 103 (1875) 29. Gervais, R., Rahman, Q.I., Schmeisser, G.: A bandlimited function simulating a duration-limited one. In: Anniversary volume on approximation theory and functional analysis, Schiftenreihe Numer. Math., Basel: Birkhäuser, 1984, pp. 355–362 30. Strohmer, T., Tanner, J.: Implementations of Shannon’s sampling theorem, a time-frequency approach. Sampling Thy. in Signal and Image Proc., 4, 1 (2005); Fritz John: Partial Differential Equations. New York: Springer-Verlag, 1991 31. Hastings, M.B.: Topology and phases in fermionic systems. J. Stat. Mech. L01001 (2008) Communicated by A. Connes
Commun. Math. Phys. 291, 347–356 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0791-7
Communications in
Mathematical Physics
Proof of the Julia–Zee Theorem Joel Spruck1 , Yisong Yang2 1 Department of Mathematics, Johns Hopkins University, Baltimore,
MD 21218, USA. E-mail: [email protected]
2 Department of Mathematics, Yeshiva University, New York, NY 10033, USA.
E-mail: [email protected] Received: 7 October 2008 / Accepted: 24 December 2008 Published online: 1 April 2009 – © Springer-Verlag 2009
Abstract: It is a well accepted principle that finite-energy static solutions in the classical relativistic gauge field theory over the (2 + 1)-dimensional Minkowski spacetime must be electrically neutral. We call such a statement the Julia–Zee theorem. In this paper, we present a mathematical proof of this fundamental structural property. 1. Introduction Consider the Maxwell equations ∂ν F µν = −J µ
(1.1)
defined over a Minkowski spacetime of signature (+ − · · · −), where Fµν = ∂µ Aν − ∂ν Aµ
(1.2)
is the electromagnetic tensor induced from the gauge vector field Aµ , µ = 0 designates the temporal index, µ = i, j, k denote the spatial indices, and J µ = (J 0 , J i ) = (ρ, j) is the current density in which ρ = J 0 expresses the electric charge density. As spatial vector fields, the electric field E = (E i ) and magnetic field B = (B i ) are given by F 0i = −E i ,
F i j = −εi jk B k .
(1.3)
In view of (1.3), the µ = 0 component of (1.1) relating electric field and charge density reads div E = ∂i E i = ρ,
(1.4)
which is commonly referred to as the Gauss law (constraint). In the static situation, we have E i = −F 0i = F0i = −∂i A0 .
(1.5)
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Thus, a trivial temporal component of the gauge field, A0 = 0,
(1.6)
implies that electric field is absent, E = 0. The condition (1.6) is also known as the temporal gauge condition, which makes the static solution electrically neutral. In their now classic 1975 paper [10], Julia and Zee studied the Abelian Higgs gauge field theory model. Using a radially symmetric field configuration ansatz and assuming a sufficiently fast decay rate at spatial infinity, they were able to conclude that a finite-energy static solution of the equations of motion over the (2 + 1)-dimensional Minkowski spacetime must satisfy the temporal gauge condition (1.6), and thus, is necessarily electrically neutral. This result, referred to here as the Julia–Zee theorem, leads to many interesting consequences. For example, it makes it transparent that the static Abelian Higgs model is exactly the Ginzburg–Landau theory [6] which is purely magnetic [9,12]. Since the work of Julia and Zee [10], it has been accepted [4,7,8,11,13,20] that, in order to obtain both electrically and magnetically charged static vortices, one needs to introduce into the Lagrangian action density the Chern–Simons topological terms [2,3], which is an essential construct in anyon physics [22,23]. See also [5]. On the other hand, it is well known that electrically and magnetically charged static solitons, called dyons by Schwinger [17] (see also the related work of Zwanziger [24, 25]), exist as solutions to the Yang–Millis–Higgs equations over (3 + 1)-dimensional spacetime [10,14,16]. Therefore, the Julia–Zee theorem is valid only in (2 + 1) dimensions. The importance of the Julia–Zee theorem motivated us to carry out this study. In Sect. 2, we make a precise statement of the Julia–Zee theorem in the context of the original Abelian Higgs model and present a rigorous proof. In Sect. 3, we extend the Julia–Zee theorem to the situation of a non-Abelian Yang–Mills–Higgs model. In Sect. 4, we prove a non-Abelian version of the theorem. Fortunately our method works almost exactly as in the simpler Abelian Higgs model. In Sect. 5, we consider further extensions and applications of our results.
2. The Julia–Zee Theorem Recall that, in normalized units, the classical Abelian Higgs theory over the (2 + 1)dimensional spacetime is governed by the Lagrangian action density 1 1 L = − Fµν F µν + Dµ φ D µ φ − V (|φ|2 ), 4 2
(2.1)
where D µ φ = ∂ µ φ + iAµ φ defines the gauge-covariant derivative, φ is a complex scalar (Higgs) field, the spacetime indices µ, ν run through 0, 1, 2, the spacetime metric takes the form η = (ηµν ) = diag(1, −1, −1), which is used to lower and raise indices, and V ≥ 0 is the potential density of the Higgs field. The associated equations of motion are Dµ D µ φ = −2V (|φ|2 )φ, ∂ν F µν = −J µ , i µ φ D φ − φ Dµφ . Jµ = 2
(2.2) (2.3) (2.4)
Proof of the Julia–Zee Theorem
349
In the static situation, the operator ∂0 = 0 nullifies everything. Hence the electric charge density ρ becomes ρ = J0 =
i (φ D 0 φ − φ D 0 φ) = −A0 |φ|2 2
(2.5)
and a nontrivial temporal component of the gauge field, A0 , is necessary for the presence of electric charge. On the other hand, the µ = 0 component of the left-hand side of the Maxwell equation (2.3) is ∂ν F 0ν = ∂i (Fi0 ) = ∂i2 A0 = A0 .
(2.6)
Consequently, the static version of the equations of motion (2.2)–(2.4) may be written as Di2 φ = 2V (|φ|2 )φ − A20 φ, i φ Di φ − φ Di φ , ∂ j Fi j = 2 A0 = |φ|2 A0 ,
(2.7) (2.8) (2.9)
in which (2.9) is the Gauss law. Moreover, since the energy-momentum (stress) tensor has the form
Tµν = −ηµ ν Fµµ Fνν +
1 Dµ φ Dν φ + Dµ φ Dν φ − ηµν L, 2
(2.10)
the Hamiltonian density is given by H = T00 =
1 1 1 1 |∂i A0 |2 + A20 |φ|2 + Fi2j + |Di φ|2 + V (|φ|2 ), 2 2 4 2
so that the finite-energy condition reads H dx < ∞. R2
(2.11)
(2.12)
With the above formulation, the Julia–Zee theorem may be stated as follows. Theorem 2.1 (The Julia–Zee Theorem). Suppose that (A0 , Ai , φ) is a finite-energy solution of the static Abelian Higgs equations (2.7)–(2.9) over R2 . Then either A0 = 0 everywhere if φ is not identically zero or A0 ≡ constant and the solution is necessarily electrically neutral. Our proof of the theorem is contained in the following slightly more general statement. Proposition 2.1. Let A0 be a solution of A0 = |φ|2 A0 over R2 . Suppose that |∇ A0 |2 dx < ∞. (2.13) R2
Then A0 =constant. Furthermore, if φ is not identically zero, then A0 ≡ 0.
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Proof. Let 0 ≤ η ≤ 1 be of compact support and define for M > 0 fixed the truncated function ⎧ if A0 > M, ⎨M A0M = A0 if |A0 | ≤ M, (2.14) ⎩ −M if A < −M. 0 Then, multiplying (2.9) by η A0M and integrating, we have R2
[η∇ A0 · ∇ A0M + A0M ∇ A0 · ∇η + η|φ|2 A0M A0 ] dx = 0.
(2.15)
Using (2.14) in (2.15), we find
{|A0 |<M}∩
+
η|φ|
supp(η)
2
A20 dx
+M
2 {|A0 |>M}∩
supp(η)
η|φ|2 dx
η|∇ A0 |2 dx supp(η) 1 1 2 2 |∇ A0 |2 dx |∇η|2 dx .
{|A0 |<M}∩
≤M
R2
(2.16)
R2
For R > 0, we now choose η to be a logarithmic cutoff function given as ⎧ ⎨1 η = 2− ⎩ 0
log |x| log R
if |x| < R, if R ≤ |x| ≤ R 2 , if |x| > R 2 .
(2.17)
2π . log R
(2.18)
Then R2
|∇η|2 dx =
Using (2.18) in (2.16) gives
{|A0 |<M}∩B R
|φ|
2
A20 dx
≤
{|A0 |<M}∩B R
|∇ A0 |2 dx
{|A0 |<M}∩B R
≤M
+
|φ|
2
A20 dx
1 2π R2 |∇ A0 |2 dx 2 1
(log R) 2
+M .
2
|φ| dx + 2
{|A0 |>M}∩B R
{|A0 |<M}∩B R
|∇ A0 |2 dx (2.19)
The right-hand side of (2.19) tends to zero as R tends to infinity. Letting M tend to infinity proves the proposition.
Proof of the Julia–Zee Theorem
351
3. A Non-Abelian Julia–Zee Theorem In this section, we consider the simplest non-Abelian Yang–Mills–Higgs theory for which the gauge group is SU (2) or S O(3). For convenience, we work with the gauge group in the adjoint representation so that the gauge field and Higgs field are all real 3-vectors expressed as Aµ = (A1µ , A2µ , A3µ )T , = (φ 1 , φ 2 , φ 3 )T . Then the Yang–Mills field curvature tensor and gauge-covariant derivative are given by Fµν = ∂µ Aν − ∂ν Aµ + Aµ × Aν ,
Dµ = ∂µ + Aν × ,
so that the Lagrangian density is written as 1 1 L = − Fµν · Fµν + Dµ · D µ − V ( ) 4 2 1 1 1 1 = − Fi j · Fi j + F0i · F0i − Di · Di + D0 · D0 − V ( ). (3.1) 4 2 2 2 As a consequence, the equations of motion, or the Yang–Mills–Higgs equations, are D µ Fµi + × Di = 0, δV ( ) = 0, D µ Dµ + δ coupled with the Gauss-law constraint
(3.2)
D µ Fµ0 + × D0 = 0.
(3.4)
(3.3)
This is the equation that governs the A0 field and is our main concern. The actual form of (3.4) is: − Di Fi0 + × D0 = 0, i = 1, 2.
(3.5)
In the static case, ∂0 = 0. So D0 = A0 × , Fi0 = ∂i A0 + Ai × A0 .
(3.6)
Inserting (3.6) into (3.5), we get A0 + ∂i (Ai × A0 ) + Ai × ∂i A0 + Ai × (Ai × A0 ) − × (A0 × ) = 0. (3.7) On the other hand, the Hamiltonian density of the theory is H = F0i · F0i + D0 · D0 − L 1 1 1 1 = Fi j · Fi j + F0i · F0i + Di · Di + D0 · D0 + V ( ), 4 2 2 2 which is positive definite. The terms containing A0 give us the A0 energy,
1 1 F0i · F0i + D0 · D0 dx E(A0 ) = 2 R2 2
1 1 2 2 |∂ dx. = i A0 + (Ai × A0 )| + |A0 × | 2 R2 2
(3.8)
(3.9)
It can be checked that the governing equation for A0 , Eq. (3.7), is the Euler–Lagrange equation of (3.9).
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Theorem 3.1 (A non-Abelian extension of the Julia–Zee Theorem). Let A0 be a solution of (3.4) with finite energy E(A0 ) < ∞. Then E(A0 ) = 0. In particular, F0i ≡ 0, D0 ≡ 0, and |A0 | ≡constant. Moreover, if the nonnegative potential density V is such that V ( ) = U (| |2 ) and ( , Ai , A0 ) is a finite-energy solution of the Yang–Mills –Higgs equations (3.2)–(3.4), then A0 ≡ 0. Otherwise the solution triplet ( , Ai , A0 ) must be trivial, i.e., | | ≡ θ, Fi j ≡ 0, E(A0 ) = 0,
(3.10)
where s = θ 2 ≥ 0 is a certain zero of the function U (s). To see that the absence of the electric sector in the non-Abelian Yang–Mills–Higgs model is implied by the above theorem, recall that the ’t Hooft electromagnetic tensor [19] (see also [15,18] for related discussion and extension) may be written as Fµν =
1 1 · Fµν − · (Dµ × Dν ). | | | |3
(3.11)
Hence E i = F0i ≡ 0 if E(A0 ) = 0. 4. Proof of the Non-Abelian Julia–Zee Theorem Let 0 ≤ η ≤ 1 be of compact support and define for M > 0 fixed the truncated vector field,
A0M
=
A0 M |A0 | A0
if |A0 | ≤ M, if |A0 | > M.
(4.1)
Then, using ηA0M as a test function, we obtain from (3.7) the expression
η ∂i A0M · ∂i A0 − A0M · ∂i (Ai × A0 ) − A0M · (Ai × ∂i A0 ) R2 −A0M · (Ai ×(Ai × A0 ))+A0M · ( × (A0 × )) +∂i η A0M · ∂i A0 dx = 0. (4.2)
Using the definition of A0M in (4.1), we see that A0M · (∂i Ai × A0 ) = 0, 2 |A0 | (Ai × A0M ) · (Ai × A0 ) = Ai × A0M in {|A0 | > M}, M 2 |A0 | M (A0M × ) · (A0 × ) = A0 × in {|A0 | > M}, M |A0 | (∂i A0M )2 in {|A0 | > M}, ∂i A0M · ∂i A0 = M |A0 | ∂i A0M · (Ai × A0M ) in {|A0 | > M}. −2A0M · (Ai × ∂i A0 ) = 2 M
Proof of the Julia–Zee Theorem
353
We then obtain from (4.2) that η{|∂i A0 + (Ai × A0 )|2 + |A0 × |2 } dx {|A0 |≤M} |A0 | + η{|∂i A0M + Ai × A0M |2 + |A0M × |2 } dx {|A0 |>M} M {∂i η A0M · ∂i A0 } dx =− R2 =− {∂i η A0M · (∂i A0 + Ai × A0 )} dx. R2
(4.3)
We again choose η according to (2.17). Using (2.18), we have |∂i A0 + (Ai × A0 )|2 + |A0 × |2 dx {|A0 ≤M}∩B R
≤M ≤M
2π log R
1
1 2
|∂i A0 + Ai × A0 | dx 2
R2
4π E(A0 ) log R
2
1 2
.
(4.4)
The right-hand side of (4.4) tends to zero as R tends to infinity. Letting M tend to infinity proves E(A0 ) = 0. To see that |A0 | =constant, we use the result F0i = 0 to deduce that ∂i |A0 |2 = 2∂i A0 · A0 = −2(Ai × A0 ) · A0 = 0. Suppose A0 = 0. Then |A0 | = a > 0 for some constant a. Note that E(A0 ) = 0 also implies that remains parallel to A0 everywhere. So there is a scalar function u such that = uA0 . Consequently, we have Di = (∂i u)A0 + u Di A0 = (∂i u)A0 + uFi0 = (∂i u)A0 .
(4.5)
Now assume that the Higgs potential density takes the form V ( ) = U (| |2 ). Iterating (4.5), we get Di Di = (u)A0 . Hence, by (3.3) and D0 = 0, we arrive at u = U (a 2 u 2 )u in R2 . In view of the finite-energy condition and (4.5), we have
1 1 2 2 2 I (u) = |∇u| + 2 U (a u ) dx < ∞. 2a R2 2
(4.6)
(4.7)
It may easily be checked that, as a solution of (4.6), u is a finite-energy critical point of the functional (4.7). However, using a standard rescaling argument with x → xσ = σ x and u(x) → u σ (x) = u(xσ ) so that dI (u σ )/dσ = 0 at σ = 1, we find U (| |2 ) dx = U (a 2 u 2 ) dx = 0, (4.8) R2
R2
which implies that U (a 2 u 2 ) ≡ 0. Since U ≥ 0, we have U (a 2 u 2 ) ≡ 0. Inserting this into (4.6), we have u = 0 in R2 . In view of Proposition 2.1, we conclude that u ≡constant. Consequently, there is a zero, s, of U (·) of the form s = θ 2 ≥ 0 such that u ≡ ± aθ or
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θ = ± A0 , a
(4.9)
which immediately gives us Di = ± aθ Di A0 = ± aθ Fi0 = 0 over R2 . Therefore, the coupled equations (3.2) and (3.3) are reduced to the pure static Yang– Mills equations Di Fi j = 0 in R2 ,
(4.10)
which is known to have only the trivial solution, Fi j = 0, over R2 , as can easily be seen from a similar rescaling argument involving x → xσ = σ x and Ai (x) → (Aσ )i (x) = σ Ai (xσ ), i = 1, 2, in the energy functional R2
|Fi j |2 dx.
(4.11)
The proof of the stated non-Abelian extension of the Julia–Zee Theorem is complete.
5. Extension and Application As an extension, consider a general non-Abelian gauge group, say the unitary group U (N ), with Lie algebra U(N ) consisting of N × N anti-Hermitian matrices. Then A, B = −Tr(AB),
A, B ∈ U(N ),
(5.1)
is the inner product over U(N ) which allows one to regard the Lie commutator, [ , ] on U(N ), as an exterior product so that A, [A, B] = 0, A, [B, C] = C, [A, B] = B, [C, A] ,
A, B, C ∈ U(N ). (5.2)
The U (N ) Yang–Mills–Higgs theory with the Higgs field represented adjointly has the Lagrangian action density 1 1 L = − Fµν , F µν + Dµ , D µ − V ( ). 4 2
(5.3)
In view of the method in Sect. 4 and the property (5.2), we may similarly show that a finite-energy static solution of the equations of motion of the Yang–Mills–Higgs theory in the (2 + 1)-dimensional Minkowski spacetime defined by (5.3) has a trivial temporal component, A0 . Furthermore, it is clear that our result applies to the models that contain several Higgs fields as well. As an application, consider the classical Abelian Chern–Simons–Higgs theory [13] defined by the Lagrangian action density 1 κ 1 λ L = − Fµν F µν + εµνα Aµ Fνα + Dµ φ D µ φ − (|φ|2 − 1)2 , 4 4 2 8
(5.4)
Proof of the Julia–Zee Theorem
355
over the (2 + 1)-dimensional Minkowski spacetime, where κ is the Chern–Simons coupling constant. The equations of motion governing static field configurations are λ (|φ|2 − 1)φ − A20 φ, 2 i − κ ε jk ∂k A0 = (φ D j φ − φ D j φ), 2 A0 = κ F12 + |φ|2 A0 . D 2j φ =
∂k F jk
(5.5) (5.6) (5.7)
Using the Julia–Zee Theorem stated in Sect. 2 and the existence theorem obtained in [1], we see that a finite-energy solution of the Chern–Simons–Higgs equations (5.5)– (5.7) exists which has a nontrivial temporal component A0 of the gauge field, hence a nontrivial electric sector is present in the theory, if and only if κ = 0, which switches on the Chern–Simons topological term in the model. In view of Theorem 3.1, similar applications may be made to non-Abelian Chern– Simons–Higgs vortex models [11,20,21]. Acknowledgments. The authors were supported in part by the NSF.
References 1. Chen, R.M., Guo, Y., Spirn, D., Yang, Y.: Electrically and magnetically charged vortices in the Chern– Simons–Higgs theory (2009, preprint) 2. Chern, S.S., Simons, J.: Some cohomology classes in principal fiber bundles and their application to Riemannian geometry. Proc. Nat. Acad. Sci. USA 68, 791–794 (1971) 3. Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974) 4. Dunne, G.: Self-Dual Chern–Simons Theories. Lecture Notes in Physics, Vol. 36, Berlin: Springer, 1995 5. Fröhlich, J., Marchetti, P.A.: Quantum field theories of vortices and anyons. Commun. Math. Phys. 121, 177–223 (1989) 6. Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity. In: Collected Papers of L. D. Landau (edited by D. Ter Haar), New York: Pergamon, 1965, pp. 546–568 7. Hong, J., Kim, Y., Pac, P.-Y.: Multivortex solutions of the Abelian Chern–Simons–Higgs theory. Phys. Rev. Lett. 64, 2330–2333 (1990) 8. Jackiw, R., Weinberg, E.J.: Self-dual Chern–Simons vortices. Phys. Rev. Lett. 64, 2334–2337 (1990) 9. Jaffe, A., Taubes, C.H.: Vortices and Monopoles. Boston: Birkhäuser, 1980 10. Julia, B., Zee, A.: Poles with both magnetic and electric charges in non-Abelian gauge theory. Phys. Rev. D 11, 2227–2232 (1975) 11. Kumar, C.N., Khare, A.: Charged vortex of finite energy in nonabelian gauge theories with Chern–Simons term. Phys. Lett. B 178, 395–399 (1986) 12. Nielsen, H.B., Olesen, P.: Vortex-line models for dual strings. Nucl. Phys. B 61, 45–61 (1973) 13. Paul, S., Khare, A.: Charged vortices in an Abelian Higgs model with Chern–Simons term. Phys. Lett. B 17, 420–422 (1986) 14. Prasad, M.K., Sommerfield, C.M.: Exact classical solutions for the ’t Hooft monopole and the Julia–Zee dyon. Phys. Rev. Lett. 35, 760–762 (1975) 15. Ryder, L.H.: Quantum Field Theory. 2nd ed., Cambridge: Cambridge U. Press, 1996 16. Schechter, M., Weder, R.: A theorem on the existence of dyon solutions. Ann. Phys. 132, 293–327 (1981) 17. Schwinger, J.: A magnetic model of matter. Science 165, 757–761 (1969) 18. Tchrakian, D.H.: The ’t Hooft electromagnetic tensor for Higgs fields of arbitrary isospin. Phys. Lett. B 91, 415–416 (1980) 19. ’t Hooft, G.: Magnetic monopoles in unified gauge theories. Nucl. Phys. B 79, 276–284 (1974) 20. de Vega, H.J., Schaposnik, F.: Electrically charged vortices in non-Abelian gauge theories with Chern– Simons term. Phys. Rev. Lett. 56, 2564–2566 (1986) 21. de Vega, H.J., Schaposnik, H.J.: Vortices and electrically charged vortices in non-Abelian gauge theories. Phys. Rev. D 34, 3206–3213 (1986) 22. Wilczek, F.: Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982)
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23. Wilczek, F.: Fractional Statistics and Anyon Superconductors. Singapore: World Scientific, 1990 24. Zwanziger, D.: Quantum field theory of particles with both electric and magnetic charges. Phys. Rev. 176, 1489–1495 (1968) 25. Zwanziger, D.: Local-Lagrangian quantum field theory of electric and magnetic charges. Phys. Rev. D 3, 880–891 (1971) Communicated by S. Zelditch
Commun. Math. Phys. 291, 357–401 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0840-2
Communications in
Mathematical Physics
AQFT from n-Functorial QFT Urs Schreiber Zentrum für Mathematische Physik, Bundesstrasse 55, Universität Hamburg, D-20146 Hamburg, Germany. E-mail: [email protected] Received: 10 October 2008 / Accepted: 26 February 2009 Published online: 23 May 2009 – © Springer-Verlag 2009
Abstract: There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses n-functors which assign to each patch a “propagator of states”. In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor (“parallel surface transport”) naturally yields a local net, whose locality derives from the 2-categorical exchange law, and which is covariant if the 2-functor is equivariant. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general Lorentzian structure, bare lightcone structure and higher dimensions. It does not, however, by itself imply anything about the existence of a vacuum state or about positive energy representations. Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . The Situation for 1-Dimensional QFT . . . . . Nets of Local Monoids . . . . . . . . . . . . . Extended 2-Dimensional Minkowskian FQFT The Main Point: AQFT from Extended FQFT . Covariance/Equivariance . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . 7.1 1-dimensional case . . . . . . . . . . . .
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7.2 Examples from parallel 2-transport . . . . . . . 7.3 2-Functors constant on one object . . . . . . . 7.4 Nets from wedge algebras . . . . . . . . . . . . 7.5 Lattice models . . . . . . . . . . . . . . . . . . 7.6 Boundary FQFT and boundary AQFT . . . . . 7.7 2-C ∗ -category codomains . . . . . . . . . . . . 7.8 Hopf spin chain models . . . . . . . . . . . . . 8. Further Issues . . . . . . . . . . . . . . . . . . . . A. 2-Vector Spaces and the Canonical 2-Representation
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1. Introduction Out of the numerous tools and concepts that physicists have used for the description of quantum field theory few are well defined beyond simple toy examples. Still, in many cases they “work”, often with dramatic success. Axiomatizations of QFT attempt to extract from the ill-defined symbols that appear in the physics literature those properties which are actually being used in structural proofs. • While the path integral itself usually is ill-defined, all that often matters is the assumption that it satisfies the gluing law [58]. Taking this law as an axiom leads to the Atiyah-Segal formulation of functorial QFT. • Similarly, while the products of physical field observables are usually ill-defined, all that often matters is the assumption that they satisfy the locality property [13]. Taking this as an axiom leads to the Haag-Kastler formulation of algebraic QFT. The power of axiomatizations is that they lead to a more robust and clearer picture. The danger of axiomatizations is that they fail to capture important phenomena. Therefore it is especially important to understand how different axiomatizations of the same situation are related. AQFT: Nets of local algebras. Nets of local operator algebras have been introduced [25] (see [26] for a review) in order to formalize the concept of the algebra of local observables in quantum field theory. One way to think of such a net is as a co-presheaf on a sub-category of open subsets of a given Lorentzian manifold X with values in algebras. These co-presheaves are required to satisfy a couple of conditions (the first two mandatory, the third and fourth usually desired but sometimes dropped, the fifth crucial for real-world physical examples): 1. (Isotony) All co-restriction morphisms are required to be inclusions of sub-algebras—this makes the co-presheaf a net. 2. (Locality/“microcausality”) The inclusions of two algebras assigned to two spacelike separated open subsets into the algebra assigned to a joint superset are required to commute with each other. 3. (Covariance) The net is covariant with respect to the action of a group G on X (for instance the Poincaré-group or the conformal group) if there is a family of algebra isomorphisms between the algebras assigned to any region and its image under the group action, compatible with the group product and the net structure. 4. (Time slice axiom) The algebra of a subset is equal to that assigned to any neighbourhood of any of its Cauchy surfaces. 5. (Vacuum state and spectrum condition) There is a state (a suitable linear functional) on the total algebra of the co-presheaf which behaves like a physical vaccum state
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(in that for instance it is translation invariant and induces a positive-energy representation of the translation group). Out of the study of these structures a large subfield of mathematical physics has developed, which is equivalently addressed as algebraic quantum field theory, or as axiomatic quantum field theory or as local quantum field theory, but usually abbreviated as AQFT. For a review of physical applications see [20]. FQFT: n-functorial cobordism representation. Remarkably, all three of the terms— algebraic, axiomatic, local—would equally well describe what is probably the main alternative parallel development: the study of representations of cobordism categories, i.e. of functors from categories whose objects are (d − 1)-dimensional manifolds and whose morphism are d-dimensional cobordisms between these to a category of vector spaces. A pedagogical introduction to this concept is in [4]. Such functors have been introduced to formalize the concept of the quantum propagator acting on the space of quantum states and imagined to arise from an integral kernel given by a path integral. While this functorial approach did not receive a canonical name so far, here we shall refer to it as functorial quantum field theory and abbreviate that as FQFT. FQFT has most famously been studied in the context of topological QFT, from which Atiyah originally deduced his sewing axioms [2]. A review is [10]. While topological FQFT is by far the most tractable and hence the best understood one, FQFT is not restricted to the topological case: equipping the cobordisms for instance with conformal structure yields conformal QFT, an observation which is the basis of Segal’s functorial axiomatization of QFT [53]. Restricting to 2-dimensional conformal cobordisms of genus 0 this yields the axioms of vertex operator algebras [28], see [35] for review and generalization. The result in [18] can be regarded as providing examples for Segal’s CFT axioms (though in that work Atiyah’s formulation of the functoriality axiom is being referred to). Similarly, ordinary non-relativistic quantum mechanics ((1+0)-dimensional QFT) is about (monoidal) representations (i.e. functors to Vect) of the (monoidal) category of 1-dimensional Riemannian cobordisms [55]. Taking this point of view on ordinary quantum mechanics seriously leads to Abramsky-Coecke’s categorical semantics of quantum protocols [1]. See [15] for an overview. In this vein, here we shall be concerned with functors on cobordisms with pseudoRiemannian structures, and with flat Lorentzian structure (Minkowski structure) in particular. In [21,22] it was suggested that the FQFT picture can and should be refined to an assignment of data of “order n” to codimension n spaces for all n, such that this assignment respects all possible gluings. Formally this should mean that for d-dimensional quantum field theory the 1-category of cobordisms is refined to a d-category of cobordisms [14,57] whose k-morphisms are k-dimensional cobordisms between (k −1)dimensional cobordisms, and that one considers d-functors from this d-category to a suitable codomain d-category. Baez and Dolan began to draw the grand picture emerging here in [7], which was recently picked up by Hopkins and Lurie [27]. This extended n-functorial description of d-dimensional QFT is only beginning to be explored. First concrete descriptions of Chern-Simons and Wess-Zumino-Witten theory in this context appeared in [21,22,55] and in various talks given by Freed and Hopkins, aspects of which have recently been made available as [23]. Much progress has been made with understanding the extended FQFT of finite group Chern-Simons theory
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(Dijkgraaf-Witten theory) [11]. The general idea (for smooth n-groups) is currently best understood not for quantum but for “classical” propagation, where it describes parallel transport in n-bundles ( (n − 1)-gerbes) with connection [8,47,49–51]. But there are numerous indications that the picture is correct, useful and compelling. In [19] we shall demonstrate that the formulation of 2-dimensional CFT and 3-dimensional TFT appearing in [18] (see [46] for a review) is secretly a 2- and 3-FQFT of this form. The relation. An obvious question, which does not seem to have been addressed before, is: What is the relation between the axioms of AQFT and FQFT? Intuitively it is clear that the locality of local nets captures the same physical aspect as the n-functoriality of n-FQFTs does: that assignments to larger patches are already determined by the assignment to their pieces. But the nature of the assignments are different. We shall demonstrate that every FQFT determines an AQFT in the sense of items 1 through 4 of the above list by postcomposing with the higher analog of the functor End : Vect iso → Algebras which sends each vector space to its algebra of endomorphisms and each isomorphism of vector space to the corresponding isomorphism of algebras. The above functor is held in high esteem, if only implicitly so, in quantum mechanics, where it encodes the passage from what is called the Schrödinger picture to the Heisenberg picture of quantum mechanics: given a unitary morphism of Hilbert spaces of the it H
e / E for H some self-adjoint operator, which sends each element form E ψ ∈ E to the element eit H ψ, its image under the above functor is the isomorphism of endomorphism algebras
End : ( E
eit H
/ E ) → ( End(E)
eit H ◦(−)◦e−it H
/ End(E) )
which sends any operator A on E to eit H Ae−it H . The situation is summarized in Table 1. Remark. The reader should beware that we do not consider or require in the present article structure related to Item 5 of the above list of AQFT characteristics, involving existence and nature of vacuum states on our local nets. In this sense our notion of AQFT for the purpose of this article is considerably weaker than what is appropriate in the context of concrete physical applications, and in particular some of our examples in Sect. 7 are formal examples in this sense, that will not extend to examples for AQFTs in a stricter physical sense that demands a suitable vacuum state. On the other hand, nothing in our discussion precludes the existence of a natural extra condition on FQFTs which would induce suitable vaccum structure on the corresponding AQFT. But discussion of this point shall not concern us here. Plan. We start in Sect. 2 by discussing everything for the very simple case of 1-dimensional QFT (quantum mechanics), which should help to set the scene. Then in Sect. 3 we quickly review those essentials of AQFT and in Sect. 4 those of FQFT which we need later on. Here we restrict to d = 2 dimensions for ease of discussion. The generalization to higher dimensions is relatively obvious and straightforward, we briefly comment on that in Sect. 8.
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Table 1. The two approaches to the axiomatization of quantum field theory together with their interpretation and relation as discussed here. The rectangular diagrams are explained in Sects. 3 and 4. The construction of the AQFT A Z from the extended FQFT Z is our main point, described in Sect. 5 algebraic QFT (also: axiomatic QFT, local QFT) AQFT
names abbreviations
functorial QFT FQFT assign
algebras (of observables) (time evolution) operators to patches, compatible with inclusion composition (gluing) Haag, Kastler Atiyah, Segal Heisenberg Schrödinger picture picture transport co-presheaf n-functor
idea axioms due to aspect of QFT formal structure t
cartoon of domain structure
O
?? ????? ? ? •? ????? •? ?? ???? ???? •• ? ?? ?
t
O
?? ****** ??? * ** ?? * ? y x ?? ****** ?? y ?? ? ?? x ?
/
/
x
x
7 c form endomorphism algebras
relation
main existing general theorems main existing nontrivial examples
?? ??? x ?? ?? y ?? _ AZ o x ?? End Z ?? ? y ??
spin-statistics theorem, PCT theorem chiral 2-d CFT
Z
results about topological invariants topological QFTs, full rational 2-d CFT
Our main definition is Def. 9 in Sect. 5, which gives the prescription for turning an FQFT 2-functor into a 2-dimensional local net of algebras. Our main result is Theorem 1, which states that this definition works. Theorem 2 says that this construction extends to a 2-functor from the 2-category of FQFT 2-functors to the category of local nets, and, similarly, Theorem 3 in Sect. 6 says that the obvious notion of equivariance on FQFT induces the right notion of covariance in AQFT. We close by discussing some examples in Sect. 7 and some further issues in Sect. 8. 2-categories. See [38] for the basics of 2-categories and 2-functors between them. For the time being we can and will entirely restrict attention to strict 2-categories and strict 2-functors between them. A review of all the basics of strict 2-categories that we need here can be found for instance in the Appendix of [51]. After we have established our
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construction for strict 2-categories the generalization to arbitrary weak 2-categories is immediate. 2. The Situation for 1-Dimensional QFT To put the following construction into perspective, it is useful to indicate what the transition from FQFT to AQFT that we are after looks like for the simple case where we are dealing with 1-dimensional quantum field theory, also known as quantum mechanics. Functorial quantum mechanics—Schrödinger picture. There are some slight variations on the theme of how to think of ordinary quantum mechanics—and in particular of possibly time dependent quantum mechanics—as a transport functor. These slight variations will have analogs also in higher dimensions, and hence are worth considering. Let X = R be the real line, thought of as the worldline of a particle and in particular thought of as equipped with the obvious trivial Minkowski structure, which regards each vector as timelike. Let P1 (X ) be the category of homotopy classes of future-directed paths in X . Hence the objects of P1 (R) are the points of R and there is a unique morphism from x to y whenever x ≤ y. In other words, P1 (X ) happens to be nothing but R regarded as a poset. There is the closely related category, 1CobRiem , whose objects are disjoint unions of points and whose morphisms are abstract 1-dimensional cobordisms equipped with a Riemannian structure. If we forget the monoidal structure on 1CobRiem (which is important, but not for our purposes here) and restrict it to just a single point, then we find t / • | t ∈ [0, ∞) , 1CobRiem BR0,+ = • where on the right we have the one-object category whose space of morphisms is the non-negative real half-line with composition given by addition of real numbers. There is a canonical projection functor P1 (R)
/ / 1CobRiem
/ y to the Riemannian cobordism • which sends the path x the same length. Now, ordinary time-independent quantum mechanics is a functor
t=(y−x)
/ • of
Z : 1CobRiem → Vect isos which sends the single object of 1CobRiem to the space of states, E, and sends the Riemannian cobordism of length t to an automorphism Z : (•
t
/ •) → ( E
exp(it H )
/ E ),
for H some endomorphism of the complex vector space E—the Hamiltonian. Here we take Vect isos to be the category whose objects are vector space and whose endomorphisms are linear isomorphisms.
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By the above, we can understand this as a functor on paths on the worldline, P1 (R), which happens to factor through BR0,+ : / Vect isos O
P1 (R) BR0,+
Z
1CobRiem .
Using the interpretation of such functors as vector bundles with connection [49], we can think of this as a vector bundle on the real line obtained from an R0,+ -equivariant vector bundle over the point. A more general situation is obtained when one considers time dependent quantum mechanics. Here the space of states and the Hamiltonian is allowed to change. There is then a 1-parameter family t → E t of spaces of states and H is no longer necessarily constant. This, then, is the case of a general functor P1 (R2 ) → Vect isos : (x
P exp(i
/ y ) → (E x
y x
H (t) dt)
/ E y) ,
where the expression on the right denotes the path-ordered exponential, which is nothing but the parallel transport with respect to the connection 1-form A = H dt. (More on that in Sect. 7.) A slightly different but very similar concept plays an important role in [55], where quantum field theories over a space X are considered, as functors from a category of cobordisms that come equipped with maps to X : The category 1CobRiem (R) of cobordisms equipped with a (smooth, say) map to the real line is not quite the same as P1 (R), but very similar. There is an obvious canonical functor P1 (R)
/ 1CobRiem (R)
which sends a path γ in R to the Riemannian cobordism of the same length equipped with the obvious map to R which coincides with γ . This way, from every “1-dimensional QFT over R” in the sense of [55], F : 1CobRiem (R) → Vect isos , one obtains an instance of ordinary time-dependent quantum mechanics by pulling back to P1 (R): / Vect isos . P1 (R) M p8 MMM p pp MMM p p MMM pp ppp F & 1CobRiem (R) Z
(In [55] Euclidean QFT is considered such that the morphisms assigned by Z are not in general invertible. While this is of no real relevance for the point of the above discussion, notice that later on, when we pass from FQFT to AQFT, we make crucial use of the fact that we assume FQFTs to assign invertible time propagators.) Depending on the precise details, the functor Z is usually demanded to factor through vector spaces with suitable extra structure. Topological vector spaces and Hilbert spaces are common choices. For our current purposes all such extra structure does not add anything to the aspects that we are interested in here and will be ignored until we come to concrete examples in Sect. 7.
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Algebraic quantum mechanics—Heisenberg picture. Given such a functor Z , we can form for each point x ∈ X the endomorphism algebra of the vector space, by sending x → End(Z (x)). In the case that there is extra structure on our vector spaces we would demand suitable endomorphisms. In the case of Hilbert spaces one usually demands all endomorphisms to be bounded operators. The endomorphism algebras thus obtained is known often as the algebra of observables. In the present case, we would be tempted to associate this algebra at time x with the entire future of x. So let S(X ) be the category whose objects are open sets Ox := {x ∈ X |x > x} and whose morphisms are inclusions Ox ⊂ O y of open subsets. Of course, due to the simplicity of the present setup, S(X ) is canonically isomorphic to the opposite of P1 (X ) itself, hence is itself just the opposite catgeory of R regarded as a poset. But for the discussions to follow it is useful to think of S(X ) as a category of open subsets of X . The crucial point now is that sending spaces of states to their algebras of endomorphisms sends the functor Z : P1 (X ) → Vect iso to a functor A Z defined by AZ / Algebras . S(X )H 9 HH ss s HH s s HH ss HH Z $ sss End Vect iso
The functor A Z sends open subsets in S(X ) to the algebras of endomorphisms of the spaces of states sitting over their boundary, and it sends inclusions of open subsets to the inclusion of the algebras which is induced from using conjugation with the propagator that is assigned to the path connecting the respective boundaries. More precisely: A Z : (O y ⊂ Ox ) → (End(Z (y))
Z (x→y)−1 ◦(−)◦Z (x→y)/
End(Z (x))).
Of course this means that all inclusions of algebras here are actually isomorphisms. But this is again just due to the simplicity of the one-dimensional example. In conclusion, since there is no content in the locality axiom in 1 dimension, this means that A Z is indeed a net of local monoids. It is this simple situation which we want to generalize from 1- to 2-dimensional QFT. 3. Nets of Local Monoids We start by considering a simple version of the relevant axioms of nets of local algebras on Minkowski space. Compare with Sect. 2.1 of [26]. Various refinements and generalizations are possible but add no further insight into the main point we want to make here. In particular, we shall ignore all extra structure that might be present on the algebras that appear below (such as them being C ∗ - or von-Neumann algebras) and even be content with regarding them just as monoids (i.e. forgetting their vector space structure). Our
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Fig. 1. A “causal subset” of 2-dimensional Minkowski space is the interior of a rectangle all of whose sides are lightlike. Such subsets are entirely fixed in particular by their left and right corners
Fig. 2. The category S(R2 ) of causal subsets of 2-dimensional Minkowski space. Objects are causal subsets, morphisms are inclusions of these
main point, that the inclusion and the locality axioms of local nets follow from taking endomorphisms on n-functors, is entirely independent of all such details. An interesting question is which extra structure on the n-functor will induce which extra structure on the local nets. While this shall not be our main concern here, the examples in Sect.7 give some indications. So let X = R2 be thought of as equipped with the standard Minkowski metric on R2 of which we will need only the induced lightcone structure on R2 , hence only the conformal class of the standard Minkowski metric. By a causal subset of X we shall mean as usual the interior of the intersection of the future of one point with the past of another. Definition 1. We denote by S(X ) the category whose objects are open causal subsets V ⊂ X of X and whose morphisms are inclusions V ⊂ V . In order to concentrate just on the properties crucial for our argument, we shall now talk about nets of local monoids (sets equipped with an associative and unital product). Write Monoids for the category of monoids and monoid homomorphisms and write Monoidsincl → Monoids for the subcategory containing only injections (monomorphisms). Definition 2. Two objects O1 , O2 in S(X ) are called spacelike separated if all pairs of points (x1 , x2 ) ∈ O1 × O2 are spacelike separated. Definition 3. A functor A : S(R2 ) → Monoids ,
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Fig. 3. Two spacelike separated causal subsets of R2
is a net of monoids on 2-dimensional Minkowski if it sends all morphisms in S(R2 ) to injections (monomorphisms) of monoids, i.e. if it factors as A
/ Monoids , S(R2 ) P 6 PPP ( llllll P( Monoidsincl This is a net of local monoids if for all spacelike separated O1 , O2 ⊂ O the corresponding algebras commute with each other in O, i.e. [A(O1 ), A(O2 )] = 0 as an identity in A(O). Notice that a monoid (possibly an algebra) A can be regarded as a one-object catea / • |a ∈ A (possibly enriched over vector spaces). As such, these gory BA := • monoids naturally form the 2-category whose objects are monoids, whose morphisms are homomorphisms and whose 2-morphisms are intertwiners. See also Appendix A. Definition 4. We write AQFT(R2 ) for the sub-2-category of the 2-functor 2-category 2Funct(S(R2 ), Cat) whose objects are local nets A, regarded as functors S(R2 )
A
/ Monoids
B(−)
/ Cat
taking values in one-object categories, whose morphisms are ordinary (as opposed to lax or pseudo) natural transformations between these, and whose 2- morphisms are modifications between those. Monoidal categories of endomorphisms of local nets. From this it is immediate that for A ∈ AQFT(R2 ) the endomorphisms EndAQFT(R2 ) (A) form a monoidal category (since it arises from a one-object 2-category). This is the monoidal catgegory defined in Definitions 8.1 and 8.5 in [26] and proven there to be monoidal in Proposition 8.30. The full subcategory (A) ⊂ EndAQFT(R2 ) (A) of local (meaning supported on some O ∈ S(R2 )) and transportable (meaning independent of support region up to isomorphism) endomorphisms is the main entity of interest in, and maybe in AQFT in general. The famous Doplicher-Roberts reconstruction theorem was motivated by the study of (A). This is discussed in great detail in [26].
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Symmetries, covariance and equivariance. Let G be a group acting on R2 and preserving the causal set structure in that the action lifts to a functor g : S(R2 ) → S(R2 ) for all g ∈ G. For A any local net we write g ∗ A : S2 (R2 )
g
/ S2 (R2 )
A
/ Monoids
for the pullback of the net along the action of g ∈ G. Definition 5. An equivariant structure on a local net A is a choice of isomorphisms A
rg
/ g∗A
for all g ∈ G such that for all g1 , g2 ∈ G we have g∗A > 1 II ∗ } IIg1 rg2 r g1 } } II }} II } I$ } } g1 g2 / (g1 g2 )∗ A . A Remark. This is 1-categorical descent [56] along the nerve of the action groupoid X//G of the category-valued presheaf Funct(S(−), Monoids). Remark. In the AQFT literature this equivariant structure is often called a covariant structure (for instance Assumption 3 on p. 14 of [26]) and is often expressed in terms of the total algebra colim S(R2 ) A (compare fact 5.10 on p. 41 of [26]). 4. Extended 2-Dimensional Minkowskian FQFT Instead of regarding causal subsets as a category under inclusion of subsets, we can think of them as living in a 2-category under composition (gluing). Definition 6. Let P2 (R2 ) be the 2-category whose objects are the points of R2 , whose morphisms are piecewise lightlike right-moving paths in R2 and whose 2-morphisms are generated from the closure of causal bigons t
O
?? **** ??? x ?? **** y ?? ? ? /
x
regarded as 2-morphisms as indicated, under gluing along pieces of joint boundary. Composition is by gluing along pieces of joint boundary, in the obvious way.
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Fig. 4. A typical 2-morphism in P2 (R2 )
Remark. The restriction that 1-morphism have to go “right” and 2-morphisms “downwards” simplifies the discussion a bit but is otherwise of no real relevance. Various generalizations of P2 (R2 ) can be considered without changing the substance of the following arguments. Just as with local nets, there are many variations of definitions of extended quantum field theories on 2-dimensional Minkowski space which one could consider. We choose to take the following simple definition. (Compare with the notion of parallel surface transport [8,50,51].) Definition 7. For any 2-groupoid C, an extended FQFT on 2-dimensional Minkowski space is a 2-functor Z : P2 (R2 ) → C. We write FQFT(R2 , C) := 2Funct(P2 (R2 ), C) for the 2-functor 2-category and FQFTisos (R2 , C) for the maximal strict 2-groupoid inside it. In concrete application C will usually be a 2-category of 2-vector spaces (which in general is not strict), as for instance those whose objects are (von Neumann) algebras, whose morphisms are bimodules over these, and whose 2-morphisms are bimodule homomorphisms [55]. We will see such an example in Sect. 7 based on some constructions summarized in Appendix A. But for the moment we do not need to make any concrete choice concerning C. The only necessary requirement for the following is actually that the 2-morphisms in C all be invertible and that horizontal composition by the images of the 1-morphisms under Z is injective. Equivariant structures. Let G be a group acting by diffeomorphisms on R2 which respects causal subsets in that the action extends to a functor g : S2 (R2 ) → S2 (R2 ) with the induced 2-functor (denoted by the same symbol) g : P2 (R2 ) → P2 (R2 ) for all g ∈ G. There is a canonical notion of what it means for a 2-functor Z : P2 (R2 ) → C to be equivariant with respect to this action [47,51]: for g ∈ G denote by g ∗ Z : P2 (R2 )
g
/ P2 (R2 )
the pullback of Z along the diffeomorphism G.
Z
/C
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Definition 8. (Equivariance of 2-functors). A G-equivariant structure on Z is choice of isomorphisms f g of 2-functors (i.e. strictly invertible pseudonatural transformations) Z
fg
/ g∗ Z
for all g ∈ G, and a choice for all g1 , g2 ∈ G of invertible 2-morphisms (i.e. modifications of pseudonatural transformations) g∗ Z > 1 II ∗ } IIg1 f g2 f g1 } } II }} Fg1 ,g2 III } } $ } / (g1 g2 )∗ Z Z f g1 g2
such that for all g1 , g2 , g3 ∈ G the tetrahedra 2-commute: g1∗ f g2
/ (g1 g2 )∗ Z ????? F w; ???? g1 ,g2 www ??? w # www f g1 f g1 g2 (g1 g2 )∗ f g3 ww w ww ww w Fg1 g2 ,g3 ∗ w / (g1 g2 g3 ) Z Z f g1∗ Z O
g1 g2 g3
g1∗ f g2
=
/ (g1 g2 )∗ Z { Fg2 ,g3 f g1 (g1 g2 )∗ f g3 . g1∗ f g2 G g // / 3GGG /// GG GG Fg1 ,g2 g3 // # / (g1 g2 g3 )∗ Z Z f g1∗ Z O GG GG GG GG
g1 g2 g3
Remark. In the case that G acts freely, this is nothing but 2-categorical descent [56] along / / X/G) with coefficients in the 2-category-valued presheaf Y := ( X 2Funct(P2 (−), C) [47]. If G does not act freely it is descent with respect to the nerve of the action groupoid of G. 5. The Main Point: AQFT from Extended FQFT We define a map from FQFTs in the sense of Definition 7 to AQFTs in the sense of Definition 3 and demonstrate, Theorem 1, that it indeed sends 2-functors Z to local nets of monoids A Z . Then we observe, Theorem 2, that this construction extends to a 2-functor from FQFTs to AQFTs on R2 . We end the section with a discussion of the properties of A Z in light of the time slice axiom. Definition 9. Given any extended 2-dimensional FQFT, i.e. a 2-functor Z : P2 (R2 ) → C, we define a functor A Z : S(R2 ) → Monoids. On objects it acts as
⎛
⎞ ⎛ ⎛ ⎞⎞ ?? ⎜ ??? ⎟ A Z : ⎝ x ?? ⎠ → EndC ⎝ Z ⎝ x ??γ y ⎠⎠ , y ??? ? γ ?? ?
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where on the right we form the monoid of 2-endomorphism a in C on the 1-morphism γ Z (x → y) in C that is the past boundary of Ox,y , γ
Z (x →y)
Z (x)
a
Z (y) . ?
γ
Z (x →y)
On morphisms A Z is defined to act as follows. For any inclusion Ox ,y ⊂ Ox,y ∈ S(R2 ) t 1 JJJ JJ tt t JJ tt JJ t JJ tt t JJ t t JJ t JJ tt t J tt y x8 2= 88 == 88 88 y x 88 == == 8 3 DD 4 5 DD zz z DD z DD zz D zzz 6 (the numbers here and in the following are just labels for various points in order to help us navigate these diagrams) we form the pasting diagram :1J tt JJJ JJ tt t JJ tt JJ t JJ tt t JJ t t JJ t JJ tt t t $ y x8 2 @ == C 88 = 88 88 y = 88 @ x == @ == == 8 3 DD 4 =5 DD zz z DD z DD f zzz D! zz 6 in P2 (R2 ). Here the obvious projections along light-like directions (for instance from x onto x → 6 yielding 3) is used. It is at this point that the light-cone structure crucially enters the construction.
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Let f be the 2-morphism obtained by whiskering (= horizontal composition with identity 2-morphisms) the indicated 2-morphism f with the 1-morphisms x → 3 and 5 → y.
f :=
y x8 88 C 88 88 y = 88 @ x == @ . == == 8 3 DD 4 =5 DD zz z DD z DD f zzz D! zz 6
For any a ∈ EndC Z (x , 4, y ), Z (x →4→y )
Z (x )
Z (y ) , >
a
Z (x →4→y )
let a be the corresponding re-whiskering by Z (x, 3, x ) from the left and by Z (y , 5, y) from the right:
Then we obtain an injection EndC (Z (x , 4, y ))
/ EndC (Z (x, 3, 6, 5, y))
by setting a → Z ( f ) ◦ a ◦ Z ( f )−1 , i.e.
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Fig. 5. The exchange law in 2-categories, which is the functoriality of horizontal composition on the Homcategories, says that the 2-dimensional order of composition of 2-morphisms is irrelevant
Remark. Notice that this prescription is essentially nothing but the one we described already for the 1-dimensional case in Sect. 2: to open subsets we assign the endomorphism algebra of the space of states assigned to one part of their boundary. To an inclusion of open subsets we then assign the inclusion of such algebras obtained by parallel transporting the algebra of the inner set into the algebra of the outer set using conjugation with the propagators that the 2-functor assigns to 2-morphisms in P2 (R2 ). The difference to the 1-dimensional case here is that this conjugation operation involves some (the obvious) re-whiskering. We will see that it is essentially this re-whiskering and the exchange law in 2-categories which lead to the locality of the net of monoids obtained this way. Now we come to our main point. Theorem 1. The functor A Z is a net of local monoids. Proof. We need to demonstrate three things: 1. that the above assignment is functorial; 2. that the above assignment satisfies the locality axiom. The first two properties turn out to be a direct consequence of 2-functoriality of Z and the exchange law in 2-categories. To see functoriality, consider a chain of inclusions / Ox,y Ox ,y r ; HH x HH xx HH x HH xx H# - xx Ox ,y in S(R2 ) and the corresponding pasting diagram
AQFT from n-Functorial QFT
x4 44 44 44 44 44 44 44
373
y r8 1 MMMM E rr M r MMM r rr MMM rr r M MMM rr rr MMM rr r & 2 ? y IIII u: x'''' ;; ~? @@@ u ~ u II '''' ;;; ~~ $ uuu '''' ;; y 3> x ?4 ? A > > ; >> AAA }} >> '''' ; >> f } l fr >> '''' 6 F 5 GGG w; 7 33 >> '''' w GG w 33 >> ' ' GG f www 33 >>' ' G w G# ww 8 GG 9 10 v; GG vv GG v v GG vv fc GG vv GG v G# vvv 11
in P2 (R2 ). The composite inclusion EndC (Z (x → 6 → y )) → EndC (Z (x → 5 → 9 → 7 → y )) → EndC (Z (x → 3 → 8 → 11 → 10 → 4 → y)) Z (x →6→y )
sends Z (x )
a
Z (y ) to >
Z (x →6→y )
The contributions from fl and fr manifestly cancel and we are left with the pasting diagram for the direct inclusion EndC (Z (x → 6 → y )) → EndC (Z (x → 3 → 8 → 11 → 10 → 4 → y)).
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This shows that / A Z (O) A Z (O K)s 9 KKK t tt KKK t t KKK tt , ttt % A Z (O ) commutes, as desired. To see locality, let Ox,y and Ox ,y be two spacelike separated causal subsets inside O(3,5 ) . The relevant pasting diagram in P2 (R2 ) is of the form 7 ? 7 >> A ;;;; >> ; y x A x ;; A >> ? >> ? y >> ; > > >> ;; >> > 8 3A 5 8 5 D = = AA z . || DDDD z | AA | zz DD AA f 1 ||| f 2 zzz DD AA z DD AA ||| D" zzz | f0 9 CC y< 9 CC y CC yy CC yy y CC y C! yyy 10
(We are displaying a very symmetric configuration only for notational convenience. The argument does not depend on that symmetry but just on the fact that Ox,y does not intersect the past of Ox ,y and vice versa.) Now given any two endomorphisms Z (x →8 →y )
Z (x→8→y)
Z (x)
a
Z (x→8→y)
Z (y)a and Z (x ) >
a
Z (y ) we can either first include >
Z (x →8 →y )
a in EndC (Z (3 → 9 → 10 → 9 → 5 )) and then a , or the other way around. Either way, the total endomorphism in EndC (Z (3 → 9 → 10 → 9 → 5 )) is
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375
This means that the inclusions of a and a in EndC (Z (3 → 9 → 10 → 9 → 5 )) commute. Theorem 2. This construction extends to a 2-functor FQFTisos (R2 , C) → AQFT(R2 ) faithful on 1-morphisms and trivial on 2-morphisms. Proof. The proof is very analogous to the proof of Theorem 3 in the next section, only slightly simpler. Extension to Cauchy covers and the time slice axiom. The above construction restricts attention to causal subsets, while the principle underlying the construction is more general. We make some remarks on this generalization and its relation to the time slice axiom. The category P2 (X ) from Definition 6—still for X = R2 , for definiteness—contains (as its 2-morphisms) more subsets of R2 than the category S(X ) from Definition 1 contains as objects: the former contains subsets bounded by any two piecewise lightlike ? ?? ?? ??? , rightbound paths with same source and target point, such as the interior of ??? ? ? ?? ? ? ? ??? while the latter contains only the causal double cones Ox,y := x ??? y , which are ?? the usual domains considered in AQFT. Definition 10. (Cauchy neighbourhoods). We write S (X ) for the category whose objects of these more general open subsets bounded by piecewise lightlike rightbound paths (morphisms are still inclusions of open subsets). For the following paragraphs we shall refer to the objects of S (X ) as local convex causal Cauchy neighbourhoods, or simply as Cauchy neighbourhoods for reasons to be discussed shortly.
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/ S (X ) as well as a natural Notice that we have an obvious inclusion S X / / S(X ) (obtained by sending an object U of S (X ) to Os(U ),t (U ) , surjection S X where s and t are the source and target objects of U when regarded as a 2-morphism in P2 (X )) such that S(X )
σ
/ S (X )
p
/ / S(X ) . 4
Id
Definition 11. (Local net on Cauchy neighbourhoods) For Z : P2 (X ) → C a 2-functor as before, let AZ : S (X ) → Monoids be the functor constructed verbatim as in Definition 9 but with objects of S(X ) generalized everywhere to objects of S (X ). Everything goes through exactly as before, and in fact our original construction of A is just the restriction of the construction of A to causal subsets. Moreover, one notices that the endomorphism monoid assigned by A to a Cauchy neighbourhood U is equal to the endomorphism monoid assigned by A to Os(U ),t (U ) , since in the definition of the inclusion morphisms (Definition 9) in the net there is in this case no re-whiskering involved in translating from the endomorphism monoid of the Cauchy neighbourhood to that of its double cone causal subset – compare the remark at the end of Definition 9. We can summarize this by Proposition 1. A Z equals the restriction of AZ along the inclusion σ and A is naturally isomorphic to the pullback of A along the projection p: . S(X _ ) J JJ JJA Z JJ JJ J% S (X ) A Z / Monoids t9 ttt p tt ~tttt A Z S(X ) To interpret this physically, recall that a Cauchy surface in a globally hyperbolic Lorentzian manifold is a codimension 1 hypersurface such that every timelike curve intersects it precisely once. Cauchy surfaces are the supports of initial data for causal time evolution in globally hyperbolic Lorentzian manifolds. Noticing that the Cauchy surfaces of causal subsets Ox,y are precisely the spacelike paths in Ox,y connecting x and y we find that p
/ / S(X ) over an object Ox,y ∈ Observation 1. The objects in the fiber of S (X ) S(X ) are precisely the convex open neighbourhoods U ⊂ Ox,y of Cauchy surfaces in Ox,y which arise as open covers U = ∪i Oi by causal subsets Oi → Ox,y . This justifies the term “Cauchy neighbourhoods” for the objects of S (X ). In light of this interpretation, Proposition 1 asserts that the local net A Z does (regarded as a co-presheaf on S (X )) not actually depend on the full interior of any given causal subset, but just on that of any of the neighbourhoods of Cauchy surfaces in that causal subset.
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There is one sense in which this statement is trivial: given any local net A : S(X ) → Monoids, we can always extend it to a net A on Cauchy neighbourhoods simply by p / / S(X ) A / S (X ) , and this pair (A, A ) will form a comsetting A : S (X )
muting diagram as in Proposition 1. But what our construction shows is that if A arises as the endomorphism co-presheaf of a 2-functor, then also this A naturally has an interpretation as an endomorphism copresheaf. For comparison, we state the usual time slice axiom in a form that exhibits its role in the context of the diagram appearing in Proposition 1. Recall for that the notion of Kan extensions of functors along morphisms out of their domain, for instance from Chapter 4 of [33]: the Kan extension is a universal solution to the problem of enlarging the domain of a functor, such as from S(X ) to S (X ).
Definition 12. (Time slice axiom for local nets of monoids) A local net of monoids A : S(X ) → Monoids, regarded as a net of submonoids of Atot = colim S(X ) A, (assumed to exist) to be written A : S(X ) → MonoidsAtot satisfies the time slice axiom if its left Kan extension Lanσ A along the inclusion of causal subsets into their Cauchy neighbourhoods coincides with their pullback along the projection from Cauchy neighbourhoods to causal subsets, i.e. if the lower triangle in S(X _ ) L LLL LLAL σ LLL & / MonoidsAtot S (X ) 8 Lanσ A r rr p rrr r r rr A S(X ) commutes. Proposition 2. We have • Lanσ A assigns to each U ∈ S (X ) the monoid ∨σ (O)⊂U A(U ) which is the monoid generated from all submonoids {A(O)}σ (O)⊂U which correspond to causal subsets inside U ; • the condition in Definition 12 therefore demands that for {Oi ∈ S(X )}i any maximal cover of a Cauchy neighbourhood (U ∈ S (X )) ⊂ σ (O) of a causal subset O ∈ S(X ) by causal subsets Oi (i.e. by all causal subsets Oi with σ (Oi ) ⊂ U ) we have A(σ (U )) = ∨i A(Oi ). Proof. To compute the left Kan extension Lanσ A notice that we can regard S(X ) and S (X ), being posets, as categories enriched over the discrete monoidal category V = {∅, {•}} with product the cartesian product of sets (there is either an inclusion O → O or not, so all Hom-sets are either empty or the singleton). Moreover, the category Monoids is tensored over V if we set the product of a monoid with the empty set to be the trivial monoid. In this case Eq. (4.24) in [33] applies which says that the left Kan extension is given by the coend Oi ∈S(X )
(Lanσ A)(U ) =
Hom S (σ (Oi ), U ) · A(Oi ) ,
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U. Schreiber
where Hom S (σ (Oi ), U ) is either empty if Oi is not a subset of U , in which case the expression Hom S (σ (Oi ), U ) · A(Oi ) is the trivial monoid, or is the singleton if Oi is a subset of U , in which case the expression is just A(Oi ) itself. This means that the coend reduces to the colimit over the A(Oi ) for Oi ⊂ U , σ (O i )⊂U
···
A(Oi ) = colimσ (Oi )⊂U A(Oi ) =: ∨i A(Oi ).
In summary we have • the idea expressed by the time slice axiom is that a net on causal subsets extends to a net on Cauchy neighbourhoods and is then determined on double cones by its value on any of the double cone’s Cauchy neighbourhoods; • without further information the only reasonable extension of a net to Cauchy neighbourhoods is by U → ∨ Oi ⊂U A(Oi ), which we identified with the universal extension in the sense of Kan extensions of functors; • but a net arising as the endomorphism co-presheaf of a 2-functor, as described here, has as such a (possibly different) natural extension obtained by applying the endomorphism construction to Cauchy neighbourhoods themselves. A net arising as an endomorphism co-presheaf A Z may fail the time slice axiom in its usual form in that A Z (U ) is not the same as ∨ Oi ⊂U A Z (Oi ), still its value on any O, which is an endomorphism monoid associated to the boundary of O, is isomorphic to the corresponding endomorphism monoid of any Cauchy neighbourhood U inside O.
6. Covariance/Equivariance We had seen definitions for equivariance (“covariance”) of local nets and of FQFT 2-functors. The following theorem says that these notions are compatible under our relation of the two. Theorem 3. Every G-equivariant structure, Definition 8, on the FQFT Z : P2 (R2 ) → C induces a G-equivariant structure, Definition 5, on the AQFT A Z obtained from it according to Definition 9. Proof. For any g ∈ G the component map of the pseudonatural transformation f g is
fg : ( x
γ
/ y ) →
Z (γ )
/ Z (y) s ssss sssss s s s ssss f g (x) f g (y) ssssfs(γ ) . s s s g s s s s s s sss u} ss Z (g(γ )) / Z (g(y)) Z (g(x)) Z (x)
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For γ the target boundary of the causal subset O, t
?? **** ??? x ?? **** O y ?? ? ? O
γ
/
x
conjugating with the components on the right defines the monoid isomorphism r g (O) : EndC (Z (γ )) → EndC (Z (g(γ )))
⎛ ⎜ r g (O) : ⎜ ⎝ Z (x)
Z (γ )
a
⎞ )
⎟ ⎟ 5 Z (y)⎠ →
Id
Z (γ )
Z (g(γ ))
/ Z (g(y)) sss sssss s s s ssss sssss f g (x)−1 f (y)−1 s s s s f g (γ )−1 p g s s s ss sssss Z (γ ) u} ss ) a Z (x) 5 Z (y) ss Z (γ ) s s s ssss sssss s s s f g (x) f g (y) ssss sssss f g (γ ) s s s sssss u} ss Z (g(γ )) / Z (g(y)) Z (g(x))
Z (g(x))
Id
.
Here f g (γ )−1 p denotes the inverse of the 2-cell f g (γ ) with respect to vertical pasting (which is the ordinary inverse up to a re-whiskering). We need to check that this construction 1. yields a morphism of nets in that it makes for all O ⊂ O the naturality squares A Z (O _ )
A Z (O)
r g (O )
/ A Z (g(O )) _
r g (O)
/ A Z (g(O))
commute; 2. produces the commuting triangles in Definition 5. This can be seen as follows:
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U. Schreiber
1. The pseudo-naturality condition on the components of f g Z (γ ) Z (O)
/ Z (y) sss sssss s s s ssss sssss f g (x) f g (y) s s s sss f g (γ ) s s s sssss u} ss Z (g(γ )) / Z (g(y)) Z (g(x)) Z (x)
=
Z (γ )
/ Z (y) s ssss sssss s s s ss ssss f g (x) f g (y) sssssf g (γ ) s s s s s sssss u} ss / Z (g(y)) Z (g(γ )) Z (g(x)) C Z (x)
Z (γ )
Z (g(O))
Z (g(γ ))
for all O implies precisely the condition r g (O)| A(O ) = r g (O ) when applied to our Definition 9 of the inclusion map A(O ) → A(O): that inclusion was obtained by conjugating with Z (y) Z (x)@ @@ ~? ~ @@ ~~ @@ ) ) ~~ Z (y Z (x @@ ~ III III . u: u: @@ ~~ I$ I$ ~~ uuu uuu Z (5) Z (3) N Z (4) NNN pp8 p NNN p Z( f ) NNN ppp NN& ppppp Z (6) Following this by the action of r g (O) amounts to conjugating with Z (y) Z (x)F ; FF xx FF x x FF ) ) xx Z (y FF Z (x x x MMM MMM FF xx qq8 qq8 MM& MM& f g (x) f g (y) F# qqq xx qqq z Z (4) Z (5) z Z (3) M zzzzzz MMM qq 8 f g (5→y) f g (x→3) q M q
MMM zzz qqq
MMM q Z( f ) Z (g(y)) Z (g(x))y zzz q
GG MMM qq
w; GG f g (3) f g (5) M& qqqq ww
GG w
GG www sss Z (6)
GG sssss www f g (6→5) GG f g (3→6)
# ww ssss
Z (g(3)) u} ssss Z (g(5))
PPP f g (6) n6 PPP
nnn
n
PPP n
n PPP
nnn P( nnn Z (g(6))
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By pseudonaturality of f g this equals conjugation with Z (y) FF x< FF x FF xx ) ) FF xx Z (y Z (x x FF x NNN NNN FF pp7 pp7 xx f g (x) f g (y) NN' NN' " xx ppp ppp z Z (3) Z (4) Z (5) zzzzz f g (x ) f g (y ) f g (x→3) f g (5→y) z zz f g (3→x ) f g (4→y ) Z (g(x)) x zzz Z (g(y)) f g (x →4) f g (y →5) FF < x FF f g (3) f g (4) f g (5) xxxx FF FF xx Z (g(x )) Z (g(y )) FF NNN NNN ppp7 ppp7 xxx FF N N N N " pp ' pp ' xx Z (x)
Z (g(3))
Z (g(4)) Z (g(5)) QQQ mm6 QQQ m m m QQQ Z (g( f )) mmm QQQ mmm QQ( m m m Z (g(6))
Since the endomorphism a to be conjugated is localized on Z (x ) → Z (y ), Z (x )→Z (4)→Z (y )
Z (x)
/ Z (x )
/ Z (3)
a
Z (y ) >
/ Z (5)
/ Z (y) ,
Z (x )→Z (4)→Z (y )
both f g (x → 3 → x ) and f g (y → 5 → y) drop out when conjugating and only conjugation with f g (x → 4 → y ) acts nontrivially. But that precisely amounts to first applying r g (O ) and then injecting into O. 2. The equivariance triangle condition in Definition 8 says precisely that r g (O) makes the required covariance triangle in Definition 5 commute: To see this it is convenient to equivalently rewrite the previous equation for r g (O) as Z (γ ) a
/ Z (y) sss sssss s s s ssss sssss f g (x) f g (y) s s s sss f g (γ ) s s s sssss u} ss Z (g(γ )) / Z (g(y)) Z (g(x)) Z (x)
=
Z (γ )
/ Z (y) s ss ssss sssss s s s ssss f g (x) f g (y) sssssf g (γ ) s s s s s sssss u} ss / Z (g(y)) Z (g(γ )) Z (g(x)) C Z (x)
Z (γ )
r g (O)(a)
Z (g(γ ))
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U. Schreiber
for all a ∈ End(Z (γ )). Accordingly, we have for the composition of two transformations
for all a ∈ End(Z (γ )). Using now the triangle of pseudonatural transformations in Definition 8 this is equivalent to
Z (γ ) a
/ Z (y) BB BB BB B f g1 (x) f g1 (y) | BBBB || BB | Fg1 ,g2 (y) | B ~|| | f (x) f (γ ) f (y) g1 g2 g1 g2 g1 g2 Z (g1 (x)) Z (g1 (y)) BB −1 || BB Fg ,g (x) | | BB 12 | B | || f g2 (g1 (x)) || BB | BB || f g2 (g1 (y)) BB || | B } ~|| Z (g1 g2 )(γ ) / Z ((g1 g2 )(y)) Z ((g1 g2 )(x)) Z (x) || | || ||
Z (γ )
=
AQFT from n-Functorial QFT
383
/ Z (y) BB BB BB B f g1 (x) f g1 (y) | BBBB || BB | F (y) | g1 ,g2 B . ~|| | f (x) f (γ ) f (y) g1 g2 g1 g2 g1 g2 Z (g1 (x)) (y)) Z (g 1 BB −1 || BB Fg ,g (x) || BB 12 | B | || f g2 (g1 (x)) || BB | BB || f g2 (g1 (y)) BB || | B } ~|| Z ((g1 g2 )(x)) Z (g1 g2 )(γ ) / Z ((g1 g2 )(y)) A Z (x) || | || ||
Z (γ )
r g2 (g1 (O))◦gg1 (O)a Z (g1 g 2 )(γ )
But in this equation we can cancel the F·,· on both sides to obtain Z (γ )
a
/ Z (y)
Z (x)
Z (γ )
f g1 g2 (x)
f g1 g2 (γ )
} Z (g1 g2 )(γ )
Z ((g1 g2 )(x))
f g1 g2 (y)
/ Z ((g1 g2 )(y))
=
/ Z (y)
Z (x)
Z (γ )
f g1 g2 (x)
f g1 g2 (γ )
Z ((g1 g2 )(x))
Z (g1 g2 )(γ )
}
f g1 g2 (y)
/ Z ((g1 g2 )(y)) B
r g2 (g1 (O))◦gg1 (O)a Z (g1 g 2 )(γ )
This shows that r g2 (g1 (O)) ◦ r g1 (O)(a) = r g1 g2 (O)(a).
7. Examples 7.1. 1-dimensional case. Before looking at concrete examples for 2-FQFTs on Minkowski space it is again helpful to first recall some simple facts in the 1-dimensional case from our perspective. We can regard ordinary quantum mechanics as given by an associated U (E)-bundle with connection on the real line (the “worldline”) for E some Hilbert space. This bundle
.
384
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is necessarily trivializable. After picking a trivialization its globally defined Lie(U (E))valued connection 1-form is A = i H dt ∈ 1 (R1 , u(E)) with t the canonical coordinate and H a self-adjoint operator on E: the Hamilton operator. The quantum time evolution operator Z : ( t0
/ t1 ) → ( E
P exp(
[t0 ,t1 ]
A)
/E)
is nothing but the parallel transport with respect to A (see for instance [49]). In general H depends on t, in which case one speaks of time dependent quantum mechanics and the above formula, with its “path ordered exponential” on the right, is what is usually referred to as the Dyson formula in quantum mechanics textbooks. In that case there is no translational invariance on the worldline. If however H is constant we have time independent quantum mechanics. In that case the quantum time evolution propagator reads Z : ( t0
/ t1 ) → ( E
P exp(
[t0 ,t1 ]
A)
/ E )=( E
exp(i(t1 −t0 )H )
/ E ).
In either case, there is a canonical equivariant structure, Definition 8, on Z with respect to the action of R on R by translations: for a ∈ R the components of the natural transformation Z
ft
/ a∗ Z
are simply fa : x → ( E x
Z (x→x+a)
/ E x+a ).
Naturality of f t and commutativity of the equivariance coherence triangle both follow directly from the functoriality of Z . The equivariant structure on the net A Z induced by this according to Sect. 6 is that which acts on each local algebra A Z (Ox ) by the Heisenberg propagation rule a → Z (x → x + a) ◦ a ◦ Z (x → x + a)−1 . 7.2. Examples from parallel 2-transport. The above shows that the dynamics of quantum mechanics (1+0-dimensional QFT) can be entirely thought of as a vector bundle (or Hilbert bundle, rather) with connection on the “worldline” R. Similarly, 2-vector 2-bundles [9,59] ( gerbes) with connection [8,47,50,51] on the “worldsheet” R2 can be regarded as giving the dynamics of (1+1)-dimensional QFT. Indeed, every parallel transport 2-functor on R2 as in [8,50,51] gives an example of a 2-FQFT in the sense of Definition 7, simply by restricting it from all 2-paths in R2 to those contained in P2 (R2 ). From each such 2-functor one obtains, by Theorem 1, a local net of monoids. Whether this local net of monoids has any covariance depends, according to Proposition 3, or whether or not the 2-functor has any equivariant structure. Whether the net of monoids obtained from the 2-functor is actually a net of algebras with certain extra structure (in particular C ∗ , von Neumann) depends on what precisely the 2-functor takes values in over 1-morphisms, because that determines what the endomorphism monoids are like.
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While not every 2-bundle on 2-dimensional base space is necessarily trivializable, we here want to restrict attention to the case that the 2-bundle is trivializable. (If not, global effects such as described in [19] will play a role, too.) Then we can assume its parallel transport 2-functor to come from globally defined differential form data. If we require the 2-functor to be strict and to take values in a 2-groupoid with a single object, which we shall denote BG, then Theorem 2.20 in [50] says that it comes precisely from a pair consisting of a 1-form and a 2-form A ∈ 1 (R2 , g), B ∈ 2 (R2 , h) with values in Lie algebras g and h which form a differential crossed module t / α / g der(g) ) such that (h F A + t∗ ◦ B = 0 , where FA ∈ 2 (R2 , g) is the curvature 2-form of A. We write Z (A,B) : P2 (R2 ) → BG for the 2-functor obtained this way. The local net A Z (A,B) obtained from this by Theorem 1 is a local net of groups. We get proper nets of local algebras by passing instead to an associated parallel 2-transport functor [51], which is induced by a 2-representation of G on 2-vector space, i.e. a 2-functor ρ : BG → 2Vect , where 2Vect denotes a 2-category of 2-vector spaces. In particular, [48], there are large classes of 2-representations which factor through the bicategory of bimodules ρ
/ 2Vect . BG H HH t: t HH tt HH t HH tt # tt Bimod More details on this are summarized in Appendix A and in [51]. The corresponding associated 2-FQFT functor Z ρ(A,B) : P2 (R2 )
Z (A,B)
/ BG
/ Bimod
/ 2Vect
sends each edge to a bimodule over some algebra. 2-Functors of this form and interpreted as 2-FQFTs have in particular been considered in [55]. Therefore the corresponding local net A Z ρ(A,B) sends each O ∈ S(R2 ) to an algebra of bimodule endomorphisms. This is reminiscent of various other constructions that have been considered in the context of AQFT. But a more detailed discussion will have to be given elsewhere. As in the 1-dimensional case, we canonically have an equivariant structure on Z and on A Z with respect to any 1-parameter group of translations which respects the lightcone structure. Let in particular R act by translation along the canonical time coordinate on R2 . Then for a ∈ R the component of the pseudonatural transformation Z
fa
/ a∗ Z
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U. Schreiber
Fig. 6. 2-Vector transport coming from a 2-connection (A, B) ∈ • (R2 , (h → g)) with values in the strict Lie 2-algebra (h → g) and the canonical representation ρ of the corresponding strict Lie 2-group G (2) on 2-vector spaces. The 2-FQFT obtained this way assigns algebras to points, bimodules to paths and bimodule homomorphisms to surfaces. The corresponding local net A Z (A,B) assigns algebras of bimodule endomorphisms
is
⎛
fa
x? ?? ⎜ ?? : ⎜ ?? ⎝ ?
?z y
⎞ ⎟ ⎟ → ⎠
Z (z) rr8 r r rrr rrr
Z (y)
,
Z ( (x,y,z,a))
Z (z + a) Z (x + a)
LLL
r8 r LLL r r LLL rrr r L& r r Z (y + a) Z (x) L LLL LLL LLL &
where (x, y, z, a) denotes the surface swept out by the path x → y → z when translating it continuously to (x + a) → (y + a) → (z + a). This surface is not part of P2 (R2 ) the way we have defined it, but is a more general 2-path in R2 on which we can evaluate our 2-functor Z , by assumption. Pseudonaturality and coherence of the assignment f a for all a ∈ R is a direct consequence of the 2-functoriality of Z , very similar to the 1-dimensional case. The induced equivariant structure on the net A Z is the local Heisenberg picture time propagation. 7.3. 2-Functors constant on one object. A simple class of examples worth looking at to get a feeling for the situation are those FQFT 2-functors Z on P2 (R2 ) which assign a fixed object V ∈ Obj(C) to each point of R2 , send all paths to the identity morphism on that object and all surfaces to the identity 2-morphism on this identity 1-morphism. The local net A Z obtained from such a 2-functor is constant. It assigns the same monoid to all causal subsets: A Z : O → End(Id V ). For this to be a local net, it must be true that End(Id V ) is a commutative monoid. And indeed it is: this is the Eckmann-Hilton argument which holds in general for 2-endomorphisms of identity 1-functors. The argument is entirely analogous (and that is of course
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387
no coincidence) to that which shows that the second homotopy group of any space is abelian. In [24] the endomorphisms of the identity on an object V in a 2-category C is interpreted as the trace of the identity on V , which in turn is interpreted in [11] as the dimension of V : A Z (O) = End(Id V ) =: Tr(Id V ) =: dim(V ). For instance (see [11]) if V = Rep(H ) is the category of representations of some group or groupoid H , regarded as a 2-vector space, then dim(V ) = Z (C(H )) is the center of the group ring of H . Another example, [24]: if C is the bicategory of bimodules, C = Bimod, and V is any algebra, then dim(V ) is the 0th Hochschild cohomology of V . Full Hochschild cohomology is obtained by taking the derived category of bimodules. Of particular interest are objects V with a representation (meaning: 2-representation!) of the Poincaré group G in two dimensions, or some related group, on them. 2-Representations of the Poincaré group have been examined for instance in [16]. The constant FQFT 2-functor on such an object canonically carries a nontrivial G-equivariant structure in the sense of Sect. 6, hence induces a covariant structure on the corresponding local net. 7.4. Nets from wedge algebras. A special case of interest of the constructions in Sect. 7.2 is the following: Denote by W := {(x 0 , x 1 ) ∈ R2 | x 1 ≤ −|x 0 |} ⊂ R2 the standard left wedge in 2-dimensional Minkowski space. Let FW : R2 → Algebras ↓ A W be a family of subalgebras of a fixed algebra A W – to be called the wedge algebra—over 2-dimensional Minkowski space with the special property that every algebra in a wedge y + W is a subalgebra of that at y: ∀x, y ∈ R2 : (x − y ∈ W ⇒ A x → A y ). Wedge algebras of this general form are well known in algebraic quantum field theory: for instance, following [52], they play a major role in [37] (see Definition 2.1.1, where of course they are equipped with more structure than considered for our purposes here). Notice that the data of a wedge algebra naturally defines a 2-functor Z FW : P2 (R2 ) → Algebras with values in the strict 2-category of algebras, algebra homomorphisms and intertwiners by the assignment ?? **** ??? / * Z FW : x ??? *** ? y → FW (x) Id FW (y) p > ?? for all causal subsets between x and y as indicated. The reader may find it useful to think of this after the further inclusion Algebras → Bimod (see also Appendix A) under which the right hand becomes FW (y)
· · · → FW (x)
Id
FW (y)
FW (y) >
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U. Schreiber
with FW (y) regarded as an FW (x)-FW (y)-bimodule in the obvious way. In either case, /y)∈ the endomorphism algebra of the image under Z FW of any 1-morphism ( x 2 P2 (R ) is seen to be a relative commutant in that End(Z FW ( x
/ y )) = (FW (x)) ∩ FW (y) ,
where (FW (y)) ⊂ A W denotes the commutant of FW (y) in A W , the algebra of elements of A W that commute with all elements of FW (y). Hence the local net A Z FW obtained from this 2-functor assigns A Z FW
? ? ??? : x ???O y → (FW (x)) ∩ FW (y). ??
Local nets of this form obtained from wedge algebras are discussed for instance in [37] (there, again, equipped with more structure than considered here, see Eq. (2.18) in view of Eq. (2.1.3b)). 7.5. Lattice models. All our definitions and constructions make sense for S(R2 ) and P2 (R2 ) replaced by their restrictions S(Z2 ) and P2 (Z2 ) along that embedding Z2 → R2 which makes addition of (1, 0) a lightlike translation. This allows to see a class of important examples without the need to worry about weak 2-categories and issues in functional analysis. Let
C := BVect =
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
V
•
φ
@ • |( V
W
φ
/ W ) ∈ Vect
⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭
be the strict 2-category obtained from the strict monoidal category of finite-dimensional vector spaces: it has a single object, its 1-morphisms are finite dimensional vector spaces with composition of morphisms being the tensor product of vector spaces, and 2-morφ / W between vector spaces. phisms are linear maps V Pick a fixed finite dimensional vector space V and consider the two 2-FQFT 2-functors
Z : P2 (Z2 ) → BVect and Z × : P2 (Z2 ) → BVect which assign V to every elementary 1-morphism in P2 (Z2 ) and which assign to every elementary square the linear map
AQFT from n-Functorial QFT
⎛
? y ?? ~ ?? ~ ⎜ ?? ~~ ⎜ ~ ?? ~ ⎜ ⎜ ~~ x Z ⎜ @z ⎜ >>> ⎜ > ⎜ >> ⎝ > y and
⎛
?y? ~~ ??? ⎜ ~ ?? ~ ⎜ ?? ~~ ⎜ ⎜ ~~ x Z× ⎜ @z ⎜ >>> ⎜ > ⎜ >> ⎝ > y
389
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ := ⎟ ⎟ ⎟ ⎠
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ := ⎟ ⎟ ⎟ ⎠
?•@ ~~ @@@V ~ @@ ~~ @ ~~ • @ Id • Id @@ ~? ~ @@ ~ ~~ V @@ ~~ V •
V ⊗V
V
?•@ ~~ @@@V ~ @@ ~~ @ ~~ Id====Id ==== •@ • @@ | " ~~? @@ ~ ~~ V @@ ~~ V •
=
•
@•
Id
V ⊗V
V ⊗V
V
=
•
θV,V
@• ,
V ⊗V
θ
V,W / W ⊗ V denotes the canonical symmetric braiding respectively, where V ⊗ W isomorphism in Vect. The monoids assigned by the corresponding local nets A Z and A Z × are algebras of the form End(V ⊗n ), where n is the total number of elementary edges in the respective boundary of a region. Given the inclusion of regions Oa,b ⊂ Ox,x
c ? AAA x: a O b :: A <<< a,b @ === y d :: @y : z ? f > z ?? ~~ ~ w
?x
we get, according to Definition 9, inclusions A Z , A Z × : End(V ⊗2 ) → End(V ⊗6 ) of endomorphism algebras given by ⎞ ⎛ ⎛ 10 0 0 00 1 0 ⎜0 1 0 0 0 0⎟ ⎜0 A ⎟ ⎜ ⎜ A B A B ⎜0 0 A B 0 0⎟ ⎜0 0 ; A Z× : → ⎜ → ⎜ A Z : ⎟ C D C D ⎜0 0 C D 0 0⎟ ⎜0 0 ⎝0 0 0 0 1 0⎠ ⎝0 C 00 0 0 01 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 B 0 0 D 0
⎞ 0 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎠ 0 1
where each entry in these matrices is an endomorphism of V . The locality of the net A Z is manifest. The algebras assigned to two elementary regions clearly commute if and only if the two regions are spacelike separated. For A Z ×
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U. Schreiber
the algebras of course also commute if the regions are spacelike separated, but here they also commute if the two regions are timelike separated. Only if two elementary regions are lightlike separated do the inclusions of algebras due to A Z × not commute. There are various variations of this example. In particular for Z × one would want to consider the case where two different vector spaces Vl and Vr and two nontrivial automorphisms Ul : Vl → Vl and Ur : V R → Vr are assigned to elementary causal subsets as follows: ⎛
y ~? ??? ~ ⎜ ?? ~~ ⎜ ?? ~~ ⎜ ? ~ ~ ⎜ x> Z× ⎜ @z ⎜ >> ⎜ >> ⎜ >> ⎝ y
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ := ⎟ ⎟ ⎟ ⎠
• ~? @@@ V ~ @@ r ~ @@ ~~ ~ ~ Ul===Ur = •@ • ==== ? @@ | " ~~ @@ ~ ~~ Vr @@ ~~ Vl •
Vl ⊗Vr
Vl
=
•
θVl ,Vr ◦Ul ⊗Ur
@• .
Vr ⊗Vl
Denote by c : End(Vr ) ⊗ End(Vl ) → End(Vr ⊗ Vl ) the canonical inclusion of algebras and by c∗ A Z ×
/ AZ×
the local sub-net of A Z × obtained by restricting along c everywhere. Then c∗ A Z × is what is called a chiral AQFT. Its structure is encoded entirely in the two independent projections onto two orthogonal lightlike curves.
c∗ A Z ×
?y? ⎛ ~~ ??? ~ ?? ~ ~ ?? ~ ⎜ ~~ ⎜ x : >> @ z → Al ⎜ ⎝ >> >> y > y
⎞ ⎛ z x @ ⎟ ⎜ ??? ⎟ ⎜ ?? ⎟ ⊗ Ar ⎜ ?? ⎠ ⎝
⎞
y
⎟ ⎟ ⎟ ⎠
= End(Vl ) ⊗ End(Vr ). Restricting attention to just one of these and then “compactifying” that to a circle leads to the models [30,32] of 2-dimensional (conformal) field theories as local nets on the circle. This important example is further expanded on in Sect. 7.6.
7.6. Boundary FQFT and boundary AQFT. AQFT on spaces with boundary has been introduced in [40] for the case of the Minkowski half-plane X = R2< . Here we briefly indicate how boundary conditions are formulated for FQFT and how we recover the picture in [40] from this point of view.
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391
We obtain the poset of causal subsets on the half plane, S(R2< ), by starting with S(R2 ) and intersecting everything with R2< . We form P2 (R2< ) by first restricting to 2paths that run entirely within R2< and then throwing in new boundary generators for 1and 2-morphisms of the form (0, 9 t + x) ss s ss ss ss (x, t) K KK KK KK KK % (0, t − x) From examples of classical parallel n-transport [47] and from the 2-functorial description of rational CFT [19] it is known that boundary conditions for n-functors Z correspond to choices of morphism from some trivial n-functor I into the restriction of the given one to the boundary: / Z |∂ X .
I
We illustrate this in the context of the last example, Z × : P2 (R2 ) → BVect, from Sect. 7, which lead to the discussion of chiral nets i ∗ A Z × ⊂ A Z × . For that purpose, let I be the 2-functor I : P2 (R2 ) → BVect which is constant on < : / BVect which the single object of BVect and consider 2-functors Z × P2 (R2< ) coincide with our Z × in the bulk. Then we have the simple but important. Proposition 3. If a morphism < b : I → Z× |∂ R2<
exists and is time independent in that its component map is constant on objects (but not < assigns the identity to all boundary paths. the 0 dimensional vector space), then Z × Proof. The components of the morphism, which is a pseudonatural transformation of 2-functors, are 2-cells in BVect of the form /• ~ ~ ~~~~ ~~~~ ~ ~ . ~~ b(t) b(t ) ~~~~ ~ ~ ~~~~ ~~~~ z /• • < •
Id
Z × ((0,t)→(0,t ))
By assumption of time independence of the boundary condition we have b(t) = b(t ) = < ((0, t) → (0, t )) must be a vector space such that there exists b(0). This means that Z × an isomorphism of vector spaces < b(0) ⊗ Z × ((0, t) → (0, t )) b(0).
392
U. Schreiber
< of a spacelike wedge on the left Minkowski half Fig. 7. The image under the boundary FQFT 2-functor Z × plane
< will specify identifications of the vector spaces V So in this case the 2-functor Z × l and Vr at the boundary
< Z× :
(0, 9 t + x) ss s ss ss ss (x, t) K KK KK KK KK % (0, t − x)
?• ~~ ~ ~~ ~~ •@ Id . @@ ~ @@ Vr @@ • Vl
→
By taking endomorphisms this defines a net of algebras on the boundary, which entirely encodes the chiral part c∗ A Z ×< of A Z ×< . This way we arrive at the picture of boundary AQFT given in [40]. Further details should be discussed elsewhere. 7.7. 2-C ∗ -category codomains. In most applications to physics one wants the algebras in a local net to be C ∗ -algebras. A natural type of 2-category in which endomorphism algebras of 1-morphisms are C ∗ -algebras is that of 2-C ∗ -categories: categories enriched in C ∗ -categories. Definition 13. A C ∗ -category (or C ∗ -algebroid: the many-object version of a C ∗ -algebra) is a category C enriched in complex Banach spaces (meaning that for all objects ρ, σ, τ of C we have that C(ρ, σ ) is a complex Banach space and that composition ◦ρ,σ,τ : C(ρ, σ ) × C(σ, τ ) → C(ρ, τ ) is a morphism of complex Banach spaces) which is equipped with an involutive antilinear functor (·)∗ : C → C op that satisfies the C ∗ -condition ∀ρ, σ ∈ Obj(C) : ∀S ∈ C(a, b) :
S ∗ ◦ S is positive in C(ρ, ρ) , S ∗ ◦ S = S2
where · : C(ρ, σ ) → C is the Banach norm.
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393
A C ∗ -algebra A is precisely the endomorphism algebra of an object ρ in a C ∗ -category, A = C(ρ, ρ). We write BA for the one object C ∗ -category whose single endomorphism algebra is A. C ∗ -categories form a strict monoidal 2-category (C ∗ Cat, ×) whose morphisms are Banach space functors (continuous on each Hom-space). Therefore one can enrich in C ∗ -categories themselves: Definition 14. A (strict) 2-C ∗ -category is a category enriched in C ∗ Cat. A discussion of aspects of 2-C ∗ -categories can be found in [60]. The canonical example of a strict 2-C ∗ -category is AmpliC ∗ ⊂ BimodC ∗ , the 2category whose objects are unital C ∗ -algebras, whose morphisms are amplimorphisms between these and whose 2-morphisms are intertwiners between those. BimodC ∗ is very similar, but is not strict. See [39] and Sect. 2 of [60]. So we have Observation 2. For Z : P2 (X ) → C a transport 2-functor with values in a 2-C ∗ -category C, the corresponding local net A Z is a net of C ∗ -algebras. 7.8. Hopf spin chain models. Recall the description of lattice models with boundary from Sect. 7.6. Consider the extreme case where there is a left and right boundary which are separated only by a single lattice spacing: a? ??
ρ?
?? ?b ρ a? Id , ?? ρ? ?? Id ?b ρ a Id
where for simplicity we are concentrating on the case that Z sends each edge to one and the same morphism ρ : a → b in C. Physically, we can think of this as a lattice model for an open string stretching from a brane of type a to a brane of type b. It is a crude lattice model, consisting of a single “string bit”. Consider another such strip, labeled by another morphism ρ¯ : b → a
394
U. Schreiber
?a ρ¯ Id b? ?? ρ¯ ? ?? Id ?a ρ¯ Id b AA A
ρ¯ AA AA
.
As the notation suggests, we want to think of ρ¯ to be conjugate to ρ, meaning that ρ and ρ¯ form an ambidextrous adjunction [36] between a and b such that the unit of the left-handed adjunction is the ∗ -adjoint of the counit of the right-handed adjunction, and vice versa (see p. 8 of [60]). Then it makes sense to think of this as a lattice model for an open string, or rather a “string bit”, as before, but now with that string taken to stretch from the b-type brane to the a-type brane. We can then consider lattice models built from the above building blocks by gluing the above strip-wise 2-functors horizontally: a? ??
ρ?
?? Id ?b ρ a? ?? ρ? ?? Id ?b ρ a
?a ρ¯ Id ?? ? ρ¯ ? ?? ?a ρ¯ Id ?? ? ρ¯ ? ?? a
a? ??
ρ?
,
?? Id ?b ρ a? ?? ρ? ?? Id ?b ρ a
?a? ?? ρ? ?? ρ¯ ?? ?b ? ρ¯ ? ?? ρ Id ? a ?? ? ρ? ?? ρ¯ ?b ?? ? ρ¯ ? ?? ρ a
, ···
The algebras assigned by the corresponding net A Z to the elementary causal bigon Oρ,ρ¯ and Oρ,ρ ¯ are A Z (Oρ,ρ¯ ) = EndC (ρ¯ ◦ ρ) and A Z (Oρ,ρ ¯ ¯ ) = EndC (ρ ◦ ρ).
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If C is a 2-C ∗ -category and ρ is an “irreducible 1-morphism generating a 2-C ∗ -category of depth two” as in Sect. 4 of 7, then these are C ∗ -Hopf algebras H and Hˆ which are duals of each other [41,60]. Due to the fact that the 2-morphisms in the above diagrams do not mix ρ and ρ, ¯ we can understand the nature of the net A Z obtained from the above 2-functor Z already by concentrating on the endomorphism algebras assigned to a horizontal zig-zag a? ??
?a? ?a? ?? ?? ρ? ρ? ρ? ?? ρ¯ ?? ρ¯ ?? . b b b
|
|
|
|
|
|
If we to restrict to evaluating the net A Z on zig-zags of even length, this gives rise to a net on the latticized real axis with the property that algebras A Z (I1 ) and A Z (I2 ) commute if the intervals I1 and I2 are not just disjoint but differ by at least one lattice spacing. Precisely these kind of 1-dimensional nets are considered in [42], where they are addressed as Hopf spin chain models. 8. Further Issues There are various immediate further questions to be addressed. We shall be content here with just briefly commenting on the following. Continuum limits of lattice models and von Neumann algebra valued nets. We have shown that 2-functorial FQFTs very generally give rise to local nets of monoids and observed that 2-functors with values in 2-C ∗ -categories give rise to local nets of C ∗ algebras. One would want to identify concretely those 2-functors which induce the celebrated local nets of von Neumann algebra factors. It is to be expected that many local nets of von Neumann algebras can be obtained from taking continuum limits of lattice models. By considering in this continuum limit the relation between lattice AQFT and lattice FQFT discussed in Sect. 7, one should be able to construct examples of the desired 2-functors. But while the idea of obtaining AQFT nets from continuum limits of lattice models seems to be straightforward and of considerable relevance, there exist to date apparently no published studies of this problem. A discussion of the problem of 2-functorial FQFT corresponding to local nets of von Neumann algebra factors will therefore have to be given elsewhere. General Lorentzian structure. AQFT was originally conceived entirely in its application to quantum field theories on Minkowski space, which is the case we have been concentrating on above. A generalization of Poincaré-covariant nets on causal subsets in Minkowski space to nets on globally hyperbolic Lorentzian spaces has later been proposed in [12]. The possibly most natural and immediate generalization to AQFT on a fixed general Lorentzian space was indicated in [44]: on a Lorentzian manifold X an AQFT net should
396
U. Schreiber
be locally local: the locality axiom should hold after restriction of the net to any globally hyperbolic subspace of X . The same should be true for the time slice axiom. No guesswork is required for generalizing the concept of Minkowskian FQFT 2functors to general Lorentzian 2-functors: the concept of the 2-functor itself makes unambiguous sense for any choice of 2-path 2-category in X . So we can use our construction of local nets from 2-functors to derive locality properties of nets on Lorentzian spaces. Doing so confirms the idea of [44]: Let (X, g) be any 2-dimensional oriented and time-oriented Lorentzian manifold. In generalization of Definition 1 consider Definition 15. A causal subset O ⊂ X is a subset of a globally hyperbolic subset of X which is the interior of a non-empty intersection of the future of one point with the past of another. Write S(X ) for the category with such causal subsets as objects and inclusion of subsets as morphisms. In generalization of Definition 6 consider Definition 16. Let P2 (X ) be the strict 2-category whose objects are the points in X , whose 1-morphisms are piecewise lightlike and right-moving paths (with respect to the chosen orientation and time-orientation of X ) and whose 2-morphisms are generated under gluing along common boundaries from closures of causal subsets. Our construction in Definition 9 immediately generalizes to a construction of a net A Z : S(X ) → Monoids from a 2-functor Z : P2 (X ) → C. All the arguments need to be done within globally hyperbolic subsets of X , where they go through literally as before. We can read off from the result of this construction the locality properties of A Z : Proposition 4. The net A Z : S(X ) → Monoids obtained from any 2-functor Z : P2 (X ) → C is locally local and satisfies the local time slice axiom: for any inclusion i : Y → X with Y globally hyperbolic we have that i ∗ A Z is a local net satisfying the time slice axiom. This concept of local locality is compatible with [12] but does not presuppose any covariance condition on the net. Higher dimensional QFT. We had considered, for ease of discussion, in Definition 4 the 2-category P2 (X ) whose 2-morphisms are generated from gluing the closures of 2-dimensional causal subsets along common boundaries. But nothing in our constructions crucially depends on gluing of causal subsets, and in fact gluing of causal subsets becomes less natural in higher dimensions. As the examples we presented in Sect. 7, where we obtained FQFT 2-functors by restricting 2-functors on a larger 2-category of 2-paths to P2 (X ), clearly indicate, the 2-category P2 (X ) can be replaced by any 2-category of 2-paths in X which is large enough that every causal subset in X can be regarded as a 2-morphism in there, so that every FQFT 2-functor can be evaluated on causal subsets. And this statement then immediately generalizes to higher dimensions. For X a d-dimensional Lorentzian manifold, we should take the category S(X ) to be that whose objects are causal subsets in X , which are those subsets that arise within any globally hyperbolic subset of X as the interior of the future of one point with the past of another point. Morphisms are inclusions.
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Fig. 8. A 3-morphism in a 3-path 3-category: a volume V , cobounding two surfaces 1 and 2 , which each cobound two paths γ1 and γ2 which each cobound two points x and y
The d-category Pd (X ) used to describe Lorentzian FQFT on X can be any sub-dgroupoid of the path d-groupoid [47] which is large enough so that every causal subset in X comes from a d-morphism in Pd (X ) and such that the obvious higher dimensional generalizations of the diagrams in Sect. 5 exist in Pd (X ). In particular, one can always use the full path d-groupoid. With such a setup, all our constructions here should have essentially straightforward generalizations to higher dimensions, leading to a construction of local nets on X from any FQFT d-functor on X . In such a context the spatial separation of two causal subsets would manifest itself not in the position of endomorphisms in a 1-dimensional string of products, but in their position in a higher dimensional algebra. A. 2-Vector Spaces and the Canonical 2-Representation In Sect. 7 we obtained examples of FQFT 2-functors from differential form data and a choice of 2-representation. Here we briefly indicate a bit of background concerning these 2-representations. For our purposes here a 2-vector space is an abelian module category, i.e an abelian category equipped with an action by a monoidal category. Notice that the category of k-vector spaces is the category of k-modules Vect k = k − Mod. Accordingly we write 2Vect = Vect Vect = Vect − Mod for the 2-category of abelian categories equipped with a (left, say) (Vect, ⊗)-action. Since Vect is symmetric monoidal, one can keep going this way and in principle define recursively the n-category nVect = (n − 1)Vect − Mod. Notice in particular that then 0Vect = k. There are other monoidal categories over which one may want to consider 2-vector spaces. For instance if we denote by Disc(k) the discrete category over the ground field (the ground field as its objects and only identity morphisms), then Disc(k) − Mod Cat(Vect)
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is the 2-category of categories internal to vector spaces, which in turn is equivalent to chain complexes concentrated in degree 0 and 1. These are the 2-vector spaces considered in [5]. Disc(k)-modules are the “right” notion for 2-vector space for higher Lie theory, but probably not [3] as models for fibers of interesting 2-vector bundles. The entirety of the 2-category of all Vect-modules is quite untractable. What is more accessible and more useful is the 2-category of 2-vector space that “have a basis”. Noticing that an ordinary vector space V has a basis if there is a set S such that V HomSet (S, k), we should define a basis for a 2-vector space V to be a category S such that V Hom(S, Vect). If S is itself Vect-enriched this says that V is a category of algebroid modules. We shall restrict attention to S having a single object, in which case we are left with modules for ordinary algebras. This way we find the bicategory Bimod of algebras, bimodules and bimodule homomorphisms sitting inside 2Vect as a sub-2-category of 2-vector spaces with basis: / 2Vect Bimod −⊗ A N
N
φ
A
BB
→ Mod A
N
−⊗ A φ
Mod > B.
−⊗ A N
Notice how Mod A is a category of modules which is itself a module category over Vect. The 2-category of Kapranov-Voevodsky 2-vector spaces [29] is the full sub 2-category of Bimod on all algebras of the form k ⊕n for n ∈ N, KV2Vect → Bimod. While Bimod is not a strict 2-category, it is a framed bicategory in the sense of [54]: there is the strict 2-category Algebras of algebras, algebra homomorphisms and intertwiners (the obvious 2-category for algebras regarded as one-object Vect-enriched categories), and the obvious inclusion / Bimod Algebras is full and faithful on all Hom-categories. Noticing that similarly groups, when regarded as one-object groupoids, live in the 2-category Groups of groups, group homomorphisms and inner automorphisms, we get a strict 2-functor Groups
/ Algebras
induced from forming for each group its group algebra. For each group H there is the 2-group AUT(H ) := Aut Groups (H ) and the canonical inclusion / Groups BAUT(H ) induces, combined with the above discussion, the canonical 2-representation of AUT(H ) given by ρcan : BAUT(H )
/ Groups
/ Algebras
/ Bimod
/ 2Vect .
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The logic of this construction generalizes to arbitrary strict 2-groups G (2) coming from t
α
crossed modules of groups (H → G → Aut(G)) (see for instance [50] for a review) and algebras obtained from a representation of H : Proposition 5. For ρ : BH → Vect a representation of H such that the action of G on H extends to algebra automorphisms of the representation algebra ρ(H ), the assignment ρ˜ : B(H → G) → Algebras given by α(g)
g
•
C • → ρ(H )
h
g
ρ(h)
! ρ(H ) =
α(g )
is a strict 2-functor. Accordingly we obtain a 2-representation B(H → G)
ρ˜
/ Algebras
/ Bimod
/ 2Vect .
All this should go through when the vector spaces here are equipped with more structure. ˆ → Hilb In particular, for G a compact, simple and simply connected group, for ρ : BG a positive-energy representation of the weight 1 central extension of its loop group and for vNBimod the bicategory of vonNeumann algebras and their bimodules composed under Connes-fusion, [55] the above should extend to a 2-representation BString(G) → vNBimod of the strict String 2-group [6]. Acknowledgement. I am grateful to David Corfield, Christoph Schweigert, Zoran Škoda, Jim Stasheff, Jamie Vicary and Konrad Waldorf for comments on earlier versions of this text, to Bruce Bartlett for discussion of aspects of some of the examples, to Maarten Bergvelt for discussion of relations between chiral nets and vertex operator algebras, to Jacques Distler for general discussion about AQFT and QFT, to Liang Kong for describing to me his work with Yi-Zhi Huang, to Peter Teichner for discussion of aspects at the beginning of Sect. 2 and to Klaus Fredenhagen and an anonymous referee for pointing out the relevance of the example in Sect. 7.4. I had very useful discussion with Roberto Conti at an Oberwolfach CFT workshop in 2007, when I started thinking about the ideas presented here. Finally I heartily thank Pasquale Zito for a pleasant visit, for very useful discussion about his thesis and about Hopf spin chain models and for teaching me about asymptotic inclusion and pointing me to the relevant references. This work was being completed while the author enjoyed a research fellowship at the Hausdorff Center for Mathematics in Bonn.
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References 1. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press, 2004, pp. 415–425[arXiv:quant-ph/0402130] 2. Atiyah, M.: Topological quantum field theory. Publ. Math. de l’IHÉS 68, 175–186 (1988) 3. Baas, N., Bökstedt, M., Kro, T.: Two-categorical bundles and their classifying spaces. http://arxiv.org/ abs/math/0612549v2[math.AT], 2008 4. Baez, J.: Quantum quandaries: a category-theoretic perspective. In: Structural Foundations of Quantum Gravity, eds. S. French, D. Rickles, J. Saatsi, Oxford: Oxford U. Press, 2006, pp. 240–265 5. Baez, J., Crans, A.: Lie 2-algebras. Theor. Appl. of Categories 12, 492–528 (2004) 6. Baez, J., Crans, A., Schreiber, U., Stevenson, D.: From loop groups to 2-groups. Homology, Homotopy Appl. 9(2), 101–135 (2007) 7. Baez, J., Dolan, J.: Higher dimensional algebra and topological quantum field theory. J. Math. Phys. 36, 6073–6105 (1995) 8. Baez, J., Schreiber, U.: Higher gauge theory. In: Contemporary Mathematics 431, Categories in Algebra, Geometry and Mathematical Physics, Providence, RI: Amer. Math. Soc., 2007, pp. 7–30 9. Bartels, T.: 2-Bundles. PhD Thesis, UC Riverside, 2006, available at http://math.ver.edu/home/baez/ thesis_toby.pdf 10. Bartlett, B.: Categorical aspects of topological quantum Field theories. M.Sc Thesis, Utrecht University, 2005, http://arxiv.org/abs/math/0512103v1[math.QA], 2005 11. Bartlett, B.: On unitary 2-representations of finite groups and TQFT. PdD Thesis, University of Sheffield, 2008 12. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle – A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31–68 (2003) 13. Buchholz, D., Haag, R.: The quest for understanding in relativistic quantum physics. J. Math. Phys. 41, 3674–3697 (2000) 14. Cheng, E., Gurski, N.: Towards an n-category of cobordisms. Theory and Applications of Categories 18(10), 274–302 (2007) 15. Coecke, B.: Kindergarten quantum mechanics. Talk at Quantum Information, Computation and Logic (Perimeter Institute, 2005), available at http://www.quxat.org/quoxic.talks/bobkinder.pdf 16. Elgueta, J.: Representation theory of 2-groups on Kapranov and Voevodsky’s 2-vector spaces. Adv. in Math. 213(1), 53–92 (2007) 17. Evans, D.E., Kawahigashi, Y.: Quantum symmetries on operator algebras. Oxford: Oxford University Press, 1998 18. Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: Uniqueness of open/closed rational CFT with given algebra of open states. Adv. Theor. Math. Phys. 12(6), 1283–1375 (2008) 19. Fjelstad, J., Schreiber, U.: Rational CFT is parallel transport, in preparation [http://www.math.unihamburg.de/home/schreiber/cc.pdf], 2008 20. Fredenhagen, K., Rehren, K., Seiler, E.: Quantum field theory: Where we are. Lecture Notes in Physics 721, 61–87 (2007) 21. Freed, D.S.: Higher algebraic structures and quantization. Commun. Math. Phys. 159, 343–398 (1994) 22. Freed, D.S.: Quantum groups from path integrals. Proceedings of Particles and Fields (Banff, 1994), 63Ð107, CRM Ser. Math. Phys., New York: Springer, 1999, available at http://arxiv.org/abs/q-alg/ 9501025v1, 1995 23. Freed, D.S.: Remarks on Chern-Simons Theory. Bull. Amer. Math. Soc. (NS) 46(2), 221–254 (2009) [arXiv:0808.2507] 24. Ganter, N., Kapranov, M.: Representation and character theory in 2-categories. Adv. in Math. 213(1), 53–92 (2007) 25. Haag, R.: Local Quantum Physics: Fields, Particles, Algebras. Berlin: Springer, 1992 26. Halvorson, H., Mueger, M.: Algebraic quantum field theory. In: J. Butterfield and J. Earman (eds.) Handbook of the Philosophy of Physics, Amsterdam: North-Holland, 2007, pp. 731–922 27. Lurie, J.: On the classication of topological field theories, [http://www-math.mit.edu/~lurie/papers/ cobordism.pdf], 2009 28. Huang, Y.-Z.: Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA 88, 9964– 9968 (1991) 29. Kapranov, M., Voevodsky, V.: 2-categories and Zamolodchikov tetrahedra equations. In: Algebraic Groups and Theor Generalization Quantum and Infinite-Dimension Methods, Proc. Symp. Pure Math. 56, Part 2, Providence, RI: Amer. Math. Soc., 1994, pp. 177–259 30. Kawahigashi, Y.: Classification of operator algebraic conformal field theories in dimensions one and two. In: Proceedings of XIV International Congress on Mathematical Physics at Lisbon (2003) [arXiv:mathph/0308029]
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31. Kawahigashi, Y.: Conformal Field Theory and Operator Algebras. In: Proceedings of ICMP, Rio de Janeiro (2006), available at http://arxiv.org/abs/0704.0097v1[math-ph], 2007 32. Kawahigashi, Y., Longo, R.: Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories. 244, Number 1, 63–97 (2004) 33. Kelly, G.M.: Basic concepts of enriched category theory. Cambridge University Press, Lecture Notes in Mathematics 64, 1982; Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1–136, available at http://www.tac.mta.ca/tac/reprints/articles/10/+10.pdf 34. Kawahigashi, Y., Longo, R.: Local conformal nets arising from framed vertex operator algebras. Adv. in Math. 206(2), 729–751 (2006) 35. Kong, L.: Open-closed field algebras. Commun. Math. Phys. 280, 207–261 (2008) 36. Lauda, A.: Frobenius algebras and ambidextrous adjunctions. Theory and Applications of Categories 16(4), 84–122 (2006) 37. Lechner, G.: On the construction of quantum field theories with factorizing S-Matrices. PhD thesis, Göttingen, 2006 38. Leinster, T.: Basic bicategories. http://arxiv.org/abs/math/9810017v2[math.CT], 1998 39. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995) 40. Longo, R., Rehren, K.-H.: Local fields in boundary conformal QFT. Rev. Math. Phys. 16, 909 (2004) 41. Müger, M.: From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories. J, Pure Appl. Alg. 180, 81–157 (2003) 42. Nill, F., Szlachány, K.: Quantum chains of Hopf algebras with order-disorder fields and quantum double symmetry. Commun. Math. Phys. 187(1), 159–200 (1997) 43. Ocneanu, A.: Quantized group, string algebras and Galois theory for algebras. In: Operator Algebras and Applications, Vol. 2, eds. Evans, D.E., Takesaki, M. Cambridge: Cambridge University Press, 1988, 119 44. Rehren, K.-H.: weblog comment, [http://golem.ph.utexas.edu/~distler/blog/archives/000987.html# c005396] 45. Roberts, J., Longo, R.: A theory of dimension. K-Theory 11(2), 103–159 (1997) 46. Runkel, I., Fjelstad, J., Fuchs, J., Schweigert, C.: Topological and conformal field theory as Frobenius algebras. Contemp. Math. 431, 225–248 (2007) 47. Sati, H., Schreiber, U., Škoda, Z., Stevenson, D.: Twisted nonabelian differential cohomology. In preparation, [ http://www.math.uni-hamburg.de/home/schreiber/nactwist.pdf], 2008 48. Schreiber, U.: The canonical 2-representation. [http://www.math.uni-hamburg.de/home/schreiber/ canrep.pdf], 2008 49. Schreiber, U., Waldorf, K.: Parallel transport and functors. To appear in Journal of Homotopy and Related Structures available at http://arxiv.org/abs/0705.0452v2[math.D6], 2008 50. Schreiber, U., Waldorf, K.: Smooth functors vs. differential forms. http://arxiv.org/abs0802.0663v2[math. DG], 2008 51. Schreiber, U., Waldorf, K.: Connections on non-abelian Gerbes and their Holonomy. http://arxiv.org/ abs/0808.1923v1[math.DG], 2008 52. Schroer, B.: Modular Wedge Localization and the d = 1+1 Formfactor Program. Annals Phys. 275, 190– 223 (1999) 53. Segal, G.: The definition of conformal field theory. In: Topology, geometry and quantum field theory, London Math. Soc. LNS 308, edited by U. Tillmann, Cambridge: Cambridge Univ. Press 2004, pp. 247–343 54. Shulman, M.: Framed bicategories and monoidal fibrations. Theory and Applications of Categories 20(18), 650–738 (2008) 55. Stolz, S., Teichner, P.: What is an elliptic object? In: Topology, geometry and quantum field theory, London Math. Soc. LNS 308, Cambridge: Cambridge Univ. Press, 2004, pp. 247–343 56. Street, R.: Categorical and combinatorial aspects of descent theory. Applied Categorical Structures 12(5-6), 537–576 (2004) 57. Verity, D.: Cobordisms and weak complicial sets. Talk given at Macquarie Univ., Australian Category Theory Seminar, February 13, 2008 58. Walker, K.: TQFTs, [http://canyon23.net/math/tc.pdf] 59. Wockel, C.: A global perspective to gerbes and their gauge stacks. http://arxiv.org/abs/0803.3692v3[math. DG], 2008 60. Zito, P.: 2-C ∗ -categories with non-simple units. PhD thesis, available at http://arxiv.org/abs/math/ 0509266v1, 2005 Communicated by Y. Kawahigashi
Commun. Math. Phys. 291, 403–441 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0769-5
Communications in
Mathematical Physics
On Action-Minimizing Retrograde and Prograde Orbits of the Three-Body Problem Kuo-Chang Chen, Yu-Chu Lin Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan. E-mail: [email protected]; [email protected] Received: 13 October 2008 / Accepted: 4 December 2008 Published online: 17 March 2009 – © Springer-Verlag 2009
Abstract: A retrograde orbit of the planar three-body problem is a relative periodic solution with two adjacent masses revolving around each other in one direction while their mass center revolves around the third mass in the other direction. The orbit is said to be prograde or direct if both revolutions follow the same direction. Let T > 0 and φ ∈ [0, 2π ) be fixed, and consider the rotating frame which rotates the inertia frame about the origin with angular velocity Tφ . In a recent work of K.-C.Chen [5], the existence of action-minimizing retrograde orbits which are T -periodic on this rotation frame were proved to exist for a large class of masses and a continuum of φ. In this paper we generalize the main result in [5], provide some quantitative estimates for admissible masses and mutual distances, and show miscellaneous examples of action-minimizing retrograde orbits. We also show the existence of some prograde and retrograde solutions with additional symmetries. 1. Introduction and Notations The planar three-body problem concerns the motion of three masses m 1 , m 2 , m 3 moving in C in accordance with Newton’s law of universal gravitation: m k x¨k =
∂ U (x), k = 1, 2, 3, ∂ xk
(1)
where xk ∈ C is the position of m k , x = (x1 , x2 , x3 ), and m2m3 m1m3 m1m2 + + U (x) = |x1 − x2 | |x2 − x3 | |x1 − x3 | is the potential energy (negative Newtonian potential). Unless specified otherwise, throughout this paper a “solution” of (1) is referred to a “classical solution” of (1). A retrograde orbit of the planar three-body problem is a relative periodic solution of (1) with two adjacent masses revolving around each other in one direction while their
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Fig. 1. Retrograde braids γ and γ
mass center revolves around the third mass in the other direction. The orbit is said to be prograde or direct if both revolutions follow the same direction. By taking an appropriate rotating (synodic) coordinate system and adding a time axis, the trajectory of a relative periodic orbit without collision traces out a pure braid within one period in the three-dimensional space-time. By a pure braid we mean a braid which begins and ends with the same space coordinates. Retrograde and prograde paths can be defined as the collection of collision-free paths which trace out certain braid types in a suitable rotating coordinate system, as illustrated in Fig. 1. Being retrograde or prograde is a topological property. The idea of using braids to describe topological types of planar periodic solutions is a natural one, see for instance [17,18]. Whether or not a topological class of paths contains a true solution of (1) is in general a difficult task to rigorously verify. The classical lunar theory and analysis of relative equilibria produce numerous solutions that belong to three basic braid types (in an appropriate coordinate frame); namely the prograde, retrograde, and trivial braid types. Among them the prograde and retrograde braids are known to carry solutions for certain masses and for a continuous family of rotating frames. Periodic solutions in other braid classes have not received much rigorous treatment. The figure-8 orbit with equal masses constructed by variational methods provides another example of allowable braid class on both inertia and rotating frames [8,9]. This is a pioneering work which utilizes minimizing methods on the three-body problem and initiates a sequel of research work in the past decade. A vast amount of numerical solutions in various braid classes was recently found for similar choreographic three-body problems [21], some of which may find analytical counterparts in [1,10] but precise determination of topological types for such orbits from equivariant problems and local deformation arguments are yet to be found. Readers are referred to [7,10,14,15,22] for further bibliographies and historical remarks. The purpose of this paper is to extend the work in [5] which endeavors to prove the existence and minimizing properties of retrograde orbits with various choices of masses. We generalize the main result in [5], provide detailed descriptions and miscellaneous samples of action-minimizing retrograde solutions as well as some prograde and retrograde solutions with additional symmetries. Here we also provide some quantitative estimates for the mutual distances of action-minimizing solutions. In what follows we provide a more precise and formal definition for retrograde and prograde paths in terms of braids. The definition is slightly more general than the definition in [5]. Fix φ ∈ [0, 2π ) and the relative period T > 0, then consider the rotating frame which rotates the inertia frame about the origin with angular velocity φ/T . The class of T -periodic loops on this rotating frame is Cφ,T := {x ∈ C(R, C3 ) : x(t + T ) = eφi x(t) ∀t ∈ R}.
On Action-Minimizing Retrograde and Prograde Orbits
405
Let := {x ∈ C3 : xi = x j for some i = j} be the variety of collision configurations. The subset ∗ Cφ,T := {x ∈ Cφ,T : x(t) ∈ C3 \ for any t ∈ R} ∗ corresponds to a of Cφ,T consists of collision-free paths. Each x = (x1 , x2 , x3 ) in Cφ,T pure braid α = (α1 , α2 , α3 ) in C × [0, T ] with three threads
αk (t) = (e
−φt T i
xk (t), t) ∈ C × R, t ∈ [0, T ], k = 1, 2, 3.
∗ This correspondence from Cφ,T to the class of pure braids in C × [0, T ] is clearly bijective. By relabeling indices if necessary, we may confine our path space to only those which begin and end with collinear configurations and with a prescribed ordering: † ∗ Cφ,T := {x ∈ Cφ,T : x(0) ∈ R3 , x3 (0) < x2 (0) < x1 (0)}. † Let B = Bφ,T be the set of pure braids corresponding to Cφ,T . Two pure braids in B are considered equivalent if one can be continuously deformed to the other among the set B. Let R and R be the equivalence classes of braids γ and γ in Fig. 1. Retrograde † are defined as those which have their braids belonging to either R or R . paths in Cφ,T In any two equivalence classes B1 , B2 of braids we may pick representatives α1 in B1 , α2 in B2 such that α1 (0) = α2 (0). The standard definition of braids multiplication t ∈ [0, T /2] α1 (2t), (α1 · α2 )(t) = α2 (2t − T ), t ∈ (T /2, T ]
induces a well-defined multiplication and group structure for equivalence classes of B. The multiplicative identity is called the trivial braid class belonging to which there is a Euler’s relative equilibrium. Paths with braids belonging to inverses P, P of braid classes R, R are called prograde paths (or direct paths) in Cφ,T . A representative γ −1 of P is depicted in Fig. 1. The concept of retrograde or prograde motion can be defined for some spatial orbits but we shall focus on planar motions in this paper. In [5] the author proved the existence and minimizing property of retrograde orbits for various choices of masses and for a continuum of φ ∈ (0, π ]. In our present work we extend the main result in [5] to a much larger class of masses and wider range of φ, see Sects. 2 and 4 for details. Sections 5 and 8 contain miscellaneous examples of actionminimizing orbits. In Sect. 9 we show some quantitative estimates of mutual distances for these orbits. Sections 3, 6, 7 are devoted to proofs of our main theorems. 2. Main Theorems In this section we state our main theorems on the existence of retrograde orbits for (1) in † Cφ,T , and the existence of retrograde and prograde orbits with additional symmetries. The variational problem described below resembles the one in [5] except that φ belongs to (0, 2π ) instead of (0, π ]. For convenience, and without loss of generality, throughout the rest of this paper we set T = 1 and m 3 = 1. From now on we drop T from the ∗ , C† , B notations for function spaces and denote Cφ,T , Cφ,T φ,T respectively by C φ , φ,T
Cφ∗ , Cφ† , Bφ .
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Let Rφ be the set of retrograde paths in Cφ† which corresponds to the braid class R, a representative of which is the braid γ in Fig. 1. Paths in Rφ are initially aligned in the order x3 (0) < x2 (0) < x1 (0), curves x1 and x2 revolve clockwise around each other while their center and x3 revolve counterclockwise around the origin. We may also define Rφ as the path component of eφti (1 + e−2π ti ), eφti (1 − e−2π ti ), −eφti in the space Cφ† . Each path in Cφ∗ can be translated to a path with mass center at the origin without altering its braid type; namely, suppose x ∈ Cφ∗ has mass center xˆ ∈ C([0, 1], C), then the new path (x1 − x, ˆ x2 − x, ˆ x3 − x) ˆ ∈ Cφ∗ and x have the same braid type. Since the center of mass is an integral of motion, it is natural to consider only paths staying inside the configuration space V := {x ∈ C3 : m 1 x1 + m 2 x2 + m 3 x3 = 0}. 1 (R, V ) Equations (1) are the Euler-Lagrange equations for the action functional A: Hloc → R ∪ {+∞} defined by 1 K (x) ˙ + U (x) dt, A(x) := 0
where K (x) ˙ =
1 m 1 |x˙1 |2 + m 2 |x˙2 |2 + m 3 |x˙3 |2 2
1 (R, V ) the action functional A can be written is the kinetic energy of the path x. On Hloc 1 1 M 1 |x˙i − x˙ j |2 + dt, (2) mi m j A(x) = M |xi − x j | 0 2 i< j
where M = m 1 + m 2 + m 3 is the total mass. By choosing a sequence of motionless paths with greater and greater mutual dis1 (R, V ) is zero, which is not tances, it can be easily seen that the infimum of A on Hloc attained. To ensure solvability of the action-minimizing problem, we fix φ ∈ (0, 2π ) and consider the following subspace of Cφ : 1 Hφ := {x ∈ Hloc (R, V ) : x(t + 1) = eφi x(t)}.
The conventional definition of inner product on the Sobolev space H 1 ([0, 1], V ) defines an inner product on Hφ as well. It can be verified that collision-free critical points of A restricted to Hφ are indeed classical solutions of (1). Any path x in Hφ satisfies x(0), x(1) ≤ max{0, cos φ} |x(0)| · |x(1)|, where ·, · denotes the standard scalar product on (R2 )3 . From this condition, the action functional A restricted to Hφ is coercive (that is, A(x) → ∞ as x H 1 → ∞, see [3, Prop. 2]) and therefore, by weak lower semicontinuity of A on Hφ and a standard argument in variational calculus, attains its infimum on Hφ .
On Action-Minimizing Retrograde and Prograde Orbits
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Consider a linear transformation σ on Hφ defined by (σ · x)(t) := x(−t).
(3)
It is an order-2 isometry on Hφ and, because of the rotational symmetry on Hφ , the action functional A is σ -invariant. By Palais’ principle of symmetric criticality [19], any collision-free critical point of A restricted to the subspace σ
Hφ
:= {x ∈ Hφ : σ · x = x}
of σ -invariant paths is also a critical point of A on Hφ , and hence solves (1). Detailed descriptions and the variational principles mentioned in the above three paragraphs can be found in [3–5]. In order to simplify the statements of our theorems, we define the following auxiliary functions: J : [0, 1) → R+ , ξ : R2+ × (0, 2) → R, and E : R2+ → (1, 2) by J (s) :=
1
0
1 dt, |1 − s e2π ti |
(4)
1
ξ(m 1 , m 2 , η) :=
η 2−η
2
3
, (m 1 + m 2 + 1) (m 1 + m 2 ) √ 2 m 1 + m 2 + 1 (m 1 + m 2 ) E(m 1 , m 2 ) := √ . 3 m 1 + m 2 + 1 (m 1 + m 2 ) + max{m 1 , m 2 } 2 1 3
2 3
(5) (6)
Theorem 1. Given φ = ηπ , η ∈ (0, 2). Let m 3 = 1 and let J , ξ = ξ(m 1 , m 2 , η), E be as in (4), (5), (6). Suppose
η
2 3
η < E(m 1 , m 2 ), m1m2 (m 1 +m 2 )ξ
(7) 2 3
+ 23 [m 1 (J (m 2 ξ ) − 1) + m 2 (J (m 1 ξ ) − 1)] < (2 − η) m 1 m 2 2 2 2 2 2 2 (8) + min (2 3 − (2 − η) 3 )m 1 m 2 , (2 3 − η 3 )m 1 , (2 3 − η 3 )m 2 .
Then the three-body problem (1) has a retrograde solution in Cφ† which minimizes the σ
action functional A in Rφ ∩ Hφ . Theorem 1 provides a criterion on the masses (m 1 , m 2 ) with which an actionminimizing retrograde solution in Cφ† for (1) exists. Solutions obtained by Theorem 1 are periodic if φ/π is rational and are quasi-periodic if φ/π is irrational. The condition (7) is fulfilled for all η ∈ (0, 1] and for some η ∈ (1, 2), depending on the values of m 1 and m 2 . The condition (8) is also valid for a wide range of (m 1 , m 2 ) when η ∈ (0, 1]. Roughly speaking, straightforward calculations show that the first term in the first line of (8) is significantly dominated by the terms in the second line for most choices of masses, and the remaining terms in the first line are generally very small since J (s) is fairly close to 1 for most s ∈ (0, 1) (see the appendix of [5]). We will see in Sect. 4 that the assumptions (7) and (8) in Theorem 1 are much less restrictive than the conditions imposed to the main result in [5]. When η ∈ (0, 1], Proposition 5 in Sect. 4 provides a precise and simple criterion for (m 1 , m 2 ) to satisfy the requirements of Theorem 1, and the theorem tells us that “most” choices of masses
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are admissible regardless of the value of η ∈ (0, 1]. Many numerical figures for these retrograde action minimizers will be presented in Sect. 5. In addition to the isometry σ given in (3), consider the linear transformation τ on Hφ defined by φ T . (9) (τ · x)(t) := e− 2 i (x2 , x1 , x3 ) t + 2 It is another order-2 isometry on Hφ . The action functional A is τ -invariant provided m 1 = m 2 = m. Let σ, τ be the group of isometries on Hφ generated by σ and τ . Following again from Palais’ principle of symmetric criticality, any collision-free critical point of A restricted to the subspace σ,τ
Hφ
:= {x ∈ Hφ : g · x = x for any g ∈ σ, τ }
of σ, τ -invariant paths is also a critical point of A on Hφ , and hence solves (1). Theorem 2. Given φ = ηπ , η ∈ (0, 2). Let J , ξ be as in (4), (5) with m 3 = 1 and m 1 = m 2 = m. Suppose √
2
η3
1 3
+
1 4ξ
η < √4 2m+1√ , 2 2m+1+ m 1 2 + 23 J (mξ ) < min 2 3 m + η 3 ,
2 m 3 2 (2 − η)
2
+ 23 .
(10) (11)
Then the three-body problem (1) has a retrograde solution in Cφ† which minimizes the σ,τ
action functional A in Rφ ∩ Hφ
.
The region of (m, η) which satisfies (11) extends the regions of admissible masses given in Theorem 1. In particular, for fixed (m 1 , m 2 ) the inequality (8) fails to hold if η is sufficiently small, but (11) holds for arbitrarily small η as long as m is not close to zero. Section 5 includes several numerical figures of the action-minimizing retrograde solutions obtained by Theorem 2, and Sect. 8 shows the region of admissible (m, η). The special case m 1 = m 2 = m 3 = 1 includes part of the retrograde revolution family discovered numerically by Broucke [2] and Hénon [12]. Let Pφ be the set of prograde paths in Cφ† corresponding to the braid class P, the multiplicative inverse of R in B. Paths in Pφ are initially aligned in the order x3 (0) < x2 (0) < x1 (0), curves x1 and x2 revolve counterclockwise around each other while their center and x3 revolve counterclockwise around the origin. We may also define Pφ as the path component of eφti (1 + e2π ti ), eφti (1 − e2π ti ), −eφti in the space Cφ† . Consider another auxiliary function ζ : R2+ × (0, 2) → R defined by ζ (m, η) :=
1 1 3
(2m + 1) (2m)
2 3
η 2+η
2
3
.
(12)
On Action-Minimizing Retrograde and Prograde Orbits
409
Theorem 3. Given φ = ηπ , η ∈ (0, 2). Let m 3 = 1, m 1 = m 2 = m, and J , ζ = ζ (m, η) be as in (4), (12). Suppose
1 2 2 2 2 1 m 1 2 (13) η3 + + J (mζ ) < min 2 3 m + η 3 , (2 + η) 3 + 2 3 . 3 4ζ 3 2 Then the three-body problem (1) has a prograde solution in Cφ† which minimizes the σ,τ
action functional A in Pφ ∩ Hφ
.
In Sect. 8 we show the region of (m, η) which satisfies (13) and several numerical figures for the action-minimizing prograde orbits obtained by Theorem 3. The special case m 1 = m 2 = m 3 = 1 includes part of the direct revolution family in [2]. 3. Proof of Theorem 1 The proof presented here is an improvement of the proof for the main result in [5, Theorem 5], where some estimates are refined here and the rotation angle φ ∈ (0, π ] is extended to φ = ηπ ∈ (0, 2π ). The lower bound estimate for action values over collision paths in [5] is derived based on the symmetry constraint. Here we refine the estimates by taking both the symmetry and topological constraints into consideration. The improvement is significant especially when φ is away from π . As observed earlier, the action functional A is coercive on Hφ and therefore it attains its infimum on every weakly closed subset of Hφ . In particular, it attains its infimum σ
on the weak closure of Rφ ∩ Hφ at some z φ . Furthermore, as mentioned in the pre σ
vious section, collision-free critical points of A on Hφ
are solutions to (1), If z φ is σ
σ
collision-free, then it certainly solves (1) since Rφ ∩ Hφ is relatively open in Hφ . To prove Theorem 1, it is sufficient to prove that z φ does not belong to the weak σ
σ
boundary ∂ Rφ ∩ Hφ of Rφ ∩ Hφ under the assumptions (7), (8). What we will prove is the inequality
inf
σ
x∈Rφ ∩Hφ
A(x) <
inf
σ
x∈∂ Rφ ∩Hφ
A(x)
(14)
under the given conditions on φ and masses. This will be accomplished by providing good estimates for both sides of (14). A good lower bound estimate for the right side of (14) can be obtained by using a key formula in [5], which is motivated by Gordon’s theorem [11], and which we describe in Proposition 4 below. Given any θ ∈ (0, π ], T > 0, consider the following path space:
T,θ := {r ∈ H 1 ([0, T ], C) : r(0), r(T ) = |r(0)||r(T )| cos θ }, ∗
T,θ := {r ∈ T,θ : r(t) = 0 for some t ∈ [0, T ]}. The symbol ·, · stands for the standard scalar product in R2 ∼ = C. Define the Keplerian action functional Iµ,α,T : H 1 ([0, T ], C) → R ∪ {+∞} by T µ 2 α |˙r| + dt. Iµ,α,T (r) := |r| 0 2
410
K.-C. Chen, Y.-C. Lin
Proposition 4. Let θ ∈ (0, π ], T > 0, µ > 0, α > 0 be constants. Then 1 3 (µα 2 θ 2 T ) 3 , 2 1 3 inf∗ Iµ,α,T (r) = (µα 2 π 2 T ) 3 . 2 r∈ T,θ
inf Iµ,α,T (r) =
(15)
r∈ T,θ
(16)
σ
Given x ∈ Rφ ∩ Hφ , according to σ -invariance and the definition of Hφ , all masses are aligned on the real axis at t = 0, and φ
φ
φ
φ
e− 2 i x(t) = e− 2 i x(−t) = e 2 i x(−t) = e− 2 i x(1 − t).
(17) φ
This tells us that x(t) and x(1 − t) are symmetric with respect to the line L φ = {r e 2 i : 2
r ∈ R} and, in particular, all masses are aligned on L φ when t = 21 . Furthermore, 2
I1,M,1 (xi − x j ) = 2I1,M, 1 (xi − x j ) 2
(18)
for each pair of i, j. Here M = m 1 + m 2 + 1 is the total mass. Suppose x ∈ ∂ Rφ , then xi (t) = x j (t) for some t ∈ [0, 21 ] and i = j. Assume for now i = 1, j = 2. Then x1 − x2 ∈ ∗1
φ 2,2
or ∗1
φ 2 ,π − 2
,
x1 − x3 , x2 − x3 ∈ 1 , φ or 1 ,π − φ . 2 2
2
2
If x1 − x3 belongs to 1 , φ , then by Proposition 4 we have 2 2
3 I1,M, 1 (x1 − x3 ) ≥ 2 2
Mφ 2
2 1 3 1 3 3 2 2 = M3φ3. 2 4
If x1 − x3 belongs to 1 ,π − φ , we claim that x1 and x3 actually collide on [0, 21 ], thus 2
2
x1 − x3 ∈ ∗1
φ 2 ,π − 2
.
Suppose not, then x1 (t)− x3 (t) = 0 for every t ∈ [0, 21 ], implying that x1 (t)− x3 (t) = 0 for all t. All paths in a small C 0 -neighborhood of x would have the same value of Deg(x1 − x3 ), the degree of x1 (t) − x3 (t) over [0, 1] on the rotating frame. The reflection symmetry (17) plus the condition x1 − x3 ∈ 1 ,π − φ force x1 − x3 to have nonzero 2 2 winding number about the origin on the rotating frame over the time interval [0, 1]. This contradicts the assumption that x ∈ ∂ Rφ , because any path y in Rφ must satisfy Deg(y1 − y3 ) = 0, and the weak limit (which is also the uniform limit) x of any sequence y (n) in Rφ must either have Deg(x1 − x3 ) = 0 or x1 (t) − x3 (t) = 0 for some t. Now, knowing that x1 (t) − x3 (t) = 0 for some t ∈ [0, 21 ], by Proposition 4 we obtain a larger lower bound estimate for I1,M, 1 (x1 − x3 ): 2
2 3 I1,M, 1 (x1 − x3 ) ≥ (Mπ ) 3 2 2
1 1 3 . 2
On Action-Minimizing Retrograde and Prograde Orbits
411
The same arguments and estimates hold for I1,M, 1 (x2 − x3 ). 2 By (2), (15), (16), and (17), A(x) =
=
2 M
2 M
1
2 1 0 2 | x˙i
− x˙ j |2 +
M |xi −x j |
dt
m 1 m 2 I1,M, 1 (x1 − x2 ) + m 1 m 3 I1,M, 1 (x1 − x3 ) + m 2 m 3 I1,M, 1 (x2 − x3 ) 2 2
2 1 2 2 2 2 2 3 3 3 1 3 + 3 (m 1 m 3 + m 2 m 3 )M 3 φ 3 ≥ M 2 m1m2 M π 2 4 =
3 2
π2 M
mi m j
i< j
1
3
2 2 2 3 m 1 m 2 + η 3 (m 1 m 3 + m 2 m 3 ) .
Now we consider the second case: x1 (t) = x3 (t) for some t ∈ [0, 21 ]. Then, as in the previous case, x1 − x3 ∈ ∗1
φ 2,2
or ∗1
φ 2 ,π − 2
,
x1 − x2 , x2 − x3 ∈ 1 , φ or 1 ,π − φ . 2 2
2
2
If x2 − x3 belongs to 1 , φ , then by Proposition 4 we have 2 2
I1,M, 1 (x2 − x3 ) ≥ 2
3 2
Mφ 2
2 1 3 1 3 3 2 2 = M3φ3. 2 4
If x2 − x3 belongs to 1 ,π − φ , then following the arguments above, x2 and x3 actually 2
2
collide on [0, 21 ], and thus
x2 − x3 ∈ ∗1
φ 2 ,π − 2
,
2 3 I1,M, 1 (x2 − x3 ) ≥ (Mπ ) 3 2 2
1 1 3 3 2 2 3 3 ≥ M φ . 2 4
If x1 − x2 belongs to 1 ,π − φ , then by Proposition 4 we have 2
3 I1,M, 1 (x1 − x2 ) ≥ 2 2
2
M(2π − φ) 2
2 1 3 2 1 3 3 2 = M 3 (2π − φ) 3 . 2 4
If x1 − x2 belongs to 1 , φ , then x1 and x2 must collide on [0, 21 ], because otherwise 2 2 the reflection symmetry (17) plus the condition x1 − x2 ∈ 1 , φ force x1 − x2 to have 2 2 an even winding number about the origin over the time interval [0, 1], contradicting the assumption that x ∈ ∂ Rφ . In this case we have x1 − x2 ∈ ∗1
φ 2,2
,
and therefore 2 3 I1,M, 1 (x1 − x2 ) ≥ (Mπ ) 3 2 2
1 2 1 3 3 2 ≥ M 3 (2π − φ) 3 . 2 4
412
K.-C. Chen, Y.-C. Lin
As in the previous case, by (2), (15), (16), and (17), A(x)
2
m 1 m 2 I1,M, 1 (x1 − x2 ) + m 1 m 3 I1,M, 1 (x1 − x3 ) + m 2 m 3 I1,M, 1 (x2 − x3 ) 2 2 2 M 1 3 2 2 2 2 2 2 1 2 3 3 3 m 1 m 2 M 3 (2π − φ) 3 + m 1 m 3 M 3 π 3 ≥ + m2m3 M 3 φ 3 M 4 2 2 4
=
3 = 2
π2 M
13
2 2 2 (2 − η) 3 m 1 m 2 + 2 3 m 1 m 3 + η 3 m 2 m 3 .
The third case, x2 (t) = x3 (t) for some t ∈ [0, 21 ], is similar to the second case. In this case we have A(x) 2 ≥ M 3 = 2
2 2 2 2 2 2 3 3 3 m 1 m 2 M 3 (2π − φ) 3 + m 1 m 3 M 3 φ 3 + m 2 m 3 M 3 π 3 4 4 2
π2 M
1 1 3 2
13
2 2 2 (2 − η) 3 m 1 m 2 + η 3 m 1 m 3 + 2 3 m 2 m 3 .
Summarizing the above estimates, and by setting m 3 = 1, we conclude that inf
σ
x∈∂ Rφ ∩Hφ
A(x) 13
2 2 2 2 2 min 2 3 m 1 m 2 + η 3 (m 1 + m 2 ), (2 − η) 3 m 1 m 2 + 2 3 m 1 + η 3 m 2 , 2 2 2 (2 − η) 3 m 1 m 2 + η 3 m 1 + 2 3 m 2
3 ≥ 2
π2 M
1 2 2 π2 3
(2 − η) 3 m 1 m 2 + η 3 (m 1 + m 2 ) M 2 2 2 2 2 2 + min (2 3 − (2 − η) 3 )m 1 m 2 , (2 3 − η 3 )m 1 , (2 3 − η 3 )m 2 .
3 = 2
(19)
This provides a lower bound estimate for the right side of (14). Now we look for a good upper bound estimate for the left side of (14). Let Q(t) := R(t) :=
1 (Mφ)
2 3
eφti , 1 2 3
2
(m 1 + m 2 ) (2π − φ) 3
e(φ−2π )ti ,
and consider an artificial path (φ)
(φ)
(φ)
x (φ) (t) = (x1 (t), x2 (t), x3 (t)) := (Q(t) + m 2 R(t), Q(t) − m 1 R(t), − (m 1 + m 2 ) Q(t)) .
(20)
On Action-Minimizing Retrograde and Prograde Orbits
413 (φ)
Then x (φ) has the initial ordering as required in the definition of Cφ† . Particles x1
and
(φ) x2 (φ) x3
revolve clockwise about their mass center Q(t) along circular paths, while Q(t) and revolve counterclockwise along circular paths about the origin, which is exactly the σ
mass center of x (φ) . Clearly x (φ) belongs to Hφ . The assumption (7) is equivalent to 3
2−η max{m 1 , m 2 } 2 2π − φ < = , √ η φ M(m 1 + m 2 ) or
max{m 1 , m 2 }
max{m 1 , m 2 }|R(t)| =
2 3
(m 1 + m 2 ) (2π − φ)
<
2 3
M φ2
1 3
= M|Q(t)|. (φ)
This inequality is equivalent to the condition that the circular path x3 never touch the (φ) (φ) line segment connecting x1 and x2 . From this observation we see that x (φ) belongs σ
to Rφ ∩ Hφ if and only if (7) holds. Straightforward calculations show that 2 2 1 1 3 3 3 3 (φ) |x˙1 |2 = Mφ2 + m 22 (2π −φ) 4 − 2m 2 φ 2 (2π −φ) 2 cos(2π t), (φ)
|x˙2 |2 =
φ M2
(m 1 +m 2 ) 3
2
3
+m
2 2 (2π −φ) 3 4 1 (m 1 +m 2 ) 3
M 3 (m 1 +m 2 ) 3 1
+ 2m 1
1
φ 3 (2π −φ) 3 M
2 2 3 (m 1 +m 2 ) 3
cos(2π t),
2 3 (φ) |x˙3 |2 = (m 1 + m 2 )2 Mφ2 ,
2 1 2 3 3 . + m 1 m 2 (2π −φ) K (x˙ (φ) ) = 21 (m 1 + m 2 ) φM 1 (m 1 +m 2 ) 3
Note that the last line is independent of time. It can be easily verified that (7) ensures both m 1 ξ and m 2 ξ are strictly less than 1. The contribution of U (x (φ) ) to the total action can therefore be written 1 U (x (φ) ) dt 0
1
=
m1m2
0
−
(φ) x2 |
+
m1
(φ) |x1
(φ) x3 |
+
m2
dt (φ) − − x3 | 2 13 2 13 2 1 φ m 1 m 2 (2π − φ) 3 m1 m2 φ + dt + = 1 −2π ti | −2π ti | M |1 − m ξ e M |1 − m 2 1ξ e 3 0 (m 1 + m 2 ) 2 13 2 m 1 m 2 (2π − φ) 3 φ = + (m 1 J (m 2 ξ ) + m 2 J (m 1 ξ )) . 1 M (m 1 + m 2 ) 3 (φ) |x1
(φ) |x2
Here ξ = ξ(m 1 , m 2 , η) is as in (5). Therefore, 2 1 2 3 m 1 +m 2 3 inf x∈R ∩H σ A(x) ≤ 3m 1 m 2 (2π −φ) + φM + m 1 J (m 2 ξ ) + m 2 J (m 1 ξ ) 1 2 φ
=
η2 π M
φ
1
2 3
2(m 1 +m 2 ) 3
3m 1 m 2 2(m 1 +m 2 )ξ
+
3(m 1 +m 2 ) 2
+ m 1 (J (m 2 ξ ) − 1) + m 2 (J (m 1 ξ ) − 1) . (21)
414
K.-C. Chen, Y.-C. Lin
The assumption (8) is easily seen to be equivalent to that the upper bound estimate in (21) is strictly less than the lower bound estimate in (19). Therefore (8) implies our desired inequality (14). This completes the proof for Theorem 1. 4. Region of Admissible Masses The region of admissible (m 1 , m 2 ) given by Theorem 1 depends on the choice of φ = ηπ ∈ (0, 2π ). The values of the function √ 2 M(m 1 + m 2 ) E(m 1 , m 2 ) = √ 3 M(m 1 + m 2 ) + max{m 1 , m 2 } 2 are strictly between 1 and 2. The region of (m 1 , m 2 ) satisfying (7) shrinks rapidly as the angle φ approaches 2π . In fact, when φ is close to 2π the braid type of the path x (φ) given in the previous section is the same as the Euler relative equilibrium E 3 which rotates with angular velocity π − φ and which begins with ordering x2 (0) < x3 (0) < x1 (0). Figure 2 shows the regions of admissible masses determined by (8) with various φ, which increases from 0.01π to 1.35π . The condition (7) is automatic when φ ∈ (0, π ]. When φ ∈ (π, 2π ), the diagonal component of the region is the region of admissible masses which satisfy both (7) and (8). This component is unbounded when φ ∈ (0, π ] and it rapidly shrinks to a bounded region as φ increases until the region diminishes when φ ≈ 1.38π . The two triangular regions which border coordinate axes with φ ≥ 1.2 are regions where (8) holds but (7) fails. The next proposition states that for any φ ∈ (0, π ], “most” choices of masses are admissible. What we mean here is that, if a pair of positive masses (m 1 , m 2 ) are randomly chosen from a large ball on the m 1 m 2 -mass plane, the odds that they fall inside the region of admissible masses is more than 0.5. For this purpose it is sufficient to show that (8) is fulfilled whenever m1 2 5 ≤ ≤ 5 m2 2 except possibly a bounded region in the m 1 m 2 -mass plane. One can easily verify that the proportion of (m 1 , m 2 ) in region {m 1 , m 2 ≥ 0, m 21 + m 22 ≤ R 2 } satisfying the above inequalities is approximately 51.55% when R is sufficiently large. In Proposition 5 we replace the ratio bound 5/2 by 3, which allows us to improve the odds of choosing “good” masses from at least 51.55% to at least 59.03%, regardless of the value of φ ∈ (0, π ]. This is far from optimal and significantly better bounds can be found when the estimates are carried out for individual values of φ. For example, in [5, Example 6] the ratio m 1 /m 2 is bounded by 1/6 and 6 when φ = π , and the coverage of admissible masses satisfying these specific bounds is approximately 78.97%. Proposition 5. For any φ = ηπ ∈ (0, π ], the inequality (8) holds whenever 2
m 1 , m 2 ≥ m :=
2
23 − η3 2 3
2 − (2 − η)
2 3
and
m1 1 ≤ ≤ 3. 3 m2
Proof. The inequality (8) is unchanged by swapping m 1 and m 2 , thus we only need to show that (8) holds whenever m ≤ m 1 ≤ m 2 ≤ 3m 1 .
On Action-Minimizing Retrograde and Prograde Orbits 100
20
90
18
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100
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Fig. 2. Regions of admissible masses with φ equal (from top left to the bottom right) 0.01π , 0.05π , 0.1π , 0.2π , 0.3π , 0.5π , 0.8π , 0.9π , 0.97π , π , 1.1π , 1.2π , 1.25π , 1.3π , 1.35π . The mass region for the first figure is 0 < m 1 , m 2 ≤ 100, the second one is 0 < m 1 , m 2 ≤ 20, others are 0 < m 1 , m 2 ≤ 8
416
K.-C. Chen, Y.-C. Lin
Let m 1 = m, m 2 = λm. Then λ ∈ [1, 3], η = φ/π ∈ (0, 1], and 2 3 η 1 , ξ(m, λm, η) = 1 2 2 ((1 + λ)m + 1) 3 (1 + λ) 3 m 3 2 − η 2 1 3 η m3 1 m ξ(m, λm, η) = . < 1 2 2 − η 1 + λ ((1 + λ)m + 1) 3 (1 + λ) 3
(22)
Putting the first line into (8), the inequality (8) can be rewritten 2
2η 3 [(J (λm ξ(m, λm, η)) − 1) + λ (J (m ξ(m, λm, η)) − 1)] 3 1 3 2 2 2 2 2 1 + min λm(2 3 − (2 − η) 3 ), 2 3 − η 3 . < (2 − η) 3 λm 1 − 1 + (1 + λ)m (23) Using (22) and the fact that J (0) = 1, J is monotonically increasing on [0, 1) (see [5, Appendix]), the first line in (23) is bounded from above by 2 λ 1 J −1 +λ J −1 3 1+λ 1+λ 3 1 2 J −1 +3 J −1 (≈ 0.2907). < 3 4 2 According to the definition of m we have 2 2 2 2 2 2 min λm(2 3 − (2 − η) 3 ), 2 3 − η 3 = 2 3 − η 3 , from which the second line in (23) is bounded from below by 1 3 2 2 2 2 2 2 1 λ 1+ 2 3 − η 3 − (2 − η) 3 λm − 1 > 2 3 − η 3 − (2 − η) 3 (1 + λ)m 3(1 + λ) 2 2 2 1 ≥ 2 3 − η 3 − (2 − η) 3 . 4 Therefore, the inequality (23) follows easily if the function 2 2 2 3 1 1 2 J −1 +3 J −1 2 3 − η 3 − (2 − η) 3 − 4 3 4 2
is positive for every η ∈ (0, 1]. The minimum of this function on (0, 1] occurs at η = 1 and the minimum value (≈ 0.0467) is indeed positive. This concludes the proof of Proposition 5. We remark here that Proposition 5 manifests what we asserted earlier: Theorem 1 is a substantial extension of the main result in [5]. It can be easily verified that, in [5, Theorem 5] the region of admissible masses is bounded for most φ ∈ (0, π ] and the region diminishes when φ is smaller than 0.38π , see Fig. 3 for examples. In [5] the angle φ is confined to φ ≤ π but it can actually be extended to φ ≈ 1.16π . In contrast, for any φ ∈ (0, π ], our theorem holds for “most” choices of masses. As φ increases beyond π , the region of admissible masses remains nonempty until φ ≈ 1.38π .
On Action-Minimizing Retrograde and Prograde Orbits
417
8
8
8
7
7
7
6
6
6
5
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8
0
1
2
3
4
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6
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2
3
4
5
6
7
8
Fig. 3. Regions of admissible masses given by [5, Theorem 5] with φ equal (from left to right) 0.5π , 0.8π , 0.97π . Masses in these figures are bounded by 0 < m 1 , m 2 ≤ 8
5. Retrograde Orbits with Various φ The purpose of this section is to show miscellaneous samples of action-minimizing retrograde solutions obtained in Theorem 1. This family of solutions are determined by three parameters: m 1 , m 2 , φ. Tables of figures here are listed in the order of increasing φ, masses m 1 ≤ m 2 are selected to illustrate deformation of action-minimizing orbits as mass parameters vary. Masses (m 1 , m 2 ) are mostly selected from the following lists, in which (m 1 , m 2 ) are aligned in the same order as the figures. (A)
(0.2, 0.8) (0.2, 0.6) (0.2, 0.4) (0.2, 0.2)
(C)
(0.01, 4) (0.01, 1)
(0.4, 0.8) (0.4, 0.6) (0.4, 0.4) (0.1, 4) (0.1, 1)
(0.6, 0.8) (0.6, 0.6)
(0.5, 4) (0.5, 1)
(0.8, 0.8) (B)
(0.8, 4) (0.8, 1)
(D)
(1, 7) (1, 5) (1, 3) (1, 1) (1, 100)
(3, 7) (3, 5) (3, 3)
(5, 7) (5, 5)
(10, 100)
(7, 7)
(50, 100)
(100, 100)
Lists (A) and (B) contain typical examples of masses with m 1 ≤ m 2 < 1 and 1 ≤ m 1 ≤ m 2 . Numerical figures corresponding to lists (C), (D) illustrate how actionminimizing orbits deform when m 1 1 ≤ m 2 or 1 m 2 . A more complete catalog of action-minimizing retrograde orbits is available at [13]. There is a total of 129 examples in this section, not all of them are covered by Theorem 1, but nonetheless we put them in for the entirety of our graphics. Seventeen examples in here fail to fulfill requirements in Theorem 1, ten of them are covered by Theorem 2, another three of them can be obtained by choosing other test paths. The remaining four examples without existence proof are: (m 1 , m 2 , φ) = (0.2, 0.6, 2π/3), (0.2, 0.4, 2π/3), (0.2, 0.4, 3π/4), (0.2, 0.4, 4π/5). All of these examples belong to class (B). For brevity we put numerical data for all of our examples, such as their initial conditions and action values, in the Appendix. In Fig. 18 we skip the case (m 1 , m 2 , φ) = (1, 100, π ) not only because this case is not covered by our theorems, but also because we don’t think such a retrograde orbit can even exist. If one looks at the trajectory of the first body on the rotating frame, the loop deform across the third body as m 1 decreases from 10 to 1, resulting in a solution in a different topological class.
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Fig. 4. Action minimizing retrograde solutions with masses (m 1 , m 2 ) = (0.5, 0.5), (1, 1), (2, 2), (4, 4) (first row), (0.5, 8), (1, 8), (2, 8), (4, 8) (second row), φ = π/4 0.8 0.6
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Fig. 5. Action minimizing retrograde solutions with masses (m 1 , m 2 ) = (0.2, 0.2), (0.4, 0.4), (0.6, 0.6), (0.8, 0.8), φ = π/2
6. Proof of Theorem 2 The proof for Theorem 2 is similar to that of Theorem 1 but with some subtle improvements on the lower bound estimates for A over collision paths. As observed in Sect. 2, the action functional A attains its infimum on the weak clo σ,τ
sure of Rφ ∩ Hφ . All we need to show is that, under the assumptions of φ and masses in Theorem 2, inf
σ,τ
x∈Rφ ∩Hφ
A(x) <
inf
σ,τ
x∈∂ Rφ ∩Hφ
A(x),
(24) σ,τ
and hence action minimizers do not fall on the weak boundary ∂ Rφ ∩ Hφ σ,τ
Hφ
of Rφ ∩
. σ,τ
Given x ∈ Rφ ∩ Hφ
, according to τ -invariance and the definition of Hφ ,
φ 1 x(t) = x(−t) = e− 2 i (x2 , x1 , x3 )(−t + ), 2
where hence φ φ 1 e− 4 i x(t) = e− 4 i (x2 , x1 , x3 )(−t + ). 2
(25)
On Action-Minimizing Retrograde and Prograde Orbits
419 2
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Fig. 6. Action minimizing retrograde solutions with (m 1 , m 2 ) in (B), φ = π/2
Combining with (17), we see that [0, 14 ] is a fundamental domain of the σ, τ -action; φ
x1 (t) and x2 (−t + 21 ) are symmetric with respect to the line L φ = {r e 4 i : r ∈ R} as well 4
as x2 (t) and x1 (−t + 21 ), and the same for the curves x3 (t) and x3 (−t + 21 ). In particular, x1 ( 41 ) and x2 ( 41 ) are symmetric with respect to L φ . Moreover, for any t ∈ [0, 41 ], 4
1 1 |x1 (t) − x2 (t)| = |x1 (−t + ) − x2 (−t + )|, 2 2 1 1 |x˙1 (t) − x˙2 (t)| = |x˙1 (−t + ) − x˙2 (−t + )|, 2 2 implying that I1,M, 1 (x1 − x2 ) = 2I1,M, 1 (x1 − x2 ), 2
where M = 2m + 1 is the total mass.
4
(26)
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Fig. 7. Action minimizing retrograde solutions with (m 1 , m 2 ) in (A), φ = 2π/3
Suppose x ∈ ∂ Rφ , then xi (t) = x j (t) for some t ∈ [0, 41 ] and i = j. One possibility is i = 1, j = 2. In this case, x1 − x2 ∈ ∗1
π 4, 2
− φ4
or ∗1
π 4, 2
+ φ4
,
x1 − x3 , x2 − x3 ∈ 1 , φ or 1 ,π − φ . 2 2
2
2
As discussed in Sect. 3, due to our topological assumption, the case x1 − x3 ∈ 1 ,π − φ 2 2 (or x2 − x3 ∈ 1 ,π − φ ) forces x1 − x3 (or x2 − x3 ) to fall into the space ∗1 φ . There2
2
2 ,π − 2
fore, for any φ ∈ (0, 2π ), I1,M, 1 (x1 − x3 ) and I1,M, 1 (x2 − x3 ) are both bounded from 2 2 below by 3 2 2 M3φ3. 4 By (2), Proposition 4, (17), (25), and (26), 1 2 1 2 M |x˙i − x˙ j |2 + dt A(x) = mi m j M 2 |x − xj| i 0 i< j
On Action-Minimizing Retrograde and Prograde Orbits 1.5
421
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Fig. 8. Action minimizing retrograde solutions with (m 1 , m 2 ) in (B), φ = 2π/3
2 2 2m I1,M, 1 (x1 − x2 ) + m I1,M, 1 (x1 − x3 ) + m I1,M, 1 (x2 − x3 ) 4 2 2 M 1 2 2 1 3 3 2 2 2m ≥ + M3φ3 3m M 3 π 3 M 4 2
=
= 3m
π2 M
13
1 2 23 m + η3 .
The second possibility is i = 1, j = 3. In this case, by the τ -invariance x2 and x3 also collide at some t ∈ [ 41 , 21 ]. Therefore, x1 − x2 ∈ 1 , π − φ or 1 , π + φ , 4 2
4
x1 − x3 , x2 − x3 ∈ ∗1
φ 2,2
4 2
or ∗1
4
φ 2 ,π − 2
.
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Fig. 9. Action minimizing retrograde solutions with (m 1 , m 2 ) in (A), φ = 3π/4
Then
1 2 1 2 M |x˙i − x˙ j |2 + dt A(x) = mi m j M 2 |x − xj| i 0 i< j 2 2 = 2m I1,M, 1 (x1 − x2 ) + m I1,M, 1 (x1 − x3 ) + m I1,M, 1 (x2 − x3 ) 4 2 2 M 1 1 2 1 2 π φ 3 1 3 3 2 2 1 3 3 2 2 1 3 2m − ≥ + M3π 3 + M3π3 3m M 3 M 2 4 4 2 2 2 2 = 3m
π2 M
13
2 2 m (2 − η) 3 + 2 3 . 2
The estimate for the third possibility i = 2, j = 3 is identical. Summarizing these estimates, we conclude that 2 13 1 2 2 2 π m inf A(x) ≥ 3m min 2 3 m + η 3 , (2 − η) 3 + 2 3 . σ,τ
M 2 x∈∂ Rφ ∩H φ
(27)
On Action-Minimizing Retrograde and Prograde Orbits
423
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Fig. 10. Action minimizing retrograde solutions with (m 1 , m 2 ) in (B), φ = 3π/4
This provides a lower bound estimate for the right side of (24). σ,τ
Observe that the path x (φ) given in (20) belongs to Hφ . It actually belongs to Rφ because (10) is identical to (7) in our case. Therefore, writing ξ = ξ(m, m, η) and by (21), inf
σ,τ
x∈Rφ ∩Hφ
A(x)
2 2 13 3m 1 m 2 3(m 1 + m 2 ) η π + + m 1 (J (m 2 ξ ) − 1) + m 2 (J (m 1 ξ ) − 1) ≤ M 2(m 1 + m 2 )ξ 2
2 2 13 3 η π (28) =m + 1 + 2J (mξ ) . M 4ξ The assumption (11) is equivalent to that of the upper bound estimate in (28) and is strictly less than the lower bound estimate in (27). Therefore (11) implies (24). This completes the proof for Theorem 2.
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Fig. 11. Action minimizing retrograde solutions with (m 1 , m 2 ) in (A), φ = 4π/5
7. Proof of Theorem 3 The proof for Theorem 3 is similar to that of Theorem 2 but a prograde test path with action value larger than that of x (φ) is inevitable. On the other hand, due to the topological nature of prograde paths, the lower bound estimate on collision paths can also be slightly improved. As discussed before, the action functional A attains its infimum on the weak closure σ,τ
of Pφ ∩ Hφ and what we need to show is that inf
σ,τ
x∈Pφ ∩Hφ
A(x) <
inf
σ,τ
x∈∂ Pφ ∩Hφ
A(x)
(29)
under the assumptions of φ and masses in Theorem 3. Let us first see how a “good” lower bound estimate for the right side of (29) can be σ,τ
obtained. The estimate (27) is valid for A on ∂ Pφ ∩ Hφ as well, but we can do a bit σ,τ
better than that. If x ∈ Pφ ∩ Hφ valid for any path in
σ,τ
Hφ .
, then clearly (17), (25), and (26) hold since they are
On Action-Minimizing Retrograde and Prograde Orbits
425
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Fig. 12. Action minimizing retrograde solutions with (m 1 , m 2 ) in (B), φ = 4π/5
1
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Fig. 13. Action minimizing retrograde solutions with (m 1 , m 2 ) in (C), φ = 4π/5
0
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K.-C. Chen, Y.-C. Lin
3
3
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Fig. 14. Action minimizing retrograde solutions with (m 1 , m 2 ) in (D), φ = 4π/5 0.6
0.6 0.6 0.6
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Fig. 15. Action minimizing retrograde solutions with (m 1 , m 2 ) in (A), φ = π
Suppose x ∈ ∂ Pφ , then xi (t) = x j (t) for some t ∈ [0, 14 ] and i = j. The estimate for the first possibility i = 1, j = 2 is the same as the one obtained in Sect. 6: A(x) ≥ 3m
π2 M
13
1 2 23 m + η3 .
As before, M = 2m + 1 denotes the total mass.
On Action-Minimizing Retrograde and Prograde Orbits
427
1 1 1 0.8
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Fig. 16. Action minimizing retrograde solutions with (m 1 , m 2 ) in (B), φ = π
1
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Fig. 17. Action minimizing retrograde solutions with (m 1 , m 2 ) in (C), φ = π
0
0.2
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0.6
428
K.-C. Chen, Y.-C. Lin 3
2.5
3 2 2 2
1.5 1
1
1
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2
3
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1
2
3
Fig. 18. Action minimizing retrograde solutions with (m 1 , m 2 ) in (D), φ = π , except the case (m 1 , m 2 ) = (1, 100) 0.6
1
0.5
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Fig. 19. Action minimizing retrograde solutions with φ = 5π/4 and (m 1 , m 2 ) = (0.5, 0.5), (1, 1), (2, 2), (4, 4)
The second possibility we consider is that x1 (t) = x3 (t) for some t ∈ [0, 41 ] and x1 (t) = x2 (t) for any t ∈ [0, 14 ]. We claim that in this case we must have x1 − x2 ∈ 1 , π + φ . 4 2
4
σ,τ
To see this, consider any sequence x (n) in Pφ ∩ Hφ
which converges weakly to
(n) x1
(n)
x ∈ ∂ Pφ . The symmetry conditions (17) and (25) force − x2 , for each n, to turn an angle π2 counterclockwise on the rotating frame, and thus turn an angle π2 + φ4 coun(n) (n) terclockwise on the inertia frame. But since x1 − x2 converges uniformly on [0, 41 ] to x1 − x2 which stays away from the origin for t ∈ [0, 41 ], x1 − x2 also turns an angle φ π π 2 counterclockwise on the rotating frame and an angle 2 + 4 counterclockwise on the inertia frame. This proves what we just claimed. Since x is τ -invariant, x2 and x3 collide at some t ∈ [ 41 , 21 ] and therefore x1 − x3 , x2 − x3 ∈ ∗1
φ 2,2
or ∗1
φ 2 ,π − 2
.
Then by (2), Proposition 4, (17), (25), and (26), 1 2 1 2 M |x˙i − x˙ j |2 + dt A(x) = mi m j M 2 |x − xj| i 0 i< j 2 2 = 2m I1,M, 1 (x1 − x2 ) + m I1,M, 1 (x1 − x3 ) + m I1,M, 1 (x2 − x3 ) 4 2 2 M 2 1 1 1 2 π φ 3 1 3 3 2 2 1 3 3 2 2 1 3 2m ≥ + M3π3 + M3π 3 3m M 3 + M 2 4 4 2 2 2 2 = 3m
π2 M
13
2 2 m (2 + η) 3 + 2 3 . 2
On Action-Minimizing Retrograde and Prograde Orbits
429
The remaining possibility is that x2 (t) = x3 (t) for some t ∈ [0, 41 ] and x1 (t) = x2 (t) for any t ∈ [0, 41 ]. The resulting lower bound estimate for A(x) is clearly the same as the second possibility. Summarizing these estimates, we conclude that inf
σ,τ
x∈∂ Pφ ∩Hφ
A(x) ≥ 3m
π2 M
13
1 2 2 2 m min 2 3 m + η 3 , (2 + η) 3 + 2 3 . 2
(30)
This provides a lower bound estimate for the right side of (29). It is sharper than (27) for the boundary of retrograde paths. Now we take a prograde test path modified from x (φ) in (20) to acquire a good upper bound estimate for the left side of (29). Let Q(t) := P(t) :=
1 2
(Mφ) 3 1
2
eφti , e(φ+2π )ti ,
2
(2m) 3 (2π +φ) 3
and consider an artificial path (φ)
(φ)
(φ)
y (φ) (t) = (y1 (t), y2 (t), y3 (t)) := (Q(t) + m P(t), Q(t) − m P(t), −2m Q(t)) .
