Reviews in Mathematical Physics - Volume 9

Z-GRADED EXTENSIONS OF POISSON BRACKETS JANUSZ GRABOWSKI∗ Institute of Mathematics, University of Warsaw ul. Banacha 2, ...

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Z-GRADED EXTENSIONS OF POISSON BRACKETS JANUSZ GRABOWSKI∗ Institute of Mathematics, University of Warsaw ul. Banacha 2, 02-097 Warsaw, Poland and Mathematical Institute, Polish Academy of Sciences ´ ul. Sniadeckich 8, P. O. Box 137, 00-950 Warsaw, Poland Received 3 June 1996 Revised 25 July 1996 A Z-graded Lie bracket { , }P on the exterior algebra Ω(M ) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M, P ), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an ‘integral’ of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dµ, dν]P = d{µ, ν}P . A naturally defined generalized Hamiltonian map is proved to be a homomorphism between olicher–Nijenhuis bracket of vector valued forms. Also relations of this { , }P and the Fr¨ graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an ‘integral’ of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

1. Introduction The classical Poisson bracket {f, g} =

X ∂f ∂g ∂f ∂g − ∂pi ∂qi ∂qi ∂pi i

(1.1)

was first introduced by Poisson in the early nineteenth century in his study of the equation of motion in celestial mechanics. About thirty years later, Jacobi discovered the famous ‘Jacobi identity’ and Hamilton, using the Poisson bracket, found that the equations of motion could be written in the form of what is now called ‘Hamilton’s equations’. Since then, Poisson brackets in more and more general form have been exploited for almost two hundred years in geometry and physics. We found a way to extend the Poisson brackets to Z-graded Lie brackets on the exterior algebra of differential forms. The aim of this paper is to present this extension together with observations on its relations to other graded Lie brackets over a Poisson manifold. The graded Lie brackets have become a topic of interest in physics in the context of ‘supersymmetries’ relating particles of differing statistics (cf. [5]). The growing ∗ Supported

by KBN, grant 2 P03A 042 10.

1 Reviews in Mathematical Physics, Vol. 9, No. 1 (1997) 1–27 c

World Scientific Publishing Company

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J. GRABOWSKI

interest in graded Lie algebras in mathematics started in the context of deformation theory (see the survey [35]) and the discovery of Schouten [36] and Nijenhuis, who observed that the standard Lie bracket of vector fields can be extended to a graded Lie bracket of multivector fields. This bracket — the Schouten–Nijenhuis bracket [ , ]S−N — satisfies the Leibniz rule and the whole algebraic structure is a prototype of what is now called a Gerstenhaber algebra (see [24]). The last notion goes back to Gerstenhaber’s work on cohomology rings of algebras [16]. The Schouten–Nijenhuis bracket detects Poisson structures: P ∈ Γ(Λ2 T M ) is a Poisson tensor if and only if [P, P ]S−N = 0. Similarly, the Fr¨ olicher–Nijenhuis bracket [ , ]F −N on the graded space Ω(M ; T M ) of vector valued differential forms detects complex structures: a nearly complex structure J ∈ Ω1 (M ; T M ) is complex (integrable) if and only if [J, J]F −N = 0, as stated in the famous theorem of Newlander and Nirenberg. It is also used to define Nijenhuis operators — an important tool in the theory of integrable systems. The Koszul–Schouten bracket [ , ]P defined on differential forms on a Poisson manifold (M, P ) plays, in turn, an important role in the theory of Poisson Lie groups, where it is used to define dressing actions. Recent papers by Lian and Zuckerman [27], Getzler [17] and several papers on string theory make also an extensive use of graded Lie brackets and their generalizations (cf. the appearance of Batalin–Vilkovisky algebras in the BRST cohomology of topological field theories). The importance of the Hamiltonian formalism and the belief that the propagation of higher-order geometric singularities can be described in terms of suitably extended Hamiltonian formalism provoke several attempts to extend the Poisson bracket defined on C ∞ (M ) by a symplectic form, or, more generally, by a Poisson tensor P , to the graded algebra Ω(M ) of differential forms (see [31, 8]). It should be stressed that for this extension the bracket degree coincides with the degree of a form, whereas the degree of a k-form with respect to the Koszul–Schouten bracket is (k−1). Michor in [31] for symplectic, and Cabras and Vinogradov in [8] for arbitrary Poisson structures, proposed several graded brackets extending the Poisson bracket { , }P of functions. Their brackets, however, fail to be either skew-symmetric (and those may be viewed as prototypes of what is now called Leibniz or Loday brackets [25]), or to satisfy the Jacobi identity. The direct skew-symmetrization of the first ones is not the right solution, since it leads again to brackets not satisfying the Jacobi identity. In this paper, we propose a true graded Lie bracket extending the Poisson bracket of functions. Our bracket fails to satisfy the Leibniz rule (in fact, we prove that Leibniz rule contradicts another natural property of the extended bracket), but it seems to be a right one, since it coincides on co-exact forms with the bracket of Michor, Cabras, and Vinogradov and it is nicely related to other graded Lie brackets on the manifold: the exterior derivative is a homomorphism into the Koszul–Schouten bracket, a generalized Hamiltonian map is a homomorphism into the Fr¨ olicher–Nijenhuis bracket and another Hamiltonian map is a homomorphism into the Schouten–Nijenhuis bracket. Moreover, our extension of the canonical

Z-GRADED EXTENSIONS OF POISSON BRACKETS

3

Poisson bracket on the cotangent bundle T ∗ M contains the Fr¨ olicher–Nijenhuis and the symmetric Schouten bracket on M (cf. [10]). We get also a graded analog of the well-known exact sequence of Lie algebra homomorphisms on a symplectic manifold (M, ω) c H 0 −→ H 0 (M ) −→ C ∞ (M ) −→ LHam(ω) −→ H 1 (M ) −→ 0 ,

(1.2)

where H 0 (M ) and H 1 (M ) are the De Rham cohomology spaces with trivial Lie brackets, C ∞ (M ) is taken with the Poisson bracket, LHam(ω) is the Lie algebra of locally Hamiltonian vector fields, H is the Hamiltonian map, and c assigns to X ∈ LHam(ω) the cohomology class of iX ω. We are also able to extend the Poisson bracket to a graded bracket on the exterior algebra generated over C ∞ (M ) by Hamiltonian vector fields, which is an ‘integral’ of the Schouten–Nijenhuis bracket. In the case of a symplectic manifold (i.e. nondegenerate P ) it is defined on all multivector fields and on ‘co-exact’ multivector fields; we get the same structure as Cabras and Vinogradov [8]. Our proofs are chosen in such a way that they can be used immediately in a more general setting of Poisson tensors for an arbitrary Lie algebroid. This fact, together with the lack of the Leibniz rule, makes some of the proofs computationally complicated. Recall that the notion of a Lie algebroid is a straightforward generalization of a Lie algebra and, what is more significant in our case, also a generalization of a tangent bundle and plays a significant role in Poisson geometry (see [4]). All generalizations are presented in the last section, so the readers, who are not familiar with the concept of a Lie algebroid, may simply concentrate on ‘classical’: vector fields, differential forms, Poisson structures, etc. In this paper, we do not give any direct applications of the introduced bracket to physics, rather concentrating on its properties and making the paper rigorously mathematical. However, our belief is that, because of its naturality and nice relations to other significant structures, the presented extension of the Poisson bracket will find its applications in geometry and physics, as the other mentioned brackets do. The paper is organized as follows. In the next section, we briefly recall main properties of the Schouten–Nijenhuis, Nijenhuis–Richardson, and Fr¨ olicher– Nijenhuis brackets. In Sec. 3, we deal with a Poisson manifold (M, P ), defining the Poisson bracket { , }P of functions and the Koszul–Schouten bracket. We also define a generalized Hamiltonian and related maps. Section 4 is devoted to the definition of the graded extension of { , }P , to the proof that it is a graded Lie bracket and to its main properties and relations to other graded Lie brackets. Extensions of the Poisson bracket { , }P to multivector fields are studied in Sec. 5. In the last section we consider these structures in a more general setting of an arbitrary Lie algebroid.

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2. Graded Lie Brackets on a Manifold

L A graded Lie bracket on a graded vector space V = n∈Z Vn (‘graded’ means always ‘Z-graded’ throughout this paper) is a bilinear operation [ , ] : V × V −→ V , being graded [Vn , Vm ] ⊂ Vn+m ,

(2.1)

[x, y] = −(−1)|xky|[y, x] ,

(2.2)

graded skew-symmetric

and satisfying the graded Jacobi identity [[x, y], z] = [x, [y, z]] − (−1)|xky| [y, [x, z]] ,

(2.3)

where |x| is the degree of x, i.e. x ∈ V|x| . Let us fix the convention that we write simply x, µ, etc., for the degrees |x| (or |X|), |µ|, etc., when no confusion arises. One sometimes writes the graded Jacobi identity in the form (−1)xz [[x, y], z] + (−1)yx [[y, z], x] + (−1)zy [[z, x], y] ,

(2.4)

which is equivalent to (2.3) for graded skew-symmetric brackets. However, for non-skew-symmetric brackets the formula (2.3) seems to be better, since it means def

that the adjoint map x 7→ adx = [x, ·] is a representation of the bracket, i.e. ad[x,y] is equal to the graded commutator def

[adx , ady ] = adx ◦ ady − (−1)xy ady ◦ adx = ad[x,y] ,

(2.5)

whereas (2.4) has no clear direct meaning. With a given smooth (C ∞ ) manifold M there are associated several natural graded Lie brackets of tensor fields. Historically the first was probably the famous Schouten–Nijenhuis bracket [ , ]S−N defined on multivector fields (see [36] for the original and [32] for a modern version). It is the unique graded extension of the usual bracket [ , ] on the space X (M) of vector fields to the exterior algebra L n n n A(M ) = n∈Z A (M ) of multivector fields (where A (M ) = Γ(Λ T M ) is the space of n-vector fields for n ≥ 0 and An (M ) = {0} for n < 0) such that (1) the degree of X ∈ An (M ) with respect to the bracket is (n − 1), (2) [X, f ]S−N = L(X)f for X ∈ A1 (M ), f ∈ A0 (M ) = C ∞ (M )

(2.6)

and L(X) being the Lie derivative along X, (3) [X, Y ∧ Z]S−N = [X, Y ]S−N ∧ Z + (−1)(k−1)l Y ∧ [X, Z]S−N

(2.7)

for X ∈ Ak (M ), Y ∈ Al (M ), i.e. ad is a representation of the Schouten–Nijenhuis bracket in graded derivations of the graded algebra A(M ).

Z-GRADED EXTENSIONS OF POISSON BRACKETS

5

Graded algebras furnished with a graded bracket satisfying (2.6) are called Gerstenhaber algebras (see [24, 25]). From (2.7), it follows that [X1 ∧ . . . ∧ Xm , Y1 ∧ · · · ∧ Yn ]S−N (2.8) P ck ∧ . . . ∧ Xm ∧ Y1 ∧ . . . ∧ Ybl ∧ . . . ∧ Yn , = k,l (−1)k+l [Xk , Yl ] ∧ . . . ∧ X where Xk , Yl ∈ X (M ) and the hats stand for omissions. Let us note that the skew-symmetric Schouten–Nijenhuis bracket has its symmetric counterpart — the symmetric Schouten bracket [ , ]S — defined on the space S(M ) of symmetric multivector fields (see [3, 10]). It is an ordinary (non-graded) Lie bracket extending the Lie bracket of vector fields, satisfying the analog of (2.6) and [X1 ∨ . . . ∨ Xm , Y1 ∨ . . . ∨ Yn ]S P ck ∨ . . . ∨ Xm ∨ Y1 ∨ . . . ∨ Ybl ∨ . . . ∨ Yn . = k,l [Xk , Yl ] ∨ . . . ∨ X

(2.9)

It is well known that the map ι : (S(M ), [ , ]S ) −→ (C ∞ (T ∗ M ), { , }PM ), ι(X1 ∨ . . . ∨ Xm ) = ι(X1 ) . . . ι(Xm ) , def

(2.10)

def

∗ (f ) for f ∈ C ∞ (M ), where Xk ∈ X (M ) and ι(Xk )(θq ) = hXk (q), θq i, ι(f ) = πM is an injective homomorphism of the symmetric Schouten bracket on M into the canonical Poisson bracket { , }PM on the cotangent bundle πM : T ∗ M −→ M . Let us denote by Ωn (M ) the space of n-forms on M , by Ω(M ) the exterior L algebra of differential forms (Ω(M ) = n∈Z Ωn (M ), with Ωn (M ) = {0} for n < 0), L and by Ω(M ; T M ) = n∈Z Ω(M ; T M ) the Ω(M )-module of vector valued forms. Clearly, we have Ω0 (M ; T M ) = X (M ) and the left and right actions of Ω(M ) on Ω(M ; T M ) are given by

ν ∧ (µ ⊗ X) = (ν ∧ µ) ⊗ X = (−1)µν (µ ⊗ X) ∧ ν ,

(2.11)

for µ, ν ∈ Ω(M ), X ∈ X (M ). We can extend usual insertion operators i(X) : Ωn (M ) −→ Ωn−1 (M ) and Lie derivatives L(X) : Ωn (M ) −→ Ωn (M ), defined for X ∈ X (M ), to insertions i (K) : Ωn (M ) −→ Ωn+k−1 (M ) and Lie differentials L(K) : Ωn (M ) −→ Ωn+k (M ), defined for K ∈ Ωk (M ; T M ), putting def

i (µ ⊗ X)ν = µ ∧ i (X)ν

(2.12)

and def

L(K) = i (K) ◦ d + (−1)k d ◦ i (K) ,

(2.13)

L(µ ⊗ X)ν = µ ∧ L(X)ν + (−1)µ dµ ∧ i (X)ν .

(2.14)

so that

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J. GRABOWSKI

We can also extend the insertion i (K) itself to the operator i (K) : Ωn (M ; T M ) −→ Ωn+k−1 (M ; T M ) , def

i (K)(µ ⊗ X) = (i (K)µ) ⊗ X .

(2.15)

The Nijenhuis–Richardson bracket [ , ]N −R is a graded Lie bracket on the graded space Ω(M ; T M ), with elements of Ωn (M ; T M ) being of degree (n − 1), defined by [K, L]N −R = i (K)L − (−1)(k−1)(l−1) i (L)K def

(2.16)

for K ∈ Ωk (M ; T M ), L ∈ Ωl (M ; T M ). We have the identity def

[ i (K), i (L)] = i (K) ◦ i (L) − (−1)(k−1)(l−1) i (L) ◦ i (K) = i ([K, L]N −R ).

(2.17)

The Fr¨ olicher–Nijenhuis bracket [ , ]F −N is a graded Lie bracket on the same space Ω(M ; T M ), but, this time, with the grading which agrees with the form degree, defined on simple tensors µ ⊗ X ∈ Ωµ (M ; T M ) and ν ⊗ Y ∈ Ων (M ; T M ) by [µ ⊗ X, ν ⊗ Y ]F −N = µ ∧ ν ⊗ [X, Y ] + µ ∧ L(X)ν ⊗ Y − L(Y )µ ∧ ν ⊗ X +(−1)µ (dµ ∧ i (X)ν ⊗ Y + i (Y )µ ∧ dν ⊗ X) .

(2.18)

The Fr¨olicher–Nijenhuis bracket extends the usual bracket of vector fields (recall that Ω0 (M ; T M ) = X (M )) and satisfies the following (cf. [19]) def

[L(K), L(L)] = L(K) ◦ L(L) − (−1)kl L(L) ◦ L(K) = L([K, L]F −N ) ,

(2.19)

def

[L(K), i (L)] = L(K) ◦ i (L) − (−1)k(l+1) i (L) ◦ L(K) = i ([K, L]F −N ) − (−1)k(l+1) L(i (L)K) .

(2.20)

Moreover, [K, µ ∧ L]F −N = L(K)µ ∧ L + (−1)kµ µ ∧ [K, L]F −N − (−1)(µ+1)(k+1) dµ ∧ i (L)K (2.21) (see [31, 10, 19]). 3. Brackets on Poisson Manifolds Let us suppose that we are given a Poisson tensor on a manifold M , i.e. a bivector field P ∈ A2 (M ) such that [P, P ]S−N = 0 .

(3.1)

Z-GRADED EXTENSIONS OF POISSON BRACKETS

7

The corresponding Poisson bracket of functions {f, g}P = hP, df ∧ dgi

(3.2)

{f, {g, h}P }P + {g, {h, f }P }P + {h, {f, g}P }P = 0

(3.3)

satisfies the Jacobi identity

(which is equivalent to (3.1)) and the Leibniz rule {f, gh}P = {f, g}P h + g{f, h}P .