(31)
Then y (φ) has the same initial ordering as x (φ) and fulfills the requirement in the defi(φ) (φ) nition of Cφ† . Particles y1 and y2 revolve counterclockwise about their mass center (φ)
Q(t) along circular paths, while Q(t) and y3 revolve counterclockwise along circular σ,τ
paths about the origin, the mass center of y (φ) . The path y (φ) belongs to Hφ , as does x (φ) . Note that η ∈ (0, 2) implies
φ 2(2π + φ)
2
3
=
η 2(2 + η)
2
3
1 1 1 3 M 3 < 2+ = . m m
Equivalently, m|P(t)| = (φ)
m 2 3
(2m) (2π + φ)
2 3
<
M φ2
1
(φ)
3
= M|Q(t)|, (φ)
which ensures that y3 stays away from the binary y1 , y2 . This implies y (φ) ∈ Pφ . Similar to the calculations for K (x˙ (φ) ), we have ⎤ ⎡ 2 13 2 3 m (2π + φ) 1 φ m 1 2 ⎦ K ( y˙ (φ) ) = ⎣(m 1 + m 2 ) + 1 2 M (m 1 + m 2 ) 3 ⎤ ⎡ 2 13 2 1⎣ m 2 (2π + φ) 3 ⎦ φ = + . 2m 1 2 M (2m) 3
430
K.-C. Chen, Y.-C. Lin
The contribution of U (y (φ) ) to the total action can be written 1 U (y (φ) ) dt 0
1
=
m2 −
(φ) y2 |
+
m
(φ) |y1
(φ) y3 |
+
m
dt (φ) − − y3 | 2 13 2 13 2 1 2 φ m (2π + φ) 3 m m φ + dt + = 1 2π ti M |1 − mζ e | M |1 − mζ e2π ti | 0 (2m) 3 2 13 2 m 2 (2π + φ) 3 φ = + 2m J (mζ ) . 1 M (2m) 3 0
(φ) |y1
(φ) |y2
Here ζ = ζ (m, η) is given in (12). Therefore, 2
inf
σ,τ
x∈Pφ ∩Hφ
A(x) ≤
3m 2 (2π + φ) 3
φ2 M
13
+ [m + 2m J (mζ )] 1 2(2m) 3
2 2 13 3 η π =m + 1 + 2J (mζ ) . M 4ζ
(32)
The assumption (13) is easily seen to be equivalent to that the upper bound estimate in (32) is strictly less than the lower bound estimate in (30). Therefore (13) implies (29). This completes the proof for Theorem 3. 8. Retrograde and Prograde Orbits with Additional Symmetries The set of admissible (m, η) for action-minimizing retrograde orbits obtained by Theorem 2, where m 1 = m 2 = m and m 3 = 1, is the region below the cliff-like curve in the first graph of Fig. 20. The nearly horizontal portion (η ≈ 1.45) which cuts out admissible (m, η) with large η is due to the inequality (10). Similarly, the other graph of Fig. 20 shows the set of admissible (m, η) for action-minimizing prograde orbits obtained by Theorem 3. The region is given by a single inequality (13). Note that these two curves match at η = 0 since (11) and (13) are the same at this limiting case. The variational problem studied throughout the paper can be also formulated as the minimization problem of the action functional over relative periodic loops in Cφ,T or † † with φ ∈ (−2π, 2π ). Retrograde paths in Cφ,T are continuously deformed to proCφ,T grade paths as φ (or η) decreases continuously from positive values to negative values. The borderline case φ = 0 is the only case without action minimizer, since it is the only case without coercivity. Figure 21 shows several numerical figures of action-minimizing prograde orbits obtained in Theorem 3. More action-minimizing prograde orbits with m 1 = m 2 can be found in [13]. Many examples of retrograde orbits with m 1 = m 2 are already included in Sect. 5. In the order of their appearance, the set of (m, η) in Fig. 21 are
(0.5, 0.5) (0.5, 0.25)
(1, 1) (1, 0.5) (1, 0.25)
(1.5, 1.25) (1.5, 1) (1.5, 0.5) (1.5, 0.25)
(2, 1.25) (2, 1) (2, 0.5) (2, 0.25)
On Action-Minimizing Retrograde and Prograde Orbits
431
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
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1
1.5
2
2.5
3
3.5
4
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 20. Admissible (m, η) given by Theorem 2 and Theorem 3 for retrograde (left) and prograde (right) σ,τ
solutions in Hφ
Numerical data for these orbits are listed on Table 18 in the Appendix. The cases (0.5, 1.25), (1, 1.25), (0.5, 1) are missing because they do not fall inside our region of admissible (m, η). Similar to Proposition 5 we can now establish simple criteria for (m, η) to satisfy the requirements in Theorem 2 and Theorem 3. The criterion in Proposition 6 for retrograde orbits is far from sharp but it is easy to verify, and it shows in particular that the region of admissible (m, η) contains a large rectangle of the form [m 0 , ∞) × (0, 1]. Proposition 7 is similar to Proposition 6. For fixed m, as η approaches zero, our retrograde or prograde orbits link to classical results by analytic continuation near infinity (see [22] for references). For fixed η, as m approaches infinity, these orbits link to satellite orbits of some restricted n-body problems, which can also be obtained by the minimization method [6]. Proposition 6. Let J be as in (4). For any φ = ηπ ∈ (0, 2π ), inequalities (10),(11) hold whenever √ 2 2 23 − η3 2 2m + 1 m≥ 1 and η ≤ √ (33) √ . 2 2m + 1 + 2m 2 3 − 21 (2 − η) 3 Proof. The first inequality in (33) is equivalent to m 1 2 2 2 2 2 m min 2 3 m + η 3 , (2 − η) 3 + 2 3 = (2 − η) 3 + 2 3 . 2 2 Clearly (10) follows from the second inequality of (33), which also implies that η2 − 4(2m + 1)η + 4(2m + 1) ≥ 0. In fact the term on the right end of (33) is a root of this quadratic polynomial in η. Let ξ be as in (5) with m 1 = m 2 = m. Then the above inequality can be also written mξ =
1 2(1 +
1 2m )
1 3
η 2−η
2
3
≤
1 . 2
432
K.-C. Chen, Y.-C. Lin
0.6
0.5 0.4
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1 0.8 0.6
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0
0
0
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−0.8 −1
−0.6 −0.8 −0.6
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0
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−1 −0.8
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−1
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−1
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1
1 0.8
1
1
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0.6 1 0.4
0.5 0.2 0
0
0
−0.5
−0.5
−1
−1
0
−0.2 −0.5 −0.4
−1 −0.6 −0.8
−1.5
−1 −1
−0.8
−0.6
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−0.2
0
0.2
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0.8
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1.5
−1
−0.5
0
0.5
1
1.5
Fig. 21. Action-minimizing prograde orbits obtained in Theorem 3 with m 1 ≤ m 2
Recall that J is increasing on [0, 1). Therefore 1 J (mξ ) ≤ J . 2 Note that the second inequality of (33) is always valid if η ∈ (0, 1]. When η ∈ (1, 2), (33) can be written 2
2
23 − η3 1 3
2 − 21 (2 − η)
2 3
≤m≤
(2 − η)2 . 2 (2 − η)2 − η2
The inequality for the two expressions in η fails if, say η ≥ 3/2. It can be easily verified that 2
2
23 −
η3 3
2 1 1 1 + 2J − (2 − η) 3 > 0 2 12
On Action-Minimizing Retrograde and Prograde Orbits
433
for η ∈ (0, 3/2) (in fact, this inequality holds whenever η < 1.8146). From this observation we have 1 2 2 2 2 1 η3 m 1 1 3 3 3 2 − − 1 (2 − η) 3 . 1 + 2J > (2 − η) > 1+ 3 2 12 2 2m Then 2
2 m η3 2 + (2 − η) 3 > 2 3 2 3
2
1 2 1 m 1 3 (2 − η) 3 1 + 2J + 1+ 2 2 2m 2
η3 η3 ≥ . (1 + 2J (mξ )) + 3 4ξ This proves (11). Proposition 7. Let J be as in (4). For any φ = ηπ ∈ (0, 2π ), inequality (13) holds whenever 2 2 2 2 2 1 1 η3 23 − η3 3 − 1 + 2J − (2 + η) 3 . (34) and 0 < 2 m≥ 1 2 3 2 12 2 3 − 21 (2 + η) 3 Proof. The first inequality (34) is equivalent to 1 m 2 2 2 2 2 m min 2 3 m + η 3 , (2 + η) 3 + 2 3 = (2 + η) 3 + 2 3 . 2 2 Let ζ be as in (12) with m 1 = m 2 = m. Then mζ is easily seen to be bounded from above by 1/2 for every η ∈ (0, 2). Following the last paragraph in the proof of Proposition 6, the inequality (34) is readily seen to imply (13). The special case m = 1 is connected to many numerical discoveries by Broucke [2] and Hénon [12]. In [12] Hénon shows the complete retrograde family for the special case m 1 = m 2 = m 3 = 1. As η decreases from 2 to a small positive number, the retrograde orbit deforms from Schubart’s rectilinear orbit [20] to a retrograde orbit with a tight binary (see [12, Fig.1-3]). Part of this family and some prograde orbits with equal masses are also obtained by Broucke [2]. It is unknown whether this prograde family is continuable to η = 2 as the retrograde family. Classical methods on existence of such solutions apply to cases with small η. Schubart’s orbit has recently received some rigorous treatments via topological [16] and variational arguments [23]. This orbit falls on the boundary of retrograde paths in † C2π with x2 bouncing back and forth between x1 and x3 . Within the class of prograde † there are numerical solutions [2, Fig.9-10] whose rigorous and retrograde paths in C2π existence proofs are yet to be found. 9. Estimating Mutual Distances This section is dedicated to providing quantitative estimates for the mutual distances of action-minimizing orbits. Action-minimizing retrograde solutions have no tight binaries, as we shall see in this section, and they are therefore different from those retrograde solutions obtained by classical perturbation arguments. A brief explanation for this fact
434
K.-C. Chen, Y.-C. Lin
along with a crude estimate of upper and lower bounds for distance ratios can be found in [5, §6]. The estimates supplied here are finer. σ
Let x ∈ Rφ ∩ Hφ be an action-minimizing retrograde solution obtained in Theorem 1. Let r i j = max |xi (t) − x j (t)|, r i j = min |xi (t) − x j (t)|. t∈[0,1]
φ 2
(35)
t∈[0,1]
Writing xi −x j in polar form ri j eiθi j , then by σ -invariance and (17), θi j ( 21 )−θi j (0) = or π − φ2 . Let φ φ π − |π − φ| = min ,π − . (36) ψ := 2 2 2
Then
1 2
0
1 M |x˙i − x˙ j |2 + dt = 2 |xi − x j |
1 2
1 2 M (˙ri j + ri2j θ˙i2j ) + dt ri j 0 2 2 2 1 1 2 2 M 2 |˙ri j |dt + r i j |θ˙i j |dt + ≥ 2r i j 0 0
≥ (r i j − r i j )2 + r i2j ψ 2 + ≥
ψ 2 r i2j ψ2 + 1
+
M 2r i j
M . 2r i j
Consequently, 1 2 1 2 M |x˙i − x˙ j |2 + dt A(x) = mi m j M |xi − x j | 0 2 i< j 2 2 2ψ r i j 1 M ≥ + mi m j . M ψ2 + 1 ri j
(37)
i< j
The elementary identity min
s∈(0,∞)
1 2ψ 2 3 4ψ 2 M 2 3 M 2 s + = ψ2 + 1 s 2 ψ2 + 1
implies that, for any pair of i < j, 2 2 2 2 13 m i m j 2ψ r i j M 4ψ M 3 A(x) ≥ + + M ψ2 + 1 ri j 2M ψ 2 + 1
(38)
m k ml .
(39)
k
For any fixed masses and φ (and hence ψ), the two positive solutions for the equation
2 2 13 2ψ 2 4ψ M M M 3 2 A(x) − s + = ψ2 + 1 s mi m j 2m i m j ψ 2 + 1
k
m k ml
(40)
On Action-Minimizing Retrograde and Prograde Orbits
435
in s are clearly lower and upper bounds for r i j . Summarizing these observations, we obtain: Proposition 8. Fix m 1 , m 2 > 0, φ ∈ (0, 2π ), and let x be an action-minimizing retro σ
grade solution of the three-body problem in Rφ ∩ Hφ given by Theorem 1. Let r i j and ψ be as in (35) and (36). Then for each pair of i < j from {1, 2, 3}, the two positive solutions of identity (40) are lower and upper bounds for r i j . The practical way of applying Proposition 8 is to choose an ideal retrograde path σ
xtest in Rφ ∩ Hφ and replace the A(x) in (40) by its upper bound A(xtest ) = Atest . The resulting bounds for r i j are of course not as sharp, but as indicated by the tables in the Appendix, the test paths selected in Sect. 3 whose action values were given in (21) provide fairly sharp upper bound estimates for A(x) for a wide range of masses and rotation angles. They often provide satisfactory approximations for the bounds given in Proposition 8. It is maybe surprising that the difference between the action value A(x) and Atest is within 0.3% (many of them differ by less than 0.1%!) among most of the examples given in this paper. A recursive application of (37) leads to better estimates for mutual distances, as explained below. Let αi j and βi j denote the lower and upper bounds for r i j obtained in Proposition 8. For example, the case (m 1 , m 2 ) = (3, 5) and φ = π gives (α12 , β12 ) ≈ (0.5320, 3.1920), (α23 , β23 ) ≈ (0.2690, 4.7163), (α13 , β13 ) ≈ (0.1809, 5.8235). Among our examples the ratios βi j /αi j range mostly from 5 to 70. The estimates αi j and βi j for mutual distances can be refined by improving (39). The inequality (39) is obtained from (38), but with Proposition 8 we may write 2 2 m k m l 2ψ 2 α 2 m i m j 2ψ r i j M M kl + + A(x) ≥ + M ψ2 + 1 ri j M ψ 2 + 1 βkl k
and rewrite (40) accordingly. Once the new lower and upper bound estimates for each r i j were obtained, still denote them by αi j and βi j , then we may repeat this process and further improve the above inequality and αi j , βi j . Repetition of this process results in finer and finer estimates for mutual distances. Numerical calculations show that this procedure often lowers the ratios βi j /αi j by about 15%. 10. Appendix: Numerical Data This Appendix contains numerical data for the orbits listed in Sect. 5 and 8. Most of the data in here are accurate to the fourth decimal place. Readers may reproduce most of the figures in Sect. 5 and Sect. 8 by using these initial data. However, a few examples are highly unstable and initial data with much greater precision are needed if one wishes to produce satisfactory numerical figures for their orbits. Our intention is to provide numerical data that are satisfactory for the purpose of action value calculations. Numerical data with higher precision can be found in [13]. On the tables of numerical data (Tables 1-16), A(x) is the numerical action value of the action-minimizing retrograde orbits, the symbol Atest stands for the action value of the test path (20) in Sect. 3, namely the upper bound for true action values of minimizers
436
K.-C. Chen, Y.-C. Lin Table 1. Initial data and action values for solutions in Fig. 4 (φ = π/4)
(m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.5, 0.5)∗ (1, 1)∗ (2, 2) (4, 4) (0.5, 8) (1, 8) (2, 8) (4, 8)
(0.9013, 0.5831, −0.7422) (0.7683, 0.3657, −1.1340) (0.6560, 0.1509, −1.6139) (0.5937, −0.0483, −2.1817) (0.8813, 0.2234, −2.2280) (0.8476, 0.1793, −2.2821) (0.7902, 0.1003, −2.3832) (0.7043, −0.0321, −2.5606)
(−0.3098i, 1.4767i, −0.5834i) (−0.6741i, 1.5652i, −0.8911i) (−1.0976i, 1.7320i, −1.2688i) (−1.5523i, 1.9812i, −1.7154i) (−3.1719i, 0.4170i, −1.7502i) (−3.0628i, 0.6070i, −1.7929i) (−2.8635i, 0.9499i, −1.8722i) (−2.5229i, 1.5130i, −2.0125i)
2.1833 5.4813 14.7629 42.3032 14.2853 23.3078 40.4350 71.8272
2.1836 5.4835 14.7690 42.3052 14.2861 23.3108 40.4530 71.8428
2.0269 5.3120 14.9345 44.1980 14.7710 24.2997 42.4866 76.0265
Table 2. Initial data and action values for solutions in Fig. 5 (φ = π/2) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.8, 0.8) (0.6, 0.6)∗ (0.4, 0.4)∗ (0.2, 0.2)∗
(0.5986, 0.1924, −0.6328) (0.6240, 0.2587, −0.5296) (0.6634, 0.3448, −0.4033) (0.7190, 0.4701, −0.2378)
(−0.3905i, 1.6392i, −0.9990i) (−0.2353i, 1.6294i, −0.8365i) (−0.0238i, 1.6138i, −0.6360i) (0.2751i, 1.5987i, −0.3748i)
4.6800 3.3097 2.0661 0.9562
4.6824 3.3108 2.0673 0.9567
4.7357 3.2839 2.0048 0.9074
Table 3. Initial data and action values for solutions in Fig. 6 (φ = π/2) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(1, 7) (3, 7) (5, 7) (7, 7) (1, 5) (3, 5) (5, 5) (1, 3) (3, 3) (1, 1)
(0.7963, 0.0831, −1.3777) (0.6888, −0.0789, −1.5144) (0.6077, −0.2013, −1.6292) (0.5523, −0.3051, −1.7300) (0.7449, 0.0956, −1.2227) (0.6177, −0.0931, −1.3875) (0.5315, −0.2280, −1.5179) (0.6782, 0.1148, −1.0225) (0.5249, −0.1148, −1.2302) (0.5792, 0.1403, −0.7195)
(−2.6569i, 0.6893i, −2.1683i) (−2.2898i, 1.3229i, −2.3910i) (−2.0423i, 1.8266i, −2.5749i) (−1.8303i, 2.2211i, −2.7353i) (−2.2097i, 0.8273i, −1.9269i) (−1.8298i, 1.5358i, −2.1895i) (−1.5863i, 2.0665i, −2.4007i) (−1.6030i, 1.0717i, −1.6121i) (−1.2207i, 1.8692i, −1.9454i) (−0.5204i, 1.6567i, −1.1363i)
22.5684 50.2584 74.8848 97.2712 17.9843 39.4550 58.1129 12.7300 27.2922 6.1754
22.5853 50.2830 74.8949 97.3335 17.9918 39.4844 58.1233 12.7406 27.2967 6.1783
23.4650 53.0809 79.6536 104.019 18.9882 42.6424 63.4343 13.7021 30.3876 6.3521
in the inequality (21), and the symbol Acoll stands for the lower bound estimate for action values over collision paths obtained in (19). There are certain cases which need to be handled with care. Theorem 1 is readily applied to cases where A(x) < Atest < Acoll . Those which fail to fulfill requirements in Theorem 1 are marked by either * or . They correspond respectively to (m 1 , m 2 )∗ :
Acoll < A(x) < Atest
(41)
(m 1 , m 2 ) :
A(x) < Acoll < Atest .
(42)
and
In our examples there are 14 cases of (41), among them there are 10 cases whose existence follow from Theorem 2 instead of Theorem 1. For such cases we may replace Acoll by A∗coll , the lower bound estimates for action values of collision paths obtained in (27). From the first graph in Fig. 20 it is quite apparent that our examples with m 1 = m 2 are all covered by Theorem 2. We list the values of A∗coll for these 10 cases in Table 17.
On Action-Minimizing Retrograde and Prograde Orbits
437
Table 4. Initial data and action values for solutions in Fig. 7 (φ = 2π/3) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.2, 0.8) (0.4, 0.8) (0.6, 0.8) (0.8, 0.8) (0.2, 0.6)∗ (0.4, 0.6) (0.6, 0.6) (0.2, 0.4)∗ (0.4, 0.4)∗ (0.2, 0.2)∗
(0.6865, 0.3135, −0.3881) (0.6281, 0.2348, −0.4391) (0.5817, 0.1696, −0.4847) (0.5434, 0.1139, −0.5259) (0.6741, 0.3309, −0.3334) (0.6111, 0.2429, −0.3902) (0.5607, 0.1729, −0.4401) (0.6554, 0.3485, −0.2705) (0.5867, 0.2501, −0.3347) (0.6231, 0.3617, −0.1970)
(−0.5258i, 1.1535i, −0.8176i) (−0.4306i, 1.3744i, −0.9273i) (−0.3613i, 1.5521i, −1.0249i) (−0.3093i, 1.7006i, −1.1130i) (−0.3076i, 1.2730i, −0.7023i) (−0.2076i, 1.5112i, −0.8237i) (−0.1468i, 1.6977i, −0.9305i) (−0.0292i, 1.4398i, −0.5701i) (0.0670i, 1.6989i, −0.7063i) (0.3719i, 1.7032i, −0.4150i)
2.5833 3.4582 4.2648 5.0204 2.1276 2.8988 3.6073 1.6369 2.2982 1.0945
2.5867 3.4628 4.2704 5.0256 2.1311 2.9047 3.6120 1.6394 2.3028 1.0961
2.5976 3.5223 4.3988 5.2344 2.1187 2.9219 3.6794 1.6090 2.2867 1.0606
Table 5. Initial data and action values for solutions in Fig. 8 (φ = 2π/3) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(1, 7) (3, 7) (5, 7) (7, 7) (1, 5) (3, 5) (5, 5) (1, 3) (3, 3) (1, 1)
(0.8193, 0.0461, −1.1421) (0.7026, −0.1211, −1.2602) (0.6259, −0.2529, −1.3595) (0.5644, −0.3579, −1.4456) (0.7528, 0.0523, −1.0141) (0.6231, −0.1428, −1.1556) (0.5364, −0.2829, −1.2674) (0.6686, 0.0602, −0.8492) (0.5140, −0.1716, −1.0268) (0.5334, 0.0653, −0.5987)
(−2.5020i, 0.7013i, −2.4073) (−2.1812i, 1.3163i, −2.6706i) (−1.9253i, 1.7876i, −2.8867i) (−1.7419i, 2.1800i, −3.0670i) (−2.0771i, 0.8440i, −2.1428i) (−1.7254i, 1.5255i, −2.4516i) (−1.4950i, 2.0334i, −2.6918i) (−1.4832i, 1.0929i, −1.7955i) (−1.1400i, 1.8655i, −2.1764i) (−0.4353i, 1.7032i, −1.2679i)
23.1241 49.1779 72.2105 93.2918 18.4808 38.7838 56.4443 13.1597 27.1144 6.5328
23.1600 49.1903 72.3221 93.4163 18.4997 38.8214 56.4514 13.1775 27.1280 6.5436
23.8356 51.4231 76.1899 98.9096 19.2760 41.3764 60.8186 14.1331 30.1992 6.9464
Table 6. Initial data and action values for solutions in Fig. 9 (φ = 3π/4) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.2, 0.8) (0.4, 0.8) (0.6, 0.8) (0.8, 0.8) (0.2, 0.6) (0.4, 0.6) (0.6, 0.6) (0.2, 0.4)∗ (0.4, 0.4) (0.2, 0.2)∗
(0.6683, 0.2823, −0.3595) (0.6102, 0.2038, −0.4071) (0.5647, 0.1387, −0.4497) (0.5274, 0.0832, −0.4885) (0.6515, 0.2974, −0.3087) (0.5879, 0.2106, −0.3615) (0.5401, 0.1406, −0.4084) (0.6282, 0.3121, −0.2505) (0.5601, 0.2154, −0.3102) (0.5901, 0.3222, −0.1825)
(−0.4735i, 1.1856i, −0.8538i) (−0.3842i, 1.4050i, −0.9704i) (−0.3179i, 1.5808i, −1.0739i) (−0.2686i, 1.7273i, −1.1669i) (−0.2599i, 1.3092i, −0.7336i) (−0.1688i, 1.5491i, −0.8620i) (−0.1072i, 1.7318i, −0.9747i) (0.0157i, 1.4811i, −0.5956i) (0.1052i, 1.7422i, −0.7389i) (0.4133i, 1.7532i, −0.4333i)
2.7193 3.5998 4.4136 5.1730 2.2416 3.0265 3.7445 1.7270 2.4074 1.1585
2.7253 3.6077 4.4216 5.1832 2.2468 3.0325 3.7517 1.7306 2.4126 1.1614
2.7567 3.7073 4.6084 5.4677 2.2506 3.0810 3.8644 1.7119 2.4186 1.1323
Table 7. Initial data and action values for solutions in Fig. 10 (φ = 3π/4) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(1, 7) (3, 7) (5, 7) (7, 7) (1, 5) (3, 5) (5, 5) (1, 3) (3, 3) (1, 1)
(0.8366, 0.0314, −1.0566) (0.7193, −0.1411, −1.1705) (0.6378, −0.2750, −1.2642) (0.5751, −0.3828, −1.3461) (0.7639, 0.0351, −0.9395) (0.6311, −0.1639, −1.0738) (0.5450, −0.3088, −1.1812) (0.6702, 0.0391, −0.7874) (0.5162, −0.1977, −0.9555) (0.5193, 0.0369, −0.5562)
(−2.4242i, 0.7065i, −2.5211i) (−2.1130i, 1.3068i, −2.8082i) (−1.8757i, 1.7738i, −3.0381i) (−1.6984i, 2.1598i, −3.2301i) (−2.0087i, 0.8507i, −2.2449i) (−1.6764i, 1.5214i, −2.5777i) (−1.4487i, 2.0154i, −2.8334i) (−1.4279i, 1.1037i, −1.8833i) (−1.0958i, 1.8597i, −2.2914i) (−0.3985i, 1.7280i, −1.3295i)
23.3797 48.5301 70.9024 91.2663 18.6978 38.3957 55.4811 13.3502 26.9694 6.6987
23.3906 48.5574 70.9183 91.3112 18.7093 38.4204 55.5211 13.3647 26.9926 6.7093
23.9582 50.5045 74.3437 96.2175 19.3712 40.6749 59.4243 14.1970 29.7414 7.2245
438
K.-C. Chen, Y.-C. Lin Table 8. Initial data and action values for solutions in Fig. 11 (φ = 4π/5)
(m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.2, 0.8) (0.4, 0.8) (0.6, 0.8) (0.8, 0.8) (0.2, 0.6) (0.4, 0.6) (0.6, 0.6) (0.2, 0.4)∗ (0.4, 0.4) (0.2, 0.2)∗
(0.6596, 0.2659, −0.3447) (0.6019, 0.1873, −0.3906) (0.5567, 0.1224, −0.4320) (0.5199, 0.0670, −0.4695) (0.6401, 0.2799, −0.2959) (0.5780, 0.1932, −0.3471) (0.5303, 0.1235, −0.3923) (0.6146, 0.2930, −0.2401) (0.5472, 0.1971, −0.2977) (0.5735, 0.3016, −0.1750)
(−0.4437i, 1.2046i, −0.8749i) (−0.3569i, 1.4232i, −0.9958i) (−0.2937i, 1.5984i, −1.1025i) (−0.2461i, 1.7444i, −1.1986i) (−0.2335i, 1.3308i, −0.7518i) (−0.1430i, 1.5691i, −0.8843i) (−0.0846i, 1.7525i, −1.0007i) (0.0413i, 1.5056i, −0.6105i) (0.1273i, 1.7677i, −0.7580i) (0.4365i, 1.7828i, −0.4439i)
2.7988 3.6824 4.4985 5.2593 2.3084 3.0984 3.8225 1.7797 2.4695 1.1956
2.8064 3.6923 4.5097 5.2748 2.3144 3.1071 3.8331 1.7840 2.4769 1.1995
2.8494 3.8150 4.7305 5.6036 2.3274 3.1737 3.9721 1.7718 2.4953 1.1741
Table 9. Initial data and action values for solutions in Fig. 12 (φ = 4π/5) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(1, 7) (3, 7) (5, 7) (7, 7) (1, 5) (3, 5) (5, 5) (1, 3) (3, 3) (1, 1)
(0.8522, 0.0231, −1.0141) (0.7315, −0.1527, −1.1255) (0.6477, −0.2887, −1.2178) (0.5847, −0.3994, −1.2971) (0.7752, 0.0253, −0.9016) (0.6384, −0.1766, −1.0323) (0.5518, −0.3241, −1.1383) (0.6750, 0.0273, −0.7569) (0.5195, −0.2126, −0.9206) (0.5133, 0.0214, −0.5347)
(−2.3680i, 0.7080i, −2.5878i) (−2.0716i, 1.3007i, −2.8900i) (−1.8437i, 1.7637i, −3.1278i) (−1.6670i, 2.1427i, −3.3301i) (−1.9592i, 0.8529i, −2.3053i) (−1.6440i, 1.5175i, −2.6552i) (−1.4210i, 2.0049i, −2.9191i) (−1.3906i, 1.1083i, −1.9343i) (−1.0697i, 1.8559i, −2.3587i) (−0.3763i, 1.7429i, −1.3666i)
23.4679 48.0762 69.9367 89.9079 18.7932 38.1218 54.8522 13.4283 26.8477 6.7864
23.5160 48.1552 70.0436 90.0063 18.8251 38.1621 54.9375 13.4702 26.8987 6.8050
24.0139 49.9255 73.1992 94.5573 19.4144 40.2327 58.5601 14.2260 29.4530 7.3865
Table 10. Initial data and action values for solutions in Fig. 13 (φ = 4π/5) (m 1 , m 2 ) Initial Position
Initial Velocity
(0.01, 4) (0.1, 4) (0.5, 4) (0.8, 4) (0.01, 1) (0.1, 1) (0.5, 1) (0.8, 1)
(−1.9263i, 0.4705i, −1.8627i) 6.5852 (−1.9048i, 0.5208i, −1.8927i) 7.5217 (−1.8128i, 0.7291i, −2.0102i) 11.5279 (−1.7543i, 0.8726i, −2.0869i) 14.3917 (−0.7236i, 0.8704i, −0.8631i) 2.2552 (−0.6678i, 0.9894i, −0.9227i) 2.7424 (−0.4991i, 1.3968i, −1.1472i) 4.6924 (−0.4177i, 1.6184i, −1.2842i) 5.9801
(0.8464, 0.1833, −0.7417) (0.8334, 0.1669, −0.7508) (0.7816, 0.0998, −0.7899) (0.7460, 0.0550, −0.8170) (0.7389, 0.3361, −0.3435) (0.7072, 0.2944, −0.3652) (0.5994, 0.1492, −0.4489) (0.5438, 0.0677, −0.5028)
A(x)
Atest
Acoll
6.5915 7.5311 11.5464 14.4082 2.2570 2.7487 4.7027 5.9982
6.5966 7.5846 11.8468 14.9185 2.2596 2.7806 4.9462 6.4401
Table 11. Initial data and action values for solutions in Fig. 14 (φ = 4π/5) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
(1, 100) (10, 100) (50, 100) (100, 100)
(1.93617, 0.00572, −2.50767) (1.82450, −0.15637, −2.60800) (1.48170, −0.71128, −2.95672) (1.22396, −1.19099, −3.29748)
(−7.06757, 0.13386, −6.31863) 137.950 138.0457 138.2022 (−6.70235, 0.73687, −6.66520) 821.639 822.4731 824.3510 (−5.46436, 2.80847, −7.62894) 3497.41 3499.1535 3511.6142 (−4.51049, 4.59515, −8.46580) 6305.41 6310.1779 6337.2166
Atest
Acoll
For most of the cases we have numerical action values A(x) of true action minimizers which are fairly close to Atest , and Acoll often stays well above action values of our test path. This suggests that action-minimizers tend to be Keplerian-like for both the pair m 1 , m 1 and the pair m 3 , m 1 + m 2 . Among the examples we have, there are 3 cases of type (42), for which the relative motions of m 1 and m 2 are less like Keplerian orbits.
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Table 12. Initial data and action values for solutions in Fig. 15 (φ = π ) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.2, 0.8) (0.4, 0.8) (0.6, 0.8) (0.8, 0.8) (0.2, 0.6) (0.4, 0.6) (0.6, 0.6) (0.2, 0.4) (0.4, 0.4) (0.2, 0.2)∗
(0.6387, 0.2122, −0.2975) (0.5822, 0.1326, −0.3390) (0.5382, 0.0673, −0.3768) (0.5016, 0.0115, −0.4104) (0.6116, 0.2231, −0.2562) (0.5511, 0.1362, −0.3021) (0.5043, 0.0663, −0.3423) (0.5758, 0.2323, −0.2081) (0.5112, 0.1369, −0.2592) (0.5231, 0.2354, −0.1517)
(−0.3249i, 1.2793i, −0.9584i) (−0.2495i, 1.4969i, −1.0977i) (−0.1966i, 1.6709i, −1.2188i) (−0.1571i, 1.8181i, −1.3288i) (−0.1277i, 1.4129i, −0.8222i) (−0.0484i, 1.6529i, −0.9724i) (0.0020i, 1.8395i, −1.1049i) (0.1322i, 1.6018i, −0.6672i) (0.2109i, 1.8702i, −0.8324i) (0.5179i, 1.9034i, −0.4843i)
3.1012 3.9938 4.8164 5.5925 2.5582 3.3703 4.1201 1.9766 2.7047 1.3372
3.1226 4.0224 4.8525 5.6305 2.5776 3.3978 4.1501 1.9912 2.7266 1.3473
3.2024 4.2254 5.1956 6.1211 2.6199 3.5267 4.3824 1.9999 2.7878 1.3331
Table 13. Initial data and action values for solutions in Fig. 16 (φ = π ) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(1, 7) (3, 7) (5, 7) (7, 7) (1, 5) (3, 5) (5, 5) (1, 3) (3, 3) (1, 1)
(0.9220, −0.0068, −0.8741) (0.7913, −0.1990, −0.9812) (0.7009, −0.3474, −1.0727) (0.6328, −0.4683, −1.1520) (0.8267, −0.0096, −0.7789) (0.6825, −0.2282, −0.9062) (0.5892, −0.3875, −1.0086) (0.7032, −0.0157, −0.6561) (0.5415, −0.2703, −0.8137) (0.5015, −0.0330, −0.4686)
(−2.1389i, 0.7162i, −2.8744i) (−1.9004i, 1.2788i, −3.2504i) (−1.7008i, 1.7190i, −3.5293i) (−1.5428i, 2.0802i, −3.7615i) (−1.7628i, 0.8653i, −2.5638i) (−1.5003i, 1.4969i, −2.9836i) (−1.3056i, 1.9644i, −3.2936i) (−1.2328i, 1.1296i, −2.1559i) (−0.9641i, 1.8481i, −2.6519i) (−0.2861i, 1.8041i, −1.5180i)
23.8531 46.2670 66.1712 84.3250 19.1647 36.9147 52.3279 13.7845 26.3570 7.1300
24.0013 46.4415 66.3543 84.5245 19.2727 37.0432 52.4590 13.8756 26.4648 7.1751
24.1111 47.3987 68.3333 87.5574 19.4900 38.3030 54.8854 14.2767 28.1940 8.0032
Table 14. Initial data and action values for solutions in figure 17 (φ = π ) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.01, 4) (0.1, 4) (0.5, 4) (0.8, 4) (0.01, 1) (0.1, 1) (0.5, 1) (0.8, 1)
(0.8967, 0.1574, −0.6386) (0.8833, 0.1396, −0.6468) (0.8272, 0.0670, −0.6816) (0.7917, 0.0183, −0.7065) (0.7274, 0.2887, −0.2960) (0.6940, 0.2451, −0.3145) (0.5871, 0.0966, −0.3902) (0.5318, 0.0139, −0.4394)
(−1.6680i, 0.5064i, −2.0088i) (−1.6576i, 0.5546i, −2.0526i) (−1.6039i, 0.7542i, −2.2150i) (−1.5567i, 0.8902i, −2.3154i) (−0.5776i, 0.9366i, −0.9308i) (−0.5304i, 1.0571i, −1.0040i) (−0.3866i, 1.4605i, −1.2672i) (−0.3200i, 1.6803i, −1.4243i)
7.6246 8.4913 12.2146 14.8662 2.6077 3.0955 5.0356 6.3229
7.6281 8.5276 12.3038 14.9676 2.6114 3.1115 5.0710 6.3670
7.6266 8.5214 12.3835 15.1680 2.6155 3.1627 5.4377 7.0081
Table 15. Initial data and action values for solutions in Fig. 18 (φ = π ) (m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(10, 100) (2.0561, −0.1833, −2.2351) (−6.26188, 0.7013, −7.5127) 746.294 747.3566 747.0770 (50, 100) (1.6666, −0.8074, −2.5932) (−5.13402, 2.6533, −8.6320) 3121.82 3122.6917 3129.5061 (100, 100) (1.3752, −1.3459, −2.9376) (−4.24437, 4.3405, −9.6090) 5616.38 5617.1174 5634.9039
Even though Theorem 1 does not apply to these cases, one can still choose test paths carefully so that their action values are between A(x) and Acoll . In other words, we can still establish existence of these orbits by selecting test paths that are sufficiently close to true action-minimizers and by making use of (19). Similar to the cases of retrograde orbits, on Table 18 of numerical data for prograde orbits displayed in Figure 21, A(x) represents numerical action values of orbits obtained in Theorem 3, Atest denotes action values of the test path (31) given by the lower bound
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K.-C. Chen, Y.-C. Lin Table 16. Initial data and action values for solutions in Fig. 19 (φ = 5π/4)
(m 1 , m 2 )
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(0.5, 0.5) (1, 1) (2, 2) (4, 4)
(0.4918, 0.0426, −0.2672) (0.5089, −0.0918, −0.4170) (0.5533, −0.2491, −0.6083) (0.6338, −0.4240, −0.8392)
(0.2015, 1.9829, −1.0922) (−0.1628, 1.8973, −1.7345) (−0.5640, 1.8414, −2.5549) (−0.9910, 1.8774, −3.5457)
3.7236 7.5066 15.9065 36.5253
3.8157 7.7055 16.2605 37.1206
3.9768 7.9717 16.5537 37.4317
Table 17. Cases of type (41) with m = m 1 = m 2 from Tables 1-12. Here A∗coll represents the lower bound in (27) (m, φ)
(0.5, π/4)
(1, π/4)
(0.2, π/2)
(0.4, π/2)
(0.6, π/2)
A∗coll
2.6222
7.3923
1.0147
2.3994
4.1143
(m, φ)
(0.2, 2π/3)
(0.4, 2π/3)
(0.2, 3π/4)
(0.2, 4π/5)
(0.2, π )
A∗coll
1.1679
2.6813
1.2396
1.2813
1.4404
Table 18. Initial data and action values for prograde solutions in Fig. 21 (m, η)
Initial Position
Initial Velocity
A(x)
Atest
Acoll
(1.5, 1.25) (2, 1.25) (1, 1) (1.5, 1) (2, 1) (0.5, 0.5) (1, 0.5) (1.5, 0.5) (2, 0.5) (0.5, 0.25) (1, 0.25) (1.5, 0.25) (2, 0.25)
(0.27877, 0.04165, −0.48063) (0.27456, 0.00280, −0.55471) (0.34357, 0.10696, −0.45053) (0.32849, 0.04507, −0.56035) (0.32038, 0.00197, −0.64470) (0.59029, 0.34678, −0.46853) (0.51250, 0.20293, −0.71543) (0.47412, 0.11700, −0.88667) (0.45209, 0.05747, −1.01911) (0.87638, 0.60624, −0.74131) (0.73855, 0.39467, −1.13322) (0.66528, 0.27024, −1.40328) (0.62206, 0.18484, −1.61379)
(2.62717i, −1.44573i, −1.77216i) (2.66332i, −1.64288i, −2.04088i) (2.30073i, −0.93097i, −1.36976i) (2.33063i, −1.20013i, −1.69575i) (2.38571i, −1.40988i, −1.95166i) (1.77698i, −0.31495i, −0.73102i) (1.85867i, −0.74190i, −1.11677i) (1.93521i, −1.01320i, −1.38302i) (2.01149i, −1.21673i, −1.58952i) (1.54970i, −0.38575i, −0.58198i) (1.64887i, −0.76113i, −0.88774i) (1.74023i, −1.00711i, −1.09968i) (1.81864i, −1.18703i, −1.26322i)
18.0564 26.5282 9.7838 16.5371 24.4253 3.0921 7.5222 13.0854 19.6905 2.3955 6.1544 11.0275 16.8920
18.1397 26.6137 9.8075 16.5716 24.4555 3.0955 7.5254 13.0924 19.6961 2.3961 6.1587 11.0366 16.9142
18.5481 27.6994 10.0834 17.5727 26.4922 3.2175 8.4324 15.3226 23.7071 2.6222 7.3923 13.9051 21.9526
in (21), and Acoll stands for the lower bound estimates for action values of collision paths obtained in (30). We list initial data to the fifth decimal place because of the instability of many action-minimizing prograde orbits. Acknowledgement. This research work is partly supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan.
References 1. Barutello, V., Ferrario, D., Terracini, S.: Symmetry groups of the planar three-body problem and actionminimizing trajectories. Arch. Ration. Mech. Anal. 190, 189–226 (2008) 2. Broucke, R.: On relative periodic solutions of the planar general three-body problem. Celes. Mech. 12, 439–462 (1975) 3. Chen, K.-C.: Binary decompositions for the planar N -body problem and symmetric periodic solutions. Arch. Ration. Mech. Anal. 170, 247–276 (2003) 4. Chen, K.-C.: Removing collision singularities from action minimizers for the N -body problem with free boundaries. Arch. Ration. Mech. Anal. 181, 311–331 (2006) 5. Chen, K.-C.: Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses. Annals of Math. 167, 325–348 (2008) 6. Chen, K.-C.: Variational constructions for some satellite orbits in periodic gravitational force fields. Amer. J. Math. (2009, to appear)
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7. Chenciner, A.: Symmetries and “Simple” Solutions of the Classical N-body Problem. Proceeding of ICMP, Lisbone, 2003, River Edge, NJ, World Scientific, 2006 8. Chenciner, A., Féjoz, J., Montgomery, R.: Rotating eights I. Nonlinearity 18, 1407–1424 (2005) 9. Chenciner, A., Montgomery, R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. 152, 881–901 (2000) 10. Ferrario, D., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155, 305–362 (2004) 11. Gordon, W.: A minimizing property of Keplerian orbits. Amer. J. Math. 99, 961–971 (1977) 12. Hénon, M.: A family of periodic solutions of the planar three-body problem, and their stability. Celes. Mech. 13, 267–285 (1976) 13. http://www.math.nthu.edu.tw/~kchen/retrograde.html 14. Meyer, K.R.: Periodic Solutions of the N -body Problem. Lecture Notes in Mathematics, Vol. 1719, Berlin: Springer-Verlag, 1999 15. Marchal, C.: The Three-Body Problem. Studies in Astronautics, 4. Amsterdam: Elsevier Science Publishers, B.V., 1990 16. Moeckel, R.: A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete Contin. Dyn. Syst. Ser. B 10, 609–620 (2008) 17. Montgomery, R.: The N -body problem, the braid group, and action-minimizing periodic solutions. Nonlinearity 11, 363–376 (1998) 18. Moore C.: Braids in classical dynamics. Phy. Rev. Lett. 70, 3675–3679 (1993) 19. Palais, R.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979) 20. Schubart, J.: Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem. Astr. Nachr. 283, 17–22 (1956) 21. Simó, C.: Choreographic solutions of the planar three body problem. http://www.maia.ub.es/dsg/3body. html 22. Szebehely, V.: Theory of Orbits – The Restricted Problem of Three Bodies. New York: Academic Press, 1967 23. Venturelli, A.: A variational proof of the existence of von Schubart’s orbit. Discrete Contin. Dyn. Syst. Ser. B 10, 699–717 (2008) Communicated by G. Gallavotti
Commun. Math. Phys. 291, 443–471 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0841-1
Communications in
Mathematical Physics
On the ‘Stationary Implies Axisymmetric’ Theorem for Extremal Black Holes in Higher Dimensions Stefan Hollands1 , Akihiro Ishibashi2,3 1 School of Mathematics, Cardiff University, Cardiff, United Kingdom.
E-mail: [email protected]
2 Cosmophysics Group, Institute of Particle and Nuclear Studies,
KEK, Tsukuba, Japan. E-mail: [email protected]
3 Perimeter Institute for Theoretical Physics, Waterloo, Canada
Received: 13 October 2008 / Accepted: 18 February 2009 Published online: 12 June 2009 – © Springer-Verlag 2009
Abstract: All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gravity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [24] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einstein’s equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain “diophantine condition,” which holds except for a set of measure zero. 1. Introduction In a recent paper [24], we proved the following two statements about stationary, asymptotically flat, analytic black hole solutions to the vacuum or electrovacuum Einstein equations with a non-degenerate (non-extremal) event horizon for general spacetime dimension n ≥ 4: (i) The event horizon is in fact a Killing horizon, and (ii) if it is rotating, then the spacetime must also be axisymmetric. Property (i) establishes the zeroth law of black hole thermodynamics as the surface gravity must be constant over a Killing horizon. Property (ii) may be viewed as a “symmetry enhancement” theorem, as it shows that such black holes must have at least one more symmetry than originally assumed. Statements (i) and (ii) are often referred to as the rigidity theorem, since they imply in particular that the horizon must be rotating rigidly relative to infinity. An alternative proof of these statements was given by Moncrief and Isenberg [35], based on previous work by these authors on symmetries of Cauchy horizons [34]. The rigidity theorem was originally proved for n = 4 dimensions by [4,13,22, 23], and it plays a critical role in the proof of the black hole uniqueness theorem [1,3,29,30,33,46] for stationary (electro-)vacuum black hole solutions in n = 4
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dimensions1 . In higher dimensions, the uniqueness theorem no longer holds as it stands. A variety of explicit stationary black hole solutions have been constructed in recent years but their complete classification is still a major open problem2 . Properties (i) and (ii) therefore place an important restriction on such black hole solutions in n > 4. The purpose of the present paper is to establish a version of the rigidity theorem also for the case of degenerate (extremal) black holes. This case corresponds to a vanishing Hawking temperature and is of particular physical importance e.g. for the investigation of the quantum properties of black holes in string theory. Neither the proof [24] nor [35] appears to carry over straightforwardly to the extremal case. To give an indication of the problems that arise, let us first recall the basic strategy of proof employed in [24]. By assumption, there is a stationary Killing vector field, t a , which is tangent to the horizon, but not null on the horizon if the latter is rotating. The key step in the proof is to construct another Killing field K a which is null on the horizon. This is obtained in turn by finding a distinguished foliation of a neighborhood of the horizon by (n − 2)-dimensional cross sections. To determine that special foliation, one needs to integrate a certain ordinary differential equation along the orbits of the projection s a of t a onto an arbitrary horizon cross-section, . If the orbits of s a close on , then the integration of this differential equation is straightforward. In n = 4, the cross section is topologically a two-sphere by the topology theorem [7,23], implying that the orbits of s a must necessarily close. But in higher dimensions, the orbits need not be closed and can in fact be dense on . Nevertheless, if the horizon is non-degenerate, then a solution to the desired ordinary differential equation can be obtained using basic methods from ergodic theory. Unfortunately, this method of constructing the desired solution does not seem to generalize straightforwardly to the case of degenerate horizons. In this paper, we therefore use a different argument which is basically as follows. First, we argue that we can decompose s a = 1 ψ1a + · · · + N ψ Na locally on , where N ≥ 1 and where the vector fields ψia commute and have closed orbits with period 2π . The constants i can be viewed as a local definition of the angular velocities of the horizon. We now make a Fourier decomposition of the quantities involved in our differential equation on the N -tori T N ⊂ generated by the N vector fields ψia . If this is done, then we can construct the desired solution to our differential equation provided the vector = (1 , . . . , N ) satisfies | · m| > || · |m|−q for some number q and for all but finitely many m ∈ Z N . We refer to this condition on the angular velocities as a “diophantine condition.” It is satisfied for all except for a set of measure zero. In summary, if the diophantine condition holds, then we can complete the proof of statements (i) and (ii)—i.e. the rigidity theorem—in the degenerate case. We are unsure whether this condition is a genuine restriction or an artefact of our method of proof. Our paper is organized as follows. In Sect. 2, we prove statement (i) and (ii) in the extremal case for vacuum black holes. In Sect. 3, we extend these results to include matter fields. The matter fields that we consider consist of a multiplet of scalar fields and abelian gauge fields with a fairly general action, including typical actions characteristic for many supergravity theories. As a by-product, we also generalize our previous 1 An alternative strategy to prove this result bypassing the rigidity theorem was recently proposed in [27]. However, this argument relies on certain restrictive extra assumptions on the geometry besides stationarity and asymptotic flatness. 2 For a partial classification see [25,26] and also [20,21].
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results in the non-extremal case [24] to such theories. The rigidity theorem for theories with an additional Chern-Simons term in the action is proved for a typical example in Appendix C. In Sect. 4 we briefly discuss further the nature of the diophantine condition. The decomposition of Einstein’s equation used in the main part of the paper is given in Appendices A and B. Our signature convention for gab is (−, +, +, · · · ). The Riemann tensor is defined by Rabc d kd = 2∇[a ∇b] kc and the Ricci tensor by Rab = Racb c . We also set 8π G = 1. 2. Proof of the Rigidity Theorem in the Vacuum Case Let (M, gab ) be an n-dimensional, smooth, asymptotically flat stationary solution to the vacuum Einstein equation containing a black hole. Thus, we assume the existence in the spacetime of a Killing field t a with complete orbits which are timelike near infinity, we assume that the infinity I = I − ∪ I + has standard topology I ± ∼ = R × S n−2 , and that the domain of outer communication J − (I + ) ∩ J + (I − ) is globally hyperbolic. Let H denote the portion of the event horizon of the black hole that lies to the future of past null infinity. We assume that H has topology R × , where is compact and connected3 , and that t a is not everywhere tangent (and hence normal) to the null generators of H . The event horizon H is mapped into itself by a one-parameter group of isometries generated by t a . Following our earlier paper [24], and work of Isenberg and Moncrief [28,34], our main aim in this section is to prove that there exists a vector field K a defined in a neighborhood of H which is normal to H and on H satisfies L L · · · L (L K gab ) = 0, m = 0, 1, 2, . . . ,
(1)
m times
where is an arbitrary vector field transverse to H . As we shall show at the end of this section, if we assume analyticity of gab and of H it follows that K a is a Killing field. We shall proceed by constructing a candidate Killing field, K a , and then proving that Eq. (1) holds for K a . This candidate Killing field is expected to satisfy the following properties: (i) K a should be normal to H . (ii) If we define S a by Sa = t a − K a
(2)
then, on H , S a should be tangent to cross-sections4 of H . (iii) K a should commute with t a . (iv) K a should have constant surface gravity on H , i.e., on H we should have K a ∇a K b = κ K b with κ constant on H , since, by the zeroth law of black hole mechanics, this property is known to hold on any Killing horizon in any vacuum solution of Einstein’s equation. We begin by choosing a cross-section , of H . By arguments similar to those given in the proof of Proposition 4.1 of [6], we may assume without loss of generality that has been chosen so that each orbit of t a on H intersects at precisely one point, so that t a is everywhere transverse to . We extend to a foliation, (u), of H by the action of the time translation isometries, i.e., we define (u) = φu (), where φu denotes the 3 If is not connected, our arguments can be applied to any connected component of . 4 Note that, since H is mapped into itself by the time translation isometries, t a must be tangent to H , so S a is automatically tangent to H . Condition (iii) requires that there exist a foliation of H by cross-sections (u) such that each orbit of S a is contained in a single cross-section.
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one-parameter group of isometries generated by t a . Note that the function u on H that labels the cross-sections in this foliation automatically satisfies Lt u = 1.
(3)
t a = na + s a ,
(4)
Next, we define n a and s a on H by
where n a is normal to H and s a is tangent to (u). It follows from the transversality of t a that n a is everywhere non-vanishing and future-directed. Note also that Ln u = 1 on H . Our strategy is to extend this definition of n a to a neighborhood of H via Gaussian null coordinates. This construction of n a obviously satisfies conditions (i) and (ii) above, and it also will be shown below that it satisfies condition (iii). However, it will, in general, fail to satisfy (iv). We shall then modify our foliation so as to produce a new foliation ˜ u) ( ˜ so that (iv) holds as well. We will then show that the corresponding K a = n˜ a satisfies Eq. (1). Given our choice of (u) and the corresponding choice of n a on H , we can uniquely define a past-directed null vector field a on H by the requirements that n a a = 1, and that a is orthogonal to each (u). Let r denote the affine parameter on the null geodesics determined by a , with r = 0 on H . Let x A = (x 1 , . . . , x n−2 ) be local coordinates on an open subset of . We extend these coordinates to an open neighborhood of H by demanding that they be constant along the orbits of n a and of a . The coordinates (u, r, x A ) that are constructed in this manner are referred to as Gaussian null coordinates. If we cover with an atlas of charts, then we obtain a corresponding atlas of Gaussian null coordinates covering an open neighborhood of H . The metric takes the form ds 2 = 2(dr − r αdu − rβ A dx A )du + γ AB dx A dx B .
(5)
βa = β A (dx A )a , γab = γ AB (dx A )a (dx B )b ,
(6)
We write
and we note that βa , γab are tensor fields that are intrinsically defined in a neighborhood of H , independent of the choice of coordinates x A on . Both these tensor fields are by definition orthogonal to n a and a , meaning βa n a = βa a = 0 and γab n a = γab a = 0. It follows from the definition of u and r that Lt u = 1, Lt r = 0,
(7)
Lt n a = 0, Lt a = 0.
(8)
Lt α = 0, Lt βa = 0, Lt γab = 0.
(9)
and that
It can also be shown that
We also have
na =
∂ ∂u
a
, a =
∂ ∂r
a ,
(10)
Theorem for Extremal Black Holes in Higher Dimensions
447
and n a and a commute in particular. Thus, we see that in Gaussian null coordinates the spacetime metric, gab , is characterized by the quantities α, βa , and γab . In terms of these quantities, if we were to choose K a = n a , then the condition (1) will hold if and only if the conditions L L · · · L (Ln γab ) = 0, m times
L L · · · L (Ln α) = 0,
(11)
m times
L L · · · L (Ln βa ) = 0, m times
hold on H . The next step in the analysis is to use the Einstein equation Rab n a n b = 0 on H , in a manner completely in parallel with the 4-dimensional case [23]. This equation is precisely the Raychaudhuri equation for the congruence of null curves defined by n a on H . It yields Ln γab = 0. Thus, the first equation in Eq. (11) holds with m = 0. However, n a in general fails to satisfy condition (iv) above. Indeed, from the form, Eq. (5), of the metric, we see that the surface gravity, κ, associated with n a is simply α, and there is no reason why α need be constant on H . Since Ln γab = 0 on H , the Einstein equation (94) of Appendix A on H yields Da α =
1 Ln βa , 2
(12)
where Da denotes the derivative operator on (u), i.e., Da α = γa b ∇b α. Thus, if α is not constant on H , then the last equation in Eq. (11) fails to hold even when m = 0. As previously indicated, our strategy is to repair this problem by choosing a new ˜ so that the corresponding n˜ a arising from the Gaussian normal coordicross-section ˜ nate construction will have constant surface gravity on H . The determination of this requires some intermediate constructions, to which we now turn. First, since we already know that Lt γab = 0 everywhere and that Ln γab = 0 on H , it follows immediately from the fact that t a = s a + n a on H that Ls γab = 0
(13)
on H (for any choice ). Thus, s a is a Killing vector field for the Riemannian metric γab on . Therefore the flow, φˆ τ : → of s a yields a one-parameter group of isometries of γab , which coincides with the projection of the flow φu of the original Killing field t a to . Furthermore, using that Lt βa = 0, it similarly follows that 1 Da α = − Ls βa 2 on H . We next define κ=
1 Area()
(14)
α dV,
(15)
where dV is the volume element on defined from γab . In our previous paper [24], we assumed that κ = 0, i.e., that the horizon is non-degenerate. Here, we assume that the horizon is degenerate, κ = 0.
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S. Hollands, A. Ishibashi
˜ of the We seek a new Gaussian null coordinate system based on a new choice initial cross section such that the corresponding fields u, ˜ r˜ , x˜ A , α, ˜ β˜a , γ˜ab satisfy all the above properties together with the additional requirement that α˜ = 0, i.e., constancy of the surface gravity. Let us determine the conditions that these new coordinates would have to satisfy. Since clearly n˜ a must be proportional to n a , we have n˜ a = f n a ,
(16)
for some positive function f . Since Lt n˜ a = Lt n a = 0, we must have Lt f = 0. Since on H we have n a ∇a n b = αn b , and since α˜ is given by n˜ a ∇a n˜ b = α˜ n˜ b ,
(17)
α˜ = Ln f + α f = −Ls f + α f = 0.
(18)
we find that f must satisfy The last equality provides an equation that must be satisfied by f on . Writing F = log f , this equation may be written alternatively in the form Ls F = α.
(19)
Lt u˜ = 1,
(20)
The new coordinate u˜ must satisfy
as before. However, in view of Eq. (16), it also must satisfy Ln u˜ = n a ∇a u˜ =
1 a 1 n˜ ∇a u˜ = . f f
(21)
Since n a = t a − s a , we find that on , u˜ must satisfy Ls u˜ = 1 − e−F .
(22)
Thus, if our new Gaussian null coordinates exist, there must exist smooth solutions to Eqs. (19) and (22), and conversely, any solution to these equations will give us the desired new set of Gaussian null coordinates. It is not difficult to show that there is always an analytic solution F to Eq. (19). To see this, we let F be a solution to the Laplace equation 1 D a Da F = − D a βa . (23) 2 This solution indeed exists, because the function on the right side has vanishing integral over by Gauss’s theorem, as has no boundary. Hence, the right side is orthogonal in L 2 (, d V ) to the one dimensional subspace of constant functions and hence in the image of the Laplace operator. By standard elliptic regularity results for the Laplace operator on a compact manifold, F is smooth and even analytic if βa and (, γab ) are analytic. Using that the Lie-derivative Ls commutes with Da , using Eq. (23), and using the Einstein equation (14), we calculate that 1 D aDa(Ls F − α) = Ls D aDaF − D aDa α = − LsD a βa − D aDa α 2 1 = D a − Ls βa − Da α = 0. 2
(24)
Theorem for Extremal Black Holes in Higher Dimensions
449
Thus F satisfies the desired Eq. (19), up to a term annihilated by D a Da , i.e. Ls F = α+C, where C ∈ R. But we have shown in our previous paper that 1 T lim α ◦ φˆ τ (x) dτ = κ = 0 (25) T →∞ T 0 from which it follows that
1 T C = lim Ls F ◦ φˆ τ (x) dτ T →∞ T 0 1 T d = lim F ◦ φˆ τ (x) dτ T →∞ T 0 dτ 1 = lim F(φˆ T (x)) − F(x) = 0. T →∞ T
(26)
Thus, we have constructed a solution F to Eq. (19). We are free to add to our solution F any function F ∗ on H with the property that Ls F ∗ = 0. We take 1 T ∗ exp[−F (x)] = lim exp[−F ◦ φˆ τ (x)] dτ, (27) T →∞ T 0 where the limit exists by the ergodic theorem [51], since φˆ τ are isometries of γab and hence in particular area-preserving. Again by the ergodic theorem, the right side can also be written as the integral over the closure of the orbit of φˆ τ . Using precisely the same arguments as below in the proof of Lemma 15 , it is possible to show that F ∗ is analytic. By replacing F with F − F ∗ if necessary, we can hence achieve that our solu∗ tion F to Eq. (19) satisfies Eq. (27) with e−F = 1. This will turn out to be convenient momentarily, as the orbit average of the source term in Eq. (22) then vanishes. We now turn to Eq. (22). We note that this equation actually has exactly the same form as Eq. (19). Also, in both cases the orbit average of the source term on the right side vanishes. However, a difference is that, for Eq. (22), we do not appear to have a differential relation analogous to (14). Hence, it does not appear to be possible to solve that equation by the same type of technique as Eq. (19). For this reason, we now turn to a different technique. For this, we first consider the abelian Lie-group G of isometries of (, γab ) that is generated by the flow φˆ τ , τ ∈ R of the vector field s a . The isometry group of any compact Riemannian manifold is known to be a compact Lie group, so it follows that the closure K of G must be contained in the isometry group. Being the closure of an abelian Lie-group, K, too, must be abelian, and hence it must be contained in a maximal torus of the isometry group of (, γab ). Hence, it must be isomorphic to an N -torus, K ∼ = T N , for some N ≥ 1. Let ψ1a , . . . , ψ Na , be the Killing fields on (, γab ) corresponding to the N commuting generators of T N . We assume them to be normalized so that their orbits close after 2π . Then we have s a = 1 ψ1a + · · · + N ψ Na ,
(28)
for some numbers (1 , . . . , N ), all of which are non-zero. If N = 1, then the orbits of s a are closed. If N > 1, then the orbits of s a are not closed, and the numbers i 5 The statement follows by establishing bounds on the derivatives of exp[−F ∗ (y)]. These bounds are obtained precisely as in (44), by considering m = 0 and replacing J (y) by exp[−F(y)] in that equation.
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S. Hollands, A. Ishibashi
are linearly independent over Z. Since the choice of commuting generators of T N is arbitrary, the vector of numbers (1 , . . . , N ) ∈ R N is only unique up to ⎞ ⎛ A11 . . . A1N N
⎜ .. ⎟ ∈ S L(N , Z). (29) i → Ai j j , ± ⎝ ... . ⎠ j=1 AN1 . . . AN N The Riemannian manifold (, γab ) may be identified with the space of null-generators of the horizon. Since this is an invariant concept, the vector of numbers (1 , . . . , N ) ∈ R N , too, is invariantly defined in terms of (M, gab ), i.e., it does not depend on our choice of up to the above ambiguity. If it was already known that the vector fields ψia were extendible to global Killing fields, then i would be the corresponding angular velocities of the horizon. That the desired solution to Eq. (22) exists is a consequence of the following lemma: Lemma 1. Let J be a smooth function on with the property that 1 T J ◦ φˆ τ (x)dτ. 0 = lim T →∞ T 0
(30)
Let = (1 , . . . , N ) ∈ R N [see Eq. (28)] satisfy the following “diophantine condition”: There exists a number q such that6 | · m| > || · |m|−q holds for all but finitely many m ∈
ZN .
(31)
Then the equation
Ls h = J,
(32)
with s a as in Eq. (28), has a smooth solution h on . Furthermore, if J is real analytic, then the same statements hold true and h is real analytic. Proof. Let us assume that J is real analytic. It is instructive to first treat the case N = 1 separately. In this case, the diophantine condition is trivially fulfilled. If T = 2π/ 1 , then φˆ T (x) = x for all x in . We define 1 T h(x) = J ◦ φˆ τ (x)τ dτ. (33) T 0 This function is analytic, and we claim that it also solves the desired differential equation. Indeed, we have 1 T Ls h(x) = Ls J ◦ φˆ τ (x)τ dτ T 0 1 T d = J ◦ φˆ τ (x)τ dτ T 0 dτ τ =T 1 T τ =− J ◦ φˆ τ (x)dτ + J ◦ φˆ τ (x) T T 0
= J (x).
τ =0
6 Note that · m = 0 if m = 0, since the entries of are linearly independent over Z.
(34)
Theorem for Extremal Black Holes in Higher Dimensions
451
We next treat the case N > 1. In that case, we have i / j ∈ / Q for i = j, and the diophantine condition is non-trivial. Let τ = (τ1 , . . . , τ N ) ∈ R N /(2π Z) N = T N and let τ ∈ Isom() be the isometry of defined as follows. For each x ∈ we let τ (x) be the point in obtained by letting x flow for parameter time τ1 along the flow line of the Killing field ψ1a of , then for parameter time τ2 along the flow line of the Killing field ψ2a etc. The order in which these flows are applied does not matter as the Killing fields mutually commute. We next define 2π 2π 1 ... eim·τ J ◦ τ(x)dτ1 . . . dτ N . (35) J (x, m) = (2π ) N 0 0 The term under the integral is analytic in (τ1 , . . . , τn ) for each fixed x, so it may be analytically continued for sufficiently small |Im τi | < ci (x), where ci (x) is positive. Because is compact, it follows that the infimum ci of ci (x) as x ranges over and as i ranges from 1, . . . , N is a positive constant. By shifting the contours of integration to Im τi = sign(m i )ci , it then follows that 2π ±ic N 2π ±ic1 1 . . . eim·τ J ◦ τ (x)dτ1 . . . dτ N , (36) J (x, m) = (2π ) N ±ic1 ±ic N √ and therefore that (setting c = N inf{ci ; i = 1, . . . , N }) |J (x, m)| ≤ e−c|m| sup{|J ◦ τ (x)| ; x ∈ , 0 ≤ Reτi ≤ 2π, |Im τi | = ci } = const. e−c|m| ,
(37)
for all m ∈ Z N , uniformly in x. We now set
h(x) = i m∈Z N \0
J (x, m) . ·m
(38)
We claim that this is the desired solution. Let us first check that this is well-defined for all x. In view of Eq. (31), we can estimate |h(x)| by pulling the absolute values inside the series (38), to obtain |h(x)| ≤
m∈Z N \0
≤
const. e−c|m| const. ≤ ·m ||
|m|q e−c|m|
m∈Z N \0
const. , ||
(39)
cq+N
where the last constant is of order (q + N ). This estimate is uniform in x ∈ . Hence, the series (38) for h(x) converges absolutely, uniformly in x. We would next like to show that h(x) is real analytic. For this, we recall that if a function ψ on Rk is real analytic near the origin in Rk , then there is an r > 0 and a K > 0 such that |∂ α ψ(y)| ≤ K |α| α! ,
(40)
for all y in an open ball of radius r around the origin. Here we use the multi-index notation α = (α1 , . . . , αk ) ∈ Nk0 , ∂α =
∂ |α| , |α| = αi , α! = αi !. 1 α k α 1 k (∂ y ) · · · (∂ y ) i
i
(41)
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S. Hollands, A. Ishibashi
This statement follows from the multi-dimensional generalization of the Cauchy integral representation of an analytic function. Conversely, if Eq. (40) holds, then ψ is analytic near the origin. Now let ψ be a real analytic function on , choose a point x0 ∈ , and let y 1 , . . . , y n−2 be Riemannian normal coordinates centered at x0 . Then there exist K , r > 0 such that Eq. (40) holds for ψ(y) for all y in a ball of radius r around the origin (here we identify a neighborhood of x0 with an open neighborhood of the origin of the Riemann normal coordinates). Furthermore, since is compact, we may choose K , r to depend only on ψ, but not on the choice of x0 . If ci > 0 are as above and c = (sign(m 1 )c1 , . . . , sign(m N )c N ) ∈ R N , we have ∂ α (J ◦ τ +ic (y)) = ∂ α (J ◦ ic ◦ τ (y)) = (∂ α ψ)(y ) ,
(42)
where the derivatives in the last expression are taken with respect to the Riemann normal coordinates centered at the image of x0 under the isometry τ , and where y is the image of y, identified with the corresponding Riemann normal coordinates. In the last step, we have used that, because τ is an isometry, it takes Riemann normal coordinates to Riemann normal coordinates. Furthermore, we have defined the real analytic function ψ on by ψ = J ◦ ic . We now apply the above estimate (40) to obtain α ∂ (J ◦ τ +ic (y)) ≤ K |α| α!, (43) for all y in a ball of radius r . As above, we next shift the contour of the τ integration in the expression for ∂α J (y, m) by ic, to arrive at 2π e−c·m 2π α im·τ α |∂ J(y, m)| = ... e ∂ (J ◦ τ +ic (y)) dτ1 . . . dτ N N (2π ) 0 0 ≤ e−c|m| K |α| α!.
(44)
Substituting this bound into the series for ∂α h(y) and bounding each term in this series by its absolute value, we obtain |∂ α h(y)| ≤ C |α| α! for some constant C > 0 and all y in a ball of radius r . Hence, h(y) is analytic, as we desired to show. We finally need to check that h(x) as defined above satisfies the desired differential equation. For this, we first note that J (x, 0) = 0. Indeed, since i / j ∈ / Q, we know that the orbit of R → T N , t → (t1 , . . . , t N ) is dense in
TN ,
mod (2π Z) N
so application of the ergodic theorem (see e.g. [51]) gives 2π 2π 1 . . . J ◦ τ (x)dτ1 . . . dτ N J (x, 0) = (2π ) N 0 0 1 T = lim J ◦ (t1 ,...,t N ) (x)dt. T →∞ T 0
(45)
(46)
On the other hand, (t1 ,...,t N ) (x) is by definition equal to φˆ t (x). Hence, in view of our assumption (30), we have J (x, 0) = 0. Next, we calculate 2π 2π 1 . . . eim·τ Ls J ◦ τ (x)dτ1 . . . dτ N (2π ) N 0 0 2π 2π ∂ 1 im·τ ∂ + · · · + J ◦ τ (x)dτ1 . . . dτ N = . . . e 1 N ∂τ1 ∂τ N (2π ) N 0 0 = −i m · J (x, m). (47)
Ls J (x, m) =
Theorem for Extremal Black Holes in Higher Dimensions
453
Using J (x, 0) = 0, we then have Ls h(x) = i
m∈Z N \0
Ls J (x, m) = J (x, m) = J ◦ τ =0 (x) = J (x). ·m N
(48)
m∈Z
This concludes our proof in the case when J is real analytic. Next, suppose J is only smooth. Then the argument in the case N = 1 is unchanged and gives a smooth solution h. In the diophantine case N > 1, we now have for any k, l ∈ N0 an estimate |l J (x, m)| ≤ const. (1 + |m|)−k
(49)
for a constant only depending on k, l, where = D a Da . It follows again from the diophantine condition that the sum (38) for h(x) and the corresponding sums for l h(x) converge uniformly for all l. Thus, |l h(x)| is uniformly bounded and hence h is in any of the Sobolev spaces W p,l (, dV ), and therefore smooth. That h(x) satisfies the desired differential equation follows as in the analytic case. The lemma shows that the desired new Gaussian null coordinates (u, ˜ r˜ , x˜ A ) and cor˜ responding foliation (˜r , u) ˜ exist under the assumptions stated there. For the rest of the paper, we assume that these hold. Now let K a = n˜ a . We have previously shown that Ln˜ γ˜ab = 0 on H , since this relation holds for any choice of Gaussian null coordinates. However, since our new coordinates have the property that α˜ = 0 on H , we clearly have that Ln˜ α˜ = 0 on H . Furthermore, for our new coordinates, Eq. (12) immediately yields Ln˜ β˜a = 0 on H . Thus, we have proven that all of the relations in Eq. (11) hold for m = 0. We next prove that the equation L˜ Ln˜ γ˜ab = 0 holds on H . Using what we already know about β˜a , γ˜ab and taking the Lie-derivative Ln˜ of the components of the Einstein ˜ r , u) equation tangent to (˜ ˜ (see Eq. (97) of Appendix A), we get 0 = Ln˜ Ln˜ L˜γ˜ab ,
(50)
˜ u), on H . Since t a = n˜ a + s˜ a , with s˜ a tangent to ( ˜ and since all quantities appearing in Eq. (50) are Lie derived by t a , we may replace in this equation all Lie derivatives Ln˜ by −Ls˜ . Hence, we obtain 0 = Ls˜ Ls˜ L˜γ˜ab ,
(51)
˜ Now, write L ab = L ˜γ˜ab . We fix x0 ∈ ˜ and view Eq. (51) as an equation holding on . ∗ ˆ ˜ → ˜ now denotes the flow of at x0 for the pullback, φτ L ab , of L ab to x0 , where φˆ τ : s˜ a . Then Eq. (51) can be rewritten as d2 ∗ φˆ L ab = 0. dτ 2 τ
(52)
1 ∗ (φˆ L ab − L ab ) = Cab , τ τ
(53)
Integration of this equation yields
where Cab is a tensor at x0 that is independent of τ . However, since φˆ τ is an isometry, each orthonormal frame component of φˆ τ∗ L ab at x0 is uniformly bounded in τ by
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S. Hollands, A. Ishibashi
˜ Consequently, the limit of Eq. (53) as τ → ∞ immedisup{(L ab L ab (x))1/2 ; x ∈ }. ately yields Cab = 0.
(54)
Thus, we have Ls˜ L˜γ˜ab = 0, and therefore Ln˜ L˜γ˜ab = L˜ Ln˜ γ˜ab = 0 on H, as we desired to show. Thus, we now have shown that the first equation in (11) holds for m = 0, 1, and that the other equations hold for m = 0, for the tensor fields associated with the “tilde” Gaussian null coordinate system, and K a = n˜ a. In order to prove that Eq. (11) holds for all m, we proceed inductively. Let M ≥ 1, and assume inductively that the first of Eqs. (11) holds for all m ≤ M, and that the remaining equations hold for all m ≤ M −1. Our task is to prove that these statements then also hold when M is replaced by M + 1. To show this, we apply the operator (L˜) M−1 Ln˜ to the Einstein equation Rab n˜ a ˜b = 0 (see Eq. (93)) and restrict to H. Using the inductive hypothesis, one sees that (L˜) M (Ln˜ α) ˜ = 0 on H , thus establishes the second equation in (11) for m ≤ M. Next, we apply the operator (L˜) M−1 Ln˜ to the components of Einstein’s equation Rab ˜b = 0 tangent to ˜ r , u) (˜ ˜ (see Eq. (96)), and restrict to H . Using the inductive hypothesis, one sees that (L˜) M (Ln˜ β˜a ) = 0 on H , thus establishes the third equation in (11) for m ≤ M. Next, we apply the operator (L˜) M Ln˜ to the components of the Einstein equation tangent ˜ r , u) to (˜ ˜ (see Eq. (97)), and restrict to H . Using the inductive hypothesis and the above results (L˜) M (Ln˜ α) ˜ = 0 and (L˜) M (Ln˜ β˜a ) = 0, one sees that the tensor field M+1 Sab ≡ (L˜) γ˜ab satisfies a differential equation of the form Ln˜ Ln˜ Sab = 0
(55)
on H. By the same argument as given above for L ab , it follows that Ln˜ Sab = 0. This establishes the first equation in (11) for m ≤ M + 1, and closes the induction loop. Thus, we have shown (1) for our choice of K a . In the analytic case, since gab and a K are analytic, so is L K gab . It follows immediately from the fact that this quantity and all of its derivatives vanish at any point of H that L K gab = 0, where defined, i.e., within the region where the Gaussian null coordinates (u, ˜ r˜ , x˜ A ) are defined. This proves existence of a Killing field K a in a neighborhood of the horizon. We may then try to extend K a further by analytic continuation. Now, analytic continuation need not, in general, give rise to a single-valued extension, so we cannot a priori conclude that there exists a Killing field on the entire spacetime. However, by a theorem of Nomizu [38] (see also [4]), if the underlying domain is simply connected, then analytic continuation does give rise to a single-valued extension. In this paper, we are assuming that the spacetime under consideration is asymptotically flat, with standard asymptotic infinity of topology I± ∼ = S n−2 × R, and that the domain of outer communication is globally hyperbolic. In this case, the topological censorship theorem [5,14,15] applies and we can conclude that the domain of outer communication is indeed simply connected. Consequently, there exists a unique, single valued extension of K a to the domain of outer communication, i.e., the exterior of the black hole (with respect to a given end of infinity). Thus, in the analytic case, we have proven the following theorem: Theorem 1. Let (M, gab ) be an analytic, asymptotically flat n-dimensional solution of the vacuum Einstein equations containing a black hole and possessing a Killing field t a with complete orbits which are timelike near infinity. Assume that the event horizon, H , of the black hole is analytic and is topologically R × , with compact and connected,
Theorem for Extremal Black Holes in Higher Dimensions
455
and that κ = 0 (where κ is defined by Eq. (15) above). Let = (1 , . . . , N ) be the angular velocities associated with projection of φτ onto , see Eq. (28). If these satisfy the diophantine condition | · m| > || · |m|−q
(56)
for some number q and for all but finitely many m ∈ Z N , then there exists a Killing field K a , defined in a region that covers H and the domain of outer communication, whose orbits are tangent to the null-generators of H . Remarks. (1) Note that the diophantine condition is trivially satisfied when N = 1, i.e., when the one-parameter group of symmetries φτ associated with t a maps the horizon generators to themselves after some fixed period T. For N > 1, the diophantine condition is non-trivial. We will discuss it in some more detail in Sect. 4. (2) If the diophantine condition is satisfied for , then it is also satisfied for A when ±A ∈ S L(N , Z). Thus, the diophantine condition is invariant under changes of the form (29), which as we discussed, constitute the only ambiguity in our definition of for the given spacetime. If we are in the situation described in Theorem 1, we can apply the same type of reasoning as in our previous paper [24] to extend the rotational Killing fields ψ˜ ia in the decomposition s˜ a = 1 ψ˜ 1a + · · · + N ψ˜ Na [see Eq. (28)] to Killing fields on the entire exterior of the spacetime, i.e., we have the following theorem. Theorem 2. Let (M, gab ) be an analytic, asymptotically flat n-dimensional solution of the vacuum Einstein equations containing a black hole and possessing a Killing field t a with complete orbits which are timelike near infinity. Assume that the event horizon, H , of the black hole is analytic and is topologically R × , with compact and connected, and that κ = 0. As above, assume that (1 , . . . , N ) [see Eq. (28)] satisfy the diophantine condition (56). If t a is not tangent to the generators of H , then there exist mutually commuting Killing fields ψ˜ 1a , . . . , ψ˜ Na (where N ≥ 1) with closed orbits with period 2π which are defined in a region that covers H and the entire domain of outer communication. Each of these Killing fields commutes with t a , and t a can be written as t a = K a + 1 ψ˜ 1a + · · · + N ψ˜ Na ,
(57)
where K a is the horizon Killing field whose existence is guaranteed by Theorem 1. Remarks. (1) If the spacetime is asymptotically flat in the standard sense with asymptotic infinity of type S n−2 as we are assuming, then there can be at most N = [(n + 1)/2] (we mean the integer part of a number) mutually commuting Killing fields including the stationary Killing field. For example, Myers-Perry black holes [37] in arbitrary n > 4 possess a stationary Killing field plus [(n − 1)/2] rotational Killing symmetries with angular velocities i , i = 1, . . . , [(n − 1)/2]. These solutions admit a regular extremal (degenerate horizon) limit for a wide range of the parameters of the solutions, for example when all the angular velocities are equally large. However, note that when a Myers-Perry hole has only a single nonvanishing angular momentum, the horizon becomes singular in the extremal limit for n = 5, and for n ≥ 6, there is no extremal limit; the angular velocity can be arbitrary large in that case. A black ring solution [9,39] in n = 5 which possesses 3 mutally commuting Killing fields also admits a regular extremal limit if it has two non-vanishing angular velocities. For more details on higher dimensional, extremal black holes see e.g. [8,10,11,31,32], and references therein.
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(2) If the black hole is non-rotating, i.e. if t a is tangent to the null generators of H , then our arguments do not apply. In the non-extremal case, one can show that the solution is static in this case [50]. The same result also holds for Einstein-Maxwell theory [50], and more generally presumably also for many of the Einstein-Matter theories described in the next section. In the non-extremal case, the uniqueness theorems [18,19] for static Einstein-Maxwell-Dilaton black hole solutions then apply. In the extremal case, it does not appear to be known whether the solution has to be static in the non-rotating case. The uniqueness of higher dimensional, static Einstein-Maxwell black hole solutions was shown in [47,48]. 3. Matter Fields So far we have focused on vacuum solutions to the Einstein equations for the sake of simplicity. In this section we generalize our results to include certain types of matter fields. We consider theories containing scalar fields φ taking values in a target space manifold X with positive definite metric f i j (φ) and vector fields Aa taking values in a vector bundle over X with positive definite vector bundle metric h I J (φ). We write the components of the scalar and vector fields as φ i and AaI respectively. We take the action to be 1 1 n √ ab i j ac bd I J S = d x −g R − f i j (φ)g ∇a φ ∇b φ − U (φ) − h I J (φ)g g Fab Fcd 2 4 +Stop , (58) I = ∇ A I −∇ A I , where U is a potential, and where S where Fab a b b a top denotes a topological term. A typical example for such a term is a Chern-Simons action. It does not affect the form of stress-energy tensor but it can modify the equation of motion for the gauge field, Eq. (61). In this section we will discard the topological term for simplicity. But we will discuss the minimal supergravity in n = 5 dimensions as an example of a theory with a Chern-Simons term in Appendix C. The above class of theories obviously includes the case of pure gravity with a cosmological constant, which corresponds to solutions with constant φ. It also includes many interesting supergravity theories in 5-(and 4)-dimensions arising from supergravity theories in 11-dimensions and string theories in 10-dimensions by appropriate dimensional reductions. In the latter case, one must include a topological term. Varying the action Eq. (58) gives the following equations of motion: I J Rab = f i j (φ)∇a φ i ∇b φ j + h I J (φ)g cd Fac Fbd 1 2 I J cd gab U (φ) − h I J (φ)Fcd F , + n−2 4 1 1 I ∇a ( f i j (φ)∇ a φ j ) − f jk|i ∇ a φ j ∇a φ k − U|i − h I J |i Fab F J ab = 0, 2 4
∇c h I J (φ)F J ca = 0,
(59) (60) (61)
and the Bianchi identities, J ∇[a Fbc] = 0,
(62)
Theorem for Extremal Black Holes in Higher Dimensions
457
where here and in the following the vertical stroke denotes the derivative with respect to a scalar field component, φ i, e.g., f jk|i = ∂ f jk (φ)/∂φ i . We now consider a stationary black hole solution in the above theory with corresponding Killing field t a , that is Lt gab = 0. We also assume that the other fields are J = 0, and that all fields are real analytic. invariant under t a , that is Lt φ i = 0, Lt Fab Which asymptotic conditions on the dynamical fields are reasonable in the above theory will in general depend on the precise choice of the potential U (φ) and the metrics f i j (φ), h I J (φ). In the vacuum case, we assumed asymptotic flatness for the metric with standard infinity I ± ∼ = S n−2 × R. This assumption was used implicitly to show that t a does not vanish on H , a fact which we needed to obtain the desired foliation (u, r ) in our construction of the Gaussian null coordinates. Asymptotic flatness was also implicitly used in the proof of Theorem 2, in combination with the topological censorship theorem [14]. Here, it was needed in order to establish that the exterior of the black hole is a simply connected manifold, which in turn is essential in order to be able to analytically extend the Killing fields K a and ψia to the full exterior of the black hole in a single valued way, cf. [24] for the details of this argument. In the present section, we will simply assume that t a is nowhere vanishing on H, and that the exterior is simply connected. As in the vacuum case, we also assume that the black hole is rotating, i.e. that t a is not everywhere tangent to the null generators of H. For the case when the orbits of t a are tangent to the generators, see Remark 2 following Theorem 2. As in the vacuum case, we distinguish between extremal and non-extremal black holes. In the non-extremal case we will show that, if the orbits of t a are not everywhere tangent to the null generators of the horizon H , then the analogues of Theorems 1 and 2 hold without any restrictions on the vector of angular velocities . This generalizes previous results in [24] to the above type of theories. In the extremal case we will show the same result under the additional assumption that the vector of angular velocities verifies the diophantine condition given in the statement of Theorem 1. Let us now explain how the desired additional Killing field K a described in Theorems 1 and 2 is constructed in the above types of theories. By analogy to the vacuum case, we must now show that L L · · · L (L K gab ) = 0, L L · · · L L K φ i = 0, m times
L L · · · L
I L K Fab
m times
= 0.
(63)
m times
Again, we first introduce a Gaussian null coordinate system (u, r, x A ) adapted to the horizon geometry, and we seek to adjust the remaining freedom in choosing this coordinate system in such a way that the desired K a is given by n a = (∂/∂u)a . I with respect to To do this, it is convenient to first decompose the components of Fab the Gaussian null coordinate system. For this, we define I Fab n a b = S I ,
I a c Fac n p b = VbI ,
I a c Fac p b = WbI ,
I I Fcd p c a p d b = Uab , (64)
where pa b projects onto the surfaces (u, r ) of constant u, r , cf. Appendix A for details. The field equations are written in terms of these variables and γab , βa , α in Appendix B. J = 0 that L S I = 0 , L V I = 0 , L W I = It immediately follows from Lt Fab t t a t a I 0 , Lt Uab = 0. Our task is now to show Eqs. (11) and
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L L · · · L Ln φ i = 0,
(65)
m times
L L · · · L Ln S I = 0, m times
L L · · · L Ln VaI = 0, m times
L L · · · L Ln WaI = 0, m times
I L L · · · L Ln Uab = 0,
(66)
m times
for a suitable choice of our Gaussian null coordinate system. First, we consider the Raychaudhuri equation for a congruence of null geodesic generators of the event horizon H, i.e. the Einstein equations contracted with n a n b : d 1 σ ab σab − f i j (Ln φ i )Ln φ j − h I J q ab VaI VbJ , θ =− θ2 − dλ n−2
(67)
where λ is an affine parameter of null geodesic generators of H and where θ and σab denote, respectively, the expansion and the shear of the null geodesic generators. Because the terms on the right-hand side are negative definite7 , we may argue as in the proof of the area theorem [23] to show that θ = 0. It then also follows that σab = 0, and Ln φ i = 0, VaI = 0,
on H.
(68)
The relations θ = 0 = σab on H are equivalent to Ln γab = 0, on H ,
(69)
which—when substituted into the Einstein equations Eqs. (94) and (105) and combined with Eqs. (68)—give Da α =
1 Ln βa , on H . 2
(70)
In the non-extremal case, we may now argue as in [24] that we can always pass to a modified system of Gaussian null coordinates with associated quantities α, ˜ β˜a , γ˜ab , φ˜ i , V˜aI , S˜ I , etc. such that α˜ is constant and non-zero over H . In the extremal case, we can use the same arguments as in the previous section to construct a modified system of Gaussian null coordinates such that α˜ = 0 on H under the assumption that the vector of angular velocities verifies the diophantine condition given in the statement of Theorem 1. We assume from now on that our Gaussian null coordinates have been chosen in this way in either case, and we drop the “tilde” from the corresponding quantities again to lighten the notation. Thus it follows that Ln βa = 0, on H . 7 Here it enters that the target space metrics h I J and f i j are positive definite.
(71)
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459
Next, from the Bianchi identities [see Eqs. (62) and (115)] and Eqs. (68), we find that I = 0, on H . Ln Uab
(72)
Using conditions (68), we immediately can show that n a (∇b φ i )F I ab = 0 on H . Then, using the results above and the equation of motion for the gauge field, Eq. (61), contracted with gab n b , we obtain Ln S I = 0, on H .
(73)
Ln L γab = 0 , on H .
(74)
At this point, we can show that
Indeed, if we take a Lie derivative Ln of Eq. (59), and contract with p c a p d b (see Eqs. (97), (99), and (108)), then we obtain Ln {Ln L γab + αL γab } = 0 , on H ,
(75)
as in the vacuum case. In the non-extremal case, Eq. (74) follows from Eq. (75) and the argument below Eq. (72) of [24]. For the extremal case, i.e. when α = 0, the same argument as given around Eq. (50) above applies. We next show that Ln WaI = 0, on H.
(76)
First, taking a Lie derivative Ln of the gauge field equation (61), and contracting with p c a (see Eq. (113)), we have Ln Ln q ab WbI + 2αLn q ab WbI + Ln L q ab VbI = 0, on H .
(77)
Second, taking a Lie derivative Ln of the Bianchi identities, Eq. (62) (see Eq. (114)) and using Ln S I = 0, we have Ln L VaI − Ln Ln WaI = 0 , on H .
(78)
Substituting this into the above equation, Eq. (77), we find Ln {Ln WaI + αWaI } = 0, on H.
(79)
Then Eq. (76) follows by the same type of argument as below Eq. (72) of [24] for the non-extremal case α = κ = 0, and the type of argument as below Eq. (50) for the extremal case α = κ = 0. Thus, we have shown Eqs. (11), (65), and (66), for m = 0, and the first of Eq. (11), for m = 1, on H. The remaining equations for all other values of m are verified by the same type of inductive argument as in [24] for the non-extremal case, and as in the previous section for the extremal case. In summary, we have verified that Theorems 1 and 2 continue to hold in the presence of matter fields described by the above action (58). In the non-extremal case, the diophantine condition stated in Theorem 1 is not required.
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4. Discussion In this paper, we have considered degenerate (extremal) stationary black hole spacetimes with Killing field t a . We showed that, if the vacuum Einstein equations hold and the spacetime is asymptotically flat, then there exists a Killing field K a that is tangent and normal to the horizon generators, i.e. the black hole horizon is a Killing horizon. We also proved that if t a is not everywhere tangent to the null generators (so that K a = t a ), then there exist N additional rotational Killing fields, where N is at least one. Our proof relied on two technical assumptions about the nature of the black hole: First we assumed that the spacetime metric is real analytic. Secondly, we had to assume that the corresponding angular velocities = (1 , . . . , N ) satisfy the “diophantine condition” (56). This condition is automatically satisfied when N = 1, in which case the spacetime isometries generated by the timelike Killing field map the horizon generators to themselves after the period T = 2π/ 1 . However, when N > 1—which can happen only in n > 4 spacetime dimensions—the diophantine condition is non-trivial. In this sense, our theorem is weaker than that obtained in our previous paper [24] for the nondegenerate case, because no assumption of that type had to be made there. We also considered a class of theories containing scalar and abelian gauge fields and derived similar results in this context. Let us make a few elementary remarks concerning the diophantine condition (56). First, it is well-known that this condition holds for all = (1 , . . . , N ) ∈ R N except for a set of Lebesgue measure zero8 . This follows immediately from the fact that the set where the condition (56) does not hold for any q is contained in the intersection ∩q q of the sets q = { = (1 , . . . , N ) ∈ R N | | · m| ≤ || · |m|−q for some m with |m| > 1}. (80) This intersection has Lebesgue measure zero, ⎛ ⎞ ∞ measure ⎝ q ⎠ = 0.
(81)
q=1
For completeness, let us briefly show this: If Br denotes the ball of radius r in R N , then we have
measure(q ∩ Br ) ≤ const. r N |m|−q−1 |m|>1
≤ const. r N
|x|>1
|x|−q−1 d N x ≤
const. r N . q +1− N
(82)
Since this goes to zero for q → ∞, the claim follows immediately. Thus, it would seem that the assumptions of our theorem are satisfied in almost the entire space of possible parameters . Unfortunately, our analysis gives no indication exactly what the true parameter space of possible values of for n-dimensional stationary black holes really is.For example, it could still happen that this space is very sparsely populated for certain 8 Since we know that the orbits t → t mod Z N are dense on T N, it follows that the entries of are linearly independent over Z. By the Schmidt-subspace theorem [49], if there is an i such that the ratios j / i are algebraic numbers for j = 1, . . . , N , then satisfies the diophantine condition. Of course, the set of for which this condition is satisfied is much larger—it has full measure.
Theorem for Extremal Black Holes in Higher Dimensions
461
types of black holes, i.e., it is theoretically possible that extremal black holes could only exist for in a set of measure zero. In that case, the statement that the assumptions are satisfied for almost all ∈ R N would have little value. Let us look at the example of a 5-dimensional black ring. It admits N = 2 rotational Killing fields and there is a regular limit in which the horizon becomes degenerate. In this limit, the angular velocities = (1 , 2 ) are non-vanishing and satisfy θ = 1 / 2 = ±1 for the first branch of solutions9 , or √ (83) θ = (1 + x)/2 x, 0 ≤ x < ∞ for the second branch (see, e.g., [8,10]). Thus, for the first branch, θ is in particular rational, and the orbits of the projection of t a onto the space of null-generators of the horizon consequently always close. For the second branch, θ varies continuously and may thus be irrational. The vector satisfies the diophantine condition for almost all extremal black hole solutions, but there is a measure zero set where it does not, corresponding to certain transcendental values of θ . However, even in those exceptional cases the black hole continues to have N = 2 rotational Killing fields and is a Killing horizon. This suggests that our theorem might be true even dropping the diophantine condition. Secondly, as we have seen, the diophantine condition is needed in Lemma 1 to control the sizes of denominators of the form | · m| when m becomes large. It appears that this condition cannot easily be lifted for generic analytic functions J (x) in this lemma. Indeed, let us suppose that we have, say N = 2, and θ = 1 / 2 is given by the series ∞
1 θ= ai
(84)
i=0
where ai is defined recursively by a0 = 1 and ai+1 = 2 K ai , with K ∈ N. This series is converging rapidly to a transcendental10 number 1 < θ < 2, as ai+1 − ai ≥ 1 and ai+1 /ai = 2 K (ai −ai−1 ) ≤ 2−K. If pk /qk denotes the k-th partial sum, 1+ pk = qk
k
ak i=1 ak−1
ai · · · ai−1
ak
,
(85)
then h.c. f.( pk , qk ) = 1. Furthermore, we have ∞ Ki
ai−1 1 ak+1 2 i≥0 2 |θ − pk /qk | = ··· ≤ = 2−K qk +1 . (86) 1+ ≤ ak+1 ai ak+2 ak+1 ak+1 i=k+1
This implies that an exponential type sum of the form considered in Lemma 1
e−c| p|−c|q| | p/q − θ |
(87)
p,q∈Z
9 It has been discussed that the limiting solutions in this branch correspond to extremal Myers-Perry black holes rather than extremal black rings [8,44] 10 By Eq. (85), θ cannot be rational. If x were not transcendental, then by a classic theorem of Liouville, we would have |θ − p/q| > const. |q|−d , where d is the degree of the algebraic number θ . This condition is not satisfied due to Eq. (86).
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S. Hollands, A. Ishibashi
cannot converge for sufficiently large K , as there are always terms of size at least e−c| pk |−c|qk | 1 ≥ ak+1 e−c|ak | ≥ 2ak (K −c log2 e)−1 ≥ 2k(K −c log2 e)−1 → ∞ (k → ∞) | pk /qk − θ | 2 (88) in this sum. In the proof of Lemma 1, e−c| p|−c|q| bounds the Fourier coefficients (35) of J (x). If it is only known that J (x) is analytic, then no better bound can be derived, and the solution to the equation Ls h(x) = J (x) consequently cannot be obtained by the method of the lemma. However, in our case J (x) = 1 − e−F(x) , where F in turn satisfies Ls F = α. It might be possible that further constraints can be derived on the Fourier coefficients of J (x) from such a relation combined with Einstein’s equations. But we have not been able to find such relations. Let us finally make a remark about the assumption of analyticity. It is known that the Einstein-Maxwell system admits extremal multi-black hole solutions which have nonsmooth—hence non-analytic—horizons [2,17,52]. Therefore when including Maxwell fields, the analyticity assumption—which is one of the key assumptions in our proof—is not entirely plausible. As shown in [13,24,40], the analyticity assumption can be partially removed when the event horizon is non-degenerate. In that case, the horizon can be shown to be isometric to a portion of a bifurcate null hypersurface [41,42], and one can use the characteristic initial value formulation for Einstein’s equations [12,36,45] on the bifurcate null hypersurface in order to extend K a defined on the horizon to the black hole interior region. This is, however, not the case for degenerate horizons, since on such a horizon with κ = 0, the completeness of the Killing parameter of K a on the horizon implies that the horizon generator is affinely complete and hence there is no bifurcate surface. Thus, the key issue when generalizing our results to the EinsteinMaxwell system is whether the diophantine condition holds, and whether the solution is analytic, including a neighborhood of the horizon. An interesting generalization of this work would be to consider vacuum spacetimes which are not asymptotically flat in the standard sense (with asymptotic infinity of topology S n−2 ), but instead for example asymptotically Kaluza-Klein, with asymptotic infinity of the form S 2 × Y , with Y a compact manifold of dimension n − 4. In the non-degenerate case, there would be no change in our analysis of the local horizon geometry, and we could construct a vector field K a in a neighborhood of the horizon H satisfying (1), i.e., the Killing equation L K gab = 0 holds on H to all orders in a Taylor expansion off of H . The same would also apply in the degenerate case if we assume the diophantine condition (56). However, in both cases it might no longer be possible to construct the desired K a globally by analytic continuation: The point is that we are only guaranteed to get a single-valued extension if the exterior part of the spacetime is simply connected [38]. Now, the topological censorship theorem [5,14,15] guarantees that any curve with endpoints in the asymptotic region can be deformed continuously to a curve in S 2 × Y . This yields a surjective map π1 (Mexterior ) ← π1 (S 2 × Y ).
(89)
Unlike in the case of an asymptotically flat spacetime with infinity S n−2 , the fundamental group π1 (S 2 × Y ) no longer needs to vanish. Nevertheless, if π1 (Y ) = 0, then it does, and we presumably again get results analogous to Theorems 1 and 2. Acknowledgements. We would like to thank P. Chrusciel, H. Reall, and Bob Wald for valuable discussions and comments. SH would like to thank M. Huxley for discussions on the diophantine condition. We also would
Theorem for Extremal Black Holes in Higher Dimensions
463
like to thank P. Figueras for comments on the extremal limits of a black ring solution. AI wishes to thank the School of Mathematics, Cardiff University for its hospitality during the time some of this research was carried out and also wishes to thank the Perimeter Institute for Theoretical Physics for its hospitality during the time other parts of this research were carried out. This research was supported in part by the Perimeter Institute for Theoretical Physics.
A. Ricci Tensor in Gaussian Null Coordinates In this Appendix (identical to Appendix A of [24]), we provide expressions for the Ricci tensor in a Gaussian null coordinate system. As explained in Sect. 2, in Gaussian null coordinates, the metric takes the form gab = 2 ∇(a r − r α∇(a u − rβ(a ∇b) u + γab , (90) where the tensor fields βa and γab are orthogonal to n a and a . The horizon, H , corresponds to the surface r = 0. We note that γ a b is the orthogonal projector onto the subspace of the tangent space orthogonal to n a and a , and that when rβa = 0, it differs from the orthogonal projector, q a b , onto the surfaces (u, r ). It is worth noting that in terms of the Gaussian null coordinate components of γab , we have q ab = (γ −1 ) AB (∂/∂ x A )a (∂/∂ x B )b . It also is convenient to introduce the non-orthogonal projector pa b , uniquely defined by the conditions that pa b n b = pa b b = 0 and that pa b is the identity map on vectors that are tangent to (u, r ). The relationship between pa b and γ a b is given by pa b = −r a βb + γ a b .
(91)
In terms of Gaussian null coordinates, we have pa b = (∂/∂ x A )a (dx A )b , from which it is easily seen that Ln pa b = 0 = L pa b . It also is easily seen that q ac γcb = pa b and that pa b q b c = q a c . We define the derivative operator Dc acting on a tensor field T a1 ...ar b1 ...bs by the following prescription. First, we project the indices of the tensor field by q a b , then we apply the covariant derivative ∇c , and we then again project all indices using q a b . For tensor fields intrinsic to , this corresponds to the derivative operator associated with the metric qab . We denote the Riemann and Ricci tensors associated with qab as Rabc d and Rab . The Ricci tensor of gab can then be written in the following form: 1 1 1 n a n b Rab = − q ab Ln Ln γab + q ca q db (Ln γab )Ln γcd + α q ab Ln γab 2 4 2 r + · 4αL L α + 8αL α + (L α)q ab Ln γab 2 +q ab L γab · −Ln α − rq cd βc Ln βd
+ (rq cd βc βd + 2α)L (r α) + rq cd βc Dd α
+2q ab Da {βb L (r α) + Db α − Ln βb } +q bc L (rβc ) · (rq e f βe β f + 2α)L (rβb ) − 4Db α + 2Ln βb + 4rq ae βe D[a βb] +2(L α)L (r 2 q ab βa βb ) + 4rq ab βa βb L α + 2rq ab βa βb L L α
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S. Hollands, A. Ishibashi
! 1 +2q ab βa L (rβb ) · 2L (r α) − rq cd βc L (rβd ) 2 −1 2 ab + 2r L r q βa (Db α−Ln βb ) +2r −1 αL (r 2 q ab βa βb ) , 1 n a b Rab = −2L α + q ca q db (Ln γcd )L γab 4 1 1 1 − q ab L Ln γab − α q ab L γab − q ab βa βb 2 2 2 r 1 + · −2L L α − q ab L γab · 2L α + q cd βc L (rβd ) 2 2 ab ab ab − q βa L βb − L {q βa L (rβb )} − q Da (L βb ) ,
(92)
(93)
1 1 n b p c a Rbc = − p b a Db α + Ln βa + βa q bc Ln γbc − p d [a p e b] Dd (q bc Ln γce ) 2 4 1 bc r (q Ln γbc )L βa + Ln L βa + 2αL βa + · 2 2 +L (rβa ) · r −1 L (r 2 q bc βb βc ) + 2L α −2 p b a Db (L α)+L (q bc βb Ln γca )−2r −1 L r 2 q cd βc p b a D[b βd] 1 − q bc L γbc · −(rq e f βe β f + 2α)L (rβa ) 2
+ 2 p d a Dd α − q bc βb Ln γca + 2rq e f βe p d a D[d β f ] −2L (αβa ) − 2r (L α)L βa + p d a Db q bc βc L (rβd ) −2 p b a q cd Dd D[b βc] − q bc (L βb )Ln γca −q bc L (rβb ) · (rq e f βe β f + 2α)L γca + p d a Dc βd + βc L (rβa ) − rq e f βc β f L γea + q bc (L γca ) · 2βb L (r α)+2Db α−Ln βb +2rq de βe D[b βd] ,
(94)
1 1 a b Rab = − q ab L L γab + q ca q db (L γab )L γcd , (95) 2 4 1 1 b p c a Rbc = − βa q bc L γbc − L βa + q bc βc L γab − p d [a p e b] Dd q bc L γce 4 2 r + · − L L βa + L q bc βc L γab 2 1 + (q cd L γcd ) −L βa + q be βe L γab , (96) 2
Theorem for Extremal Black Holes in Higher Dimensions
465
1 p c a p d b Rcd = −L Ln γab − αL γab + p c a p d b Rcd − p c (a p d b) Dc βd − βa βb 2 1 cd (q Ln γcd )L γab + (q cd L γcd )Ln γab +q cd L γd(a Ln γb)c − 4 r + · −2αL L γab − p e a p f b Dc (q cd βd L γe f ) 2 1 − (q cd L γcd ) (rq e f βe β f + 2α)L γab + 2 p e (a p f b) De β f 2 −2(L α)L γab − r −1 {L (r 2 q e f βe β f )}L γab −rq e f βe β f L L γab − 2L { p c (a p d b) Dc βd } −2β(a L βb) − r (L βa )L βb − rq ce q d f βc βd (L γae )L γb f " # +2q cd βd L (rβ(a ) L γb)c + 2 p e (a p f b) q cd (Dd βe ) L γ f c + q cd (rq e f βe β f + 2α)(L γca )L γdb .
(97)
B. Gravity Coupled to Matter Fields We start from the action √ 1 1 I J S = dn x −g R − f i j (φ)g ab ∇a φ i ∇b φ j −U (φ)− h I J (φ)g ac g bd Fab , Fcd 2 4 (98) I = ∇ A I − ∇ A I , and where f (φ) and h (φ) are positive definite metrics where Fab a b b a ij IJ on the spaces of scalar fields, φ i , and gauge fields, AaI , respectively. Variation of S gives
T gab , n−2 1 1 I F J ab = 0 , ∇a ( f i j (φ)∇ a φ j ) − f jk|i ∇ a φ j ∇a φ k − U|i − h I J |i Fab 2 4 Rab = Tab −
∇c h I J (φ)F J ca = 0,
(99) (100) (101)
where the stress-energy tensor, Tab , is given by 1 Tab = f i j (φ)∇a φ i ∇b φ j − gab f i j (φ)∇ c φ i ∇c φ j + 2U (φ) 2 ! 1 cd I J I F J cd , +h I J (φ) g Fac Fbd − gab Fcd 4
(102)
and T = T c c , and where here and in the following the vertical bar denotes the derivative with respect to a scalar field, φ i , e.g., f jk|i = ∂ f jk (φ)/∂φ i . In terms of the tensor
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S. Hollands, A. Ishibashi
I , and the metric variables α, β , γ introduced in the context of fields, S I, VaI , WaI Uab a ab Gaussian null coordinates, the right-hand side of Eq. (99) can be decomposed as follows: T gab = f i j (φ)(Ln φ i )Ln φ j n a n b Tab − n−2 +h I J (φ) · q ab VaI VbJ + 2r · αS I S J + β a VaI S J 4α +r 2 · β b βb S I S J − r · ·T, (103) n−2 T gab = f i j (φ)(Ln φ i )L φ j n a b Tab − n−2 2 +h I J (φ) · q ab VaI WbJ + r · β a WaI S J + ·T, n−2 (104) T gbc = f i j (φ)(Ln φ i ) p b a Db φ j n b p c a Tbc − n−2 J +h I J (φ) · −S I VaJ + q bc VbI Uac I J +r · −2αS I WaJ + β b Uab S − β b VbI WaJ 2βa − r 2 · β c βc S I WaJ − r · ·T , (105) n−2 T gab = f i j (φ)(L φ i )L φ j + h I J (φ) · q ab WaI WbJ , a b Tab − (106) n−2 T gbc = f i j (φ) p c a (L φ i )Dc φ j b p c a Tbc − n−2 J + h I J (φ)· S I WaJ +q bc WbI Uac −r · β b WbI WaJ , (107) T p c a p d b Tcd − gcd = f i j (φ) p c a p d b (Dc φ i )Dd φ j n−2 I J + g I J (φ) · 2V(aI Wb)J + q cd Uca Udb J J + r · 2αWaI WbJ − β c (WaI Ubc + WbI Uac ) 2 · γab · T, + r 2 · β c βc WaI WbJ + (108) n−2
where
1 1 I J T = U (φ) − h I J (φ) · − S I S J + q ab VaI WbJ + q ab q cd Uca Udb 2 4 a I J c ab I J −r · h I J (φ) · β Wa S − 2β q Wa Ubc + αq ab WaI WbJ −
r2 · h I J (φ) · β c βc q ab − β a β b WaI WbJ . 2
The tensors q ab and pa b are defined in Appendix A.
(109)
Theorem for Extremal Black Holes in Higher Dimensions
467
Similarly, the equation for scalar fields, Eq. (100), are explicitly written as 0 = U|i 1 1 +h I J |i − S I S J + q ab VaI WbJ + q ab q cd Uca Udb 2 4 J +r · β a WaI S J − β c q ab WaI Ubc + αq ab WaI WbJ ! r 2 c + · β βc q ab − β a β b WaI WbJ 2 1 1 − f i j · 2Ln L φ j + (q ab Ln γab )L φ j + (q ab L γab )Ln φ j 2 2 +2αL φ j + q ab βa Db φ j + q ab Da Db φ j +r · α(q ab L γab )L φ j + 2L (αL φ j ) + 2β c βc L φ j 1 + (q ab L γab )q cd βc Dd φ j + L (q ab βb ) p c a Dc φ j 2 + q ab βa Db (L φ j ) + Da (q ab βb L φ j )
r2 c ab j c j · β βc (q L γab )L φ + 2L (β βc L φ ) + 2 1 + f jk|i − 2 f ki| j · (Ln φ j )L φ k + q ab (Da φ j )Db φ k 2 +r · α(L φ j )L φ k + q ab βa (L φ j )Db φ k r2 · β c βc (L φ j )L φ k . + 2
(110)
I , are given by The equations of motion for the gauge fields, Fab
0 = h I J |i S J (L φ i ) − q ab (Da φ i )WbJ
! 1 +h I J −Da (q ab WbJ ) + L S J − β a WaJ + S J (q ab L γab ) 2 ! 1 +r · −h I J |i (L φ i )β a WaJ + h I J −L (β a WaJ )+ β c WcJ (q ab L γab ) , 2 (111) i J ab i J 0 = h I J |i −(Ln φ )S − q (Da φ )Vb ! 1 J ab J ab J +h I J −Ln S − S (q Ln γab ) − Da (q Vb ) 2 J + 2αWbJ +r · h I J |i (Ln φ i )β a WaJ − h I J |i (Da φ i )q ab βb S J − β c Ubc 1 +h I J Ln (β a WaJ ) + (β c WcJ )(q ab Ln γab ) 2
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! J − Da (q ab βb S J ) + q ab Da (β c Ubc ) − 2Da (αq ab WbJ ) +r 2 · −h I J |i (Da φ i )q ab β c βc WbJ − βb β c WcJ + h I J Da β c βc q ab WbJ − q ab βb β c WcJ ,
(112)
1 J 0 = h I J Ln (q ab WbJ ) + q ad WdJ (q bc Ln γbc ) + Dc (q bc q ad Ubd ) 2 ! 1 J + 2αq ab WbJ + L (q ab VbJ ) + q ab VbJ (q cd L γcd ) + q ab βb S J − q ab β c Ubc 2 J +h I J |i (Ln φ i )q ab WbJ + (L φ i )q ab VbJ + q ad q bc (Db φ i )Ucd J +r · h I J 2Dc (q b[c q a]d βb WdJ ) + L (q ab βb S J ) − L (q ab β c Ubc ) + 2L (αq ab WbJ ) + 2β c βc q ab WbJ − 2β a β b WbJ ! 1 de i J h I J (q L γde )+h I J |i (L φ ) · q ab βb S J −q ab β c Ubc +2αq ab WbJ + 2 + 2h I J |i q c[a q b]d (Db φ i )βc WdJ +r 2 · h I J L β c βc q ab WbJ − q ab βb β c WcJ ! 1 de i ab c J J (113) + h I J (q L γde ) + h I J |i (L φ ) · q β βc Wb − βb Wc . 2 I = 0, are written as The Bianchi identities ∇[a Fbc]
Ln WaI − L VaI + p c a Dc S I = 0, I − 2 p c [a p d b] Dc VdI Ln Uab I L Uab − 2 p c [a p d b] Dc WdI p d [a p e b p f c] Dd UeIf
(114)
= 0,
(115)
= 0,
(116)
= 0.
(117)
C. Chern-Simons Theories in n = 5 Here we outline how the rigidity theorem can be proved in the presence of an additional Chern-Simons term in the action. For simplicity and concreteness, we restrict attention to the example of minimal supergravity theory in n = 5 dimensions11 . This theory has a metric and a single gauge field with field strength tensor Fab = ∇a Ab − ∇b Aa ; we set the Fermi-fields equal to zero. Its action is 2 abcde 5 √ ab S = d x −g R − Fab F − √ (118) Aa Fbc Fde . 3 3 11 In the n = 5 minimal supergravity theory described by (118) (and also in other supergravity theories), it is common to consider solutions possessing a covariantly constant spinor field. By forming a suitable bi-linear combination of this spinor field, one obtains an everywhere non-spacelike Killing vector field, which in particular must be null at the event horizon. Therefore, the event horizon in minimal supergravity theories in 5-dimensions must be a Killing horizon for such solutions [16,43].
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The last term in this action is a Chern-Simons term, with abcde the unique antisymmetric tensor satisfying abcde abcde = −5!. The resulting Einstein equations (i.e., varying gab ) are precisely the same as those given previously in Eqs. (99) and (102) with φ i = 0 and h I J = δ I J , as the stress-energy tensor is not modified by the addition of the Chern-Simons term, whereas the equations of motion for the gauge field (i.e., varying Aa ) are modified to 1 ∇b F ab + √ abcde Fbc Fde = 0. 2 3
(119)
We decompose this equation by contracting the free index into n a , a , and pa b , respectively. The first term of the left-hand side of the above equation is given by Eqs. (111), (112), and (113), respectively (with h I J = δ I J ), and the second term is given respectively by abc a abcde Fbc Fde = −4// Wa Ubc , abcde abc (dr )a Fbc Fde = 4// Va Ubc − r · Uab Ucd βe q de ,
(120) (121)
abc pa e ebcd f Fbc Fde = −4// (SUbc − 2Vb Wc ) ebc −q a f β f Ubc We + pa e q d f β f (Ubc Wd +2Wb Ucd ) , +4r · //
(122) abc = q ad q be q c f n p q where we have introduced // pqde f . We will now outline how to prove the rigidity theorem in the presence of the Chern-Simons term. For brevity, we only outline the main changes compared to the case without Chern-Simons term described in Sect. 3. Recall that in our proof we need to use the equation of motion for the gauge field only when we show the following equations:
L L · · · L (Ln S) = 0, L L · · · L (Ln Wa ) = 0. m times
(123)
m times
First we note that since Va = 0 on H from the Raychaudhuri equation, Eq. (121) must vanish on H . Thus, when contracted with (dr )a , the Chern-Simons term in Eq. (119) is irrelevant to the equation of motion at least on H . We can then show that the first of Eq. (123) is satisfied for m = 0, so Ln S = 0 on H . We can then easily show that the Lie-derivative Ln of Eq. (122) vanishes on H and hence does not contribute to the Liederivative Ln of Eq. (119) contracted with p c b on H . Then, from these results, we find that Eq. (77) also holds in the presence of a Chern-Simons term. By the same argument as after Eq. (77), we then conclude that the second of Eq. (123) holds for m = 0. Next, taking the Lie-derivative Ln of Eq. (119) contracting it with a = (du)a and using the results derived so far (in particular Ln W a = 0 on H ), we can show that the first of Eq. (123) holds for m = 1. Furthermore, taking Ln L of Eq. (119) contracted with p c b , and using what we already know (in particular Ln L S = 0 on H ), we can show that the second of Eq. (123) holds for m = 1. Then, using the inductive method as in a similar manner for the vacuum case, we can show that Eqs. (123)—as well as Eqs. (11), (65), and (66)—hold for all m = 0, 1, 2, . . . . Thus, our rigidity theorems also applies to the theory described by the action (118).
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References 1. Bunting, G.L.: Proof of the Uniqueness Conjecture for Black Holes. PhD Thesis, Univ. of New England, Armidale, N.S.W., 1983 2. Candlish, G.N., Reall, H.S.: On the smoothness of static multi-black hole solutions in higher dimensional Einstein-Maxwell theory. Class. Quant. Grav. 24, 6025–6039 (2007) 3. Carter, B.: Axisymmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971) 4. Chru´sciel, P.T.: On rigidity of analytic black holes. Commun. Math. Phys. 189, 1–7 (1997) 5. Chrusciel, P.T., Galloway, G.J., Solis, D.: Topological censorship for Kaluza-Klein space-times. http:// arxiv.org/abs/0808.3233v1[gr-qc], 2008 6. Chru´sciel, P.T., Wald, R.M.: Maximal hypersurfaces in asymptotically stationary space-times. Commun. Math. Phys. 163, 561 (1994) 7. Chru´sciel, P.T., Wald, R.M.: On the topology of stationary black holes. Class. Quant. Grav. 11, L147 (1994) 8. Elvang, H., Rodriguez, M.J.: Bicycling black rings. JHEP 0804, 045 (2008) 9. Emparan, R., Reall, H.S.: A rotating black ring in five dimensions. Phys. Rev. Lett. 88, 101101 (2002) 10. Emparan, R., Reall, H.S.: Black holes in higher dimensions. Living Rev. Rel. 11, 6 (2008) 11. Figueras, P., Kunduri, H.K., Lucietti, J., Rangamani, M.: Extremal vacuum black holes in higher dimensions. Phys. Rev. D 78, 044042 (2008) 12. Friedrich, H.: On the global existence and the asymptotic behavior of solutions to the Einstein-MaxwellYang-Mills equations. J. Diff. Geom. 34, 275 (1991) 13. Friedrich, H., Racz, I., Wald, R.M.: On the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys. 204, 691–707 (1999) 14. Galloway, G.J., Schleich, K., Witt, D.M., Woolgar, E.: Topological censorship and higher genus black holes. Phys. Rev. D 60, 104039 (1999) 15. Galloway, G.J., Schleich, K., Witt, D., Woolgar, E.: The AdS/CFT correspondence conjecture and topological censorship. Phys. Lett. B 505, 255 (2001) 16. Gauntlett, J.P., Gutowski, J.B., Hull, C.M., Pakis, S., Reall, H.S.: All supersymmetric solutions of minimal supergravity in five dimensions. Class. Quant. Grav. 20, 4587 (2003) 17. Gibbons, G.W., Horowitz, G.T., Townsend, P.K.: Higher dimensional resolution of dilatonic black hole singularities. Class. Quant. Grav. 12, 297 (1995) 18. Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness of (dilatonic) charged black holes and black p-branes in higher dimensions. Phys. Rev. D 66, 044010 (2002) 19. Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness and non-uniqueness of static black holes in higher dimensions. Phys. Rev. Lett. 89, 041101 (2002) 20. Harmark, T.: Stationary and axisymmetric solutions of higher-dimensional general relativity. Phys. Rev. D 70, 124002 (2004) 21. Harmark, T., Olesen, P.: On the structure of stationary and axisymmetric metrics. Phys. Rev. D 72, 124017 (2005) 22. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972) 23. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-time. Cambridge: Cambridge University Press, 1973 24. Hollands, S., Ishibashi, A., Wald, R.M.: A higher dimensional stationary rotating black hole must be axisymmetric. Commun. Math. Phys. 271, 699–722 (2007) 25. Hollands, S., Yazadjiev, S.: Uniqueness theorem for 5-dimensional black holes with two axial Killing fields. Commun. Math. Phys. 283, 749–768 (2008) 26. Hollands, S., Yazadjiev, S.: A Uniqueness theorem for 5-dimensional Einstein-Maxwell black holes. Class. Quant. Grav. 25, 095010 (2008) 27. Ionescu, A. D., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. http:// arxiv.org/abs/0711.0040v2[gr-qc], 2008 28. Isenberg, J., Moncrief, V.: Symmetries of cosmological Cauchy horizons with exceptional orbits. J. Math. Phys. 26, 1024–1027 (1985) 29. Israel, W.: Event horizons in static vacuum space-times. Phys. Rev. 164, 1776–1779 (1967) 30. Israel, W.: Event horizons in electrovac vacuum space-times. Commun. Math. Phys. 8, 245–260 (1968) 31. Kunduri, H.K., Lucietti, J., Reall, H.S.: Near-horizon symmetries of extremal black holes. Class. Quant. Grav. 24, 4169 (2007) 32. Kunduri, H.K., Lucietti, J.: A classification of near-horizon geometries of extremal vacuum black holes. http://arxiv.org/abs/0806.2051v2[hep-th], 2008 33. Mazur, P.O.: Proof of uniqueness of the Kerr-Newman black hole solution. J. Phys. A 15, 3173–3180 (1982) 34. Moncrief, V., Isenberg, J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387–413 (1983)
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35. Moncrief, V., Isenberg, J.: Symmetries of higher dimensional black holes. Class. Quant. Grav. 25, 195015 (2008) 36. Müller zum Hagen, H.: Characteristic initial value problem for hyperbolic systems of second order differential systems. Ann. Inst. Henri Poincaré 53, 159–216 (1990) 37. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space-times. Annals Phys. 172, 304 (1986) 38. Nomizu, K.: On local and global existence of Killing vector fields. Ann. Math. 72, 105–120 (1960) 39. Pomeransky, A.A., Sen’kov, R.A.: Black ring with two angular momenta. http://arxiv.org/abs/hep-th/ 0612005, 2006 40. Racz, I.: On further generalization of the rigidity theorem for spacetimes with a stationary event horizon or a compact Cauchy horizon. Class. Quant. Grav. 17, 153 (2000) 41. Racz, I., Wald, R.M.: Extensions of spacetimes with Killing horizons. Class. Quant. Grav. 9, 2643–2656 (1992) 42. Racz, I., Wald, R.M.: Global extensions of spacetimes describing asymptotic final states of black holes. Class. Quant. Grav. 13, 539–552 (1996) 43. Reall, H.S.: Higher dimensional black holes and supersymmetry. Phys. Rev. D 68, 024024 (2003) [Erratum-ibid. D 70, 089902 (2004)] 44. Reall, H.S.: Counting the microstates of a vacuum black ring. JHEP 0805, 013 (2008) 45. Rendall, A.: Reduction of the characteristic initial value problem to the Cauchy problem and its application to the Einstein equations. Proc. Roy. Soc. Lond. A427, 211–239 (1990) 46. Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) 47. Rogatko, M.: Uniqueness theorem of static degenerate and non-degenerate charged black holes in higher dimensions. Phys. Rev. D 67, 084025 (2003) 48. Rogatko, M.: Classification of static charged black holes in higher dimensions. Phys. Rev. D 73, 124027 (2006) 49. Schmidt, W.M.: Norm form equations. Ann. of Math. (2) 96, 526–551 (1972) 50. Sudarsky, D., Wald, R.M.: Extrema of mass, stationarity, and staticity, and solutions to the Einstein Yang-Mills equations. Phys. Rev. D 46, 1453–1474 (1992) 51. Walters, P.: An Introduction to Ergodic Theory. New York: Springer-Verlag, 1982 52. Welch, D.L.: On the smoothness of the horizons of multi-black hole solutions. Phys. Rev. D 52, 985 (1995) Communicated by G. W. Gibbons
Commun. Math. Phys. 291, 473–490 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0802-8
Communications in
Mathematical Physics
A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation Claus Köstler1 , Roland Speicher2, 1 Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall,
1409 West Green Street, Urbana, IL 61801, USA. E-mail: [email protected]
2 Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston,
ON K7L 3N6, Canada. E-mail: [email protected] Received: 16 October 2008 / Accepted: 27 January 2009 Published online: 4 April 2009 – © Springer-Verlag 2009
Abstract: We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen “exchangeability” (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables (xi )i∈N , we prove that invariance of the joint distribution of the xi ’s under quantum permutations is equivalent to the fact that the xi ’s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi ’s. 1. Introduction The de Finetti theorem states that an infinite family of random variables whose distribution is invariant under finite permutations (such a family is called exchangeable) is independent and identically distributed with respect to the conditional expectation onto the tail algebra of the random variables. Since the implication in the other direction is fairly elementary one has the equivalence between exchangeability and conditional independence. See, e.g., [Kal] for an exposition on the classical de Finetti theorem. In a noncommutative context classical random variables are replaced by, typically noncommuting, operators on Hilbert spaces. The expectation with respect to a probability measure is then replaced by a state on the algebra generated by these operators. Of course, the notion of invariance of mixed moments still makes sense. Thus one can ask what exchangeability means in such a context. It turns out that in the noncommutative world there are actually many quite different possibilities for exchangeable random variables. It was shown in [Koe1] that they all possess some kind of factorization property; but, as one sees from the variety of examples, one cannot expect that exchangeability implies some fixed kind of independence. Indeed, both independence and freeness Research supported by Discovery and LSI grants from NSERC (Canada) and by a Killam Fellowship from the Canada Council for the Arts.
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provide basic examples for exchangeable random variables. (See also [Leh,Koe2] for more on this.) However, if one moves into the noncommutative realm, one should also take into account that invariance under permutations is a commutative concept and should be replaced by its noncommutative analogue. To provide such noncommutative analogues of actions of groups was one of the motivations for the creation of the theory of quantum groups, which has been developed very extensively within the last 20 years or so. In particular, Wang introduced in [Wan] the noncommutative analogue of the permutation group Sn , namely the quantum permutation group As (n). So if one considers noncommuting random variables, it is natural to replace the requirement of invariance under permutations by the stronger requirement of invariance under quantum permutations. Classical (commuting) independent random variables do not satisfy this stronger form of exchangeability any more and, as we will show in our main theorem, this noncommutative version of exchangeability singles out again a very special situation - namely freeness with amalgamation. In the same way as classical exchangeability is equivalent to conditional independence, quantum exchangeability is equivalent to freeness with amalgamation. Thus our noncommutative de Finetti theorem is another instance of the general philosophy that freeness plays in the noncommutative world the same role as independence plays in the commutative world. Note that freeness is not a hidden assumption in our de Finetti theorem, but it is a consequence of replacing the permutation group by its noncommutative counterpart. Here is the statement of our noncommutative de Finetti theorem. All relevant notions will be defined in Sects. 2 and 4. Theorem 1.1. Let (A, ϕ) be a W ∗ -probability space and consider an infinite sequence of selfadjoint elements (xi )i∈N in A. Assume that the xi (i ∈ N) generate A as a von Neumann algebra. Then the following two statements are equivalent: (a) The joint distribution of (xi )i∈N with respect to ϕ is invariant under quantum permutations. (b) The sequence (xi )i∈N is identically distributed and free with respect to the ϕ-preserving conditional expectation E onto the tail algebra of the (xi )i∈N . Our paper is organized as follows. In the next section we collect the preliminaries. On one side, we present the definition of the quantum permutation group and the notion of invariance under quantum permutations. On the other side, we recall the basic definitions and relevant results about free independence with amalgamation. In Sect. 3, we will prove the “easy” implication of our de Finetti theorem, namely that freeness with amalgamation implies invariance under quantum permutations. This is actually not as elementary as in the classical case (where it follows directly from the fact that independence is a rule for expressing mixed moments in terms of moments of the single random variables) and we will have to use some of the basic theory of freeness for this proof. In Sect. 4, we will define the tail algebra of our sequence of random variables, and show some basic properties of the corresponding conditional expectation. Section 5 will finally give the proof of the other implication of our de Finetti theorem, Theorem 1.1. The paper closes with an example which shows that, as in the classical case (see [DF]), one needs infinitely many random variables in our de Finetti theorem: quantum exchangeability of finitely many random variables does not necessarily imply freeness with amalgamation. We would like to mention that a recent preprint of Curran [Cur], which was inspired
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by an earlier version of the present paper, gives an exhaustive treatment of the situation concerning finite quantum exchangeable sequences. 2. Preliminaries 2.1. Noncommutative probability spaces and distributions of random variables. Here we recall the basic notions of non-commutative probability spaces and distributions of random variables. Definition 2.1. 1) A noncommutative probability space (A, ϕ) consists of a unital algebra A and a unital linear functional ϕ. 2) A W ∗ -probability space (A, ϕ) is a von Neumann algebra A together with a faithful normal state ϕ on A. Note that for a W ∗ -probability space we do not require that our state ϕ is a trace. Definition 2.2. Let (A, ϕ) be a non-commutative probability space and (xi )∈∈N a sequence in A. The joint distribution of (xi )i∈N is given by the collection of all moments ϕ(xi(1) · · · xi(n) ) for all n ∈ N and all i(1), . . . , i(n) ∈ N. 2.2. Quantum permutation group. Wang introduced in [Wan] the following noncommutative version of the permutation group Sn . Definition 2.3. The quantum permutation group As (n) is defined as the universal unital C ∗ -algebra generated by elements u i j (i, j = 1, . . . , n) such that we have • each u i j is an orthogonal projection: u i∗j = u i j = u i2j for all i, j = 1, . . . , n, • the elements in each row and column of u = (u i j )i,n j=1 form a partition of unity, i.e., are orthogonal and sum up to 1: for each i = 1, . . . , n and k = l we have u ik u il = 0
and
u ki u li = 0;
and for each i = 1, . . . , n we have n
u ik = 1 =
k=1
n
u ki .
k=1
Note that the above requirements imply in particular that the matrix u = (u i j )i,n j=1 ∈ Mn (As (n)) is an orthogonal matrix in the sense that for each i, j = 1, . . . , n we have n k=1
u ik u jk = δi j 1
and
n
u ki u k j = δi j 1.
k=1
As (n) is a compact quantum matrix group, in particular a compact quantum group, in the sense of Woronowicz [Wor], with comultiplication : As (n) → As (n) ⊗ As (n) n determined by (u i j ) = k=1 u ik ⊗ u k j , counit ε : As (n) → C determined by ε(u i j ) = δi j , and antipode S : As (n) → As (n)op determined by S(u i j ) = u ji . (Note that , ε, and S are algebra morphisms and thus determined by the above relations.) That As (n) is the right noncommutative version of the permutation group Sn can be seen from the facts that the abelianization of As (n) yields a Hopf algebra morphism onto the
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group algebra CSn and that, by a theorem of Wang [Wan], As (n) is the biggest Hopf algebra coacting on a space of n points. For more information on As (n), see [BC,BBC]. For n = 1, 2, 3 the quantum permutation group is the same as the usual permutation group, i.e., in these cases As (n) is isomorphic as a Hopf algebra to CSn , see [BBC]. For n ≥ 4, however, the quantum version is strictly larger than the classical one and genuinely non-commutative; this can be seen, for example, by finding infinite-dimensional non-commutative representations of the C ∗ -algebra As (n). Here is a representation in the case n = 4: ⎛ ⎞ q1 1 − q1 0 0 q1 0 0 ⎟ ⎜1 − q1 u=⎝ , 0 0 q2 1 − q2 ⎠ 0 0 1 − q2 q2 where q1 and q2 are arbitrary projections. Since the C ∗ -algebra generated by q1 and q2 can be chosen non-commutative and infinite dimensional (take for example q1 and q2 as free non-trivial projections), As (4), too, must be non-commutative and infinite dimensional. Definition 2.4. Consider a noncommutative probability space (A, ϕ) and a sequence of random variables (xi )i∈N in A. We say that the joint distribution (with respect to ϕ) of this sequence is invariant under quantum permutations or that the sequence is quantum exchangeable if, for any k ∈ N, the natural coaction of As (k) on the k-tuple (x1 , . . . , xk ), given by xi → x˜i :=
k
u i j ⊗ x j ∈ As (k) ⊗ A,
j=1
does not change the distribution, i.e., the joint distribution of the k-tuple (x1 , . . . , xk ) with respect to ϕ is the same as the joint distribution of the k-tuple (x˜1 , . . . , x˜k ) with respect to id ⊗ ϕ. More explicitly, this means: for all k, n ∈ N and all 1 ≤ i(1), . . . , i(n) ≤ k we have ϕ(xi(1) · · · xi(n) )1 =
k
u i(1) j (1) · · · u i(n) j (n) · ϕ(x j (1) · · · x j (n) )
(1)
j (1),..., j (n)=1
for any matrix u = (u i j )i,k j=1 whose entries are bounded operators on some Hilbert space and satisfy the defining relations of As (k) from Definition 2.3. One should note that the quantum group structure of As (n) guarantees that the mapping (xi ) := x˜i , extended as an algebra homomorphism to the algebra generated by the xi , is indeed a coaction in the sense that (id ⊗ ) = ( ⊗ id) and (ε ⊗ id) = id. For a permutation σ ∈ Sk the permutation matrix (ei j )i,k j=1 with ei j = δσ (i) j provides an example of a matrix u as required for (1); this gives ϕ(xi(1) · · · xi(n) ) =
k
δσ (i(1)) j (1) · · · δσ (i(n)) j (n) ϕ(x j (1) · · · x j (n) )
j (1),..., j (n)=1
= ϕ(xσ (i(1)) · · · xσ (i(n)) ), which is just the invariance of the distribution of (xi )i∈N under the permutation σ . Thus invariance under quantum permutations includes in particular invariance under permutations; quantum exchangeable random variables are in particular exchangeable.
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2.3. Freeness with amalgamation. Here we collect the basic definitions and needed facts about freeness. For general introductions on free probability theory, see [VDN,NS,HP]. In the classical de Finetti theorem we do not get ordinary independence of the random variables, but have to condition this over the tail algebra. In the same spirit, in our noncommutative de Finetti theorem, we cannot hope for ordinary freeness with respect to the state ϕ, but must expect that we have to condition this with respect to the tail algebra of the random variables. Voiculescu introduced such a conditional version of freeness (called operator-valued freeness or freeness with amalgamation) from the very beginning and developed its basic theory in [Voi]. In [Spe] this concept was treated from the combinatorial point of view and it was shown that the theory of free cumulants extends to the operator-valued frame. As our proof of the “easy” direction of Theorem 1.1 relies on free cumulants, we will below recall the relevant facts about operator-valued free cumulants. Let us first give the definition of an operator-valued probability space and freeness. This will be done in a general, algebraic context. Recall that a conditional expectation E : A → B (for unital algebras B ⊂ A) is a linear map which satisfies E[b] = b for all b ∈ B and the bimodule property E[b1 ab2 ] = b1 E[a]b2
for all b1 , b2 ∈ B and for all a ∈ A.
Definition 2.5. 1) An operator-valued probability space (A, E : A → B) consists of a unital algebra A, a unital subalgebra B ⊂ A and a conditional expectation E : A → B. Elements in A are called (operator-valued) random variables. 2) For a unital algebra B we denote by B X the B-valued polynomials in the formal variable X ; these are linear combinations of elements of the form b0 X b1 X · · ·bn−1 X bn for all n = 0, 1, 2, . . . and all b0 , . . . , bn ∈ B. (For n = 0, this is just b0 .) Elements from B do not commute with X (with the exception of 1 · X = X = X · 1). For p ∈ B X and a ∈ A (for some algebra A which contains B as a subalgebra) we denote by p(a) ∈ A the element which one gets by replacing the variable X by a. 3) Let (A, E : A → B) be an operator-valued probability space and (xi )i∈N a sequence of random variables in A. We say that the sequence is identically distributed (with respect to E) if for each p ∈ B X the expression E[ p(xi )] does not depend on i ∈ N. In the case of an ordinary noncommutative probability space, i.e., B = C and E = ϕ, the bi in the definition of B X = C X are superfluous and C X are ordinary polynomials; in this case “identically distributed” means that for each n ∈ N the ordinary moment ϕ(xin ) does not depend on i. Definition 2.6. Let (A, E : A → B) be an operator-valued probability space and I an arbitrary index set. Random variables (ai )i∈I are called free with respect to E (or free with amalgamation over B) if we have for all n ∈ N, all i(1), . . . , i(n) ∈ I with i(1) = i(2) = · · · = i(n) and all B-valued polynomials p1 , . . . , pn ∈ B X with E[ pm (ai(m) )] = 0 (m = 1, . . . , n) that also E[ p1 (ai(1) ) · · · pn (ai(n) )] = 0. The special case where B is C (and thus E a unital linear functional ϕ : A → C) gives the usual definition of freeness.
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2.4. Operator-valued free cumulants. The combinatorial theory of operator-valued freeness [Spe] relies on the notions of non-crossing partitions and free cumulants. We will now recall these notions. Definition 2.7. 1) A partition π of a set S is a decomposition π = {V1 , . . . , Vr } of S into disjoint, non-empty subsets Vi . The elements Vi are called the blocks of π . We denote the partitions of S by P(S). In the case S = {1, . . . , n}, we write P(n). 2) For π, σ ∈ P(n) we say that π ≤ σ if each block of π is contained in a block of σ . 3) Let S be an ordered set. A partition π ∈ P(S) is called non-crossing if there do not exist two different blocks V, W of π such that we have s1 < t1 < s2 < t2 and s1 , s2 ∈ V and t1 , t2 ∈ W . The set of non-crossing partitions of S is denoted by N C(S), or N C(n) in the case of S = {1, . . . , n}. If one draws partitions by connecting elements belonging to the same block by half-circles below the numbers 1, . . . , n, then the partition is non-crossing if and only if one does not get crossings between different blocks in such a drawing. Another characterization of a non-crossing partition is the following recursive description: π ∈ P(S) is non-crossing if at least one of the blocks of π , say V , is an interval (i.e., consists of consecutive numbers) and if, after removing this block V , π \V is a non-crossing partition of S\V . Definition 2.8. Let (A, E : A → B) be an operator-valued probability space. 1) A map ρ : An → B (for n ∈ N) is called a B-functional if it is n-linear and if we have for all b0 , . . . , bn ∈ B and all a1 , . . . , an ∈ A that ρ(b0 a1 b1 , a2 b2 , . . . , an−1 bn−1 , an bn ) = b0 ρ(a1 , b1 a2 , . . . , bn−2 an−1 , bn−1 an )bn . 2) Let, for each k ∈ N, a B-functional ρk : Ak → B be given. Then, for n ∈ N and π ∈ N C(n) we define a B-functional ρπ : An → B recursively as follows. If π is the maximal element 1n ∈ N C(n), which has only one block, then we put for all a1 , . . . , an ∈ A, ρ1n [a1 , . . . , an ] = ρn (a1 , . . . , an ). Otherwise, let V = (i + 1, . . . , i + r ) be an interval of π . Then, for a1 , . . . , an ∈ A, ρπ [a1 , . . . , an ] = ρπ \V [a1 , . . . , ai−1 , ai · ρr (ai+1 , . . . , ai+r ), ai+r +1 , . . . , an ] . As an illustration of this definition consider π = {{1, 10}, {2, 5, 9}, {3, 4}, {6}, {7, 8}} ∈ N C(10), 1
2
3
4
5
6
7
8
9 10
The corresponding ρπ is ρπ [a1 , . . . , a10 ] = ρ2 (a1 · ρ3 (a2 · ρ2 (a3 , a4 ), a5 · ρ1 (a6 ) · ρ2 (a7 , a8 ), a9 ) , a10 ).
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Definition 2.9. Let (A, E : A → B) be an operator-valued probability space. The corresponding operator-valued free cumulants (κnE )n∈N are defined recursively by the moment-cumulant formulas: for each n ∈ N and all a1 , . . . , an ∈ A we have E[a1 · · · an ] =
π ∈N C(n)
κπE [a1 , . . . , an ].
(2)
Note that in the moment-cumulant formula (2) the right-hand side is of the form κnE (a1 , . . . , an ) plus products of lower order terms; thus this can indeed recursively be solved for the κnE . There is a quite a lot one can say about the structure of the formulas for the κnE , but we will not need this here and refer for more information on this to [NS,Spe]. Here are as examples just the first three cumulants: κ1E (a1 ) = E[a1 ],
κ2E (a1 , a2 ) = E[a1 a2 ] − E[a1 ] · E[a2 ] ,
and κ3E (a1 , a2 , a3 ) = E[a1 a2 a3 ] − E[a1 ] · E[a2 a3 ] − E [a1 · E[a2 ] · a3 ] −E[a1 a2 ] · E[a3 ] + 2E[a1 ] · E[a2 ] · E[a3 ]. The main result which we will use about free cumulants is that they characterize freeness via the property “vanishing of mixed cumulants”. Theorem 2.10 ([Spe]). Let (A, E : A → B) be an operator-valued probability space and consider, for some index set I , random variables (ai )i∈I . Then the following are equivalent: (1) The random variables (ai )i∈I are free with respect to E. (2) We have the vanishing of mixed operator-valued free cumulants: For all n ≥ 2, all i(1), . . . , i(n) ∈ I , and all b1 , . . . , bn−1 ∈ B we have κnE (ai(1) b1 , . . . , ai(n−1) bn−1 , ai(n) ) = 0 whenever there are 1 ≤ k, l ≤ n such that i(k) = i(l). As a consequence of this we have: freeness of the ai implies that κπE [ai(1) , . . . , ai(n) ] can only be non-zero when all the i-indices belonging to the same block of π are equal. It will be convenient to have a notation at hand which encodes that information. Notation 2.11. For n ∈ N and an n-tuple i = (i(1), . . . , i(n)) we denote by ker i ∈ P(n) that partition of 1, . . . , n which is determined by k and l are in the same block
⇔
i(k) = i(l).
With this notation we have: if (ai )i∈I are free with respect to E, then κπE [ai(1) , . . . , ai(n) ] can only be non-zero for ker i ≥ π . Note that ker i is in general a possibly crossing partition.
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3. Operator-Valued Free Random Variables are Invariant under Quantum Permutations We will now first prove the “easy” direction of our de Finetti theorem, namely that random variables which are free with respect to a conditional expectation E are invariant under quantum permutations with respect to any ϕ which is compatible with E. In contrast to the other direction this can be done in a purely algebraic frame, thus we will treat this implication in the context of an arbitrary non-commutative probability space. Note also that this implication does actually not require that our sequence is infinite. Proposition 3.1. Let (A, ϕ) be a noncommutative probability space, B ⊂ A a unital subalgebra, and E : A → B a conditional expectation such that ϕ = ϕ ◦ E. Consider a sequence (xi )i∈N in A which is identically distributed and free with respect to E. Then the joint distribution of the sequence (xi )i∈N with respect to ϕ is invariant under quantum permutations. Proof. Fix n, k and i = (i(1), . . . , i(n)) with 1 ≤ i(1), . . . , i(n) ≤ k. We have k
u i(1) j (1) · · · u i(n) j (n) · ϕ x j (1) · · · x j (n)
j (1),..., j (n)=1
=
k
u i(1) j (1) · · · u i(n) j (n) · ϕ E[x j (1) · · · x j (n) ]
j (1),..., j (n)=1
=
k
⎛ u i(1) j (1) · · · u i(n) j (n) · ϕ ⎝
j (1),..., j (n)=1
=
π ∈N C(n)
k
π ∈N C(n) j (1),..., j (n)=1
⎞ κπE [x j (1) , . . . , x j (n) ]⎠
u i(1) j (1) · · · u i(n) j (n) · ϕ κπE [x j (1) , . . . , x j (n) ] .
Now we note that, by the vanishing of mixed cumulants for free variables, the term κπE [x j (1) , . . . , x j (n) ] is only non-vanishing if ker j ≥ π , where j = ( j (1), . . . , j (n)). Furthermore, by the identical distribution with respect to E of our random variables, for any j with ker j ≥ π the term κπE [x j (1) , . . . , x j (n) ] has the same value, which we denote by κπE . Thus we can continue the above calculation as follows: k j (1),..., j (n)=1
=
π ∈N C(n)
u i(1) j (1) · · · u i(n) j (n) · ϕ(x j (1) · · · x j (n) ) ϕ κπE
u i(1) j (1) · · · u i(n) j (n) .
j (1),..., j (n)=1,...,k ker j ≥π
The sum over j (1), . . . , j (n) with ker j ≥ π means that we sum for each block of π independently over one j-variable. Since π is non-crossing at least one of its blocks is an interval, i.e., of the form { p, p + 1, p + 2, . . . , p + s} for some 1 ≤ p ≤ p + s ≤ k. Then we have j ( p) = j ( p + 1) = · · · = j ( p + s), the sum over this variable is independent
Noncommutative de Finetti Theorem
481
of the other sums; and it only involves k
u i( p) j u i( p+1) j · · · u i( p+s) j .
j=1
Because of the orthogonality of different elements in the same row of u = (u i j )i,k j=1 , the term u i( p) j u i( p+1) j · · · u i( p+s) j is zero for any j unless i( p) = i( p + 1) = · · · = i( p + s). In the latter case, u i( p) j u i( p+1) j · · · u i( p+s) j = u i( p) j and the sum over j just gives 1. In this way we are left with the same problem as before but with the positions p, p + 1, . . . , p + s removed. For π we have just removed one of its interval blocks. Since π is non-crossing, we can now find another interval block in the new partition and repeat the above argument. In this way we can do all the summations over the blocks of π in an inductive way. In each step the i-indices must agree on the considered block of π to get a non-vanishing contribution. If they do then the summation over the j-index for this block gives 1. So in the end we get that
1, ker i ≥ π . u i(1) j (1) · · · u i(n) j (n) = 0, otherwise j (1),..., j (n)=1,...,k ker j ≥π
Thus, by recalling that κπE is equal to κπE [xi(1) , . . . , xi(n) ] for any i with ker i ≥ π , we have k
u i(1) j (1) · · · u i(n) j (n) · ϕ(x j (1) · · · x j (n) ) =
j (1),..., j (n)=1
π ∈N C(n) ker i ≥π
ϕ κπE
⎛
⎜ = ϕ⎜ ⎝
π ∈N C(n) ker i ≥π
⎞ ⎟ κπE ⎟ ⎠
⎛
⎜ = ϕ⎜ ⎝
π ∈N C(n) ker i ≥π
⎞ ⎟ κπE [xi(1) , . . . , xi(n) ]⎟ ⎠
= ϕ E[xi(1) · · · xi(n) ]
= ϕ xi(1) · · · xi(n) . 4. Properties of the Conditional Expectation onto the Tail Algebra In order to make the step from quantum exchangeability to freeness with amalgamation we need some more analytic structure. Consider a W ∗ -probability space (A, ϕ) and suppose (xi )i∈N is a sequence of selfadjoint random variables in A that generates A as a von Neumann algebra.
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Notation 4.1. The tail algebra of the sequence (xi )i∈N is given by Atail :=
∞
v N (xk | k ≥ n),
n=1
where v N (xk | k ≥ n) ⊂ A is the von Neumann algebra generated by all xk with k ≥ n. Atail is a von Neumann subalgebra of A. If our sequence is exchangeable, then there exists a unique ϕ-preserving conditional expectation E : A → Atail . This is clear if ϕ is a trace, in which case one does of course not need the exchangeability. The general case, which allows non-tracial states, is treated in [Koe1] and we adapt a proof from therein for the convenience of the reader. Proposition 4.2. Suppose the sequence (xi )i∈N ⊂ A is exchangeable. Then there exists a ϕ-preserving conditional expectation E from A onto Atail . Proof. Recall our assumptions that xi = xi∗ for all i ∈ N and that A is generated by (xi )i∈N . Now exchangeability implies the stationarity of (xi )i∈N and thus the existence of an endomorphism α of A such that ϕ ◦ α = ϕ and α(xi ) = α(xi+1 ). Let A I := v N (xi |i ∈ I ) for I ⊂ N and suppose a, b ∈ |I |<∞ A I . Consequently we can assume a ∈ A I and b ∈ A J such that there exists N ∈ N with I ∩ (J + N ) = ∅. We infer from exchangeability that ϕ(bα n (a)) = ϕ(bα n+1 (a)) for all n ≥ N . This establishes the limit lim ϕ(bα n (a))
n→∞
on the weak*-dense *-algebra |I |<∞ A I . Approximation ensures now the existence of this limit for a, b ∈ A, using the norm density of the functionals {ϕ(b ·)|b ∈ A} and the boundedness of the set {α n (a)|n ≥ 0}. We conclude from this that the pointwise limit of the sequence (α n )n∈N (in the weak operator topology) defines a linear map Q : A → A such that Q(A) ⊂ Atail . It is easily seen that the linear map Q enjoys ϕ = ϕ ◦ Q and Q(a) ≤ afor a ∈ A. Thus Q is a conditional expectation from A onto Atail , if we can ensurethat Q(a) = a for all a ∈ Atail (see, e.g., [Tak]). To this end let a ∈ Atail and b ∈ |I |<∞ A I . We infer from Atail ⊂ α N (A) and A[N ,∞) ⊂ α N (A) for all N ∈ N that there exists some N ∈ N such that a ∈ α N (A) and b ∈ A[0,N −1] . We approximate a ∈ A in the weak operator topology by a sequence (ak )k∈N ⊂ |I |<∞ α N (A I ) and conclude further from the definition of Q and from exchangeability that ϕ(bQ(a)) = lim ϕ(bQ(ak )) = lim lim ϕ(bα n (ak )) = lim ϕ(bak ) = ϕ(ba). k
k
n
k
This shows that Q(a) = a for all a ∈ Atail . Thus Q is the conditional expectation of A onto Atail with respect to ϕ, which we denote from now on by E.
Noncommutative de Finetti Theorem
483
Our main goal will be to show that quantum exchangeability implies freeness with respect to this conditional expectation E : A → Atail . Let us first check that quantum exchangeability with respect to ϕ extends to the same property with respect to E. Proposition 4.3. Let (A, ϕ) be a W ∗ -probability space, (xi )i∈N a sequence of selfadjoint elements in A, and E the conditional expectation onto the corresponding tail algebra Atail . Assume that the joint distribution of (xi )i∈N with respect to ϕ is invariant under quantum permutations. Then the same is true for the joint distribution of (xi )i∈N with respect to E, i.e., for each k ∈ N and u = (u i j )i,k j=1 the generating matrix of As (k), we have for all n ∈ N, all 1 ≤ i(1), . . . , i(n) ≤ k and all b2 , . . . , bn ∈ Atail that E[xi(1) b2 xi(2) · · · bn xi(n) ] k
=
u i(1) j (1) · · · u i(n) j (n) · E[x j (1) b2 x j (2) · · · bn x j (n) ].
(3)
j (1),..., j (n)=1
More generally, for any p1 , . . . , pn ∈ Atail X we have E[ p1 (xi(1) ) p2 (xi(2) ) · · · pn (xi(n) )] k
=
u i(1) j (1) · · · u i(n) j (n) · E[ p1 (x j (1) ) p2 (x j (2) ) · · · pn (x j (n) )]. (4)
j (1),..., j (n)=1
Proof. Fix n, k, i(1), . . . , i(n). Because of ϕ = ϕ ◦ E, (3) will follow if we can show that
ϕ b1 xi(1) b2 xi(2) · · · bn xi(n) =
k
u i(1) j (1) · · · u i(n) j (n) · ϕ b1 x j (1) b2 x j (2) · · · bn x j (n)
(5)
j (1),..., j (n)=1
for all b1 , . . . , bn ∈ Atail . This will follow if we can show Eq. (5) for b1 , . . . , bn of the form xr (1) · · · xr ( p) for p ∈ N and all r (1), . . . , r ( p) ≥ k + 1. (Note that those b1 , . . . , bn are not from the tail algebra, but we can use them to approximate elements from Atail . Indeed, by Kaplansky’s theorem, these approximations can be done on a norm bounded set, where the multiplication of elements is continuous in the strong operator topology.) Fix such a choice of b1 , . . . , bn and let N be the maximum of all indices appearing in the product b1 xi(1) b2 xi(2) · · · bn xi(n) (written as a product in x’s). We extend now the u from Mk (As (k)) to a matrix u˜ = (u˜ i j )i,N j=1 ∈ M N (As (k)) according to
u i j , if 1 ≤ i, j ≤ k . u˜ i j = δi j , otherwise Then this u˜ satisfies the defining relations of As (N ) and Eq. (1) for k = N and u = u˜ yields exactly Eq. (5). (Note that a priori we also get factors u˜ r j corresponding to the factors xr of the b’s, however, all those will just give a δr j contribution, as in this case r ≥ k + 1; thus the b’s from the left side of the equation will just reproduce on the right side of the equation. Also the summation over the j (m)-indices for the xi(m) will a priori be from 1 to N , but the factor u˜ i(m) j (m) restricts this to the range from 1 to k, since i(m) ≤ k.)
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Equation (4) follows from (3) by multilinearity and by checking that we have compatibility of our formulas under multiplying two xi together and under inserting a factor b ∈ Atail . But this is clear from the relations of the u i j ; the first compatibility follows from k
u i j (r ) u i j (r +1) =
j (r ), j (r +1)=1
and the second one from
k
j=1 u i j
k
u i j (r ) ,
j (r )=1
= 1.
It is clear that the same arguments work also for the case of exchangeability. Since we will use this version in the proof of Proposition 4.5, let us state it here explicitly for later use. Proposition 4.4. Let (A, ϕ) be a W ∗ -probability space, (xi )i∈N a sequence of selfadjoint elements in A, and E the conditional expectation onto the corresponding tail algebra Atail . Assume that the joint distribution of (xi )i∈N with respect to ϕ is invariant under classical permutations. Then the same is true for the joint distribution of (xi )i∈N with respect to E, i.e., for each k ∈ N, we have for all n ∈ N, all 1 ≤ i(1), . . . , i(n) ≤ k and all b2 , . . . , bn ∈ Atail that E[xi(1) b2 xi(2) · · · bn xi(n) ] = E[xσ (i(1)) b2 xσ (i(2)) · · · bn xσ (i(n)) ] for each permutation σ ∈ Sk . More generally, for any p1 , . . . , pn ∈ Atail X we have E[ p1 (xi(1) ) p2 (xi(2) ) · · · pn (xi(n) )] = E[ p1 (xσ (i(1)) ) p2 (xσ (i(2)) ) · · · pn (xσ (i(n)) )]
(6)
for each permutation σ ∈ Sk . In the next section we will show how the quantum exchangeability of E will imply freeness with respect to E. For this we will need as an important ingredient the following factorization property of E. This is actually a consequence of the classical exchangeability property with respect to ϕ and was shown in [Koe1] for more general situations. To establish this desired factorization property it is crucial to work with an infinite sequence of random variables (see also Remark 5.3). In order to make the present paper self-contained we provide the proof for this factorization in our case. For more details and generalizations one should see [Koe1]. A related finite version of that result was also considered in Lemma 2.6 of [AL]. Proposition 4.5. Let (A, ϕ) be a W ∗ -probability space and (xi )i∈N a sequence of selfadjoint elements in A whose joint distribution is invariant under classical permutations. Let E be the conditional expectation onto the tail algebra Atail of the sequence (see Proposition 4.2). Then E has the following factorization property: for all n ∈ N, all polynomials p1 , . . . , pn ∈ Atail X and all i(1), . . . , i(n) ∈ N we have E p1 (xi(1) ) · · · pl (xi(l) ) · · · · pn (xi(n) ) (7) = E p1 (xi(1) ) · · · E[ pl (xi(l) )] · · · pn (xi(n) ) whenever i(l) is different from all the other i(r ) (r = l).
Noncommutative de Finetti Theorem
485
Note that exchangeability with respect to ϕ does not imply a factorization property for ϕ, but only for the conditional expectation E. This is of course responsible for the fact that we get freeness with respect to E and not with respect to ϕ in our noncommutative de Finetti theorem. Proof. By L 2 (A, ϕ) we will denote the GNS Hilbert space corresponding to ϕ, equipped with the inner product a, b = ϕ(a ∗ b). The exchangeability of our sequence implies that we can define on L 2 (A, ϕ) an isometric shift α given by α(xi(1) · · · xi(n) ) = xi(1)+1 · · · xi(n)+1 . Restricted to A ⊂ L 2 (A, ϕ), this shift maps A into itself and acts there as an endomorphism. Let us denote the fixed point algebra of this shift by Aα := {a ∈ A | α(a) = a}. Clearly, Aα ⊂ Atail . We want to show that also Atail ⊂ Aα . For this, fix b ∈ Atail and consider, for m ∈ N and r (1), . . . , r (m) ∈ N, the moment ϕ(xr (1) · · · xr (m) b). By approximating b with noncommutative polynomials in {xk | k > max(r (1), . . . , r (m))} and using the exchangeability of the (xi )i∈N , one sees that ϕ(xr (1) · · · xr (m) b) = ϕ(xr (1) · · · xr (m) α(b)) for all m ∈ N, r (1), . . . , r (m) ∈ N. But then one also has ϕ(ab) = ϕ(aα(b)) for all a ∈ A, and hence b = α(b). Since this is true for any b ∈ Atail we actually have that Aα = Atail . Thus Proposition 4.2 entails that the ϕ-preserving conditional expectation E α from A onto Aα exists and equals the conditional expectation E onto the tail algebra. We recall next that the mean ergodic theorem of von Neumann implies that we have for all a ∈ A, m 1 i α (a) = E α [a] = E[a], m→∞ m
lim
i=1
in the strong operator topology (see, e.g., [Koe1]). Now let us consider the situation in our proposition. By Proposition 4.4 we have exchangeability of E according to (6); this means that we have in our situation E p1 (xi(1) ) · · · pl (xi(l) ) · · · pn (xi(n) ) = E p1 (xi(1) ) · · · pl (xi ) · · · pn (xi(n) ) for any i > N := max{i(1), . . . , i(n)}. But then we also have E p1 (xi(1) ) · · · pl (xi(l) ) · · · pn (xi(n) ) N +m 1 E p1 (xi(1) ) · · · pl (xi ) · · · pn (xi(n) ) m i=N +1 N +m 1 = E p1 (xi(1) ) · · · pl (xi ) · · · pn (xi(n) ) . m
=
i=N +1
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But N +m m 1 i 1 pl (xi ) = α ( pl (x N )) m m i=N +1
i=1
converges by the mean ergodic theorem to E[ pl (x N )] = E[ pl (xi(l) )] and thus we get E p1 (xi(1) ) · · · pl (xi(l) ) · · · · pn (xi(n) ) = E p1 (xi(1) ) · · · E[ pl (xi(l) )] · · · pn (xi(n) ) . Note again for the convergence argument that multiplication on norm bounded sets is continuous in the strong operator topology. 5. Invariance under Quantum Permutations Implies Freeness over the Tail Algebra We will now provide the proof of the implication “a ⇒ b” of Theorem 1.1. Throughout this section E will denote the conditional expectation as introduced in Proposition 4.2. Let us first address the identical distribution of the xi with respect to E. For this we actually need only the classical exchangeability. Proposition 5.1. Let (A, ϕ) be a W ∗ -probability space and consider a sequence of selfadjoint elements (xi )i∈N in A. Assume that the joint distribution of (xi )i∈N with respect to ϕ is invariant under classical permutations. Then the sequence (xi )i∈N is identically distributed with respect to the conditional expectation E onto the tail algebra of (xi )i∈N . Proof. This is just a special case of Proposition 4.4.
Now we will address the freeness property. For this one needs, as in the classical case, an infinite sequence of random variables. One should, however, note that the only way in which this infinity enters is via the factorization property of Proposition 4.5 (which relied in the end on the mean ergodic theorem). If this factorization property is assumed then it is feasible that a more elaborated version of the following arguments is also applicable to finite sequences of random variables. We will check that x1 , x2 , . . . are free with respect to E by verifying the defining relations for freeness. So let us consider n ∈ N and polynomials p1 , . . . , pn ∈ Atail X such that E[ pi (x1 )] = 0 for all i = 1, . . . , n. Then we have to show that for all i(1) = i(2) = · · · = i(n), E[ p1 (xi(1) ) · · · pn (xi(n) )] = 0. We will prove this (for fixed n and p1 , . . . , pn ) by induction over the number of blocks of ker i, starting from the biggest number and going down in steps of one. To get started consider the biggest number of blocks, which is n. Then all i(1), . . . , i(n) are different and, by an iterated application of the factorization property, Proposition 4.5, we have E[ p1 (xi(1) ) · · · pn (xi(n) )] = E[ p1 (xi(1) )] · · · E[ pn (xi(n) )] = 0. Now assume, for some r , we have proved that E[ p1 (xi(1) ) · · · pn (xi(n) )] = 0 whenever i(1) = i(2) = · · · = i(n) and ker i has at least r + 1 blocks. We want to show the same for the case that ker i has r blocks.
Noncommutative de Finetti Theorem
487
By Proposition 4.3, we have E[ p1 (xi(1) ) · · · pn (xi(n) )] k
=
u i(1) j (1) · · · u i(n) j (n) · E[ p1 (x j (1) ) · · · pn (x j (n) )]
j (1),..., j (n)=1
=
k
u i(1) j (1) · · · u i(n) j (n) · E[ p1 (x j (1) ) · · · pn (x j (n) )].
(8)
π ∈P (n) j (1),..., j (n)=1 ker j =π
Let us first observe that j’s where two neighboring indices are the same do not contribute; this follows from the fact that u i(s) j (s) u i(s+1) j (s+1) = 0 if j (s) = j (s + 1) because i(s) = i(s + 1). Thus we only have to sum over j = ( j (1), . . . , j (n) in the above sum for which j (1) = j (2) = · · · = j (n). But for those our induction hypothesis applies and thus we see that in the summation over π ∈ P(n) we can restrict to π which have at most r blocks. Since E[ p1 (xi(1) ) · · · pn (xi(n) )] is invariant under permutations, we can fix specific (different) i-values for the r blocks of ker i; let us take 1, 3, 5, . . . , 2r − 1 for them. The above Eq. (8) has to be true for any k and any k × k matrix u the entries of which satisfy the defining relations of As (k). Let us choose k = 2r and u = (u i j )i,2rj=1 of the form ⎛ ⎞ q1 1 − q1 0 0 ... 0 0 ⎜1 − q 1 q1 0 0 ... 0 0 ⎟ ⎜ ⎟ 0 q2 1 − q2 . . . 0 0 ⎟ ⎜ 0 ⎜ 0 0 1 − q2 q2 ... 0 0 ⎟ ⎟, (9) u=⎜ ⎜ . . . . . .. ⎟ .. ⎜ . ⎟ . . . . . . . . . . . ⎜ ⎟ ⎝ 0 1 − qr ⎠ 0 0 0 ... qr 0 0 0 0 . . . 1 − qr qr where q1 , . . . , qr are arbitrary projections. With this choice of u we have that for a non-vanishing u i j the j-value determines the i-value (since we only have the odd numbers as possible i-values), i.e., we have ker j ≤ ker i; thus in the sum (8) we can restrict to π ≤ ker i. But since we also restricted to π with at most r blocks, we are just left with the one possibility π = ker i, i.e., with the above u we can continue (8) as follows: E[ p1 (xi(1) ) · · · pn (xi(n) )] =
2r
u i(1) j (1) · · · u i(n) j (n) j (1),..., j (n)=1 ker j =ker i
⎛ ⎜ =⎜ ⎝
2r
· E[ p1 (x j (1) ) · · · pn (x j (n) )] ⎞
⎟ u i(1) j (1) · · · u i(n) j (n) ⎟ ⎠ · E[ p1 (xi(1) ) · · · pn (xi(n) )] .
j (1),..., j (n)=1 ker j =ker i
In the last step we have used the fact that, because of the identical distribution of the xi ’s with respect to E, the term E[ p1 (x j (1) ) · · · pn (x j (n) )] depends only on ker j.
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If we can show that 2r
u i(1) j (1) · · · u i(n) j (n) j (1),..., j (n)=1 ker j =ker i
(10)
is different from 1, then this implies that E[ p1 (xi(1) ) · · · pn (xi(n) )] has to vanish, and we are done. Note that if ker i is non-crossing then the sum (10) is actually equal to 1 for any u satisfying the relations of the quantum permutation group. However, if ker i is noncrossing then the condition i(1) = i(2) = · · · = i(n) implies that ker i must have at least one singleton, i.e., one i-index appears only once and then the factorization property (7) gives right away that E[ p1 (xi(1) ) · · · pn (xi(n) )] = 0. Thus we can restrict to crossing ker i when considering (10). Note also that if all the u i j in (10) commute, then we will actually get 1 (independent of whether ker i is crossing or non-crossing); this shows that invariance under usual permutations is (clearly) not strong enough to imply freeness. We have to invoke “some real quantum permutations” (i.e., some appropriate non-commutative quotient of the quantum permutation group). So we should choose the q1 , . . . , qr in (9) as non-commuting. However, it turns out that it suffices to take just two of them as non-commuting. Since ker i is crossing we can choose two blocks which have a crossing. For those two blocks we choose some non-commuting projections p and q, whereas for all the other blocks we choose their projections as 1. Then the sum (10) reduces to ( pq)s + ( p(1 − q))s + ((1 − p)q)s + ((1 − p)(1 − q))s or to ( pq)s p + ( p(1 − q))s p + ((1 − p)q)s (1 − p) + ((1 − p)(1 − q))s (1 − p) for some s ≥ 2. It is clear that this is not equal to 1 for generic projections p and q. Actually one has that these expressions are equal to 1 if and only if p and q commute, see the following lemma. This finishes the proof of the implication “a ⇒ b” in Theorem 1.1. Lemma 5.2. Consider an orthogonal projection p and an orthogonal projection q. Let s ≥ 2. Then the following are equivalent. (a) We have ( pq)s + ( p(1 − q))s + ((1 − p)q)s + ((1 − p)(1 − q))s = 1. (b) We have ( pq)s p + ( p(1 − q))s p + ((1 − p)q)s (1 − p) + ((1 − p)(1 − q))s (1 − p) = 1. (c) p and q commute. Proof. It is clear that “ p and q commute” implies the two equations. So it only remains to see that each of these equations implies that p and q commute. Consider the first equation. Multiply this by p from the left and by q from the right to k obtain ( pq)s = pq. Thus we have ( pq)s = pq for all k ≥ 1. Since, by von Neumann’s k alternating projection theorem, the powers ( pq)s converge to the projection p ∧ q, we
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conclude p ∧ q = pq. As p ∧ q is symmetric in p and q, exchanging the roles of the two projections yields pq = qp. In the second case we multiply from the left and from the right by p to obtain ( pq)s p + ( p(1 − q))s p = p. By using the estimates pqp ≥ pqpqp ≥ · · · ≥ ( pq)s p one deduces from this that pqpqp+ p(1−q) p(1−q) p = p, and by elementary expansion of this that ( pqp)2 = pqp. As before this yields pqp = p ∧ q, and thus, by exchanging the roles of p and q, that pqp = qpq. But this implies that (qp − pq)∗ (qp − pq) = ( pq − qp)(qp − pq) = pqp − pqpq − qpqp + qpq = 0. Hence pq = qp. Remark 5.3. As already mentioned before, our noncommutative de Finetti theorem is not true for a finite number of random variables. To infer freeness from quantum exchangeability one needs, as in the classical case, infinitely many variables. To prove that claim we will in the following present an example, which can be considered as an analogue to classical urn models without replacement. (For more precise information on this analogy with urn models we refer to the recent paper [Cur].) Consider the quantum permutation group As (n) itself, with defining matrix u = (u i j )i,n j=1 . Then, by a fundamental result of Woronowicz [Wor], there exists a normalized Haar functional ψ : As (n) → C. Consider now the G N S representation of As (n) with respect to ψ; this gives a W ∗ -probability space (A, ψ), where A is the weak closure of As (n). The defining invariance property of the Haar functional implies that each column of u = (u i j )inj=1 is invariant under quantum permutations from As (n). To be concrete, let us consider the first column, u 11 , . . . , u n1 . Actually, what we have here are just n projections pi := u i1 which are a partition of unity; thus they generate the abelian algebra Cn and the state ψ on this is determined by ψ( pi ) = 1/n for all i = 1, . . . , n. One can check quite easily directly that these p1 , . . . , pn are quantum exchangeable with respect to ψ. (The more abstract formulation in terms of the Haar functional is not needed for this example, but it opens the way to further investigations; see for example the result that, as in the classical case, all finite quantum exchangeable sequences are conditionally of this form [Cur].) However, we claim now that there does not exist a conditional expectation E onto a von Neumann subalgebra B ⊂ Cn with ψ ◦ E = ψ, such that p1 , . . . , pn are identically distributed and free with respect to E. Assume the contrary. Since p1 p2 = 0, we would have 0 = E[ p1 p2 ] = E[ p1 ]E[ p2 ] = E[ p1 ]E[ p1 ]. Since E[ p1 ] is selfadjoint this implies that E[ p1 ] = 0, and thus also ψ( p1 ) = ψ(E[ p1 ]) = 0. However, we have ψ( p1 ) = 1/n and thus arrive at a contradiction. Acknowledgement. We thank Franz Lehner for some helpful comments on an earlier version of the manuscript. We also thank the referee for his very careful reading of the manuscript and his suggestions which improved the presentation.
References [AL] [BC]
Accardi, L., Lu, Y.G.: A continuous version of de finetti’s theorem. Ann. Probab. 21, 1478–1493 (1993) Banica, T., Collins, B.: Integration over quantum permutation groups. J. Funct. Anal. 242, 641–657 (2007)
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[BBC] Banica, T., Bichon, J., Collins, B.: Quantum permutation groups: a survey. Banach Center Publ. 78, 13–34 (2007) [Cur] Curran, S.: Quantum exchangeable sequences of algebras. Preprint. (electronic) http://arXiv.org/abs/ 0812.3428v1[math.OA], 2008 [DF] Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Probab. 8, 745–764 (1980) [Leh] Lehner, F.: Cumulants in noncommutative probability theory. IV. Noncrossing cumulants: De Finetti’s theorem and L p -inequalities. J. Funct. Anal. 239, 214–246 (2006) [HP] Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. Math. Surveys and Monogr. 77, Providence, RJ: Amer. Math. Soc., 2000 [Kal] Kallenberg, O.: Probabilistic Symmetries and Invariance Principles. Probability and Its Applications. Berlin-Heidelberg-New York: Springer-Verlag, 2005 [Koe1] Köstler, C.: A noncommutative extended de Finetti theorem. Preprint. (electronic) http://arXiv.org/ abs/0806.3621v1[math.OA], 2008 [Koe2] Köstler, C.: On Lehner’s ‘free’ noncommutative analogue of de Finetti’s theorem. Preprint. (electronic) http://arXiv.org/abs/0806.3632v1[math.OA], 2008 [NS] Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, no. 335. Cambridge: Cambridge University Press, 2006 [Spe] Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer. Math. Soc., Vol. 132(627), pp. x+88, 1998 [Tak] Takesaki, M.: Theory of Operator Algebras II. Encyclopaedia of Mathematical Sciences. Berlin-Heidelberg-NewYork: Springer, 2003 [Voi] Voiculescu, D.: Operations on certain non-commutative operator-valued random variables. Astérisque, 232, 243–275 (1995) [VDN] Voiculescu, D., Dykema, K., Nica, A.: Free Random Variables. Providence, RI: Amer. Math. Soc., 1992 [Wan] Wang, S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195, 195–211 (1998) [Wor] Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111, 613–665 (1987) Communicated by Y. Kawahigashi
Commun. Math. Phys. 291, 491–512 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0859-4
Communications in
Mathematical Physics
The Metric Entropy of Endomorphisms Lin Shu LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China. E-mail: [email protected] Received: 17 October 2008 / Accepted: 17 April 2009 Published online: 26 June 2009 – © Springer-Verlag 2009
Abstract: We prove that, for a C 2 non-invertible but non-degenerate map on a compact Riemannian manifold without boundary, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents. This generalizes Ledrappier-Young’s entropy formula [5] (for negative Lyapunov exponents of diffeomorphisms) to the case of endomorphisms. 1. Introduction and Statement of the Results Let M be a C ∞ compact connected Riemannian manifold without boundary and let f : M → M be a C 2 diffeomorphism preserving a Borel probability measure µ. Let h µ ( f ) denote the (measure theoretic) entropy of ( f, µ) and, for µ-a.e. x, let λ1 (x) < λ2 (x) < · · · < λr (x) (x) be the distinct Lyapunov exponents of f at x and let ⊕i≤r (x) E i (x) be the corresponding decomposition of Tx M. It is well known by Ledrappier and Young [5] that λi (x)γi (x) dµ, (1.1) hµ( f ) = λi (x)>0
where γi (x) denotes, roughly speaking, the dimension of µ “in the direction of the subspace E i (x)”. (A similar formula for negative Lyapunov exponents holds by considering f −1 in (1.1).) The equality is of high importance in the study of dimension theory of dynamical systems [2] and serves as a milestone in establishing the exact dimensional property of hyperbolic measures [1]. It is interesting to know the relations between entropy, Lyapunov exponents and dimensions when f is non-invertible. Note that, in this case, the unstable manifolds (on which the dimensions {γi }λi >0 in (1.1) are defined) may not exist. Any equation relating This work is supported by National Basic Research Program of China (973 Program) (2007 CB 814800).
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entropy and positive exponents (see Qian-Xie [14] and also [12,13]) must involve some notion of fractal dimension γi to be interpreted with the help of the inverse limit space of (M, f ) (which is a nature extension of (M, f ) to form an invertible system). An alternative way (to generalizing (1.1) for endomorphisms) is to consider the relation between entropy and negative exponents since stable manifolds on M (see (1.3)) exist even when f is non-invertible. In [7], Liu proved (under some mild assumptions on degenerate points of the map) that h µ ( f ) ≤ Fµ ( f ) − λi (x)m i (x) dµ, (1.2) λi (x)<0
where Fµ ( f ) := Hµ (| f −1 ), being the partition of M into single points, is called the folding entropy of ( f, µ) and m i (x) is the multiplicity of λi (x). (The inequality was conjectured by Ruelle in [11] for the purpose of studying positivity of entropy production in nonequilibrium statistical mechanics. Later in [8], Liu also characterized the measures obtaining the equality in (1.2).) It was asked by Liu in [7] whether there holds the generalized formula h µ ( f ) = Fµ ( f ) − λi (x)γi (x) dµ, λi (x)<0
where γi (x) := δi (x) − δi−1 (x) with δi (x) (δ0 (x) = 0) being the supposed dimension of µ on the stable manifold corresponding to λi (x). In this paper, we give a positive answer to this question. Assume a C 2 map f : M → M such that Tx f is non-degenerate at every x ∈ M (i.e. detTx f = 0 at every x ∈ M). Such an f is called a C 2 non-degenerate endomorphism. Let µ be an invariant probability measure of f . Let be the set of points that are regular in the sense of Oseledec [10]. For x ∈ , let λ1 (x) < λ2 (x) < · · · < λr (x) (x) be the distinct Lyapunov exponents of f at x with multiplicities m 1 (x), · · · , m r (x) (x), respectively. Then there exists a sequence of subspaces of Tx M, {0} = V0 (x) ⊂ V1 (x) ⊂ · · · ⊂ Vr (x) (x) = Tx M, such that lim
n→∞
1 log |Tx f n v| = λi (x) n
for all v ∈ Vi (x)\Vi−1 (x), 1 ≤ i ≤ r (x). Set I = {x ∈ : λi (x) ≥ 0 for all 1 ≤ i ≤ r (x)}. For x ∈ I , define W s (x) = {x}. For x ∈ \I , let s(x) = max{i : λi (x) < 0}. For 1 ≤ i ≤ s(x), define W i (x) = {y ∈ M : lim sup n→∞
1 log d( f n (x), f n (y)) ≤ λi (x)} n
(1.3)
and set W s (x) := W s(x) (x). (Here d is the metric on M.) Each W i (x) is called the i th stable manifold of f at x. Let V i (x) denote the arc connected component of W i (x) which contains x. It is a C 1,1 immersed submanifold of M with dimension j≤i m j (x) [6]. Denote by dxi (or simply d i ) the metric on the V i (x) inherited form M.
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Let Bµ (M) denote the completion of the Borel σ -algebra of M with respect to µ so that (M, Bµ (M), µ) constitutes a Lebesgue space. Let i = {x ∈ : i ≤ s(x)}. A measurable partition ξ of (M, Bµ (M), µ) is said to be subordinate to the W i -manifolds of ( f, µ) on i if for µ-a.e. x ∈ i one has ξ(x) ⊂ W i (x) (ξ(x) denotes the element of ξ that contains x) and ξ(x) contains an open neighborhood of x in V i (x) (with respect to the submanifold topology of V i (x)). Fix i with µ(i ) > 0. For x ∈ i , let B i (x, ρ) denote the d i ball in V i (x) centered at x of radius ρ. Let ξ be a measurable partition subordinate to W i . (See Sec. 2.4 for ξ an outline of a construction of such a partition.) Denote by {µx } a system of canonical conditional measures of µ associated with ξ . For x ∈ i , define ξ
δ i (x, ξ ) = lim inf
log µx B i (x, ρ) log ρ
δ i (x, ξ ) = lim sup
log µx B i (x, ρ) . log ρ
ρ→0
and
ρ→0
ξ
As we will show later, δ i (x, ξ ) and δ i (x, ξ ) are constants along the forward orbit of x and there is a measurable function δi on i (independent of ξ ) such that for µ-a.e. x ∈ i , δi (x) = δ i (x, ξ ) = δ i (x, ξ ). The function δi is called the dimension of µ on W i manifolds. We state the main result of this paper Theorem 1.1. Let f : M → M be a C 2 non-invertible but non-degenerate endomorphism and let µ be an f -invariant Borel probability measure on M. Let λ1 (x) < λ2 (x) < · · · < λs(x) (x) be the distinct negative Lyapunov exponents at x and let δi : i → R be the dimension of µ on W i . Then there exist measurable functions γi : i → R with 0 ≤ γi (x) ≤ m i (x) such that at µ-a.e. x, γ j (x) δi (x) = j≤i
for i = 1, . . . , s(x) and h µ ( f ) = Fµ ( f ) −
λi (x)γi (x) dµ(x).
i≤s(x)
Since the functions λi (·), δi (·), s(·) are constants along the forward orbit of a point, Theorem 1.1 can be proved by a decomposition of µ into ergodic components. More precisely, it will follow from Theorem 1.2. Let f : M → M be a C 2 non-invertible but non-degenerate endomorphism and let µ be an f -ergodic Borel probability measure on M. Let λ1 < λ2 < · · · < λs be the distinct negative Lyapunov exponents of f with multiplicities m 1 , . . . , m s ,
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respectively, and let δi be the dimension of µ on W i -manifolds. Then for 1 ≤ i ≤ s there are numbers γi with 0 ≤ γi ≤ m i such that δi = γj j≤i
for i = 1, 2, · · · , s and h µ ( f ) = Fµ ( f ) −
λi γi .
i≤s s The main idea of the proof of Theorem 1.2 is adapted from that of [5]. Let {ξ i }i=1 be a sequence of measurable partitions of (M, Bµ (M), µ) with each ξ i subordinate to the W i -manifolds to be specified in Sec. 2.4. Put ξ 0 := , with being the partition of M into single points. Set γi = δi − δi−1 for 1 ≤ i ≤ s, where δ0 = 0. Theorem 1.2 will follow immediately by adding the formulas: i) Hµ (ξ i | f −1 ξ i ) − Hµ (ξ i−1 | f −1 ξ i−1 ) = −λi (δi − δi−1 ) for 1 ≤ i ≤ s, ii) Hµ (ξ s | f −1 ξ s ) = h µ ( f ).
Note that each Hµ (ξ i | f −1 ξ i ), for 1 ≤ i ≤ s, is independent of the ξ i chosen and ii) has been showed in [8] (see Prop. 2.8). The main difficulty in establishing i) comes from the fact that f is non-invertible. This, unfortunately, can not be completely removed as in [14], especially in showing “≤” of the equality, by lifting the system to its inverse limit space since the lift does not help to split stable manifolds W i into distinct manifolds corresponding to different past paths. The trick to handle “≤” of i) is to set up a special Lyapunov coordinate chart right in M using only the positive semi orbits { f n x, n ≥ 0}, from which one can define as in [5] a transversal metric from (i − 1)th stable manifolds to the i th stable manifolds; in the meanwhile, we carry out the estimations of corresponding transversal dimensions in the inverse limit space of (M, f ), regarding it as a probabilistic disintegration of the original system. We point out that the classical Lyapunov chart (in the inverse limit space) of an endomorphism involves the behavior of the inverse orbits of points and hence is not suitable for us to define the transversal metric mentioned above. We also emphasize that in contrast with [5], even in the case i = 1 of i), we need to use the idea of transversal dimension instead of comparing geometric balls with Bowen balls directly as in the invertible case. This crucial step of argument implicitly tells how folding contributes to the distribution of the dimension of the measure along stable manifolds when the map is not invertible. Although most techniques used in this paper are based on those of [5] for the study of entropy formulas for diffeomorphisms, Theorem 1.1 and Theorem 1.2 do have their own interest. In showing the existence of the dimension of µ on W s manifolds, this (together with [14]) paves the way for proving the existence of pointwise dimension of every hyperbolic measure invariant under a C 2 non-degenerate endomorphism almost everywhere [15]. This also motivates a new Lyapunov dimension formula for endomorphisms [15] (which is different from the classical one for diffeomorphisms [3]). We arrange the paper as follows. In the preliminaries, we give some basic notions and properties about the (forward) Lyapunov metric, the inverse limit space of (M, f ), entropies in this space and partitions subordinate to the stable manifolds. The proof of the main result will appear in Sec. 3, divided into two parts, namely the “≥” direction and the “≤” direction of the equation i) above.
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2. Preliminaries In the sequel, if it is not specified, µ is assumed to be ergodic. Let λ1 < · · · < λr be the distinct Lyapunov exponents of ( f, µ) with m i being the multiplicity of λi . We will assume λ1 < 0 and let λ1 < · · · < λs < 0 denote all the negative exponents. 2.1. Forward Lyapunov metric. The following notion of forward Lyapunov metric first appeared in Katok-Strelcyn [4] and was formally explored in detail in Liu-Qian [6] for random maps. We slightly modify that of [6] for our use. Write Rdim M = Rm 1 × · · · × Rm s × Ri>s m i . For v ∈ Rdim M , let (v 1 , . . . , v s , v s+1 ) be its coordinates with respect to this splitting. Define |v| = max{|v i |i }, where |·|i is the Euclidean norm on Rm i for i ≤ s and Ri>s m i for i = s +1, respectively. For x ∈ , let Vs (x) be as in the Introduction. There exists a splitting of Vs (x) into E 1 (x) ⊕ · · · ⊕ E s (x) such that for each 1 ≤ i ≤ s, Tx f E i (x) = E i ( f x) and 1 log |Tx f n v| = λi for 0 = v ∈ E i (x). n→∞ n lim
Put f 0 = Id. For n, l ≥ 0 and 1 ≤ i ≤ s, define E i (x, n) = E i ( f n x), T l (x, n) = T f n x f l ,
H (x, n) = Tx f n (Vs (x)⊥ ), Sil (x, n) = T l (x, n)| Ei (x,n) ,
U l (x, n) = T l (x, n)| H (x,n) . Let λ0 = min{|λi | : λi = 0} and let 0 < ε < min{λ0 /100, |λi − λ j |/100, i = j} be given. Lemma 2.1. (cf. [6, Chap. III, Lemma 1.1]). There exists a measurable function ι : × Z+ → (0, +∞) such that for each x ∈ and n, l ∈ Z+ , we have ε
ε
i) ι(x, n)−1 e(λi − 2 )l |u| ≤ |Sil (x, n)u| ≤ ι(x, n)e(λi + 2 )l |u|, u ∈ E i (x, n), 1 ≤ i ≤ s; ε ε ii) ι(x, n)−1 e− 2 l |v| ≤ |U l (x, n)v| ≤ ι(x, n)e(λr + 2 )l |v|, v ∈ H (x, n); ε iii) γ (E i (x, n+l), E j (x, n+l)) for i = j, γ (Vs ( f n+l x), H (x, n+l)) ≥ ι(x, n)−1 e− 2s l , where γ (·, ·) denotes the angle between the two associated subspaces of T f n+l x M; ε iv) ι(x, n + l) ≤ ι(x, n)e 2s l . Proof. The proof mimics that of [6, Chap. III, Lemma 1.1]. We give it for completeness. Set k0 = 0 and let ki = ij=1 m j for i = 1, . . . , r . Denote by ρ (1) ≤ · · · ≤ ρ (m) the m Lyapunov exponents, where m = kr . Then ρ ( j) = λi for ki−1 + 1 ≤ j ≤ ki , i = 1, . . . , r . Let x ∈ and let n, l ∈ Z+ . By Oseledec theorem [10], we can choose a basis {w j }mj=1 i of Tx M such that {w j }kj=k ⊂ E i (x, 0), for i = 1, . . . , s, {w j }mj=ks +1 ⊂ H (x, 0) i−1 +1 and
lim
t→+∞
1 log |Tx f t w j | = ρ ( j) for each w j . t
(2.1)
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Moreover, one has for every two non-empty disjoint subsets P, Q ⊂ {1, 2, . . . , m}, 1 log γ (Tx f t E P , Tx f t E Q ) = 0, t
lim
t→+∞
(2.2)
where E P and E Q denote respectively the subspaces of Tx M spanned by {wi }i∈P and {w j } j∈Q . As a consequence of (2.1) and (2.2), we have i) there exists c(x, n) > 0 such that for each w j and t ∈ Z+ , c(x, n)−1 e(ρ
( j) − 1 ε)t 8s
≤ |Tx f n+t w j | ≤ c(x, n)e(ρ
( j) + 1 ε)t 8s
;
ii) the quantity
A(x, n) := inf inf γ (Tx f n+t E P , Tx f n+t E Q )e 4m t P,Q t∈Z+
ε
is positive and satisfies A(x, n + t) ≥ A(x, n)e− 4m t . In particular, if we define
l (0) (x, n) = inf inf γ (Tx f n+t E P , Tx f n+t E Q )e 4m t , P,Q t∈Z+
where P = Q ∈ {{ki−1 + 1, . . . , ki }, i = 1, . . . , s, {ks + 1, . . . , m}}, then it is an everywhere positive measurable function on × Z+ . Now let w = mj=1 α j Tx f n w j ∈ E i (x, n) and k, l ∈ Z+ . First, by using i) twice for t = k, k + l respectively, we have ε
ε
c(x, n)−2 e−(λi + 8s )l− 4s k |Tx f n+k+l w j | ≤ |Tx f n+k w j | ε
ε
≤ c(x, n)2 |Tx f n+k+l w j |e−(λi − 8s )l+ 4s k . (2.3) Note that for a vector space E with an inner product < ·, · > and · , the induced norm, if u, v ∈ E satisfies γ (u, ±v) ≥ q −1 , then u + v ≤ 4q u + v . We have by using ii) inductively for m − 1 times that for t ∈ Z+ , m
m m |α j | · |Tx f n+t w j | ≤ 4 A(x, n)−1 | α j Tx f n+t w j |.
j=1
j=1
If follows from (2.3) and (2.4) that m m α j Tx f n+k+l w j ≤ |α j |· Tx f n+k+l w j |T k+l (x, n)w| = j=1 j=1 ⎛ ⎞ m ε ε ≤⎝ |α j | · Tx f n+k w j ⎠ · c(x, n)2 e(λi + 8s )l+ 4s k j=1
ε ε ≤ T k (x, n)w (4m A(x, n)−m c(x, n)2 )e(λi + 8s )l+ 2s k
(2.4)
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497
and
m k+l n+k+l = (x, n)w α T f w T j x j j=1 ≥4
−m
A(x, n)
m
m
|α j | · Tx f n+k+l w j
j=1
⎞ ⎛ m ε ε ≥ 4−m A(x, n)m c(x, n)−2 e(λi − 8s )l− 4s k ⎝ |α j | · Tx f n+k w j ⎠ j=1 ε
ε
≥ |T k (x, n)w|(4−m A(x, n)m c(x, n)−2 )e(λi − 8s )l− 2s k . Thus the function
k+l |T (x, n)w| −(λi + ε )l− ε k + 8s 2s e l (x, n) := max sup , k, l ∈ Z , 0 = w ∈ E i (x, n) |T k (x, n)w|
k |T (x, n)w| −(λi − ε )l+ ε k + 8s 2s e , k, l ∈ Z , 0 = w ∈ E i (x, n) sup |T k+l (x, n)w| (i)
is finite at each point x ∈ . Similar argument gives that
k+l |T (x, n)w| −(λr + ε )l− ε k (s+1) + 8s 2s e (x, n) := max sup , k, l ∈ Z , 0 = w ∈ H (x, n) l |T k (x, n)w|
k |T (x, n)w| ε l+ ε k 8s 2s , k, l ∈ Z+ , 0 = w ∈ H (x, n) e sup |T k+l (x, n)w| is also finite at each point of x of . Finally, define ι(x, n) = max{l (0) (x, n)−1 , l (i) (x, n), i = 1, 2, . . . , s + 1}. It is easy to verify that it is a measurable function on × Z+ fulfilling the requirement of the lemma. Let x ∈ and n ∈ Z+ . Lemma 2.1 allows us to introduce an inner product < ·, · >x,n on T f n x M such that
u, u x,n =
n
e2(λi −ε)l Sil (x, n − l)−1 u, Sil (x, n − l)−1 u
l=1 +∞
+
e−2(λi +ε)l Sil (x, n)u, Sil (x, n)u , u, u ∈ E i (x, n), 1 ≤ i ≤ s,
l=0
(2.5)
v, v x,n =
n
e−2εl U l (x, n − l)−1 v, U l (x, n − l)−1 v
l=1 +∞
+
l=0
e−2(λr +ε)l U l (x, n)v, U l (x, n)v , v, v ∈ H (x, n)
(2.6)
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L. Shu
and E i (x, n), 1 ≤ i ≤ s and H (x, n) are orthogonal with respect to < ·, · >x,n . Then we define a norm · x,n on T f n x M such that 1
u x,n = [< u, u >x,n ] 2 , u ∈ E i (x, n), 1 ≤ i ≤ s, 1
v x,n = [< v, v >x,n ] 2 , v ∈ H (x, n),
w x,n = max wi x,n , where w = w 1 + · · · + w s+1 ∈ (⊕i≤s E i (x, n)) ⊕ H (x, n).
x.
The sequence of norms { · x,n }+∞ n=0 is called a forward Lyapunov metric at the point
Let ι ∈ N be such that ι := {x ∈ : ι(x, 0) ≤ ι } = Ø. For fixed n ∈ N, the inner product < ·, · >x,n depends continuously on x ∈ ι . We have by (2.5), (2.6) that
Lemma 2.2. (cf. [6, Chap. III, Lemma 1.3]). Let x ∈ ι . The sequence of norms { · +
x,n }+∞ n=0 satisfies for each n ∈ Z , i) eλi −ε u x,n ≤ Si1 (x, n)u x,n+1 ≤ eλi +ε u x,n , u ∈ E i (x, n), 1 ≤ i ≤ s; ii) e−ε v x,n ≤ U 1 (x, n)v x,n+1 ≤ eλr +ε v x,n , v ∈ H (x, n); 1 1 iii) s+1 |w| ≤ w x,n ≤ Aeεn |w|, w ∈ T f n x M, where A = (4ι )s+1 (1 − e−ε )− 2 . Proof. For each u ∈ E i (x, n), we calculate that
Si1 (x, n)u x,n+1 n+1 = e2(λi −ε)l Sil (x, n + 1 − l)−1 Si1 (x, n)u, Sil (x, n + 1 − l)−1 Si1 (x, n)u l=1
+
+∞
e
−2(λi +ε)l
Sil (x, n
+ 1)Si1 (x, n)u, Sil (x, n
+ 1)Si1 (x, n)u
1 2
l=0
= e2(λi −ε)
n
e2(λi −ε)l Sil (x, n − l)−1 u, Sil (x, n − l)−1 u
l=1
+e2(λi −ε) u, u + e2(λi +ε)
+∞
e−2(λi +ε)l Sil (x, n)u, Sil (x, n)u
1 2
.
l=1
From this we easily deduce using (2.5) that eλi −ε u x,n ≤ Si1 (x, n)u x,n+1 ≤ eλi +ε u x,n . Similarly, we have for v ∈ H (x, n), n 1
U (x, n)v x,n+1 = e−2ε e−2εl U l (x, n − l)−1 v, U l (x, n − l)−1 v l=1
+e
−2ε
v, v + e
2(λr +ε)
+∞ l=1
e
−2(λr +ε)l
1 2 l l U (x, n)v, U (x, n)v ,
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499
from which we deduce that e−ε v x,n ≤ U 1 (x, n)v x,n+1 ≤ eλr +ε v x,n . Now let w ∈ T f n x M and write w = w 1 + . . . ws+1 with wi ∈ E i (x, n) for 1 ≤ i ≤ s and w s+1 ∈ H (x, n). Then clearly |w| ≤
s+1
|wi | ≤
i=1
s+1
wi x,n ≤ (s + 1) w x,n ,
i=1
which implies the first inequality in iii). For the other one, we have by i) and iv) of Lemma 2.1 that n ε i e2(λi −ε)l ι(x, n − l)2 e−2(λi − 2 )l |wi |2
w x,n ≤ l=1
+
+∞
1 e
−2(λi +ε)l
2 2(λi + 2ε )l
ι(x, n) e
2
|w |
i 2
l=0
≤
n
e−εl (ι )2 e
ε s (n−l)
+
+∞
l=1
1 e−εl (ι )2 e
ε sn
2
|wi |
l=0
≤ 2ι (1 − e
−ε − 21
)
e
ε 2n
|w |. i
Similarly, we have 1
ε
w s+1 x,n ≤ 2ι (1 − e−ε )− 2 e 2 n |w s+1 |. Note that ε
min{γ (E i (x, n), E j (x, n)), γ (E i (x, n), H (x, n)), i = j, i, j = 1, . . . , s} ≥ (ι e 2s n )−1 .
Hence we have
w n ≤
s+1
ε
1
wi n ≤ [2ι (1 − e−ε )− 2 ]e 2 n (|w 1 | + · · · + |w s+1 |)
i=1
≤ [2ι (1 − e
−ε − 21
)
]e
ε 2n
(4ι e
ε 2s n
s+1 ) wi s
i=1
s+1
≤ (4ι )
(1 − e
−ε − 21 εn
)
e |w|.
Put l(x, n) = Cι(x, n)s+1 for some large C (which depends on λi , ε and the exponential map). Then l(x, n + l) ≤ l(x, n)eεl and A in Lemma 2.2 can be replaced by l(x, 0). Moreover, we have
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L. Shu
Lemma 2.3 (cf. [6, Chap. III, Lemma 1.4]). For x ∈ and l ≥ 0, the map ◦ f ◦ exp f l x : Fx,l := exp−1 f l+1 x
w ∈ T f l x M : w x,l ≤ l(x, 0)−1 e−2εl → T f l+1 x M is well defined and satisfies Lip · (T Fx,l ) ≤ l(x, 0)e2εl , Lip · (Fx,l − T0 Fx,l ) ≤ ε, where Lip · is defined with respect to · x,l and · x,l+1 . It follows that there is some λ (which depends on {λi } and ε) such that
Fx,l (w) x,l+1 ≤ eλ w x,l , if w x,l ≤ e−λ−2ε l(x, 0)−1 e−2εl .
(2.7)
Hence the set Fx,l {w : w x,l ≤ e−λ−2ε l(x, 0)−1 e−2εl } is contained in the domain of Fx,l+1 . For later use, we say that the above reduced domain of Fx,l is well controlled. Let W i (x) be the i th stable manifold of f at x, defined as in (1.3). We have Theorem 2.4 (cf. [6, Chap. III, Theorem 3.1]). For x ∈ , and 1 ≤ i ≤ s, there exists a sequence of j≤i m j -dimensional discs {W i (x, n)}n≥0 with each W i (x, n) ⊂ V i ( f n x) such that i) W i (x, n) = expx Graph(h x,n ) for some C 1,1 map h x,n : Oi (x, n) → (⊕ j>i E j (x, n)) ⊕ H (x, n) tangent to ⊕ j≤i E j (x, n) at f n x, where Oi (x, n) is an open subset of ⊕ j≤i E j (x, n) which contains {u ∈ ⊕ j≤i E j (x, n) : u x,n ≤ l(x, 0)−1 e−2εn } such that h x,n (0) = 0, Lip · h x,n < e−ε and Lip · T h x,n < De2εn for some constant D; ii) f W i (x, n) ⊂ W i (x, n + 1); iii) d i ( f l y, f l z) ≤ l(x, 0)eεn e(λi +2ε)l d i (y, z), for y, z ∈ W i (x, n), l ∈ N. −n W i (x, n). iv) W i (x) = ∪+∞ n=0 f 2.2. Local coordinates on stable manifolds. For x ∈ , let L x,0 : Tx M → Rm 1 × · · · × Rm s × Ri>s m i be a linear map taking E i (x), 1 ≤ i ≤ s, and Vs (x)⊥ onto {0} ji m j , 1 ≤ i ≤ s, and {0} j≤s m j × R j>s m j respectively and satisfying
L x,0 u, L x,0 v = u, vx,0 , for every u, v ∈ Tx M. The closed disk in Rm of radius ρ centered at 0 is denoted by Rm (ρ) and R(ρ) := m R 1 (ρ) × · · · × Rm s (ρ) × Ri>s m i (ρ). We write R (i) (ρ) = Rm 1 (ρ) × · · · × Rm i (ρ) and R r −(i) (ρ) = Rm i+1 (ρ) × · · · × Rm s (ρ) × R j>s m j (ρ). Put Φx,0 := expx ◦L −1 x,0 . Then there is some universal constant K such that for all −1 z, z ∈ R(l(x, 0) ), K −1 d(Φx,0 z, Φx,0 z ) ≤ |z − z | ≤ l(x, 0)d(Φx,0 z, Φx,0 z ). For x ∈ , 1 ≤ i ≤ s, let Wxi (x, 0) be the subset of R(l(x, 0)−1 ), the Φx,0 image of which is the component of W i (x) ∩ Φx,0 R(l(x, 0)−1 ) containing x. For 0 < τ < 1/2, s (x, 0). As in [5, (8.2)], we have denote Wxs (x, 0) ∩ R(τl(x, 0)−1 ) by Wx,τ
The Metric Entropy of Endomorphisms
501
s (x, 0), Lemma 2.5 For x ∈ , there exists τ , 0 < τ < 1/2 such that for all y ∈ Φx,0 Wx,τ −1 i i) Wxi (y, 0) := Φx,0 W (y) ∩ Wxs (x, 0) is a graph of a function from R (i) (l(x, 0)−1 ) to R r −(i) (l(x, 0)−1 ) with norm of derivative less than some constant (< 1); ii) Wxi (y, 0) ⊂ Wxi+1 (y, 0); s (x, 0), then either W i (y, 0) = W i (z, 0) or W i (y, 0) ∩ W i (z, 0) = iii) if z ∈ Φx,0 Wx,τ x x x x Ø.
Based on this, exactly as in [5, (8.3)], one can further introduce a map πx,0 : s (x, 0) → Ri≤s m i to straighten {W i (y, 0) : y ∈ Φ W s (x, 0)} out into planes Wx,τ x,0 x,τ x such that i) πx,0 Wxi (y, 0) lies on a j≤i m j dimensional plane parallel to R j≤i m j × {0} × · · · × {0} and Wxi (y, 0) = Wxi (z, 0), then πx,0 Wxi (y, 0) and πx,0 Wxi (z, 0) lie on distinct planes; s (x, 0) and its image with Lip(π ), ii) πx,0 is a Lipeomorphism between Wx,τ x,0 −1 Lip(πx,0 ) ≤ N x (depending only on l(x, 0) and other constants determined by f ). 2.3. Inverse limit space of (M, f ). Consider M Z endowed with the product topology. Define M := x = (xn )+∞ −∞ : x n ∈ M, f x n = x n+1 , n ∈ Z . Denote by θ the left shift transformation on M. The pair (M, θ ) is called the inverse limit space of (M, f ). Let p be the natural projection from M to M, i.e., p(x) = x0 , ∀ x ∈ M. Then p ◦ θ = f ◦ p on M. For each invariant probability measure µ on M, denote by µ the unique invariant probability measure on M that satisfies µ ◦ p −1 = µ. Then µ is ergodic whenever µ is. Given two partitions ξ1 and ξ2 of M, we say that ξ1 refines ξ2 (ξ1 > ξ2 ) if ξ1 (x) ⊂ ξ2 (x) at µ-a.e. x ∈ M. Denote by the join of two partitions. Proposition 2.6 ([8, Prop. 4.1.1.]). There exists a sequence of measurable partitions ξ 1 > ξ 2 > · · · > ξ s of (M, Bµ (M), µ) with the following properties: i) ξi > f −1 ξ i and ξ i is subordinate to the W i manifolds of ( f, µ); ∞ n −1 i ii) n=0 θ ( p ξ ) is the partition of M into single points. 2.4. Outline of the construction of ξ i . We give a sketch of the construction of ξ i for the reader’s convenience. It is similar to [5, Lemma 9.1.1]. (See also [8].) There exists a measurable set S with the following properties: a) µ(S) > 0; b) S is the disjoint union of a continuous family of embedded discs Dα , where each Dα is an open neighborhood of xα in V s (xα ); c) For µ-a.e. x ∈ M, there is an open neighborhood Ux of x in V s (x) such that for each n ≥ 0, either f n Ux ∩ S = Ø or f n Ux ⊂ Dα for some α;
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L. Shu
d) There is γ > 0 such that i) the d s -diameter of every Dα in S is less than γ ; ii) if x, y ∈ S are such that y ∈ V s (x) and d s (x, y) > γ , then x, y lie on distinct Dα -discs. For i = 1, 2, . . . , s, let ξˆi (x) =
V i (x) ∩ Dα , M\S,
if x ∈ Dα ; if x ∈ S.
−n ξˆi . Then ξ i satisfies our requirement. Let ξ i = +∞ n=0 f i The ξ has the following characterization. Let x ∈ . Then y ∈ ξ i (x) if and only if y ∈ W i (x) and for all n ≥ 0, 1) f n y ∈ S if and only if f n x ∈ S; 2) d s ( f n x, f n y) ≤ γ whenever f n x ∈ S. 2.5. A local coordinate map of S. As in [5, (8.4)], for properly chosen S, one can introduce a continuous (coordinate) map π : S → Rm 1 +···+m s (slightly modifying πw,0 for some density point w of a set {x : l(x, 0) ≤ l0 } ∩ S with positive measure) such that i) for each α with Dα ⊂ S, π | Dα and (π | Dα )−1 are Lipschitz map with Lipschitz constant less than some constant N0 (which depends on l0 ); ii) π = (π1 , . . . , πs ) satisfies for x, y ∈ S with x ∈ Dα and i ≤ s − 1, π j (x) = π j (y) for j = i + 1, . . . , s if and only if y ∈ V i (x) ∩ Dα . 2.6. Entropies in (M, θ ). Let {ξ i }i≤s be as in Proposition 2.6. Denote ξ i = p −1 ξ i . Due to the generating property of ξ i (cf. ii) of Proposition 2.6), a similar argument as in [5, Lemma 9.3.1] gives Proposition 2.7 Let P be a measurable partition of M with Hµ (P) < ∞. Then for µ-a.e. x, 1 ξi lim − log µx (P ∨ ξ i )0n (x) = Hµ (ξ i | f −1 ξ i ), n→∞ n
0 ξi x denotes where µx is the disintegration of µ with respect to ξ i and P ∨ ξ i n n the element of the partition j=0 θ j P ∨ ξ i that contains x. Proposition 2.8 [8, Prop. 4.1.2]. h µ ( f ) = Hµ (ξ s | f −1 ξ s ). 3. Proof of the Main Result Put ξ 0 := , with being the partition of M into single points, and δ 0 = δ 0 = 0. Let ξ 1 > · · · > ξ s be the partitions given in Proposition 2.6. We first show δ i − δ i−1 ≤
1 (Hµ (ξ i | f −1 ξ i ) − Hµ (ξ i−1 | f −1 ξ i−1 )), i = 1, . . . , s. −λi
(3.1)
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503
The comparison of δ i and δ i−1 will be done in (M, θ ) using Prop. 2.7. Note that M is not of finite dimension; the classical Borel density lemma does not apply. Hence a special treatment as in [14] will be adopted to overcome this. Let X be an abstract Lebesgue space, m a Borel probability measure on X , and ξ a ξ measurable partition of X . We denote by m x the disintegrated measure of m with respect to the partition ξ on ξ(x), the element of ξ that contains x. Since Tx f is non-degenerate for any x ∈ M, there are ρ0 , ρ1 > 0 such that, for any x ∈ M, f | B(x,ρ0 ) : B(x, ρ0 ) → M is a diffeomorphism to the image which contains B( f x, ρ1 ). First, we have by Proposition 2.7 and definitions of δ i , δ i that Lemma 3.1 (cf. [5, Lemma 10.2]). There exists a partition P with Hµ (P) < ∞ and a measurable function n 0 : M → N such that for µ-a.e. x, the following five properties are satisfied for all n ≥ n 0 (x): (3.2) L x0 (n) := B i−1 x0 , en(λi +2ε) ⊂ ξ i−1 (x0 ), ξ i−1
log µx0 L x0 (n) ≤ δ i−1 (x0 ) + ε, n(λi + 2ε) 1 ξ i−1 − log µx Pn0 (x) ≥ Hµ (ξ i−1 | f −1 ξ i−1 ) − ε, n 0 i p ξ ∨ P (x) ⊂ B i (x0 , en(λi +2ε) ), n
(3.3) (3.4) (3.5)
0 1 ξi − log µx ξ i ∨ P (x) ≤ Hµ (ξ i | f −1 ξ i ) + ε. (3.6) n n Proof Notice that (3.2), (3.3), (3.4) and (3.6) can be easily satisfied by the definitions of ξ i−1 , δ i−1 and Proposition 2.7. For the lemma, it suffices to find some partition P (with Hµ (P) < ∞) fulfilling (3.5). Let S = S ∩ {x : l(x, 0) ≤ l0 } for some l0 such that µ(S ) > 0. For x ∈ p −1 S , define n − (x) = inf{n > 0 : θ −n (x) ∈ p −1 S }. Then let φ : M → (0, +∞) be
− min{K −1l0−2 e−(λ+4ε)(n (x)+1) , ρ0 }, φ(x) = −1 −λ−4ε min{K e , ρ0 },
if x ∈ p −1 S , if x ∈ p −1 S .
We can follow [8, Lemma 4.2.1] (to use Mañé’s idea [9]) to construct a partition P with Hµ (P) < ∞ such that pP(x) ⊂ B(x, φ(x)) for almost every x. Indeed, take numbers C > 0 and r0 > 0 such that for any 0 < r ≤ r0 , there exists a measurable partition αr of M which satisfies diam αr (x) ≤ r for all x ∈ M and |αr | ≤ Cr −dim M , where |αr | denotes the number of elements of αr . Put Un = {x ∈ M : e−(n+1) < φ(x) ≤ e−n }. Define a partition P of M by requiring P > {Un : n ≥ 0} and P|Un = { p −1 A : A ∈ αrn }|Un , where rn = e−(n+1) . Then clearly, diam pP(x) ≤ φ(x) for any x ∈ M, and, by the µ-integrability of log φ, one has Hµ (P) < ∞.
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L. Shu
Consider y ∈ (ξ i ∨ P)0n (x) for n large. Let k < n be the largest integer such that x−k ∈ S . Since θ −k y ∈ P(θ −k x), |Φx−1 (y−k )| ≤ l0 φ(x−k ) ≤ K −1l0−1 e(λi −4ε)(n−k) . −k ,0 Note that y−k ∈ ξ i (x−k ) and x−k ∈ S , which by our construction of ξ i actually implies y−k ∈ W i (x−k , 0). Hence we have by iii) of Theorem 2.4 that d i (x0 , y0 ) ≤ l0 e(λi +2ε)k d i (x−k , y−k ) ≤ en(λi +2ε) . Proof of (3.1). Let P and n 0 be as in Lemma 3.1. For any 0 < γ < 1, fix N large enough such that if A = {x : n 0 (x) ≤ N } then µ(A) > 1 − γ . We first claim (by using an idea of Qian-Xie [14]) that there is N1 ∈ N such that if we define 1 ξ i−1 µ (L x0 (n)), ∀ n ≥ N1 }, 4 x0 √ then the set A2 := A1 ∩ A has measure greater than 1 − 7 γ . Let {µC : C ∈ ξ i−1 } be a canonical system of conditional measures of µ with respect to the partition ξ i−1 . Denote by µ/ i−1 the corresponding induced measure on the factor ξ √ i−1 space of M with respect to ξ . Put A = {C ∈ ξ i−1 : µC (A) ≥ 1 − γ }, then √ µ(A) = µC (A) dµ/ i−1 ≤ 1 − γ (1 − µ/ i−1 (A)), ξ i−1
A1 = {x : µx
( p −1 L x0 (n) ∩ A) ≥
ξ
which gives µ/
ξ i−1
(A) ≥ 1 −
ξ
√ γ . For each C ∈ A fixed, define
K C := {y ∈ M : µC∩ p
−1 {y}
(A) ≥
1 }. 2
√ Put C := p(C). Then µC (K C ) ≥ 1 − 2 γ and µC ( p −1 L y (n) ∩ A) ≥
1 C µ (L y (n) ∩ K C ). 2
and K ⊂ K C of measure µC (K ) ≥ By Borel density lemma, there exists nˆ = n(C) ˆ C C √ 1 − 3 γ such that µC (L y (n) ∩ K C ) ≥
1 C µ (L y (n)), ∀ n ≥ n, ˆ y ∈ K C . 2
Thus we can define a measurable function nˆ : A → Z+ such that the above equation holds true. Let N1 be a large number such that the set ˆ ≤ N1 } A1 := {C ∈ A : n(C) √ has measure µ/ i−1 (A1 ) ≥ 1 − 2 γ . Therefore, if C ∈ A1 and y ∈ K C ∩ p(A ∩ C), ξ
then for n ≥ N1 ,
µC ( p −1 L y (n) ∩ A) ≥
1 C µ (L y (n)), 4
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505
i.e. p −1 K C ∩ A ∩ C ⊂ A1 . Thus µ(A1 ) ≥ µC (A1 ) dµ/ i−1 ξ A1 ≥ µC ( p −1 K C ∩ A ∩ C) dµ/
ξ i−1
A1
√ √ ≥ (1 − 4 γ )(1 − 2 γ ) √ ≥ 1 − 6 γ,
√ and µ(A2 ) ≥ 1 − 7 γ as claimed. For µ-a.e. x ∈ A2 , there is n(x) ≥ max{N , N1 } such that for n = n(x), in addition to (3.2)-(3.6), x0 = p(x) satisfies ξ i−1
µx
( p −1 L x0 (n) ∩ A)
ξ i−1 µx0 (L x0 (n))
≥
1 , 4
(3.7)
ξi
log µx0 B i (x0 , en(λi +2ε) ) ≥ δ i (x0 ) − ε, n(λi + 2ε) 1 < ε. n log 4
(3.8)
Fix such an x and n = n(x) ≥ N . We have by (3.7) and (3.3) that ξ i−1
µx
1 ξ i−1 µ (L x0 (n)) 4 x0 1 ≥ en(λi +2ε)(δi−1 (x0 )+ε) . 4
( p −1 L x0 (n) ∩ A) ≥
(3.9)
Hence the same argument as in [5, (10.2)] gives ξi
µξx0 B i (x0 , en(λi +2ε) ) = µx p −1 B i (x0 , en(λi +2ε) ) i
≥ #{distinct (P ∨ ξ i )0n -atoms intersecting p −1 L x0 (n) ∩ A} ×minimal measure of such an atom 1 n(λi +2ε)(δi−1 (x0 )+ε) n(Hµ (ξ i−1 | f −1 ξ i−1 )−ε) −n(Hµ (ξ i | f −1 ξ i )+ε) ≥ e e e , 4 where the last inequality holds by (3.4), (3.6) and (3.9). Comparing this with (3.8), we have δ i (x0 ) − δ i−1 (x0 ) − 2ε ≤
Hµ (ξ i | f −1 ξ i ) − Hµ (ξ i−1 | f −1 ξ i−1 ) + 3ε . −λi + 2ε
We need two more lemmas from [5] to show δ i − δ i−1 ≥
1 (Hµ (ξ i | f −1 ξ i ) − Hµ (ξ i−1 | f −1 ξ i−1 )), i = 1, . . . , s. (3.10) −λi
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Lemma 3.2 [5, Lemma 11.3.2]. Let (, ν) be an abstract probability space and m a probability measure on × Rq written m(dw, ds) = m w (ds)ν(dw). Let γ ≥ 0 be such that at m-a.e. (w, s) γ ≤ lim inf ρ→0
log m w (B q (s, ρ)) , log ρ
then γ ≤ lim inf ρ→0
log m(B q (s, ρ)) , m − a.e. log ρ
Lemma 3.3 [5, Lemma 4.1.3]. Let (X, µ) be a Lebesgue space and let p : X → Rn be −1 a measurable map. Disintegrate µ (with respect to the partition { p {t}}t∈Rn ) to get a family of probability measures {µt }t∈Rn . Let α be a partition of X with Hµ (α) < +∞. For t ∈ Rn and A ∈ α, define g A (t) = µt (A), gA (t) =
1 µ( p −1 B(t, ))
p −1 B(t,)
g A ( p (x)) dµ,
g∗A (t) = inf gA (t). >0
Let g, g and g∗ : X → R be given by g(x) =
χ A (x)g A ( p (x)),
A∈α
g (x) =
χ A (x)gA ( p (x)) and
A∈α
g∗ (x) =
χ A (x)g∗A ( p (x)),
A∈α
where χ A denotes the characterization function of the set A. Then g → g almost everywhere on X and − log g∗ dµ ≤ Hµ (α) + log C + 1, where C = C(n) is as in the Besicovitch Covering lemma. As mentioned in the introduction, the main difference between the proof of the inequality (3.10) with that of [5] is the case i = 1, where folding entropy emerges. The idea is to regard W 1 locally as a transversal space over point, with transversal metric defined with the help of the forward Lyapunov metric and estimations of the corresponding dimensions being done in the inverse limit space (this is the only new feature in the proof of the case i ≥ 2 since folding does not interfere the value of transversal dimension there).
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Proof of (3.10) Let β > 0 be given. Fix ε > 0. Assume that ε ≤ β/4, min{λ0 /100, |λi − λ j |/100, i = j}. We divide the proof into several steps. a) Construction of partitions. Pick up a set E ⊂ S ∩ {l ≤ l0 } with µ(E) > 0. For x ∈ p −1 E, let n + (x) and n − (x) be the smallest positive integers k, l such that θ k (x) ∈ p −1 E and θ −l (x) ∈ p −1 E respectively. Define φ : M → (0, +∞) by
if x ∈ p −1 E, min{τ K −1 e−λ−7ε , ρ0 }, φ(x) = + (x),n − (x)}+1) −1 −2 −(λ+7ε)(max{n , ρ0 }, if x ∈ p −1 E, min{τ K l(x, 0) e where λ is as in (2.7) and τ is some reduction constant involved in the construction π . Since µ almost everywhere and log φ is µ-integrable as of the map φ is defined + − p −1 E n (x) dµ = p −1 E n (x) dµ = 1, we can follow Lemma 3.1 to find a partition ζ with Hµ (ζ ) < ∞ such that diam p(ζ (x)) ≤ φ(x) for any x ∈ M. k + It follows from (2.7) that exp−1 xk p(θ ζ (x)) for k ≤ n (x) stays inside the well controlled domain of the map Fx0 ,k . Let ξ 1 > · · · > ξ s be the partitions given in Proposition 2.6. Let P be a finite entropy partition that refines ζ , { p −1 S, M\ p −1 S} and { p −1 E, M\ p −1 E} with E ⊂ S to be specified later. Define ηi = ξ i P − , i = 0, 1, · · · , s,
where P − =
+∞
j=0 θ
− j P.
Then as in [8] (cf. [5, Lemma 3.2.1]), we have
Claim 3.4. For 0 ≤ i ≤ s, Hµ (θ ηi |ηi ) = Hµ (ξ i | f −1 ξ i ). We present the proof for completeness. For 0 ≤ i ≤ s, we have Hµ (θ ηi |ηi ) = h µ (θ, ηi ) = h µ (θ, ξ i ∨ θ −n P − ) = Hµ (ξ i ∨ θ −n P − | θ −1 ξ i ∨ θ −(n+1) P − ) = Hµ (ξ i | θ −1 ξ i ∨ θ −(n+1) P − ) + Hµ (P − | θ n ξ i ∨ θ −1 P − ), where the first term is ≤ Hµ (ξ i | θ −1 ξ i ) (= Hµ (ξ i | f −1 ξ i )) and the second term goes to 0 as n → +∞ since θ n ξ i goes to the partition of M into single points. On the other hand, h µ (θ, ηi ) = h µ (θ, ξ i ∨ P) ≥ h µ (θ, ξ i ) = Hµ (ξ i | f −1 ξ i ), since Hµ (P) < +∞. This proves the claim. b) A transversal metric. For 1 ≤ i ≤ s, we define a transversal metric on ηi (x)/ηi−1 for µ-a.e. x. Let π be as in Sec. 2.5. Define π : ∪n≥0 f −n E → Ri≤s m i by π (x) = π( f r0 (x) (x)), where r0 (x) = inf{n ≥ 0 : f n (x) ∈ E} is the first hitting time of x to E. For x, y ∈ ∪n≥0 f −n E satisfying r0 (x) = r0 (y), we define dτi (x, y) = | πi (x) − πi (y)|,
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where π = ( π1 , . . . , πs ). Denote by Bτi (x, ρ) the corresponding ball centered at x with radius ρ in the metric dτi . Since P refines { p −1 E, M\ p −1 E}, dτi induces a metric on ηi (x)/ηi−1 by letting dτi (y, z) := dτi (y0 , z 0 ), ∀ y, z ∈ ηi (x). Let Bτi (x, ρ) = {y ∈ ηi (x) : dτi (x, y) ≤ ρ}. Bτ1 (x, ρ) is roughly It is clear that Bτi (x, ρ) = ηi (x) ∩ p −1 Bτi (x0 , ρ). In the case i = 1, th 1 r (x ) a branch of r0 (x0 ) preimages of B ( f 0 0 (x0 ), ρ), refined in the inverse limit space. Claim 3.5. There exists N (depends on N0 and l0 ) such that if x ∈ p −1 E, n is the smallest positive integer such that x−n ∈ E and y ∈ θ n (ηi (θ −n x)), then dτi (θ −n x, θ −n y) ≤ N 2 e(−λi +4ε)n dτi (x, y). For x, y above, we have by construction of ηi that exp−1 xk yk for −n ≤ k ≤ 0 belongs to the well controlled domain of the map Fx−n ,k+n (see the remark after Lemma 2.3). Then it follows by Lemma 2.2 and Lemma 2.3 (cf. [5, Lemma 2.3.1]) that (−λi +3ε)n
exp−1
exp−1 x−n y−n x−n ,0 ≤ e x0 y0 x−n ,n ,
which, by iii) of Lemma 2.2, implies (−λi +4ε)n | exp−1 | exp−1 x−n y−n | ≤ l0 e x0 y0 |.
Hence |πx−n ,0 y−n | ≤ N1 e(−λi +4ε)n |πx0 ,0 y0 | for some constant N1 depends on N0 and l0 . The claim follows by noticing that there is N2 (depends on N0 and l0 ) such that for x, y ∈ E, N2−1 |πx,0 x − πx,0 y| ≤ |π(x) − π(y)| ≤ N2 |πx,0 x − πx,0 y|. c) Proof of (3.10) for the case i=1. It is equivalent to show ξ1
log µx p −1 B 1 (x0 , ρ) 1 lim inf ≥ − (Hµ (ξ 1 | f −1 ξ 1 ) − Fµ ( f )), µ − a.e.(3.11) ρ→0 log ρ λ1 Pick up x ∈ M. (The point x will be subjected to a finite number of a.e. assumptions.) Let n ∈ N. Let · · · < r− j < · · · < r−2 < r−1 < 0 ≤ r0 be such that xr− j ∈ E. For k ≤ 0, let j be such that r−( j+1) < k ≤ r− j and define Bτ1 (θ −k x, N 2 j e(λ1 −4ε)(n+r− j ) ), a(x, k) = where N is the constant in Claim 3.5. We show 1 η lim inf − log µx 1 a(x, 0) ≥ (1 − β)(Hµ (ξ 1 | f −1 ξ 1 ) − Fµ ( f ) − β). n→∞ n
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For this, we calculate that for p < n, η µx 1 a(x, 0)
= =
η
p−1
µθ 1−k x a(x, k)
η · µθ 1− p x a(x, p) η1 µ a(x, k + 1) k=0 θ −(k+1) x η p−1 η µθ 1−k x ((θ η1 )(θ −k x)) 1 µθ −k x a(x, k) η1 µθ −k x (θ (a(x, k + 1)) ∩ (θ η1 )(θ −k x)) k=0
η
· µθ 1− p x a(x, p),
where the second equation holds by the invariance of µ and the uniqueness of the disη integration of a measure. Let p = [n(1 − ε)] + 1. Notice that µθ 1− p x a(x, p) ≤ 1. Hence 1 1 η − log µx 1 a(x, 0) ≥ − n n −
1 n
[n(1−ε)]
η
log
η
µθ 1−k x (θ (a(x, k + 1)) ∩ (θ η1 )(θ −k x))
k=0 [n(1−ε)]
µθ 1−k x a(x, k)
η
log µθ 1−k x ((θ η1 )(θ −k x))
k=0
=: (I) + (II). It is clear that (II) will converge (for µ-a.e. x) to (1 − ε)
η
− log µx 1 ((θ η1 )(x)) dµ(x) = (1 − ε)Hµ (θ η1 |η1 ),
which is (1 − ε)Hµ (ξ 1 | f −1 ξ 1 ) by Claim 3.4. For (I), we first observe that a(x, k) ∩ (θ η1 )(θ −k x) ⊂ θ (a(x, k + 1)).
(3.12)
This is automatically true when k = −r−( j+1) + 1 for any j ≥ 0 since dτ1 is defined by pushing points forward to E. When k = −r−( j+1) + 1 for some j, it suffices to show for y ∈ M with y0 ∈ E and let r > 0 be the smallest integer such that y−r ∈ E, then for z ∈ (θ r η1 )(y), dτ1 (θ −r y, θ −r z) ≤ N 2 e(−λ1 +4ε)r dτ1 (y, z), which is true by Claim 3.5. Now we have by (3.12) that (I) ≥ −
1 n
[n(1−ε)] k=0
η
log
µθ 1−k x a(x, k)
η
µθ 1−k x (a(x, k) ∩ (θ η1 )(θ −k x))
.
(3.13)
Since η0 refines η1 , we have by the notion of the disintegration of measures that η
µθ 1−k x (a(x, k) ∩ (θ η1 )(θ −k x)) =
a(x,k)
η
η
µz 0 (θ η1 )(θ −k x) dµθ 1−k x (z).
(3.14)
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For > 0, define g, g , g∗ : M → R by η
g(y) = µ y0 (θ η1 )(y), 1 g (y) = η1 1 µ y Bτ (y, )
η
Bτ1 (y,)
η
µz 0 (θ η1 )(y) dµ y1 (z),
g∗ (y) = inf g (y). >0
We see that
− log g dµ = Hµ (θ η0 |η0 ).
(3.15)
For this, it suffices to show η0 (y) ∩ (θ η1 )(y) = η0 (y) ∩ (θ η0 )(y).
(3.16)
Notice that for z to belong to η0 (y) ∩ (θ η1 )(y), a necessary and sufficient condition is that z −1 = y−1 and z 0 = y0 ; this is equivalent to say that z ∈ η0 (y) ∩ (θ η0 )(y). So (3.16) is true and the integration of − log g is Hµ (θ η0 |η0 ), which is the folding entropy. Next we have that Claim 3.6. g (y) → g, µ-a.e. and − log g∗ dµ < +∞. η
To see this, one considers one η1 -element at a time. Fix x. Substitute (η1 (x), µx 1 ) for (X, µ) in Lemma 3.3, and let p : η1 (x) → Rm 1 be π ◦ p with p and π defined as in the Preliminaries and let α = η1 /(θ η1 ). Then g, g , g∗ as defined above agree η with the corresponding functions in Lemma 3.3. Therefore g (y) → g, µx 1 -a.e. and η1 we have − log g∗ dµx ≤ Hµη1 (θ η1 ) + log C + 1. Integrating over M, this gives x − log g∗ dµ ≤ Hµ (θ η1 |η1 ) + log C + 1 < ∞. Let (x, n, k) = e(λ1 −4ε)(n+r− j ) . We have by (3.13), (3.14) and the definition of the functions g, g , g∗ that (I) ≥ −
1 n
[n(1−ε)]
− log g(x,n,k) (θ −k x).
k=0
By Claim 3.6, there is a measurable function (·) : M → R+ such that for µ-a.e. y ∈ M, if ≤ (y), then
− log g (y) ≤ − log g(y) + ε.
Also since − log g∗ dµ < +∞, there is a number 0 such that if A = {y : (y) ≥ 0 }, then M\A − log g∗ dµ ≤ ε. Moreover, for 0 small we have by (3.15) that A − log g dµ > Fµ ( f ) − ε. For µ almost every x, if n is sufficiently large, then (x, n, k) ≤ 0 for all k ≤ [n(1 − ε)]. To see this, first, by Birkhoff’s ergodic theorem, there is N (x) such that for n ≥ N (x), #{0 ≤ i ≤ n : x−i ∈ E} ≤ 2nµ(E). If n ≥ N (x), then (x, n, k) = e(λ1 −4ε)(n+r− j ) N 2 j ≤ (e(λ1 −4ε)ε N 4µ(E) )n , which is less than 0 for n sufficiently large if we take E such that e(λ1 −4ε)ε N 4µ(E) < 1.
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Thus, we have lim inf (I) n→∞ ⎛
⎞ [n(1−ε)] [n(1−ε)] 1 1 (− log g(θ −k x) + ε) − − log g∗ (θ −k x)⎠ ≥ lim inf ⎝− n→∞ n n −k −k k=0,θ (x)∈A k=0,θ (x)∈ A ≥ (1 − ε) − − log g(x) dµ − ε − − log g∗ (x) dµ A
M\A
≥ (1 − ε)(−Fµ ( f ) − 3ε) ≥ (1 − β)(−Fµ ( f ) − β). So η
lim inf ρ→0
η
log µx 1 p −1 Bτ1 (x0 , ρ) log µx 1 p −1 Bτ1 (x0 , e(λ1 −4ε)(n+r0 ) ) = lim inf n→∞ log ρ (λ1 − 4ε)n Hµ (ξ 1 | f −1 ξ 1 ) − Fµ ( f ) − β . ≥ (1 − β) −λ1 + β
By iii) of Theorem 2.4 and the Lipschitz property of π , we have B 1 (x0 , e(λ1 −4ε)n ) ∩ p(η1 (x)) ⊂ Bτ1 (x0 , e(λ1 −4ε)(n+r0 ) ) for some depending on x0 and N . It follows that η
lim inf ρ→0
log µx 1 p −1 B 1 (x0 , ρ) Hµ (ξ 1 | f −1 ξ 1 ) − Fµ ( f ) − β ≥ (1 − β) . log ρ −λ1 + β
The above inequality also holds for ξ 1 , in place of η1 , by Lemma 3.2. Finally, letting β tend to zero gives (3.11). d) To prove (3.10) for the case i ≥ 2, one considers Bτi (x0 , ρ) in place of Bτ1 (x0 , ρ) in the above argument to obtain η
lim inf ρ→0
log µx i p −1 Bτi (x0 , ρ) Hµ (ξ i | f −1 ξ i ) − Hµ (ξ i−1 | f −1 ξ i−1 ) − β ≥ (1 − β) . log ρ −λi + β (3.17) ξ i ∨P −
Hence by Lemma 3.2, the same inequality holds for µx0 (taking the place of which refines {S, M\S}, {E, M\E}. In other words, if we let the transversal dimension of ζi /ζi−1 is greater than the righthand side of (3.17). Hence exactly the same argument as in [5, (11.4)] gives
η µx i p −1 ) for any P ζi = ξ i ∨ P − , then
δ i − δ i−1 ≥ (1 − β)
Hµ (ξ i | f −1 ξ i ) − Hµ (ξ i−1 | f −1 ξ i−1 ) − β . −λi + β
The desired inequality (3.10) follows immediately.
Acknowledgements. The author is very grateful to the anonymous referee for a careful reading of the manuscript, pointing out a mistake in the original proof of Lemma 3.1, and giving some important remarks which led to this revised version. She also thanks Professor Peidong Liu for constant encouragement.
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References 1. Barreira, L., Pesin, Y., Schmeling, J.: Dimension and product structure of hyperbolic measures. Ann. Math. 149, 755–783 (1999) 2. Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985) 3. Farmer, J., Ott, E., Yorke, J.: The dimension of chaotic attractors. Physica 7D, 153–180 (1983) 4. Katok, A., Strelcyn, J.-M.: Invariant Manifold, Entropy and Billiards; Smooth Maps with Singularities Lec. Not. Math. 1222, Berlin-Heidelberg-New York: Springer Verlag, 1986 5. Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122, 509–539 (1985); The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122, 540–574 (1985) 6. Liu, P.-D., Qian, M.: Smooth Ergodic Theory of Random Dynamical Systems. Lecture Notes in Mathematics 1606, Berlin: Springer-Verlag, 1995 7. Liu, P.-D.: Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 240, 531–538 (2003) 8. Liu, P.-D.: Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents. Commun. Math. Phys. 284, 391–406 (2008) 9. Mané, R.: A proof of Pesin’s formula. Ergod. Th. Dynam. Syst. 1, 95–102 (1981) 10. Oseledeˇc, V.-I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–221 (1968) 11. Ruelle, D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85, 1–23 (1996) 12. Qian, M., Zhang, Z.-S.: Ergodic theory for axiom A endomorphisms. Ergod. Th. & Dynam. Sys. 15, 161–174 (1995) 13. Qian, M., Zhu, S.: SRB measures and Pesin’s entropy formula for endomorphisms. Trans. Amer. Math. Soc. 354, 1453–1471 (2002) 14. Qian, M., Xie, J.-S.: Entropy formula for endomorphisms: relations between entropy, exponents and dimension. Disc. Contin. Dyn. Syst. 21, 367–392 (2008) 15. Shu, L.: Dimension theory for invariant measures of endomorphisms. Manuscript Communicated by G. Gallavotti
Commun. Math. Phys. 291, 513–532 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0858-5
Communications in
Mathematical Physics
Constructing Locally Connected Non-Computable Julia Sets Mark Braverman1, , Michael Yampolsky2, 1 Microsoft Research New England, One Memorial Drive, Cambridge,
MA 02142, USA
2 Mathematics Department, University of Toronto, 100 St. George Street,
Toronto, Ontario, M5S 3G3, Canada. E-mail: [email protected] Received: 18 October 2008 / Accepted: 14 April 2009 Published online: 12 July 2009 – © Springer-Verlag 2009
Abstract: A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. 1. Preliminaries In this paper, we will assume that the reader is familiar with the concept of computability of a subset of Rn and its applications to Julia sets of rational functions. We refer the reader to our paper [BY08a] and the book [BY08b] for an introduction to computability over the reals, as it applies to the study of Julia sets. A detailed treatment of computability over the reals is found in [Wei00]. We will denote f c (z) = z 2 + c, and Pθ (z) = z 2 + e2πiθ z , two parameterizations of the quadratic family. The latter is more convenient in studying quadratics with a neutral fixed point. We denote Jc , Jθ and K c , K θ the Julia sets and the filled Julia sets respectively. When a polynomial Pθ with θ ∈ (0, ∞) has a Siegel disk at the origin, we label it θ . Consider a conformal isomorphism φ : D → fixing 0. The conformal radius of the Siegel disk θ is the quantity r (θ ) = |φ (0)|. For all other θ ∈ [0, ∞) we set r (θ ) = 0. This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow. The second author’s research is supported by NSERC operating grant.
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Informally, the Julia set Jθ is computable, if given arbitrarily good approximations of the parameter θ , a Turing Machine can output images of Jθ with an arbitrarily high resolution. The parameter θ is provided to the machine via an oracle, which the machine can query with an arbitrarily high precision. We have shown in [BBY07] that for polynomials Pθ computability of the Julia set is equivalent to the computability of the conformal radius r (θ ) of the Siegel disc: Theorem 1.1. The set Jθ is computable by a Turing Machine with an oracle for θ if and only if the number r (θ ) is computable with an oracle for θ . A number r ∈ R is called right-computable if there exists a Turing Machine which computes a sequence of rationals rn r. As we have shown in [BY08a]: Theorem 1.2. Let a real number r ∈ (0, sup r (θ )). θ∈T
Then r = r (θ ) is the conformal radius of a Siegel disk θ with a computable parameter θ if and only if r is right-computable. In [BY08b] (Theorem 6.16) it has been shown that there exist parameters θ for which the Julia set Jθ is non-computable and locally connected. However the proof was nonconstructive. On the other hand, by Theorem 1.2 any right-computable number that is not computable gives rise to a non-computable Julia set with a computable parameter θ . The goal of the present note is to strengthen both results by showing that non-computable locally-connected examples can be generated constructively. Main Theorem. There exists a computable value of θ for which the Julia set Jθ is non-computable and locally connected. As we will see from the discussion in the next section, locally connected Siegel Julia sets Jθ have very simple explicit topological models. This makes the above result very surprising: the picture of Jθ is topologically simple, the parameter θ can be constructively produced (in fact, computable in polynomial time, assuming a widely believed conjecture), yet no algorithm to draw Jθ exists. 2. Locally Connected Quadratic Julia Sets 2.1. Local connectedness of sets in C. Recall, that a topological space X is locally connected if for each point x ∈ X there exists a sequence of neighborhoods Ui (x) x such that: (1) Ui (x) is open and connected in X ; (2) ∩Ui (x) = {x}. We remark that the condition (1) can be weakened: (1a) Ui (x) is connected in X and contains an open neighborhood around x.
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The main significance of local connectedness in the study of quadratic Julia sets comes from the following construction. Consider a quadratic polynomial f c (z) = z 2 + c with a connected Julia set. The Riemann mapping ˆ \ Kc → C ˆ \U :C is uniquely determined by the normalization (∞) = ∞ and (z) ∼ z for z → ∞. It then coincides with the Böttcher coordinate of f c (z) at infinity: ( f c (z)) = ((z))2 .
(2.1)
ˆ \ U, Eq. (2.1) implies As the map z → z 2 preserves the polar coordinate grid on C that the preimages of polar coordinate lines under form an invariant grid for f c . In particular, each radial curve Rθ ≡ −1 ({r e2πiθ | r ∈ (1, ∞)}) is mapped onto the curve Rθ by f c , with θ ≡ 2θ mod Z. These curves are known as the external rays of Jc . For a fixed angle θ , as r → 1+, the points r e2πiθ approach the Julia set Jc . We say that a ray Rθ lands at a point z ∈ Jc if lim −1 (r e2πiθ ) = z.
r →1+
In this case, the point z is accessible from infinity. The equipotential curve Er for r > 0 is the preimage Er ≡ −1 ({e2πiθ·r | θ ∈ T}). It is mapped to E 2r by f c . It is well-known that a connected Julia set may fail to be locally connected. In particular, the following theorem was proved by Doaudy and Sullivan [Sul83], and independently by Lyubich [Lyu86]: Proposition 2.1. If a polynomial f c has a periodic point of Cremer type, then its Julia set is not locally connected. Moreover, if f c has a cycle of Siegel disks, and Jc is locally connected, then necessarily the critical point 0 of f c is contained in the boundary of one of the periodic Siegel disks. In the case when the Julia set Jc is locally connected, a key to its topological structure is given by the Theorem of Carathéodory. Recall that a set K ⊂ C is full if its complement is connected in C: Caratheodory’s Theorem. For a connected compact and full set K ⊂ C denote the Riemann mapping ˆ \K →C ˆ \ U with (∞) = ∞ and (∞) = 1. :C Then the following conditions are equivalent: • • • •
the set K is locally connected; the set J = ∂ K is locally connected; the inverse mapping −1 extends continuously to a map S 1 → J ; every radial ray −1 ({r e2πiθ | r > 1}) lands at a point of J .
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As an immediate corollary we have the following: Corollary 2.2. Assume that the Julia set of f c is connected and locally connected. Then the inverse Böttcher map −1 continuously extends to a surjection ψ : S 1 → Jc which is a semi-conjugacy ψ(z 2 ) = f c (ψ(z)). The parametrization γc : θ → z = exp(2πiθ ) → ψ(z) ∈ Jc is known as the Carathéodory loop of Jc . 2.2. Locally connected Siegel Julia sets. As was first shown by Herman [Her85], there exist quadratic polynomials in the family Pθ (z) = z 2 + e2πiθ z with a Siegel disk θ at the origin, such that the critical point pθ = −e2πiθ /2 ∈ / ∂θ . By Proposition 2.1 in this case Jc is not locally connected. There exist, however, topologically well-behaved examples with Siegel disks as we will see in the following section. Assume now that Pθ has a Siegel disk with the critical point pθ = −e2πiθ /2 in the boundary. Assume further that this point is accessible from infinity. In this case, Jθ \{ pθ } has two connected components; we denote L 0 the one which does not contain θ . A limb of generation n is a component of Pθ−n (L 0 ). There exist various natural ways of labeling limbs of generation n. For instance, denote R1 and R2 the two external rays which land at pθ , and set
= R1 ∪ R2 ∪ { pθ }. Then we have two well-defined branches of the inverse map Pθ−1 mapping C \ Pθ ( ) to one of the components of C \ . Let us denote ψ0 the inverse branch which fixes θ , and ψ1 the other one. We can then distinguish the limbs of the same generation by the order in which the two inverse branches were applied, so for σ¯ ∈ {0, 1}n , we have L σ¯ = ψσn ◦ · · · ◦ ψσ1 (L 0 ). Theorem 2.3. The Julia set Jθ is locally connected if and only if the following three properties hold: (I) ∂θ is a Jordan curve, and contains pθ ; (II) the point pθ is accessible from infinity; (III) there exists a positive function s : N → R with s(n) −→ 0 such that the diameter n→∞
of each limb of generation n is bounded from above by s(n). The necessity of Condition (III) is not difficult to see. If there existed a non-trivial accumulation set of an infinite sequence of limbs (a “ghost limb”) then all its points would have to correspond to a single external ray Rθ , in violation of Carathéodory’s Theorem. As for the sufficiency of Conditions (I)-(III), the limbs themselves can be used to construct a basis of connected neighborhoods. For more details, see e.g. [Yam99].
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1
Fig. 1. On the left is the filled Julia set of the quadratic polynomial Pθ with θ =
. The critical 1 1 + ··· point pθ is on the boundary of the Siegel disk; the two external rays landing at pθ and the initial limb L 0 are also indicated. On the right is an illustration of the topological model of Jθ 1+
Note that, if Jθ is locally connected, then by Theorem 2.3 (II) and Carathéodory’s Theorem the conformal linearizing coordinate φθ : U → θ extends continuously to the boundary. Hence the restriction Pθ : ∂θ → ∂θ is conjugated by a homeomorphic change of coordinates φθ : S 1 → ∂θ to an irrational rotation of the circle. As pθ ∈ ∂θ , we obtain the following: Proposition 2.4. If Jθ is locally connected, then ∂θ = Postcrit(Pθ ). If Jθ is locally connected, then a topological model for the dynamics of Pθ : Jθ → Jθ can be constructed similarly to what is done in [Thu and Dou93]. However, if we are interested in constructing a topological model of Jθ without the dynamics, the exercise becomes rather trivial. We can, for instance, replace the Siegel disk itself, as well as its every preimage, with a round circle. Each of the circles has a countable set of circles attached to its boundary, at a dense set of points. Putting them together has to be done so that there are no intersections not only of the circles themselves, but of the closures of infinite chains of circles. 3. Examples of Computable and Locally Connected Siegel Julia Sets: Parameters of Bounded Type An irrational angle θ is said to be of type bounded by B if it is represented by an infinite continued fraction with positive terms θ = [a0 , a1 , a2 , . . .] such that sup ai ≤ B. The union of all numbers of a bounded type are Diophantine numbers of exponent 2; a zero measure subset of T. As we have shown in [BBY07]:
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Theorem 3.1. If θ is of a bounded type, then r (θ ) is a computable real. Moreover, all such r (θ ) can be computed by a single algorithm with a single parameter – an upper bound B on the coefficients of the continued fraction of r (θ ). To outline the proof of this below, we recall that Siegel quadratic Julia of bounded type sets may be constructed by means of quasiconformal surgery (cf. [Dou88]) on a Blaschke product f γ (z) = e2πiτ (γ ) z 2
z−3 . 1 − 3z
This map homeomorphically maps the unit circle T onto itself with a single (cubic) critical point at 1. The angle τ (γ ) can be uniquely selected in such a way that the rotation number of the restriction ρ( f γ |T ) = γ . For each n, the points {1, f γ (1), f γ2 (1), . . . , f γqn+1 −1 (1)} form the n th dynamical partition of the unit circle. The following result is due to Swiatek and Herman (for the proof see e.g. Theorem 3.1 of [dFdM99]): Theorem 3.2 (Universal real a priori bound). There exists an explicit constant B > 1 independent of γ and n such that the following holds. Let γ ∈ R \ Q and n ∈ N. Then any two adjacent intervals I and J of the n th dynamical partition of f γ are B-commensurable: B −1 |I | ≤ |J | ≤ B|I |. Proposition 3.3 ([Her86]). For each bounded type γ = [a0 , . . . , ak , . . .] the Blaschke product f γ is K 1 -quasisymmetrically conjugate to the rotation Rγ : x → x + γ mod Z. 2 The quasisymmetric constant may be taken as K 1 = (2 sup ai )10B . Let us now consider the mapping which identifies the critical orbits of f γ and Pγ by : f γi (1) → Pγi (cγ ). We have the following (see, for example, Theorem 3.10 of [YZ01]): Theorem 3.4 (Douady, Ghys, Herman, Shishikura). The mapping extends to a K -quasiconformal homeomorphism of the plane C which maps the unit disk D onto the Siegel disk γ . The constant K may be taken as the quasiconformal dilatation of any global quasiconformal extension of the K 1 -qs conjugacy of Proposition 3.3. In particular, K ≤ 2K 1 . Elementary combinatorics implies that each interval of the n th dynamical partition contains at least two intervals of the (n + 2)nd dynamical partition. This in conjunction with Theorem 3.2 implies that the size of an interval of the (n + 2)nd dynamical partition of f γ is at most τ n , where B . τ= B+1
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Hence, setting n = {Pγi (cγ ), i = 0, . . . , qn+2 }, by Theorem 3.4, dist H (n , ∂γ ) < K τ n .
(3.1)
We quote: Lemma 3.5 (see [BY08a]). Let U be a simply-connected bounded subdomain of C containing the point 0 in the interior. Suppose V ⊂ U is a simply-connected subdomain of U , and ∂ V ⊂ B(∂U, ). Let r (U, 0) denote the conformal radius of U with respect to 0, and similarly for r (V, 0). Then √ r (U, 0) − r (V, 0) ≤ 4 r (U, 0) . Moreover, denote F(x) = 4x/(1 + x)2 . Then r (V, 0) ≤ r (U, 0)F
ρ(V, 0) . ρ(U, 0)
In combination with (3.1), the above lemma yields an algorithm for computing the value of r (θ ) with an arbitrary desired precision. This finishes the sketch of proof of Theorem 3.1. The existence of locally connected Siegel Julia sets was first demonstrated by Petersen [Pet96]. A different proof was also given by the second author in [Yam99]: Theorem 3.6 ([Pet96]). If θ is an irrational number of bounded type, then the Julia set Jθ is locally connected. A work of Petersen and Zakeri [PZ04] later extended this result to a class of rotation numbers θ having full measure in S 1 . 4. Control of the Postcritical Set of a Siegel Quadratic 4.1. Cylinder renormalization. Cylinder renormalization is the tool which we will use to gain control of the postcritical set of Pθn in the above discussion. It was introduced by the second author in [Yam02], and applied to maps with Siegel disks in [Yam08]. We refer the reader to these two works for a more detailed description. To define the procedure, we start with an analytic map f defined in a neighborhood W of the origin, and of the form f (z) = e2πiθ z + o(z), where θ is some Brjuno number. Recall that { pn /qn } denote its rational convergents. Fix some n ≥ 0. Assume that there exists a simple arc l ⊂ W which connects a fixed point a of the iterate f qn to 0, and has the property that f qn (l) is again a simple arc whose only intersection with l is at the two endpoints. Let C f be the topological disk in C \ {0} bounded by l and f qn (l). We say that C f is a fundamental crescent if the inverse branch f −qn |C f mapping f qn (l) to l is defined and univalent, and the quotient of C f ∪ f −qn (C f ) \ {0, a} by the iterate f qn is conformally isomorphic to C/Z.
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Fig. 2. Schematics of cylinder renormalization
For a point z in the fundamental crescent, consider the first return map R f (z) given by the smallest iterate f i (z) which is again contained in C f , assuming such an i exists. It will, of course, exist, and will be locally constant for all z in the intersection of C f with the Siegel disk f . Let us now select a conformal isomorphism κ : C f ∪ f −qn (C f ) \ {0, a} / f qn −→ C/Z, which sends the puncture at {0} to the “upper” end +i · ∞ of C/Z. Its composition with the exponential map χ (z) = exp(2πiκ(z)) maps the quotient of the crescent to the complex plane punctured at the origin. Consider the map h = χ ◦ R f ◦ χ −1 . It is not difficult to see that it is an analytic function defined in a neighborhood of the origin. Moreover, filling in the removable singularity at 0, we have: 1
h = exp(2πiθ )z + o(z), with θ = G n+1 (θ ),
where G(θ ) = θ is the Gauss map. How well-defined is h? First, and most crucially, the Liouville’s Theorem implies that the only flexibility we have in the choice of χ is in post-composing it with a homothety around 0. A different choice of C f could, a priori produce a different h. However, Proposition 4.1. Every other fundamental crescent C f with the same endpoints as C f , and such that C f ∪ C f is a topological disk, produces the same renormalized map h (defined up to a change of coordinates by a homothety). Now, let us suppose that θ is of bounded type, and the Siegel disk f is contained in the domain W of f . Further, let the boundary of f contain a unique critical point of f . Then h is also going to have a single critical point on the boundary of its Siegel disk. Let us uniquely specify χ by putting this point at 1. We then call the map h a cylinder renormalization of f with period qn . The boundary of the Siegel disk of h is obtained by a conformal “blow-up” of an arc of the boundary of f . The cylinder renormalization acts as a zoom-in into the postcritical set.
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Theorem 4.2. Let θ be of a bounded type. Let Pθ be as above. There exists a sequence gn , n ∈ N of cylinder renormalizations of Pθ with the following properties: (I) For each n, the map gn is a cylinder renormalization of Pθ with period qn . Thus gn has a Siegel disk with rotation number G n+1 (θ ) centered at the origin, whose boundary is a quasicircle, containing the critical point 1. (II) Denoting Cn the fundamental crescent of the respective renormalization, we have dn = sup dist(z, θ ) → 0. z∈Cn
Moreover, dn is K -commensurable with Cn ∩ ∂θ for a universal K > 0. Hence dn < Ab−n for some A > 0, and b > 1. (III) Finally, there exists k ∈ N such that for all n 1 and for n 2 ≥ n 1 + k, the map gn 2 is a cylinder renormalization of gn 1 . Another useful assumption on the rotation number θ is that of a “high type”: that all of the digits of the continued fraction of θ are uniformly large. A recent result of Inou and Shishikura [IS07], which we quote below, implies that if the lower bound on the digits is sufficiently high, then all cylinder renormalizations of Pθ belong to a compact family. W {0, 1}, an Theorem 4.3. There exist N0 ∈ N, a pair of topological disks W open set V in the Banach space of analytic maps in W with the sup-norm, and a compact subset Y V such that the following is true: • Let θ = [a1 , a2 , . . .] ∈ (0, 1) \ Q with ai ≥ N0 . For every f ∈ V with f (0) = e2πiθ we have the following. The map f is cylinder renormalizable with period 1 = q0 , and the corresponding cylinder renormalization g(z) = exp(2πi G(θ ))z + o(z) ∈ Y. . Moreover, g analytically extends to the larger domain W • Further, consider the quadratic polynomial f = Pθ (z). Set gn to be the sequence of cylinder renormalizations of f as in Theorem 4.2. Then there exists j ∈ N such that g j |W ∈ Y. As an easy corollary, note that: Corollary 4.4. Let g(z) and W be as in the above theorem. Then the critical orbit
g n (1) ⊂ W.
n≥0
Proof. Indeed, the theorem implies that there exists an infinite sequence of cylinder renormalizations of the restriction g|W . Hence, iterates (g|W )n (1) are defined for arbitrarily large values of n. In what follows we will mostly work with rotation numbers θ of a bounded and high type (that is sup ai < ∞ and inf ai ≥ N0 ).
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5. Modifying the Conformal Radius of a Siegel Disk Let us recall, that for an irrational θ ∈ T the Yoccoz’s Brjuno function (θ ) is defined as follows. Inductively set θ1 = θ and θn+1 = {1/θn }. In this way, θn = [rn , rn+1 , rn+2 , . . .]. Then, (θ ) =
∞
θ1 θ2 · · · θn log
n=1
1 . θn+1
Yoccoz [Yoc95] has shown that the sum (θ ) + log r (θ ) is bounded from below independently of θ . Buff and Chéritat [BC06] have greatly improved this result by showing that: Theorem 5.1 ([BC06]). The function υ : θ → (θ ) + log r (θ )
(5.1)
extends to R as a 1-periodic continuous function. It is conjectured that: Conjecture 5.2. The function υ defined by (5.1) is computable. We note that Marmi, Moussa, and Yoccoz [MMY97] have conjectured that υ is Hölder with exponent 1/2. This is known to be stronger than Conjecture 5.2 (see [BY08b] for details). We will require the following technical lemma (for a proof see [BY08a or BY08b]). Lemma 5.3. Fix N ∈ N. For any given initial segment I m 0 > 0, write ω = [a0 , a1 , . . . , an , N , N , N , . . . ]. Then uniformly compute m > m 0 , an integer t and an integer β = [a0 , a1 , . . . , an , N , N , . . . , N , M, N , N , . . .], where M position, we have r (ω) − 2ε < r (β) < r (ω) − ε, (β) > (ω),
= [a0 , a1 , . . . , an ] and for any ε > 0, we can M such that if we write is located in the n + m th
(5.2) (5.3)
and for any γ = [a0 , a1 , . . . , an , N , N , . . . , N , M, N , . . . , N , cn+m+t+1 , cn+m+t+2 , . . .], (γ ) > (ω) − 2−n . (5.4)
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6. Admissible Perturbations of Siegel Julia Sets In the proof of the Main Theorem we will use a sequence of perturbations from the last section to “fool” all Turing Machines attempting to compute Jθ . In this section we will develop the necessary machinery that would guarantee that the resulting limiting set is locally connected. Let N0 be as in Theorem 4.3, and fix N > N0 . An admissible irrational number θ = [a1 , a2 , . . .] ∈ T has all of the coefficients ai ≥ N0 , and a j = N for all sufficiently large values of j. One consequence of the renormalization picture we have described above is the following: Proposition 6.1. There exists B = B(N ) such that the following holds. Suppose θ is an admissible number. Then there exists k0 ∈ N such that for all k ≥ k0 , denoting gk the cylinder renormalization of Pθ from Theorem 4.2, the boundary of the Siegel disk ∂gk is a B-quasicircle. In fact, denoting by fˆ the fixed point of Rcyl with rotation number [N , N , N , . . .] whose existence is postulated in the Theorem on Hyperbolicity of Renormalization, we see that the boundary of the Siegel disk of Pθ at small scales converges to that of fˆ in Hausdorff distance. Definition 6.1. Now let α be a Brjuno number such that Jα is locally connected. We will say that Jβ is an admissible 2−n -perturbation of Jα if the following properties hold: (1) The Julia set Jβ is locally connected. (2) dist H (∂α , ∂β ) < 2−n . β
(3) For each n and each σ¯ ∈ {0, 1}n denoting L ασ¯ and L σ¯ the limbs of Jα and Jβ respectively, we have β
| diam(L ασ¯ ) − diam(L σ¯ )| < 2−n . (4) Consider the Riemann mapping α : U → α normalized by α (0) = 0, α (0) = 1, and a similarly defined β . Then sup |α − β | < 2−n . z∈U
(5) Similarly, denote ˆ \U→C ˆ \ Kα α : C the Böttcher map of Pα , and β the Böttcher map of Pβ . Let γα± be the angles of the two external rays of Jα , which land at the critical point pα , and similarly for γβ± . Then ±
±
||β (te2πiγβ ) − α (te2πiγα )|| < 2−n in the spherical norm for t ∈ [1, ∞). In particular, | pα − pβ | < 2−n .
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We have: Theorem 6.2. Let βn be a sequence of Brjuno numbers such that, for each n, Jβn is a 2−n -admissible perturbation of Jβn−1 . Assume that βn → α, which is another Brjuno number. Then Jα is locally connected. Proof. By Property (1) of an admissible perturbation and Theorem 2.3 the boundary of βn is a Jordan curve. By Carathéodory’s Theorem, the Riemann mapping βn extends continuously to S 1 = ∂U. By Properties (2) and (4) of an admissible perturbation, the sequence βn ⇒ α . U
Applying Carathéodory’s Theorem to α , we see that α is a Jordan curve. By Properties (2) and (5) the critical point pα ∈ ∂α , and is bi-accessible from infinity. By Property (3), Theorem 2.3, and considerations of continuity the diameters of the limbs of Jα of generation n shrink to zero uniformly with n. By Theorem 2.3 the proof is complete. We now formulate the following key consequence of the result of Inou and Shishikura (cf. the discussion in [BC05]): Proposition 6.3. Consider an admissible number α = [Iα , N , N , N , . . .], where Iα is some initial segment of the continued fraction. There exists a Turing Machine which takes as inputs the segment Iα and a natural number which outputs δ > 0 and M ∈ N such that the following holds: Let β be a perturbation of α of the form β = [Iα , N , N , . . . , N , A1 , A2 , . . . , Ak , N , N , N , . . .], where m ≥ M and Ai ≥ N ,
m
and such that |rα − rβ | < δ. Then dist H (∂α , ∂β ) < 2− . Proof. The boundary of α is obtained by taking the closure of the critical orbit {Pαn (1)}. By simple considerations of continuity, there exists k0 ∈ N such that for every m ≥ k0 , ∂α ⊂ B(∂β , 2− ). By (3.1) the value of k0 can be obtained constructively, given Iα . Let τ be any number larger than . For the map Pα select Cn as in Theorem 4.2, (II). Consider the arc n = ∂α ∩ Cn
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of the boundary of the Siegel disk trapped inside the fundamental crescent. By the inverse branch (Pα )−1 , fixing the Siegel disk, it is rotated around the boundary. An inspection shows: ⎛ ⎞ ⎛ ⎞ qn n−1
q
⎝ (Pα )− j (n−1 )⎠ ⎝ (Pα )− j (n )⎠ ⊃ ∂α . j=0
j=0
Denote Wn ⊂ Cn the lift of the domain W from Theorem 4.3. Note that by Corollary 4.4, n ⊂ Wn . By Theorem 4.2, (II), for any ν > 0, we can constructively find k1 ∈ N such that Wn B(α , 2−ν ) for n ≥ k1 . An application of the Koebe Distortion Theorem to pull-backs ⎛ ⎞ ⎛ ⎞ qn n−1
q
⎝ Yn ≡ ⎝ (Pα )− j (Wn−1 )⎠ (Pα )− j (Wn )⎠ j=0
j=0
implies the existence of k2 ∈ N such that, for n ≥ k2 , Yn ⊂ B(α , 2−τ ). The explicit bound in the Koebe Distortion Theorem again allows us to obtain k2 constructively. Set k = k2 + |Iα |. Now denote Cn , Wn , and Yn the corresponding objects for Pβ . By considerations of continuity, for every d we can constructively find m(d) n ≥ k, such that for m ≥ m(d) the domains Cn , Cn−1 are 2−d -perturbations of Cn , Cn−1 in the Hausdorff sense. By Corollary 4.4, we have
β ∩ Cn ⊂ Wn . Select m τ large enough so that for m > m τ the previous inclusions hold, and Yn ⊂ B(Yn , 2−2τ ). Then β ⊂ B(α , 2−τ ). Thus by moving the perturbation far enough to the right in the continued fraction of α, we can guarantee that ∂β does not extend outside a small neighborhood of α . It remains to ensure that ∂β does not have decorations which grow deep into α . The easiest way to see this is to note that by Theorem 3.4, ∂α is a B-quasicircle for
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some explicit B ∈ N. Hence, for every δ > 0, we can constructively find τ > 2 such that setting Uτ = B(α , 2−τ ), we have r (Uτ , 0) − rα < δ, so that r (Uτ , 0) − rβ < 2δ. By Proposition 3.5 applied to uniformization of Uτ , we can find δ small enough, so that ∂β ⊂ B(∂Uτ , 2− ). Taking these δ and τ , and m > m τ we have ∂β ⊂ B(α , 2− ) ∩ B(∂Uτ , 2− ) ⊂ B(∂α , 2− ). We now state: Proposition 6.4. Consider an admissible number α = [Iα , N , N , N , . . .]. There exists a Turing Machine which takes as inputs the segment Iα and a natural number which outputs δ > 0 and M ∈ N such that the following holds: Let β be a perturbation of α of the form β = [Iα , N , N , . . . , N , A1 , . . . , Ak , N , N , N , . . .], where m ≥ M and Ai ≥ N ,
m
and such that |rα − rβ | < δ. Then Jβ is an admissible 2− -perturbation of Jα . Proof. Constructively selecting M and δ to satisfy property (1) is straightforward. Property (2) is proved in Proposition 6.3. The stronger Property (4) follows by Proposition 6.3 and Theorem 3.4. and (4) Properties (3) and (5) follow from an explicit geometric estimate on the size of a limb of Jα of generation n, given in [Yam99], pp. 254–255.
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7. Proof of the Main Theorem We will now use the machinery developed in Sect. 6 to prove the Main Theorem. Proof overview. By Theorem 1.1 we know that computability of r (θ ) is equivalent to that of Jθ . It is thus sufficient for us to construct a parameter value θ for which Jθ is locally connected, and no Turing Machine computes r (θ ). We will do this via a diagonalization argument. Let us make a definition: Definition 7.1. We will say that a parameter θ fools a Turing Machine M(n) if there exists m ∈ N such that on this input M(m) terminates, but does not output a number r with the property |r − r (θ )| < 2−m . Less formally, the machine M fails to compute r (θ ) by computing a wrong output for some particular choice of the precision parameter m. We construct a parameter θ that fools all oracle TMs attempting to compute r (θ ). Let us first recall briefly the approach to producing non-computable Julia sets of our paper φ φ [BY06]. We would begin by enumerating all oracle Turing Machines M1 , M2 , . . .. Assuming that a machine in our list computes the conformal radius r (θ ), we can fool it by modifying the parameter θ far enough in its continued fraction, so that the conformal radius of the Siegel disk decreases by a sufficiently large amount. Applying this idea successively to all of the machines in our list, we end up with a sequence of parameters θk = [Ik , N , N , . . .], such that the finite sequences I1 ⊂ I2 ⊂ . . . are nested, and such that every machine fails to compute r (θk ) starting from some k. A passage to the limit, carefully made, produces a parameter with a non-computable Julia set. Using the bounds of Proposition 6.4, we can ensure that the perturbations are carried out so that the limit is locally connected (see [BY08b]). Turning this argument into a constructive one, however, meets a logical obstacle. When we attempt to fool the machine φ Mk , we cannot just “simulate it on φk , see what it does, and modify the parameter to φ φ fool it” – the machine Mk may not terminate. In fact, we cannot know whether Mk terminates without solving the Halting Problem, which is undecidable. To bypass this difficulty, we run the machines in parallel, and modify the parameter φ when one of them does output a value r . When Mk outputs an answer, then we can modify the parameter to fool it. If it never outputs an answer, then it never gets fooled, but this is not a problem, since a machine that never halts cannot be computing r (θ ). φ
φ
Proof of the Main Theorem. We enumerate all oracle Turing Machines M1 , M2 , . . .. We will show how to construct a parameter value θ such that none of the listed machines correctly computes the conformal radius r (θ ). At every step k of the construction we will maintain a finite initial segment Ik of the continued fraction expansion of the final parameter θ , and the current parameter θk = [Ik , N , N , . . .]. The segment Ik will be an extension of Ik−1 so that the chain I1 ⊂ I2 ⊂ . . . converges to the continued fraction expansion of θ = limk→∞ θk . Also at every step we will maintain a finite status string Sk ∈ {0, 1}∗ . The status string φ attaches a status to each machine M j the algorithm is currently considering. Sk [ j] = 1 φ
means that M j is fooled by the current parameter θk ; Sk [ j] = 0 means that it is undeφ
termined, whether θk fools M j . We also define Sk∗ ∈ {0, 1}ω as the infinite {0, 1}-string obtained from Sk by appending an infinite sequence of 0’s. The limit S ∗ of the sequence
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{Sk∗ } is defined as the termwise limit (if one exists). The value S ∗ [ j] at the limit will be 1 φ φ if the parameter θ fools M j , and S ∗ [ j] will be 0 if M j fails to terminate on θ . In either φ
case, M j will fail to compute r (θ ). We define an order relationship Sk ≺ Sl on the status sequences to be the lexicographical order: Sk ≺ Sl if ∃ j : Sk∗ [ j] = 0,
Sl∗ [ j] = 1, and ∀ (i < j) Sk∗ [i] = Sl∗ [i].
In our construction, the status strings will satisfy S1 S2 S3 . . . . Note that this means, in particular, that for each j, Sk∗ [ j] changes its value at most 2 j − 1 times as k grows. This implies that the limit S ∗ = lim Sk∗ k→∞
exists. At step k we will be working on fooling the first k machines, thus |Sk | = k for all k. For each j ≤ k such that Sk [ j] = 0 we will maintain a value δ j which will either be a positive number, or undefined (in which case we will write δ j = ⊥). This is the amount by which we are willing to drop the conformal radius in order to fool the j th machine. We will further require that if i > j then δ j > 16i− j δi (if defined), that is, the defined terms in the sequence {δ j } decrease geometrically. At each step we will also be maintaining an integer Mk that specifies beyond which location of the continued fraction expansion we are allowed to change θk . We will now describe the initialization and the step in the execution of our construction. Initialization (iteration k = 0). We set I0 = [N ], S0∗ [ j] = 0, δ j = ⊥ for all j. Step k ≥ 1. We start by computing δ j for j ≤ k. We compute values of δ j such that currently Sk−1 [ j] = 0 and δ j = ⊥ (that is, the j th machine has not been fooled, and the value of δ j is currently undefined). We compute δ j > 0 in increasing order of j, 1 ≤ j ≤ k. We choose δ j that satisfies δ j < 16−(i− j) δi for all i < j. We compute a δ > 0 and M j as in Proposition 6.4 with = j. That is, for each β such that (1) β is obtained from θk−1 by changing finitely many positions Am , for m > M j , in its continued fraction expansion from N to a bigger number; and (2) |r (θk−1 ) − r (β)| < δ, the set Jβ is an admissible 2− j -perturbation of Jθk−1 . We choose δ j < δ/4; we also choose Mk such that Mk > 2k and Mk > M j for all j ≤ k. φ Next, for each j ≤ k with Sk−1 [ j] = 0 we simulate the machine M j on θk−1 with precision parameter −[log2 (δ j /8)] + 1 for at most k steps of execution. There are two cases:
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φ
Case 1. No machine Mi terminates with an answer that is δi /4-close to r (θk−1 ). In this case Ik is obtained from Ik−1 by adding Mk digits N , thus θk = θk−1 remains unchanged. φ For each j such that the machine M j terminates (with a wrong answer) we set Sk [ j] = 1, for all other j’s we let Sk [ j] = Sk−1 [ j]. The numbers δ j remain unchanged. φ
Case 2. At least one machine Mi terminates with an answer that is δi /4-close to r (θk−1 ). Let j be the smallest such i. We use Lemma 5.3 to compute Ik that extends Ik−1 and has at least Mk digits N immediately after Ik−1 so that δ j /2 < r (θk−1 ) − r (θk ) < δ j and (5.3), (5.4) hold .
(7.1)
By Lemma 5.3 we can perform the perturbation in such a way that (θk−1 ) < (θk ) < (θk−1 ) + C · δ j
(7.2)
for some constant C, as long as r (θk ) is bounded away from 0. We then update Sk [ j] to 1 and Sk [i] for i > j to 0. We also set δi = ⊥ for j < i ≤ k. The intuition behind the last step is that once we have decreased the conformal radius by a large value δ j , all smaller drops intended to fool machines further down the line become irrelevant. It is evident from the construction that Sk−1 Sk , and hence the limit S = lim Sk exists. By construction, the sequence θk converges to a computable limit θ . We need to see that r (θ ) is non-computable and that Jθ is locally connected. We first show that Jθ is locally connected. For each j with S[ j] = 1, denote by k j the index of the last iteration when S[ j] was set to 1 (recall that there can be at most 2 j − 1 such iterations). Then, by the construction, k j is increasing with j and θ = lim j→∞ θk j . In addition, by the construction, for each j, Jθk j is an admissible 2− j perturbation of Jθk j−1 . Hence by Theorem 6.2 the limit Jθ is locally connected. We first note that r (θ ) = lim r (θk ). Indeed, it is not difficult to show (see Proposition 5.10 of [BY08b]) that r (θ ) ≥ r = lim r (θk ).
(7.3)
By (7.2), the values of the Yoccoz-Brjuno function ∞ > (θ ) > (θk ) − C · (r (θk ) − r ), and (θk ) ≥ (θk−1 ), and hence (θ ) ≥ lim (θk ).
(7.4)
As the function υ : θ → (θ ) + log r (θ ) is continuous, (7.3) and (7.4) imply that r (θ ) = r. φ Next, we show that r (θ ) is non-computable. Suppose that the machine Mk computes r (θ ) for some k. There are two cases.
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Case 1. S[k] = 1. This means that for some δ and , an execution of Mk on θ with precision δ/8 terminates in steps and outputs an answer that is at least δ/4 away from the correct one. Further changes in θ happen beyond position M > 2 in the continued φ fraction expansion, and do not affect the execution of Mk with precision δ/8. In addition, by the construction, further changes in θ will only change the conformal radius by at ∞ φ δ/16i < δ/8. Thus Mk when run on θ with precision δ/8 will terminate most i=1 after steps and output an answer that is at least δ/8 from r (θ ). Contradiction. Case 2. S[k] = 0. In our construction, the value of δk is changed finitely many times. Let δ be the last value of δk that is attained after some step 1 . This means that after step φ 1 in our construction, no entries S[ j] for j ≤ k are updated. We claim that Mk does not terminate when executed with precision parameter δ/8 on θ . Suppose for contradiction that it does terminate in 2 steps. Let > max(1 , 2 ). Then on iteration of our conφ struction we run Mk on θ−1 for > 2 steps. The execution is identical to the execution φ of Mk on θ for steps, since |θ−1 − θ | < 2−−2 . Hence, by our assumption it must terminate and output a radius value. But if this were the case, our construction would set S[k] to 1, contradicting the assumption that S[k] is not set to 1 after step 1 < . Let us make a final observation: Remark 7.1. Assuming Conjecture 5.2, there is a poly-time algorithm for computing such a parameter θ . In other words, θ can be computed with precision 2−n in time bounded by a polynomial in n. The modifications in the proof to make the algorithm work in polynomial time follow the strategy outlined in §5.2.3 of [BY08b] and we will not repeat them here. 8. Computing Jθ is as Hard as Solving the Halting Problem In this final section we prove that computing the Julia set Jθ constructed in the previous section is as hard as solving the Halting Problem. By Theorem 1.1, the computability of r (θ ) is equivalent to that of Jθ . We thus proceed to formulate the following: Theorem 8.1. Let θ be the computable parameter from Main Theorem, constructed by the algorithm given in Sect. 7. (I) There exists an oracle Turing Machine, which, given an oracle for the Halting Problem, computes r (θ ). (II) Conversely, there exists an oracle Turing Machine, which, given an oracle for the conformal radius r (θ ), solves the Halting Problem. Proof of Part (I) By Theorem 1.2 the conformal radius r (θ ) is right-computable. Since every right-computable number is computable with an oracle for the Halting Problem, the statement follows. Proof of Part (II) For a TM M denote by T (M) the amount of time M takes to halt. If M does not halt, then set T (M) := ∞. We will show that using an oracle for r (θ ) we can compute a function B(n) satisfying the following: B(n) ≥
max
M a TM with |M| ≤ n and T (M) < ∞
T (M).
(8.1)
In other words, B(n) is a bound on the longest finite amount of time a Turing Machine of description length n can take before halting.
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It is clear that if one can compute a function B(n) satisfying (8.1), one can also solve the Halting Problem: on an input M with |M| = m it follows from the definition of B(m) that M will either halt within the first B(m) steps or will never halt. Thus to decide the Halting Problem on M it would suffice to simulate it for B(m) steps and see if it terminates until then. Denote by M1 the machine constructed in the previous section, which produces the parameter θ . Denote by In the initial segment of θ the machine M1 has generated n steps into the execution. We define a machine M2 as follows. The machine M2 takes a description of a TM M as an input. It then runs M until it terminates. If M terminates in K steps, the machine M2 simulates M1 for K iterations to obtain a value θ K = [I K , N , N , N , . . .]. M2 (M) then outputs r (θ K ). If M does not halt, then neither does M2 (M). The description size of the machine M2 is some constant C2 , so that the total description size of M2 (M) is C2 + |M|. At each stage of the execution of M1 , we have the current value of the radius r (θk ), as well as the values of the drops δ j in r (θk ) that we will use to fool the machine M j if it hasn’t been fooled yet. Using the oracle φ for the final value of r (θ ) and seeing whether r (θ ) ≈ r (θ1 ) or r (θ ) ≈ r (θ1 ) − δ1 , we can check whether the drop by δ1 has occurred. If it has, we will run M1 until the drop occurs, to obtain an updated value for δ2 . We will then use the oracle for r (θ ) to check whether a drop by δ2 has occurred. We can continue this process to evaluate the function D(n) := the number of steps M1 takes before no more drops by δ1 , . . . , δn occur.
(8.2)
We claim that B(n) := D(2n+C2 +1 ) satisfies (8.1). The function B(n) obviously can be computed from D(n), and hence it can be computed from an oracle for r (θ ). Let M be a TM with |M| = n. Suppose, for contradiction, that T (M) > B(n). The machine M2 (M) can be viewed as an oracle TM M2 (M)φ that ignores its oracle and outputs the conformal radius r M = r (θT (M) ) on all inputs. The description size of M2 (M)φ is bounded by n+C2 , and hence M2 (M)φ = φ M L , where L < 2n+C2 +1 is the index of M2 (M)φ in the enumeration of all oracle TMs used by M1 . By the definition of D(L), we have |r (θk ) − r (θ )| < δ/4,
(8.3)
for all k ≥ D(L), where δ is the current value of δ L at time D(L). Further, we know that the value of δ L will not change after time D(L). Equation (8.3) implies that |r M − r (θk )| < δ/4
(8.4)
for all k ≥ D(L). By the assumption we have T (M) > B(n) ≥ D(L), thus when M1 φ will simulate M L , the simulation will take more than D(L) steps to complete. Hence
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the first simulation of M L by M1 that will terminate will be a simulation with precision φ δ. By (8.4) output of M L will be consistent with the value of r (θk ) within an error of δ/4. Thus, by the definition of M1 , it will cause a drop of δ in the value of r (θk ) to fool φ M L . This contradicts (8.4). Hence B(n) ≥ T (M), which completes the proof. References [BBY07] [BC05] [BC06] [BY06] [BY08a] [BY08b] [dFdM99] [Dou88] [Dou93] [Her85] [Her86]
[IS07] [Lyu86] [MMY97] [Pet96] [PZ04] [Sul83] [Thu] [Wei00] [Yam99] [Yam02] [Yam08] [Yoc95] [YZ01]
Binder, I., Braverman, M., Yampolsky, M.: Filled Julia sets with empty interior are computable. J. FoCM 7, 405–416 (2007) Buff, X., Chéritat, A.: Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive. Comptes Rendus Math. 341(11), 669–674 (2005) Buff, X., Chéritat, A.: The Brjuno function continuously estimates the size of quadratic siegel disks. Ann. Math. 164(1), 265–312 (2006) Braverman, M., Yampolsky, M.: Non-computable Julia sets. J. Amer. Math. Soc. 19(3), 551–578 (2006) Braverman, M., Yampolsky, M.: Computability of Julia sets. Moscow Math. J. 8(2), 185–231 (2008) Braverman, M., Yampolsky, M.: Computability of Julia Sets. Algorithms and Computations in Math. Vol.23 Berlin-Heidelberg-New York: Springer, 2001 de Faria, E., de Melo, W.: Rigidity of critical circle mappings I. J. Eur. Math. Soc. (JEMS) 1(4), 339–392 (1999) Douady, A.: Disques de Siegel et anneaux de Herman. Séminaire Bourbaki, Vol. 1986/87, Astérisque 152-153(4), 151–172 (1988) Douady, A.: Descriptions of compact sets in c. In: Topological Methods in Modern Mathematics (Stony Brook, NY, 1991) Houston, TX: Publish or Perish, 1992 Herman, M.: Are there critical points on the boundaries of singular domains? Commun. Math. Phys. 99(4), 593–612 (1985) Herman, M.: Conjugaison quasi symétrique des homéomorphismes du cercle à des rotations, and Conjugaison quasi symétrique des difféomorphismes du cercle à des rotations et applications aux disques singuliers de Siegel. Available from http://www.math.kyoto-u.ac.jp/~mitsu/ Herman/index.html, 1986 Inou, H., Shishikura, M.: Renormalization for parabolic fixed points and their perturbations. Preprint, 2007 Lyubich, M.: Dynamics of rational transformations: topological picture. Russ. Math. Surv. 41(4), 43–117 (1986) Marmi, S., Moussa, P., Yoccoz, J.-C.: The Brjuno functions and their regularity properties. Commun. Math. Phys. 186, 265–293 (1997) Petersen, C.: Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math. 177, 163–224 (1996) Petersen, C., Zakeri, S.: On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. Math. 159(1), 1–52 (2004) Sullivan, D.: Conformal dynamical systems. In: Geometric Dynamics (Palis, ed.), Lecture Notes Math., Vol. 1007, Berlin- Heidelberg-New York: Springer-Verlag, 1983, pp. 725–752 Thurston, W.: On the combinatorics and dynamics of iterated rational maps. Preprint Weihrauch, K.: Computable Analysis. Berlin: Springer-Verlag, 2000 Yampolsky, M.: Complex bounds for renormalization of critical circle maps. Erg. Th. & Dyn. Syst. 19, 227–257 (1999) Yampolsky, M.: Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Études Sci. 96, 1–41 (2002) Yampolsky, M.: Siegel disks and renormalization fixed points. In: Holomorphic Dynamics and, Renormalization, Fields Institute Communications 53, Providence, RI: Amer. Math. Soc., 2008 Yoccoz, J.-C.: Petits diviseurs en dimension 1. S.M.F., Astérisque 231, 3–242 (1995) Yampolsky, M., Zakeri, S.: Mating Siegel quadratic polynomials. J. Amer. Math. Soc. 14(1), 25–78 (2001)
Communicated by G. Gallavotti
Commun. Math. Phys. 291, 533–542 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0799-z
Communications in
Mathematical Physics
Regular Strongly Typical Blocks of O q Anders Frisk, Volodymyr Mazorchuk Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden. E-mail: [email protected]; [email protected] Received: 27 October 2008 / Accepted: 19 January 2009 Published online: 1 April 2009 – © Springer-Verlag 2009
Abstract: We use the technique of Harish-Chandra bimodules to prove that regular strongly typical blocks of the category O for the queer Lie superalgebra qn are equivalent to the corresponding blocks of the category O for the Lie algebra gln . 1. Introduction and the Main Result For n ∈ N let qn denote the queer Lie superalgebra and Oq denote the category O for qn . The category Oq decomposes into a direct sum of blocks, which can be typical or atypical. Atypical blocks are very complicated and may contain infinitely many simple objects. Typical blocks are much easier and are always equivalent to module categories over finite-dimensional algebras. In [Fr] it was shown that the finite-dimensional algebras describing typical blocks are always properly stratified in the sense of [Dl]. Among all typical blocks one separates strongly typical ones, which are described ([Fr]) by quasi-hereditary algebras in the sense of [CPS,DR]. A very general conjecture about the combinatorial structure of the category Oq is given in [Br, 4.8]. In the special case of regular strongly typical blocks this conjecture says that multiplicities of simple highest weight supermodules in Verma supermodules for qn are given by Kazhdan-Lusztig combinatorics. For this special case a much stronger conjecture was formulated in [Fr, 3.9], namely that strongly typical blocks of Oq are equivalent to the corresponding blocks of the category O for the Lie algebra gln . A strong evidence for this conjecture, established in [Fr], was a similarity between the quasi-hereditary structures in both cases. Moreover, [Fr, 3.9] contains also an explicit conjecture for the structure of all typical blocks. The aim of this paper is to prove the conjecture from [Fr, 3.9] (and hence the conjecture from [Br, 4.8] as well) for regular strongly typical blocks. There is a natural restriction functor from Oq to the category O for the Lie algebra gln . However, unlike the case of most of the other Lie superalgebras, this restriction functor does not induce an equivalence in a straightforward way. The problem is that the
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highest weights of Verma supermodules over Oq are not one-dimensional (because the Cartan subsuperalgebra of q is not commutative). Subsequently, under restriction Verma supermodules are not mapped to the corresponding Verma modules but rather to direct sums of Verma modules (see Proposition 2). This suggests that the naive restriction functor is a direct sum of several copies of some “smaller” functor, which defines the desired equivalence of categories. This is exactly what we prove in this paper. The main idea of the proof is to realize the induction functor (the left adjoint to the restriction) as a tensor product with some Harish-Chandra bimodule. This requires several definitions and some technical work as we are forced to go beyond the original categories and instead work with the so-called thick version of the category O. Unfortunately, along the way we use some properties of Harish-Chandra bimodules, which require an additional assumption of regularity of the blocks we work with. The main result of the paper is the equivalence of blocks of categories O for qn and gln , see Theorem 1. This extends earlier results of Penkov, Serganova and Gorelik (see [PS,Go2]) to the case of the algebra qn and verifies conjectures from [Fr, 3.9] and [Br, 4.8] in our setup. At the same time Theorem 1 is a refinement (in the case of qn ) of the main result of [Pe]. The paper is organized as follows: In Sect. 2 we give all necessary definitions and formulate our main result. In Sect. 3 we collect auxiliary technical results about HarishChandra bimodules. The main result is proved in Sect. 4. We conclude the paper with some remarks in Sect. 5.
2. Definitions, Preliminaries and Formulation of the Main Result For all undefined notions we refer the reader to [Fr]. Let N and N0 denote the set of all positive and nonnegative integers, respectively, and fix n ∈ N. Set Nn = {1, . . . , n}. For an algebraically closed field k of characteristic zero let g = gln denote the general linear Lie algebra of n × n matrices over k. Let q = qn denote the queer Lie superalgebra over k, which consists of block matrices of the form M(A, B) =
A B , B A
A, B ∈ gln .
The even and the odd spaces q0 and q1 consist of the matrices M(A, 0) and M(0, B), respectively, and we have q = q0 ⊕ q1 . For a homogeneous element X ∈ q we denote by X ∈ Z/2Z the degree of X . Then the Lie superbracket in q is given by [X, Y ] = X Y − (−1) X Y Y X , where X, Y ∈ q are homogeneous. For i, j ∈ Nn let E i j ∈ gln denote the corresponding matrix unit. We have the Cartan subsuperalgebra h of q, which is the linear span of Hi = M(E ii , 0) and M(0, E ii ), i ∈ Nn . The superalgebra h inherits from q the decomposition h = h0 ⊕ h1 . Let {εi : i ∈ Nn }, denote the basis of h∗0 , which is dual to the basis {Hi : i ∈ Nn } of h0 . Then = {εi − ε j : i, j ∈ Nn , i = j} is the root system of q with the corresponding Weyl group W ∼ = Sn . We also have the standard set + = {εi − ε j : i, j ∈ Nn , i < j} + of positive roots. Set = {εi + ε j : i, j ∈ Nn , i < j}. The group W acts on h∗0 in the usual way and via the dot action w · λ = w(λ + ρ) − ρ, where ρ is the half of the sum of all positive roots. Let (·, ·) denote the standard nondegenerate W -invariant bilinear form, given by (εi , ε j ) = δi, j .
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For α ∈ set qα = {v ∈ q : [H, v] = α(H )v for all H ∈ h0 }. Then qα = qα0 ⊕ qα1 , where both components are one-dimensional, and we further have the decomposition q=h⊕ qα . α∈
This induces the natural triangular decomposition q = n− ⊕ h ⊕ n+ with respect to our choice + of positive roots. Elements in h∗0 are called weights and are written λ = (λ1 , . . . , λn ) with respect to the basis {εi }. For λ, µ ∈ h∗0 we write λ ≤ µ provided that µ − λ ∈ N0 + . For λ ∈ h∗0 we denote by W λ the integral Weyl group of λ (the set of all w ∈ W such that wλ ∈ λ + Z+ ). A weight λ is called • • • • •
integral provided that λi ∈ Z for any i ∈ Nn ; dominant provided that wλ ≤ λ for any w ∈ W λ ; regular provided that wλ = λ for any w ∈ W λ ; + typical provided that (α, λ) = 0 for all α ∈ ; strongly typical provided that it is typical and λi = 0 for all i.
Strongly typical weights were originally defined as the highest weights for which the corresponding highest weight modules are not annihilated by the ghost element T , introduced in [Go1]. The above definition follows from the explicit description of T for the superalgebra qn , see for example [Go3, Theorem 10.4]. The algebra q0 is identified with the Lie algebra g = gln in the obvious way, and g inherits from q the triangular decomposition g = n− ⊕ h ⊕ n+ . A q-supermodule M is called a weight supermodule if it is a weight (that is h-diagonalizable) module with respect to g. We consider the category SM of all q-supermodules, where all morphisms are homogeneous maps of degree zero. This is an abelian category with usual kernels and cokernels. Let : SM → SM denote the autoequivalence, which changes the parity. Let further M denote the category of all g-modules, which is also abelian with usual kerq nels and cokernels. Let Resg : SM → M denote the functor of restriction from q to g, which sends a q-supermodule M = M0 ⊕ M1 to the g-module M0 . We denote by q q Indg : M → SM the left adjoint of Resg . q g Let O and O denote the BGG categories O for q and g, respectively (see [BGG]). These are full subcategories in the respective categories of finitely generated (super)module, which consist of weight (super)modules, which are locally U (n+ )- and U (n+ )-finite, respectively. Let U (q) and U (g) denote the universal enveloping (super)algebra of q and g, respectively. Let, further, Z (q) and Z (g) denote the (super)center of U (q) and U (g), respectively. The action of the (super)center gives rise to the following block decomposition of Oq and Og , indexed by central characters: Oq = ⊕χ Oχq , Og = ⊕χˆ Oχgˆ . For any χ we have the inclusion functor inclχ : Oχ → O and the projection functor projχ : O → Oχ , which are both left and right adjoint to each other. Similarly for χ. ˆ Throughout the paper we fix a regular dominant strongly typical weight λ for q. Denote by λˆ the corresponding g-weight (note that we formally have λ = λˆ as elements in h∗0 , however, it is convenient to use different notation to specify the algebra we work
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with). Let χ = χλ and χˆ be the central characters for q and g, which correspond to λ and λˆ , respectively. We also denote by mχ the kernel of χ and by mχˆ the kernel of χ. ˆ If λ is integral, then the block Oχgˆ in indecomposable. If, in addition, n is odd, the block q
Oχ is indecomposable as well. If n is even, then we have a decomposition Oχq ∼ = O˜ χq ⊕ O˜ χq
q such that O˜ χ is indecomposable for integral λ. To make our notation independent of the q q parity of n we will denote by O˜ χ the whole of Oχ for odd n. Abusing notation we will q denote the inclusion and projection functors between O˜ χ and Oq in the same way as above. We have the following pair of functors: q
q O˜ χ m
G:=projχˆ ◦Resg ◦inclχ q
- Og χˆ
F:=projχ ◦Indg ◦inclχˆ
From the definitions we have that (F, G) is an adjoint pair of functors. The main result of this paper is the following theorem: Theorem 1. There is a direct summand F1 of F and a direct summand G1 of G such that the functors F1 and G1 are mutually inverse equivalences of categories. Before we proceed it is necessary to say why the original functors F and G are not q equivalences of categories. Both O˜ χ and Oχgˆ are equivalent to categories of modules q over finite-dimensional quasi-hereditary algebras (see [BGG] for Og and [Fr] for O˜ χ ). χˆ
Moreover, simple objects in both categories are naturally indexed by elements from W . Quasi-hereditary structure on both categories comes with the collection of standard modules. Standard modules in Oχgˆ are the usual Verma modules M(µ), ˆ µˆ ∈ W · λˆ (observe that q here we have the dot action of W ). In O˜ χ standard modules are Verma supermodules. They are constructed as follows: For µ ∈ h∗0 let V (µ) be a simple h-supermodule of weight µ. The supermodule V (µ) is unique if n is odd and satisfies (V (µ)) ∼ = V (µ). If n is even then there are exactly two simple h0 -supermodules of weight µ, namely (V (µ)) and V (µ) (we denote by V (µ) the supermodule, which will give rise to the q Verma supermodule in O˜ χ ). We have V (µ) = V (µ)0 ⊕ V (µ)1 and dimk (V (µ)0 ) = dimk (V (µ)1 ) = 2 (n−1)/2 =: k. Letting n+ act trivially on V (µ) and inducing the obtained module up to U (q) we obtain the Verma supermodule (V (µ)) (note that these supermodules were called Weyl modules in [Go3]). The weight µ is a highest weight of (V (µ)) and has both even and odd q dimension k. The standard modules in O˜ χ are (V (µ)), µ ∈ W λ (observe that here we have the usual action of W ).
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Proposition 2. For every w ∈ W we have ˆ ˆ ⊕ M(w · λ) ˆ ⊕ · · · ⊕ M(w · λ) ˆ =: kM(w · λ), G (V (wλ)) ∼ = M(w · λ) k summands
FM(w · λˆ ) ∼ = (V (wλ)) ⊕ · · · ⊕ (V (wλ)) =: k (V (wλ)). k summands
Proof. We will need the following combinatorial lemma: Lemma 3. Let P denote the multiset { α∈I α : I ⊂ + }. (i) For any w ∈ W we have ({w · λ} + P) ∩ W λ = {wλ}, ({wλ} − P) ∩ W · λ = {w · λ}. (ii) For any w ∈ W there is a unique element x ∈ P such that we have w · λ + x = wλ. Proof. The second equality of (i) follows from the first one, so we will prove only the first equality. Fix w ∈ W . Then we have ρ − wρ =
1 1 α− β= 2 2 + + α∈
β∈w
α=
α∈+ ,w−1 α∈+
α
α∈+ ∩w−
(here − = −+ ), which is an element of P. This yields wλ − w · λ ∈ P and hence wλ ∈ {w · λ} + P. For any α ∈ + we either have α ∈ w+ or α ∈ w− . This implies that ρ + w P = wρ + P (bijection of multisets) and hence w · λ + P = wλ + wρ − ρ + P = wλ + w P. Using the latter equality we obtain (w · λ + P) ∩ W λ = (wλ + w P) ∩ W λ = w(λ + P) ∩ W λ = w((λ + P) ∩ W λ). We have λ ≤ λ + µ for every µ ∈ P. On the other hand, for a dominant regular λ and w ∈ W the inequality λ ≤ wλ forces w to be the identity element. Hence the multiset (λ + P) ∩ W λ consists of exactly one element. The claim of the lemma follows. Fix w ∈ W . Consider the module N = U (g) ⊗U (g) M(w · λˆ ). By the PBW Theorem
for Lie superalgebras ([Ro]), the algebra U (g) is a free U (g)-module with basis g1 . In particular, it follows that the module N is free over U (n− ) of finite rank. Hence N has a filtration whose subquotients are Verma supermodules. As every Verma supermodule is free over U (n− ) by definition, we obtain that the highest weight elements of subquotients in any Verma filtration of N are images of the elements from the space
(n+ )1 ⊗ v, where v is a highest weight vector of M(w · λˆ ). Hence the corresponding highest weights belong to w · λˆ + P. By Lemma 3(i), the only weight from W λ, which intersects w · λˆ + P is wλ = w λˆ . This yields that FM(w · λˆ ) is isomorphic to the direct sum of several copies of (V (wλ)), say k copies.
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Similarly, the restriction of (V (wλ)) to U (g) is a U (n− )-free module of finite rank, and hence has a Verma filtration. The highest weight elements of subquotients of this Verma filtration are images of elements from (n− )1 ⊗ v, where v is a highest weight vector of (V (wλ)). Hence the corresponding highest weights have form wλ − P. By Lemma 3(i), the only weight from W · λˆ , which intersects wλ − P is w · λˆ = w · λ. This yields that G (V (wλ)) is isomorphic to the direct sum of several copies of M(w · λˆ ), say m copies. By adjunction we have ˆ (V (wλ))) = HomU (g) (M(w · λ), ˆ G (V (wλ))), HomU (g) (FM(w · λ), which yields k = m as Verma (super)modules have trivial endomorphism ring. Finally, the multiplicity m equals the even multiplicity of the weight w · λˆ = w · λ in the space (n− )1 ⊗ V (wλ). By Lemma 3(ii), the multiplicity of the weight w · λ − wλ in (n− )1 equals 1. Since dimk (V (wλ)0 ) = dimk (V (wλ)1 ) = k, it follows that m = k and the proof is complete. From Proposition 2 it follows that in order to prove Theorem 1 we have to decompose the functors F and G into a direct sum of k copies of isomorphic functors. For this we will need the technique of Harish-Chandra bimodules. 3. Harish-Chandra U(q) − U(g)-Bimodules This section is inspired by and based on [Go2, 3.1.2]. Let M, N be U (g)-modules. Then the space HomC (M, N ) has the natural structure of a U (g)-bimodule and contains the subbimodule L(M, N ), consisting of all elements, the adjoint action of g on which is locally finite (see [Ja, Kapitel 6]). Similarly one defines L(M, N ) in the case M and N are U (q)-modules and in the case M is a U (g)-module and N is a U (q)-module. As U (q) is a finite extension of U (g) ([Ro]), in all cases we can impose the condition that the adjoint action of the Lie algebra g on elements from L(M, N ) is locally finite. Fix r ∈ N. Let m denote the maximal ideal of U (h), generated by the elements H − λˆ (H ), H ∈ h. Consider the U (h)-module U (h)/mr . Let n+ act on U (h)/mr triviˆ as follows (see [So]): ally and construct the thick Verma module M r (λ) M r (λˆ ) = U (g)
U (h)/mr .
U (h⊕n+ )
Since λˆ is regular, by [So] we have ˆ = U (g)mr , L(M r (λ), ˆ M r (λ)) ˆ ∼ AnnU (g) (M r (λ)) = U (g)/U (g)mrχˆ . χˆ
(1)
Our main result in this section is the following statement, which generalizes [Go2, 3.1.2] to our setup: Proposition 4. There is an isomorphism of U (q) − U (g)-bimodules as follows: ˆ Indqg M r (λ)). ˆ U (q)/U (q)mrχˆ ∼ = L(M r (λ),
Regular Strongly Typical Blocks of Oq
539
Proof. Consider the homomorphism ϕ of U (q) − U (g)-bimodules, defined as follows: q ϕ : U (q) → L(M r (λˆ ), Indg M r (λˆ )), where ϕ(u)(m) = u ⊗ m. By (1) we have AnnU (g) (M r (λˆ )) = U (g)mrχˆ , and hence the map ϕ induces a U (q) − U (g)-bimodule homomorphism ϕ : U (q)/U (q)mrχˆ → L(M r (λˆ ), Indqg M r (λˆ )). Since U (q) is free over U (g), we conclude that ϕ is injective. Let us prove that ϕ is surjective.
By the PBW theorem we have U (q) ∼ = q1 ⊗ U (g). By Kostant separation theorem (see [Ko]), there is a submodule H of the adjoint g-module U (g) such that U (g) ∼ = H ⊗ Z (g). This gives us the following isomorphism of adjoint g-modules: U (q)/U (q)mrχˆ ∼ q1 ⊗ H ⊗ Z (g)/mrχˆ . =
On the other hand, since q1 is finite-dimensional, we also have the following isomorphism of adjoint g-modules: PBW L(M r (λˆ ), Indqg M r (λˆ )) = L(M r (λˆ ), q1 ⊗ M r (λˆ )) ˆ M r (λ)) ˆ (by [Ja, 6.8]) = q1 ⊗ L(M r (λ), (by (1)) = q1 ⊗ U (g)/U (g)mrχˆ = q1 ⊗ H ⊗ Z (g)/mrχˆ . The claim of the proposition follows. 4. Proof of the Main Result We will need the following lemma: Lemma 5. There exists r ∈ N such that mrχˆ M = 0 for any M ∈ Oχgˆ . Proof. The category Oχgˆ is equivalent to the category of modules over some finitedimensional algebra ([BGG]). Hence it has a projective generator Q of finite length, say r , and every object in Oχgˆ is a quotient of a direct sum of some copies of Q. As mχˆ annihilates all M(w · λˆ ), we have that mχˆ annihilates all simple objects and hence that mrχˆ annihilates Q. The claim follows. Now we have to define the thick Verma supermodule (V r (λ)). Let V r (λ) denote the indecomposable projective cover of the simple object V (λ) in the category Frλ of all finite-dimensional h-supermodules, annihilated by mr . Let n+ act on V r (λ) trivially and define the thick Verma supermodule (V r (λ)) as follows: V r (λ). (V r (λ)) = U (q) U (h⊕n+ )
Let Fˆ rλ denote the category of all finite-dimensional h-modules, annihilated by mr . Then U (h)/mr is the unique (up to isomorphism) indecomposable projective module in Fˆ rλ .
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q ˆ ∼ Lemma 6. We have projχ ◦ Indg M r (λ) = k (V r (λ)).
Proof. By adjunction it is enough to show that the kernel of mrχˆ on the U (g)-module (V r (λ)) is isomorphic to kM r (λˆ ). 0
h Lemma 7. We have Resh V r (λ)0 ∼ = k U (h)/mr . h Proof. The restriction functor is left adjoint to the coinduction functor Coindh ∼ = h h n Indh , the latter being exact ([Fr, Prop. 22]). Hence Resh maps V r (λ) to a projech tive module in Fˆ rλ , that is Resh V r (λ)0 ∼ = k U (h)/mr . Let kλ denote the simple object in Fˆ rλ . We have h
k = dim HomFˆ r (Resh V r (λ)0 , kλ ) λ
h
(by adjunction) = dim HomFrλ (V r (λ), Coindh kλ ) h
(by [Fr, Proposition 22]) = dim HomFrλ (V r (λ), n Indh kλ ) h
(by projectivity) = length(n Indh kλ ) (by PBW) = k.
The claim of Lemma 6 follows from Lemma 7 by the same arguments as in the proof of Proposition 2. Proof of Theorem 1. Choose r as given by Lemma 5. Then the functor F is a direct q summand (defined by the projection on O˜ χ ) of the functor U (q)/U (q)mrχˆ −. U (g)
By Proposition 4, the latter functor is isomorphic to the functor L(M r (λˆ ), Indqg M r (λˆ )) −. U (g)
q ˆ decomposes into a direct By Lemma 6 we have that the module projχ ◦ Indg M r (λ) r sum of k copies of (V (λ)). Hence the additivity of the functor L(X, − ) implies that the functor F decomposes into a direct sum of k copies of some functor F1 . By adjunction, the adjoint G decomposes into a direct sum of k copies of some functor G1 such that (F1 , G1 ) forms an adjoint pair. From Proposition 2 we have
ˆ ∼ ˆ F1 M(w · λ) = (V (wλ)), G1 (V (wλ)) ∼ = M(w · λ) for all w ∈ W . As all Verma (super)modules are not annihilated by F1 and G1 , respectively, it follows that the adjunction morphisms IdOg → G1 F1 and F1 G1 → IdO˜ q are χ χˆ nonzero. As the endomorphism ring of a Verma (super)module is trivial, it follows that these adjunction morphisms are isomorphisms.
Regular Strongly Typical Blocks of Oq
541
q As any simple object in both Oχgˆ and O˜ χ is a unique quotient of some Verma (super)module, we conclude that the adjunction morphisms are isomorphisms, when q evaluated on all simple objects. As any object in Oχgˆ and O˜ χ has finite length, a standard induction on the length implies that the adjunction morphisms are isomorphisms of functors. This completes the proof.
5. Concluding Remarks Remark 8. For the superalgebra pq(n) = q(n)/(M(Id, 0)), where Id is the identity matrix, the notion of a strongly typical weight coincides with that for q(n), see [Go3, 10.4 and (35)]). The main result of the paper transfers to pq(n) in a straightforward way for example using the following argument: By the definition, the element M(Id, 0) acts as q the scalar χ (M(Id, 0)) on the block Oχ . When χ (M(Id, 0)) = 0, we obtain that the block q Oχ coincides with the corresponding block for pq(n). Remark 9. For two other Q-type superalgebras, namely sq(n) = {M(A, B) : tr(B) = 0} and psq(n) = sq(n)/(M(Id, 0)) the notion of a strongly typical weight is more complicated and is not given by linear conditions for n > 2 (see [Go3, 10.4 and (35)]). More precisely, for these two superalgebras a weight λ is strongly typical provided that n
λ1 · · · λi−1 λi+1 · · · λn = 0.
(2)
i=1
From [Go3, A.3] it follows that for such strongly typical weights the corresponding simple U (h)-supermodules (which are determined by simple supermodules over some Clifford algebra) are projective in the category of weight supermodules. Using this and following the proof of [Fr, Theorem 12] one shows that for strongly typical weights the corresponding blocks of the category O are described by quasi-hereditary algebras. Now one easily checks that the main result of the paper and all proofs transfer mutatis mutandis to both sq(n) and psq(n) with respect to the definition of strongly typical weights, given by (2). Remark 10. To extend the main result to singular weights one has to develop the theory of Harish-Chandra bimodules for superalgebras in the similar way as done for Lie algebras in [BG]. Remark 11. Blocks which are typical but not strongly typical are described by properly stratified rather than by quasi-hereditary algebras (see [Fr]). Hence the results of this paper do not extend to such blocks. In [Fr, 3.9] it is conjectured that such blocks are tensor products of strongly typical blocks with k[x]/(x 2 )-mod. Remark 12. As one of the referees pointed out, a natural question is whether the result can be extended to larger categories than O. Acknowledgements. For the second author the research was partially supported by the Swedish Research Council. We would like to thank the referees for several corrections and very useful suggestions, which led to improvements in the original version of the paper.
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References [BG] [BGG] [Br] [CPS] [Dl] [DR] [Fr] [Go1] [Go2] [Go3] [Ja] [Ko] [Pe] [PS] [Ro] [So]
Bernstein, J., Gelfand, S.: Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. Comp. Math. 41(2), 245–285 (1980) Bernstrein, I., Gelfand, I.M., Gelfand, S.I.: A certain category of g-modules. Funk. Anal. I Pril. 10(2), 1–8 (1976) Brundan, J.: Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n). Adv. Math. 182(1), 28–77 (2004) Cline, E., Parshall, B., Scott, L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988) Dlab, V.: Properly stratified algebras. C. R. Acad. Sci. Paris Sér. I Math. 331(3), 191–196 (2000) Dlab, V., Ringel, C.: Quasi-hereditary algebras. Ill. J. Math. 33(2), 280–291 (1989) Frisk, A.: Typical blocks of the category for the queer lie superalgebra. J. Algebra Appl. 6(5), 731–778 (2007) Gorelik, M.: On the ghost centre of Lie superalgebras. Ann. Inst. Fourier (Grenoble) 50(6), 1745–1764 (2000) Gorelik, M.: Strongly typical representations of the basic classical Lie superalgebras. J. Amer. Math. Soc. 15(1), 167–184 (2002) Gorelik, M.: Shapovalov determinants of Q-type Lie superalgebras. IMRP Int. Math. Res. Pap. 2006, Art. ID 96895, 71 pp Jantzen, J.C.: Einhüllende Algebren halbeinfacher Lie-Algebren. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 3. Berlin: Springer-Verlag, 1983 Kostant, B.: Lie group representations on polynomial rings. Amer. J. Math. 85, 327–404 (1963) Penkov, I.: Generic representations of classical Lie superalgebras and their localization. Monatsh. Math. 118(3-4), 267–313 (1994) Penkov, I., Serganova, V.: Representations of classical lie superalgebras of type I. Indag. Math. (N.S.) 3(4), 419–466 (1992) Ross, L.: Representations of graded Lie algebras. Trans. Amer. Math. Soc. 120, 17–23 (1965) Soergel, S.: Équivalences de certaines catégories de g-modules. C. R. Acad. Sci. Paris Sér. I Math. 303(15), 725–728 (1986)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 291, 543–577 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0804-6
Communications in
Mathematical Physics
Non-Perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles Ivan G. Avramidi, Guglielmo Fucci New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA. E-mail: [email protected]; [email protected] Received: 28 October 2008 / Accepted: 9 January 2009 Published online: 7 April 2009 – © Springer-Verlag 2009
Abstract: We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group G × U (1) over a Riemannian manifold M without boundary. The total connection on the vector bundle naturally splits into a G-connection and a U (1)-connection, which is assumed to have a parallel curvature F. We find a new local short time asymptotic expansion of the offdiagonal heat kernel U (t|x, x ) close to the diagonal of M × M assuming the curvature F to be of order t −1 . The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the G-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature F, more precisely, on t F. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in F in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature F. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion. 1. Introduction The heat kernel is one of the most powerful tools in quantum field theory and quantum gravity as well as mathematical physics and differential geometry (see for example [10– 12,18–20,22,23] and further references therein). It is of particular importance because the heat kernel methods give a framework for manifestly covariant calculation of a wide range of relevant quantities in quantum field theory like one-loop effective action, Green’s functions, effective potential, etc. Unfortunately the exact computation of the heat kernel can be carried out only for exceptional highly symmetric cases when the spectrum of the operator is known exactly, (see [17,19,20] and the references in [8,13–15]). Although these special cases are very important, in quantum field theory we need the effective action, and, therefore, the heat kernel for general background fields. For this reason various approximation schemes have
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been developed. One of the oldest methods is the Minackshisundaram-Pleijel short-time asymptotic expansion of the heat kernel as t → 0 (see the references in [2,18,23]). Despite its enormous importance, this method is essentially perturbative. It is an expansion in powers of the curvatures R and their derivatives and, hence, is inadequate for large curvatures when t R ∼ 1. To be able to describe the situation when at least some of the curvatures are large one needs an essentially non-perturbative approach, which effectively sums up in the short time asymptotic expansion of the heat kernel an infinite series of terms of certain structure that contain large curvatures (for a detailed analysis see [4,9] and reviews [10,12]). For example, the partial summation of higher derivatives enables one to obtain a non-local expansion of the heat kernel in powers of curvatures (high-energy approximation in physical terminology). This is still an essentially perturbative approach since the curvatures (but not their derivatives) are assumed to be small and one expands in powers of curvatures. On another hand to study the situation when curvatures (but not their derivatives) are large (low energy approximation) one needs an essentially non-perturbative approach. A promising approach to the calculation of the low-energy heat kernel expansion was developed in non-Abelian gauge theories and quantum gravity in [3–8,13–15]. While the papers [3,4,6,7] dealt with the parallel U (1)-curvature (that is, constant electromagnetic field) in flat space, the papers [5,8,13] dealt with symmetric spaces (pure gravitational field in absence of an electromagnetic field). The difficulty of combining the gauge fields and gravity was finally overcome in the papers [14,15], where homogeneous bundles with parallel curvature on symmetric spaces was studied. In this paper we compute the heat kernel for the covariant Laplacian with a large parallel U (1) curvature F in a Riemannian manifold (that is, strong covariantly constant electromagnetic field in an arbitrary gravitational field). Our aim is to evaluate the first three coefficients of the heat kernel asymptotic expansion in powers of Riemann curvature R but in all orders of the U (1) curvature F. This is equivalent to a partial summation in the heat kernel asymptotic expansion as t → 0 of all powers of F in terms which are linear and quadratic in Riemann curvature R. 2. Setup of the Problem Let M be a n-dimensional compact Riemannian manifold without boundary and S be a complex vector bundle over M realizing a representation of the group G ⊗ U (1). Let ϕ be a section of the bundle S and ∇ be the total connection on the bundle S (including the G-connection as well as the U(1)-connection). Then the commutator of covariant derivatives defines the curvatures [∇µ , ∇ν ]ϕ = (Rµν + i Fµν )ϕ,
(2.1)
where Rµν is the curvature of the G-connection and Fµν is the curvature of the U (1)connection (which will be also called the electromagnetic field). In the present paper we consider a second-order Laplace type partial differential operator, L = −,
= g µν ∇µ ∇ν .
(2.2)
The heat kernel for the operator L is defined as the solution of the heat equation (∂t + L ) U (t|x, x ) = 0,
(2.3)
Non-Perturbative Heat Kernel Asymptotics
545
with the initial condition U (0|x, x ) = P(x, x )δ(x, x ),
(2.4)
where δ(x, x ) is the covariant scalar delta function and P(x, x ) is the operator of parallel transport of the sections of the bundle S along the geodesic from the point x to the point x. The spectral properties of the operator L are described in terms of the spectral functions, defined in terms of the L 2 traces of some functions of the operator L , such as the zeta-function ζ (s) = Tr L −s , and the heat trace (2.5) Tr exp(−tL ) = dvol tr U diag (t), M
where dvol = g 1/2 d x is the Riemannian volume element with g = det gµν and tr denotes the fiber trace. Here and everywhere below the diagonal value of any two point quantity f (x, x ) denotes the coincidence limit as x → x , that is, f diag = f (x, x).
(2.6)
It is well known [18] that the heat kernel has the asymptotic expansion as t → 0 (see also [2,10,11,23]), ∞ σ (x, x ) k U (t|x, x ) ∼ (4π t)−n/2 P(x, x )1/2 (x, x ) exp − t ak (x, x ), (2.7) 2t k=0
where σ (x, x ) is the geodesic interval (or the world function) defined as one half the square of the geodesic distance between the points x and x and (x, x ) is the Van Vleck-Morette determinant. The coefficients ak (x, x ) are called the off-diagonal heat kernel coefficients. The heat kernel diagonal and the heat trace have the asymptotic expansion as t → 0 [2,23], U diag (t) ∼ (4π t)−n/2
∞
diag
,
(2.8)
t k Ak ,
(2.9)
t k ak
k=0
Tr exp(−tL ) ∼ (4π t)−n/2
∞ k=0
where diag
ak and
Ak =
= ak (x, x)
diag
dvol tr ak
(2.10)
.
(2.11)
M
The coefficients Ak are called the global heat kernel coefficients; they are spectral invariants of the operator L .
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I. G. Avramidi, G. Fucci diag
The diagonal heat kernel coefficients ak are polynomials in the jets of the metric, the G- connection and the U (1)-connection; in other words, in the curvature tensors and their derivatives. Let us symbolically denote the jets of the metric and the G-connection by R(n) = ∇(µ1 · · · ∇µn R a µn+1 b µn+2 ) , ∇(µ1 · · · ∇µn Ra µn+1 ) , (2.12) and the jets of the U (1) connection by F(n) = ∇(µ1 · · · ∇µn F a µn+1 ) .
(2.13)
Here and everywhere below the parentheses indicate complete symmetrization over all indices included. By counting the dimension it is easy to describe the general structure of the coeffidiag cients ak . Let us introduce the multi-indices of nonnegative integers i = (i 1 , . . . , i m ),
j = ( j1 , . . . , jl ).
(2.14)
|j| = j1 + · · · + jl .
(2.15)
Let us also denote |i| = i 1 + · · · + i m , Then symbolically diag
ak
=
k N N −l
N =1 l=0 m=0
i,j≥0 |i|+|j|+2N =2k
C(k,l,m),i,j F( j1 ) · · · F( jl ) R(i1 ) · · · R(im ) ,
(2.16)
where C(k,l,m),i,j are some universal constants. The lower order diagonal heat kernel coefficients are well known [2,11,18] diag
a0
diag
a1
= 1,
=
1 R, 6
(2.17) (2.18)
1 2 1 1 1 R + R − Rµν R µν + Rαβµν R αβµν 30 72 180 180 1 1 1 Fµν F µν . + Rµν Rµν + Rµν i F µν − (2.19) 12 6 12 To avoid confusion we should stress that the normalization of the coefficients ak differs from the papers [2,10,11]. In the present paper we study the case of a parallel U (1) curvature (covariantly constant electromagnetic field), i.e. diag
a2
=
∇µ Fαβ = 0.
(2.20)
That is, all jets F(n) are set to zero except the one of order zero, which is F itself. In this case Eq. (2.16) takes the form diag
ak
=
k N N −l
N =1 l=0 m=0
i≥0 |i|+2N =2k
C(k,l,m),i F l R(i1 ) · · · R(im ) ,
where C(k,l,m),i are now some (other) numerical coefficients.
(2.21)
Non-Perturbative Heat Kernel Asymptotics
547
Thus, by summing up all powers of F in the asymptotic expansion of the heat kernel diagonal we obtain a new (non-perturbative) asymptotic expansion U diag (t) ∼ (4π t)−n/2
∞
diag
t k a˜ k
(t),
(2.22)
k=0 diag
where the coefficients a˜ k diag
a˜ k
(t) =
(t) are polynomials in the jets R(n) k N
N =1 m=0
i≥0 |i|+2N =2k
(k) f (m,i) (t) R(i1 ) · · · R(im ) ,
(2.23)
(k)
and f (m,i) (t) are some universal dimensionless tensor-valued analytic functions that depend on F only in the dimensionless combination t F. For the heat trace we obtain then a new asymptotic expansion of the form Tr exp(−tL ) ∼ (4π t)−n/2
∞
t k A˜ k (t),
(2.24)
k=0
where A˜ k (t) =
diag
dvol tr a˜ k
(t).
(2.25)
M
This expansion can be described more rigorously as follows. We rescale the U (1)-curvature F by ˜ F → F(t) = t −1 F,
(2.26)
so that t F(t) = F˜ is independent of t. Then the operator L (t) becomes dependent on t (in a singular way!). However, the heat trace still has a nice asymptotic expansion as t → 0, Tr exp[−tL (t)] ∼ (4π t)−n/2
∞
t k A˜ k ,
(2.27)
k=0
where the coefficients A˜ k are expressed in terms of F˜ = t F(t), and, therefore, are independent of t. Thus, what we are doing is the asymptotic expansion of the heat trace for a particular case of a singular (as t → 0) time-dependent operator L (t). Let us stress once again that the Eq. (2.23) should not be taken literally; it only reprediag sents the general structure of the coefficients a˜ k (t). To avoid confusion we list below the general structure of the low-order coefficients in more detail:
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I. G. Avramidi, G. Fucci diag
(t) = f (0) (t),
diag
(t) = f (1,1) αβµν (t)Rαβµν + f (1,2) µν (t)Rµν ,
diag
(t) = f (1,1) αβµνσρ (t)∇(α ∇β) Rµνσρ + f (1,2) αβµν (t)∇(α ∇β) Rµν
a˜ 0 a˜ 1 a˜ 2
(2.28)
(1)
(1)
(2)
(2.29)
(2)
(2) αβγ δµνσρ (2) αβµν + f (2,1) (t)Rαβγ δ Rµνσρ + f (2,2) (t)Rαβ Rµν (2)
+ f (2,3) αβµνσρ (t)Rαβ Rµνσρ ,
(2.30) (i)
with obvious enumeration of the functions. It is the universal tensor functions f (l,m) (t) (i)
that are of prime interest in this paper. Our main goal is to compute the functions f (l,m) (t) diag
diag
diag
for the coefficients a˜ 0 (t), a˜ 1 (t) and a˜ 2 (t). Of course, for t = 0 (or F = 0) the coefficients a˜ k (t) are equal to the usual diagonal heat kernel coefficients diag
a˜ k
diag
(0) = ak
.
(2.31) diag
Therefore, by using the explicit form of the coefficients ak given by (2.19) we obtain (i) the initial values for the functions f ( j,k) . Moreover, by analyzing the corresponding diag
diag
terms in the coefficients a3 and a4 (which are known, [2,18,22]), one can obtain (i) partial information about some lower order Taylor coefficients of the functions f ( j,k) (t): 1 2 t Fµν F µν + O(t 3 ), 12 1 α β (1) f (1,1) αβ µν (t) = δ[µ δν] + O(t), 6 1 (1) f (1,2) µν (t) = ti F µν + O(t 2 ), 6 1 αβ µ ν (2) αβµν g δ[σ δρ] + O(t), f (1,1) σρ (t) = 30 1 β) (2) f (1,2) αβ µν (t) = − ti F (α [ν δµ] + O(t 2 ), 15 f (0) (t) = 1 −
(2)
1 1 [γ σ] gµ[α gβ]ν g σ [γ g δ]ρ − δ gβ][ν g δ][ρ δµ] 180 180 [α 1 γ δ σ ρ + δ[α δβ] δ[µ δν] + O(t), 72 1 α β (2) f (2,2) αβ µν (t) = δ δ + O(t), 12 [µ ν]
(2.32) (2.33) (2.34) (2.35) (2.36)
f (2,1) αβ γ δ µν σρ (t) =
(2)
f (2,3) αβµν σρ (0) = −
(2.37) (2.38)
1 1 1 [µ β] µ ν α β ti F αβ δ[σ δρ] − ti F µν δ[σ δρ] + δ[σ ti F ν][α δρ] + O(t 2 ). 36 30 9 (2.39)
This information can be used to check our final results.
Non-Perturbative Heat Kernel Asymptotics
549
Notice that the global coefficients A˜ k (t) have exactly the same form as the local ones; the only difference is that the terms with the derivatives of the Riemann curvature do not contribute to the integrated coefficients since they can be eliminated by integrating by parts and taking into account that F is covariantly constant. Moreover, we study even more general non-perturbative asymptotic expansion for the off-diagonal heat kernel and compute the coefficients of zero, first and second order in the Riemann curvature. We will show that there is a new non-perturbative asymptotic ˜ so that t F is fixed) expansion of the off-diagonal heat kernel as t → 0 (and F = t −1 F, of the form U (t|x, x ) ∼ P(x, x )1/2 (x, x )U0 (t|x, x )
∞
t k/2 bk (t|x, x ),
(2.40)
k=0
where U0 is an analytic function of F such that for F = 0, σ (x, x ) −n/2 U0 (t|x, x ) = (4π t) exp − . 2t F=0
(2.41)
Here bk (t|x, x ) are analytic functions of t that depend on F only in the dimensionless combination t F. Of course, for t = 0 they are equal to the usual heat kernel coefficients, that is, b2k (0|x, x ) = ak (x, x ),
b2k+1 (0|x, x ) = 0.
(2.42)
Moreover, we will show below that the odd-order coefficients vanish not only for t = 0 and any x = x but also for any t and x = x , that is, on the diagonal, diag
b2k+1 (t) = 0.
(2.43)
Thus, the heat kernel diagonal has the asymptotic expansion (2.22) as t → 0 with diag
a˜ k
diag
(t) = (4π t)n/2 U0
diag
(t)b2k (t).
(2.44)
3. Geometric Framework Our goal is to study the heat kernel U (t|x, x ) in the neighborhood of the diagonal as x → x . Therefore, we will expand all relevant quantities in covariant Taylor series near the diagonal following the methods developed in [2,10–12]. We fix a point, say x , on the manifold M and consider a sufficiently small neighborhood of x , say a geodesic ball with a radius smaller than the injectivity radius of the manifold. Then, there exists a unique geodesic that connects every point x to the point x . In order to avoid a cumbersome notation, we will denote by Latin letters tensor indices associated to the point x and by Greek letters tensor indices associated to the point x . Of course, the indices associated with the point x (resp. x ) are raised and lowered with the metric at x (resp. x ). Also, we will denote by ∇a (resp. ∇µ ) covariant derivative with respect to x (resp. x ). We recall below the definition of some of the two-point functions that we will need in our analysis. First of all, the world function σ (x, x ) is defined as one half of the square of the length of the geodesic between the points x and x . It satisfies the equation σ =
1 a 1 u ua = uµuµ, 2 2
(3.1)
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I. G. Avramidi, G. Fucci
where u a = ∇a σ,
u µ = ∇µ σ .
(3.2)
The variables u µ are nothing but the normal coordinates at the point x . The Van Vleck-Morette determinant is defined by 1
1
(x, x ) = g − 2 (x) det[−∇a ∇ν σ (x, x )]g − 2 (x ).
(3.3)
This quantity should not be confused with the Laplacian = g µν ∇µ ∇ν . Usually, the meaning of will be clear from the context. We find it convenient to parameterize it by (x, x ) = exp[2ζ (x, x )].
(3.4)
ηµ b = ∇b ∇ µ σ ,
(3.5)
Next, we define the tensor and the tensor γ a µ inverse to ηµ a by γ a µ ηµ b = δba ,
ηµ b γ b ν = δνµ .
(3.6)
This enables us to define new derivative operators by ∇¯ µ = γ a µ ∇a .
(3.7)
These operators commute when acting on objects that have been parallel transported to the point x (in other words the objects that do not have Latin indices). In fact, when acting on such objects these operators are just partial derivatives with respect to normal coordinate u, ∂ ∇¯ µ = µ . ∂u
(3.8)
1 Dµ = ∇¯ µ − i Fµα u α . 2
(3.9)
We also define the operators
Obviously, they form the algebra Dµ , Dν = i Fµν ,
Dµ , u ν = δµ ν .
(3.10)
Next, the parallel displacement operator P(x, x ) of sections of the vector bundle S along the geodesic from the point x to the point x is defined as the solution of the equation u a ∇a P(x, x ) = 0 ,
(3.11)
P(x, x) = I ,
(3.12)
with the initial condition
where I is the identity endomorphism of the bundle S. Finally, we define the two-point quantity Aµ = P −1 ∇¯ µ P .
(3.13)
Non-Perturbative Heat Kernel Asymptotics
551
We recall, here, that we consider the case of a covariantly constant electromagnetic field, i.e. ∇µ Fαβ = 0 .
(3.14)
In this case we find it useful to decompose the quantity Aµ as 1 Aµ = − i Fµα u α + A¯µ . (3.15) 2 By using this machinery we can rewrite the heat kernel as follows. First of all, the heat kernel can be presented in the form U (t|x, x ) = exp (−tL ) P(x, x )δ(x, x ) ,
(3.16)
which can also be written as 1 U (t|x, x ) = P(x, x ) 2 (x, x ) exp(−t L˜ )δ(u) ,
(3.17) uµ
(recall that u µ where δ(u) is the usual delta-function in the normal coordinates depends on x and x and u = 0 when x = x ) and L˜ is an operator defined by 1 1 L˜ = P −1 (x, x )− 2 (x, x )L 2 (x, x )P(x, x ) .
(3.18)
As is shown in [2,11] the operator L˜ can be written in the form 1 1 L˜ = − 2 (∇¯ µ + Aµ )−1 X µν (∇¯ ν + Aν ) 2
= − ∇¯ µ + Aµ − ζµ X µν ∇¯ ν + Aν + ζν
= − Dµ + A¯µ − ζµ X µν Dν + A¯ν + ζν ,
(3.19)
where ζµ = ∇¯ µ ζ . Now, by using these equations and by recalling the formula in (3.15), one can rewrite the operator in (3.19) in another way as follows:
(3.20) L˜ = − X µν Dµ Dν + Y µ Dµ + Z , where X µν = ηµ a ηνa ,
(3.21)
Y µ = (∇¯ µ X µν ) + 2X µν A¯µ ,
(3.22)
Z = A¯µ X µν A¯ν − ζµ X µν ζν + (∇¯ µ X µν )A¯ν + (∇¯ µ X µν )ζν + X µν ∇¯ µ A¯ν + X µν ∇¯ µ ζν .
(3.23)
4. Perturbation Theory Our goal is now to develop the perturbation theory for the heat kernel. We need to identify a small expansion parameter ε in which the perturbation theory will be organized as ε → 0. First of all, we assume that t is small, more precisely, we require t ∼ ε2 . Also, since we will work close to the diagonal, that is, x is close to x , we require that u µ ∼ ε. This will also mean that ∇¯ ∼ ε−1 and ∂t ∼ ε−2 . Finally, we assume that F is large, that is, of order F ∼ ε−2 . To summarize, t ∼ ε2 ,
u µ ∼ ε,
F ∼ ε−2 .
(4.1)
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I. G. Avramidi, G. Fucci
4.1. Covariant Taylor expansion. The Taylor expansions of the quantities introduced above have the form (up to the fifth order) [2,11], X µν = g µν +
1 µ ν α β 1 1 R α β u u − ∇ α R µ β ν γ u α u β u γ + ∇ α ∇β R µ γ ν δ u α u β u γ u δ 3 6 20
1 µ R αλβ R λ γ ν δ u α u β u γ u δ + O(u 5 ), 15 1 1 1 ζ = Rαβ u α u β − ∇α Rβγ u α u β u γ + ∇α ∇β Rγ δ u α u β u γ u δ 12 24 80 1 µ ν α β γ δ 5 Rµανβ R γ δ u u u u + O(u ), + 360 1 1 1 1 2 = 1 + Rαβ u α u β − ∇α Rβγ u α u β u γ + ∇α ∇β Rγ δ u α u β u γ u δ 12 24 80 1 1 Rαβ Rγ δ u α u β u γ u δ + Rµανβ R µ γ ν δ u α u β u γ u δ + O(u 5 ), + 288 360 1 1 1 A¯µ = − Rµα u α + Rµανβ i F ν γ u α u β u γ + ∇α Rµβ u α u β 2 24 3 1 1 + Rµανβ Rν γ u α u β u γ − ∇α ∇β Rµγ u α u β u γ 24 8 1 ν λ α β γ δ Rµανβ R γ λδ i F u u u u u + O(u 6 ) . − 720 +
(4.2)
(4.3)
(4.4)
(4.5)
We would like to stress that all coefficients of such expansions are evaluated at the point x . Also note that the expansion for A¯µ is valid in the case of a covariantly constant electromagnetic field. 4.2. Perturbation Theory for the Operator L . Now, we expand the operator L˜ in a formal power series in ε (recall that D ∼ ε−1 and u ∼ ε) to obtain L ∼−
∞
Lk ,
(4.6)
k=0
where Lk are operators of order εk−2 . In particular, L0 = D2 , L1 = 0, µν µ Lk = X k Dµ Dν + Yk Dµ + Z k ,
k ≥ 2,
(4.7) (4.8) (4.9)
where D2 = g µν Dµ Dν , µν
µ
(4.10)
and X k , Yk and Z k are some tensor-valued polynomials in normal coordinates u µ . µν Note that X k are homogeneous polynomials in normal coordinates u µ and F of µ order εk . Similarly, Yk ∼ εk−1 and Z k ∼ εk−2 . Of course, here the terms Fuu are counted as of order zero. That is, they have the form
Non-Perturbative Heat Kernel Asymptotics
553 µν
µν
X k = P(1), k (u), µ
µαβ
µ
Yk = P(2), k−1 + Fαβ P(3), k+1 (u), Z k = P(4), k−2 +
αβ Fαβ P(5), k (u) +
αβρσ Fαβ Fρσ P(6), k+2 (u),
(4.11) (4.12) (4.13)
where P( j), k (u) are homogeneous tensor valued polynomials of degree k. By using the covariant Taylor expansions in (4.2), (4.5) and (4.3) we find the explicit expression of the coefficients: µν
µν
α β αβ u u , µ µ µ Y2 = E 2 α u α + G 2 αβγ u α u β u γ , Z 2 = H2 αβ u α u β + L 2 ,
X 2 = C2
µν
µν
α β γ αβγ u u u , µ µ Y3 = E 3 αβ u α u β , Z 3 = H3 α u α ,
X 3 = C3
µν
α β γ δ αβγ δ u u u u , µ µ = E 4 αβγ u α u β u γ + G 4 αβγ δ u α u β u γ u δ u , H4 αβ u α u β + L 4 αβγ δ u α u β u γ u δ + O4 αβγ δκ u α u β u γ u δ u u κ , µ Y4
Z4 =
µν
X 4 = C4
(4.14) (4.15) (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) (4.22)
where µν
C2
αβ
=
1 µ ν R (α β) , 3
1 µ E 2 α = − R µ α − Rµ α , 3 1 µ G 2 αβγ = − R µ (α ν β i Fγ )ν , 12 1 H2 αβ = − Rµ(α i F µ β) , 24 1 L 2 = R, 6 1 µν C3 αβγ = − ∇(α R µ β ν γ ) , 6 µ
1 1 2 ∇(α R µ β) − ∇ µ Rαβ + ∇(α Rµ β) , 3 6 3 1 1 µ = ∇µ R α − ∇α R, 3 6 1 µ 1 R (α|λ|β R λ γ ν δ) + ∇(α ∇β R µ γ ν δ) , = 15 20 1 1 µ ν 1 R (α β Rγ )ν − R µ (α ν β R|ν|γ ) = − R µ ν(α|λ| R ν β λ γ ) − 15 60 4 1 3 1 + ∇(α ∇ µ Rβγ ) − ∇(α ∇β R µ γ ) − ∇(α ∇β Rµ γ ) , 10 20 4
(4.23)
E 3 αβ = H3 α µν
C4
αβγ δ
µ
E4
αβγ
(4.24)
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I. G. Avramidi, G. Fucci µ
G4
αβγ δ
H4 αβ
L 4 αβγ δ
O4 αβγ δκ
1 µ R (α|ν|β R ν γ λ δ i F|λ|) , 40 1 1 1 1 = Rµ(α Rµ β) − Rµα R µ β − ∇(α ∇|µ| Rµ β) + Rµν R µ α ν β 4 30 4 60 1 1 3 + Rµλγ α R µλγ β + Rαβ + ∇α ∇β R, 60 40 40 1 1 Rµ(α|λ|β R λ γ µν i F|ν|δ) = − Rµ(α ν β R µ γ i F|ν|δ) − 80 80 1 − Rµ(α R µ β ν γ i F|ν|δ) , 24 1 Rµ(α ν β R µ γ λ δ i F|ν| i F|λ|κ) . = (4.25) 576 =
Here and everywhere below the parentheses denote the complete symmetrization over all indices enclosed; the vertical lines indicate the indices excluded from the symmetrization.
4.3. Perturbation theory for the heat semigroup. Now, by using the perturbative expansion (4.6) of the operator L˜ and recalling that D2 ∼ ε−2 and t ∼ ε2 , we see that the operator tD2 is of zero order and the operator tLk , k ≥ 2, is of (higher) order εk . Therefore, we can consider the terms tLk with k ≥ 2 as a perturbation. In order to evaluate the heat semigroup we utilize the Volterra series for the exponent of two non-commuting operators. Let X be an operator and Y be a perturbation (say of order one in a small parameter). Then exp(X + Y ) = T exp X,
(4.26)
where T =I+
∞ k=1 0
τk
1
τ2 dτk−1 · · ·
dτk 0
dτ1 Y˜ (τ1 )Y˜ (τ2 ) · · · Y˜ (τk )
(4.27)
0
and Y˜ (τ ) = eτ X Y e−τ X .
(4.28)
By using the above series for the operator in (4.6) we obtain exp(−t L˜ ) = T (t) exp(tD2 ),
(4.29)
where T (t) is an operator defined by a formal perturbative expansion T (t) ∼
∞ k=0
Tk (t),
(4.30)
Non-Perturbative Heat Kernel Asymptotics
555
with Tk (t) being of order εk . Explicitly, up to terms of fifth order we obtain T0 (t) = I,
(4.31)
T1 (t) = 0, 1 T2 (t) = t dτ1 V2 (tτ1 ),
(4.32) (4.33)
0
1 T3 (t) = t
dτ1 V3 (tτ1 ),
(4.34)
0
1 T4 (t) = t
τ2
1 dτ1 V4 (tτ1 ) + t
2
0
dτ1 V2 (tτ1 )V2 (tτ2 ),
dτ2 0
(4.35)
0
and Vk (s) = es D Lk e−s D . 2
2
(4.36)
4.4. Perturbation theory for the heat kernel. As we already mentioned above the heat kernel can be computed from the heat semigroup by using Eq. (3.17). By using the heat semigroup expansion from the previous section we now obtain the heat kernel in the form U (t|x, x ) ∼ P(x, x )1/2 (x, x )U0 (t|x, x )
∞
t k/2 bk (t|x, x ) ,
(4.37)
k=0
where U0 (t|x, x ) = exp(tD2 )δ(u),
(4.38)
bk (t|x, x ) = t −k/2 U0−1 (t|x, x )Tk (t)U0 (t|x, x ) .
(4.39)
and
Thus, the calculation of the heat kernel coefficients reduces to the evaluation of the zero-order heat kernel U0 (t|x, x ) and to the action of the differential operators Tk (t) on it. The zero order heat kernel U0 (t|x, x ) can be evaluated by using the algebraic method developed in [3,4]. First, the heat semigroup exp(tD2 ) can be represented as an average over the (nilpotent) Lie group (3.10) with a Gaussian measure
1 µ 2 −n/2 ν µ J (t) dk exp − k Mµν (t)k + k Dµ , (4.40) exp(tD ) = (4π t) 4 Rn
where
J (t) = det
tiF sinh(tiF)
1/2 (4.41)
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I. G. Avramidi, G. Fucci
and M(t) is a symmetric matrix defined by M(t) = iF coth(tiF).
(4.42)
We would like to stress, at this point, that here and everywhere below all the functions of the 2-form F are analytic and should be understood in terms of a power series in F. Then by using the relation
one obtains
exp(k µ Dµ )δ(u) = δ(u + k),
(4.43)
1 U0 (t|x, x ) = (4π t)−n/2 J (t) exp − u µ Mµν (t)u ν , 4
(4.44)
which is nothing but the Schwinger kernel for an electromagnetic field on Rn [21]. To obtain the asymptotic expansion of the heat kernel diagonal we just need to set x = x (or u = 0). At this point, we notice the following interesting fact. The operators tLk , t Vk (tτ ) and Tk (t) are differential operators with homogeneous polynomial coefficients (in u µ ) of order εk . Recall that u ∼ ε, t ∼ ε2 and F ∼ ε−2 , so that t F and Fuu are counted as of order zero. Since the zero order heat kernel U0 is Gaussian, then the off-diagonal coefficients bk (t|x, x ) are polynomials in u. The point we want to make now is the following. Lemma 1. The off-diagonal odd-order coefficients b2k+1 are odd order polynomials in u µ , that is, they satisfy b2k+1 (t|x, x ) = −b2k+1 (t|x, x ), (4.45) u→−u
and, therefore, vanish on the diagonal, diag
b2k+1 (t) = 0.
(4.46)
Proof. We discuss the transformation properties of various quantities under the reflection of the coordinates, u → −u. First, we note that the operator D changes sign, and, therefore, the operator L0 = −D2 is invariant. Next, from the general form of the operator Lk discussed above we see that Lk → (−1)k Lk . Therefore, the same is true for the operator Vk (tτ ), that is, Vk → (−1)k Vk . Now, the operator Tk (t) has the following general form: Tk = t
k
1 [k/2] m=1 0
τm−1
dτ1 · · ·
dτm 0
Cm, j V j1 (tτ1 ) · · · V jm (tτm ),
(4.47)
|j|=k
where the summation goes over multiindex j = ( j1 , . . . , jm ) of integers j1 , . . . , jm ≥ 2 such that |j| = j1 + · · · + jm = k, and Cm, j are some numerical coefficients. Therefore, the operator Tk transforms as Tk → (−1)k Tk . Since the zero-order heat kernel U0 is invariant under the reflection of coordinates u → −u, we finally find that the coefficients bk transform according to bk → (−1)k bk . Thus, b2k are even polynomials and b2k+1 are odd-order polynomials.
Non-Perturbative Heat Kernel Asymptotics
557
By using this lemma and by setting x = x we obtain the asymptotic expansion of the heat kernel diagonal U diag (t) ∼ (4π t)−n/2 J (t)
∞
diag
t k b2k (t),
(4.48)
k=0
where the function J (t) is defined in (4.41). Thus, we obtain diag
a˜ k
diag
(t) = J (t)b2k (t).
(4.49)
4.5. Algebraic framework. As we have shown above the evaluation of the heat semigroup is reduced to the calculation of the operators Vk (s) defined by (4.36), which reduces, in turn, to the computation of general expressions es D u ν1 · · · u νn Dµ1 · · · Dµm e−s D = Z ν1 (s) · · · Z νn (s)Aµ1 (s) · · · Aµm (s), 2
2
(4.50)
where Z ν (s) = es D u ν e−s D , 2
Aµ (s) = e
s D2
2
Dµ e
−s D 2
(4.51) .
(4.52)
[Z µ (s), Z ν (s)] = 0.
(4.53)
Obviously, the operators Aµ and Z ν form the algebra [Aµ (s), Z ν (s)] = δµν ,
[Aµ (s), Aν (s)] = i Fµν ,
The operators Aµ (s) and Z ν (s) can be computed as follows. First, we notice that Aµ (s) satisfies the differential equation ∂s Aµ (s) = AdD2 Aµ (s),
(4.54)
with the initial condition Aµ (0) = Dµ . Hereafter AdD2 is an operator acting as a commutator, that is, AdD2 Aµ (s) ≡ [D2 , Aµ (s)] .
(4.55)
The solution of Eq. (4.54) is Aµ (s) = exp(sAdD2 )Dµ ,
(4.56)
which can be written in terms of series as Aµ (s) =
∞ k k s
AdD2 Dµ . k!
(4.57)
k=0
Now, by using the algebra (3.10) we first obtain the commutator [D2 , Dµ ] = −2i Fµα Dα ,
(4.58)
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I. G. Avramidi, G. Fucci
and then, by induction, k
AdD2 Dµ = (−2i)k Fµα1 F α1 α2 · · · F αk−1 αk Dαk = [(−2i F)k ]µα Dα . (4.59) By substituting this result in the series (4.57) we finally find that Aµ (s) = µ α (s)Dα ,
(4.60)
(s) = exp(−2si F) .
(4.61)
where
Similarly, for the operators Z ν (s) we find Z µ (s) = exp(sAdD2 )u µ =
∞ k k s
AdD2 u µ . k!
(4.62)
k=0
Now, by using the commutators in (3.10), we find AdD2 u µ = D2 , u µ = 2Dµ ,
(4.63)
and then, by induction, we obtain, for k ≥ 2,
k AdD2 u µ = 2[(−2i F)k−1 ]µα Dα .
(4.64)
Thus the operator Z µ (s) in (4.52) takes the form Z µ (s) = u µ − 2sDµ + 2
∞ k s k=2
k!
[(−2i F)k−1 ]µα Dα .
(4.65)
This series can be easily summed up to give Z µ (s) = u µ + µα (s)Dα ,
(4.66)
where (s) =
1 − exp(−2si F) sinh(si F) = 2 exp(−si F) . iF iF
(4.67)
Now, by using (4.61) and (4.67) we obtain −1 (s) =
1 1 iF [coth(si F) + 1] = [M(s) + iF] . 2 2
(4.68)
We will need the symmetric and the antisymmetric parts of −1 (s). By recalling that the matrix F is anti-symmetric it is easy to show 1 Mµν (s) . 2 1 −1 [µν] (s) = i Fµν , 2
−1 (µν) (s) =
(4.69) (4.70)
Here and everywhere below the square brackets denote the complete antisymmetrization over all indices included.
Non-Perturbative Heat Kernel Asymptotics
559
For future reference we also notice that −1 (s)T (s) = −1 (s) = exp(2si F),
(4.71)
Finally, we define another function T 1 (s) = (s)−1 (s) = −1 (s) = [M(s) − i F] . 2
(4.72)
It is useful to remember that the functions , F and are dimensionless. 4.6. Flat connection. Next, we transform the operators Z µ to define new (time-dependent) derivative operators by ν Dµ (s) = −1 µν (s)Z (s) .
(4.73)
By using the explicit form of the operators Z µ and Dµ we have ρ Dµ (s) = Dµ + −1 µρ (s)u 1 = ∇¯ µ + Mµρ (s)u ρ . 2
(4.74)
Since the operators Z µ commute, the operators Dµ (s) obviously commute as well. In other words the connection Dµ is flat. Therefore, it can also be written as Dµ (s) = e−(s) ∇¯ µ e(s) ,
(4.75)
where (s) =
1 µ u Mµν (s)u ν . 4
(4.76)
Now, we can rewrite the operators Aµ (s) and Z µ (s) in (4.60) and (4.66) in terms of the operators Dµ (s) ρ Aµ (s) = µ α (s) Dα (s) − −1 , (s)u αρ Z µ (s) = µα (s)Dα (s).
(4.77)
5. Evaluation of the Operator T The perturbative expansion of the operator T is given by Eq. (4.30), with the operators Tk being integrals of the operators Vk (s) and their product. Thus, according to (4.33)-(4.35), to compute the operator T up to the fourth order we need to compute the operators V2 (s), V3 (s), V4 (s) and V2 (s1 )V2 (s2 ).
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I. G. Avramidi, G. Fucci
5.1. Second order. Now, by using the explicit expression for L2 given by Eqs. (4.9), (4.16) and (4.23), utilizing the results of Sect. 3, exploiting Eqs. (4.77), (B.2) and (B.3), using Eqs. (4.61), (4.67), (4.71) and (4.72) after some straightforward but cumbersome calculations we obtain 1 γδ σγδ ρσ γ δ σ V2 (s) = R + N(2) Dσ + P(2) Dγ Dδ + W(2) Dσ Dγ Dδ + Q (2) Dρ Dσ Dγ Dδ , (5.1) 6 where 1 σ N(2) = Rµ α − R µ α ασ µη u η , (5.2) 3 1 1 γδ P(2) = R µ α ν β α(γ |β|δ) µκ νσ u κ u σ − Mµν 3 2 1 ν ρ(γ δ) δν + 7ν δ) − Rν β β(γ ν δ) , (5.3) + R ρ 24 1 σ δγ W(2) = − R µ α ν β α(σ |β|δ δν γ ) + 7ν γ ) µκ u κ , (5.4) 12 1 µ ν α(ρ |β|σ δ ρσ δγ Q (2) = R α β µ δν γ ) + 3ν γ ) . (5.5) 12 Note that all these coefficients as well as the operators Dµ depend on the time variable s. We will indicate explicitly the dependence of various quantities on the time parameter only in the cases when it causes confusion, in particular, when there are two time parameters. 5.2. Third order. Similarly, by using the explicit expression for L3 given by (4.9), (4.19) and (4.24), utilizing the results of Sect. 3, exploiting Eqs. (4.77), (B.2) and (B.3), using Eqs. (4.61), (4.67), (4.71) and (4.72) after some straightforward but cumbersome calculations we obtain σρ
σρι
σ V3 (s) = N(3) Dσ + P(3) Dσ Dρ + W(3) Dσ Dρ Dι σρι
σρικ
+Q (3) Dσ Dρ Dι D + Y(3)
Dσ Dρ Dι D Dκ ,
(5.6)
where 1
∇α R + 2∇µ Rµ α ασ , 6 1
= − ∇ µ Rαβ − 2∇α R µ β + 4∇α Rµ β α(γ |β|δ) µκ u κ , 6 1 1 µ ν α(σ |β|γ |ρ|δ) κ µκ ν u u − Mµν = − ∇α R β ρ 6 2
(5.8)
1 µ ∇ Rαβ − 2∇(α R µ β) + 4∇(α Rµ β) α(σ |β|γ µ δ) , 6
(5.9)
σ N(3) =− γδ
P(3)
σγδ
W(3)
+ ρσ γ δ
Q (3)
ρσ γ δ
Y(3)
(5.7)
1 ∇α R µ β ν α(ρ |β|σ ||γ νδ) µκ u κ , 3
(5.10)
1 = − ∇(α R µ β ν η) α(ρ |β|σ |η|γ µδ ν) . 6
(5.11)
=
Non-Perturbative Heat Kernel Asymptotics
561
Here again, for simplicity, we omitted the dependence of the coefficient functions and the derivatives on the time variable s. 5.3. Fourth order. 5.3.1. Operator V4 (s). By taking into account the definition of L4 in (4.9) by using Eqs. (4.20)-(4.22), (4.77), (B.2) and (B.3), and the explicit form of the functions and , we obtain σρ
σρι
σρι
V4 (s) = P(4) Dσ Dρ + W(4) Dσ Dρ Dι + Q (4) Dσ Dρ Dι D σρικ
+Y(4)
σρικλ
Dσ Dρ Dι D Dκ + S(4)
Dσ Dρ Dι D Dκ Dλ ,
(5.12)
where σρ
P(4) =
σρι
W(4) = σρι
Q (4) =
σρικ
=
σρικλ
=
Y(4)
S(4)
1 Rµν R µ α ν β + Rµνλα R µνλ β − 2R µ α Rµβ α(ρ |β|σ ) 60 1 Rαβ + 3∇α ∇β R α(ρ |β|σ ) + 40 1 + Rµα Rµ β + ∇α ∇µ Rµ β α(ρ |β|σ ) , (5.13) 4 1 6∇α ∇ µ Rβγ + 15∇α ∇β Rµ γ + 15R µ α ν β Rγ ν − 9∇α ∇β R µ γ 60 −R µ α ν β Rγ ν − 4R µ ναλ R ν β λ γ α(σ |β|ρ |γ |ι) µξ u ξ , (5.14) 1 20R µ αλβ R λ γ ν δ + 15∇α ∇β R µ γ ν δ α(σ |β|ρ |γ |ι |δ|) 300 1 ξ ς × µξ νς u u − Mµν 2 1 α µβνγ (σ ρ ι R νR α β γ 3δµ ) + µ ) + 240 1 Rλ µ ν α R λβνγ α (σ β ρ γ ι 3δµ ) + 13µ ) + 240 1 − Rν α R µβνγ α (σ β ρ γ ι δµ ) + 5µ ) 24 1 α β µγ 3∇ ∇ R − 2∇ α ∇ µ R βγ − 5∇ α ∇ β Rµγ α (σ β ρ γ ι µ ) , + 20 (5.15) 1 α β µγ νδ (σ ρ ι κ) ξ − ∇ ∇ R α β γ δ ν µξ u 10 1 Rλ αµβ R λγ νδ α (σ β ρ γ ι δ 3δν κ) + 13ν κ) µξ u ξ , − (5.16) 120 1 α β µγ νδ (σ ρ ι κ λ) α β γ δ µ ν ∇ ∇ R 20 1 + R ηαµβ Rη γ νδ α (σ β ρ γ ι δ 62(µ κ δν) λ) 2880 +125µ κ ν λ) + 5δµ κ δν λ) .