(3.4)

The standard examples of Poisson brackets in mechanics are associated with a phase space T ∗ M or, more generally, with a symplectic manifold (M, ω), where P = ω −1 in the sense that the mapping P# : T ∗ M −→ T M, P# (µ) = i (µ)P , def

(3.5)

is the inverse of (3.6) ω [ : T M −→ T ∗ M, ω [ (X) = −i (X)ω . P ∂ P In canonical coordinates, with ω = k dpk ∧ dqk we associate P = k ∂pk ∧ ∂q∂k and the Poisson bracket (1.1). More general (degenerate) Poisson structures appear, for instance, in the process of Poisson reduction (e.g. The Kostant–Kirillov–Souriau bracket on the dual space of a Lie algebra), or, in the process of passing to semi-classical limits of quantum groups, when one encounters Poisson Lie structures (see [9]). It is well known that assigning to a function f its Hamiltonian vector field def P H (f ) = P# (df ) gives a Lie bracket homomorphism: def

HP ({f, g}P ) = [HP (f ), HP (g)]

(3.7)

{f, g}P = L(HP (f ))g .

(3.8)

and that

Let us note that we will usually write Pµ instead of P# (µ) and Hµ instead of HP (µ), when it is clear what a Poisson tensor P we have in mind. It is known [11, 28, 13, 20, 26] that the Poisson structure P defines not only the Poisson bracket { , }P of functions, but also a Lie bracket [ , ]P on 1-forms, given by [µ, ν]P = L(Pµ )ν − L(Pν )µ − dhP, µ ∧ νi = i (Pµ )dν − i (Pν )dµ + dhP, µ ∧ νi ,

(3.9)

where h , i is the pairing between forms and multivector fields. In particular, [df, dg]P = d{f, g}P and P# is a Lie bracket homomorphism: [Pµ , Pν ] = P[µ,ν]P .

(3.10)

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J. GRABOWSKI

The bracket [ , ]P on 1-forms can be extended to a graded Lie bracket on Ω(M ), as it was observed by Koszul [21] (see also [26] and [20]), with n-forms being of degree (n − 1), by the formula [µ, ν]P = (−1)µ (∂P (µ ∧ ν) − ∂P µ ∧ ν − (−1)µ µ ∧ ∂P ν) ,

(3.11)

def

where ∂P = [i (P ), d] = i (P ) ◦ d − d ◦ i (P ) is the Poisson homology operator of Koszul and Brylinski (cf. [37]). The following is essentially due to Koszul [21]. Theorem 1. [21] On the exterior algebra Ω(M ) of differential forms, Eq. (3.11) defines a graded Lie bracket, called the Koszul–Schouten bracket, with n-forms being of bracket degree (n − 1). This bracket satisfies the Leibniz rule [µ, θ ∧ ν]P = [µ, θ]P ∧ ν + (−1)(µ−1)θ θ ∧ [µ, ν]P ,

(3.12)

where µ ∈ Ωµ (M ) etc., and the exterior derivative is a derivation of the bracket d[µ, ν]P = [dµ, ν]P + (−1)µ−1 [µ, dν]P .

(3.13)

[df, dg]P = d{f, g}P , [df, g]P = {f, g}P , [f, g]P = 0

(3.14)

Moreover,

for functions f, g ∈ C ∞ (M ) = Ω0 (M ), and the mapping ΛP : Ω(M ) −→ A(M ) ,

(3.15)

defined by def

def

ΛP (f ) = f, ΛP (µ1 ∧ . . . ∧ µm ) = Pµ1 ∧ . . . ∧ Pµm

(3.16)

for f ∈ C ∞ (M ) and µk ∈ Ω1 (M ), is a homomorphism of the Koszul–Schouten into the Schouten–Nijenhuis bracket: ΛP ([µ, ν]P ) = [ΛP (µ), ΛP (ν)]S−N .

(3.17)

It is easy to see that the mapping ΛP is invertible if and only if P is nondegenerate. In this case, the inverse (ΛP )−1 is given by (ΛP )−1 (X1 ∧ . . . ∧ Xm ) = ω [ (X1 ) ∧ . . . ∧ ω [ (Xm ) ,

(3.18)

where Xk ∈ X (M ), ω = P −1 is the symplectic form associated with P and ω [ (X) = −i(X)ω (cf. (3.5) and (3.6)). In [31], Michor defines a ‘generalized Hamiltonian mapping’ on a symplectic manifold, using the unique extension P# : Ω(M ) −→ Ω(M ; T M ) of (3.5) into a derivation of degree −1 on Ω(M ) with values in the Ω(M )-module Ω(M ; T M ) and def

putting HP = P# ◦ d. We will continue writing in most cases Pµ instead of P# (µ), Hµ instead of HP (µ), etc.

Z-GRADED EXTENSIONS OF POISSON BRACKETS

9

The Michor’s construction is actually valid on any Poisson manifold (see [8, 25, 1, 15]). The mapping P# : Ω(M ) −→ Ω(M ; T M ) is characterized by the following (1)

Pf

= 0 for f ∈ C ∞ (M ) ,

(2)

Pµ

= i(µ)P for µ ∈ Ω1 (M ) , P = k (−1)k+1 µ1 ∧ . . . ∧ µ ck ∧ . . . ∧ µm ⊗ Pµk

(3) Pµ1 ∧...∧µm

(3.19)

for µk ∈ Ω1 (M ). As we mentioned already, P# is a derivation of degree −1: Pµ∧ν = Pµ ∧ ν + (−1)µ µ ∧ Pν

(3.20)

and we have the generalized Hamiltonian map def

HP : Ω(M ) −→ Ω(M ; T M ), HP (µ) = P# (dµ).

(3.21)

Let us denote def

< µ, ν >P = (−1)µ+1 (i (P )(µ ∧ ν) − i (P )µ ∧ ν − µ ∧ i (P )ν) .

(3.22)

Lemma 1. (1) hµ, νiP ∈ Ωµ+ν−2 . (2) hµ, νiP = −(−1)

(µ+1)(ν+1)

(3.23) hν, µiP .

(3.24)

(3) hµ, ν ∧ θiP = hµ, νiP ∧ θ + (−1)µν ν ∧ hµ, θiP .

(3.25)

(4) hµ, νiP = i (Pµ )ν .

(3.26)

Proof. Parts (1) and (2) are trivial. To prove the rest, let us assume that X cjk Xj ⊗ Xk , Xk ∈ X (M ), cjk = −ckj ∈ R . (3.27) P = j,k

Then, hµ, νiP =

X

X cjk i (Xj )µ ∧ i (Xk )ν = i ( cjk i (Xj )µ ⊗ Xk )ν

j,k

(3.28)

j,k

and it is easy to see that hµ, νiP is a derivation of degree (µ − 2) with respect to ν, which proves (3). Since both hµ, ·iP and i (Pµ ) are derivations of degree (µ − 2), due to (2) and by induction, it is now sufficient to prove (4) for µ, ν ∈ Ω1 (M ). We have, according to (3.28), X cjk hXj , µi hXk , νi = hµ, νiP , (3.29) i (Pµ )ν = j,k

where h , i is the obvious pairing, and the lemma follows.

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Theorem 2. [1] The Koszul–Schouten bracket (3.11) may also be written in the form [µ, ν]P = i (Hµ )ν − (−1)µ L(Pµ )ν = i (Hµ )ν − (−1)µ i (Pµ )dν + d i (Pµ )ν

(3.30)

= hdµ, νiP − (−1) hµ, dνiP + dhµ, νiP . µ

Proof. Since both the Koszul–Schouten bracket on the left-hand side and the bracket defined by the right-hand side of (3.30) are graded skew-symmetric and are graded derivations of degree (µ − 1) with respect to ν, it is sufficient to check (3.30) for functions and 1-forms. Both sides are clearly 0 for functions, so let us assume that µ ∈ Ω1 (M ), ν = f ∈ C ∞ (M ). Then, in view of (3.11), [µ, f ]P = f ∂P µ − ∂P (f µ) = f i (P )dµ − i (P )(df ∧ µ + f dµ) = i (P )(µ ∧ df ) = i (Pµ )df , which agrees with (3.30), since i (Hµ )f = i (Pµ )f = 0. For 1-forms µ, ν ∈ Ω1 (M ), according to 3.9 and Lemma 1, [µ, ν]P = i (Pµ )dν − i (Pν )dµ + dhP, µ, νi = hµ, dνiP − hν, dµiP + dhµ, νiP = hdµ, νiP + hµ, dνiP + dhµ, νiP

and the theorem follows. 4. Graded Extensions of Poisson Brackets

There were several attempts to extend the Poisson bracket of functions to a graded Lie bracket on Ω(M ) such that the degree of a differential form coincides with the degree with respect to { , }P . The next assumption is that the exterior derivative is a homomorphism of the extended { , }P into the Koszul–Schouten bracket. The latter implies (and is equivalent in the case of a nondegenerate Poisson structure; see Corollary 1 and Theorem 6) that the generalized Hamiltonian map (3.21) should olicher–Nijenhuis bracket. be a homomorphism of the extended { , }P into the Fr¨ For symplectic manifolds (but with a straightforward generalization to arbitrary Poisson manifolds), Michor considered in [31] the brackets def

def

{µ, ν}1 = i (Hµ )dν, {µ, ν}2 = L(Hµ )ν .

(4.1)

The first is graded skew-symmetric but it does not satisfy the graded Jacobi identity, while the second satisfies the Jacobi identity but it is not skew-symmetric (it is a prototype of a Loday bracket [25]). The direct skew-symmetrization def

{µ, ν}3 =

1 (L(Hµ )ν − (−1)µν L(Hν )µ) 2

(4.2)

Z-GRADED EXTENSIONS OF POISSON BRACKETS

11

turns out again not to satisfy the Jacobi identity. All the brackets differ by something exact and H{µ,ν}1 = H{µ,ν}2 = H{µ,ν}3 = [Hµ , Hν ]F −N .

(4.3)

Similar brackets for general Poisson structures, described in [8], are again either not skew-symmetric or do not satisfy the graded Jacobi identity. The graded Jacobi identity is satisfied only modulo exact forms and therefore all these brackets define (in fact, the same) graded Lie brackets on the space Ω(M )/B(M ) of co-exact forms (B(M ) denotes, clearly, exact forms). In this section, we show the proper form of a graded extension of the Poisson bracket { , }P which is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. Before we define the graded extension of the Poisson bracket { , }P of functions, let us start with some lemmata concerning the mapping P# : Ω(M ) 3 µ 7→ Pµ ∈ Ω(M ; T M ). Lemma 2. For µ, ν ∈ Ω(M ), we have Pi (Pµ )ν = (−1)µ (i (Pµ )Pν + (−1)µν i (Pν )Pµ ) .

(4.4)

Proof. Being a graded commutator of derivatives of degrees −1 and (µ − 2), def

[P# , i (Pν )] = P# ◦ i (Pµ ) − (−1)µ i (Pµ ) ◦ P#

(4.5)

is a graded derivative of degree (µ − 3) on Ω(M ) with values in the Ω(M )-module Ω(M ; T M ), vanishing on functions. We will show that def

[P# , i (Pν )](ν) = Pi (Pµ )ν − (−1)µ i (Pµ )Pν = (−1)µ(ν+1) i (Pν )Pµ .

(4.6)

First, observe that def

ν 7→ Fµ (ν) = (−1)µ(ν+1) i (Pν )Pµ

(4.7)

is also a graded derivation of degree (µ − 3) on Ω(M ) with values in the Ω(M )module Ω(M ; T M ), vanishing on functions. Indeed, Fµ (ν ∧ θ) = (−1)µ(ν+θ+1) i (Pν∧θ )Pµ = (−1)µ(ν+θ+1) i (Pν ∧ θ + (−1)ν ν ∧ Pθ )Pµ = (−1)µ(ν+θ+1)+µθ i (Pν )Pµ ∧ θ + (−1)µ(ν+θ+1)+ν ν ∧ i (Pθ )Pµ = Fµ (ν) ∧ θ + (−1)(µ−3)ν ν ∧ Fµ (θ) . Hence, inductively, it is sufficient to check (4.6) for ν being a 1-form. Thus, let us take ν ∈ Ω1 (M ) and µ = µ1 ∧ . . . ∧ µm with µj ∈ Ω1 (M ). We have X (−1)j+1 µ(j) ⊗ Pµj (4.8) Pµ = j

12

J. GRABOWSKI

and i (Pµ )ν =

X

(−1)j+1 hP, µj ∧ νiµ(j) ,

(4.9)

j def

cj ∧ . . . ∧ µm . Hence, where µ(j) = µ1 ∧ . . . ∧ µ X Pi (Pµ )ν = (−1)j+l hP, µj ∧ νiµ(l,j) ⊗ Pµl l<j

+

X

(−1)j+l+1 hP, µj ∧ νiµ(j,l) ⊗ Pµl ,

(4.10)

j
where, for 1 ≤ l < j ≤ m, def

µ(l,j) = µ1 ∧ . . . ∧ µbl ∧ . . . ∧ µ cj ∧ . . . ∧ µm .

(4.11)

On the other hand, i (Pµ )ν = 0, since Pν ∈ Ω0 (M ; T M ), and X (−1)m(1+1) i (Pν )Pµ = i (Pν ) (−1)l+1 µ(l) ⊗ Pµl =

X

l

(−1)j+l hP, ν ∧ µj iµ(j,l) ⊗ Pµl

(4.12)

j
+

X (−1)j+l+1 hP, ν ∧ µj iµ(l,j) ⊗ Pµl . l<j

We see from (4.10) and (4.12) that Pi (Pµ )ν = i (Pν )Pµ , since ν ∧ µj = −µj ∧ ν. The following generalization of (3.10) will be used in the sequel. Lemma 3. For µ, ν ∈ Ω(M ), we have [Pµ , Pν ]F −N = Pd i(Pµ )ν + (−1)(µ+1)(ν+1) i (Pν )Pdµ − i (Pµ )Pdν = Hi (Pµ )ν + (−1)(µ+1)(ν+1) i (Pν )Hµ − i (Pµ )Hν .

(4.13)

Proof. Both sides of Eq. (4.13) vanish if µ or ν is a function. For µ, ν ∈ Ω1 (M ) Eq. (4.13) is equivalent to (3.10). Indeed, the left-hand side reduces to the usual bracket of vector fields and the right-hand side equals, according to (4.4), Pd i(Pµ )ν − Pi (Pν )dµ + Pi (Pµ )dν = P[µ,ν]p ,

(4.14)

since i (Pdµ )Pν = i (Pdν )Pµ = 0. Hence, inductively, it is sufficient to prove that if (4.13) holds for a form ν ∈ Ω(M ), then it holds also for ν 0 ∧ ν with ν 0 ∈ Ω1 (M ). We have, on one hand (cf. 2.21), [Pµ , Pν 0 ∧ν ]F −N = [Pµ , ν ∧ Pν 0 − ν 0 ∧ Pν ]F −N = L(Pµ )ν ∧ Pν 0 + (−1)(µ−1)ν ν ∧ [Pµ , Pν 0 ]F −N − (−1)νµ dν ∧ i (Pν 0 )Pµ − L(Pµ )ν 0 ∧ Pν + (−1)µ ν 0 ∧ [Pµ , Pν ]F −N + (−1)νµ dν 0 ∧ i (Pν )Pµ ,

(4.15)

Z-GRADED EXTENSIONS OF POISSON BRACKETS

13

and on the other hand, P# (d i(Pµ )(ν 0 ∧ ν)) + (−1)(µ+1)ν i (Pν 0 ∧ν )Hµ − i (Pµ )Pd(ν 0 ∧ν) ) = P# ((d i(Pµ )ν 0 ) ∧ ν + (−1)µ+1 i (Pµ )ν 0 ∧ dν + (−1)µ dν 0 ∧ i (Pµ )ν + (−1)µ+1 ν 0 ∧ d i(Pµ )ν) + (−1)(µ+1)ν ν ∧ i (Pν 0 )Hµ − (−1)(µ+1)ν ν 0 ∧ i (Pν )Hµ − i (Pµ )(Hν 0 ∧ ν + dν 0 ∧ Pν − Pν 0 ∧ dν + ν 0 ∧ Hν ) = Pν 0 ∧ (i (Pµ )dν + (−1)µ+1 d i (Pµ )ν) − (i (Pµ )dν 0 + (−1)µ+1 d i (Pµ )ν 0 ) ∧ Pν + (Hi (Pµ )ν 0 + i (Pν 0 )Hµ − i (Pµ )Hν 0 ) ∧ ν + (−1)µ ν 0 ∧ (Hi (Pµ )ν + (−1)(µ+1)(ν+1) i (Pν )Hµ − i (Pµ )Hν ) + (−1)µ+1 Pi (Pµ )ν 0 ∧ dν + dν 0 ∧ ((−1)µ Pi (Pµ )ν − i (Pµ )Pν ) = (−1)µ ν 0 ∧ [Pµ , Pν ]F −N + [Pµ , Pν 0 ]F −N ∧ ν + (−1)µ+1 i (Pν 0 )Pµ ∧ dν + (−1)µν dν 0 ∧ i (Pν )Pµ + Pν 0 ∧ L(Pµ )ν − L(Pµ )ν 0 ∧ Pν , where we used (4.4) and the inductive assumption. Since the last coincides with (4.15), the lemma follows. Corollary 1. The space Z(M ) of closed forms is a graded Lie subalgebra of the Koszul–Schouten algebra (Ω(M ), [ , ]P ) and the map P# : (Z(M ), [ , ]P ) −→ (Ω(M ; T M ), [ , ]F −N )

(4.16)

is a homomorphism of graded Lie algebras: P[µ,ν]P = [Pµ , Pν ]F −N f or dµ = 0, dν = 0 .

(4.17)

Proof. According to (3.30) and (4.13), [µ, ν]P = d i(Pµ )ν for µ, ν being closed and P[µ,ν]P = Hi (Pµ )ν = [Pµ , Pν ]F −N .

(4.18)

Let us define a bracket operation on differential forms on a given Poisson manifold (M, P ), by putting def

{µ, ν}P = L(Hµ )ν + dL(Pµ )ν = i (Pdµ )dν + d i (Pµ )dν − (−1)µν d i(Pν )dµ = hdµ, dνiP + dhµ, dνiP + (−1)µ dhdµ, νiP .

(4.19)

14

J. GRABOWSKI

The notation is justified by the fact that on functions the bracket (4.19) coincides with the Poisson bracket (3.8). The bracket (4.19) is manifestly graded {µ, ν}P ∈ Ωµ+ν (M ) ,

(4.20)

{µ, ν}P = −(−1)µν {ν, µ}P ,

(4.21)

graded skew-symmetric

and differs from the Michor’s brackets (4.1) by exact forms. Theorem 3. We have, for µ, ν ∈ Ω(M ), (1)H{µ,ν}P = [Hµ , Hν ]F −N .

(4.22)

(2)P{µ,ν}P = [Pµ , Hν ]F −N + (−1)µ [Hµ , Pν ]F −N . R−N

(3)H[µ,ν]P = [Hµ , Hν ]

F −N

+ [Hµ , Pν ]

(4.23) µ+1

+ (−1)

F −N

[Pµ , Hν ]

.

(4.24)

(4)P[µ,ν]P = [Pµ , Pν ]F −N + (−1)µ i (Hµ )Pν − (−1)µ(ν+1) i (Hν )Pµ . (4.25) Proof. The proof depends on obvious calculations with the use of Lemmata 2 and 3. We present only the calculations for the identities (4.22) and (4.23) which will be used in the sequel. Putting µ := dµ and ν := dν in (4.13), we get [Hµ , Hν ]F −N = Hi (Hµ )dν

(4.26)

H{µ,ν}P = Pd{µ,ν}P = Hi (Hµ )dν .

(4.27)

and, by definition,

To prove (4.23), we can write, using (4.13) and (4.4), P{µ,ν}P = Pi (Pdµ )dν + Hi (Pµ )dν − (−1)µν Hi (Pν )dµ = (−1)µ+1 (i (Hµ )Hν + (−1)(µ+1)(ν+1) i (Hν )Hµ ) + [Pµ , Hν ]F −N − (−1)(µ+1)ν i (Hν )Hµ − (−1)µν ([Pν , Hµ ]F −N − (−1)µ(ν+1) i (Hµ )Hν ) = [Pµ , Hν ]F −N + (−1)µ [Hµ , Pν ]F −N .

Theorem 4. The graded skew-symmetric bracket (4.19) satisfies the graded Jacobi identity {{µ, ν}P , θ} = {µ, {ν, θ}P }P − (−1)µν {ν, {µ, θ}P }P .

(4.28)

Proof. According to (2.5), the Jacobi identity (4.28) is equivalent to L(H{µ,ν}P ) + dL(P{µ,ν}P ) = [L(Hµ ) + dL(Pµ ), L(Hν ) + dL(Pν )] .

(4.29)

15

Z-GRADED EXTENSIONS OF POISSON BRACKETS

It is easy to see that the right-hand side of (4.29) equals [L(Hµ ), L(Hν )] + d[L(Pµ ), L(Hν )] + (−1)µ d[L(Hµ ), L(Pν )] ,

(4.30)

where the brackets are the graded commutators of derivations, and further, in view of (2.20), to L([Hµ , Hν ]F −N ) + dL([Pµ , Hν ]F −N + (−1)µ [Hµ , Pν ]F −N ) .

(4.31)

According to (4.22) and (4.23), the last equals L(H{µ,ν}P ) + dL(P{µ,ν}P )

(4.32)

and the theorem follows.

Remark. One can prove the identities (4.24) and (4.25) using the methods similar to those of the above proof and the fact that we already know that the Koszul–Schouten bracket satisfies the graded Jacobi identity. Then, we can obtain (4.22) and (4.23) as consequences. However, this clever method fails for generalizations, when the Lie derivatives L(K) do not determine K. Thus we used a direct method in proving Lemma 2, which has the advantage that it is valid in general. Properties of the extended Poisson bracket are collected in the following. Theorem 5. For the bracket (4.19), we have (1) Given µ, ν ∈ Ω(M ) and f ∈ C ∞ (M ), {f, µ}P = L(Hf )µ and d{µ, ν}P = {dµ, ν}P = (−1)µ {µ, dν}P .

(4.33)

In particular, the graded subspace Z(M ) of closed forms is a commutative Lie ideal of (Ω(M ), { , }P ). (2) Given f0 , . . . , fn , g0 , . . . , gm ∈ C ∞ (M ), {f0 df1 ∧ . . . ∧ dfn , g0 dg1 ∧ . . . ∧ dgm }P

(4.34)

= {f0 , g0 }P df1 ∧ . . . ∧ dfn ∧ dg1 ∧ . . . ∧ dgm X dk . . . ∧ dfn ∧ dg0 ∧ . . . dg cl . . . ∧ dgm − (−1)k+l {fk , gl }P df0 ∧ . . . df k,l>0

−f0

X

dk . . . ∧ dfn ∧ dg0 ∧ . . . dg cl . . . ∧ dgm (−1)k+l d{fk , gl }P df1 ∧ . . . df

k>0,l

−g0

X

dk . . . ∧ dfn ∧ dg1 ∧ . . . dg cl . . . ∧ dgm . (−1)n+k+l d{fk , gl }P df0 ∧ . . . df

l>0,k

(3) Instead of the classical Leibniz rule, we have {µ, θ ∧ ν}P = {µ, θ}P ∧ ν + (−1)µθ θ ∧ {µ, ν} +(−1)θ+µ+1 L(Pµ )θ ∧ dν + (−1)θ(µ+1) dθ ∧ L(Pµ )ν

(4.35)

16

J. GRABOWSKI

and the generalized Leibniz rule {µ, θ ∧ ν ∧ β}P = {µ, θ ∧ ν}P ∧ β + (−1)µθ θ ∧ {µ, ν ∧ β}P +(−1)νβ {µ, θ ∧ β}P ∧ ν − {µ, β}P ∧ ν ∧ β

(4.36)

−(−1)µθ θ ∧ {µ, ν}P ∧ β − (−1)µ(θ+ν) θ ∧ ν ∧ {µ, β}P . Proof. The part (1) follows directly from definition (4.19). To prove (2), let us dk ∧ . . . ∧ dfn , denote φ = df0 ∧ . . . ∧ dfn , γ = dg0 ∧ . . . ∧ dgm , φ(k) = df0 ∧ . . . ∧ df dk ∧ . . . ∧ dfn , etc. We have φ(0,k) = df1 ∧ . . . ∧ df {f0 df1 ∧ . . . ∧ dfn , g0 dg1 ∧ . . . ∧ dgm }P = {f0 φ(0) , g0 γ (0) }P = i (Pφ )γ + d i (Pf0 φ(0) )γ − (−1)nm d i (Pg0 γ (0) )φ P P = i ( (−1)k φ(k) ⊗ Hfk )γ + d(f0 i ( (−1)k+1 φ(0,k) ⊗ Hfk )γ) k k>0 P −(−1)nm d(g0 i ( (−1)l+1 γ (0,l) ⊗ Hgl )φ l>0 P = (−1)k+l {fk , gl }P φ(k) ∧ γ (l) k,l

P

+ d(f0

(−1)k+l+1 {fk , gl }P φ(0,k) ∧ γ (l) )

k>0,l

P

−(−1)nm d(g0 =

P k,l

−

(−1)k+l+1 {gl , fk }P γ (0,l) ∧ φ(k) )

l>0,k

(−1)k+l {fk , gl }P φ(k) ∧ γ (l) −

P

(−1)k+l {fk , gl }P φ(k) ∧ γ (l)

k>0,l

P

(−1)k+l {fk , gl }P φ(k) ∧ γ (l)

l>0,k

−f0

P

(−1)k+l d{fk , gl }P φ(0,k) ∧ γ (l)

k>0,l

P

−(−1)n g0

(−1)k+l d{fk , gl }P φ(k) ∧ γ (0,l)

l>0,k

= {f0 , g0 }P φ(0) ∧ γ (0) − −f0

P

P

(−1)k+l {fk , gl }P φ(k) ∧ γ (l)

k,l>0

(−1)k+l d{fk , gl }P φ(0,k) ∧ γ (l)

k>0,l

−g0

P

(−1)n+k+l d{fk , gl }P φ(k) ∧ γ (0,l) .

l>0,k

We get the identity (4.35) by obvious calculations. Then, using (4.35), we get {µ, θ ∧ ν ∧ β}P − (−1)µθ θ ∧ {µ, ν ∧ β}P − {µ, θ ∧ ν}P ∧ β + (−1)µθ θ ∧ {µ, ν}P ∧ β = (−1)νβ ((−1)µ+θ+1 L(Pµ )θ ∧ dβ + (−1)(µ+1)θ dθ ∧ L(Pµ )β) ∧ ν) = (−1)νβ ({µ, θ ∧ β}P − (−1)µθ θ ∧ {µ, β}P − {µ, θ}P ∧ β) ∧ ν , which is equivalent to (4.36).

Z-GRADED EXTENSIONS OF POISSON BRACKETS

17

The next theorem shows certain relations between the extended Poisson bracket and other graded Lie brackets. Theorem 6. (1) The exterior derivative d : (Ω(M )), { , }P ) −→ (Ω(M ), [ , ]P )

(4.37)

is a homomorphism of { , }P into the Koszul–Schouten bracket: d{µ, ν}P = [dµ, dν]P .

(4.38)

Moreover, we have the exact sequence of graded Lie algebra homomorphisms d

0 −→ Z(M ) −→ (Ω(M ), { , }P ) −→ (Z(M ), [ , ]P ) −→ H(M ) −→ 0 , (4.39) where on the closed forms Z(M ) (on the left) and on the De Rham cohomology space H(M ) we put the trivial brackets. (2) The generalized Hamiltonian map HP : (Ω(M ), { , }P ) −→ (Ω(M ; T M ), [ , ]F −N )

(4.40)

is a homomorphism of { , }P into the Fr¨ olicher–Nijenhuis bracket: H{µ,ν}P = [Hµ , Hν ]F −N .

(4.41)

(3) The total Hamiltonian map GP : (Ω(M ), { , }P ) −→ (A(M ), [ , ]S−N ) ,

(4.42)

def

given by GP = ΛP ◦ d, (see (3.16)) is a homomorphism of { , }P into the Schouten– Nijenhuis bracket: G{µ,ν}P = [Gµ , Gν ]SN .

(4.43)

Proof. The part (1) follows easily from (3.30) and (4.19). Note that Z(M ) is a commutative Lie ideal with respect to { , }P by Theorem 5 (1). The part (2) just repeats (4.22), and (3) follows immediately, when we combine (4.38) with (3.17). Let us remark that the exact sequence (4.39) generalizes (1.2). Indeed, we have Z 0 (M ) = H 0 (M ) and locally Hamiltonian vector fields LHam(ω) form the Lie algebra isomorphic to (Z 1 (M ), [ , ]P ) in the case of a symplectic form ω = P −1 . Also the exact sequences considered by Michor [31], using the generalized Hamiltonian map, may be obtained from (4.39) in the symplectic case, since we can pass to the same bracket on co-exact forms, and the mapping Ωm (M ) 3 µ 7→

(−1)m [i (Hµ )Ω] ∈ H m+1 (M ) m

(4.44)

18

J. GRABOWSKI

equals, in the symplectic case, Ωm (M ) 3 µ 7→ [dµ] ∈ H m+1 (M ) .

(4.45)

The equation (4.34) shows that our bracket is given by a bilinear differential operator of order 2, and Eq. (4.36) that we have only a generalized Leibniz rule. In fact, we cannot expect the classical Leibniz rule {µ, θ ∧ ν}P = {µ, θ}P ∧ ν + (−1)µθ θ ∧ {µ, ν}P ,

(4.46)

as shown in the following. Theorem 7. For a non-trivial Poisson tensor P there is no graded Lie bracket { , } on Ω(M) extending the Poisson bracket of functions and satisfying simultaneously (4.38) and the Leibniz rule (4.46). Proof. Combining (4.38) and (4.46), we get easily (−1)µ+θ {µ, θ}P ∧ dν + (−1)µθ dθ ∧ {µ, ν}P = (−1)θ [dµ, θ]P ∧ dν + (−1)µ(θ+1) dθ ∧ [dµ, ν]P .

(4.47)

Putting now θ := dθ in 4.47, we get {µ, dθ}P ∧ dν = (−1)µ d{µ, θ}P ∧ dν

(4.48)

{µ, dθ}P = (−1)µ d{µ, θ}P = (−1)µ {dµ, θ}P .

(4.49)

and hence

(cf. (4.33)), since dim(M ) > 1 for non-trivial P . Now, putting θ := θ ∧ ν in (4.49) and using the Leibniz rule (4.46), we get (−1)µν dθ ∧ {µ, ν}P + (−1)θ+µ {µ, θ}P ∧ dν = 0 .

(4.50)

Hence, dθ ∧ {µ, dν}P = 0 and, consequently, {µ, dν}P = d{µ, ν}P = 0 for all µ and ν being functions. This implies P = 0 . It is interesting that we have not only the homomorphism (4.40) but also an embedding of the Fr¨ olicher–Nijenhuis bracket into the extended canonical Poisson bracket on the corresponding cotangent bundle. Namely, let πM : T ∗ M −→ M be the cotangent bundle over M and let PM be the canonical Poisson structure on T ∗ M associated with the canonical symplectic form ωM . Recall that we have the homomorphism of Lie algebras (2.10). Theorem 8. The mapping J ∗ : (Ω(M ; T M ), [ , ]F −N ) −→ (Ω(T ∗ M ), { , }PM ) ,

(4.51)

Z-GRADED EXTENSIONS OF POISSON BRACKETS

19

def

∗ given on simple tensors by J ∗ (µ⊗X) = ι(X)πM (µ), is an injective homomorphism of graded Lie algebras. Composing J ∗ with the mappings HPM and GPM (cf. 4.40 and 4.42), we get the injective homomorphisms of graded Lie algebras:

H : (Ω(M ; T M ), [ , ]F −N ) −→ (Ω(T ∗ M ; T T ∗M ), [ , ]F −N ) , def,

H = HPM ◦ J ∗ ,

(4.52)

and G : (Ω(M ; T M ), [ , ]F −N ) −→ (A(T ∗ M ), [ , ]S−N ) , G = GPM ◦ J ∗ . def

(4.53)

For the proof, depending on obvious direct calculations with the use of (4.34), we refer to [15]. Let us only remark that, due to the embeddings (4.51) and (2.10), we can regard (Ω(T ∗ M ), { , }PM ) as a common generalization of the Fr¨olicher– Nijenhuis and the symmetric Schouten brackets. Also (Ω(T ∗ M ; T T ∗M ), [ , ]F −N ), in the presence of (4.52), may be viewed as such a common generalization, which was first observed by Dubois–Violette and Michor [10]. On the other hand, the embedding (4.53) implies that one can regard the Fr¨ olicher–Nijenhuis algebra (over M ) as a subalgebra of the Schouten algebra (over T ∗ M ), so that (4.52) may serve as a definition of [ , ]F −N . Let us finish this section with results showing that our extension of the Poisson bracket behaves well also with respect to the tangent lifts. The tangent lifts (vertical vT and complete dT ) of tensors over a manifold M to tensors over its tangent bundle τM : T M −→ M have been considered, for example, in [38, 14, 15] (for definitions and basic facts we refer to [14]). In particular, ∗ vT = τM : Ω(M ) −→ Ω(T M ) def

(4.54)

is a homomorphism of exterior algebras and the complete lift dT : Ω(M ) −→ Ω(T M ) is a vT -derivation of degree 0: dT (µ ∧ ν) = dT (µ) ∧ vT (ν) + vT (µ) ∧ dT (ν) ,

(4.55)

commuting with the exterior derivative dT (dµ) = ddT (µ) .