(5.17)
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I. G. Avramidi, G. Fucci
5.3.2. Operator V2 (s1 )V2 (s2 ). Next, we need to compute the product of two operators V2 (s) depending on different times s1 and s2 by using Eq. (5.1). To simplify the notation we denote the derivatives Dµ (sk ) depending on different times sk simply by Dµ(k) . To present the product V2 (s1 )V2 (s2 ) in the “normal” form we need to move all derivative (1) operators Dµ to the right and all coordinates u ν to the left. In order to perform this (1) task we need the commutator of the derivative operator Dµ with the coefficients of the operator V2 (s2 ). First, by using the commutators listed in Appendix B we obtain the relevant commutators (1) ι Dµ(1)1 · · · Dµ(1)n , N(2) (s2 ) = n f ι (µ1 (s2 )Dµ(1)2 · · · Dµn ) , (5.18) ιη (1) Dµ(1)1 · · · Dµ(1)n , P(2) (s2 ) = n(n − 1)g ιη (µ1 µ2 (s2 )Dµ(1)3 · · · Dµn ) (1)
Dµ(1)1 · · ·
where
ιηκ Dµ(1)n , W(2) (s2 )
+ nh ιη (µ1 (s2 )Dµ(1)2 · · · Dµn ) , = np ιηκ (µ1 (s2 )Dµ(1)2 · · ·
Dµ(1)n ) ,
1 f ι λ = Rµ β − R µ β βι µλ , 3 1 g ιη λκ = R µ (α ν β) αι βη µλ νκ , 3 2 ιη h λ = R µ (α ν β) αι βη µκ νλ u κ , 3 1 ιηκ p λ = − R µ α ν β α(ι |β|η δν κ) + 7ν κ) µλ . 12
(5.19) (5.20)
(5.21)
(5.22) (5.23)
Next, by using the expression for the operator V2 (s) in (5.1) and the non-vanishing commutators in (5.18)-(5.19) we obtain V2 (s1 )V2 (s2 ) =
1 2 1 R + R [V2 (s1 ) + V2 (s2 )] + L(s1 , s2 ), 36 6
(5.24)
where L(s1 , s2 ) =
4 4
µ ···µn ν1 ···νk
1 C(n,k)
(s1 , s2 )Dµ(1)1 · · · Dµ(1)n Dν(2) · · · Dν(2) , 1 k
(5.25)
k=1 n=0
and ρ
α C(0,1) = N(2) (s1 ) f ρ α (s2 ), αρ
ρ
α ια C(1,1) = 2N(2) (s1 )N(2) (s2 ) + 2P(2) (s1 ) f ρ ι (s2 ), αβρ
αβ
καβ
ρ
C(2,1) = 2P(2) (s1 )N(2) (s2 ) + 3W(2) (s1 ) f ρ κ (s2 ), αβγρ
αβγ
ρ
αβγ δρ
αβγ δ
ρ
λαβγ
C(3,1) = 2W(2) (s1 )N(2) (s2 ) + 4Q (2) (s1 ) f ρ λ (s2 ), C(4,1) = 2Q (2) (s1 )N(2) (s2 ), ρσ C(0,2)
=
αβ α N(2) (s1 )h ρσ α (s2 ) + 2P(2) (s1 )g ρσ αβ (s2 ),
(5.26)
Non-Perturbative Heat Kernel Asymptotics αρσ
563 αβγ
ρσ
αβ
ρσ
αβγ
α C(1,2) = 2N(2) (s1 )P(2) (s2 ) + 2P(2) (s1 )h ρσ β (s2 ) + 6W(2) (s1 )g ρσ βγ (s2 ), αβρσ
αβ
αβγ δ
C(2,2) = 2P(2) (s1 )P(2) (s2 ) + 3W(2) (s1 )h ρσ γ (s2 ) + 12Q (2) (s1 )g ρσ γ δ (s2 ), αβγρσ
C(3,2)
αβγ δρσ
C(4,2)
ρσ υ C(0,3) αρσ υ C(1,3) αβρσ υ C(2,3) αβγρσ υ C(3,3) αβγ δρσ υ C(4,3) αρσ υχ C(1,4) αβρσ υχ C(2,4) αβγρσ υχ C(3,4) αβγ δρσ υχ C(4,4)
αβγ
ρσ
αβγ δ
ρσ
αβγ δ
= 2W(2) (s1 )P(2) (s2 ) + 4Q (2) (s1 )h ρσ δ (s2 ), = 2Q (2) (s1 )P(2) (s2 ), α = N(2) (s1 ) p ρσ υ α (s2 ),
(5.27)
ρσ υ
µα
ρσ υ
µαβ
α = 2N(2) (s1 )W(2) (s2 ) + 2P(2) (s1 ) p ρσ υ µ (s2 ), αβ
= 2P(2) (s1 )W(2) (s2 ) + 3W(2) (s1 ) p ρσ υ µ (s2 ), αβγ
ρσ υ
αβγ δ
ρσ υ
µαβγ
= 2W(2) (s1 )W(2) (s2 ) + 4Q (2)
(s1 ) p ρσ υ µ (s2 ),
= 2Q (2) (s1 )W(2) (s2 ), = = = =
(5.28)
ρσ υχ α N(2) (s1 )Q (2) (s2 ), αβ ρσ υχ P(2) (s1 )Q (2) (s2 ), αβγ ρσ υχ W(2) (s1 )Q (2) (s2 ), αβγ δ ρσ υχ Q (2) (s1 )Q (2) (s2 ) .
(5.29)
6. Generalized Hermite Polynomials Thus, we reduced the calculation of the asymptotic expansion of the heat kernel to the calculation of the derivatives Dµ (s) of the zero order heat kernel U0 (t|x, x ) given by (4.44). The needed derivatives of the zero order heat kernel can be expressed in terms of the following symmetric tensors: Hµ1 ···µn (s) = U0−1 (t|x, x )Dµ1 (s) · · · Dµn (s)U0 (t|x, x )
(6.1)
and ν1 ···νm µ1 ···µn (s1 , s2 ) = U0−1 (t|x, x )Dν(1) · · · Dν(1) Dµ(2)1 · · · Dµ(2)n U0 (t|x, x ), (6.2) m 1 (k)
where we denoted as before Dµ = Dµ (sk ). (1) (2) We recall that the derivatives Dµ and Dν do not commute! Also, U0 is a scalar function that depends on x and x only through the normal coordinates u µ . The derivative operator Dµ (s) is defined by (4.74), and, when acting on a scalar function is equal to ∂ 1 + Mµν (s)u ν ∂u µ 2 ∂ = e−(s) µ e(s) , (6.3) ∂u where the tensor Mµν (s) is defined by (4.42) and the function (s) is a quadratic form defined by (4.76). Therefore, by using the explicit form of the zero order heat kernel (4.44) we see that the tensors Hµ1 ···µn (s) can be written in the form Dµ (s) =
Hµ1 ···µn (s) = exp{(t) − (s)}
∂ ∂ · · · µ exp{(s) − (t)}, ∂u µ1 ∂u n
(6.4)
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I. G. Avramidi, G. Fucci
The tensors Hµ1 ···µn (s) are polynomials in u µ . They differ from the usual Hermite polynomials of several variables (see, for example, [16]) by some normalization. That is why, we call them just Hermite polynomials. The generating function for Hermite polynomials H(ξ, s) =
∞ 1 µ1 ξ · · · ξ µn Hµ1 ...µn (s) n!
(6.5)
n=0
can be computed as follows:
∂ H(ξ, s) = exp {(t) − (s)} exp ξ µ µ exp {(s) − (t)}, ∂u
1 α ξ αβ (s) ξ β + 2u σ , = exp 2
(6.6)
where (s) = =
1 [M(s) − M(t)] 2 iF sinh[(t − s)i F] 1 . 2 sinh(ti F) sinh(si F)
(6.7)
By expanding the exponent in ξ we obtain the Hermite polynomials explicitly. They can be read off from the expression n
ξ µ1 · · · ξ µn Hµ1 ···µn (s) =
2 (2k)! n
k=0
2k k! 2k
ξ α αβ (s)ξ β
k ρ n−2k ξ ρσ (s)u σ . (6.8)
For convenience some low-order Hermite polynomials are given explicitly in tensorial form in Appendix A. Similarly, the tensors ν1 ···νm µ1 ···µn (s1 , s2 ) can be written in the form ν1 ···νm µ1 ···µn (s1 , s2 ) = exp [(t) − (s1 )]
(6.9)
∂ ∂ ∂ ∂ · · · ν exp [(s1 ) − (s2 )] µ · · · µ exp [(s2 ) − (t)] . ν m 1 1 ∂u ∂u ∂u ∂u n µ They are obviously polynomial in u as well. We call them Hermite polynomials of second kind. The generating function for these polynomials is defined by ×
(ξ, η, s1 , s2 ) =
∞ m,n=0
1 ν1 ξ · · · ξ νm ηµ1 · · · ηµn ν1 ···νm µ1 ···µn (s1 , s2 ), (6.10) m!n!
and can be computed as follows:
∂ (ξ, η, s1 , s2 ) = exp {(t) − (s1 )} exp ξ µ µ exp {(s1 ) − (s2 )} ∂u ∂ (6.11) × exp ην ν exp {(s2 ) − (t)} , ∂u
1 1 α ξ αβ (s1 )(ξ β + 2u β ) + ηµ µν (s2 )(ην + 2u ν ) + ξ ρ ρσ (s2 )ησ . = exp 2 2
Non-Perturbative Heat Kernel Asymptotics
565
Notice that (ξ, η, s1 , s2 ) = H(ξ, s1 )H(η, s2 ) exp ξ ρ ρσ (s2 )ησ .
(6.12)
This enables one to express all Hermite polynomials of second kind (n) (s1 , s2 ) in terms of the Hermite polynomials H(m) (s1 ), H(l) (s2 ), and the matrix (s2 ). Namely, they can be read off from the expression m n (6.13) k k k=0 k
×ξ ν1 · · · ξ νm−k Hν1 ···νm−k (s1 )ηµ1 · · · ηµn−k Hµ1 ···µn−k (s2 ) ξ ρ ρσ (s2 )ησ .
ξ ν1 · · · ξ νm ηµ1 · · · ηµn ν1 ···νm µ1 ···µn (s1 , s2 ) =
min(m,n)
k!
7. Off-diagonal Coefficients bk By using the machinery developed above, we can now write the coefficients of the asymptotic expansion of the heat kernel in terms of generalized Hermite polynomials. We define the following quantity: 1
b2,(1) (t|x, x ) = 0
γδ σγδ σ dτ N(2) (tτ )Hσ (tτ ) + P(2) (tτ )Hγ δ (tτ ) + W(2) (tτ )Hσ γ δ (tτ ) ρσ γ δ +Q (2) (tτ )Hρσ γ δ (tτ ) .
(7.1)
Then, by referring to the formulas (5.1), (5.6), (5.12) and (5.24) and by using the following formula for multiple integrals: τn
b
dτn−1 · · ·
dτn a
τ2
a
a
1 dτ1 f (τ1 ) = (n − 1)!
b dτ (b − τ )n−1 f (τ ),
(7.2)
a
we obtain 1 R + b2,(1) (t|x, x ), (7.3) 6 1 γδ σγδ −1/2 σ b3 (t|x, x ) = t dτ N(3) (tτ )Hσ (tτ ) + P(3) (tτ )Hγ δ (tτ ) + W(3) (tτ )Hσ γ δ (tτ )
b2 (t|x, x ) =
0 ρσ γ δ ιρσ γ δ +Q (3) (tτ )Hρσ γ δ (tτ ) + Y(3) (tτ )Hιρσ γ δ (tτ ) ,
b4 (t|x, x ) =
1 2 1 R + Rb2,(1) (t|x, x ) 72 6 1 ι ικ +t −1 dτ P(4) (tτ )Hι (tτ ) + W(4) (tτ )Hικ (tτ ) 0
(7.4)
566
I. G. Avramidi, G. Fucci ικλη
+Q ικλ (4) (tτ )Hικλ (tτ )+Y(4) +
4 4
τ2
1
k=1 n=0 0
µ ···µn ν1 ···νk
1 dτ1 C(n,k)
dτ2
ικληγ
(tτ )Hικλη (tτ )+S(4)
(tτ )Hικληγ (tτ )
(tτ1 , tτ2 ) µ1 ···µn ν1 ···νk (tτ1 , tτ2 ) .
0
(7.5) 8. Diagonal Coefficients bk In order to obtain the diagonal values bk (t) of the coefficients bk (t|x, x ) we just need to set u = 0 in Eqs. (7.3), (7.4) and (7.5). For the rest of this section we will employ the usual convention of denoting the coincidence limit by square brackets, that is, diag
[ f (u)]diag = f (0).
(8.1)
By inspection of the equation defining the generalized Hermite polynomials in Appendix A one can easily notice that, in the coincidence limit, all the ones with an odd number of indices vanish identically, namely diag =0. (8.2) Hµ1 ···µ2n+1 By using the last remark we have the following expression for the coincidence limit of (7.3), i.e. diag
b2 (t) =
1 diag R + b2,(1) (t), 6
(8.3)
where diag b2,(1) (t)
1 =
diag γδ ρσ γ δ dτ P(2) (tτ )Hγ δ (tτ ) + Q (2) (tτ )Hρσ γ δ (tτ ) .
(8.4)
0
By using the explicit form of the coefficients P(2) , Q (2) and the generalized Hermite polynomials in Appendix A, we obtain b2,(1) (t) = J(1) αβ µν (t)R µ α ν β + J(2) µν (t)Rµν + J(3) µν (t)Rµν , diag
(8.5)
where J(1)
αβ
1 µν (t) =
dτ
1 − αγ βδ Mµν γ δ 6
0
αρ βσ δ 1 γ γ + δν + 3ν µ (ρσ δγ ) , 4 1 1 µν J(2) (t) = dτ δ (ν δ + 7 (ν δ µ)γ γ δ , 24
(8.6)
(8.7)
0
J(3) µν (t) =
1 0
dτ [µ γ ν]δ γ δ .
(8.8)
Non-Perturbative Heat Kernel Asymptotics
567
Here all functions in the integrals depend on tτ . Next, we introduce the following matrices: 1 exp[(t − 2s)i F] − exp(−ti F) , 2 sinh(ti F) coth(ti F) cosh[(t − 2s)i F] B(s) = (s) (s)(s)T = − , iF i F sinh(ti F) 1 3 (s) = −1 (s) − (s) (s) − (s) 4 4 iF 1 3i F coth(ti F) + cosh[(t − 2s)i F] . = 8 sinh(ti F)
A(s) = (s) (s) =
(8.9) (8.10)
(8.11)
Then, by using the relation (s) (s)(s)T = T (s) (s) = AT (s) ,
(8.12)
we obtain J(1)
αβ
1 µν (t)
=
1 dτ − B αβ (tτ ) (µν) (tτ ) 3
0
1 (α Aµ (tτ )Aβ) ν (tτ ) + 3A(µ α (tτ )Aν) β (tτ ) , 6 1 1 1 µν J(2) (t) = dτ A(µν) (tτ ) = δ µν , 3 6 +
(8.13)
(8.14)
0
J(3)
µν
1 (t) = −
dτ A
[µν]
1 (tτ ) = − 2
[µν]
1 − coth(ti F) ti F
.
(8.15)
0
Unfortunately the integral J(1) αβ µν can not be computed explicitly, in general. As we already mentioned above all odd order coefficients b2k+1 have zero diagonal values. We see this directly for the coefficient b3 , which is given by (7.4). That is, by recalling the formulas in (5.7) through (5.11) and the remark (8.2) we have diag
b3 (t) = 0 .
(8.16)
Finally, we evaluate the diagonal values of fourth order coefficient b4 given by (7.5). It can be written as follows: diag
b4 (t) =
1 2 1 diag diag diag R + Rb2,(1) (t) + b4,(2) (t) + b4,(3) (t) . 72 6
(8.17)
By noticing that for odd n + k, the diagonal values of the coefficients C(n,k) vanish, µ1 ···µn ν1 ···νk diag C(n,k) = 0,
(8.18)
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I. G. Avramidi, G. Fucci
and by using the explicit form of Hermite polynomials and the generating function (6.12) we obtain diag b4,(2) (t)
=t
−1
diag ι dτ P(4) (tτ ) ι (tτ ) + 3 Q ικλ (ι (tτ ) κλ) (tτ ) (4) (tτ )
1 0
ικληγ
+15S(4) diag b4,(3) (t)
τ2
1 =
dτ2 0
(tτ ) (ι κλ (tτ ) ηγ ) (tτ ) ,
(8.19)
diag ια dτ1 2 P(2) (τ1 ) f ρ ι (τ2 ) (2) αρ
0 αβ
+2 P(2) (τ1 )
diag
g (ρσ ) αβ (τ2 ) (2) ρσ
diag diag λαβγ αβ ρσ (2) P +12Q (2) (τ1 ) f ρ λ (τ2 ) (1) + 2 P (τ ) (τ ) αβ γρ (2) 2 (2) 1 αβγ δ
+12Q (2) (τ1 )g (ρσ ) γ δ (τ2 )
(1)
(2)
(2) αβ (2) ρσ + 2 αρ βσ
diag µα (2) +6 P(2) (τ1 ) p (ρσ ν) µ (τ2 ) (2) αρ σ ν diag αβγ δ ρσ (1) (1) (1) (2) (2) 3 αβ γ δ (2) +2Q (2) (τ1 ) P(2) (τ2 ) ρσ + 12 αβ γρ δσ µαβγ (2) (2) (2) (2) (2) +4Q (2) (τ1 ) p (ρσ ν) µ (τ2 ) 9 (1) + 6 αρ βσ γ ν αβ γρ σ ν diag αβ ρσ νχ (1) (2) (2) (2) (2) + P(2) (τ1 ) Q (2) (τ2 ) 3 αβ (2) + 12 ρσ νχ αρ βσ νχ αβγ δ ρσ νχ (1) (1) (1) (2) (2) (2) (2) +Q (2) (τ1 )Q (2) (τ2 ) 9 αβ γ δ (2) ρσ νχ + 72 αβ ργ σ δ νχ (2) (2) (2) (2) , (8.20) +24 αρ βσ γ ν δχ where the superscript on the matrix denotes its dependence on either tτ1 or tτ2 . diag We see that the scalar curvature appears only in the term b2,(1) (t). Now, the term diag
b4,(2) (t) only contains derivatives of the curvature and quantities which are quadratic in the curvature with some of their indices contracted. It has the following form: diag
b4,(2) (t) =
1 (1) (2) Bαβ (t)Rµνλ α R µνλβ + Aλαγβ (t)Rµ λ ν α R µγ νβ + Aαµβγ νδ (t)Rη αµβ R ηγ νδ 60 1 1 (3) Bαβ (t)Rµ α R µβ + Bαβ (t)Rµν R µανβ + Aαµβγ (t)R α ν R µβνγ − 60 30 1 (4) +Aαµβγ (t)Rν α R µβνγ + Bαβ (t)Rν α Rνβ 4 1 (5) (6) Bαβ (t)R αβ +Aαβµγ νδ (t)∇ α ∇ β R µγ νδ + Aαβµν (t)∇ α ∇ β R µν + 40 3 1 (7) + Bαβ (t)∇ α ∇ β R + Aαβµν (t)∇ α ∇ β Rµν + Bαβ (t)∇ α ∇µ Rµβ . (8.21) 40 4
Non-Perturbative Heat Kernel Asymptotics
569
Here the tensors a (i) (s) are functions that only depend on F (but not on the Riemann curvature) defined by a (1) λαγβ (s) = a (2) α µ βγ ν δ (s) =
a (3) αµβγ (s) = a (4) αµβγ (s) = a (5) αβ µ γ ν δ (s) =
a (6) αβµν (s) = a (7) αβµν (s) =
3 13 B(αγ Aβ)λ + B(αγ A|λ|β) , (8.22) 80 80 1 µν 1 Bα(β Bγ δ) 31( )(µν) + 65 µν − M Bα(β Bγ δ) 480 10 187 31 Bα(β Aγ (µ Aν) δ) + B(βγ A(ν δ) Aα µ) + 480 240 25 + B(βγ A(ν δ) Aµ) α 96 1 Bα(β Aγ (µ Aδ) ν) + B(βγ Aδ) (ν Aα µ) , (8.23) + 96 3 1 B(αβ Aγ )µ + B(αβ A|µ|γ ) , (8.24) 80 80 1 5 − B(αβ Aγ )µ − B(αβ A|µ|γ ) , (8.25) 8 8 3 3 − Bα(β Bγ δ) M µν (t) + Bα(β A(µ γ Aν) δ) 40 10 3 + B(βγ A(µ δ) Aν) α , (8.26) 10 9 3 B(αβ A|µ|ν) − Bµ(α Aβν) , (8.27) 20 10 3 − B(αβ A|µ|ν) . (8.28) 4
All functions here are evaluated at the time s (unless specified otherwise). diag The term b4,(3) (t) only contains quantities which are quadratic in the curvature with none of their indices contracted. It has the form αβµν γ δρσ µν αβρσ µν αβ b4,(3) (t) = D(1) R + D(2) + D(3) αβµνγ δρσ (t)R µναβρσ (t)R R µναβ (t)R R diag
(4)
(5)
(6)
+Dµναβρσ (t)Rµν R αβρσ + Dµναβ (t)Rµν R αβ + Dµναβ (t)Rµν Rαβ ,
(8.29)
where D(i) µ1 ···µn (t) are some tensor-valued functions that depend on t F. They have the form D(i) µ1 ···µn (t)
τ2
1 =
dτ2 0
dτ1 dµ(i)1 ···µn (tτ1 , tτ2 ) .
(8.30)
0
To describe our results for the tensors d (k) we define new tensors E( p)µ ν = δµ ν + p µ ν , Sαβρσ ικ = Bβσ αι ρκ − Aβ(ι A|σ |κ) Mαρ 3 − β (η σ χ E(1)ρ ) αι κη χ 4 3 + β ( σ λ α η E(3)ρ χ ) ι κλ ηχ , 2
(8.31)
(8.32)
570
I. G. Avramidi, G. Fucci
Vγ δρσ ικηχ (tτ1 , tτ2 ) = ηχ (tτ1 ) Bδσ γ ι ρκ (tτ2 ) + 2 Aδι Aσ κ γ η ρχ (tτ2 )
1
− ικ ηχ (tτ1 ) Bδσ Mγρ (tτ2 ) 4
− ικ (tτ1 ) Aδ(χ A|σ |η) Mγρ (tτ2 )
3 2 κη (tτ1 ) χ ωτ (tτ2 ) + κ (tτ1 ) ηω χ τ (tτ2 ) − 4 3 τ) ( ω × δ σ E(1)ρ γ ι (tτ2 ) 3
+ ικ ηχ (tτ1 ) ( τ ωλ ) (tτ2 ) 16
8
ι κτ ηω χ λ (tτ2 ) +8 ικ (tτ1 ) η τ χ ωλ (tτ2 ) + 3 λ) ( τ ω (tτ2 ) . (8.33) δ σ γ E(3)ρ Then the tensors d (k) have the form
1 (ι κ) β ν Mαµ (tτ1 ) Bδσ γ ι ρκ (tτ2 ) 9 1
(tτ2 ) + Bβν Mαµ (tτ1 ) Bδσ −1 (γρ) 9 1 −1 + β (ι ν κ) Mαµ (tτ1 ) A(δ|ι| Aσ )κ Mγρ (tτ2 ) 9 1
+ (β ι Aν)η Mαµ (tτ1 ) σ (η δ E(1)ρ χ ) γ ι χ (tτ2 ) 12
1
− Bδσ Mγρ (tτ1 ) ικ ηχ (tτ2 ) 24
+4 δ (ω σ λ) Mγρ (tτ1 ) ωι λκ ηχ (tτ2 ) β (ι ν κ α η E(3)µ χ ) (tτ2 ) 1 + β (ι ν κ α η E(3)µ χ ) (tτ1 )Vγ δρσ ικηχ (tτ1 , tτ2 ), (8.34) 3
1 (2) dµναβρσ (tτ1 , tτ2 ) = β (ι σ κ) Mαρ (tτ1 ) µι Aνκ (tτ2 ) 9 1
− Bβσ Mαρ (tτ1 )A(µν) (tτ2 ) 9
1 − A(µν) (tτ1 ) Bβσ Mαρ (tτ2 ) 9 1 (ι κ η + β σ α E(3)ρ χ ) A(µν) ικ ηχ (tτ2 ) 12
1 (ι κ) − β σ Mαρ (tτ1 ) Aνι κη E(7)µ η (tτ2 ) 36
1 + β (ι σ κ α η E(3)ρ χ ) (tτ1 ) − κη (tτ1 ) Aνχ µι (tτ2 ) 3
(1) dαβµνγ δρσ (tτ1 , tτ2 ) = −
Non-Perturbative Heat Kernel Asymptotics
+
571
1
1 A(µν) ικ ηχ (tτ1 )+ ικ (tτ1 ) Aν(η χ ) E(7)µ (tτ2 ) 2 4
1 ν (ι E(7)µ κ) (tτ1 )Sαβρσ ικ (tτ2 ), 36
1 (ι (3) ν E(7)µ κ) (tτ1 ) αι Aβκ (tτ2 ) (tτ1 , tτ2 ) = − dµναβ 36 +
2 + A(µν) (tτ1 )A(αβ) (tτ2 ) 9
1 + ν (ι E(7)µ κ) (tτ1 ) Aβι κσ E(7)α σ (tτ2 ), 144
(8.35)
(8.36)
1 (ι κ) β σ Mαρ (tτ1 ) µι Aνκ (tτ2 ) 3
2 − ν (ι µ κ) (tτ1 ) Bβσ αι ρκ (tτ2 ) 3 1
+ Bβσ Mαρ (tτ1 )Aµν (tτ2 ) 3
1 + Aµν (tτ1 ) Bβσ Mαρ (tτ2 ) 3
2 + β (ι σ κ) Mαρ (tτ1 ) µ η Aνι κη (tτ2 ) 3
2 (ι κ) + ν µ (tτ1 ) Aβι Aσ κ Mαρ (tτ2 ) 3 1 + ν ( µ λ) (tτ1 ) β (ι σ κ E(1)ρ η) α λι κη (tτ2 ) 2
1 + β (ι σ κ α η E(3)ρ ) (tτ1 ) 2 κη (tτ1 ) Aν µι (tτ2 ) 2
− Aµν ικ η (tτ1 ) − 4 ικ (tτ1 ) µ λ Aν ηλ (tτ2 ) 1
Aµν ικ η (tτ2 ) + 4 ν (ω µ λ) (tτ1 ) − 4
ωι λκ η (tτ2 ) × β (ι σ κ α η E(3)ρ ) (tτ2 ) , (8.37)
1 (5) β (γ E(7)α δ) (tτ1 ) µγ Aνδ (tτ2 ) dµναβ (tτ1 , tτ2 ) = 12
2 + ν (γ µ δ) (tτ1 ) αγ Aβδ (tτ2 ) 3 2 2 − A(αβ) (tτ1 )Aµν (tτ2 ) − A(αβ) (tτ2 )Aµν (tτ1 ) 3 3
1 (ι − β E(7)α κ) (tτ1 ) µ Aνκ ι (tτ2 ) 6 1 − ν (ι µ κ) (tτ1 ) Cβ (η α ) ιη κ (tτ2 ), (8.38) 6
(4)
dµναβρσ (tτ1 , tτ2 ) = −
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I. G. Avramidi, G. Fucci
(6) dµναβ (tτ1 , tτ2 ) = −2 ν (γ µ δ) (tτ1 ) αγ Aβδ (tτ2 ) + 2Aµν (tτ1 )Aαβ (tτ2 )
+4 ν (γ µ δ) (tτ1 ) α σ Aβδ γ σ (tτ2 ) . (8.39) 9. Conclusions In this paper we studied the heat kernel expansion for a Laplace operator acting on sections of a complex vector bundle over a smooth compact Riemannian manifold without boundary. We assumed that the curvature F of the U (1) part of the total connection (the electromagnetic field) is covariantly constant and large, so that t F ∼ 1, that is, F is of order t −1 . In this situation the standard asymptotic expansion of the heat kernel as t → 0 does not apply since the electromagnetic field can not be treated as a perturbation. In order to calculate the heat kernel asymptotic expansion we use an algebraic approach in which the nilpotent algebra of the operators Dµ plays a major role. In this approach the calculation of the asymptotic expansion of the heat kernel is reduced to the calculation of the asymptotic expansion of the heat semigroup and, then, to the action of differential operators on the zero-order heat kernel. Since the zero-order heat kernel has the Gaussian form the heat kernel asymptotics are expressed in terms of generalized Hermite polynomials. The main result of this work is establishing the existence of a new non-perturbative asymptotic expansion of the heat kernel and the explicit calculation of the first three coefficients of this expansion (both off-diagonal and the diagonal ones). As far as we know, such an asymptotic expansion and the explicit form of these modified heat kernel coefficients are new. We presented our result as explicitly as possible. Unfortunately, some of the integrals of the tensor-valued functions cannot be evaluated explicitly in full generality. They can be evaluated, in principle, by using the spectral decomposition of the two-form F, F=
[n/2]
Bk E k ,
F2 = −
[n/2]
k=1
Bk2
k,
(9.1)
k=1
where Bk are the eigenvalues, E k are the (2-dimensional) eigen-two-forms, and k = −E k2 are the corresponding eigen-projections onto 2-dimensional eigenspaces. Then for any analytic function of ti F we have f (ti F) =
[n/2] k=1
1 f (t Bk ) ( 2
k
+ i Ek ) +
[n/2] k=1
1 f (−t Bk ) ( 2
k
− i E k ).
(9.2)
However, this seems impractical in the general case of n dimensions. It would simplify substantially in the following cases: i) there is only one eigenvalue (one magnetic field) in a corresponding two-dimensional subspace, that is, F = B1 E 1 (which is essentially 2-dimensional), and ii) all eigenvalues are equal so that F 2 = −I (which is only possible in even dimensions). We plan to study this problem in a future work. The work carried on in this paper can find useful applications in various fields of theoretical physics and mathematics. For instance, our results can be applied to the study of the heat kernel asymptotic expansion on Kähler manifolds. The complex structure on Kähler manifolds is a parallel antisymmetric two-tensor which plays the role of the covariantly constant electromagnetic field. This subject is also interesting, in particular, in connection with String Theory.
Non-Perturbative Heat Kernel Asymptotics
573
Appendix A. Hermite Polynomials The Hermite polynomials are defined by
∂ 1 α ∂ 1 α β β · · · µ exp Hµ1 ···µn = exp − u αβ u u αβ u 2 ∂u µ1 ∂u n 2 ∂ ∂ ν1 νn ··· · 1. = + µ1 ν1 u + µn νn u ∂u µ1 ∂u µn
(A.1)
They can be computed explicitly as follows. First, let H(n) (ξ ) = ξ µ1 · · · ξ µn Hµ1 ···µn
(A.2)
and B = ξµ
∂ , ∂u µ
A = ξ µ µν u ν .
(A.3)
Then H(n) (ξ ) = (A + B)n · 1 .
(A.4)
C = [B, A] = ξ µ µν ξ ν .
(A.5)
Finally, let
Obviously, the operators A, B, C form the Heisenberg algebra [B, A] = C, [A, C] = [B, C] = 0 . Lemma 2. There holds, n
2 n−2k (2k)! n n − 2k n (A + B) = C k An−2k−m B m . 2k k! 2k m
(A.6)
k=0 m=0
Proof. Notice that et (A+B) is the generating functional for (A + B)n . Now, by using the Baker-Hausdorff-Campbell formula t2
et (A+B) = e 2 C et A et B , expanding both sides in t and computing the Taylor coefficients of the right hand side we obtain Eq. (A.6).
By using this result we obtain an explicit expression for (A.4), n
H(n) (ξ ) = ξ µ1 · · · ξ µn Hµ1 ···µn =
2
k=0
n! C k An−2k . 2k k!(n − 2k)!
(A.7)
By setting A = 0 we immediately obtain the (diagonal) values of Hermite polynomials at u = 0, diag = 0, (A.8) Hµ1 ···µ2n+1 diag (2n)! Hµ1 ···µ2n = n (µ1 µ2 · · · µ2n−1 µ2n ) . (A.9) 2 n!
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I. G. Avramidi, G. Fucci
We list below a few low order Hermite polynomials needed for our calculation H(0) = 1,
(A.10)
Hµ1 = µ1 α u α ,
(A.11)
Hµ1 µ2 = (µ1 µ2 ) + µ1 α µ1 β u α u β , Hµ1 µ2 µ3 = 3 (µ1 µ2 µ3 )α u α + µ1 α µ2 β µ3 γ u α u β u γ ,
(A.12) (A.13)
Hµ1 µ2 µ3 µ4 = 3 (µ1 µ2 µ3 µ4 ) + 3 (µ1 µ2 µ3 |α| µ4 )β u α u β + µ1 α µ2 β µ3 γ µ4 δ u α u β u γ u δ , α
(A.14) α β γ
Hµ1 µ2 µ3 µ4 µ5 = 15 (µ1 µ2 µ3 µ4 µ5 )α u + 5 (µ1 µ2 µ3 |α| µ4 |β| µ5 )γ u u u + µ1 α µ2 β µ3 γ µ4 δ µ5 η u α u β u γ u δ u η ,
(A.15)
Hµ1 µ2 µ3 µ4 µ5 µ6 = 15 (µ1 µ2 µ3 µ4 µ5 µ6 ) + 45 (µ1 µ2 µ3 µ4 µ5 |α| µ6 )β u α u β +15 (µ1 µ2 µ3 |α| µ4 |β| µ5 |γ | µ6 )δ u α u β u γ u δ + (µ1 |α| µ2 |β| µ3 |γ | µ4 |δ| µ5 |η| µ6 )ι u α u β u γ u δ u η u ι .
(A.16)
We list below some of the generalized Hermite polynomials of the second kind. Now we have two sets of Hermite polynomials that depend on the quadratic forms at two different times, s1 and s2 . Let us define H(n) (s1 ) = ξ µ1 · · · ξ µn Hµ1 ···µn (s1 ), H(n) (s2 ) = ηµ1 · · · ηµn Hµ1 ···µn (s2 ),
(A.17) (A.18)
(s2 ) = ξ α αβ (s2 )ηβ .
(A.19)
and
Then from Eq. (6.13) we obtain the quantities (m,n) that we need in our calculations (0,1) (s1 , s2 ) = (1,1) (s1 , s2 ) = (2,1) (s1 , s2 ) = (3,1) (s1 , s2 ) = (4,1) (s1 , s2 ) = (0,2) (s1 , s2 ) = (1,2) (s1 , s2 ) =
H(1) (s2 ), (s2 ) + H(1) (s1 )H(1) (s2 ), 2 (s2 )H(1) (s1 ) + H(1) (s2 )H(2) (s1 ), 3 (s2 )H(2) (s1 ) + H(1) (s2 )H(3) (s1 ), 4 (s2 )H(3) (s1 ) + H(1) (s2 )H(4) (s1 ) , H(2) (s2 ), 2 (s2 )H(1) (s2 ) + H(2) (s2 )H(1) (s1 ),
(A.20) (A.21) (A.22) (A.23) (A.24) (A.25) (A.26)
(2,2) (s1 , s2 ) = 2 2 (s2 ) + 4 (s2 )H(1) (s2 )H(1) (s1 ) + H(2) (s2 )H(2) (s1 ),
(A.27)
(3,2) (s1 , s2 ) = 6 (s2 )H(1) (s1 ) + 6 (s2 )H(1) (s2 )H(2) (s1 ) +H(2) (s2 )H(3) (s1 ),
(A.28)
(4,2) (s1 , s2 ) = 12 (s2 )H(2) (s1 ) + 8 (s2 )H(1) (s2 )H(3) (s1 ) +H(2) (s2 )H(4) (s1 ),
(A.29)
2
2
Non-Perturbative Heat Kernel Asymptotics
575
(0,3) (s1 , s2 ) = H(3) (s2 ), (1,3) (s1 , s2 ) = 3 (s2 )H(2) (s2 ) + H(3) (s2 )H(1) (s1 ),
(A.30) (A.31)
(2,3) (s1 , s2 ) = 6 2 (s2 )H(1) (s2 ) + 6 (s2 )H(2) (s2 )H(1) (s1 ) +H(3) (s2 )H(2) (s1 ),
(A.32)
(3,3) (s1 , s2 ) = 6 (s2 ) + 18 (s2 )H(1) (s2 )H(1) (s1 ) +9 (s2 )H(2) (s2 )H(2) (s1 ) + H(3) (s2 )H(3) (s1 ),
(A.33)
(4,3) (s1 , s2 ) = 24 (s2 )H(1) (s1 ) + 36 (s2 )H(1) (s2 )H(2) (s1 ) +12 (s2 )H(2) (s2 )H(3) (s1 ) + H(3) (s2 )H(4) (s1 ), (0,4) (s1 , s2 ) = H(4) (s2 ), (1,4) (s1 , s2 ) = 4 (s2 )H(3) (s2 ) + H(4) (s2 )H(1) (s1 ),
(A.34) (A.35) (A.36)
(2,4) (s1 , s2 ) = 12 2 (s2 )H(2) (s2 ) + 8 (s2 )H(3) (s2 )H(1) (s1 ) +H(4) (s2 )H(2) (s1 ),
(A.37)
3
2
3
2
(3,4) (s1 , s2 ) = 24 (s2 )H(1) (s2 ) + 36 (s2 )H(2) (s2 )H(1) (s1 ) +12 (s2 )H(3) (s2 )H(2) (s1 ) + H(4) (s2 )H(3) (s1 ), 3
2
(A.38)
(4,4) (s1 , s2 ) = 24 (s2 ) + 96 (s2 )H(1) (s2 )H(1) (s1 ) + 72 (s2 )H(2) (s2 )H(2) (s1 ) +16 (s2 )H(3) (s2 )H(3) (s1 ) + H(4) (s2 )H(4) (s1 ). (A.39) 4
3
2
The coincidence limit of the quantities (m,n) ,with m + n odd, vanishes identically
(m,n) (s1 , s2 )
diag
= 0,
if (m + n) is odd.
(A.40)
By recalling the coincidence limits of the Hermite polynomials we obtain the following diag (1,1) (s1 , s2 ) diag (3,1) (s1 , s2 ) diag (0,2) (s1 , s2 ) diag (2,2) (s1 , s2 ) diag (4,2) (s1 , s2 ) diag (1,3) (s1 , s2 ) diag (3,3) (s1 , s2 ) diag (2,4) (s1 , s2 ) diag (4,4) (s1 , s2 )
= (s2 ),
(A.41)
= 3 (s1 ) (s2 ),
(A.42)
= (s2 ),
(A.43)
= (s1 ) (s2 ) + 2 2 (s2 ),
(A.44)
= 3 (s1 ) (s2 ) + 12 (s1 ) (s2 ),
(A.45)
= 3 2 (s2 ),
(A.46)
2
2
= 9 (s1 ) (s2 ) + 6 (s2 ), 2
3
(A.47)
= 3 (s1 ) 2 (s2 ) + 12 3 (s2 ),
(A.48)
= 9 (s1 ) (s2 ) + 72 (s1 ) (s2 ) + 24 (s2 ) . (A.49) 2
2
3
4
Appendix B. Commutators Lemma 3. Let Dµ and u ν be operators satisfying the algebra [Dµ , u ν ] = δµν ,
[Dµ , Dν ] = [u µ , u ν ] = 0.
(B.1)
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I. G. Avramidi, G. Fucci
Then
Dµ1 · · · Dµn , u ρ = n δ ρ (µ1 Dµ2 · · · Dµn ) , Dµ1 · · · Dµn , u ρ u σ = n(n − 1)δ ρ (µ1 δ σ µ2 Dµ3 · · · Dµn ) +2n u (ρ δ σ ) (µ1 Dµ2 · · · Dµn ) . Proof. Let X (ξ ) = ξ µ Dµ and ϕ ρ (t) = et X (ξ ) , u ρ = et X (ξ ) u ρ e−t X (ξ ) − u ρ et X (ξ ) .
(B.2) (B.3)
(B.4)
Then e
t X (ξ ) ρ −t X (ξ )
u e
∞ k k t
AdX (ξ ) u ρ . = k!
(B.5)
k=0
By using the commutation relation in (B.1) we have [X (ξ ), u ρ ] = ξ ρ ,
(B.6)
et X (ξ ) u ρ e−t X (ξ ) = u ρ + tξ ρ .
(B.7)
ϕ ρ (t) = tξ ρ et X (ξ ) .
(B.8)
and, therefore,
Thus
By expanding in Taylor series both sides of the last equation we obtain ∞ t k+1 µ1 ξ · · · ξ µk+1 D(µ1 · · · Dµk+1 ) , u ρ (k + 1)! k=0
=
∞ k+1 t k=0
k!
ξ µ1 · · · ξ µk+1 δ ρ (µ1 Dµ2 · · · Dµk+1 ) .
(B.9)
Now by equating the same powers of t in both series we obtain the claim (B.2). The second relation can be proved in a similar manner. We introduce, in this case, the following generating function: ϕ ρσ (t) = et X (ξ ) , u ρ u σ . (B.10) By the same argument used in the proof of the first relation we obtain that ϕ ρσ (t) = et X (ξ ) , u ρ u σ = 2tξ (ρ u σ ) et X (ξ ) + t 2 ξ ρ ξ σ .
(B.11)
Now, as before, by expanding the last equation in Taylor series and equating the same powers of t we obtain the claim (B.3).
Non-Perturbative Heat Kernel Asymptotics
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References 1. Avramidi, I.G.: The covariant technique for calculation of the heat kernel asymptotic expansion. Phys. Lett. B 238, 92–97 (1990) 2. Avramidi, I.G.: The covariant technique for calculation of one-loop effective action. Nuc. Phys. B 355, 712–754 (1991); Erratum: Nucl. Phys. B 509, 557–558 (1998) 3. Avramidi, I.G.: A new algebraic approach for calculating the heat kernel in gauge theories. Phys. Lett. B 305, 27–34 (1993) 4. Avramidi, I.G.: Covariant methods for calculating the low-energy effective action in quantum field theory and quantum gravity, http://arXiv.org/abs/gr-qc/9403036v1, 1994 5. Avramidi, I.G.: The heat kernel on symmetric spaces via integrating over the group of isometries. Phys. Lett. B 336, 171–177 (1994) 6. Avramidi, I.G.: Covariant algebraic calculation of the one-loop effective potential in non-Abelian gauge theories and a new approach to stability problem. J. Math. Phys. 36, 1557–1571 (1995) 7. Avramidi, I.G.: Covariant algebraic method for calculation of the low-energy heat kernel. J. Math. Phys. 36, 5055–5070 (1995); Erratum: J. Math. Phys. 39, 1720 (1998) 8. Avramidi, I.G.: A new algebraic approach for calculating the heat kernel in quantum gravity. J. Math. Phys. 37, 374–394 (1996) 9. Avramidi, I.G.: Covariant approximation schemes for calculation of the heat kernel in quantum field theory. In: Quantum Gravity, eds. V. A. Berezin, V. A. Rubakov, D. V. Semikoz, Singapore: World Scientific, 1997, pp. 61–78 10. Avramidi, I.G.: Covariant techniques for computation of the heat kernel. Rev. Math. Phys. 11, 947– 980 (1999) 11. Avramidi, I.G.: Heat Kernel and Quantum Gravity, Lecture Notes in Physics, Series Monographs, LNP 64 Berlin: Springer-Verlag, 2000 12. Avramidi, I.G.: Heat kernel approach in quantum field theory. Nucl. Phys. Proc. Suppl. 104, 3–32 (2002) 13. Avramidi, I.G.: Heat kernel asymptotics on symmetric spaces, Proc. Midwest Geometry Conference, Published in Global. J. Pure Appl. Math. 1, 1–17 (2008) 14. Avramidi, I.G.: Heat kernel on homogeneous bundles. Int. J. Geom. Meth. Mod. Phys. 5, 1–23 (2008) 15. Avramidi, I.G.: Heat kernel on homogeneous bundles over symmetric spaces. To appear Commun. Math. Phys. DOI:10.1007/s00220-008-0639-6, 2008 16. Bateman, H., Erdeyi, A.: Higher Transcendental Functions. New-York: McGraw-Hill, vol. 2, 1953 17. Camporesi, R.: Harmonic analysis and propagators on homogeneous spaces. Phys. Rep. 196, 1–134 (1990) 18. Gilkey, P.B.: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. Boca Raton, FL: CRC Press, 1995 19. Hurt, N.E.: Geometric Quantization in Action: Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory, Dordrecht: Reidel Publishing, 1983 20. Kirsten, K.: Spectral Functions in Mathematics and Physics. Boca Raton, FL: CRC Press, 2001 21. Schwinger, J.S.: On gauge invariance and vacuum polarization. Phys. Rev. 82, 664–679 (1951) 22. Van de Ven, A.E.M.: Index free heat kernel coefficients. Class. Quant. Grav. 15, 2311–2344 (1998) 23. Vassilevich, D.V.: Heat kernel expansion: user’s manual. Phys. Rep. 388, 279–360 (2003) Communicated by A. Connes
Commun. Math. Phys. 291, 579–590 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0803-7
Communications in
Mathematical Physics
Geometric Construction of the r-Map: From Affine Special Real to Special Kähler Manifolds D. V. Alekseevsky1 , V. Cortés2 1 The University of Edinburgh and Maxwell Institute for Mathematical Sciences,
JCMB, The King’s buildings, Edinburgh, EH9 3JZ, UK. E-mail: [email protected]
2 Department Mathematik und Zentrum für Mathematische Physik, Universität Hamburg,
Bundesstraße 55, D-20146, Hamburg, Germany. E-mail: [email protected]; [email protected] Received: 29 October 2008 / Accepted: 23 December 2008 Published online: 21 April 2009 – © Springer-Verlag 2009
Abstract: We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N = T M carries a canonical structure of (affine) special Kähler manifold. This gives an intrinsic description of the r -map as the map M → N = T M. On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when M is any Hessian manifold. Contents 1. 2.
Hessian Geometry and Affine Special Real Geometry . . . . . . . . . . . . 581 Geometric Structures on the Tangent Bundle . . . . . . . . . . . . . . . . . 585
Introduction Projective (or local) very special real geometry is the scalar geometry of five-dimensional supergravity coupled to vector multiplets [GST,dWvP,ACDV]. We will usually omit the word “very”. It is locally the geometry of a nondegenerate hypersurface H ⊂ Rn+1 defined by a homogeneous cubic polynomial h. In relation with string compactifications the polynomial h could be, for instance, the cubic form α∧α∧α h([α]) = X
on the second cohomology of a Calabi-Yau 3-fold X . In this paper we are concerned with affine (or rigid) very special real geometry, which is the scalar geometry of five-dimensional rigid vector multiplets [CMMS1]. The
580
D. V. Alekseevsky, V. Cortés
Lagrangian of rigid vector multiplets is encoded in a (not necessarily homogeneous) cubic polynomial and the metric of the scalar manifold is the Hessian of this polynomial. In the first part of the paper we shall provide an intrinsic definition of the notion of an affine special real manifold and study, in particular, the geometry of such manifolds: Definition 1. An affine special real manifold (M, g, ∇) is a pseudo-Riemannian manifold (M, g) endowed with a flat torsion-free connection ∇ such that the tensor field S = ∇g is totally symmetric and ∇-parallel. We relate the intrinsic definition to the description in the physical literature in terms of a cubic prepotential. In fact, we show that any simply connected affine special real manifold (M, g, ∇) of dimension n admits an affine immersion ψ onto a domain V ⊂ Rn , such that g is the pull back of the Hessian of a cubic polynomial h, see Corollary 1. The pair (V, h) is unique up to affine tranformations of Rn . We calculate the curvature tensor of a special real manifold (and, more generally, of a Hessian manifold, see Corollary 4) and find a simple expression in terms of the tensor S. As an application, we obtain that a special real manifold with a positive definite metric has nonnegative Ricci curvature, see Corollary 7. Dimensional reduction of (local/rigid) supersymmetric theories from 5 to 4 spacetime dimensions induces a correspondence between the respective scalar geometries, which is known as the (local/rigid) r-map [dWvP,CMMS1]. The relevant scalar geometry in 4 space-time dimensions is (projective/affine) special Kähler geometry, see [C] for a survey. The (local/rigid) r-map associates a (projective/affine) special Kähler manifold to any (projective/affine) special real manifold. The r-map is explicitly known in terms of the prepotentials, which locally define special real and special Kähler geometry. In the affine case, for instance, the r-map associates to the real cubic polynomial h(x 1 , . . . , x n ) defining the special real manifold the holomorphic prepotential F(z 1 , . . . , z n ) = 2i1 h(z 1 , . . . , z n ) defining the corresponding special Kähler manifold in terms of special holomorphic coordinates z 1 , . . . , z n [CMMS1]. However, an intrinsic geometric construction of the affine and projective r-maps is missing. In the second part of the paper we shall give such a construction in the affine case1 . We show that the tangent bundle N = T M of any (affine) special real manifold (M, g, ∇) carries the structure of an (affine) special Kähler manifold (N , J, g N , ∇ N ). More precisely, the special Kähler structure (J, g N , ∇ N ) on N is canonically associated to the geometric data (g, ∇) on M, see Theorem 2. Recall that a special Kähler manifold (N , J, g N , ∇ N ) is a (pseudo-)Kähler manifold (N , J, g N ) endowed with a connection ∇ N which is special (i.e. ∇ N J is symmetric), torsion-free, symplectic (with respect to the Kähler form) and flat. The map r : {special real manifolds} → {special Kähler manifolds}
(0.1)
(M, g, ∇) → (N , J, g , ∇ ), N
N
which associates to the special real manifold (M, g, ∇) the special Kähler manifold (N = T M, J, g N , ∇ N ) is our geometric construction of the r-map. As an application, we prove that there is no compact simply connected special real manifold with a positive definite metric, see Theorem 5. 1 The projective case is the subject of work in progress.
Geometric Construction of the r-Map
We show that our r-map extends naturally to a map Kähler manifolds with a torsion-free, r : {Hessian manifolds} → symplectic and special connection (M, g, ∇) → N , J, g N , ∇ N .
581
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A Hessian manifold (M, g, ∇) is a pseudo-Riemannian manifold such that S = ∇g is totally symmetric (but not necessarily ∇-parallel). The flatness of the connection ∇ N is lost when the r-map is applied to Hessian manifolds which are not special real. In fact, ∇ N is flat if and only if (M, g, ∇) is special real. Moreover, the manifold (N , J, g N , ∇ N ) associated to a Hessian manifold (M, g, ∇) by the r-map is again Hessian if and only if (M, g, ∇) is special real, see Corollary 9. Finally, we characterise the image of the maps (0.1) and (0.2) in Theorems 3 and 4. We calculate the curvature tensors of the Levi-Civita connection of the metric g N and of ∇ N , which have a simple expression in terms of the symmetric tensor S = ∇g and ∇ S, respectively, see Corollaries 8, 10 and 11. In particular, it follows from those theorems that a special Kähler manifold (N , J, g N , N ∇ ) of real dimension 2n can be locally obtained from the r-map if and only if it admits locally n holomorphic2 Killing vector fields which span a Lagrangian distribution and which are ∇ N -parallel along this distribution. We prove that under some assumptions a simply connected n-dimensional Hessian manifold admits a canonical realisation as an improper affine hypersphere in Rn+1 equipped with the Blaschke metric and the induced connection. 1. Hessian Geometry and Affine Special Real Geometry Definition 2. A Hessian manifold (cf. [S]) (M, g, ∇) is a pseudo-Riemannian manifold (M, g) with a flat torsion-free connection ∇ such that S = ∇g is a symmetric three-form (cubic form). An affine (very) special real manifold (M, g, ∇) is a Hessian manifold (M, g, ∇) with ∇-parallel cubic form S. Example 1. Let h be a smooth function in affine space V ∼ = Rn . We will say that h is 2 nondegenerate at a point x0 ∈ V if the Hessian ∂ h(x0 ) is nondegenerate, where ∂ is the flat connection in V . We denote by V (x0 ) ⊂ V the connected component of x0 in {det ∂ 2 h = 0} ⊂ V . The domain V (x0 ) is equipped with the pseudo-Riemannian metric g = ∂ 2 h. Then (V (x0 ), g, ∂) is a Hessian manifold. Indeed S = ∂g = ∂ 3 h is completely symmetric. It is an affine special real manifold if and only if the cubic form S is constant. Proposition 1. Any Hessian manifold (respectively, affine special real manifold) (M, g, ∇) is locally isomorphic to a domain (V (x0 ), g, ∂) associated with a smooth function h (respectively, cubic polynomial h), as in Example 1. The polynomial is given by 1 1 Si jk x i x j x k + bi j x i x j . h= 6 2 Here x i are ∇-affine coordinates g = gi j d x i d x j , gi j = Si jk x k + bi j and S = Si jk d x i d x j d x k . For any Hessian manifold the n linearly independent gradient vector fields grad(x i ) commute and the coordinate vector fields ∂∂x i are also commuting gradient vector fields. 2 Recall that a real vector field X on a complex manifold (N , J ) is called holomorphic if L J = 0. X
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Proof. Since S = ∇g is totally symmetric, there exists locally a smooth function h such that g = ∇ 2 h. Moreover, if ∇ S = 0, then the function h is a cubic polynomial in affine local coordinates x i . Then ∂ 2 h = ∇ 2 h = g = gi j d x i d x j and ∂ 3 h = ∇ 3 h = ai jk x k + bi j and Si jk = ai jk . This shows that S= Si jk d x i d x j d x k , where gi j = h coincides with the above expression up to terms of degree less or equal to 1 in the coordinates x i , which do not contribute to g and S. Now we check that the vector fields grad(x i ) = g i j ∂ j commute:
g i j ∂ j , g kl ∂l = g i j ∂ j g kl ∂l − g kl ∂l g i j ∂ j = −g i j g kk Sk l j gl l ∂l + g kl g ii Si j l g j j ∂ j = −S kli ∂l + S ilk ∂l = 0,
(1.1)
by the complete symmetry of S. (Here and in later calculations we use the Einstein summation convention.) The coordinate vector field ∂∂x i is the gradient of the function ∂h .
∂xi Corollary 1. Let (M, g, ∇) be a simply connected Hessian manifold of dimension n. Then there exists an affine immersion ψ : (M, ∇) → (Rn , ∂) onto some domain ψ(M) ⊂ Rn , unique up to affine transformations of Rn . The gradients grad(x i ) of the coordinate functions x i = ψ i span a canonical n-dimensional commutative Lie algebra of vector fields. Conversely, a pseudo-Riemannian manifold (M, g) with n pointwise linearly independent gradient vector fields grad(x i ) is canonically extended to a Hessian manifold (M, g, ∇). If (M, g, ∇) is an affine special real manifold, then, moreover, there exists a unique cubic polynomial h on Rn without linear and constant terms such that g = ψ ∗ ∂ 2 h. Proof. Since M is simply connected, there exists a ∇-parallel coframe (ξ 1 , · · · , ξ n ). The one-forms ξ i are closed and, hence, exact, i.e. ξ i = d x i for globally defined functions x i . These functions define an affine immersion ψ = (x 1 , · · · , x n ) since ∇d x i = 0. Given a pseudo-Riemannian manifold (M, g) with n pointwise linearly independent gradient vector fields grad(x i ), there is a unique flat torsion-free connection ∇ such that ∇d x i = 0. Then ∇g is completely symmetric by (1.1). On any domain U ⊂ M such that ψ|U : U → ψ(U ) is a diffeomorphism there exists a smooth function h U such that gi j |U = ∂i ∂ j h U . The function h U is unique up to the addition of an affine function in the local coordinates x i . In the special real case h U is a cubic polynomial, which can be canonically chosen by the requirement that the linear and constant terms vanish: h U = 16 ai jk x i x j x k + 21 bi j x i x j . The coefficients ai jk , bi j are independent of U , since M is connected and h U = h V on overlaps U ∩ V = ∅. Therefore h = 16 ai jk x i x j x k + 21 bi j x i x j is canonically associated to the immersion ψ and satisfies g = ψ ∗ ∂ 2 h.
Remark. Shima and Yagi proved that the domain ψ(M) is convex if g is positive definite, see [SY]. Corollary 2. A pseudo-Riemannian manifold (M, g) admits the structure of an affine special real manifold if it admits an atlas with affine transition functions such that the metric coefficients gi j = ai jk x k + bi j are affine functions and the coefficients ai jk are symmetric. Then gi j = ∂i ∂ j h, where h = 16 ai jk x i x j x k + 21 bi j x i x j .
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Corollary 3. Let (M, g, ∇) be a simply connected Hessian manifold with transitive action of G = Aut(M, g, ∇). Then the affine immersion ψ : M → ψ(M) ⊂ Rn of Corollary 1 is a G-equivariant covering M = G/H → ψ(M) = Gx0 = G/G x0 over the (open) orbit Gx0 of a point x0 ∈ ψ(M) with respect to an affine action α of G on Rn . In particular, there is no nonconstant G-invariant function on the domain ψ(M) and at most one (up to scaling) relative invariant with character χ (a) = (det A)−2 , where α(a)x = Ax + b, a ∈ G. If such a relative invariant δ : ψ(M) → R exists, it is given (up to a constant factor) by the formula δ ◦ ψ = det(gi j ) (which in general defines only a local relative invariant). For an affine special real manifold, the (globally defined) relative invariant δ is a polynomial of degree n. Recall that given a pseudo-Riemannian metric g and a connection ∇ on manifold M, the conjugate connection ∇¯ is defined by X g(Y, Z ) = g(∇ X Y, Z ) + g(Y, ∇¯ X Z ), where X, Y, Z are vector fields on M. Notice that ∇¯ X = D X + Sˆ X∗ , where D is the LeviCivita connection, Sˆ = D − ∇ and Sˆ X∗ is the metric adjoint of the endomorphism Sˆ X . Proposition 2. Let (M, g, ∇) be a Hessian manifold with cubic form S = ∇g. Then Sˆ X = 21 g −1 ◦ S X = Sˆ X∗ . The conjugate connection is flat and torsion-free and we have the following formulas: ∇¯ X = D X + Sˆ X , ∇ X = D X − Sˆ X . Proof. The connection ∇ + 21 g −1 ◦ S is torsion-free, by the symmetry of S. We check that it preserves the metric g: 1 1 1 ∇ X g + (g −1 ◦ S X ) · g = S X − g(g −1 ◦ S X ·, ·) − g(·, g −1 ◦ S X ·) = S X − S X = 0. 2 2 2 ˆ Hence 1 g −1 ◦ S X = This shows that ∇ + 21 g −1 ◦ S is the Levi-Civita connection D = ∇ + S. 2 Sˆ X = Sˆ X∗ . It is clear that the conjugate connection ∇¯ X = D X + Sˆ X has zero torsion. It ¯ For this we write ∇¯ = ∇ + 2 Sˆ and compute: remains to calculate its curvature R. ¯ ˆ R(X, Y ) = R ∇ (X, Y ) + 2d ∇ S(X, Y ) + 4[ Sˆ X , SˆY ] ∇ˆ ˆ = 2(d S(X, Y ) + 2[ S X , SˆY ]), 1 1 1 ∇ X Sˆ = ∇ X (g −1 ◦ S) = − g −1 ◦ S X g −1 ◦ S + g −1 ◦ ∇ X S 2 2 2 1 = −2 Sˆ X Sˆ + g −1 ◦ ∇ X S. 2 Therefore ˆ ˆ − (∇Y S)X ˆ Y ) = (∇ X S)Y = −2[ Sˆ X SˆY ], d ∇ S(X, since ∇ S is symmetric. Thus R¯ = 0.
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Corollary 4. Under the assumptions of the proposition, the following formulas are satisfied: ˆ S] ˆ = 0. R D (X, Y ) = −[ Sˆ X , SˆY ], d D Sˆ = 0, d ∇ Sˆ + 2[ S, Proof. The first two equations are obtained by taking the sum and difference of the equations ˆ S], ˆ 0 = R¯ = R D + d D Sˆ + [ S, ∇ D Dˆ ˆ S]. ˆ 0 = R = R − d S + [ S, ˆ The third equation follows from ∇¯ = ∇ + 2 S: ˆ S] ˆ = 2(d ∇ Sˆ + 2[ S, ˆ S]). ˆ 0 = R¯ = R ∇ + 2d ∇ Sˆ + 4[ S,
The following results are analogues of the corresponding results in special Kähler geometry, see [BC1]. Theorem 1. Let (M, g, ∇) be a simply connected Hessian manifold such that ∇ preserves the metric volume form. Then there exists a realisation of (M, g, ∇) as an improper affine hypersphere ϕ : M → Rn+1 , unique up to unimodular affine transformations. Moreover, any automorphism of (M, g, ∇) has a unique extension to a unimodular affine transformation of Rn+1 preserving ϕ(M). Proof. By the fundamental theorem of affine geometry a simply connected pseudoRiemannian manifold (M, g, ∇) with a torsion-free connection ∇ admits a Blaschke immersion ϕ : M → Rn+1 as a hypersurface with Blaschke metric g and induced connection ∇ if and only if the conjugate connection ∇¯ is torsion-free and projectively flat, and if the metric volume form is ∇-parallel, see [DNV]. Moreover, the Blaschke immersion is unique up to unimodular affine transformations and is an improper affine hypersphere if and only if the connection ∇ is flat. The assumptions of the fundamental theorem are satisfied in virtue of Proposition 2 and ∇ is flat by Definition 2. If ψ : M → M is an automorphism, then, due to the unicity statement in the fundamental theorem, there exists a unimodular affine transformation α : Rn+1 → Rn+1 such that α ◦ ϕ = ϕ ◦ ψ. It is unique since any affine transformation which fixes an affine frame is the identity and an affine frame in Rn+1 is determined by a frame in Tϕ( p) M and the affine normal which is invariant under unimodular affine tranformations.
Corollary 5. If G = Aut (M, g, ∇) acts transitively on a simply connected Hessian manifold (M, g, ∇) and ∇ preserves the metric volume form, then the Blaschke immersion ϕ : (M, g, ∇) → Rn+1 is a covering M = G/H → ϕ(M) = Gx0 = G/G x0 over the orbit Gx0 of a point x0 ∈ ϕ(M) with respect to an affine action of G on Rn+1 . Moreover, ϕ(M) ⊂ Rn+1 is an improper affine hypersphere. Corollary 6. Let (M, g, ∇) be a Hessian manifold with complete and positive definite metric g and such that ∇ preserves the metric volume form. Then g is flat and D = ∇. In particular, any homogeneous Hessian manifold with positive definite metric and volume preserving connection ∇ is finitely covered by the product of a flat torus and a Euclidean space.
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Proof. By the previous theorem, the universal covering of (M, g, ∇) can be realised as an improper affine hypersphere with complete and positive definite Blaschke metric. By the Calabi-Pogerelov theorem, see [NS] and references therein, such a hypersurface is a paraboloid and the Blaschke metric is flat. Now the existence of the finite covering follows from Bieberbach’s theorem.
2. Geometric Structures on the Tangent Bundle Now we show that the tangent bundle π : N = T M → M of a Hessian (pseudo-) Riemannian manifold (M, g, ∇) has a natural (pseudo-) Kähler structure and the tangent bundle of an affine special real manifold has a natural special (pseudo-) Kähler structure. We recall that an (affine) special (pseudo-) Kähler structure (g, J, ∇) on a manifold N is given by a (pseudo-) Kähler structure (g, J ) and a flat torsion-free symplectic (∇ω = ∇g ◦ J = 0) connection such that ∇ J is a symmetric (1,2)-tensor, see [C] and references therein. Let T N = T h N ⊕ T v N be the decomposition of the tangent bundle of N = T M into vertical and horizontal subbundles with respect to the flat connection ∇ on the Hessian manifold (M, g, ∇). We have a canonical isomorphism Tξ N = Tξh ⊕ Tξv ∼ = Tπ(ξ ) M ⊕ Tπ(ξ ) M. Local affine coordinates x i on M induce canonical coordinates (x i , u i ) on N such that any vector ξ ∈ T M is written as ξ = u i ∂∂x i . Then ∂i := ∂∂x i , ∂i := ∂u∂ i forms a local frame of T h and T v , respectively. For a vector field X = X i ∂∂x i on M we denote by X h = X i ∂∂x i , X v = X i ∂u∂ i the horizontal and vertical lifts of X , respectively. Then we have the formulas: [X h , Y h ] = [X, Y ]h , [X v , Y v ] = 0, [X h , Y v ] = (∇ X Y )v . The canonical isomorphism 1 = ∂u∂ i ⊗ d x i : Tξh ∼ = Tξv defines the = Tπ(ξ ) M ∼ complex structure
0 −1 , J := 1 0 which is integrable since it has constant coefficients in the coordinates (x i , u i ). We define a natural extension of the metric by
g 0 . g N := 0 g Proposition 3 (cf. [S0]). For any Hessian (pseudo-) Riemannian manifold (M, g, ∇) the pair (g N , J ) is a (pseudo-) Kähler structure on N = T M. Proof. It is sufficient to check that the 2-form
0 −g N = −gik (x)d x i ∧ du k ω=g ◦J = g 0 is closed. Indeed, ∂gik i d x ∧ d x j ∧ du k = 0 ∂x j due to the total symmetry of ∇g = ∂g.
dω =
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Proposition 4. With respect to the coordinates x I = (x i , u i ), the Christoffel symbols
I = ( IKJ ) of the metric g N are given by
i =
Sˆi 0
0 Sˆi
, i =
− Sˆi 0
0 Sˆi
= J i .
If (M, g, ∇) is special real, that is ∇ S = 0, then the curvature R N of the Levi Civita connection D N is given by R N (X h , Y h ) = R N (X v , Y v )
D
R (X, Y ) 0 [ Sˆ X , SˆY ] = = − D 0 R (X, Y ) 0
0 { Sˆ X , SˆY } . R N (X h , Y v ) = −{ Sˆ X , SˆY } 0
0 , [ Sˆ X , SˆY ]
(2.1) (2.2)
Corollary 7. Let (M, g, ∇) be an affine special real manifold. Then the Ricci curvature of the (pseudo-) Kähler manifold (N , g N , J ) is given by ric(X h + Y v , X h + Y v ) = 2tr Sˆ X2 + 2tr SˆY2 . If the metric g is Riemannian then the Riemannian metric g N has nonnegative Ricci curvature. The Ricci curvature is strictly positive if and only if the map X → S(X, ·, ·) has trivial kernel. Proof. Since the Ricci form of any Kähler manifold is given by ρ(X, Y ) = ric(J X, Y ) = 1 2 tr J R(X, Y ) the formulas for the Ricci curvature follow from the previous proposition. Since Sˆ X is symmetric with respect to g, tr Sˆ X2 ≥ 0 if g is definite.
Now we extend the (1,2)-tensor field Sˆ on M considered as tensor on Tξh N ∼ = Tπ(ξ ) M N to a (1,2)-tensor Sˆ on N such that Sˆ JN = Sˆ N J = −J Sˆ N . Then it is given by Sˆ XNh =
Sˆ X 0
0 − Sˆ X
, Sˆ XNv =
0 − Sˆ X
− Sˆ X 0
.
(2.3)
We define a connection ∇ N on N by ∇ N = D N − Sˆ N . Lemma 1. Let (N , g N , J ) be the Kähler manifold associated to a Hessian manifold (M, g, ∇). Then the above connection ∇ N has the following Christoffel symbols with respect to the coordinates x I = (x i , u i ):
0 0 0 0 . , i =
i = 0 2 Sˆi 2 Sˆi 0
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ˆ be an affine special real manifold. Then N , g N , J, Theorem 2. Let (M, g, ∇ = D − S)
∇ N = D N − Sˆ N is an affine special (pseudo-) Kähler manifold. Proof. Due to Proposition 3 it suffices to show that the connection ∇ N is a) torsion-free, b) symplectic, c) special (i.e. ∇ N J is symmetric) and d) flat. The properties a), b) and c) are valid for any Hessian manifold (M, g, ∇). Indeed the Levi Civita connection D N is torsion-free and preserves ω and J . Therefore: a) follows from the symmetry Sˆ XN Y = SˆYN X for X, Y ∈ T N . b) follows from the symmetry ω( Sˆ XN Y, Z ) = ω( Sˆ XN Z , Y ) for all X, Y, Z ∈ T N . c) follows from the symmetry of Sˆ XN Y , since ∇ N J = [ Sˆ XN , J ] = −2J Sˆ XN . Now we check that the curvature R I J = ∂ I J − ∂ J I + [ I , J ] of ∇ N vanishes. Using the formula (1.2) with ∇ S = 0 and the previous lemma we get:
∂ 0 0 0 0 , ∂i J = i J = 0. , ∂
= ∂i j = i j ˆ ˆ ˆ ˆ 0 −4 Si S j −4 Si S j 0 ∂u Now it is easy to check that R I J = 0.
The same calculation shows: Corollary 8. The curvature of the connection ∇ N of the Kähler manifold (N , g N , J ) associated to a Hessian manifold (M, g, ∇) is given by:
0 0 0 0 , , R(X v , Y v ) = 0, R(X h , Y v ) = R(X h , Y h ) = 0 PX,Y − PY,X PX,Y 0 where PX,Y Z = g −1 (∇ X S)(Y, Z , ·). The Ricci curvature of ∇ N is given by: ric(X h , Y h ) = −tr PX,Y , ric(X v , Y v ) = ric(X h , Y v ) = 0. Corollary 9. Let (g N , J, ∇ N ) be the geometric structures on N = T M associated to a Hessian manifold (M, g, ∇). Then the following are equivalent: (i) ∇ N is flat. (ii) N , g N , ∇ N is Hessian. (iii) (M, g, ∇) is special real. Proof. The equivalence of (i) and (iii) follows from the previous corollary and it is clear that (ii) implies (i). It remains to check that (i) implies (ii). The complete symmetry of the tensor field ∇ N g N follows from the symmetry Sˆ XN Y = SˆYN X , since ∇ XN g N (Y, Z ) = − Sˆ XN · g N (Y, Z ) = g N Sˆ XN Y, Z + g N Y, Sˆ XN Z = g N Sˆ XN Y, Z − ω N J N Y, Sˆ XN Z = g N Sˆ XN Y, Z − ω N J N Sˆ XN Y, Z = 2g N Sˆ XN Y, Z .
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Corollary 10. Under the assumptions of Theorem 2 the exterior covariant derivatives of the endomorphism valued one-form Sˆ N are given by:
N N 0 [ Sˆ X , SˆY ] , d ∇ Sˆ N (X h , Y h ) = d ∇ Sˆ N (X v , Y v ) = −2 0 [ Sˆ X , SˆY ]
N 0 { Sˆ X , SˆY } d ∇ Sˆ N (X h , Y v ) = 2 , −{ Sˆ X , SˆY } 0 N d D Sˆ N = 0.
(2.4)
Proof. The proof follows from (2.1)–(2.3), R ∇ = 0 and the formulas N
N N d ∇ Sˆ N = R N − R ∇ − [ Sˆ N , Sˆ N ], N N d D Sˆ N = R N − R ∇ + [ Sˆ N , Sˆ N ].
Corollary 11. Under the assumptions of Theorem 2 the curvature of the Levi-Civita connection of g N is given by R N = −[ Sˆ N , Sˆ N ]. Definition 3. The map which to any affine special real manifold (M, g, ∇) associates the affine special Kähler manifold r(M) := (N , g N , J, ∇ N ) is called the (affine) r-map. Corollary 12. Let (M, g, ∇) be a special real manifold which locally admits a homogeneous Hesse potential h = 16 Si jk x i x j x k . Then the Kähler manifold r(M) is not flat. Proof. By (2.1) R N = 0 is equivalent to Sˆ X SˆY = 0 for all X, Y ∈ T M. This is impossible since, by Proposition 1, locally we can identify the Hessian manifold with a domain (V (x0 ), g, ∂) and 2g ◦ Sˆ X = S(X, ·, ·) = ∂ 2 h(X ) = g X is invertible for all X ∈ V (x0 ).
Flat special Kähler manifolds were classified in [BC2]. By the corollary, they cannot be obtained from a homogeneous cubic polynomial by the r-map. Theorem 3. Let (M, g, ∇) be a Hessian manifold of dimension n and (N = T M, g N , J, ∇ N ) the corresponding (pseudo-) Kähler manifold with the connection defined above. Then (i) The decomposition T N = T v + T h = T v + J T v is a decomposition into two orthogonal integrable Lagrangian distributions which are totally geodesic and flat with respect to ∇ N . The horizontal distribution is also totally geodesic with respect to D N . (ii) For any leaf L = M(ξ ), ξ ∈ N , of the horizontal distribution there exists an involution σ = σ L ∈ Aut(N , g N , J, ∇ N ) which preserves the vertical and horizontal foliations and such that L = N σ . (iii) The group generated by products σ L ◦ σ L preserves each fiber T p M, p ∈ M, and acts as the translation group of the fiber T p M.
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Proof. (i) follows from the formulas for the Christoffel symbols and curvature of D N and ∇ N and from the formulas for g N = gi j (x)(d x i d x j +du i du j ), J and ω = g N ◦ J . In each coordinate domain one can check directly that the reflection σU : (x, u) → (x, −u+2u 0 ) with respect to an open domain U = {u = u 0 } ⊂ L in a leaf L has the properties claimed in (ii). The reflections σU coincide on overlaps and, hence, define the global reflection σ L . (iii) follows from the fact that the product of two central symmetries in the affine space T p M is a parallel translation.
The converse can be stated as follows. Theorem 4. Let (N , J, g N ) be a 2n-dimensional pseudo-Kähler manifold which admits a free holomorphic and isometric action of the vector group Rn with (metrically) nondegenerate Lagrangian orbits such that the projection π : N → N /Rn = M is a trivial (principal) bundle. Then there exists an induced pseudo-Riemannian metric g and flat connection ∇ on M such that (M, g, ∇) is Hessian and N is identified with T M with the pseudo-Kähler structure (J, g N ) induced from (M, g, ∇) by Proposition 3. If moreover (N , J, g N , ∇ N ) is special Kähler and the Killing vector fields Ui of the above action are ∇ N -parallel along the Lagrangian orbits, then (M = N /Rn , g, ∇) is special real and (N , J, g N , ∇ N ) is obtained from (M = N /Rn , g, ∇) by the r-map. Proof. We denote by Ui the commuting vector fields on N which are the generators of the action of Rn . The holomorphicity of the Ui implies that the vector fields Ui , X j = −J U j commute. They are linearly independent since the distribution T v N := span{Ui |i = 1, · · · , n} is nondegenerate and Lagrangian. There exist local coordinates (x i , u j ) such that Ui = ∂u∂ i and X j = ∂ ∂x j . In these coordinates g N = gi j d x i d x j + du i du j , where the functions gi j = gi j (x) depend only on the x i , since the Ui are Killing vector fields. Since Ui is a holomorphic Killing vector field, it is also symplectic with respect to the Kähler form. This implies that X i = −J Ui is a gradient vector field, hence the one-form g(X i , ·) is closed and ∂∂x i g jk is completely symmetric. This shows that (M, g, ∇) is a Hessian manifold, where g = (gi j (x)) is the metric on M which makes N → M a pseudo-Riemannian submersion and ∇ is the flat connection induced by the flat connection on the leaves of the distribution T h := J T v with parallel vector fields ∂ i . Identifying M with a section of the trivial bundle N → M we can identify ∂x
N with T M. It is clear that the pseudo-Kähler structure on N = T M is obtained by Proposition 3 from the Hessian manifold (M, g, ∇). In the special Kähler case the assumption ∇UNi U j = 0 implies ∇ JNUi (J U j ) = ∇ JNUi J U j + J ∇ JNUi U j = −J ∇UNj J Ui + J ∇UNj J Ui = 0. Using the fact that for a special Kähler manifold Sˆ N := D N − ∇ N = − 21 J ∇ N J , one can easily check that ∇ N is the connection from Lemma 1. Now Corollary 8 shows that ∇ S = 0, i.e. (M, g, ∇) is special real. As an application we prove the following non-existence result.
Theorem 5. There is no compact simply connected special real manifold of positive dimension.
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Proof. Let (M, g, ∇) be a compact simply connected special real manifold. We first prove that the metric of the special Kähler manifold (N = T M, g N , J, ∇ N ) is complete. As in the proof of Corollary 1, we have a ∇ parallel coframe ξ i = d x i , i = 1, . . . , n. We denote by ∂i the dual global frame on M and by u i the corresponding globally defined functions on N = T M, which are the coordinates of a vector with respect to the frame ∂i . There exists a free isometric action of Rn on N = T M given by u i → u i + ci , whose orbits are the tangent spaces. The quotient of (N , g N ) by the lattice of integral translations is a compact, hence, complete Riemannian manifold. Thus (N , g N ) is complete as the universal covering of a complete Riemannian manifold. By [BC1], (N , g N , ∇ N ) can be realised as an improper affine hypersphere with complete Riemannian metric. In fact, one can check that the integrability conditions3 for the existence of an affine hypersphere immersion ϕ : N → R2n+1 follow from the properties of a special Kähler manifold. By the Calabi-Pogorelov theorem, see [NS] and references therein, the metric g N admits a quadratic Hesse potential and the Levi-Civita connection of g N coincides with ∇ N . Using the formulas (2.3), this implies that the Levi-Civita connection of the special real manifold coincides with ∇ and, hence, is flat. Since there is no compact simply connected flat Riemannian manifold of positive dimension we obtain the theorem.
References [ACDV] [BC1] [BC2] [C] [CMMS1] [dWvP] [DNV] [GST] [NS] [S0] [S] [SY]
Alekseevsky, D.V., Cortés, V., Devchand, C., Van Proeyen, A.: Flows on quaternionic-Kähler and very special real manifolds. Commun. Math. Phys. 238, 525–543 (2003) Baues, O., Cortés, V.: Realisation of special Kähler manifolds as parabolic spheres. Proc. Am. Math. Soc. 129(8), 2403–2407 (2001) Baues, O., Cortés, V.: Abelian simply transitive groups of symplectic type. Annales de l’Institut Fourier 52(6), 1729–1751 (2002) Cortés, V.: Special Kähler manifolds: a survey. Rend. Circ. Mat. Palermo (2) Suppl. no. 66, 11–18 (2001) Cortés, V., Mayer, C., Mohaupt, T., Saueressig, F.: Special geometry of Euclidean supersymmetry I: vector multiplets. J. High Energy Phys. 2004, no. 3, 028, 73 pp de Wit, B., Van Proeyen, A.: Special geometry, cubic polynomials and homogeneous quaternionic spaces. Commun. Math. Phys. 149, 307–333 (1992) Dillen, F., Nomizu, K., Vrancken, L.: Conjugate connections and Radon’s theorem in affine differential geometry. Monatsh. Math 109, 221–235 (1990) Günaydin, M., Sierra, G., Townsend, P.K.: The geometry of N = 2 Maxwell-Einstein supergravity and Jordan algebras. Nucl. Phys. B 242, 244–268 (1984) Nomizu, K., Sasaki, T.: Affine differential geometry. Cambridge Tracts in Mathematics 111, Cambridge: Cambridge University Press, 1994 Shima, H.: Homogeneous Hessian manifolds. Ann. Inst. Fourier 30(3), 91–128 (1980) Shima, H.: The geometry of Hessian structures. Hackensack, NJ: World Scientific Publishing, 2007 Shima, H., Yagi, K.: Geometry of Hessian manifolds. Diff. Geom. Appl. 7, 277–290 (1997)
Communicated by G.W. Gibbons
3 These are: ∇ N is flat, torsion-free and preserves the metric volume and the conjugate connection ∇ ¯ is (projectively) flat and torsion-free.
Commun. Math. Phys. 291, 591–597 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0835-z
Communications in
Mathematical Physics
Map Lattices Coupled by Collisions Gerhard Keller1 , Carlangelo Liverani2 1 Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstr. 1 1 , 2
Erlangen 91052, Germany. E-mail: [email protected]
2 Dipartimento di Matematica, II Università di Roma (Tor Vergata),
Via della Ricerca Scientifica, Roma 00133, Italy. E-mail: [email protected] Received: 21 November 2008 / Accepted: 8 March 2009 Published online: 22 May 2009 – © Springer-Verlag 2009
Abstract: We introduce a new coupled map lattice model in which the weak interaction takes place via rare “collisions”. By “collision” we mean a strong (possibly discontinuous) change in the system. For such models we prove uniqueness of the SRB measure and exponential space-time decay of correlations. 1. Introduction During the last years many results have been published on coupled map lattices. Much of this work deals with the weak coupling situation; see [3] and references therein to have a more precise idea of all the related work and results. In most of this considerable body of work the weak coupling is described by a diffeomorphism of the state space close to identity. Only recently couplings close to identity but not diffeomorphic could be investigated in a mathematically rigorous way. This setting (coupling close to identity) models a weak interaction and is reminiscent of a situation in which a collection of systems is interacting via a weak potential. Yet, a collection of systems can also have interactions that are weak only on average. The typical example of such a situation are interactions that can be strong but are rare, such as in the case of collisions in a rarefied gas. Examples of such a situation that has attracted some attention lately are models of the type introduced in [9] and recently used in [5,6] to argue for a derivation of the Fourier Law from microscopic dynamics. In this note we consider a simple system of coupled dynamics of the latter type. Namely a lattice of piecewise expanding interval maps with strong but rare interactions. Of course, this model is very far from a system of interacting disks, yet it is interesting that the available techniques can be applied to a case with strong but rare interactions. Given the present efforts in trying to devise a setting for Anosov maps with discontinuities that shares the same good properties of the BV setting for piecewise expanding maps [1,4] it is reasonable to hope that in the future the present results could be extended to more realistic situations.
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What we prove is that such a system, if the local maps are sufficiently expanding, has a unique SRB measure with exponential mixing properties in time and space. These are the same type of results we proved for systems with weak interactions of the “coupling close to identity” type in [8]. 2. The System and the Results Consider = [0, 1]Z and a map τ : [0, 1] → [0, 1], piecewise smooth and expanding, that is |τ | ≥ α > 1 and (ττ )2 bounded. The uncoupled system is described by the product d
dynamics T0 : → defined by T0 (x)p = τ (xp ) (p ∈ Zd ). The strength of the interaction will be expressed by a parameter ε > 0 which measures how rare the interactions are. To be precise, let {ei } be the standard base of Rd and V := {ei , −ei } ⊂ Rd , and, for each ε > 0 sufficiently small, fix a set of disjoint open intervals {Aε,v }v∈V ⊂ [0, 1] such that |Aε,v | = ε. Consider the coupling1 xp+v if xp ∈ Aε,v and xp+v ∈ Aε,−v for some v ∈ V (ε (x))p = otherwise. xp
The dynamics we are interested in is then defined by Tε = ε ◦ T0 . Remark 2.1. The interpretation of ε is quite obvious: nearby systems can interact only if the coordinates belong to a small set (the “collision” set). If this is the case, then the coordinates undergo a violent change. We use · for the bounded variation norm and | · | for the total variation norm of a measure. That is, calling D the space of differentiable local functions on and D1 := {ϕ ∈ D : |ϕ|∞ ≤ 1}, for each complex Borel measure µ on we define |µ| := sup |µ(ϕ)|, ϕ∈D1
µ := sup sup |µ(∂xp ϕ)|. p∈Zd ϕ∈D1
(2.1)
To study the statistical properties of the map Tε we will follow the well established path of studying its action on the space of measures. That is we will investigate the operator Tε∗ .2 To this end we will follow the path laid down in [8] and restrict our study to measures belonging to the space B := {µ < ∞}, see [7] for a detailed explanation of the meaning and properties of this choice. We will then use exactly the same strategy as developed in [8]. The following lemma is proven in [7, Prop. 4]; see also Lemma 2.2 of [8]. Lemma 2.2. There exists a constant B0 > 0 such that for each µ ∈ B, |T0∗ µ| ≤ |µ| T0∗ µ ≤ 2α −1 µ + B0 |µ|. 1 This special case is considered for simplicity. Yet, one can easily treat the more general case in which xp+v is replaced by φv (xp , xp+v ), for some set of invertible smooth maps φv : [0, 1]2 → [0, 1]2 . 2 As usual for each map : → the operator ∗ is defined by ∗ µ(ϕ) = µ(ϕ ◦ ), for each measure µ and function ϕ.
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The proof of the next lemma is provided in Sect. 3. Lemma 2.3. There exists a constant B1 > 0 such that for each µ ∈ B, |∗ε µ| ≤ |µ|,
∗ε µ ≤ (2 + 2d)µ + B1 −1 ε |µ|, where ε is the minimal distance between the intervals Aε,v (v ∈ V ). The above lemmas imply a Lasota-Yorke type inequality for Tε , namely: there exists a constant B > 0 such that Tε∗ µ
|Tε∗ µ| ≤ |µ|, ≤ (4 + 4d)α
−1
µ +
(2.2) B −1 ε |µ|.
(2.3)
α The second of these inequalities is relevant in the case λ := 4+4d > 1 which we will 3 assume from now on. The only thing left is to define a one-site decoupling ε,p of the dynamics and to show that we can decouple at a single site paying only a small price. Namely, we define ⎧ ⎪ if q − p ∈ V ∪ {0} ⎨(ε (x))q (ε,p (x))q = (ε (x))q (2.4) if v := q − p ∈ V, and xq ∈ Aε,−v ⎪ ⎩x otherwise. q
A moment of thought will show that ε,p differs from ε only insofar as the coordinate xp is now independent of the other coordinates. Lemma 2.4. For each p ∈ Zd , |∗ε,p µ| ≤ |µ|,
(2.5)
∗ε,p µ ≤ (2 + 2d)µ + B1 −1 ε |µ|,
(2.6)
|∗ε µ − ∗ε,p µ|
≤ 4dε µ.
(2.7)
The proof of this lemma can be found in Sect. 3. Equations (2.2), (2.3) and Lemma 2.4 are all that is needed to prove our main theorem. Theorem 2.5. If the local map τ is mixing, if α > 4 + 4d and if lim inf ε 0 ε > 0, then there is some ε0 > 0 such that for each ε ∈ (0, ε0 ) there exists a unique invariant element µinv ∈ B for the dynamics Tε . In addition, for all local smooth observables, µinv enjoys exponential decay of space-time correlations, and it is the SRB measure of the system. (For more precise statements and discussions of these assertions see Sect. 4 of [8].) Proof. The proof follows very closely the arguments in [8], replacing some of the estimates from that paper by the above lemmas. For the reader’s convenience we sketch the complete argument. The main notational difference to [8] is that our µ is denoted by Var(µ) there and that the lattice Zd is called there. 3 With some more work one can certainly weaken the requirement on the expansion constant α. Since the goal of this note is to show that this type of maps can be treated by transfer operator methods, we decided to restrict ourselves to the simplest possible setting.
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The basic idea is to define, for a suitable power (Tε∗m ) of Tε∗ , an extension (T ε,m , B) of the linear system (Tε∗m , B) and to prove that this extension has a spectral gap. So we introduce the following notation: B := Xp∈Zd Bp , where Bp := {µ ∈ B : ∂xp ϕ = 0 ⇒ µ(ϕ) = 0}.
(2.8)
Elements of B are denoted by µ = (µp )p∈Zd , and µ := supp∈Zd µp is a norm on B. We need to lift measures µ ∈ B 0 := {µ ∈ B : µ(1) = 0} to B. To this end we fix an enumeration S : N → Zd which has the property that i ≤ j implies |S(i)| ≤ |S( j)| and we define, for each p = p + S(0) and q = p + S(i) in Zd and ϕ ∈ D, (2.9)
p,q ϕ(x) := ϕ(x) d xp . . . d xp+S(i−1) − ϕ(x) d xp . . . d xq . Observe that ∗p,q µ = πZd \B i−1 µ ⊗ m ⊗Bp
i−1
p
− πZd \Bpi µ ⊗ m ⊗Bp , where Bp = {p, p + i
j
S(1), . . . , p + S( j)} and π B µ denotes the marginal of the measure µ on I B . It is easily checked that ∗p,q µ ∈ Bq so that (µ) := ( ∗0,q µ)q∈Zd defines a lift from B to B. A projection P from B to B is only partially defined by P(µ) := p∈Zd µp . It is well defined on the subspace of all µ for which µp = 0 except for finitely many p, and it extends to a continuous linear operator from B to B(θ ) which, for 0 < θ < 1, is the completion of B under the norm µθ := sup θ | | π µ with ranging over all finite “boxes” ⊂ Zd . Note however, that P( (µ)) = µ for all µ ∈ B 0 . For µ ∈ B define T ε µ := (Tε∗ µp )p∈Zd . In general, if ε = 0, then T ε µ ∈ B, because Tε∗ µp ∈ Bp . But H (T ε µ) := ( p∈Zd ∗p,q Tε∗ µp )q∈Zd belongs to B, and the same is m true for T ε,m µ := H (T ε µ) for each m ∈ N. As in [8], it is readily checked that P T ε,m = Tε∗m P, so T ε,m : B → B is the wanted extension of Tε∗m : B 0 → B 0 . For details see [8, Lemma 3.1] where it is also shown that T ε,m µ ≤ const m d sup Tε∗m µp .
(2.10)
p∈Zd
Writing m = m 1 + m 2 , separate m 1 - and m 2 -fold applications of Eq. (2.3) yield −1 −1 −m 1 µp + |Tε∗m 2 µp |) Tε∗m µp ≤ B −1 ε (1 − λ ) (λ
where also Eq. (2.2) and the fact that |µ| ≤ µ are used. The next step, after setting Tε,p := ε,p ◦ T0 , is to show that ∗m 2 Tε,p µ − Tε∗m 2 µ ≤ const m 2 ε −1 ε µ.
(2.11)
(2.12)
For m 2 = 1 this is an immediate consequence of Eqs. (2.7) and (2.3), and with a simple telescoping argument, as in the proof of [7, Theorem 4.3], one treats the case m 2 > 1. The benefit of working with measures µp ∈ Bp is that ∗m 2 µp | ≤ const σ0m 2 µp , |Tε,p
(2.13)
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where σ0 ∈ (0, 1) is the mixing rate of the Perron-Frobenius operator of the local map τ . This is proved precisely like the corresponding estimate on pp. 40/41 of [8]. Combining Eqs. (2.11)– (2.13) yields
m2 −1 −1 Tε∗m µp ≤ B −1 λ−m 1 + const m 2 ε −1 µp , (2.14) ε (1 − λ ) ε + const σ0 1
and setting σ1 := max{λ−1 , σ0 } 4 < 1 and m 1 = m 2 , there is, for m large enough, some ε(m) > 0 such that for |ε| < ε(m), Tε∗m µp ≤ σ1m µp .
(2.15)
1
Now choose m so large that σ := (const m d ) m σ1 < 1 for the constant from Eq. (2.10). We obtain then the exact analogue of (3.19) in [8], T ε,m µ ≤ σ m µ.
(2.16)
This implies that for each smooth local function ϕ, depending only on L variables and µ, ν ∈ B, the following holds: k k ∗km Tε µ(ϕ) − Tε∗km ν(ϕ) ≤ [P T ε,m (µ − ν)](ϕ) ≤ LT ε,m (µ − ν) |ϕ|∞ ≤ Lσ km (µ − ν) |ϕ|∞ ≤ 4Lσ km µ − ν |ϕ|∞ , which clearly implies the existence of a space-time exponentially mixing invariant measure. For more details see the last lines of Sect. 3 and Sect. 4 of [8]. 3. Proofs Proof of Lemma 2.3. Let ϕ be a smooth local function and let p ∈ Zd . Then, setting ε,v,p = {x : xp ∈ Aε,v , xp+v ∈ Aε,−v } and ε,p = ∪v∈V ε,v,p , we have ∂xp (ϕ ◦ ε ) (x) if x ∈ cε,p (∂xp ϕ) ◦ ε (x) = ∂xp+v (ϕ ◦ ε ) (x) if x ∈ ε,v,p . Therefore, for µ ∈ B, ∗ε µ(∂xp ϕ)
= =
(∂xp ϕ) ◦ ε dµ
cε,p
∂xp (ϕ ◦ ε ) dµ +
v∈V
ε,v,p
∂xp+v (ϕ ◦ ε ) dµ.
(3.1)
In order to estimate these integrals against the variation of µ in directions p and p + v we must modify the test function ϕ ◦ ε in such a way that it becomes continuous in xp or xp+v , respectively (see the characterization of · in [7], Sect. 3.3). For the first integral, let ϕ∗ be a function that, for each fixed x =p , is piecewise linear (but not necessarily continuous!) in xp interpolating between the requirements ϕ∗ |ε,p = 0 and ϕ∗ (x) = ϕ(ε (x)) for each x = (x =p , xp ) ∈ ∂p ε,p , where ∂p ε,p denotes the set of those points in the boundary of ε,p where the boundary is normal to the xp -direction. Thus, given x =p , the partial derivative ∂xp ϕ∗ (x) exists for Lebesguea.e. xp , and ϕ˜ := 1cε,p · (ϕ ◦ ε ) − ϕ∗ is Lipschitz continuous in xp , |ϕ| ˜ ∞ ≤ 2|ϕ|∞ ,
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ϕ| ˜ ∂p ε,p = 0, and |∂xp ϕ∗ |∞ ≤ C −1 ε |ϕ|∞ , where ε is the minimal distance between intervals Aε,v (v ∈ V ). Hence, ∂xp (ϕ ◦ ε ) dµ = ∂xp ϕ˜ dµ + ∂xp ϕ∗ dµ ≤ 2|ϕ|∞ µ + C −1 ε |ϕ|∞ |µ|. cε,p
Similarly, for each v ∈ V , let ϕv be a function that, for each fixed x =p+v , is piecewise constant in xp+v and such that ϕv |ε,p,v = 0 and ϕv (x) = ϕ(ε (x)) for each x = (x =p+v , xp+v ) ∈ ∂p+v ε,p,v . Then ϕ˜v := 1ε,p,v · (ϕ ◦ ε ) + ϕv is Lipschitz continuous in xp+v , |ϕ˜ v |∞ ≤ |ϕ|∞ and ∂xp+v ϕv (x) = 0 for x ∈ ∂p+v ε,p,v . Hence ∂xp+v (ϕ ◦ ε ) dµ = ∂xp+v ϕ˜v dµ − ∂xp+v ϕv dµ ≤ |ϕ|∞ µ. ε,v,p
Observing Eq. (3.1), this yields ∗ε µ ≤ (2 + 2d)µ + C −1 ε |µ|. This proves the second inequality of the lemma. The first one is trivial.
Proof of Lemma 2.4. The first two inequalities of this lemma are proved exactly as in Lemma 2.3. We turn to the third one. Let Aε = ∪v∈V Aε,v and notice that ε (x) = ε,p (x) for all x ∈ such that xp ∈ Aε . Thus, for all smooth local functions ϕ and all µ ∈ B, |∗ε µ(ϕ) − ∗ε,p µ(ϕ)|
= 1 Aε,v (xi )1 Aε,−v (xp+v ) ϕ ◦ ε (x) − ϕ ◦ ε,p (x) µ(d x) . v∈V
Next, since ϕ ◦ ε depends only on finitely many variables, say the variables in the finite set 0 ⊂ Zd , we can consider the marginal of µ on [0, 1] , = 0 ∪ {p + v}v∈V ∪{0} . Such a marginal is absolutely continuous with respect to Lebesgue measure, and its density h is a function of bounded variation with |h| BV ≤ µ, (see [7] for details). Hence,
|∗ε µ(ϕ) − ∗ε,p µ(ϕ)| ≤ 2|ϕ|∞ 1 Aε,v (xp )1 Aε,−v (xp+v )|h(x)|d x, v∈V
[0,1]
and, since |(|h|)| BV ≤ |h| BV (see e.g. the proof of Lemma 2.3 in [2]), we can consider the marginal h v (xp , xp+v ) := |h(x)|d xq ∈{p,p+v} . As |h v | BV ≤ |(|h|)| BV ≤ |h| BV ≤ µ, the usual Sobolev inequalities imply
|∗ε µ(ϕ) − ∗ε,p µ(ϕ)| ≤ 2|ϕ|∞ 1 Aε,v (x)1 Aε,−v (y)h v (x, y)d xd y v∈V
[0,1]2
≤ 2(2d)|ϕ|∞ µε. Acknowledgement. We are indebted to the ESI where, during the Workshop on Hyperbolic Dynamical Systems with Singularities (2008), this work was started. L.C. thanks the ENS, Paris, where he was invited during part of this work. Also we like to thank the Institut Henri Poincare - Centre Emile Borel where, during the trimester Mécanique statistique, probabilités et systèmes de particules (2008), this work was finished. Finally, G.K. acknowledges the support by the DFG grant KE 514/7-1, and L.C. by the MIUR grant PRIN 2007B3RBEY.
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References 1. Baladi, V., Gouezel, S.: Good Banach spaces for piecewise hyperbolic maps via interpolation. http://arxtiv. org/abs/0711.1960v1[math.DS], 2007 2. Bardet, J.-B., Gouëzel, S., Keller, G.: Limit theorems for coupled interval maps. Stoch. and Dyn. 7, 17–36 (2007) 3. Chazottes, J.-R., Fernandez, B. (eds.): Lectures from the School-Forum (CML 2004), (Paris, June 21–July 2, 2004), Lecture Notes in Physics 671. Berlin: Springer, 2005 4. Demers, M., Liverani, C.: Stability of statistical properties in two-dimensional piecewise hyperbolic Maps. Trans. Amer. Math. Soc. 360, 4777–4814 (2008) 5. Gaspard, P., Gilbert, T.: Heat conduction and Fourier’s Law by consecutive local mixing and thermalization. Phys. Rev. Lett. 101, 020601 (2008) 6. Gaspard, P., Gilbert, T.: Heat conduction and Fourier’s law in a class of many particle dispersing billiards. New J. Phys. 10, 103004 (2008) 7. Keller, G., Liverani, C.: A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps. In: Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Eds.: Chazottes J.-R., Fernandez B., Lecture Notes in Physics 671, Berlin: Springer Verlag, 2005, pp. 115–151 8. Keller, G., Liverani, C.: Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Commun. Math. Phy. 262(1), 33–50 (2006) 9. Bunimovich, L., Liverani, C., Pellegrinotti, S., Suhov, Y.: Ergodic systems of n balls in a billiard table. Commun. Math. Phy. 146, 357–396 (1992) Communicated by A. Kupiainen
Commun. Math. Phys. 291, 599–644 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0876-3
Communications in
Mathematical Physics
On the Self-Similar Asymptotics for Generalized Nonlinear Kinetic Maxwell Models A. V. Bobylev1 , C. Cercignani2 , I. M. Gamba3 1 Department of Mathematics, Karlstad University, Karlstad,
SE-651 88 Sweden. E-mail: [email protected]
2 Politecnico di Milano, Milano, Italy. E-mail: [email protected];
[email protected]
3 Department of Mathematics, The University of Texas at Austin, Austin,
TX 78712-1082, U.S.A. E-mail: [email protected] Received: 16 November 2006 / Accepted: 20 May 2009 Published online: 12 August 2009 – © Springer-Verlag 2009
Abstract: Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economics, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. A new application of multi-linear models to economics and social dynamics is discussed. Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Maxwell Models of the Boltzmann Equation and Their Generalizations . . . . . . . . . . . . . . . . . . 3. Main Equations and Statement of the Problem . . . . . 4. Existence and Uniqueness of Solutions . . . . . . . . . 5. Large Time Asymptotics and Self-Similar Solutions . . 6. Existence of Self-Similar Solutions . . . . . . . . . . . 7. Properties of Self-Similar Solutions . . . . . . . . . . . 8. Main Results for Fourier Transformed Maxwell Models with Multiple Interactions . . . . . . . . . . . . . . . . 9. Distribution Functions, Moments and Power-Like Tails 10. Applications . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The classical elastic Boltzmann equation with Maxwell-type interactions is well-studied in literature (see [4,15] and references therein). This is a mathematical model of a rarefied gas with binary collisions where the collision frequency is independent of the velocities of colliding particles. Maxwell models of granular gases were introduced relatively recently in [6] (see also [2] for the one dimensional case). Soon after, these models became very popular among people studying granular gases (see, for example, the book [14] and references therein). There are two obvious reasons for this to happen. One is due to the fact that the inelastic Maxwell-Boltzmann equation can be essentially simplified by the Fourier transform similarly to the elastic one [5,6], and the other one is that solutions to the spatially homogeneous inelastic Maxwell-Boltzmann equation have a nontrivial self-similar asymptotics approaching a corresponding self-similar solution that has a power-like tail for large velocities. This latter property was conjectured in [17] and later proved in [8,10,11]. It is actually remarkable that such an asymptotics is absent in the elastic case. This is due to the fact that, loosely speaking, the elastic Boltzmann equation has too many conservation laws. On the other hand, this self-similar asymptotics was also proved in the elastic case for initial data with infinite energy [7] using other mathematical tools from those of [8]. In another development, exact self-similar solutions [11] for elastic Maxwell mixtures, in the asymptotic limit of heavy particles in equilibrium with a cold background interacting with light particles dissipating the total kinetic energy, also exhibit power-like tails definitely suggesting self-similar asymptotics for such dissipative systems. Finally we mention recent publications [1,16,24], where one dimensional Maxwell-type models were introduced for applications to economics and again the self-similar asymptotics and power-like tail were found. All these discussed models describe qualitatively different processes in physics or economics, however their solutions have a lot in common from the mathematical point of view. It is also clear that some further generalizations are possible: models for multiple, not just binary, interactions still assuming the constant Maxwell-type rate of interactions. A natural question that arises is whether multi-linear models have similar properties. The answer to this question is affirmative, as we shall see below. It becomes clear that there must be some general mathematical properties of Maxwell models, which, in turn, can explain properties of any particular model. Essentially, there must be just one main theorem, from which one can deduce all the above discussed facts and their possible generalizations. The goal of this paper is to consider Maxwell models from a very general point of view and to establish their key properties that lead to the self-similar asymptotics. The paper is organized as follows. In Sect. 2 we focus on the classical homogeneous Boltzmann equation for Maxwell-type pair interactions and the stochastic N -particle model, introduced by M. Kac [20], related to this equation. Then we consider a generalization of the N -particle model which includes multi-particle interactions. It is shown that certain natural assumptions formally lead to a class of equations which can be considered as the most general Maxwell-type model. We confine ourselves to the case of isotropic solutions when the phase space variable (i.e. the velocity for the classical Boltzmann equation) is a d-dimensional vector with d ≥ 2. Then the Fourier transform leads to equations, which are the main object of our study in this paper (see Sect. 3). The same equations can be obtained by Laplace transform if the phase variable is a nonnegative real number (such a case is important for applications to economics as considered in Sect. 10). The concept of the generalized multi-linear Maxwell model in Fourier
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space is introduced in Sect. 3. Such models and their generalizations are studied in detail in Sects. 4-9. The concept of an L-Lipschitz nonlinear operator, one of the most important for our approach, is explained in Sect. 3 (Definition 3.1). It is proved (Theorem 3.2) that all multi-linear Maxwell models satisfy the L-Lipschitz condition. This property of the models constitutes a basis for the general theory. The existence and uniqueness of solutions to the initial value problem is proved in Sect. 4 (Theorem 4.2). Then we study in Sect. 5 the large time asymptotics under very general conditions that are fulfilled, in particular, for all our models. It is shown that the L-Lipschitz condition leads to self-similar asymptotics provided the corresponding self-similar solution does exist. The existence and uniqueness of self-similar solutions is proved in Sect. 6 (Theorem 6.1). This result can be considered, to some extent, as the main theorem for general Maxwell-type models. Then, in Sect. 7, we go back to the multilinear models of Sect. 3 and study more specific properties of their self-similar solution. We explain in Sect. 8 how to use our theory for applications to any specific model: it is shown that the results can be expressed in terms of just one function µ( p), p > 0, that depends on the spectral properties of the specific model. General properties of the selfsimilar solutions, such as positivity, power-like tails, and more, are studied in Sect. 9. This study also includes the case of one dimensional models, where the Laplace (instead of Fourier) transform is used. A new application of multi-linear Maxwell models to economics or social dynamics is presented in Sect. 10. It is shown how the general theory can predict a time dependent distribution of wealth among participants of economic games with simple rules and with arbitrary number of players. For brevity, we did not include in this paper more traditional applications to Boltzmann type equations. Such applications were included in the original version of this manuscript posted on the preprint arXiv:math-ph/0608035v1, and are discussed in [9]. 2. Maxwell Models of the Boltzmann Equation and Their Generalizations We consider a spatially homogeneous d-dimensional (d = 2, 3, . . .) rarefied gas of particles having a unit mass. Let f (v, t), where v ∈ Rd and t ∈ R+ denote respectively the velocity and time variables, be a one-particle distribution function with the usual normalization f (v, t) dv = 1. (2.1) Rd
We also assume that the collision frequency is independent of the velocities of colliding particles (Maxwell-type interactions) and that the total scattering cross section is finite. Hence, we can choose such units of time such that the corresponding classical Boltzmann equation reads f t = Q + ( f ) − f,
(2.2)
where Q + ( f ) is the gain term of the collision integral (e.g. see [15]). We shall need only some general properties of Q + ( f ): Q + ( f ) ≥ 0 if f ≥ 0, and [Q + ( f )](v) dv = 1, (2.3) Rd
for any f satisfying (2.1). Therefore, the operator Q + transforms f to another probability density. The structure of Eq. (2.2) leads to the well-known probabilistic interpretation
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by M. Kac [20]. Accordingly, we can forget for awhile the rarefied gas and consider stochastic dynamics of N particles with phase coordinates (velocities) vi (t) ∈ Rd , i = 1, . . . , N . A bit simplified Kac rules of the dynamics are: on each time-step t = 2/N choose randomly a pair of integers 1 ≤ i < l ≤ N and perform a transformation (vi , vl ) → (vi , vl ) which corresponds to a collision of two particles with pre-collisional velocities vi and vl . It is easy to show (at least formally) that these rules lead, under the additional assumption of molecular chaos [20], to Eq. (2.2) in the limit N → ∞. A similar stochastic model, with a slightly different but same type of transformation (vi , vl ) → (vi , vl ), leads to the inelastic version of Eq. (2.2) [6,14]. It is clear that the same stochastic model admits other possible generalizations. For example we can also include multiple interactions and interactions with a background (thermostat). This type of model will formally correspond to a version of Eq. (2.2) for some Q + ( f ). For example, we take (1)
(2)
(M)
Q + ( f ) = α1 Q + ( f ) + α2 Q + ( f ) + · · · + α N Q + ( f ),
(2.4)
( j)
where Q + , j = 1, . . . , M, are j-linear positive operators describing interactions of j ≥ 1 particles, α j ≥ 0 are relative probabilities of such interactions. It is assumed that each
Rd
( j)
[Q + ( f )](v) dv = 1;
and that
M
α j = 1,
(2.5)
j=1
so condition (2.1) always holds. ( j) Next, we focus on what properties of operators Q + are needed to make them consistent with Maxwell-type interactions. We postulate the main property of multi-particle systems with such interactions in the following way: Temporal evolution of the system is invariant under scaling transformations of the phase space. That is, if St is the evolution operator of the above discussed N -particle system such that St {v1 (0), . . . , v M (0)} = {v1 (t), . . . , v M (t)},
t ≥ 0,
then St {λv1 (0), . . . , λv M (0)} = {λv1 (t), . . . , λv M (t)}
(2.6)
for any constant λ > 0. It is easy to see that this assumption leads to the following ( j) property of Q + ( j = 1, 2, . . . , M): ( j)
( j)
Q + (Aλ f ) = Aλ Q + ( f ),
Aλ f (v) = λd f (λv),
λ > 0.
(2.7)
Note that the transformation Aλ is consistent with the normalization (2.1). This property shows that it is convenient to use the Fourier Transform ˆ f (v, t)e−ik·v dv, k ∈ Rd , f (k, t) = F( f ) =
(2.8)
Rd
since the resulting equation fˆt = Qˆ + ( fˆ) − fˆ,
Qˆ + ( fˆ) =
M j=1
( j) α j Qˆ + ( fˆ),
(2.9)
Asymptotics for Generalized Nonlinear Maxwell Models
603
is invariant under scaling transformations k → λk, k ∈ Rd . Finally, it is natural to assume (a least in the case when v ∈ Rd is a velocity of the particle) that all interactions are invariant under rotations in Rd . Then the general problem can be simplified if we confine ourselves to a class of isotropic distribution functions f (|v|, t). In particular, denoting u(x, t) = dv f (|v|, t)e−ik·v , x = |k|2 , (2.10) Rd
we consider a particular form of Eq. (2.9) for u(x, t). The resulting equation (see Sect. 3) is the main mathematical object studied in this paper. All the above considerations remain valid for d = 1, the only differences are that, first, Eq. (2.2) should be considered as the one-dimensional Kac equation [20], and second, rotations in R1 = R should be replaced by reflections. An interesting one-dimensional system, presented in Sect. 9, is based on the above discussed multi-particle stochastic model with non-negative phase variables v = R+ , for which the Laplace transform u(x, t) =
∞
f (v, t)e−xv dv,
x ≥ 0,
(2.11)
0
leads to exactly the same class of equations for u(x, t), described in Sect. 3. 3. Main Equations and Statement of the Problem We consider Eq. (2.9) for the case of isotropic solutions fˆ(k, t) = u(|k|2 , t). The oper( j) ator Qˆ + is a linear combination of n-linear operators Qˆ + , 1 ≤ j ≤ M, acting on the 2 x = |k| variable and invariant under dilations {xλ = λx, λ > 0} in R+ . A general class of such operators acting on u(x) can be written in the form ( j) Qˆ + (u) =
∞
∞
da1 . . .
0
da j Q j (a1 , . . . , a j )
0
j
u(ai x),
i=1
where Q j (a1 , . . . , a j ) can be an generalized function of j non-negative variables. In ( j) our case both u(x) and Qˆ + (u) are related by Fourier (or Laplace) transforms to some d probability densities in R (or R+ ) (see Eqs. 2.1, (2.5), (2.10), (2.11)). Hence u(0) = 1, 0
∞ ∞
Q j (a1 , . . . , a j ) da1 . . . da j = 1,
0
moreover u ∈ C(R+ ). Finally we note that the original (before Fourier/Laplace trans( j) ( j) forms) operators Q + were positive, i.e., Q + ( f ) ≥ 0 if f ≥ 0. To satisfy this condition it is sufficient to assume that Q j (a1 , . . . , a j ) ≥ 0 in the above formulas. This follows directly from the fact that the product of two transforms is the transform of a convolution of originals. These arguments explain our choice of equations below. We slightly change the notation and consider the following equation for u(x, t): u t + u = (u),
x ≥ 0, t ≥ 0,
(3.1)
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A. V. Bobylev, C. Cercignani, I. M. Gamba
where (u) =
M
α j ( j) (u),
j=1
( j) (u) =
M
α j = 1, α j ≥ 0,
j=1
∞
...
0
∞
A j (a1 , . . . , a j ) 0
j
(3.2) u(ak x) da1 . . . da j ,
j = 1, . . . , M.
k=1
We assume that
∞
A j (a) = A j (a1 , . . . , a j ) ≥ 0,
∞
da1 . . .
0
da j A(a1 , . . . , a j ) = 1, (3.3)
0 j
where A j (a) = A j (a1 , . . . , a j ) is a generalized density of a probability measure in R+ for any j = 1, . . . , M. We also assume that all A j (a) have compact support, i.e., A j (a1 , . . . , a j ) ≡ 0 if
j
ak2 > R 2 ,
j = 1, . . . , M,
(3.4)
k=1
for sufficiently large 0 < R < ∞. In fact a much weaker assumption that
∞ 0
∞
...
A j (a1 , . . . , a j ) 0
j
p
ak da1 . . . da j < ∞,
j = 1, . . . , M,
(3.5)
k=1
for all p > 0 is needed for most of our results (the assumption (3.4) is used only in Sect. 10). Classical models of elastic or inelastic particle interactions of Maxwell type are particular cases of Eq. (3.1) with, for example 1 1 M = 2, α1 = ds H (s), α2 = ds G(s), A1 (a1 ) =
1 α1
A2 (a1 , a2 ) =
0 1
0
1 α2
0
ds H (s)δ[a1 − c(s)],
1
(3.6)
ds G(s)δ[a1 − a(s)]δ[a2 − b(s)],
0
where the interaction law is determined by the functions a(s), b(s) and c(s). Then, it is clear that Eq. (3.1) can be considered as a generalized Fourier transformed isotropic Maxwell model with multiple interactions provided u(0, t) = 1. The case M = ∞ in Eqs. (3.2) can be treated in the same way. Therefore, the general problem we consider below can be formulated in the following way: We consider the initial value problem u t + u = (u), u |t=0 = u 0 (x), x ≥ 0, t ≥ 0,
(3.7)
in the Banach space B = C(R+ ) of continuous functions u(x) with the norm u = sup |u(x)|. x≥0
(3.8)
Asymptotics for Generalized Nonlinear Maxwell Models
605
It is usually assumed that u 0 ≤ 1 and that the operator is given by Eqs. (3.2). On the other hand, there are just a few properties of (u) that are essential for existence, uniqueness and large time asymptotics of the solution u(x, t) of the problem (3.7). Therefore, in many cases the results can be applied to more general classes of operators in Eqs. (3.7) and more general functional space, for example B = C(Rd ) (anisotropic models). That is why we study below the class (3.2) of operators as the most important example, but simultaneously indicate which properties of are relevant in each case. In particular, most of the results of Sect. 3-6 do not use a specific form (3.2) of and, in fact, are valid for a more general class of operators. Following this way of study, we first consider the problem (3.7) with given by Eqs. (3.2) and point out the most important properties of . We simplify notations and omit in most of the cases below the argument x of the function u(x, t). The notation u(t) (instead of u(x, t)) means then the function of the real variable t ≥ 0 with values in the space B = C(R+ ). Remark 1. We shall omit below the argument x ∈ R+ of functions u(x), v(x), etc., in some cases when this does not cause a misunderstanding. In particular, inequalities of the kind |u| ≤ |v|, for functions u(x) and v(x), should be understood as a point-wise control in absolute value, i.e. “|u(x)| ≤ |v(x)| for any x ≥ 0”. We first start by giving the following general definition for operators acting in a unit ball of a Banach space B denoted by U = {u ∈ B : u ≤ 1}.
(3.9)
Definition 3.1. The operator = (u) is called an L-Lipschitz operator if there exists a linear bounded operator L : B → B such that the inequality |(u 1 ) − (u 2 )|(x) ≤ (L|u 1 − u 2 |) (x), x ≥ 0 ;
(3.10)
holds for any pair of functions u 1,2 in U . Remark 2. Note that the L-Lipschitz condition (3.10) holds, by definition, at any point x ∈ R+ . Thus, condition (3.10) is much stronger than the classical Lipschitz condition (u 1 ) − (u 2 ) < Cu 1 − u 2 if u 1,2 ∈ U,
(3.11)
which obviously follows from (3.10) with the constant C = L B , the norm of the operator L in the space of bounded operators acting in B. In other words, the terminology “L-Lipschitz condition” means the point-wise Lipschitz condition with respect to a specific linear operator L. A generalization to the case B = C(Rd ) is obvious: we just need to change x ≥ 0 to x ∈ Rd in Eqs. (3.7), (3.8) and (3.10). The next lemma shows that the operator (u) defined in Eqs. (3.2), which satisfies (1) = 1 (mass conservation) and maps U into itself, satisfies an L-Lipschitz condition, where the linear operator L is the one given by the linearization of near unity. We assume without loss of generality that the kernels A j (a1 , . . . , a j ) in Eqs. (3.2) are symmetric with respect to any permutation of the arguments (a1 , . . . , a j ), j = 2, 3, . . . , M.
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Theorem 3.2. The operator (u) defined in Eqs. (3.2) maps U into itself and satisfies the L-Lipschitz condition (3.10), where the linear operator L is given by ∞ Lu(x) = da K (a)u(ax), (3.12) 0
with K (a) =
M
jα j K j (a),
j=1
where
K j (a) =
∞
...
0
∞
A j (a, a2 , . . . , a j ) da2 . . . da j and
0
M
(3.13) αj = 1
j=1
for symmetric kernels A j (a1 , a2 , . . . , a j ), j = 2, . . .. Proof. First, the operator (u) in (3.2)-(3.4) maps B into itself and also satisfies (u) ≤
M
α j u j ,
j=1
M
α j = 1.
(3.14)
j=1
Hence, (u) ≤ 1 if u ≤ 1,
(3.15)
and then (U ) ⊂ U , so it maps U into itself. Since (1) = 1, we introduce the linearized operator L : B → B such that formally (1 + εu) = 1 + εLu + O(ε2 ).
(3.16)
By using the symmetry of kernels A j (a), j = 2, 3, . . . , M, one can easily check that L is given by Eqs. (3.12) and (3.13). In order to prove the L-Lipschitz property (3.10) for the operator given in Eqs. (3.2), we make use of the multi-linear structure of the integrand associated with the definition of (u). Indeed, from the elementary identity ab − cd =
a+c b+d (b − d) + (a − c), 2 2
we estimate |u 1 (a1 x)u 1 (a2 x) − u 2 (a1 x)u 2 (a2 x)| ≤ |u 1 (a1 x) − u 2 (a1 x)| + |u 1 (a2 x) − u 2 (a2 x)|, provided u 1,2 ≤ 1. Then we obtain |
(2)
(u 1 ) −
(2)
∞
(u 2 )| ≤ 2
x ≥ 0, a1,2 ≥ 0,
da K 2 (a)|u 1 (ax) − u 2 (ax)|
0
in the notation of Eqs. (3.2), (3.13). It remains to prove that ∞ | ( j) (u 1 ) − ( j) (u 2 )| ≤ j K j (a)|u 1 (ax) − u 2 (ax)| da 0
(3.17)
Asymptotics for Generalized Nonlinear Maxwell Models
607
for 3 ≤ j ≤ M (the case j = 1 is trivial). This problem can be obviously reduced to an elementary inequality j j j xk − yk ≤ |xk − yk |, j = 3, . . . , (3.18) k=1 k=1 k=1 provided |xk | ≤ 1, |yk | ≤ 1, k = 1, . . . , j. Since this is true for j = 2, we can use the induction. Let a = x j+1 , c = y j+1 , b =
j
xk , d =
k=1
then
j
yk ,
k=1
j+1 j+1 xk − yk = |ab − cd| ≤ |a − c| + |b − d| k=1 k=1 ≤ |x j+1 − y j+1 | +
j
|xk − yk |,
k=1
and the inequality (3.18) is proved for any j ≥ 3. Then we proceed exactly as in case j = 2 and prove the estimate (3.18) for arbitrary j ≥ 3. Inequality (3.10) follows directly from the definition of operators and L.
Corollary. The Lipschitz condition (3.11) is fulfilled for (u) given in Eqs. (3.2) with the constant C = L =
M
jα j ,
j=1
M
α j = 1,
(3.19)
j=1
where L is the norm of L in B. Proof. The proof follows directly from the inequality (3.10) and Eqs. (3.12), (3.13).
It is also easy to prove that the L-Lipschitz condition holds in B = C(Rd ) for “gainoperators” in the Fourier transformed Boltzmann equations for both elastic and inelastic Maxwell models. 4. Existence and Uniqueness of Solutions The aim of this section is to state and prove, with minimal requirements, the existence and uniqueness results associated with the initial value problem (3.7) in the space B = C(R+ ). In fact, this existence and uniqueness result is an application of the classical Picard iteration scheme and holds for any operator which satisfies the usual Lipschitz condition (3.11) and transforms the unit ball U into itself. We include its proof for the sake of completeness. Lemma 4.1 (Picard Iteration scheme). If the conditions in (3.15) and (3.11) are fulfilled then the initial value problem (3.7) with arbitrary u 0 ∈ U has a unique solution u(t) such that u(t) ∈ U for any t ≥ 0.
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Proof. Consider the integral form of Eq. (3.1), t e−(t−τ ) [u(τ )] dτ, u(t) = u 0 e−t +
(4.1)
0
and apply the standard Picard iteration scheme t u (n+1) (t) = u 0 e−t + e−(t−τ ) [u (n) (τ )] dτ,
u (0) = u 0 .
(4.2)
0
Consider a finite interval 0 ≤ t ≤ T and denote |||u|||T = sup u(t). 0≤t≤T
Then u (n+1) (t) ≤ u 0 e−t + (1 − e−t )|||(u (n) )|||t , and therefore, by induction u (n) (t) ≤ 1 for all n = 1, 2, . . . , and t ∈ [0, T ], since u 0 ≤ 1 and (u) satisfies the inequality (3.15). If, in addition, (u) satisfies the Lipschitz condition (3.11), then it is easy to verify that |||u n+1 − u n |||T ≤ (1 − e−T )C|||u n − u n−1 |||T ,
n = 1, 2, . . . ,
and therefore, the iteration scheme (4.2) converges uniformly for any 0 ≤ t ≤ T provided C(1 − e−T ) < 1 ⇒ T < ln
C if C > 1 C −1
(4.3)
(T can be taken arbitrarily large if C ≤ 1). It is easy to verify that u(t) = lim u (n) (t), n→∞
0 ≤ t ≤ T,
is a solution of Eqs. (3.7), (4.1), satisfying the inequality u(t) ≤ 1,
0 ≤ t ≤ T.
(4.4)
The Lipschitz condition (3.11) is also sufficient to show that this solution is unique in a class of functions satisfying the inequality (4.4) on any interval 0 ≤ t ≤ ε. Since the length T of the initial time-interval does not depend on the initial conditions (see Eq. (4.3)), then we can proceed by taking the next interval T ≤ t ≤ 2T and so on. Thus we obtain the global in time solution u(t) ∈ U , where U is the closed unit ball in B, of the Cauchy problem (3.7).
This proof of existence and uniqueness for the Cauchy problem (3.7)-(3.8) is quite standard (see any textbook on ODEs) and therefore we omit some details. The next statement shows that the L-Lipschitz condition (3.10) implies pointwise stability.
Asymptotics for Generalized Nonlinear Maxwell Models
609
Theorem 4.2. Consider the Cauchy problem (3.7) with u 0 ≤ 1 and assume that the operator : B → B: (a) maps the closed unit ball U ⊂ B into itself, and (b) satisfies a L-Lipschitz condition (3.10) for some positive bounded linear operator L : B → B. Then, for any t ≥ 0, i) there exists a unique solution u(t) of the problem (3.7) such that u(t) ≤ 1; ii) any two solutions u(t) and w(t) of problem (3.7) with initial data in the unit ball U satisfy the pointwise in x inequality |u(t) − w(t)| ≤ exp{t (L − 1)}(|u 0 − w0 |).
(4.5)
Proof. The proof of i) follows directly from Lemma 4.1 (with the Lipschitz constant C = L). For the proof of ii), let u(t) and w(t) be two solutions of this problem such that u(0) = u 0 , w(0) = w0 , u 0 ≤ 1, w0 ≤ 1.
(4.6)
Then the function y(t) = u(t) − w(t) satisfies the equation yt + y = g(x, t) = (u) − (w), Hence, y(t) = e−t y0 +
t
y|t=0 = u 0 − w0 = y0 .
e−(t−τ ) g(x, τ ) dτ,
0
and, applying Theorem 3.2 for the use of the L-Lipschitz condition (3.10), we obtain t −t e−(t−τ ) L|y(τ )| dτ. (4.7) |y(t)| ≤ |y0 |e + 0
Clearly, |y(t)| ≤ y∗ (t), where y∗ (t) satisfies the equation t e−(t−τ ) Ly∗ (τ ) dτ. y∗ (t) = |y0 |e−t +
(4.8)
0
Since the linear operator L : B → B is positive and bounded, then Eq. (4.8) has a unique solution y∗ (t) = e−t (1−L) |y0 | = e−t
∞ n t n=0
n!
L n |y0 |,
so estimate (4.5) follows and the proof of the theorem is completed.
Theorem 3.2 and the inequality (3.14) show that the operator given in Eqs. (3.2) satisfies all conditions of the theorem. Remark 3. The above consideration is, of course, a simple generalization of the proof of the usual Gronwall inequality for the scalar function y(t). The essential difference is, however, that y(t) is a “vector” y(x, t) with values in the Banach space B = C(R+ ) and, consequently, the estimate (4.5) for the functions u(x, t) and w(x, t) holds at any point x ∈ R+ .
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Remark 4. We stress that estimates (4.7)-(4.8) do not depend on specific properties of the operator L beyond that of being positive and bounded. The Banach space B = C(R+ ) can also be replaced, for example, by B = C(Rd ) (that is the case of non-isotropic models) provided some obvious changes in Eqs. (3.7), (3.8) and (3.10) are made. At this point we remind the reader that the initial value problem (3.7) appeared as a generalization of the initial value problem (2.9) for a characteristic function ϕ(x, t), i.e., for the Fourier transform of a probability measure, see Eqs. (2.9), (2.10)). It is important therefore to show that the solution u(x, t) of the problem (3.7) is a characteristic function for any t > 0 provided this is so for t = 0. The answer to such and similar questions is given in the following statement. Lemma 4.3. Let U ⊂ U ⊂ B be any closed convex subset of the unit ball U (i.e., u = (1 − θ )u 1 + θ u 2 ∈ U for any u 1,2 ∈ U and θ ∈ [0, 1]). If u 0 ∈ U in Eq. (3.7) and U is replaced by U in condition (a) of Theorem 4.2, the theorem holds and u(t) ∈ U for any t ≥ 0. Proof. The only important point which should be changed in the proof is the consideration of Eqs. (4.2) in the proof of Lemma 4.1. We need to verify that, for any u 0 ∈ U and v(τ ) ∈ U , 0 ≤ τ ≤ t, t u(t) ˆ = u 0 e−t + e−(t−τ ) v(τ ) dτ ∈ U . (4.9) 0
In order to see that (4.9) holds, we note that v(τ ) = [u (n) (τ )] in Eqs. (4.2) is, by construction, a continuous function of τ ∈ [0, t] and that t e−t + e−(t−τ ) dτ = 1. 0
Therefore we can approximate u(t) ˆ by an integral sum uˆ m (t) = u 0 e−t +
m k=1
γk (t)v(τk ),
m
γk (t) = 1 − e−t .
(4.10)
k=1
Then u m (t) ∈ U as a convex linear combination of elements of U . Taking the limit m → ∞ to the sequence generated in (4.10), with U is a closed subset of U , it follows that u(t) ˆ ∈ U , so (4.9) holds. Hence, the corresponding sequence as in Eqs. (4.2) also satisfies u (n) (t) ∈ U for all n ≥ 0. The rest of the proof continues as the one of Lemma 4.1, so that the result remains true for any closed convex subset U ⊂ U and so does Theorem 4.2. Thus the proof of Lemma 4.3 is completed.
Remark 5. It is well-known (see, for example, the textbook [18]) that the set U ⊂ U of Fourier transforms of probability measures in Rd (Laplace transforms in the case of R+ ) is convex and closed with respect to uniform convergence. On the other hand, it is easy to verify that the inclusion (U ) ⊂ U , where is given in Eqs. (3.2), holds in both cases of Fourier and Laplace transforms. Hence, all results obtained for Eqs. (3.1), (3.2) can be interpreted in terms of “physical” (positive and satisfying condition (2.1)) solutions of corresponding Boltzmann-like equations with multi-linear structure of any order.
Asymptotics for Generalized Nonlinear Maxwell Models
611
We also note that all results of this section remain valid for operators satisfying conditions (3.2) with a more general condition such as M
α j ≤ 1,
α j ≥ 0,
(4.11)
j=1
so that (1) < 1 and so the mass may not be conserved. The only difference in this case is that the operator L satisfying conditions (3.12), (3.13) is not a linearization of (u) near unity. Nevertheless Theorem 3.2 remains true. The inequality (4.11) is typical for Fourier (Laplace) transformed Smoluchowski-type equations where the total number of particles is decreasing in time (see [22,23] for related work). 5. Large Time Asymptotics and Self-Similar Solutions In this section we study in more detail the solutions to the initial value problem (3.7)-(3.8) constructed in Theorem 4.2 and, in particular, their long time behavior. First of all, we note that the operator given in Eqs. (3.2) has the followings properties. Main properties of the operator : (a) maps the unit ball U of the Banach space B = C(R+ ) into itself, that is (u) ≤ 1 for any u ∈ C(R+ )
such that u ≤ 1.
(b) is a L-Lipschitz operator with L given by ∞ L u(x) = K (a)u(a x) da,
x ≥ 0,
(5.1)
0
where K (a) ia a generalized density of a positive measure in R+ satisfying, ∞ 0< K (a)a p da < ∞, for any p ≥ 0. (5.2) 0
That means
∞
|(u 1 ) − (u 2 )|(x) ≤ (L|u 1 − u 2 |)(x) =
K (a)|u 1 (ax) − u 2 (ax)| da,
0
(5.3) for all x ≥ 0 and for any two functions u 1,2 ∈ C(R+ ) such that u 1,2 ≤ 1. (c) is invariant under dilations: eτ D (u) = (eτ D u), D = x
∂ , eτ D u(x) = u(xeτ ), τ ∈ R. ∂x
(5.4)
No specific information about beyond these three conditions will be used in sects. 5 and 6. It was already shown in Theorem 4.2 that the conditions (a) and (b) guarantee existence and uniqueness of the solution u(x, t) to the initial value problem (3.7)-(3.8). The property (b) yields the estimate (4.5) that is very important for large time asymptotics,
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as we shall see below. Property (c) suggests a special class of self-similar solutions to Eq. (3.7). Note that the operator L in property (b) has a general form of linear positive operator invariant under dilations, i.e. its specific form is connected with property (c) Next, we recall the usual meaning of the notation y = O(x p ) (often used below): y = O(x p ) if and only if there exists a positive constant C such that |y(x)| ≤ Cx p
for any x ≥ 0.
(5.5)
In order to study long time stability properties to solutions whose initial data differs in terms of O(x p ), we will need some spectral properties of the linear operator L. The integral formula for L (see condition (b) ) allows to extend L to a much wider functional classes than B = C(R+ ). We assume below that this obvious extension is already done. Definition 5.1. Let L be the integral operator given in Eq. (5.3), then ∞ 0 < λ( p) = K (a)a p da < ∞, p ≥ 0, Lx p = λ( p)x p ,
(5.6)
0
and the spectral function µ( p) is defined by µ( p) =
λ( p) − 1 . p
(5.7)
An immediate consequence of properties (a) and (b), as stated in (5.3), is that one can obtain a criterion for a point-wise in x estimate of the difference of two solutions to the initial value problem (3.7). Lemma 5.2. Let u 1,2 (x, t) be any two classical solutions of the problem (3.7) with satisfying (a) and (b), and initial data satisfying the conditions |u 1,2 (x, 0)| ≤ 1,
|u 1 (x, 0) − u 2 (x, 0)| ≤ C x p , x ≥ 0
(5.8)
for some positive constant C and p. Then |u 1 (x, t) − u 2 (x, t)| ≤ Cx p e−t (1−λ( p)) ,
for all t ≥ 0.
(5.9)
Proof. The existence and uniqueness of u 1,2 (x, t) follow from Theorem 4.2. Estimate (4.5) (a consequence from the L-Lipschitz condition!) yields |u 1 (x, t) − u 2 (x, t)| ≤ e−t e Lt w(x), with w(x) = |u 1 (x, 0) − u 2 (x, 0)|. (5.10) The operator L from (5.3) is positive, and therefore monotone. Hence we obtain et L w(x) =
tj j
that completes the proof.
j!
L j w(x) ≤ C et L x p = Ceλ( p) t x p ,
Asymptotics for Generalized Nonlinear Maxwell Models
Corollary 1. The minimal constant C for which condition (5.8) is satisfied is |u 1 (x, 0) − u 2 (x, 0)| u 1 (x, 0) − u 2 (x, 0)| , = C0 = sup p p x x x≥0 and the following estimate holds: u 1 (x, t) − u 2 (x, t) ≤ e−t (1−λ( p)) u 1 (x, 0) − u 2 (x, 0) p p x x
613
(5.11)
(5.12)
for any p > 0. Proof. It follows directly from Lemma 5.2.
We note that a result similar to Lemma 5.2 was first obtained in [8] for the inelastic Boltzmann equation. Its corollary in the form similar to (5.12) for Eq. (2.10) was stated later in [10] and was interpreted there as “the contraction property of the Boltzmann operator” (note that the left hand side of Eq. (5.12) can be understood as a non-expansive distance between any two solutions). However, independently of the terminology, the key reason for estimates (5.9)-(5.12) is the L-Lipschitz property of the operator as defined in (5.4). It is remarkable that the large time asymptotics of u(x, t), satisfying the problem (3.7) with such , can be explicitly expressed through spectral characteristics of the linear operator L. Hence, in order to study the large time asymptotics of u(x, t) in more detail, we distinguish two different kinds of asymptotic behavior: 1) convergence to stationary solutions, 2) convergence to self-similar solutions provided condition (c), of the main properties on , is satisfied. Case 1) is relatively simple. Any stationary solution u(x) ¯ of the problem (3.7) satisfies the equation (u) ¯ = u, ¯
u¯ ∈ C(R+ ), u ¯ ≤ 1.
(5.13)
If the stationary solution u(x) ¯ does exist (note, for example, that (0) = 0 and (1) = 1 for given in Eqs. (3.2)) then the large time asymptotics of some classes of initial data u 0 (x) in (3.7) can be studied directly on the basis of Lemma 5.2. It is enough to assume that |u 0 (x) − u(x)| ¯ satisfies (5.8) with p such that λ( p) < 1. Then u(x, t) → u(x) ¯ as t → ∞, for any x ≥ 0. This simple consideration, however, does not answer at least two questions: A) What happens with u(x, t) if the inequality (5.8) for |u 0 (x) − u(x)| ¯ is satisfied with such p that λ( p) > 1? B) What happens with u(x, t) for large x ? (Note that the estimate (5.9) becomes trivial if x → ∞.) In order to address these questions we consider a special class of solutions of Eq. (3.7), the so-called self-similar solutions. Indeed property (c) of shows that Eq. (3.7) admits a class of formal solutions u s (x, t) = w(x eµ∗ t ) with some real µ∗ . It is convenient for our goals to use a terminology that slightly differs from the usual one.
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Definition 5.3. The function w(x) is called a self-similar solution associated with the initial value problem (3.7) if it satisfies the problem µ∗ Dw + w = (w), D = x
∂ , ∂x
w ≤ 1.
(5.14)
Note that the convergence of solutions u(x, t) of the initial value problem (3.7) to a stationary solution u(x) ¯ can be considered as a special case of the self-similar asymptotics with µ∗ = 0. Under the assumption that self-similar solutions exist (the existence is proved in the next section), we prove a fundamental result on the convergence of solutions u(x, t) of the initial value problem (3.7) to self-similar ones (sometimes called in the literature self-similar stability). Lemma 5.4. We assume that i) for some µ∗ ∈ R, there exists a classical (continuously differentiable if µ∗ = 0) solution w(x) of the problem (5.14); ii) the initial data u(x, 0) = u 0 in the problem (3.7) satisfies u 0 = w + O(x p ),
u 0 ≤ 1, for p > 0 such that µ( p) < µ∗ , (5.15)
where µ( p) defined in (5.7) is the spectral function associated to the operator L. Then |u(xe−µ∗ t , t) − w(x)| = O(x p )e− pt (µ∗ −µ( p)) ,
(5.16)
and therefore lim u(xe−µ∗ t , t) = w(x),
t→∞
x ≥ 0.
(5.17)
Proof. By assumption, the function u 2 (x, t) = w(x eµ∗ t ) satisfies Eq. (3.7). Let u 1 (x, t) be a solution of the problem (3.7) such that u 1 (x, 0) = u 0 (x). Then, by Lemma 5.2 and by assumption ii) we obtain |u 1 (x, t) − w(xeµ∗ t )| ≤ C x p e−(1−λ( p)) t , for some constant C > 0 and all x ≥ 0, t ≥ 0. In this inequality we can change x to x˜ e−µ∗ t , then |u 1 (xe−µ∗ t , t) − w(x)| ≤ C x p e−( pµ∗ +1−λ( p)) t , where the tildes are omitted. Note that u 1 (x, t) = u(x, t) in the formulation of the lemma and that pµ∗ + 1 − λ( p) = p(µ∗ − µ( p)) in the notation (5.7). Hence, the estimate (5.16) is proved. Equation (5.17) follows from (5.16) since µ∗ < µ( p). So the proof is completed.
Remark 6. Lemma 5.4 shows how to find a part of the domain of attraction of any selfsimilar solution provided the self-similar solution is itself known. It is remarkable that this part of the domain of attraction can be expressed in terms of just the spectral function µ( p), p > 0, defined in (5.7). Generally speaking, the equality (5.17) can be also fulfilled for some other values of p with µ( p) > µ∗ in Eq. (5.15), but, at least, it always holds if µ( p) < µ∗ .
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615
We shall need some properties of the spectral function µ( p), p > 0. Note that λ(0) − 1 , p
µ( p) ≈
p → 0+ ,
(5.18)
i.e. µ( p) > 0 (µ( p) < 0), for small p > 0, if λ(0) > 1 (λ(0) < 1). In the particular case of given by (3.2) and L by (3.12) and (3.13), we obtain λ(0) =
∞
K (a) da =
M
0
j=1
α j , j ≥ 1,
M
α j = 1, α j ≥ 0.
(5.19)
j=1
Note that, for this case λ(0) = 1 if and only if M = 1, i.e. for a linear operator (3.2). It is easy to see that the problem (5.18) with a linear operator given by (3.2) has no solutions (the condition w ≤ 1 is important!), except for the trivial ones w = 0, 1. Having in mind applications to nonlinear operators from (3.2), we assume below that λ(0) > 1. Lemma 5.5. The spectral function µ( p) given in Eqs. (5.7) and (5.6), with any K (a) from property (b), is analytic for Re p > 0. It also has the following properties for real valued p > 0, provided λ(0) > 1: i) µ( p) is positive and unbounded as p → 0+ , with asymptotic behavior given by (5.18); ii) there is not more than one point 0 < p0 < ∞, where the spectral function µ( p) achieves its minimum. Proof. The regularity (analyticity) of µ( p) in the half-space Re p > 0 follows from standard considerations of Laplace transforms in R+ . Next, i) is obvious from (5.18). The statement ii) follows, first, from the convexity of λ( p) since ∞ λ ( p) = K (a)a p (ln a)2 da ≥ 0, (5.20) 0
and from the identity µ ( p) =
ψ( p) , p2
ψ( p) = pλ ( p) − λ( p) + 1.
(5.21)
We note that ψ( p) in (5.21) is a monotone increasing function of p,(ψ = pλ ≥ 0) and therefore it has not more than one zero, at say, p = p0 > 0. Then p = p0 is also a minimum point for µ( p) since from Eq. (5.18) µ( p) → +∞ as p → 0 and thus µ ( p) < 0 for p → 0. This completes the proof.
The following corollaries are readily obtained from Lemma 5.5 under its assumptions. Corollary 2. The spectral function µ( p) is always monotone decreasing in the interval (0, p0 ), and µ( p) ≥ µ( p0 ) for 0 < p < p0 . This implies that there exists a unique inverse function p(µ) : (µ( p0 ), +∞) → (0, p0 ), monotone decreasing in its domain of definition. Proof. It follows immediately from Lemma 5.5, part ii) and its proof.
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µ(p)
(a)
µ(p)
0
(b)
p0
p
p
0 µ(p)
µ(p)
(c)
(d)
p0 0
0
p
p
p0
Fig. 1. Possible profiles of the spectral function µ( p)
Corollary 3. There are precisely four different kinds of qualitative behavior of µ( p) shown on Fig. 1 (the intermediate case with µ( p0 ) = 0 is considered as a coincidence of Fig. 1 (c) and Fig. 1(d)). Proof. There are two options: µ( p) is a monotone decreasing function (Fig. 1 (a)) or µ( p) has a minimum at p = p0 (Fig. 1 (b,c,d)). In the first case µ( p) > 0 for all p > 0 since µ( p) > −1/ p, and therefore lim p→∞ µ( p) ≥ 0. The asymptotics of λ( p) (5.6) is clear: ∞ (1) λ( p) −−−→ λ∞ ∈ R+ if 1+ K (a) da = 0; (5.22) p→∞ ∞ (5.23) (2) λ( p) −−−→ ∞ if 1+ K (a) da > 0. p→∞
In the case (1) when µ( p) → ∞ as p → 0, two possible pictures (with and without minimum) are shown on Fig. 1 (b) and Fig. 1 (a) respectively. In case (2), from Eq. (5.6) it is clear that λ( p) grows exponentially for large p, therefore µ( p) → ∞ as p → ∞. Then the minimum always exists and we can distinguish two cases: µ( p0 ) < 0 (Fig. 1 (d) and µ( p0 ) > 0 (Fig. 1 (c))
Figure 1 gives a clear graphic representation of the domains of attraction of selfsimilar solutions (Lemma 5.4): it is sufficient to draw the line µ( p) = µ∗ = constant, and to consider a p such that the graph of µ( p) lies below this line. Therefore, the following corollary follows directly from the properties of the spectral function µ( p), as characterized by the behaviors in Fig. 1, where we assume that µ( p0 ) = 0 for p0 = ∞, for the case shown on Fig. 1 (a).
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617
Corollary 4. Any self-similar solution u s (x, t) = w(xeµ∗ t ) with µ( p0 ) < µ∗ < ∞ has a non-empty domain of attraction, where p0 is the unique (minimum) critical point of the spectral function µ( p). Proof. We use Lemma 5.4 part ii) on any initial state u 0 = w + O(x p ) with p > 0 such that µ( p0 ) ≤ µ( p) < µ∗ . In particular, Eqs. (5.16) and (5.17) show that the domain of attraction of w(xeµ∗ t ) contains any solution to the initial value problem (3.7) with the initial state as above.
The inequalities of the kind u 1 − u 2 = O(x p ) for any p > 0 such that µ( p) < µ∗ , for any fixed µ∗ ≥ µ( p0 ) play an important role for the self-similar stability. We can use specific properties of µ( p) in order to express such inequalities in a more convenient form. Lemma 5.6. Let the conditions of Lemma 5.5 be satisfied. Then, for any given µ∗ ∈ (µ( p0 ), ∞) and u 1,2 (x) such that u 1,2 < ∞, the following two statements are equivalent: i) There exists p > 0 such that u 1 − u 2 = O(x p ),
with µ( p) < µ∗ .
(5.24)
ii) There exists ε > 0 such that u 1 − u 2 = O(xp(µ∗ )+ε ),
(5.25)
where p(µ) is the inverse to µ( p) function, as defined in Corollary 2. Proof. Let property i) hold, then recall from Corollary 2 that µ( p) is monotone on the interval 0 < p ≤ p0 , so its inverse function p(µ) is defined uniquely. It is clear, as it can be seen in from Fig. 1, that the condition µ( p) < µ∗ is satisfied only for some p > p(µ∗ ), therefore inequality (5.25) with ε = p − p(µ∗ ) follows directly from (5.24). Conversely, if ii) holds, first note that for any pair of uniformly bounded functions (note that u 1,2 < ∞ by assumption) which satisfy inequality |u 1 (x) − u 2 (x)| < C xq ,
C = const.,
for some q > 0, then the same inequality holds with any p such that 0 < p < q and perhaps another constant. Therefore, if the condition (5.25) is satisfied, then one can always find a sufficiently small 0 < ε1 ≤ ε such that taking p = p(µ∗ ) + ε1 , condition (5.24) is fulfilled. This completes the proof.
Finally, to conclude this section, we show a general property of the initial value problem (3.7) for any non-linear operator satisfying conditions (a) and (b) given in the beginning of this section. This property gives the control to the point-wise difference of any two rescaled solutions to (3.7) in the unit sphere of B, whose initial states differ by O(x p ). It is formulated as follows. Lemma 5.7. Consider the problem (3.7), where satisfies the conditions (a) and (b). Let u 1,2 (x, t) be two solutions satisfying the initial conditions u 1,2 (x, 0) = u 1,2 0 (x) such that 1 2 p u 1,2 0 ≤ 1, u 0 − u 0 = O(x ),
p > 0,
(5.26)
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then, for any real µ∗ , µ∗ (x, t) = u 1 (xe−µ∗ t , t) − u 2 (xe−µ∗ t , t) = O(x p )e− pt[µ∗ −µ( p)]
(5.27)
and therefore lim µ∗ (x, t) = 0,
t→∞
x ≥ 0,
(5.28)
for any µ∗ > µ( p). Proof. The proof is a repetition of the arguments that led to Eq. (5.16). In particular, we obtain from Eqs. (4.5), (5.9) the estimate |u 1 (x, t) − u 2 (x, t)| = x p et[λ( p)−1]
u 10 − u 20 xp
and then change x to xe−µ∗ t . This leads to Eq. (5.27) in the notation (5.7) and this completes the proof.
Remark 7. There is an important point to understand here: Lemmas 5.2 and 5.7 hold for any operator that satisfies just the two properties (a) and (b) stated at the beginning of this section. It says that, in some sense, a distance between any two solutions with initial conditions satisfying Eqs. (5.26) tends to zero as t → ∞, i.e. non-expansive distance. Such terminology and corresponding distances were introduced in [19] for the elastic Maxwell-Boltzmann with finite initial energy, and used in specific forms of inelastic binary Maxwell-Boltzmann models in [3,24]. It should be pointed out, however, that this contraction property may not say much about large time asymptotics of u(x, t), unless the corresponding self-similar solutions are known. Therefore one must study the problem of existence of self-similar solutions, which is considered in the next section. 6. Existence of Self-Similar Solutions A goal is to study problem (5.14). Theorem 6.1 below shows a criterion for existence and uniqueness of self-similar solutions, in the sense of Definition 5.3 for any operator that satisfies just conditions (a) and (b). Then Theorem 6.2 follows, showing a general criterion to self-similar asymptotics of the problem (3.7) for any operator satisfying conditions (a), (b) and (c). We consider Eq. (5.14) written in the form µ∗ xw (x) + w(x) = g(x),
g = (w), µ∗ ∈ R,
(6.1)
and, assuming that w < ∞, transform this equation to the following integral form. 1−µ∗
Multiplying the equation by the integrating factor µ∗ −1 x µ∗ , integrating and changing coordinates in the resulting right-hand side integral, it is easy to verify that the resulting integral equation reads 1 g(xτ µ∗ ) dτ. (6.2) w(x) = 0
We prove the following result formulated in terms of the spectral function µ( p) from (5.7).
Asymptotics for Generalized Nonlinear Maxwell Models
619
Theorem 6.1. Consider Eq. (6.1) with arbitrary µ∗ ∈ R and the operator satisfying the conditions (a) and (b) from Sect. 5. Assume that there exists a continuous function w0 (x), x ≥ 0, such that i) ii)
w0 ≤ 1 and
1
g0 (xτ µ∗ ) dτ = w0 (x) + O(x p ),
g0 = (w0 ),
(6.3)
0
with some p > 0 satisfying the inequality µ( p) < µ∗ . Then, there exists a classical solution w(x) of Eq. (6.1) such that w ≤ 1, w(x) = w0 (x) + O(x p ),
with the same p > 0.
(6.4)
The solution is unique in the class of continuous functions satisfying conditions w(x) = w0 (x) + O(x p1 ),
w < ∞,
(6.5)
with any positive p1 such that µ( p1 ) < µ∗ . Proof. The existence is proven by the following iteration procedure. We choose an initial approximation w0 ∈ U such that w0 ≤ 1 and consider the iteration scheme 1 wn+1 (x) = gn (xτ µ∗ ) dτ, gn = (wn ), n = 0, 1, . . . . (6.6) 0
Then, because of property (a) of , wn ≤ 1 for all n ≥ 1 and 1 |wn+1 (x) − wn (x)| ≤ |gn (xτ µ∗ ) − gn−1 (xτ µ∗ )| dτ,
n ≥ 1.
0
By using the inequality (5.3) (i.e. property (b) of ), we control the right hand side of the above inequality by 1 |wn+1 (x) − wn (x)| ≤ L(|wn − wn−1 |)(xτ µ∗ ) dτ 0
=
1
dτ 0
∞
K (a)|wn − wn−1 |(axτ µ∗ ) da.
(6.7)
x ≥ 0, C = constant.
(6.8)
0
Next, by assumption ii), initially |w1 (x) − w0 (x)| ≤ Cx p ,
Then, recalling the definition for λ( p) given in (5.6), we can control the right-hand side of (6.7) by 1 ∞ λ( p) p p x K (a)a da τ pµ∗ dτ = x p , pµ∗ > pµ( p) > −1. 1 + pµ∗ 0 0 Therefore, we estimate the left hand side of (6.7) by |wn+1 (x) − wn (x)| ≤ Cγ n ( p, µ∗ ) x p ,
γ ( p, µ∗ ) =
λ( p) . 1 + pµ∗
(6.9)
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Then, λ( p) > 0 and µ( p) = there exists a point-wise limit
λ( p)−1 p
< µ∗ imply 0 < γ ( p, µ∗ ) < 1. Therefore,
w(x) = lim wn (x)
(6.10)
n→∞
satisfying the inequality |w(x) − w0 (x)| ≤
∞
|wn+1 − wn (x)| ≤
n=0
C xp. 1 − γ ( p, µ∗ )
(6.11)
Estimate (6.9) with γ < 1 shows that the convergence wn (x) → w(x) is uniform on any interval 0 ≤ x ≤ R, for any R > 0. Therefore w(x) is a continuous function, moreover w ≤ 1 since wn ≤ 1 for all n = 0, 1, . . .. The next step is to prove that the limit function w(x) from (6.10) satisfies Eqs. (6.1), or equivalently, (6.2). We note that ∞ K (a)|wn+1 (ax) − wn (ax)| da |gn+1 (x) − gn (x)| ≤ (6.12) 0 ≤ C λ( p)γ n ( p, µ∗ ) x p . Therefore gn (x) → g(x), where g(x) ∈ C(R+ ) and g ≤ 1, since gn ≤ 1 for all n. In addition, from the continuity of the operator (u) for u ≤ 1 follows that g = (w), and the transition to the limit in the right hand side of Eq. (6.2) is justified since gn ≤ 1. Hence, w(x) satisfies Eq. (6.2). When µ∗ = 0 one also needs to check that Eq. (6.1) is satisfied. We note that, for any continuous and bounded w0 (x), all functions wn (x), n ≥ 1, are differentiable for x > 0 and their derivatives wn (x) satisfy the equations (see Eqs. (6.1), (6.6)) µ∗ x wn (x) = wn (x) + gn−1 (x),
n = 1, 2, . . . .
Hence, − wn ) = (wn − wn+1 ) + (gn − gn−1 ), µ∗ x (wn+1
and, by using inequalities (6.9), (6.12), we obtain − wn | ≤ C γ p−1 ( p, µ∗ ) (γ ( p, µ∗ ) + λ( p)) x p , |µ∗ | |wn+1
n ≥ 1.
Therefore the sequence of derivatives {wn (x), n = 1, . . .} converges uniformly on any interval ε ≤ x ≤ R. Hence, the limit function w(x) from (6.10) is differentiable for x > 0, w (x) = O(x p−1 ),
p > 0,
(6.13)
and the equality (6.1) is also satisfied for µ∗ = 0. Finally we note that the condition of convergence 0 < γ ( p, µ∗ ) < 1 is equivalent to the condition µ( p) < µ∗ (see Eq. (5.7)) that has already appeared in Lemma 5.4. It remains to prove the statement concerning the uniqueness of the solution to (6.1). If there are two solutions w 1,2 satisfying (6.5), with p1 = p 1,2 respectively, then the integral equation (6.2) yields 1 |w 1 (x) − w 2 (x)| ≤ |g 1 (xτ µ∗ ) − g 2 (xτ µ∗ )| dτ, g 1,2 = (w 1,2 ). (6.14) 0
Asymptotics for Generalized Nonlinear Maxwell Models
621
Since w 1,2 ≤ 1, we obtain |w 1 (x) − w 2 (x)| ≤ C xq ,
q = min( p 1 , p 2 ),
where obviously µ(q) < µ∗ . Then we again apply the inequality (5.3) to the integral in Eq. (6.2) and get the new estimate |w1 (x) − w 2 (x)| ≤ C1 xq ,
C1 = γ (q, µ∗ )C < C.
By repeating the same considerations as in the existence argument, it follows that |w 1 (x) − w 2 (x)| ≤ Cγ n (q, µ∗ )xq ,
with γ (q, µ∗ ) < 1,
for any integer n ≥ 0. Therefore w1 (x) ≡ w 2 (x) and the proof is completed.
Now we can combine Lemma 4.1 with Theorem 6.1 and prove the general statement related to the self-similar asymptotics for the problem (3.7). Having in mind applications to operators from (3.2), we assume below that λ(0) > 1 in Eqs. (5.6), (5.7). Then Lemma 5.5 and its corollaries show that the spectral function µ( p) has a qualitative behavior shown on Fig. 1. In particular, there exists a p0 > 0 such that µ( p) > µ( p0 ) = inf p>0 µ( p), including formally the case p0 = ∞ (Fig. 1a). Theorem 6.2. Let u(x, t) be a solution of the problem (3.7) with initial data u 0 ≤ 1 and satisfying conditions (a), (b) with λ(0) > 1 in the notation of Eqs. (5.6), and (c) from Sect. 5. We assume that 1) There exists a continuous function y(x), x ≥ 0, such that 1 y ≤ 1 g(xτ µ( p) ) = y(x) + O(x p+ε1 ) dτ,
g = (y),
(6.15)
0
for some p ∈ (0, p0 ) and ε1 > 0; 2) The initial state satisfies u 0 (x) = y(x) + O(x p+ε2 )
(6.16)
for the same p and some ε2 > 0. Then, i) there exists a self-similar solution u p (x, t) = w(x eµ( p)t ) of Eq. (3.7) such that w ≤ 1,
w(x) = y(x) + O(x p+ε1 ) ;
(6.17)
ii) lim u(x e−µ( p)t , t) = w(x),
t→∞
x ≥ 0,
(6.18)
where the convergence is uniform on any bounded interval in R+ and u(x e−µ( p)t , t) − w(x) = O(x p+ε e−β( p,ε)t ),
(6.19)
with β( p, ε) = ( p + ε)( µ( p) − µ( p + ε) ) > 0 and ε = min{ε1 , ε2 }. iii) If the condition 1) for y(x) holds simultaneously with two different values p1,2 ∈ (0, p0 ), with p1 > p2 , then w(x) = const. (trivial self-similar solutions) for p2 .
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Proof. Since λ(0) > 1, we can apply Lemma 5.6 and see that the condition 1) for y(x) coincides with the conditions of Theorem 6.1 for w0 (x) provided the obvious changes of notation are made in (6.3). The w(x) in i) is the function constructed in Theorem 6.1 for w0 = y and µ∗ = µ( p). We note that, in accordance with Eqs. (6.16) and (6.17), u 0 (x) − w(x) = O(x p+ε ),
with ε = min{ε1 , ε2 }.
Then, statement ii) follows directly from Lemma 5.4 since µ( p) is monotone decreasing on (0, p0 ). It remains to prove iii). If there are two different values p1 > p2 for which Eq. (6.15) is satisfied, then there are two different self-similar solutions wi (x eµ( pi )t ), i = 1, 2. It follows from (6.17) that w1,2 ≤ 1, with w1 (x) = w2 (x) + O(x p2 +ε ), where ε > 0 can be made as small as we want. Then, recalling that w1,2 (x) are initial data for two different self-similar solutions, apply Eq. (5.16) with w(x) = w2 (x), u(x, t) = w1 (x eµ( p1 )t ), µ∗ = µ( p2 ) and p = p2 + ε to obtain w1 (x e(µ( p1 )−µ( p2 ))t ) − w2 (x) = O(x p2 +ε e−β( p2 ,ε)t ) in the notation of Eq. (6.19). Hence, w2 (x) = w1 (0) = constant since w1 (x) is continuous at x = 0, and this completes the proof of Theorem 6.2.
Thus we obtain a general criterion of self-similar asymptotics of u(x, t) for a given initial condition u 0 (x). It is important that the case of non-trivial (w = const. in (6.18)) asymptotics can appear only if p ∈ (0, p0 ) is the maximal number for which Eq. (6.15) is satisfied. For brevity, we do not consider the case with λ(0) ≤ 1 (a criterion for selfsimilar asymptotics in this case can also be obtained by a combination of Theorem 6.1 and Lemma 5.4 ), since we do not know any specific application of this case. On the other hand, it was already noted above that all conditions of Theorem 6.2 (in particular λ(0) > 1) for operators from (3.2) are satisfied in the non-linear case (N ≥ 2). Applications to these type of operators are considered in Sects. 8 and 9. 7. Properties of Self-Similar Solutions We consider the integral equation (6.2) written as w = µ∗ (w) =
1 0
g(xτ µ∗ ) dτ s,
g = (w), µ∗ ∈ R.
(7.1)
The following two properties of w(x) are independent of the specific form (3.2) of : Lemma 7.1. i) If there exists a closed subset U ⊂ U of the unit ball U in B, as given in (3.15), such that µ∗ (U ) ⊂ U for any µ∗ ∈ R, and for some function w0 ∈ U the conditions of Theorem 6.1 are satisfied, then w ∈ U , where w is constructed by the iterative scheme as defined in (6.6). ii) If the conditions of Theorem 6.1 for are satisfied and, in addition, (1) = 1, then the solution w∗ = 1 of Eq. (7.1) is unique in the class of functions w(x) satisfying conditions w < ∞,
w(x) = 1 + O(x p ),
µ( p) < µ∗ .
(7.2)
Asymptotics for Generalized Nonlinear Maxwell Models
623
Proof. The first statement follows from the iteration scheme (6.6) with w0 ∈ U . Then, by assumption i), w0 ∈ U for all integers n ≥ 1, and wn → w ∈ U . The second statement follows from the obvious fact that w∗ = 1 satisfies Eq. (7.1) with any µ∗ ∈ R provided (1) = 1 and from the uniqueness of w(x) stated in Theorem 6.1. This completes the proof.
The statement ii) can be interpreted as a necessary condition for existence of “nice” non-trivial (w = const.) solutions of Eq. (7.1): if there exists a non-trivial solution w(x) of Eq. (7.1) with (1) = 1 and with some µ∗ ∈ R, such that w = 1,
w = 1 + O(x p ),
p > 0,
then
µ∗ ≤ µ( p).
(7.3)
Let us consider now the specific class (3.2)–(3.3) of operators , with functions u(x) satisfying the condition u(0) = 1. That is, u(0, t) = 1 for the solution u(x, t) of the problem (3.7). Since the operators (3.2) are invariant under dilation transformations (5.4) (property (c), Sect. 5), the problem (3.7) with the initial condition u 0 (x) satisfying u(0) = 1, u 0 = 1;
u 0 (x) = 1 − βx p + · · · , x → 0,
(7.4)
can be always reduced to the case β = 1 by the transformation x = xβ 1/ p . Moreover, the whole class of operators (3.2), with different kernels A j (a1 , . . . , a j ), j = 1, 2, . . ., is invariant under transformations x˜ = x p , p > 0. The result of such a transformation acting on is another operator of the same class (3.2) with ker˜ p) (see the end of this nels A˜ j (a1 , . . . , a j ) and the corresponding spectral function µ( section). Therefore, we fix the initial condition (7.4) with β = 1 and transform the function (7.4) and Eq. (3.7) to new variables x˜ = x p . Then, we omit the tildes and reduce the problem (3.7), with initial condition (7.4) to the case β = 1, p = 1. We study this case in detail and formulate afterwards the results in terms of initial variables. Our goal now is to apply the general theory (in particular, Theorem 6.2) to the particular case where the initial data u 0 (x) satisfies, u 0 = 1, u 0 (x) = 1 − x + O(x1+ε ), x → 0,
(7.5)
with some ε > 0. We also assume that the spectral function µ( p), given by Eqs. (5.6), (5.7) and (3.13), corresponds to one of the four cases shown on Fig. 1 with a unique minimum (infimum) achieved at p0 > 0. Let us consider a typical function u 0 = e−x satisfying (7.5) and investigate the criterion (6.15) from Theorem 6.2 for this particular function. Theorem 7.2. Whenever µ(1) > −1, the estimate µ( p) (e−x ) − e−x = 0(x p+ε ),
ε > 0,
(7.6)
holds for positive p if and only if p ≤ 1. Proof. In order to prove (7.6), we assume that µ( p) > −1,
(7.7)
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and investigate the structure of µ (e−x ) for any µ > −1. Its explicit formula reads ⎡ ⎤ j M α j j A j (a1 , . . . , a j )Iµ ⎣x ak ⎦ da1 . . . da j , (7.8) µ (e−x ) = R+
j=1
k=1
with Iµ (y) =
1
µ
e−yτ dτ,
M
µ ∈ R, y > 0,
0
α j = 1.
(7.9)
j=1
The following lemma holds. Lemma 7.3. If µ > −1, y ≥ 0, then 0 < Iµ (y) ≤ 1, and Iµ (y) = e
−y
µy µ2 1+ + rµ (y), 1+µ 1+µ
where 0 ≤ rµ (y) ≤ B(y, µ) with ⎧ 2 −1 ⎪ ⎨y (2µ + 1) 2 B(y, µ) = 2y (− ln y + y) 1 ⎪ ⎩ (2−|µ|−1 ) |µ| y |µ|
if µ > − 21 , if µ = − 21 ; if µ ∈ −1, − 21 .
(7.10)
(7.11)
Proof. We consider the integral Iµ (y), integrate twice by parts and obtain Eq. (7.10) with 1 µ 2 rµ (y) = y e−yτ τ 2µ dτ. 0
If µ > −1/2 then the estimate (7.11) is obvious. Otherwise we transform ry (y) into the form ∞ 1 1 1 |µ| 1 rµ (y) = y e−τ τ 1− |µ| dτ, µ≤− , |µ| 2 y so that the estimate (7.11) with µ ∈ (−1, − 21 ) is also clear. Finally, in the case µ = − 21 we obtain r−1/2 (y) = 2y E(y), 2
with E(y) =
∞ y
e−τ dτ. τ
Hence, rewriting
1 1 e−τ dτ dτ E(y) = dτ + + (e−τ − 1) τ τ τ 1 y y y dτ + C, = − ln y + (1 − e−τ ) τ 0 ∞
Asymptotics for Generalized Nonlinear Maxwell Models
625
where C is the well-known integral ∞ −τ 1 ∞ e dτ −τ (e − 1) e−τ ln τ dτ = −γ , C= dτ + = τ τ 1 0 0 where γ = (1) 0.577 is the Euler constant. Therefore, since C < 0 and (1−e−τ ) ≤ τ , then the term E(y) is estimated by E(y) ≤ − ln y + y. Hence, we obtain the estimate (7.11) for µ = −1/2 and the proof is completed.
We can characterize now possible values of p > 0 for which Eq. (7.6) holds. From the previous lemma we obtain Iµ (y) = 1 −
y + O(y 1+ε ), 1+µ
ε > 0, y → 0,
provided µ > −1. Therefore µ (e−x ) − e−x = θ (µ)x + O(x1+ε ), where j M 1 θ (µ) = 1 − α j j A j (a1 , . . . , a j ) ak da1 . . . da j . 1+µ R+ j=1 k=1 We recall that kernels A j (a1 , . . . , a j ), j = 1, . . . , M, are assumed to be symmetric functions of their arguments. Then ∞ 1 λ(1), λ( p) = θ (µ) = 1 − K (a)a p da, (7.12) 1+µ 0 where K (a) is given in Eqs. (3.13). Recalling the definition of µ( p) in (5.7), we obtain µ( p) (e−x ) − e−x = θ [µ( p)]x + O(x1+ε ),
(7.13)
where θ [µ( p)] =
µ( p) − µ(1) . 1 + µ( p)
(7.14)
It follows from Eqs. (7.13), (7.14) that the condition (7.6) is fulfilled if and only if p ≤ 1. Thus, Theorem 7.2 is proved.
Hence all conditions of Theorem 6.2 for y = u 0 = e−x are fulfilled for p satisfying µ( p) > −1 p ≤ 1 and 0 < p < p0 , where p0 > 0 is a unique critical point of µ( p) (Fig. 1). However, Theorem 6.2 states that only a maximal point pmax of this set can lead to a non-trivial self-similar solution w(x) = const. Such a point pmax = 1 does exist only in the case when p0 > 1. Then, automatically µ( p) > µ( p0 ) > −
1 > −1 p0
for all p > 0.
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Finally, it remains to show that the iteration scheme wn+1 = µ∗ (wn ),
w0 = e−x , n = 0, 1, . . . ,
µ∗ = µ(1),
(7.15)
leads to a non-trivial solution w(x) = lim wn (x) = const.,
x ≥ 0.
n→∞
(7.16)
According to Theorem 6.1, w(x) is a continuously differentiable function on [0, ∞) and w(x) = w0 (x) + O(x1+ε ).
(7.17)
Therefore w (0) = w0 (0) = −1 and so w(x) = const. We shall study w(x) in more detail in this section. By Theorem 6.1, w ∈ C1 (R+ ) and satisfies the equation µ∗ x w (x) + w(x) = (x),
µ∗ = µ(1).
(7.18)
The differentiability of w(x) was proved in Theorem 6.1 only for µ = 0, but the proof can be easily extended to the case µ = 0 since w0 = e−x in Eqs. (7.15) has a bounded and continuous derivative. From (7.15) and (7.17) it is clear that the limit function w satisfies 0 ≤ w(x) ≤ 1,
w(0) = 1, w (0) = −1;
(7.19)
and, by considering a sequence of derivatives in Eqs. (7.15), it is easy to see that w (x) ≤ 0,
|w (x)| ≤ 1.
(7.20)
Then, estimates from Theorem 6.1 and Lemma 7.3 yield that w(x) = e−x + O(xπ(µ∗ ) ), where
⎧ ⎪ ⎨2 π(µ∗ ) = 2 − ε with any ε > 0 ⎪ ⎩ 1 |µ∗ |
(7.21)
for µ∗ > − 21 , for µ∗ = − 21 , for − 1 < µ∗ < − 21 .
(7.22)
Hence, we collect all essential properties of w(x) in the following statement. Theorem 7.4. The limiting function w(x) constructed in (7.16) satisfies Eq. (7.1) with µ = µ(1) and Eqs. (7.18), where is given in Eqs. (3.2), µ( p) is defined in Eqs. (5.6), (5.7) and (3.13). The conditions (7.19), (7.20), (7.21) are fulfilled for w(x). Moreover 1 ≥ w(x) ≥ e−x ,
lim w(x) = 0,
x→∞
and there exists a generalized non-negative function R(τ ), τ ≥ 0, such that ∞ ∞ ∞ R(τ )e−τ x dτ , R(τ ) dτ = R(τ )τ dτ = 1. w(x) = 0
0
0
(7.23)
(7.24)
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627
Proof. It remains to prove (7.23) and (7.24). First we note that Eq. (7.1) is obtained as the integral form of Eq. (6.1). Then, the identity µ x v (x) + v(x) = (v) + (x), where (x) = µ x v (x) + v(x) − (v),
(7.25)
is fulfilled for any function v(x), and the integral form of this identity reads 1 v(x) = µ (v) + (xτ µ ) dτ. 0
Hence, if (x) ≤ 0 then v ≤ µ (v) and vice-versa. We intend to prove that (x) ≤ 0 for v = e−x . If so, then wn+1 (x) ≥ wn (x) at any x ≥ 0 in the sequence (7.15) generated by the corresponding iteration scheme with w0 = e−x , and obviously w(x) ≥ e−x . Indeed, by substituting v = e−x in Eqs. (7.25) we obtain, for µ = µ(1), (x) =
M
α j j (x),
j=1
where using (7.9),
⎛
j (x) =
j
R+
A j (a1 , . . . , a j )P ⎝x,
j
⎞ ak ⎠ da1 , . . . , da j ,
with
k=1
P(x, s) = e−x [1 − (s − 1)x] − e−sx ≤ 0. We note that P(x, s) ≤ 0 for any real s and x, since ey ≤ 1 + y for any real y. Then n (x) ≤ 0, and so also (x) ≤ 0. Hence, the inequality in (7.23) is proved. In order to prove the limiting identity (7.23) we denote w∞ = lim w(x). x→∞
Such a limit exists since w(x) is a monotone function. From Theorem 7.4, the nice properties of w(x) allow to take the limit in both sides of Eq. (7.1). Then w∞ =
M
n αn w∞ ,
n=1
M
αn = 1, αn ≥ 0,
n=1
and therefore we obtain ∞
n−1 αn w∞ (1 − w∞ ) = 0.
n=2
This equation has just two non-negative roots: w∞ = 0 and w∞ = 1. The root w∞ = 1 is possible only if w(x) = 1 for all real x. Since by (7.16) this is not the case, then w∞ = 0.
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A. V. Bobylev, C. Cercignani, I. M. Gamba
It remains to prove the integral representation (7.24). In order to do this we denote by U the set of Laplace transforms of probability measures in R+ , i.e., u ∈ U if there exists a generalized function F(τ ) ≥ 0 such that ∞ ∞ F(τ )e−xτ dτ, F(τ ) dτ = 1. u(x) = 0
0
Then e−x ∈ U (with F = δ(τ − 1)) and it is easy to check that µ (U ) ⊂ U for any real µ. The set U is closed with respect to uniform convergence in R+ (see, for example, [18]). Thus, according to Lemma 7.1, [i], w ∈ U . On the other hand, it is already known from (7.19) that w (0) = −1. Hence, the corresponding function R(τ ) has a unit first moment [18]. This completes the proof of Theorem 7.4.
The integral representation (7.24) is important for the properties of the corresponding distribution functions satisfying Boltzmann-type kinetic equations. Now it is easy to return to initial variables with u 0 given in Eq. (7.4) and to describe the complete picture of the self-similar relaxation for the problem (3.7). In order to explain how this picture is obtained we formulate below a related result to this section. Lemma 7.5. Let u(x, t) be a solution of the problem (3.7) with a non-linear operator from (3.2) (N ≥ 2) and initial condition u 0 (x) satisfying (7.5). Then u(x, t) has a non-trivial self-similar asymptotics lim u(x e−µ(1)t , t) = w(x),
t→∞
x ≥ 0,
(7.26)
provided p0 > 1, where p0 > 0 is a critical point of µ( p) given in Eqs. (5.6), (5.7) and (3.11). The function w(x) is described in Theorem 7.4. Proof. Since all conditions of Theorem 6.2 are satisfied for y(x) = e−x , it remains to make sure that w(x) = const. This fact, in turn, follows from Theorem 7.4 and completes the proof.
In order to generalize this statement to more general initial conditions (7.4) with fixed β = 1 and some p = p1 > 0, we just need to pass from x to x p1 in Eqs. (3.2) and (3.7). Then a new spectral function reads (see Eq. (3.13)) µ( ˜ p) =
1 µ( p1 p), p
where µ( p) is the original spectral function. Hence we can apply Lemma 7.5 with µ( ˜ p) and then reformulate the result. This leads to a scenario described in Section 8. 8. Main Results for Fourier Transformed Maxwell Models with Multiple Interactions We consider the Cauchy problem (3.7) with a fixed non-linear operator (3.2) (M ≥ 2) and study the time evolution of u 0 (x) satisfying the conditions u 0 = 1; u 0 = 1 − x p + O(x p+ε ), x → 0,
(8.1)
Asymptotics for Generalized Nonlinear Maxwell Models
629
with some positive p and ε. Then there exists a unique classical solution u(x, t) of the problem (3.7), (8.1) such that, for all t ≥ 0, u(·, t) = 1;
u(x, t) = 1 + O(x p ), x → 0.
(8.2)
We explain below the simplest way to analyze this solution, in particular in the case of self-similar asymptotics. Step 1. Consider the linearized operator L given in Eqs. (3.12)–(3.13) and construct the spectral function µ( p) given in Eqs. (5.6)–(5.7). The resulting µ( p) will be of one of four kinds described qualitatively on Fig. 1. Step 2. Find the value p0 > 0 where the minimum (infimum) of µ( p) is achieved. Note that p0 = ∞ just for the case described on Fig. 1 (a), otherwise 0 < p0 < ∞. Compare p0 with the value p from Eqs. (8.1). If p < p0 then the problem (3.7), (8.1) has a self-similar asymptotics (see below). The above consideration shows that two different cases are possible: (1) p ≥ p0 provided p0 < ∞; (2) 0 < p < p0 , that is, µ( p) is monotone decreasing for all 0 < p < p0 . In Case (1) a behavior of u(x, t) for large t may depend strictly on initial conditions. The only general conclusion that can be drawn for the initial data (8.1) with p ≥ p0 is the following: lim u(xe−µt , t) = 1,
t→∞
x ≥ 0,
(8.3)
for any µ > µ( p0 ). This follows from Lemma 5.7 with u (1) = u, u (2) = 1 and from the fact that any such function u 0 (x) satisfies the condition u 0 = 1 + O(x p0 ). Case (2) with 0 < p < p0 in Eqs. (8.1) is more interesting. In this case (assume that p ∈ (0, p0 ) is fixed) there exists a unique self-similar solution u s (x, t) = ψ(xeµ( p)t )
(8.4)
satisfying Eqs. (7.1) at t = 0. We again use Lemma 5.7 with u 1 = u and u 2 = u s and obtain for the solution u(x, t) of the problem (3.7), (8.1): ⎧ ⎪ if µ > µ( p) ⎨1 −µt lim u(xe , t) = ψ(x) if µ = µ( p) (8.5) t→∞ ⎪ ⎩0 if µ( p) > µ > µ( p + δ), with sufficiently small δ > 0. The third equality follows from the fact that u 0 (x) − ψ(x) = O(x p+ε ) and from the equality (see Eqs. (7.23)) lim ψ(x) = 0.
x→∞
We note that ψ(x) = w(x p ), where w(x) has all properties described in Theorem 7.4. The equalities (8.5) explain the exact meaning of the approximate identity, u(x, t) ≈ ψ(xeµ( p)t ),
t → ∞, xeµ( p)t = const.,
(8.6)
that we call self-similar asymptotics. We collect the results in the following statement:
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Proposition 8.1. The solution u(x, t) of the problem (3.7), (8.1), with given in Eqs. (3.2), satisfies either one of the following limiting identities: (1) Eq. (8.3) if p ≥ p0 for the initial data (8.1), (2) Eqs. (8.5) provided 0 < p < p0 . The convergence in Eqs. (8.3), (8.5) is uniform on any bounded interval 0 ≤ x ≤ R, and u(xeµ( p)t , t) − ψ(x) = O(x p+ε )e−β( p,ε)t ,
β( p, ε) = ( p + ε) (µ( p) − µ( p + ε) ),
for 0 < p < p0 and 0 < ε < p0 − p. Proof. It remains to prove the last statement. It follows in both cases from the estimate (5.27) for the remainder term in Lemma (5.7). This completes the proof.
It is interesting that our considerations are the same for both positive and negative values of µ( p). There are, however, certain differences if we want to consider the “pure” large time asymptotics, i.e., the limits (8.3), (8.5) with µ = 0. Then we can conclude that (1) (2)
lim u(x, t) = 1
t→∞
if p ≥ p0 and µ( p0 ) < 0, or 0 < p < p0 and µ( p) < 0;
lim u(x, t) = ψ(x) if 0 < p < p0 and µ( p) = 0.
t→∞
It seems probable that u(x, t) → 0 for large t in all other cases, but our results, obtained on the basis of Lemma 5.7, are not sufficient to prove this. Remark 8. We mention that the self-similar asymptotics becomes more transparent in logarithmic variables ˆ y = ln x, u(x, t) = u(y, ˆ t), ψ(x, t) = ψ(y, t) Then Eq. (8.6) reads ˆ + µ( p)t), u(y, ˆ t) ≈ ψ(y
t → ∞, y + µ( p)t = const.,
(8.7)
i.e., the self-similar solutions are simply nonlinear waves (note that ψ(−∞) = 1, ψ(+∞) = 0) propagating with constant velocities c p = −µ( p) to the right if c p > 0 or to the left if c p < 0. If c p > 0 then the value u(−∞, t) = 1 is transported to any given point y ∈ R when t → ∞. If c p < 0 then the profile of the wave looks more natural for the functions u˜ = 1 − u, ˆ ψ˜ = 1 − ψ. Thus, Eq. (3.7) can be considered in some sense as the equation for nonlinear waves. The self-similar asymptotics (8.7) means a formation of the traveling wave with a universal profile for a broad class of initial conditions. It is a purely non-linear phenomenon, it is easy to see that such asymptotics cannot occur in the particular case (M = 1 in Eqs. (3.2)) of the linear operator .
Asymptotics for Generalized Nonlinear Maxwell Models
631
9. Distribution Functions, Moments and Power-Like Tails We have described above the general picture of the behavior of the solutions u(x, t) to the problem (3.7), (8.1). On the other hand, Eq. (3.7) (in particular, its special cases in [3-11]) was obtained as the Fourier transform of the kinetic equation. Therefore we need to study in more detail the corresponding distribution functions. We assume in this section that u 0 (x) in the problem (3.7) is an isotropic characteristic function of a probability measure in Rd , i.e., f 0 (|v|)e−ik·v dv, k ∈ Rd , x = |k|2 , (9.1) u 0 (x) = F[ f 0 ] = Rd
where f 0 is a generalized positive function normalized such that u 0 (0) = 1 (distribution function). Let U be a closed unit ball in the B = C(R+ ) as defined in (3.9). Then, as was already mentioned at the end of Sect. 4, the set U ⊂ U of isotropic characteristic functions is a closed convex subset of U . Moreover, (U ) ⊂ U if is defined in Eqs. (3.2). Hence, we can apply Lemma 4.3 and conclude that there exists a distribution function f (v, t), v ∈ Rd , satisfying Eq. (2.1), such that u(x, t) = F[ f (·, t)],
x = |k|2 ,
(9.2)
for any t ≥ 0. A similar conclusion can be obtained if we assume the Laplace (instead of Fourier) transform in Eqs. (9.1). Then there exists a distribution function f (v, t), v > 0, such that ∞ u(x, t) = L[ f (·, t)] = f (v, t)e−xv dv, u(0, t) = 1, x ≥ 0 , t ≥ 0, (9.3) 0
where u(x, t) is the solution of the problem (3.7) constructed in Theorem 4.2 and Lemma 4.3. We remind the reader that the point-wise convergence u n (x) → u(x), x ≥ 0, where {u n , n = 1, 2, . . .} and u are characteristic functions (or Laplace transforms) is sufficient for the weak convergence of the corresponding probability measures [18]. Hence, all results of pointwise convergence related to self-similar asymptotics can be easily re-formulated in corresponding terms for the related distribution functions (or, equivalently, probability measures). The approximate equation (8.6) in terms of distribution functions (9.2) reads d
1
1
f (|v|, t) e− 2 µ( p) t F p (|v|e− 2 µ( p) t ), t → ∞, |v|e− 2 µ( p) t = const.,
(9.4)
where F p (|v|) is a distribution function such that for x = |k|2 , ψ p (x) = F[F p ],
(9.5)
with ψ p given in Eq. (8.4) (the notation ψ p is used in order to stress that ψ defined in (8.4), depends on p). The factor 1/2 in Eqs. (9.4) is due to the notation x = |k|2 . Similarly, for the Laplace transform (9.3), we obtain f (v, t) e−µ( p)t p (ve−µ( p)t ),
t → ∞, ve−µ( p)t = const.,
(9.6)
where ψ p (x) = L[ p ].
(9.7)
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A. V. Bobylev, C. Cercignani, I. M. Gamba
In the space of distributions, the approximate relation is weak in the sense of distributions, i.e. the classical approximation concept of real valued expression obtained after integrating by test functions. The positivity and some other properties of F p (|v|) follow from the fact that ψ p (x) = w p (x p ), where w p (x) satisfies Theorem 7.4. Hence ∞ ∞ ∞ p ψ p (x) = R p (τ )e−τ x dτ , dτ R p (τ ) dτ = R p (τ )τ dτ = 1, (9.8) 0
0
0
where R p (τ ), τ ≥ 0, is a non-negative generalized function (of course, both ψ p and R p depend on p). We stress here that if u 0 (x) is a characteristic function given in Eq. (9.1), and condition (8.1) is fulfilled, then p ≤ 1 in Eqs. (8.1). In addition, the case p > 1 is impossible for the non-negative initial distribution f 0 in Eqs. (9.1) since u (0) = −Cd f 0 (|v|)|v|2 dv, (9.9) Rd
where Cd > 0 is a constant factor that depends only on the space dimension for the problem. Hence, the self-similar asymptotics (9.4) for any initial data f 0 ≥ 0 occurs if p0 > 1 (see Step 2 at the beginning of Sect. 8). Otherwise it occurs for p ∈ (0, p0 ) ⊂ (0, 1). Therefore, for any spectral function µ( p) (Fig. 1), the approximate relation (9.4) holds for sufficiently small 0 < p ≤ 1. It follows from Eq. (9.9) that, if (8.1) holds, then m2 = f 0 (|v|)|v|2 dv < ∞ Rd
for p = 1, and m 2 = ∞ for p < 1. Similar conclusions can be made for the Laplace transforms (9.3) since in that case ∞ f (v, t)v dv, u (0, t) = − 0
therefore the first moment of f plays the same role as the second moment in the case of Fourier transforms. The positivity of F(|v|) in Eqs. (9.5) and (9.7) follows from the integral representation (9.8) with p ≤ 1. It is well-known that F −1 (e−|k| ) > 0, 2p
L−1 (e−x ) > 0 2p
for any 0 < p ≤ 1 (the so-called stable distributions [18]). Thus, Eqs. (9.8) explain the connection of the self-similar solutions of generalized Maxwell models with stable distributions. We can use standard formulas for the inverse Fourier (Laplace) transforms and denote (d = 1, 2, . . . is fixed) 1 2p M p (|v|) = e−|k| +ik·v dk, (2π )d Rd (9.10) a+i∞ 1 −x p +xv N p (v) = e dx, 0 < p ≤ 1. 2πi a−i∞
Asymptotics for Generalized Nonlinear Maxwell Models
633
Then the self-similar solutions F p and p (distribution functions) given in the right-hand sides of Eqs. (9.6) and (9.8) respectively, satisfy ∞ −d − 1 F p (|v|) = R p (τ )τ 2 p M p (|v|τ 2 p ), dτ, 0
p (v) =
∞
(9.11) R p (τ )τ
− 1p
N p (vτ
− 1p
) dτ,
v ≥ 0, 0 < p ≤ 1.
0
That is, they admit an integral representation through stable distributions. Note that M1 (|v|) is the standard Maxwellian in Rd . The functions N p (v) (9.10) are studied in detail in the literature [18,21]. Thus, for given 0 < p ≤ 1, the kernel R p (τ ), τ ≥ 0, is the only unknown function that is needed to describe the distribution functions F p (|v|) and p (v). Therefore we study R p (τ ) and its s-moments in more detail. It was already noted in Sect. 7 that the general problem (7.1), (7.3), with given 0 < p < p0 , can be reduced to the case p = 1 by the transformation of variables x˜ = x p . We assume therefore that such transformation is already made and consider the case p = 1. Then the equation for R(τ ) = R1 (τ ) can be obtained (see Eqs. (7.24)) by applying the inverse Laplace transform to Eq. (7.18). Then we obtain, with µ∗ = µ(1), − µ(1)
∂ τ R(τ ) + R(τ ) = Z (R) = L−1 [(w)], ∂τ
(9.12)
where (see Eqs. (3.2)) Z (R) =
M
α j Z j (R),
j=1
M
α j = 1, α j ≥ 0,
j=1
j A j (a1 , . . . , a j ) ∗ τ da1 , . . . , da j , R j a a . . . a a 1 2 j k R+ k=1
Z j (R) = j ∗
Rk (τ ) = R1 ∗ R2 ∗ . . . ∗ R j ,
R1 ∗ R2 =
τ
R1 (τ )R2 (τ − τ ) dτ .
0
k=1
Let us denote the s-moment of R, ∞ R(τ )τ s dτ, ms =
s ≥ 0,
(9.13)
0
then, in order to obtain an identity for these m s moments, we multiply Eq. (9.12) by τ s and obtain, after integration in τk = ak −1 τ , the following identity: (µ(1)s + 1)m s ⎞s ⎞ ⎛ ⎛ j j M ⎝ = αj A ak τk ⎠ R(τk )dτ1 . . . dτ j ⎠ da1 . . . da j . j (a1 . . . a j ) ⎝ j j j=1
R+
R+ k=1
k=1
(9.14)
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Next, recalling the notations from (3.13), (5.6), that is µ( p) = (λ( p) − 1) p −1 , with (5.7), ⎛ ⎞ ∞ j M p K (a)a p da = α j j A j (a1 , . . . , a j ) ⎝ ak ⎠ da1 . . . da j , λ( p) = 0
R+
j=1
k=1
then, subtracting λ( p) m s from both sides of identity (9.14), the s-moment equation associated to Eq. (9.12) can be written in the form (sµ(1) − λ(s) + 1)m s = s (µ(1) − µ( p)) m s =
M
α j I j (s),
(9.15)
j=2
where I j (s) =
⎛
A(a 1, . . . , a j ) ⎝ j
R+
⎛ (s)
g j (y1 , . . . , y j ) = ⎝
j
(s) g (a1 τ1 , . . . , a j τ j ) j j
R+
⎞ R(τk ) dτ1 . . . dτ j ⎠ da1 . . . da j ,
k=1
⎞s yk ⎠ −
k=1
j
j
yks ,
j = 1, . . . , M.
(9.16)
k=1
(s)
We note that g1 = 0 for any s ≥ 0 and that m 0 = m 1 = 1 (see Eqs. (9.8)). Our aim is to study the moments m s , s > 1, on the basis of Eq. (9.15). The approach is related to the one used in [24] for a simplified version of Eq. (9.15) with M = 2. The main results are formulated below in terms of the spectral function µ( p) (see Fig. 1) under the assumption that p0 > 1. Proposition 9.1. [i] If the equation µ(s) = µ(1) has the only solution s = 1, then m s < ∞ for any s > 0. [ii] If this equation has two solutions s = 1 and s = s∗ > 1, then m s < ∞ for s < s∗ and m s = ∞ for s > s∗ . [iii] m s∗ < ∞ only if I j (s∗ ) = 0 in Eq. (9.15) for all j = 2, . . . , M. Proof. The proof is based on the following inequality that controls, from above and below, the right-hand side of Eq. (9.15), 0≤
M
α j I j (s) ≤ C M (s)m 1 m s−1
(9.17)
j=2
with s > 1 and some positive constant C M (s) to be determined. Then, as (9.17) holds, combining with identity (9.15), we obtain, for m s ≥ 0, ms ≤
C M (s) m s−1 , s[µ(1) − µ(s)]
with m 0 = m 1 = 1.
In the case [i] µ(1) > µ(s) (see Fig. 1) for all s > 1. The same is true in the case [ii] for s < s∗ . It is clear from Eq. (9.13) that m s > 0 since R(τ ) ≥ 0 and m 1 = 1. This means that the inequality (9.17) cannot be satisfied for s > s∗ , therefore moments of orders s > s∗ cannot exist. The statement [iii] follows directly from Eq. (9.15).
Asymptotics for Generalized Nonlinear Maxwell Models
635
We note that Proposition 9.1 relates to moments (9.13) of R(τ ) = R1 (τ ) in Eqs. (9.11). A similar statement can be formulated for moments of R p (τ ), with any p > 0 (by using the change of variables x˜ = x p ), but we shall need below just the case p = 1 as described in Proposition 9.1. Hence, it remains to prove the inequality (9.17). The proof is based on the following elementary inequality. Lemma 9.2. In the notation of Eqs. (9.16), s−1 s 0 ≤ g (s) j (y1 , . . . , y j ) ≤ 2
j−1
γk ψ(yk+1 , Yk ),
s > 1,
(9.18)
k=1
where ψ(y1 , y2 ) = y1s−1 y2 + y2s−1 y1 , γk = max(k, k s−1 ),
Yk = max(y1 , . . . , yk ),
k = 1, . . . , j; j = 2, 3, . . . .
Proof. If j = 2, we assume without loss of generality that y1 ≤ y2 and reduce the upper estimate problem to the inequality (x) = (1 + x)s − 1 − xs − 2s−1 s(x + xs−1 ) ≤ 0,
x=
y1 ≤ 1. y2
Its proof is obvious since, (0) = 0, (x) ≤ 0. The lower estimate in Eqs. (9.18) is similarly reduced for j = 2 to the inequality g(θ ) = 1 − θ s − (1 − θ )s ≥ 0,
θ=
x ≤ 1. x+y
Then, its proof follows from the fact that g(0) = g(1) = 0, g (θ ) ≤ 0. We proceed by induction. It is easy to see that ⎛ ⎞ j (s) (s) (s) ⎝ g j+1 (y1 , . . . , y j+1 ) = g j (y1 , . . . , y j ) + g2 yk ⎠ , y j+1 , j = 3, . . . . k=1
Then the lower estimate (9.18) becomes clear for any j ≥ 2. By applying the upper estimate (9.18) for g2 (s) we obtain ⎛ ⎞ j g (s) (y1 , . . . , y j+1 ) ≤ g (s) + 2s−1 s ψ ⎝y j+1 , yk ⎠ , j+1
j
k=1
and note that ψ(x, y) is an increasing function of y. Clearly, j
yk ≤ jY j ,
Y j = max(y1 , . . . , y j ),
k=1
and therefore (s)
(s)
g j+1 (y1 , . . . , y j+1 ) ≤ g j (y1 , . . . , y j ) + 2s−1 sγ j ψ(y j+1 , Y j ).
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This is precisely what is needed to get the upper estimate (9.18) by induction. This completes the proof of Lemma 9.2.
In order to complete the proof of Proposition 9.1, we just need to prove the inequal(s) ity (9.17). First substitute the estimates (9.18) applied to g j (a1 τ1 , . . . , a j τ j ) into the right-hand side of Eq. (9.15). Then the lower estimate (9.17) becomes clear since the (s) functions g j are non-negative. The upper estimate (9.17) follows after we note that max(a1 τ1 , . . . , a j τ j ) ≤ a¯ j max(τ1 , . . . , τ j ), a¯ j = max(a1 , . . . , a j ), j = 1, . . . , M; and we can estimate (9.15) as follows. First we show that, for s > 1, (s) g j (a1 τ1 , . . . , a j τ j )
≤s2
s−1
a¯ sj
j−1
γk ψ(τk+1 , max(τ1 , . . . , τk ) ),
k=1
as it follows from Eqs. (9.18). Then the internal integral in Eqs. (9.16) is controlled by a sum of integrals having the following structure: j R+
j
R(τi ) ψ(τk+1 , max(τ1 , . . . , τk ) ) dτ1 . . . dτ j
i=1
= k! 0≤τ1 ≤···≤τk <∞
[m 1 τks−1
+ m s−1 τk ]
k
R(τi )
dτ1 . . . dτk
i=1
≤ 2k! m 1 m s−1 ,
k = 1, . . . , j − 1,
in the notation of Eqs. (9.13), with m 0 = 1. Hence, we obtain from (9.16), ⎛ ⎞ j−1 s ⎝ ⎠ I j (s) ≤ m 1 m s−1 2 s k! γk A(a1 , . . . , a j ) a¯ sj da1 . . . da j , j = 2, . . . , M, k=1
j R+
and the upper estimate (9.17) follows from conditions (3.2)–(3.5). Thus, the proof of Proposition 9.1 is completed.
Now we can draw some conclusions concerning the moments of the distribution functions (9.11). We denote ∞ ∞ s m s ( p ) = p (v)v dv, m s (R p ) = R p (τ )τ s dτ, m 2s (F p ) =
0
0
Rd
F p (|v|)|v|2s dv,
s > 0, 0 < p ≤ 1,
and use similar notations for N p (v) and M p (|v|) in Eqs. (9.11). Then, by formal integration of Eqs. (9.11), we obtain m s ( p ) = m s (N p )m s/ p (R p ), m 2s (F p ) = m 2s (M p )m s/ p (R p ), where M p and N p are given in Eqs. (9.10).
Asymptotics for Generalized Nonlinear Maxwell Models
637
First we consider the case 0 < p < 1. It follows from general properties of stable distributions that the moments m s (N p ) and m 2s (M p ), 0 < p < 1, are finite if and only if s < p (see [18]). On the other hand, m 0 (R p ) = m 1 (R p ) = 1, therefore m s (R p ) is finite for any 0 ≤ s ≤ 1. Hence, in this case m s ( p ) and m 2s (F p ) are finite only for s < p. The remaining case p = 1 is less trivial since all moments of functions |v|2 , v ∈ Rd , M1 (|v|) = (4π )−d/2 exp − 4 N1 (v) = δ(v − 1),
v ∈ R+ ,
are finite. Therefore everything depends on moments of R1 in Eqs. (9.12) with p = 1. It remains to apply Proposition 9.1. Hence, the following statement is proved for the moments of the distribution functions (9.4), (9.6). Proposition 9.3. [i] If 0 < p < 1, then m 2s (F p ) and m s ( p ) are finite if and only if 0 < s < p. [ii] If p = 1, then Proposition 9.1 holds for m s = m 2s (F1 ) and for m s = m s (1 ). Remark 9. Proposition 9.3 can be interpreted in other words: the distribution functions F p (|v|) and p (v), 0 < p ≤ 1, can have finite moments of all orders in the only case when two conditions are satisfied (1) p = 1, and (2) the equation µ(s) = µ(1) (see Fig. 1) has the unique solution s = 1. In all other cases, the maximal order s of finite moments m 2s (F p ) and m s ( p ) is bounded. This fact means, roughly speaking, that the distribution functions F p and p have power-like tails for large values of their arguments. For the sake of the reader’s convenience, we end this section with some general comments on distribution functions related to generalized Maxwell models. Coming back to solutions f (|v|, t) of Eqs. (2.2), (2.4) discussed at the beginning of this section, we observe that their self-similar asymptotics is described by Eqs. (9.4) (Eqs. (9.6) in the one-dimensional case), where the parameter 0 < p ≤ 1 is related to initial data (9.1) in the following way: u 0 (x) = 1 − αx p + O(x p+ε ),
x → 0,
0 < p ≤ 1.
(9.19)
We proved that the sufficient condition for such asymptotics is the inequality p < p0 , where p0 is a minimum point of the corresponding spectral function µ( p) defined in (5.7). The case p < 1 in (9.19) corresponds to infinite initial energy (the second moment of f 0 (|v|)), for which the corresponding self-similar solution F p (|v|) from (9.11) has also infinite energy. The most important case p = 1 (finite initial energy) leads to a self-similar solution F p (|v|), which has a power-like tail provided there exists a second root s > 1 of the equation µ(s) = µ(1). The existence of such a root depends on the specific form of µ( p) (see Fig. 1). For example, it does not exist in the case (a) and always exists in the cases (c) and (d).
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A. V. Bobylev, C. Cercignani, I. M. Gamba
A dissipative Boltzmann equation corresponds to the case (b) with µ(1) < 0, therefore the second root exists and we have a power-like tail. The classical (elastic) Boltzmann Equation also relates to the case (b), however with µ(1) = 0 (energy conservation). Therefore the second root does not exist in this case and the “self-similar” solution (usual stationary Maxwellian) has bounded moments of all orders. More detailed information on self-similar solutions is contained in Eqs (9.11) and in Proposition 9.3 in this section. The rate of convergence in Eqs. (9.4), (9.6) is characterized in terms of uniform convergence of the corresponding characteristic functions in Proposition 8.1 in the previous section. 10. Applications The main feature of multi-particle stochastic models discussed in Sect.2 was independence of two scales (in time and in phase variable respectively). Maxwell molecules, for which the frequency of pair collisions is independent of velocities of colliding particles, are the classical example of such a system. Perhaps it is not very easy to find another example in physics, but at least one example of such a system from “real life” is obvious. Imagine that the phase variable are goods, such as moneys or wealth, whereas the particles are players participating in various economical “games.” Then, a realistic assumption is that a scaling transformation of the phase variable, such as a change of currency in this case, does not influence a behavior of players. A phase state of j ≥ 1 players (identical particles) is characterized by a vector j V j = (v1 , . . . , v j ) ∈ R+ of their individual capitals (velocities). The game of these j partners is understood as a random linear transformation ( j-particle collision) ˆ j, V j = GV
V j = (v1 , . . . , v j ), V j = (v1 , . . . , v j ),
(10.1)
where Gˆ is a square j × j matrix with non-negative random elements, V j and V j are understood as vector-columns. The matrix Gˆ must satisfy certain conditions to ensure that the model does not depend on numeration of identical particles. The simplest class of such matrices is a two-parameter family a if i = k (10.2) Gˆ = {gik , i, k = 1, . . . , j}, gik = b otherwise, with arbitrary non-negative a and b. The parameters (a, b) can be fixed or randomly distributed in R2+ with some probability density B j (a, b). The corresponding transformation vk = avk + b
j
vi ,
k = 1, . . . , j,
(10.3)
i=1,(i=k)
can be easily interpreted in terms of economics; the interpretation is discussed below. In order to derive a kinetic equation we can consider the stochastic model from Sect.2 with large phase vector VN (t) = (v1 (t), . . . , v N (t)) ∈ R+N (in terminology of “particles” and “collisions”). We use the standard approach of Kac [20] and assume that VN (t) undergoes random jumps caused by collisions. Intervals between two successive jumps have the Poisson distribution with the average t N = /N , = const. Then
Asymptotics for Generalized Nonlinear Maxwell Models
639
we introduce the N -particle distribution function F(VN , t) and consider a weak form of the Kac Master equation [20], N d F(VN , t)(VN ) d VN = F(VN , t)(VN ) − (VN ) d VN , (10.4) dt R+N R+N where (VN ) is a nice test function; the average · is taken over all possible jumps. We assume that 1) jumps are caused by interactions of 1 ≤ j ≤ M < N particles (the case j = 1 is understood as an interaction with background) and 2) relative probabilities of interactions which involve 1, 2, . . . , M particles are given respectively by non-negative real numbers β1 , β2 , . . . , β M such that β1 + β2 + · · · + β M = 1.
(10.5)
Then (VN ) =
M
β j (VN ) j ,
(10.6)
j=1
where . . . j is the average over all j-particle interactions, with 1 ≤ j ≤ M. Assuming that the interactions are described by Eqs. (10.3) with various random parameters a and b (if j ≥ 2), we obtain for j = 1, 2, ∞ N 1 (VN )1 = B1 (a) (v1 , . . . , avk , . . . , v N ) da, N 0 k=1
(VN )2 = ×
∞
∞
0
2 N (N − 1)
∞
(v1 , . . . , avk + bvl , . . . , bvk + avl , . . . , v N ) db da,
1≤k
∞ ∞
B1 (a) da =
0
B2 (a, b)
0
0
B2 (a, b) db da = 1.
0
Those terms with j ≥ 3, where it is assumed that a and b in Eqs. (9.3) are distributed with probability density B j (a, b) in R2+ , are defined in similar way. Then we introduce a one-particle distribution function (setting v1 = v) ∞ ∞ ∞ F(VN , t) dv2 , . . . dv N , f (v, t) dv = 1, (10.7) f (v, t) = 0
0
and its Laplace transform
0
∞
u(x, t) =
f (v, t)e−xv dv ,
x ≥ 0.
(10.8)
0
It remains to substitute (VN ) = e−xv1 into Eqs. (10.4), (10.6) and to assume formally that (“molecular chaos”) F(VN , t) ≈
N k=1
This assumption is postulated below.
f (vk , t),
N → ∞.
(10.9)
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A. V. Bobylev, C. Cercignani, I. M. Gamba
Then we obtain in the limit N → ∞, ∞ ∂ 1 u(x, t) + u(x, t) = β1 B1 (a)u(ax, t) da ∂t 0 ∞ ∞ M + jβ j B2 (a, b)u(ax)u j−1 (bx, t) db da . j=2
0
0
(10.10) We assume without loss of generality that =
M
jβ j .
(10.11)
j=1
Then Eq. (10.10) is a particular case of Eqs. (3.1)–(3.4) with αj =
jβ j , 1 ≤ j ≤ M;
A1 (a) = B1 (a),
A j (a1 , a2 , . . . , a j ) = B2 (a1 , a2 )
A2 (a1 , a2 ) = B2 (a1 , a2 ), (10.12)
j
δ(ak − a2 ), 3 ≤ j ≤ M.
k=3
Hence, all our results can be applied to this system. The spectral function reads λ( p) − 1 , p ⎫ ⎧ ∞⎨ ∞ M ⎬ λ( p) = αj B2 (a, b)[a p + ( j − 1)b p ] db da, (10.13) α1 a p B1 (a) + ⎭ 0 ⎩ 0 µ( p) =
j=2
and its knowledge is sufficient for our goals. A typical example of the application in terms of particles-players and velocities-goods is discussed below. Assume a typical initial condition f |t=0 = δ(v − 1)
⇐⇒
u |t=0 = e−x ,
(10.14)
in the notation of Eq. (10.8). Thus, each player has one unit of currency at t = 0 (full equality). The game of j ≥ 1 players is played in three steps: (1) the participants collect all their goods and form a sum = v1 + v2 + · · · + v j ; (2) the sum is multiplied by a random number θ ≥ 0 distributed with given probability density q(θ ) in R+ ; (3) the resulting sum = θ = v1 + · · · + v j is given back to the players in accordance with the following rule: a part of it 1 = (1 − γ ) is divided proportionally to initial contributions, whereas the rest 2 = γ is divided among all players equally, with some fixed or random parameter 0 ≤ γ ≤ 1.
Asymptotics for Generalized Nonlinear Maxwell Models
641
Simple algebra shows that this “game” is equivalent to Eq. (10.3) with j −1 γθ a =θ 1− γ , b= , j ≥1 0 ≤ γ ≤ 1. j j
(10.15)
The meaning of the parameter θ is clear: something was bought (or produced) for the value and then sold for = θ (with gain if θ > 1 or loss if θ < 1). An interesting example arises from assuming the following probability density for θ : q(θ ) = qδ(θ ) + (1 − q)δ(θ − s),
s > 1,
(10.16)
where 0 ≤ q ≤ 1 characterizes a risk of complete loss. The parameter γ can be interpreted as a fixed control parameter to give more chances to losers, and hence γ is introduced in the game in order to prevent large differences between affluent and destitute in the future. In particular, the theory we present here can explain exactly how these differences depend on the parameter γ . In order to clarify this point we make one more simplification by assuming that only games with some fixed number j ≥ 2 of players are allowed. Then Eq. (10.10) for such a model reads
∞ γθ j −1 ut + u = γ x u j−1 x dθ q(θ )u θ 1 − j j 0
j −1 j j−1 γ s = qu (0) + (1 − q)u s 1 − γ x u x , u |t=0 = e−x , j j (10.17) provided Eqs. (10.14), (10.16) hold. The corresponding spectral function (10.13) reads # "
p p 1 j −1 γ µ( p) = − 1 . (10.18) (1 − q)s p 1 − γ + ( j − 1) p j j Then, if we assume that j (1 − q) > 1 (which is always the case if q < 1/2), µ( p) > 0 for small p > 0, and we have a typical example of functions shown on Fig. 1. In this case µ(1) = s(1 − q) − 1, s > 1,
j −1 µ (1) = 1 + (1 − q)s log s − 1 + ψ j γ , j x + (1 − x) log(1 − x). ψ j (x) = x log j −1
(10.19)
Next, we apply to Eq. (10.18) our criteria for self-similar asymptotics. According to the analysis of the spectral function µ( p), for p = 1, presented in the previous sections, the condition µ (1) < 0 would mean that the large time asymptotics of u(x, t) is described by the self-similar solution w[x exp(µ(1)t)] studied in previous sections. It is easy to verify that µ (1) is a monotone decreasing function of γ ∈ [0, 1] and µ (1) = 1 + (1 − q)s(log s − 1 − log j) if γ = 1.
642
A. V. Bobylev, C. Cercignani, I. M. Gamba
Then the inequality 1 −1 j > s exp (1 − q)s
guarantees that there exists a real number γ∗ > 0 such that we do have self-similar asymptotics for all γ∗ < γ ≤ 1. This number is a unique root of equality
j −1 γ∗ + 1 + (1 − q)s(log s − 1) = 0 ψj j in the notation of Eq. (10.19). There is also the second important number γ∗∗ , which indicates an appearance of power-like tails for the probability distribution function f (v, t) defined in (10.7) whose Laplace transform satisfies Eqs. (10.17). We note that “long” tails (i.e. values of f (v, t) that decay no faster than negative powers v, for large v) correspond to relatively big differences between affluent and destitute. We can expect existence of such tails for γ∗ < γ < γ∗∗ < 1, provided they disappear for γ close to γmax = 1. We already know that power-like tails do exist if the equation µ( p) = µ(1) has the second root p > 1. Therefore, conditions of appearance of the second root can be easily understood from comparison of Fig. 1 with Eq. (10.18). It is clear that ⎧
j −1 ⎪ ⎪ γ >1 ⎨∞ if s 1 − j
µ( p) −−−→ j − 1 p→∞ ⎪ ⎪ γ ≤ 1. ⎩0 if s 1 − j Hence, the second root exists (see Fig. 1 c,d) if
1 j γ∗ < γ < γ∗∗ = 1− , j −1 s
s > 1,
independently of all other parameters. If γ > γ∗∗ , then the second root still exists, provided µ(1) < 0 (loss of total wealth), since µ( p) has a profile shown on Fig. 1b. Note that γ∗∗ ≤ 1 only if s ≤ j, i.e., the second root always exists if s > j. Finally we note that Eq. (10.17) can be simplified in the limit j = ∞, i.e. for games with a very large number of participants. Assuming that u(x, t) = 1 − α(t)x + O(x1+ε ),
x → 0, t > 0,
we consider Eq. (10.17), where we formally obtain $ % & ' ( ) u j (0) = 1 u s 1 − j−1 j γ x −−−→ u (1 − γ ) sx and
u j−1
%
& % γs j x ≈ 1−
j→∞
& j−1 αγ s −−−→ exp(−α γ sx), j x j→∞
where α = α(t) = −u x (0, t).
(10.20)
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Hence, the formal limit of Eq. (10.18) at j = ∞ reads u t + u = q + (1 − q)u[(1 − γ )sx]e−α(t)γ sx ,
(10.21)
with α(t) from Eq. (10.20). As always, we assume that α(0) = 1 and find α(t) by considering Eq. (10.21) for small x. Then we obtain ( ) αt + α − (1 − q) (1 − γ ) s + γ s α = 0, and therefore α(t) = exp{[(1 − q)s − 1] t}. Thus, Eq. (10.21) becomes linear for a class of initial data satisfying the condition α(0) = 1. For brevity we omit simple conclusions about this equation. Acknowledgements. The first author was supported by grant 621-2006-3404 from the Swedish Research Council (NFR). The research of the second author was supported by MUIR of Italy. The third author has been partially supported by NSF under grant DMS-0507038 and an FRG grant DMS-0757450. Support from the Institute for Computational Engineering and Sciences at the University of Texas at Austin is also gratefully acknowledged.
References 1. Ben-Abraham, D., Ben-Naim, E., Lindenberg, K., Rosas, A.: Self-similarity in random collision processes. Phys. Rev. E 68, R050103 (2003) 2. Baldassarri, A., Marini Bettolo Marconi, U., Puglisi, A.: Influence of correlations on the velocity statistics of scalar granular gases. Europhys. Lett. 58, 14–20 (2002) 3. Bisi, M., Carrillo, J.A., Toscani, G.: Decay rates towards self-similarity for the Ernst-Brito conjecture on large time asymptotics of the inelastic Maxwell model. J. Stat. Phys. 124(2-4), 625–653 (2006) 4. Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Math. Phys. Rev. 7, Soviet Sci. Rev., Sect. C Math. Phys. Rev. 7, Chur: Harwood Academic Publ., 1988, pp. 111–233 5. Bobylev, A.V.: The Fourier transform method for the Boltzmann equation for Maxwell molecules. Sov. Phys. Dokl. 20, 820–822 (1976) 6. Bobylev, A.V., Carrillo, J.A., Gamba, I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98(3-4), 743–773 (2000) 7. Bobylev, A.V., Cercignani, C.: Self-similar solutions of the Boltzmann equation and their applications. J. Stat. Phys. 106, 1039–1071 (2002) 8. Bobylev, A.V., Cercignani, C.: Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Stat. Phys. 110, 333–375 (2003) 9. Bobylev, A.V., Cercignani, C., Gamba, I.M.: Generalized kinetic Maxwell type models of granular gases. In: Mathematical models of granular matter Series: Lecture Notes in Mathematics, Vol. 1937, G. Capriz, P. Giovine, P. M. Mariano (eds.) Berlin-Heidelberg-New York: Springer, pp. 23–58, 2008 10. Bobylev, A.V., Cercignani, C., Toscani, G.: Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials. J. Stat. Phys. 111, 403–417 (2003) 11. Bobylev, A.V., Gamba, I.M.: Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails. J. Stat. Phys. 124(2–4), 497–516 (2006) 12. Carlen, E.A., Carvalho, M.C., Gabetta, E.: Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Comm. Pure Appl. Math. LIII, 370–397 (2000) 13. Carlen, E.A., Carvalho, M.C., Gabetta, E.: On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. J. Funct. Anal. 220(2), 362–387 (2005) 14. Poschel, T., Brilliantov, N. (eds.): Granular Gas Dynamics. Berlin: Springer, 2003 15. Cercignani, C.: The Boltzmann Equation and Its Applications. N.Y.: Springer-Verlag, 1988 16. Cordier, S., Pareschi, L., Toscani, G.: On a kinetic model for a simple market economy. J. Stat. Phys. 120, 253–277 (2005) 17. Ernst, M.H., Brito, R.: Scaling solutions of inelastic Boltzmann equations with overpopulated high energy tails. J. Stat. Phys. 109, 407–432 (2002)
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18. Feller, W.: An Introduction to Probability Theory and Applications. Vol.2, N.-Y.; Wiley, 1971 19. Gabetta, E., Toscani, G., Wennberg, W.: Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation. J. Stat. Phys. 81, 901–934 (1995) 20. Kac, M.: Probability and Related Topics in the Physical Sciences. London-New York: Interscience, 1959 21. Lukacs, E.: Characteristic Functions. London: Griffin, 1970 22. Menon, G., Pego, R.: Approach to self-similarity in Smoluchowski’s coagulation equations. Comm. Pure Appl. Math. 57(9), 1197–1232 (2004) 23. Menon, G., Pego, R.: Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence. SIAM Rev. 48(4), 745–768 (2006) 24. Pareschi, L., Toscani, G.: Self-similarity and power-like tails in nonconservative kinetic models. J. Stat. Phys. 124(2-4), 747–779 (2006) Communicated by H. Spohn
Commun. Math. Phys. 291, 645–658 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0813-5
Communications in
Mathematical Physics
Rational Misiurewicz Maps are Rare Magnus Aspenberg Mathematisches Seminar, Christian-Albrechts Universität zu Kiel, Ludewig-Meyn Str. 4, 24098 Kiel, Germany. E-mail: [email protected]; [email protected] Received: 12 March 2007 / Revised: 20 January 2008 / Accepted: 27 December 2008 Published online: 30 July 2009 – © Springer-Verlag 2009
Abstract: We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree d ≥ 2.
Introduction The notion of Misiurewicz maps has its origin from the paper [16] from 1981 by M. Misiurewicz. The (real) maps studied in this paper have, among other things, no sinks and the omega limit set ω(c) of every critical point c does not contain any critical point. In particular, in the quadratic family f a (x) = 1 − ax 2 , where a ∈ (0, 2), a Misiurewicz map is a non-hyperbolic map where the critical point 0 is non-recurrent. D. Sands showed in 1998 [19] that these maps have Lebesgue measure zero, answering a question posed by Misiurewicz in [16]. In this paper we state a corresponding theorem for rational maps on the Riemann sphere. A generalisaton of this result in degree 2 can be found in [2], where also it is shown that the set of Misiurewicz maps has full Hausdorff dimension (i.e. equal to the dimension of the parameter space). In the complex case, there have been some variations on the definition of Misiurewicz maps. (The sometimes used definition of being a postcritically finite map is far too narrow to adopt here.) In [21], S. van Strien studies Misiurewicz maps with a definition similar to the definition in [16], (allowing super-attracting cycles but no sinks). In [8] ´ by J. Graczyk, G. Swiatek, and J. Kotus, a Misiurewicz map is roughly a map for which every critical point c has the property that ω(c) does not contain any critical point, (allowing sinks but not super-attracting cycles). In this paper we allow attracting cycles, and only care about critical points on the Julia set (suggested by J. Graczyk). The author gratefully acknowledges funding from the Swedish Research Council and the Research Training Network CODY of the European Commission for his stay at Département de Mathématiques at Université Paris-Sud, Orsay.
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ˆ Let f (z) be a rational function of a given degree d ≥ 2 on the Riemann sphere C. Let Crit ( f ) be the set of critical points of f , J ( f ) the Julia set of f and F( f ) the Fatou set of f . Definition 0.1. A Misiurewicz map f is a non-hyperbolic rational map that has no parabolic cycles and such that ω(c) ∩ Crit ( f ) = ∅ for every c ∈ Crit ( f ) ∩ J ( f ). We prove the following. Theorem A. The set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree d ≥ 2. The Misiurewicz maps are a special type of Collet-Eckmann maps, which - on the contrary - have positive Lebesgue measure in the parameter space of rational maps of any fixed degree d ≥ 2, (see [1]). Hence, Theorem A shows that for a typical (nonhyperbolic) Collet-Eckmann map, the critical set is recurrent. 0.1. Two definitions of Misiurewicz maps. Let Md be the set of Misiurewicz maps of degree d according to the definition above. Define P k ( f, c) = f n (c) and P k ( f ) = P k ( f, c). c∈Crit ( f )
n>k
Set P 0 ( f, c) = P( f, c). The set P( f ) is the postcritical set of f . We will also use the notion postcritical set for P k ( f ) for some suitable k ≥ 0. Let SupCrit ( f ) be the set of critical points in super-attracting cycles. For B(z, r ) = {w : |w − z| < r }, let B(c, δ). Uδ = c∈Crit ( f )\SupCrit ( f )
We introduce another equivalent definition of Misiurewicz maps as follows. ˜ d be the set of non-hyperbolic rational maps of degree d withDefinition 0.2. Let M out parabolic periodic points such that for which every critical point c satisfies ω(c) ∩ (Crit ( f ) \ SupCrit ( f )) = ∅. Let ˜ d : P k ( f ) ∩ Uδ = ∅} and Mδ = ∪k≥0 Mδ,k . Mδ,k = { f ∈ M
(1)
˜ d = Md . If f ∈ Mδ,k then we say that f is (δ, k)Then it is easy to see that M Misiurewicz. If f ∈ Mδ then we say that f is δ-Misiurewicz . So for every Misiurewicz map f , there are constants δ > 0 and k ≥ 0 such that P k ( f ) ∩ Uδ = ∅. 0.2. Some preliminaries and proof outline. Let Rd be the space of rational maps of degree d. Every rational map of degree d can be written in the form R(z) =
P(z) a0 + a1 z + . . . + ad z d = , Q(z) b0 + b1 z + . . . + bd z d
(2)
where ad and bd are not both zero and where R and Q have no common zeros. Without loss of generality we may assume that bd = 1. The case ad = 0, bd = 0 is treated analogously. Hence, the set of rational functions of degree d is a 2d + 1-dimensional complex manifold and subset of the projective space P2d+1 (C). Now, simply take the measure on the coefficient space in one of the two charts ad = 1 or bd = 1. There also
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is a coordinate independent measure on the space of rational maps of a given degree d, induced by the Fubini-Study metric (see [9]). The Lebesgue measure on any of the two charts is mutually absolutely continuous to the Fubini-Study measure. A family of rational maps Ra for a ∈ V ⊂ Cm , where V is open and connected, is normalized if any two functions Ra and Rb , a, b ∈ V , are conformally conjugate if and only if a = b. Introduce an equivalence relation ∼ on the parameter space, saying that f ∼ g if and only if f = T −1 ◦ g ◦ T , for some Möbius transformation T . The set of Möbius transformations forms a 3-dimensional complex manifold, and therefore the space Rd / ∼ of normalised rational maps has dimension 2d − 2. We will always work in this normalised space unless otherwise stated. Although Rd / ∼ is a space of equivalence classes rather than rational functions, locally it can be viewd as a smooth manifold of normalised rational maps. Indeed, a Möbius transformation close to the identity has the form T (z) =
z+γ , αz + β
where (α, β, γ ) is close to (0, 1, 0). Then T −1 ◦ f ◦ T is a 3-dimensional smooth family of conformally conjugate rational maps in a neighbourhood of (0, 1, 0). Moreover, these conformal conjugacy classes form a foliation locally around f . Hence there exists a smooth manifold M ⊂ Rd of normalised rational maps, where dim(M) = 2d − 2 and M contains f . To fix M, let it be transversal to the conformal conjugacy classes around f . Moreover, since there is no natural metric in M, we fix a chart on M and measure distances with respect to this chart. The metrics obtained in this way for different charts are of course mutally absolutely continuous and it will be of no importance for the proof which chart we choose (since we only care about positive measure sets or zero measure sets and not of the actual measure). Hence, to prove Theorem A, by Fubini’s Theorem, it suffices to consider families of normalised maps. If fact, we will consider 1-dimensional slices of normalised so-called rigid families in a neighbourhood of a starting function R = R0 . To outline the ideas of the proof, let R0 (z) = R(z) = P(z)/Q(z) be the starting unperturbed rational map of degree d = max (deg(P), deg(Q)) and assume that R0 (z) is a Misiurewicz map. Let c j (a) be the critical points in Crit (Ra ). We will study the iterates of a critical point c j = c j (a) dependent on the parameter a ∈ B(0, r ) ⊂ B(0, r ), where B(0, r ) is a 1-dimensional disk of radius r > 0, and B(0, r ) is the disk of radius r > 0 in the full (2d − 2)-dimensional parameter space (contained in the manifold M described above). In B(0, r ) we consider the so-called k-Whitney disks D0 = B(a0 , r0 ), where r0 /|a0 | = k, for some fixed 0 < k < 1. A Whitney disk is a k-Whitney disk for some 0 < k < 1. We sometimes write R(z, a) = Ra (z). Put ξn, j (a) = R n (v j (a), a), where v j (a) = R k j (c j (a), a) and where k j is chosen so that v j (0) = R k j (c j (0), 0) has no critical points in its forward orbit. (A priori there can be finite chains of critical points mapped onto each other. Therefore we assume that v j (a) is the last critical value.) For simpler notation, we sometimes drop the index j and write only ξn, j (a) = ξn (a). We want to avoid the situation when critical points split under perturbation. This is however a rare event in the parameter space. Indeed, by classical theory (see e.g. [6], Theorem, p. 7) the set of parameters where critical points of higher multiplicity exist is an analytic (discriminant) set, which has codimension 1. Hence we can assume that all critical points are non-degenerate, i.e. they do not split.
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We also make the following convention. Choosing δ > 0, we always assume that the parameter disk B(0, r ) is chosen so that the critical points ci (a) move inside B(ci , δ/100) as a ∈ B(0, r ). We use the spherical metric and the spherical derivative unless otherwise stated. The proof consists of taking an arbitrarily small parameter disk B(0, r ) and proving that the set of (δ, k )-Misiurwicz maps has Lebesgue density < 1 at a = 0 for given δ > 0 and k ≥ 0 (Theorem B). To prove this, for some fixed 0 < k < 1, we take any k-Whitney disk D0 ⊂ B(0, r ), and show that ξn (D0 ) grows to the large scale under bounded distortion (Lemma 3.5). After this the eventually onto property for rational maps is used to get that ξn+m (D0 ) eventually covers some fixed neighbourhood U = Uδ/2 of the critical points for some m ≤ N˜ , where N˜ only depends on the large scale. We −1 (U ) is a certain portion of D0 . If this holds then pullback U to D0 and show that ξn+m for every D0 ⊂ B(0, r ), and every r > 0 sufficiently small, the Lebesgue density of (δ, k )-Misiurewicz maps will be < 1 at a = 0. 1. Transversality To prove Theorem A we will divide the parameter space of rational functions into so-called “rigid” analytic families. These families cannot be quasiconformal conjugacy classes of Misiurewicz maps, unless they are conformal conjugacy classes. In every rigid family, we prove a one-dimensional slice-version of Theorem A, namely Theorem B (see below). Recall that a compact set , which is invariant under f , is hyperbolic if there are constants C > 0 and λ > 1 such that for any z ∈ and any n ≥ 1, |( f n ) (z)| ≥ Cλn . We will use the following result by Mañé (see [13]). ˆ → C ˆ be a rational map and ⊂ J ( f ) be Theorem 1.1 (Mañé’s Theorem I). Let f : C a compact invariant set not containing critical points or parabolic points. Then either is a hyperbolic set or ∩ ω(c) = ∅ for some recurrent critical point c of f . Theorem 1.2 (Mañé’s Theorem II). If x ∈ J ( f ) is not a parabolic periodic point and does not intersect ω(c) for some recurrent critical point c, then for every ε > 0, there is a neighborhood U of x such that • For all n ≥ 0, every connected component of f −n (U ) has diameter ≤ ε. • There exists N > 0 such that for all n ≥ 0 and every connected component V of f −n (U ), the degree of f n |V is ≤ N . • For all ε1 > 0 there exists n 0 > 0, such that every connected component of f −n (U ), with n ≥ n 0 , has diameter ≤ ε1 . An alternative proof of Mañé’s Theorem can also be found by L. Tan and M. Shishikura in [20]. A corollary of Mañé’s Theorem II is that a Misiurewicz map cannot have any Siegel disks, Herman rings or Cremer points (see [13 or 20]). In particular, a Misiurewicz map has no indifferent cycles.
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1.1. Expansion near the postcritical set. By Mañé’s Theorem, the Misiurewicz condition gives rise to expansion of the derivative in a (closed) neighborhood of the postcritical set. More precisely, the postcritical set P k (R) for a Misiurewicz map R(z) is hyperbolic for some (smallest) k > 0. Put = P k (R) for this k. Since is hyperbolic there exists ˆ such that for each fixed a ∈ B(0, r ), a holomorphic motion h : × B(0, r ) → C, h(z, a) = h a : → a is quasiconformal and for fixed z ∈ the map h(z, a) is holomorphic in a ∈ B(0, r ) (see [7], Theorem III.1.6). Moreover, h a ◦ R0 (z) = Ra ◦ h a (z), for all z ∈ .
(3)
Since is hyperbolic there is some N ≥ 1 such that |(R0N ) (z)| ≥ λ0 for some λ0 > 1 for all z ∈ . Now take a neighborhood N of such that we have expansion of |(RaN ) (z)| for all z ∈ N . Thus, for some C1 > 0 and λ1 > 1, |(RaN ) (z)| ≥ C1 λ1 , j
(4)
whenever R k (z) ∈ N for k = 0, 1, . . . , j, for all a ∈ B(0, r ). Assume moreover N so that U10δ ∩ N = ∅ and that N is closed. We get immediately the following lemma. Lemma 1.3. There exists some λ > 1 and r > 0, such that whenever Rak (z) ∈ N for k = 0, 1, . . . , j and a ∈ B(0, r ), we have |(Ra ) (z)| ≥ Cλ j . j
(5)
Take some δ > 0 such that {z : dist(z, ) ≤ 11δ } ⊂ N . Moreover, choose r > 0 small enough so that {z : dist(z, a ) ≤ 10δ } ⊂ N , for all a ∈ B(0, r ). Hence a is well inside N for all a ∈ B(0, r ). Further, there will be more conditions on δ in Sect. 3 (so that we might have to diminish δ ). Define Pδ = {z : dist(z, ) < δ }. 1.2. The transversality condition. Recall that every critical point c j of R0 eventually k
maps onto , i.e. R0 j (c j ) = v j ∈ , for some (smallest) k j > 0. These v j (a) move holomorphically in a. We want to compare these functions with the holomorphic motion of the starting critical value v j = v j (0). Put x j (a) = v j (a) − h a (v j ).
(6)
Definition 1.4. If x j is not identically equal to zero in B then we say that the corresponding critical point c j (or the function x j itself) has finite order contact in B. If it is clear in what set B c j has finite order contact, we just say c j has finite order contact. If x j ≡ 0 then we say that c j (or x j ) has infinite order contact. To every 1-dimensional ball B(0, r ) ⊂ B(0, r ) we can associate a direction v = (α1 , . . . , α2d−2 ) ∈ P(C2d−3 ), such that the plane in which B(0, r ) lies can be written as {k(α1 , . . . , α2d−2 ) : k ∈ C}.
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Definition 1.5. Let I be the set of indices j for which c j has finite order contact in B(0, r ). Let v be a direction for which all x j , j ∈ I , has finite order contact in B(0, r ). Then we call this B(0, r ) a rigid family around R0 . More generally, we also say that a slice B around R0 is rigid around R0 if x j has finite order contact in B for all j ∈ I . The main result which we will prove is the following. Theorem B. Let Ra , a ∈ B(0, r ) be a normalised and rigid family of rational maps of degree d and assume that R0 is a Misiurewicz map. Then for every δ > 0 and k ≥ 0, the set of (δ, k)-Misiurewicz maps in B(0, r ) has Lebesgue density strictly less than one at a = 0. Since ∪k M1/k,k = Md , Theorem A will follow from Theorem B because almost every direction v has that the corresponding B(0, r ) is a rigid family. This fact we shall prove next. 2. Rigidity of Misiurewicz Maps The main result we shall prove in this section is that a Misiurewicz map cannot carry an invariant line field on its Julia set. This will imply that not all x j ≡ 0, and we thereby reach an important transversality criteria. If all x j (a) ≡ 0 then it means that every critical point c j (a) in the Julia set is mapped onto the hyperbolic set a by (3). Moreover, since all the functions ξn, j (a) would form a normal family, we get that every Ra would be a Misiurewicz map, quasiconformally conjugate to R0 , by Theorem 4.2 in [14]. Before stating the next lemma, a Lattés map is a type of postcritically finite rational map for which the Julia set is equal to the whole Riemann sphere. These type of maps were first introduced by S. Lattés [10]. We also need the two following results; the first is Teichmüller’s module theorem (see [11]): Theorem 2.1. Let G be an annular domain which separates 0 and z 1 from z 2 and ∞. Then mod (G) ≤ log
|z 1 | + |z 2 | |z 2 | + C( ), |z 1 | |z 1 |
where the function C is bounded by 2 log 4. The second is Lemma 2.2 from [5]: Lemma 2.2. Let D ⊂ C be a simply connected domain and let F : C → D = {|z| > 1}, F(∂D) ⊂ ∂D, be p-valent (i.e. degree p). Then if ρ denotes the hyperbolic metric, {w ∈ D : ρD (F(z 0 ), w) ≤ C} ⊂ F({z ∈ D : ρD (z, z 0 ) ≤ 1}) ⊂ {w ∈ D : ρD (F(z 0 , w)) ≤ 1}, where C depends on p. Lemma 2.3. If f is a Misiurewicz map, then its Julia set cannot carry an invariant line field unless f is a Lattés map.
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ˆ then it is already (albeit implicitly) in the literature that the Julia Proof. If J (R0 ) = C set has measure zero (see e.g. [17] by Przytycki, [5] by Carleson, Jones and Yoccoz for polynomials, and the more recent work [15] by Mihalache for the general case). Here is an ad hoc proof of this fact (contributed by the referee): First note that since is hyperbolic it has measure zero, and so has all its preimages. Now take some x ∈ J (R0 ) not in the aforementioned set. The forward orbit of x cannot converge to since it is hyperbolic, hence there are infinitely many n such that R0n (x) stays at a distance s > 0 from the postcritical set P 0 (R0 ) and converges to some point y. By the definition of P 0 (R0 ), a disk D = B(y, s/2) centered at y of radius s/2 can be pulled back under R0n to x univalently (for large enough n). Hence we have bounded distortion on D = B(y, s/4) and hence x cannot be a Lebesgue density point of J (R0 ) if J (R0 ) is not the whole sphere. Since a line field cannot be supported on a set of measure zero the lemma is proved ˆ in the case J ( f ) = C. ˆ If J ( f ) = C then we follow an argument in [14] (cf. also [21] Theorem 3.3). The quasiconformal conjugacy induces a dilatation µ(z). Let E be the support of µ and assume that E has positive Lebesgue measure. Take a density point z 0 ∈ E such that z 0 is a Lebesgue point for µ(z). Let θ (z) be the angle of the invariant line field induced by µ(z). Take a limit point x of f n (z 0 ), so that f n k (z 0 ) → x. By Mañé’s Theorem, the degree of f n k restricted to the component Wk of f −n k (B(x, η)) containing z 0 is uniformly bounded by some constant N < ∞. Moreover, Mañé’s Theorem implies that diam(Wk ) → 0 as k → ∞. Since z 0 is a point of density for E, we have lim
k→∞
m(E ∩ Wk ) → 1. m(E)
From Lemma 2.2 [5] it follows that for every C > 0 there is some constant q only depending on N and C such that f n k ({w ∈ Wk : ρWk (z, z 0 ) ≤ C }) ⊃ {z ∈ B(x, η) : ρ B(x,η) (z, f n k (z 0 )) ≤ q}. Put Wk = {w ∈ Wk : ρWk (z, z 0 ) ≤ C }. Now, from Teichmüller’s modulus theorem it follows that if G k = Wk \ Wk has sufficiently large modulus (which is obtained by choosing C sufficiently small), then there is a ball Bk with boundary Ck , such that Ck ⊂ G k . For every k, we have f n k (Bk ) ⊃ {z ∈ B(x, η) : ρ B(x,η) (z, f n k (z 0 )) ≤ q}. Let Ak : Bk → D be a linear normalisation of Bk , where D is the unit disk. Then f k = f n k ◦ A−1 k : D → B(x, η) is a normal family and of course f k (D) ⊃ {z ∈ B(x, η) : ρ B(x,η) (z, f n k (z 0 )) ≤ q}. Hence there is a subsequence k such that f k converges uniformly to a non-constant limit function g. Since z 0 is a Lebesgue point for θ , the family of line fields (Ak )∗ (θ ) tends to a constant line field θ in D. Since g is a non-constant holomophic function, the pushforward g∗ (θ ) of the constant line field θ is a line field, apart from a discrete set of singularities. Obviously, there is a disk D ⊂ D, where g is univalent and hence we ˆ where the invariant linefield is smooth. have an open set in C Lemma 3.16 in [14] now implies that f has to be a Lattés map or E has measure zero. The lemma follows.
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Recall that B(0, r ) is a 2d − 2-dimensional ball in Rd / ∼, locally described by the manifold M defined in the introduction (Subsect. 0.2). Lemma 2.4. Assume that Ra , a ∈ B(0, r ), is a family of normalised rational maps and R0 is a Misiurewicz map. Then at least one critical point on the Julia set has finite order contact in B(0, r ). Proof. If x j ≡ 0 for all critical points c j it means that the family B(0, r ) is structurally stable. By Theorem E in [12] there can be no components of structurally stable Misiurewicz maps, since a Misiurewicz map cannot carry an invariant line field according to Lemma 2.3. Hence we have the following important transversality criteria: Transversality Criteria. There is at least one critical value v j (0) ∈ , where is a holomorphically moving hyperbolic set, for which x j (a) = v j (a) − h a (v j (0)) is not identically equal to zero. Proof that Theorem B implies Theorem A. Since for any critical point c j which has finite order contact, the set x j = 0 is an analytic set of codimension 1, we get that the set of directions v ∈ P(C2d−3 ) for which x j ≡ 0 in the corresponding B(0, r ) has Lebesgue measure zero. Hence the set of directions for which B(0, r ) is rigid has full measure. By Fubini’s Theorem, Theorem A follows. 3. Distortion Lemmas This section is devoted to the distortion results, which are used to get control of ξn (D0 ) for Whitney disks D0 ⊂ B(0, r ) up to the large scale. One of the main results in this section is Proposition 3.4, which shows a close relation between the space derivative and parameter derivative. The idea of comparing these two quantities was first introduced by M. Benedicks and L. Carleson in [4] (see also [3]). By the transversality criteria there is some non-trivial rigid family B(0, r ) ⊂ B(0, r ) such that for some non-empty set I of indices, x j (a) for j ∈ I , is not identically equal to zero in B(0, r ) (and also in B(0, r )). Fix this rigid family B(0, r ) throughout this section. Also fix some x j , where j ∈ I , and put x(a) = x j (a). We have x(a) = K 1 a k + · · · ,
(7)
for some K 1 = 0. Define µn (a) = µn, j (a) = h a (R0n (v j )). Then in particular x(a) = ξ0 (a) − µ0 (a). The hyperbolic set and its neighbourhood N will be the backbone in the expansion and distortion estimates. Let us first state an elementary property, saying that two points close two each other inside N repel each other uniformly up to some large scale. By the definition of N there exist constants N > 0 and λ > 1 such that |(RaN ) (z)| ≥ λ for all z ∈ N and a ∈ B(0, r ) for some λ > 1. Hence to every z ∈ N there is some radius r (z) > 0 such that |RaN (z) − RaN (w)| ≥ λ|z − w|,
(8)
for all w ∈ N satisfying |z − w| ≤ r (z) (possibly diminishing λ > 1 slightly). Since N is compact and r (z) is continuous there is a constant r˜ > 0 such that (8) holds for all z, w ∈ N provided |z − w| ≤ r˜ . For simplicity assume that N = 1. The following lemma, which will be needed in the subsequent lemma, is a variant of Lemma 15.3 in [18] (see also [1], Lemma 2.1).
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Lemma 3.1. Let u n ∈ C be complex numbers for 1 ≤ n ≤ N . Then N N |u n | − 1. (1 + u n ) − 1 ≤ exp n=1
(9)
n=1
In the following, by (Ran ) (µ0 (a)) or (Ran ) (v(a)) we mean (Ran ) (z) evaluated at z = µ0 (a) or z = v(a) respectively. Lemma 3.2 (Main Distortion Lemma). For every ε > 0, there are arbitrarily small constants δ > 0 and r > 0 such that the following holds. Let a, b ∈ B(0, r ) and suppose that |ξk (t) − µk (t)| ≤ δ , for t = a, b and all k ≤ n. Then n
(R ) (v(a), a) < ε. (10) − 1 (R n ) (v(b), b) The same statement holds if one replaces v(s) = ξ0 (s), s = a, b, by µ0 (t), t = a, b in (10). Proof. The proof goes in two steps. Let us first show that n
(Rt ) (µ0 (t)) (R n ) (ξ (t)) − 1 ≤ ε1 , 0 t
(11)
where ε1 = ε1 (δ ) is close to 0. We have n−1 n−1
Rt (µ j (t)) − Rt (ξ j (t)) ≤ Cδ |Rt (µ j (t)) − Rt (ξ j (t))| R (ξ (t)) j=0
t
j
j=0
≤
Cδ max |Ra (z)|
n−1
|µ j (t) − ξ j (t)|
j=0
≤C
n−1
λ j−n |µn (t) − ξn (t)| ≤ C(δ ),
j=0
where we used Eq. (8). By Lemma 3.1, (11) holds if δ is small enough. Secondly, we show that n
(Rt ) (µ0 (t)) (R n ) (µ (s)) − 1 ≤ ε2 , 0 s where ε2 = ε2 (δ ) is close to 0. Put λt, j = Rt (µ j (t)). Since λt, j are all analytic in t we have λt, j = λ0, j (1 + c j t l + · · · ). Moreover, since n ≤ −C log |x(t)| = −C log |t|, n−1 λ0, j (1 + c j t l + · · · ) (Rtn ) (µ0 (t)) λt, j 1 + cnt l + · · · = = . = n
l (Rs ) (µ0 (s)) λs, j λ0, j (1 + c j s + · · · ) 1 + cns l + . . . j=0
Both the last numerator and denominator in the above equation can be estimated by 1 + O((log |t|)|t|l ) and 1 + O((log |s|)|s|l ), which both can be made arbitrarily close to 1 if r > 0 is small enough. From this the lemma follows.
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Lemma 3.3. Let ε > 0. If δ > 0 is sufficiently small, then for every 0 < δ
< δ
there exist r > 0 such that the following holds. Let a ∈ B(0, r ) and assume that |ξk (a) − µk (a)| ≤ δ , for all k ≤ n and |ξn (a) − µn (a)| ≥ δ
. Then ξn (a) (R n ) (µ (a))x (a) − 1 ≤ ε. 0 a Proof. First we note that by Lemma 3.2 we have ξn (a) = x(a)(Ran ) (µ0 (a)) + µn (a) + E n (a), where, for instance |E n (a)| ≤ |ξn (a) − µn (a)|/1000 independently of n and a if δ is small enough. Put Ra (µ j (a)) = λa, j . Differentiating with respect to a we get ⎛ ⎞ n−1 n−1 λ
(a) + E (a) µ a, j (12) λa, j ⎝x (a) + x(a) + n n−1 n ⎠ . ξn (a) = λa, j j=0 λa, j j=0
j=0
We claim that only the x (a) is dominant in (12) if n is large so that δ
≤ |ξn (a) − µn (a)| ≤ δ . This means that, by Lemma 3.2, (1 − ε1 )δ
≤ |x(a)|
n−1
|λa, j | ≤ (1 + ε1 )δ < 1,
j=0
where ε1 > 0 is arbitrarily small provided r > 0 is small enough. Since λn , for some λ > 1, taking logarithms and rearranging we get (1 − ε1 )
n−1
log |λa, j | ≤ − log |x(a)| ≤ (1 + ε1 )
j=0
n−1
n−1
log |λa, j |,
j=0 |λa, j |
≥
(13)
j=0
if | log δ
| | log |x(a)||, which is true if the perturbation r > 0 is chosen sufficiently small compared to δ
. Since |λa, j | ≥ λ > 1, this means that |x(a)|
n−1 |λ | a, j j=0
|λa, j |
≤ |x(a)|nC ≤ −C|x(a)| log |x(a)|.
Finally −|x(a)| log |x(a)|/|x (a)| → 0 as a → 0. Now, |E n (a)| is uniformly bounded in B(0, r ). Therefore, |E n (a)| is also uniformly bounded on compact subsets of B(0, r ) by Cauchy’s Formula. By diminishing r > 0 slightly we can assume that both |E n (a)| and |E n (a)| are uniformly bounded on B(0, r ). Hence, the last two terms in (12) tend to zero as n → ∞, since also |µ n (a)| is uniformly bounded. We have proved that n−1
ξ (a) − x (a) λa, j ≤ ε|ξn (a)|, n j=0 if |ξn (a) − µn (a)| ≤ δ and n ≥ N for some N . Choose the perturbation r sufficiently small so that this N is at most the number n in (13). Since λa, j = Ra (µ j (a)), the proof is finished.
Rational Misiurewicz Maps are Rare
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Combining Lemma 3.3 and Lemma 3.2 we immediately get the following important result. Proposition 3.4. Let ε > 0. If δ > 0 is small enough and 0 < δ
< δ , there is an r > 0 such that the following holds. Take any a ∈ B(0, r ) and assume that |ξk (a)−µk (a)| ≤ δ , for all k ≤ n and |ξn (a) − µn (a)| ≥ δ
. Then ξn (a) (14) (R n ) (v(a))x (a) − 1 ≤ ε. a Lemma 3.5. If r > 0 is sufficiently small, there exists a number 0 < k < 1 only depending on the function x, and a number S = S(δ ), such that the following holds for any k-Whitney disk D0 = B(a0 , r0 ) ⊂ B(0, r ): There is an n > 0 such that the set ξn (D0 ) ⊂ N and has diameter at least S. Moreover, we have low argument distortion, i.e.
ξk (a) ≤ 1 , − 1 (15) ξ (b) 100 k for all a, b ∈ D0 and all k ≤ n. Proof. Choose n maximal such that ξk (a0 ) ∈ N for all k ≤ n and |ξn (a0 ) − µn (a0 )| ≤ (δ + δ
)/2.
(16)
Proposition 3.4 holds for all a ∈ B(0, r ) satisfying δ
≤ |ξn (a) − µn (a)| ≤ δ .
(17)
x (a)
has bounded distortion on Whitney disks, for parameters a, b ∈ D0 satisfySince ing (17) we have good control of the geometry:
ξn (a) (18) ξ (b) − 1 ≤ 1/100. n Hence, ξn is almost linear in D0 = B(a0 , r0 ) if (17) is satisfied for all a ∈ D0 . Write r0 = k0 |a0 |. Assuming that (17) holds for all a ∈ D0 , then the diameter d of the set ξn (D0 ) can be estimated by d ≥ |ξn (a0 )||k0 a0 | ≥ (1/2)|(Ran0 ) (µ0 (a0 ))||x (a0 )||k0 a0 |. If (17) does not hold for all parameters in D0 (so that the Whitney disk D0 is too large), then we may have to diminish r0 (and hence k0 ). However, it is clear that we can beforehand fix δ
> 0 sufficiently small and k0 > 0 so that (17) holds for all a ∈ D0 , given the condition (16) on n in the beginning of the proof. ˆ then by Lemma 3.2 If M0 is the supremum of |Ra (z)| over all a ∈ B(0, r ) and z ∈ C and (16), (δ + δ
)/(2M0 ) ≤ |ξn (a0 ) − µn (a0 )| ≤ 2|(Ran0 ) (µ0 (a0 ))||x(a0 )|. Thus for some constant C > 0 depending on δ
and M0 , |(Ran0 ) (µ0 (a0 ))| |k0 a0 ||x (a0 )| d ≥ C . ≥ C δ
|(Ran0 ) (µ0 (a0 ))| |x(a0 )| Hence, the diameter d of the set ξn (D0 ) is greater than some S = C δ = S(δ ). Also, by (18), we have bounded argument distortion for all a, b ∈ D0 .
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4. Conclusion and Proof of Theorem B After ξn (D0 ) has grown to the large scale, then after a finite number of iterates ξn+m (D0 ) will cover U , where m is uniformly bounded by some constant N˜ and U = Uδ/2 is a neighbourhood of the critical points (not belonging to superattracting cycles) for f 0 . We then pull back those points in U under ξn+m to the disk D0 and show that they correspond to a certain fraction of parameters. This will imply that the Lebesgue density of (δ, n + m − 1)-Misiurewicz maps in the whole disk B(0, r ) is strictly less than 1. We have the following elementary lemma. Lemma 4.1. For every d > 0 there is an r > 0 such that the following holds. Let D be a set which contains a disk of radius d centered at the Julia set of R. Then there is some integer N˜ only depending on R and d such that inf{m ∈ N : R m (D) ⊃ U )} ≤ N˜ . Proof. First cover J (R) with the collection of open disks Dz of diameter d centered at any point z of J (R). Since R n is not normal on the Julia set, we get that for every Dz , there is a smallest number n = n(z) such that R0n (D j ) ⊃ U . Note that n(z) is constant in some neighbourhood of z since R n is a continuous function. Since J (R) is compact there is some uniform N˜ such that n(z) ≤ N˜ . The lemma is proved. Since Ran (Dz ) moves continuously in a, there is an r > 0 such that the same statement holds for Ra instead of R0 , if a ∈ B(0, r ). Now let us prove Theorem B. Suppose that f 0 is a δ-Misiurewicz map and put U = Uδ/2 . Choosing r > 0 small enough we can ensure that for every critical point c j (0) ∈ Crit ( f 0 ) \ SupCrit ( f 0 ) we have c j (a) ∈ Uδ/100 , for all a ∈ B(0, r ). Let D0 = B(a0 , r0 ) ⊂ B(0, r ) be any Whitney disk obtained from Lemma 3.5, where B(0, r ) is rigid. Suppose that c j has finite order contact in the full parameter ball B(0, r ) for all indices j ∈ I ⊂ {1, . . . , 2d − 2}. By Lemma 2.4, I = ∅. Since B(0, r ) is rigid, we get from Lemma 3.5 that for every j ∈ I there is some n j such that set ξn j , j (D0 ) contains a ball of diameter d = S/2. Put D j = ξn j , j (D0 ). Let D j = ξn j , j (D1 ), D
j = ξn j , j (D2 ), where D1 = B(a0 , r0 /2), D2 = B(a0 , r0 /4) and D0 = B(a0 , r0 ). Now we have two cases. (Recall that F(R0 ) and J (R0 ) is the Fatou set and Julia set respectively for the function R0 .) Case I. D j ⊂ F(R0 ) for all j ∈ I . Then there is a perturbation r > 0 such that for all a ∈ B(0, r ) we have D
j = ξn j , j (D2 ) ⊂ F(Ra ). In this case we have two choices: (1) The index set I = {1, . . . , 2d − 2}, i.e. all c j have finite order contact. In this case D2 is a family of hyperbolic rational maps. (2) The index set I is a strict subset of {1, . . . , 2d − 2}. In this case D2 is a family of new Misiurewicz maps, such that every critical point c j , j ∈ I belong to an attracting basin. The other critical points c j , j ∈ / I , belongs to the Julia set for all a ∈ B(0, r ) ⊃ D2 and hence every critical point on the Julia set has infinite order contact in B(0, r ). Hence, there is a small ball D˜ ⊂ B(0, r ) in the full (2d − 2)-dimensional parameter space such that D˜ are all quasiconformally conjugate Misiurewicz maps. This contradicts the fact that Misiurewicz maps cannot carry an invariant line field, by Lemma 2.3.
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Hence if D j ⊂ F(R0 ) for all j ∈ I , the first choice above must occur. Since hyperbolic maps are not Misiurewicz maps by definition, by Lemma 3.5 we conclude that µ({a ∈ D0 : Ra is a Misiurewicz map}) ≤ Cµ(D0 ), for some 0 < C < 1. Case II. D j ⊂ F(R0 ) does not hold for all j ∈ I . Then there is some j ∈ I such that D j ∩ J (R0 ) = ∅. Hence ξn j , j (D j ) must contain a disk K of diameter S/4 centered at J (R0 ). Now fix j and put D j = D and ξn j , j = ξn . Use Lemma 4.1 with d = S/4. We get some number N˜ such that R0m (K ) ⊃ U , for some m ≤ N˜ . For small r > 0 we also have Ram (K ) ⊃ U for a ∈ B(0, r ). Since m ≤ N˜ , where N˜ is uniformly bounded, f m has bounded degree. By bounded distortion (Lemma 3.5), we get −1 (U )) ≥ Cµ(D0 ), µ(ξn+m
for some 0 < C < 1. −1 (U )) ⊂ D0 , f a cannot be a (δ, n +m −1)-Misiurewicz For any parameter a ∈ µ(ξn+m map, unless ξn+m (a) ∈ B(c j (0), δ/2) ⊂ Uδ/2 , and c j (a) has become superattracting, i.e. c j (a) ∈ SupCrit ( f a ) (see Subsect. 0.1). But c j (a) ∈ SupCrit ( f a ) only for a zero measure set of parameters. It follows that the set Mδ,n+m−1 of (δ, n + m − 1)-Misiurewicz maps in D0 has Lebesgue measure at most Cµ(D0 ) for some constant 0 < C < 1. Since the sets Mδ,k are nested (i.e. Mδ,k ⊂ Mδ,k+1 for all k ≥ 0) and since n + m can be made arbitrarily large if r > 0 is small enough, we conclude that for every k ≥ 0 there is an r > 0, such that the set of (δ, k)-Misiurewicz maps in D0 ⊂ B(0, r ) has Lebesgue measure at most Cµ(D0 ). It follows that the Lebesgue density of the set of (δ, k)-Misiurewicz maps at a = 0 is strictly less than 1 and Theorem B is proved.
Acknowledgements. I am grateful to Michael Benedicks for many valuable comments and discussions, especially on the transversality condition. I am grateful to Jacek Graczyk for communicating the ideas on a suggested significant generalisation of the original result, where no sinks were allowed. I wish to thank Duncan Sands, Nicolae Mihalache and Neil Dobbs for interesting remarks and discussions on a preliminary version. I thank Dierk Schleicher for very interesting discussions at an early stage of this paper. I want to express my gratitude to the referee for many useful remarks, comments and suggestions of improvements. Finally, I want to express my warm thanks to Nan-Kuo Ho for useful comments and her encouragement during the writing of this paper. I dedicate this paper to her. This paper was written at the Department of Mathematics at Université Paris-Sud. I gratefully acknowledge the hospitality of the department.
References 1. Aspenberg, M.: The Collet-Eckmann condition for rational maps on the Riemann sphere. Ph. D. thesis, Stockholm, 2004, http://www.diva-portal.org/diva/getDocument?Urn_nbn_se_kth-diva-3788-2fulltext. pdf 2. Aspenberg, M., Graczyk, J.: Dimension and measure for semi-hyperbolic rational maps of degree 2. C. R. Acad. Sci. Paris Ser. I 347, 395–400 (2009) 3. Benedicks, M., Carleson, L.: On iterations of 1 − ax 2 on (−1, 1). Ann. of Math. (2) 122(1), 1–25 (1985) 4. Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. of Math. (2) 133(1), 73–169 (1991) 5. Carleson, L., Jones, P.W., Yoccoz, J.-C.: Julia and John. Bol. Soc. Brasil. Mat. (N.S.) 25(1), 1–30 (1994) 6. Chirka, E.M.: Complex Analytic Sets, Volume 46 of Mathematics and its Applications (Soviet Series). Dordrecht: Kluwer Academic Publishers Group, 1989, Translated from the Russian by R. A. M. Hoksbergen
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7. de Melo, W., van Strien, S.: One-Dimensional Dynamics. Volume 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Berlin: Springer-Verlag, 1993 ´ 8. Graczyk, J., Swiatek, G., Kotus, J.: Non-recurrent meromorhic functions. Fund. Math. 182, 269–281 (2004) 9. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library, New York: John Wiley & Sons Inc., 1994, Reprint of the 1978 original 10. Lattés, S.: Sur l’itération des substitutions rationnelles et les fonctions de Poincaré. C. R. Acad. Sci. Paris 166, 26–28 (1918) 11. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane. New York: Springer-Verlag, Second edition, 1973, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126 12. Mañé, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Scient. de l’Ec. Norm. Sup. 16(2), 193–217 (1983) 13. Mañé, R.: On a theorem of Fatou. Bol. Soc. Brasil. Mat. (N.S.) 24(1), 1–11 (1993) 14. McMullen, C.T.: Complex Dynamics and Renormalization. Volume 135 of Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 1994 15. Mihalache, N.: La condition de Collet-Eckmann pour les orbites critiques recurrents. Ph.D. thesis, 2006, Orsay 16. Misiurewicz, M.: Absolutely continuous invariant measures for certain maps of an interval. Publ. Math. de l’ IHÉS 53, 17–51 (1981) 17. Przytycki, F.: On measure and Hausdorff dimension of Julia sets of holomorphic Collet-Eckmann maps. In: International Conference on Dynamical Systems (Montevideo, 1995), Volume 362 of Pitman Res. Notes Math. Ser., Harlow: Longman, 1996, pp. 167–181 18. Rudin, W.: Real and Complex Analysis. New York: McGraw-Hill Book Co., Third edition, 1987 19. Sands, D.: Misiurewicz maps are rare. Commun. Math. Phys. 197(1), 109–129 (1998) 20. Shishikura, M., Lei, T.: An alternative proof of Mañé’s theorem on non-expanding Julia sets. In: The Mandelbrot set, theme and variations, Volume 274 of London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, 2000, pp. 265–279 21. van Strien, S.: Misiurewicz maps unfold generically (even if they are critically non-finite). Fund. Math. 163(1), 39–54 (2000) Communicated by A. Kupiainen
Commun. Math. Phys. 291, 659–689 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0882-5
Communications in
Mathematical Physics
Quenched and Annealed Critical Points in Polymer Pinning Models Kenneth S. Alexander , Nikos Zygouras Department of Mathematics KAP 108, University of Southern California, Los Angeles, CA 90089-2532, USA. E-mail: [email protected]; [email protected] Received: 12 May 2008 / Accepted: 8 June 2009 Published online: 6 August 2009 – © Springer-Verlag 2009
Abstract: We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u + Vn which the chain encounters when it visits a special state 0 at time n. The disorder (Vn ) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when u exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length n has the form n −c ϕ(n) with c ≥ 1 and ϕ slowly varying. Comparing to the corresponding annealed system, in which the Vn are effectively replaced by a constant, it was shown in [1,4,13] that the quenched and annealed critical points differ at all temperatures for 3/2 < c < 2 and c > 2, but only at low temperatures for c < 3/2. For high temperatures and 3/2 < c < 2 we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case c = 3/2 we show that the gap is positive provided ϕ(n) → 0 as n → ∞, and for c > 3/2 with arbitrary temperature we provide an alternate proof of the result in [4] that the gap is positive, and extend it to c = 2. 1. Introduction A polymer pinning model is described by a Markov chain (X n )n≥0 on a state space , containing a special point 0 where the polymer interacts with a potential. The space-time trajectory of the Markov chain represents the physical configuration of the polymer, with the n th monomer of the polymer chain located at (n, X n ), or alternatively, one can view X n as the location of the n th monomer, with n being just an index; these are mathematically equivalent. We denote the distribution of the Markov chain in the absence of the potential, started from 0, by P X and we assume that it is recurrent and has an excursion length distribution (from the 0 state) with power-law decay: ϕ(n) (1.1) P X (E = n) = c , n ≥ 1. n The research of the first author was supported by NSF grant DMS-0405915.
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Here E denotes the length of an excursion from 0, c ≥ 1, and ϕ(·) is a slowly varying function, that is, a function satisfying ϕ(κn)/ϕ(n) → 1 as n tends to infinity, for all κ > 0. When the chain visits 0 at some time n, it encounters a potential of form u + Vn , with the values Vn typically modeling variation in monomer species. This (quenched) pinning model is described by the Gibbs measure β,u,V
dµ N
(x) =
1 β H u (x,V) e N d P X (x), ZN
(1.2)
where x = (xn )n≥0 is a path, V = (Vn )n≥0 is a realization of the disorder, and H Nu (x, V) =
N (u + Vn )δ0 (xn )
(1.3)
n=0
and the normalization
u Z N = Z N (β, u, V) = E X eβ HN (x,V)
is the partition function. The disorder V is a sequence of i.i.d. random variables with mean zero, variance one and finite exponential moments. We assume these variables are Gaussian here to avoid certain technical complications that might obscure the main ideas (see the comments at the end of the paper); we believe the result should be valid for more general distributions with finite exponential moments, but we have not proved this. We denote the distribution of the sequence V by P V . The parameter u ∈ R is thus the mean value of the potential, and β > 0 is the inverse temperature. One would like to understand how the presence of the random potential affects the path properties of the Markov chain, and in particular how the case with disorder differs from the homogeneous case Vn ≡ 0. These effects can be quantified via the free energy N and the contact fraction. To be more precise, letting L N = L N (x) = n=0 δ0 (x n ) denote the local time at 0, it is proved in [2] that there exists a nonrandom Cq (β, u) such that LN = Cq (β, u), lim E µβ,u,V P V − a.s. N →∞ N N Cq (β, u) is called the quenched contact fraction. We will say that the polymer is pinned at (β, u) if Cq (β, u) > 0 and depinned if Cq (β, u) = 0. Monotonicity in u is clear q q so there exists u c (β) such that the polymer is pinned for u > u c (β) and depinned for q u < u c (β). Note that when c < 2 the Markov chain is null recurrent and the set of paths with any given positive contact fraction is exponentially rare, so pinning requires a compensating energy gain from the potential to offset this entropy cost. Pinning can also be described in terms of the quenched free energy f q (β, u) given by β f q (β, u) = lim
N →∞
1 log Z N (β, u, V); N
(1.4)
the fact that the free energy exists and is nonrandom (off a null set of disorders) is proved q q in [2]. The free energy is 0 if u < u c (β) and strictly positive if u > u c (β). The free energy and contact fraction are related by Cq (β, u) =
∂ f q (β, u). ∂u
Quenched vs. Annealed Critical Points
661
The effect of disorder is studied by comparing the quenched pinning model to its annealed version, obtained by averaging the Gibbs weight over the disorder: β,u
dν N =
1 −1 eβ(u+β log MV (β))L N d P X , ZN
EV
(1.5)
where MV (β) = E V (eβV1 ) is the moment generating function. The annealed model is thus equivalent to a quenched model with Vn ≡ 0 and u replaced by u + β −1 log MV (β), and it is readily shown (see [7]) that the critical point u ac (β) in the annealed model is the point where the exponent in (1.5) is 0, which in the Gaussian case means u c (β) = −β/2. It is therefore natural to define the variable by u=−
β + , 2
q
giving critical points ac (β) = 0 and c (β). We then have (cf. (1.5)) E V Z N (β, u, V) = E X eβ L N .
(1.6)
The annealed contact fraction and free energy are given by β f a (β, u) = lim
N →∞
1 1 log E V Z N (β, u, V) = lim log E X eβ L N N →∞ N N
and
Ca (β, u) = lim E ν β,u N →∞
N
LN N
=
(1.7)
∂ f a (β, u) ∂u
respectively. Since E V (log Z N (β, u, V)) ≤ log E V (Z N (β, u, V)), we have f q ≤ f a q and therefore c (β) ≥ 0. It is proved in [1] that 2−c 1 Ca (β, u) ∼ (β ) c−1 ϕˆc−1 as β 0 (1.8) β for c < 2, while for c > 2 the transition is discontinuous: Ca (β, u) →
1 E X (E)
>0
as β 0.
(1.9)
Here ϕˆc−1 is a slowly varying function related to ϕ; see the proof of Lemma 3.1 below. This means that the annealed specific heat exponent (which is, roughly speaking, the exponent α such that the free energy decreases as 2−α as → 0) is (2c − 3)/(c − 1). A strong effect of disorder is evident when the specific heat exponent and/or critical point differ between quenched and annealed systems. In the physics literature, the disorder is said to be relevant if these specific heat exponents differ. Predictions from that literature were confirmed rigorously when it was shown that the disorder is relevant for c > 3/2, i.e. when the specific heat exponent is positive [9], and (for small β) irrelevant for 1 < c < 3/2 [1]. In [1] the quenched and annealed critical points were also proved q equal ( c (β) = 0) for small β when c < 3/2, and recently in [4] it was proved that q c (β) > 0 for all β > 0 when 3/2 < c < 2 and when c > 2, as well as for large β with arbitrary c > 1. Alternate proofs of these results from [1] appear in [14]. Very recently
662
K. S. Alexander, N. Zygouras q
in [8] it was proved that for c = 3/2 and ϕ asymptotically constant, c (β) > 0 for all β > 0, as well. In [1] the following was proved for 3/2 < c < 2 and β sufficiently small. In contrast to (1.8), which has an infinite derivative at = 0, we have the linear bound Cq (β, u) ≤
2 , β
so if we define 0 = 0 (β) by 2−c 2 = (β ) c−1 ϕˆc−1 β
1 , β
we see that Cq (β, u) is forced to be smaller than Ca (β, u) for (roughly) < 0 , and in fact Cq (β, u) = o(Ca (β, u)) as → 0. On the other hand, given > 0 there is a K = K ( ) such that Cq (β, u) < for all > K 0 . − 1 (1.10) C (β, u) a Up to a constant, then, the value 0 (β) ∼ K 1 β
1/(2c−3)
ϕˆc− 3 2
1/2 1 β
separates those (small) values of for which the disorder significantly reduces the contact fraction, from those (larger) values for which it does not. Here ϕˆc− 3 is another 2 slowly varying function related to ϕ. It should be noted that our 0 (β) here is essentially the 1 (β) defined in [1], while 0 (β) in [1] denotes a quantity which is asymptotically a constant multiple of our 0 (β) here. Since we only care about the order of magnitude here, the difference is not significant. The quenched and annealed polymers, then, must behave quite differently for 0 . It is useful to describe this heuristically in terms of strategies, by which we mean classes of qualitatively similar paths. For > 0 the strategy of the annealed polymer is essentially to alter its excursion length distribution (compared to P X ) so that the mean becomes 1/Ca (β, u). The altered distribution which minimizes the relative entropy has the form να (E = n) =
e−αn P X (E = n) , n ≥ 1, E X [e−α E ]
with α chosen to give the desired mean E να (E) = 1/Ca (β, u); this is achieved for α = β f a (β, u) [9]. This is the limiting distribution (as N → ∞) for the excursion length in the annealed polymer [7]. If the quenched polymer is pinned for 0 , it must employ a substantially different strategy, since the quenched contact fraction must be a small fraction of the annealed one. One candidate for such an alternate strategy consists of locating those rare “rich” segments in which the average disorder value is exceptionally large, and making long excursions from one rich segment to another. Such strategies have been studied in some related contexts ([3,9]). Whether such a strategy can be successful (that is, can produce positive free energy) depends on the tradeoff between the energy gain in rich segments and the entropic cost of the excursions between such segments. In this sense, such strategies are loosely related to the Imry-Ma argument
Quenched vs. Annealed Critical Points
663
of [10], which considered the energy gain in “favorable” regions of a disordered system as compared to the energy cost of forming domain walls to separate such regions. The significant use, or not, of alternate strategies (based on rich segments, or otherwise) can q be quantified, at least heuristically, by whether or not c (β) is o( 0 ) as β → 0, since alternate strategies are required for pinning when 0 . Our main result here says that alternate strategies are not used significantly: there exists 0 such that the quenched polymer is not pinned when < 0 0 . In combination with (1.10) this says (still heuristically) that the ability of the quenched polymer to mimic the annealed one breaks down entirely as passes down through order 0 . An alternate description of 0 is as follows. Consider a block of monomers extending one annealed correlation length, that is, having length M ≈ (β f a (β, u))−1 . The fluctuaM tions in the average V = i=1 Vi of the disorder over such a block are of typical order −1/2 M . If this typical fluctuation is at least of the order of , then blocks with average potential u + V < u ac (that is, + V < 0) will be relatively common. In such “bad” blocks it will typically not be energetically advantageous for the quenched polymer to be pinned. It is easily shown using the asymptotics established in [1] that for = 0 , M −1/2 and are of the same order as β → 0, while M −1/2 if 0 , and M −1/2 if 0 . Thus as 0, 0 is essentially the order at which “bad” blocks of length M start to become common. The question of whether the annealed and quenched critical points are different has concerned the physics community, with sometimes-disagreeing predictions. Based on nonrigorous expansions and renormalization techniques, it was claimed in [6] that when c = 3/2 and ϕ is asymptotically constant the two critical points are equal, while in [5] it was claimed that they are different and a prediction on the gap between them was provided. The question was also studied numerically in [11]. The above-mentioned result q from [8], that in this case c (β) > 0 for all β > 0, settled this matter. The following is our main result. Theorem 1.1. Suppose that V = (Vn )n≥1 is a sequence of i.i.d. standard Gauusian random variables. Then, writing u = − β2 + , (i) if (1.1) holds with c > 3/2, then there exist 0 , 1 > 0 such that for all β, > 0 q satisfying < 0 0 (β) and β < 1 , we have Cq (β, u) = 0; therefore u c (β) > a u c (β). If 3/2 < c < 2 then there is a constant K such that for all sufficiently q small β we have 0 0 < u c (β) − u ac (β) < K 0 . q (ii) if (1.1) holds with c = 3/2 and ϕ(n) → 0 as n → ∞, then u c (β) > u ac (β) for all β > 0. Theorem 1.1(i) improves on the recent result in [4] which establishes a positive lower q bound for c (β). The lower bound in [4], however, is o( 0 ) and therefore does not rule out the significant use of alternate strategies. Our proof is very different from [4] as well. Theorem 1.1(ii) improves a result in [4] which requires ϕ(n) = o((log n)−η ) for some η > 1/2. The condition ϕ(n) → 0 is equivalent to ϕˆc−1 (t) → ∞ as t → ∞, for ϕˆc−1 of (1.8) (see [1].) For c = 3/2 this is equivalent to the contact fraction having an infinite derivative (as a function of ) at = 0; see Lemma 3.1 below. In [1] it is proved that for the marginal case c = 3/2 the disorder is irrelevant, i.e. critical points and critical exponents are the same for quenched and annealed, as long as the slowly varying function ϕ(·) satisfies the condition ∞ n=1
1 < ∞. n(ϕ(n))2
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K. S. Alexander, N. Zygouras
There is a gap between such ϕ and those covered by Theorem 1.1(ii) and [8], for which the question of disorder relevance remains open. This gap includes for example ϕ(n) ∼ (log n)ν for 0 < ν ≤ 1/2. 2. Notation and Idea of the Proof Idea of the proof. We begin with an informal outline of the proof and the introduction of some preliminary notation. We use δ ∗ = δ ∗ ( ) as an alternate notation for the annealed contact fraction. The annealed correlation length is defined to be (β f a (β, u))−1 . The annealed free energy of (1.7) is given by the variational formula β f a (β, u) = sup(β δ − δ IE (δ −1 )), δ≥0
and δ ∗
is the value where this sup occurs [2]. Here IE is the large-deviations rate function of the excursion length variable E. For c > 1 we have β δ ∗ → (c − 1) ∧ 1 as β → 0; β f a (β, u) this is proved in [1] for c < 2 and extends readily to c ≥ 2. Therefore the annealed correlation length is asymptotically proportional to M = M( ) =
1 . β δ ∗ ( )
(2.1)
In order to show that the quenched free energy is zero, we need to show that the quenched partition function increases at most subexponentially. To do so we need to divide the paths into classes, and control the contribution to the partition function from each class. For a path x = (xn )n≤N , an excursion is called long if it exceeds a certain scale R = R( ) (to be determined), and short otherwise. We can view excursions as open intervals in the time axis; the closed intervals between long excursions are called occupied segments, and the union of the occupied segments forms the skeleton of the path x, denoted S(x, R), or just S(x) if no confusion is likely. The skeleton contact fraction of x is the fraction of indices i ∈ S(x) with xi = 0. We will show that attention can effectively be restricted to skeletons in which all occupied segments have length at least M. A path x has sparse returns if the skeleton contact fraction is less than δ2 = 2 δ ∗ ( ), (with 2 small, to be determined) and dense returns, otherwise. As we will see, sparsereturn paths are exponentially rare, and even in the annealed model their contribution to the partition function does not grow exponentially. This annealed contribution is an upper bound for the quenched case. More precisely, for a skeleton J we define |J | to be the number of sites in J , m(J ) + 1 to be the number of occupied segments in J , W(J ) = {x : S(x) = J }, W− (J , δ2 ) = {x : S(x) = J and x has sparse returns}, and W+ (J , δ2 ) = {x : S(x) = J and x has dense returns}.
Quenched vs. Annealed Critical Points
665
Ideally we would like to show that log P X (W− (J , δ2 ) | W(J )) ≤ −K |J |/R for some K , K , so the contribution to the partition function from W− (J , δ2 ) has logarithm at most β δ2 |J | −
K |J | + log P X (W(J )). R
(2.2)
The sum of the first two terms in (2.2) is negative, if K is large and we choose 1 R = R( ) = β δ . The same is true for the sum of the first three terms, if we discard (in 2 an appropriate sense) short occupied segments to ensure that |J |/m(J ) is large. Therefore in this case the whole expression in (2.2) would be negative. We cannot actually do exactly this; we need to incorporate coarse-graining to group together similar skeletons J first, and there is a positive term proportional to m(J ) in (2.2), but the idea is the same. In contrast to sparse returns, the contribution from paths with dense returns cannot be handled by comparison to the annealed system. In this case we will use semiannealed estimates. That is, we will first compute the conditional expectation of the contribution J to the partition function from W+ (J , δ2 ) given a certain average value V over the skeleton J (or more precisely, over a coarse-grained approximation to J .) This conditional expectation is easily shown to be
J β2 L2 J β( +V )L N − 2|JN| V X β H Nu (x,V) X E e =E ; W+ (J , δ2 ) V ; W+ (J , δ2 ) . E e (2.3) The quadratic term −β 2 L 2N /2|J | in the exponent in (2.3) reflects the fact that conditionJ
ing on V reduces the exponential moment (under E V ) of H Nu (x, V), and this reduction increases with the skeleton contact fraction L N /|J |; for dense-return paths the reduction becomes large enough to be useful in establishing that the partition function grows at most subexponentially. An annealed estimate at this point would amount to taking the J expectation with respect to V in (2.3). But this would cancel the essential quadratic term. Instead, letting DJ (x) denote the contact fraction within J , we will find a function g(J , δ) and a set TN of disorders such that lim inf N P V (TN ) > 0 and such that for every J disorder in TN and every path in W+ (J , δ2 ), we have β( + V )L N ≤ g(J , DJ (x)). Then also J
J
β( + V )L N ≤ λβ( + V )L N + (1 − λ)g(J , DJ (x))
(2.4)
for every 0 < λ < 1. The next step is to perform an annealed estimate for the (semiannealed) partition function which has the right side of (2.3) replaced by its upper J bound from (2.4). The logarithm of the exponential moment of λβV L N will then be 2 2 2 λ β L N /2|J | which now does not fully cancel the quadratic term −β 2 L 2N /2|J | and will result in the desired control. By means of this estimate we will only be able to say that the partition function on W+ (J , δ2 ) increases subexponentially on the set TN . But since TN has uniformly positive probability and the quenched free energy is nonrandom off a null set of disorders, necessarily the quenched free energy will be zero. As noted in [9], for technical convenience the partition function Z N in (1.4) can be replaced by the constrained partition function
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K. S. Alexander, N. Zygouras
u Z 0N = E X eβ HN (x,V) δ0 (x N ) ,
(2.5)
as both give the same free energy and contact fraction. Notation. Throughout the paper, K i and i represent constants which depend only on c and ϕ from (1.1). Define R( ) =
1 , β δ2
(2.6)
and δ2 = 2 δ ∗ ( ) with 2 to be specified, satsfying 2 < 1/2 so that 2M( ) < R( ). For a path x and A ⊂ R we define the local time of x in A and the corresponding contact fraction: L A = L A (x) =
δ0 (xn ),
D A = D A (x) =
n∈A
L A (x) , |A|
where |A| denotes the number of sites in A. We abbreviate L [0,N ] as L N . For a set A of nonnegative integers, we define the average disorder V
A
=
1 Vn . |A| n∈A
For a general subset B of R, we define V
B
=V
B∩Z
m(n) = E X (E; E ≤ n) =
. For n ≥ 1 we let
n
k −c ϕ(k).
k=1
We denote the length of the i th excursion from 0 for a path x by Ei = Ei (x) for i ≥ 1. Let , 1 , 2 sets of paths. We use the notation u Z N () = E X eβ HN (x,V) δ (x) , u Z N (1 |2 ) = E X eβ HN (x,V) δ1 (x) | 2 ,
(2.7)
N and similarly for Z 0N . For a ≤ b we can replace n=0 with bn=a in the definition (1.3) of the Hamiltonian, and we may restrict to a set of paths as in (2.7); we denote the u (x, V) and Z resulting Hamiltonian and partition function by H[a,b] [a,b] (), respectively, suppressing the dependence on (β, u, V) in the latter notation. Definition 2.1. An R-skeleton (or just a skeleton if confusion is unlikely) in [0, N ] is a m (a , b ), with m ≥ 0, 0 ≤ a < b < · · · < a < b ≤ N and set of form [0, N ] \ ∪i=1 i i 1 1 m m bi − ai ≥ R for i = 1, . . . , m. In this context we use the notation b0 = 0, am+1 = N . We denote a generic skeleton by J , and m(J ) denotes the number of open intervals in [0, N ] \ J . The intervals [bi−1 , ai ], 1 ≤ i ≤ m + 1, are called the occupied segments of J . An occupied segment is short if its length is at most M( ), and long otherwise.
Quenched vs. Annealed Critical Points
667
[0, a1 ] and [bm , N ] are the initial and final occupied segments, respectively, and all other occupied segments are called central. For a skeleton J we then define W(J ) = W(J , δ) = W+ (J , δ) = W− (J , δ) =
{x : S(x) = J }, {x : S(x) = J , DJ (x) = δ}, {x : S(x) = J , DJ (x) > δ}, {x : S(x) = J , DJ (x) ≤ δ}.
A skeleton J and a value δ > 0 are called compatible if W(J , δ) = φ. For a skeleton in [0, N ], a compatible δ is always a rational number with denominator at most N . Definition 2.2. A lifted skeleton, generically denoted Jˆ , is a skeleton in which all central occupied segments are long. To each skeleton J there corresponds a lifted skeleton ˆ ), obtained by deleting from J all short central occupied segments. We define the L(J ˆ ˆ lifted skeleton of x to be S(x) = L(S(x)), and form classes of paths according to the contact fraction in this lifted skeleton: ˆ + (Jˆ , δ) = {x : S(x) ˆ W = Jˆ , DJˆ (x) > δ},
ˆ − (Jˆ , δ) = {x : S(x) ˆ W = Jˆ , DJˆ (x) ≤ δ}. We then define T (δ) =
Jˆ
D(δ) =
Jˆ
ˆ − (Jˆ , δ) = {x : D ˆ (x) ≤ δ}, W S (x)
(2.8)
ˆ + (Jˆ , δ) = {x : D ˆ (x) > δ}. W S (x)
(2.9)
A path in T (δ2 ) is said to have sparse returns, and a path in D(δ2 ) is said to have dense returns. When we will deal with paths having dense returns, we will need to use a coarse graining (CG) scheme, which we introduce now. Definition 2.3. We fix 3 , to be specified, such that 3 R( ) is an integer. A CG block is an interval of form [(k − 1) 3 R( ), k 3 R( )] with k ≥ 1. A CG point is an endpoint of a CG block. We assume that N is a CG point. A CG skeleton is a skelm (a , b ) in which all a , b are CG points. We denote a generic CG eton [0, N ] \ ∪i=1 i i i i skeleton by J ∗ , and write w(J ∗ ) for the number of CG blocks comprising J ∗ . Given m (a , b ) we let a ∗ and b∗ denote the smallest CG point a skeleton J = [0, N ] \ ∪i=1 i i i i greater than ai and the largest CG point less than bi , respectively, and we define the m (a ∗ , b∗ ), which is the union of all CG blocks that CG skeleton L∗ (J ) = [0, N ] \ ∪i=1 i i intersect J . If J is an R( )-skeleton then L∗ (J ) is a (1 − 2 3 )R( )-skeleton. We let S ∗ (x) = L∗ (S(x)). A lifted CG skeleton is a CG skeleton of form L∗ (Jˆ ), where Jˆ is a lifted skeleton; we denote a generic lifted CG skeleton by Jˆ ∗ . We again form classes of paths according to the contact fraction in the lifted CG skeleton: W ∗ (J ∗ ) = {x : S ∗ (x) = J ∗ }, W ∗ (J ∗ , δ) = {x : S ∗ (x) = J ∗ , DJ ∗ (x) = δ}, ˆ ∗ (Jˆ , δ) = {x : S(x) = Jˆ , D ∗ ˆ (x) = δ}. W L (J )
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K. S. Alexander, N. Zygouras
To deal with paths having sparse returns, we need a different coarse-graining scheme, as follows. Definition 2.4. With R = R( ), fix some (small) 4 > 0 such that (1 + 4 )l1 = R for some integer l1 , and let l0 = max{k : (1 + 4 )k < M( )/4}. Define intervals Il0 = [0, (1 + 4 )l0 ], Ik = ((1 + 4 )k−1 , (1 + 4 )k ], l0 < k ≤ l1 , Il1 +k = (R + (k − 1) 4 R, R + k 4 R], k ≥ 1. + We write n − k and n k for the smallest and largest integers, respectively, in Ik . A semi-CG skeleton is a skeleton in which each occupied segment has length in {n +k , k ≥ l0 }. We m (a , b ) denote a generic semi-CG skeleton by J s . Given a skeleton J = [0, N ] \ ∪i=1 i i s + + we let ai = bi−1 + min{n k : bi−1 + n k ≥ ai } and define the semi-CG skeleton Ls (J ) = m (a s , b ), which is the smallest semi-CG skeleton containing J . Note that [0, N ] \ ∪i=1 i i Ls (J ) is determined by specifying for each occupied segment of J (i) its exact starting point, and (ii) the value of k for which Ik contains the segment’s length. Also, if J is an R-skeleton then Ls (J ) is a ((1 − 4 )R)-skeleton. We let S s (x) = Ls (S(x)). A lifted semi-CG skeleton is a semi-CG skeleton of form Ls (Jˆ ), where Jˆ is a lifted skeleton; we denote a generic lifted semi-CG skeleton by Jˆ s . We once more form classes of paths according to the contact fraction in the lifted semi-CG skeleton:
W s (J s ) = {x : S s (x) = J s }, s ˆs W− (J , δ) = {x : Sˆ s (x) = Jˆ s , DSˆ (x) (x) ≤ δ}. ˆ ∗ (Jˆ , δ) in Definition 2.3, the condition that the density of Note that in contrast to W ˆ returns be at most δ here is applied to the density in S(x). 3. Paths with Dense Returns Recall that a path is said to have dense returns if its skeleton contact fraction is greater than δ2 = 2 δ ∗ ( ). Let also α0 = α0 (β ) be given by E X e−α0 E = e−β . The following result on the concavity of the contact fraction shows the relevance of our hypotheses on c and ϕ. Lemma 3.1. (i) Suppose that P X satisfies (1.1) with c > 3/2. There exists 5 as follows. For every K > 0 there exists > 0 such that 0 < β < 5 and < 0 (β) imply δ ∗ ( ) ≥ K /β. (ii) Suppose that P X satisfies (1.1) with c = 3/2 and ϕ(n) → 0 as n → ∞. Then δ ∗ ( )/ → ∞ as → 0. For 3/2 < c < 2, for small β the condition < 0 (β) will imply the condition 0 < β < 5 , while for large β the reverse implication will hold. In other words, for small β the hypothesis is that < 0 (β), and for large β the hypothesis is that β is small.
Quenched vs. Annealed Critical Points
669
Proof of Lemma 3.1. Case 1. Suppose that E X (E) < ∞, so that c ≥ 2. Then the transition is first order, with 1/E X (E) ≤ δ ∗ ( ) ≤ 1 for all > 0 (see [7], Theorem 2.1), while 0 (β) ≤ β for all β > 0. Hence β 1 δ ∗ ( )β ≥ X ≥ , X E (E) 0 (β) E (E) and the result follows immediately, with 5 = ∞. Case 2. Suppose c < 2. We can extend ϕ from Z+ to [1, ∞) by piecewise linearity; the result is still slowly varying. Define ϕ(x) = 1/ϕ(x 1/(c−1) ) and let ϕ ∗ be a slowly varying function conjugate to ϕ. ϕ ∗ is characterized (up to asymptotic equivalence) by the fact that ϕ ∗ (xϕ(x)) ∼
1 as x → ∞; ϕ(x)
(3.1)
see [12]. Then define ϕ(x) ˆ = ϕˆc−1 (x) = ϕ ∗ (x)−1/(c−1) and G β ( ) =
δ ∗ ( ) 2 δ ∗ ( )β = β , 2 2β
so G β ( 0 (β)) = 1. From [1] we have δ ∗ ( ) ∼ K 2 (β )−(2c−3)/(c−1) ϕˆ β
1 β
as β → 0,
so there exists 6 such that β < 6 implies δ ∗ ( ) 1 1 1 K 2 (β )−(2c−3)/(c−1) ϕˆ ≤ ≤ 2K 2 (β )−(2c−3)/(c−1) ϕˆ . (3.2) 2 β β β Under the assumptions in (ii) the exponent in (3.2) is 0, and we have ϕ(x) → 0 as x → ∞ so ϕ(x) → ∞, so ϕ ∗ (x) → 0, so ϕ(x) ˆ → ∞. Therefore G β ( ) → ∞ as → 0 and (ii) is proved. Thus suppose 3/2 < c < 2. Since 0 (β) ≤ β, there exists β0 such that β < β0 implies β 0 (β) < 6 . Then for β < β0 and < 0 (β), by (3.2)
1 G β ( ) 1 β −(2c−3)/(c−1) ϕˆ β
. ≥ G β ( ) = (3.3) G β ( 0 ) 4 β 0 ϕˆ 1 β 0
With a reduction in 6 if necessary, we then have β −(2c−3)/2(c−1) G β ( ) ≥ , β 0 which exceeds K for small / 0 , proving (ii) for β < β0 . For β ≥ β0 we can use (3.2) to conclude that if β is less than some 7 then we have ∗ 1 2 1 2 δ ( ) −(2c−3)/(c−1) ≥ β0 K 2 (β ) ≥ K, G β ( ) ≥ β0 ϕˆ 2β 4 β proving (i) for β ≥ β0 .
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K. S. Alexander, N. Zygouras
Case 3. It remains to consider c = 2 with E X (E) = ∞. Here to obtain a substitute for (3.2) we need to consider the asymptotics of δ ∗ ( ) as β → 0. First observe that for fixed a > 1, for large n, ϕ(n) log a ≤ ϕ(n)
n a −1≤k≤n−1
1 ≤2 k n
2 −1≤k≤n−1
ϕ(k) ≤ 2m(n), k
(3.4)
so that for sufficiently large s we have ϕ(s) ≤ m(s),
(3.5)
and hence for small t, 1 − M E (−t) = ≥
∞ ϕ(k) (1 − e−tk ) 2 k k=1 t ϕ(k)
2
1≤k≤1/t
k
1 1 = tm 2 t
(3.6)
and ϕ(k) ϕ(k) + k k2 1≤k≤1/t k>1/t 1 1 + 2tϕ ≤ tm t t 1 . ≤ 3tm t
1 − M E (−t) ≤ t
(3.7)
Let α± = α± (β ) be given by 1 1 1 = e−β , 1 − 3α− m = e−β , 1 − α+ m 2 α+ α− so that α− ≤ α0 ≤ α+ for small β , by (3.6) and (3.7). As β → 0 we have 1 1 1 1 1 ∼ ∼ β or α+ m , 2 α+ α+ m 1 2β α+
(3.8)
and hence 2β 2β ∼ α+ (β ) ∼ ∗ , 1 1 m α1+ m β (the second equivalence being a consequence of (3.8) and the definition (3.1)), and then ∗ 1 1 1 ∼ . (3.9) m α+ m β
Quenched vs. Annealed Critical Points
671
The same holds similarly for α− in place of α+ , and hence also for α0 . Also, as β → 0, ∞
1 ϕ(k) = (log M , ) (−α ) ∼ M (−α ) = e−α0 k E 0 0 E δ ∗ ( ) k k=1
which analogously to (3.6) leads to 1 1 1 1 1 1 ≤m ≤ ∗ +ϕ ≤ 2m , m 3 α0 δ ( ) α0 α0 α0 where the last inequality follows from (3.5). This and (3.9) (with α0 in place of α+ ) show that for small β , ∗ 1 1 1 1 1 ∗ 1 ≤ ∗ ≤3 . (3.10) 4 m β δ ( ) m β Since m is slowly varying, so is (1/m)∗ , so we can use (3.10) in place of (3.2) to prove (i) for c = 2 with E X [E] = ∞ in the same manner as we did for 3/2 < c < 2. Recall that R( ) and M( ) are defined in (2.6) and (2.1). Note that R( ) β δ ∗ ( ) 1 = = > 1. M( ) β δ2 2
(3.11)
Let 23 < λ < 1 and let K 3 > 2 be specified (see Lemma 3.8.) For fixed 2 , we take 3 < 1 small enough so that 4 3 1 < . 2 2 By (3.11) a CG block is at most an h 1 /4 fraction of a long occupied segment: h1 =
3 R( ) =
h1 M( ). 4
(3.12)
(3.13)
Let K 4 satisfy
and
1 4 < K 4 2 4
(3.14)
1 1 (1 − λ) λ − 3 2 K 4 ≥ K 3 . 8 2
(3.15)
By Lemma 3.1, for sufficiently small 0 and β , for < 0 0 (β) we have δ ∗ ( ) ≥ K 4
. β
(3.16)
For a lifted CG skeleton Jˆ ∗ we define 1 ψλ (Jˆ ∗ , v, δ) = λβ( + v)δ| Jˆ ∗ | − β 2 δ 2 | Jˆ ∗ |, (3.17) 2 3 g(Jˆ ∗ , δ) = β 2 δ 2 | Jˆ ∗ | − log P X (W ∗ (Jˆ ∗ , δ)). (3.18) 4 Observe that if β( + v)δ| Jˆ ∗ | ≤ g(Jˆ ∗ , δ) then ψ1 (Jˆ ∗ , v, δ) ≤ ψλ (Jˆ ∗ , v, δ) + (1 − λ)g(Jˆ ∗ , δ).
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K. S. Alexander, N. Zygouras
m (a , b ) be a lifted skeleton, with m ≥ 2. Then Lemma 3.2. Let Jˆ = [0, N ] \ ∪i=1 i i
| Jˆ | ≤ | L∗ (Jˆ ) | ≤ (1 + h 1 ) | Jˆ |. Proof. The first inequality is clear. Regarding the second one we have | L∗ (Jˆ ) | =
m+1 ∗ (ai∗ − bi−1 ) i=1
≤ (a1 − b0 ) + (am+1 − bm ) + 2 3 R( ) +
m
((ai − bi−1 ) + 2 3 R( ))
i=2
≤ (a1 − b0 ) + (am+1 − bm ) +
m
(3.19)
((ai − bi−1 ) + 4 3 R( ))
i=2
m+1 4 3 4 3 R( ) |Jˆ |, (ai − bi−1 ) = 1 + ≤ 1+ M( ) 2 i=1
where the last equality follows from (3.11).
The next lemma gives a uniform lower bound for the size of a set TN of disorders in which the averages over skeletons are uniformly well-controlled. Lemma 3.3. There exists ρ = ρ( 3 ) > 0 as follows. For the event Jˆ ∗ ∗ ∗ ˆ ˆ ( Vi )i≤N : βδ( + V )| J | ≤ g(J , δ) , TN = Jˆ ∗ δ≥(1−h 1 )δ2
we have P V (TN ) ≥ ρ for all large N . Here the second intersection is over δ compatible with Jˆ ∗ . Proof. By (3.16) and Lemma 3.2, for δ ≥ (1 − h 1 )δ2 ≥ 21 2 δ ∗ ( ) we have βδ ≥
1 1 2 βδ ∗ ( ) ≥ K 4 2 , 2 2
(3.20)
while |Jˆ ∗ | ≥ 2 3 R( ) for all Jˆ ∗ . Hence by Chebyshev’s inequality and (3.14) we have Jˆ ∗ V ∗ ∗ ˆ ˆ βδ( + V )| J | > g(J , δ) P Jˆ ∗ ˆ ∗ βδ | Jˆ ∗ |−g(Jˆ ∗ ,δ) V βδV |J | E e ≤e 1 2 2 ˆ∗ ∗ ˆ = exp βδ | J | − β δ |J | P X (W ∗ (Jˆ ∗ , δ)) 4 1 ∗ ˆ ≤ exp 1 − 2 K 4 β δ|J | P X (W ∗ (Jˆ ∗ , δ)) 8 ≤ exp (−2β δ 3 R( )) P X (W ∗ (Jˆ ∗ , δ)) ≤ exp(− 3 )P X (W ∗ (Jˆ ∗ , δ)). We now sum over Jˆ ∗ and δ and take ρ = 1 − e− 3 .
(3.21)
Quenched vs. Annealed Critical Points
673
The next step is to separate the contribution to the partition function from the short segments of the skeletons from that of the long segments. Before doing this we need some more definitions. We use x[a,b] to denote a generic path (xi )a≤i≤b . When confusion is unlikely, given a path x = x[0,N ] , we also let x[a,b] denote the segment of x from a to b. Definition 3.4. For b − a ≥ R( ), we will denote by Q[a,b] the set of all paths x[a,b] such that (i) xa = xb = 0. (ii) The excursion starting from a and the excursion ending at b (which may be the same excursion) are long. (iii) All the occupied segments are short. The normalized partition function over the set Q[a,b] is Q [a,b] = For a lifted skeleton Jˆ = [0, N ]\
1 pb−a
Z [a,b] ( Q[a,b] ).
m
i=1 (ai , bi )
Q(Jˆ ) =
m
we define
Q [ai ,bi ] ,
i=1
which can be viewed as a factor in the total contribution to the overall partition function ˆ ) = Jˆ . Finally we let Y[a,b),r denote the set of paths x from skeletons J with L(J satisfying xa = 0 and having no excursions longer than r and starting in [a, b], and 0 0 0 . If Y[a,b),r = Y[a,b),r ∩ {xb = 0}. We abbreviate Y[0,n),R as Yn,R , and Y[0,n),R as Yn,R 0 [a, b] is an occupied segment in a path x, then necessarily x ∈ Y[a,b) . Observe that Z 0N (D(δ2 )) =
Z 0N (W+ (J , δ2 ))
Jˆ J :Lˆ (J )=Jˆ
=
Jˆ
≤
Z 0N (W+ (Jˆ , δ2 ))Q(Jˆ )
ˆ ∗ (Jˆ , δ) | W ˆ ∗ (Jˆ , δ))P X (W ˆ ∗ (Jˆ , δ)), Q(Jˆ )Z 0N (W
Jˆ δ≥(1−h 1 )δ2
(3.22) where the last sum is over δ compatible with L∗ (Jˆ ). To control the growth of (3.22), we will need some estimates for quantities related to those appearing on the right side. Let us start with P X (W ∗ (Jˆ ∗ )). Define Ik∗ = [(k − 1) 3 R( ), k 3 R( )].
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K. S. Alexander, N. Zygouras
Proposition 3.5. Let 8 > 0. Then there exists K 5 such that, provided β is sufficiently m (ai∗ , bi∗ ) a lifted CG skeleton, for small (depending on 3 , 8 ), for Jˆ ∗ = [0, N ]\ i−1 ∗ the positive integers ki , i given by ai = ki 3 R( ), bi∗ = i 3 R( ), we have P X (W ∗ (Jˆ ∗ )) ≤
m i=1
K5 . (i − ki )(1− 8 )c
(3.23)
Proof. Write R for R( ). We sum over the starting and ending points for the long excursions, within the CG blocks:
P X W ∗ (Jˆ ∗ ) =
P X (W(Jˆ ))
Jˆ :L∗ (Jˆ )=Jˆ ∗ m+1
≤
(ai ∈Ik∗ , bi ∈I∗ +1 )i≤m i=1 i
i=1
i
m
≤
m 0 xb = 0) P X (Y[b pbi −ai i−1 i−1 ,ai ),R
P X (xai = 0 | Y[bi−1 ,ai ),R ) pbi −ai
(ai ∈Ik∗ , bi ∈I∗ +1 )i≤m i=1 i
i
m
=
P X (xai = 0 | Y[bi−1 ,ai ),R )
(ai ∈Ik∗ , bi ∈I∗ +1 )i≤m i=1 i
ϕ(bi − ai ) . (bi − ai )c
i
(3.24) We have (i − ki ) 3 R ≤ bi − ai ≤ (i − ki + 2) 3 R, and provided 3 R is large enough (depending on 8 ), i.e. β is small enough, ϕ ( 3 R(i − ki )) ≤ (i − ki ) 8 c ϕ( 3 R). Therefore we can bound (3.24) by
m
P X (xai = 0 | Y[bi−1 ,ai ),R )
(ai ∈Ik∗ , bi ∈I∗ +1 )i≤m i=1 i
=
i
m
P X (xai = 0 | Y[bi−1 ,ai ),R )
i=1 ai ∈Ik∗ , bi−1 ∈I∗ m
2ϕ( 3 R) − ki )(1− 8 )c
( 3 R)c (i
i−1 +1
i
=
2ϕ( 3 R) ( 3 R)c (i − ki )(1− 8 )c
E X (L Ik∗ | Y[bi−1 ,ai∗ ),R ) i
i=1 bi−1 ∈I∗
2ϕ( 3 R) − ki )(1− 8 )c
( 3 R)c (i
i−1 +1
≤
m i=1
3 R
max ∗
bi−1 ∈I
i−1 +1
E X (L Ik∗ | Y[bi−1 ,ai∗ ),R ) i
2ϕ( 3 R) . − ki )(1− 8 )c
( 3 R)c (i
(3.25)
Quenched vs. Annealed Critical Points
675
We now need the bound E X (L Ik∗ | Y[bi−1 ,ai∗ ),R ) ≤ E X (L 3 R | Y 3 R,R )
max
bi−1 ∈I∗
i−1 +1
i
=
P X (L 3 R ≥ k | Y 3 R,R )
k
≤
P X (max Ei ≤ 3 R | Y 3 R,R ) i≤k
k
k 1 − P X (E > 3 R | E ≤ R) = k
≤
X e−k P (E > 3 R|E ≤R)
k
1 > 3 R | E ≤ R) K 6 ( 3 R)c−1 . ≤ ϕ( 3 R)
≤
P X (E
Inserting this bound into (3.25) we obtain (3.23).
(3.26)
Lemma 3.6. Let 0 < θ < 1. Provided 2 is sufficiently small, and β is sufficiently small (depending on 2 ), for all a, b with b − a ≥ R( ) we have E V [ Q [a,b] ] ≤
1 . 1−θ
Proof. We write R, M for R( ), M( ), respectively. Let Q k[a,b] , k ≥ 0, be the contribution to Q [a,b] from paths which have k short occupied segments, so that Q 0[a,b] = 1. Let us first estimate the contribution Q 1[a,b] =
1 pb−a
a+R≤ j1 < j2 ≤b−R j2 ≤ j1 +M
β Hu (x,V) p j1 −a E X e [ j1 , j2 ] δ{x j2 =0} | x j1 = 0 pb− j2 .
Let W[a,b] denote the last time that the path visits zero in the interval [a, b]. Using the symmetry over the indices j1 , j2 we get that Q 1[a,b] ≤
≤
2 pb−a 2 pb−a
a+R≤ j1 < j2 ≤b−R j2 ≤ j1 +M; j1 ≤ b+a 2
a+R≤ j1 < j2 ≤b−R j2 ≤ j1 +M; j1 ≤ b+a 2
β Hu (x,V) p j1 −a E X e [ j1 , j2 ] δ{x j2 =0} | x j1 = 0 pb− j2 β Hu (x,V) p j1 −a E X e [ j1 , j1 +M] δ{W[ j1 , j1 +M] = j2 } | x j1 = 0
·
P X (E
pb− j2 . > M − j2 + j1 )
(3.27)
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K. S. Alexander, N. Zygouras
Recalling that M < R, we have b − j2 ≥ (b − a)/4, and thus pb− j2 ≤ 2 p(b−a)/4 , for all j2 appearing in the sum. Therefore (3.27) yields E V [Q 1[a,b] ] ≤
4 pb−a ·
p(b−a)/4 P X (E > M)
u p j1 −a E X E V eβ HM (x,V) ; W[0,M] = j2 − j1
a+R≤ j1 < j2 ≤b−R j2 ≤M; j1 ≤ b+a 2
≤
u (x,V) 4c+2 X V β HM E e E P X (E > M)
p j1 −a
a+R≤ j1 ≤ b+a 2
4c+2 X β L M e P X (E > R). (3.28) E P X (E > M) From [1] we have E X eβ L M < e K 7 for some constant K 7 . Hence provided β is small (depending on 2 ), from (3.11) we have ≤
P X (E > R) P X (E > M) c−1 K7 M ≤ 2K 8 e R
E V [ Q 1[a,b] ] ≤ K 8 e K 7
= 2K 8 e K 7 2c−1 . We now take 2 small enough so the last quantity is at most θ . For y ∈ [a, b] and k ≥ 1 let Aiy denote the event that the i th long excursion starting at or after a ends at y, and let Q k[a,b] (Aiy ) denote the contribution to Q k[a,b] from paths in Aiy . For k = 1 we have Q 1[a,b] (A2b ) = Q 1[a,b] since all contributing paths have the second excursion ending at b, and similarly for k = 0 we have Q 0[a,b] (A1b ) = Q 0[a,b] = 1. Thus what we have shown is that E V [ Q 1[a,b] (A2b ) ] ≤ θ E V [ Q 0[a,b] (A1b ) ]. The same argument applied to the interval [a, y] in place of [a, b] gives E V [ Q k[a,b] (A2y ) ] ≤ 1 θ E V [ Q k−1 [a,b] (A y ) ] for all k ≥ 2 and all y, so summing over y and then iterating over k k V gives E [ Q [a,b] ] ≤ θ k . Then E V [ Q [a,b] ] ≤
k≥0
E V [ Q k[a,b] ] ≤
k≥0
θk =
1 . 1−θ
In bounding Z 0N (D(δ2 )) via (3.22), the crucial estimate will be on the partition function
ˆ ∗ (Jˆ , δ) | W ˆ ∗ (Jˆ , δ)) = exp Y (δ, Jˆ ) + Y (δ, Jˆ ) , Z 0N (W where Y (δ, Jˆ ) = log E V
ˆ ∗ (Jˆ , δ) | W ˆ ∗ (Jˆ , δ)) V L∗ (Jˆ ) Z 0N (W
Quenched vs. Annealed Critical Points
677
and Y (δ, Jˆ ) = log
EV
ˆ ∗ (Jˆ , δ) | W ˆ ∗ (Jˆ , δ)) Z 0N (W . ˆ ∗ (Jˆ , δ) | W ˆ ∗ (Jˆ , δ)) V L∗ (Jˆ ) Z 0N (W
Recalling (3.17) and (3.18), define ∗ ˆ Yλ (δ, Jˆ ) = ψλ (L∗ (Jˆ ), V L (J ) , δ) + (1 − λ)g(L∗ (Jˆ ), δ).
Lemma 3.7. For all 0 < λ < 1, all lifted skeletons Jˆ and all δ ≥ (1 − h 1 )δ2 compatible ∗ ˆ with L∗ (Jˆ ) we have Y (δ, Jˆ ) ≤ ψ1 (L∗ (Jˆ ), V L (J ) , δ), and on the set TN we have Y (δ, Jˆ ) ≤ Yλ (δ, Jˆ ). L∗ (J )
= v for some v, (Vn )n∈L∗ (J ) is multivariate normal Proof. Conditionally on V with easily calculated mean and covariance; as noted in [1] it follows readily that ˆ ˆ ∗ (Jˆ , δ) | W ˆ ∗ (Jˆ , δ)) V L∗ (Jˆ ) eY (δ,J ) = E V Z 0N (W ⎤ ⎞ ⎡ ⎛ β 2 L 2ˆ 1 ∗ ˆ J ⎠ ˆ∗ ˆ W (J , δ)⎦ . = E X ⎣exp ⎝ β( + V J ) L Jˆ − (3.29) 2 | L∗ (Jˆ ) | ˆ ∗ (Jˆ , δ) of paths, we have By Lemma 3.2, on the set W 2 2 1 β L Jˆ ∗ ˆ β( + V L (J ) ) L Jˆ − 2 | L∗ (Jˆ ) |
1 ∗ ˆ = β( + V L (J ) )δ | L∗ (Jˆ ) | − β 2 δ 2 | L∗ (Jˆ ) | 2 ∗ ˆ = ψ1 (L∗ (Jˆ ), V L (J ) , δ), and it is immediate from the definitions that on the set TN of disorders, we have ∗ ˆ ∗ ˆ ψ1 (L∗ (Jˆ ), V L (J ) , δ) ≤ ψλ (L∗ (Jˆ ), V L (J ) , δ) + (1 − λ)g(L∗ (Jˆ ), k(δ)),
which with (3.29) yields the result.
Equation (3.22) and Lemma 3.7 together show that on the set TN , Z 0N (D(δ2 )) is bounded above by
ˆ ∗ (Jˆ , δ)), (3.30) Z 0N ,λ = Q(Jˆ ) exp Yλ (δ, Jˆ ) + Y (δ, Jˆ ) P X (W Jˆ
δ≥(1−h 1 )δ2
where the second sum is over δ compatible with L∗ (Jˆ ). We will show that E V [ Z 0N ,λ ]
increases at most polynomially in N . Now Q(Jˆ ) and exp Yλ (δ, Jˆ ) + Y (δ, Jˆ ) are
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K. S. Alexander, N. Zygouras
L∗ (Jˆ ) independent functions of V for fixed Jˆ , and E V (exp(Y (δ, Jˆ )) | V ) = 1, so we have
E V [ Z 0N ,λ ] =
ˆ ∗ (Jˆ , δ)). E V [ Q(Jˆ ) ] E V exp Yλ (δ, Jˆ ) P X (W
Jˆ δ≥(1−h 1 )δ2
(3.31) Moreover, recalling w(L∗ (Jˆ )) from Definition 2.3, we have the following estimate. Lemma 3.8. Given K 3 > 0, provided 0 is sufficiently small, for all δ ≥ (1 − h 1 )δ2 and < 0 0 we have
−(1−λ) E V exp Yλ (δ, Jˆ ) ≤ exp −K 3 w(L∗ (Jˆ )) P X W ∗ (L∗ (Jˆ ), δ) . Proof. Using (3.20) and (3.15), for K 9 = 41 (1 − λ) λ − 21 we obtain
E V exp Yλ (δ, Jˆ ) 1 2 3 ∗ ˆ 2 2 ∗ ˆ 2 2 ∗ ˆ = exp λβδ | L (J ) | + (λ − 1)β δ |L (J )| + (1 − λ)β δ |L (J )| 2 4
−(1−λ) ·P X W ∗ (L∗ (Jˆ ), δ)
−(1−λ) = exp λβδ − 2K 9 β 2 δ 2 | L∗ (Jˆ ) | P X W ∗ (L∗ (Jˆ ), δ)
−(1−λ) ≤ exp −( 2 K 9 K 4 − λ)β δ| L∗ (Jˆ ) | P X W ∗ (L∗ (Jˆ ), δ)
−(1−λ) 1 ∗ ˆ ≤ exp − 2 K 9 K 4 3 w(L (J )) P X W ∗ (L∗ (Jˆ ), δ) 2
−(1−λ) ≤ exp −K 3 w(L∗ (Jˆ )) P X W ∗ (L∗ (Jˆ ), δ) . (3.32) One can think of K 3 as the “cost per CG block” of an occupied segment, averaged over the disorder, on the set TN of disorders where Lemma 3.7 applies. Lemma 3.8 says that this cost can be made arbitrarily large by taking 0 small. By contrast, the annealed system has a bounded gain per block, because the negative term in the exponent on the right side of (3.29) is absent. We can now conclude the following. Lemma 3.9. Provided 2 and then 0 are chosen sufficiently small, for sufficiently small β , E V [ Z 0N ,λ ] grows at most linearly in N .
Quenched vs. Annealed Critical Points
679
Proof. From (3.31), Lemma 3.6 (with θ = 1/2) and Lemma 3.8 (with K 3 ≥ 2 to be specified), since there are at most N values of δ compatible with a given Jˆ ∗ , we have E V [ Z 0N ,λ ] ≤
ˆ 2m(J ) exp −K 3 w(Jˆ ∗ )
Jˆ ∗ δ≥(1−h 1 )δ2 Jˆ :L∗ (Jˆ )=Jˆ ∗
−(1−λ) X ∗ ˆ∗ X
ˆ ∗ (Jˆ , δ)) P (W ·P W (J , δ)
λ ˆ∗ ≤ 2w(J ) exp −K 3 w(Jˆ ∗ ) P X W ∗ (Jˆ ∗ , δ) δ≥(1−h 1 )δ2
Jˆ ∗
≤N
∞ l=1
λ P X W ∗ (Jˆ ∗ ) .
e−K 3 l/2
{Jˆ ∗ :w(Jˆ ∗ )=l}
Since λ > 2/3, we can take 8 so that (1 − 8 )λc > 1. For fixed l a CG skeleton Jˆ ∗ with w(Jˆ ∗ ) = l can be characterized by a sequence of l − 1 positive integers, the j th integer giving the number of CG blocks from the j th CG block in Jˆ ∗ to the ( j + 1)st CG block in Jˆ ∗ . Therefore by Proposition 3.5 we can take K 3 such that ⎛ ⎞l−1 ∞
λ 1 λ(l−1) ⎝ ⎠ P X W ∗ (Jˆ ∗ ) ≤ K 5 ≤ e K 3 l/4 . 1+ j (1− 8 )λc {Jˆ ∗ :w(Jˆ ∗ )=l}
j=2
Then E V [ Z 0N ,λ ] ≤ N
∞
e−K 3 l/4 ≤ 3N e−K 3 /4 .
l=1
The following is straightforward from Lemma 3.9, Cheyshev’s inequality and the Borel-Cantelli lemma. Proposition 3.10. Provided 2 and then 0 are chosen sufficiently small, for sufficiently small β , with P V probability one, we have lim sup N →∞
1 log Z 0N ,λ = 0. N
4. Paths with Sparse Returns We estimate Z 0N (T (δ2 )) using the following variant of (3.22): E V [Z 0N (T (δ2 ))] =
E V [Q(Jˆ )] E V [Z 0N (W− (Jˆ , δ2 ))]
Jˆ s {Jˆ :Ls (Jˆ )=Jˆ s }
≤
Jˆ s
s ˆs E V [Z 0N (W− (J , δ2 ))]
max
{Jˆ :Ls (Jˆ )=Jˆ s }
E V [Q(Jˆ )]
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K. S. Alexander, N. Zygouras
=
s ˆs E V Z 0N W− (J , δ2 ) | W s (Jˆ s ) P X (W s (Jˆ s ))
Jˆ s
max
{Jˆ :Ls (Jˆ )=Jˆ s }
E V [Q(Jˆ )]. (4.1)
The last maximum is easily bounded: by Lemma 3.6 with θ = 1/2 we have max
{Jˆ :Ls (Jˆ )=Jˆ s }
ˆs
E V [Q(Jˆ )] ≤ 2m(J ) .
(4.2)
By straightforward computation (cf. (1.5), (1.6)), for a lifted skeleton Jˆ , we have the annealed bound
s ˆs E V Z 0N W− (J , δ2 ) | W s (Jˆ s )
s ˆs ≤ exp β δ2 | Jˆ s | P X W− (J , δ2 ) | W s (Jˆ s ) , (4.3) so we need to show that, on the right side, the exponential decay of the probability overcomes the growth of the exponential factor. We truncate and tilt the excursion length distribution to obtain a measure να,R on paths, given by να,R (E = k) =
EX
eαk P X E = k E ≤ R , k ≥ 1. eα E | E ≤ R
(4.4)
(Strictly speaking, να,R specifies a distribution only for excursion lengths, not for paths, but since the only relevant feature of the paths is their returns to 0, we will mildly abuse notation and view να,R as a distribution on paths.) We then have the following. 0 from Definition 3.4, we have Lemma 4.1. For all β, χ , n, R positive, for Yn,R , Yn,R 0 να,R (xn = 0) , E X e−βχ L n Yn,R = e−αn X P ( xn = 0 | Yn,R )
where α = α(βχ , R) satisfies
eβχ = E X eα E | E ≤ R .
(4.5)
Proof. We compute 0 E X e−βχ L n Yn,R E X e−βχ L n δ{xn =0} | Yn,R = P X ( xn = 0 | Yn,R ) −βχ k X P E1 + · · · + Ek = n | Yn,R ke = P X ( xn = 0 | Yn,R ) eαn P X E1 + · · · + Ek = n | Yn,R e−αn = X k P ( xn = 0 | Yn,R ) E X eα E | E ≤ R k να,R ( xn = 0 ) . = e−αn X P ( xn = 0 | Yn,R )
Quenched vs. Annealed Critical Points
681
The ratio of return probabilities which appears in Lemma 4.1 is difficult to bound uniformly in n, R. The purpose of our semi-CG skeletons is to allow replacement of the return probabilities at time n by expected numbers of returns in an interval Ii (see Definition 2.4.) These are more readily estimated, as follows. Lemma 4.2. Let l0 be as in Definition 2.4. There exists K 10 (depending on 4 ) such that, provided R is sufficiently large, for all α > 0 and i ≥ l0 , E να,R ( L Ii ) X E ( L Ii | Yn i+ ,R
)
≤ K 10 .
(4.6)
Proof. For i = l0 the lemma (with K 10 = 1) follows from the fact that excursion lengths are stochastically larger under να,R than under P X (· | Yn i+ ,R ), for n i+ from Definition 2.4. Hence we fix i > l0 , R and α and define n = n i+ ∧ R. Let si = n i+ − n i− ∈ ( 4 n − 2, 4 n] and let ri = si /4 + 1 ≥ 4 n/4. For the numerator of (4.6), using again the stochastic domination of excursion lengths we have E να,R ( L Ii ) ≤ E να,R ( L Ii | xn − = 0 ) i
≤ E να,R (L si ) ≤ 4E να,R (L ri ) ≤ 4E X ( L ri | Yri ,R ).
(4.7)
For the denominator or (4.6), let J2 = (n i− , n i− + 2ri ], which is roughly the first half of Ii , and let η J2 = inf{t ≥ 0 : xt ∈ J2 }, with η J2 = ∞ if there is no such t. If we condition in the denominator also on a return to 0 in J2 , then we get a lower bound similar to (4.7). More precisely, we have E X ( L Ii | Yn i+ ,R ) ≥ E X (L Ii | Yn i+ ,R , η J2 = j)P X ( η J2 = j | Yn i+ ,R ) j∈J2
≥ E X ( L ri | Yri ,R )P X (η J2 ∈ J2 | Yn i+ ,R ),
(4.8)
which with (4.7) shows that E να,R ( L Ii ) X E ( L Ii | Yn i+ ,R
)
≤
P X (η
J2
4 , ∈ J2 | Yn i+ ,R )
(4.9)
so we need a lower bound for the probability on the right side of (4.9). Define the interval J1 = n i− − n, n i− ∩ [0, ∞). Note that J1 and J2 are adjacent. Due to the truncation of excursion lengths, there is always a visit to 0 in J1 , provided we count the visit at time 0 when 0 ∈ J1 , and considering the first such visit we obtain P X (η J2 ∈ J2 | Yn i+ ,R ) ≥ min P X (η J2 ∈ J2 | Yn i+ ,R ∩ {x j = 0}). j∈J1
(4.10)
If E X (E) < ∞, it follows easily from the SLLN that the right side of (4.10) is near 1 provided n i+ is large, so we assume E X (E) = ∞, which means c ≤ 2.
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K. S. Alexander, N. Zygouras
For j ≥ 0 let U j and W j be the starting and ending points, respectively, for the first excursion starting in [ j, ∞) of length at least ri . If U j > n i− then there is no excursion which jumps over the interval J2 , so η J2 ∈ J2 . Hence for j ∈ J1 , P X (η J2 ∈ J2 | Yn i+ ,R ∩ {x j = 0}) = P X (η J2 ∈ J2 | Y∞,R ∩ {x j = 0}) ≥ P X (U j > n i− | Y∞,R ∩ {x j = 0}) P X (U j = k, W j ∈ J2 | Y∞,R ∩ {x j = 0}) + k∈[ j,n i− ]
≥ P X (U j > n i− | Y∞,R ∩ {x j = 0}) P X (W j ∈ J2 | Y∞,R ∩ {x j = 0} ∩ {U j = k}) + k∈[ j,n i− ]
·P X (U j = k | Y∞,R ∩ {x j = 0}) = P X (U j > n i− | Y∞,R ∩ {x j = 0}) + min P X (E ∈ (n i− − k, n i− − k + 2ri ] | E ≥ ri ) k∈[ j,n i− ]
·P X (U j ≤ n i− | Y∞,R ∩ {x j = 0}) ≥ min
k∈[ j,n i− ]
P X (E ∈ [n i− − k + ri , n i− − k + 2ri ]) . P X (E ≥ ri )
(4.11)
Since n i− − k + 2ri ≤ 2n, provided n is large (depending on 4 ), we have that the last ratio in (4.11) is bounded below by 1 − c ri (2n)−c ϕ(2n) 1 − c 4 c ϕ(2n) ≥ 11. ≥ 2 2 2 ϕ(ri ) ri1−c ϕ(ri ) With K 10 = 4/K 11 , the lemma follows from this together with (4.9), (4.10) and (4.11). Lemma 4.3. Let K 12 > 0, let α = K 12 /R( ) and let χ be given by (4.5). Provided 2 is sufficiently small (depending on K 12 ) and β is sufficiently small (depending on 2 ), we have βχ δ2 ≤
1 α(βχ , R( )). 2
(4.12)
Proof. We write α for α(βχ , R), δ ∗ for δ ∗ ( ), M for M( ) and R for R( ). We have eαk ≤ 1 +
e K 12 − 1 αk for all k ≤ R, K 12
so
e K 12 − 1 eβχ = E X eα E | E ≤ R ≤ 1 + α E X (E | E ≤ R), K 12
Quenched vs. Annealed Critical Points
683
and therefore, for large R, e K 12 − 1 αm(R)δ2 . K 12
(4.13)
K 12 m(R) 1 ≤ . R β 4(e K 12 − 1)
(4.14)
βχ δ2 ≤ 2 Hence we need to show that m(R)δ2 =
Case 1. When E X (E) < ∞, (4.14) is true whenever 2 is small, since m(R) ≤ E X (E). Case 2. Suppose 3/2 ≤ c < 2. Then as β → 0, for some K 13 , K 14 we have c−1 M 1 m(R) M c−1 2 and ∼ K 13 ∼ K 14 , ϕ R M 2 β ϕ(M) the first being uniform in 2 < 1 and the second being proved in [1]. Hence for β small (so that M is large) and 2 < 1 we have M (c−1)/2 ϕ(M)−1 ≤ K 15 2 m(R)δ2 ≤ K 15 2c−1 ϕ , 2 and (4.14) follows for small 2 . Case 3. Suppose c = 2 and E X (E) = ∞. For β small and 2 < 1 we obtain using (3.10) that ∗
1 1 1 m β m β 1 1/2 1/2 ∗
δ m(R)δ2 = 2 m ≤ 4 ≤ 8 2 , (4.15) 1 ∗ 1 2 2 β δ ∗ m
β
and (4.14) follows for small 2 . Here we used the fact that the rightmost ratio in (4.15) converges to 1 as β → 0 by the definition of the conjugate. The next lemma shows that cost per length R( ), in occupied segments, of having sparse returns can be made arbitrarily large by taking 2 small. This cost appears as the constant K 16 . Lemma 4.4. There exists a constant C = C(R( )) as follows. For every K 16 > 0, provided 2 is small enough (depending on K 16 and 4 ), for all lifted semi-CG skeletons Jˆ s , for K 10 from Lemma 4.2,
ˆs ˆ s ˆs P X W− (J , δ2 ) | W s (Jˆ s ) ≤ C(4K 10 )m(J )+1 e−K 16 | J |/R( ) . Proof. We write R, M for R( ), M( ). Let α = K 16 /(1 − 4 )R( ). m (a , b ) for which P X (W s (Jˆ s )) > 0, and for paths in Fix Jˆ s = [0, N ]\ ∪i=1 i i W s (Jˆ s ) let [bi−1 , bi−1 + Ti ] denote the i th occupied segment, for 1 ≤ i ≤ m + 1. Thus, given W s (Jˆ s ), bi−1 is deterministic and Ti is random, but lying in Ii = Iki ∩ Z for some particular ki = ki (Jˆ s ), for i ≤ m, and Tm+1 = N − bm . For notational convenience we define tm+1 = N − bm and the one-point interval Im+1 = {tm+1 }. Suppose ti ∈ Ii for all i ≤ m + 1. If there is at least one long occupied segment in Jˆ s m+1 then it follows analogously to Lemma 3.2 that i=1 ti ≥ (1− 4 )|Jˆ s |− M/4 ≥ |Jˆ s |/2.
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If there is no long occupied segment, then m = 1 and t1 + t2 < 2M, and we have α|Jˆ s | < 3α M ≤ 6 2 K 16 ≤ 6K 16 . Thus in all cases we have ! m+1 α ˆs exp − ti ≤ e2K 16 e−α|J |/4 . (4.16) 2 i=1
The Ti ’s are conditionally independent given W s (Jˆ s ); in fact
P X Ti = ti for all i ≤ m W s (Jˆ s ) =
m
(4.17)
P X Ti = ti W s (Jˆ s )
i=1
P X xti = 0, Yti ,R P X (E = bi − bi−1 − ti ) . = X x = 0, Y X t t,R P (E = bi − bi−1 − t) t∈I P i=1 m
i
For sites t ∈ 4 is small,
Ii
we have bi −bi−1 −t ≥ (1− 4 )R, while |Ii | ≤ 4 R. Therefore provided max
s,t∈Ii
P X (E = bi − bi−1 − s) ≤ 2, P X (E = bi − bi−1 − t)
which with (4.17) shows that m
2P X xti = 0, Yti ,R . P X Ti = ti for all i ≤ m W s (Jˆ s ) ≤ P X x t = 0, Yt,R t∈I i=1
(4.18)
(4.19)
i
Let χ be given by (4.5). We obtain using (4.16), (4.19) and Lemmas 4.1 and 4.3 that
s ˆs (J , δ2 ) | W s (Jˆ s ) P X W− L S s (x) ≤ δ2 W s (Jˆ s ), Ti = ti for all i ≤ m ··· PX = |S(x)| tm ∈Im t1 ∈I1
·P X Ti = ti for all i ≤ m W s (Jˆ s ) ≤
···
m+1
t1 ∈I1
tm ∈Im i=1
m+1
eβχ δ2 ti E X e−βχ L ti xti = 0, Yti ,R 2P X xti = 0, Yti ,R · X x = 0, Y t t,R t∈I P i
=
t1 ∈I1
···
e−αti /2
tm ∈Im i=1
να,R (xti = 0) X P (xti = 0 | Yti ,R )
2P X xti = 0, Yti ,R · X x = 0, Y t t,R t∈Ii P m+1 2 t∈I να,R (xt = 0)P X (Yt,R ) i 2K 16 −α|Jˆ s |/4 . ≤e e X (x = 0 | Y )P X (Y ) P t t,R t,R t∈I i=1
i
(4.20)
Quenched vs. Annealed Critical Points
685
Now the event Yt,R is nonincreasing in t, so P X (Yn − ,R ) 1 1 P X (Ys,R ) ki ≤ X = X ≤ X . max X + + − P (Y ) P (Y ) P (Y | Y ) P (Y + s,t∈Ii t,R n k ,R n k ,R n ,R n −n − ,R ) i
i
ki
ki
(4.21)
ki
Since n +ki − n − ki ≤ 4 R, it is easily shown that provided 4 is sufficiently small (and also 2 ≤ 4 , to cover the case of ki = l0 which occurs if the initial segment is short), the denominator on the right side of (4.21) is at least 1/2, and therefore max
s,t∈Ii
P X (Ys,R ) ≤ 2. P X (Yt,R )
(4.22)
Further, for fixed t, P X (xt = 0 | Y j,R ) takes the same value for all j ≥ t. With (4.20), (4.22) and Lemma 4.2 this shows that
s ˆs P X W− (J , δ2 ) | W s (Jˆ s ) m+1 4 t∈I να,R (xt = 0) i 2K 16 −α|Jˆ s |/4 e ≤e X t∈I P (x t = 0 | Yt,R ) i=1
=4
i
m+1 2K 16 −α|Jˆ s |/4
e
e
m E να,R ( L Ii ) να,R (xtm+1 = 0) P X (xtm+1 = 0 | Ytm+1 ,R ) E X ( L Ii | Yn + ,R ) i=1
≤
m −α|Jˆ s |/4 4m+1 K 10 e
e2K 16 . P X (xtm+1 = 0 | Ytm+1 ,R )
(4.23)
For large R, since limt→∞ P X (xt = 0 | Yt,R ) = E X (E | E ≤ R)−1 > 0, there exists C(R) > 0 such that P X (xt = 0 | Yt,R ) ≥ C(R)−1 for all t for which P X (xt = 0 | Yt,R ) > 0. Since P X (W s (Jˆ s )) > 0, tm+1 must be such a value of t, so from (4.23),
ˆs s ˆs (J , δ2 ) | W s (Jˆ s ) ≤ C(R)(4K 10 )m+1 e−K 16 |J |/R . P X W− Proposition 4.5. Provided 2 is sufficiently small, and β is sufficiently small (depending on 2 ), E V [ Z 0N (T (δ2 )) ] is bounded in N .
(4.24)
Proof. Write R for R( ). From (4.1)–(4.3), and from Lemma 4.4 with K 16 ≥ 2 to be specified, for C(R) from that lemma, provided β is sufficiently small we have E V [Z 0N (T (δ2 ))]
ˆs s ˆs 2m(J ) exp β δ2 | Jˆ s | P X W− (J , δ2 ) | W s (Jˆ s ) P X (W s (Jˆ s )) ≤ Jˆ s
K 16 ˆs | Jˆ s | (4K 10 )m(J )+1 P X (W s (Jˆ s )) β δ2 − R Jˆ s K 16 ˆ s ˆs ≤ C(R) exp − | J | (4K 10 )m(J )+1 P X (W s (Jˆ s )). 2R ≤ C(R)
Jˆ s
exp
(4.25)
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K. S. Alexander, N. Zygouras
We use notation from the proof of Lemma 4.4. In particular, for a lifted semi-CG skeleton m (a , b )for which P X (W s (Jˆ s )) > 0, and for paths in W s (Jˆ s ), let Jˆ s = [0, N ]\ ∪i=1 i i [bi−1 , bi−1 + Ti ] denote the i th occupied segment, for i ≤ m + 1. Ti is then required to lie in Ii = Iki ∩ Z for some particular ki = ki (Jˆ s ), for i ≤ m. Analogously to (3.26), provided β is small (so |Ii | is large) we have E X [L Ii | xbi−1 = 0, Y[bi−1 ,ais ),R ] ≤ ζ (|Ii |),
(4.26)
where ζ (s) =
K 17 s c−1 . ϕ(s)
For i ≥ 2 and ti ∈ Ii we have ζ (|Ii |) 2ζ ( 4 (ti ∧ R)) ≤ , |Ii | 4 (ti ∧ R)
(4.27)
so bounding the maximum by twice the average we obtain P X (W s (Jˆ s )) ≤
m i=1 ti ∈Ii
≤
m i=1
⎛
P X ( xbi−1 +ti = 0 | xbi−1 = 0, Y[bi−1 ,ais ),R ) pbi −bi−1 −ti !
E [L Ii | xbi−1 = 0, Y[bi−1 ,ais ),R ] max pbi −bi−1 −ti X
ti ∈Ii
⎞ 2ζ (|Ii |) ⎝ ≤ ⎝2 pb1 −b0 −t1 ⎠ pbi −bi−1 −ti ⎠ |Ii | i=2 t1 ∈I1 ti ∈Ii ⎛ ⎞ ⎛ ⎞ m ζ ( 4 (ti ∧ R)) ⎝4 pbi −bi−1 −ti ⎠. (4.28) ≤ ⎝2 pb1 −b0 −t1 ⎠ 4 (ti ∧ R)
⎞
t1 ∈I1
m
i=2
⎛
ti ∈Ii
Note that for i = 1, when (4.27) need not be valid, we have used the bound |Ii | in place of (4.26). Using (4.28) we then bound the sum in (4.25) by
4K 10
∞
m=0 (bi )i≤m (ti )i≤m+1 m ζ ( 4 (ti ∧ R)) × 4 (ti ∧ R) i=2
(16K 10 ) !
m
m+1 i=1
m
! e
−K 16 ti /2R
!
pbi −bi−1 −ti .
(4.29)
i=1
Here 0 = b0 < b1 < · · · < bm ≤ N , and for fixed b0 , . . . , bm the third sum includes those ti for which tm+1 = N − bm , bi − bi−1 − ti ≥ (1 − 4 )R for all 1 ≤ i ≤ m, and ti ≥ (1 − 4 )M ≥ M/2 for all 2 ≤ i ≤ m. The second sum includes those (bi )i≤m for
Quenched vs. Annealed Critical Points
687
which such ti exist. Now provided β is small so that M, R are large (depending on 4 ) we see that (4.29) is bounded above by ⎛ ⎞2 ∞ (16K 10 )m ⎝ e−K 16 t/2R ⎠ 4K 10 + 4K 10 m=1
t≥1
⎞m−1 ⎛ ζ ( 4 (t ∧ R)) ×⎝ e−K 16 t/2R ⎠ ⎝ 4 (t ∧ R) ⎛
t≥M/2
⎞m pn ⎠ .
(4.30)
n≥(1− 4 )R
Now
2R , K 16
(4.31)
K 18 ϕ(R) , R c−1
(4.32)
e−K 16 t/2R ≤
t≥1
and for some K 18 ,
pn ≤
n≥(1− 4 )R
and for some K 19 , K 20 K 21 , provided β is small (depending on 2 , 4 ), ζ ( 4 (t ∧ R)) e−K 16 t/2R = 4 (t ∧ R)
t≥M/2
≤
M/2≤t≤R
K 19 4
t>R
max
M/2≤t≤R
K 20 ≤ 4 ϕ(R) ≤
ζ ( 4 t) −K 16 t/2R ζ ( 4 R) −K 16 t/2R e + e 4 t 4 R
ζ ( 4 t)e−K 16 t/2R +
4 R K 16
c−1 +
2ζ ( 4 R) K 16 4
2ζ ( 4 R) K 16 4
K 21 R c−1 . (c−1)∧1 2−c 4 ϕ(R)K 16
(4.33)
Therefore for some C (R) and K 22 , (4.30) is bounded by
4K 10 + C (R)
∞
K 22 K 10 (c−1)∧1
m=1
42−c K 16
!m ,
which is finite provided K 16 is taken sufficiently large (depending on 4 .) Taking 2 sufficiently small (depending on K 16 ) ensures that Lemma 4.4 can be applied with this K 16 to obtain (4.25). The reason the coarse-graining scheme in Definition 2.4 is different from the one in Definition 2.3 is that we need to avoid making a choice of 4 (specifying the fineness of the coarse-graining scheme) that depends on 2 (which, via δ2 , determines sparse vs. dense returns.) Having K 16 large when Lemma 4.4 is applied in the proof of Proposition 4.5 requires taking 2 small, while on the right side of (4.30) we need K 16 depending on 4 . The coarse-graining scheme in Definition 2.4 avoids any circularity in the choices, allowing us to specify 4 and then 2 depending on 4 .
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Proposition 4.6. Provided 2 is sufficiently small, and β is sufficiently small (depending on 2 ), we have lim sup N →∞
1 log Z 0N ( T (δ2 ) ) = 0. N
Proof. This follows immediately by Chebyshev’s inequality and the previous proposition. Proof of Theorem 1.1. Let ρ be as in Lemma 3.3. Then ρ ≤ P V (TN )
≤ P V Z 0N (D(δ2 )) < Z 0N ,λ
≤ P V eβ fq (β, )N /2 < Z 0N ,λ + P V Z 0N (D(δ2 )) < eβ fq (β, )N /2
≤ P V eβ fq (β, )N /2 < Z 0N ,λ + P V Z 0N < 2eβ fq (β, )N /2
+ P V Z 0N (T (δ2 )) > eβ fq (β, )N /2 . If f q (β, ) > 0 then as N tends to infinity the right hand side of the above inequality tends to zero, by Proposition 3.10, Proposition 4.6 and the fact that N1 log Z 0N tends to β f q (β, ), P V –a.s. This is a contradiction so f q (β, ) = 0. Regarding the Gaussian assumption, the most difficult point to remove it appears to L∗ (J ) be (3.29), particularly for large β, due to the conditioning on V . The assumption is also used in (3.21), (3.28) and (3.32).
References 1. Alexander, K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008) 2. Alexander, K.S., Sidoravicius, V.: Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16, 636–669 (2006) 3. Bodineau, T., Giacomin, G.: On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 (2004) 4. Derrida, B., Giacomin, G., Lacoin, H., Toninelli, F.L.: Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys. 287, 867–887 (2009) 5. Derrida, B., Hakim, V., Vannimenus, J.: Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992) 6. Forgacs, G., Luck, J.M., Nieuwenhuizen, Th.M., Orland, H.: Exact critical behavior of two-dimensional wetting problems with quenched disorder. J. Stat. Phys. 51, 29–56 (1988) 7. Giacomin, G.: Random Polymer Models. Imperial College Press, London, 2007 8. Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. (2009, to appear) 9. Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006) 10. Imry, Y., Ma, S.-K.: Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399–1401 (1975) 11. Naidenov, A., Nechaev, S.: Adsorption of a random heteropolymer at a potential well revisited: location of transition point and design of sequences. J. Phys. A: Math. Gen. 34, 5625–5634 (2001)
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12. Seneta, E.: Regularly Varying Functions. Lecture Notes in Math. 508, Berlin: Springer-Verlag, 1976 13. Toninelli, F.L.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18(4), 1569–1587 (2007) 14. Toninelli, F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008) Communicated by M. Aizenman
Commun. Math. Phys. 291, 691–761 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0878-1
Communications in
Mathematical Physics
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications N. Kitanine1 , K. K. Kozlowski2 , J. M. Maillet2 , N. A. Slavnov3 , V. Terras2, 1 LPTM, Université de Cergy-Pontoise et CNRS,
Cergy-Pontoise, France. E-mail: [email protected]
2 Laboratoire de Physique, ENS Lyon et CNRS, Lyon, France.
E-mail: [email protected]; [email protected]; [email protected]
3 Steklov Mathematical Institute, Moscow, Russia. E-mail: [email protected]
Received: 29 July 2008 / Accepted: 13 April 2009 Published online: 30 July 2009 – © Springer-Verlag 2009
Abstract: We investigate the asymptotic behaviour of a generalised sine kernel acting on a finite size interval [ −q ; q ]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener–Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models. Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . Problem to Solve and Main Results . . . . . . . . . . . . 2.1 Generalised sine kernel: assumptions and notations . 2.2 The main results . . . . . . . . . . . . . . . . . . . . 2.3 Comparison with known results . . . . . . . . . . . . The Initial Riemann–Hilbert Problem . . . . . . . . . . . 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Riemann–Hilbert problem associated to the GSK Transformations of the Original RHP . . . . . . . . . . . 4.1 The first step χ → . . . . . . . . . . . . . . . . . 4.2 The second step → ϒ . . . . . . . . . . . . . . . 4.3 Parametrix around −q . . . . . . . . . . . . . . . . . 4.4 Parametrix around q . . . . . . . . . . . . . . . . . . 4.5 The last transformation ϒ → . . . . . . . . . . . Asymptotic Solution of the RHP . . . . . . . . . . . . . 5.1 Asymptotics of the jump matrices . . . . . . . . . .
3. 4.
5.
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On leave of absence from: LPTA, Université Montpellier II et CNRS, France
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692 694 694 695 699 699 700 700 703 704 705 706 708 709 710 710
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
5.2 Asymptotic expansion for . . . . . . . . . . . . . . . . . . . . . 5.3 The functions f ± to the leading order . . . . . . . . . . . . . . . . . 5.4 Integral bounds for the resolvent . . . . . . . . . . . . . . . . . . . 5.5 Asymptotic expansion of the resolvent . . . . . . . . . . . . . . . . 6. Leading Asymptotic Behaviour of log det [I + V ] . . . . . . . . . . . . . 6.1 The leading asymptotics from the γ -derivative method . . . . . . . 6.2 The leading asymptotics from the q-derivative method . . . . . . . 6.3 The first corrections to the leading asymptotics of the Fredholm determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Applications to Truncated Wiener–Hopf Operators . . . . . . . . . . . . 7.1 The Akhiezer–Kac formula . . . . . . . . . . . . . . . . . . . . . . 7.2 The resolvent of truncated Wiener–Hopf operators . . . . . . . . . . 8. Asymptotics of Multiple Integrals . . . . . . . . . . . . . . . . . . . . . 8.1 Leading asymptotic behaviour of In [Fn ] . . . . . . . . . . . . . . . 8.2 Study of sub-leading corrections . . . . . . . . . . . . . . . . . . . 8.2.1 General strategy. . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 γ -derivatives of log det[I + V ]sub . . . . . . . . . . . . . . . . . 8.2.3 Application of the density procedure and proof of Theorem 8.1. 8.2.4 Asymptotic expansion of Insub [Fn ] and proof of Theorem 8.2. . 9. More General Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Some Properties of Confluent Hypergeometric Function . . . . . . . . . B. Three Preparatory Lemmas . . . . . . . . . . . . . . . . . . . . . . . . C. The Density Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Form of the Sub-leading Terms in Insub . . . . . . . . . . . . . . . . . .
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1. Introduction The sine kernel S (λ, µ) =
sin x2 (λ − µ) , π (λ − µ)
(1.1)
is a very important object in mathematical physics. In particular, the Fredholm determinant of the integral operator I − S acting on some interval J ⊂ R appears in random matrix theory [21]. In the bulk scaling limit, det J [I − S] stands for the probability [22] that a matrix belonging to the Gaussian unitary ensemble has no eigenvalues in x J . The kernel (1.1) also appears in the theory of quantum integrable systems. In particular, the determinant det J I + γ S , γ being a parameter, describes various zero–temperature correlation functions of the impenetrable Bose gas [33,40]. In all these interpretations of the sine kernel, one is interested in the large x behaviour of its Fredholm determinant. The first attempt to analyze the x → +∞ asymptotics of det J [I − S] goes back to Gaudin and Mehta [21,22]. In 1973, Des Cloizeaux and Mehta [18] showed that log det [ −1 ;1 ] [I − S] = −
1 x2 − log x + O (1) , 8 4
x → +∞ .
(1.2)
Three years later, using Widom’s formula [44] for the asymptotics of Toeplitz determinants supported on an arc, Dyson [19] gave a heuristic derivation of the constant terms
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
693
c0 and proposed a recursive method to compute the subleading coefficients c1 , c2 , . . . in the asymptotic expansion: 1 c1 c2 x2 − log x + c0 + + 2 + ... . (1.3) 8 4 x x However, the forementioned results were heuristic. It was only in 1994 that Widom [45] managed to prove rigorously the first term in the asymptotics (1.2): log det [ −1 ;1 ] [I − S] = −
d x log det [ −1 ;1 ] [I − S] = − + o (1) . (1.4) dx 4 One year later, this analysis was extended to the multiple interval case [46]. While Widom studied the asymptotic behaviour of the Fredholm determinant by operator techniques, Deift, Its and Zhou applied the Riemann–Hilbert problem (RHP) formulation for integrable integral operators [30] to the sine kernel acting on a union of intervals ∪ J and proved the existence of the asymptotic expansion (1.3). However, their method did not allow them to obtain an estimate for the constant c0 , as they inferred the asymptotic expansion of log det J [I − S] from that of d log det [I − S] . (1.5) dx The first proofs of Dyson’s heuristic formula for c0 appeared in the independent, and based on completely different methods, works of Ehrhardt [20] and Krasovsky [38] and more recently in [15]. We would like to point out that there is a very nice connection of the sine kernel to the Painlevé V equation [33], as Px solves this equation. The link between Painlevé V and Px was also investigated in [13] in the framework of RHP. It was shown that one can deduce this Painlevé equation directly from the RHP data. This article is devoted to the study a generalisation of the sine kernel (1.1). This kernel, that we will refer to as the generalised sine kernel (GSK), is of the form √ γ F (λ) F (µ) V (λ, µ) = e+ (λ) e− (µ) − e− (λ) e+ (µ) , (1.6) 2iπ (λ − µ) Px = x
where e± (λ) = e±[i x p(λ)+g(λ)]/2 .