(4.56)

The identity (4.55) is also true for multivector fields and, for X ∈ X (M ) and a differential form µ, dT (i (X)µ) = i (dT (X))(dT (µ)) , vT (i (X)µ) = i (vT (X))(dT (µ)) (see [14]).

(4.57)

20

J. GRABOWSKI

The complete tangent lift preserves the Schouten–Nijenhuis bracket: [dT (X), dT (Y )]S−N = dT ([X, Y ]S−N ) ,

(4.58)

so the complete lift dT (P ) of a Poisson tensor P on M is a Poisson tensor on T M (see [14, 6, 7]). Theorem 9. (1) The complete lift of differential forms on a Poisson manifold (M, P ) preserves the graded extensions of Poisson brackets: dT ({µ, ν}P ) = {dT (µ), dT (ν)}dT (P ) ,

(4.59)

where dT (P ) is the Poisson tensor on T M being the complete lift of P . (2) The following cotangent Poisson lift J P = J ∗ ◦ HP : (Ω(M ), { , }P ) −→ (Ω(T ∗ M ), { , }PM ) , def

(4.60)

where PM is the canonical Poisson tensor on the cotangent bundle T ∗ M , is a homomorphism of graded Lie algebras: J P ({µ, ν}P ) = {J P (µ), J P (ν)}PM .

(4.61)

Proof. According to (4.19) and the fact that the complete lift of forms commutes with the exterior derivative, it suffices to show that

For P =

P

dT (hµ, νiP ) = hdT (µ), dT (ν)idT (P ) .

(4.62)

⊗ Xk , we have (cf. (4.57)) X cjk (dT (Xj ) ⊗ vT (Xk ) + vT (Xj ) ⊗ dT (Xk )) dT (P ) =

(4.63)

j,k cjk Xj

j,k

and

X cjk i (Xj )µ ∧ i (Xk )ν) dT (hµ, νiP ) = dT ( =

X

j,k

cjk (dT (i (Xj )µ) ∧ vT (i (Xk )ν)

j,k

+ vT (i (Xj )µ) ∧ dT (i (Xk )ν)) X cjk (i (dT (Xj ))dT (µ) ∧ i (vT (Xk ))dT (ν) = j,k

+ i (vT (Xj ))dT (µ) ∧ i (dT (Xk ))dT (ν)) = hdT (µ), dT (ν)idT (P ) , where we used (4.55) and (4.57). The second part is trivial, since both HP and J ∗ are homomorphisms of graded Lie algebras (see 4.41 and Theorem 8).

21

Z-GRADED EXTENSIONS OF POISSON BRACKETS

5. Extensions of Poisson Brackets to Multivector Fields We know already from Theorem 1 that the mapping ΛP : Ω(M ) −→ A(M ) is a homomorphism of the Koszul–Schouten into the Schouten–Nijenhuis bracket. Moreover, if the Poisson tensor P is nondegenerate and ω = P −1 is the associated symplectic form, then ΛP (ω) = P and ΛP (i(P )µ) = i(ω)ΛP (µ).

(5.1)

Using the Koszul–Schouten bracket on sections of the cotangent bundle T ∗ M, we can develop a contravariant version of differential calculus exchanging the role of forms and multivector fields (see [30, 15, 26, 37]). For example, the ‘exterior derivative’ dP : An (M ) −→ An+1 (M ) is given by the Cartan formula X (−1)k+1 Pµk (X(µ1 , . . . , µ ck , . . . , µn+1 )) dP X(µ1 , . . . , µn+1 ) = k

+

X

(−1)k+l X([µk , µl ]P , µ1 , . . . , µ ck , . . . , µbl , . . . , µn+1 ).

k
It is well known that dP (X) = [P, X]S−N and ΛP (dµ) = −dP (ΛP (µ)).

(5.2)

The Ω(M )-module Ω(M ; T M ) of vector valued form is replaced by the A(M )module A(M ; T ∗ M ) of 1-form valued miltivector fields (or multivector valued 1-form) and we can define, instead of insertions i(K)µ with K ∈ Ω(M ; T M ), µ ∈ Ω(M ), insertions i(κ)X for κ ∈ A(M ; T ∗ M ) and X ∈ A(M ) in the natural way. We have even an analog of the Fr¨ olicher–Nijenhuis bracket on A(M ; T ∗ M ) (see [15]). Using the Poisson tensor P, we can define a mapping λP : Ω(M ) −→ A(M ; T ∗ M ),

(5.3)

def

putting λP (f ) = 0 for f ∈ C ∞ (M ) and X def d (−1)k+1 Pµ1 ∧ . . . ∧ P λP (µ1 ∧ . . . ∧ µm ) = µk ∧ . . . ∧ Pµm ⊗ µk

(5.4)

k

for µk ∈ Ω1 (M ). It is easy to see that λP is a ΛP -derivation of degree −1: λP (µ ∧ ν) = λP (µ) ∧ ΛP (ν) + (−1)µ ΛP (µ) ∧ λP (ν)

(5.5)

(cf. (3.20)). Fixing P, we will write usually λµ instead of λP (µ) and Λµ instead of ΛP (µ). Theorem 10. ΛP ({µ, ν}P ) = i(λdµ )dP (Λν ) + (−1)µ dP (i (λdµ )Λµ ) − dP (i (λµ )dP (Λν )).

(5.6)

22

J. GRABOWSKI

In particular, Ker(ΛP ) is a Lie ideal of (Ω(M ), { , }P ). Proof. First, let us see that Λi(Pµ )ν = −i(λµ )Λν .

(5.7)

Indeed, both sides of (5.7) vanish for µ or ν being a function, so let us assume that µ = µ1 ∧ . . . ∧ µm , ν = ν1 ∧ . . . ∧ νn , with µk , νl ∈ Ω1 (M ). Then, Λi(Pµ )ν X d c = (−1)k+l hP, µk ∧ νl iPµ1 ∧ . . . P µk . . . ∧ Pµ1 ∧ Pν1 ∧ . . . Pνl . . . ∧ Pνn k,l

X d − i ( (−1)k+1 Pµ1 ∧ . . . P µk . . . ∧ Pµm ⊗ µk )Pν1 ∧ · · · ∧ Pνn = −i (λµ )Λν . k

Now, the theorem is a direct consequence of (4.19) and (5.2). def

On the image AP (M ) = ΛP (Ω(M )) of ΛP , which is the exterior algebra generated over C ∞ (M ) by Hamiltonian vector fields and a graded Lie subalgebra of (A(M ), [ , ]S−N ), let us define the Poisson bracket { , }P , putting def

{Λµ , Λν }P = Λ{µ,ν}P .

(5.8)

On functions we get the original Poisson bracket; on hamiltonian vector fields we get 0, but it is not clear in general that the bracket (5.8) is well defined. Theorem 11. The bracket on AP (M ), given by (5.8), is well defined and we have the formula {f0 Hf1 ∧ . . . ∧ Hfn , g0 Hg1 ∧ . . . ∧ Hgm }P = {f0 , g0 }P Hf1 ∧ . . . ∧ Hfn ∧ Hg1 ∧ . . . ∧ Hgm X d c (−1)k+l {fk , gl }P Hf0 ∧ . . . H − fk . . . ∧ Hfn ∧ Hg0 ∧ . . . Hgl . . . ∧ Hgm

(5.9)

k,l>0

− f0

X

d c (−1)k+l [Hfk , Hgl ] ∧ Hf1 ∧ . . . H fk . . . ∧ Hfn ∧ Hg0 ∧ . . . Hgl . . . ∧ Hgm

k>0,l

− g0

X

d c (−1)n+k+l [Hfk , Hgl ] ∧ Hf0 ∧ . . . H fk . . . ∧ Hfn ∧ Hg1 ∧ . . . Hgl . . . ∧ Hgm ,

l>0,k

where fk , gl ∈ C ∞ (M ) and Hfk , etc., are the corresponding Hamiltonian vector fields. Moreover, the map − dP : (AP (M ), { , }P ) → (A(M ), [ , ]S−N ), − dP (X) = −[P, X]S−N is a homomorphism of graded Lie algebras.

(5.10)

Z-GRADED EXTENSIONS OF POISSON BRACKETS

23

If P is nondegenerate, then AP (M ) = A(M ) and the bracket (5.8), defined on all multivector fields, is an ‘integral’ of the Schouten–Nijenhuis bracket and can be written in terms of the associated symplectic form ω = P −1 as follows: {X, Y }P = hdP (X), dP (Y )iω + (−1)x dP hdP (X), Y iω + dP hX, dP (Y )iω ,

(5.11)

where def

hX, Y iω = (−1)x+1 (i(ω)(X ∧ Y ) − i(ω)X ∧ Y − X ∧ i(ω)Y ).

(5.12)

Proof. It follows from (5.6) that Λ µ = 0 implies Λ{µ,ν}P = 0, so the bracket (5.8) is well defined. We get now (5.9) from (4.34), since Λf0 df1 ∧...∧dfn = f0 Hf1 ∧ . . . ∧ Hfn and H{f,g}P = [Hf , Hg ]. The last part is a direct consequence of (4.19), (5.1) and (5.2).

(5.13)

6. Generalizations All the above can be done in a more general setting. We just replace the tangent bundle τM : T M → M , furnished with the Lie bracket [ , ] on its sections (vector fields), with an arbitrary Lie algebroid over M . Let us recall that a Lie algebroid (see [34] and [29]) over a manifold M is a triple (τ, [ , ]τ , aτ ), where τ : E → M is a vector bundle over M, [ , ]τ is a Lie bracket on sections Γ(E) of τ , and aτ : E → T M is a vector bundle morphism (called the anchor map), such that (1) The anchor map induces on sections a Lie algebra homomorphism: aτ ([X, Y ]τ ) = [aτ (X), aτ (Y )].

(6.1)

(2) For any f ∈ C ∞ (M ) and X, Y ∈ Γ(E), we have the following Leibniz rule [X, f Y ]τ = f [X, Y ]τ + aτ (X)(f )Y.

(6.2)

The tangent bundle itself, with τ = τM , aτ = idT M and the usual Lie bracket on Γ(T M ) = X (M ) is a canonical Lie algebroid. Another significant example is a Lie algebroid structure on the cotangent bundle πM : T ∗ M → M over a Poisson manifold, with the Lie bracket on sections of T ∗ M (1-forms) given by the bracket [ , ]P (cf. (3.9)) and the anchor map P# (cf. (3.5)). Given a Lie algebroid, we can generalize the standard calculus of differential forms and vector fields. We replace the exterior algebras of multivector fields and differential forms by the exterior algebras M def def (6.3) Φk (τ ), Φk (τ ) = Γ(Λk E), Φ(τ ) = k∈Z

24

J. GRABOWSKI

and def

Φ(π) =

M

Φk (π), Φk (π) = Γ(Λk E ∗ ), def

(6.4)

k∈Z

associated with the Lie algebroid bundle τ : E −→ M and its dual π : E ∗ −→ M. The exterior derivative dτ : Φk (π) −→ Φk+1 (π) is defined by the Cartan formula (cf. (5.2)) X ck , . . . , Xn+1 )) (−1)k+1 aτ (Xk )(µ(X1 , . . . , X dτ µ(X1 , . . . , Xn+1 ) = X

k

ck , . . . , X cl , . . . , Xn+1 ) . (−1)k+l µ([Xk , Xl ]τ , X1 , . . . , X

(6.5)

k
We have d2τ = 0 and, for X ∈ Γ(E), one defines the insertion operator i(X) : def

Φn (π) −→ Φn−1 (π) and the Lie derivative L(X) = i(X) ◦ dτ + dτ ◦ i(X) in the obvious way. All the standard formulae, like dτ (µ ∧ ν) = dτ (µ) ∧ ν + (−1)µ µ ∧ dτ (ν),

(6.6)

L(X) ◦ L(Y ) − L(Y ) ◦ L(X) = L([X, Y ]τ ),

(6.7)

or

hold and, quite parallel to the standard definitions, we can define the (generalized) Schouten–Nijenhuis, Nijenhuis–Richardson, and Fr¨ olicher–Nijenhuis brackets (see [15]). The Schouten–Nijenhuis bracket [ , ]S−N is defined on Φ(τ ) and satisfies the −R and the formulae analogous to (2.6) and (2.7). The Nijenhuis–Richardson [ , ]N τ F −N brackets are defined on the space Fr¨ olicher–Nijenhuis [ , ]τ def

Φ1 (π) =

M

Φk1 (π), Φk1 (π) = Γ(Λk E ∗ ⊗ E), def

(6.8)

k∈Z

with formally the same definitions as (2.16) and (2.18) in the classical case. One important difference in the general case is the fact that the space Φk (π) of ‘kforms’ is usually not generated, as a C ∞ (M )-module, by exact ‘forms’, as in the case Φk (π) = Ω(M ). This is possible, since the exterior derivative dτ depends on the anchor aτ and the Lie bracket [ , ]τ which may be quite degenerated. It makes inductive proofs harder and it is also the reason to why the definition of the generalized Fr¨ olicher–Nijenhuis bracket via the identity (2.20) is not possible: L(K) : Φ(π) −→ Φ(π) may be trivial for a non-trivial K ∈ Φ1 (π). For the Lie algebroid structure on the cotangent bundle πM : T ∗ M −→ M over a Poisson manifold (M, P ), the exterior derivative, we get, is exactly dP = [P, · ]S−N , as was proved independently in various contexts (see [2, 26] and [18]), and the generalized Schouten–Nijenhuis bracket is, in this case, the Koszul–Schouten bracket (3.11) ([21, 26, 24, 22, 23]).

Z-GRADED EXTENSIONS OF POISSON BRACKETS

25

In general, a Poisson tensor for a Lie algebroid (τ, [ , ]τ , aτ ) is a tensor P ∈ = 0. The formula (3.8) defines then a Lie algebroid Φ2 (τ ) satisfying [P, P ]S−N τ structure on the dual bundle π : E ∗ −→ M, with the anchor aπ = aτ ◦ P# (cf. [26] and [30]). Hence, we have Lie algebroid structures on both: the original bundle τ : E −→ M and the dual bundle π : E ∗ −→ M which form a nice structure of a triangular Lie bialgebroid in the sense of Mackenzie and Xu [30]. As in the classical case, a (generalized) Poisson tensor defines analogously to (3.2) a Poisson bracket { , }P on functions on M. We can also define the mapping P# : Φ(π) −→ Φ1 (π) and the generalized Hamiltonian map HP : Φ(π) −→ Φ1 (π) similarly to (3.19) and (3.21) and to define an extension of { , }P to a graded Lie bracket on Φ(π). Most of the results of Secs. 4 and 5 can be obtained mutatis mutandis, since the proofs can be almost immediately adapted to the general case. Let us summarize some of these results in the following. Theorem 12. Given a Poisson tensor P for a Lie algebroid (τ, [ , ]τ , aτ ), the formulae def

[µ, ν]P = i(Hµ )ν − (−1)µ L(Pµ )ν

(6.9)

and def

{µ, ν}P = L(Hµ )ν + dτ L(Pµ )ν

(6.10)

define graded Lie algebra structures on Φ(π), with elements of Φk (π) being of degree (k − 1) with respect to the bracket [ , ]P , and of degree k with respect to { , }P . The maps: ), ΛP : (Φ(π), [ , ]P ) −→ (Φ(τ ), [ , ]S−N τ dτ : (Φ(π), { , }P ) −→ (Φ(π), [ , ]P ), H : (Φ(π), { , }P ) −→ (Φ1 (π), [ , P

−N ]F ), τ

(6.11) (6.12) (6.13)

are homomorphisms of graded Lie algebras and we have the following exact sequence of homomorphisms of graded Lie algebras d

τ (Z(π), [ , ]P ) −→ H(π) −→ 0, 0 −→ Z(π) −→ (Φ(π), { , }P ) −→

(6.14)

where the dτ -closed elements Z(π) of Φ(π) and the space of dτ -cohomology H(π) are taken with the trivial brackets. Moreover, the equation def

{ΛP (µ), ΛP (ν)}P = ΛP ({µ, ν}P )

(6.15)

defines properly a graded Lie bracket on the graded Lie subalgebra def

ΦP (τ ) = ΛP (Φ(π))

(6.16)

). For this bracket, the mapping X 7→ −[P, X]S−N is a homomorof (Φ(τ ), [ , ]S−N τ τ . phism into the generalized Schouten–Nijenhuis bracket [ , ]S−N τ

26

J. GRABOWSKI

References [1] J. V. Beltr´ an and J. Monterde, “Graded Poisson structures on the al-gebra of differential forms”, Comment. Math. Helv. 70 (1995) 383–402. [2] K. H. Bhaskara and K. Viswanath, “Calculus on Poisson manifolds”, Bull. London Math. Soc. 20 (1988) 68–72. [3] K. H. Bhaskara and K. Viswanath, Poisson Algebras and Poisson Manifolds, Pitman Research Notes in Math. 174, Longman Sci., 1988. [4] A. Coste, P. Dazord and A. Weinstein, “Groupo¨ıdes symplectiques”, Publ. du D´epartement de Math´ematiques de de l’Universit´e Clause–Bernard de Lyon I (1987) 1–65. [5] L. Corwin, Yu. Neeman and S. Sternberg, “Graded Lie algebras in mathematics and physics”, Rev. Mod. Phys. 47 (1975) 573–603. [6] T. J. Courant, “Dirac manifolds”, Trans. Amer. Math. Soc. 319 (1990) 631–661. [7] T. J. Courant, “Tangent Dirac structures”, J. Phys. A23 (1990) 5153–5160. [8] A. Cabras and A. M. Vinogradov, “Extensions of the Poisson bracket to differential forms and multi-vector fields”, J. Geom. Phys. 9 (1992) 75–100. [9] V. G. Drinfel’d, “Quantum groups”, Proc. ICM, Berkeley, Amer. Math. Soc., 1986, pp. 789–829 [10] M. Dubois-Violette and P. W. Michor, “A common generalization of the Fr¨ olicherNijenhuis bracket and the Schouten bracket for symmetric multivector fields”, Indag. Math. N. S. 6 (1995) 51–66. [11] B. Fuchssteiner, “The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems”, Prog. Theor. Phys. 68 (1982) 1082–1104. [12] A. Fr¨ olicher and A. Nijenhuis, “Theory of vector-valued differential forms, Part 1”, Indag. Math. 18 (1956) 338–359. [13] I. M. Gelfand and I. Ya. Dorfman, “Hamiltonian operators and the classical Yang– Baxter equation”, Funct. Anal. Appl. 16 (4) (1982) 241–248. [14] J. Grabowski and P. Urba´ nski, “Tangent lifts of Poisson and related structures”, J. Phys. A28 (1995) 6743–6777. [15] J. Grabowski and P. Urba´ nski, “Tangent and cotangent lifts and graded Lie algebras asssociated with Lie algebroids” (to appear). [16] M. Gerstenhaber, “The cohomology structure of an associative ring”, Ann. Math. 78 (1963) 267–288 [17] E. Getzler, “Batalin–Vilkovisky algebras and two-dimensional topological field theories”, Comm. Math. Phys. 159 (1994) 265–285. [18] J. Huebschmann, “Poisson cohomology and quantization”, J. Reine Angew. Math. 408 (1990) 55–113. [19] I. Kol´ aˇr, P. W. Michor and J. Slov´ ak, Natural Operations in Differential Geometry, Springer Verlag, 1993. [20] M. V. Karasev, “Analogues of the objects of Lie group theory for nonlinear Poisson brackets”, Mat. USSR - Izv. 28 (1987) 497–527. [21] J. -L. Koszul, “Crochet de Schouten–Nijenhuis et cohomologie”, Elie Cartan et les math´ematiques d’aujourd’hui, Ast´ erisque hors s´erie 1985, pp. 257–271. [22] I. Krasilshchik, “Schouten bracket and canonical algebra, II”, Global Analysis, Lect. Notes in Math. 1334, 1988, pp. 79–100. [23] I. Krasilshchik, “Supercanonical algebras and Schouten brackets”, Mathematical Notes 49 (1991) 70–76. [24] Y. Kosmann–Schwarzbach, “Exact Gerstenhaber algebras and Lie bialgebroids”, Acta Applicandae Math. 41 (1995) 153–165. [25] Y. Kosmann–Schwarzbach, “From Poisson algebras to Gerstenhaber algebras”, (preprint 1995). [26] Y. Kosmann-Schwarzbach and F. Magri, “Poisson-Nijenhuis structures”, Ann. Inst. Henri Poincar´e, Phys. Th´eor. 53 (1990) 35–81.

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27

[27] B. H. Lian and G. J. Zuckerman, “New perspectives on the BRST-algebraic structure in string theory”, Comm. Math. Phys. 154 (1993) 613–646. [28] F. Magri and C. Morosi, “A geometrical characterization of integrable hamiltonian systems through the theory of Poisson-Nijenhuis manifolds”, Universit` a degli studi di Milano 19 (1984). [29] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lecture Notes Ser. 124, Cambridge Univ. Press, 1987. [30] K. Mackenzie and P. Xu, “Lie bialgebroids and Poisson grupoids”, Duke Math. J. 73 (1994) 415–452. [31] P. W. Michor, “A generalization of Hamiltonian mechanics”, J. Geom. Phys. 2 (1985) 67–83. [32] P. W. Michor, “Remarks on the Schouten–Nijenhuis bracket”, Suppl. Rend. Circ. Mat. di Palermo, Serie II 16 (1987) 207–215. [33] A. Nijenhuis and R. Richardson, “Deformations of Lie algebra structures”, J. Math. Mech. 171 (1967) 89–106. [34] J. Pradines, Fibr´es vectoriels doubles et calcul des jets non holonomes, Notes polycopi´ees, Amiens, 1974. [35] C. Roger, “Alg´ebres de Lie gradu´ees et quantification”, Symplectic Geometry and Mathematical Physics, ed. P. Donato et al., Progress in Math. 99, Birkh¨ auser, 1991. ¨ [36] J. A. Schouten, “Uber Differentialkonkomitanten zweier kontravarianter Gr¨ oßen”, Indag. Math. 2 (1940) 449–452. [37] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Math. 118 Birkh¨ auser, 1994. [38] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc., 1973.

POISSON SPACES WITH A TRANSITION PROBABILITY N. P. LANDSMAN∗ Department of Applied Mathematics and Theoretical Physics University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Received 11 March 1996 Revised 21 June 1996 The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ, σ) = δρσ , unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck’s constant). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math. 144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys. 82 (1982) 497–509), we give axioms guaranteeing that P is the space of pure states of a unital C ∗ -algebra. We give an explicit construction of this algebra from P.

1. Introduction Section 1.1 motivates the axiomatic study of state spaces (rather than operator algebras) in the foundations of quantum mechanics. In 1.2 we review the work of Alfsen et al. on the structure of state spaces of C ∗ -algebras. In 1.3 we discuss the concept of a transition probability space, and in 1.4 it is shown how the pure state space of a C ∗ -algebra is an example of such a space. Section 1.5 recalls the concept of a Poisson manifold, and introduces (uniform) Poisson spaces generalizing this concept. Poisson structures may be intertwined with transition probabilities, leading to the notion of unitarity, and to the central idea of this paper, a Poisson space with a transition probability. In Sec. 2 we introduce our axioms on pure state spaces, and formulate the theorem relating these axioms to pure state spaces of C ∗ -algebras. Section 3 outlines the proof of this theorem, which essentially consists of the reconstruction of a C ∗ algebra from its pure state space, endowed with the structure of a uniform Poisson space with a transition probability. This reconstruction is of interest in its own right. Some longer proofs and other technical comments appear in Sec. 4. ∗

E.P.S.R.C. Advanced Research Fellow 29

Review of Mathematical Physics, Vol. 9, No. 1 (1997) 29–57 c

World Scientific Publishing Company

30

N. P. LANDSMAN

In this paper functions and functionals are real-valued, unless explicitly indicated otherwise. Hence C(X) stands for C(X, R), etc. Similarly, vector spaces (including the various algebras appearing in this paper) are generally over R, unless there is an explicit label C denoting complexification. An exception to this rule is that we use the standard symbols H for a complex Hilbert space, and B(H) (K(H)) for the set of all bounded (compact) operators on H. The self-adjoint part of a C ∗ -algebra AC is denoted by A; we denote the state space of AC by S(A) or S(AC ), and its pure state space by P(A) or P(AC ). Here the ‘pure state space’ is the space of all pure states, rather than its w∗ -closure. 1.1. Algebraic aspects of mechanics At face value, quantum mechanics (Hilbert space, linear operators) looks completely different from classical mechanics (symplectic manifolds, smooth functions). The structure of their respective algebras of observables, however, is strikingly similar. In quantum mechanics, one may assume [46, 22] that the observables A form the self-adjoint part of some C ∗ -algebra AC . The associative product does not map A into itself, but the anti-commutator A ◦ B = 12 (AB + BA) and the (scaled) commutator [A, B]~ = i(AB − BA)/~ do; in conjunction, they give A the structure of a so-called Jordan–Lie algebra [26, 22]. This is a vector space V equipped with two bilinear maps ◦ and [ , ] : V × V → V , such that ◦ is symmetric, [ , ] is a Lie bracket (i.e., it is anti-symmetric and satisfies the Jacobi identity), and the Leibniz property [A, B ◦ C] = [A, B] ◦ C + B ◦ [A, C]

(1.1)

holds; in other words, the commutator is a derivation of the Jordan product. Moreover, one requires the associator identity (A ◦ B) ◦ C − A ◦ (B ◦ C) = k[[A, C], B]

(1.2)

for some k ∈ R. This implies the Jordan identity A2 ◦ (A ◦ B) = A ◦ (A2 ◦ B) (where A2 = A ◦ A), which makes (V, ◦) a Jordan algebra [22, 28]); accordingly, the symmetric product ◦ is referred to as the Jordan product. Note that for V = A and [A, B] = [A, B]~ one has k = ~2 /4. Conversely, a Jordan–Lie algebra A for which k > 0 (cf. [22] for comments on the case k < 0), and which in addition is a so-called JB-algebra, is the self-adjoint part of a C ∗ -algebra AC . Here a JB-algebra [9, 28] is a Jordan algebra which is a Banach space, and satisfies k A ◦ B k ≤ k A k k B k, k A2 k=k A k2 , and k A2 k ≤ k A2 + B 2 k for all A, B ∈ A; the first axiom can actually be derived from the other two; alternatively, 2 k. the last two axioms may be replaced by k A k2 ≤ k A2 + B√ ∗ The associative C -product is given by A·B = A◦B −i k[A, B] (the · is usually omitted); the associativity follows from the Leibniz property, (1.2), and the Jacobi identity. For the construction of the norm and the verification of the axioms for a C ∗ -algebra, see [58, 47] and Sec. 3.8 below.

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In classical mechanics, one takes the Jordan–Lie algebra to consist of all smooth functions on the phase space, equipped with the operations of pointwise multiplication f ◦ g = f g and Poisson bracket [f, g] = {f, g} (the latter coming from a symplectic structure, or from a more general abstract Poisson structure [55, 39]). The identity (1.2) is then satisfied with k = 0. A Jordan–Lie algebra for which k = 0 in (1.2) is called a Poisson algebra. Thus from an algebraic point of view the only difference between classical and quantum mechanics is that in the former the Jordan product ◦ is associative, whereas in the latter the more general identity (1.2) is satisfied for some k > 0. From an axiomatic point of view, it is rather difficult to justify (1.2), and it is hard to swallow that the non-associativity of ◦ should be the defining property of quantum mechanics. Historically, the commutator hardly played a role in algebraic quantum axiomatics, all attention being focused on the Jordan structure [43, 49, 9, 28, 22]. Whereas the Jordan identity may be justified by the need to have a spectral theory, the step from the Jordan- to the full C ∗ -structure has had to be justified algebraically by an appeal to the need to combine different physical systems using a well-behaved tensor product [11, 27]. This gives the commutator a different status from its classical counterpart (viz. the Poisson bracket), which describes the way observables lead to flows (i.e., dynamics). 1.2. State spaces and the work of Alfsen, Shultz, and Hanche-Olsen A transparent way of analyzing and justifying algebras of observables is the study of their state spaces. A state on a JB-algebra A is defined as a linear functional ω on A satisfying ω(A2 ) ≥ 0 for all A ∈ A and kωk = 1; in case that A has an identity I this implies that ω(I) = 1. The idea is that the algebraic structure of A is encoded in certain (geometric) properties of its state space S(A), so that A may be reconstructed from S(A), equipped with these properties. The most basic property of S(A) is that it is a convex set, which is compact in the w∗ -topology if A is a JB-algebra with unit. The description of quantum mechanics in terms of general compact convex state spaces is closely tied to the so-called operational approach, and is invariably interpreted in terms of laboratory procedures such as filtering measurements [48, 40, 41, 42, 37, 14, 35]. For C ∗ -algebras (which are special instances of complexified JB-algebras) this type of study culminated in [5], where axioms were given which guarantee that a given compact convex set K (assumed to be embedded in a locally convex Hausdorff vector space) is the state space of a C ∗ -algebra with unit (also cf. [4, 12, 8]). In order to motivate our own approach, we need to explain these axioms to some extent. Firstly, a face F is defined as a convex subset of K with the property that ρ and σ are in F if λσ + (1 − λ)ρ ∈ F for some λ ∈ (0, 1). A face F is called norm-exposed [7] if it equals F = {ρ ∈ K|hf, ρi = 0} for some f ∈ A+ b (K). Here (K) its subspace Ab (K) is the space of all bounded affine functions on K, and A+ b of positive functions. A(K) will stand for the space of continuous affine functions on K [6, 12].

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A face F is said to be projective [6] if there exists another face F ] such that F and F ] are norm-exposed and affinely independent [3], and there exists a map (a socalled affine retraction) : K → K with image the convex sum of F and F ] , leaving its image pointwise invariant, and having the technical property of transversality (cf. [6, 3.8] or [4]) (alternative definitions are possible [6]). The first axiom of [5] is Axiom AHS1. Every norm-exposed face of K is projective. A face consisting of one point is called a pure state, and the collection of pure states forms the so-called extreme boundary ∂e K of K. The smallest face containing a subset S ⊂ K is denoted by F (S), and we write F (ρ, σ) for F ({ρ, σ}). Two pure states ρ, σ are called inequivalent if F (ρ, σ) is the line segment {λσ + (1 − λ)ρ | λ ∈ [0, 1]}. Otherwise, they are called equivalent. The second axiom is Axiom AHS2. If pure states ρ and σ 6= ρ are equivalent, then F (ρ, σ) is normexposed and affinely isomorphic to the state space of the C ∗ -algebra M2 (C) of 2 × 2 matrices over C. Moreover, each pure state is norm-exposed. The state space S(M2 (C)) is affinely isomorphic to the unit ball B 3 in R3 . Concretely, we identify a state on M2 (C) with a density matrix on C2 , which may be parametrized as 1 1 + x y + iz (1.3) 2 y − iz 1 − x where x, y, z ∈ R. The positivity of this matrix then corresponds to the constraint x2 + y 2 + z 2 ≤ 1 (see [5]). From the point of view of quantum logic (cf. e.g. [54, 14, 31]), Axiom AHS1 allows one to define an orthomodular lattice, whose elements are the projective faces of K [6, §4]. Axiom AHS2 not only allows one to prove that this lattice has the covering property [8, 6.15], but also eventually implies that the co-ordinatizing field of the lattice is C (cf. Sec. 4.1). In the finite-dimensional case Axioms AHS1 and AHS2 are sufficient to construct a C ∗ -algebra AC whose state space is K; as a Banach space A = A(K) with the sup-norm. To cover the general case, more axioms are needed. Axiom AHS3. The σ-convex hull of ∂e K is a split face of K. P Here the σ-convex hull in question consists of all sums i λi ρi , where ρi ∈ P ∂e K, λi ∈ [0, 1], i λi = 1, and the sum converging in the norm topology (regarding K as a subset of the dual of the Banach space A(K)). A face F of K is split if there exists another face F 0 such that K = F ⊕c F 0 (direct convex sum). Let C ⊂ ∂e K consist of all pure states in a given equivalence class, and let F (C) be the σ-convex hull of C (this coincides with the smallest split face containing any member of C). Then Ab (F (C))C can be made into a von Neumann algebra (with predual F (C)C ) on the basis of axioms 1–3 [8, §6], [5, §6]. Axiom AHS3 is used to show that this is an atomic (type I) factor, i.e., B(HC ) for some Hilbert space HC . The remaining axioms serve to combine all the A(F (C)) into A(K) in such a way that one obtains the self-adjoint part of a C ∗ -algebra. The Jordan product A ◦ B (or, equivalently, A2 ) is constructed using the non-commutative spectral

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theory defined by K [6, 7]. This product then coincides with the anti-commutator in Ab (F (C)) ' B(HC ). In principle this could map A ∈ A(K) into A2 ∈ Ab (K) (that is, not necessarily in A(K)). Hence Axiom AHS4. If A ∈ A(K) then A2 ∈ A(K). This is not the formulation of the axiom given in [8, 5], but by [6, 9.6], [8, 7.2] it is immediately equivalent to the version in the literature. Finally, the commutator, already defined on each A(F (C)), needs to be well-defined on all of A(K). This is guaranteed by Axiom AHS5. K is orientable. Roughly speaking, this means that one cannot transport a given face F (ρ, σ) ' B 3 (cf. Axiom AHS2) in a continuous way around a closed loop so that it changes its orientation (cf. [5, §7] for more detail; also Sec. 4.3 below). It is remarkable that A(K) is automatically closed under the commutator, given the axioms. It is proved in [5] that a compact convex set is the state space of a unital C ∗ -algebra iff Axioms AHS1–AHS5 are satisfied. Even if one is happy describing quantum mechanics with superselection rules in terms of C ∗ -algebras, from a physical perspective one should not necessarily regard the above axioms as unique, or as the best ones possible. The notion of a projective face (or, equivalently, a P -projection [6]) is a complicated one (but cf. [11] for a certain simplification in the finite-dimensional case, and [35] for an analogous interpretation in terms of filters in the general case). One would like to replace the concept of orientability by some statement of physical appeal. Most importantly, the comparison of classical and quantum mechanics seems facilitated if one could start from the space of pure states ∂e K as the basic object. Moreover, from an ontological rather than an epistemological point of view one would prefer a formulation in terms of pure states as well, and the same comment applies if one is interested in an individual (as opposed to a statistical) interpretation of quantum mechanics. 1.3. Transition probability spaces Clearly, the extreme boundary ∂e K of a given compact convex set K as a topological space does not contain enough information to reconstruct K. However, one can equip ∂e K with the additional structure of a so-called transition probability, as first indicated by Mielnik [41] (also cf. [50]). Namely, given ρ, σ ∈ ∂e K one can define p by p(ρ, σ) = inf{f (ρ)|f ∈ Ab (K), 0 ≤ f ≤ 1, f (σ) = 1} .