(1.7)
We will be more specific about the functions F, p and g later on. Various particular cases of the kernel (1.6) already appeared in the literature. These particular kernels were mostly used for the description of correlation functions of matrix models or quantum integrable models equivalent to free fermions (see e.g. [8–10,27– 29,31,36,41,43]). In the present paper we consider a rather general case, only based on the analytic properties of the functions F, p and g. The GSK (1.6) plays a crucial role in the study of correlation functions of (non free-fermion) quantum integrable systems [35]. It is also useful for the asymptotic analysis of truncated Wiener–Hopf operators with Fischer–Hartwig singularities [37]. We investigate here the large x asymptotic behaviour of the Fredholm determinant of the GSK in the framework of RHP. Our work is a natural extension of an unpublished analysis by Deift, Its and Zhou of the sine kernel I + γ S by RHP. This kernel was also analysed by RHP in [8]. This article is organized as follows. In Sect. 2, we announce the main results of the paper, namely,
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• the large x asymptotic behaviour of the Fredholm determinant of the integral operator I + V , cf. (1.6); • the asymptotic resolvent of some Wiener–Hopf operators connected to (1.6); • the asymptotic behaviour of coupled multiple integrals involving a cycle of kernels V (1.6) versus some holomorphic symmetric functions. The proof of the asymptotic behaviour of log det[I + V ] is given in the core of the paper (Sects. 3, 4, 5 and 6). More precisely, in Sect. 3, we recast the problem into a certain RHP. In Sect. 4, we transform this initial RHP into a RHP that can easily be solved asymptotically. This asymptotic solution is presented in Sect. 5 and used in Sect. 6 to obtain the leading and the first subleading terms of log det[I + V ] in the x → +∞ limit. In Sect. 7, we apply these results to truncated Wiener–Hopf operators. We show how one can use the asymptotic resolvent of the generalised sine kernel to construct asymptotic resolvents of truncated Wiener–Hopf operators acting on [ −x ; x ], with x large. This asymptotic resolvent is used to reproduce the low magnetic field behaviour of the so-called dressed charge arising in the theory of quantum integrable models solvable by the Bethe ansatz [6]. Section 8 is devoted to the study of the asymptotic behaviour of some particular type of coupled multiple integrals which can be obtained in terms of the GSK. This is in fact our main motivation to study the GSK: indeed, from the knowledge of the asymptotic behaviour of this type of multiple integrals one can obtain the asymptotic behaviour of quantum integrable models correlation functions, as it is done in [35]. Finally, in Sect. 9, we consider the case of further modifications of the GSK, in particular those useful for the correlation functions of the integrable Heisenberg spin chains [35]. Some properties of confluent hypergeometric functions and proofs of several lemmas are gathered in the Appendices.
2. Problem to Solve and Main Results 2.1. Generalised sine kernel: assumptions and notations. Let I + V be the integral operator with kernel (1.6) and acting on L 2 ([ −q ; q ]). We assume that there exists some open relatively compact neighbourhood U of [ −q ; q ] such that the functions p, F and g, as well as the parameter γ , satisfy the following properties: • F and g are holomorphic on U , the closure of U ; • p is holomorphic and injective on U , p ([ −q ; q ]) ⊂ R, and p stabilizes the upper half plane H+ (resp. the lower half plane H− ), i.e. p (U ∩ H± ) ⊂ H± ; • γ ∈ D0,r = {λ ∈ C : |λ| < r }, where r is such that |r F| < 1 and arg (1 + γ F) ∈ ] −π ; π [ on U . We study the large x expansion of the Fredholm determinant of I + V under these assumptions. This will be done by asymptotically solving a certain matrix RHP. It will become clear in the next section that the assumption p ([ −q ; q ]) ⊂ R is tantamount to imposing the associated RHP to be of oscillatory nature. Moreover, the case p (U ∩ H± ) ⊂ H∓ is obtained by the negation (γ , g (λ)) → (−γ , −g (λ)). Note that γ plays here the role of a regularisation parameter; in particular it should be stressed that our method does not allow to reach the |γ F| = 1 case corresponding to (1.3) which requires a different analysis [15,20,38].
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Before presenting the main result of this article, let us introduce some convenient notations. First, we define two auxiliary functions used in the article: −1 log (1 + γ F (λ)), 2iπ ⎧ ⎫ ⎨q ν (λ) − ν (µ) ⎬ κ(λ; q) ≡ κ(λ) = exp dµ . ⎩ ⎭ λ−µ ν (λ) =
(2.1) (2.2)
−q
Note that κ is a function of the two parameters λ and q, although we will sometimes omit the dependence on the second parameter. Finally, we will use the following simplified notations for the values of the functions p and ν and of their derivatives at the points ±q: p± = p(λ) ν± = ν(λ)
λ=±q λ=±q
p± = p (λ)
,
ν± = ν (λ)
,
λ=±q
λ=±q
,
,
etc. , etc.
(2.3) (2.4)
2.2. The main results. We now give the asymptotic behaviour of the Fredholm determinant in the x → +∞ limit: Theorem 2.1. Let V be the GSK (1.6) with p, g, F and γ satisfying the assumptions of Sect. 2.1. Then, in the x → +∞ limit, log det[I + V ] behaves as log det[I + V ] = log det[I + V ](0) + o(1),
(2.5)
with log det[I + V ] ⎡ + log ⎣
(0)
q = −i x
q
ν(λ) p (λ) −q
dλ − (ν+2 ⎤
2 + ν− ) log x
−
ν(λ)g (λ) dλ
−q
q
G(1, ν+ ) G(1, ν− ) κ ν+ (q; q) ⎦ 1 ν (λ)ν(µ) − ν(λ)ν (µ) + dλ dµ ,
ν 2
2 2 λ−µ ν− κ ν− (−q; q) 2qp+ + 2qp− −q (2.6)
in which we have used the notations of Sect. 2.1. The Barnes G-function [2,3] admits the integral representation: ⎧ ⎫ ⎨ z (z − 1) z ⎬ z
(z) + tψ (t) dt , (z) > −1, ψ(z) = , G (z + 1) = (2π ) 2 exp − ⎩ ⎭ 2
(z) 0
(2.7) and we denote G(1, z) ≡ G(1 + z)G(1 − z). Using the perturbation theory for singular integral equations one can refine the theorem and obtain sub-leading corrections. Although, in principle, nothing opposes deriving the next sub-leading corrections, the computations become more and more involved. We have proved the structure of the first corrections to Eq. (2.6).
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Proposition 2.1. Let V be the GSK (1.6) with the conditions of Sect. 2.1. The leading asymptotics log det [I + V ](0) of log det [I + V ] as defined in Theorem 2.1 has non-oscillating and oscillating corrections. Let 0 < δ < q be such that the disks D±q,δ of radius δ centered at ±q fulfill D±q,δ ⊂ U . Let ε = 2 sup∂ Dq,δ ∪∂ D−q,δ | (ν)|. Then the first non-oscillating corrections are of the form 1 N1 , (2.8) +O x x 2(1−ε) with ν2 ν−σ νσ d σ d N1 = i 2σ νσ log x + σ − . log u σ + pσ p dq dq pσ q σ =± σ The first oscillating corrections are of the form O1 1 , +O x2 x 3(1−ε) and the leading oscillating coefficient is given by u + σ ν− ν+ O1 = x 2σ (ν+ +ν− ) eiσ x( p+ − p− ) , u− (2q)2 p+ p−
(2.9)
(2.10)
(2.11)
σ =±1
where we have introduced
(1 − ν+ ) (2qp+ )ν+ 2 u+ = e ,
(1 + ν+ ) κ(q; q) −2
(1 + ν− ) ν− (2qp− u − = e g(−q) ) κ(−q; q) .
(1 − ν− ) g(q)
(2.12) (2.13)
Remark 2.1. The GSK depends only on the combination i x p(λ)+g(λ) (see (1.7)). Therefore the Fredholm determinant and its asymptotics can only depend on this combination. This observation allows us to obtain the complete asymptotic expansion depending on the function g(λ) from the asymptotic expansion of the Fredholm determinant I + V corresponding to g = 0. Namely, it is enough to replace in the obtained formulae p(λ) by p(λ) − xi g(λ) and then expand into negative powers of x. It is quite interesting to apply the latter proposition in order to obtain the first few terms of the asymptotic expansion of det [I + V ]. The reason why we draw the reader’s attention to these asymptotics is because they present a very interesting structure: the leading oscillating terms in the asymptotic expansion are just given by the sum of the leading asymptotics evaluated at ν shifted by 1 or −1. This structure of the asymptotics seems to restore, at least partly, the original periodicity ν → ν + n, n ∈ Z, of the Fredholm determinant of I + V . Corollary 2.1. Let I+V be the GSK as above, det [I + V ](0) [ν] the leading asymptotics of its Fredholm determinant just as in Theorem 2.1, N1 and O1 as in Proposition 2.1. Note that we have emphasized the structure of det [I + V ](0) [ν] as a functional of ν.
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Then the oscillating corrections O1 can be reproduced from the non-oscillating part via the shift of ν by ±1: det [I + V ](0) [ν]
O1 = det [I + V ](0) [ν + 1] + det [I + V ](0) [ν − 1] . x2
(2.14)
This structure of the first terms of the large x asymptotic expansion for det [I + V ] leads us to raise the following conjecture on the structure of the asymptotic series : Conjecture 2.1. The asymptotic expansion of the Fredholm determinant det [I + V ] of the GSK restores the periodicity ν → ν + n, n ∈ Z, of the determinant. In particular, this asymptotic expansion contains all the Z-periodized terms with respect to ν of the leading asymptotics det [I + V ](0) [ν]. Thus, all the oscillating terms can be deduced from the non-oscillating ones. More precisely, let C1 (log x) [ν] C M (log x) [ν] (0) + ··· + A [ν] (x) ∼ det [I + V ] [ν] 1 + + ... x xM (2.15) stand for the formal asymptotic series corresponding to the non-oscillating part of the asymptotic series for log det [I + V ]. There Ck (X ) [ν] are polynomials of degree k in X whose coefficients are functionals in ν. Moreover each of the Ck ’s has no oscillating exponents of the type e±i x p± . Then the formal asymptotic series for det [I + V ] is given by det [I + V ] ∼
A [ν + n] (x) .
(2.16)
n∈Z
This conjecture is supported by (2.14) and also by the results of [42] where several sub-leading corrections to the asymptotics of the Fredholm determinant of the pure sine-kernel were computed. The first application of the asymptotic behaviour of the GSK we consider in this article concerns the asymptotic inversion of truncated Wiener–Hopf operators. We will prove in Sect. 7 the following proposition: Proposition 2.2. Let I + K be a truncated Wiener–Hopf operator on ] −x ; x [, acting on functions g ∈ L 2 (R) as x [(I + K ) .g] (t) = g (t) +
K t − t g t dt .
(2.17)
−x
The kernel K is defined by its Fourier transform F: K (t) = F −1 [F] (t) , and we suppose that there exists δ > 0 such that • F admits an analytic continuation to {z : |(z)| ≤ δ}; • ξ → F (ξ ± iδ) ∈ L 1 (R); • the analytic continuation of 1 + F never vanishes for |(z)| ≤ δ.
(2.18)
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Then the resolvent I − R of I + K fulfills
dξ dη F(ξ ) 4π 2 i
R (λ, µ) = R
×
α+ (η) i x(ξ −η) α+ (ξ ) −i x(ξ −η) e e − α− (ξ ) α− (η)
ei(µη−λξ ) + O e−2δx , ξ −η
(2.19)
where α(λ) is given by ⎧ ⎨
1 α (λ) = exp − ⎩ 2iπ
R
⎫ dµ ⎬ log(1 + F(µ)) . µ − λ⎭
(2.20)
Our main motivation to study the asymptotics of log det [I + V ] comes from the theory of one dimensional quantum integrable models. Indeed, the generating function of the zero temperature two-point correlation functions (at distance x) of different quantum integrable models [35] has a series expansion in terms of cycle integrals of the type In [Fn ] =
dn z (2iπ )n
([ −q ;q ])
q −q
dn λ Fn (2iπ )n
{λ} {z}
n j=1
ei x ( p(z j )− p(λ j ))
. z j − λ j z j − λ j+1
(2.21)
Here the function Fn is holomorphic in some open neighbourhood of [ −q ; q ]2n and symmetric in the n variables {λ} (we set λn+1 ≡ λ1 ) and in the n variables {z};
([ −q ; q ]) is a counter clockwise closed contour around [ −q ; q ] inside this neighbourhood. In Sect. 8, using the above results for the GSK, we prove the following asymptotic expansion of In [Fn ] in the x → +∞ limit : Proposition 2.3. Let Fn and In [Fn ] be as above. Then for x → +∞, 1 In [Fn ] = 2iπ +
q
dλ i x p (λ) + ∂ Fn
−q
bn − cn log 2qpσ x Fn
σ =±
n−1
q
+
+
n (2π )2 n
σ =± p=1−q n−1 q
dλ
{λ}n {λ + }, {λ}n−1
n
=0
{σ q} {σ q}n {σ q}n {σ q} p , {λ}n− p − Fn Fn {σ q}n {σ q} p , {λ}n− p p (n − p) (q − σ λ)
dλdµ {λ + } , {λ} p−1 , {µ}n− p ∂ Fn {λ + } , {λ} p−1 , {µ}n− p (n − p) (λ − µ)
2 (2π )2 p=1 −q {µ + } , {µ} p−1 , {λ}n− p −∂ Fn {µ + } , {µ} p−1 , {λ}n− p
=0
+ o (1) ,
(2.22)
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699
with (−1)n−1 ∂ n ν02 cn = (n − 1)! ∂γ n
γ =0
(−1)n−1 ∂ n log G (1, ν0 ) , bn = , (n − 1)! ∂γ n γ =0
i ν0 = log(1 + γ ), 2π
(2.23)
and where {λ}n denotes the set formed by n copies of the same parameter λ. Moreover, in Sect. 8 we will also describe the form of the sub-leading corrections to this result.
2.3. Comparison with known results. There are several results in concern the literature ing the asymptotic behaviour of the Fredholm determinant det I + γ S . This determinant corresponds to the GSK with p = id, F = 1 and g= 0. It is clear that we reproduce the answer concerning the leading asymptotics of det I + γ S analyzed in [7 and 4]. d log det I + γ S satisfies the fifth Painlevé equation. The As observed in [33], x dx authors of [33] used this property to obtain an asymptotic expansion of log det I + γ S . This fact was also exploited by the authors of [42] in order to derive the first few terms in the sub-leading asymptotics of the latter quantity. Their result reads x
ν3 d log det[I + γ S] = −4i xν0 − 2ν02 − i 0 dx x 2 −4i x ν02
(−ν0 ) 2 e
(−ν 0) 4ν0 4i x +i , (4x) e − 4x
(ν0 )
(ν0 ) (4x)4ν0
(2.24)
with ν0 given in (2.23) and q = 2. It is straightforward to see that in such a limit N1 = iν03 and 2 −2iq x (ν ) 2 ν02
e (−ν ) 0 0 O1 → + e2iq x (2q x)4ν0 , (2.25)
(ν0 ) (2q)2 (2q x)4ν0 (−ν0 ) which reproduces the oscillating terms (2.24) after setting q = 2 and taking the q derivative.
3. The Initial Riemann–Hilbert Problem The GSK (1.6) belongs to a special algebra of integral operators, the so-called integrable integral operators. This algebra was first singled out in [30] and then studied more thoroughly in [13]. It is well known that many properties of these integrable operators can be obtained from the solution of a certain RHP. In this section, we formulate our problem in terms of a RHP that we then asymptotically solve.
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3.1. Notations. An important property of completely integrable integral operators is that their resolvent still lies in the same algebra. However, before presenting the formula for the resolvent we introduce some quite useful vector notations. Namely, let √
γ F (λ) e+ (λ) , E L (λ) | = F (λ) −e− (λ) , e+ (λ) , (3.1) | E R (λ) = e− (λ) 2iπ so that the kernel V has a simple expression in terms of | E R (λ) and E L (λ) |: V (λ, µ) =
E L (λ) | E R (µ) . λ−µ
(3.2)
Observe that E L (λ) | E R (λ) = 0,
(3.3)
and, hence, the kernel V is not singular at λ = µ . Let | F R (λ) be the solution to the integral equation: q | F (µ) + R
V (λ, µ) | F R (λ) dλ = | E R (µ) ,
(3.4)
−q
and F L (λ) | be the solution to the corresponding dual equation. It is convenient to write | F R (λ) as well as its dual F L (λ) | in a form similar to | E R (λ) and E L (λ) |: √
γ F (λ) f + (λ) R , F L (λ) | = F (λ) − f − (λ) , f + (λ) . | F (λ) = f (λ) − 2iπ (3.5) Then the resolvent of the kernel V defined by I − R = (I + V )−1 reads: √ γ F(λ)F(µ) F L (λ) | F R (µ) = f + (λ) f − (µ) − f + (µ) f − (λ) . R(λ, µ) = λ−µ 2iπ (λ − µ) (3.6) 3.2. The Riemann–Hilbert problem associated to the GSK. Proposition 3.1. Let V be the GSK (1.6) understood as acting on L 2 ([ −q ; q ]), and such that det [I + V ] = 0. Then, there exists a 2 × 2 matrix χ (λ) such that | F R (λ) = χ (λ) | E R (λ) ,
F L (λ) | = E L (λ) |χ −1 (λ) .
The matrix χ (λ) is the unique solution of the RHP: • χ is analytic C \ [ −q ; q ] ; on 11 log λ2 − q 2 for λ → ±q; • χ (λ) = O 11 10 • χ (λ) → I2 = ; 01 λ→∞ • χ+ (λ) G χ (λ) = χ− (λ) for λ ∈ ] −q ; q [ .
(3.7)
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
−q
701
+ q
− Fig. 1. Original contour for the RHP
The jump matrix G χ for this RHP reads 1 − γ F (λ) γ F (λ) e+2 (λ) = I + 2iπ | E R (λ) E L (λ) |. G χ (λ) = 2 (λ) 1 + γ F (λ) −γ F (λ) e−
(3.8)
Finally, χ and its inverse can be expressed in terms of | F R (λ) and of its dual F L (λ) |: q χ (λ) = I2 − −q
| F R (µ) E L (µ) | dµ, χ −1 (λ) = I2 + µ−λ
q
−q
| E R (µ) F L (µ) | dµ. µ−λ
(3.9) 11 , is to be understood entrywise. MoreWe emphasize that the big O symbol, O 11 over, χ± (µ) stands for the non-tangential limit of χ (λ) when λ approaches a point µ belonging to the jump curve from the left, resp. right, side of the contour (see Fig. 1).
Proof. The unicity of the solution to this RHP is proved along the same lines as in [39]. The proof of existence of the solution is based on the equivalence between RHP and singular integral equations which, in the case of the above RHP, implies q χ (λ) = I2 + −q
dµ χ+ (µ) | E R (µ) E L (µ) |, λ ∈ C \ [ −q ; q ]. λ−µ
(3.10)
The solution to this equation can be expressed in terms of the resolvent kernel I − R of I + V , q χ (λ) = I2 + −q
dµ R | E . (I − R) (µ) E L (µ) |. λ−µ
(3.11)
In its turn, the resolvent kernel exists as det [I + V ] = 0. Moreover, the explicit construction of the resolvent through a Fredholm series shows that (λ, µ) → R (λ, µ) R (µ) = | E R . (I − R) (µ). The estimate × U . Hence, so is | F is analytic in U |χ | = O log λ2 − q 2 , λ → ±q follows from the integral representation (3.11) supplemented with the fact that both E L | and | F R are smooth on [ −q ; q ]. Applying (3.9) to |E R (λ) and E L (λ)| we obtain Eqs. (3.7). Hereby one can easily check that due to the orthogonality condition (3.3) the transform (3.7) is continuous across [−q, q]. It is also possible to express logarithmic derivatives of det [I + V ] either in terms of the resolvent R of I + V or in terms of χ . Indeed, we have
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Lemma 3.1. The derivative of log det [I + V ] with respect to x is related to the following trace involving the matrix χ :
" ! dλ p(λ) tr ∂λ χ (λ) σ3 χ −1 (λ) , 4π
∂x log det [I + V ] =
(3.12)
([ −q ;q ])
1 0 and ([ −q ; q ]) defined as in (2.21), whereas its derivatives with 0 −1 respect to γ and q are expressed in terms of the resolvent as with σ3 =
q
dλ R (λ, λ) , ∂q log det [I + V ] = R (q, q) + R (−q, −q) . γ
∂γ log det [I + V ] = −q
(3.13) Proof. The last two equations are easily proved by the multiple integral series expansion of log det [I + V ]. We shall only focus on the equation relating the x derivative of log det [I + V ] to χ . Clearly q ∂x log det [I + V ] =
[∂x V. (I − R)] (λ, λ) dλ ,
(3.14)
−q
with ∂x V (λ, µ) = −
([ −q ;q ])
p (z) dz E L (λ) |σ3 | E R (µ) . 4π (z − λ) (z − µ)
(3.15)
So that, using the representation (3.6) of the resolvent R in terms of F L | and | F R and the fact that F L (µ) | σ3 | F R (λ) = tr σ3 | F R (λ) F L (µ) | , we get ∂x log det [I + V ] = −
dz p (z) 4π
+tr
⎪ ⎩
×
dλ
E L (λ) |σ3 | E R (λ) (z − λ)2
−q
([ −q ;q ])
⎧ ⎪ ⎨
q
dz p (z) 4π
([ −q ;q ])
q dλdµ | F R (λ) E L (λ) |
−q
| 1 1 σ3 − λ−z λ−µ
ER
⎫ ⎪ (µ) (µ) | ⎬ . ⎪ (µ − z)2 ⎭ FL
(3.16)
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703
Using the integral expressions (3.9) for χ and χ −1 , we obtain
dz p (z) 4π
∂x log det [I + V ] = − ⎧ ⎪ ⎨ +tr =
([ −q ;q ])
dz p(z) 4π
q dλ
E L (λ) |σ3 | E R (λ)
−q
q dµ (χ (µ) − χ (z)) σ3
(z − λ)2 |
E R (µ)
⎪ ⎩ −q
([ −q ;q ]) dz p (z) tr ∂z χ (z) σ3 χ −1 (z) , 4π
⎫ ⎪
F L (µ) | ⎬
(µ − z)2
⎪ ⎭ (3.17)
([ −q ;q ])
where we used (3.7).
It is worth noticing that formula (3.12) is particularly effective when p is a rational function as then the contour of integration can be deformed to the poles of p (including the pole at ∞). The integrals can then be easily calculated. In particular, in the case p (λ) = λ, we have the following result: Corollary 3.1. Let χ1 be the first non-trivial coefficient of the expansion of χ around ∞, i.e. χ1 1 χ (λ) = I2 + . (3.18) +o λ λ Then i ∂x log det [I + V ] | p=id = − tr {χ1 σ3 }. 2
(3.19)
Proof. −tr (σ3 χ1 ) is the residue of the pole at infinity of λ → λ tr ∂λ χ (λ) σ3 χ −1 (λ) . In this way, we recover one of the formulae derived for the sine kernel [13], but also for more general kernels as in [27,28,31]. We emphasize that (3.19) is valid not only for the sine kernel as it was originally derived, but also for the generalised sine kernel with p = id. 4. Transformations of the Original RHP In this section we perform several transformations on the RHP for χ so as to implement Deift–Zhou’s steepest descent method [14]. The first substitution maps the RHP for the matrix χ into a RHP for a matrix whose jump matrix has 1 on its lower diagonal entry. This jump matrix is then easily factorized into upper/lower triangular matrices. This factorization allows us to define another RHP for an unknown matrix ϒ whose jump matrices are already exponentially close to identity uniformly away from the endpoints ±q. It remains to construct the parametrices at q and −q. These parametrices enable us to define a matrix satisfying a RHP with jump matrices uniformly I2 + o (1) when x → +∞.
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4.1. The first step χ → . Let ⎧ q ⎫ ⎨ ν (µ) ⎬ λ − q ν(λ) α (λ) = exp . dµ = κ (λ) ⎩ µ−λ ⎭ λ+q
(4.1)
−q
Then clearly, α (λ) solves the scalar RHP α− (λ) = α+ (λ) (1 + γ F (λ)), λ ∈ [ −q ; q ] , α(λ) → 1 at λ → ∞.
(4.2)
The functions κ (λ) and ν (λ) were already introduced in (2.2) and (2.1). In the following, we shall also use another representation for the function α(λ) : $ α (λ) = κ p (λ)
p(λ) − p+ p(λ) − p−
%ν(λ)
,
(4.3)
where κ p is defined as q ν (λ) log κ p (λ; q) ≡ log κ p (λ) = −q
p (µ) ν (µ) − dµ . p (λ) − p (µ) λ − µ
(4.4)
We specify that we chose the principal branch of the logarithm, i.e. arg ∈ ] −π ; π [. Due to our assumptions on γ , F and p, Morera’s theorem implies that the functions ν, log κ and log κ p are holomorphic on U . Moreover we have | (ν (λ)) | < 1/2, ∀λ ∈ U . Indeed ν (λ) =
1 i log |1 + γ F (λ)| − arg (1 + γ F (λ)) , 2π 2π
(4.5)
and we have assumed that arg (1 + γ F) ∈ ] −π ; π [. We use the function α to transform the RHP for χ . Let us define the matrix (λ) according to
α (λ) 0 (λ) = χ (λ) . 0 α −1 (λ)
(4.6)
This new matrix (λ) satisfies the following RHP: • is analyticon C\ [ −q ; q ]; ±σ (ν ) 1 1 2 λ − q 2 3 ± log λ2 − q 2 • |(λ)| = O 11 • (λ) → I2 ;
for λ → ±q ;
λ→∞
• + (λ) G (λ) = − (λ) for λ ∈ [ −q ; q ]. Here the new jump matrix G reads G (λ) =
1 + P(λ)Q(λ) P(λ)ei x p(λ) Q(λ)e−i x p(λ) 1
,
(4.7)
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Fig. 2. Contours + and − associated with the RHP for ϒ
and γ F(λ) sin π ν(λ) g(λ) α −2 (λ) eg(λ) = −2ieiπ ν(λ) e , 1 + γ F(λ) + α+2 (λ) sin π ν(λ) 2 γ F(λ) 2 α− (λ) e−g(λ) = 2ieiπ ν(λ) α− (λ) . Q(λ) = − 1 + γ F(λ) eg(λ) P(λ) =
(4.8) (4.9)
The solution of this RHP for exists as it can be constructed from χ . Moreover it is unique as seen by arguments similar to those providing uniqueness of the solution to the RHP for χ . 4.2. The second step → ϒ. As already mentioned, the jump matrix G admits an explicit factorization into a product of upper and lower triangular matrices: G = M+ M− . The matrices M± are given by 1 P (λ) ei x p(λ) , M+ (λ) = 0 1
(4.10)
M− (λ) =
1 0 , Q (λ) e−i x p(λ) 1
(4.11)
and can be continued to U ∩H+ , resp. U ∩H− , where we recall that H± is the upper/lower half plane and U is the domain of holomorphy of all the functions appearing in the RHP. Then we draw two new contours ± in p (U ) and define a new matrix ϒ(λ) according to Fig. 2. As readily checked, ϒ (λ) is continuous across ] −q ; q [ and thus holomorphic in the interior of + ∪ − . We have thus removed the cut along [ −q ; q ] and replaced it with cuts along + ∪ − . The matrix ϒ solves the following RHP: • ϒ is analytic C\ ; + ∪ −± (ν in ± ) |λ ∓ q|∓ (ν± ) |λ ∓ q| 11 log |λ ∓ q| , λ −→ ±q; • ϒ(λ) = O 11 |λ ∓ q|∓ (ν± ) 0 λ∈H I ± (ν± ) |λ ∓ q| 0 11 log |λ ∓ q| , λ −→ ±q; • ϒ(λ) = O 11 |λ ∓ q|± (ν± ) |λ ∓ q|∓ (ν± ) λ∈H I I
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Fig. 3. Contours in the RHP for P
11 |λ ∓ q|±σ3 (ν± ) log |λ ∓ q| , λ −→ ±q; 11 λ∈H I I I • ϒ(λ) → I2 ; λ→∞ ϒ+ (λ) M+ (λ) = ϒ− (λ) for λ ∈ + , • −1 ϒ+ (λ) M− (λ) = ϒ− (λ) for λ ∈ − ,
• ϒ(λ) = O
where the domains H I , H I I , H I I I are shown on Fig. 2. Clearly, the solution of the RHP for ϒ exists and is unique. Hence, the matrices ϒ and χ are in a one to one correspondence. −1 Note that, except in some vicinities of q and −q, the jump matrices M+ and M− for ϒ are exponentially close to the identity matrix. Therefore, to study the asymptotic solution of the RHP, it is enough to study the local problems in the vicinities of q and −q. 4.3. Parametrix around −q. We first present the parametrix P on a small disk D−q,δ ⊂ U of radius δ and centered at −q, that is an exact solution of the RHP: { ∪ − } ; • P is analytic D−q,δ \ +− (ν on − ) |λ + q| (ν− ) |λ + q| 11 • P(λ) = O log |λ + q| , λ −→ −q ; 11 |λ + q| (ν− ) 0 λ∈H I |λ + q|− (ν− ) 0 11 log |λ + q| , λ −→ −q ; • P(λ) = O − (ν− ) |λ 11 + q| (ν− ) λ∈H I I |λ + q| 11 |λ + q|−σ3 (ν− ) log |λ + q| , λ −→ −q ; • P(λ) = O 11 λ∈H I I I 1 , uniformly for λ ∈ ∂ D−q,δ , • P(λ) = I2 + O x 1−ε for λ ∈ + ∩ D−q,δ , P+ (λ) M+ (λ) = P− (λ) • −1 P+ (λ) M− (λ) = P− (λ) for λ ∈ − ∩ D−q,δ . Here ε = 2 sup | (ν)| < 1. The canonically oriented contour ∂ D−q,δ is depicted ∂ D−q,δ
in Fig. 3. The RHP for P admits a class of solutions. Each element of this class is related to another one through a left multiplication by a holomorphic matrix that is uniformly I2 + O 1/x 1−ε on ∂ D−q,δ . In order to construct the solution P to this problem, we first focus on the simpler case where the functions F, g and κ p are constant. Then the solution
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to the RHP for Pconst can be obtained by the differential equation method [11,12,26]. This leads to the solution −νσ3 iπ ν Pconst (λ) = (λ) L (λ) ζ−q e 2 . (4.12) Here ζ−q = x ( p (λ) − p− ), ν = i log (1 + γ F) /2π ,
ib12 1 + ν, 1; iζ −ν, 1; −iζ−q −q
, (λ) = ν, 1; iζ−q −ib21 1 − ν, 1; −iζ−q
(4.13)
and finally b12 (λ) = −i b21 (λ) = −i
sin [π ν] 2 (1 + ν) π κ 2p [x ( p+ − p (λ))]2ν π κ 2p [x ( p+ − p (λ))]2ν sin [π ν] 2 (ν)
ei x p− +g ,
(4.14)
e−i x p− −g .
(4.15)
(a, c; z) denotes Tricomi confluent hypergeometric function (CHF) of the second kind (see Appendix A). It solves the differential equation zy + (c − z) y − ay = 0 .
(4.16)
Recall that has a cut along R− . Note that this choice for the cut of implies the use of the principal branch of the logarithm: −π < arg (z) < π . The expression for the piecewise constant matrix L depends on the region of the complex plane. Namely, ⎧ I2 −π/2 < arg p (λ) − p− < π/2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 0 π/2 < arg p (λ) − p− < π , −2iπ ν 0e L (λ) = (4.17) ⎪ ⎪ ⎪ −2iπ ν ⎪ ⎪ ⎪ 0 ⎪ ⎩ e −π < arg p (λ) − p− < −π/2. 0 1 The reader can check using the monodromy properties of Tricomi CHF (A.4) and (A.5) that the jump condition in the case of constant functions F and g are satisfied by the matrix Pconst . Moreover the asymptotic expansion for (a, c; z) at z → ∞ allows one to check that Pconst has the correct behaviour at infinity. We recall that this parametrix also appeared recently in the work [32]. In order to extend this result to the case of arbitrary holomorphic functions F (λ), g (λ) and κ p (λ), it is enough to add the λ dependency in all places where these functions appear. One ends up with the following solution to the RHP for P : −ν(λ)σ3 iπ ν(λ) e 2 . P (λ) = (λ) L (λ) ζ−q Here ζ−q = x ( p (λ) − p− ),
−ν (λ) , 1; −iζ−q
(λ) = −ib21 (λ) 1 − ν (λ) , 1; −iζ−q
(4.18)
ib12 (λ) 1 + ν (λ) , 1; iζ−q , ν (λ) , 1; iζ−q (4.19)
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with b12 (λ) = −i b21 (λ) = −i
sin [π ν (λ)] 2 (1 + ν (λ)) π κ 2p (λ) [x ( p+ − p (λ))]2ν(λ) π κ 2p (λ) [x ( p+ − p (λ))]2ν(λ) sin [π ν (λ)] 2 (ν (λ))
eg(λ)+i x p− = −iν (λ) u (λ; x) , e−i x p− −g(λ) = −i
ν (λ) , u (λ; x)
(4.20) (4.21)
and finally u (λ; x) =
(1 + ν (λ)) κ p (λ) x ν(λ) [ p+ − p (λ)]ν(λ)
(1 − ν (λ))
−2
ei x p− +g(λ) .
(4.22)
In the above formulae we have explicitly stressed the dependence of the functions b12 , b21 and ν on λ. Finally, the matrix L (λ) is given by (4.17) with ν replaced by the function ν (λ). This construction originates from the observation that the replacements F → F (λ), g → g (λ) and κ p → κ p (λ) preserve the jump conditions as the latter hold pointwise. Of course, once the parametrix P is guessed it is not a problem to check directly that it solves the RHP in question. The asymptotic behaviour is inferred from (A.6), whereas the jump conditions can be verified thanks to (A.4) and (A.5). Furthermore, due to the definition of the matrix L, the solution is continuous across the line arg p − p =π (λ) − and thus analytic in the whole domain λ ∈ C; p (λ) − p− < 0 . & around q reads 4.4. Parametrix around q. The RHP for the parametrix P & { • P is analytic
− } ; on D q,δ \ + ∪+ (ν + ) |λ − q|− (ν+ ) |λ − q| 1 1 & log |λ − q| , λ −→ q ; • P(λ) =O 11 |λ − q|− (ν+ ) 0 λ∈H I + (ν+ ) |λ − q| 0 1 1 & log |λ − q| , λ −→ q ; • P(λ) =O + (ν+ ) |λ 11 − q|− (ν+ ) λ∈H I I |λ − q| 1 1 σ (ν ) & |λ − q| 3 + log |λ − q| , λ −→ q ; • P(λ) =O 11 λ∈H I I I 1 & uniformly for λ ∈ ∂ Dq,δ ; • P(λ) = I2 + O x 1−&ε &+ (λ) M+ (λ) = P &− (λ) for λ ∈ + ∩ Dq,δ , P • −1 &+ (λ) M− &− (λ) for λ ∈ − ∩ Dq,δ , P (λ) = P & around q and & ε = 2 sup∂ Dq,δ | (ν)| < 1. The solution of the RHP for the parametrix P can be formally obtained from the one at −q through the transformation q → −q and ν → −ν on the solution to the RHP for P. Indeed, the two RHP are identical modulo this negation. Just as for the parametrix around −q, we focus on the solution & (λ) = & (λ) & P L (λ) ζqν(λ)σ3 e− where ζq = x [ p (λ) − p+ ], and
ν (λ) & (λ) =
, 1; −iζq −i b˜21 (λ) 1 + ν (λ) , 1; −iζq
iπ ν(λ) 2
,
(4.23)
i b˜12 (λ) 1 − ν (λ) , 1; iζq . −ν (λ) , 1; iζq (4.24)
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& Fig. 4. Contours in the RHP for P
Here sin[π ν(λ)] 2 (1 − ν(λ)) b˜12 (λ) = i ˜ x), [x( p(λ) − p− )]2ν(λ) eg(λ)+i x p+ = iν(λ)u(λ; π κ 2p (λ) b˜21 (λ) = i
π κ 2p (λ) e−g(λ)−i x p+ sin[π ν(λ)] 2 (−ν(λ)) [x( p(λ) − p− )]2ν(λ)
=i
ν(λ) , u(λ; ˜ x)
and
(1 − ν (λ)) u˜ (λ; x) =
(1 + ν (λ))
ν(λ) 2 x ν(λ) p (λ) − p− ei x p+ +g(λ) . κ p (λ)
(4.25)
Just as for the parametrix around −q, the matrix L˜ (λ) depends on the quadrant of the complex plane: ⎧ I2 −π/2 < arg [ p (λ) − p+ ] < π/2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎨ 1 π/2 < arg [ p (λ) − p+ ] < π , 0 e2iπ ν(λ) L˜ (λ) = (4.26) ⎪ ⎪ ⎪ 2iπ ν(λ) ⎪ ⎪ ⎪ 0 ⎪ ⎩ e −π < arg [ p (λ) − p+ ] < −π/2. 0 1 4.5. The last transformation ϒ → . Let ⎧ &−1 (λ) for λ ∈ Dq,δ , ⎨ ϒ(λ)P (λ) = ϒ(λ)P −1 (λ) for λ ∈ D−q,δ (4.27) , ⎩ ϒ(λ) for λ ∈ C\ D q,δ ∪ D −q,δ . Introduce the curve C = { + ∪ − } ∩ D q,δ ∪ D −q,δ . Then is continuous across C \ {q, −q}. Since is holomorphic in a vicinity of C , we have that is holomorphic & and ϒ around the in Dq,δ ∪ D−q,δ \ {q, −q}. This, in turn, due to the estimates for P, P points ±q, ensures that the singularities at these points are of a removable type. Hence is holomorphic on the disks Dq,δ ∪ D−q,δ . Finally, we see that satisfies the following RHP:
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Fig. 5. Contour appearing in the RHP for
• is analytic in C\ (cf Fig. 5) ; • ⎧ (λ) = I2 + O (1/λ) for λ → ∞; + (λ) M+ (λ) = − (λ) for λ ∈ + , ⎪ ⎪ ⎨ −1 , + (λ) M− (λ) = − (λ) for λ ∈ − • + (λ) P (λ) = − (λ) for λ ∈ ∂ D−q,δ , ⎪ ⎪ ⎩ & (λ) = − (λ) for λ ∈ ∂ Dq,δ . + (λ) P The solution to the RHP for , exists and is unique as seen by standard arguments. ∪ The jump matrices for are uniformly exponentially close to I2 in x on − + ε−1 on ∂ Dq,δ ∪ ∂ D−q,δ , with ε = 2 sup∂ Dq,δ ∪ ∂ D−q,δ | (ν)|. As and uniformly I2 + O x
a consequence, I2 is the unique solution of the RHP, up to uniformly O x ε−1 corrections. In addition, using the equivalence between singular integral equations and RHP, the asymptotic expansion of can be obtained by a Neumann series. This will be done in the upcoming section. 5. Asymptotic Solution of the RHP In this section we asymptotically solve the above RHP for . We derive an asymptotic expansion into negative powers of x for the jump matrices for , and use it to prove the existence of an asymptotic series for . The corresponding asymptotic series for χ follows readily. One can finally infer the asymptotic behaviour of the resolvent of the GSK up to any order in 1/x. 5.1. Asymptotics of the jump matrices. Denote the jump matrices for by I2 + (λ). Then the matrix (λ) has the asymptotic expansion in the limit x → +∞: (λ) =
M (m) (λ; x) −M−1+ε , + o x xm
(5.1)
m=1
with ε = 2 sup∂ Dq,δ ∪ ∂ D−q,δ | (ν)|. The explicit form of the matrices (n) (λ; x) depends on the position of λ in the con , whereas the asymptotic expansion tour : they vanish to any order in 1/x on + ∪ −
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
for on ∂ Dq,δ ∪ D−q,δ follows promptly from CHF (A.6). More explicitly, for any n ∈ N∗ , ⎧ (n) (λ; x) ⎪ ⎪ ⎪ (−) n ⎪ ⎪ ⎪ ⎪ p (λ) − p− ⎪ ⎨ (n) (λ; x) (n) (λ; x) = (+) ⎪ ⎪ ⎪ ⎪ [ p (λ) − p+ ]n ⎪ ⎪ ⎪ ⎪ ⎩ 0
711
the asymptotic expansion of Tricomi
for λ ∈ ∂ D−q,δ , for λ ∈ ∂ Dq,δ ,
(5.2)
. for λ ∈ + ∪ − −n We have separated the jump matrices into their pole parts p (λ) − p± and regular (n) , with and parts (n) (−) (+) ⎛ ⎞ n u(λ; x) n 1 − i ⎜ 0 ⎟ (−1)n (−ν(λ))2n ν(λ) (5.3) (n) (λ; x) = ⎝ ⎠ n (−) 0 (ν(λ))2n n! 1 ν(λ)u(λ; x)
for λ ∈ ∂ D−q,δ , and
⎛
1 ⎜ (n) (λ; x) = ⎝ n (+) n! − ν(λ)u(λ; ˜ x) in
⎞ n u(λ; ˜ x) 0 (−1)n (ν(λ))2n ν(λ) ⎟ (5.4) ⎠ 0 (−ν(λ))2n 1
for λ ∈ ∂ Dq,δ . Here we use the standard notation (ν)n = (ν + n) / (ν) and u (λ; x), resp. u˜ (λ; x), have been defined in (4.22), resp. (4.25). Thus, the matrices (n) depend on x, but their entries are a O(x ε ). 5.2. Asymptotic expansion for . Using the equivalence between RHP and singular integral equations we can express in terms of its boundary value from the “+” side of the contour 1 ds . (5.5) (λ) = I2 + + (s) (s) 2iπ λ−s
In its turn + (λ) belongs to L 2 ( ) and fulfills the linear singular integral equation of Cauchy type + + (z) = I2 + C [+ ] (z) .
(5.6)
Recall that the Cauchy operator on L 2 ( ) is defined as + [g](z) = lim+ C [g](t) and C [g](t) = C t→z
1 2iπ
g(s) ds , t ∈ . (5.7) t −s
The notation t → z + stands for the non-tangential limit of t approaching z from the “+” side of the contour . Recall that the Cauchy operator is bounded: i.e. there exists
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- + a constant c2 such that, for any function g ∈ L 2 ( ), one has -C [g] - ≤ c2 g, where . is the canonical L 2 ( ) norm. The matrix + can be asymptotically approximated by the following series: (k)
Proposition 5.1. Let + be defined recursively according to (k)
+ =
k
! " (k− p) ( p) (0) + with + = I2 . C +
(5.8)
p=1
Then, for any integer M > 0, there exists a constant C(M) > 0 such that M−1 C(M) − p ( p) -+ − x + - ≤ M(1−ε) . x p=0
(5.9)
Proof. Let us prove this statement by induction on M. For M = 1 we have that + − I2 = -C + [(+ − I2 ) ] + C + []- ≤ c2 + − I2 + c2 .
(5.10) Therefore, for x large, + − I2 ≤
c2 C(1) ≤ 1−ε . 1 − c2 x
(5.11)
Let us now suppose that the result holds up to M. Then, ⎡ ⎤- M M k - - - + ⎣ −k (k− p) ( p) ⎦C = x −k (k) − x -+ − + + + - p=1 k=0 k=1 ⎡ . ⎛ ⎞ ⎤/ M− M M p - + −k (k) (k) −k ⎠ ( p) − p ⎦⎣ ⎝ C =x − x x − + + + - + p=1 k=1 k=0 ≤ c2 + C M x −M−1+ε +
M
c2 -( p) - x − p C(M − p) x ( p−M−1)(1−ε)
p=1
C(M + 1) ≤ (M+1)(1−ε) , x
(5.12)
for some constants C M and C(M + 1). We used the fact that all ( p) are in L 2 ( ) and that + is bounded by virtue of (5.11). Let us now extend this result for points λ being uniformly away from the contour . Define the matrices (0) (z) = I2 ,
( p) (z) =
p
! " ( p−k) (k) (z), p > 0, C +
(5.13)
k=1
(z; M) =
M
x − p ( p) (z),
p=0
which are analytic away from . Then we have the following result:
(5.14)
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Proposition 5.2. Let K be any compact subset of C\ . Then, ∀k ∈ N, ∀M ∈ N , k! C(M) lgth( ) k , λ ∈ K. (5.15) ∂λ (λ) − ∂λk (λ; M − 1) ≤ d(K , )k+1 x M(1−ε) Here |.| denotes the usual max norm || ≡ maxi, j i, j , d(., .) is any distance on C and lgth( ) is the length of the curve . Proof. Let k ∈ N, M ∈ N , then M−1 k − p k ( p) ∂ (λ) − x ∂ (λ) λ λ p=0 ⎫ ⎧ p M−1 k! ⎬ ⎨ ( p−l) ds −p (l) ≤ (s) (s) − x (s) (s) + + k+1 ⎩ 2iπ ⎭ − s) (λ p=1 l=1 ≤
k! C(M) lgth( ) , M(1−ε) x d k+1 (K , )
due to (5.9).
5.3. The functions f ± to the leading order. We now perform the transformations from back to χ . The solution to the RHP of Proposition 3.1 reads χ (λ) = (λ) χ (0) (λ) .
(5.16)
We call χ (0) the zeroth order solution (i.e. obtained for = I2 ). In the vicinities of the endpoints of [ −q ; q ], χ (0) is given as P(λ) M+−1 (λ) α(λ)−σ3 , λ ∈ D−q,δ ∩ {0 < arg[ p(λ) − p− ] < π/2}, (0) χ (λ) = & M+−1 (λ) α(λ)−σ3 , λ ∈ Dq,δ ∩ {π/2 < arg[ p(λ) − p+ ] < π }. P(λ) Similarly, on [ −q ; q ], and uniformly away from the endpoints, χ (0) (λ) = M+−1 (λ) α+ (λ)−σ3 ,
λ ∈ ] −q + δ ; q − δ [ .
In the (λ) = 0+ limit and for (λ) ∈ [ −q ; q ], 2iπ ν e+ (λ) e = (α+ e− )−σ3 M+−1 α+−σ3 , e− (λ) 1 so that, for λ ∈ ] −q + δ ; q + δ [ , / . ! "σ (0) f + (λ) iπ ν(λ) −1 iπ ν(λ) 3 1 , α+ (λ) e+ (λ) e =e (0) 1 f − (λ) where we have explicitly written all the dependence on λ.
(5.17)
(5.18)
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When λ ∈ [ −q ; −q + δ ], we should multiply the latter expression by P. Using the decomposition (A.7) of Humbert CHF into a sum of two Tricomi CHF we get . / . /σ3
(0) iπ ν e+ (λ)
(1 + ν) −ν, 1; −iζ−q f + (λ)
2 , (5.19) =e (0)
(1 − ν) ν, 1; iζ−q κ p (λ) ζqν f − (λ) with ζq = x [ p+ − p (λ)] and ζ−q = x p (λ) − p− . Analogously, for λ ∈ [ q − δ ; q ], / .
ν σ3 (0) e+ (λ)ζ−q iπ ν
(1 − ν) ν, 1; iζq f + (λ) 2 . (5.20) = e (0)
(1 + ν) −ν, 1; −iζq κ p (λ) f − (λ) (0)
Note that the piecewise expressions for the functions f ± (λ) are in fact analytic in a vicinity of their respective domain of validity, although they have been obtained by taking the limit of λ approaching a point in [ −q ; q ] from the upper half plane. More precisely, the formula (5.19) holds on D−q,δ , (5.20) on Dq,δ , and (5.18) on the connected component of the interior of π containing [ δ − q ; q − δ ]. This observation follows from (3.3), but of course it can be checked by a direct computation based on the expression for the matrix χ in the lower half plane. 5.4. Integral bounds for the resolvent. We now introduce a function R (0) (λ, µ) and show that it is a good approximation of the resolvent in the sense that (5.21) tr R − R (0) = O x ε−1 . Such estimates are necessary for the integration of the γ -derivative of log det [I + V ]. Definition 5.1. Let τ (λ) denote the solution of the RHP given in Subsect. 4.5 whose . jumps are on circles of radius τ and on the corresponding curves + and − (0)
(0)
We can then write the solution of the RHP for χ as χ (λ) = τ χτ . There χτ do not depend explicitly on τ . The radius τ only determines which patch we should use for the (0) (0) definition of the matrix χτ . Moreover the whole combination τ χτ does not depend on the radius τ at all. Hence, we can represent the exact resolvent as ! "−1 −1 (λ) (µ) τ τ R (λ, µ) = E L (λ) | χτ(0) (λ) χτ(0) (µ) | E R (µ) . (5.22) λ−µ There, without altering the value of R (λ, µ), we can choose different values of τ depending on the point we consider. This is quite useful as we can take one value of the radius τ in order to have estimates around ±q and another one to perform estimates in the bulk [ δ − q ; q − δ ]. This will become clearer during the proof of the proposition below. Definition 5.2. Let us fix δ, q > δ > 0 and define what we call the diagonal zeroth order resolvent γ F (λ) (0) (0) (0) (0) R (0) (λ, λ) = ∂λ f + (λ) f − (λ) − ∂λ f − (λ) f + (λ) , (5.23) 2iπ (0)
where the functions f ± (λ) are given by (5.18) for λ ∈ [ δ − q ; q − δ ], (5.19) for λ ∈ [ −q ; δ − q [ and (5.20) for λ ∈ ] q − δ ; q ]. Similarly, | F R;(0) (λ) and (0) F L;(0) (λ) | are defined in terms of the same functions f ± .
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We stress that the radius τ previously introduced to build the exact solution τ (λ)χτ (λ) and δ appearing in the definition are, a priori, unrelated. Proposition 5.3. Let R (λ, µ) be the exact resolvent of the generalised sine kernel. Then (5.24) tr R − R (0) = O x ε−1 , where the O is uniform in γ ∈ D0,r . Proof. According to the preceding observations we have, for λ ∈ [ −q ; −q + δ [ ∪ ] q − δ ; q ], R (λ, λ) = R (0) (λ, λ) + F L;(0) (λ) |2δ (λ) ∂λ 2δ (λ) | F R;(0) (λ) ,
(5.25)
R (λ, λ) = R (0) (λ, λ) + F L;(0) (λ) |δ/2 (λ) ∂λ δ/2 (λ) | F R;(0) (λ) ,
(5.26)
and
for λ ∈ [ δ − q ; q − δ ]. The advantage of using two different matrices for the corrections of R (λ, λ) with respect to the zeroth order resolvent R (0) (λ, λ) is that the corrections are always analytic on the whole domain where they are considered. One does not need to take into account that δ (λ) has a jump across λ = ± (q − δ). This might be problematic as, for instance, the integral of ∂λ (λ) on [ −q ; δ − q ] might be ill-defined. Moreover the uniform estimates that we have derived for the matrix (λ) only hold uniformly away from the jump contour. As we will only integrate the terms containing 2δ on [ −q ; δ − q [ ∪ ] q − δ ; q ], we will be in this situation. The same holds for the terms involving δ/2 . However, we would not be able to use the uniform estimates (5.2) for ∂λ δ when integrating it on [ −q ; δ − q ], as we would not always be uniformly away from the boundary of the jump contour for δ . With this way of understanding the corrections we have ⎛ ⎞ δ−q q ⎜ ⎟ L;(0) + tr R − R (0) = ⎝ (λ) |2δ (λ) ∂λ 2δ (λ) | F R;(0) (λ) ⎠ dλ F −q
q−δ
q−δ
+
dλ F L;(0) (λ) |δ/2 (λ) ∂λ δ/2 (λ) | F R;(0) (λ) .
(5.27)
δ−q
Let us start by the bulk part of the integral, i.e. the part on [ δ − q ; q − δ ]. From (0) the explicit form for f ± on [ δ − q ; q − δ ] given in (5.18) we see that these functions are uniformly O (1). Moreover, the uniform estimates for the matrices δ/2 (λ) for λ uniformly away from the jump contour guarantee that (5.28) F L;(0) (λ) |δ/2 (λ) ∂λ δ/2 (λ) | F R;(0) (λ) = O x ε−1 ,
the O x ε−1 being uniform in γ , at least for γ small enough. The situation at the boundaries is a little more complex. We only consider the right boundary as the other case is treated similarly. We still have that 2δ (λ) = I2 +O x ε−1
(0) and ∂λ 2δ (λ) = O x ε−1 uniformly on [ q − δ ; q ]. However the functions f ± (λ)
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are no longer uniformly a O (1) on this interval. We should thus estimate the following integral: q σ,σ =±q−δ
(0)
f σ(0) (λ) f σ (λ) G σ,σ (λ) dλ
(5.29)
with G σ,σ (λ) = O x ε−1 being related to the entries of δ/2 (λ) ∂λ δ/2 (λ). The sit uation being
similar for all the possible choices of σ and σ , we explain the mechanism for σ, σ = (+, +). The asymptotics of Humbert CHF guarantees that (a, 1; ±it) =
c± (1 + o (1)) t → +∞ |t|a
(5.30)
for some computable constants c± depending on a. These constants are continuous with respect to a belonging to an open neighbourhood of ν ([ q − δ ; q ]), and so is the o (1) term. Hence, there exists an a independent constant C such that (1 + |t|) (a) (a, 1; ±it) ≤ C .
(5.31)
Indeed the latter function is continuous on R and has a finite limit at ∞. Moreover the constant C can be chosen in such a way that the estimate holds for a belonging to some small vicinity of ν ([ q − δ ; q ]). Hence, by explicitly extracting the x ε−1 factor coming from G +,+ (λ) we get that, for some constant C , (0) (0) f + (λ) f + (λ) G +,+ (λ) ≤ C x ε−1 ϕx ( p (λ) − p+ ) ,
(5.32)
with ϕx (t) = x 2 (ν(λ)) (1 + x |t|)−2 (ν(λ)) . The function ϕx (t) fulfills |ϕx ( p (λ) − p+ )| ≤ C˜ | p (λ) − p+ |−2 (ν(λ))
(5.33)
as, for any α ∈ R, t → t α / (1 + t α ) is bounded. The latter function is integrable on the case | (ν)| < 1/2). Thus the integrals in (5.29) do even[ q − δ ; q ] (we consider
tually yield O x ε−1 contributions. One can prove, in a very similar way, the estimates for the Hilbert–Schmidt norm of the resolvent. Namely, Proposition 5.4. Under the assumptions of the previous proposition, - R − R (0) - = O x ε−1 2
with .2 being the Hilbert–Schmidt norm.
(5.34)
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5.5. Asymptotic expansion of the resolvent. We now prove that the asymptotic expansion for can be used to obtain an asymptotic expansion for the diagonal of the resolvent R(λ, λ). We derive point-wise bounds for the latter as this quantity appears in the q-derivative of the Fredholm determinant: ∂q log det [I + V ] = R (q, q) + R (−q, −q) .
(5.35)
We need to estimate the error when we replace the exact resolvent R by the approximate one R (0) . The magnitude for the error term follows from the following result: Proposition 5.5. Let χ (0) be the solution of the RHP for χ up to the leading order in x, that is to say the one obtained from = I2 and corresponding to the contour with disks D±q,δ having radius δ. Define the leading vectors F L;(0) | and | F R;(0) as F L;(0) (λ) | = E L (λ) | χ (0) (λ)−1 ,
| F R;(0) (λ) = χ (0) (λ) | E R (λ) , (5.36)
and the leading order of the resolvent by R (0) (λ, µ) =
F L;(0) (λ) | F R;(0) (µ) . λ−µ
(5.37)
k R ( p) (λ, λ) xε , + O xp x (k+1)(1−ε)
(5.38)
Then R (λ, λ) = R (0) (λ, λ) +
p=1
for λ uniformly away from and belonging to [ −q ; q ]. Here, R (0) (λ, λ) = −F L;(0) (λ) | ∂λ F R;(0) (λ), ˆ ( p) (λ) | F R;(0) (λ) , p > 0, R ( p) (λ, λ) = − F L;(0) (λ) |
(5.40)
. (k+1)(1−ε)
(5.41)
(5.39)
in which −1 (λ; k) ∂λ (λ; k) =
k p=1
ˆ ( p) (λ) x − p + O
1
x
Proof. Clearly, R(λ, µ) =
F L;(0) (λ) | F R;(0) (µ) −1 (λ) (µ) − I2 R;(0) + F L;(0) (λ) | |F (µ) , λ−µ λ−µ → −F L;(0) (λ) | ∂λ F R;(0) (λ)− F L;(0) (λ) |−1 (λ) ∂λ (λ)| F R;(0) (µ) .
λ→µ
The corrections to the leading order for the resolvent are given here by the second term. The inversion operator on M2 (C): u → u −1 is continuously differentiable around the identity I2 . Thus there exists an open neighbourhood matrix I2 - W of the identity and a constant C > 0 such that, ∀ A, B ∈ W , one has - A−1 − B −1 - ≤ C A − B. Here . denotes any matrix norm. The matrices (λ) and (λ; k) belong to W for x
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sufficiently large, as they both go to I2 in the x → +∞ limit for λ uniformly away from , and we get, from Proposition 5.2, - −1 - (λ) ∂λ (λ) − −1 (λ; k) ∂λ (λ; k)≤ C (λ) − (λ; k) ∂λ (λ) + C ( (λ) + (λ) − (λ; k)) ∂λ (λ) − ∂λ (λ; k) & (k) C ≤ (k+1)(1−ε) , x & for some constant C(k). Thus, uniformly away from and on the real axis, one has L;(0) (λ) |−1 (λ) ∂λ (λ) − −1 (λ; k) ∂λ (λ; k)| F R;(0) (λ) F xε =O . (5.42) x (k+1)(1−ε) (0)
In the last equality, we used the fact that f ± are at most of order O(x ε ) on the real axis, as follows from their behaviour around ±q. 6. Leading Asymptotic Behaviour of log det [I + V ] In this section, we prove the result of Theorem 2.1; that is to say, we compute the leading asymptotic behaviour det[I + V ](0) of det[I + V ] up to o(1) corrections in the x → +∞ limit. More precisely, we show that ⎡ ⎤ q σ νσ (σ q; q) G(1, ν ) κ σ ⎦ log det [I + V ](0) = 2 dλ ν(λ) log [e− (λ)] + log ⎣
νσ2 2qp x σ =± σ −q 1 + 2
q dλ dµ −q
ν (λ) ν(µ) − ν(λ) ν (µ) . λ−µ
(6.1)
This result will be obtained by two different methods based on the integration of Eqs. (3.13). The first one, which uses the derivative of the Fredholm determinant over γ , is based on the uniformness of the asymptotic expansion for the resolvent for γ small enough. It is worth mentioning that this way is technically quite involved. The second method deals with the derivative of the Fredholm determinant over q. Although we have not been able to provide a full rigorous proof for it, we would like to draw the reader’s attention to this method as it is much more direct and simple. 6.1. The leading asymptotics from the γ -derivative method. Due to Proposition 5.3, the proof of the leading asymptotics of the Fredholm determinant from the first equation (3.13), q ∂γ log det [I + V ] = −q
dλ R (λ, λ) , γ
(6.2)
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719
only necessitates the use of R (0) (λ, λ) defined in (5.37). Recall that R (0) (λ, λ) has different leading asymptotics in the bulk ] −q ; q [ and near the boundary. Let δ > 0 be sufficiently small. Then ⎧ (0) ⎪ ⎪ Rq (λ, λ) λ ∈ [ q − δ ; q ] , R (0) (λ, λ) ⎨ (0) = Rbk (λ, λ) λ ∈ [ −q + δ ; q − δ ] , (6.3) ⎪ γ ⎪ ⎩ R (0) (λ, λ) λ ∈ [ −q ; −q + δ ] , −q
where
0 1 F(λ) p+ − p(λ) ν(λ) = , 2∂λ log e+ (λ) − 2∂λ log κ p (λ) 2iπ (1 + γ F(λ)) p(λ) − p− (0) R−q (λ, λ) = −νϕ ν; x p − p− 2ν log x − 2∂λ ν log ( p+ − p) − 2∂λ log κ p + ν [ψ (1 + ν) + ψ (1 − ν)] + g + i xνp τ ν; x p − p− + νν ρ ν; x p − p− , Rq(0) (λ, λ) = −νϕ (ν; x [ p+ − p]) 2ν log x − 2∂λ ν log ( p − p− ) − 2∂λ log κ p + ∂λ ν [ψ (1 + ν) + ψ (1 − ν)] + g (0) Rbk (λ, λ)
+ i xνp τ (ν; x [ p+ − p]) − νν ρ (ν; x [ p+ − p]). Here ψ(z) =
d dz
log (z) and we have introduced the shorthand notations
ϕ (ν; t) = (−ν, 1; −it) (ν, 1; it) , ρ (ν; t) = (∂1 ) (ν, 1; it) (−ν, 1; −it) + (∂1 ) (−ν, 1; −it) (ν, 1; it) , τ (ν; t) = − (−ν, 1; −it) (ν, 1; it) + (∂z ) (−ν, 1; −it) (ν, 1; it) + (−ν, 1; −it) (∂z ) (ν, 1; it) . Moreover, in order to lighten the above expressions and similar ones in the following, we omit the explicit dependence on the argument λ of the different functions involved (like ν, p, and their derivatives ν , p , etc.). We can now split the integration contour into three parts q R −q
(0)
dλ = (λ, λ) γ
−q+δ −q
dλ (0) R (λ, λ) + γ −q
q−δ
−q+δ
dλ (0) R (λ, λ) + γ bk
q
q−δ
dλ (0) R (λ, λ) . γ q (6.4)
The bulk integral is carried out straightforwardly. The integrals over the vicinities of the endpoints are more involved. Consider, for instance, the integration over [ −q ; −q + δ ]. Using the asymptotic series for Humbert CHF (A.9) and Eqs. (A.10), (A.11) we get that ϕ (a; t) − eiπa ,
4ia eiπa 2 log t − ψ (1 − a) − ψ (1 + a) − ,
(1 − a) (1 + a) 1+t 2ia , τ (a; t) + eiπa 1 − 1+t
ρ (a; t) +
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are uniformly Riemann integrable on R+ in the sense of the definition of Lemma B.1 (see Appendix B). Using the integration Lemma B.1 as well as the estimates for the integrals of τ and ϕ (A.10), (A.11), we find
−q+δ
R
(0)
−q+δ
(λ, λ) dλ = −
−q
−q
dλ eiπ ν 2ν log x − 2∂λ ν log ( p+ − p)
(ν) (1 − ν)
+ν (ψ (ν) + ψ (−ν)) + g
dλ eiπ ν ν 4iν 2 log x ( p− p− ) − ψ (−ν)−ψ (ν) +
(ν) (1−ν) 1 + x ( p − p− )
−q+δ
+ −q
2iν −1 dλ p ν x ( p − p− ) + 1
−q+δ
+ ix −q
+
ν− eiπ ν− {2 − ψ (ν− ) − ψ (−ν− )} + o (1) .
(1 − ν− ) (ν− )
(6.5)
Here the o (1) is with respect to the successive limits xδ → +∞ and δ → 0. The two terms proportional to log x compensate each other. The remaining part of the first three lines of (6.5) is an O (δ) and can thus be dropped. The integral in the last two lines of (6.5) is evaluated thanks to the second integration Lemma B.2 (see Appendix B). We get,
−q+δ
R
(0)
−q+δ
(λ, λ) dλ = −i
−q
−q
eiπ ν− ν− eiπ ν p dλ +
(ν) (1 − ν)
(ν− ) (1 − ν− )
× −2 log x ( p (δ − q) − p− ) + 2 − ψ (ν− ) − ψ (−ν− ) + o (1) .
(6.6)
The integration over [ q − δ ; q ] can be treated similarly. The result reads
q q−δ
R (0) (λ, λ) dλ = −i
q q−δ
ν+ eiπ ν+ eiπ ν p +
(ν) (1 − ν) (ν+ ) (1 − ν+ )
× {−2 log [x ( p+ − p (q − δ))] + [2 − ψ (ν+ ) − ψ (−ν+ )]} + o (1) ,
(6.7)
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721
so that, q R
(0)
q (λ, λ) dλ =
−q
−q
γF i x p + g − 2∂λ log κ p dλ 2iπ (1 + γ F)
q−δ % $ γF dλ γνF p+ − p q−δ p+ − p ∂λ − log + ν log p − p− 1 + γ F iπ iπ (1 + γ F) p − p− δ−q δ−q
γ ν− F− 2 log x ( p (δ − q) − p− ) − 2 + ψ (ν− ) + ψ (−ν− ) 2iπ (1 + γ F− ) γ ν+ F+ {2 log [x ( p (q − δ) − p+ )] − 2 + ψ (ν+ ) + ψ (−ν+ )} + o (1), + 2iπ (1 + γ F+ ) (6.8)
+
eiπ ν γF = − . Using the integral representation
(ν) (1 − ν) 2iπ (1 + γ F) for the Barnes G-function (2.7), it is not a problem to see that ⎧ q ⎛ ⎞⎫ q ⎨ G (1, ν+ ) G (1, ν− ) ⎠⎬ dλ = ∂γ R (0) (λ, λ) ν ∂λ log e− dλ + log ⎝ ν 2 +ν 2 ⎩ γ x ( p+ − p− ) + − ⎭ −q −q where we used
q
q ∂γ ν ∂λ log κ p dλ − 2
+2 −q
∂λ ∂γ ν ν log
−q
p+ − p p − p−
dλ.
(6.9)
Now we should recast the last line as a derivative with respect to γ . We have q $ 2
∂γ ν∂λ log κ p − ∂λ ∂γ ν ν log
−q
p+ − p p − p−
% dλ
2 2 κ ν+ (q; q) p+ − p− 2ν+ p+ − p− 2ν− = ∂γ log κ ν− (−q; q) 2qp+ 2qp− σ ∂γ νσ log κ (σ q; q) − σ νσ ∂γ log κ (σ q; q) + σ =± q $
−2 −q
% %$ q −λ ∂γ ∂λ ν log κ + ν log dλ. q +λ
(6.10)
It remains to apply the identity q ∂γ −q
ν (λ) ν (µ) − ν (µ) ν (λ) dλ dµ = −2 2 (λ − µ)
% %$ q $ q −λ ∂γ ∂λ ν log κ + ν log dλ q +λ
−q
+ σ ∂γ νσ log κ (σ q; q) − σ νσ ∂γ log κ (σ q; q) . σ =±
(6.11)
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Indeed, we have for the r.h.s. of (6.11) q RHS = −q
ν+ ∂γ ν − ν ∂γ ν+ ν− ∂γ ν − ν∂γ ν− + q −λ q +λ
q ν (µ) ∂γ ∂λ ν (λ)
+
1 = 2
−q q
ν (µ) ∂γ ν (λ)
+ −q
dλ
1 1 + λ − µ + i0 λ − µ − i0
νσ ∂γ ν + ν∂γ νσ
−q σ =± q
dλ dµ
1 1 + q − σ λ + i0 q − σ λ − i0
1 (λ − µ + i0)2
+
1 (λ − µ − i0)2
dλ
dλ dµ.
(6.12)
There we have regularized all the integrands and then performed an integration by parts. On the other hand, one has for the l.h.s. of (6.11), q LHS = −q
∂γ ν (µ) ∂λ ν (λ) + ν (µ) ∂γ ∂λ ν (λ)
1 1 + λ − µ + i0 λ − µ − i0
dλ dµ. (6.13)
Taking the last integral by parts we arrive at (6.12). Thus, the l.h.s. of (6.8) is presented as a derivative with respect to γ . Since the asymptotic expansion is uniform in γ we can integrate this result from 0 to γ . As log det [I + V ] |γ =0 = 0 we get the desired result. 6.2. The leading asymptotics from the q-derivative method. The method we use here is based on the second equation in (3.13), ∂q log det [I + V ] = R (q, q) + R (−q, −q).
(6.14)
For the purpose of this sub-section, we assume that | (ν (λ))| < 1/4. Indeed we are then able to use the pointwise estimates for the resolvent established in Proposition 5.5. Such a restriction on | (ν (λ))| could be relaxed by much more refined estimates. Recall that one has for λ uniformly away from the boundary corresponding to disks of radius δ, C (q) C (q) (6.15) R (λ, λ) − R (0) (λ, λ) ≤ 1−2ε ≤ 1−2e , with e = 2sup | ν| , x x U so that the q anti-derivative of R (q, q) + R (−q, −q) − R (0) (q, q) − R (0) (−q, −q) will be a o (1) in the x → +∞ limit. Equation (6.3) allows us to determine the value of R (0) (λ, λ) = −F L;(0) (λ) | ∂λ F R;(0) (λ) γ F(λ) (0) (0) f + (λ) f − (λ) {∂λ log f + − ∂λ log f − } = 2iπ at both endpoints q and −q.
(6.16)
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Consider, for instance, R (0) (−q, −q). We have, for λ ∈ D−q,δ ,
R (0) (λ, λ) = −ν(λ) −ν, 1; −iζ−q ν, 1; iζ−q {2∂λ log e+ (λ) −2∂λ log κ p (λ) + ν(λ) log ζq + ν (λ) [ψ (1 + ν) + ψ (1 − ν)] 0
1 (∂z ) ν, 1; iζ−q (∂z ) −ν, 1; −iζ−q
+
−i x p (λ) ν, 1; iζ−q −ν, 1; −iζ−q 0
1 (∂1 ) ν, 1; iζ−q (∂1 ) −ν, 1; −iζ−q
+
−ν (λ) , (6.17) ν, 1; iζ−q −ν, 1; −iζ−q where, so as to lighten the formula, we have omitted the argument λ of ν(λ) when ν appears as an argument of another function (here ψ or ). The symbol ∂z stands for the derivative of a CHF with respect to its variable, whereas ∂1 stands for the derivative with respect to its first argument. Recall also that ζ−q = x p (λ) − p− and ζq = x [ p+ − p (λ)]. It is remarkable that the last two terms involving derivatives of CHF vanish in the λ → −q limit. The resulting expression can be further simplified thanks to the identities: q −λ λ+q log κ p (λ) = log κ(λ) + ν(λ) log , (6.18) − log p+ − p(λ) p(λ) − p− (6.19) ν(λ) ν (λ) [ψ(1 + ν) + ψ(1 − ν)] = ∂λ log G(1, ν) + 2ν(λ) ν (λ), Thus, we obtain R (0) (−q, −q) = −2ν− ∂λ log e+ (λ)
λ=−q 2 ν−
+ 2ν− ν− log x
2 p− + ν− ν− log(2qp− )− +2ν− q p− −2ν− ν− + 2ν− ∂λ log κ(λ) ,
(6.20)
λ=−q
where we have used the notations (2.3), (2.4). The final aim is to integrate (6.14) over the variable q. One should keep in mind that the function κ(λ) ≡ κ(λ; q) is actually a function of the two parameters λ and q. Therefore, one should replace partial λ derivatives at λ = ±q by total q derivatives thanks to d [log κ(−q; q)] = −∂λ log κ(λ; q) dq
λ=−q
+ ∂q log κ(λ; q)
.
λ=−q
(6.21)
Then R (0) (−q, −q) is almost a total q derivative: R (0) (−q, −q) = −2ν− ∂λ log e+ (λ)
⎤ ⎡ G ν d (1, ) − ⎦ + log ⎣
ν 2 λ=−q dq 2qp x −
d (ν+ − ν− ) −2ν− log κ (−q; q) + ν− . dq q
−
(6.22)
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Similar calculations based on the expressions (5.20) for f ± around q lead to ⎡ ⎤ G ν d (1, + ) ⎦ log ⎣ R (0) (q, q) = −2ν+ ∂λ log e+ (λ) +
ν 2 dq λ=q 2qp x + +
d (ν+ − ν− ) + 2ν+ log κ (q; q) − ν+ . dq q Hence, we have ∂q log det [I + V ] = 2
νσ ∂λ log e− (λ)
σ =±
(6.23) ⎤
⎡ λ=σ q
+
G (1, ν+ ) G (1, ν− ) ⎦ d log ⎣
ν 2
ν 2 dq 2qp x − 2qp x +
−
d log κ(σ q; q) (ν+ − ν− )2 +2 σ νσ − + o (1) . dq q σ =±
+
(6.24)
It remains to express the last line as a total q-derivative thanks to Lemma B.3. After an integration with respect to q of (6.24) we arrive to ⎤ ⎡ q ν+ (q; q) G(1, ν ) G(1, ν ) κ + − ⎦ log det [I + V ] = 2 dλ ν(λ) log [e− (λ)] + log ⎣
ν+2
ν−2 ν − κ (−q; q) 2qp+ x 2qp− x −q 1 + 2
q dλdµ −q
ν (λ) ν(µ) − ν(λ) ν (µ) + C +o(1), λ−µ
(6.25)
where C is a q-independent integration constant still to be determined. One can give arguments that this constant should be also γ -independent. Indeed, the asymptotic expansion of the Fredholm determinant, being a functional of the holomorphic function γ F(λ) in U , can depend on this function either in the integral form with integration over [−q; q], or through the values of γ F and of its derivatives at the ends of the integration contour −q and q. In both cases the result should depend on q. Hence, the q-independent constant C can not depend on γ F(λ) and, thus, it is γ -independent. We can then fix the constant C by setting γ = 0 in the asymptotic formula. This yields C = 0. A rigourous proof of this equality within this q-derivative method is however still missing. Indeed, although the above statement (about the functional form of the asymptotic expansion of the Fredholm determinant) is clear in the case of onedimensional oscillatory integrals without saddle point, its generalisation to the needed series of multiple oscillatory integrals would require additional work. 6.3. The first corrections to the leading asymptotics of the Fredholm determinant. We close this section by deriving the sub-leading corrections from the x-derivative (3.19) of log det [I + V ]. This will constitute the proof of Proposition 2.1. In order to prove the claim of Proposition 2.1, one has to derive the first two subleading corrections for the matrix . As one might expect the computations are, by far, simpler than those necessary to fix the constant. We also would like to point out that one can obtain the sub-leading asymptotic of det [I + V ] by the q-derivative method. However, the computations are quite involved, so we omit the presentation of this method.
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
725
We derive the first term in the 1/x expansion of log det [I + V ] thanks to (3.12): ! " dλ p(λ) tr ∂λ χ (λ) σ3 χ −1 (λ) , ∂x log det [I + V ] = (6.26) 4π
([ −q ;q ])
where we chose the contour ([ −q ; q ]) to lie outside of the contour but still in the region of holomorphy for p. There the solution for the RHP for χ has a simple form: χ (λ) = (λ) α −σ3 (λ).
(6.27)
In order to derive the first correction to the leading asymptotics, it is enough to consider the first two terms in the asymptotic expansion for (λ): (1) (λ) (2) (λ) 3(ε−1) + . (6.28) (λ) = I2 + + O x x x2 There, as follows from (5.15), the O is uniform on the whole contour ([ −q ; q ]). Thus ! " ∂λ α (λ) 1 dλ dλ ∂x log det [I + V ] = − p (λ) + p (λ) tr σ3 ∂λ (1) (λ) 2π α (λ) x 4π +
1 x2
([ −q ;q ])
([ −q ;q ])
! " dλ p(λ) tr σ3 ∂λ (2) (λ) − (1) (λ)∂λ (1) (λ) + O x 3(ε−1) . 4π
([ −q ;q ])
(6.29) The first term in this expansion will yield the leading correction. Indeed, −
∂λ α (λ) dλ p (λ) = 2π α (λ)
([ −q ;q ])
dλ p (λ) log α (λ) = −i 2π
([ −q ;q ])
q
dλ p (λ) ν (λ).
−q
(6.30) Here we shrunk the contour to [ −q ; q ] and used the jump condition for α. In order to evaluate the higher order corrections in (6.29) we need to derive the" ! (1) expressions for the matrices (1) = C (1) and (2) = C + (1) + (2) outside of . An elementary computation of residues yields: ∂λ
(1)
(1)
(λ) = −
(+) (q; x)
(λ − q)2 p+
(1)
−
(−) (−q; x) (λ + q)2 p−
,
(6.31)
as well as ∂λ (2) (λ) − (1) (λ) ∂λ (1) (λ) (2) (1) (1) ∂λ (σ ) (σ q; x) + (σ ) (σ q; x) ∂λ (σ ) (σ q; x) =− 2 (λ − σ q) pσ σ =± ! "2 (2) (1) 2(σ ) (σ q; x) + (σ ) (σ q; x) pσ 2 + − 2 pσ (λ − σ q)2 (λ − σ q)3 2 pσ σ =± ! " (1) (1) (+) (q; x) , (−) (−q; x) + (6.32) (λ − q)(λ + q) . 2q p+ p−
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
Thus the 1/x term in (6.29) gives the coefficient of log x in (2.6). Indeed,
" " ! 1 ! dλ (1) (1) p (λ) tr σ3 ∂λ (1) (λ) = tr σ3 (−) (−q; x) + σ3 (+) (q; x) 4π 2i
([ −q ;q ])
2 . = − ν+2 + ν−
(6.33)
We now focus on the last term in (6.29). It yields, after an x integration, the first correction to (2.6). A straightforward computation leads to
! " dλ p (λ) tr σ3 ∂λ (2) (λ) − (1) (λ) ∂λ (1) (λ) 2iπ
([ −q ;q ])
" p+ − p− ! (1) (1) tr σ x) , x) (q; (−q; 3 (+) (−) 4q 2 p− p+ " 1 ! (2) (1) (1) ∂ ∂ . tr σ q; x) q; x) + q; x) − (σ (σ (σ 3 λ λ (σ ) (σ ) (σ ) p σ =± σ
=
(6.34) The first term corresponds to the oscillating correction: ! " u˜ (q; x) u (−q; x) (1) (1) tr σ3 (+) (q; x) , (−) (−q; x) = 2ν+ ν− − . (6.35) u (−q; x) u˜ (q; x) The last term gives the non-oscillating one: dν 3 (2) tr σ3 ∂λ (σ ) (σ q; x) = −2 σ , dq
(6.36)
and 1 (1) (1) tr σ q; x) ∂ (σ 3 λ (σ ) (σ ) (σ q; x) p σ =± σ 2 2 2 2 ν ∂λ (log u˜ (λ; x)) + ν− ∂λ (log u (λ; x)) p+ + p λ=q λ=−q − d ν2 2ν 2 ν−σ d p σ σ σ log u 2σ ν − , (6.37) = log x + σ + σ σ p dq νσ dq pσ p− σ =± σ =
with
(1 − ν+ ) (2qp+ )ν+ 2 ,
(1 + ν+ ) κ(q; q) −2
(1 + ν− ) ν− (2qp− u − = e g(−q) ) κ(−q; q) .
(1 − ν− )
u + = e g(q)
(6.38) (6.39)
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727
Putting all this together we obtain q ∂x log det [I + V ] = −i
dλ ν (λ) p (λ) −
−q
2 ν+2 + ν− log x − 1 νσ2 dνσ − 2i x x2 p dq σ =± σ
d ν−σ νσ i νσ2 d σ − log u σ + pσ − 2 x σ =± pσ dq dq pσ q i ( p+ − p− ) ν− ν+ u + 2(ν+ +ν− ) i x( p+ − p− ) u − −2(ν+ +ν− ) i x( p− − p+ ) x e − x e + x2 u− u+ (2q)2 p+ p− i x( p+ − p− ) e 1 +O . (6.40) , x 3(1−ε) x 3(1−ε) The first two terms reproduce the already known answer for the leading asymptotics. The remaining ones reproduce the first oscillating and non-oscillating corrections as given in Proposition 2.1. Note that for the oscillating terms, one only should integrate the exponent with respect to x as all the other terms will give subdominant contributions. 7. Applications to Truncated Wiener–Hopf Operators Truncated Wiener–Hopf operators appear in many domains of mathematical physics such as scattering or diffusion processes. Moreover, many observables (dressed energy, momentum or dressed charge) related to quantum integrable models are solutions of integral equations of truncated Wiener–Hopf type (2.17). Let us recall that a truncated Wiener–Hopf operator can be interpreted as an integral operator I + K on L 2 (R) such that it acts on L 2 (R) functions according to x (I + K ) .ϕ (t) = ϕ (t) +
dt K t − t ϕ t dt .
(7.1)
0
The kernel K is characterized in terms of its Fourier transform F: 1 K (t) = F −1 [F] (t) , with F −1 [F](t) = dξ F(ξ ) e−itξ , ∀F ∈ L 1 (R). 2π R
(7.2) The study of truncated Wiener–Hopf operators is equivalent to a 2 × 2 matrix RHP. Another facet of this equivalence is the correspondence between a truncated Wiener– Hopf operator and the GSK acting on R in which p = id and g = 0. Indeed, it is easy to see that & ◦ F [ϕ] , K .ϕ = F −1 ◦ V
(7.3)
& acts in L 2 (R) with a kernel where V i x(ξ −η) − 1 &(ξ, η) = F (ξ ) e . V 2iπ (ξ − η)
(7.4)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
The operator identity
& F, I + K = F −1 . I + V
(7.5)
& is trace-class and F ±1 are continuous, ensures the equality together with the facts that V between the Fredholm determinants: & . det [I + K ] = det I + V (7.6) & is related to The kernel V V (ξ, η) =
F (ξ ) F (η)
sin (x (ξ − η) /2) π (ξ − η)
(7.7)
by a similarity transformation. Hence, det [I + K ] L 2 (0,x) = det [I + V ] L 2 (R) .
(7.8)
7.1. The Akhiezer–Kac formula. Our study of the generalised sine kernel allows us to recover the Akhiezer–Kac formula describing the large x behaviour of Fredholm determinants of truncated Wiener–Hopf operators: Theorem 7.1 (Akhiezer–Kac [1,34]). Let I + K be a truncated Wiener–Hopf operator as above and such that • F is analytic in an open neighbourhood U of R; • F goes sufficiently fast to 0 at ±∞; • 1 + F (ξ ) does not vanish on U . Then ∞ log det [I + K ] = x τ (0) + E[F] + o(1), with E[F] =
t τ (t) τ (−t) dt,
(7.9)
0
in which τ (t) =
1 2π
log(F(ξ ) + 1) e−itξ dξ.
(7.10)
R
Proof. The large x asymptotics of det [I + K ] follows from (7.8) after taking the q → +∞ limit in the leading asymptotics for the corresponding generalised sine kernel (2.6). This limit may seem a little heuristic as we did not specify any estimates in q for the small o terms with respect to the leading asymptotics. However, the validity of such a limit may either be seen by refining all the estimates obtained in the previous section or by considering the RHP for χ (3.1) on the whole real line from the very beginning. We shall make the second approach more explicit in the forthcoming Subsect. 7.2. Here we formally take the q → +∞ limit in the leading asymptotics of Theorem 2.1. One should notice that, in the asymptotic formula (2.6), all the terms evaluated at the endpoints vanish due to the fact that ν (±q) log q → 0 when q → +∞, which is a
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729
consequence of the sufficiently fast decrease of F at infinity. Hence, the only constant contribution E[F] to the asymptotics of log det [I + K ] is given by the integral 1 2
E [F] = lim
q→+∞
q −q
ν (λ) ν (µ) − ν (λ) ν (µ) dλ dµ. λ−µ
(7.11)
Let us recast the constant E [F] in a more standard form. We have E [F] = −
1 8π 2
i = 16π 2
dξ dη R
log (F(ξ )+1) log (F(η)+1) − log (F(η)+1) log (F(ξ ) + 1) ξ −η
dξ dη dx dy R
1 1 + τ (x)τ (y) (x − y) ei xη+i yξ . ξ − η + i0 ξ − η − i0
Let H be the Heaviside function, then E [F] = −
1 8π
1 =− 4
dη dx dy τ (x) τ (y) (x − y) eiη(x+y) H (y) − eiη(x+y) H (−y)
R
+∞
1 dy τ (y) τ (−y) (−2y) + 4
0
0 dy τ (y) τ (−y) (−2y) −∞
+∞ = dy τ (y) τ (−y) y , 0
which ends the proof of Theorem 7.1.
It happens that this correspondence between truncated Wiener–Hopf operators and generalised sine kernels can be pushed further so as to obtain the asymptotic behaviour of Fredholm determinants of truncated Wiener–Hopf operators with symbols having Fischer–Hartwig type discontinuities. Considering the GSK for finite q corresponds to the asymptotic behaviour of a determinant whose symbol has two jumps. The case of symbols having general Fischer–Hartwig type singularities is studied in [16,17,37]. The results for the case of Toeplitz, Hankel and Toeplitz + Hankel determinants with Fisher–Hartwig singularities appeared recently in [16,17].
7.2. The resolvent of truncated Wiener–Hopf operators. Proposition 7.1. Let I + K be a truncated Wiener–Hopf operator on ] −x ; x [ , x [(I + K ).g] (t) = g(t) +
K (t − t ) g(t ) dt , with K (t) = F −1 [F](t).
−x
(7.12)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
Suppose that there exists δ > 0 such that • F admits an analytic continuation to {z : |(z)| ≤ δ}; • ξ → F (ξ ± iδ) ∈ L 1 (R); • the analytic continuation of 1 + F does not vanish on U . Then the resolvent I − R of I + K fulfills R(λ, µ) = R
dξ dη F(ξ ) 4iπ 2
α+ (η) i x(ξ −η) α+ (ξ ) −i x(ξ −η) ei(µη−λξ ) e e + O e−2δx , − × α− (ξ ) α− (η) ξ −η (7.13)
where ⎧ ⎫ ⎨ ν (µ) ⎬ α(λ) = exp dµ , ⎩ µ−λ ⎭ R
and ν(λ) =
i log (1 + F(λ)) . 2π
(7.14)
Proof. The GSK associated to I +K through the transformation F −1 ◦[I + V ]◦F = I +K acts on the whole real axis with the kernel V (ξ, η) = F(ξ )
ei(ξ −η)x − ei(η−ξ )x . 2iπ (ξ − η)
(7.15)
Just as for the leading asymptotics of log det [I + K ] (see Sect. 7.2), one can obtain the leading asymptotic of the resolvent of V just by taking formally the limit q → +∞ in all the expressions derived in the first part of the article. Note that in this process all power law corrections vanish: they are computed as contour integrals around ±q and, since F approaches 0 sufficiently fast at infinity, the residues at ±q vanish in the q → +∞ limit. However, in order to justify this limit, one should also check that all the uniform estimates still hold for q → +∞. An alternative way is to consider from the very beginning a RHP for χ on the whole real axis R. This is actually much simpler than the RHP on a finite interval. Then it is enough to perform the first two transformations described in Sect. 4 so as to obtain jump matrices that are already uniformly close to I2 up to exponentially small corrections in −1 (4.11), so that x. Moreover, the jump matrices for this RHP are given by M+ and M− they approach the identity matrix at λ → ∞ just as fast as F goes to zero at infinity. As expected, there is no need for parametrices, and the corrections are immediately exponentially decreasing with x. It means that, up to uniformly exponentially small corrections, the resolvent R V of V is given by (0)
R V (ξ, η) =
F (ξ ) 2iπ (ξ − η)
α+ (η) i x(ξ −η) α+ (ξ ) −i x(ξ −η) , e e − α− (ξ ) α− (η)
(7.16)
where as usual α (λ) is given by (7.14). Note that the integral in (7.14) is well defined by virtue of the assumptions made on F.
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731
We should now take the Fourier/inverse Fourier of R V in order to get R. To this end, we must justify that the sub-leading corrections do admit a Fourier transform in two variables. Recall the exact expression for the resolvent: (0)
R V (λ, µ) = R V (λ, µ) + F L;(0) (λ) |
−1 (λ) (µ) − I2 R;(0) |F (µ) . λ−µ
(7.17)
Here, the matrix is defined in terms of + (λ) , the limiting value of on when λ approaches a point of from the “+”’ side of the contour: −2 dz 0 1 F (z) α+ (z) 2i x z + (z) e (λ) = I2 + 00 λ−z 1 + F (z) +
−
+
2 dz 0 0 F (z) α− (z) 2i x z + (z) e . 10 λ−z 1 + F (z)
(7.18)
The L 1 integrability of F as well as the asymptotic condition (λ) −→ I2 guarantee λ→+∞
that the integrals are well defined. Thus one readily infers from (7.18) the asymptotics of on the real axis: −2δx e−2δx C e , (7.19) (λ) = I2 + +O λ λ2 where C is some constant matrix and where we have explicitly extracted the exponential (0) decay in x of the matrix C. Hence using the boundedness of f ± on the real axis we obtain that, c being some constant, −2δx e F (λ) F (λ) (0) +o . (7.20) R V (λ, µ) = R V (λ, µ) + e−2δx c λµ λµ Hence the corrections admit a Fourier transform in λ and an inverse Fourier transform in µ as oscillatory integrals. Therefore, taking the Fourier transform does not change the nature of the corrections. It is clear that, up to a similarity transformation, a Wiener–Hopf operator on [ −a ; b ] has the same generalised sine kernel as the same operator acting on [ 0 ; a + b ]. Therefore our method works for any interval, of course up to a similarity transformation on the resolvent (7.16) of V . We chose here to present this less standard form of Wiener–Hopf operators as it fits better the forthcoming application. We apply our asymptotic inversion formula for truncated Wiener–Hopf operators acting on a symmetric interval [ −x ; x ] to re-derive some formulas concerning the low magnetic field behaviour of the so-called dressed charge [6]. This function, traditionally denoted Z (λ), describes the intrinsic magnetic moment of an elementary excitation above the ground state in the XXZ spin-1/2 model. It satisfies the following integral equation: Z (λ) +
x dµ K (λ − µ) Z (µ) = 1, with K (λ) =
−x
sin 2ζ . 2π sinh(λ + iζ ) sinh(λ − iζ ) (7.21)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
K is often called the Lieb kernel and ζ ∈ ] 0 ; π [ is some real parameter describing the coupling of the model. The large parameter x is a function of the external longitudinal magnetic field; it goes to infinity when the magnetic field vanishes. For the study of Z , one should distinguish two domains in the interval [ −x ; x ]: the bulk, i.e. the region |λ| x, and the boundaries λ ∼ ±x. While the asymptotic value of Z (λ) in the bulk (|λ| x) is enough to describe the intrinsic magnetic moment of elementary excitations, the value of Z at the boundaries (±x) determines the critical exponents of the two-point functions of the model [5,23–25]. As we will see, the bulk and the boundary behaviour of the dressed charge differ fundamentally. First, let us note that, setting directly x = +∞ in (7.21), one can solve explicitly the integral equation for Z by taking the Fourier transform: one obtains in this case that Z (λ) is equal to a constant value Z (λ) = π/ [2 (π − ζ )] on the whole real axis. Let us now consider the limit x → ∞ in (7.21) in a more accurate way, namely, 2 be the Fourier taking x large but finite, and use the method described above. Let K transform of K , 2 (ξ ) ≡ F [K ] (ξ ) = sinh [ξ (ζ − π/2)] . K sinh (ξ ζ /2)
(7.22)
Then by virtue of Proposition 7.1, x Z (λ) = 1 − =1− R
R (λ, µ) dµ
(7.23)
−x
2 (ξ ) α+ (0) i(x−λ)ξ dξ K α+ (ξ ) −i(x+λ)ξ . e e − 2iπ ξ α− (ξ ) α− (0)
(7.24)
First let us study the bulk limit, i.e. |λ| x. Using the jump equation satisfied by 2(λ) α+ (λ) = α− (λ), we recast the integrand as α± : 1 + K 2 (ξ ) α+ (0) i(x−λ)ξ α− (ξ ) −i(x+λ)ξ dξ K e e − 2 (ξ ) (ξ − i0+ ) α+ (ξ ) 2iπ 1 + K α− (0) R 2(ξ + iζ /2) 1 e(x−λ)(iξ −ζ /2) dξ α+ (0) K = − 2 (0) 2(ξ + iζ /2) α+ (ξ + iζ /2) 2iπ 1 + K ξ + ζ /2 1+K R 2(ξ − iζ /2) K e−(x+λ)(iξ +ζ /2) α− (0) − . (7.25) 2(ξ − iζ /2) α− (ξ − iζ /2) ξ − ζ /2 1+K
Z (λ) = 1 −
Here we have separated the integral into two parts and then moved the contour to the upper/lower half-plane. This gives a pole contribution from ξ = i0. The integral appearing in (7.25) is clearly a O e−(x−|λ|)ζ /2 . So that, in the bulk, Z (λ) ∼
1 π , = 2 (0) 2 (π − ζ ) 1+K
(7.26)
up to exponentially small corrections. As expected, we recover the value of Z obtained in the case of an infinite interval. Note that the corrections become larger and larger as we approach any of the endpoints ±x.
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Let us now study the behaviour of the dressed charge at the boundaries. Since the kernel K is even, so is Z . We can thus focus on a single boundary, say λ = x. We have, Z (x) = 1 − R
2 dξ 2(ξ ) α+ (0) − K (ξ ) α− (ξ ) e−2i xξ . K 2(ξ ) α− (0) 2iπ (ξ − i0+ ) α− (ξ ) 1 + K
(7.27)
As before,
the integral of the second term gives an exponentially small contribution O e−xζ . The integral of the first term is explicitly computable. Using once again the jump equation satisfied by α± (ξ ), we have, Z (x) = 1 − α+ (0) R
= 1 − α+ (0) R
−1
(ξ ) dξ α+−1 (ξ ) − 1 + 1 − α− + O e−xζ 2iπ ξ − i0
dξ α+−1 (ξ ) − 1 + O e−xζ 2iπ ξ − i0
= α+ (0) + O e−xζ .
(7.28)
−1 We have computed the remaining integral by residues, since α± (ξ ) − 1 = O ξ −1 for ξ → ∞ in the respective half plane of holomorphy. 2(ξ ) = K 2(−ξ ), and then it follows from the For an even kernel like the Lieb one K −1 2(0) = integral representation (7.14) of α that α+ (ξ ) = α− (−ξ ). This means that 1 + K −2 α+ (0). Hence, for x large enough, 3 4 1 π , (7.29) Z (x) ∼ = 2(0) 2 (π − ζ ) 1+K and the value of Z (λ) at the boundary is the square root of its value in the bulk up to exponentially small corrections. In the limit x → +∞ this correspondence becomes exact. 8. Asymptotics of Multiple Integrals We have already mentioned that the asymptotic expansion of the Fredholm determinant of the GSK can be used for the asymptotic analysis of correlation functions of quantum integrable models. For a relatively wide class of integrable systems, the correlation functions can be presented as series of multiple integrals of a special type [35]. These series can be summed up to Fredholm determinants for the models equivalent to free fermions. In the general case, such a reduction to determinants is not known. However, the asymptotic behaviour of individual terms of the series can be derived from the asymptotics of the Fredholm determinant of the GSK. In the present section we consider this problem. More precisely, our purpose is to derive the large x asymptotic behaviour of the following type of integrals (cycle integrals): In [Fn ] =
dn z (2iπ )n
([ −q ;q ])
q −q
dn λ Fn (2iπ )n
{λ} {z}
n j=1
ei x ( p(z j )− p(λ j ))
. z j − λ j z j − λ j+1
(8.1)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
In this expression, Fn is a holomorphic function of 2n variables λ1 , . . . , λn , z 1 , . . . , z n in U n × W n , in which U and W are open neighbourhoods of [ −q ; q ], and ([ −q ; q ]) denotes a closed counter-clockwise contour in W surrounding [ −q ; q ] with index 1. We moreover assume that Fn is symmetric separately in the n variables λ1 , . . . , λn and in the n variables z 1 , . . . , z n . Finally, we agree upon λn+1 ≡ λ1 . 8.1. Leading asymptotic behaviour of In [Fn ]. Let us first suppose that the function Fn is of the special (factorized) type Fn(ϕ,φ)
{λ} {z}
=
n
[ϕ(λi ) φ(z i )] ,
(8.2)
i=1
where ϕ is a one-variable holomorphic function in U and φ is a one-variable holomorphic function in W , non-vanishing on W . For two such functions ϕ and φ we introduce the associated GSK V (ϕ,φ) given by (1.6) provided the identification F(λ) = ϕ(λ)φ(λ) and e g(z) = φ(z) is made. Then, the integral (8.1) can be expressed in terms of log det I + V (ϕ,φ) as q n ! " (ϕ,φ) In Fn = dn λ V (ϕ,φ) (λk , λk+1 ) k=1
−q
=
" ! (−1) ∂γn log det I + V (ϕ,φ) (n − 1)! n−1
γ =0
.
(8.3)
In this specific case, it is straightforward to obtain the leading asymptotic behaviour of the multiple integral (8.3) in the large x limit thanks to the results of the previous sections. This remark leads us to the following definition: Definition 8.1. Let U , W be two open neighbourhoods of [ −q ; q ], and let H(U ) (resp. H(W )) be the set of holomorphic functions on U (resp. on W ). Let also p U,W (ϕ ,φ ) S&n = Fn ; p ∈ N, (ϕ , φ ) ∈ H(U ) × H(W ) and φ = 0 , [ −q ;q ]
=1
(8.4) (ϕ,φ)
in which Fn denotes a pure factor function of 2n variables defined in terms of ϕ and φ as in (8.2). We define the linear functional In(0) on S&nU,W as ! " (−1)n−1 "(0) ! ∂γn log det I + V (ϕ,φ) In(0) Fn(ϕ,φ) = (n − 1)!
γ =0
,
(8.5)
5p
(ϕ ,φ ) . Here V (ϕ,φ) denotes the general =1 Fn (0) ϕ(λ)φ(λ) and eg(z) = φ(z), and log det I +V (ϕ,φ)
and by imposing linearity on functions
ised sine kernel (1.6) with F(λ) = denotes the leading asymptotics of the Fredholm determinant log det I + V (ϕ,φ) as in Theorem 2.1.
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
735
(ϕ,φ) (0) , to obtain an explicit It is easy, using !the expression (2.6) of log det I + V " (ϕ,φ)
(0)
Fn
expression for In
! " Fn(ϕ,φ) =
q
In(0)
−q
:
dλ n ϕ (λ) φ n−1 (λ) i x p (λ) φ(λ) + φ (λ) 2iπ
bn − cn log 2qpσ x [ϕ(σ q) φ(σ q)]n + σ =±
n−1 n [ϕ(σ q) φ(σ q)]n − [ϕ(σ q) φ(σ q)] p [ϕ(λ) φ(λ)]n− p + 2 dλ 4π σ =± p (n − p) (q − σ λ) q
p=1−q n−1 q
+
n 8π 2
p=1−q
dλ dµ ∂λ [ϕ(λ) φ(λ)] [ϕ(λ) φ(λ)] p−1 [ϕ(µ) φ(µ)]n− p (n − p) (λ − µ)
−∂µ [ϕ(µ) φ(µ)] [ϕ(µ) φ(µ)] p−1 [ϕ(λ) φ(λ)]n− p ,
(8.6)
where bn and cn are given by (2.23). The x → +∞ asymptotics of the Fredholm determinant are uniform in γ to any fixed order n in ∂γn . This means that ! " ! " (8.7) In Fn(ϕ,φ) = In(0) Fn(ϕ,φ) + o (1) . (0)
In the next proposition we show that In can be extended into a linear functional on the (ϕ,φ) ) that are symspace of holomorphic functions Fn (not necessarily of the form Fn metric in n variables λ1 , . . . , λn and n variables z 1 , . . . , z n separately. This extension (0) of In , as we prove below, is the good way to evaluate cycle integrals (8.1) with such arbitrary symmetric functions Fn . Proposition 8.1. Let U and W be open neighbourhoods of [ −q ; q ], and let Symn (U, W ) be the set of holomorphic functions Fn on U n × W n of 2n variables λ1 , . . . , λn , z 1 , . . . , z n , symmetric in the n variables λ1 , . . . , λn and in the n variables z 1 , . . . , z n separately. Then In(0) extends to a continuous linear functional on Sym n (U, W ) endowed with the topology of the sup norm convergence on compact sets. ! " (ϕ,φ) (0) contains at most first order derivatives of the functions ϕ and φ. Proof. In Fn ◦
Now recall that, for any compacts K , P such that1 K ⊂ P and P ⊂ U , ≤ ck φ0;P , ∀ k ∈ N, ∃ ck such that ∀ φ ∈ H (U ) , -φ (k) 0;K
(8.8)
where .0;K = supz∈K |.| is the sup norm with support on the compact K . In conse(0) quence, In is continuous on S&nU,W . The latter is dense in SnU,W , with p U,W (ϕ ,φ ) Sn = Fn ; p ∈ N, (ϕ , φ ) ∈ H(U ) × H(W ) . (8.9) =1 ◦
1 Here P is the interior of P.
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras (0)
Hence In extends by density to a continuous linear functional on SnU,W . Due to the density Theorem C.1 (see Appendix C), we have that SnU,W is dense in Symn (U, W ). (0) Therefore In extends to a linear functional on Symn (U, W ). Corollary 8.1. Let U and W be open neighbourhoods of [ −q ; q ], and let Fn ∈ Symn (U, W ). Then, In(0) [Fn ]
1 = 2iπ +
q
dλ i x p (λ) + ∂ Fn
−q
n−1
q
+
{λ}n {λ + }, {λ}n−1
bn − cn log 2qpσ x Fn
σ =±
+
n (2π )2 n
dλ
n
=0
{σ q} {σ q}n {σ q}n {σ q} p , {λ}n− p − Fn Fn {σ q}n {σ q} p , {λ}n− p p (n − p) (q − σ λ)
σ =± p=1−q n−1 q
dλdµ {λ + } , {λ} p−1 , {µ}n− p ∂ Fn {λ + } , {λ} p−1 , {µ}n− p (n − p) (λ − µ)
2 (2π )2 p=1 −q {µ + } , {µ} p−1 , {λ}n− p −∂ Fn {µ + } , {µ} p−1 , {λ}n− p
=0
.
(8.10)
There {λ}n denotes the set formed by n copies of the same parameter λ. Proof. Apply Theorem C.1 to (8.6).
Finally, we have the following large x asymptotic behaviour for integrals of the form (8.1) (which seems hardly attainable through a direct analysis of the multiple integrals): Theorem 8.1. Let U and W be open neighbourhoods of [ −q ; q ], and let Fn ∈ Symn (U, W ). Then, when x → +∞, the integral In [Fn ] (8.1) behaves as n log x (0) , (8.11) In [Fn ] = In [Fn ] + O x (0)
the explicit expression of In [Fn ] being given in Corollary 8.1. The whole difficulty of the proof is to show that the small o (1) in (2.5) is preserved by the density procedure formulated in Theorem C.1. This is nontrivial since the series converging to Fn may not converge absolutely. We need therefore, so as to prove this theorem, to study more precisely the sub-leading corrections and to see how they pass through all the steps described above. This will be done in the next subsection. 8.2. Study of sub-leading corrections. In this subsection, we study the behaviour of the sub-leading corrections to log det[I + V ](0) when the above procedure is applied. In particular, we will show that they indeed remain subleading, which will prove Theorem 8.1. In fact we will prove an even more general result:
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737
Theorem 8.2. Let U and W be open neighbourhoods of [ −q ; q ], and let Fn ∈ Symn (M) (U, W ). For any positive integer M, there exists a continuous linear functional In such that n log x when x → +∞. (8.12) In [Fn ] = In(M) [Fn ] + O x M+1 (M)
The explicit expression for In can be obtained by some perturbative computations that become more and more involved with the growth of M. We will nevertheless (M) obtain the general structure for In , showing that it can be decomposed in terms of nonoscillating and oscillating contributions, with oscillating factors of the form eimx( p+ − p− ) , m ∈ Z∗ : In(M) [Fn ] = In(0) [Fn ] +
M M 1 (N ; nosc) 1 (N ; osc) I I + [F ] [Fn ] , (8.13) n n N x xN n
N =1
= In(0) [Fn ] +
M N =1
+
m∈Z∗ |m|≤M/2
N =2
1 (N ; nosc) I [Fn ] xN n
ei xm( p+ − p− )
M N =2|m|
1 (N ; m) I [Fn ] , xN n
(8.14)
In(N ; nosc) [Fn ] (resp. In(N ; osc) [Fn ]) being given in terms of the function Fn and of its derivatives up to order N (resp. up to order N − 2) evaluated at ±q or integrated from −q to q. 8.2.1. General strategy. In the previous subsection, we have defined the functional (0) In from the leading asymptotic part log det[I + V ](0) (2.6) of the GSK. More precisely, we have seen in Corollary 8.1 that ∂γn log det [I + V ](0) |γ =0 yields the functional (0)
(−1)n−1 (n − 1)! In [Fn ] after the density procedure, as explained in Proposition 8.1 (0) and Theorem C.1, is applied. In order to estimate the corrections to In [Fn ] for the large x behaviour of cycle integrals In [Fn ] of length n (8.1), we have to take into account the corrections log det [I + V ]sub to log det[I + V ](0) , log det [I + V ] = log det [I + V ](0) + log det [I + V ]sub ,
(8.15)
and to analyze the effect of the density procedure on the n th γ -derivative of the subleading part ∂γn log det [I + V ]sub |γ =0 . We will show in particular that it preserves the small o(1) with respect to the x → +∞ limit, i.e. that ∂γn log det [I + V ]sub |γ =0 can only generate o(1) corrections. In the spirit of Definition 8.1, we therefore introduce: Definition 8.2. Let U , W be two open neighbourhoods of [ −q ; q ]. We define the linear functional Insub on S&nU,W as ! " (−1)n−1 "sub ! Insub Fn(ϕ,φ) = ∂γn log det I + V (ϕ,φ) (n − 1)!
γ =0
,
(8.16)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
5p (ϕ ,φ ) (ϕ,φ) and by imposing linearity on functions =1 Fn . Here, as in Definition 8.1, Fn denotes a factorized function of 2n variables defined in terms of ϕ and φ as in (8.2), and V (ϕ,φ) denotes the generalised sine kernel (1.6) with F(λ) = ϕ(λ)φ(λ) and eg(z) = φ(z). According to the scheme presented in the previous subsection, the next steps will be: ! " (ϕ,φ) : this means in particular to • to obtain a convenient representation for Insub Fn obtain the form of n th γ -derivatives of log det[I + V ]sub in terms of the functions F and g, to set g(z) = log φ(z) and F(z) = ϕ(z) φ(z), and to estimate this result in the large x limit; • to apply the density procedure: one should first extend by density and continuity the functional Insub to the space SnU,W ; then, for any holomorphic function Fn in 2n variables λ1 , . . . , λn , z 1 , . . . , z n , symmetric separately in the variables λ and in the 5N (ϕ ,φ ) variables z, one has to consider a sequence (ϕk , φk ) such that k=1 F n k k → Fn sub so as to be able to define and characterize In [Fn ] and to see how it behaves in the large x limit; • to refine the procedure in order to get an asymptotic expansion of Insub [Fn ]. 8.2.2. γ -derivatives of log det[I + V ]sub . As in Sect. 6.3, we will obtain the corrections to log det[I + V ](0) through the x-derivative path, starting from formula (3.12) that we recast as q ∂x log det [I +V ] = −i
dλ p (λ) ν (λ) +
−q
dλ p(λ) tr [∂λ (λ)] σ3 −1 (λ) . 4π
([ −q ;q ])
(8.17) Here, as in Sect. 6.3, we have chosen the counter-clockwise contour ([ −q ; q ]) to lie in U and to encircle , which means that χ (λ) = (λ) α −σ3 (λ) on ([ −q ; q ]). Integrating this equation with respect to x, we obtain ⎧ ⎪ ⎨ dλ p(λ) tr [∂λ (λ)] σ3 −1 (λ) = dx ⎪ 4π ⎩ +∞ ([ −q ;q ]) ⎫
⎪ (1) ⎬ z; x σ3 tr 1 dz . + ⎪ x 2iπ (λ − z)2 ⎭ x
log det [I + V ]sub
(8.18)
The convergence of this integral will be proved later on. We recall that the second term in (8.17) produces also, when integrated over x, the log x term appearing in definition (2.6) of log det [I + V ](0) (see (6.33)). We have therefore subtracted the corresponding contribution (second term of (8.18)) in the definition of log det [I + V ]sub . In order to obtain the n th γ derivatives of this expression, we have to compute the γ -derivatives of ∂λ (λ) and of −1 (λ), which in their turn follow from those of the jump matrix (λ).
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
739
• γ -derivatives of . In order to determine the n th γ -derivative at γ = 0 of the jump matrix (z), it is convenient to express it in the following form: & (z) κ σ3 (z). (8.19) (z) = κ −σ3 (z) & (z) depends on γ only through the combination γ F(z), whereas Here, the matrix 6q κ ±σ3 (z) depends on γ through the combination −q dµ [ν (z) − ν (µ)] /(z − µ). & (z) at γ = 0. It is given as It is easy to compute the multiple γ -derivative of ! " & (z) = Adeσ3 g(z)/2 ∂γn 0 (z) · F n (z). (8.20) ∂γn γ =0
γ =0
In this expression, Ad X [Y ] stands for the usual adjoint action of the matrix X on the matrix Y , and 0 denotes the jump matrix at F ≡ 1 and g ≡ 0. It remains to compute the γ -derivatives of κ ±σ3 (z). They follow from the Faa-diBruno formula: ⎤ ps ⎡ n q s n n n! (±σ3 )s=1 ps ⎣ (−1)s F (z) − F s (µ) ⎦ n ±σ3 ∂γ κ (z) dµ = . n 7 2iπ s z−µ γ =0 p1 ,..., pn =0 s=1 ps ! −q n s=1 sps =n
s=1
(8.21) Therefore, gathering these informations and applying Leibnitz’s rule, we obtain that ∂γn
(z)
γ =0
=
n
n
p
q
Cn Cn− p p! q! n 7 ( ps )! (qs )!
p+ p0 +q=n p1 ,..., pn =0 q1 ,...,qn =0 n sp = p n sq =q p0 ≥1 s=1 s s=1 s=1 s
0
× Adeσ3 g(z)/2 (−σ3 ) ×F (z) p0
n s=1
=
p0 +sps =n p0 ≥1
n p s=1 s
⎡ s ⎣ (−1) 2iπ s
q −q
1
· ∂γp0 0 (z)
γ =0
· (σ3 )
n q s=1 s
⎤ ps +qs F s (z) − F s (µ) ⎦ dµ z−µ
⎤ ps ⎡ q n s (z) − F s (µ) F ⎣ dµ⎦ . δ (z; x) F p0 (z) z−µ
({ pi })
s=1 −q
(8.22) Note that, in the first line, we have extended for convenience the sum over parameters ps and qs up to n, since anyway, due to the constraint, ps ≤ p and ps = 0 if s > p (resp. qs ≤ q and qs = 0 if s > q). In the last line, we have changed the order of summations and incorporated all the F independent prefactors into the definition of the matrix δ({ pi }) (z; x). More precisely, p q n ({ pi }) Cn Cn− p p! q! (−1)s ps δ (z; x) = n 7 2iπ s n sq =q p+q=n− p0 s=1 s=1 (qs )! ( ps − qs )! s s=1 0 1 × Adeσ3 g(z)/2 (−σ3 )s=1 ( ps −qs ) · ∂γp0 0 (z) n
· (σ3 )s=1 qs . n
γ =0
(8.23)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
From the properties of the jump matrix 0 (z),
it is easy to see that the diagonal entries of the matrices δ({ pi }) (z; x) are a O x −1 , whereas their off-diagonal ones are a O (log p0 x/x) uniformly on the contours . • γ -derivatives of ∂λ . Let us recall the integral representation for ∂λ (λ), which is a direct consequence of (5.5), dz 1 + (z) (z) . ∂λ (λ) = − (8.24) 2iπ (λ − z)2
Therefore, the γ -derivatives of ∂λ (λ) can be directly obtained from the ones of (λ) and of + (λ).
Recall that + (λ) satisfies the following integral equation on M2 L 2 ( ) :
(8.25) I − C [+ ] = I2 , is defined by where the operator C
+ C [M] ≡ C [M] , ∀M ∈ M2 L 2 ( ) .
(8.26)
This matrix Cauchy operator is invertible, at least for x large enough. Indeed, using the continuity of the scalar Cauchy operator: - + ∃ c2 > 0 such that, ∀g ∈ L 2 ( ) , -C [g]- L 2 ( ) ≤ c2 g L 2 ( ) , (8.27) - - - one gets that the operator norm -C - fulfills: - - -C - ≤ c2 M2 ( L 2 ( )) −→ 0.
(8.28)
x→+∞
being a holomorphic function of γ we have that, for x large enough, Moreover, C ) is invertible and that its inverse is also a holomorphic function of γ . In (I − C particular, (8.25) implies
−1 ∂ γ + = I − C ◦ ∂γ C (8.29) [+ ] . ({ pi })
A straightforward induction shows that there exist some coefficients cr that n
({ p }) −1 p1 I − C ∂γn + = cr i ◦ ∂ C γ
∈ Z such
r p =n r =1 i=1 i
−1 p2 −1 pr ◦ I − C ◦ · · · ◦ I − C ◦ ∂ C ◦ ∂ C [+ ] . γ γ
(8.30)
This expression simplifies at γ = 0 as |γ =0 = 0 and + |γ =0 = I2 . Hence, ∂γn +
γ =0
=
n
r p =n r =1 i=1 i
∂γp1 C ◦ ∂γp2 C ◦ · · · ◦ ∂γpr C [I2 ] .
({ pi })
cr
(8.31)
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741
(i)
Fig. 6. Encased Contours (in the case p = id)
We can slightly deform the different contours , so as to regularize the explicit integral representation for the above chain of operators. Namely, recalling the construction of the jump contours for the matrices occurring in the different transformations applied to the RHP for χ , we are able to write ∂γn +
γ =0
=
n
({ pi })
cr
∂γp1 C (1) ◦ ∂γp2 C (2) ◦ · · · ◦ ∂γpr C (r ) [I2 ] .
r p =n r =1 i=1 i
(8.32) (i)
(i−1)
is at small but non vanishing There, the contours are such that the − side of (i) distance from the + side of , with the exception of a finite number of points of inter(i) ±1 − I2 section (cf. Fig. 6). The matrix corresponding to the contour is equal to M± (i) & on ± , to P − I2 on ∂ D−q,δi , and to P − I2 on ∂ Dq,δi . We emphasize that, already in (8.32), one can use the integral representation for the Cauchy operators without turning to boundary values. Indeed, the integrand appearing in (8.32) is already integrable on ×(r ) (1) (r ) ≡ × · · · × . Finally, one infers from (8.32), from (8.22) and from the integral representation for ∂λ (8.24) that, when λ ∈ ([ −q ; q ]), there exist some recursively computable ({ p }) coefficients c˜r i such that ∂γn ∂λ (λ)
γ =0
=−
n
({ p i })
c˜r
r =1 p 0 ≥1, p 1 ,..., p n ≥0 r =1 p¯ =n
dr z δ({ pri }) (zr ; x) . . . δ({ p1i }) (z 1 ; x) r 7 (2iπ )r (λ − z 1 )2 (z −1 − z ) ×(r )
×
×
⎧ r ⎨
=1
⎩
F p 0 (z )
n
⎡ ⎣
=2
q
m=1 −q
⎤ p m ⎫ F m (z ) − F m (µ) ⎦ ⎬ . ⎭ z − µ
(8.33)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras ×(r )
(1)
In this expression, the integration is performed over the skeleton = × · · · × (r ) , and the second summation is performed over integers p j , 1 ≤ ≤ r , 1 ≤ j ≤ n, r with 1 ≤ p 0 ≤ n and 0 ≤ p j ≤ n for j ≥ 1, and such that =1 p¯ = n, in which we have introduced the notation p¯ = p 0 + nj=1 j p j . • γ -derivatives of −1 . All the above observations also hold for the inverse matrix Indeed −1 + satisfies the integral equation ! " t ∇ −1 , (8.34) −1 + C = I (λ) 2 + +
−1 (λ).
in which t
∇ + C [M] ≡ C [∇ M] ,
∀M ∈ M2 (L 2 ( )),
(8.35)
and the matrix ∇ is defined2 by the equation I2 + ∇ = (I2 + )−1 . In other words, ∇ is the adjugate of (we recall that we consider 2 × 2 matrices and that det [I + ] = 1). Hence, one easily sees that, for n ≥ 1, n ! " ({ p }) dt z ({ p1i }) (z 1 ; x) . . . ({ pti }) (z t ; x) ∂γn −1 (λ) = c˜t i t 7 (2iπ )t γ =0 t=1 t p¯ =n − z (λ ) (z −1 − z ) ×(t) 1 =1 ×
⎧ t ⎨
=1
⎩
p 0 ≥1
F p 0 (z )
n
⎡ ⎣
q
m=1 −q
=2
⎤ p m ⎫ F m (z ) − F m (µ) ⎦ ⎬ , ⎭ z − µ
(8.36)
where ({ p i }) (z; x) is the adjugate matrix of δ({ p i }) (z; x). • γ -derivatives of log det [I + V ]sub . From the expressions (8.18), (8.33) and (8.36), ({ p }) it is easy to see that there exist some combinatorial coefficients Cr,t i ∈ Z (with ({ p }) C1,01i = −δ p10 ,n ) such that ∂γn log det [I + V ]sub
γ =0
×
1≤r +t≤n r ≥1, t≥0
({ p })
6x
([ −q ;q ])
=1
⎩
F p 0 (z )
dr +t z (2iπ )r +t
×(r +t)
dx trr,t j ({z j }; x )[g]
(λ − z 1 )2 (λ − zr +1 ) ×
dλ p(λ) 4π
({ p })
+∞
⎧ r +t ⎨
Cr,t i
p 0 ≥1, p 1 ,..., p r +t ≥0 r +t p¯ =n =1
×
=
n
r 7
=2 ⎡ q
⎣
m=1 −q
(z −1 − z )
r7 +t =r +2
(z −1 − z )
⎤ p m ⎫ F m (z ) − F m (µ) ⎦ ⎬ , dµ ⎭ z − µ
(8.37)
2 We stress that this matrix ∇ has nothing to do with the differential operator usually denoted by ∇.
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
in which, in the terms t = 0, the empty product (λ − zr +1 )
743
7r +t
=r +2 (z −1 − z ) should ({ p j }) be understood as 1. In this expression, trr,t ({z j }; x )[g] corresponds to the following
trace:
({ p }) tr1,01 j (z; x )[g]
= tr
σ3 ∂γn
$ % 1 0 (z; x ) − (1) (z; x ) x 0
γ =0
({ p }) trr,t j ({z j }; x )[g] = tr δ({ pri }) (zr ; x ) . . . δ({ p1i }) (z 1 ; x ) σ3 ({ pr +1i }) ({ pr +ti }) × (zr +1 ; x ) . . . (zr +t ; x ) if r + t > 1.
,
(8.38)
(8.39)
(1)
In (8.38), 0 (z; x ) corresponds to the first term in the asymptotic expansion (5.1) of 0 (z). Remark 8.1. We have gathered in the term r = 1, t = 0 the contribution of the second term in (8.18), as well as the term that would correspond to the contribution of only one jump matrix in the first term of (8.18). Note that, in this term r = 1, t = 0, the only non-zero contribution comes from the diagonal elements of , hence from the sequence ({ p }) p10 = n, p1 j = 0 for j ≥ 1 (indeed we have C1,01i = −δ p10 ,n ). Remark 8.2. It is easy to see from these expressions that the integrals over x are convergent. Indeed, it follows from the asymptotic expansion of the matrices 0 that $ % n 1 log x (8.40) ∂γn 0 (z; x) − (1) (z; x) = O 0 x x2 γ =0 uniformly on the integration contour, so that an integration of the trace (8.38) is convergent. We emphasize that the trace (8.39) is at least O (log x)n /x 2 uniformly on the integration contour: indeed, each of the matrices δ({ p j }) or ({ p j }) is uniformly a
p 0 O (log x) /x ; in addition, the trace (8.39) involves a product of at least two such matrices since r + t ≥ 2. These estimates guarantee that the integrals over x in (8.37) are well defined. 8.2.3. Application of the density procedure and proof of Theorem 8.1. In order to be able to apply the density procedure, we should express more explicitly the functional sub (ϕ,φ) dependence of ∂γn log det I + V (ϕ,φ) |γ =0 on Fn . sub n The F-dependence of ∂γ log det [I + V ] |γ =0 has already been explicitly extracted ({ p })
in (8.37), and all the g-dependence is contained in the traces trr,t j ({z j }; x )[g]. Using the structure of the matrices δ({ p i }) (z; x) and ρ ({ p i }) (z; x), one can be more precise concerning this g-dependence. Indeed, it follows from (8.23) that there exist some coef[{ p },{ }] z j ; x which are piecewise smooth on the integration contour ficients Dr, t j j such that r +t
({ p j }) [{ p j },{ j }] trr,t z j ; x exp ({z j }; x)[g] = Dr, t g (z ) . 1 ,...,r +t ∈{±1,0} i =0
=1
(8.41)
744
N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
[{ p },{ }] z j ; x are at least O (log x)n /x 2 uniformly Note that these coefficients Dr, t j j on the integration contour. Integrating these coefficients with respect to x, and defining &r,[{ tp j },{ j }] D
({ p }) z j ; x = Cr,t j
x
[{ p },{ j }]
dx Dr, t j
z j ; x ,
(8.42)
+∞
(i) which are at least O (log x)n /x uniformly on the integration contours , one gets dλ dr +t z n sub ∂γ log det [I + V ] p(λ) = 4π (2iπ )r +t γ =0 r +t 1≤r +t≤n p¯ =n ([ −q ;q ]) r ≥1, t≥0 =1 p 0 ≥1
×(r +t)
&r,[{ tp j },{ j }] z j ; x D × r r7 +t 7 2 1 ,..., (z −1 − z ) (z −1 − z ) 5r +t ∈{±1,0} (λ − z 1 ) (λ − z r +1 )
×
⎧ r +t ⎨ =1
⎩
i =0
F p 0 (z ) e g(z )
n
⎡ ⎣
=2
q
m=1 −q
=r +2
⎤ p m ⎫ ⎬ F m (z ) − F m (µ) dµ⎦ . ⎭ z − µ
(8.43)
We stress that each e±g(z ) may only appear in combination with at least one F (z ) (as p 0 ≥ 1): F (z ) e±g(z ) . This guarantees that the functional above is continuous with respect to the sup norm on the space of symmetric functions in n variables z and n variables λ. Before applying the density procedure, let us introduce one more useful notation. (m) Define the finite difference operator ðz (µ) by its action on pure product functions k k k−m (z) F m (µ). ð(m) z (µ) · F (z) = F (z) − F
This action naturally extends to symmetric functions of n variables $ % $ % p¯ i p¯ i (m) = Fn {z i } ðz (µ) · Fn {z i } i=1,...,r i=1,...,r $ % −Fn {z k } p¯k , {µ}m , {z } p¯ −m . k=
(8.44)
(8.45)
n sp , with 5r +t p¯ = n. We have moreover We recall here that p¯ = p 0 + s=1 s =1 p¯ {z } used the notation , which means that the variable z is repeated p¯ times, and {z i } p¯i i=1,...,r , which means that the variable z 1 is repeated p¯ 1 times, z 2 is repeated p¯ 2 times, …, zr is repeated p¯r times. The purpose of introducing such a finite difference operator is to recast products of functions F m (z ) − F m (µ) appearing in (8.43) into a more compact form. Namely, ⎤ p m ⎤ p m ⎡ q ⎡ q (m) n n m (z ) − F m (µ) ðz (µ) · F m (z ) F ⎣ ⎣ dµ⎦ dµ⎦ = z − µ z − µ m=1 −q m=1 −q ⎫ ⎧ q p m n n p m ⎬ dµ ,m, j ⎨ = ð(m) (µ ) · F p¯ − p 0 (z ) . ,m, j z ⎭ z − µ ,m, j ⎩ m=1 j=1−q
m=1 j=1
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
745
Therefore, setting e g(z) = φ(z) and F(z) = ϕ(z) φ(z), we get ∂γn
! log det I + V
(ϕ,φ)
"sub γ =0
=
dλ p(λ) 4π
1≤r +t≤n r +t p¯ =n ([ −q ;q ]) r ≥1, t≥0 =1 p 0 ≥1
dr +t z (2iπ )r +t
×(r +t)
&r,[{ tp j },{ j }] z j ; x D × r r7 +t 7 2 1 ,..., (z −1 − z ) (z −1 − z ) 5r +t ∈{±1,0} (λ − z 1 ) (λ − z r +1 ) i =0 =2 =r +2 ⎫ ⎧ p m ⎨q r +t r +t n ⎬ dµ ,m, j × ϕ p¯ (z ) φ p¯ + (z ) . (8.46) ð(m) (µ ) · ,m, j ⎭ ⎩ z − µ ,m, j z
=1 m=1 j=1 −q
=1
It follows immediately from the density procedure formulated in Theorem C.1 that Insub can be extended into a linear functional on Sym n (U, W ). Its action on a holomorphic function Fn ∈ Symn (U, W ) is given as Insub [Fn ]
(−1)n−1 = (n − 1)!
dλ p(λ) 4π
1≤r +t≤n r +t p¯ =n ([ −q ;q ]) r ≥1, t≥0 =1 p 0 ≥1
dr +t z (2iπ )r +t
×(r +t)
&r,[{ tp j },{ j }] z j ; x D × r r7 +t 7 2 1 ,..., (z −1 − z ) (z −1 − z ) 5r +t ∈{±1,0} (λ − z 1 ) (λ − z r +1 ) i =0 =2 =r +2 ⎫ ⎧ . / p m ⎨q r +t n ⎬ {z } p¯ 1≤ ≤r +t dµ ,m, j (m) × ð (µ ,m, j ) · Fn . {z } p¯ + 1≤ ≤r +t ⎭ ⎩ z − µ ,m, j z
=1 m=1 j=1 −q
(8.47) The sum appearing in (8.47) is finite, and since each integrand is a O (logn x/x), Insub [Fn ] is itself a O (logn x/x). Hence Theorem 8.1 follows directly, since In [Fn ] = In(0) [Fn ] + Insub [Fn ] .
(8.48)
8.2.4. Asymptotic expansion of Insub [Fn ] and proof of Theorem 8.2. In order to prove the existence of an asymptotic series for In [Fn ], i.e. for Insub [Fn ], we should be more &r,t , i.e. show that they themselves admit precise on the structure of the coefficients D an asymptotic expansion. Let us recall that these coefficients are obtained from the traces (8.41) involving the matrices δ({ p j }) (z ; x) and ({ p j }) (z ; x). The latter being obtained from the jump matrix 0 . (i) Clearly, all terms corresponding to an integration on the contours ± yield exponentially small corrections. Thus in what concerns the proof of an asymptotic expansion we can only focus on integrations along the contours ∂ D±q,δi . We decompose ×(r +t) the relevant contour ∂ Dq,δ ∪ ∂ D−q,δ into sums of elementary skeletons ×(r +t) ∂ Dσ q,δ ≡ ∂ Dσ1 q,δ1 × · · · × ∂ Dσr +t q,δr +t , where each σi takes values in {±1}:
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
(r +t)
dr +t z = (2iπ )r +t σ =± i
(r +t) ∂ Dσ q,δ
−∞ dr +t z . r +t + O x (2iπ )
(8.49)
The matrices δ({ p j }) (z; x) (8.23) admit an asymptotic expansion into inverse powers of x on ∂ Dq,δ ∪ ∂ D−q,δ . This fact follows from the asymptotic expansion of 0 (z). The latter is obtained by taking adequate a-derivatives at a = 0 or 1 of the asymptotics
series (A.6) for (a, 1; z) when z → ∞. This is licit as, for fixed M, the O z −M−1 estimate in the asymptotic series (A.6) is uniform with respect to a and since we perform a finite number of derivatives with respect to a. This asymptotic expansion takes the following form: ! ({ p },k) " ⎧ j ⎪ Ad X (z; ) δ σ [i x p +g(z)]/2 M + + ⎪ + 3 5 log p0 x e ⎪ ⎪ , + O ⎪ ⎨ x M+1 x k ( p (z) − p+ )k ({ p j }) (z; x) = k=1 ! ({ p },k) " δ j ⎪ ⎪ (z; X − ) M Adeσ3 [i x p− +g(z)]/2 δ− ⎪ 5 log p0 x ⎪ ⎪ ⎩ , +O x M+1 x k ( p (z) − p− )k k=1
z ∈ ∂ Dq,δ , z ∈ ∂ D−q,δ ,
(8.50) the corrections being uniform on the contours. There the diagonal entries of the matrices ({ p j },k) (z; X ± ) are some constants (i.e. x and z independent), whereas the off-diagonal δ± ones are polynomials of degree p0 in the variable X ± = log ±x ( p (z) − p∓ ) . An exactly similar structure holds for ({ p j }) (z; x) as it is adjunct to δ({ p j }) (z; x). Hence, on the skeleton ∂ Dσ q,δ = ∂ Dσ1 q,δ1 × · · · × ∂ Dσr +t q,δr +t , the trace (8.41) can be expanded in the following form: ({ p })
trr,t j ({z j }; x)[g] =
M+1 N =2
1 xN
exp ×
N
Dr, t
k1 ,...,kr +t =1 1 ,...,r +t ∈{±1,0} ki =N i =0
r +t 5
g (z ) + i x pσ
=1 r7 +t =1
p (z ) − pσ
k
% $ { p j }, {i } ({z i } ; log x) {σi }, {ki }
+O
logn x x M+2
.
(8.51)
There we have explicitly factored out the dependence on the oscillating factor exp i x pσ . In this expression, the coefficients Dr, t are piecewise smooth on the integration contour and are polynomials of degree | | p 0 in log x. We set, for N ≥ 2, % { p j }, {i } ({z i } ; log x) {σi }, {ki } % $ x
dx ({ p j }) { p j }, {i } {z i } ; log x = Cr,t D N r, t {σi }, {ki } (x ) (0)
&r, t x 1−N D
$
+∞
(8.52)
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
when
5
747
pσ = 0, and
e
ix
5
pσ
M
x
−k
k=N
=
({ p }) Cr,t j
x
+∞
% $ n log x (k) { p j }, {i } & Dr, t ({z i } ; log x) + O {σi }, {ki } X M+1 dx
(x )
D N r, t
% $ 5 { p j }, {i } ({z i } ; log x) ei x pσ {σi }, {ki }
(8.53)
5 otherwise. Note at this i = 0, there exist some integer 5 stage that, due to the constraint ix pσ i xm( p − p ) + − m = 0 such that e =e . We then insert the result of integration into the expression for Insub [Fn ], rearrange the asymptotic expansion into decreasing powers of x and separate the oscillating and non-oscillating parts. We obtain
Insub [Fn ]
n M M 1 (N ; nosc) 1 (N ; osc) log x , I I = [Fn ] + [Fn ] + O xN n xN n x M+1 N =1
N =2
(8.54)
where
In(N ; nosc) [Fn ] =
N +1
(0)
Ir, t
r,t σ1 ,...,σr +t =± k1 ,...,kr +t =1 { p j },{i } pσ =0 ki =N +1
% $ { p j }, {i } (log x) [Fn ] , {σi }, {ki } (8.55)
In(N ; osc) [Fn ] =
N
r,t σ1 ,...,σr +t =± s=2 { p j },{i } pσ =0 s
×
ix
e
r5 +t =1
pσ
(s) Ir, t
k1 ,...,kr +t =1 ki =s
$
% { p j }, {i } (log x) [Fn ] . {σi }, {ki }
(8.56)
In these expressions, r,t { p j },{i }
≡
p 0 ,..., p n 1≤r +t≤n 1 ,...,r +t ∈{±1,0} r +t p¯ =n r ≥1, t≥0 p 0 ≥1, =1 i =0
(8.57)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
and the functional Ir, t is given by (s) Ir, t
% $ (−1)n−1 { p j }, {i } (log x) [Fn ] = {σi }, {ki } (n − 1)!
dλ p(λ) 4π
([ −q ;q ])
dr +t z (2iπ )r +t
×(r +t)
∂ Dσ q,δ
% $ (s) { p j }, {i } &r, D r +t t {σ }, {k } ({z i } ; log x) 1 i i ×
k r r7 +t 7 (λ − z 1 )2 (λ − zr +1 ) (z −1 − z ) (z −1 − z ) =1 p(z ) − pσ =2 =r +2 ⎫ ⎧ q . / p r +t n m ⎬ ⎨ {z } p¯ 1≤ ≤r +t dµ ,m, j (m) × ð (µ ,m, j ) · Fn . {z } p¯ + 1≤ ≤r +t ⎭ ⎩ z − µ ,m, j z =1 m=1 j=1 −q
(8.58) 5 &r, t being polynomials of degree The coefficients D | | p 0 in log x, this ends the proof of Theorem 8.2 concerning the existence of the asymptotic expansion of cyclic integrals to any order in 1/x. As it is presented, the form of this asymptotic expansion may look quite involved. ×(r +t) Note however that the integrals over the contours ∂ Dσ q,δ in (8.58) can be computed; they are expressible in terms of partial derivatives of the function Fn at ±q (see Appen(N ; nosc) dix D). It is proved in Appendix D that the non-oscillating term In [Fn ] of order N can be expressed in terms of derivatives of Fn of total order not higher than N , whereas the order of such derivatives does not exceed N − 2 in the case of In(N ; osc) [Fn ]. This property is useful in [35], when we sum up the asymptotic behaviour of a whole class of cycle integrals of the form (8.1) to obtain the asymptotic behaviour of correlation functions. To perform this summation we use the knowledge of the number of partial derivatives applied to Fn . We finally point out that the integral over λ produces derivatives of the function p(λ) evaluated at ±q.
9. More General Kernels In the applications to quantum integrable models, one sometimes needs to use some modified versions of the GSK. Consider the operator I + Vθ acting on [ −q ; q ] with kernel Vθ (λ, µ) =
F (λ) F (µ) θ (λ) θ (µ)
e+ (λ) e− (µ) − e− (λ) e+ (µ) , 2iπ [θ (λ) − θ (µ)]
(9.1)
where e± and F are defined as in (1.7). We assume in addition that θ is a biholomorphism of U onto its image, that θ ([ −q ; q ]) ⊂ R and θ (U ∩ H± ) ⊂ H± . Then the asymptotic behaviour of log det[I + Vθ ] when x → ∞ follows from Theorem 2.1. More precisely, we have the following corollary:
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
749
Corollary 9.1. Let Vθ be as above. Then q log det [I + Vθ ] = 2
dλ ν(λ) log [e− (λ)]
−q
⎤ (σ q)νσ2 Kσ νσ (σ q; q) G(1, ν ) θ σ ⎦ log ⎣ + νσ2 (θ (q) − θ (−q)) pσ x σ =± ⎡
1 + 2
q dλ dµ −q
ν (λ) θ (µ) ν(µ) − ν(λ) θ (λ) ν (µ) + o (1) , θ (λ) − θ (µ)
(9.2)
where
⎧ q ⎫ ⎨ ν(λ) − ν(µ) ⎬ −1 ν(λ) = log (1 + γ F(λ)) , K(λ; q) = exp θ (µ) dµ , ⎩ θ (λ) − θ (µ) ⎭ 2iπ
(9.3)
−q
= [∂ p(λ)] | and, as before, p± λ λ=±q , ν± = ν (±q).
Proof. The change of variables θ (λ) = ξ maps the kernel Vθ on the one of the GSK 8 e+ ◦ θ −1 (ξ )e− ◦ θ −1 (η) − e− ◦ θ −1 (ξ ) e+ ◦ θ −1 (η) V (ξ, η)= F ◦ θ −1 (ξ )F ◦ θ −1 (η) . 2iπ (ξ −η)
This kernel acts on [ θ (−q) ; θ (q) ] which is, a priori, a non symmetric interval. However, it is enough to apply the transformation λ → λ−(θ (q) + θ (−q)) /2 so as to recover the symmetry of the interval. Then, it remains to enforce the inverse transformations on the asymptotic formula for the Fredholm determinant of V . Let us write explicitly the asymptotics (9.2) in the case of the kernel Vsh (λ, µ) = γ
F(λ) F(µ)
e+ (λ) e− (µ) − e− (λ) e+ (µ) , e± (λ) = e±[i x p(λ)+g(λ)]/2 , 2iπ sinh(λ − µ)
as it plays a crucial role in the analysis of the asymptotic behaviour of the two-point functions in the massless phase of the XXZ Heisenberg chain [35]. In this case, Eq. (9.2) reads q log det [I + Vsh ] = 2
+
⎧ ⎨
−q
G(1, νσ ) log ν 2 ⎩ sinh(2q) pσ x σ σ =±
q
ν (λ) ν(µ) − ν(λ) ν (µ) tanh (λ − µ) −q ⎫ q νσ − ν(λ) ⎬ + σ νσ dλ + o (1) . (9.4) tanh (σ q − λ) ⎭
1 dλ ν(λ) log [e− (λ)] + 2
dλ dµ
−q
It is clear that the last equation can be used in order to obtain an analog of the asymptotic expansion for multiple integrals of the type (8.1) where the rational functions z − λ are replaced by the hyperbolic sinh(z − λ). Namely, let
750
N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
Insh
[Fn ] =
dn z (2iπ )n
([ −q ;q ])
q −q
dn λ Fn (2iπ )n
n ei x ( p(z j )− p(λ j )) {λ}
. {z} sinh z j − λ j sinh z j − λ j+1 j=1 (9.5)
Then under the conditions of Corollary 8.1 one has the following asymptotic estimate Insh
1 [Fn ] = 2iπ
q
dλ i x p (λ) + ∂ Fn
−q
bn − cn log sinh(2q) pσ x Fn
{λ}n {λ + }, {λ}n−1
n
=0
{σ q} {σ q}n σ =± {σ q}n {σ q} p , {λ}n− p q − F F n−1 n {σ n n q} {σ q} p , {λ}n− p n dλ + p (n − p) tanh (q − σ λ) (2π )2 σ =± p=1
+
−q
+
n
n−1 q
dλdµ {λ + } , {λ} p−1 , {µ}n− p ∂ Fn {λ + } , {λ} p−1 , {µ}n− p (n − p) tanh (λ − µ)
2 (2π )2 p=1 −q {µ + } , {µ} p−1 , {λ}n− p −∂ Fn {µ + } , {µ} p−1 , {λ}n− p
+ o(1).
(9.6)
=0
10. Conclusion We have obtained in this article the leading asymptotic expansion of the Fredholm determinant of the GSK. As we have mentioned, our main motivation is to apply this result to the asymptotic analysis of the correlation functions of quantum integrable models, using in particular the asymptotic study of multiple integrals performed in Sect. 8. This is done in [35]. Another development is to extend the above analysis so as to handle truncated Wiener–Hopf operators with symbols having Fischer–Hartwig type discontinuities. The corresponding results are published in [37]. The results for the case of Toeplitz, Hankel and Toeplitz + Hankel determinants with Fisher-Hartwig singularities appeared recently in [16,17]. Let us also point out some unsolved problems. One of them concerns the derivation of the asymptotics of the Fredholm determinant of the GSK via the method based on its derivative over endpoint q. It would be important to obtain a complete justification of this method, since it is rather powerful and at the same time relatively simple. Another problem is to prove the conjecture on the Z−ν periodicity for the asymptotic expansion of the Fredholm determinant. If this property does hold, then all oscillating corrections can be obtained from the non-oscillating ones via a simple shift of ν by integer numbers. This could lead to a much simpler way to compute sub-leading corrections for such determinants. Acknowledgements. We are very grateful to A. R. Its for useful and numerous discussions. J. M. M., N. S. and V. T. are supported by CNRS. N. K., K. K. K., J. M. M. and V. T. are supported by the ANR program
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
751
GIMP ANR-05-BLAN-0029-01. N. K. and V. T. are supported by the ANR program MIB-05 JC05-52749. We acknowledge support from the French-Russian Exchange Program GDRI-471 of CNRS and RFBR-0901-93106-CNRSa. N. S. is also supported by the Program of RAS Mathematical Methods of the Nonlinear Dynamics, RFBR-08-01-00501a, Scientific Schools 795.2008.1. N. K and N. S. would like to thank the Theoretical Physics group of the Laboratory of Physics at ENS Lyon for hospitality, which makes this collaboration possible.
A. Some Properties of Confluent Hypergeometric Function For generic parameters (a, c) the Tricomi confluent hypergeometric function (a, c; z) is one of the solutions to the differential equation zy + (c − z) y − ay = 0.
(A.1)
It satisfies the properties: • Differentiation: a [(a − c + 1)(a + 1, c; z) − (a, c; z)] z 1 = [(a − c + z)(a, c; z) − (a − 1, c; z)]. z
(a, c; z) =
(A.2)
• Monodromy:
(a, 1; ze2miπ ) = (a, 1; z) 1 − meiπa(+1) + meiπa(−1) +
2πimeiπa+z (1 − a, 1; −z),
2 (a)
(A.3)
where = sgn ((z)). In particular, 2πie−iπa+z (1 − a, 1; −z), (z) < 0,
2 (a) (A.4) iπa+z 2πie (1 − a, 1; −z), (z) > 0. −
2 (a) (A.5)
(a, 1; ze2iπ ) = (a, 1; z)e−2iπa +
(a, 1; ze−2iπ ) = (a, 1; z)e2iπa
• Asymptotic expansion: (a, c; z) ∼
∞ n=0
(−1)n
3π (a)n (a − c + 1)n −a−n 3π z < arg(z) < , , z → ∞, − n! 2 2 (A.6)
with (a)n = (a + n) / (a). We have the following recombination between the Tricomi CHF (a, c; z) and the Humbert CHF (a, c; z): (a, c; z) =
(c) iπ (a−c)+z
(c) iaπ e e (a, c; z) + (c − a, c; −z) ,
(c − a)
(a)
(A.7)
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
where = sgn ((z)), and (a, c; z) =
∞ (a)n z n . (c)n n!
(A.8)
n=0
Such a recombination formula allows to obtain the asymptotic expansion of the Humbert CHF: iπ a M e
(c) (a)n (a − c + 1)n (a, c; z) = + O |z|−a−M−1 n
(c − a) z n! (−z) n=0
+
(c) z a−c e z
(a)
N n=0
(c − a)n (1 − a)n z a−1−c−N + O z . (A.9) e n!z n
One can estimate integrals involving a product of two CHF as below, either by using Laplace–type integral representations for the functions (a, c; z) and (a, c; z) or applying the method given in [37]. The latter uses Erdelyi’s representation of Laplace transforms of products of CHF in terms of a Lauricella function adjoint to some asymptotic expansion of a Lauricella function. In any case, the result reads: +∞ dt e−iπa ϕ (a; t) − 1 = −2ia,
(A.10)
0
+∞ 2ia = 2ia − a [ψ (a) + ψ (−a)] , dt e−iπa τ (a; t) + 1 + t +1
(A.11)
0
and the Riemann integrability of the integrands is part of the conclusion. We recall the definition of the functions τ (a; t) and ϕ (a; t): ϕ (ν; t) = (−ν, 1; −it) (ν, 1; it) , (A.12) τ (ν; t) = − (−ν, 1; −it) (ν, 1; it) + (∂z ) (−ν, 1; −it) (ν, 1; it) + (−ν, 1; −it) (∂z ) (ν, 1; it) . (A.13) B. Three Preparatory Lemmas Here we prove three preparatory integration lemmas used in Sect. 6. Lemma B.1. Let R (u, t) be a function of two variables defined on I × R+ , where I is an open interval of R containing 0. Suppose that the partial applications u → R (u, t) are C 1 (I ) for all but finitely many t’s and that t → R (u, t) is Riemann integrable uniformly in u, i.e.: ∀ρ > 0, ∀M > 0, ∀u 0 ∈ I, ∃υ > 0 such that +∞ ! " u ∈ ] −υ + u 0 ; υ + u 0 [ ∩ I, k ∈ {0, 1} ⇒ dt ∂1k R (u, t) − ∂1k R (u 0 , t) ≤ ρ. M
(B.1)
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
753
Then for g ∈ C 1 (I ), δ
+∞ xg (t) R (t, xt) dt = g (0) R (0, t) dt + o (1) ,
0
(B.2)
0
where the small o (1) is with respect to the successive limit xδ → +∞ and δ → 0. Proof. One has δ
δ x (g (t) R (t, xt) − g (0) R (0, xt)) dt =
0
xδ ∂ y [g (y) R (y, t)] dt.
dy
(B.3)
xy
0
Consider a function b g : (y, a, b) →
dt∂ y [g (y) R (y, t)]
(B.4)
a
on the compact set [ 0 ; δ ] × R+ × R+ . g is clearly continuous on the interior and the uniform Riemann-integrability of R (y, t) guarantees that it is continuous in an neighbourhood of (∗, +∞, ∗), (∗, ∗, +∞) and (∗, +∞, +∞). Hence, |g| is bounded, say by B, as a continuous function on a compact set. Thus, δ xδ dy ∂ y [g (y) R (y, t)] dt ≤ δ B, (B.5) xy
0
which ends the proof of Lemma B.1.
Lemma B.2. Let g ∈ C 1 (I ) for some open interval I containing 0, then δ
g (t) xdt = g (0) log xδ + o (1) , 1 + xt
(B.6)
0
where o (1) stands with respect to the successive limits xδ → +∞ and δ → 0. Proof. We have δ
xdt = g (0) log (xδ + 1) + g (t) 1 + xt
0
δ
t dt
0
dy
g (y) x 1 + xt
(B.7)
0
δ = g (0) log δx + o (1) +
xδ + 1 . dyg (y) log xy + 1
(B.8)
0
But,
δ δ + 1/x dyg (y) log ≤ sup g × (δ − log (δ + 1/x) /x) −→ 0 , (B.9) δ→0 y + 1/x [ 0 ;δ ] 0
which ends the proof of Lemma B.2.
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
Lemma B.3. Let κ be defined in terms of ν as in (2.2), and set 1 H (q) = 2
q dλdµ −q
ν (λ) ν(µ) − ν (µ) ν(λ) + ν log κ(q; q). λ−µ =±
(B.10)
Then, 2ν+
(ν+ − ν− )2 dH (q) d d log κ (q; q) − 2ν− log κ (−q; q) − = . (B.11) dq dq q dq
Proof. Using (2.2), one can express the derivative of H (q) as dH (q) ν (q) − ν (µ) d = + log κ (q; q) . ν (q) dµ ν (q) dq q − µ dq =± =± q
−q
(B.12) Thus, proving (B.11) amounts to establishing the equality (ν+ − ν− )2 d ν (q) − ν (µ) log κ (q; q) − = . ν (q) ν (q) dµ dq q q − µ =± =± q
−q
(B.13) The latter follows from an integration by parts:
d log κ (q; q) dq =± ⎧ ⎫ q ⎨ ⎬ − ν − ν ν ν − ν − q) (µ) + − (µ = + dµ ν ν + ⎩ ⎭ 2q (µ − q)2 =± −q ⎧ ⎫ q ⎨ (µ) − ν ⎬ ν − ν − ν + 2qν ν ν + − − + + dµ = ν ν + ⎩ ⎭ 2q −2q µ − q =± ν (q)
−q
=
(ν+ − ν− )2 + ν q =±
This ends the proof.
q dµ −q
ν (µ) − ν . µ − q
C. The Density Theorem
{λ} Theorem C.1. Let U , W be two open neighbourhoods of [ −q ; q ], and let Fn {z} be a holomorphic function on U n × W n , symmetric separately in the n variables λ and
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
755
in the n variables z. Then,
for any compact subsets K (resp. P) of U (resp.W) there exists a sequence ϕ p , φ p p∈N in H (K ) × H (P) such that Fn
{λ} {z}
=
+∞ n
ϕ p (λi ) φ p (z i )
uniformly on K n × P n .
(C.1)
p=0 i=1
Proof. Let K and P be as above.
Let X = K n × P n / ∼, where the relation ∼ is defined as follows: (λ, z) ∼ λ , z if there exists a couple of permutations (σ, π) ∈ Gn × Gn such that (λσ , z π ) = λ , z , where λσ stands for λσ (1) , . . . , λσ (n) . Since Gn × Gn is a discrete group, its action on K n × P n is by definition proper, i.e. ∀L ⊃ K n × P n , (C.2) (σ, π) ∈ Gn × Gn : L σ,π ∩ L = ∅ is discrete. This ensures that X is a compact Hausdorff topological space. Moreover the space C (X, C) of continuous functions on X is canonically identified with the space of continuous functions on K n × P n that are symmetric in the first or the last n variables. (ϕ,φ) of the form Define the subspace S of C (X, C) as the subset of functions Fn Fn(ϕ,φ)
{λ} {z}
=
n
ϕ (λi ) φ (z i ),
(C.3)
i=1
where (ϕ, φ) ∈ H (K ) × H (P), and let S be the C ∗ -algebra generated by S. We have that S and hence S separates points in X . Indeed, let (λ, z) and (µ, y) be any two representatives in K n × P n of two distinct points in X . Thus • there exists λi ∈ K such that exactly p of the n coordinates of the n-tuple λ are equal to λi , whereas exactly q of the n coordinates of the n-tuple µ are equal to λi , with p = q; • or there exists z i ∈ P such that exactly p of the n coordinates of the n-tuple z are equal to z i , whereas exactly q of the n coordinates of the n-tuple y are equal to z i , with p = q. The situation is similar in the case of the first n and last n variables, therefore we only treat the first case. By Lagrange interpolation there exists a polynomial Q such that, for any coordinate λk of λ and any coordinate µk of µ satisfying λk = λi and µk = λi , Q (λk ) = Q (µk ) = 1 and Q (λi ) = 2.
(C.4)
The function Fn(Q,1)
{λ} {z}
=
n
Q λp ∈ S
(C.5)
p=1
separates the projections of (λ, z) and (µ, y) on X . Thus S is a C ∗ -subalgebra of C (X ; C) that separates points. It then follows by the Stone-Weierstrass theorem that S = C (X ; C). Let Fn be holomorphic on U n × W n and symmetric in the first and in the last n vari◦
◦
ables. There exists compact sets K ⊂ U and P ⊂ W such that K ⊂ K and P ⊂ P .
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras ◦
Here, K stands for the interior of K . Thus the restriction of Fn to K n × Pn also &p belongs to C (X ; C), with X = K n × Pn / ∼, and therefore there exists & ϕp, φ p∈N in C (K ; C) × C (P ; C) such that n +∞ {λ} &p (z i ) = & ϕ p (λi ) φ (C.6) Fn uniformly on K n × Pn . {z} p=0 i=1
In particular the sequence converges uniformly to Fn on (∂ K )n × (∂ P )n , the latter set being compact. Therefore we have N p=0∂ K
dn µ (2iπ )n
−→
N →+∞
∂ K
n n N &p (yi ) ϕ p (µi ) φ dn y & = ϕ p (λi ) φ (z i ) (2iπ )n (µi − λi ) (yi − z i )
dn µ (2iπ )n
uniformly in (λ, z) ∈
p=0 i=1
i=1
∂ P
dn y Fn ({µ} | {y}) = Fn ({λ} | {z}), n (2iπ )n 7 (µi − λi ) (yi − z i )
∂ P
Kn
i=1
×
ϕ p (λ) = ∂ K
(C.7)
Pn.
Moreover,
ϕ p (µ) dµ & and φ p (z) = 2iπ µ − λ
∂ P
&p (y) dy φ 2iπ y − z
(C.8)
are holomorphic in K , resp. P.
D. Form of the Sub-leading Terms in Insub In this Appendix, we focus on the general structure of the sub-leading asymptotics of cyclic integrals. We show that the 1/x N term in the non-oscillating part can be obtained as an action of at most N partial derivatives of the function Fn followed by an evaluation at ±q or by an integration over [−q; q]. In principle, the contour integrals defining (8.58) can be computed to the end. However, the result is quite intricate, and we do not need, for the further applications, the formula in its whole generality. Indeed, we are interested in a particular sub-class of such integrals. More precisely we shall focus on the sub-class that is susceptible to produce the highest possible derivatives of the function Fn . Here, by highest derivative we mean the total degree of all the partial derivatives that might act on the integrand. This subclass is identified in the upcoming lemma. Lemma D.1. Let r, t ∈ N with r + t ≥ 1 label negations σ1 , . . . , σr +t ∈ {±}. Also introduce sufficiently small numbers 0 < δ1 < · · · < δr +t < q as
well as positive integers k1 , . . . , kr +t . Finally, let G ∈ H D σ1 q,δ1 × · · · × D σr +t q,δr +t and ({σ },{k }) Gr, t i i [G]
= ×(r +t)
∂ Dσ q,δ
×
r +t r dr +t z 1 1 r +t z −1 − z z −1 − z (2iπ ) =2
r +t
1
=1
(z − σ q)k
=r +2
G({z}) ,
(D.1)
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications ×(r +t)
with ∂ Dσ q,δ
757
≡ ∂ Dσ1 q,δ1 × · · · × ∂ Dσr +t q,δr +t . ({σ },{k })
Then, the integral Gr, t i i [G] can be computed as some combinatorial sum involving derivatives of G at the points σi q, the maximal order of such derivatives being equal to r +t
ki − nr − n t + δr,0 + δt,0 − 2.
(D.2)
i=1
Here nr , resp. n t , is the number of times the sequence (σ1 , . . . , σr ), resp. (σr +1 , . . . , σr +t ), changes sign, and δr,0 , δt,0 denote the usual Kronecker symbols. Proof. Let us prove the claim by induction on r + t. First, for r + t = 1, (D.2) is obviously satisfied. Indeed, 1 (σ,k) (σ,k) G1,0 ∂zk−1 G (σ q). [G] = G0,1 [G] = (k − 1)!
(D.3)
Let us now assume that the result holds for any function G up to some value of r + t. We will prove that it also holds for r + t + 1. ! " ({σ },{k }) ({σ˜ },{k˜ }) ˜ {σ˜ i }, {k˜i } are obtained Note first that Gr, t i i [G] = Gt, r i i G˜ , in which G, from G, {σi }, {ki } by a reordering of the variables. Hence, it is enough to prove the claim for Gr +1, t . We will have to distinguish two cases, depending on whether r + 1 = 1 or r + 1 > 1. In the case r + 1 = 1, it is easy to see that ! " 1 ({σ },{k }) (σ2 ,...,σt+1 ,k2 ,...,kt+1 ) k1 −1 t+1 } {z G1, t i i [G] = G0, ∂ , (D.4) G σ q, 1 i z i=2 t 1 (k1 − 1)! ({σ },{k })
which means that G1, t i i [G] can be expressed in terms of derivatives of G of maximal 5 t+1 k − 1 − n + δ order t t,0 + (k1 − 1), hence the result. i=2 i Let us now consider the case r + 1 > 1. We have ({σ },{k }) Gr +1,i t i [G]
1 = (k1 − 1)!
×(r +t)
∂ Dσ q,δ
×
r +t+1 =r +3
=− ×(r +t)
∂ Dσ q,δ
×
k 1 −1 k=0
r +t+1 =2
r +1 dz 1 2iπ z −1 − z =3
r +t+1 G({z}) 1 1 k1 −1 ∂ z z −1 − z z1 − z2 (z − σ q)k 1 =2 r +t+1 =2
dz 2iπ
r +1 =3
1 z −1 − z
r +t+1 =r +3
1 z −1 − z
z 1 =σ1 q
r +t+1
1
=2
(z − σ q)k
1 1 k r +t+1 } {z ∂ . G σ q, 1 i i=2 k! (z 2 − σ1 q)k1 −k z 1
(D.5)
At this point one should distinguish between the two possible cases: σ1 σ2 = 1 or σ1 σ2 = −1. We first assume σ1 σ2 = 1 (i.e. that there is no change of sign between σ1
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N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras
and σ2 ), and set Gˆ +k (z 2 , . . . , zr +t+1 ) = ∂zk1 G (z 1 , . . . , zr +t+1 ) |z 1 =σ1 q . Then ({σ },{ki })
Gr +1,i t
[G] = −
k 1 −1 k=0
1 ({σi }r2+t+1 ,{k2 +k1 −k}∪{ki }r3+t+1 ) ! ˆ + " Gk . G k! r,t
(D.6)
The latter can be expressed in terms of derivatives 5 of G of maximal order 5r +t+1 +t+1 k + k2 + k1 − k + i=3 ki − nr − n t + δt,0 − 2 = ri=1 ki − 2 − nr +1 − n t + δt,0 . We now assume that σ1 σ2 = −1. This leads to ({σ },{ki })
Gr +1,i t
[G] = −
k 1 −1 k=0
1 ({σi }r2+t+1 ,{ki }r2+t+1 ) ! ˆ − " G Gk , k! r,t
(D.7)
where the function Gˆ − k (z 2 , . . . , z r +t+1 ) =
∂zk1 G (z 1 , . . . , zr +t+1 ) |z 1 =σ1 q
(D.8)
(z 2 + σ2 q)k1 −k
is holomorphic inside the integration contour ∂ Dσ2 q,δ2 × · · · × ∂ Dσr +t q,δr +t . Once again, the result will be expressedin terms of derivatives of G andthe maximal order of these 5r +t+1 5r +t+1 derivatives will be k1 − 1 + i=2 ki − n r − n t + δt,0 − 2 = i=1 ki − n r +1 − n t + δt,0 − 2, which ends the proof of Lemma D.1. Remark D.1. The integral can be explicitly computed using the recurrence formulas (D.6) and (D.7). In particular, in the simplest case σ1 = · · · = σr and σr +1 = · · · = σr +t , we have ({σ },{ki })
Gr, t i
[G] = (−1)r +t−δr,0 −δt,0
r +t 1 ur +t ∂ . . . ∂zu11 G ({z}) u ! zr +t
u 1 ,...,u r +t =1 u ∈
z i =σi q
, (D.9)
in which the parameters u are summed over sets defined as ⎧ ⎫ −1 ⎨ ⎬
= 0, . . . , kj − uj −1 , ⎩ ⎭ j=1 j=1 ⎧ ⎫ r + r + −1 ⎨ ⎬
r + = 0, . . . , kj − uj −1 , (1 ≤ < r ), (D.10) ⎩ ⎭ j=r +1 j=r +1 ⎧ ⎫ ⎧ ⎫ r r −1 r +t r +t−1 ⎨ ⎬ ⎨ ⎬
r = kj − u j − 1 , r +t = kj − u j − 1 . (D.11) ⎩ ⎭ ⎩ ⎭ j=1
j=1
j=r +1
j=r +1
Corollary D.1. The subleading terms of order N in the asymptotic expansion (8.54) for the cycle integral In [Fn ] are obtained in terms of derivatives of the function Fn . More (N ; nosc) [Fn ] involves derivatives of Fn of total order precisely, the non-oscillating term In (N ; osc) [Fn ] involves derivatives of Fn at most equal to N , whereas the oscillating one In of total order at most equal to N − 2.
Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications
759 ×(r +t)
Proof. In order to apply Lemma D.1 to the integral over ∂ Dσ q,δ in (8.58), let us set % $ (s) { p j }, {i } & k Dr, t ({z i } ; log x) r +t {σi }, {ki } z − σ q G ({z}) = p(z ) − pσ (λ − z 1 )2 (λ − zr +1 ) =1 ⎫ ⎧ q . / p m ⎨ r +t n ⎬ {z } p¯ 1≤ ≤r +t dµ ,m, j (m) × ð (µ ,m, j ) · Fn . {z } p¯ + 1≤ ≤r +t ⎭ ⎩ z − µ ,m, j z =1 m=1 j=1 −q
(D.12) ×(r +t)
The poles at z 1 = λ and zr +1 = λ being outside of the skeleton ∂ Dσ q,δ , this function ×(r +t)
is indeed holomorphic in a vicinity of the polydisc Dσ q,δ . 5 Applying the result of Lemma D.1 to this function and using the fact that ki = N +1 (N ; nosc) in (8.55), it follows immediately that the expression of In [Fn ] cannot involve derivatives of the function Fn of order higher than N . This maximal order of derivatives corresponds to t = 0 and ∀ i σi = σ with σ = ± in (8.55). 5 (N ; osc) Similarly, as in (8.56) k i ≤ N , In [Fn ] cannot 5 involve derivatives 5 of Fn of order higher than N − 1. Moreover, due to the constraints = 0 and pσ = 0, it follows that the variables σi have to take both values + and −, which means that either (N ; osc) t ≥ 1 or nr ≥ 1 in (D.2) (we recall that r ≥ 1). Hence In [Fn ] cannot involve derivatives of Fn of order higher than N − 2.
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14. Deift, P.A., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Intl. Math. Res. 6, 285–299 (1997) 15. Deift, P.A., Its, A.R., Krasovsky, I., Zhou, X.: The Widom-Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202(1), 26–47 (2007) 16. Deift, P., Its, A.R., Krasovsky, I.: Toeplitz and Hankel determinants with singularities: announcement of results, http://arXiv.org/abs/0809.2420v1[math.FA], 2008 17. Deift, P., Its, A.R., Krasovsky, I.: Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher–Hartwig singularities, http://arXiv.org/abs/0905.0443v1[math.FA], 2009 18. des Cloizeaux, J., Mehta, M.L.: Asymptotic behaviour of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14, 1648–1650 (1973) 19. Dyson, F.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys. 47, 171–183 (1976) 20. Ehrhardt, T.: Dyson’s constant in the asymptotics of the fredholm determinant of the sine kernel. Commun. Math. Phys. 262, 317–341 (2006) 21. Gaudin, M.: Sur la loi limite de l’espacement des valeurs propres d’une matrice aléatoire. Nucl. Phys. 25, 447–458 (1961) 22. Gaudin, M., Mehta, M.L.: On the density of eigenvalues of a random matrix. Nucl. Phys 18, 420–427 (1960) 23. Haldane, F.D.M.: General relation of correlation exponents and spectral properties of one-dimensional Fermi systems: Application to the anisotropic s = 1/2 Heisenberg chain. Phys. Rev. Lett. 45, 1358 (1980) 24. Haldane, F.D.M.: Demonstration of the “Luttinger liquid” character of Bethe-ansatz soluble models of 1-D quantum fluids. Phys. Lett. A 81, 153 (1981) 25. Haldane, F.D.M.: Luttinger liquid theory of one-dimensional quantum fluids: I. Properties of the Luttinger Model and Their Extension to the General Interacting 1D Spinless Fermi Gas. J. Phys. C: Solid State Phys. 14, 2585 (1981) 26. Its, A.R.: Asymptotic behaviour of the solutions to the nonlinear Schrödinger equation, and isomonodromic deformations of systems of linear differential equations. Dokl. Akad. Nauk SSSR 261(1), 14–18 (1981) 27. Its, A.R., Izergin, A.G., Korepin, V.E.: Long-distance asymptotics of temperature correlators of the impenetrable Bose gas. Commun. Math. Phys. 130, 471–488 (1990) 28. Its, A.R., Izergin, A.G., Korepin, V.E.: Temperature correlators of the impenetrable Bose gas as an integrable system. Commun. Math. Phys. 129, 205–222 (1990) 29. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Temperature correlations of quantum spins. Phys. Rev. Lett. 70, 1704 (1993) 30. Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 1003–1037 (1990) 31. Its, A.R., Izergin, A.G., Korepin, V.E., Varguzin, G.G.: Large time and distance asymptotics of the correlator of the impenetrable bosons at finite temperature. Physica D 54, 351 (1991) 32. Its, A.R., Krasovsky, I.: Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump. Contemp. Maths. (AMS series) 458, Providence, RI: Amer. Math. Soc., 2008 pp. 215–247 33. Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica D 1, 80–158 (1980) 34. Kac, M.: Toeplitz matrices, translation kernels and related problem in probability. Duke Math. J. 21, 501–510 (1954) 35. Kitanine, N., Kozlowski, K.K., Maillet, J.-M., Slavnov, N.A., Terras, V.: Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions. J. Stat. Mech. (2009) P04003, doi:10.1088/17425468/2009/04/R04003 36. Korepin, V.E., Slavnov, N.A.: The time dependent correlation function of an impenetrable Bose gas as a Fredholm minor. I. Commun. Math. Phys. 129(1), 103–113 (1990) 37. Kozlowski, K.K.: Truncated Wiener-Hopf operators with Fischer-Hartwig singularities. http://arXiv.org/ abs/0805.3902v1[math.FA], 2008 38. Krasovsky, I.V.: Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. 2004, 1249–1272 (2004) 39. Kuijlaars, A.B.J., McLaughlin, K.T.-R., Van Assche, W., Vanlessen, M.: The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]. Adv. in Math. 188, 337–398 (2004) 40. Lenard, A.: Momentum distribution in the ground state of the one-dimensional system of impenetrable bosons. J. Math. Phys. 5, 930–943 (1964) 41. Lenard, A.: One-dimensional impenetrable bosons in thermal equilibrium. J. Math. Phys. 7, 1268– 1272 (1966) 42. McCoy, B.M., Tang, S.: Connection formulae for Painlevé V functions. II. The delta function Bose gas problem. Physica D 20, 187–216 (1986)
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43. McCoy, B.M., Perk, J.H.H., Shrock, R.E.: Time-dependent correlation functions of the transverse Ising chain at the critical magnetic field. Nucl. Phys. B 220, 35–47 (1983) 44. Widom, H.: The strong Szegö limit theorem for circular arcs. Indiana Univ. Math. J. 21, 277–283 (1971) 45. Widom, H.: The asymptotics of a continuous analogue of orthogonal polynomials. J. Approx. Th. 77, 51–64 (1994) 46. Widom, H.: Asymptotics for the Fredholm determinant of the Sine Kernel on a Union of Intervals. Commun. Math. Phys. 171, 159–180 (1995) Communicated by L. Takhtajan
Commun. Math. Phys. 291, 763–798 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0800-x
Communications in
Mathematical Physics
Normal Forms for Semilinear Quantum Harmonic Oscillators Benoît Grébert, Rafik Imekraz, Éric Paturel Laboratoire de Mathématiques Jean Leray UMR 6629, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 3, France. E-mail: [email protected]; [email protected]; [email protected] Received: 27 October 2008 / Accepted: 19 January 2009 Published online: 18 April 2009 – © Springer-Verlag 2009
Abstract: We consider the semilinear harmonic oscillator ¯ iψt = (− + |x|2 + M)ψ + ∂2 g(ψ, ψ), x ∈ Rd , t ∈ R, where M is a Hermite multiplier and g a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on M related to the non resonance of the linear part, this normal form is integrable when d = 1 and gives rise to simple (in particular bounded) dynamics when d ≥ 2. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions. Résumé (Formes normales de Birkhoff pour l’oscillateur harmonique quantique non linéaire): Dans cet article nous considérons l’oscillateur harmonique semi-linéaire: ¯ iψt = (− + x 2 + M)ψ + ∂2 g(ψ, ψ), x ∈ Rd , t ∈ R, où M est un multiplicateur de Hermite et g est une fonction régulière globalement d’ordre au moins trois. Nous montrons qu’une telle équation admet, au voisinage de zéro, une forme normale de Birkhoff à n’importe quel ordre et que, sous des hypothèses génériques sur M liées à la non résonance de la partie linéaire, cette forme normale est complètement intégrable si d = 1 et donne lieu à une dynamique simple (et en particulier bornée) pour d ≥ 2. Ce résultat nous permet de démontrer l’existence presque globale et de contrôler les normes de Sobolev d’indice grand des solutions de l’équation non linéaire ci-dessus avec donnée initiale petite.