(1.4)

For later use, we notice that it follows that p(σ, ρ) = 1 − sup{f (σ)|f ∈ Ab (K), 0 ≤ f ≤ 1, f (ρ) = 0} .

(1.5)

For the moment we denote ∂e K by P. By construction, p : P × P → [0, 1]

(1.6)

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satisfies ρ = σ ⇒ p(ρ, σ) = 1. Moreover, we infer from (1.5) that p(ρ, σ) = 0 ⇐⇒ p(σ, ρ) = 0 .

(1.7)

If K has the property that every pure state is norm-exposed, then, as is easily verified, p(ρ, σ) = 1 ⇒ ρ = σ, so that p(ρ, σ) = 1 ⇐⇒ ρ = σ .

(1.8)

Any function p on a set P with the properties (1.6), (1.7), and (1.8) is called a transition probability, and (P, p) is accordingly called a transition probability space. (In its abstract form these concepts are due to von Neumann [44], who in addition required p to satisfy (1.9) below; also cf. [40, 59, 13, 14, 45]). A transition probability is called symmetric if p(ρ, σ) = p(σ, ρ) ∀ρ, σ ∈ P .

(1.9)

A subset S ⊂ P is called orthogonal if p(ρ, σ) = 0 for all pairs ρ 6= σ in S. A P basis B of P is an orthogonal subset for which ρ∈B p(ρ, σ) = 1 for all σ ∈ P (here the sum is defined as the supremum of all finite partial sums). A basic theorem is that all bases of a given symmetric transition probability space have the same cardinality [40]; this cardinality is the dimension of P. One imposes the requirement Every maximal orthogonal subset of P is a basis.

(1.10)

A transition probability space is called irreducible if it is not the union of two (nonempty) orthogonal subsets. A component C is a subset of P with the property that p(ρ, σ) = 0 for all ρ ∈ C and all σ ∈ P\C. Thus a transition probability space is the disjoint union of its irreducible components [13]. An irreducible component of P is called a sector. This agrees with the terminology in algebraic quantum mechanics, where P is the pure state space of a C ∗ -algebra (of observables) [46]. If one defines a topology on P through the metric d(σ, ρ) = l.u.b.{|p(ρ, τ )−p(σ, τ )|, τ ∈ P} [13], then the topological components coincide with the components just defined. However, a different topology may be defined on P, and therefore we shall use the term ‘sector’ as referring to ‘component’ in the first (probabilistic) sense. Two points lying in the same sector of P are called equivalent (and inequivalent in the opposite case). Any subset Q ⊂ P has an orthoplement Q⊥ = {σ ∈ P | p(ρ, σ) = 0 ∀ρ ∈ Q}. One always has Q ⊆ Q⊥⊥ ; a subset Q is called orthoclosed if Q = Q⊥⊥ . Any set of the type Q⊥ (hence in particular Q⊥⊥ ) is orthoclosed. In particular, one may choose P an orthogonal subset S, in which case [40, 59] S ⊥⊥ = {ρ ∈ P| σ∈S p(ρ, σ) = 1}. (Clearly, if S = B is a basis then B ⊥⊥ = P.) Not every orthoclosed subset is necessarily of this form, however there exist examples of orthoclosed subsets which do not have any basis [59, 14]. To exclude pathological cases, one therefore adds the axiom [59, 14]: If Q ⊆ P is orthoclosed then every maximal orthogonal subset of Q is a basis of Q.

(1.11)

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Definition 1. A well-behaved transition probability space is a pair (P, p) satisfying (1.6)–(1.11). Of course, (1.7) and (1.10) follow from (1.9) and (1.11), respectively. The simplest example of a well-behaved transition probability space is given by putting the ‘classical’ transition probabilities (1.12)

p(ρ, σ) = δρσ

on any set P. One can associate a certain function space A(P) with any transition probability space P. Firstly, for each ρ ∈ P define pρ ∈ `∞ (P) by pρ (σ) = p(ρ, σ).

(1.13)

Secondly, the normed vector space A00 (P), regarded as a subspace of `∞ (P) (with PN sup-norm), consists of all finite linear combinations of the type i=1 ci pρi , where ci ∈ R and ρi ∈ P. The closure of A00 (P) is called A0 (P). Thirdly, the double dual of A0 (P) will play a central role in what follows, so that we use a special symbol: A(P) = A0 (P)∗∗ .

(1.14)

Since A0 (P) ⊆ `0 (P), one has A(P) ⊆ `0 (P)∗∗ = `∞ (P). The space A(P) is the function space intrinsically related to a transition probability space P. In the case (1.12) one immediately finds A(P) = `∞ (P). (Following a seminar the author gave in G¨ ottingen, 1995, A. Uhlmann informed him that in his lectures on quantum mechanics A00 (P) had long been employed as the space of observables.) 1.4. Transition probabilities on pure state spaces Using the results in [8] (in particular, the so-called ‘pure state properties’) as well as Theorem 2.17 in [6], it is not difficult to show that the pure state space of a unital JB-algebra (where every pure state is indeed norm-exposed) is a symmetric transition probability space. If one further specializes to the pure state space P(A) of a unital C ∗ -algebra AC , from (1.4) one may derive the explicit expression 2

p(ρ, σ) = 1 − 14 kρ − σk ,

(1.15)

p(ρ, σ) = |(Ωρ , Ωσ )|2

(1.16)

which coincides with

if ρ and σ are equivalent (where Ωρ is a unit vector implementing ρ in the corresponding GNS representation, etc.), and equals 0 if they are not; cf. [25, 46, 50]. This will be proved in Sec. 4.2. The notion of equivalence between pure states used here may refer either to the one defined between Eqs. (1.10) and (1.11) in the context of transition probability

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spaces, or to the unitary equivalence of the GNS-representations defined by the states in question in the context of C ∗ -algebras; these notions coincide. In fact, P(A) has the following decomposition into sectors (see [46], which on this point relies on [25]): P(A) = ∪α PHα ,

(1.17)

where Hα is isomorphic to the irreducible GNS-respresentation space of an arbitrary state in the projective Hilbert space PHα . All states in a given subspace PHα are equivalent, and any two states lying in different such subspaces are inequivalent. We regard the self-adjoint part A of AC as a subspace of C(P(A)) (where P(A) is equipped with the w∗ -topology inherited from S(A)) through the Gel’fand transˆ form A(ρ) = ρ(A), for arbitrary A ∈ A and ρ ∈ P(A). Similarly, an operator A ∈ B(H) is identified with a function Aˆ ∈ C(PH) through the canonical inclusion PH ⊂ S(B(H)) (where PH carries the w∗ -topology relative to this inclusion). Under these identifications, for each ρ ∈ P(A) the irreducible representation πρ (A) is unitarily equivalent to the restriction of A to the sector containing ρ; every irreducible representation of A is therefore given (up to unitary equivalence) by the restriction of A to one of its sectors. In any case, one recovers the usual transition probabilities of quantum mechanics. If AC = K(H) (or MN (C) = B(CN )), the pure state space P(K(H)) is the projective Hilbert space PH (or PCN ). One may then equally well interpret Ωρ (etc.) in (1.16) as a lift of ρ ∈ PH to the unit sphere SH in H. In particular, it follows that the pure state space of a unital C ∗ -algebra is a well-behaved transition probability space. The space A(P(A)) can be explicitly identified. Let πra be the reduced atomic representation of AC [30]; recall that πra is the direct sum over irreducible respresentations πra = ⊕ρ πρ (on the Hilbert space Hra = ⊕ρ Hρ ), where one includes one representative of each equivalence class in P(A). For the weak closure one obtains πra (AC )− = ⊕ρ B(Hρ ). The Gel’fand transform maps πra (A)− into a subspace of `∞ (P(A)). It will be shown in Sec. 3.4 that this subspace is precisely A(P(A)); we write this as A(P(A)) = π ˆra (A)− .

(1.18)

The isomorphism between πra (A)− and A(P(A)) thus obtained is isometric and preserves positivity (since the Gel’fand transform does). For any well-behaved transition probability space P one can define a lattice L(P), whose elements are the orthoclosed subsets of P (including the empty set ∅, and P itself). The lattice operations are: Q ≤ R means Q ⊆ R, Q ∧ R = Q ∩ R, and Q ∨ R = (Q ∪ R)⊥⊥ . The zero element 0 is ∅. Note that the dimension of L(P) as a lattice equals the dimension [31] of P as a transition probability space. It is orthocomplemented by ⊥, and is easily shown to be a complete atomic orthomodular lattice [59, 13, 14] (cf. [31] for the general theory of orthomodular lattices). In our approach, this lattice plays a somewhat similar role to the lattice F (K) of projective faces of K (or, equivalently, of P -projections [6]; note that for C ∗ -algebras L(∂e K)

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is not necessarily isomorphic to F (K)). However, it seems to us that both the definition and the physical significance of L(P) are more direct. If P is a classical transition probability space (see 1.12) then L(P) is the distributive (Boolean) lattice of subsets of P. If P = P(A) is the pure state space of a C ∗ -algebra AC then L(P(A)) may be shown to be isomorphic (as an orthocomplemented lattice) to the lattice of all projections in the von Neumann algebra πra (AC )− . For general compact convex sets it is not clear to what extent ∂e K as a transition probability space equipped with the w∗ -topology determines K. If, however, K = S(A) is the state space of a unital C ∗ -algebra AC (with self-adjoint part A), then one can reconstruct A as a JB-algebra, and hence the state space S(A), from the pure state space P(A) as a transition probability space (with transition probabilities given by (1.15)), equipped with the w∗ -uniformity (this is the uniformity [33] U generated by sets of the form {(ρ, σ) ∈ P × P| |hρ − σ, Ai| < ε} for some ε > 0 and A ∈ A; the physical interpretation of such uniformities has been discussed by Haag, Kastler, and Ludwig, cf. [57] for a very clear discussion). The essential step in this reconstruction is the following reformulation of a result of Shultz [50] (whose formulation involved πra (AC )− rather than A(P(A))) and Brown [16]: if A is the self-adjoint part of a unital C ∗ -algebra then A = A(P(A)) ∩ Cu (P(A)) ,

(1.19)

where Cu (P(A)) is the space of uniformly continuous functions on P(A), and, as before, A has been identified with a subspace of C(P(A)) through the Gel’fand transform. Note that to recover AC as a C ∗ -algebra from the pure state space P(A), one in addition needs an orientation of P(A), see [5,50] and Sec. 4.3. For certain C ∗ -algebras (called perfect, cf. [50, 2]) one can replace Cu (P(A)) by C(P(A)) (with respect to the w∗ -topology). These include B(H) and K(H), for any Hilbert space H. 1.5. Poisson spaces with a transition probability Our goal, then, is to give axioms on a well-behaved transition probability space P which enable one to construct, by a unified procedure, a C ∗ -algebra or a Poisson algebra, which has P as its space of pure states, and reproduces the original transition probabilities. Moreover, even if one is not interested in these axioms and takes quantum mechanics (with superselection rules) at face value, the structure laid out in this paper provides a transparent reformulation of quantum mechanics, which may prove useful in the discussion of the classical limit [36]. We first have to define a number of concepts, which play a foundational role in both classical and quantum mechanics. Apart from transition probabilities, Poisson brackets play a central role in dynamical theories. Recall that a Poisson manifold [55, 39] is a manifold P with a Lie bracket { , } : C ∞ (P ) × C ∞ (P ) → C ∞ (P ), such that C ∞ (P ) equipped with this Lie bracket, and pointwise multiplication as the Jordan product ◦, is a Poisson algebra. Symplectic manifolds are special instances of Poisson manifolds; in the symplectic case the Hamiltonian vector fields span the

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tangent space at every point of P . Recall from classical mechanics [39] that any H ∈ C ∞ (P ) defines a so-called Hamiltonian vector field XH by XH f = {H, f }; the flow of XH is called a Hamiltonian flow; similarly, one speaks of a Hamiltonian curve. The most important result in the theory of Poisson manifolds states that a Poisson manifold P admits a decomposition into symplectic leaves [55, 39]. This means that there exists a family Sα of symplectic manifolds, as well as continuous injections ια : Sα → P , such that P = ∪α ια (Sα ) (disjoint union), and {f, g}(ια (σ)) = {ι∗α f, ι∗α g}α (σ) ,

(1.20)

for all α and all σ ∈ Sα . Here { , }α is the Poisson bracket associated to the symplectic structure on Sα [39], and (ι∗α f )(σ) = f (ια (σ)), etc. We will need a generalization of the notion of a Poisson manifold, which is inspired by the above decomposition. Definition 2. A Poisson space P is a Hausdorff topological space together with a linear subspace A ⊂ C(P ) and a collection Sα of symplectic manifolds, as well as continuous injections ια : Sα → P, such that: • P = ∪α ια (Sα ) (disjoint union); • A separates points; • A ⊆ CL∞ (P ), where CL∞ (P ) consists of all f ∈ C(P ) for which ι∗α f ∈ C ∞ (Sα ) for each α; • A is closed under Poisson brackets. The last requirement means, of course, that the Poisson bracket, computed from the symplectic structure on the Sα and the above decomposition of P through (1.20), maps A × A into A. In the context of Poisson spaces, each subspace ια (Sα ) of P is called a symplectic leaf of P . This terminology is sometimes applied to the Sα themselves as well. In general, this decomposition falls under neither foliation theory nor (Whitney) stratification theory (cf. [51] for this theory in a symplectic context). If the ambient space P carries additional structure, such as a uniformity, or a smooth structure, one can refine the above definition in the obvious way; such refinements will play an important role in what follows. Definition 3. A uniform Poisson space is a Poisson space P in which the topology is defined by a uniformity on P, and which satisfies Definition 2 with C(P ) replaced by Cu (P ). Here Cu (P ) is the space of uniformly continuous functions on P ; it follows that elements of CL∞ (P ) are now required to be uniformly continuous. Similarly, a smooth Poisson space is a Poisson space for which P is a manifold, and C(P ) in Definition 2 is replaced by C ∞ (P ). Hence CL∞ (P ) = C ∞ (P ). By the symplectic decomposition theorem, a smooth Poisson space with A = C ∞ (P ) is nothing but a Poisson manifold.

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In any case, CL∞ (P ) is the function space intrinsically related to a (general, uniform, or smooth) Poisson space P . The pure state space P(A) of a C ∗ -algebra AC is a uniform Poisson space in the following way. We refer to (1.17) and subsequent text. Firstly, it follows directly from the definition of the w∗ -uniformity on P(A) that ˆ A ∈ A, is in Cu (P(A)); hence A ⊂ Cu (P(A)), as required. As is well known, each A, ∗ a C -algebra separates the points of its pure state space (cf. [30]). Secondly, it is not difficult to show that the natural manifold topology on a projective Hilbert space PH coincides with the w∗ -topology it inherits from the canonical inclusion PH ⊂ S(B(H))∗ . It follows that the inclusion map of any sector PHα (equipped with the manifold topology) into P(A) (with the w∗ -topology) is continuous. Thirdly, there is a unique Poisson structure { , } on P(A) such that \ ˆ B} ˆ = i(AB {A, − BA) .

(1.21)

This Poisson bracket is defined by letting the sectors PHα of P(A) coincide with its symplectic leaves, and making each PHα into a symplectic manifold by endowing it with the (suitably normalized) Fubini–Study symplectic form [53, 38, 18, 19, 20, 39]. The reason that this structure is uniquely determined by (1.21) is that in an irreducible representation π(AC ) on a Hilbert space H the collection of differentials [ A ∈ A} is dense in the cotangent space at each point of PH. Note that the {dπ(A), precise choice of Hα in its unitary equivalence class does not affect the definition of this Poisson structure, since it is invariant under unitary transformations. Since AC is a C ∗ -algebra, A is closed under the right-hand side of (1.21), and therefore under the Poisson bracket on the left-hand side as well. We now return to general Poisson spaces. If P is simultaneously a (general, uniform, or smooth) Poisson space and a transition probability space, two function spaces are intrinsically associated with it: CL∞ (P) and A(P), respectively. The space naturally tied with both structures in concert is therefore AL (P) = A(P) ∩ CL∞ (P) .

(1.22)

Since elements of AL (P) are smooth on each symplectic leaf of P, they generate a well-defined Hamiltonian flow, which, of course, stays inside a given leaf. Definition 4. A (general, uniform, or smooth) Poisson space which is simultaneously a transition probability space is called unitary if the Hamiltonian flow on P defined by each element of AL (P) preserves the transition probabilities. That is, if ρ(t) and σ(t) are Hamiltonian curves (with respect to a given H ∈ AL (P)) through ρ(0) = ρ and σ(0) = σ, respectively, then p(ρ(t), σ(t)) = p(ρ, σ) for each t for which both flows are defined. We now come to the central concept of this work.

(1.23)

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Definition 5. A (general, uniform, or smooth) Poisson space with a transition probability is a set P which is a well-behaved transition probability space and a unitary (general, uniform, or smooth) Poisson space, for which A = AL (P). This definition imposes two closely related compatibility conditions between the Poisson structure and the transition probabilities: firstly, it makes a definite choice for the space A appearing in the definition of a Poisson space, and secondly it imposes the unitarity requirement. If (P, p) is a classical transition probability space (that is, p is given by (1.12)), then any Poisson structure is unitary. This is, indeed, the situation in classical mechanics, where P is the phase space of a physical system. The best-known example is, of course, P = R2n with canonical symplectic structure. The pure state space P(A) of a C ∗ -algebra is a uniform Poisson space with a transition probability. Indeed, we infer from (1.18) that A(P) ⊂ CL∞ (P(A)), so that AL (P(A)) as defined in (1.22) coincides with A as given in (1.19). Moreover, the flow of each A ∈ A on a given symplectic leaf (= sector) PHα of P(A) is the projection of the flow Ψ(t) = exp(−itA)Ψ on Hα . Since A is self-adjoint, exp(−itA) is a unitary operator, and the transition probabilities (1.16) are clearly invariant under such flows. 2. Axioms for Pure State Spaces As remarked above, a direct translation of the Axioms AHS1–AHS5 for compact convex sets to axioms on their extreme boundaries is difficult. Nevertheless, we can work with a set of axioms on a set P, some of which are similar to AHS1–AHS5. In particular, AHS2 can be directly translated: Definition 6. A well-behaved transition probability space P is said to have the two-sphere property if for any two points ρ, σ (with ρ 6= σ) lying in the same sector of P, the space {ρ, σ}⊥⊥ is isomorphic as a transition probability space to the two-sphere S 2 , with transition probabilities given by p(z, w) = 12 (1 + cos θ(z, w)) (where θ(z, w) is the angular distance between z and w, measured along a great circle). Here the orthoclosed space {ρ, σ}⊥⊥ = ρ ∨ σ may be regarded as an element of the lattice L(P). If ρ and σ lie in different sectors of P, then ρ ∨ σ = {ρ, σ}; this follows from repeated application of De Morgan’s laws [31] and ρ⊥⊥ = ρ (etc.). To understand the nature of the two-sphere property, note that a two-sphere S 2 with radius 1 may be regarded as the extreme boundary of the unit ball B 3 ⊂ R3 , seen as a compact convex set. As we saw in Sec. 1.2, B 3 ' S(M2 (C)). Restricted to the extreme boundary, the parametrization (1.3) leads to a bijection between P(M2 (C)) ' PC2 and S 2 . Under this bijection the transition probabilities (1.16) on PC2 are mapped into the ones stated in Definition 6. In other words, the two-sphere property states that there exists a fixed reference 2 ' PC2 , equipped with the standard Hilbert space transition protwo-sphere Sref

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2 babilities p = pC2 given by (1.16), and a collection of bijections Tρ∨σ : ρ ∨ σ → Sref , defined for each orthoclosed subspace of the type ρ ∨ σ ⊂ P (where ρ and σ 6= ρ lie in the same sector of P), such that

pC2 (Tρ∨σ (ρ0 ), Tρ∨σ (σ 0 )) = p(ρ0 , σ 0 )

(2.1)

for all ρ0 , σ 0 ∈ ρ ∨ σ. Now consider the following axioms on a set P: Axiom 1. P is a uniform Poisson space with a transition probability; Axiom 2. P has the two-sphere property; Axiom 3. The sectors of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space; Axiom 4. The space A (defined through Axiom 1 by (1.22)) is closed under the Jordan product constructed from the transition probabilities; Axiom 5. The pure state space P(A) of A coincides with P. The meaning of Axiom 4 will become clear as soon as we have explained how to construct a Jordan product on A(P), for certain transition probability spaces P. This axiom turns A into a JB-algebra, which is contained in C(P). Hence each element of P defines a pure state on A by evaluation; Axiom 5 requires that all pure states of A be of this form (note that, by Axiom 1, A already separates points). Axioms 2 and 4 are direct analogues of Axioms AHS2 and AHS4, respectively (also cf. the end of Sec. 4.2). The ‘bootstrap’ Axiom 5 restricts the possible uniformities on P; it is somewhat analogous to Axiom AHS3. In the previous section we have seen that the pure state space of a unital ∗ C -algebra satisfies Axioms 1–5. The remainder of this paper is devoted to the proof of the following Theorem. If a set P satisfies Axioms 1–5 (with P as a transition probability space containing no sector of dimension 3), then there exists a unital C ∗ -algebra AC , whose self-adjoint part is A (defined through Axiom 1). This AC is unique up to isomorphism, and can be explicitly reconstructed from P, such that (1) P = P(A) (i.e., P is the pure state space of A); (2) the transition probabilities (1.4) coincide with those initially given on P; (3) the Poisson structure on each symplectic leaf of P is proportional to the Poisson structure imposed on the given leaf by (1.21); (4) the w∗ -uniformity on P(A) defined by A is contained in the initial uniformity on P; (5) the C ∗ -norm on A ⊂ AC is equal to the sup-norm inherited from the inclusion A ⊂ `∞ (P). The unfortunate restriction to transition probability spaces without 3-dimensional sectors (where the notion of dimension is as defined after (1.10), i.e., as the cardinality of a basis of P as a transition probability space) follows from our method of proof, which uses the von Neumann co-ordinatization theorem for Hilbert lattices

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[23, 54, 32]. In view of the parallel between our axioms and those in [5], however, we are confident that the theorem holds without this restriction. To make progress in this direction one has to either follow our line of proof and exclude the possibility of non-Desarguesian projective geometries (cf. [23, 24] in the present context), or abandon the use of Hilbert lattices and develop a spectral theory of well-behaved transition probability spaces, analogous to the spectral theory of compact convex sets of Alfsen and Shultz [6, 7]. Despite considerable efforts in both directions the author has failed to remove the restriction. The theorem lays out a possible mathematical structure of quantum mechanics with superselection rules. Like all other attempts to do so (cf. [43, 44, 49, 37, 14]), the axioms appear to be contingent. This is particularly true of Axiom AHS2 and of our Axiom 2, which lie at the heart of quantum mechanics. One advantage of the axiom schemes in [5] and the present paper is that they identify the incidental nature of quantum mechanics so clearly. If P is merely assumed to be a Poisson space with a transition probability (i.e., no uniformity is present), then the above still holds, with the obvious modifications. In that way, however, only perfect C ∗ -algebras [50, 2] can be reconstructed (cf. Sec. 1.4). 3. From Transition Probabilities to C ∗ -algebras The proof of the theorem above essentially consists of the construction of a C -algebra AC from the given set P. In summary, we can say that in passing from pure states to algebras of observables one has the following correspondences. ∗

Pure state space transition probabilities Poisson structure unitarity

Algebra of observables Jordan product Poisson bracket Leibniz rule

To avoid unnecessary interruptions of the argument, some of the more technical arguments are delayed to Chapter 4. 3.1. Identification of P as a transition probability space This identification follows from Axiom 1 (of which only the part stating that P be a well-behaved transition probability space is needed) and Axiom 2, as a consequence of the following result. Proposition 1. Let a well-behaved transition probability space P (with associated lattice L(P)) have the two-sphere property. If P has no sector of dimension 3, then P ' ∪α PHα as a transition probability space (for some family {Hα } of complex Hilbert spaces), where each sector PHα is equipped with the transition probabilities (1.16). This statement is not necessarily false when P does have sectors of dimension 3 (in fact, we believe it to be always true in that case as well); unfortunately our proof does not work in that special dimension.

POISSON SPACES WITH A TRANSITION PROBABILITY

43

In any case, it is sufficient to prove the theorem for each sector separately, so we may assume that P is irreducible (as a transition probability space). Even so, the proof is quite involved, and will be given in Sec. 4.1. 3.2. Spectral theorem For each orthoclosed subset Q of a well-behaved transition probability space P, define a function pQ on P by X

dim(Q)

pQ =

pei ;

(3.1)

i=1

here {ei } is a basis of Q; it is easily seen that pQ is independent of the choice of this basis (cf. [59]). Definition 7. Let P be a well-behaved transition probability space. A spectral resolution of an element f ∈ `∞ (P) is an expansion (in the topology of pointwise convergence) X λj pQj , (3.2) f= j

where λj ∈ R, and {Qj } is an orthogonal family of orthoclosed subsets of P (cf. (3.1)) P for which j pQj equals the unit function on P. Proposition 2. If P = ∪α PHα (with transition probabilities (1.16)) then any f ∈ A00 (P) has a unique spectral resolution. By the previous section this applies, in particular, to a transition probability space P satisfying Axioms 1 and 2. Proof. Firstly, the case of reducible P may be reduced to the irreducible one PN by grouping the ρi in f = i=1 ci pρi into mutually orthogonal groups, with the property that (∪ρ)⊥⊥ is irreducible if the union is over all ρi in a given group. Thus we henceforth assume that P is irreducible, hence of the form P = PH with the transition probabilities (1.16). If P is finite-dimensional the proposition is simply a restatement of the spectral theorem for Hermitian matrices. In the general case, let f be as above, and Q := P {ρ1 , . . . , ρN }⊥⊥ . If σ ∈ Q then f (σ) = j λj pQj (σ) for some λj and mutually orthogonal Qj ⊂ Q, as in the previous paragraph. If σ ∈ Q⊥ this equation trivially holds, as both sides vanish. Let us assume, therefore, that σ lies neither in Q nor in Q⊥ . Define ϕQ (σ) by the following procedure: lift σ to a unit vector Σ in H, project Σ onto the subspace defined by Q, normalize the resulting vector to unity, and project back to PH (this is a Sasaki projection in the sense of lattice theory [14,31]). In the Hilbert space case relevant to us, the transition probabilities satisfy p(σ, ρ) = p(σ, ϕQ (σ))p(ϕQ (σ), ρ)

(3.3)

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for ρ ∈ Q and σ ∈ / Q⊥ . We now compute f (σ) by using this equation, followed by the use of the spectral theorem in Q, and subsequently we recycle the same equation in the opposite direction. This calculation establishes the proposition for σ ∈ / Q⊥ . If P is a classical transition probability space (see (1.12)) then a spectral theorem obviously holds as well; it simply states that a function f with finite support {σi } P is given by f = i f (σi )pσi . 3.3. Jordan structure (1.16)), Proposition 3. If P = ∪α PHα (with transition probabilities P f = j λj pQj is the spectral resolution of f ∈ A00 (P), and f 2 is defined by f 2 = P 2 j λj pQj , then the product ◦ defined by f ◦ g = 14 ((f + g)2 − (f − g)2 )

(3.4)

turns A00 (P) into a Jordan algebra. Moreover, this Jordan product ◦ can be extended to A0 (P) by (norm-) continuity, which thereby becomes a JB-algebra (with the supnorm inherited from `∞ (P)). Finally, the bidual A(P) is turned into a JB-algebra by extending ◦ by w∗ -continuity. The bilinearity of (3.4) is not obvious, and would not necessarily hold for arbitrary well-behaved transition probability spaces in which a spectral theorem (in the sense of Proposition 2) is valid. In the present case, it follows, as a point of principle, from the explicit form of the transition probabilities in PH. The quickest way to establish bilinearity, of course, is to look at a function pQ (where Q lies in a sector PH of P) as the Gel’fand transform of a projection operator on H. Given bilinearity, the claims of the proposition follow from the literature. The extension to A0 (P) by continuity, turning it into a JB-algebra, is in [6, Thm. 12.12] or [8, Prop. 6.11]. For the the extension to A(P) see Sec. 3 of [9] and Sec. 2 and Prop. 6.13 of [8]. (There is a spectral theorem in A(P), which is a so-called JBW algebra, as well, cf. [6, 7, 9], but we will not need this.) The norm in A(P) is the sup-norm inherited from `∞ (P) as well; this establishes item 5 of the Theorem. If P is classical, A(P) = `∞ (P), and the Jordan product constructed above is given by pointwise multiplication. This explains why the latter is used in classical mechanics. 3.4. Explicit description of A(P) Proposition 4. Let P = ∪α PHα (with transition probabilities (1.16)), and regard self-adjoint elements A = ⊕α Aα of the von Neumann algebra MC = ⊕α B(Hα ) ˆ = ρ(Aα ). Denote as functions Aˆ on P in the obvious way: if ρ ∈ PHα then A(ρ) ˆ A ∈ M, by M. ˆ Then the subspace of `∞ (P) consisting of all such A, ˆ . A(P) = M

(3.5)

POISSON SPACES WITH A TRANSITION PROBABILITY

45

Note that the identification of A ∈ M with Aˆ ∈ `∞ (P) is norm-preserving relative to the operator norm and the sup-norm, respectively. Also, it is clear that this proposition proves (1.18). Proof. Inspired by [1, 19], we define a (locally non-trivial) fiber bundle B(P), whose base space B is the space of sectors, equipped with the discrete topology, and whose fiber above a given base point α is B(Hα )sa ; here Hα is such that the sector α is PHα . Moreover, P itself may be seen as a fiber bundle over the same base space; now the fiber above α is PHα . We will denote the projection of the latter bundle by pr. A cross-section s of B(P) then defines a function sˆ on P by sˆ(ρ) = [s(pr(ρ))](ρ). The correspondence s ↔ sˆ is isometric if we define the norm of a cross-section of B(P) by ksk = supα∈B ks(α)k (where the right-hand side of course contains the operator norm in B(Hα )), and the norm of sˆ as the sup-norm in `∞ (P). It follows directly from its definition that the space A00 (P) consists of section s of B(P) with finite support, and such that s(α) has finite rank for each α. Its closure A0 (P) contains all sections such that the function α → ks(α)k vanishes at infinity, and s(α) is a compact operator. It follows from elementary operator algebra theory that the dual A0 (P)∗ may be realized as the space of sections for which s(α) is of trace-class and α → ks(α)k1 (the norm here being the trace-norm) is in `1 (B). The bidual A(P) then consists of all sections of B(P) for which α → ks(α)k is in `∞ (B) (here the crucial point is that K(H)∗∗ = B(H)). Eq. (3.5) is then obvious. For later use, we note that A0 (P) and even A00 (P) are dense in A(P) in the topology of pointwise convergence. This is because firstly K(H) is dense in B(H) in the weak operator topology [30] (as is the set of operators of finite rank), hence certainly in the coarser topology of pointwise convergence on P, and secondly the topology of pointwise convergence on `∞ (B) is contained in the w∗ -topology (`∞ (B) being the dual of `1 (B), which in turn is the dual of `0 (B)); recall that any (pre-) Banach space is w∗ -dense in its double dual (e.g., [30]). ˆ ↔ M the Jordan product constructed Under the correspondence A(P) = M in the previous section is then simply given by the anti-commutator of operators in M. 3.5. Algebra of observables By Axiom 1, the space of observables A is defined by (1.22). We now use Axiom 3, which implies that each symplectic leaf of P is a projective Hilbert space PHα . For the moment we assume that each leaf PHα has a manifold structure ˆ where A ∈ B(Hα )sa , are smooth (such as its usual relative to which all functions A, manifold structure). Then A(P) ∩ Cu (P) ⊂ CL∞ (P) by the explicit description of A(P) just obtained. It then follows from (1.22) that A = A(P) ∩ Cu (P) .