764
B. Grébert, R. Imekraz, E. Paturel
1. Introduction, Statement of the Results The aim of this paper is to prove a Birkhoff normal form theorem for the semilinear harmonic oscillator equation ¯ iψt = (− + |x|2 + M)ψ + ∂2 g(ψ, ψ) (1.1) ψ|t=0 = ψ0 on the whole space Rd (d ≥ 1) and to discuss its dynamical consequences. Here g is a smooth function, globally of order p ≥ 3 at 0, and ∂2 g denotes the partial derivative of g with respect to the second variable. The linear operator M is a Hermite multiplier. To define it precisely (at least in the case d = 1, see Sect. 3.2 for the multidimensional case), let us introduce the quantum harmonic oscillator on Rd , denoted by T = − + |x|2 . When d = 1, T is diagonal in the Hermite basis (φ j ) j∈N¯ : ¯ T φ j = (2 j − 1)φ j , j ∈ N, Hn (x) −x 2 /2 e φn+1 = √ , n ∈ N, 2n n! ¯ denotes N \ {0} and where Hn (x) is the n th Hermite polynomial relative to the where N 2 weight e−x : √ 2 e−x Hm (x)Hn (x)d x = 2n n! πδnm . R
In this basis (and for d = 1), a Hermite multiplier is an operator given by Mφ j = m j φ j , where (m j ) j∈N¯ is a bounded sequence of real numbers, that will be chosen in the following classes: for any k ≥ 1, we define the class Wk = {(m j ) j∈N¯ | for each j, m j =
m˜ j with m˜ j ∈ [−1/2, 1/2]} jk
that we endow with the product probability measure. In this context the linear frequencies, i.e. the eigenvalues of T + M = −d 2 /d x 2 + x 2 + M are given by ωj = 2 j − 1 + m j = 2 j − 1 +
m˜ j , jk
¯ j ∈ N.
(1.2)
Let H˜ s = { f ∈ H s (Rd , C)|x → x α ∂ β f ∈ L 2 (Rd ) for any α, β ∈ Nd satisfying 0 ≤ |α| + |β| ≤ s},
(1.3)
where H s (Rd , C) is the standard Sobolev space on Rd . We note that, for any s ≥ 0, the domain of T s/2 is H˜ s (see for instance [Hel84] Prop. 1.6.6) and that for s > d/2, H˜ s is an algebra. If ψ0 ∈ H˜ s is small, say of norm , local existence theory implies that (1.1) admits a unique solution in H˜ s defined on an interval of length c − p+2 . Our goal is to prove
Normal Forms for Semilinear Quantum Harmonic Oscillators
765
that for M outside an exceptional subset, given any integer r ≥ 1 and provided that s is large enough and is small enough, the solution extends over an interval of length c −r . Furthermore we control the norm of the solution in H˜ s -norm (d ≥ 1) and localize the solution in the neighborhood of a torus (only in the case d = 1, cf. Theorem 3.4 and Theorem 3.10). Precisely we have Theorem 1.1. Let r, k ∈ N be arbitrary integers. There exists a set Fk ⊂ Wk whose measure equals 1 such that if m = (m j ) j∈N¯ ∈ Fk and if g is a C ∞ function on a neighborhood of the origin in C2 , satisfying g(z, z¯ ) ∈ R and vanishing at least at order 3 at the origin, there is s0 ∈ N such that for any s ≥ s0 , f there are 0 > 0, c > 0, such that for any ∈ (0, 0 ), for any ψ0 in H˜ s with ||ψ0 ||s ≤ , the Cauchy problem (1.1) with initial datum ψ0 has a unique solution ψ ∈ C 1 ((−T , T ), H˜ s ) with T ≥ c −r . Moreover, for any t ∈ (−T , T ), one has ψ(t, ·) H˜ s ≤ 2 .
(1.4)
¯ = λ 2 |ψ| p+1 with p ≥ 1 and without Hermite mulFor the nonlinearity g(ψ, ψ) p+1 tiplier (M = 0), we recover the Gross-Pitaevskii equation iψt = (− + |x|2 )ψ + λ|ψ| p−1 ψ, t ∈ R, x ∈ Rd .
(1.5)
In this case, the global existence in the energy space H˜ 1 has been proved for1 1 ≤ d+2 p < (d−2) + without smallness assumption on the Cauchy data in the defocusing case (λ < 0) and for small Cauchy data in the focusing case (λ > 0) (see [Car02] and also [Zha05]). But nothing is known for nonlinearities of higher order, neither about conservation of the H˜ s -norm for s > 1. Our result states that, avoiding resonances by adding a generic linear term Mψ (but M = 0 is not allowed), we recover almost global existence for solutions of the Gross-Pitaevskii equation with a nonlinearity of arbitrary high order and small Cauchy data in H˜ s for s large enough. In some sense, this shows that the instability for Gross-Pitaevskii that could appear in that regime are necessarily produced by resonances. More precisely, we can compare with the semi-classical cubic Gross-Pitaevskii in R3 which appears in the study of Bose-Einstein condensates (for a physical presentation see [PS03]) i hu t = −h 2 u + |x|2 u + h 2 |u|2 u, t ∈ R, x ∈ R3 ,
(1.6)
where h is a small parameter. The scaling relation between the ψ solution of the (1.5) and the u solution of (1.6) is given by 1 x u(t, x) = √ ψ(t, √ ). h h We note that for multi indices α, β ∈ N3 , with y = α β 2 y ∂ ψ 2
L (R 3 )
1 We use the convention
√x , h
2 = h |β|−|α|−1/2 x α ∂ β u L 2 (R3 ) .
d+2 = +∞ for d = 1, 2, and (d − 2)+ = d − 2 for d ≥ 3. (d−2)+
(1.7)
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Thus the smallness of ψ0 in H˜ s imposed in Theorem 1.1, i.e. ψ0 H˜ s ≤ C , reads |β|+|α|≤s
2 h |β|−|α|−1/2 x α ∂ β u 0 L 2 (R3 ) ≤ C 2 .
Taking = h 1/6 with h small enough, this allows the derivatives of order greater than 1 to have large L 2 -norm when h is small: β 2 ∂ u 0 2
L (R 3 )
= O(h −|β|+5/6 ),
i.e. the initial data has to be small in L 2 but may have large oscillations. Then, Theorem 1.1 states that, avoiding the resonances by adding a generic linear term (which, in the preceding scaling, stays of order h), the same estimates remain true for the solution u(t, .) with |t| = O(h −r/6 ), r being chosen arbitrarily from the principle. Notice that the role of the linear operator M is to remove the resonances between the free modes (see (1.2)). The fully resonant case M = 0 is beyond the scope of the paper. To prove Theorem 1.1 we use the Birkhoff normal form theory. This technique has been developed by Bourgain [Bou96], Bambusi [Bam03], Bambusi-Grébert [BG06] for semilinear PDEs (typically semilinear Schrödinger equation or semilinear wave equation) on the one dimensional torus and by Bambusi-Delort-Grébert-Szeftel [BDGS07] for the semilinear Klein-Gordon equation on the sphere S d (or a Zoll manifold). These cases were concerned with compact domains. In our work the domain is Rd , the potential x 2 guarantees that the spectrum remains pure point, but the free modes of the harmonic oscillator are not so well localized. For general reference on Hamiltonian PDEs and their perturbations, see the recent monographs [Cra00,Kuk00,Bou05,KP03]. We also note that in [Kuk93], a KAM-like theorem is proved for (1.1) in one dimension and with special nonlinearities. Let us describe roughly the general method. Consider a Hamiltonian system whose Hamiltonian function decomposes in a quadratic part, H0 (associated to the linear part of the equation), and a perturbative nonlinear part P (at least cubic): H = H0 + P. We assume that H0 is diagonal in a Hilbert basis (φ j ) j≥1 of the phase space P : H0 = j ω j ξ j η j for (ξ, η) ∈ P and ω = (ω j ) j≥1 is the vector of free frequencies (the eigenvalues of the linear part). In the harmonic oscillator case, the Hilbert basis is given by the Hermite functions and P = 2 × 2 . The heuristic idea could be summarized as follows: if the free modes do not interact linearly (i.e. if ω is non resonant), and if they do not interact too much via the nonlinear term, then the system will remain close to an integrable one, up to a nonlinear term of very high order, and thus the solutions will exist and stay under control during a very long time. More precisely, by a Birkhoff normal form approach we prove (cf. Theorem 2.23 which is our main theorem) that H ∼ H0 + P , where H0 is no more quadratic but remains integrable (in the case d = 1) and P is at least of order r , where r can be chosen arbitrarily large as soon as we work in a sufficiently small neighborhood of the origin. To guarantee the second condition, i.e. that the free modes do not interact too much via the nonlinear term, we have to control the integral of the product of three or more modes: aj = φ j1 (x) · · · φ jk (x)d x, (1.8) D
Normal Forms for Semilinear Quantum Harmonic Oscillators
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where D is the space domain (Rd in our case) and j is a multi-index in Nk , k being smaller than the fixed order r and larger than 3. It turns out that, in our case, this control cannot be as good as in the cases of compact domains studied previously. Let us consider ordered multi-indices j, i.e. such that j1 ≥ j2 ≥ · · · ≥ jk . In [BDGS07,Gré07,Bam07] the following control was used: there exists ν > 0, and for any N ≥ 1 there exists C N > 0 such that for all ordered j, N j3 |a j | ≤ C N j3ν . (1.9) j3 + j1 − j2 In the case of the harmonic oscillator, this estimate is false (cf. [Wan08] where an equivalent is computed for four modes) and we are only able to prove the following: there exists ν > 0, and for any N ≥ 1 there exists C N > 0 such that for all ordered j, √ N j3ν j2 j3 . (1.10) |a j | ≤ C N 1/24 √ j2 j3 + j1 − j2 j1 The difference could seem minimal but it is technically important: j3 µ ∼ C j3 for an uniform constant C providing µ > 1 and similarly j1 j3 + j1 − j2 √ j2 j3 µ √ √ ∼ C j2 j3 for µ > 1. In the first case, the extra term j3 can be j1 j j +j −j 2 3
1
2
absorbed by changing the value of ν in (1.9) (ν = ν + 1). This is not possible in the second case. In some sense the perturbative nonlinearity is no longer short range (cf. [Wan08]). Actually in the case studied in [Bou96,Bam03,BG06], the linear modes (i.e. the eigenfunctions of the linear part) are localized around the exponentials eikx , i.e. the eigenfunctions of the Laplacian on the torus. In particular the product of eigenfunctions is close to another eigenfunction which makes the control of (1.8) simpler. In the harmonic oscillator case, the eigenfunctions are not localized and the product of eigenfunctions has more complicated properties. Notice that, in the case of the semilinear Klein-Gordon equation on the sphere, the control of (1.8) is more complicated to obtain, but an estimate of type (1.9) is proved in [DS04] for the Klein-Gordon equation on Zoll manifolds. From the point of view of a normal form, the substitution of (1.9) by (1.10) has the following consequence: Consider a formal polynomial Q(ξ, η) ≡ Q(z) =
k
a j z j1 . . . z jl
l=0 j∈Nl
with coefficients a j satisfying (1.9). In [Gré07] or [Bam07], it is proved that its Hamiltonian vector field X Q is then regular from2 Ps = 2s × 2s to Ps for all s large enough (depending on ν). In our present case, i.e. if a j only satisfy (1.10), which defines the class T ν , then we prove that X Q is regular from Ps to Ps for all s < s − 1/2 + 1/24 and s large enough. This “loss of regularity” would of course complicate an iterative procedure, but it is bypassed in the following way: the nonlinearity P is regular in the sense that X P maps Ps to Ps continuously for s large enough (essentially because the space H˜ s is an algebra for s > d/2). On the other hand, we build at each step a 2 here l 2 = {(z ) | l 2s |z |2 < ∞} and corresponds to functions ψ = z φ in H˜2s . l l l l s
768
B. Grébert, R. Imekraz, E. Paturel
canonical transform which preserves the regularity. Indeed, at each iteration, we compute the canonical transformation as the time 1 flow of a Hamiltonian χ , and the solution of the so called homological equation gives rise to an extra term in (1.10) for the coefficient of the polynomial χ : √ N j3ν j2 j3 |a j | ≤ C N 1/24 . (1.11) √ j2 j3 + j1 − j2 j1 (1 + j1 − j2 ) Using such an estimate on the coefficients (in the class3 denoted T ν,+ in Sec. 2.2), we prove in Proposition 2.13 that X χ is regular from Ps to Ps for all s large enough. Furthermore, we prove in Proposition 2.18 that the Poisson bracket of a polynomial in T ν with a polynomial in T ν,+ is in T ν for some ν larger than ν. So an iterative procedure is possible in Ps . This smoothing effect of the homological equations was already used by S. Kuksin in [Kuk87] (see also [Kuk93,Pös96]). Notice that this is, in some sense, similar to the local smoothing property for Schrödinger equations with potentials superquadratic at infinity studied in [YZ04]. Our article is organized as follows: in Sect. 2 we state and prove a specific Birkhoff normal form theorem adapted to the loss of regularity that we explained above. In Sect. 3, we apply this theorem to the 1−d semilinear harmonic oscillator equation (Subsect. 3.1) and we generalize it to cover the multidimensional case (Subsect. 3.2). 2. The Birkhoff Normal Form 2.1. The abstract model. To begin with, we give an abstract model of infinite dimensional Hamiltonian system. In Sect. 3 we will verify that the nonlinear harmonic oscillator ¯ = N\{0} can be described in this abstract framework. Throughout the paper, we denote N 2 2 ¯ and Z = Z \ {0}. We work in the phase space Ps ≡ Ps (C) := s (C) × s (C), where, ¯ 2s 2 for s ∈ R+ , 2s (C) := {(a j ) j≥1 ∈ CN | j≥1 j |a j | < +∞} is a Hilbert space for 2 2s 2 ¯ the standard norm: a s = j≥1 | j| |a j | . We denote Ps (R) := {(ξ, ξ ) ∈ Ps (C)} the “real” part of Ps (C). We shall denote a general point of Ps by z = (ξ, η) with z = (z j ) j∈Z¯ , ξ = (ξ j ) j∈N¯ , η = (η j ) j∈N¯ and the correspondence: z j = ξ j , z − j = η j ¯ Finally, for a Hamiltonian function H , the Hamiltonian vector field X H for all j ∈ N. is defined by ∂H ∂H . − , X H (z) = ∂ξk k∈N¯ ∂ηk k∈N¯ Definition 2.1. Let s ≥ 0, we denote by Hs the space of Hamiltonian functions H defined on a neighborhood U of the origin in Ps ≡ Ps (C), satisfying H (ξ, ξ¯ ) ∈ R (we say that H is real) and H ∈ C ∞ (U, C) and X H ∈ C ∞ (U, Ps ), as well as every homogeneous polynomial Hk appearing in the Taylor expansion of H at 0: Hk ∈ C ∞ (U, C) and X Hk ∈ C ∞ (U, Ps ). 3 Actually in Sect. 2.2, instead of T ν and T ν,+ , we consider more general classes T ν,β and T ν,β,+ , where the parameter β plays the role of the exponent 1/24 in (1.10) and (1.11).
Normal Forms for Semilinear Quantum Harmonic Oscillators
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Remark 2.2. This property, for Hamiltonians contributing to the nonlinearity, will in particular force them to be semilinear perturbations of the harmonic oscillator. In particular the Hamiltonian vector fields of functions F, G in Hs are in 2s (C) × and we can define their Poisson bracket by
2s (C)
{F, G} = i
∂ F ∂G ∂ F ∂G − . ∂ξ j ∂η j ∂η j ∂ξ j j≥1
Notice that since for P ∈ Hs , the vector field X P is a C ∞ function from a neighborhood of Ps to Ps we have Lemma 2.3. Let P ∈ Hs such that P vanishes up to order r + 1 at the origin, that is: ¯ k, ∀k ≤ r + 1, ∀ j ∈ Z
∂k P (0) = 0. ∂z j1 . . . ∂z jk
Then there exists ε0 > 0 and C > 0 such that, for z ∈ Ps satisfying ||z||s ≤ ε0 , we have ||X P (z)||s ≤ C||z||rs . Our model of integrable system is the harmonic oscillator H0 = ωjξjηj, j≥1 ¯
where ω = (ω j ) j≥1 ∈ RN is the frequency vector. We will assume that these frequencies ¯ grow at most polynomially, i.e. that there exist C > 0 and d¯ ≥ 0 such that for any j ∈ N, ¯
|ω j | ≤ C| j|d ,
(2.1)
in such a way that H0 be well defined on Ps for s large enough. The perturbation term is a real function, P ∈ Hs , having a zero of order at least 3 at the origin. Our Hamiltonian function is then given by H = H0 + P and Hamilton’s canonical equations read
∂P ξ˙ j = −iω j ξ j − i ∂η , j ≥1 j ∂P , j ≥ 1. η˙ j = iω j η j + i ∂ξ j
(2.2)
Our theorem will require essentially two hypotheses: one on the perturbation P (see Definition 2.6) and one on the frequency vector ω that we describe now. ¯ k with k ≥ 3, we define µ( j) as the third largest integer among | j1 |, . . . , | jk |. For j ∈ Z Then we set S( j) := | ji1 | − | ji2 |, where | ji1 | and | ji2 | are respectively the largest integer and the second largest integer among | j1 |, . . . , | jk |. In particular, if the multi-index j is ordered, i.e. if | j1 | ≥ . . . ≥ | jk |, then µ( j) := | j3 | and S( j) = | j1 | − | j2 |. In [Bam03,BG06,Gré07,Bam07] the non resonance condition on ω reads
770
B. Grébert, R. Imekraz, E. Paturel ¯
¯ there are Definition 2.4. A frequency vector ω ∈ RN is non-resonant if for any r ∈ N, ¯ r and any 1 ≤ i ≤ r , one has γ > 0 and δ > 0 such that for any j ∈ N ω j + · · · + ω j − ω j − · · · − ω j ≥ γ (2.3) r 1 i i+1 µ( j)δ except in the case { j1 , . . . , ji } = { ji+1 , . . . , jr }. In the harmonic oscillator case4 , we are able to work with a slightly refined non-resonance condition. ¯ ¯ Definition 2.5. A frequency vector ω ∈ RN is strongly non-resonant if for any r ∈ N, ¯ r and any 1 ≤ i ≤ r , one has there are γ > 0 and δ > 0 such that for any j ∈ N ω j + · · · + ω j − ω j − · · · − ω j ≥ γ 1 + S( j) (2.4) r 1 i i+1 µ( j)δ except if { j1 , . . . , ji } = { ji+1 , . . . , jr }. This improvement of the non-resonance condition is similar to the modification to the standard second Melnikov condition introduced first by S. Kuksin in [Kuk87] (see also [Kuk93 and Pös96]. ¯ k with k ≥ 3, we have already defined µ( j) and 2.2. Polynomial structure. For j ∈ Z S( j), we now introduce B( j) = | ji2 ji3 |1/2 , C( j) = | ji1 |, where | ji1 |, | ji2 | and | ji3 | are respectively the first, the second and the third largest integer among | j1 |, . . . , | jk |. We also define A( j) =
B( j) . B( j) + S( j)
(2.5)
In particular, if the multi-index j is ordered, i.e. if | j1 | ≥ . . . ≥ | jk |, then A( j) =
| j2 j3 |1/2 | j2 j3 |1/2 + | j1 | − | j2 |
and C( j) = | j1 |. Definition 2.6. Let k ≥ 3, β ∈ (0, +∞) and ν ∈ [0, +∞) and let a j z j1 . . . z jl Q(ξ, η) ≡ Q(z) =
(2.6)
¯k j∈Z ν,β
be a formal homogeneous polynomial of degree k on Ps (C). Q is in the class Tk ¯ k, any N ≥ 1 there exists a constant c N > 0 such that for all j ∈ Z |a j | ≤ c N
µ( j)ν A( j) N . C( j)β
if for
(2.7)
We will also use 4 The following holds, more generally, if the frequency vector is non-resonant as in Definition 2.4 and satisfies the asymptotic: ωl ∼ l n with n ≥ 1.
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771
Definition 2.7. Let k ≥ 3, β ∈ [0, +∞) and ν ∈ [0, +∞) and let Q(ξ, η) ≡ Q(z) = a j z j1 . . . z jl ¯k j∈Z ν,β,+
be a formal homogeneous polynomial of degree k on Ps (C). Q is in the class Tk for any N ≥ 1 there exists a constant c N > 0 such that for all j ∈ Z¯ k , |a j | ≤ c N
µ( j)ν A( j) N . C( j)β (1 + S( j))
if
(2.8) ν,β
The best constants c N in (2.7) define a family of semi-norms for which Tk Fréchet space.
is a
Remark 2.8. Notice that the formula (2.6) does not give a unique representation of polynomials on Ps . However, since the estimates (2.7) and (2.8) are symmetric with respect to the order of the indexes j1 , . . . , jk , this non-uniqueness does not affect Definitions 2.6 and 2.7. Remark 2.9. In the estimate (2.7), the numerator allows an increasing behaviour with respect to µ( j) that will be useful to control the small divisors. The denominator imposes a slightly decreasing behaviour with respect to the largest index C( j) and a highly decreasing behaviour for monomials having their two modes of largest indexes that are ν,β,+ not of the same order. This control is slightly better in Tk . ν,β
⊂ Hs for Remark 2.10. We will see in Proposition 2.13 that, if β > 1/2, then Tk s ≥ ν + 1. Unfortunately β is not that large in the harmonic oscillator case, where the ν,β best we obtain is β = 1/24. Thus P ∈ Tk does not imply P ∈ Hs . Nevertheless, ν,β as we will see in Proposition 2.13, a polynomial in Tk is well defined and continuous on a neighborhood of the origin in Ps (C) for s large enough. As a comparison, in [Gré07,Bam07], our estimate (2.7) is replaced with |a j | ≤ C N
µ( j) N +ν , (µ( j) + S( j)) N
(2.9)
which is actually better than (2.7), since it implies the Hs regularity. This type of control on the coefficients a j was first introduced in [DS04] in the context of multilinear forms. Definition 2.11. Let ν ≥ 0 and β ≥ 0. A function P is in the class T ν,β if – there exists s0 ≥ 0 such that, for any s ≥ s0 there exists Us , a neighborhood of the origin in Ps such that P ∈ C ∞ (Us , C). – P has a zero of order at least 3 in 0. k T ν,β . – for each k ≥ 3 the Taylor expansion of degree k of P at zero belongs to ⊗l=3 l We now define the class of polynomials in normal form: Definition 2.12. Let k = 2m be an even integer. A formal homogeneous polynomial Z of degree k on Ps is in normal form if it reads Z (z) = b j z j1 z − j1 . . . z jm z − jm , (2.10) ¯m j∈N
i.e. Z depends only on the actions Il := zl z −l = ξl ηl .
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B. Grébert, R. Imekraz, E. Paturel
The aim of the Birkhoff normal form theorem is to reduce a given Hamiltonian of the form H0 + P with P in Hs to a Hamiltonian of the form Z + R, where Z is in normal form and R remains very small, in the sense that it has a zero of high order at the origin. We now review the properties of polynomials in the class T ν,β . ¯ ν ∈ [0, +∞), β ∈ [0, +∞), s ∈ R with s > ν + 1, and let Proposition 2.13. Let k ∈ N, ν,β P ∈ Tk+1 . Then (i) P extends as a continuous polynomial on Ps (C) and there exists a constant C > 0 such that for all z ∈ Ps (C), |P(z)| ≤ C z k+1 s . (ii) For any s < s + β − 21 , the Hamiltonian vector field X P extends as a bounded function from Ps (C) to Ps (C). Furthermore, for any s0 ∈ (ν +1, s], there is C > 0 such that for any z ∈ Ps (C), X P (z) s ≤ C z s z (k−1) . s0
(2.11)
ν,β,+
(iii) Assume moreover that P ∈ Tk+1 with β > 0, then the Hamiltonian vector field X P extends as a bounded function from Ps (C) to Ps (C). Furthermore, for any s0 ∈ (ν + 1, s], there is C > 0 such that for any z ∈ Ps (C), X P (z) s ≤ C z s z (k−1) . s0
(2.12)
ν,β
(iv) Assume finally that P ∈ Tk+1 and P is in normal form in the sense of Definition 2.12, then the Hamiltonian vector field X P extends as a bounded function from Ps (C) to Ps (C). Furthermore, for any s0 ∈ (ν, s], there is C > 0 such that for any z ∈ Ps (C), X P (z) s ≤ C z s z (k−1) . s0
(2.13)
Remark 2.14. Since homogeneous polynomials are their own Taylor expansion at 0, ν,β,+ ν,β assertions (iii) and (iv) imply that every element of Tk+1 , and every element of Tk+1 s in normal form is in H . ν,β
Proof. (i) Let P be an homogeneous polynomial of degree k + 1 in Tk+1 and for z ∈ Ps (C) write P(z) =
¯ k+1 j∈Z
a j z j1 . . . z jk+1 .
(2.14)
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773
One has, using first (2.7) and then that A( j) ≤ 1, C( j) ≥ 1, k+1 µ( j)ν N A( j) |z ji | C( j)β
|P(z)| ≤ C
¯ k+1 j∈Z
≤C
¯ k+1 j∈Z
≤C
¯ k+1 j∈Z
⎛ ≤C⎝ ¯ l∈Z
i=1
k+1 µ( j)ν s | ji | |z ji |
k+1 s i=1 | ji | i=1
k+1
k+1
1
s−ν i=1 | ji |
| ji |s |z ji |
i=1
⎞ k+1 2
1 |l|2s−2ν
⎠
z k+1 s
where, in the last inequality, we used k + 1 times the Cauchy-Schwarz inequality. Since s > ν + 1/2, the last sum converges and the first assertion is proved. (ii) We have to estimate the derivative of polynomial P with respect to any of its variables. Because of (2.7), given any N , we get µ( j, l)ν ∂ P A( j, l) N |z j1 | . . . |z jk |, ∂z ≤ C N (k + 1) C( j, l)β l ¯k j∈Z
where the quantities µ( j, l), C( j, l) and A( j, l) are computed for the k + 1-tuple made of j1 , . . . , jk , l. Furthermore ||X P (z)||2s ≤ C
¯ l∈Z
⎞2 |l|s µ( j, l)ν ⎝ A( j, l) N |z j1 | . . . |z jk |⎠ C( j, l)β ⎛
¯k j∈Z
⎛ ⎞2 ⎜ |l|s µ( j, l)ν ⎟ ≤ C(k!)2 A( j, l) N |z j1 | . . . |z jk |⎠ ⎝ C( j, l)β ¯ l∈Z
≤ C ||z||2(k−3) s0
¯k j∈Z >
⎛
⎝
¯ l∈Z
| j1 |≥| j2 |≥| j3 |
⎞2 |l|s µ( j, l)ν A( j, l) N |z j1 ||z j2 ||z j3 |⎠ , C( j, l)β (2.15)
¯ k> denotes the set of ordered k-uples ( j1 , . . . , jk ) such that | j1 | ≥ | j2 | ≥ · · · ≥ where Z | jk |. We used the following result in the last inequality: Lemma 2.15. Given any s ≥ 0, s0 > ¯ j∈Z
1 2
and z ∈ 2s+s0 we have
| j|s |z j | ≤ Cs0 ||z||s+s0 .
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B. Grébert, R. Imekraz, E. Paturel
Proof. This result is a simple consequence of Cauchy-Schwarz inequality: ⎞1 ⎛ 2 1 1 s s+s0 ⎠ ⎝ | j| |z j | = | j| |z j | ≤ ||z||s+s0 . | j|s0 | j|2s0 ¯ j∈Z
¯ j∈Z
¯ j∈Z
Before continuing with the proof of assertion (ii) of Proposition 2.13, we give two technical lemmas which give an estimate of A( j, l). ¯ we have Lemma 2.16. Given any ordered k-tuple j ∈ Z¯ k> and l ∈ Z, |l|A( j, l) ≤ 2| j1 |. Proof. It is straightforward if |l| ≤ 2| j1 |, since A( j, l) ≤ 1. If not, the order is the following: |l| > 2| j1 | > | j1 | ≥ | j2 | and √ √ |l| | j1 j2 | |l| | j1 j2 | ≤ |l|A( j, l) = √ ≤ 2| j1 |, |l|/2 | j1 j2 | + |l| − | j1 | and the lemma is proved.
Lemma 2.17. Given any ordered k-uple j ∈ Z¯ k> and l ∈ Z¯ we have
| 2 |l|+| |j1j2|−| if |l| ≤ | j2 |, √ j2 | ˜ A( j, l) ≤ A( j1 , j2 , l) := |l j | 2 2 √|l j |+|| j |−|l|| if |l| ≥ | j2 |. 2
1
Proof. If |l| > 2| j1 |, A( j, l) reads:
√ | j1 j2 | . A( j, l) = √ | j1 j2 | + |l| − | j1 |
We can write: | j1 j2 | + |l| − | j1 | = |l j2 | + |l| − | j1 | − | j2 |( |l| − | j1 |) |l| − | j1 | = |l j2 | + |l| − | j1 | − | j2 | √ √ |l| + | j1 | |l| − | j1 | ≥ |l j2 | + |l| − | j1 | − | j1 | √ √ |l| + | j1 | √ 2 ≥ |l j2 | + √ (|l| − | j1 |). 2+1 Hence,
√ √ √ | j1 j2 | |l j2 | 1+ 2 ≤ 2√ . A( j, l) ≤ √ √ | j1 j2 | + |l| − | j1 | |l j2 | + |l| − | j1 | 2 If | j2 | ≤ |l| ≤ 2| j1 |, then B( j, l)2 = | j2 | min(|l|, | j1 |) ∈ |l 2j2 | , |l j2 | , therefore A( j, l) ≤
√ |l j2 |
√
|l j2 | . ≤ 2√ √ √ |l j2 | + ||l| − | j1 || 1/ 2 |l j2 | + ||l| − | j1 ||
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775
Finally, if |l| ≤ | j2 | we get
√ |l j2 | | j2 | , A( j, l) = √ ≤2 |l| + | j1 | − | j2 | |l j2 | + | j1 | − | j2 |
and this ends the proof of Lemma 2.17.
To continue with the proof of assertion (ii) of Proposition 2.13, we define 0 < ε < s − s − 21 , and N = s + 1 + ε. In view of (2.15), we may decompose: ||X P (z)||2s ≤ C (2.16) (T1 (l) + T2 (l))2 , ¯ l∈Z
with
T1 (l) =
| j1 |≥| j2 |≥| j3 |,| j2 |>|l|
T2 (l) =
| j1 |≥| j2 |≥| j3 |,| j2 |≤|l|
Since A( j, l) ≤ 1 and N > 2.17: T1 (l) ≤ C
1 2
|l|s |µ( j, l)|ν A( j, l) N |z j1 ||z j2 ||z j3 |, max(| j1 |, |l|)β |l|s |µ( j, l)|ν A( j, l) N |z j1 ||z j2 ||z j3 |. max(| j1 |, |l|)β
+ s + ε, we may estimate T1 (l) using Lemmas 2.16 and
1
1
| j1 |s− 2 −ε |z j1 || j2 |ν+ 2 +ε |z j2 |
˜ j1 , j2 , l) 2 +ε |z j1 ||z j2 ||z j3 | | j1 |s | j2 |ν A(
| j1 |≥| j2 |≥| j3 |,| j2 |>|l|
≤ C||z||s0
| j1 |≥| j2 |,| j2 |>|l|
≤ C||z||s0
1 1
|l| 2 +ε
|l|
1 2 +ε
1
1
||z||s ||z||ν+1+2ε ,
hence T1 (l) is an 2 -sequence, whose 2 -norm is bounded above by C||z||2s0 ||z||s if we assume that s0 > ν + 1 + 2ε. Concerning T2 (l), using Lemmas 2.16 and 2.17, we obtain 1 s ν ˜ N −s |z j1 ||z j2 ||z j3 | T2 (l) ≤ C +β | j1 | | j2 | A( j1 , j2 , l) s−s |l| | j1 |≥| j2 |≥| j3 |,| j2 |≤|l| √ 1+ε ||z||s0 |l j2 | ≤ C s−s +β | j1 |s |z j1 || j2 |ν |z j2 | 1 + || j1 | − |l|| |l| | j1 |≥| j2 |,| j2 |≤|l| ⎞ ⎛ | j1 |s |z j1 | ||z||s0 | j2 |ν+(1+ε)/2 |z j2 |⎠ . ≤ C s−s +β−(1+ε)/2 ⎝ (1 + || j1 | − |l||)1+ε |l| ¯ j2 ∈Z
¯ j1 ∈Z
The last sum in j1 is a convolution product of the 2 -sequence | j1 |s |z j1 | and the 1 -sequence (1+| j1 |)1+ε and thus a 2 -sequence with respect to the index l, whose 1
2 -norm is bounded by ||z||s . Choosing ε > 0 in such a way that s −s +β −(1+ε)/2 > 0, the sequence T2 (l) is in 2 , with a norm bounded by ||T2 || ≤ C||z||s0 ||z||ν+(1+ε)/2 ||z||s ≤ C||z||2s0 ||z||s ,
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with s0 > ν + (1 + ε)/2. Collecting the estimates for T1 and T2 , we obtain the desired inequality. (iii) We define 0 < ε < 1/12 and N = s + 21 + ε. We have, as in (ii), this first estimate ||X P (z)||2s ≤
C||z||2(k−3) s0
⎛
⎝
¯ l∈Z
| j1 |≥| j2 |≥| j3 |
⎞2 |l|s µ( j, l)ν N A( j, l) |z j1 ||z j2 ||z j3 |⎠ . C( j, l)β (1 + S( j, l)) (2.17)
As in (ii), we may also decompose the sum on j1 , j2 and j3 into two pieces, T1+ (l) collecting all the terms with | j2 | > |l|, and T2+ (l) collecting those with | j2 | ≤ |l|. Following (ii), since C( j, l) ≥ 1 and 1 + S( j, l) ≥ 1, we obtain for T1+ : T1+ (l) ≤ C |l|s | j2 |ν A( j, l) N |z j1 ||z j2 ||z j3 | | j1 |≥| j2 |≥| j3 |,| j2 |>|l|
≤C
|l|1/2+ε | j1 |s−1/2−ε | j2 |ν A( j, l) N −(s−1/2−ε) |z j1 ||z j2 ||z j3 |
| j1 |≥| j2 |≥| j3 |,| j2 |>|l|
≤ C||z||s0
˜ j1 , j2 , l) N −(s−1/2−ε) |z j1 ||z j2 | |l|1/2+ε | j1 |s−1/2−ε | j2 |ν A(
| j1 |≥| j2 |,| j2 |>|l|
≤ C||z||s0
|l|s−N | j1 |s−1/2−ε | j2 |ν+N −(s−1/2−ε) |z j1 ||z j2 |
| j1 |≥| j2 |,| j2 |>|l|
≤ C||z||s0 ≤ C||z||s0
1 |l|
1 2 +ε
1 1
|l| 2 +ε
| j1 |s−1/2−ε | j2 |ν+N −(s−1/2−ε) |z j1 ||z j2 |
| j1 |≥| j2 |,| j2 |>|l|
||z||s ||z||ν+1+2ε ,
hence T1+ (l) is a 2 -sequence, whose 2 -norm is bounded above by C||z||2s0 ||z||s if s0 > ν + 1 + 2ε. The estimate on T2+ will need all factors assigned in the definition of T ν,β,+ :
| j1 |s | j2 |ν ˜ j1 , j2 , l) N −s |z j1 ||z j2 ||z j3 | A( max( j1 , l)β (1 + || j1 | − |l||) | j1 |≥| j2 |≥| j3 |,| j2 |≤|l| √ ε |l j2 | 1 | j1 |s |z j1 || j2 |ν |z j2 | ≤ C||z||s0 β 1+|| j1 | − |l|| |l| (1+|| j1 | − |l||)
T2+ (l) ≤ C
| j1 |≥| j2 |,| j2 |≤|l|
≤ C||z||s0
1 |l|β−ε/2
| j2 |ν+ε/2 |z j2 |
¯ j2 ∈Z
¯ j1 ∈Z
| j1 |s |z j1 | . (1 + || j1 | − |l||)1+ε
Once again, the last sum in j1 is a convolution product of the 2 sequence | j1 |s |z j1 | and the 1 sequence (1+| j1 |)1+ε . Choosing ε > 0 in such a way that β − ε/2 > 0, the sequence 1
T2+ (l) is in 2 , with a norm bounded by
||T2 || ≤ C||z||s0 ||z||ν+(1+ε)/2 ||z||s ≤ C||z||2s0 ||z||s ,
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777
with s0 > ν + (1 + ε)/2. Collecting the estimates for T1+ and T2+ , we obtain the stated inequality. (iv) Let k + 1 = 2m. As in (ii), we obtain ||X P ||2s ≤ C
⎛ ⎝
¯ l∈Z
⎞2 ν µ( j, j, l, l) |l|s |zl | |z j ||z − j1 | . . . |z jm−1 ||z − jm−1 |⎠ , C( j, j, l, l)β 1 m−1
¯> j∈N
(2.18) using the same convention for µ( j, j, l, l) and C( j, j, l, l) as for µ( j, l) and C( j, l): as an example, µ( j, j, l, l) is the third biggest integer among | j1 |, | j1 |, . . . | jm−1 |, | jm−1 |, |l| and |l|, that is, if j is ordered, either µ( j, j, l, l) = | j1 |, and in this case C( j, j, l, l) = |l|, or µ( j, j, l, l) = |l| and in this case C( j, j, l, l) = | j1 |. Notice that A( j, j, l, l) = 1 does not help for this computation. The sum over j can be decomposed into two parts :
|l|s |zl |
¯ m−1 j∈N > , j1 ≤l
≤
µ( j, j, l, l)ν |z j ||z − j1 | . . . |z jm−1 ||z − jm−1 | C( j, j, l, l)β 1
|l|s |zl |
¯ m−1 j∈N > , j1 ≤l
≤ |l|s−β |zl |
| j1 |ν |z j1 ||z − j1 | . . . |z jm−1 ||z − jm−1 | |l|β
j1ν |z j1 ||z − j1 |||z||0
2(m−2)
j1
≤ |l|
s−β
2(m−2)
|zl |||z||2ν/2 ||z||0
,
and ¯ m−1 j∈N > , j1 >l
≤ |l|s |zl |
|l|s |zl |
µ( j, j, l, l)ν |z j ||z − j1 | . . . |z jm−1 ||z − jm−1 | C( j, j, l, l)β 1 | j1 |ν−β |z j1 ||z − j1 | . . . |z jm−1 ||z − jm−1 |
¯ m−1 j∈N > 2(m−2)
≤ |l|s |zl |||z||2(ν−β)/2 ||z||0
.
Inserting these two estimates in (2.18) we get (2.12).
ν,β
The second essential property satisfied by polynomials in Tk following
is captured in the
Proposition 2.18. Let k1 , k2 ≥ 2, ν1 , ν2 ≥ 0 and β > 0. The map (P, Q) → {P, Q} ν ,β,+ ν ,β ν ,β defines a continuous map from Tk11+1 × Tk22+1 to Tk1 +k2 for ν = 2(ν1 + ν2 ) + 1. ν ,β,+
Proof. We assume that P ∈ Tk11+1 we write
P(z) =
ν ,β
and Q ∈ Tk22+1 are homogeneous polynomials and ¯ k1 +1 j∈Z
a j z j1 . . . z jk1 +1
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and Q(z) =
¯ k2 +1 i∈Z
bi z i1 . . . z ik2 +1 .
In view of the symmetry of the estimate (2.7) with respect to the involved indices, one easily obtains {P, Q}(z) = c j,i z j1 . . . z jk1 z i1 . . . z ik2 ¯ k1 +k2 ( j,i)∈Z
with |c j,i | ≤ c N ,N
¯ l∈Z
ν2 µ( j, l)ν1 N µ(i, l) A( j, l) A(i, l) N . C( j, l)β (1 + S( j, l)) C(i, l)β
Therefore it remains to prove that, for each M ≥ 1, there exist N , N ≥ 1, C > 0 such that for all j ∈ Z¯ k1 and all i ∈ Z¯ k2 , ¯ l∈Z
ν2 µ( j, l)ν1 µ( j, i)ν N µ(i, l) N A( j, l) A(i, l) ≤ C A( j, i) M C( j, l)β (1 + S( j, l)) C(i, l)β C(i, j)β
(2.19)
with ν = 2(ν1 + ν2 ) + 1. In order to simplify the notations, and because it does not change the estimates of (2.19), we will assume that k1 = k2 = k. We can also assume by symmetry that – all the indices are positive: j1 , . . . , jk , i 1 , . . . , i k ≥ 1, – j and i are ordered: j1 ≥ . . . ≥ jk and i 1 ≥ . . . ≥ i k . We begin with two technical lemmas whose proofs are postponed at the end of this proof. ¯ k2 and l ∈ Z¯ ¯ k1 , i ∈ Z Lemma 2.19. There is a constant C > 0 such that for any j ∈ Z we have A( j, l)2 A(i, l)2 ≤ C A(i, j).
(2.20)
¯ k2 and l ∈ Z¯ ¯ k1 , i ∈ Z Lemma 2.20. There is a constant C > 0 such that for any j ∈ Z we have max(µ( j, l)A(i, l)2 , µ(i, l)A( j, l)2 ) ≤ Cµ(i, j)2 . Using these lemmas, in order to prove (2.19), it suffices to prove ¯ l∈Z
1 C( j, l)β (1 +
A(i, l)2 µ( j, i) ≤C . S( j, l)) C(i, l)β C(i, j)β
Noticing that C(i, l)C( j, l) ≥ C(i, j)l, it suffices to verify that ¯ l∈Z
A(i, l)2 ≤ Cµ( j, i). (1 + S( j, l))l β
(2.21)
Normal Forms for Semilinear Quantum Harmonic Oscillators
Decompose the sum in two parts, I1 = have I1 =
l> j2
l> j2
779
and I2 =
l≤ j2 .
For the first sum we
A(i, l)2 1 ≤ ≤ C, (1 + S( j, l))l β (1 + |l − j1 |)l β ¯ l∈Z
while for the second one I2 =
l≤ j2
A(i, l)2 A(i, l)2 ≤ . (1 + S( j, l))l β lβ l≤ j2
In this last sum, if j2 < µ(i, j), then I2 ≤ j2 ≤ µ(i, j). On the other hand, if µ(i, j) ≤ j2 , then we decompose the I2 sum in two parts, I2,1 = l<2i1 and I2,2 = l≥2i1 . Since i 1 ≤ µ(i, j) = max(i 1 , j3 ) we have I2,1 =
A(i, l)2 ≤ 2i 1 ≤ 2µ(i, j). lβ
l≤2i 1
Finally, when l ≥ 2i S(i, l) ≥ l/2 and B(i, l)2 = i 1 i 2 ≤ i 2 l/2 ≤ µ(i, j)l/2 √1 we have−1/2 and thus A(i, l) ≤ 2µ(i, j)l which leads to I2,2 =
2i 1 ≤l≤ j2
1 A(i, l)2 ≤ Cµ(i, j) ≤ Cµ(i, j). β l l 1+β ¯ l∈N
Proof of Lemma 2.19. The estimate (2.20), being symmetric with respect to i and j, we can assume that j1 ≥ i 1 . We consider three cases, depending on the position of l with respect to i 1 and j1 . First case l ≥ j1 . We have S(i, l) = |i 1 − l| ≥ |i 1 − j1 | ≥ S(i, j) and B(i, l) = (i 1 i 2 )1/2 ≤ B(i, j). Therefore A(i, l) =
B(i, l) B(i, j) B(i, l) ≤ ≤ = A(i, j), B(i, l) + S(i, l) B(i, l) + S(i, j) B(i, j) + S(i, j)
and using A( j, l) ≤ 1, (2.20) is proved. Second case l ≤ i 1 . Similarly as in the first case, we have S( j, l) ≥ S(i, j) and B( j, l) = ( j2 max( j3 , l))1/2 ≤ ( j2 max( j3 , i 1 ))1/2 ≤ B(i, j) and thus A( j, l) ≤ A(i, j). Third case i 1 < l < j1 . That is the most complicated case and we have to distinguish whether i 1 ≥ j2 or not. Subcase 1. i 1 ≥ j2 . We have B(i, l) ≤ B(i, j), thus if S(i, l) = |i 1 − l| ≥ 21 |i 1 − j1 | = 1 1 2 S(i, j) we obtain A( j, l) ≤ 2 A(i, j) and (2.20) holds true. Now if S(i, l) < 2 S(i, j)
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B. Grébert, R. Imekraz, E. Paturel
then S( j, l) ≥ 21 S(i, j) since S(i, l)+S( j, l) ≥ S(i, j). Furthermore, if B( j, l) ≤ B(i, j) then A( j, l) =
B( j, l) B(i, j) B( j, l) ≤2 ≤2 = 2 A(i, j), B( j, l) + S( j, l) B( j, l) + S(i, j) B(i, j) + S(i, j)
and (2.20) holds. If B( j, l) > B(i, j), then using B( j, l)2 = j2 l = j2 i 1 + j2 (l − i 1 ) ≤ B(i, j)2 + j2 S(i, l) ≤ B(i, j)2 +
1 B(i, j)S(i, j), 2
we deduce A( j, l)2 ≤
B( j, l)2 (B(i, j)+ 21 S(i, j))2
≤2
B(i, j)2 + B(i, j)S(i, j) ≤ 2(A(i, j)2 + A(i, j)), (B(i, j) + S(i, j))2
thus (2.20) is also satisfied in this case, since A(i, j) ≤ 1. Subcase 2. i 1 ≤ j2 . We still have B(i, l) ≤ B(i, j), thus if furthermore S(i, j) ≤ 2S(i, l), then A(i, l) ≤ 2 A(i, j) and (2.20) is true. So we assume 2S(i, l) < S(i, j) which implies S(i, j) ≤ 2S( j, l) since S(i, l) + S( j, l) ≥ S(i, j). If furthermore l ≤ j3 , B( j, l) = B( j) ≤ B(i, j) and thus A( j, l) ≤ 2 A(i, j) and (2.20) is again true. So we assume j3 ≤ l and we have B( j, l)2 = l j2 = i 1 j2 + j2 (l − i 1 ) ≤ B(i, j)2 + j2 S(i, l). If S(i, l) ≤ l/2 then we deduce B( j, l)2 ≤ 2B( j, l)2 and (2.20) is satisfied. It remains to consider the case S(i, l) > l/2 which implies i 1 < l/2 and thus A(i, l) ≤ Let n ≥ 1 such that
l 2n+1
≤ i1 ≤
l 2n ,
i1 i1 ≤2 . i 1 + l/2 l
(2.22)
we get from (2.22)
A(i, l) ≤
1 2n−1
.
(2.23)
On the other hand (l j2 )1/2 (l j2 )1/2 ≤ 2 (l j2 )1/2 + S(i, j) (i 1 j2 )1/2 + S(i, j)
(2.24)
1 (i 1 j2 )1/2 (l j2 )1/2 ≥ . (i 1 j2 )1/2 + S(i, j) 2n+1 (i 1 j2 )1/2 + S(i, j)
(2.25)
A( j, l) ≤ 2 and A(i, j) ≥
Combining (2.23), (2.24) and (2.25) we conclude A(i, l)A( j, l) ≤ 8A(i, j).
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781
Proof of Lemma 2.20. The estimate (2.21) being symmetric with respect to i and j, we can assume j1 ≥ i 1 . If furthermore i 1 ≥ j2 then one easily verifies that µ(i, l) ≤ µ(i, j) and µ( j, l) ≤ µ(i, j) and estimate (2.21) is satisfied. In the case j1 ≥ j2 ≥ i 1 we still have µ(i, l) ≤ µ(i, j) but µ( j, l) could be larger than µ(i, j). Actually if µ( j, l) ≤ 2µ(i, j), estimates (2.21) is still trivially satisfied. Therefore it remains to consider the case where µ( j, l) > 2µ(i, j). Remark that in this case i 1 ≤ µ(i, j) ≤ µ(2j,l) ≤ l/2 and thus S(i, l) = |i 1 − l| ≥ l/2 which leads to A(i, l) ≤
(2i 2 )1/2 (i 1 i 2 )1/2 (2µ(i, j))1/2 ≤ ≤ . S(i, l) l 1/2 l 1/2
Using this last estimate one gets µ( j, l)A(i, l)2 ≤ l A(i, l)2 ≤ 2µ(i, j)2 . We end this section with a proposition concerning Lie transforms of homogeneous δ,β,+ , i.e. time 1 flow of the Hamiltonian vector field X χ . polynomials χ ∈ Tl δ,β,+
Proposition 2.21. Let χ be a real homogeneous polynomial in Tl with δ ≥ 0, β > 0, l ≥ 3 take s > s1 := δ + 3/2 and denote by φ the Lie transform associated with χ . We have (i) φ is an analytic canonical transformation from an open ball B of center 0 and radius in Ps into the open ball B2 in Ps satisfying φ(z) − z s ≤ Cs z 2s for any z ∈ Bε .
(2.26)
In particular if F ∈ Hs with s > s1 then F ◦ φ ∈ Hs . Furthermore, if F is real then F ◦ φ is real too. ν,β (ii) Let P ∈ Tn ∩ Hs , ν ≥ 0, n ≥ 3 and fix r ≥ n an integer. Then P ◦ φ = Q r + Rr , where: – Q r is a polynomial of degree at most r , belonging to T ν ,β ∩ Hs with ν = 2r −n ν + (2r −n − 1)(2β + 1), – Rr is a Hamiltonian in T ν ,β ∩ Hs with ν = 2r −n+1 ν + (2r −n+1 − 1)(2β + 1), having a zero of order r + 1 at the origin. δ,β,+
, by Proposition 2.13(iii), X χ ∈ C ∞ (Ps , Ps ) for s > s1 = Proof. (i) Since χ ∈ Tl δ + 3/2. In particular, for s > s1 , the flow t generated by the vector field X χ transports an open neighborhood of the origin in Ps into an open neighborhood of the origin in Ps . Notice that since χ is real, t transports the “real part” of Ps , {(ξ, ξ¯ ) ∈ Ps }, into itself. Furthermore one has for z ∈ Ps small enough t t (z) − z = X χ (t (z))dt , 0
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B. Grébert, R. Imekraz, E. Paturel
and since χ has a zero of order 3 at least, one gets by Proposition 2.13(iii), t t t 2 (z) − z ≤ Cs (z) dt . χ s 0
s
Then, by a classical continuity argument, there exists > 0 such that the flow Bε z → tχ (z) ∈ B2ε is well defined and smooth for 0 ≤ t ≤ 1. Furthermore, the Lie transform φ = 1 satisfies (2.26). On the other hand, by simple composition we get that if F ∈ Hs with s > s1 , then F ◦ φ ∈ C ∞ (B , C). In view of the formula X F◦φ (z) = (Dφ(z))−1 X F (φ(z)), we deduce that X F◦φ ∈ C ∞ (B , Ps ). We now have to check the properties concerning the Taylor polynomials of F ◦ φ. Denoting by Fk (resp. (F ◦ φ)k ) the homogeneous polynomial of degree k appearing in the Taylor expansion of F (resp F ◦ φ), and putting [ j+1] Fk[0] = Fk , Fk = {Fk , χ }, we have [ j] (F ◦ φ)k (z) = Fk (z), j≥0,k ≥0,k + j (l−2)=k
since χ is itself a homogeneous polynomial of degree l. It is then sufficient to prove that the Poisson bracket of a homogeneous polynomial Fk in Hs with χ stays in Hs . Using the (constant) symplectic form ω on Ps , we get {Fk , χ }(z) = ω(X Fk , X χ ), and so {Fk , χ } ∈ C ∞ (B , C). Moreover 1 (X Fk − t∗ (X Fk ))(z). t→0 t
X {Fk ,χ } (z) = [X Fk , X χ ] = lim
Since t is the flow of the regular Hamiltonian χ ∈ C ∞ (B , R), the Cauchy Lipschitz theorem implies that the mapping (t, z) → t∗ (X Fk )(z) is in C ∞ ([−1, 1] × B , Ps ). Now, X {Fk ,χ } is nothing else but the time derivative of this mapping at time 0, hence X {Fk ,χ } ∈ C ∞ (B , Ps ) and the claim is proved. (ii) By a direct calculus one has dk P ◦ t (z) t=0 = P [k] (z) k dt with the same notation P [k+1] = {P [k] , χ } and P [0] = P. Therefore applying Taylor’s formula to P ◦ t (z) between t = 0 and t = 1 we deduce P ◦ φ(z) =
1 r −n 1 [k] 1 P (z) + (1 − t)r P [r −n+1] (t (z))dt. n! (r − n)! 0
(2.27)
k=0
Notice that P [k] (z) is a homogeneous polynomial of degree n + k(l − 2) and, by Proposik k tion 2.18, P [k] (z) ∈ T 2 ν+(2 −1)(2δ+1),β . Moreover P [k] (z) is a homogeneous polynomial in the Taylor expansion of P ◦ φ ∈ Hs , hence it is in Hs . Therefore (2.27) decomposes ν ,β in the sum of a polynomial of degree r in Tr , and a function in Hs having a zero of degree r + 1 at the origin.
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783
2.3. The Birkhoff normal form theorem. We start with the resolution of the homological equation and then state the normal form theorem. Lemma 2.22. Let ν ∈ [0, +∞) and assume that the frequency vector of H0 is strongly non resonant (see Definition 2.5). Let Q be a homogeneous real polynomial of degree k ν,β in Tk , there exist ν > ν, and Z and χ two homogeneous real polynomials of degree ν ,β
k, respectively in Tk
ν ,β,+
and Tk
, which satisfy
{H0 , χ } + Q = Z
(2.28)
{Z , I j } = 0 ∀ j ≥ 1,
(2.29)
and
and thus Z is in normal form. Furthermore, Z and χ both belong to Hs for s > ν + 1. ¯ k1 and l ∈ N ¯ k2 with k1 + k2 = k we denote Proof. For j ∈ N ξ ( j) η(l) = ξ j1 . . . ξ jk1 ηl1 . . . ηlk2 . One has {H0 , ξ ( j) η(l) } = −i( j, l)ξ ( j) η(l) with ( j, l) := ω j1 + . . . + ω jk1 − ωl1 − . . . − ωlk2 . ν,β
Let Q ∈ Tk
, Q=
a jl ξ ( j) η(l) ,
¯k ( j,l)∈N
¯ k means that j ∈ N ¯ k1 and l ∈ N ¯ k2 with k1 + k2 = k. Let us define where ( j, l) ∈ N b jl = i( j, l)−1 a jl , c jl = 0 when { j1 , . . . , jk1 } = {l1 , . . . , lk2 }
(2.30)
c jl = a jl , b jl = 0 when { j1 , . . . , jk1 } = {l1 , . . . , lk2 }.
(2.31)
and
As ω is strongly non-resonant, there exist γ and α such that |( j, l)| ≥ γ
1 + S( j, l) µ( j, l)α
¯ k with { j1 , . . . , jk1 } = {l1 , . . . , lk2 }. Thus, in view of Definitions 2.6 for all ( j, l) ∈ N and 2.7, the polynomial χ= b j,l ξ ( j) η(l) , ¯k ( j,l)∈N
784
B. Grébert, R. Imekraz, E. Paturel ν ,β,+
belongs to Tk
while the polynomial Z= c j,l ξ ( j) η(l) ¯k ( j,l)∈N
ν ,β
belongs to Tk with ν = ν +α. Notice that in this non-resonant case, (2.29) implies that Z depends only on the actions and thus is in normal form. Furthermore by construction they satisfy (2.28) and (2.29). Note that the reality of Q is equivalent to the symmetry relation: a¯ jl = al j . Taking into account that l j = − jl , this symmetry remains satisfied for the polynomials χ and Z . Finally, χ and Z belong to Hs , since they are homogeneous polynomials (they are their own Taylor expansions) and as a consequence of Proposition 2.13 (iii) and (iv) respectively. We can now state the main result of this section: Theorem 2.23. Assume that P is a real Hamiltonian belonging to Hs for all s large enough and to the class T ν,β for some ν ≥ 0 and β > 0. Assume that ω is strongly non resonant (cf. Definition 2.5) and satisfies (2.1) for some d¯ ≥ 0. Then for any r ≥ 3 there exists s0 and for any s ≥ s0 there exists Us , Vs neighborhoods of the origin in Ps and τs : Vs → Us a real analytic canonical transformation which is the restriction to Vs of τ := τs0 and which puts H = H0 + P in normal form up to order r , i.e. H ◦ τ = H0 + Z + R with (a) Z is a real continuous polynomial of degree r with a regular vector field (i.e. Z ∈ Hs ) which only depends on the actions: Z = Z (I ). (b) R ∈ Hs is real and X R (z) s ≤ Cs z rs for all z ∈ Vs . (c) τ is close to the identity: τ (z) − z s ≤ Cs z 2s for all z ∈ Vs . Proof. The proof is close to the proof of the Birkhoff normal form theorem stated in [Gré07 or Bam07]. The main difference has been already pointed out: we have here to check the Hs regularity of the Hamiltonian functions at each step, independently of the fact that they belong to T ν,β (here P ∈ T ν,β does not imply P ∈ Hs ). Having fixed some r ≥ 3, the idea is to construct iteratively for k = 3, . . . , r , a neighborhood Vk of 0 in Ps (s large enough depending on r ), a canonical transformation τk , defined on Vk , an increasing sequence (νk )k=3,...,r of positive numbers and real Hamiltonians Z k , Pk+1 , Q k+2 , Rk such that Hk := H ◦ τk = H0 + Z k + Pk+1 + Q k+2 + Rk ,
(2.32)
satisfying the following properties: (i) Z k is a polynomial of degree k in T νk ,β ∩ Hs having a zero of order 3 (at least) at the origin and Z k depends only on the (new) actions: {Z k , I j } = 0 for all j ≥ 1. νk ,β (ii) Pk+1 is a homogeneous polynomial of degree k + 1 in Tk+1 ∩ Hs . ν ,β s k ∩ H having a zero of order k + 2 (iii) Q k+2 is a polynomial of degree r + 1 in T at the origin. (iv) Rk is a regular Hamiltonian belonging to Hs and having a zero of order r + 2 at the origin.
Normal Forms for Semilinear Quantum Harmonic Oscillators
785
First we fix s > νr + 3/2 to be sure to be able to apply Proposition 2.13 at each step (νr will be defined later on independently of s). Then we notice that (2.32) at order r proves Theorem 2.23 with Z = Z r and R = Pr +1 + Rr (since Q r +2 = 0). Actually, since R = Pr +1 + Rr belongs to Hs and has a zero of order r + 1 at the origin, we can apply Lemma 2.3 to obtain X R (z) s ≤ Cs z rs
(2.33)
on V ⊂ Vr a neighborhood of 0 in Ps . At the initial step (which for convenience we will denote the k = 2 step), the Hamiltonian H = H0 + P has the desired form (2.32) with τ2 = I , ν2 = ν, Z 2 = 0, P3 being the Taylor polynomial of P of degree 3, Q 4 being the Taylor polynomial of P of degree r + 1 minus P3 and R2 = P − P3 − Q 4 . We show now how to go from step k to step k + 1. We look for τk+1 of the form τk ◦ φk+1 , φk+1 being the Lie transform associated to a homogeneous polynomial χk+1 of degree k + 1. We decompose Hk ◦ φk+1 as follows: Hk ◦ φk+1 = H0 + Z k + {H0 , χk+1 } + Pk+1 +H0 ◦ φk+1 − H0 − {H0 , χk+1 } +Z k ◦ φk+1 − Z k +Pk+1 ◦ φk+1 − Pk+1 +Q k+2 ◦ φk+1 +Rk ◦ φk+1 . ν ,β,+
k Using Lemma 2.22 above, we choose χk+1 in Tk+1
(2.34) (2.35) (2.36) (2.37) (2.38) (2.39)
in such a way that
Zˆ k+1 := {H0 , χk+1 } + Pk+1
(2.40)
ν ,β
k . We put then Z k+1 = Z k + Zˆ k+1 , is a homogeneous real polynomial of degree k+1 in Tk+1 which obviously has degree k + 1 and a zero of order 3 (at least) at the origin, and the right-hand side of line (2.34) becomes H0 + Z k+1 . We just recall that νk = νk + α, where α is determined by ω, independently of r and s. By Proposition 2.21, the Lie transform associated to χk+1 is well defined and smooth on a neighborhood Vk+1 ⊂ Vk and, for z ∈ Vk+1 satisfies
φk+1 (z) − z s ≤ C z 2s . Then from Proposition 2.18, Proposition 2.21 and formula (2.27), we find that (2.36), (2.37), (2.38) and (2.39) are regular Hamiltonians having zeros of order k + 2 at the origin. For instance concerning (2.36), one has by the Taylor formula for any z ∈ Vk+1 , 1 Z k ◦ φk+1 (z) − Z k (z) = {Z k , χk+1 }(z) + (1 − t){{Z k , χk+1 }, χk+1 }(tχk+1 (z)) dt, 0
and {Z k , χk+1 } is a polynomial having a zero of order 3 + degree(χk+1 ) − 2 = k + 2 while the integral term is a regular Hamiltonian having a zero of order 2k + 1. Thus if 2k + 1 ≥ r + 2 this last term contributes to Rk+1 and if not, we have to use a Taylor formula at a higher order.
786
B. Grébert, R. Imekraz, E. Paturel
Therefore the sum of (2.36), (2.37), (2.38) and (2.39) decomposes in P˜k+2 + Q˜ k+3 + R˜ k+1 with P˜k+2 , Q˜ k+3 and R˜ k+1 satisfying respectively the properties (ii), (iii) and (iv) at rank k + 1 (with νk+1 = kνk + νk + k + 2). Concerning the term (2.35), one has to proceed differently since H0 does not belong to the Hs . First notice that by the homological equation (2.40) one has {H0 , χk+1 } = Z k+1 − Z k − Pk+1 . By construction Z k and Pk+1 belong to Hs . On the other hand, by ν ,β
k Lemma 2.22, Z k+1 ∈ Tk+1 and is in normal form (i.e. it depends only on the action variables). Thus by Proposition 2.13 assertion (iv), one concludes that Z k+1 ∈ Hs . Therefore we have proved that {H0 , χk+1 } ∈ Hs . Now we use the Taylor formula at order one to get
H0 ◦ φk+1 (z) − H0 (z) = 0
1
{H0 , χk+1 }(tχk+1 (z)) dt.
But we know from the proof of Proposition 2.21 that tχk+1 : Vk+1 → Ps for all t ∈ [0, 1]. Therefore H0 ◦ φk+1 − H0 ∈ Hs and thus (2.35) defines a regular Hamiltonian. Finally we use again the Taylor formula and the homological equation to write H0 ◦ φk+1 (z) − H0 (z) − {H0 , χk+1 }(z) 1 = (1 − t){Z k+1 − Z k − Pk+1 , χk+1 }(tχk+1 (z)) dt 0
ν ,β
ν ,β,+
k k and , since Z k+1 − Z k − Pk+1 belongs to Tk+1 and χk+1 ∈ Tk+1 , we conclude by ν ,β k+1 . Finally we use ProposiProposition 2.18 that H0 ◦ φk+1 − H0 − {H0 , χk+1 } ∈ T tion 2.21 to decompose it in Pˆk+2 + Qˆ k+3 + Rˆ k+1 with Pˆk+2 , Qˆ k+3 and Rˆ k+1 satisfying respectively the properties (ii), (iii) and (iv) at rank k + 1. The proof is achieved defining Pk+2 = Pˆk+2 + P˜k+2 , Q k+3 = Qˆ k+3 + Q˜ k+3 and Rk+1 = Rˆ k+1 + R˜ k+1 .
3. Dynamical Consequences 3.1. Nonlinear harmonic oscillator in one dimension. We recall the notations of the 2 introduction. The quantum harmonic oscillator T = − ddx 2 + x 2 is diagonalized in the Hermite basis (φ j ) j∈N¯ : ¯ T φ j = (2 j − 1)φ j , j ∈ N, Hn (x) −x 2 /2 e φn+1 = √ , n ∈ N, 2n n! where Hn (x) is the n th Hermite polynomial relative to the weight e−x : √ 2 e−x Hm (x)Hn (x)d x = 2n n! πδnm . 2
R
In this basis, the Hermite multiplier is given by Mφ j = m j φ j ,
(3.1)
Normal Forms for Semilinear Quantum Harmonic Oscillators
787
where (m j ) j∈N¯ is a bounded sequence of real number. For any k ≥ 1, we define the class Wk = {(m j ) j∈N¯ | for each j, m j =
m˜ j with m˜ j ∈ [−1/2, 1/2]} jk
(3.2)
that we endow with the product probability measure. In this context the frequencies, i.e. the eigenvalues of T + M = −d 2 /d x 2 + x 2 + M are given by ωj = 2 j − 1 + m j = 2 j − 1 +
m˜ j , jk
¯ j ∈ N.
Proposition 3.1. There exists a set Fk ⊂ Wk whose measure equals 1 such that if m = (m j ) j∈N¯ ∈ Fk then the frequency vector (ω j ) j≥1 is strongly non-resonant (cf. Definition 2.5). Proof. First remark that it suffices to prove that the frequency vector (ω j ) j≥1 is non resonant in the sense of Definition 2.4. Actually, if we prove that (2.3) is satisfied for given constants δ and γ , then if S( j) < r µ( j), ω j + · · · + ω j − ω j − · · · − ω j ≥ r 1 i i+1 and thus (2.4) is satisfied with δ = δ + 1 and γ =
γ γ 1 + S( j) ≥ , r + 1 µ( j)δ +1 µ( j)δ γ r +1 .
Now if S( j) ≥ r µ( j) then use
ω j + · · · + ω j − ω j − · · · − ω j ≥ S( j) − (r − 2)µ( j), r 1 i i+1
(3.3)
to conclude that ω j + · · · + ω j − ω j − · · · − ω j ≥ 2 S( j) ≥ γ r 1 i i+1 r r +1
1 + S( j) , µ( j)δ +1
provided γ is small enough. The proof that there exists a set Fk ⊂ Wk whose measure equals 1 such that if m = (m j ) j∈N¯ ∈ Fk then the frequency vector (ω j ) j≥1 is non-resonant is exactly the same as the proof of Theorem 5.7 in [Gré07]. So we do not repeat it here (see also [BG06]). In Eq. (1.1) with d = 1, the Hamiltonian perturbation reads g(ξ(x), η(x))d x, P(ξ, η) = R
(3.4)
where g ∈ C ∞ (C2 , C), ξ(x) = j≥1 ξ j φ j (x), η(x) = j≥1 η j φ j (x) and ((ξ j ) j≥1 , (η j ) j≥1 ) ∈ Ps . We first check that P belongs to Hs for s large enough. Lemma 3.2. Let P be given by (3.4) with g ∈ C ∞ (U, C), U being a neighborhood of 0 in C2 , g real i.e. g(z, z¯ ) ∈ R and g having a zero of order at least 3 at the origin. Then P ∈ Hs for all s > 1/2.
788
B. Grébert, R. Imekraz, E. Paturel
Proof. One computes
∂P (ξ, η) = ∂1 g(ξ(x), η(x))φ j (x)d x ∂ξ j R
and
∂P (ξ, η) = ∂2 g(ξ(x), η(x))φ j (x)d x. ∂η j R In the same way, we have ∂ l+r P (ξ, η) ∂ξ j1 . . . ∂ξ jl ∂ηk1 . . . ∂ηkr ∂1l ∂2r g(ξ(x), η(x))φ j1 (x) . . . φ jl (x)φk1 (x) . . . φkr (x)d x. = R
(3.5)
Since g is a C ∞ function, all these partial derivatives are continuous from Ps to C, and the corresponding differentials (ξ, η) → Dl+r P(ξ, η) are continuous from Ps to the space of l + r -linear forms on Ps . We get moreover 2 2 2s X P (ξ, η) s = | j| ∂1 g(ξ(x), η(x))φ j (x)d x j≥1
R
2 2s + | j| ∂2 g(ξ(x), η(x))φ j (x)d x . j≥1
R
Therefore, to check that z → X P (z) is a regular function from a neighborhood of the origin in Ps into Ps , it suffices to check that the functions x → ∂1 g(ξ(x), η(x)) and x → ∂2 g(ξ(x), η(x)) are in H˜ s provided ξ(x) and η(x) are in H˜ s . So it remains to prove that functions of the type x → |x|i ∂1l+1 ∂2m g(ξ(x), η(x))(ξ (l1 ) (x))α1 · (ξ (lk1 ) (x))αk1 (η(m 1 ) (x))β1 · · · (η(m k2 ) (x))βk2 are in L 2 (R) for all 0 ≤ i + l + m ≤ s, 0 ≤ i + l j ≤ s, 0 ≤ i + m j ≤ s. But this is true because – g is a C ∞ function, ξ and η are bounded functions and thus x → ∂1l+1 ∂2m g(ξ(x), η(x)) is bounded. – H˜ s is an algebra for s > 1/2 and thus x → |x|k ξ (l) (x)η(m) (x) ∈ L 2 (R) for all 0 ≤ k + l + m ≤ s. – |∂1 g(ξ(x), η(x))|, |∂2 g(ξ(x), η(x))| ≤ C(|ξ(x)| + |η(x)|)2 for some uniform constant C > 0 and thus x → |x|k ∂1 g(ξ(x), η(x)) ∈ L 2 (R) for all 0 ≤ k ≤ s. There remains to prove the same properties concerning the Taylor homogeneous polynomial Pm of P at any order m, computed at (0, 0). From (3.5), we get 1 Pm = ∂1l ∂2r g(0, 0) ξ j1 φ j1 (x) . . . ξ jl φ jl (x)ηk1 φk1 (x) . . . ηkr φkr (x)d x, m! R l+r =m
j,k
hence Pm can be computed directly from formula (3.4), replacing g by its Taylor homogeneous polynomial gm of order m: 1 l r gm (ξ(x), η(x)) = ∂1 ∂2 g(0, 0)ξ(x)l η(x)r , m! l+r =m
and this gives the statement, since gm satisfies the same properties as g.
Normal Forms for Semilinear Quantum Harmonic Oscillators
789
The fact that P belongs to the class T ν,β is directly related to the distribution of the φ j ’s. Actually we have 1 . For each k ≥ 1 and for each N ≥ 0 Proposition 3.3. Let ν > 1/8 and 0 ≤ β ≤ 24 k ¯ there exists c N > 0 such that for all j ∈ N , ν φ j . . . φ j d x ≤ c N µ( j) A( j) N . (3.6) 1 k C( j)β R
As a consequence, any P of the general form (3.4) is in the class T ν . The proof will be done in the multidimensional case in the next section (cf. Proposition 3.6). We can now apply our Theorem 2.23 to obtain Theorem 3.4. Assume that M ∈ Fm defined in Proposition 3.1 and that g ∈ C ∞ (C2 , C) is real, i.e. g(z, z¯ ) ∈ R and has a zero of order at least 3 at the origin. For any r ≥ 3 there exists s0 (r ) an integer such that for any s ≥ s0 (r ), there exist ε0 > 0 and C > 0 such that if ψ0 H˜ 2s = ε < ε0 the equation ¯ iψt = (− + x 2 + M)ψ + ∂2 g(ψ, ψ), x ∈ R, t ∈ R
(3.7)
with Cauchy data ψ0 has a unique solution ψ ∈ C 1 ((−T , T ), H˜ 2s ) with T ≥ C −r .
(3.8)
Moreover (i) ψ(t, ·) H˜ 2s ≤ 2 for any t ∈ (−T , T ). (ii) j≥1 j 2s ||ξ j (t)|2 − |ξ j (0)|2 | ≤ ε3 for any t ∈ (−T , T ), where |ξ j (t)|2 , j ≥ 1 are the actions of ψ(t, ·) = ξ j (t)φ j . (iii) There exists a torus T0 ⊂ H˜ 2s such that, dist2s (ψ(t, ·), T0 ) ≤ Cεr1 /2
for |t| ≤ −r2 ,
where r1 + r2 = r + 3 and dist2s denotes the distance on H˜ 2s associated with the norm · H˜ 2s . Proof. Let ψ0 = ξ j (0)φ j and denote z 0 = (ξ(0), ξ¯ (0)). Notice that if ψ0 ∈ H˜ 2s with ψ0 H˜ 2s = then z 0 ∈ Ps and z 0 s = . Denote by z(t) the solution of the Cauchy problem z˙ = X H (z), z(0) = z 0 , where H = H0 + P is the Hamiltonian function associated to Eq. (3.7) written in the Hermite decomposition ψ(t) = ξ j (t)φ j , z(t) = (ξ(t), ξ¯ (t)). We note that, since P is real, z remains a real point of Ps for all t and that ψ(t) H˜ 2s = z(t) s. Then we denote by z = τ −1 (z), where τ : Vs → Us is the transformation given by Theorem 2.23 (so that z denotes the normalized coordinates) associated to the order r + 2 and s ≥ s0 (r + 2) given by the same theorem. We note that, since the transformation τ is generated by a real Hamiltonian, z (t) is still a real point. Let ε0 > 0 be such that B2ε0 ⊂ Vs and take 0 < ε < ε0 . We assume that z(0) s = ψ0 H˜ s = ε. For z = (ξ, η) ∈ Ps we define
790
B. Grébert, R. Imekraz, E. Paturel
N (z) := 2
∞
j 2s I j (ξ, η),
j=1
where we recall that I j (ξ, η) = ξ j η j . We notice that for a real point z = (ξ, ξ¯ ) ∈ Ps , N (z) = z 2s . Thus in particular we have5 N (z(t)) = z(t) 2s
2 and N (z (t)) = z (t)s .
Using that Z depends only on the normalized actions, we have N˙ (z ) = {N , H ◦ τ } ◦ τ −1 (z) = {N , R}(z ).
(3.9)
Therefore as far as z(t) s ≤ 2ε, and thus z(t) ∈ Vs , by assertion (c) of Theorem 2.23, z (t) ≤ Cε and using (3.9) and assertion (b) of Theorem 2.23 (at order r + 2) we get s t N (z (t)) − N (z (0)) ≤ {N , R}(z (t ))dt ≤ Ct z (t)r +3 ≤ Ctεr +3 . s 0
In particular, as far as z(t) s ≤ 2ε and |t| ≤ Cε−r , N (z (t)) − N (z (0)) ≤ Cε3 . Therefore using again assertion (c) of Theorem 2.23, we obtain |N (z(t)) − N (z(0))| ≤ Cε3 which, choosing ε0 small enough, leads to z(t) s ≤ 3/2 ε as long as z(t) s ≤ 2ε and |t| ≤ Cε−r . Thus (3.8) and assertions (i) follow by a continuity argument. To prove assertion (ii) we recall the notation I j (z) = I j (ξ, η) = ξ j η j for the actions associated to z = (ξ, η). Using that Z depends only on the actions, we have {I j ◦ τ −1 , H }(z) = {I j , H ◦ τ } ◦ τ −1 (z) = {I j , R}(z ). Therefore, we get in the normalized coordinates d ∂R ∂R I j (ξ , η ) = −iξ j + iη j , dt ∂η j ∂ξ j and thus d ∂R ∂ R j 2s I j (ξ , ξ¯ ) = j 2s −ξ j + ξ¯ j dt ∂η ∂ξ j
j
j
j
⎛
⎞1/2 ⎛ ⎞ 2 2 1/2 ∂ R ∂ R + ⎠ , ≤⎝ j 2s (|ξ j |2 +|ξ j |2 )⎠ ⎝ j 2s ∂η j ∂ξ j j
j
5 That is precisely at this point that we need to work with real Hamiltonians. The Birkhoff normal form theorem is essentially algebraic and does hold for complex Hamiltonians.
Normal Forms for Semilinear Quantum Harmonic Oscillators
which leads to j
j
2s
d I j (z ) ≤ z X R (z ) ≤ z r +3 . dt s s s
Thus, recalling that I j (ξ , ξ¯ ) = |ξ j |2 , we get j 2s |ξ j (t)|2 − |ξ j (0)|2 ≤ ε3 for any |t| ≤ Cε−r .
791
(3.10)
(3.11)
j≥1
On the other hand, using (i) and assertion (c) of Theorem 2.23, for any |t| ≤ Cε−r , one has j 2s |ξ j (t)|2 ) − |ξ j (t)|2 ≤ j 2s (|ξ j (t)| + |ξ j (t)|) ξ j (t) − ξ j (t) ≤ Cε3 . j≥1
j≥1
Combining this last relation with (3.11), assertion (ii) follows. To prove (iii), let I¯j = I j (0) be the initial actions in the normalized coordinates and define the smooth torus 0 := z ∈ Ps : I j (z) = I¯j , j ≥ 1 and its image in H˜ s , T0 = {u ∈ H˜ s : u =
ξ j φ j with τ (ξ, ξ¯ ) ∈ 0 }.
We have ⎡
ds (z(t), T0 ) ≤ ⎣
⎤1/2 2 j 2s I j (t) − I¯j ⎦ ,
j
where ds denotes the distance in Ps associated to · s . Notice that for a, b ≥ 0, √ √ a − b ≤ |a − b|. Thus, using (3.10), we get [ds (z(t), T0 )]2 ≤
j 2s |I j (t) − I j (0)|
j
≤ |t|
j 2s | I˙ j (t)|
j
1 z X R (z ) s s r 1
1 ≤ C r r +3 ≤ C r +3−r1 ,
1 ≤
which gives (ii).
(3.12)
792
B. Grébert, R. Imekraz, E. Paturel
3.2. Multidimensional nonlinear harmonic oscillator. 3.2.1. Model The spectrum of the d-dimensional harmonic oscillator T = − + |x|2 = − + x12 + · · · + xd2 is the sum of d-copies of the odd integers set, i.e. the spectrum of T equals Nd with 2N \ {0, 2, · · · , d − 2} if d is even Nd = (3.13) 2N + 1 \ {1, 3, · · · , d − 2} if d is odd. For j ∈ Nd we denote the associated eigenspace E j which dimension is d j = {(i 1 , · · · , i d ) ∈ (2N + 1)d | i 1 + · · · + i d = j}. We denote { j,l , l = 1, · · · , d j }, the basis of E j obtained by the d-tensor product of Hermite functions: j,l = φi1 ⊗ · · · φid with i 1 + · · · + i d = j. The Hermite multiplier M is defined on the basis ( j,l ) j∈Nd ,l=1,··· ,d j of L 2 (Rd ) by M j,l = m j,l j,l ,
(3.14)
where (m j,l ) j∈Nd ,l=1,··· ,d j is a bounded sequence of real numbers. The linear part of (1.1) reads H0 = − + x 2 + M. H0 is still diagonalized by ( j,l ) j∈Nd ,l=1,··· ,d j and the spectrum of H0 is σ (H0 ) = { j + m j,l | j ∈ Nd , l = 1, · · · , d j }.
(3.15)
For simplicity, we will focus on the case m j,l = m j for all l = 1, · · · , d j . In this case we have σ (H0 ) = { j + m j | j ∈ Nd } and, as a consequence of Proposition 3.1, Proposition 3.5. There exists a set Fk ⊂ Wk whose measure equals 1 such that if m = (m j ) j∈N¯ ∈ Fk , then the frequency vector (ω j,i ) j∈Nd ,i=1,··· ,d j satisfies the fol¯ there are γ > 0 and δ > 0 such that for any j ∈ Nd r , any lowing: for any r ∈ N, l ∈ {1, · · · , d j1 } × · · · × {1, · · · , d jr } and any 1 ≤ i ≤ r , one has ω j
1 ,l1
1 + S( j) + · · · + ω ji ,li − ω ji+1 ,li+1 − · · · − ω jr ,lr ≥ γ µ( j)δ
(3.16)
except if { j1 , . . . , ji } = { ji+1 , . . . , jr }. Concerning the product of eigenfunctions we have, Proposition 3.6. Let ν > d/8. For any k ≥ 1 and any N ≥ 1 there exists c N > 0 such that for any j ∈ Nd k , any l ∈ {1, · · · , d j1 } × · · · × {1, · · · , d jk }, ν ≤ c N µ( j) A( j) N . . . . d x (3.17) j ,l j ,l k k 1 d 1 1 R C( j) 24 Notice that this condition does not distinguish between modes having the same energy.
Normal Forms for Semilinear Quantum Harmonic Oscillators
793
Proof. We use the approach developed in [Bam07], Sect. 6.2. The basic idea lies in the following commutator lemma: Let A be a linear operator which maps D(T k ) into itself and define the sequence of operators A N := [T, A N −1 ],
A0 := A,
then ([Bam07], Lemma 7) for any j1 = j2 in Nd , any 0 ≤ l1 ≤ d1 , 0 ≤ l2 ≤ d2 and any N ≥ 0, |A j2 ,l2 , j1 ,l1 | ≤
1 |A N j2 ,l2 , j1 ,l1 |. | j1 − j2 | N
Let A be the operator given by the multiplication by the function = j3 ,l3 · · · jk ,lk , then by an induction argument Cα,N D α , AN = 0≤|α|≤N
where
Cα,N =
Vα,β,N (x)D β ,
0≤|β|≤2N −|α|
and Vα,β,N are polynomials of degree less than 2N − |α| − |β|. Therefore one gets 1 (3.18) d j1 ,l1 . . . jk ,lk d x ≤ | j − j | N A N j2 ,l2 L 2 R
1
≤C ≤C
2
1 | j1 − j2 | N
||Vα,β,N D β φ D α j2 ,l2 || L 2
0≤|α|≤N 0≤|β|≤2N −|α|
1 | j1 − j2 | N
j
2 ,l2
|α|
||||ν0 +2N −|α| ,
0≤|α|≤N
where we used in the last estimate (in this proof, f s = f H˜ s (Rd ) = |x|s f L 2 (Rd ) + f H s (Rd ) and f H s (Rd ) is the standard Sobolev norm) ∀ν0 > d/2
|| f g|| L 2 ≤ Cν0 || f ||ν0 ||g|| L 2 .
We now estimate ||||s . First notice that, since T j,l = j j,l , one has for all s ≥ 0, j,l ≤ C j s/2 . (3.19) s Then we recall that the Hermite eigenfunctions are uniformly bounded, and in fact (see [Sze75 or KT05]) ||φ j || L ∞ ≤ C j −1/12 ,
(3.20)
and thus, since j,l = φi1 ⊗ · · · φid with i 1 + · · · + i d = j, we deduce || j,l || L ∞ ≤ Cd j −1/12
(3.21)
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with Cd = Cd 1/12 . Thus, using the tame estimates for the standard Sobolev norms (see for instance [Tay91]), we get uv s ≤ C( u s v L ∞ + v s u L ∞ ), combined with (3.19) and (3.21), we get for j3 ≥ · · · ≥ jk , s/2
s ≤ C j3 . Inserting (3.19) and (3.22) in (3.18) we get 1 d j1 ,l1 . . . jk ,lk d x ≤ C | j − j | N 1 2 R ≤C
1 | j1 − j2 | N
(3.22)
0≤|α|≤N 0≤|β|≤2N −|α|
|α|/2 ν0 /2+N −|α|/2 j3
j2
0≤|α|≤N
1 N +ν0 /2 ≤C j | j1 − j2 | N 3 =C
|α|/2 (ν0 +|β|)/2) j3
j2
j2 j3
N /2
1 ν /2 j 0 ( j2 j3 ) N /2 . | j1 − j2 | N 3
√ Now remark that if j2 j3 ≤ | j1 − j2 | then the last estimate implies N /2 ≤ C j ν0 /2 √ ( j2 j3 ) . . . d x = Cµ( j)ν/2 A( j) N , jk ,lk 3 d j1 ,l1 ( j2 j3 + | j1 − j2 |) N R (3.23) √ while if j2 j3 > | j1 − j2 | then A( j) ≥ 1/2 and thus (3.23) is trivially true. On the other hand, using (3.21) one has −1/12 j1 ,l1 . . . jk ,lk d x ≤ C j1 = C C( j)−1/12 . Rd
Combining this estimate with (3.23) one gets for all N ≥ 1, ν ≤ c N µ( j) A( j) N . . . d x jk ,lk 1 d j1 ,l1 R C( j) 24 with ν =
ν0 4.
3.2.2. Result We first generalize the normal form theorem to a context adapted to the multidimensional case. We follow the presentation of Sect. 2 and only focus on the new features. Let s ≥ 0; we consider the phase space Qs = Ls × Ls with Ls = {(a j,l ) j∈Nd , 1≤l≤d j |
j∈Nd
| j|
2s
dj l=1
|a j,l |2 < ∞}
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that we endow with the standard norm and the standard symplectic structure as for Ps in Sect. 2.1. Writing ψ = ξ j,l j,l , ψ¯ = η j,l j,l with (ξ, η) ∈ Qs , we note that ψ ∈ H˜ 2s if and only if ξ ∈ Ls . The linear part of the multidimensional version of the linear part of (1.1) reads H0 (ξ, η) =
dj 1 ω j,l ξ j,l η j,l . 2 j∈Nd l=1
For j ≥ 1, we define J j (ξ, η) =
dj
ξ j,l η j,l .
l=1
Using notations of Sect. 2.1, we define the class Tνk of homogeneous polynomials of degree k on Qs , d
d
j1
Q(ξ, η) ≡ Q(z) =
j∈Nd k l1 =1
...
jk
a j,l z j1 ,l1 . . . z jk ,lk
lm =1
such that for each N ≥ 1, there exists a constant C > 0 such that for all j, l, |a j,l | ≤ C
µ( j)ν A( j) N . C( j)1/24
Then, following Definition 2.11 we define a corresponding class Tν of C ∞ Hamiltonians on Qs having their Taylor polynomials in Tνk . Similarly, following Definition 2.1, we also define Hds the class of real Hamiltonians P satisfying P, Pk ∈ C ∞ (Us , C) and X P , X Pk ∈ C ∞ (Us , Qs ) for some Us ⊂ Qs a neighborhood of the origin and for all k ≥ 1 (as before Pk denotes the Taylor polynomial of P of degree k). In Eq. (1.1), the Hamiltonian perturbation reads P(ξ, η) = g(ξ(x), η(x))d x, (3.24) Rd
where g is C ∞ on a neighborhood of 0 in C2 , ξ(x) = j≥1 ξ j φ j (x), η(x) = η φ (x) and ((ξ ) , (η ) ) ∈ P . As in the one dimensional case (cf. j j≥1 j j≥1 s j≥1 j j Lemma 3.2), P belongs to Hds for s large enough (s > d/2) and using Proposition 3.6, P belongs to the class Tν . Therefore one has Lemma 3.7. Let P given by (3.24) with g smooth, real and having a zero of order at least 3 at the origin. Then P ∈ Hs ∩ Tν for all s > d/2 and for ν > d/8. We also need a d-dimensional definition of normal form homogeneous polynomial: Definition 3.8. Let k = 2m be an even integer; a formal polynomial Z homogeneous of degree k on Qs is in normal form if it reads Z (ξ, η) =
j∈Nd
for all (ξ, η) ∈ Qs .
d
d
j1 k
l1 ,l1 =1
...
jk
lk ,lk =1
a j,l,l ξ j1 ,l1 η j1 ,l1 . . . ξ jk ,lk η jk ,lk
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One easily verifies that if Z is in normal form then Z commutes with each d j ξ j,l η j,l since for instance J j = l=1 {ξ j1 ,l1 η j1 ,l1 , ξ j1 ,l1 η j1 ,l1 + ξ j1 ,l1 η j1 ,l1 } = 0. Modifying slightly the proof of Theorem 2.23 we get Theorem 3.9. Let P be a real Hamiltonian belonging in Tν ∩ Hds for some ν ≥ 0 and for all s sufficiently large and let ω be a weakly non-resonant frequency vector in the sense of (3.16). Then for any r ≥ 3 there exists s0 and for any s ≥ s0 there exists U, V neighborhoods of the origin in Qs and τ : V → U a real analytic canonical transformation which puts H = H0 + P in normal form up to order r , i.e. H ◦ τ = H0 + Z + R, with (i) Z is a real continuous polynomial of degree r which belongs to Hds and which is in normal form in the sense of Definition 3.8. In particular Z commutes with all J j , j ≥ 1, i.e. {Z , J j } = 0 for all j ≥ 1. (ii) R is real and belongs to Hds , furthermore X R (z) s ≤ Cs z rs for all z ∈ Vs . (iii) τ is close to the identity: τ (z) − z s ≤ Cs z 2s for all z ∈ Vs . Proof. The only new point when comparing with Theorem 2.23, is that in assertion (ii) we obtain {Z , J j } = 0 for all j ≥ 1 instead of {Z , I j } = 0 for all j ≥ 1. Actually, in view of (3.16), we adapt Lemma 2.22, and in particular (2.30) and (2.31), in such a way χ ∈ Tν,+ and Z is in normal form in the sense of Definition 3.8. On the other hand, we also verify, following the lines of the proof of assertion (iv) of Proposition 2.13, that a homogeneous polynomial of degree k +1 in normal form Z ∈ Tν satisfies X Z (z) s ≤ C z ks for all z in a neighborhood of the origin. In particular, if Z ∈ Tν is in normal form, it automatically belongs to Hds (this point was crucial in the proof of Theorem 2.23). Notice that the normal form H0 + Z is no longer, in general, integrable. The dynamical consequences are the same as in Theorem 3.4 (i) and (ii) but we have to replace I j by J j in the second assertion. Actually the J j play the rôle of almost actions: they are almost conserved quantities. Theorem 3.10. Assume that m ∈ Fk defined in Proposition 3.5 and that g is C ∞ on a neighborhood of 0 in C2 , g is real, i.e. g(z, z¯ ) ∈ R and g vanishes at least at order 3 at the origin. For each r ≥ 3 and s ≥ s0 (r ), there exists ε0 > 0 and c > 0 such that for any ψ0 in H˜ s , any ∈ (0, 0 ), the equation ¯ iψt = (− + x 2 + M)ψ + ∂2 g(ψ, ψ), x ∈ Rd , t ∈ R with Cauchy data ψ0 has a unique solution ψ ∈ C 1 ((−T , T ), H˜ s ) with T ≥ c −r . Moreover for any t ∈ (−T , T ), one has ψ(t, ·) H˜ s ≤ 2
Normal Forms for Semilinear Quantum Harmonic Oscillators
and
797
j s |J j (t) − J j (0)| ≤ ε3 ,
j≥1
where J j (t) = j,l (·).
d j
l=1 |ξ j,l |
2,
j ≥ 1 are the “pseudo-actions” of ψ(t, ·) =
j,l
ξ j,l (t)
Proof. Just remark that as in the proof of Theorem 3.4, defining N (z) := 2 j∈Nd j s d j J j (ξ, η) = 2 j∈Nd j s l=1 ξ j,l η j,l , one has N (z) = z 2s/2 for all real points z = (ξ, ξ¯ ). On the other hand, using that Z commutes with J j , we have {N ◦ τ −1 , H }(z) = {N , H ◦ τ } ◦ τ −1 (z) = {N , R}(z ). Therefore, in the normalized variables, we have the estimate | N˙ | ≤ C N (r +1)/2 and the theorem follows as in the proof of Theorem 3.4. Acknowledgements. It is a great pleasure to thank Dario Bambusi and Didier Robert for many helpful discussions. We thank both referees for useful suggestions.
References [Bam03] [Bam07] [BDGS07] [BG06] [Bou96] [Bou05] [Car02] [Cra00] [DS04] [Gré07] [Hel84] [KP03] [KT05] [Kuk87] [Kuk93] [Kuk00] [PS03]
Bambusi, D.: Birkhoff normal form for some nonlinear pdes. Commun. Math. Physics 234, 253–283 (2003) Bambusi, D.: A birkhoff normal form theorem for some semilinear pdes. In: Hamiltonian Dynamical Systems and Applications, Berlin-Heidelberg-New York: Springer, 2007, pp. 213– 247 Bambusi, D., Delort, J.-M., Grébert, B., Szeftel, J.: Almost global existence for hamiltonian semilinear klein-gordon equations with small cauchy data on zoll manifolds. Comm. Pure Appl. Math. 60(11), 1665–1690 (2007) Bambusi, D., Grébert, B.: Birkhoff normal form for pdes with tame modulus. Duke Math. J. 135, 507–567 (2006) Bourgain, J.: Construction of approximative and almost-periodic solutions of perturbed linear schrödinger and wave equations. Geom. Func. Anal. 6, 201–230 (1996) Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Ann. Math. Stud., Vol. 158, Princeton, NJ: Princeton University Press, 2005 Carles, R.: Remarks on nonlinear schrödinger equations with harmonic potential. Ann. Henri Poincaré 3(4), 757–772 (2002) Craig, W.: Problèmes de petits diviseurs dans les équations aux dérivées partielles. Panoramas et Synthéses, no. 9, Paris: Société Mathématique de France, 2000 Delort, J.M., Szeftel, J.: Long–time existence for small data nonlinear klein–gordon equations on tori and spheres. Internat. Math. Res. Notices 37, 1897–1966 (2004) Grébert, B.: Birkhoff normal form and Hamiltonian PDEs. In: Partial Differential Equations and Applications, Sémin. Congr., Vol. 15, Paris: Soc. Math. France, 2007, pp. 1–46 Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques. Astérisque, Vol. 112, Paris: Société Mathématique de France, 1984, with an English summary Kappeler, T., Pöschel, J.: KAM & KdV, Berlin-Heidelberg-New York: Springer, 2003 Koch, H., Tataru, D.: L p eigenfunction bounds for the hermite operator. Duke Math. J. 128(2), 369–392 (2005) Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21, 192–205 (1987) Kuksin, S.B.: Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Berlin: SpringerVerlag, 1993 Kuksin, S.B.: Analysis of Hamiltonian PDEs. Oxford: Oxford University Press, 2000 Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation. International Series of Monographs on Physics, Vol. 116, Oxford: The Clarendon Press/Oxford University Press, 2003
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Pöschel, J.: A kam-theorem for some nonlinear pdes. Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV Ser. 15, 23, 119–148 (1996) Szeg˝o, G.: Orthogonal Polynomials. Fourth ed. American Mathematical Society, Colloquium Publications, Vol. XXIII, Providence, RI: Amer. Math. Soc., 1975 Taylor, M.E.: Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics, Vol. 100, Boston, MA: Birkhäuser Boston Inc., 1991 Wang, W.-M.: Pure point spectrum of the floquet hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations. Commun. Math. Physics 277, 459–496 (2008) Yajima, K., Zhang, G.: Local smoothing property and strichartz inequality for schrödinger equations with potentials superquadratic at infinity. J. Diff. Eqs. 202(1), 81–110 (2004) Zhang, J.: Sharp threshold for blowup and global existence in nonlinear schrödinger equations under a harmonic potential. Comm. Part. Diff. Eqs. 30(10–12), 1429–1443 (2005)
Communicated by G. Gallavotti
Commun. Math. Phys. 291, 799–812 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0838-9
Communications in
Mathematical Physics
Non-Abelian Vortices on Compact Riemann Surfaces J. M. Baptista Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands. E-mail: [email protected] Received: 29 October 2008 / Accepted: 6 February 2009 Published online: 19 May 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We consider the vortex equations for a U (n) gauge field A coupled to a Higgs field φ with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface , their moduli space of solutions is closely related to a moduli space of τ -stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix φ, we show that the vortex solutions are entirely characterized by the location in of the zeros of det φ and by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.
1. Introduction 1.1. The context. The simplest and earliest-known type of vortex equations appeared in the classical abelian Higgs model. It involves just one U (1) gauge field and one complex scalar Higgs field. The solutions to these equations are generally known as Nielsen-Olesen vortices [1], and the moduli space of such solutions was first described by Taubes for vortices living in the complex plane [13], and by Bradlow for vortices living in a compact Kähler manifold [6]. After that several far-reaching generalizations of the vortex equations have been studied. In one of these the gauge group can be any compact connected Lie group and the Higgs field can have values in any Kähler manifold equipped with a hamiltonian action of the group. Among these generalizations, perhaps the simplest ones to deal with and explicitly describe solutions, are the ones that have a torus as the gauge group and a Higgs field with values on a toric manifold [4]. These are abelian generalizations, of course. The non-abelian generalizations, in their turn, are more difficult to analyse, but at the same time seem to possess a richer structure and present the greatest number of novel features, as for instance the presence of vortex internal spaces. A great deal of effort has therefore been dedicated to studying various types of non-abelian vortex equations, and we suggest for example [7,14] for succinct
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reviews — one from a mathematical and one from a physical perspective — and for references to the many original articles. In this paper our intent is to focus on what is probably the simplest non-abelian model for vortices: that with one U (n) gauge field and one linear Higgs field having values on the space of complex n × n square matrices. We will study these vortex equations when they are defined on compact Riemann surfaces of large volume and will give an explicit and rigorous description of their moduli space of solutions. Since an apparent feature of the literature is that the existent rigorous accounts of non-abelian vortex moduli spaces seem to be rather hard to be made explicit, and vice-versa, we thought that this study could be of some interest. The data that we need to start with are a compact Riemann surface and a complex vector bundle E → of rank n over that surface. Recall that, as a C ∞ vector bundle, E is completely characterized by its degree d, which we assume to be positive. Fixing an hermitian metric h on E, the variables of our field theory are then the unitary connections A on the bundle and the sections φ of the direct sum ⊕n E → of n copies of E. Observe that locally φ can be regarded as a function on with values on the complex n × n matrices. The fact that these matrices are square introduces significant simplifications to the problem, for one can take inverses and determinants at will, just as in the n = 1 abelian case; it is nevertheless possible to go a long way in studying the same non-abelian model with a different number of copies of E [5,8]. The energy functional of the model is the natural Yang-Mills-Higgs functional 1 e2 2 A 2 E(A, φ) = |φ φ † − τ 1|2 , |F | + |d φ| + (1) A 2 2 2e where FA is the curvature of the connection, d A φ is the covariant derivative and e and τ are positive real parameters of the theory. (Here the hermitian conjugate φ † is of course defined with respect to the hermitian metric h on E, so that in a unitary trivialization of E it is represented by the hermitian conjugate matrix of φ.) As is well known a Bogomolny-type argument then shows that this energy is minimized by the fields (A, φ) that solve the vortex equations ∂¯ A φ = 0, ∗FA − ie2 (φ φ † − τ 1) = 0,
(2)
where ∂¯ A φ is the anti-holomorphic part of the covariant derivative and ∗ is the Hodge operator on . If these equations have any solutions at all, then the energy functional at this minimum will have the value E(A, φ) = 2π τ d. The vortex equations as written in (2) were first studied in [5], where they were related to the problem of finding τ -stable holomorphic n-pairs on . That paper was part of a much wider effort to analyse various types of non-abelian vortex equations and, through the use of Hitchin-Kobayashi correspondences, relate them to various types of stability conditions for vector bundles equipped with sections [7]. A second wave of interest, this time in the physics literature, came after the articles [2,11] arrived independently at the same non-abelian equations. The first one gave a brane-theoretical description of the vortex moduli spaces; the second was concerned with their applications to confinement in QCD. There was then a sequence of articles providing alternative and more direct constructions of these moduli spaces, other related non-abelian moduli spaces and studying their physical implications (closer to the perspective of this paper, see for example [9,3,10], but otherwise also the many others
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referred to in [14]). These constructions were carried out mostly for vortices in the complex plane = C, and while this choice introduces several topological simplifications, it also prevents the direct use of the Hitchin-Kobayashi correspondences of [5,7], for these have so far been proved only for compact . This means that for = C the moduli space constructions in the physical literature are not yet completely rigorous, though they certainly are very useful and almost surely true. In the present paper we introduce a novel way to characterize non-abelian vortex solutions — in terms of the locations and internal structures of the zeros of det φ — which is applicable to both compact and non-compact . In the compact case we can then make use of the correspondences of [5,7] to confidently describe the vortex moduli spaces. 1.2. The main result. We now describe the main result of the paper. As a first point, observe that associated to E there is the natural C ∞ complex line-bundle det E → . This determinant bundle also has degree d, its transition functions are the determinants of the transition functions of E, and a complex structure on E induces a complex structure on det E. Now, if (A, φ) is a solution of the vortex equations, then it is well known that the connection A determines a complex structure on E such that the section φ : → ⊕n E becomes holomorphic [7]. This follows from the first vortex equation. This φ then determines a holomorphic section det φ of the determinant bundle det E with the induced complex structure. But being holomorphic, the section det φ either vanishes everywhere or has exactly d zeros, counting multiplicities. In this paper we concentrate on the latter case, i.e. on vortex solutions such that det φ does not vanish identically. Now suppose that z j ∈ is one of these isolated points where det φ vanishes. Making use of local trivializations of ⊕n E that are holomorphic with respect to the complex structure induced by A, the section φ can be regarded as a holomorphic function around z j with values on the n × n square matrices. We want to characterize the behaviour of φ around the point z j where the determinant vanishes, and for this we introduce the following two definitions. Definition 1.1. A vortex internal structure In is a set of data consisting of an integer k0 ≥ 0 and a sequence (V1 , . . . , Vl ) of non-zero proper subspaces of Cn such that V j+1 ∩ V j⊥ = {0} for all indices j = 1, . . . , l − 1. The order of the internal structure In is the non-negative integer n k0 + l dimC Vl . Definition 1.2. Given a subspace V of Cn consider the orthogonal decomposition Cn = V ⊕ V ⊥ . Calling V and ⊥ V the associated projections, for any complex scalar z one defines the elementary linear transformation n n TV (z) := zV + ⊥ V : C −→ C .
It is clear that the determinant of TV (z) is z dim V and that, for z = 0, the inverse TV (z)−1 is z −1 V + ⊥ V. The use of these definitions, and a key point in our results, is that the matrix function φ(z) can then be uniquely factorized around z j as φ(z) = A(z) (z − z j )k0 TVl (z − z j ) · · · TV1 (z − z j )
(3)
for some internal structure In = (k0 , V1 , . . . , Vl ) with order equal to the multiplicity of the vanishing of det φ at the point z j . Here A(z) is some holomorphic matrix function that is invertible around z j . Since the structure In is independent of the chosen
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trivialization of E, we thus have a canonical way to associate to each zero of det φ a correspondent algebraic internal structure. It then turns out that these internal structures completely determine the vortex solution up to gauge transformations. More precisely we have the following result. Theorem 1.3. Let E → be a complex vector bundle of rank n and degree d over a compact Riemann surface, and assume that (Vol ) > 2π d/(e2 τ ). Now pick any finite set {(z 1 , In1 ), . . . , (zr , Inr )} of distinct points on the surface and associated inter nal structures such that rl=1 order(Inl ) = d. Then there is a solution (A, φ) of the non-abelian vortex equations (2), unique up to gauge equivalence, such that det φ has j zeros exactly at the points z j and φ factorizes around each z j with internal structure In . Furthermore, all solutions of (2) with det φ not identically zero are obtained in this way. The condition of large volume of is required in the τ -stability results of [5,7], which are essential in our proof of the theorem above. At the same time, observe that a simple integration over of the second vortex equation shows that no solutions exist if (Vol ) < 2π d/(n e2 τ ). Thus the general picture that arises is that for small volumes of there are no vortex solutions; then for Vol in the interval between 2π d/(n e2 τ ) and 2π d/(e2 τ ) there is a less well known, and possibly complicated, moduli space of solutions; and finally for large volumes of (or big e2 , or big τ ) the moduli space can be neatly described as above. The layout of the paper is the following. In Sect. 2 we study holomorphic matrix functions on the plane, proving the local factorization (3) and a generalization thereof. In Sect. 3 we extend this to holomorphic sections of ⊕n E → and, using the correspondences of [5,7], relate them to vortex solutions, thereby proving Theorem 1.3. Finally in Sect. 4 we look at the space of all vortex internal structures of fixed order k, and argue that it is a compact and connected space. We end up by comparing our description with the special cases k = 1, 2 already studied in the literature. 2. Holomorphic Matrix Functions on the Plane Matrix functions φ(z) defined on the plane = C were studied at length in [9], where they were called moduli matrices. The word moduli appears because what we really want to study are the equivalence classes of φ(z)’s related by the equivalence relation φ(z) ∼ V (z) φ(z), with V (z) any matrix function that is invertible for all z. This equivalence can be called a complex gauge transformation or, in the language of [9], a V-transformation. Our approach here is rather different from the one in [9]. In those articles the classes of functions φ(z) are characterized by the coefficients of the various polynomials that appear in the matrix; here we characterize them by the position in C of the zeros of det φ(z) and by the factorization of φ(z) around each of these zeros. This last method seems to make easier the generalization to compact . The first step is the following local factorization result. Proposition 2.1. Let φ(z) be a holomorphic function of one complex variable with values on the square n × n matrices. Suppose that φ(z) is defined in a neighbourhood of the origin z = 0 and that the function det φ(z) has an isolated zero of order k at this point. Then there exists a unique internal structure In (φ) = (k0 , V1 , . . . , Vl ) such that φ(z) can be written around the origin as φ(z) = A(z) z k0 TVl (z) · · · TV1 (z),
(4)
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where A(z) is a holomorphic matrix function that is invertible around z = 0. Clearly the order of this In (φ) is precisely k. Corollary 2.2. Any internal structure In can be obtained as In (φ) for an appropriate holomorphic function φ(z) defined around the origin. Corollary 2.3. Two holomorphic functions φ1 (z) and φ2 (z) determine the same internal structure if and only if φ2 (z) = A(z) φ1 (z) for some invertible matricial function A(z). Proof. We start by showing how to obtain the structure In (φ) from the function φ(z). The integer k0 is defined as the minimal order of the zeros at the origin of the n 2 entries of the matrix φ(z), or in other words it is the only integer such that φ1 (z) := z −k0 φ(z) is holomorphic and does not vanish at the origin. Observe that det φ1 (z) has a zero of order k − nk0 at z = 0. If this order is zero, then one takes l = 0 and no vector spaces appear on In (φ). If on the other hand k − nk0 > 0, one defines V1 = ker φ1 (0) as the first proper subspace of Cn . To continue with the procedure, define the second function φ2 (z) := φ1 (z) TV−1 (z). 1 By applying this transformation to vectors in V1 and in V1⊥ , it is clear that φ2 (z) is well defined and holomorphic around the origin, including at the origin itself. Again, if φ2 (0) is invertible one takes l = 1 and the sequence of vector spaces terminates. If not, one defines V2 = ker φ2 (0) and the sequence of subspaces continues. The fact that φ2 (0) is injective on V1⊥ implies that V2 ∩ V1⊥ = {0}. Moreover, it follows from Definition 1.2 that det φ2 (z) vanishes at z = 0 with order k − nk0 − dim V1 . Decomposing once more Cn = V2 ⊕ V2⊥ one can go on with the procedure until k − nk0 − lj=1 dim V j vanishes for some l. When this happens the sequence of vector subspaces terminates and the linear transformation φl+1 (z) := φl (z) TV−1 (z) = φ(z) z −k0 TV−1 (z) · · · TV−1 (z) 1 l l will be invertible at the origin. This shows the existence of In (φ) and of the decomposition (4). For the uniqueness part, suppose that φ(z) given by (4) had another decomposition associated to a second set of data (k0 , V1 , . . . , Vl ). Then the function z −k0 φ(z) would be locally given by
A(z) TVl (z) · · · TV1 (z) = z k0 −k0 A (z) TV (z) · · · TV1 (z). l
(5)
The essential point now is that the conditions V j+1 ∩ V j⊥ = {0} imply that for all s ≤ l the kernel of ⊥ TVl (0) · · · TVs (0) = ⊥ Vl · · · Vs
is Vs . This is obvious for s = l and then clear by induction. Thus applying this fact to the left-hand side of (5), we see that z −k0 φ(z) is well defined and has kernel V1 at the origin. But then looking at the right-hand side, this can be true only if k0 = k0 and V1 = V1 . (z) we would conclude that also Arguing in the same way for the function z k0 φ(z)TV−1 1 V2 = V2 , and so forth for the other V j ’s. This finishes the proof of the proposition.
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As for the first corollary, it is enough to note that given an internal structure In = (k0 , V1 , . . . , Vl ) we can just define φ(z) := z k0 TVl (z) · · · TV1 (z). By the uniqueness of the local factorization it is then obvious that for this choice In (φ) = In . The second corollary is a direct consequence of the decomposition (4).
Having understood the local internal structures associated to each zero of det φ, we will now see how the set of these zeros and internal structures effectively characterizes any matrix function defined globally on C. Theorem 2.4. Let {z 1 , . . . , zr } be any finite set of distinct points in the complex plane and let {In1 , . . . , Inr } be any set of internal structures as defined in 1.1. Then there exists a holomorphic function φ(z) with values on the n × n matrices such that det φ(z) has j zeros exactly at the points z j with associated internal structure In . This function φ(z) is unique up to left multiplication by globally invertible matrix functions. Proof. The case r = 1 follows directly from Proposition 2.1. Arguing by induction, suppose now that det φ(z) has zeros at the points z 1 , . . . , zr −1 with associated internal structures In1 , . . . , Inr −1 , and that at the point z = zr the structure of φ(z) is given by ⊥ = {0}, (k0 , V1 , . . . , Vl−1 ). Given any vector subspace Vl ⊂ Cn such that Vl ∩ Vl−1 ˜ we will construct a new function φ(z) with the same structure as φ(z) at the points z 1 , . . . , zr −1 and, at the point z = zr , internal structure (k0 , V1 , . . . , Vl−1 , Vl ). This will suffice to prove the existence part of the theorem. To start with, by Proposition 2.1 we have that φ(z) can be written around zr , and hence globally, as φ(z) = A(z) (z − zr )k0 TVl−1 (z − zr ) · · · TV1 (z − zr ) for some A(z) invertible around zr . Now define the new function ˜ φ(z) := TVl (z − zr ) A(zr )−1 φ(z). ˜ Since φ(z) is related to φ(z) by left multiplication by a matrix invertible around z 1 , . . . , ˜ zr −1 , the internal structures of φ(z) and φ(z) at these points are the same. Furthermore, observe that the matrix B(z) = TVl (z − zr ) A(zr )−1 A(z) has vanishing determinant at z = zr with order dim Vl and that the kernel of B(zr ) is exactly Vl . It then follows from Proposition 2.1 that B(z) can be written as C(z) TVl (z − zr ) for some C(z) invertible around zr . In particular one can also rewrite ˜ φ(z) = C(z) (z − zr )k0 TVl (z − zr ) TVl−1 (z − zr ) · · · TV1 (z − zr ). ˜ Appealing again to Proposition 2.1 one concludes that the internal structure of φ(z) at z = zr is (k0 , V1 , . . . , Vl ), as we wanted. For the uniqueness, suppose that both φ(z) and φ (z) satisfy the conditions of the theorem. Then both φ(z) and φ (z) can be factorized as in (4) around any of the z j ’s j with the same internal structure In ; only the matrices A(z) in (4) will possibly differ. But then it is clear that the matrix φ (z)φ −1 (z) is well defined and invertible around z j . Since this happens for all of the z j ’s we conclude that φ (z)φ −1 (z) is globally well defined and invertible. Finally the tautology φ (z) = [φ (z)φ −1 (z)] φ(z) concludes the proof.
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3. Non-Abelian Vortices on Compact Riemann Surfaces In the first proposition we extend the results of Sect. 2 to compact , i.e. to holomorphic sections of holomorphic bundles over . After that we use the Hitchin-Kobayashi-type correspondences of [5,7] to relate these holomorphic sections to the actual vortex solutions over , thereby proving Theorem 1.3. Finally at the end of the section we state a result that gives a more practical interpretation of the constant τ that appears in the vortex equations. The proof is omited, because it is almost identical to the n = 1 case proved in [6]. Proposition 3.1. Let be a compact Riemann surface, let {z 1 , . . . , zr } be any set of distinct points in the surface and let {In1 , . . . , Inr } be any set of internal structures. Then there exists a holomorphic vector bundle E → of rank n and a section φ of ⊕n E j such that det φ has zeros exactly at the points z j with respective internal structure In . This pair (E, φ) is unique up to isomorphisms of holomorphic bundles. Moreover, the j degree of the bundle E is equal to the sum of the orders of the In ’s. Proof. It is always possible to take a connected open set V0 of that contains all the z j ’s and is simultaneously the domain of a complex chart of the surface. Now, using ˜ on the open set V0 with zeros Theorem 2.4, construct a holomorphic matrix function φ(z) j at the points z j and respective internal structure In . Considering the complementary set ˜ to the intersection V0 ∩ V1 V1 = \{z 1 , . . . , zr }, we have that the restriction of φ(z) — which we call ψ — is a n × n invertible matrix on this set, and so can be taken as the transition function for some holomorphic vector bundle E → of rank n that is trivial over V0 and V1 . Finally, taking simultaneously the constant function 1n×n on V1 and the function φ˜ on V0 , we have that over the intersection of the sets they satisfy the ˜ compatibility condition φ(z) = ψ(z) 1n×n , and so these two functions define a global section φ of the direct sum ⊕n E. It is clear that this φ has the required properties. To prove uniqueness, suppose that (E , φ ) was another pair with the required properties. Then taking local holomorphic trivializations of E and E and representing the sections φ and φ by local holomorphic matrix functions, Corollary 2.3 implies that the matrix φ φ −1 is holomorphic and invertible throughout the whole domain of trivialization. Moreover, if we pick different trivializations of E and E related to the initial ones by transition functions s and s , then the matrix φ φ −1 obviously transforms as s φ φ −1 s −1 , which implies that the matrices φ φ −1 actually define a global, holomorphic and invertible section of Hom(E, E ) → , or in other words an isomorphism : E → E . Since clearly (φ) = φ , the uniqueness is proved. Finally, to justify the last statement, just observe that the section det φ of the determinant bundle det E → vanishes exactly at the points z j with multiplicity equal to j the order of the respective In . Well-known properties of line-bundles then imply that j det E, and hence E, have degree equal to the sum of the orders of the In ’s. After studying the holomorphic part of the problem, i.e. after constructing the holomorphic vector bundles and the holomorphic sections with the required zeros and internal structures, we are now in position to relate them to the actual solutions of the vortex equations. Proof of Theorem 1.3. In short, the key point here comes from the results of [5,7], which guarantee that under the volume assumption (Vol ) > 2π d/(e2 τ ) each pair (E, φ)
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constructed in Proposition 3.1 is in fact a τ -stable n-pair, and hence there is a complex gauge transformation that takes it to a solution of the vortex equations. To understand more clearly what this means, suppose that we are given a C ∞ vector j bundle E and a finite set of pairs (z j , In ) satisfying the conditions of Theorem 1.3. Then by the proposition above there is a complex structure on E and a holomorphic section j φ of ⊕n E such that det φ vanishes at the z j ’s with internal structure In . Using the fixed hermitian metric h on E, this complex structure on its turn defines a natural connection A on E — the so-called Chern connection — such that ∂¯ A φ = 0. We thus have a solution (A, φ) of the first vortex equation. Observe now that while this first equation is invariant under complex gauge transformations, i.e. gauge transformations with the complexification U (n)C = S L(n, C) as the gauge group, the second vortex equation is certainly not invariant. This entitles us to ask whether a complex gauge transformation can take our pair (A, φ), which a priori only satisfies the first equation, to a solution of also the second equation, and hence to a full vortex solution. The answer is that this is possible whenever the initial holomorphic (E, φ) is a τ -stable n-pair, and that in this case the required complex gauge transformation is unique up to real gauge transformations. We thus see how the results of [5,7] give a vortex solution for each (E, φ) constructed in Proposition 3.1. To end the proof of Theorem 1.3 there are a few more points that should be checked. The first is to note that complex gauge transformations on (A, φ) do not change the holomorphic structure on E induced by A (up to equivalence) and that, furthermore, they transform φ through holomorphic isomorphisms of the bundles. This means that in the local factorization (3) the only thing that changes under complex gauge transformations is the matrix A(z), and hence the vortex solutions that were obtained in the last paragraph by means of complex transformations have exactly the same internal j structures In at the zeros z j as the solutions φ constructed in Proposition 3.1. The second thing to check is the uniqueness. If we were given two vortex solutions with the same zeros of det φ and internal structures, then, as described in the Introduction, we would get two complex structures on E and respective holomorphic sections with the same zeros and In ’s. From the uniqueness in Proposition 3.1, however, it follows that these two holomorphic pairs (E, φ) would in fact be isomorphic. Finally from the uniqueness in the results of [5,7] for the required complex gauge tranformations, we get that the initial vortex solutions must have been related by a real gauge transformation. The last point to check is the final assertion of Theorem 1.3. Here the justification comes from the fact that, as described in the Introduction, to any given vorj tex solution on with det φ = 0 we can associate a finite set of pairs (z j , In )’s. Using these pairs to construct our own vortex solution according to the prescriptions of this section, the uniqueness part in Theorem 1.3, which is already proved, assures us that the given vortex solution is in fact gauge equivalent to the constructed one.
Another interesting property of the non-abelian vortex solutions, proved as in [6], is the following. Proposition 3.2. Let (A, φ) be a solution of the vortex equations (2) on a C ∞ hermitian bundle (E, h) of rank n. Then the norm of each of the n components φ j of the section φ : → ⊕n E satisfies the majoration |φ j (z)|2h ≤ τ for all z in .
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4. On the Space of Vortex Internal Structures 4.1. Connectedness and compactness. In the Introduction to this paper we defined an internal structure In as a set consisting of an integer plus a sequence of vector subspaces of Cn satisfying a non-intersection condition. The advantages of this characterization are that it is simple enough, general for all n and, through the local factorization (3), directly related to the actual behaviour of the vortex solutions. A significant disadvantage is that it does not reveal transparently the main geometric and topological properties of the space of all internal structures, i.e. of the internal configuration space of the vortex. For example a priori one could think that the non-intersection condition would imply that these internal spaces are non-compact, a fact that is not true. In this final section of the paper we will spend some time looking at these matters, and will conclude that in a natural topology the space In,k of all internal structures of fixed order k is in fact compact and connected. To begin with, observe that one can divide the space In,k into disjoint strata according to the dimensions of the vector spaces at each point In = (k0 , V1 , . . . , Vl ). Calling k j the dimension of V j , the fixed order condition means that nk0 + k1 + · · · + kl = k, and then each vector of integers (k0 , k1 , . . . , kl ) satisfying this equality labels exactly one of these strata. Now inside each stratum, by definition, the dimensions of V j are fixed, so the only degrees of freedom are the spacial orientations of these subspaces inside Cn , and these are parametrized by the grassmannian Gr(k j , n). Note however that due to the non-intersection conditions not all spacial orientations are allowed, and the degrees of freedom are parametrized only by a dense open subset of the grassmannian. This implies that in general the different strata are not necessarily compact, although as we will see the full space In,k is. Now, the complex dimension of a stratum labelled by (k0 , . . . , kl ) is of course l j=1 k j (n − k j ), where each term in the sum is the dimension of a grassmannian. One can then show that, under the constraint imposed by the fixed order, this sum is strictly maximized with the choice k0 = 0, l = k and k j = 1 for j ≥ 1, where it has value k(n − 1). This means that the highest dimensional stratum in In,k , i.e. the generic part of the internal space, can be described as a choice of k lines in Cn , or in other words as a dense open subset of the k-fold cartesian product of CPn−1 . Moreover, we will see how the complement in ×k CPn−1 of this dense open subset in fact describes all the remaining strata of In,k , although different points in the complement can represent the same non-generic point of In,k . More precisely we have the following result. Proposition 4.1. There is a natural surjective map ϕ : ×k CPn−1 → In,k that is one-toone on the dense open subset of ×k CPn−1 defined by the multiplets (L 1 , . . . , L k ) that satisfy the non-intersection condition L j+1 ∩ L ⊥j = {0}. Since this map is surjective, the set of internal structures In,k equipped with the quotient topology is a compact and connected space. Proof. The argument for the proof is quite simple and goes as follows. Let L = (L 1 , . . . , L k ) be a point in ×k CPn−1 . Irrespective of whether L satisfies or not the non-intersection condition, it makes sense to consider the linear transformations TˆL (z) := TL k (z) · · · TL 1 (z) for all complex z. It is clear that TˆL (z) is holomorphic and that det TˆL (z) has a single zero at z = 0 of vanishing order k. Thus applying Theorem 2.4 there is a unique internal
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structure In = (k0 , V1 , . . . , Vl ) of order k such that TˆL (z) = z k0 TVl (z) · · · TV1 (z), and this defines the map ϕ. Now call U0 the set of L’s in ×k CPn−1 that satisfy the non-intersection condition, i.e. the set of L’s that are proper internal structures. It is clear that ϕ restricted to U0 is injective with image the generic stratum of In,k ; this is a tautology, for in fact ϕ restricted to U0 is the identity. Outside U0 , on the other hand, the story is different, for if L j+1 ∩ L ⊥j is non-zero then L j+1 ⊆ L ⊥j , and it is easy to check that in this case TL j+1 (z) TL j (z) = TL j+1 ⊕L j (z).
(6)
This equality is a special instance of the more general Lemma 4.2. It means that if L j+1 and L ⊥j have non-zero intersection the map ϕ only depends on the direct sum L j+1 ⊕ L j , and so is clearly not injective. The identities (6) and (7) can also be applied recursively to show that for any vector subspace V of Cn there exist lines L 1 , . . . , L dim V such that TV (z) = TL dim V (z) · · · TL 1 (z). In fact any set of orthogonal lines such that V = L 1 ⊕ · · · ⊕ L dim V will do the job. This shows that ϕ is a surjective map. Lemma 4.2. Let V1 and V2 be any two subspaces of Cn . Then calling W the intersection V2 ∩ V1⊥ , the linear transformations of Definition 1.2 satisfy the algebraic identity TV2 (z) TV1 (z) = TV2 ∩ W ⊥ (z) TW ⊕V1 (z).
(7)
Observe moreover that the two subspaces on the right-hand side satisfy the usual nonintersection condition, i.e. V2 ∩ W ⊥ has zero intersection with (V1 ⊕ W )⊥ . Proof. It is clear from Definition 1.2 that the identity above is equivalent to the three separate identities: (i) V2 V1 = W ⊥ ∩V2 W ⊕V1 ; ⊥ ⊥ ⊥ (ii) ⊥ V2 V1 = W ⊥ ∩V W ⊕V1 ; 2
⊥ ⊥ ⊥ (iii) ⊥ V2 V1 + V2 V1 = W ⊥ ∩V W ⊕V1 + W ⊥ ∩V2 W ⊕V1 . 2
Since the proofs of these equalities are rather cumbersome and similar to each other, here we will restrict ourselves to prove (i). The first step is to note that the general identity of subspaces (A + B)⊥ = A⊥ ∩ B ⊥ implies that each of the decompositions Cn = V1 ⊕ W ⊕ (W ⊥ ∩ V1⊥ ) =
V2⊥
⊥
⊕ W ⊕ (W ∩ V2 )
(8) (9)
is orthogonal. Having this in mind, suppose now that v is a vector in W . Then we have that W ⊥ ∩V2 W ⊕V1 (v) = W ⊥ ∩V2 (v) = 0 = V1 (v),
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and so both sides of (i) annihilate v. If on the other hand v belongs to the subspace W ⊥ ∩ V1⊥ = (W ⊕ V1 )⊥ , we have that W ⊕V1 (v) = 0 = V1 (v), and so also in this case both sides of (i) annihilate v. Finally suppose that v sits in V1 . It is clear that W ⊕V1 (v) = v = V1 (v), and hence we only have to check that V2 (v) = W ⊥ ∩V2 (v). But V1 being orthogonal to W , we can use the decomposition (9) to write v = v2 + w2 with v2 ∈ V2⊥ and w2 ∈ W ⊥ ∩ V2 . Then obviously W ⊥ ∩V2 (v) = w2 . Now since V2 (v2 ) = 0 and w2 ∈ V2 , we finally have that V2 (v) = V2 (w2 ) = w2 , which concludes the proof of (i). 4.2. The case k = 2: comparison with the literature. The space of vortex internal structures has been described previously in the literature in the cases k = 1 and k = 2. The case k = 1 is simple enough: it follows directly from Definition 2 that In,1 CPn−1 , as was known. The case k = 2 has been studied in [3,12] for n = 2 and in [10] for general n. Here we will compare our results with those of [10], finding in the end that they are consistent. We start with our results. According to the previous subsection the highest dimensional stratum of In,2 has dimension 2(n − 1) and isomorphic to U0 = (CPn−1 × CPn−1 )\S, where S is the submanifold defined by L 2 ∩ L ⊥ 1 = {0}, or in other words by L 2 ⊥ L 1 . This is the domain U0 , where the map ϕ of Proposition 4.1 is injective. Outside U0 , i.e. on S, the line L 2 is perpendicular to L 1 , and so it follows from (6) that in this case ϕ(L 1 , L 2 ) = L 1 ⊕ L 2 ⊆ Cn , i.e. the internal structure associated to (L 1 , L 2 ) ∈ S is the 2-dimensional vector space L 1 ⊕ L 2 . Since these vector spaces are parametrized by the grassmannian Gr(2, n) we recognize that, in informal terms, In,2 is isomorphic to CPn−1 × CPn−1 with the submanifold S collapsed into the grassmannian Gr(2, n). (Incidentally, identifying the orthogonal space L ⊥ with the tangent space to CPn−1 at the point L, it is manifest that the submanifold S is itself isomorphic to the projectivization of the tangent bundle T CPn−1 , which is a bundle over CPn−1 with fibre CPn−2 .) In the paper [10], on the other hand, the space of vortex internal structures for k = 2 was described in the following terms. Let M be the space of 2 × (n + 1) matrices of rank 2. We can write every element M ∈ M in the form T ψ2 v2 M = = [ ψ v ], ψ1T v1 with ψ1 , ψ2 ∈ Cn and v ∈ C2 . Consider now the left action of the group C∗ × S L(2, C) on M defined by (λ, A) · [ ψ v ] = [ λAψ Av ].
(10)
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Then according to [10] there is an isomorphism In,2 M/ C∗ × S L(2, C).
(11)
To compare this result with our previous description we will study in detail the equivalence classes in this quotient. Firstly, under the action of C∗ × S L(2, C) one can distinguish two types of orbits in M: those with v = 0 and those with v = 0. The points in the orbits with v = 0 have linearly independent ψ1 and ψ2 , for M must have rank 2. When C∗ × S L(2, C) acts on these points the only invariant is the subspace V = spanC {ψ1 , ψ2 }, and these subspaces are exactly parametrized by the grassmannian Gr(2, n). Thus decomposing M = Mv=0 ∪ Mv =0 we see that Mv=0 / C∗ × S L(2, C) = Gr(2, n), which is part of the internal space described in our results. Now we want to prove that there is also an isomorphism : Mv =0 / C∗ × S L(2, C) −→ (CPn−1 × CPn−1 )\S,
(12)
and so conclude that there exists a bijection between the full internal spaces In,2 as described here and in [10]. So consider the equivalence classes of the points [ ψ v ] with v = 0. Since S L(2, C) acts transitively on C2 \{0}, any such equivalence class has a representative of the form T ψ2 1 . (13) ψ1T 0 Observe that here ψ1 = 0, since the rank of the matrix must be 2. The representative (13), however, is not unique; on the one hand because it can be acted upon by the C∗ subgroup, and on the other hand because the vector [1 0]T has a one dimensional stabilizer under the S L(2, C)-action. All in all, the representative is unique up to transformations of the form T T λ(ψ2 + aψ1 )T 1 ψ2 1 1 a ψ2 1 · , (14) − → λ = 0 1 λ ψ1T 0 ψ1T 0 ψ1T 0 where λ ∈ C∗ and a ∈ C. Now notice that the complex lines in Cn , L 1 := spanC {ψ1 } , L 2 := spanC {ψ1 + ⊥ L 1 (ψ2 )} ,
(15)
are well defined and are invariants of the transformations (14). Moreover, it is not difficult to recognize that these lines actually distinguish the orbits of these transformations, i.e. that two matrices of the form (13) that define different lines cannot be related by a transformation (14), and hence lie on different C∗ × S L(2, C) orbits. To sum up, through the representative (13) and the definition (15) we have constructed an injective map that takes any orbit in Mv =0 / C∗ × S L(2, C) to a pair of complex lines (L 1 , L 2 ). These lines are clearly non-orthogonal, and it is manifest that any pair of non-orthogonal lines can be obtained through (15) for an appropriate choice of ψ. This finally shows that the injective map that we constructed has image (CPn−1 × CPn−1 )\S, and hence is the isomorphism (12) that we were seeking.
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As a curiosity, the inverse of the isomorphism can be written down very simply as ⊥ L 1 (w2 ) 1 , (L 1 , L 2 ) −→ equivalence class of L 1 (w2 ) 0 where w2 is any non-zero vector in L 2 . Furthermore, picking any non-zero vector w1 in T 2 L 1 and decomposing w2 = αw1 + ⊥ L 1 (w2 ), with α the complex scalar w¯1 w2 /|w1 | , even the full isomorphism ˜ : In,2 = (CPn−1 × CPn−1 ))/ ϕ −→ M/ C∗ × S L(2, C) can be written down in the single expression ˜ : equiv. class of (L 1 , L 2 ) −→ equiv. class of
⊥ L 1 (w2 ) w1
√ α 0
n−1 . To check that ˜ is really well defined and does not depend on for all L 1 , L 2 in CP √ the sign choice in α, one must of course make use of the C∗ × S L(2, C)-equivalences. We also note that in general the internal spaces In,k , although well-defined compact topological spaces, are not necessarily smooth manifolds outside of the highest dimensional stratum. This has been illustrated at length in [10] in the case √ k = 2. In the discussion above a reflection of this singularity is the appearance of α in the ˜ isomorphism .
Acknowledgement. It is a pleasure to thank Minoru Eto, Muneto Nitta, Keisuke Ohashi and Norisuke Sakai for kindly hosting me during a visit to TITech, Tokyo, three years ago. I am grateful to them and to David Tong for explaining me their work on non-abelian vortices. I am partially supported by the Netherlands Organisation for Scientific Research (NWO) through the Veni grant 639.031.616. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Abrikosov, A.: On the magnetic properties of superconductors of second group. Sov. Phys. JETP 5, 1174 (1957) [Zh. Eksp. Teor. Fiz. 32, 1442 (1957)]; Nielsen, H., Olesen, P.: Vortex-line models for dual strings. Nucl. Phys. B61, 45 (1973) 2. Auzzi, R., Bolognesi, S., Evslin, J., Konishi, K., Yung, A.: Nonabelian Superconductors: Vortices and Confinement in N = 2 SQCD. Nucl. Phys. B673, 187–216 (2003) 3. Auzzi, R., Shifman, M., Yung, A.: Composite non-abelian flux tubes in N = 2 SQCD. Phys. Rev. D73, 105012 (2006) 4. Baptista, J.M.: Vortex equations in abelian gauged sigma-models. Commun. Math. Phys. 261, 161–194 (2006) 5. Bertram, A., Daskalopoulos, G., Wentworth, R.: Gromov invariants for holomorphic maps from Riemann surfaces to grassmannians. J. Amer. Math. Soc. 9, 529–571 (1996) 6. Bradlow, S.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990) 7. Bradlow, S., Daskalopoulos, G., García-Prada, O., Wentworth, R.: Stable augmented bundles over Riemann surfaces. In: Vector bundles in algebraic geometry, London Math. Soc. Lecture Note Ser. 208, Cambridge: CUP, 1995, pp. 15–67 8. Eto, M., Evslin, J., Konishi, K., Marmorini, G., Nitta, M., Ohashi, K., Vinci, W., Yokoi, N.: On the moduli space of semilocal strings and lumps. Phys. Rev. D76, 105002 (2007); Shifman, M., Yung, A.: Non-abelian semilocal strings in N = 2 supersymmetric QCD. Phys. Rev. D73, 125012 (2006); Popov, A.: Non-abelian vortices on Riemann surfaces: an integrable case. Lett. Math. Phys. 84, 139–148 (2008)
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9. Eto, M., Isozumi, Y., Nitta, M., Ohashi, K., Sakai, N.: Moduli space of non-abelian vortices. Phys. Rev. Lett. 96, 161601 (2006); Eto, M., Isozumi, Y., Nitta, M., Ohashi, K., Sakai, N.: Solitons in the Higgs phase – the moduli matrix approach –. J. Phys. A39, R315–R392 (2006); Eto, M., Fujimori, T., Gudnason, S., Konishi, K., Nitta, M., Ohashi, K., Vinci, W.: Constructing non-abelian vortices with arbitrary gauge groups. Phys. Lett. B669, 98–101 (2008) 10. Eto, M., Konishi, K., Marmorini, G., Nitta, M., Ohashi, K., Vinci, W., Yokoi, N.: Non-abelian vortices of higher winding numbers. Phys. Rev. D74, 065021 (2006) 11. Hanany, A., Tong, D.: Vortices, instantons and branes. JHEP 0307, 037 (2003) 12. Hashimoto, K., Tong, D.: Reconnection of non-abelian cosmic strings. JCAP 0509, 004 (2005) 13. Taubes, C.H.: Arbitrary N -vortex solutions to the first order Ginzburg-Landau equations. Commun. Math. Phys. 72, 277–292 (1980) 14. Tong, D.: Quantum vortex strings: a review. http://arXiv.org/abs/0809.5060v2[hep-th], 2008 Communicated by G. W. Gibbons
Commun. Math. Phys. 291, 813–843 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0890-5
Communications in
Mathematical Physics
Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding William Matthews1 , Stephanie Wehner2 , Andreas Winter1,3 1 Department of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.
E-mail: [email protected], [email protected], [email protected]
2 Institute for Quantum Information, Caltech, Pasadena, CA 91125, USA.
E-mail: [email protected]
3 Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3,
Singapore 117542, Singapore Received: 31 October 2008 / Accepted: 4 June 2009 Published online: 13 August 2009 – © Springer-Verlag 2009
Abstract: We consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements. Let the bias of a POVM be defined as the total variational distance between the outcome distributions for the two states to be distinguished. The performance of a set of measurements can then be defined as the ratio of the bias of this POVM and the largest bias achievable by any measurements. We first provide lower bounds on the performance of various POVMs acting on a single system such as the isotropic POVM, and spherical 2 and 4-designs, and show how these bounds can lead to certainty relations. Furthermore, we prove lower bounds for several interesting POVMs acting on multipartite systems, such as the set of local POVMS, POVMs which can be implemented using local operations and classical communication (LOCC), separable POVMs, and finally POVMs for which every bipartition results in a measurement having positive partial transpose (PPT). In particular, our results show that a scheme of Terhal et. al. for hiding data against local operations and classical communication [31] has the best possible dimensional dependence. 1. Introduction Suppose we are given one of two possible quantum states ρ0 and ρ1 chosen with probabilities π0 and π1 , respectively. The goal of ambiguous state discrimination is to output a guess ρb for the given state such that the average probability of error is minimized. To obtain this guess, we may thereby perform a measurement providing us with an outcome b ∈ {0, 1}. In this paper, we study the task of state discrimination when we are only allowed to perform a restricted set of measurements. To state our results, let us first explain the notion of a measurement (POVM) more formally. Consider a measurable space (X, F), that is F is a σ -algebra of subsets of the set X , where we will identify F with the possible outcomes of the measurement. A POVM is a function M : F → B+ (H) such that M(X ) = 1, where we use B + (H) to denote the set of positive Hermitian operators on a finite dimensional Hilbert space H. That is, M(A) is the measurement operator
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associated with outcome event A ∈ F. With regard to the problem of state discrimination, suppose that when performing the POVM M we guess that the state is ρ1 if we observe outcome A ∈ F, and ρ0 otherwise. The probability of error can then be written as Perr = π0 Tr(M(A)ρ0 ) + π1 Tr(M(X \ A)ρ1 ). Let us now consider which outcomes A ∈ F we should associate with ρ1 and ρ0 respectively in order to minimize the probability of error. We will use B sa (H) to denote the space of Hermitian operators, and D(H) = {ρ ∈ B + (H) | Tr(ρ) = 1} to denote the space of density operators on H. Note that for any Hermitian operator ξ ∈ B sa (H), the function νM [ξ ] : F → R defined by νM [ξ ](A) := Tr(M(A)ξ ) is a signed measure on (X, F). It has a Hahn–Jordan decomposition [2] νM [ξ ] = νM [ξ ]+ − νM [ξ ]− , where νM [ξ ]+ and νM [ξ ]− are positive measures on (X, F). For all A ∈ F these measures can be written as νM [ξ ]+ (A) := νM [ξ ](A ∩ X + ), νM [ξ ]− (A) := νM [ξ ](A ∩ X − ),
(1) (2)
where X + and X − are the positive and negative parts of the Hahn decomposition of X with respect to νM [ξ ]. Note that we can rewrite the probability of error using the function νM [ξ ] as Perr = π1 − Tr M(A)(π1 ρ1 − π0 ρ0 ) = π1 − νM [ξ ](A) with ξ = π1 ρ1 − π0 ρ0 . (3) In light of the Hahn–Jordan decomposition, it is now clear that the smallest probability of error is attained by letting A = X + correspond to the guess ρ1 , yielding 1 (νM [ξ ](X + ) − νM [ξ ](X − ) + νM [ξ ](X )) 2 1 1 = (1 − (νM [ξ ](X + ) − νM [ξ ](X − ))) = (1 − νM [ξ ]), 2 2
Perr = π1 − νM [ξ ](X + ) = π1 −
(4) (5)
where νM [ξ ] := |νM [ξ ](X + )| + |νM [ξ ](X − )| is the total variation of the signed measure νM [ξ ]. Defining the bias to be 1 − 2Perr , we have shown that the largest bias that can be attained, based on the outcomes of M, is νM [ξ ]. Hence if we are only able to implement the POVMs in some set fixed set M, the best bias that can be achieved is given by ξ M := sup νM [ξ ]. M∈M
(6)
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When the measurable space (X, F) has countable or finite X = {x1 , x2 , . . .} with F containing all subsets of X , the total variation of a signed measure ν is simply ν = |ν({x})|. x∈X
In this case, νM [ξ ] is just the 1 norm of the vector (Tr M(x1 )ξ, Tr M(x2 )ξ, . . .), and the operators M(xi ) are often called the elements of the POVM. The Holevo–Helstrom theorem [23] tells us that when we are allowed to perform any POVM, the choice of measurement that maximises the bias is the two–outcome POVM with elements equal to the projectors onto the postive and negative eigenspaces of ξ . It is not hard to see that this achieves a bias equal to the trace norm of ξ , ξ 1 . A natural indicator for the performance of a restricted set of POVMs M is thus given by the ratio ξ M . ξ 1
(7)
A. Results. We first show in Sect. 2 that (6) is a norm for every sufficiently rich set M. We furthermore make a connection to general norms in vector spaces, showing that indeed any norm on trace class operators can be interpreted as a norm of the above type. We then turn to a number of particular examples, where we especially highlight the problem of determining bounds on the ratio (7). In Sect. 3, we investigate the particular case where M consists of only one (necessarily informationally complete) POVM, finding the best upper and lower bounds on the ratio (7) for any such measurement. These bounds are attained for the isotropic (unitary invariant) POVM. We also analyse the situation for POVMs originating from 2- and 4-designs. In Section 4, we look at the situation that the system under consideration is bi- or multipartite, and that the POVMs are restricted to classes respecting the partition: local measurements, with or without classical communication between the parties, and extensions of this class. The existence of data hiding [14,22,31] states yields bounds on the ratio (7) in one direction. Here, we show that in the bipartite case, these bounds are optimal up to a constant factor by analysing the tensor product of two isotropic local POVMs: it turns out that the resulting measurement attains almost the same bias. Hence, the hiding states of [31] are already (near) optimal in the sense that we cannot hope to construct states which are less well distinguishable under LOCC operations. Finally, we make a connection to Sanchez-Ruiz’ “certainty relations” for mutually unbiased bases [29] in Sect. 5, which we show holds more generally for any 2-design POVM, and – even in a stronger form – for 4-designs. We also show how our results for bipartite systems imply a universal lower bound on the information accessible by LOCC from any pure state ensemble. Several appendices contain the proofs of more technical results in the main text. 2. First Observations on Norms and Dual Norms Before turning to the essential observations that we will need later on, we first explain some basic concepts. We follow the terminology of Rockafellar [3] when referring to some elementary concepts from convex analysis. For norms · a and · b defined on a space V , we write · a ≥ · b if xa ≥ xb for all x in V . At the heart of the Helstrom-Holevo Theorem [23] on optimal state discrimination lies the duality between
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the operator norm · and the trace norm · 1 : For operators α, A on a Hilbert space H, these are dual to each other with α1 = sup | Tr(α † B)|, B≤1
A = sup | Tr(β † A)|. β1 ≤1
In finite dimension, which we shall assume throughout this paper, the suprema are easily seen to be maxima. The duality persists when we restrict to Hermitian (self-adjoint) operators α = α † , A = A† : α1 = A =
max
Tr(α B),
max
Tr(β A).
B=B † , B≤1 β=β † , β1 ≤1
These equations are direct consequences of the singular value decomposition in the general, and of the spectral theorem in the Hermitian case. The role of the Hilbert-Schmidt inner product, A, B := Tr A† B which makes the real vector space of Hermitian operators, B sa (H), a Euclidean space, becomes more evident in geometrical language by saying that the unit balls B1 ( · 1 ) = α = α † : α1 ≤ 1 , B1 ( · ) = A = A† : A ≤ 1 , are polar to each other. To explain this notion, note that the unit ball of any norm N on a finite dimensional real vector space, K := B1 (N ) = {x : N (x) ≤ 1}, is a topologically compact, convex and symmetric set (i.e. K = −K ), containing the origin 0 in its interior. Any such body K conversely determines a norm 1 : t > 0 and t x ∈ K , x Kˇ = inf t and it is immediately verified that K = B1 ( · Kˇ ) and N = · Kˇ (unconventionally, we write · K for the norm with unit ball Kˇ rather than that with unit ball K , as it simplifies the notation later). That is, norms and convex, compact, symmetric bodies of non–empty interior are equivalent descriptions. Now, the polar of K in a Euclidean vector space with inner product ·, · is defined to be Kˇ := {y : ∀x ∈ K x, y ≤ 1}. It is easy to verify that if K is symmetric, convex and compact, and contains the origin in its interior, then Kˇ has the same properties, and Kˇˇ = K .
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By the above discussion, K is the unit ball of · Kˇ , Kˇ is the unit ball of · K and one has the important, but elementary, formulas y K = maxx, y, x∈K
x Kˇ = maxx, y, y∈ Kˇ
which are the abstract versions of the equations above. We are now ready to make a series of observations. First, we need to show that Eq. (6) really does constitute a norm for trace class operators, i.e. for operators with a finite, well-defined, trace. First we note that, for any POVM M, νM [ξ ] is a seminorm on trace class operators ξ , which we give the shorthand ξ M := νM [ξ ]. For sets of POVMs M have defined · M = supM∈M · M which, being a supremum over seminorms, is also a seminorm. Clearly, ·M is a norm iff for all ξ = 0, there is a POVM M ∈ M such that νM [ξ ] > 0. We call a set of POVMs which satisfies this property “informationally complete”. It is often said that a set of POVMs is informationally complete iff knowledge of the statistics of the POVMs in the set is sufficient to reconstruct any unknown state (operationally, we think of having an unlimited number of copies of the state on which we can perform the measurements). It is not hard to see that this is true of a set M iff span{M(E) : M ∈ M, E ∈ FM } =: S = B(H). If there is a ξ such that ξ M = 0, then we must have Tr M(E)ξ = 0 for all POVMs and events, so S = B(H). Conversely, if S = B(H), then there is an operator ξ in the orthogonal complement of S and ξ M = 0. Therefore, the two definitions of informational completeness coincide. We now show that we can restrict ourselves to POVMs with 2 outcomes. Intuitively, since we decide between two options (e.g. ρ and σ above), we can group the outcomes of each POVM in two. It is then not difficult to verify that Definition 1. For any separating set M of POVMs we define the set of two–outcome POVMs {(M(A), 1 − M(A)) : ∀ A ∈ FM } , M2 := M∈M
where FM is the set of measurable subsets of outcomes for M, satisfies · M = · M2 and we define M := cl conv {2E − 1 : (E, 1 − E) ∈ M2 } = cl conv {2M(A) − 1 : A ∈ FM }, where cl conv S denotes the closure of the convex hull of S. Lemma 2. M is a compact symmetric convex body, contained in the operator interval [−1; 1] = {X : −1 ≤ X ≤ 1} and containing ±1, and has a non–empty interior, such that ξ M = max | Tr(ξ E)| =: ξ M . E∈M
Proof. From the discussion in the Introduction, it is clear that for a particular choice of ξ , the bias for any POVM M ∈ M is equal to the bias for the two–outcome POVM (M(X + ), M(X − )).
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Note that M has a non-empty interior (and then contains the origin in its interior) if and only if the collection M is informationally complete, which is the case if and only if M2 is informationally complete. Mathematically the information-completeness is expressed by M, spanning the whole operator space. Furthermore, note that from our discussion above we have that Remark 3. The symmetric convex body M defines two norms, one on the observables and effects, the other on the trace class operators, via 1 EM : t > 0 and t M ∈ M , (8) = inf ˇ t ξ M = max Tr(ξ E). (9) E∈M
The first has exactly M as its unit ball, the second has as its unit ball the polar of M, i.e. ˇ = {ξ : ∀M ∈ M Tr(ξ M) ≤ 1} . M The norm · M (= · M ) is dual to · M ˇ: ξ M = max Tr(ξ E) : EM ˇ ≤1 , EM ˇ = max {Tr(ξ E) : ξ M ≤ 1} . Putting everything together, we can now see that Theorem 4. The norms · M associated to sets of POVMs are in one-to-one correspondence with full-dimensional symmetric compact convex bodies ±1 ∈ M ⊆ [−1; 1]. As a consequence, any norm | · | ≤ · 1 can be written as | · | = · M for some set of POVMs. Proof. First, starting with a set of POVMs M defining norms · M , Lemma 2 describes how to construct M, such that · M = · M . Conversely, starting with a full-dimensional symmetric compact convex body M ⊆ [−1; 1], we can construct a set of POVMs M = {(M, 1 − M) : M ∈ M and M ≥ 0} for which · M = · M . We formalise the connection with the state discrimination problem in the following theorem. Theorem 5. Let M be a set of POVMs on a given Hilbert space, and let M2 and M be defined as above. For any two states ρ and σ , consider the minimum error probability PEM of discriminating between these (a priori equiprobable states). Then, PEM =
inf
(M,1−M)∈M2
1 1 1 1 − | Tr((ρ − σ )M)| = − ρ − σ M . 2 2 2 4
That is, 21 ρ − σ M is the bias achievable in discriminating ρ from σ when only measurements in M are allowed. In finite dimension, which is the case we stick to in this paper, the operators also form a finite-dimensional space, and all these norms are “equivalent” in the sense that there are λ , µ > 0 such that λ · 1 ≤ · M ≤ µ · 1 .
(10)
Distinguishability of Quantum States Under Restricted Families of Measurements
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By using the above correspondences and dualities, we see that this is equivalent to λ [−1; 1] ⊆ M ⊆ µ [−1; 1]. (M)) to denote the largest λ
(11)
(smallest µ ) in these equations.
We will use λ1 (M) (µ1 The numbers λ1 and µ1 are called the constants of domination of the norm · M (with respect to · 1 ). In the following, our goal is to bound these constants of domination for various interesting classes of POVMs. These constants are especially interesting, since we know from Theorem 5 that they allow us to bound the bias that we can achieve when trying to distinguish two states ρ and σ with a restricted set of measurements. Note that µ1 (M) is trivially 1 since for ρ ≥ 0, ρM = ρ1 = Tr(ρ). Thus, we are motivated to restrict to traceless operators in Eq. (10). This is also the setting for which bounds on the constants of domination give us a bound on the bias of distinguishing two a priori equiprobable states ρ and σ . Let λ(M) and µ(M) be the largest and smallest numbers λ and µ , respectively, such that ∀ξ with Tr(ξ ) = 0 λξ 1 ≤ ξ M ≤ µξ 1 .
(12)
Equivalently, in the dual picture we have to go to the quotient modulo multiples of the identity, R1: λ[−1; 1]/R1 ⊆ M/R1 ⊆ µ[−1; 1]/R1 ,
(13)
where, for a set of operators X , X/R1 = {x − 1 Tr x/ Tr 1 : x ∈ X }. The following lemma characterizes λ1 (µ1 ) and λ (µ), and their respective relations. Lemma 6. For a set M of POVMs with associated convex body M, the constants of domination can be expressed as the solutions of the following optimisation problems: 1 λ(M) ≤ λ1 (M) = inf sup ξ M ≤ inf sup ξ M = λ(M), ξ 1 =1 ξ 1 =1 M∈M 2 M∈M Tr(ξ )=0
1 = µ1 (M) = sup sup ξ M ≥ sup sup ξ M = µ(M). ξ 1 =1 M∈M
ξ 1 =1 Tr(ξ )=0
M∈M
Here, for the purpose of λ and µ, ξ may be thought of as ξ = 21 (ρ − σ ) for orthogonal states ρ, σ . Proof. The optimisation problems are an immediate consequence of the definitions, and we already argued that µ1 (M) = 1. To lower bound λ1 (M) we proceed as follows: Given any ξ of trace norm 1, we can write it as ξ = (1 − p)ρ − pσ = (1 − p)(ρ − σ ) + (1 − 2 p)σ = 2(1 − p)ξ0 + (1 − 2 p)σ, with orthogonal states ρ and σ , and ξ0 = 21 (ρ − σ ). W.l.o.g. 0 ≤ p ≤ 1/2, otherwise use −ξ . Now let X 0 ∈ M be optimal for ξ0 , i.e. ξ0 M = Tr(ξ0 X 0 ), and test ξ with X = (1 + X 0 )/2 ∈ M. Note X ≥ 0, so ξ M ≥ Tr(ξ X ) = 2(1 − p) Tr(ξ0 X ) + (1 − 2 p) Tr(σ X ) 1 − 2p = (1 − p) Tr(ξ0 X 0 ) + Tr(σ X ) 2 1 1 1 ≥ Tr(ξ0 X 0 ) = ξ0 M ≥ λ(M), 2 2 2 concluding the proof.
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What is the relation of the constants of domination for different sets M and M ? Clearly, if M ⊆ M , then λ(M) ≤ λ(M ) and µ(M) ≤ µ(M ). More interesting relations are obtained by using the convex structure. For this purpose we look at convex combinations of POVMs in the sense of direct sums as follows. Let M : F → B sa (H) and M : G → B sa (H) be two POVMs for measurable spaces (X, F) and (Y, G). Let (X ∪ Y, K) be the direct sum of (X, F) and (Y, G), i.e. K = {A : A ∩ X ∈ F and A ∩ Y ∈ G, ∀A ∈ P(X ∪ Y )} (where P denotes powerset). For p ∈ [0, 1], define the direct convex combination (1 − p)M ⊕ pN : X ⊕ Y → B sa (H) by specifying that, for all A ∈ K, ((1 − p)M ⊕ pN)(A) = (1 − p)M(A ∩ X ) + pN(A ∩ Y ). If we have two sets of POVMs, M1 and M2 , then their direct sum convex combination is defined naturally as (1 − p)M1 ⊕ pM2 := {(1 − p)M1 ⊕ pM2 : ∀M1 ∈ M1 , M2 ∈ M2 }. More generally, we can look at convex combinations of any finite or even countable number of POVMs and sets of POVMs. These constructions have a straightforward operational interpretation: implementing p(E k )k ⊕ (1 − p)(F ) means tossing a biased coin, with p being the probability of heads, then measuring (E k ) if heads showed, and (F ) for tails. The coin toss is part of the measurement result. Lemma 7. Let Mi be sets of POVMs and pi ≥ 0 probabilities, and R = i pi Mi . Denote the corresponding convex bodies of operators Mi and R. Then, R= pi Mi , i
and consequently λ(R) ≥
pi λ(Mi ), µ(R) ≤
i
pi µ(Mi ).
i
Proof. The first relation is by inspection. For the inequalities, note that since we have λ(Mi )[−1; 1]/R1 ⊆ Mi /R1 ⊆ µ(Mi )[−1; 1]/R1 , we clearly get pi λ(Mi )[−1; 1]/R1 ⊆ pi Mi /R1 ⊆ pi µ(Mi )[−1; 1]/R1 . i
i
i
In particular, since [−1; 1] is invariant under unitary conjugation, i.e. U [−1; 1]U †
= [−1; 1], the constants of domination also have this invariance and so we obtain immediately Proposition 8. For a probability measure d p(U ) on the unitary group on H, and any symmetric, convex body with non–empty interior, ±1 ∈ M ⊆ [−1; 1],
λ d p(U )U MU † ≥ λ(M),
µ d p(U )U MU † ≤ µ(M). In other words: symmetrisation makes M “look more like [−1; 1]”.
Distinguishability of Quantum States Under Restricted Families of Measurements
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3. Single POVMs Let us look now at the constants of domination λ and µ in the case that M consists of a single, informationally complete POVM M. We denote the constants of domination λ(M) and µ(M). A. Isotropic POVM. Given a D dimensional Hilbert space H, let X denote the sphere of normalised pure states in H and let F be the Borel measurable subsets of X . We define the isotropic POVM MU : F → B + (H) by MU (A) = D |ψψ|dψ A
with dψ the unitarily invariant probability measure on (X, F). If we take any POVM M : (X, F) → B + (H) we can construct a new POVM M which corresponds to a measurement where a random unitary is drawn according to the Haar measure, and recorded, and then M is measured. M takes outcomes in the product measure space (X, F) × (SU(D), G) (where G is the set of Borel measurable subsets of the Lie group SU(D)). M is defined on ‘rectangles’ A × B, A ∈ F, B ∈ G by † M (A × B) = U M(A)U d p(U ), B
where p denotes the Haar measure on (SU(D), G). Now, the total variation of the signed measure νM [ξ ] is
Tr U M(Ai ) U † ξ d p(U ), (Tr M(Ai ))D νM [ξ ] = sup
Tr (M(Ai )) SU (D) (Ai ) i
where the supremum is over finite partitions of X into measurable sets Ai . For any D D M(A), TrM(A) j=1 p j |ψ j ψ j |, where j=1 p j = 1 M(A) has a spectral decomposition and p j ≥ 0. Therefore,
Tr U M(A) U † ξ d p(U ) D
Tr M(A) SU(D)
⎛ ⎞
D
† ⎠
Tr ⎝U (14) =D p |ψ ψ |U ξ j j j
d p(U )
SU(D)
j=1
⎛ ⎞
D
† †
⎝ =D (15) p j V j ξ V j ⎠
d p(U )
Tr U |00|U SU(D)
j=1
≤D (16)
Tr U |00|U † ξ d p(U ) = ξ MU . SU(D)
Given this, and the fact that i (Tr M(Ai )) = 1 for any partition (Ai ), νM [ξ ] can be no larger than ξ MU . This bound is attained, for example, whenever the POVM is ‘rank–one’ in the following sense:
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Definition 9. Call a POVM M : F → B + (H) with outcomes in the measurable space (X, F) ‘rank–one’ if there is a countable partition (Ai ) of X into Ai ∈ F such that rank M(Ai ) = 1 for all Ai . As a consequence of Proposition 8, we arrive at the following theorem. Theorem 10. The supremum of λ(M) over all single POVMs M in dimension D is attained by the isotropic POVM MU . In addition the infimum of µ(M) over all rank–one POVMs is µ(MU ).
1 2 1 a k b k + = ± o(1) , λ(MU ) = min 1 − 1≤a≤D/2 k D k=0,...a−1 D D D π b=D−a =0,...b−1
(17) 1 (18) µ(MU ) = . 2 Proof. Since randomising a POVM over unitary transformations (which we record) cannot decrease λ we can, without loss of generality, perform the symmetrization over the Haar measure described above without decreasing λ. From our discussion of such symmetrized POVMs it is clear that none has a larger value of λ than the isotropic POVM. The statement about µ for rank–one POVMs follows from the fact that the symmetrized version of such a POVM has the same bias as the isotropic POVM. For the constants of domination for MU , first note that for any operator ξ , (19) νMU [ξ ] = D dψ| Tr(|ψψ|ξ )|. Note that since ξ is Hermitian, we may again take ξ = (1 − p)ρ − pσ for orthogonal operators ρ and σ . For Eq. (17), we then have by the unitary invariance of the uniform POVM and the triangle inequality, λ(MU ) is attained as ξ MU for an operator of the form 1 1 P− Q, with a projector P of rank a, and Q = 1 − P, b = D − a, ξ= 2a 2b where we may even take P to be the projector onto the subspace spanned by the first a computational basis vectors (again invoking unitary invariance). For this choice of operator, according to Eq. (19), and letting p := a/D,
a D
1
1 2 2
ξ MU = D dψ |ψ j | − |ψ j | 2a 2b
j=1 j=a+1
1 k+ = 1− , (20) p k (1 − p) k D k=0,...a−1 =0,...b−1
by Lemma 24 in Appendix B. It is quite natural to conjecture that the minimal choice of ranks is a = D/2 and b = D/2. In this case we have
D/2 k D/2 k + 1 ξ MU = 1 − k D k=0,...D/2−1 D D =
=0,...D/2−1
2 ±O πD
1 D
,
(21)
Distinguishability of Quantum States Under Restricted Families of Measurements
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for large D. The analysis of the asymptotics is elementary but lengthy, and is here restricted to a few hints: We lose only terms of order O(1/D) by focusing on even D, for which the formula evaluates to
D/2−1 D/2−1 1 −2k 2k 1 −k− k + = , λ(MU ) = 1 − 2 2 k k D D k=0
k,=0
where we have used the following identity from Lemma 25, proved by induction on k: k
2−k−
=0
k+ = 1.
(22)
Then a simple application of Stirling’s formula (with explicit error bounds) yields Eq. (21). However, since we have not been able to prove that the minimum value of the expression 20 occurs for this choice of ranks, we instead follow a different route: From the proof of Lemma 24 in Appendix B, we observe that for general a and b,
a d
1
1 ξ MU = E
Xj − X j
, 2b
2a
j=1
j=a+1
with independent X j ≥ 0, each distributed according to a rescaled χ22 law. By definition, their expectation and variance are EX j = 1 and Var X j = 1, respectively (also, all higher moments are finite). Thus, by the central limit theorem,
a 1 1 , X j ≈ Y0 ∼ N 1, 2a 4a j=1
d 1 1 , X j ≈ Y1 ∼ N 1, 2b 4b j=a+1
where Y0 and Y1 are normal distributed with means µ and variance ν as indicated by N (µ, ν), and the approximation signs indicate convergence in probability as a, b → ∞. (Note that since the third moment of the X j is finite, this convergence is uniform in a and b, thanks to the Berry-Esséen theorem which bounds the rate of convergence in the central limit theorem – see e.g. [13].) 1 1 Since Y0 − Y1 =: Z ∼ N 0, 4a , we obtain asymptotically + 4b ∞ 1 1 1 1 2 1 −x 2 /2 + + , ξ MU ∼ E|Z | = dx|x|e = √ 4a 4b 2π −∞ π 4a 4b which is minimized for a = b = D/2, yielding λ(MU ) ∼ π2D , as advertised. For Eq. (18), note that by the triangle inequality, µ(M) of any POVM M is attained for an extremal traceless ξ such that ξ 1 = 1. These are easily seen to be of the form ξ = 21 |φ1 φ1 |− 21 |φ2 φ2 | for orthogonal pure state vectors |φ1 , |φ2 . By unitary invariance of the uniform POVM, any such ξ will in fact yield the same value, so we may take ξ = 21 |11| − 21 |22|, so that by Eq. (19),
1
D
µ(MU ) = ξ MU = dψ |ψ1 |2 − |ψ2 |2 = , 2 2 once more by Lemma 24 in Appendix B, applied with a = b = 1.
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Note that in terms of the bias the above translates to 1 2 1 ρ − σ 1 ≥ ρ − σ M ≥ − o(1) ρ − σ 1 . 2 D π B. Almost optimal performance of 4-designs. The results of the previous section provide the motivation to look at POVMs made from t-designs, as these are structures approximating the full random POVM better and better as t → ∞. We thus intuitively expect to obtain a similar value for λ as we obtained for the random POVM for larger t. On the k th tensor power H⊗k of a Hilbert space H, there is a natural unitary representation of the permutation group of order k, Sk , which permutes the k tensor factors. That is, for any π in Sk , Uπ
k
|ψ j =
j=1
k
|ψπ −1 ( j) .
j=1 (k)
We denote the projector onto the completely symmetric subspace of H⊗k by Psym . It has and can be expressed as an average over the action unitary representation rank D+k−1 k just described (see, for example, [1]), 1 (t) = Uπ . Psym k! π ∈Sk
Definition 11. A (weighted) spherical t-design is an ensemble ( pk , Pk )nk=1 of 1-dimensional projectors Pk and probabilities pk such that 1 1 (t) pk Pk⊗t = D+t−1 Psym = D+t−1 Uπ . t! t t k π ∈St Note that the isotropic POVM is an ∞-design. We call a t-design proper if all the probabilities are equal, pk = 1/n. Note that any t-design is automatically also a t -design for all t < t. In particular, k pk Pk = D1 1, so it makes sense to associate a POVM with every t-design of the form (E k )nk=1 , with E k = Dpk Pk , which, as before, we also call a (weighted or proper) t-design. It turns out that 4-designs already achieve essentially the same worst–case bias as the isotropic POVM (in the sense that the dimensional dependence is the same). This was discovered by Ambainis and Emerson [5], who showed, invoking a beautiful moment inequality by Berger, that if M4 is a 4-design POVM then 1 1 ρ − σ 2 ≥ √ ρ − σ 1 . (23) 3 3 D We briefly review their argument, including the Berger inequality, as we need to return to this later on in Sect. 4. ρ − σ M4 ≥
Lemma 12 (Berger [12]). For a real random variable S, E|S| ≥
(ES 2 )3/2 . (ES 4 )1/2
Distinguishability of Quantum States Under Restricted Families of Measurements
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Proof. That is just Hölder’s inequality, which states that for real random variables f and g, and 1p + q1 = 1, 1/ p 1/q E|g|q . E( f g) ≤ E| f | p Here it is applied with f = |S|2/3 , g = |S|4/3 and p = 3/2, q = 3.
Proof (of Eq. (23) – see [5]). For traceless ξ , consider the random variable S which takes value D Tr(ξ Pk ) with probability pk . Then clearly E|S| = ξ M4 , and Berger’s inequality can be used. The moments are easy calculations, using the fact that the POVM is a 4-design. First, the second moment, ES 2 = pk D 2 (Tr(ξ Pk ))2 k
=
pk D 2 Tr ((ξ ⊗ ξ )(Pk ⊗ Pk ))
k
= D 2 Tr (ξ ⊗ ξ ) =
2 P (2) D(D + 1) sym
D2 D Tr ((ξ ⊗ ξ )(1 + F)) = Tr(ξ 2 ), D(D + 1) D+1
where F is the swap operator, that is F = Us , where s is the non–identity element of S2 , and we have made use of Tr(ξ ) = 0. Similarly, ES 4 = pk D 4 (Tr(ξ Pk ))4 k
=
pk D 4 Tr ((ξ ⊗ ξ ⊗ ξ ⊗ ξ )(Pk ⊗ Pk ⊗ Pk ⊗ Pk ))
k
24 (4) Psym D(D + 1)(D + 2)(D + 3) D4 = Tr ξ ⊗4 Uπ D(D + 1)(D + 2)(D + 3)
= D 4 Tr ξ ⊗4
π ∈S4
D 3 6(Tr(ξ 2 ))2 + 3 Tr(ξ 4 ) ≤ = 9(Tr(ξ 2 ))2 . (D + 1)(D + 2)(D + 3) D+1 D3
The equality in the last line comes from the fact that there are 3 elements of the permutation group S4 with a 4–cycle, each giving rise to a term equal to Tr(ξ 4 ), and likewise, 6 elements which have 2 2–cycles, each yielding a term equal to (Tr(ξ 2 ))2 . All other elements of the group have at least one fixed point so the corresponding terms contain a factor of Tr(ξ ), which is zero. The final inequality is just an application of the Cauchy–Schwartz inequality. Thus, ξ M4 = E|S| ≥
1 1 1 Tr(ξ 2 ) = ξ 2 ≥ √ ξ 1 . 3 3 3 D
√ In other words: λ(M4 ) ≥ 1/(3 D).
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It is not known how to construct spherical 4-designs efficiently in general though 2 . To see this, note that there must exist a weighted 4-design of cardinality at most D+3 4 (4) (4) the normalised projector Psym /(Tr Psym ) lies in the convex hull of normalised symmet 2 − 1 dimensional real subspace of ric product states, which are a subset of the D+3 4 trace–one hermitian operators on the symmetric subspace. Carathéodory’s theorem [3] tells us that any point in the convex hull of a subset S of a n dimensional space can be written as a convex combination of n + 1 points from S. Constructions are known for a real vector space of small dimensions [21]. Ambainis and Emerson [5] construct approximate 4-designs which perform almost as well as Eq. (23).
C. Performance of 2-designs. Unfortunately, we have as yet been unable to give the bias for 3-design POVMs, but here we show how to bound it for 2-designs. Consider first a proper 2-design with associated POVM (E k = Dn Pk )nk=1 . I.e., 2 1 1 (1 + F) = P (2) , Pk ⊗ Pk = n D(D + 1) D(D + 1) sym k
(2)
with the projector Psym onto the symmetric subspace of C D ⊗ C D and the swap operator F. Such POVMs are always informationally complete – this will also follow from the theorem below. An example of a 2-design is a complete set of D + 1 mutually unbiased bases, which are known to exist if the dimension D is a prime power [10,32]). Let (|ψsb )s=1...D : b = 0, . . . , D , be the basis vectors of the D +1 mutually unbiased bases, where |ψsb is the s th basis vector of the bth basis. Then the set of basis state projectors Psb = |ψsb ψsb | forms a proper spherical 2-design [25]. It is conjectured that in all dimensions there exist spherical 2-designs with the minimum number n = D 2 of elements [28], giving rise to so-called symmetric informationally complete (SIC) POVMs. These are only known to exist up to dimension D = 45 [28] by numerical results, and for even fewer dimensions up to D = 19 by mathematical construction. Zauner’s conjecture states that in every dimension there exists a SIC-POVM of a particularly beautiful group symmetric form [33]. We refer to [7,15] for more information. Let M2 be any 2-design POVM. Our objective is to prove the relation. Theorem 13. For any traceless Hermitian operator ξ , ξ M2 ≥
1 1 ξ 1 . 2 D+1
In other words, for any proper 2-design POVM as above, λ(M2 ) ≥
(24) 1 1 2 D+1 .
Proof. Since this is a homogeneous relation, we may w.l.o.g. assume that ξ 1 = 2, meaning that we can write ξ = ρ − σ with two orthogonal density operators ρ and σ . 1 Thus, what we need to show is νM2 [ρ] − νM2 [σ ] ≥ D+1 .
Distinguishability of Quantum States Under Restricted Families of Measurements
827
For this, we use Proposition 21 in Appendix A, Ineq. (A1), for the vectors p and q defined as pk = Tr(ρ E k ) =
D D Tr(ρ Pk ), qk = Tr(σ E k ) = Tr(σ Pk ). n n
Namely, νM2 [ρ] − νM2 [σ ] = p − q1 D2 ≥1−n (Tr(ρ Pk ))(Tr(σ Pk )) n2 k 1 Tr ((ρ ⊗ σ )(Pk ⊗ Pk )) . = 1 − D2 n k
Now, the last sum can be evaluated as follows, using the property of spherical 2-design: 1 1 Tr(ρ Pk σ Pk ) = Tr ((Pk ⊗ Pk )(ρ ⊗ σ )) n n k
k
1 Tr ((1 + F)(ρ ⊗ σ )) D(D + 1) 1 1 (Tr(ρ) Tr(σ ) + Tr(ρσ )) = . = D(D + 1) D(D + 1) =
Inserting this above, we conclude νM2 [ρ] − νM2 [σ ]1 ≥ 1 − D 2 as advertised.
1 1 = , D(D + 1) D+1
Theorem 14. For a POVM M2 which is a weighted 2-design the conclusion of 1 Theorem 13 still holds: λ(M2 ) ≥ 21 D+1 . Proof. The idea is to break down the probabilities pk into smaller but approximately equal values. This increases the number of outcomes of the POVM, but makes it be approximated better and better by a proper 2-design, to which we can apply Theorem 13. In detail, assume that our weighted 2-design is discrete, with n elements; choose an integer N 1, and for each k let Nk = N pk and k = N pk − Nk . Define a new weighted 2-design with the same projectors Pk = Pk and “uniformised” weights βk =
k /N 1/N
for = 0, for = 1, . . . , Nk .
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W. Matthews, S. Wehner, A. Winter
Then, applying the same proof as in Theorem 13 to this refined 2-design (which has N + n outcomes), we get νM [ρ] − νM [σ ]1 = p − q1 ≥ 1 − (N + n)
2 βk D 2 (Tr(ρ Pk ))(Tr(σ Pk ))
k
N +n ≥ 1 − D2 βk Tr ((ρ ⊗ σ )(Pk ⊗ Pk )) N k
D2
N +n Tr [(1 + F)(ρ ⊗ σ )] D(D + 1) N D n 1 =1− 1+ → , D+1 N D+1 =1−
where we have used βk ≤ 1/N in the third line.
Note that the factor of 1/(D + 1) in the bound (24) is essentially best possible (up to a constant independent of D), as the example of D + 1 mutually unbiased bases shows. Indeed, if the two states ρ and σ are distinct elements of one of the bases, then the measured output distributions for all the D other bases are the same, namely uniform, 2 while in their proper basis the trace distance remains 2, so νM [ρ] − νM [σ ]1 = D+1 , 1 and hence λ(M) ≤ D+1 . Similarly, for a SIC-POVM with D 2 operators D1 Pk it is easily verified that two states from the POVM, i.e. for instance ρ = P1 and σ = P2 , have trace norm difference 2D 2 ρ − σ 1 = D+1 , while νM [ρ] − νM [σ ]1 = D+1 , so λ(M) ≤ D1 . 4. Local POVMs Consider now a multipartite system H = H1 ⊗ H2 ⊗ · · · ⊗ Hn , of local Hilbert spaces H j of dimension d j . (The total space’s dimension is denoted D = d1 d2 · · · dn in this section.) This partition suggests various classes of POVMs due to restrictions of locality. For instance, let LO be the class of all local operations, i.e. tensor product measurements: ( j) (1) (n) LO = E k1 ⊗ · · · ⊗ E kn : (E k j ) POVM on H j . More generally, LOCC is the class of measurements that can be implemented by local operations and classical communication between the parties. SEP are the separable POVMs, i.e. (1) ( j) (1) (n) (n) Ek ⊗ · · · ⊗ Ek : E k ≥ 0, Ek ⊗ · · · ⊗ Ek = 1 . SEP = k
Finally, there is the class of PPT POVMs: denoting the transpose operation (with respect to any basis) by T , it is ⎧ ⎫ ⎞ ⎛ ⎨ ⎬ PPT = (E k ) POVM : ∀k∀I ⊂ [n] ⎝ T⊗ id⎠ E k ≥ 0 , ⎩ ⎭ i∈I
i ∈ I
Distinguishability of Quantum States Under Restricted Families of Measurements
829
i.e. all POVM elements have to be PPT with respect to every bipartition of the n-party system. It is not hard to see that LO ⊂ LOCC ⊂ SEP ⊂ PPT, and all inclusions are known to be strict, at least if the dimension is large enough (see [18] and [19]). The corresponding symmetric convex bodies of operators are denoted LO ⊂ LOCC ⊂ SEP ⊂ PPT. These are interesting examples of POVM classes since we know due to so-called quantum data hiding [14,26,31] that ξ M for them can be much smaller than · 1 . Indeed, it was shown in these references that in a bipartite system Cd ⊗ Cd , the states 1+F 1−F σ = d(d+1) and α = d(d−1) , i.e. the (normalised) projectors onto the symmetric and antisymmetric subspace, respectively, obey % % %1 % 2 % ρ − 1σ% = . %2 % 2 PPT d +1 (In [14] more general statements of this type for n-partite systems can be found.) Con2 . The next result shows that this bound is quite sharp: sequently, λ(PPT) ≤ d+1 Lemma 15. For any operator ξ on an n-partite system, ξ SEP ≥ In particular, λ(SEP) ≥
2 √1 ; 2n/2 D
2 2 1 ξ 2 ≥ n/2 √ ξ 1 . 2n/2 2 D for a bipartite system, we find λ(SEP) ≥
√1 . D
Proof. Gurvits and Barnum [20] have shown that for a bipartite system, within the set of Hermitian operators, the unit ball of the Hilbert-Schmidt norm centred on the identity operator contains only separable operators. More generally they proved in an n-partite system, that the ball of radius 21−n/2 around the identity is fully separable [20]. It follows immediately that all the POVMs in the set (E, 1 − E) : 2E − 12 ≤ 21−n/2 are separable. It is easy to see that the corresponding symmetric convex body (see Lemma 2) is the ball of radius 21−n/2 in the Hilbert-Schmidt norm around the origin and so this is a subset of SEP. From this inclusion, and the fact that the Hilbert–Schmidt norm is self–dual, ξ SEP = max Tr (Eξ ) ≥ E∈SEP
max
E2
concluding the proof, if we recall ξ 1 ≤
√
≤21−n/2
Tr (Eξ ) =
Dξ 2 .
2 2n/2
ξ 2 ,
We now come to the main technical result of the present section, showing that this order of magnitude goes through all the way to LO, indeed, a particular tensor product POVM on a bipartite system is already almost as good as the class of all separable POVMs, in terms of the constant of domination. Note that Proposition 8 gives us the local POVM with the largest λ: namely, by symmetrising over all unitaries U = U A ⊗ U B , drawn from the product of the local Haar measures, we find that for any tensor product POVM MA ⊗ MB , we have λ (MUU ) ≥ λ (MA ⊗ MB ), where MUU denotes the tensor product of the isotropic POVMs on the two subsystems.
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W. Matthews, S. Wehner, A. Winter
Theorem 16. For any two states ρ and σ on a bipartite Hilbert space H A ⊗ H B , let ξ = ρ − σ . Then, 1 1 ξ 2 ≥ √ ξ 1 , ξ MUU ≥ √ 153 153D
√ where D = d A d B is the Hilbert space dimension. Consequently, λ (MUU ) ≥ 1/ 153D. Proof. We do exactly the same as in Subsect. 3B, only that we have now a POVM on H A ⊗ H B of the form (Ddϕdψ|ϕϕ| ⊗ |ψψ|), so S is the variable S = D Tr((|ϕϕ| ⊗ |ψψ|)ξ ), and the bias of the estimation based on the outcome is E|S|, as before in Subsect. 3 B. We use Berger’s inequality, Lemma 12 again, for which we need the second and fourth moment. Because now we randomise independently over H A and H B , we get 22 d 2A d B2 AA BB Tr (sym ES 2 = ⊗ sym )(ξ AB ⊗ ξ AB ) , d A (d B + 1)d B (d B + 1) 242 d 4A d B4 ES 4 = d A (d A + 1)(d A + 2)(d A + 3)d B (d B + 1)(d B + 2)(d B + 3) AAAA BBBB × Tr (sym ⊗ sym )(ξ AB ⊗ ξ AB ⊗ ξ AB ⊗ ξ AB ) , where the superscripts remind one of the systems these operators act on. Expanding the projectors into the permutations of two, respectively four, elements, we get dAdB ES 2 = Tr(ξ A2 ) + Tr(ξ B2 ) + Tr(ξ 2 ) , (25) (d A + 1)(d B + 1) where ξ A = Tr B (ξ ) and ξ B = Tr A (ξ ), because we get terms with 1 A A ⊗ 1 B B , 1 A A ⊗ F B B , F A A ⊗ 1 B B and F A A ⊗ F B B . The fourth moment is considerably more complex: looking at ES 4 =
d 3A d B3 (d A + 1)(d A + 2)(d A + 3)(d B + 1)(d B + 2)(d B + 3) × Tr (UπA A A A ⊗ UσB B B B )ξ ⊗4 ,
(26)
π,σ ∈S4
wesee that we need to calculate – or at least reasonably upper bound – the trace terms Tr (UπA A A A ⊗ UσB B B B )ξ ⊗4 . In Appendix C, Lemma 26 we show that Tr (UπA A A A ⊗ UσB B B B )ξ ⊗4 π,σ ∈S4
≤ 153(Tr(ξ 2 ))2 + 126(Tr(ξ 2 ))(Tr(ξ A2 )) + 126(Tr(ξ 2 ))(Tr(ξ B2 )) + 9(Tr(ξ A2 ))2 + 9(Tr(ξ B2 ))2 + 30(Tr(ξ A2 ))(Tr(ξ B2 )) 2 ≤ 153 Tr(ξ 2 ) + Tr(ξ A2 ) + Tr(ξ B2 ) .
Distinguishability of Quantum States Under Restricted Families of Measurements
Plugging this into Eq. (26), we find 3
2 dAdB 4 153 Tr(ξ 2 ) + Tr(ξ A2 ) + Tr(ξ B2 ) . ES ≤ (d A + 1)(d B + 1)
831
(27)
Now we conclude as in the single-system case: by virtue of Eqs. (25) and (27), ξ MUU = E|S| & (ES 2 )3 ≥ ES 4 1 ≥√ Tr(ξ 2 ) + Tr(ξ A2 ) + Tr(ξ B2 ) 153 1 1 ξ 2 ≥ √ ξ 1 , ≥√ 153 153D and we are done.
Remark 17. From the proof we see that, just as in the single-system case of Subsect. 3 B, it is enough for the local measurements to be 4-designs. Corollary 18. The constants of domination, for locality-restricted measurements on a d × d-system, are in the following relations: 1 2 . (28) ≤ λ (MUU ) ≤ λ(LO) ≤ λ(LOCC) ≤ λ(SEP) ≤ λ(PPT) ≤ √ d +1 153d For separable measurements we have the even tighter bounds, 1 2 ≤ λ(SEP) ≤ λ(PPT) ≤ . d d +1
(29)
Proof. The first inequality in (28) is just Theorem 16, the chain is by inclusion of the sets of POVMs, with the last bound following from the data hiding states αd and σd , the (appropriately normalised) projections onto the (anti-)symmetric subspace of Cd ⊗ Cd – see [31,14] and [26]. By Lemma 15 finally, λ(SEP) ≥ √1 = d1 . D
Remark 19. The first inequality (28) in Corollary 18 proves a conjecture about the optimal bias achievable with LOCC measurements ([26, Conjecture 7]. Compare also with [31], where a bias of order 1/d 2 was proven using a particular informationally complete measurement, and it was suggested there that better POVMs might exist. This result shows that in a very strong sense the original data hiding states, the symmetric and anti-symmetric subspace projections, are essentially optimal: up to a constant factor they achieve the best available bias, which is (1/d). Remark 20. The 2 -bound in Theorem 16 has another notable consequence for data hiding: observing that for orthogonal states ρ and σ , ρ − σ 2 = Tr(ρ 2 ) + Tr(σ 2 ) ≥ max {ρ2 , σ 2 } , we conclude that data hiding states have to be highly mixed. If one of them has rank bounded by r , say, Theorem 16 places a lower bound of 1/13r on the bias achievable by LOCC measurements.
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Indeed, all known constructions of data hiding states endow them with considerable entropy (comparable to or larger than the size of the “shares”), see [14,22,31]. Our bound tells us that this has to be so to guarantee security of the scheme. We intend to return to this issue on a separate occasion.
5. Certainty Relations The results on λ(MU ) for the isotropic POVM, tensor products of isotropic POVMs, and 2-designs have nice interpretations as “certainty relations” in the sense of SanchezRuiz [29]. Namely, for a complete set of D + 1 mutually unbiased bases in C D with associated basis measurements Bk , he shows that for any pure state ϕ = |ϕϕ|, D+1 S2 νBk [ϕ] ≤ (D + 1) log D − log(D − 1), ≤ 2 D
(D + 1) log
(30)
k=0
where S2 νBk [ϕ] = − log x |x|ϕ|4 is the Rényi entropy of order 2 for the orthonormal basis {|1, . . . |D}. The right hand side of Eq. (30) is referred to as a certainty relation, and intuitively states that for the chosen measurements there exists no pure state that will lead to maximum entropy for all measurements simultaneously. It quantifies the fact (quite natural, after a moment of thought) that not all the tomographic data from measuring those bases is equally informative in the sense of Shannon. The certainty relation of [29] also holds for the Shannon entropy. Let M be the POVM formed by measuring in one of the D + 1 bases at random. Using the concavity of the log, the certainty relation can then be rewritten as log (D(D + 1)) − S2 (νM [ϕ]) ≥
1 log(D − 1). D+1
From our results in the previous section, we can infer similar certainty relations. First of all, from Theorem 13 we get the following more general but weaker bound for any proper 2-design POVM with n outcomes: log n − S2 (νM [ϕ]) ≥ log n − S (νM [ϕ]) = D (νM [ϕ]νM [1/D]) 1 ≥ νM [ϕ − 1/D]2 2 ln 2 d −1 1 1 1 ≥ ≥ , 2 4 ln 2 D(D + 1) 6 ln 2 (D + 1)2 where D(··) is the classical relative entropy and the second inequality follows from the Pinsker inequality D(µν) ≥ 2 ln1 2 µ − ν2 (see [4], for example, for definitions of the relative entropy between measures). For uni- and bipartite 4-designs, in particular the isotropic POVMs, we get considerably better bounds, due to the appearance of the Hilbert-Schmidt norm. Consider any ensemble of quantum states, ρ = x px ρx . For the Shannon mutual information between the preparation variable X (distributed according to px ) and the measurement
Distinguishability of Quantum States Under Restricted Families of Measurements
833
outcome given by U, I (X : MU ) =
px D νMU [ρx ]νMU [ρ]
x
≥
px
x
≥
px
x
=
1 18 ln 2
1 = 18 ln 2
% 1 % %νM [ρx ] − νM [ρ]%2 U U 2 ln 2 1 ρx − ρ22 18 ln 2 px Tr(ρx2 ) − Tr(ρ 2 )
x
SL (ρ) −
px SL (ρx ) .
(31)
x
In other words, we get a lower bound on the accessible information of the ensemble in terms of so-called “linear entropies” SL (ρ) = 1 − Tr(ρ 2 ). In the above derivation we have used the well-known relation between mutual information and relative entropy, the Pinsker inequality and Eq. (23). A particularly interesting case is that of a pure state ensemble ρx = |ϕx ϕx |: all the SL (ρx ) are zero, so we get a positive lower bound for the accessible information Iacc ({ px , ϕx }) ≥ I (X : MU ) ≥
1 1 − Tr(ρ 2 ) , 18 ln 2
which is a small but positive constant, depending only on ρ. It turns out that the best possible lower bound on the accessible information in terms solely of ρ is known: it is the so-called subentropy Q(ρ) of Jozsa, Robb and Wootters [24], attained on a particular ensemble decomposition of ρ, named after Ebenezer Scrooge. Incidentally, for this ensemble all complete (i.e., rank-1) POVMs have the same information gain. It is largest on the maximally mixed state, and bounded by 1−γ ln 2 ≈ .6099, where γ is Euler’s constant [24]. LOCC (·), that is the For bipartite systems we furthermore obtain a lower bound for Iacc accessible information when we are restricted to performing LOCC measurements. This bound is obtained by using Theorem 16 to lower bound I (X : MUU ) – the mutual information when the locally unitarily invariant continuous POVM is used. This quantity is studied as a lower bound on the locally accessible information in [30] (where it is denoted L ({ px , ϕx })). Unlike the subentropy, this quantity depends on the ensemble (rather than the ensemble average alone) even when it is a pure state ensemble. However, in [30] it is interpreted differently as the average of the mutual information over all complete product basis measurements. Since some measurements of this form cannot be performed by LOCC, the authors (unnecessarily) restrict their claim that it is a lower bound on the locally accessible information to bipartite systems of 2 × n dimensions (where it is known that any complete product basis measurement can be performed by LOCC). This is unnecessary because, as described in Sect. 4, I (X : MUU ) is also the mutual information yielded by the protocol where Alice and Bob independently measure according to the unitarily invariant continuous POVM and share their results (which is clearly accomplished by LOCC). As noted in [30], this bound is saturated by Scrooge ensembles.
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No general closed form is known for I (X : MUU ) (although some special cases are derived in [30]) so it is useful to note that by using the same derivation as in (31), but invoking Theorem 16, we get that for an arbitrary ensemble on a bipartite system, 1 LOCC px SL (ρx ) . (32) Iacc SL (ρ) − ({ px , ρx }) ≥ I (X : MUU ) ≥ 306 ln 2 x It is worth noting that in the case of an ensemble of pure states this lower bound, unlike I (X : MUU ), depends only on the ensemble average. Hence we get a lower bound of 1 LOCC 1 − Tr(ρ 2 ) Q LOCC (ρ) := inf Iacc ({ px , ϕx }) ≥ 306 ln 2 ρ= x px ϕx on the LOCC-subentropy of ρ. 6. Conclusion We have introduced a formalism of norms on states/density operators linked to their (pairwise) distinguishability by a given, restricted, class of measurements. This allows us to study the relation between these norms in convex geometric terms. We went on to investigate the constants of domination for the resulting norms with respect to the well-known trace norm: for a single measurement we looked at the isotropic POVM, 4- and 2-designs. Furthermore, we considered several classes of locally restricted measurements, such as LOCC or PPT POVMs. The results here have strong connection to data hiding: indeed, we proved that up to a constant factor the hiding states of [31] achieve already the best possible bias. We leave many questions open, such as the eventual determination of the locally accessible information and better bounds on the constants of domination. More importantly, one ought to be able to obtain more information on the geometry of the convex bodies M and the unit balls of · M – here we only compared them with the trace and the Hilbert-Schmidt norms, but it would be interesting to get more insight into their geometric shape. It is an intriguing open question regarding single measurements where to place 3-design POVMs relative to 2- and 4-designs. Acknowledgements. AW thanks the members of the Pavia Quantum Information group for an enjoyable afternoon in October 2007, where he had occasion to discuss some of the questions of the present paper, when they were still in a nascent state. In particular the feedback of G. M. D’Ariano, G. Chiribella and M. F. Sacchi, and their suggestions regarding the use of symmetry, are gratefully acknowledged. Ashley Montanaro provided the pointer to the paper by Ambainis and Emerson, and provided the example mentioned in Appendix A. WM would like to thank Dan Shepherd for a useful discussion about groups and diagrams. WM was supported by the U.K. EPSRC. SW was supported by NSF grant number PHY-04056720. AW was supported by the U.K. EPSRC through the “QIP IRC” and an Advanced Fellowship, by a Royal Society Wolfson Merit Award and by the European Commission through IP “QAP”. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.
Appendix A: An 1 -Inequality for Probability Vectors and Density Operators n pi = 1, and Proposition 21. For probability vectors p, q in Rn (i.e. pi ≥ 0 and i=1 likewise for qi ), p − q1 ≥ 1 − n p · q,
(A1)
Distinguishability of Quantum States Under Restricted Families of Measurements
835
where on the left is the statistical distance between the distributions, namely the 1 -norm of their difference, and on the right we have the usual Euclidean inner product of vectors. Corollary 22 (Quantum case). Ineq. (A1) has a straightforward quantum generalisation: for any two density operators ρ and σ on an n-dimensional Hilbert space, ρ − σ 1 ≥ 1 − n Tr(ρσ ),
(A2)
where now on the left is the trace norm, and on the right is the Hilbert-Schmidt inner product on operator space. This actually follows from the classical case, as follows: ρ is diagonalised in some basis, with a probability vector p along the diagonal. Denote the dephasing operation in this basis by E – it is a CPTP map with E(ρ) = ρ. Denoting σ = E(σ ), which is now diagonalised in the same basis, with a probability vector q along the diagonal, we now have 1 1 ρ − σ 1 ≥ ρ − σ 1 and Tr(ρσ ) = Tr(ρσ ), 2 2 so all we need to prove is 1 ρ − σ 1 ≥ 1 − n Tr(ρσ ). 2 But because of 1 1 ρ − σ 1 = p − q1 and Tr(ρσ ) = p · q, 2 2 this is precisely (A1). Proof of Proposition 21. We use the well-known relation between trace distance and fidelity [16]: √ 1 p − q1 ≥ 1 − pi qi , 2 i
hence we are done once we show √ pi qi ≥ 1 − n pi qi , 2 1− i
√
i
which – introducing the shorthand ti = pi qi – is equivalent to 1 1 2 ti ≤ + n ti . 2 2 i i Now, for fixed s = i ti ≤ 1, the right hand side here is minimal for t1 = . . . = tn = ns , in which case it reduces to 21 + 21 s 2 , which is indeed always ≥ s. Remark 23. Ineq. (A1) becomes false when introducing a factor c < 1 on the left hand side. for sufficiently large n. Ashley Montanaro [personal communication] pointed out to us the following class of examples: 1−x 1−x 1−x Consider p = x, 0, 1−x and q = 0, x, , . . . , , . . . , n−2 n−2 n−2 n−2 , which have c p − q1 = 2cx, whereas 1 − n p · q = 1 −
n n−2 (1 −
x)2 ∼ 2x + x 2 for large n.
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Appendix B: An Integral Over the Unit Sphere Lemma 24. Let P and Q be mutually orthogonal projectors of rank a and b, respectively, in Cd . Then, for the uniform distribution on the unit vectors |ψ = dj=1 ψ j | j ∈ Cd ,
a a+b
1
1
1 1 2 2
|ψ j | − |ψ j | E Tr(ψ P) − Tr(ψ Q) = d dψ 2a 2b 2b
2a j=1
j=a+1
1 k+ =1− , p k (1 − p) k a + b k=0,...a−1 =0,...b−1
where p = a/(a + b). d Proof. Introduce a random Gaussian vector |ϕ ∼ NCd (0, 1) [11], i.e. |ϕ = √1 j=1 2d (α j + iβ j )| j with independent Gaussian distributed real and imaginary parts α j , β j ∼ N (0, 1) of zero mean and unit variance. In particular, Eϕ|ϕ = 1. Now, using this and the unitary invariance of the distribution of |ϕ, we see
1
1 1 1 Tr(ψ Q)
= Eϕ ϕ|ϕEψ
Tr(ψ P) − Tr(ψ Q)
E
Tr(ψ P) − 2a 2b 2a 2b
1 1 = Eϕ
Tr(ϕ P) − Tr(ϕ Q)
2a 2b
1 a 1 = Eα j ,β j ∼N (0,1)
(α 2j + β 2j ) 2
2a j=1
a+b
1 2 2 − (α j + β j ) 2b
j=a+1
1 1 1 = E X,Y
X − Y
. 2 2a 2b The sums of squares of Gaussian components occurring here are well-studied, and known under the name of χ 2 -distributions: a 2 (α 2j + β 2j ) =: X ∼ χ2a , j=1
a+b
2 (α 2j + β 2j ) =: Y ∼ χ2b ,
j=a+1
their probability density being given by 1 (x/2)a−1 e−x/2 dx, 2(a − 1)! 1 (y/2)b−1 e−y/2 dy. Pr{Y ∈ [y; y + dy]} = 2(b − 1)!
Pr{X ∈ [x; x + dx]} =
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This allows us to evaluate the latter expectation as follows, denoting the indicator function of a set {. . .} as 1{. . .}:
1 1 1
1
E X,Y X − Y = E X,Y dr 1{X/2a ≤ r ≤ Y/2b} 2 2a 2b 2 + dr 1{Y/2b ≤ r ≤ X/2a} 1 ∞ dr (E 1{X ≤ 2ar, Y ≥ 2br } + E 1{X ≥ 2ar, Y ≤ 2b}) = 2 0 ∞ 1 = dr (Pr{X ≤ 2ar } Pr{Y ≥ 2br } 2 0 + Pr{X ≥ 2ar } Pr{Y ≤ 2br }) 1 ∞ 1 ∞ dr Pr{X ≥ 2ar } + dr Pr{Y ≥ 2br } = 2 0 2 0 ∞ dr Pr{X ≥ 2ar } Pr{Y ≥ 2br }. − 0
Using the
χ2
densities, the probabilities under the integrals are easily evaluated:
Pr{X ≥ 2ar } = e−ar
a−1 (ar )k k=0
k!
, Pr{Y ≥ 2br } = e−br
b−1 (br ) =0
!
.
This finally gives
1
1 1
E Tr(ψ P) − Tr(ψ Q)
= E X,Y 2a 2b 2
1
X − 1 Y
2a 2b ∞ (ar )k (br ) 1 1 = + − dr e−r (a+b) 2 2 k!! 0 k=0,...a−1 =0,...b−1
a k b k + 1 , =1− k a + b k=0,...a−1 a + b a+b
=0,...b−1
where we have used the integral for the Gamma function.
We will also need the following small lemma: k Lemma 25. Let Sk denote l=0 2−(k+l) k+l l . We claim that for integers k ≥ 0, Sk = 1. n−1 Proof. Using the well known ’addition formula’ mn = n−1 m + m−1 , Sk+1 =
k+1 l=0
2−(1+k+l)
k+1 k +l k +l + 2−(1+k+l) l l −1
k 1 2k + 1 k +l +1 Sk + 2−(2k+2) + 2−(2+k+l) 2 k+1 l l=0
2k + 1 1 2k + 2 1 1 + Sk+1 − 2−(2k+2) = Sk + 2−(2k+2) k+1 k+1 2 2 2 =
(B1)
l=0
(B2) (B3)
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so Sk+1 = Sk + 2
−(2k+2)
2k + 1 2k + 2 2 − = Sk , k+1 k+1
where the final equality is due to the addition formula and the symmetry To complete the proof we note that S0 = 1.
2k+1 k+1
=
2k+1 k−1 .
Appendix C: Upper Bounds on Certain Traces Lemma 26. Let ξ be a traceless Hermitian operator on a bipartite Hilbert space H A ⊗ (4) (4) H B . Let Psym A and Psym B denote the projector onto the completely symmetric subspace
⊗4 of H⊗4 A and H B , respectively. Then, with the shorthands t := Tr(ξ 2 ), a := Tr(ξ A2 ) and b := Tr(ξ B2 ), where ξ A = Tr B (ξ ) and ξ B = Tr A (ξ ), 1 (4) (4) Tr Psym A ⊗ Psym B ξ ⊗4 ≤ 2 153t 2 + 126ta + 126tb + 9a 2 + 9b2 + 30ab . 4! (C1)
The proof is conceptually simple but a little long. We write the projection operators as averages over the unitary operators which permute the four subsystems. Defining, for permutations π ∈ S4 , the representation UπA :=
4
j∈{1,...,d}m
i=1
A | ji π(i) ji |iA ,
where {| jiA }1≤ j≤d is an orthonormal basis for the i th copy of H A in H⊗4 A , and defining UπB similarly: 1 (4) (4) A B ⊗4 . Tr U ⊗ U ξ Tr Psym A ⊗ Psym B ξ ⊗4 = π σ 24!2 π ∈S4 ,σ ∈S4
Clearly (π, σ ) → UπA ⊗ UσB is a representation of S4 × S4 . S4 × S4 has a subgroup consisting of all the elements of the form (g, g), which we’ll denote by R. R
If (π , σ ) = r −1 (π, σ )r for some r ∈ R, we write (π , σ ) ∼ (π, σ ) and note that the corresponding terms are equal since Tr UπA ⊗ UσB ξ ⊗4 = Tr (UπA ⊗ UσB )(UgA ⊗ UgB )ξ ⊗4 (UgA−1 ⊗ UgB−1 ) = Tr UπA ⊗ UσB ξ ⊗4 . Essentially, conjugation by an element of R corresponds to a permutation of the identical ξ operators, and therefore leaves the term unchanged. R
The set of all 24!2 terms is partitioned by the equivalence relation ∼ with the terms in each subset all equal to each other. We shall refer to these subsets as the R-conjugacy classes of S4 × S4 . Clearly, the R-conjugacy classes form a finer partition of S4 × S4 than the normal conjugacy classes.
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By demonstrating an appropriate upper-bound for the terms in each R-conjugacy class, and calculating the size of each class, we will prove the upper bound (C1). Tensor Diagrams. Let us establish an orthonormal basis {|i A } ({|i B }) for H A (H B ). In this basis, we can write ξ in component form thus ξi,k,lj = k| A ⊗ l| B ξ |i A ⊗ | j B . We would like to demonstrate upper bounds for terms of the form aπ(1),bσ (1) aπ(2),bσ (2) aπ(3),bσ (3) aπ(4),bσ (4) Tr UπA ⊗ UσB ξ ⊗4 = ξa1,b1 ξa2,b2 ξa3,b3 ξa4,b4 , (C2) where the ai and bi (i ∈ {1, 2, 3, 4}) are dummy variables to be contracted over according to the Einstein summation convention. Using indices in our calculations would be rather messy and confusing. Instead we use the ingenious tensor diagrams of Penrose [27]: We denote our bipartite Hermitian operator ξ by The “terminals” of this diagram correspond to indices like so ξi,k,lj = Joining the terminals with “wires” denotes contraction of the corresponding indices: r,m ξr,k,lj ξ p,q =
ξ A := Tr B (ξ ) = Tr(ξ ) = Tr(ξ A2 )
=
, ξ B := Tr A (ξ ) =
= 0, = a,
Tr(ξ 2 ) =
= t,
Tr(ξ B2 )
= b.
=
In an effort to keep the diagrams tidy and compact, we sometimes use a pair of vertical grey lines, one with wires entering from the right and the other with a matching set of wires entering from the left. A diagram with this feature is to be read as equivalent to the diagram one obtains by identifying the grey lines parallel to join the matching wires. It should not be confused with the bars drawn across wires (by Penrose and others) to denote (anti-)symmetrization. Here is an example showing how a diagram corresponds to a particular term of the form (C2): j,q l, p
i,m = ξl,q ξk,m ξ k,n j, p ξi,n .
In Fig. 1 we provide a table with a diagram representative of each of the R-conjugacy classes organised by the conjugacy class of S4 × S4 which contains it. The size of each R-conjugacy class is written to the right of the corresponding diagram. An upper bound is given and diagrams which are identically 0 (by virtue of having a factor of Tr(ξ ) = 0) are drawn in a lighter shade of grey. Proofs of upper bounds. We give bounds for the terms shown in the upper-right triangle of Fig. 1. Bounds for those terms below the diagonal follow from these by exchanging the roles of the parties. We will make repeated use of the Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product,
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Lemma 27. | Tr(A† B)|2 ≤ (Tr(A† A))(Tr(B † B)). Let P denote a positive semidefinite hermitian operator. We have the inequality Tr(P 2 ) ≤ (Tr(P))2 (by the spectral decomposition of P for example). From this fact and the Cauchy-Schwarz inequality it follows that Lemma 28. If P and Q are both positive semidefinite, then Tr(P Q) ≤ (Tr(P))(Tr(Q)). Third, since the partial transpose map is selfadjoint, Lemma 29. The quantities t, a and b are unchanged if we replace ξ with ξ . Proof of of Lemma 26. We go through the types one by one. (2,2):(2,2)
= (Tr(ξ 2 ))2 = t 2 . To show that the same bound applies to , we note that it can be written as Tr (Tr A (Z ))2 , where
Z = (ξ ⊗ 1C )(1 A ⊗ ||)(ξ ⊗ 1C ) d and | = i=1 |i B ⊗|iC . Since Z = ((ξ ⊗ 1C )(1 A ⊗ |)) ((ξ ⊗ 1C )(1 A ⊗ |))† , it is positive semidefinite, and as such Tr (Tr A (Z ))2 ≤ (Tr(Z ))2 . The result follows by noting that Tr(Z ) = t. (2,2):(1,1,1,1) = (Tr(ξ A2 ))2 = a 2 . = ab.
(2,1,1):(2,1,1) ξ2
is positive semidefinite, and applying Lemma 28, we get (4):(4) Noting that = Tr(ξ 4 ) ≤ (Tr(ξ 2 ))2 = t 2 . The partial-transpose of ξ , ξ , has the diagrammatic representation
(we choose to take the transpose on
Bob’s system). Substituting, this for ξ in
results in the diagram
= (Tr(ξ ))4 , so Lemma 29 shows that the same bound applies here. The Cauchy-Schwarz inequality yields = Tr (ξ )2 (ξ 2 ) ≤ 1/2
4 2 2 (Tr(ξ )) (Tr (ξ ) ) = , which can be seen to be ≤ t 2 because of the previous two bounds. (4):(2,1,1) = Tr (Tr B (ξ 2 ))ξ A2 ≤ (Tr(ξ 2 ))(Tr(ξ A2 )) = ta, by Lemma
2 28. = Tr (ξ(ξ A ⊗ 1 B )ξ(ξ A ⊗ 1 B )) ≤ Tr ξ(ξ A ⊗ 1 B )ξ = , where we have used the Cauchy-Schwarz inequality.
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Bob (4)
6
(3,1)
(2,1,1)
3 12
24
24 6
(2,2)
24
6
(4)
8
(t + ta) ⁄ 2
(1,1,1,1)
24
6
12
t2
2
6
1 6
a2
ta
t2 6
(3,1)
36
48
18
36
6
24
24
24
24
8
24
24
t(t+b)/2
a(t+b)/2
(ta + tb) ⁄ 2
(t2 + tb) ⁄ 2
24 8 8
Alice
8
48
64
24
48
8
12
24
3
6
3
6
(2,2)
t2
t(t+a)/2
12
a2
ta
t2
6
t2 3
18
24
9
18
3
24
24
6
6
6
12
(2,1,1)
b(t+a)/2
12 24
tb
ab 6
tb
24
6
36
48
18
36
6
6
8
3
6
1
3
6
1
b2
b2
(1,1,1,1)
1
6
8
Fig. 1. Sizes and upper-bounding expressions of the R-conjugacy classes. The faded diagrams are identically zero (because they contain a factor of Tr(ξ ))
1/2 (4):(3,1) ≤ . Using the results for these two diagrams and the arithmetic-geometric mean inequality we can bound this expression by t (t + a)/2, as was claimed.
is given
by substituting ξ into the previous diagram, so by Lemma 29 the previous bound applies.
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= Tr(Tr B (ξ 2 ))2 ≤ t 2 . For the other diagram we use the Cauchy-Schwarz inequality: ≤ Tr
(4):(2,2)
Tr
1/2
=
≤ t 2.
(2,2):(2,1,1) = ta. For the other diagram in this class it is useful to define Y B := Tr A (ξ(ξ A ⊗ 1 B )). We define Y A similarly but with the roles of the parties reversed. Tr(Y B2 ) = Tr ((Tr A (ξ(ξ A ⊗ 1 B ))) · Y B ) = Tr (ξ(ξ A ⊗ 1 B )(1 A ⊗ Y B )) = Tr (ξ(ξ A ⊗ Y B )) ≤ (Tr(ξ 2 ))(Tr(ξ A2 ))(Tr(Y B2 )), and therefore Tr(Y B2 ) ≤ (Tr(ξ 2 ))(Tr(ξ A2 )) = ta. Similarly Tr(Y A2 ) ≤ tb. Hence,
= Tr(Y B2 ) ≤ ta.
= Tr(ξ A4 ) ≤ (Tr(ξ A2 ))2 = a 2 . (3,1):(3,1) = Tr (ξ A ⊗ 1 B )ξ 2 (1 A ⊗ ξ B ) = Tr ξ 2 (ξ A ⊗ ξ B ) . Using the Cauchy-Schwarz inequality we upper bound this by (Tr(ξ 4 ))(Tr(ξ A2 ))(Tr(ξ B2 )), which in turn is bounded by (Tr(ξ 2 )) (Tr(ξ A2 ))(Tr(ξ B2 )) ≤ (ta + tb)/2, using arith(4):(1,1,1,1)
metic-geometric mean inequality at the end.
is given by substituting
ξ
into the previous diagram, so by Lemma 29 the same applies. bound 2 2 = Tr (Tr B (ξ ))Y A ≤ (Tr Tr B (ξ 2 )) (Tr(Y A2 )) ≤ (3,1):(2,2) t (t + b)/2. (3,1):(2,1,1)
= Tr(ξ A2 Y A ) ≤
(Tr(ξ A4 ))(Tr(Y A2 )) ≤ a(t + b)/2.
Now, collecting terms according to the multiplicities found in the table of Fig. 1, we conclude the proof. Remark 30. Note that for every pair of conjugacy classes of permutations, all the types falling into the corresponding box in Fig. 1 share the same upper bound. References 1. 2. 3. 4.
Fulton, W., Harris, J.: Representation Theory: A First Course. Berlin Heidelberg NewYork: Springer, 1991 Dudley, R.M.: Real Analysis and Probability. Cambridge: Cambridge University Press, 2002 Rockafellar, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press, 1997 Cover, T., Thomas, J.: Elements of Information Theory (Second Edition). NewYork: John Wiley and Sons (2006) 5. Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: Proc. 22nd Annual IEEE Conference on Computational Complexity (CCC’07), 129–140, available at http:// arxiv.org/abs/quant-ph:0701126v2, 2007
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6. Appleby, D.M., Dang, H.B., Fuchs, C.A.: Physical Significance of Symmetric Informationally-Complete Sets of Quantum States. http://arxiv.org/abs/:0707.2071v1[quant-ph], 2007 7. Appleby, D.M.: SIC-POVMs and the Extended Clifford Group. In J. Math. Phys. 46, 052107 (2005) 8. Audenaert, K.M.R., Calsamiglia, J., Munoz-Tapia, R., Bagan, E., Masanes, Ll., Acín, A., Verstraete, F.: Discriminating States: The Quantum Chernoff Bound. Phys. Rev. Lett. 98, 160501 (2007); Nussbaum, M., Szkoła, A.: The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Stat. 37, no. 2, 1040–1057 (2009) 9. Ballester, M.A., Wehner, S., Winter, A.: State Discrimination with Post-Measurement Information. In: IEEE Trans. Inf. Theory 54, no. 9, 4183–4198 (2008) 10. Bandyopadhyay, S., Boykin, P.O., Roychowdhuri, V., Vatan, F.: A New Proof for the Existence of Mutually Unbiased Bases. Algorithmica 34, 512–528 (2002) 11. Bennett, C.H., Hayden, P., Leung, D., Shor, P.W., Winter, A.: Remote preparation of quantum states. IEEE Trans. Inf. Theory 51(1), 56–74 (2005) 12. Berger, B.: The Fourth Moment Method. SIAM J. Comput. 24(6), 1188–1207 (1997) 13. Bolthausen, E.: An Estimate of the Remainder in a Combinatorial Central Limit Theorem. Zeits. für Wahrsch. Und Verw. Geb. 66, 387–405 (1984) 14. Eggeling, T., Werner, R.F.: Hiding Classical Data in Multipartite Quantum States. Phys. Rev. Lett. 89(9), 097905 (2002) 15. Flammia, S.: On SIC-POVMs in Prime Dimensions. J. Phys. A. Math. Gen. 39, 10901–10907 (2006) 16. Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory 45(4), 1216–1227 (1999) 17. Grassl, M.: On SIC-POVMs and MUBs in Dimension 6. J. Phys. A: Math. Gen. 39, 13483–13493 (2006) 18. Bennett, C., DiVincenzo, D., Fuchs, C., Mor, T., Rains, E., Shor, P., Smolin, J., Wootters, W.: Quantum nonlocality without entanglement. Phys. Rev. A 59(2), 1070–1091 (1999) 19. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of Mixed States: Necessary and Sufficient Conditions. Phys. Lett. A 223, 1–8 (1996) 20. Gurvits, L., Barnum, H.: Largest separable balls around the maximally mixed bipartite quantum state. Phys. Rev. A 66, 062311 (2002); Gurvits, L., Barnum, H.: Separable balls around the maximally mixed multipartite quantum states. Phys. Rev. A 68, 042312 (2003) 21. Hardin, R.H., Sloane, N.J.A.: McLaren’s Improved Snub Cube and Other New Spherical Designs in Three Dimensions. Discrete and Comput. Geom. 15, 429–441 (1996) 22. Hayden, P., Leung, D., Shor, P.W., Winter, A.: Randomizing Quantum States: Constructions and Applications. Commun. Math. Phys. 250(2), 371–391 (2004) 23. Helstrom, C.W.: Quantum Detection and Estimation Theory. New York: Academic Press, (1976); Holevo, A.S.: Statistical decision theory for quantum systems. J. Multivariate Anal. 3(4), 337–394, (1973) 24. Jozsa, R., Robb, D., Wootters, W.K.: Lower bound for accessible information in quantum mechanics. Phys. Rev. A 49(2), 668–677 (1994) 25. Klappenecker, A., Roetteler, M.: Mutually Unbiased Bases are complex spherical 2-designs. In: Proc. ISIT 2005, Piscataway, NJ: IEEE, 2005, pp. 1740–1744 26. Matthews, W., Winter, A.: On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states. Commun. Math. Phys. 285(1), 161–174 (2009); http://arxiv.org/abs/0710.4113v2[quantph], 2008 27. Penrose, R., Rindler, W.: Spinors and Space-Time, Vol. 1: Two-spinor calculus and relativistic fields. Cambridge: Cambridge University Press, 1986 28. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric Informationally Complete Quantum Measurements. J. Math. Phys. 45, 2171 (2004) 29. Sanchez-Ruiz, J.: Entropic uncertainty and certainty relations for complementary observables. Phys. Lett. A 173(3), 233–239 (1993); Improved bounds in the entropic uncertainty and certainty relations for complementary observables. Phys. Lett. A 201(2–3), 125–131 (1995) 30. Sen De, A., Sen, U., Lewenstein, M.: Distillation protocols that involve local distinguishing: Composing upper and lower bounds on locally accessible information. Phys. Rev. A 74, 052332 (2006) 31. Terhal, B.M., DiVincenzo, D.P., Leung, D.: Hiding Bits in Bell States. Phys. Rev. Lett. 86(25), 5807–5810 (2001); DiVincenzo, D.P., Leung, D., Terhal, B.M.: Quantum data hiding. IEEE Trans. Inf. Theory 48(3), 580–599 (2002) 32. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989) 33. Zauner, G.: Quantum Designs – Foundations of a Non-Commutative Theory of Designs (in German). Ph.D. thesis, Universität Wien (1999) Communicated by M.B. Ruskai
Commun. Math. Phys. 291, 845–861 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0763-y
Communications in
Mathematical Physics
On Affine Orbifold Nets Associated with Outer Automorphisms Feng Xu Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA. E-mail: [email protected] Received: 3 November 2008 / Accepted: 6 November 2008 Published online: 5 March 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We construct solitons in affine orbifold nets associated with outer automorphisms, and we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to settle a number of questions concerning such orbifold constructions. 1. Introduction Let A be a completely rational conformal net (cf. Section 2.3 and Def. 2.6). Let be a finite group acting properly on A (cf. Definition (2.4)). It is proved in [27] that the fixed point subnet (the orbifold) A is also completely rational, and by [16] A has finitely many irreducible representations which are divided into two classes: the ones that are obtained from the restrictions of a representation of A to A which are called untwisted representations, and the ones which are twisted (cf. definition after Th. 2.7). It follows from Th. 2.7 that a twisted representation of A always exists if A = A. The motivating question for this paper is to construct these twisted representations of A . It turns out that all representations of A are closely related to the solitons of A (cf. Section 3.3 and Prop. 2.8). Solitons are representations of A0 , the restriction of A to the real line identified with a circle with one point removed. Every representation of A restricts to a soliton of A0 , but not every soliton of A0 can be extended to a representation of A. The construction of the soliton depends on the net A and the action of . In this paper we consider the orbifold net associated with SU (n)k with an outer automorphism. When k = 1, this case is already covered by the general results of [6] and in the framework of Vertex Algebras by [3]. Our work will build upon the results of [6 and 3]. The main difference between the case considered in this paper and those of [14] is that it is not an easy question to determine the index of solitons, hence the strategy of Supported in part by NSF.
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adding up all indices to see if it agrees with the index formula in [14] does not work in the present case (cf. Cor. 4.6 for the list of indices when n = 3). Here we use a result of [1] to count the number of irreducible solitons. This allows us to show that the list of known solitons constructed in Sect. 3.2 is in fact all the irreducible solitons and hence all irreducible representations of the fixed point subnet (cf. Th. 3.9) can be determined. We expect that this idea will work in other cases. Though our main results Th. 3.9 and Cor. 3.10 are expected from various partial results (cf. [24]) , they have not appeared before, and we give some applications of these results. The rest of this paper is as follows: after preliminary sections on nets and related concepts, we construct solitons using the ideas of [6]. By a counting argument, we prove that all irreps of the fixed point nets have been constructed in Th. 3.9. In Section 4 we consider applications of our main results in Section 3 to the properties of certain fusion matrices which have been studied from different points of view in [18 and 2] where these properties are postulated motivated by considerations from boundary conformal field theories. In Prop. 4.5 we explicitly determine these fusion matrices for the first non-trivial case when n = 3, using a result in [4] about the set of lines in Euclidean space which mutually have the angles π/3 or π/2. The result agrees with formulas in [18 and 2]. As a corollary we determine the set of indices of twisted solitons in Cor. 4.6. It is an interesting question to extend these results to n ≥ 4. 2. Conformal Nets on S1 In this section we introduce some basic concepts and notations which will be used later. We refer the reader to Section 2 of [14] for more details. By an interval of the circle we mean an open connected non-empty subset I of S 1 such that the interior of its complement I is not empty. We denote by I the family of all intervals of S 1 . A net A of von Neumann algebras on S 1 is a map I ∈ I → A(I ) ⊂ B(H) from I to von Neumann algebras on a fixed Hilbert space H that satisfies: A. Isotony. If I1 ⊂ I2 belong to I, then A(I1 ) ⊂ A(I2 ). The net A is called local if it satisfies: B. Locality. If I1 , I2 ∈ I and I1 ∩ I2 = ∅, then [A(I1 ), A(I2 )] = {0}, where brackets denote the commutator. The net A is called Möbius covariant if in addition it satisfies the following properties C, D, E, F: C. Möbius covariance. There exists a strongly continuous unitary representation U of the Möbius group Möb (isomorphic to PSU (1, 1)) on H such that U (g)A(I )U (g)∗ = A(g I ), g ∈ Möb, I ∈ I. If E ⊂ S 1 is any region, we shallput A(E) ≡ E⊃I ∈I A(I ) with A(E) = C if E has empty interior (the symbol denotes the von Neumann algebra generated).
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Note that the definition of A(E) remains the same if E is an interval, namely: if {In } is an increasing sequence of intervals and ∪n In = I , then the A(In )’s generate A(I ) (consider a sequence of elements gn ∈ Möb converging to the identity such that gn I ⊂ In ). D. Positivity of the energy. The generator of the one-parameter rotation subgroup of U (conformal Hamiltonian) is positive. E. Existence of the vacuum. There exists a unit U -invariant vector ∈ H (vacuum vector), and is cyclic for the von Neumann algebra I ∈I A(I ). By the Reeh-Schlieder theorem is cyclic and separating for every fixed A(I ). The modular objects associated with (A(I ), ) have a geometric meaning itI = U ( I (2π t)) ,
J I = U (r I ) .
Here I is a canonical one-parameter subgroup of Möb and U (r I ) is a antiunitary acting geometrically on A as a reflection r I on S 1 . This implies Haag duality: A(I ) = A(I ),
I ∈I,
where I is the interior of S 1 I . F. Irreducibility. I ∈I A(I ) = B(H). Indeed A is irreducible iff is the unique U -invariant vector (up to scalar multiples). Also A is irreducible iff the local von Neumann algebras A(I ) are factors. In this case they are III1 -factors in Connes classification of type III factors (unless A(I ) = C for all I ). By a conformal net (or diffeomorphism covariant net) A we shall mean a Möbius covariant net such that the following holds: G. Conformal covariance. There exists a projective unitary representation U of Diff(S 1 ) on H extending the unitary representation of Möb such that for all I ∈ I we have U (g)A(I )U (g)∗ = A(g I ), g ∈ Diff(S 1 ), U (g)xU (g)∗ = x, x ∈ A(I ), g ∈ Diff(I ), where Diff(S 1 ) denotes the group of smooth, positively oriented diffeomorphism of S 1 and Diff(I ) the subgroup of diffeomorphisms g such that g(z) = z for all z ∈ I . Let G be a simply connected compact Lie group. By Th. 3.2 of [7], the vacuum positive energy representation of the loop group LG (cf. [19]) at level k gives rise to an irreducible conformal net denoted by AG k . By Th. 3.3 of [7], every irreducible positive energy representation of the loop group LG at level k gives rise to an irreducible covariant representation of AG k . 2.1. Genus 0 S, T -matrices. Next we will recall some of the results of [21] and introduce notations. Let {[λ], λ ∈ L} be a finite set of all equivalence classes of irreducible, covariant, finite-index representations of an irreducible local conformal net A. We will denote the conjugate of [λ] by [λ¯ ] and identity sector (corresponding to the vacuum representaν = [λ][µ], [ν] . Here µ, ν denotes tion) by [1] if no confusion arises, and let Nλµ the dimension of the space of intertwiners from µ to ν (denoted by Hom(µ, ν)). We will denote by {Te } a basis of isometries in Hom(ν, λµ). The univalence of λ and the statistical dimension of (cf. Section 2 of [11]) will be denoted by ωλ and d(λ) (or dλ )) respectively.
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Let ϕλ be the unique minimal left inverse of λ, define: Yλµ := d(λ)d(µ)ϕµ (µ, λ)∗ (λ, µ)∗ ,
(1)
where (µ, λ) is the unitary braiding operator (cf. [11]). We list two properties of Yλµ (cf. (5.13), (5.14) of [21]) which will be used in the following: Lemma 2.1. Yλµ = Yµλ = Yλ∗µ¯ = Yλ¯ µ¯ . Yλµ =
ν Nλµ
k
ωλ ωµ d(ν). ων
We note that one may take the second equation in the above lemma as the definition of Yλµ . Define a := i dρ2i ωρ−1 . If the matrix (Yµν ) is invertible, by the proposition on p.351 i of [21] a satisfies |a|2 = λ d(λ)2 . Definition 2.2. Let a = |a| exp(−2πi c80 ), where c0 ∈ R and c0 is well defined mod 8Z. Define matrices S := |a|−1 Y, T := CDiag(ωλ ),
(2)
c0 . C := exp −2πi 24 Then these matrices satisfy (cf. [21]): where
Lemma 2.3. SS † STS S2 T Cˆ
= TT † = id, = T −1 ST −1 , ˆ = C, = Cˆ T = T,
where Cˆ λµ = δλµ¯ is the conjugation matrix. Moreover ν Nλµ =
Sλδ Sµδ S ∗ νδ S1δ
(3)
δ
is known as the Verlinde formula. We will refer to the S, T matrices as defined above as genus 0 modular matrices of A since they are constructed from the fusion rules, monodromies and minimal indices which can be thought of as genus 0 chiral data associated to a Conformal Field Theory. ν is called the The commutative algebra generated by λ’s with structure constants Nλµ fusion algebra of A. If Y is invertible, it follows from Lemma 2.3, (3) that any nontrivial S for some µ. irreducible representation of the fusion algebra is of the form λ → Sλµ 1µ
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2.2. The orbifolds. Let A be an irreducible conformal net on a Hilbert space H and let be a finite group. Let V : → U (H) be a unitary representation of on H. If V : → U (H) is not faithful, we set := /kerV . Definition 2.4. We say that acts properly on A if the following conditions are satisfied: (1) For each fixed interval I and each g ∈ , αg (a) := V (g)aV (g ∗ ) ∈ A(I ), ∀a ∈ A(I ); (2) For each g ∈ , V (g) = , ∀g ∈ . We note that if acts properly, then V (g), g ∈ commutes with the unitary representation U of Möb. Define B(I ) := {a ∈ A(I )|αg (a) = a, ∀g ∈ } and A (I ) := B(I )P0 on H0 , where H0 := {x ∈ H|V (g)x = x, ∀g ∈ } and P0 is the projection from H to H0 . Then U restricts to an unitary representation (still denoted by U ) of Möb on H0 . The following is proved in [27]: Proposition 2.5. The map I ∈ I → A (I ) on H0 together with the unitary representation (still denoted by U ) of Möb on H0 is an irreducible Möbius covariant net. The irreducible Möbius covariant net in Prop. 2.5 will be denoted by A and will be called the orbifold of A with respect to . We note that by definition A = A . 2.3. Complete rationality . We first recall some definitions from [16] . Recall that I denotes the set of intervals of S 1 . Let I1 , I2 ∈ I. We say that I1 , I2 are disjoint if I¯1 ∩ I¯2 = ∅, where I¯ is the closure of I in S 1 . When I1 , I2 are disjoint, I1 ∪ I2 is called a 1-disconnected interval in [28]. Denote by I2 the set of unions of disjoint 2 elements in I. Let A be an irreducible Möbius covariant net as in Section 2.1. For E = I1 ∪ I2 ∈ I2 , let I3 ∪ I4 be the interior of the complement of I1 ∪ I2 in S 1 , where I3 , I4 are disjoint intervals. Let ˆ A(E) := A(I1 ) A(I2 ), A(E) := A(I3 ) A(I4 ) . ˆ Note that A(E) ⊂ A(E). Recall that a net A is split if A(I1 ) A(I2 ) is naturally isomorphic to the tensor product of von Neumann algebras A(I1 ) ⊗ A(I2 ) for any disjoint intervals I1 , I2 ∈ I. A is strongly additive if A(I1 ) A(I2 ) = A(I ), where I1 ∪ I2 is obtained by removing an interior point from I . Definition 2.6. [16]. A is said to be completely rational if A is split, strongly addiˆ tive, and the index [A(E) : A(E)] is finite for some E ∈ I2 . The value of the index ˆ [A(E) : A(E)] (it is independent of E by Prop. 5 of [16]) is denoted by µA and is called ˆ the µ-index of A. If the index [A(E) : A(E)] is infinity for some E ∈ I2 , we define the µ-index of A to be infinity. Note that by [17] every irreducible, split, local conformal net with finite µ-index is automatically strongly additive. The following theorem is proved in [27]: Theorem 2.7. Let A be an irreducible Möbius covariant net and let be a finite group acting properly on A. Suppose that A is completely rational. Then: (1) A is completely rational or µ-rational and µA = | |2 µA ;
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(2) There are only a finite number of irreducible covariant representations of A (up to unitary equivalence), and they give rise to a unitary modular category as defined in II.5 of [22] by the construction as given in Section 1.7 of [26]. Suppose that A and satisfy the assumptions of Th. 2.7. Then A has only finitely number of irreducible representations λ˙ and ˙ 2 = µA = | |2 µA . d(λ) λ˙
2.4. Restriction to the real line: Solitons. Denote by I0 the set of open, connected, non-empty, proper subsets of R, thus I ∈ I0 iff I is an open interval or half-line (by an interval of R we shall always mean a non-empty open bounded interval of R). Given a net A on S 1 we shall denote by A0 its restriction to R = S 1 {−1}. Thus A0 is an isotone map on I0 , that we call a net on R. In this paper we denote by J0 := (0, ∞) ⊂ R. A representation π of A0 on a Hilbert space H is a map I ∈ I0 → π I that associates to each I ∈ I0 a normal representation of A(I ) on B(H) such that π I˜ A(I ) = π I ,
I ⊂ I˜,
I, I˜ ∈ I0 .
A representation π of A0 is also called a soliton. As A0 satisfies half-line duality, namely A0 (−∞, a) = A0 (a, ∞), a ∈ R, by the usual DHR argument [5] π is unitarily equivalent to a representation ρ which acts identically on A0 (−∞, 0), thus ρ restricts to an endomorphism of A(J0 ) = A0 (0, ∞). ρ is said to be localized on J0 and we also refer to ρ as a soliton endomorphism. Clearly a representation π of A restricts to a soliton π0 of A0 . But a representation π0 of A0 does not necessarily extend to a representation of A. If A is strongly additive, and a representation π0 of A0 extends to a DHR representation of A, then it is easy to see that such an extension is unique, and in this case we will use the same notation π0 to denote the corresponding DHR representation of A. 2.5. Induction and restriction. Let A be a Möbius covariant net and B a subnet. Given a bounded interval I0 ∈ I0 we fix the canonical endomorphism γ I0 associated with B(I0 ) ⊂ A(I0 ). Then we can choose for each I ∈ I0 with I ⊃ I0 a canonical endomorphism γ I of A(I ) into B(I ) in such a way that γ I A(I0 ) = γ I0 and λ I1 is the identity on B(I1 ) if I1 ∈ I0 is disjoint from I0 , where λ I ≡ γ I B(I ). We then have an endomorphism γ of the C ∗ -algebra A ≡ ∪ I A(I ) (I bounded interval of R). Given a DHR endomorphism ρ of B localized in I0 , the α-induction αρ of ρ is the endomorphism of A given by αρ ≡ γ −1 · Adε(ρ, λ) · ρ · γ , where ε denotes the right braiding unitary symmetry (there is another choice for α associated with the left braiding). αρ is localized in a right half-line containing I0 , namely αρ is the identity on A(I ) if I is a bounded interval contained in the left complement
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of I0 in R. Up to unitary equivalence, αρ is localizable in any right half-line, thus αρ is normal on left half-lines, that is to say, for every a ∈ R, αρ is normal on the C ∗ -algebra A(−∞, a) ≡ ∪ I ⊂(−∞,a) A(I ) (I bounded interval of R), namely αρ A(−∞, a) extends to a normal morphism of A(−∞, a). We have the following Prop. 3.1 of [17]: Proposition 2.8. αρ is a soliton endomorphism of A0 . 2.6. Loop groups of type A. We denote L SU (n) the group of smooth maps f : S 1 → SU (n) under pointwise multiplication. The diffeomorphism group of the circle DiffS 1 is naturally a subgroup of Aut(L SU (n)) with the action given by reparametrization. In particular the group of rotations RotS 1 U (1) acts on L SU (n). The Lie algebra of L SU (n), denoted by Lsu(n), consists of smooth maps S 1 to su(n). We will denote from n elements of Lsu(n) by its Fourier series g(z) = n gn z , and L 0 su(n) the subspace of Lsu(n) which are polynomials in z = exp(2πiθ ), 0 ≤ θ ≤ 1. We will be interested in the projective unitary representation π : L SU (n) → U (H ) that are both irreducible and have positive energy. This means that π should extend to L SU (n) Rot S 1 so that H = ⊕n≥0 H (n), where the H (n) are the eigenspace for the action of RotS 1 , i.e., rθ ξ = exp(inθ ) for θ ∈ H (n) and dim H (n) < ∞ with H (0) = 0. By I.7 of [25] the space of finite energy vectors are C ∞ vectors for the action of L 0 su(n), and by I.9 of [25] H remains irreducible when restricting to subgroups generated by exp(i X ), X = X ∗ ∈ L 0 su(n). We will use LSU (n) to denote the central extension of L SU (n) by S 1 as constructed in Chapter 4 of [19]. It follows from [19 and 13] that for fixed level k which is a positive integer, there are only a finite number of such irreducible representations indexed by the finite set
k P++ = λ∈ P|λ= λi i , λi ≥ 0 , λi ≤ k , i=1,...,n−1
i=1,...,n−1
where P is the weight lattice of SU (n) and i are the fundamental weights. We will use 0 or simply 1 to denote the trivial representation of SU (n). For λ, µ, ν ∈ (δ) (δ) (δ∗) (δ) (δ) k , define N ν = P++ /S0 , where Sλ is given by the Kac-Peterson k Sλ Sµ Sν λµ δ∈P++ formula (cf. Eq. (4) below for an equivalent formula): (δ) Sλ = c εw exp (iw(δ) · λ2π/n) , w∈Sn
(δ)
where εw = det(w) and c is a normalization constant fixed by the requirement that Sµ ν are non-negative integers. is an orthonormal system. It is shown in [13], p. 288 that Nλµ k with structure Moreover, define Gr (Ck ) to be the ring whose basis are elements of P++ ν k ¯ constants Nλµ . The natural involution ∗ on P++ is defined by λ → λ = the conjugate of λ as a representation of SU (n). ()
()
We shall also denote S0 by S1 . Define dλ =
(λ)
S1
( ) S1 0
(δ)
. We shall call (Sν ) the S-matrix
of L SU (n) at level k. The irreducible positive energy representations of L SU (n) at level k give rise to an irreducible conformal net A (cf. [14]) and its covariant representations.
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We will use λ = (λ1 , . . . , λn−1 ) to denote irreducible representations of A and also the corresponding endomorphism of M = A(I ). Recall from [14] that A(I ) is generated as a von Neumann algebra by πk0 ( f ), ∀ f ∈ L SU (n), f I = e, where e denotes the identity element of SU (n). All the sectors [λ] with λ irreducible generate the fusion ring of A. The following form of the Kac-Peterson formula for the S matrix will be used later: Sλµ t (µ + ρ) ch λ (x1 , . . . , xn−1 , 1) , = exp (4) S1µ n(k + n) with the representation of SU (n) where ch λ is the character associated finite irreducible µ labeled by λ, and xi = exp −2πi k+ni , µi = i≤ j≤n−1 (µ j + 1), 1 ≤ i ≤ n − 1, t (λ) = 1≤i≤n−1 iλi . The following result is proved in [25] (see Corollary 1 of Chapter V in [25]). (k)
Theorem 2.9. Each λ ∈ P++ has finite index with index value dλ2 . The fusion ring (k) generated by all λ ∈ P++ is isomorphic to Gr (Ck ). In the case of SU (2)k , we will label irreducible representations by a half integer called spin 0 ≤ i ≤ k/2. Here are some examples of fusion rules: 1 1 k−1 1 ×i = i − ⊕ i+ , 0≤i ≤ ; 2 2 2 2 k−2 . (5) 1 × i = (i − 1) ⊕ i ⊕ (i + 1), 0 ≤ i ≤ 2 2.7. Twisted loop group. Let τ be the order two outer automorphism of SU (n) given ¯ ∀A ∈ SU (n) where A¯ is the complex conjugate of A. On the Lie algebra by τ (A) = A, su(n) τ is given by τ (X ) = − X¯ , ∀X ∈ su(n). (We identify su(n) with n × n Hermitian matrices.) It is convenient in this paper to think of the twisted loop group L τ SU (n) as a subgroup of L SU (n). We make the following definition:
Definition 2.10. L τ SU (n) := f ∈ L SU (n), f θ + 21 = τ ( f (θ )), 0 ≤ θ ≤ 21 ;
L τ su(n) := f ∈ Lsu(n), f θ + 21 = τ ( f (θ )), 0 ≤ θ ≤ 21 . Let su(n) = so(n) ⊕ g1 , where g1 is the eigenspace for τ with eigenvalue −1. Note that g1 is an irreducible representation of so(n) under the adjoint action (cf. Section 8 of [13]). Assume that t = t0 ⊕ t1 , where t ⊂ su(n) is the subset of diagonal matrices. Note that both t0 , t1 are nontrivial subspaces. The projective irreducible representations of L τ SU (n) are similar to that of L SU (n). For each fixed positive integer k we again have finitely many projective unitary irreducible representations of L τ SU (n) S 1 . We shall use Lτ SU (n) (resp. Lτ su(n)) to denote the central extensions of L τ SU (n) (resp. L τ su(n) ) so that the projective unitary irreducible representations of L τ SU (n) S 1 at level k are irreducible representations of Lτ SU (n) S 1 . These representations correspond to irreducible integrable highest weight modules of Lτ su(n) at level k. We refer the reader to Chapter II of [24] for more details. The following simple observation follows from Chapter 10 of [13]: Lemma 2.11. The number of irreducible representations of Lτ SU (n) at level k is equal to the number of irreducible representations λ of LSU (n) at level k such that λ = λ¯ .
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Lemma 2.12. Let B be the subnet of A SU (n)k such that B(I ) is generated as a von Neumann algebra by πk0 (g exp(i X )g ∗ ), ∀g ∈ Spin(n), X = X ∗ ∈ Lt, X (z) = 0, ∀z ∈ I . Then B = A. Proof. Note that B is indeed a subnet of A since the modular group associated with B(I )= A(I ) preserves B(I ). To show that B(I ) = A(I ) it is enough to show that I = gyg ∗ z n + gy ∗ g ∗ z −n ∈ Lsu(n) with y ∈ t and g ∈ Spin(n). Then I A(I ). Let Y πk0 (exp iY ) ∈ I B(I ). Since the adjoint actions of Spin(n) on g0 , g1 are irreducible, and t0 , t1 are nontrivial subspaces, it follows that g0 (resp. g1 ) is the linear span of g Xg ∗ , ∀g ∈ Spin(n), X = 0 ∈ t0 (resp. g Xg ∗ , ∀g ∈ Spin(n), X = 0 ∈t1 ). By Trotter’s product formulas (cf. P. 295 of [20]) we conclude that πk0 (exp iY ) ∈ I B(I ) for any Y = Y ∗ ∈ L 0 su(n). Since πk0 is irreducible as a representation of the group generated by exp(iY ), ∀Y = Y ∗ ∈ L 0 su(n), the lemma is proved. 3. Affine Orbifold Nets Associated with Outer Automorphisms Let k be a positive integer (level). Unless otherwise stated we will write A = A SU (n)k . By identifying R2n = (x, y) → x + i y ∈ Cn , where x, y are column vectors with n real entries, we have the natural inclusion SU (n)k ⊂ Spin(n)k . Define J := (I dn , −I dn ) ∈ ¯ and J. = , S O(2n) and lift it to Spin(2n). Note that for A ∈ SU (N ), JAJ = A, where is the vacuum vector for vacuum representations of LSpin(2n)k . It follows that Ad J generates a proper Z2 action of A SU (n)k . This is the action corresponding to the outer automorphism τ of SU (n). Suppose that the vacuum representation of A decompose as 1 ⊕ σ as representations of AZ2 , where 1 stands for the vacuum representation. Motivated by Lemma 8.3 of [17], we make the following definition: Definition 3.1. Let ρ be an irreducible soliton of A which restricts to a DHR representation of AZ2 , and let ρ+ be an irreducible component of ρ AZ2 . ρ is called τ -twisted if ε(ρ+ , σ )ε(σ, ρ+ ) = −1, where ε(., .) is the braiding operator (cf. [11]). 3.1. Constructions of solitons. Definition 3.2. L R SU (n) :=
f ∈ L SU (n)| f (0) = f (1) = e, f (n) (0) = f (n) (1) = 0, ∀n ≥ 1 ;
L R T0 := exp(i f (e2πiθ )), f = f ∗ ∈ Lt| f (n) (0) = f (n) (1) = 0, ∀n ≥ 0 ;
L I T0 := exp(i f ), f = f ∗ ∈ Lt, I ⊂ R, f I = 0 . The following is a special case of the covering homomorphism in Prop. 4.6 of [6]: Definition 3.3. ϕ : L R T0 → L τ T is a homomorphism defined by ϕ(g)(θ ) = g(2θ ), ϕ(g) θ + 21 = τ (ϕ(g)(θ )) , 0 ≤ θ ≤ 1/2. ϕ lifts to a homomorphism from central extensions of L R T0 to central extensions of Lτ T and by abuse of notations we will use ϕ to denote the lift.
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3.1.1. Level 1 case. When the level k = 1, we have the conformal inclusion Spin(n)2 ⊂ SU (n)1 , and by Lemma 5.1 of [27] we have AZ2 = A Spin(n)2 . By 4.3 of [27] A Spin(n)2 is completely rational, and its irreducible representations are in one to one correspondence to irreducible representations of LSpin(n)2 . Denote by T the subgroup of diagonal matrices in SU (n) and LT the subgroup of LSU (n)1 . The conformal net AT associated with LT is the same as A SU (n)1 by p. 28 of [29], and is a special case of the conformal net associated with lattices as defined in Definition 3.7 of [6]. Hence the τ -twisted irreducible solitons of A are given by Sect. 4.1 of [6]. These τ -twisted solitons are finitely direct sums of irreducible representations of LSpin(n)2 . (In particular they furnish a representation of Spin(n)), and they are in one to one correspondence to irreducible representations of Lτ SU (n) at level 1 (cf. [3,8] and references therein.) We summarize these results in the following: Lemma 3.4. The list {πτ } of irreducible τ -twisted solitons of A is given as follows: they are in one to one correspondence with the list of irreducible representations {π } of Lτ SU (n) at level 1, and we have πτ (g f g ∗ ) = π(g)π(ϕ( f ))π(g ∗ ), ∀ f ∈ L I T0 ⊂ LR T0 , ∀g ∈ Spin(n). gxg ∗ z n
(6)
gx ∗ g ∗ z −n
Lemma 3.5. Let X = + ∈ L σ su(n) with g ∈ Spin(n), x ∈ t. Then there exists a sequence of gm ∈ LR SU (n) such that π(exp(i X )) = s − lim π(ϕ(gm )), m→∞
where π is a direct sum of finitely many irreducible representations of Lτ SU (n)1 . The same is true if π is replaced by π ⊗ π . . . ⊗ π , where there are k tensor products of π . Proof. The proof is similar to the proof of Prop. 4.10 of [6]. By Prop. 1.2.3 in Chapter 4 of [23] we can choose a sequence xm (θ ) of smooth complex valued functions on [0, 1] such (n) (n) that xm θ + 21 = xm (θ ), xm (0) = xm (1) = 0, ∀n ≥ 0, ||xm − 1|| 1 ≤ 1/m, m ≥ 1 2 (cf. Section 1.2 of [23] for the definition of norm ||.|| 1 ). It follows by definition that 2 exp(i xm X ) = ϕ(exp(i f m )), where f m (θ ) = xm (θ/2)X (θ/2), 0 ≤ θ ≤ 1. Note that π is a direct sum of finitely many irreducible representations of LSpin(n)2 , by Prop. 1.3.2 in Chapter 4 of [23] we have that π(ϕ(exp(i f m ))) → π(exp(i X )) strongly. When π is replaced by π ⊗ π . . . ⊗ π , where there are k tensor products of π, the same argument as in Prop. 1.3.2 in Chapter 4 of [23] works, provided that one replaces the generator L 0 by 1≤i≤k id ⊗ . . . ⊗ L 0 . . . ⊗ id, where in the summation L 0 appears in the i th tensor. 3.2. General level case. Let π1 be a direct sum of all level 1 irreducible representations of Lτ SU (n) and let π be k tensor products of π1 . Note that π is a representation of Lτ SU (N ) at level k with positive energy. By Lemma 3.4 π gives a soliton of A⊗k SU (n)1 ⊃ A SU (n)k , and by restriction, a soliton πτ of A SU (n)k . We have πτ (g f g ∗ ) = π(g)π(ϕ( f ))π(g ∗ ), ∀ f ∈ L I T0 , I ⊂ R, ∀g ∈ Spin(n), where ϕ is defined in Definition 3.3, and we have identified f with its image in A(I ). ⊗k Note that A⊗k SU (n)1 = AT is a net associated with L(T × T . . . × T ), where there are k products, and is a net associated with a lattice as in Definition 3.7 of [6]. By Prop. Z2 where Z is generated by 4.8 of [6] π restricts to a DHR representation of (A⊗k 2 SU (n)1 ) τ ⊗ τ . . . ⊗ τ, hence πτ restricts to a DHR representation of AZSU2 (n)k .
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Proposition 3.6.
πτ (A I )) = π (Lτ SU (n)) .
I ⊂R
Proof. Since by Lemma 2.12 A(I ) is generated by g f g ∗ , ∀ f ∈ L I T0 , ∀g ∈ Spin(n), by definition we have πτ (A I )) ⊂ π(Lτ SU (n)) . I ⊂R
Since L τ SU (n) is connected (cf. Lemma 4.2 of [24]), it is sufficient to check that for X ∈ L τ su(n), π(exp(i X )) ∈ I ⊂R πτ (A I ). As in the proof of Lemma 2.12, by Trotter’s product formula (cf. p. 295 of [20]) and irreducibility of the actions of Spin(n) n ∗ ∗ −n ∗ on so(n) and g1 , it is sufficient to check that for X = gx z g + gx z g ∈ L σ su(n) with g ∈ Spin(n), x ∈ t, we have π(exp(i X )) ∈ I ⊂R πτ (A I ), and this follows from Lemma 3.5. Corollary 3.7. (1) Each irreducible representation ρ of Lτ SU (n) at level k gives an irreducible soliton ρτ of A SU (n)k such that ρ1 ρ2 as representations of Lτ SU (n) at level k iff ρ1τ ρ2τ as solitons of A SU (n)k ; (2) ρτ τ ρτ as solitons of A SU (n)k , and ρτ restricts to a DHR representation of AZSU2 (n)k . Proof. By [13] all irreducible representations of Lτ SU (n) at level k appear in π, and (1) follows from Prop. 3.6. For (2), first we note that ρτ comes from an irreducible component of πτ in Prop. 3.6. Since πτ restricts to a DHR representation of AZSU2 (n)k , it follows that ρτ restricts to a DHR representation of AZSU2 (n)k . By construction ρτ (τ (g f g ∗ )) = ρ ϕ(τ (g f g ∗ )) = ρ R1/2 ϕ(g f g ∗ ) , ∀g ∈ Spin(n), f ∈ L I T0 , where R1/2 (g)(θ ) = g θ + 21 . Since rotations are implemented on ρ (cf. p. 246 in [13] for a formula for the infinitesimal generator of rotations), (2) follows.
Let ρτ be an irreducible soliton of A as given by Cor. 3.7. By (2) of Cor. 3.9 and Cor. 4.9 of [16] ρτ decomposes into a direct sum of two distinct irreducible representations ρ+ , ρ− of AZ2 . By the same argument as Prop. 4.17 of [6] it follows that ε(ρ+ , σ )ε(σ, ρ+ ) = −1, and so we have the following lemma: Lemma 3.8. The irreducible solitons as given in Cor. 3.7 are τ -twisted as in Definition 3.1. 3.3. Counting of all irreducible τ -twisted solitons. Now we apply induction and restriction for general orbifolds in Section 4 of [16] to AZ2 ⊂ A. Recall that the vacuum representation of A decompose as 1 ⊕ σ as representations of AZ2 , where 1 stands for the vacuum representation. Let ρτ be a τ -twisted irreducible soliton of A as given by Cor. 3.7. By (2) of Cor. 3.9 and Cor. 4.9 of [16] ρτ decomposes into a direct sum of two distinct irreducible representations ρ+ , ρ− of AZ2 , and [αρ+ ] = [αρ+ ] = [ρτ ]. Similarly for an irreducible representation λ of A, we have that if [λ] = [λ¯ ], then λ
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decomposes as a direct sum of two distinct irreducible representations λ+ , λ− of AZ2 , ¯ then λ and λ¯ restrict to the same representation and [αλ+ ] = [αλ− ] = [λ]. If [λ] = [λ], Z ¯ Denote by a the number of (denoted by λ) of A 2 , and we have [αλ ] = [λ] + [λ]. irreducible τ -twisted solitons of A,, b the number of irreducible representations λ of A such that λ = λ¯ , and c the number of irreducible representations λ of A such that λ = λ¯ . By Th. 4.16 of [1], we have a + b + c = 2b + c. Hence a = b. Note that by Lemma 2.11 the number of irreducible representations of Lτ SU (n) at level k is b. By Lemma 3.8 it follows that Cor. 3.7 gives all irreducible τ -twisted representations of A. We summarize these results in the following: Theorem 3.9. (1) ρτ as given in Cor. 3.7 gives all the irreducible τ -twisted representations of A. These representations are in one to one correspondence with irreducible representations of Lτ SU (n) at level k; (2) The list of all irreducible representations of AZ2 are as follows: For an irreducible representation λ of A, we have that if [λ] = [λ¯ ], then λ decomposes as a direct sum of two distinct irreducible representations λ+ , λ− of AZ2 ; If [λ] = [λ¯ ], then λ and λ¯ restrict to the same representation (denoted by λ) of AZ2 ; ρ+ , ρ− where ρτ corresponds to irreducible representations of Lτ SU (n) at level k. By Th. 2 and Th. 3.9, we have proved the following: Corollary 3.10. The list of irreducible representations of AZ2 as in (2) of Th. 3.9 give rise to a unitary modular tensor category as defined in II.5 of [22] by the construction as given in Section 1.7 of [26]. 4. Examples of Fusion Rules Theorem 3.9 and Corollary 3.10 give strong constraints on the fusion rules related to AZ2 . In this section we give some examples of fusion rules by using the results of Section 2. The ideas are similar to Sect. 9 of [14]. ρ Denote by Nλρ21 = λρ1 , ρ2 , where λ is an irreducible representation of A, and ρ1 , ρ2 are irreducible τ -twisted representations of A. (µ)
Lemma 4.1. There exists a complex valued matrix ψρ , where ρ denotes irreducible τ -twisted representations of A, and µ = µ¯ labels an irreducible representation of A, such that Sλµ ρ Nλρ21 = ψ (µ) ψ (µ)∗ . S1µ ρ1 ρ2 µ,µ=µ¯
Proof. First we assume that λ = λ¯ . We have λρ1 , ρ2 = λαρ1+ , αρ2+ = (λ+ + λ− )ρ1+ , ρ2+ . Using the Verlinde formula we have: Sλ+ µ˙ Sρ1+ µ˙ Sρ∗ µ˙ Sλ− µ˙ Sρ1+ µ˙ Sρ∗ µ˙ 2+ 2+ (λ+ + λ− )ρ1+ , ρ2+ = + . S13µ˙ S13µ˙ µ˙
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By Lemma 9.1 of [14], we have that Sρ+ µ˙ = 0 if µ˙ comes from restriction of the representation µ = µ¯ of A, and Sλ+ µ˙ = −Sλ− µ˙ if µ˙ comes from restriction of a τ -twisted representation of A. If µ = µ, ¯ by (4) of Lemma 9.1 in [14] we have Sλ+ µ+ Sρ1+ µ+ Sρ∗2+ µ+ 3 S1µ + (µ)
Set ψρ
=
√
2
Sρ+ µ+ S1µ+
=
Sλ− µ− Sρ1+ µ− Sρ∗2+ µ− 3 S1µ −
=
Sλµ Sρ1+ µ+ Sρ∗2+ µ+ 2 S1µ S1µ +
.
and the lemma follows. The case when λ = λ¯ is similar.
4.1. n=3 case. Lemma 4.1 determines the spectrum of the square matrix Nλ whose entries are non-negative integers. In this section we determine (up to permutation) Nλ for the first non-trivial case n = 3. Our results agree with the ansatz given by [18] based on heuristic arguments. Lemma 4.2. d = (dρ )ρ is the unique (up to scalar multiplication) Perron-Frobenius eigenvector of Nv with eigenvalue Sv1 /S11 , where v denotes the vector representation. Proof. We note that the matrix Nv is irreducible. In fact since ρ1 ρ¯2 λ for some λ, and λ ≺ v m for some integer m, it follows that the (ρ1 , ρ2 )th entry of Nv is positive. Hence by [9] the lemma is proved. We shall refer to the equation Nv d =
Sv1 d S11
(7)
as the Perron-Frobenius equation. Note that by Lemma 4.1, Nv = Nv¯ . Since every irrep λ can be written as polynomials in v, v, ¯ it is sufficient to determine Nv . Let M = Nv − I . By Lemma 4.1 and Eq. (4) the spectrum of M is given as follows. If (i+1) k = 2m −1 is odd, then the spectrum of M is given by 2 cos( πm+1 ), 0 ≤ i ≤ m −1, and it is the same as the fusion matrix N1/2 associated with SU (2)m−1 , where 1/2 denotes the spin 1/2 representation. If k = 2m − 2 is even, then the spectrum of Nv is given by (i+1) 2 cos( 2π2m+1 ), 0 ≤ i ≤ m − 1, and it is the same as the fusion matrix N1 associated with SU (2)2m−1 acting on the set of integer spin representations of SU (2)2m−1 , where 1 denotes the spin 1 representation of SU (2)2m−1 . Our goal in this section is to show that up to permutation Nv = N1/2 + I when k = 2m − 1 and Nv = N1 when k = 2m − 2. First note that since ||M|| < 2, the entries of M can take only 1, −1, 0, and since M = Nv −I, only diagonal entries on M can be −1. By the known spectrum of M we have tr (M) = 0, tr (M 2 ) = 2m − 2 when k = 2m − 1, and tr (M) = −1, tr (M 2 ) = 2m − 1 when k = 2m − 2. Also since ||M 2 || < 4 each row of M contains at most three nonzero entries. Denote by k1 , k2 , k3 respectively the number of rows of M with one, two, three nonzero entries respectively. Then we have k1 + k2 + k3 = n, k1 + 2k2 + 3k3 = tr (M 2 ) = 2m − 2 when k = 2m − 1, and k1 + k2 + k3 = n, k1 + 2k2 + 3k3 = tr (M 2 ) = 2m − 1 when k = 2m − 2. Hence k1 = k3 + 2 when k is odd and k1 = k3 + 1 when k is even. For simplicity we enumerate the τ -twisted solitons by 1, . . . , m We associate a graph G to these solitons
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with vertices 1, . . . , m and connect i and j (i = j) by the (i, j)th entry of M. By Lemma 4.2 G is connected. If k3 = 0, then k1 = 2 or k1 = 1 depending on if k is odd or even. Permute solitons if necessary, we may assume that the first row contains only one nonzero entry. Using Eq. (7) we must have 1 in the first row of M, and it is not on the diagonal. Assume that 2 is the vertex connected to 1 on G and use Eq. 7 and the fact that G is connected; we conclude that unless m = 2, 2 is connected to a new vertex 3. Continue this argument we have shown that up to permutation Nv = N1/2 + I when k is odd, and Nv = N1 when k is even. When k is odd and if all diagonal entries of M are zeros, then 2I − M is a positive definite matrix with all diagonal entries equal to 2. Hence we can find a basis { 1 , . . . m } in Rm such that ( i , j ) = 0 or −1 if i = j and ( i , i ) = 2, 1 ≤ i ≤ m. It follows that G is a connected Coexter graph , and G must be A − D − E graph (cf. p. 60 of [12] or π Section 1.4 of [10]). Since G has norm 2 cos( m+1 ) with m vertices, by inspecting Table 1.4.5 of [10] we conclude that up to permutation M = N1/2 . When k is even, tr (M) = −1, there is at least one −1 on the diagonal of M. For the rest of this section we assume that k3 > 0 and there is at least one nonzero entry on the diagonal of M. Since one can easily determine M for m ≤ 3, we will also assume that m ≥ 4. We will derive contradictions from these assumptions. The basic idea is contained in Remark (2) on p. 23 of [10]. Introduce new rows numbered by 1 , 2 , . . . , m and we use M1 to denote a symmetric 2m × 2m matrix whose (i, j )th entry is the (i, j)th entry of M, and all other entries of M1 are equal to zero. We associate a graph G 1 to M1 whose vertices are 1, 2, . . . , m, 1 , 2 , . . . , m and i, j are connected by the absolute value of the (i, j)th entry of M. Since G is connected, and by our assumption there is at least one nonzero entry on the diagonal of M, it follows that G 1 is connected. Let P = 2I − M1 . Note that P is positive def in R2m such that the inner product inite, and we can find a basis ε1 , . . . εm , ε1 , . . . εm are two orthogomatrix of this basis is P. By definition ε1 , . . . , εm and ε1 , . . . , εm nal sets, and the angles between the lines spanned by the elements in the basis are is either π/3 or π/2, and by Th. 3.5 of [4] we conclude that ε1 , . . . , εm , ε1 . . . , εm contained in a direct sum of root systems of A − D − E. Since G 1 is connected, we is contained in one root system. If this root sysconclude that ε1 , . . . εm , ε1 , . . . εm tem is E 6 or E 8 , then we have m = 3 or m = 4. One can easily rule out these two cases using k1 = k3 + 2 or k1 = k3 + 1 and Eq. (7). If this root system is A2m+1 , since the elements of this system are of the form ei − e j , 1 ≤ i, j ≤ 2m + 1 (cf. Definition 3.1 of [4]), it follows that M can not have a row with three nonzero en is contained in the tries. Hence to finish the proof we assume that ε1 , . . . εm , ε1 , . . . εm root system of D2m . The root system of type D2m consists of vectors ±ei ± e j , 1 ≤ i = j ≤ 2m, where ei , 1 ≤ i ≤ 2m is an orthonormal basis in R2m . By assump as a subset of ±e ± e , 1 ≤ i = j ≤ tion we can identify ε1 , . . . εm , ε1 , . . . εm i j 2m. The following lemma will be used repeatedly in the following, and its proof follows directly from definitions: Lemma 4.3. (1) If a vertex on G 1 is connected to three different vertices i 1 , i 2 , i 3 , then εi1 , εi2 , εi3 must contain (up to multiplication by −1) e j + ek , e j − ek for some 1 ≤ j = k ≤ 2m; (2) If two vertices i , j of G 1 are such that εi = e j + ek , εj = e j − ek , then i is connected to vertex l in G 1 iff j is connected to vertex l; (3) If εi1 = e j +ek , εi2 = e j −ek , ε j1 = e p +eq , ε j2 = e p −eq , then { j, k}∩{ p, q} = ∅.
On Affine Orbifold Nets Associated with Outer Automorphisms
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Definition 4.4. A vertex i of G is called type 0 if the (i, i)th entry of M is zero, and type 1 otherwise. Case (1). If a type 1 vertex is connected to two different type 0 vertices on G , by Lemma 4.3 the two type 0 vertices are not connected. By permuting basis elements if necessary, we may assume that (εi , εj ) = 0, |(εi , εi )| = 1, 1 ≤ i = j ≤ 3. Since is contained in the root system of D , ε , . . . ε and ε , . . . , ε ε1 , . . . , εm , ε1 , . . . , εm 2m 1 1 m m are two orthogonal sets, it follows that we may assume that (up to multiplication by −1) ε1 = e3 + e5 , ε2 = e1 + e2 , ε3 = e1 − e2 and ε1 = e1 + e5 , ε2 = e3 + e4 , ε3 = e3 − e4 . But then e5 is in the subspace spanned by ε1 , ε2 , ε3 and also in the subspace spanned by is a basis. ε1 , ε2 , ε3 , contradicting the fact that ε1 , . . . εm , ε1 , . . . εm Case (2). If two type 1 vertices are connected on G, assume that one such vertex is connected to either a type 0 or type 1 vertex. The first case is ruled out by Lemma 4.3 and Case (1), and the second case is impossible by Lemma 4.3 and the fact that G is connected and m ≥ 4. Case (3). If two type 1 vertices are connected to the same type 0 vertex on G, since m ≥ 4 and G is connected, by case (1) and (2) we assume that the type 0 vertex is connected to one additional vertex on G, but one checks easily that this is impossible by Lemma 4.3. Now consider a subgraph G obtained from G by deleting type 1 vertices and edges with one endpoint a type 1 vertex. Since G is connected, by Case (1) G is connected. Moreover, if i is a type 0 vertex, define ηi := √1 (εi + εi ). Then (ηi , ηi ) = 2, and it 2 follows that two vertices i 1 , i 2 on G are connected by −(ηi , η j ) edges. Hence G is a connected Coexter graph, and by p. 60 of [12] we know that G is an A − D − E graph. Since we assume that k3 ≥ 1, G must be type D or E. In the case k1 = k3 + 2, we must have no type 1 vertex attached to the end points of G , contradicting our assumption that a type 1 vertex exists and must be connected to one type 0 vertex by Case (2). In the case k1 = k3 + 1, we must have exactly one type 1 vertex connected to one endpoint of either type D or type E graph, and these cases can be directly ruled out by tedious calculations using the Perron-Frobenius equation. Here we give a different approach. By fusion rules of SU (2)2m−1 in Eq. (5) , Nm−1 can be written as a polynomial of N1 2 with integer coefficients, and we have Nm−1 = N1 + I . The spectrum of Nm−1 is given π(i+1) by 2 cos 2m+1 , 0 ≤ i ≤ m − 1. Since N1 has the same spectrum as Nv , it follows that there is a symmetric matrix M with integer entries such that M 2 = Nv + I, and M has the same spectrum as Nm−1 . Since Nv has one 1 and m − 1 2’s on the diagonal, it follows that M has one row with one nonzero entry ±1, and m − 1 rows with two nonzero entries which are ±1. Now associate a graph with vertices 1, . . . , m to M so that the i th and the j th vertex are connected by the absolute value of the (i, j)th entry of M (If i = j and the (i, i)th entry of M is ±1, we connect i to itself by a loop.) As in the case with M this graph is connected, and it follows that it is a line segment with one loop attached to an endpoint. Since G has a trivalent vertex, it follows that there exist four different vertices i, j, k, l such that the (i, j), (i, k), (i, l) entries of Nv are 1, but one checks easily that this is impossible since M 2 = Nv + I . We have proved the following: Proposition 4.5. (1) When k = 2m − 1, we can label the τ -twisted irrep of A by inte( j−1) gers 1, 2, . . . , m such that the (i, j)th entry of Nv is given by δi j + 21 (i−1) 2 , 2 , where half integers 0 ≤ k/2 ≤ m − 1/2 label the spin of irreps of SU (2)m−1 ;
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(2) When k = 2m − 2, we can label the τ -twisted irrep of A by integers 1, 2, . . . , m such that the (i, j)th entry of Nv is given by δi j + 1(i − 1), ( j − 1) , where integers 0 ≤ k ≤ m − 1 label the spin of irreps of SU (2)2m−1 . By Prop. 4.5 the Perron-Frobenius eigenvector d is up to multiplication by a positive constant δ and possible permutations equal to (d(i−1)/2 )1≤i≤m (resp. (di−1 )1≤i≤m )) when k = 2m − 1 (resp. when k = 2m − 2). By Prop. 3.1 of [1] ρτ dρ2τ = λ dλ2 , and by Eq. (4) we can determine δ uniquely. When k = 2m − 1, δ = k = 2m − 2, δ =
√
3(2m+1) . π 2π 4 sin( 2m+1 ) sin( 2m+1 )
√
3(m+1) , π 2 sin2 ( 2m+2 )
and when
We have therefore proved the following
Corollary 4.6. (1) When k = 2m − 1, the set of indices of τ -twisted solitons are given 3(m+1) sin(iπ/(m+1)) by { 4 sin 4 ( π ) sin(π/(m+1)) , 1 ≤ i ≤ m}; 2m+2
(2) When k = 2m − 2, the set of indices of τ -twisted solitons are given by 3(2m+1)
π 16 sin2 ( 2m+1 ) sin2
2π 2m+1
sin((2i−1)π/(2m+1)) , 1 sin(π/(2m+1))
≤i ≤m .
Acknowledgement. The author would like to thank Prof. V. G. Kac for stimulating discussions and providing references and useful suggestions. The paper would not have been written without his help. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Böckenhauer, J., Evans, D.E., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210(3), 733–784 (2000) 2. Birke, L., Fuchs, J., Schweigert, C.: Symmetry breaking boundary conditions and WZW orbifolds. Adv. Theor. Math. Phys. 3, 671–726 (1999) 3. Bakalov, B., Kac, V.G.: Twisted modules over lattice vertex algebras. In: Lie theory and its applications in physics V, eds Doebner, H.-D., Dobrev. V.K., River Edge, NJ: World Sci. World Sci. Publ., 2004, pp. 3–26 4. Cameron, P.J., Goethals, J.-M., Seidel, J.J., Shult, E.E.: Line graphs, root systems, and elliptic geometry. J. Algebra 43(1), 305–327 (1976) 5. Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971); II. 35, 49–85 (1974) 6. Dong, C., Xu, F.: Conformal nets associated with lattices and their orbifolds. Adv. Math. 206(1), 279–306 (2006) 7. Gabbiani, F., Fröhlich, J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993) (2) (2) 8. ten Kroode, F., van de Leur, J.: Level one representations of the twisted affine algebras An and Dn . Acta Appl. Math. 27(3), 153–224 (1992) 9. Gantmacher, F.R.: The theory of matrices. Vol. 2. Newyork: Chelsea, 1960 10. Goodman F., de la Harpe, P., Jones, V.F. R.: Coxeter graphs and towers of algebras. Mathematical Sciences Research Institute Publications, 14. New York: Springer-Verlag, 1989 11. Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148, 521–551 (1992) 12. Humphreys, J.: Introduction to Lie algebras and representation theory. GTM 9. Berline-Heidelberg- New York: Springer-Verlag, 1972 13. Kac V.G.: Infinite Dimensional Lie Algebras. 3rd Edition. Cambridge: Cambridge University Press, 1990 14. Kac, V., Longo, R., Xu, F.: Solitons in affine and permutation orbifolds. Commun. Math. Phys. 253(3), 723– 764 (2005) 15. Kac, V.G., Wakimoto, M.: Modular and conformal invariance constraints in representation theory of affine algebras. Adv. in Math. 70, 156–234 (1988)
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16. Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001) 17. Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004) 18. Petkova, V.B., Zuber, J.-B.: Boundary conditions in charge conjugate sl(N) WZW theories. http://arxiv. org/abs/hep-th/0201239v4, 2002 19. Pressley, A., Segal, G.: Loop Groups. Oxford: Oxford University Press, 1986 20. Reed, M., Simon, B.: Methods of modern mathematical physics, I:Functional analysis. Newyork: Academic Press, 1980 21. Rehren, K.-H.: Braid group statistics and their superselection rules. In: The Algebraic Theory of Superselection Sectors. D. Kastler, ed., Singapore: World Scientific, 1990 22. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. Berlin, New York: Walter de Gruyter, 1994 23. Toledano Laredo, V.: Fusion of Positive Energy Representations of L Spin 2n . Ph.D. dissertation, University of Cambridge, 1997 24. Verrill, R.: Positive energy representations of L σ SU (2r ) and orbifold fusion. Ph.D. dissertation, University of Cambridge, 2001 25. Wassermann, A.: Operator algebras and Conformal field theories III. Invent. Math. 133, 467–538 (1998) 26. Xu, F.: 3-manifold invariants from cosets. J. Knot Theory Ram 14(1), 21–90 (2005) 27. Xu, F.: Algebraic orbifold conformal field theories. PNAS. USA 97(26) 14069–14073, 2000 28. Xu, F.: Jones-Wassermann subfactors for disconnected intervals. Commun. Cont. Math. 2, 307–347 (2000) 29. Xu, F.: Algebraic coset conformal field theories. Commun. Math. Phys. 211, 1–43 (2000) Communicated by Y. Kawahigashi
Commun. Math. Phys. 291, 863–894 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0893-2
Communications in
Mathematical Physics
Universality in the Two Matrix Model with a Monomial Quartic and a General Even Polynomial Potential M. Y. Mo School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. E-mail: [email protected] Received: 4 November 2008 / Accepted: 21 May 2009 Published online: 6 August 2009 – © Springer-Verlag 2009
Abstract: In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measure Z n−1 exp (−n (tr(V (M1 ) + W (M2 ) − τ M1 M2 )) dM1 dM2 , 4
in the case where W = y4 and V is a general even polynomial. We studied the correlation kernel for the eigenvalues of the matrix M1 in the limit as n → ∞. We extended the results of Duits and Kuijlaars in [14] to the case when the limiting eigenvalue density for M1 is supported on multiple intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained in [14] with theta functions and then showing that this parametrix is well-defined for all n by studying the theta divisor.
1. Introduction 1.1. 2 matrix models and biorthogonal polynomials. The 2-matrix Hermitian models are matrix models with the probability measure Z n−1 exp (−n (tr(V (M1 ) + W (M2 ) − τ M1 M2 )) dM1 dM2 ,
(1.1)
defined on the space of pairs (M1 , M2 ) of the n × n Hermitian matrix. The constant Z n is the normalization constant of the measure, τ ∈ R\{0} is a coupling constant and dM1 , dM2 are the standard Lebesgue measures on the space of Hermitian matrices. In (1.1), V and W are called potentials of the matrix model. In this paper, we shall consider V to 4 be a general even polynomial and W to be the monomial W (y) = y4 . The funding of this research is provided by the EPSRC grant EP/D505534/1.
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Let x1 , . . . , xn and y1 , . . . , yn be the eigenvalues of the matrices M1 and M2 respectively, then the eigenvalues of the matrix model (1.1) are distributed according to P( x , y) =
Z˜ n−1
n
2 −n
(xi − xk ) (yi − yk ) e 2
n j=1
V (x j )+W (y j )−τ x j y j
,
(1.2)
i
where Z˜ n is a normalization constant and x = (x1 , . . . , xn ), y = (y1 , . . . , yn ). The two-matrix model was introduced in [27,31] as a generalization of the one matrix model to study critical phenomena in physical systems. The 2 matrix model is needed to represent all conformal models in statistical physics [9]. It is also a powerful tool in the studies of random surfaces as the large N expansion of the partition function Z˜ n is expected to be the generating function of discretized surface [29]. Since its introduction, the 2 matrix model has become a very active research area [1–6,13,14,17,18,20,19,21– 23,28,30] and one of the major problems is to obtain rigorous asymptotics for the eigenvalue statistics. A good review of the subject can be found in [15,16]. A particular important object in the studies of eigenvalue statistics is the correlation function Rnm,l (x1 , . . . , xm , y1 , . . . , yl ), Rnm,l (x1 , . . . , xm , y1 , . . . , yl ) n n (n!)2 = P( x , y) dx j dyk . (n − m)!(n − l)! Rn−m Rn−l j=m+1
(1.3)
k=l+1
In [18,32], a connection between biorthogonal polynomials and the correlation functions of 2 matrix models (1.3) was found. Let pk (x) and ql (y) be monic polynomials of degrees k and l such that dxdypk (x)ql (y)e−n(V (x)+W (y)−τ x y) = h k δkl , (1.4) R R
for some constants h k , then these polynomials exist and are unique [5,17]. These polynomials are known as biorthogonal polynomials. Let us define some integral transforms of the biorthogonal polynomials by Q k (x) = e−nV (x) qk (y)e−n(W (y)−τ x y) dy, R (1.5) −nW (y) −n(V (x)−τ x y) Pk (y) = e pk (x)e dx, R
and define the kernels to be n K 11 (x1 , x2 ) =
n−1 1 p j (x1 )Q j (x2 ), hj j=0
n K 12 (y, x) =
n−1 j=0
n K 21 (y1 , y2 ) =
n K 22 (x, y) =
n−1 1 P j (x)q j (y), hj j=0
1 p j (y)q j (x), hj
n−1 1 P j (y1 )Q j (y2 ) − e−n(V (x)+W (y)−τ x y) . hj j=0
(1.6)
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865
Then the correlation function Rnm,l (1.3) has the following determinantal expression. Rnm,l (x1 , . . . , xm , y1 , . . . , yl ) = det
n (x , x ) K 11 i j n (y , x ) K 21 i j
m
i, j=1 l,m i, j=1
n (x , y ) K 12 i j n (y , y ) K 22 i j
m,l
li, j=1 .
(1.7)
i, j=1
Upon averaging over the variables yk , we see that the m-point correlation function n (x , x ), Rnm,0 (x1 , . . . , xm ) for the eigenvalues of the matrix M1 is given by the kernel K 11 i j n m (xi , x j ) i, j=1 . Rnm,0 (x1 , . . . , xm ) = det K 11
(1.8)
The purpose of this paper is to provide a rigorous asymptotic expression for the kernel n as n → ∞ for W (y) = y 4 and V (x) being a general even polynomial. K 11 4 n can be expressed Due to a generalized Christoffel-Darboux formula, the kernels K 11 in terms of a finite sum of the biorthogonal polynomials (See [5,19,33 and 8]). This n into finding the asympreduces the problem of finding an asymptotic expression for K 11 totics of the biorthogonal polynomials. 1.2. Riemann-Hilbert problem. The biorthogonal polynomials pk (x) in (1.4) satisfies a Riemann-Hilbert problem [6,30] similar to the one that is satisfied by orthogonal polynomials [26]. This allows the implementation of the Deift-Zhou steepest descent method ([7,10–12]). Let w j (x) be the weights defined by w j (x) = e
−nV (x)
j
R
y e
−n
y4 4 −τ x y
dy,
j = 0, 1, 2.
(1.9)
Assuming n is divisible by 3, consider the following Riemann-Hilbert problem: 1. Y (z) is analytic in C\R, ⎛ 1 w0 (z) w1 (z) 1 0 ⎜0 2. Y+ (z) = Y− (z) ⎝ 0 0 1 0 0 0 ⎛ n z 0 ⎜ −n −1 ⎜ 0 z 3 3. Y (z) = I + O(z ) ⎝ 0 0 0 0
⎞ w2 (z) 0 ⎟ , z ∈ R, 0 ⎠ 1 ⎞ 0 0 0 0 ⎟ ⎟ , z → ∞. − n3 0 ⎠ z 0
(1.10)
n
z− 3
This Riemann-Hilbert problem has a unique solution given by the biorthogonal polynomial pk (x) and some other polynomials, together with their Cauchy transforms [30]: ⎞ ⎛ pn (z) C( pn w0 )(z) C( pn w1 )(x) C( pn w2 )(z) ⎜ p (0) (z) C( p (0) w )(z) C( p (0) w )(z) C( p (0) w )(z)⎟ ⎟ ⎜ n−1 0 n−1 1 n−1 2 Y (z) = ⎜ n−1 ⎟ , (1.11) (1) (1) (1) (1) ⎝ pn−1 (z) C( pn−1 w0 )(z) C( pn−1 w1 )(z) C( pn−1 w2 )(z)⎠ (2) (2) (2) (2) pn−1 (z) C( pn−1 w0 )(z) C( pn−1 w1 )(z) C( pn−1 w2 )(z)
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where pn−1 (z), j = 0, 1, 2 are some polynomials of degree n − 1 and C( f ) is the Cauchy transform of the function f , f (s) 1 C( f )(x) = ds. 2πi R s − x This representation as a Riemann-Hilbert problem allows the authors of [14] to compute the asymptotics of the biorthogonal polynomials by extending the Deift-Zhou steepest descent method ([7,10–12]). We will now explain the main ideas behind these analyses.
1.3. Rigorous results in the “one-cut regular” case. Until recently, most results in the asymptotics of biorthogonal polynomials have been obtained through heuristic argument (See [19,20]). For a long time, the only rigorous result was the case when both W (y) and V (x) are quadratic polynomials [17]. In the recent work by Duits and Kuijlaars [14], (see also [13], Chap. 5, which was later made into the publication [14]), the Deift-Zhou steepest descent method ([10–12], see also [7]) was successfully applied to obtain the 4 asymptotics of biorthogonal polynomials with W (y) = y4 and V (x) an even polynomial in the case when the limiting eigenvalue density for M1 is supported on a single interval. The main idea in [14] is to transform and approximate the Riemann-Hilbert problem (1.10) via the use of suitable equilibrium measures and then solve the approximated Riemann-Hilbert problem explicitly to obtain asymptotic formula for the biorthogonal polynomials. The results in [14] was obtained in the case when one of these equilibrium measures is supported on a single interval. This case is called the “one-cut regular case” in [14]. 1.3.1. Equilibrium measure. We will now state some results on the equilibrium measures used in [14]. Let I (νi , ν j ) be the following energy function: 1 I (νi , ν j ) = dνi (x)dν j (y), log (1.12) |x − y| where the integral is performed on the supports of the measures νi and ν j . Then the equilibrium measures µ1 , µ2 and µ3 are the measures that minimize the following energy function E V (ν1 , ν2 , ν3 ). Definition 1 (Definition 2.2 in [14] (see also Definition 5.2.1 in [13])). The equilibrium measure (µ1 , µ2 , µ3 ) is the triplet of measures that minimizes the following energy function: E V (ν1 , ν2 , ν3 ) =
3 j=1
I (ν j , ν j ) −
2 j=1
I (ν j , ν j+1 ) +
3 4 4 V (x) − τ 3 |x| 3 4
dν1 (x) (1.13)
among non-negative Borel measures ν1 , ν2 and ν3 that satisfy the following properties: 1. All the measures ν j , j = 1, 2, 3 have finite logarithmic energies; 2. ν1 and ν3 are measures supported on R with ν1 (R) = 1 and ν3 (R) = 13 ; 3. ν2 is a measure supported on iR with ν2 (iR) = 23 ;
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4. Let σ be the unbounded measure on iR given by √ 3 4 1 dσ (z) = τ 3 |z| 3 |dz|, z ∈ iR, 2π
(1.14)
where |dz| is the unit arc length on iR, then ν2 satisfies the constraint ν2 ≤ σ . Let U ν (x) be the logarithmic potential of the measure ν, U ν (x) = − log |x − s|dν(s);
(1.15)
then it was shown in [14] that the logarithmic potentials of µ1 and µ2 satisfy the following properties: 3 4 4 2U µ1 (x) = U µ2 (x) − V (x) + τ 3 |x| 3 + l, x ∈ Sµ1 , 4 3 4 4 2U µ1 (x) ≥ U µ2 (x) − V (x) + τ 3 |x| 3 + l, x ∈ R\Sµ1 , 4
(1.16)
for some constant l, where Sµ1 is the support of µ1 . The properties of the equilibrium measures µ1 , µ2 and µ3 were studied in [14] and we have the following Theorem 1 (Theorem 2.3 in [14] (see also Theorem 5.2.2 in [13])). Let V be an even polynomial and τ > 0. Then there is a unique minimizer (µ1 , µ2 , µ3 ) of E(ν1 , ν2 , ν3 ) in (1.13) that satisfies the conditions in Definition 1. Let us denote the support of the Borel measure ν by Sν , then we have 1. Sµ1 consists of finitely many disjoint intervals g+1
Sµ1 = ∪ j=1 [λ2 j−1 , λ2 j ],
(1.17)
where λ j ∈ R and the points are ordered such that λ j < λk if j < k. Moreover, µ1 is absolutely continuous with respect to the Lebesgue measure, and on each interval [λ2 j−1 , λ2 j ], it has a continuous density of the form µ1 (z) = ρ1 (z)dz = ψ j (z) (λ2 j − z)(z − λ2 j−1 ), z ∈ [λ2 j−1 , λ2 j ], (1.18) where ψ j (z) is analytic and non-negative on [λ2 j−1 , λ2 j ]. 2. Let σ be the measure in (1.14), then Sµ2 = iR and there exists a constant c > 0 such that the support Sσ −µ2 of σ − µ2 is given by Sσ −µ2 = iR\(−ic, ic).
(1.19)
Moreover, σ − µ2 has an analytic density on Sσ −µ2 that vanishes as a square root at ±ic. 3. Sµ3 = R and µ3 has a density which is analytic in R\{0}. 4. For j = 1, 2, 3, we have µ j (A) = µ j (−A) for any Borel set A. Remark 1. In particular, by 4, above, we see that Sµ1 is symmetric under the map z → −z, that is, we have λk = −λ2g+3−k . We then have the following definition of regularity. (See Definition 2.5 in [14].)
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Definition 2. The potential V (x) is regular if the following conditions are satisfied; 1. The inequality in (1.16) is strict outside of Sµ1 ; 2. The density ρ1 (z) vanishes like a square-root at the end points of Sµ1 ; 3. The density ρ1 (z) does not vanish in the interior of Sµ1 . It is known that a generic potential V (x) is regular [14]. The “one-cut regular case” is the case when Sµ1 consists only of a single interval and that V (x) is regular. In [14], rigorous asymptotics of biorthogonal polynomials was obtained for this case. The asymptotics of the biorthogonal polynomials were then used to obtain an asymptotic expression for n in (1.6). In this paper we will extend these results to the case when µ is the kernel K 11 1 supported on any number of intervals. We will call this the multi-cut case. 1.3.2. Model Riemann-Hilbert problem. By using the equilibrium measures µ1 , µ2 and µ3 that minimize (1.13), the authors of [14] were able to transform and approximate the Riemann-Hilbert problem (1.11) by an explicitly solvable one. To state their results, let us assume V (x) is regular. Let us denote an interval in the support Sµ1 (1.17) of µ1 by ˜ j:
j and a gap between the intervals by
j = [λ2 j−1 , λ2 j ], j = 1, . . . , g + 1, ˜ j = [λ2 j , λ2 j+1 ], j = 1, . . . , g,
˜ 0 = (−∞, λ1 ],
˜ g+1 = [λ2g+2 , ∞).
We will define α j to be the constants g+1 αk = µ1 ∪ j=k+1 j , k = 0, . . . , g, αg+1 = 0.
(1.20)
(1.21)
Remark 2. Note that, since V (x) is an even polynomial, by Theorem 1, we have µ1 (A) = µ1 (−A) for any Borel set A. Therefore the constants αk in (1.21) satisfy the symmetry αk = 1 − αg+1−k .
(1.22)
This symmetry condition is used in the proof of Theorem 9 to simplify the expression of the outer parametrix. However, this condition is not essential to the construction of the outer parametrix and the result of this paper will not be affected without this condition. The steepest descent method then allows the approximation of (1.10) by the following ‘model Riemann-Hilbert problem’ (see Sect. 8 of [14] and 5.10 of [13]): 1. M(z) is analytic in C\ R ∪ Sσ −µ2 , 2. M+ (z) = M− (z)J M (z), z ⎛ 1 ⎜ 0 3. M(z) = I + O(z −1 ) ⎜ ⎝0 0
∈ R ∪ Sσ −µ2 , ⎞ 0 0 0 1 z3 0 0 ⎟ ⎟ 1 0 , 0 1 0 ⎠ 0 Aj 1 0 0 z− 3
uniformly as z → ∞ in the j th quadrant, 1 4. M(z) = O (z − λ j )− 4 , z → λ j , j = 1, . . . , 2g + 2, 1 M(z) = O (z ∓ ic)− 4 , z → ±ic,
(1.23)
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1
where Sσ −µ2 is oriented upward, the branch of z 3 is chosen such that z 3 ∈ R for z ∈ R+ and the branch cut is chosen to be the negative real axis. The A j are given by (with 2πi ω=e 3 ) ⎞ ⎛ 2 i ⎝1 −ω −ω ⎠ 1 −1 −1 , A1 = √ 3 1 −ω2 −ω ⎞ ⎛ 2 i ⎝−ω −1 ω ⎠ −1 −1 1 , A3 = √ 3 −ω −1 ω2
⎛ i ⎝ω 1 1 1 A2 = − √ 3 ω2 1 ⎛ 2 i ⎝1 −ω 1 −1 A4 = √ 3 1 −ω
⎞ ω2 1 ⎠, ω ⎞ ω 1 ⎠. ω2
The jump matrices J M (z) in (1.23) are given by the following: ⎞ 0 1 0 0 ⎜−1 0 0 0⎟ , z∈ J M (z) = ⎝ 0 0 0 1⎠ 0 0 −1 0 ⎛ 0 0 e−2nπiαk 2nπiα ⎜ 0 k e 0 J M (z) = ⎜ ⎝ 0 0 0 0 0 −1 ⎛
⎛
Sµ1 ,
1 ⎜0 J M (z) = ⎝ 0 0
⎞ 0 0 0 0 −1 0⎟ , z ∈ Sσ −µ2 , 1 0 0⎠ 0 0 1 (1.24)
⎞ 0 0⎟ ⎟, z ∈
˜ k , k = 0, . . . , g + 1. 1⎠ 0
1.3.3. Universality results. By using the solution to the model Riemann-Hilbert problem (1.23), universality results were obtained for the one-cut regular case. We have the following result on the global eigenvalue distribution of the matrix M1 . Theorem 2. Let (µ1 , µ2 , µ3 ) be the equilibrium measures that minimize the functional (1.13). Then as n → ∞ and n ≡ 0(mod3), we have lim
n→∞
1 n K (x, x) = ρ1 (x), n 11
(1.25)
uniformly for x ∈ R, where ρ1 is the density of µ1 in Definition 2. As explained in [14], the requirement n ≡ 0(mod3) is not essential and is only imposed to minimize the technicality. We also have the following result on the local distribution of the eigenvalues. Theorem 3. Let (µ1 , µ2 , µ3 ) be the equilibrium measures that minimize the functional (1.13). Then as n → ∞ and n ≡ 0(mod3), we have the following: 1. Let x ∗ be a point in the interior of the support Sµ1 . Then we have sin (π(u − v)) 1 u v n ∗ ∗ lim x + K ,x + = , n→∞ nρ1 (x ∗ ) 11 nρ1 (x ∗ ) nρ1 (x ∗ ) π(u − v) uniformly for u, v in compact subsets of R.
(1.26)
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2. Let ϕ j > 0 be such that ρ1 (z) =
1 ϕj |z − λ j | 2 + O z − λ j , π
as z → λ j , j = 1, . . . , 2g + 2 inside of Sµ1 , where λ j are defined as in (1.17). Then we have the following: ⎛ 1
⎞ u
v
n ⎝ j j ⎠ lim 2 K 11 λ j + (−1) 2 , λ j + (−1) 2 nϕ j 3 nϕ j 3 nϕ j 3
n→∞
=
Ai(u)Ai (v) − Ai(v)Ai (u) , u−v
(1.27)
uniformly for u, v in compact subsets of R, where Ai is the Airy function. Recall that the Airy function is the unique solution to the differential equation v
= zv that has the following asymptotic behavior as z → ∞ in the sector −π + ≤ arg(z) ≤ π − , for any > 0: 3 2 2 3 1 Ai(z) = √ 1 e− 3 z 1 + O(z − 2 ) , −π + ≤ arg(z) ≤ π − , z → ∞, 2 πz 4 (1.28) 3
where the branch cut of z 2 in the above is chosen to be the negative real axis. Provided a solution M(z) of (1.23) exists and is uniformly bounded in n away from the singularities, the analysis in [14] that leads to Theorems 2 and 3 in the one-cut case can be carried out with the parametrix M(z) in the multi-cut case. The main contribution of this paper is to construct this parametrix in the multi-cut case and to show its existence.
2. Statement of Results Although the steepest descent analysis in [14] already covers the general case without the 1-cut assumption, solution to the model Riemann-Hilbert problem (1.23) must be obtained to complete the Riemann-Hilbert analysis and to extend the universality results to the general multi-cut case. The main difficulties are to show that a solution of the model Riemann-Hilbert problem exists for all n and to find an explicit expression of it. This involves the study of the theta divisor, which is the set of points in which a theta function vanishes. (See Sect. 4.1 for a more detailed description of the theta function.) This is a difficult problem with very few results available. In this paper we managed to construct the solution of the model Riemann-Hilbert problem (1.23) with the use of theta functions. And by using results from [24] and [25], we were able to show the existence of the solution M(z) to (1.23) for all n. This allows us to extend the universality results in [14] to the case when V (x) is a general even polynomial and obtain universality results in the multi-cut case. To be precise, the analysis in [14] can be applied as long as a solution M(z) to the Riemann-Hilbert problem (1.23) exists and satisfies the following conditions.
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Theorem 4. Let ε > 0 be a fixed small number independent on n. Let Bε, j and Bε,±ic be small discs of radius ε centered at λ j and ±ic respectively. Let K ⊂ C be a compact subset in C and let T be the set 2g+2 T = K\ ∪ j=1 Bε, j ∪ Bε,ic ∪ Bε,−ic . (2.1) Suppose the solution M(z) of the Riemann-Hilbert problem (1.23) and its inverse M −1 (z) exist and satisfy the following conditions: 1. Both M(z) and M −1 (z) are bounded in n uniformly inside T for any compact subset K; 2. For any r > max{c, λ2g+2 } independent on n, there exist constants C ljk and ljk , 1 ≤ j, k, l ≤ 4, independent on n, such that, for z > |r |, 1 | (M(z)) jk | < C ljk z 3 , for z in the l th quadrant, 1 | M −1 (z) | < ljk z 3 , for z in the l th quadrant. jk
n is given by Then as n → ∞ and n ≡ 0mod3, the asymptotic behavior of the kernel K 11 (1.25), (1.26) and (1.27). The main new result of this paper is the construction of this parametrix and the proof of its existence.
Theorem 5. There exists a unique solution M(z) to the Riemann-Hilbert problem (1.23). Moreover, it satisfies the conditions in Theorem 4. The explicit expression of the parametrix is given in terms of theta functions and meromorphic 1-forms (see Theorem 9 in Sect. 4). From Theorem 4 and Theorem 5, we see that the universality results Theorem 2 and Theorem 3 are true in the multi-cut regular case. n is Theorem 6. Suppose V (x) is regular, then the asymptotic behavior of the kernel K 11 given by (1.25), (1.26) and (1.27).
In many applications of the Deift-Zhou steepest descent method, theta functions are needed to solve a model Riemann-Hilbert problem and the solvability of these model Riemann-Hilbert problems is important to guarantee the validity of these asymptotic formulas for all n, as n → ∞. We believe the techniques and results in this paper will not only be valuable to the random matrix community studying 2 matrix models, but it will also be important to many other problems in which the Deift-Zhou steepest descent method is applicable. 3. The Riemann Surface Associated with the Equilibrium Measures We first need to construct a Riemann surface from the equilibrium measures and use the theta function and meromorphic differentials on this Riemann surface to construct a parametrix for the Riemann-Hilbert problem (1.23). The Riemann surface is realized as a four-sheeted covering of the Riemann sphere. Define four copies of the Riemann sphere by L j , j = 1, . . . , 4: L1 = C\Sµ1 , L2 = C\(Sµ1 ∪ Sσ −µ2 ), L3 = C\(Sσ −µ2 ∪ Sµ3 ), L4 = C\(Sµ3 ), where C is the Riemann sphere obtained by adding the point z = ∞ to C.
(3.1)
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M. Y. Mo
Fig. 1. The sheet structure of the Riemann surface L
The Riemann surface L is constructed as follows: L1 is connected to L2 via Sµ1 , L2 is connected to L3 via Sσ −µ2 and L3 is connected to L4 via Sµ3 , as shown in Fig. 1. Let us define the functions F j (z) by 1 dµ j (s). F j (z) = (3.2) Sµ j z − s Then we have the following result Lemma 1 (Lemma 5.1 in [14], (see also Lemma ∪4j=1 L j → C defined by ⎧ −F1 (z) + V (z), ⎪ ⎪ 4 1 ⎪ ⎪ ⎪ F1 (z) − F2 (z) + τ 3 z 3 , ⎪ ⎪ 4 1 ⎪ ⎪ ⎪ ⎨ F1 (z) − F2 (z) − τ 43 (−z) 31 , ξ(z) = F2 (z) − F3 (z) − τ 3 (−z) 3 , ⎪ 4 1 ⎪ ⎪ F2 (z) − F3 (z) + τ 3 z 3 , ⎪ ⎪ 4πi 4 1 ⎪ ⎪ ⎪ F3 (z) + e 3 τ 3 z 3 , ⎪ ⎪ 2πi 4 1 ⎩ F3 (z) + e 3 τ 3 z 3 ,
5.4.1 in [13])). The function ξ : z z z z z z z
∈ L1 ; ∈ L2 , ∈ L2 , ∈ L3 , ∈ L3 , ∈ L4 , ∈ L4 ,
Rez Rez Rez Rez Imz Imz
> 0; < 0; > 0; < 0; > 0; < 0;
(3.3)
has an extension to a meromorphic function (also denoted by ξ ) on L. The meromorphic function has a pole of order degV − 1 at infinity on the first sheet, and a simple pole at the other point at infinity. We shall denote the restriction of ξ(z) to the sheet Lk by ξk (z).
3.1. Canonical homology basis on the Riemann surface. The Riemann surface L is of genus g. In order to define theta functions on this Riemann surface, a canonical homology basis has to be chosen. Let us define the canonical homology basis as in Fig. 2. The figure should be understood as follows. The top left rectangle denotes the first sheet L1 , the top right rectangle denotes L2 , the lower left one denotes L3 and the lower right one denotes L4 . A b-cycle is a loop in L1 around the branch cuts that is symmetric with respect to the real axis, while an a-cycle a j consists of a path in the upper half plane in L1 that goes from j+1 to j ( j is defined in (1.20)), together with a path in the lower half plane in L2 that goes from j to j+1 . The loop formed by these 2 paths is
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873
Fig. 2. The a and b cycle of Riemann surface L
an a-cycle. We will also choose these 2 paths such that their projection on the complex z-plane are mapped onto each other under complex conjugation. We can now define the basis of holomorphic differentials that is dual to this canonical homology basis. Let ω j be holomorphic 1-forms on L such that ω j = δ jk , 1 ≤ j, k ≤ g. (3.4) ak
The 1-forms ω j are known as holomorphic 1-forms that are dual to the canonical homology basis (a, b). Let the b-period of these 1-forms be i j ω j = i j , (3.5) bi
then the g × g matrix with entries i j is symmetric and Im() > 0. The matrix i j will be used in Sect. 4.1 to define theta functions associated to the curve L, which are essential to our construction. 4. Construction of the Outer Parametrix In this section we will construct the solution to the Riemann-Hilbert problem (1.23). The construction will be done in three steps. First let A = (A1 , . . . , Ag )T be a g-dimensional constant vector. Then in Sect. 4.1, we will use theta functions on the Riemann surface L to construct a vector (z, A) that has the following jump discontinuities: = − (z, A)J (z, A), z ∈ R ∪ Sσ −µ2 , + (z, A)
(4.1)
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M. Y. Mo
where J (z) is the following matrix: ⎛
0 1 0 ⎜ = ⎝1 0 0 J (z, A) 0 0 0 0 0 1 ⎛ e−2πi Ak ⎜ 0 =⎜ J (z, A) ⎝ 0 0
⎞ ⎛ ⎞ 0 1 0 0 0 0⎟ 0 0 1 0⎟ =⎜ , z ∈ Sµ1 , J (z, A) ⎝0 1 0 0⎠ , z ∈ Sσ −µ2 , 1⎠ 0 0 0 0 1 ⎞ (4.2) 0 0 0 e2πi Ak 0 0⎟ ⎟, z ∈
˜ k , k = 0, . . . , g + 1. 0 0 1⎠ 0 1 0
will have poles at various points on the gaps
˜ j in As we shall see, the vector (z, A) C but is otherwise bounded. Then in Sect. 4.2, we will use meromorphic differentials of the third kind to construct vectors j (z), j = 1, . . . , 4 that have the following jump discontinuities: j,+ (z) = j,− (z)J j (z), z ∈ R ∪ Sσ −µ2 ,
(4.3)
where the jump matrices J j (z) are given by ⎛
0 1 ⎜−1 0 J j (z) = ⎝ 0 0 0 0 ⎛ β e ( j )k ⎜ 0 J j (z) = ⎜ ⎝ 0 0
⎞ 0 0⎟ , z ∈ Sµ1 , J j (z) = J M (z), z ∈ Sσ −µ2 , 1⎠ 0 ⎞ 0 0 0 e−(β j )k 0 0⎟ ⎟, z ∈
˜ k , k = 0, . . . , g + 1, 0 0 1⎠ 0 1 0 0 0 0 1
(4.4)
where β j is a constant g-dimensional row vector and β j k is its k th component. The vectors β j will be determined by requiring that the vectors T = j,1 (z)1 (z, A), . . . , j,4 (z)4 (z, A) N j (z, A) only have fourth root singularities at the points λ j and ±ic, where j,k (z) and k (z, A) th T are the k components of these vectors. Let α be the vector α = (α1 , . . . , αg ) , then the matrix N (z) whose entries are given by ⎧ ⎨ N j,k z,
+ n α , Imz > 0; N jk (z) = β j δ ⎩ (−1) 4,k N j,k z, , Imz < 0, 2πi + n α βj 2πi
will satisfy the jump conditions in (1.24). We then normalize the behavior of the matrix N (z) at infinity (Sect. 4.3) to obtain the outer parametrix.
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4.1. Theta function on the Riemann surface L. The theta function θ : Cg −→ C associated to the Riemann surface L and the choice of canonical homology of basis Sect. 3.1 is defined by θ (s ) := eiπ n·n +2iπ s·n . (4.5) n∈Zg
The theta function has the following quasi-periodic property, which will be important to the construction of the outer parametrix. Proposition 1. The theta function is quasi-periodic with the following properties: = θ (s ), θ (s + M) = exp 2πi − M, s − M, M θ (s + M) θ (s ), 2
(4.6)
where ·, · denotes the inner product in Cg given by X, Y = X Y T , where X and Y are g-dimensional row vectors. We will now define the Abel map on L. The Abel map u : L → Cg is defined by T u(x) = u 1 (x), . . . , u g (x) =
x
ω1 , . . . ,
x0
x
T ωg
,
(4.7)
x0
where x0 is a point on L. We will choose x0 so that x0 is the point on L1 that projects to λ2g+2 in C. We will denote this point by λ12g+2 . The composition of the theta function with the Abel map is then a multi-valued function from L to C. It is either identically zero or it has g zeros on L. If a theta function is not identically zero, then there is no meromorphic function that has poles only at the zeroes of this theta function. Theorem 7. Let A = (A1 , . . . , Ag )T be a g-dimensional vector such that the function is not identically zero. Let d1 , . . . , dl be its zeroes with multiplicities k j . θ (u(z) + A) Then there is no non-constant meromorphic function f (z) that has poles only at the points d j with order less than or equal to k j and holomorphic elsewhere. This is a consequence of the Riemann-Roch theorem. In general, for a given g + l points (counting multiplicity) on a Riemann surface, there are l independent non-constant meromorphic functions with poles exactly at these points. This can be thought of as an extension of Liouville’s theorem. Let φ(z) be the anti-holomorphic involution on L defined by φ(z) : z → z, where z and z are on the same sheet.
(4.8)
Then by the definition of the canonical homology basis in Fig. 2, we see that under the involution φ, we have φ(b j ) = −b j , φ(a j ) ∼ a j ,
j = 1, . . . , g,
where the symbol ∼ means that φ(a j ) is homologous to a j .
(4.9)
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M. Y. Mo
In particular, if we consider the holomorphic 1-forms ω j (φ(x)) on L, we have ω j (φ(x)) = ω j (x) = ω j (x) = δ jk . ak
φ(ak )
ak
Hence by the uniqueness of holomorphic 1-forms that is dual to the canonical homology basis (a, b), we have ω j (φ(x)) = ω j (x). By computing the b-periods of ω j (φ(x)) and making use of (4.9), we obtain the following. Lemma 2. The period matrix of L is purely imaginary. In particular, by (4.5), we see that θ (0) is real and positive, hence θ (u(x)) has g zeros on L. Let us denote these zeros by ι1 , . . . , ιg . ˜ l be the intervals in Ll that projects onto the gaps
˜ j in (1.20), then as Remark 3. Let
j we shall see in Corollary 3 of Proposition 5 of Sect. 5, there is exactly one point ι j that ˜ 2 for j = 1, . . . , g. We shall label the ι j such that ι j ∈
˜1 ∪
˜ 2. ˜1 ∪
belongs to
j
j
j
j
4.1.1. Jump properties of ratios of theta functions. We would like to express the function θ (u(x)) as a meromorphic function on C with jump discontinuities and use it to construct in (4.1). To do so, we need to define the contour of integration in the the vector (z, A) Abel map (4.7) such that the integral can be defined without ambiguity. We will define the contour of integration as in Fig. 3. For a point z in L1 (L2 ), the contour of integration 1 goes from λ2g+2 to z in L1 (L2 ) without intersecting (−∞, λ2g+2 ) and the branch cuts on the imaginary axis. For a point z in the upper (lower) half plane in L3 , the contour of integration 3 consists of 2 parts. The first part lies in L2 , goes from λ2g+2 to a point iζ (−iζ ) on the branch cut in the imaginary axis without intersecting (−∞, λ2g+2 ) in L2 , and enters the branch cut from the left-hand side of this point. The second part lies in the upper (lower) half plane in L3 , goes from the right-hand side of iζ (−iζ ) to the point z. For a point z in the
Fig. 3. The contour of integration for the Abel map u
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upper (lower) half plane in L4 , the contour of integration consists of 3 parts. The first part lies in L2 , goes from λ2g+2 to a point −iζ (iζ ) on the branch cut in the imaginary axis without intersecting (−∞, λ2g+2 ) in L2 and enters the branch cut from the left-hand side of this point. The second part lies in the lower (upper) half plane in L3 , goes from the right-hand side of −iζ (iζ ) to the origin. The last part lies in the upper (lower) half plane in L4 , goes from the origin to the point z. The choice of the point ±iζ in the construction is immaterial as long as it lies on the branch cut on the imaginary axis. Let z j be the point on L j that projects to z in C and A be a g-dimensional constant on the complex z-plane by vector. We can now define four functions θ j (u(z) + A) = θ (u(z j ) + A). θ j (u(z) + A)
(4.10)
These functions will have jump discontinuities in the complex z-plane. By using the periodicity of the theta function (4.6), we can compute their jump discontinuities. Proposition 2. Let A = (A1 , . . . , Ag )T be a g dimensional vector. The functions are analytic in C\(R ∪ Sσ −µ2 ). On R ∪ Sσ −µ2 , they satisfy the following θ j (u(z) + A) conditions: 1 (u(z) + A) 2 (u(z) + A), = θ∓ θ± z ∈ j, l (u(z) + A)e = θ− θ+l (u(z) + A)
(−1)l 2πi
j = 1, . . . , g + 1,
u j (z)+A j +
jj 2
˜ j, , z∈
j = 1, . . . , g, l = 1, 2,
l (u(z) + A), = θ− ˜0 ∪
˜ g+1 , l = 1, 2, θ+l (u(z) + A) z∈
(4.11)
2 (u(z) + A) 3 (u(z) + A), = θ∓ θ± z ∈ Sσ −µ2 , 3 (u(z) + A) 4 (u(z) + A), = θ∓ z ∈ R, θ± l (u(z) + A), = θ− z ∈ Sσ −µ2 , l = 1, 4. θ+l (u(z) + A)
and θ 2 (u(z) + A) across Proof. Let us first consider the discontinuities of θ 1 (u(z) + A)
j . Let π : L → C be the projection of L onto the Riemann sphere. Suppose z is a j point in j . Let z ∈ C and define the points z ± ∈ L j to be j
j
π(z ±i ) = z ± i, z ±i ∈ L j .
(4.12)
We will now choose > 0 to be real and positive and let z ∈ j . From the definition of the integration contour in Fig. 3 and the canonical homology basis in Fig. 2, we have j the following relation between the points z ±i as → 0: 1 2 ) = u(z ∓i ), z ∈ g+1 , u(z ±i g 1 2 u(z ±i ) = u(z ∓i ) + ω,
2 = u(z ∓i )+
k= j ak g k
e , z ∈ j ,
(4.13) j = 1, . . . , g,
k= j
where ek is a vector with 1 in the k th entry and zero elsewhere and ω is the vector ω = (ω1 , . . . , ωg )T .
(4.14)
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M. Y. Mo
Fig. 4. The contour
From this and the periodicity of the theta function (4.6), we obtain = θ∓2 (u(z) + A), θ±1 (u(z) + A) z ∈ j,
j = 1, . . . , g + 1.
(4.15)
˜ j . Again, from the definition of the integration contour Let us now consider a point z in
and the canonical homology basis, we have, as → 0, the following: u(z li ) = u(z l−i ) + (−1)l+1 ω, bj
=
u(z l−i ) + (−1)l+1 e j ,
˜ j, z∈
j = 1, . . . , g,
(4.16)
where l = 1, 2. From this and the periodicity of the theta function, we see that
l l (u(z) + A)e = θ− (−1) 2πi u j (z)+A j + θ+l (u(z) + A)
jj 2
˜ j, , z∈
j = 1, . . . , g. (4.17)
1 2 From the definition of the contour,1it is clear that θ (u(z)+ A) and θ (u(z)+ A) integration are analytic across R\ λ1 , λ2g+2 and that θ (u(z) + A) is analytic across Sσ −µ2 . and θ 2 (u(z) + A) on Sσ −µ2 . Let us now consider the discontinuities of θ 3 (u(z) + A) Let z be a point on Sσ −µ2 , from the definition of the contours, it follows immediately that
= θ−3 (u(z) + A), z ∈ Sσ −µ2 . θ+2 (u(z) + A)
(4.18)
on the minus side of Sσ −µ2 . Let us now consider the boundary value of θ 2 (u(z) + A) For small and positive → 0, we have u(z 2 + ) + ω = u(z 3 − ), (4.19)
where is the close loop on L depicted in Fig. 4. Since this loop is contractible, we have = θ+3 (u(z) + A), z ∈ Sσ −µ2 . θ−2 (u(z) + A)
(4.20)
Finally, the conditions = θ∓4 (u(z) + A), θ±3 (u(z) + A) z ∈ R, = θ−4 (u(z) + A), θ+4 (u(z) + A) z ∈ Sσ −µ2 follow directly from the definition of the contour of integration.
(4.21)
Universality in the Two Matrix Model
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that satisfies the jump discontinuities From this we can construct a vector (z, A) in (4.2). T θ 1 u(z)+ A θ 4 u(z)+ A defined by (z, A) = Corollary 1. The vector (z, A) ,..., θ(u(z))
θ(u(z))
satisfies the jump discontinuities (4.2). will Let π be the projection from L onto C. Then the first (second) entry of (z, A) have a simple pole at the point π(ι j ) when ι j is on the first (second) sheet of L, where ι1 , . . . , ιg are the zeroes of θ (u(z)). Apart from these poles, it is bounded in C. 4.2. Meromorphic differentials. We will now introduce the meromorphic differentials and use them to construct four vectors j (z) that satisfy the jump conditions (4.4). Most of the properties of these differentials stated here can be found in standard textbooks such as [24]. First recall that a meromorphic 1-form with simple poles only is called a meromorphic 1-form of the third kind and such a 1-form with prescribed a-period and pole structure exists and is uniquely defined. Proposition 3. Let d1 , . . . , dk be k distinct points on a Riemann surface L. Let c1 , . . . , ck be complex numbers with kj=1 ck = 0 and let A1 , . . . , Ag be arbitrary complex numbers. Then there exists a unique meromorphic 1-form of the third kind d on L, holomorphic on L\{d1 , . . . , dk } such that Resz=d j d = c j , j = 1, . . . , k, d = A j , j = 1, . . . , g.
(4.22)
aj
A meromorphic 1-form with all a-periods zero is called a normalized meromorphic 1form. The periods of a normalized meromorphic 1-form can be related to the values of the Abel map at its poles. Theorem 8 (See e.g. [24], p. 65, III.3). Let η be a meromorphic differential of the third kind with simple poles at the points di ∈ L and η˜ be a holomorphic differential. Let i ˜ i be their periods and i η=, η = i+g , ai bi (4.23) i ˜ ˜ i+g . η˜ = , η˜ = ai
bi
Then the Riemann bilinear relation is the following: g
˜ i i+g − ˜ g+i i = 2πi
i=1
where x0 is an arbitrary point on L.
di
Resdi (η)
di x0
η, ˜
(4.24)
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4.2.1. Jump properties of meromorphic 1-forms. We will now define four meromorphic 1-forms of the third kind and use them to construct the vector j (z) in (4.3). First let us define a local coordinate w near ∞2 , the point on L2 , L3 and L4 that projects onto ∞ in the Riemann sphere. ⎧ 1 − ⎪ ⎨ z 3 , in the first and fourth quadrants of L2 ; 1 w = ω2 z − 3 , in the second quadrant of L2 ; ⎪ ⎩ − 13 ωz , in the third quadrant of L2 , ⎧ 2 −1 ⎪ ⎨ ω z 3 , in the first quadrant of L3 ; (4.25) 1 w = z − 3 , in the second and third quadrants of L3 ; ⎪ 1 ⎩ −3 ωz , in the fourth quadrant of L3 , 1
w=
ωz − 3 , in the first and second quadrants of L4 ; 1 ω2 z − 3 , in the third and fourth quadrants of L4 ,
2πi
1
where ω = e 3 and the branch of z 3 is chosen such that arg z ∈ (−π, π ). One can check that w is indeed holomorphic in L in a neighborhood of ∞2 . Let us now define four meromorphic 1-forms of the third kind d j , j = 1, . . . , 4 by the following properties. Definition 3. The normalized meromorphic 1-forms d j are holomorphic in L\{±it, λ11 , . . . , λ12g+2 , ι1 , . . . , ιg , ∞1 , ∞2 }, where ±it are the points in L2 that project onto ±ic and ιk are the zeros of θ (u(x)). At these points they have simple poles with residues 1 Resλ1 d j = Res±it d j = − , k = 1, . . . , 2g + 2, k 2 Resιk d j = 1, k = 1, . . . , g, Res∞1 d 1 = 0, Res∞2 d 1 = 2, Res∞1 d 2 = 3, Res∞2 d 2 = −1, Res∞1 d 3 = 2, Res∞2 d 1 = 0, Res∞1 d 4 = 1, Res∞2 d 4 = 1,
(4.26)
provided none of the ιl is equal to λ1k for some k. If some ιl is equal to λ1k for some k, then the residue at ιl will be 21 . These 1-forms are then uniquely defined. We will denote the b-period of these 1-forms by β j ,
T βj =
d j , . . . , b1
d j
,
j = 1, . . . , 4.
(4.27)
bg
˜ k (see Remark 3). To avoid ambiguity in the b-periods, let π(ιk ) be the projection of ιk on
˜ k at any Then the b-periods are computed as integrals on b-cycles in L1 that intersect
˜ k at any point x < π(ιk ) if π(ιk ) = λ2k . If π(ιk ) = λ2k , then the b-cycle can intersect
point x = λ2k in L1 .
Universality in the Two Matrix Model
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We will now define four functions in the Riemann surface L. First let ± k ∈ L be the images of k under the maps ξ1,± (z), that is, ! "
± (4.28) k = (z, ξ )|z ∈ k , ξ = ξ1,± (z) , k = 1, . . . , g + 1. Let z 0 be a point in − as long g+1 . The exact choice of z 0 is immaterial to the construction #z 1 1 as z 0 = λ2g+1 or λ2g+2 . Let us consider the functions j (z) given by j (z) = z 0 d j . The path of integration in the right hand side integral is defined in the same way as the ones for the Abel map, except that every path now starts at z 0 . Let z k be the point on Lk that projects to z in C. As before, we will now define four
functions e j (z) on the complex z-plane by k
e j (z) = e j (z ) . k
k
(4.29)
Then these functions have the following jump discontinuities in the complex z-plane. Proposition 4. The functions e j (z) are analytic in C\(R ∪ Sσ −µ2 ). On R ∪ Sσ −µ2 , they satisfy the following conditions: l
e j,± (z) = ∓e j,∓ (z) , z ∈ k , k = 1, . . . , g + 1, l l l−1 ˜ k , k = 1, . . . , g, l = 1, 2, e j,+ (z) = e j,− (z)+(−1) (β j )k , z ∈
l l ˜0∪
˜ g+1 , l = 1, 2, e j,+ (z) = e j,− (z) , z ∈
1
2
e j,± (z) = ±e j,∓ (z) , z ∈ Sσ −µ2 , 2
3
(4.30)
e j,± (z) = e j,∓ (z) , z ∈ R, 3
4
e j,+ (z) = e j,− (z) , z ∈ Sσ −µ2 , l = 1, 4, where β j k is the k th component of the vector β j . l
l
Proof. The proof follows a similar argument as the ones used in the proof of Proposition 2. First let us consider the jump discontinuities on k . Let z be a point in k and define the points z l±i as in (4.12) in the proof of Proposition 2. First consider the boundary values e j,+ (z) and e j,− (z) . Choose integration contours + and − from z 0 1 and z 2 to the points z i −i as in Fig. 5. Let = + − − . Then can be deformed into g the sum l=k al of the a-cycles and a loop 2g+2 around the point λ12g+2 in L\ pole , where pole is the set of poles of d j (see Fig. 6), 1
2
pole = {±it, λ11 , . . . , λ12g+2 , ι1 , . . . , ιg , ∞1 , ∞2 }. Therefore we have
g 1 2 exp j (z i ) = exp j (z −i )+ d j + l=k
al
2g+2
(4.31)
d j ,
2 ) , z ∈ k , k = 1, . . . , g, = − exp j (z −i
(4.32)
where the last equality follows from the fact that d j has residue − 21 at the point λ12g+2 .
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M. Y. Mo
Fig. 5. The contours ± for e
1j,+ (z)
and e
Fig. 6. The contours deformation of the loop for e
2j,− (z)
1j,+ (z)
and e
2j,− (z)
Let us now consider the boundary values e j,− (z) and e j,+ (z) on k . Since z 0 ∈ − g+1 , the integration contours − and + can now be chosen to lie in the lower (upper) half of L1 (L2 ). The loop = + − − can now be deformed into the sum plane g − l=k al of the a-cycles. However, such a deformation will necessarily go pass the poles λ12k , . . . , λ12g+1 and ιk , . . . , ιg of d j (Recall that by Remark 3, there is exactly ˜ 1 ∪
˜ 2 ). Since the residues of d j at these points are given one point ιk that belongs to
k k 2(g−k) 1 by − 2 from the λl and g − k from the ιl (when all ιl and λ1m are distinct), the total residue at these points is zero. It is clear from Definition 3 that, when some ιl coincide with the λ1m , the total residue at these points remains unchanged. Hence we have 1
exp
1 j (z −i )
2
2 = exp j (z i ) − d j , 2 = exp j (z i ) , z ∈ k , k = 1, . . . , g.
Universality in the Two Matrix Model
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˜ k . Let z ∈
˜ k . For the boundary Let us now consider the boundary values on the gaps
1j,± (z) 1 , we choose ± to be integration contours that go from z 0 to z ±i in L1 values e 1 1 without intersecting (−∞, λ2g+2 ) except at z 0 and z ±i . Let the loop be = + −− , then for k = 1, . . . , g, can be deformed into the b-cycle bk without passing any pole of d j , except possibly ιk . (Recall the definition of the b-periods of d j in Definition 3.) When ιk is not equal to λ12k or λ12k+1 , d j has integer residue at ιk , and when ιk is equal to either λ12k or λ12k+1 , the deformation from to bk will not have to go past ιk . This implies ˜ k , k = 1, . . . , g. e j,+ (z) = e j,− (z)+(β j )k , z ∈
1
1
For k = 0, the loop can be deformed into a small loop around the point ∞1 . Since the 1-forms d j have integer residues at ∞1 , we have ˜ 0. e j,+ (z) = e j,− (z) , z ∈
1
1
If k = g + 1, then the loop will be contractible in L\ pole . Hence we have ˜ g+1 . e j,+ (z) = e j,− (z) , z ∈
1
1
˜ k , let us consider ± to be inteOn the other hand, for the boundary values e j,± (z) on
2 gration contours that go from z 0 to z ±i in L2 without intersecting (−∞, λ12g+2 ) except 2 . Let k = 1, . . . , g, and let π(ι ) be the projection of ι onto the z-plane. at z 0 and z ±i k k Then depending on the relative positions of z, π(ιk ) and λ12k , the loop = + − − can be deformed into −bk , together with small loops around the poles ιk , . . . , ιg and λ12k+1 , . . . , λ12g+2 in L\ pole ; or it can be deformed into the sum of −bk and small loops around the poles ιk+1 , . . . , ιg and λ12k+1 , . . . , λ12g+2 in L\ pole . In either case, the total residue of d j at these points will be an integer. Therefore we have 2
˜ k , k = 1, . . . , g. e j,+ (z) = e j,− (z)−(β j )k , z ∈
2
2
Similarly, for k = 0, the loop can be deformed into a small loop around the points ∞2 and ±it. Since the total residue of the 1-form d j at these points is an integer, we have ˜ 0. e j,+ (z) = e j,− (z) , z ∈
2
2
If k = g + 1, then the loop will be contractible in L\ pole . Hence we have ˜ g+1 . e j,+ (z) = e j,− (z) , z ∈
2
2
We now consider the boundary values e j,− (z) and e j,+ (z) at Sσ −µ2 , let z ∈ Sσ −µ2 . Let us again denote by + and − contours of integration from z 0 to z − in L3 and z + in L2 . Then depending on whether z is in the upper or lower half plane, the loop = + − − can be deformed into a small loop around the pole it or −it in L\ pole . (See Fig. 4. The loop in this case is the same except that it begins and ends at z 0 instead of λ12g+2 .) Since the residue of d j around it or −it is − 21 , we have 2
3
e j,+ (z) = −e j,− (z) , z ∈ Sσ −µ2 . 3
2
(4.33)
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M. Y. Mo
The rest of the jump discontinuities in (4.30) now follow directly from the definition of the integration contours as in the proof of Proposition 2. From this we have the following T 1 4 Corollary 2. The vector j (z) defined by j (z) = e (z) , . . . , e (z) satisfies the jump conditions (4.4). 4.3. The outer parametrix. We will now define the functions N j (z) on L as follows: β θ u(z) + 2πij + n α N j (z) = e j (z) θ (u(z)) = e j (z) j (z),
j = 1, . . . , 4,
(4.34)
T where α = α1 , . . . , αg and the numbers α j are the constants defined in (1.21). Since the function e j (z) has zeroes at the points ι j , while j (z) has simple poles at ι j for j = 1, . . . , g, the function N j (z) is analytic at these points. Let us denote by N kj (z) the projection of N j (z) onto the k th -sheet, that is, N kj (z) = N j (z k ) = N j (z, ξk (z)), where ξk (z) is the function ξ(z) on Lk . Let N (z) be the 4 × 4 matrix whose elements are given by N jk (z) =
N kj (z), Imz > 0; (−1)δ4,k N kj (z), Imz < 0.
(4.35)
Then by Corollaries 1 and 2, we see that the matrix N (z) satisfies the jump conditions (1.24). By normalizing its behavior at infinity, we can obtain the outer parametrix. Theorem 9. Let N (z) be the 4 × 4 matrix defined in (4.35). Suppose we have βj βj θ u(∞1 ) + + n α θ u(∞2 ) + + n α = 0. 2πi 2πi
(4.36)
Let the constants L j be L 1 = N1−1 (∞1 ), L 2 = lim N2−1 (w)w −1 , w→0
L 3 = lim N3−1 (w), L 4 = lim N4−1 (w)w, w→0
(4.37)
w→0
where w is the local coordinate near ∞2 defined in (4.25) and the limits in L 2 , L 3 and L 4 are taken as z → ∞2 in the first quadrant of L2 . Then the matrix ⎛
1 ⎜0 ⎜ S ∞ (z) = ⎜0 ⎝ 0
0 √i 3
0 0
0
√i 3
0
0
0
⎞
iκ ⎟ √ 3⎟
diag (L 1 , L 2 , L 3 , L 4 ) N (z) 0⎟ ⎠
√i
3
(4.38)
Universality in the Two Matrix Model
885
satisfies the Riemann-Hilbert problem (1.23), where κ is the following in the expansion of N2 (z) at z = ∞2 : w −1 + κ2,0 − κw + O(w 2 ) , N2 (z) = L −1 2 as z → ∞2 in the first quadrant of L2 . Remark 4. The constants L j , j = 1, . . . , 4 can be represented as θ u(∞1 ) limw→0 e− 2 (w) w −1 θ u(∞2 ) − 1 (∞1 ) , L2 = L1 = e , β1 β2 θ u(∞1 ) + 2πi + n α θ u(∞2 ) + 2πi + n α limw→0 e− 4 (w) w θ u(∞2 ) θ u(∞2 ) − 3 (w) , L4 = L 3 = lim e , β3 β4 w→0 θ u(∞2 ) + 2πi + n α θ u(∞2 ) + 2πi + n α (4.39) where the limits are taken as z → ∞2 in the first quadrant of L2 . Proof. Since N (z) satisfies the jump discontinuities of (1.23), the matrix M(z)N −1 (z) does not have any jump discontinuities in C. Moreover, this matrix does not grow faster 2 than z 3 at z = ∞ and has at worst square root singularities at the points λ j and ±it. Since it has no jump discontinuities, all these singularities are removable and therefore we have M(z) = H N (z) for some constant matrix H . To determine the constant matrix H , we will have to study the behavior of N (z) as z → ∞. The behavior of M(z) is given by the following: ⎛ 2 2 2 ⎞ 1 + O(z −1 ) O(z − 3 ) O(z − 3 ) O(z − 3 ) ⎜ O(z −1 ) ∗ ∗ ∗ ⎟ ⎟, M(z) = ⎜ (4.40) ⎝ O(z −1 ) ∗ ∗ ∗ ⎠ O(z −1 ) ∗ ∗ ∗ where the 3 × 3 lower right block is given by ⎞ ⎛ i 1 1 2 ωi 13 √ z 3 (1 + O(z −1 )) −√ z (1 + O(z −1 )) − ω√ i z 3 (1 + O(z −1 )) 3 3 3 ⎟ ⎜ i 2 2 ⎜ √ (1 + O(z − 23 )) − √i (1 + O(z − 3 )) − √i (1 + O(z − 3 )) ⎟ ⎠ (4.41) ⎝ 3 3 3 1 1 1 1 1 1 2 ωi − 3 √i z − 3 (1 + O(z − 3 )) − ω √ i z − 3 (1 + O(z − 3 )) − √ z (1 + O(z − 3 )) 3
3
3
for z → ∞ in the first quadrant. From the relation between the local coordinate w and z in (4.25) and the jump discontinuities of N (z) near ∞2 , we see that, if we can show that the functions N j (z) behave as −1 1 N1 (z) = L −1 1 (1 + O(z )), z → ∞ ,
N2 (z) = O(z −1 ), z → ∞1 ,
N1 (z) = O(w 2 ), z → ∞2 , w −1 − κw + O(w 2 ) , z → ∞2 , N2 (z) = L −1 2
N3 (z) = O(z −1 ), z → ∞1 ,
2 2 N3 (z) = L −1 3 (1 + O(w )), z → ∞ ,
N4 (z) = O(z −1 ), z → ∞1 ,
2 N4 (z) = L −1 4 w(1 + O(w)), z → ∞ ,
(4.42)
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M. Y. Mo
when z → ∞2 in the first quadrant of L2 , then the matrix in (4.38) will be the unique solution of the Riemann-Hilbert problem (1.23). The asymptotic behavior of N1 (z) and N4 (z) follows immediately from the definition of the functions N j (z) (4.34), the constants L j (4.37) and behavior of the 1-forms d j (4.26). We will now prove the equations in (4.42) for N2 (z) and N3 (z). Let the involution on L be (z, ξ(z)) = (−z, ξ(−z)). To simplify the notation, we shall simply denote (z, ξ(z)) by −z. Let us consider the functions N j (−z) for j = 2, 3. The singularity structure of this function is the same as N j (z). By Proposition 2 and 4 and the expression of N j (z) (4.34), we see that the functions N2 (z) and N3 (z) satisfy the following jump discontinuities on L: N j,+ (z) = −N j,− (z), z ∈ +k , k = 1, . . . , g + 1, ˜ lk , k = 1, . . . , g, l = 1, 2, N j,+ (z) = e(−1) 2πinαk N j,− (z), z ∈
l
N j,+ (z) = −N j,− (z), z ∈
(4.43)
Sσ+−µ2 ,
˜l ˜ where ± k is defined in (4.28), k is the interval on Ll that projects to k . That is $ % ˜ lk = (z, ξ )|z ∈
˜ k , ξ = ξl (z) .
The intervals Sσ±−µ2 are defined to be ! " Sσ±−µ2 = (z, ξ )|z ∈ Sσ −µ2 , ξ = ξ3,± (z) . On the other hand, from (4.43), we see that the function N j (−z) has the following jump discontinuities: N j,+ (−z) = −N j,− (−z), z ∈ − k , k = 1, . . . , g + 1, N j,+ (−z) = e(−1)
l+1 2πinα
g+1−k
˜ lk , k = 1, . . . , g, l = 1, 2, (4.44) N j,− (−z), z ∈
N j,+ (−z) = −N j,− (−z), z ∈ Sσ−−µ2 . g+1 g+1 Note that the union of the contours ∪k=1 +k and ∪k=1 − k divides L into 2 disjoint sets, which are the first sheet L1 and the union of the other sheets L2 ∪L3 ∪L4 . Similarly, the contour Sσ+−µ2 ∪ Sσ−−µ2 divides L into the sets L1 ∪ L2 and L3 ∪ L4 . Let Nˆ j (z) be & N j (−z), z ∈ L1 ∪ L3 ∪ L4 ; Nˆ j (z) = (4.45) −N j (−z), z ∈ L2 . Since the constants αk satisfy the symmetry αk = 1 − αg+1−k (1.22), from (4.43), (4.44) and (4.45) we see that the function N˜ j (z) =
Nˆ j (z) N j (z)
(4.46)
is either a meromorphic function on L with poles exactly at the g zeros of β θ u(z) + 2πij + n α or it is a constant. By the assumption of the theorem, this theta function is not identically zero. Hence by Theorem 7, we see that N˜ j (z) must be a constant K j . By using the jump discontinuities (4.43), (4.44) of the N j (z) near z = ∞, and
Universality in the Two Matrix Model
887 1
the relation between the coordinate z 3 and w, we have the following behavior of N j (z) near ∞2 : 1 1 N2 (z) = L −1 z 3 + κ2,0 − κz − 3 + O(w 2 ) , 2 (4.47) − 31 2 1 + κ N3 (z) = L −1 z + O(w ) , 3,0 3 1
as z → ∞2 in the first quadrant of L2 , where the branch cut of z 3 is chosen to be the negative real axis. On the other hand, since −z → ∞2 in the third quadrant when z → ∞2 in the first quadrant, the functions Nˆ j (z) have the following behavior: 1 1 z 3 − κ2,0 − κz − 3 + O(w 2 ) , Nˆ 2 (z) = L −1 2 (4.48) − 31 2 −1 + κ Nˆ 3 (z) = L −1 z + O(w ) , 3,0 3 as z → ∞2 in the first quadrant. On the other hand, since N˜ j (z) in (4.46) is a constant K j , we also have 1 1 3 + κ2,0 − κz − 3 + O(w 2 ) , z Nˆ 2 (z) = K2 L −1 2 (4.49) 1 Nˆ 3 (z) = K3 L −1 1 + κ3,0 z − 3 + O(w 2 ) , 3 as z → ∞2 in the first quadrant. By comparing (4.48) and (4.49), we obtain (4.42). This concludes the proof of the theorem. It remains to show that the condition (4.36) is correct and we will do it in Sect. 5. 4.4. Comparison to the results in the one matrix model. The construction used in this paper can be applied to the case of the Hermitian one matrix model. In that case, the relevant Riemann surface is a hyper-elliptic curve that is represented as a two sheeted cover of the Riemann sphere and the Riemann-Hilbert problem for the outer parametrix is given as follows: 1. M(z) is analytic in C\[λ1 , λ2g+2 ], 2. M+ (z) = M− (z)J M (z), z ∈ [λ1 , λ2g+2 ], 3. M(z) = I + O(z
−1
),
(4.50)
z → ∞,
where
⎧
−2πinα j ⎪ e 0 ⎪ ⎪ , z ∈ (λ2 j , λ2 j+1 ), ⎪ ⎨ 0 e2πinα j
J M (z) := ⎪ 0 1 ⎪ ⎪ z ∈ [λ2 j−1 , λ2 j ], ⎪ ⎩ −1 0 ,
j = 1, . . . , g, (4.51) j = 1, . . . , g + 1.
Then for an appropriate choice of canonical homology basis, the outer parametrix can be written in the following form: β #z θ (−1)k u(z) + n α + 2πij k d (z) , j, k = 1, 2, (4.52) N jk (z) = e z0 j θ (u(z))
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M. Y. Mo
where θ (u(z)) is the hyper-elliptic theta function associated with the curve and the choice of canonical homology basis. The 1-form d j is a normalized meromorphic 1-form with # z
d
pole structure such that e z0 j has simple zeroes at the zeroes of θ (u(z)) and ∞l for l = j and square root singularities at the points λi , i = 1, . . . , 2g + 2 (as a function on L). The vector β j is the b-period of the 1-form d j . The superscript k in d kj denotes the projection onto the k th -sheet of the Riemann surface. The construction can be modified such that instead of the theta function θ (u(z)), a shifted theta function θ ((−1)k u(z) + d jk ) is used in the denominator of the entries of N (z). This will affect the choice of the 1-form d j in (4.52). In this case, a suit# able choice of shifts d jk can be made such that the functions e expressions #z
e
z0
d kj (z)
admit explicit
#z #z γ + γ −1 γ − γ −1 d 1 (z) d 2 (z) , e z0 2 = −e z0 1 = , 2 2i 1 g 4 z − λ2 j γ (z) = . z − λ2 j−1
d 11 (z)
=e
#z
z z0
z0
d 22 (z)
=
j=1
This gives us the formula for the outer parametrix derived in [12]. In the four-sheeted Riemann surface case, the meromorphic 1-forms are more complicated and such explicit formula is difficult to find. 5. The Non-Vanishing of the Theta Function β We will now prove that the normalization constants θ u(∞k ) + 2πij + n α , j = 1, . . . , 4 and k = 1, 2 do not vanish for any n ∈ N. Then the solution S ∞ (z) of the Riemann-Hilbert problem (1.23) constructed in Theorem 9 exists and is well-defined. This is achieved by first representing the Riemann surface L as a Schottky double of a bordered Riemann surface L and use the results in Chapter 6 of [25] to show that the theta function θ ( A) is real and positive whenever A is real. We will then use the Riemann bilinear relation Theorem 8 to show that the vectors β j defined in Definition 3 are purely imaginary, k while the vectors u(∞ ) arereal. Since α is a real vector, this will imply that the expresβ sions θ u(∞k ) + 2πij + n α are real and positive. This also implies that the functions β θ u(z) + 2πij + n α do not vanish identically on L as they do not vanish at the points ∞k . Finally, we will show that the solution S ∞ (z) to the Riemann-Hilbert problem (1.23) satisfies the conditions in Theorem 4. First let us define a contour ! that divides the Riemann surface L into 2 halves. Let ! be the set of points that is fixed under the map φ in (4.8). That is, ! = {x ∈ L| φ(x) = x} .
(5.1)
Then ! is a disjoint union of g + 1 closed curves ! j , j = 0, . . . , g on L, given by the following: g ˜ kj , j = 1, . . . , g, (5.2) ˜ k0 ∪
˜ kg+1 , ! j = ∪2k=1
! = ∪ j=0 ! j , !0 = ∪2k=1
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Fig. 7. The dash lines indicate the loops ! j
˜ k is the contour on Lk that projects to
˜ j . That is where
j $ % ˜ kj = (z, ξ )|z ∈
˜ j , ξ = ξk (z) .
In other words, the contours ! j are the closed loops on L that start from the branch point λ12 j , going through the interval [λ2 j , λ2 j+1 ] on L1 , then enters L2 at λ12 j+1 and go back to λ12 j through the interval [λ2 j , λ2 j+1 ] on L2 . The contour !0 starts at λ11 , goes to −∞ on the real axis on L1 , then from +∞ to λ12g+2 on the real axis on L1 , from which it enters L2 and goes to +∞ along the real axis on L2 , then goes back from −∞ on L2 to λ11 along the real axis (see Fig. 7). Note that the images of the cuts j on L1 and L2 do not belong to !. For example, let x = (z, ξ1,+ (z)) be a point on 1j , then φ(x) = (z, ξ1,+ (z)) = (z, ξ1,− (z)) = x. Similarly, the images of the real axis on L3 and L4 do not belong to ! either. The curve ! divides the Riemann surface L into 2 halves, L+ and L− , each of which is an open Riemann surface with boundary !. The Riemann surface L± consists of the upper (lower) half planes of L1 and L4 and the lower (upper) half plane of L2 and L3 . The Riemann surface L can now be thought of as a union of L+ , L− and !. Moreover, the a-cycles defined in Fig. 2 are homologous to the contours ! j . That is, we have !j ∼ aj,
j = 1, . . . , g.
(5.3)
We can think of L as the Riemann surface formed by gluing two copies of L+ along the boundary ! with an anti-holomorphic involution φ that fixes ! and maps L+ onto L− . A Riemann surface formed in this way is called a Schottky double. Since L is a Schottky double, we can apply the results in Chapter 6 of [25] to the theta function of L. Let us define the torus Sχ as in Proposition 6.2 of [25]. Definition 4. Let χ = (χ1 , . . . , χg )T ∈ (Z/2Z)g and let J0 be the Jacobian variety of L, J0 = Cg /#, # = Zg + Zg . The torus Sχ in J0 is defined by & ' 1 g Sχ = s ∈ J0 , | s = ς + χ , ς ∈ R . . 2
(5.4)
(5.5)
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Note that this definition is different from the one in [25] because the theta function in [25] is defined differently. We can now apply the results in [25]. The first result tells us where the zeros ι j of the function θ (u(x)) are located. Proposition 5 (Proposition 6.4 of [25]). For any point x0 ∈ !0 , s ∈ Sχ , the function θ (u(x) − u(x0 ) − s) either vanishes identically or has modulo 2, 1 + χk zeros on !k , where χk is the k th component of the vector χ . As a corollary, we have the following concerning the locations of the zeros ι j . Corollary 3. The function θ (u(x)) has g zeros ι1 , . . . , ιg such that ιk ∈ !k , k = 1, . . . , g. Proof. Let us take x0 = λ12g+2 , χ = 0 and s = 0 in Proposition 5, then u(x0 ) = 0 and by the paragraph after Lemma 2 of Sect. 4.1, we see that θ (u(x)) is not identically zero and hence by Proposition 5, it has 1 zero on each of the contours !k , k = 1, . . . , g. The next result shows that the theta function does not vanish when its argument is real. Proposition 6 (Proposition 6.16 of [25]). Let Sˆ 0 be the universal covering of S0 , " ! (5.6) Sˆ 0 = s ∈ Cg , | s = ς, ς ∈ Rg . Then the theta function θ (s ) is real and positive for s ∈ Sˆ 0 . That is, θ (s ) is real and positive for all s ∈ Rg . Let us now prove that the periods β j of d j in Definition 3 are purely imaginary. This, 6 will imply the non-vanishing of the theta functions together with Proposition β θ u(∞k ) + 2πij + n α for k = 1, 2 and j = 1, . . . , 4. Lemma 3. The periods β j of the 1-forms d j defined in Definition 3 are purely imaginary. Proof. First note that, by Corollary 3, all the points ιl and λl1 are invariant under the ˜ j = d j (φ(x)) has the same poles involution φ. Hence the meromorphic 1-form d ˜ j are zero. We have and residues as d j (x). Let us show that the a-periods of d d j (φ(x)) = d j (x). (5.7) ak
φ(ak )
From Fig. 2, we see that the curve φ(ak ) consists of a path from the lower half plane in L1 that goes from k+1 to k , and another path in the upper half plane of L2 that goes from k to k+1 . There are 3 poles of d j between the loops ak and φ(ak ): ιk , λ12k and λ12k+1 (see Fig. 8). From (4.26), we see that d j has a combined residue of 0 at these points, and hence we can deform φ(ak ) onto ak without affecting the value of (5.7). Therefore, by (5.7), we see that d j (φ(x)) = 0, j = 1, . . . , g. ak
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˜ 1 . The deformations for other cycles are Fig. 8. The deformation from the cycle φ(a1 ) to a1 when ι1 ∈
1 similar
By the uniqueness of the normalized 1-form, this implies d j (φ(x)) = d j . Now we use (4.9) for the b-periods, since the relations for the b-cycles in (4.9) are exact and not up to deformation, we have d j (φ(x)) = − d j (x) = − β j k , bk
bk
where β j k is the k th component of the vector β j . On the other hand, since d j (φ(x)) = d j , the above is also equal to β j k . This implies the proposition. From Proposition 6 and Lemma 3, we obtain
Theorem 10. There exists δ > 0, independent on n, such that θ u(∞k ) + δ, for k = 1, 2 and j = 1, . . . , 4 and all n ∈ N.
βj 2πi
+ n α >
Proof. Let us consider the normalized 1-form dk that has simple poles at λ12g+2 and ∞k with residues -1 and 1 for k = 1, 2. Then by similar argument used in the proof of Lemma 3, we see that dk (φ(x)) = dk , and hence the b-periods of dk are all purely imaginary. Now by the Riemann bilinear formula (4.24) and the definition of the Abel map (4.7), we see that 2πi u l (∞k ) − u l (λ12g+2 ) = 2πiu l (∞k ) = dk , k = 1, 2, l = 1, . . . , g. bl
Hence u(∞k ), k = 1, 2 are real. β Therefore, by Proposition 6 and Lemma 3, we see that θ u(∞k ) + 2πij + n α > 0. By the periodicity of the theta function (4.6), we see that the theta function is in fact
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a map from T × Rg → C, where T is the torus T = Rg /Zg . By Proposition 6, the restriction of the theta function on the compact set T × {0, 0, . . . , 0} is real and positive, and hence there exists δ > 0 such that θ (t) > δ for all t ∈ T . This then implies the theorem. This implies that the function S ∞ (z) in Theorem 9 exists. We will now show that it satisfies the conditions in Theorem 4. Corollary 4. The function S ∞ (z) in (4.38) and its inverse (S ∞ (z))−1 satisfy the conditions in Theorem 4. Proof. Let us first show that the function N (z) in (4.35) is bounded in n uniformly in T , where T is defined in (2.1). Since the entries of N (z) are restrictions of the functions N j (z) in (4.34) on different sheets of the Riemann surface, we only need to show 4 ξ (T ) in L that projects onto T . From that N j (z) is bounded inside the set Tˆ = ∪l=1 l the periodicity property of the theta function (4.6), we see that N j (z) in (4.34) can be written as β θ u(z) + 2πij + γn (5.8) N j (z) = e j (z) , j = 1, . . . , 4, θ (u(z)) where γn is a finite vector given by (γn )l = nαl − [nαl ], l = 1, . . . , g, where [x] is the biggest integer that is smaller than x. From (5.8) and the fact that θ (u(x)) is not identically zero, we see that N j (z) is bounded in n uniformly in Tˆ . We will now show that the constants L 1 , . . . , L 4 in (4.39) are bounded in n. From the singularity behavior of the meromorphic 1-forms d j in (4.26), we see that following constants e− 1 (∞ ) , 1
−1
lim e− 2 (w)w ,
w→0
lim e− 3 (w) ,
w→0
lim e− 4 (w)w ,
w→0
in (4.39) are all bounded and non-zero. Since they are all independent on n, they are also bounded away from infinity and zero as n → ∞. Since θ (u(x)) will only vanish at the points ιl that belong to !l and neither ∞1 nor ∞2 belongs to !l for l = 1, . . . , g, the constants θ (u(∞k )), k = 1, 2 are non-zero. Moreover, from the definition of the Abel map (4.7), we see that u(∞1 ) and u(∞2 ) are both finite and hence θ (u(∞1 )) 2 and θ (u(∞ and are independent on n. Let us now consider the )) are both bounded βj k factors θ u(∞ ) + 2πi + n α for k = 1, 2 and j = 1, . . . , 4. By Theorem 10, there exists δ > 0, independent on n such that these constants are greater than δ. On the other hand, from the periodicity of the theta function (4.6) and the fact that the period matrix is purely imaginary, (Lemma 2) while the vector α in (1.21) is real, we see that βj k θ u(∞ ) + 2πi + n α is bounded in n as n → ∞. Hence the constants L 1 , . . . , L 4 in (4.38) and (4.39) are bounded away from infinity and zero as n → ∞. Finally, by considering the asymptotic expansion of N j (z) in the local parameter w in (4.25) at z = ∞ and making use of (5.8), we see that κ in (4.38) is bounded in n as n → ∞. Since all the constants L j and κ are bounded in n as n → ∞ and that all the N j (z) are bounded in n uniformly in Tˆ , we see that S ∞ (z) is also bounded in n
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uniformly in T . To see that this is also the case for the inverse (S ∞ (z))−1 , let us consider the determinant of S ∞ (z). Since S ∞ (z) is a solution to the Riemann-Hilbert problem (1.23), the determinant det (S ∞ (z)) has no jump discontinuity in C and it behaves as 1 1 + O(z − 3 ) as z → ∞. From the expression of N (z) in (4.35), we see that at λ j , only the first and second columns of N (z) have fourth-root singularities, while at the points ±ic, only the second and the third columns of N (z) have fourth-root singularities. Therefore the determinant of S ∞ (z) can at worst have square-root singularities at these points. Since det (S ∞ (z)) has no jump discontinuities in C, we see that det (S ∞ (z)) cannot have square-root singularities at these points. Hence det (S ∞ (z)) is holomorphic in the whole complex plane. By Liouville’s theorem, this implies that det (S ∞ (z)) = 1. Since the entries of (S ∞ (z))−1 are degree 3 polynomials in the entries of S ∞ (z) divided by det (S ∞ (z)) = 1, we see that the entries of (S ∞ (z))−1 are also bounded in n uniformly in T . Finally, by considering the asymptotic expansion of N j (z) in the local parameter w in (4.25) at z = ∞ and making use of (5.8), it is easy to see that Condition 2 in Theorem 4 is satisfied for S ∞ (z) and its inverse. We can now use Theorem 4 to conclude that Theorem 2 and Theorem 3 are true for the multi-cut case. Acknowledgement. The author is indebted to M. Duits and A. B. J. Kuijlaars for many fruitful discussions and for inviting me to Katholieke Universiteit Leuven in which this work initiated. I am also grateful to them for providing me with an early copy of [13] and for encouraging me to complete this paper. I am also grateful to M. Bertola for showing me the relevant results in [25] which led to the completion of Sect. 5.
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