(3.6)

It is easily shown that A is closed (in the sup-norm). This follows from the fact that A(P) is closed, plus the observation that the subspace of functions in `∞ (P)

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N. P. LANDSMAN

which are uniformly continuous with respect to any uniformity on P, is closed; this generalizes the well-known fact that the subspace of continuous functions relative to any topology on P is sup-norm closed (the proof of this observation proceeds by the same ε/3-argument). Note that A0 (P) is not necessarily a subspace of A; it never is if the C ∗ -algebra AC to be constructed in what follows is antiliminal [21]. We can construct a Jordan product in A by the procedure in Sec. 3.3. By Proposition 3 and Axiom 4, this turns A into a JB-algebra. At this stage we can already construct the pure state space P(A); the first claim of the Theorem then holds by Axiom 5. We may regard the restriction of A to a given sector PHα as the Gel’fand transform of a Jordan subalgebra of B(Hα )sa . This subalgebra must be weakly dense in B(Hα )sa , for otherwise Axiom 5 cannot hold. Let us now assume that some PHα have an exotic manifold structure such that A(P) ∩ Cu (P) is not contained in CL∞ (P), so that A ⊂ A(P) ∩ Cu (P) is a proper inclusion (rather than the equality (3.6)). It follows from Axiom 5 that the statement in the previous paragraph must still hold. This weak density suffices for the results in Secs. 3.7 and 3.8 to hold, and we can construct a C ∗ -algebra AC with pure state space P. The proper inclusion above would then contradict (1.19). Hence such exotic manifold structures are excluded by the axioms. 3.6. Unitarity, Leibniz rule, and Jordan homomorphisms It is instructive to discuss a slightly more general context than is strictly necessary for our purposes. Proposition 5. Let P be a Poisson space with a transition probability in which every f ∈ A00 (P) has a unique spectral resolution (in the sense of Definition 7). Assume that for each H ∈ AL (P) (cf. (1.22)) the map f → {H, f } is bounded on AL (P) ⊂ `∞ (P) (with sup-norm). If a Jordan product ◦ is defined on AL (P) through the transition probabilities, in the manner of Proposition 3, then ◦ and the Poisson bracket satisfy the Leibniz rule (1.1). The boundedness assumption holds in the case at hand (cf. the next section); it is mainly made to simplify the proof. The proposition evidently holds when AL (P) is a Poisson algebra, for which the assumption is violated. Proof. Writing δH (f ) for {H, f }, the boundedness of δH implies that the series P∞ n (f )/n! converges uniformly, and defines a uniformly continuous αt (f ) = n=0 tn δH one-parameter group of maps on AL (P) (cf. [15]). On the other hand, if σ(t) is the Hamiltonian flow of H on P (with σ(0) = σ), then αt as defined by αt (f ) : σ → f (σ(t)) must coincide with the definition above, for they each satisfy the same differential equation with the same initial condition. In particular, the flow in question must be complete. Moreover, it follows that the Leibniz rule (yet to be established) is equivalent to the property that αt is a Jordan morphism for each t; this, in turn, can be rephrased by saying that αt (f 2 ) = αt (f )2 for all f ∈ AL (P).

POISSON SPACES WITH A TRANSITION PROBABILITY

47

P Let f ∈ A00 (P) ∩ AL (P), so that f = k λk pek , where all ek are orthogonal P (cf. Sec. 3.2). Unitarity implies firstly that αt (f ) = k λk pek (−t) , and secondly that the ek (−t) are orthogonal. Hence αt (f ) is given in its spectral resolution, so P that αt (f )2 = k λ2k pek (−t) . Repeating the first use of unitarity, we find that this equals αt (f 2 ). Hence the property holds on A00 (P). Now A00 (P) is dense in A(P) in the topology of pointwise convergence in `∞ (P). But fλ → f pointwise clearly implies αt (fλ ) → αt (f ) pointwise. This, plus the w∗ -continuity of the Jordan product [9] proves the desired result. 3.7. Poisson structure Item 3 of the Theorem follows from Axiom 3, the penultimate paragraph of Sec. 3.5, and the following Proposition 6. Let PH, equipped with the transition probabilities (1.16), be a unitary Poisson space for which the Poisson structure is symplectic, and for which A is the Gel’fand transform of a weakly dense subspace of B(Hα )sa . Then the Poisson structure is determined up to a multiplicative constant, and given by (1.21) times some ~−1 ∈ R. Proof. Axiom 3 implies that each sector PH (for some H) is a symplectic space. Unitarity (in our sense) and Wigner’s theorem (cf. [54, 14, 50] for the latter) imply that each Aˆ ∈ A generates a Hamiltonian flow on PH which is the projection d ˆ ˆ ˆ B}(ψ) ˆ B(exp(itC(A))ψ) of a unitary flow on H. Therefore, {A, = dt t=0 for some ˆ self-adjoint operator C, depending on A (here exp(itC(A))ψ is by definition the projection of exp(itC(A))ψ to PH, where ψ is some unit vector in H which projects \ to ψ ∈ PH). The right-hand side equals i(CB − BC)(ψ). Anti-symmetry of the −1 left-hand side implies that C = ~ A for some ~−1 ∈ R. By the weak density assumption, the collection of all differentials dAˆ spans the fiber of the cotangent bundle at each point of PH. Thus the Poisson structure is completely determined. This shows that the symplectic structure on each leaf is ~ωF S , where ωF S is the Fubini–Study structure [53, 38, 18, 19, 20, 39]. (A closely related fact is that the K¨ ahler metric associated to ωF S is determined, up to a multiplicative constant, by its invariance under the induced action of all unitary operators on H, cf. [1, 39].) The multiplicative constant is Planck’s constant ~, which, as we see, may depend on the sector. To satisfy Axiom 4, ~−1 must be nonzero in every sector whose dimension is greater than 1. In one-dimensional sectors the Poisson bracket identically vanishes, so that the value of ~ is irrelevant. The Poisson structure on P is determined by the collection of symplectic structures on the sectors of P, for the Poisson bracket {f, g}(ρ) is determined by the restrictions of f and g to the leaf through ρ; cf. (1.20). The choice (1.21) for the Poisson bracket on A corresponds to taking ~ a sectorindependent constant (put equal to 1). In general, we may regard ~ as a function on P(A), which is constant on each sector. If Aˆ denotes an element of A, the restriction

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of Aˆ to a sector PHα corresponds to an operator Aα ∈ B(Hα )sa (cf. Sec. 3.5). The sector in which ρ ∈ P(A) lies is called α(ρ). With this notation, and denoting AB − BA by [[A, B]] (recall that [A, B] denotes the Lie bracket in a Jordan–Lie algebra) the Poisson bracket on A is then given by ˆ B}(ρ) ˆ {A, =

i \ [[Aα(ρ) , Bα(ρ) ]](ρ). ~(ρ)

(3.7)

The sector-dependence of ~ cannot be completely arbitrary, however; Axiom 1 implies that ~ must be a uniformly continuous function on P. For suppose ~ is ˆ B ˆ ∈ A in such a way that Aα and not uniformly continuous. We then take A, Bα are independent of α in a neighbourhood of a point σ of discontinuity of ~, with [[Aα(σ) , Bα(σ) ]] 6= 0. Then the real-valued function on P(A) defined by ρ → ˆ B}(ρ) ˆ ~(ρ){A, is certainly uniformly continuous near σ, since its value at ρ is equal \ ˆ B} ˆ is uniformly continuous as well. , Bα(ρ) ]](ρ). But, by assumption, {A, to i[[Aα(ρ) ˆ ˆ Because of the factor ~, the product ~{A, B} cannot be uniformly continuous. This leads to a contradiction. 3.8. C ∗ -structure We now turn A into a Jordan–Lie algebra, and thence into the self-adjoint part of a C ∗ -algebra AC (cf. Sec. 1.1). On each leaf, the associator equation (1.2) is identically satisfied by the Poisson bracket (3.7). However, the ‘constant’ k ≡ ~2 /4 may depend on the leaf. Therefore, we have to rescale the Poisson bracket so as to undo its ~-dependence. From (3.7) this is obviously accomplished by putting [f, g](ρ) = ~(ρ){f, g}(ρ). With the Jordan product ◦ defined in Sec. 3.3, Eq. (1.2) is now satisfied. Hence we define a product · : A × A → AC by 1 f · g = f ◦ g − i[f, g] , 2

(3.8)

and extend this to AC × AC by complex linearity. As explained in Sec. 1.1, this product is associative. Indeed, in the notation introduced in the previous section one simply has ˆ \ Bα(ρ) (ρ) , Aˆ · B(ρ) = Aα(ρ)

(3.9)

where the multiplication on the right-hand side is in B(Hα(ρ) ). By Axiom 1 (in particular, closure of A under the Poisson bracket), Axiom 4, and the uniform continuity of ~(·), AC is closed under this associative product. Let A be a JB-algebra, and AC = A ⊕ iA its complexification. As shown in [58], one may construct a norm on AC , which turns it into a so-called JB ∗ -algebra [28]; the involution is the natural one, i.e., (f + ig)∗ = f − ig for f, g ∈ A. Now given a JB ∗ -algebra AC whose Jordan product ◦ is the anti-commutator of some associative product ·, it is shown in [47] that (AC , ·) is a C ∗ -algebra iff (AC , ◦) is JB ∗ -algebra.

POISSON SPACES WITH A TRANSITION PROBABILITY

49

Hence one can find a norm on AC (whose restriction to its self-adjoint part A, realized as in (1.19), is the sup-norm) such that it becomes a C ∗ -algebra equipped with the associative product (3.8). Since the unit function evidently lies in A(P) (cf. (3.5)) as well as in Cu (P), it lies in A (cf. (3.6)). In conclusion, the unital C ∗ -algebra mentioned in the theorem has been constructed. An alternative argument showing that A is closed under the commutator (Poisson bracket) is to combine the results of section 4.3 below and [5, §7]. This avoids the rescaling of the Poisson bracket by ~(·), but relies on the deep analysis of [5]. It is also possible to have + instead of − in (3.8). This choice produces a (+) (−) C ∗ -algebra AC which is canonically anti-isomorphic to AC ≡ AC . Moreover, in (+) (−) some cases AC is isomorphic to AC in a non-canonical fashion. Choose a faithful representation π(AC ) on some Hilbert space H, and choose a basis {ei } in H. Then P P define an anti-linear map J : H → H by J i ci ei = i ci ei , and subsequently a linear map j on π(AC ) by j(A) = Jπ(A)∗ J. If j maps π(AC ) into itself, it defines (−) (+) an isomorphism between AC and AC . In [5] (or [50]) this sign change would correspond to reversing the orientation of K (or P). 3.9. Transition probabilities and uniform structure Recall Mielnik’s definition (1.4) of the transition probability in the extreme boundary ∂e K of a compact convex set [41]. By Axiom 5, the extreme boundary of the state space K = S(A) of A is P. Hence P acquires transition probabilities by (1.4), which are to be compared with those originally defined on it. In Sec. 4.2 we show that these transition probabilities coincide, and this proves item 2 of the Theorem. It is immediate from the previous paragraph that A(P(A)) = A(P). The ∗ w -uniformity appearing in (1.19) is the weakest uniformity relative to which all elements of A are uniformly continuous. It then follows from (1.19) and (3.6) (in which the uniformity is the initially given one) that the initial uniformity on P must contain the w∗ -uniformity it acquires as the space of pure states of AC . This proves item 4. This completes our construction, as well as the proof of the theorem. 4. Proofs 4.1. Proof of Proposition 1 The strategy of the proof is to characterize the lattice L(P) (cf. Sec. 1.4), and then use the so-called co-ordinatization theorem in lattice theory [14, 32] to show that L(P) is isomorphic to the lattice L(H) of closed subspaces of some complex Hilbert space H (see [54, 14, 31, 32] for extensive information on this lattice; an equivalent description is in terms of the projections in the von Neumann algebra B(H)). It is known that L(P) is complete, atomic, and orthomodular [59, 13, 14] if P is a well-behaved transition probability space; hence it is also atomistic [14, 31]. Using

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the connection between the center of an orthomodular lattice and its reducibility [31], it is routine to show that the irreducibility of P as a transition probability space (which we assume for the purpose of this proof) is equivalent to the irreducibility of L(P) as a lattice. Hence L(P) is also irreducible. Lemma 1. L(P) has the covering property (i.e., satisfies the exchange axiom). See [14, 31, 32] for the relevant definitions and context. Proof. Consistent with previous notation, we denote atoms of L(P) (hence points of P) by ρ, σ, and arbitrary elements by Q, Qi , R, S. Let n = dim(Q) (as a transition probability space); for the moment we assume n < ∞. We will first use induction to prove that if ρ ∈ / Q, the element (ρ ∨ Q) ∧ Q⊥ is an atom. To start, note that if Q1 ≤ Q2 for orthoclosed Q1 , Q2 of the same finite dimension, then Q1 = Q2 , for an orthoclosed set in P is determined by a basis of it, which in turn determines its dimension. This implies that dim(ρ ∨ Q) > dim(Q) if ρ ∈ /Q (take Q1 = Q and Q2 = ρ ∨ Q). Accordingly, it must be that (ρ ∨ Q) ∧ Q⊥ > ∅, for equality would imply that dim(ρ ∨ Q) = dim(Q). For n = 1, Q is an atom. By assumption, ρ ∨ Q is S 2 , hence (ρ ∨ Q) ∧ Q⊥ is the anti-podal point to Q in ρ ∨ Q, which is an atom, as desired. Now assume n > 1. Choose a basis {ei }i=1,...,dim(Q) of Q; then Q = ∨ni=1 ei . Put R = ∨n−1 i=1 ei ; then R < Q hence Q⊥ < R⊥ , so that (ρ ∨ Q) ∧ Q⊥ ≤ (ρ ∨ Q) ∧ R⊥ . The assumption (ρ ∨ Q) ∧ Q⊥ = (ρ ∨ Q) ∧ R⊥ is equivalent, on use of Q = R ∨ en , De Morgan’s laws ⊥ [31], and the associativity of ∧, to ((ρ ∨ Q) ∧ R⊥ ) ∧ e⊥ n = (ρ ∨ Q) ∧ R , which implies ⊥ ⊥ that (ρ ∨ Q) ∧ R ≤ en . This is not possible, since the left-hand side contains en . Hence ∅ < (ρ ∨ Q) ∧ Q⊥ < (ρ ∨ Q) ∧ R⊥ .

(4.1)

It follows from the orthomodularity of L(P) that if R ≤ S and R ≤ Q, then (S ∨ Q) ∧ R⊥ = (S ∧ R⊥ ) ∨ (Q ∧ R⊥ ).

(4.2)

Since R < Q and R ≤ ρ ∨ R, one has ρ ∨ Q = (ρ ∨ R) ∨ Q. Now use (4.2) with S = ρ ∨ R to find (ρ ∨ Q) ∧ R⊥ = ((ρ ∨ R) ∨ Q) ∧ R⊥ = ((ρ ∨ R) ∧ R⊥ ) ∨ (Q ∧ R⊥ ) . By the induction hypothesis (ρ ∨ R) ∧ R⊥ is an atom (call it σ), so the right-hand side equals σ ∨ en . The equality σ = en would imply that ρ ∈ Q, hence σ 6= en . But then (4.1) and the S 2 -assumption imply 0 < dim((ρ ∨ Q) ∧ Q⊥ ) < 2, so that (ρ ∨ Q) ∧ Q⊥ must indeed be an atom. It follows that dim(ρ ∨ Q) = dim(Q) + 1. Hence any S ⊂ P satisfying Q ≤ S ≤ ρ ∨ Q must have dim(S) equal to dim(Q) or to dim(Q) + 1. In the former case, it must be that S = Q by the dimension argument earlier. Similarly, in the latter case

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the only possibility is S = ρ ∨ Q. All in all, we have proved the covering property for finite-dimensional sublattices. A complicated technical argument involving the dimension theory of lattices then shows that the covering property holds for all x ∈ L(P); see Sec. 13 in [31] and Sec. 8 in [32]. We have, therefore, shown that L(P) is a complete atomistic irreducible orthomodular lattice with the covering property. If L(P) is in addition infinitedimensional, one speaks of a Hilbert lattice (recently, there has been a major breakthrough in the theory of such lattices [52, 29], but since the infinite-dimensionality is used explicitly in this work we derive no direct benefit from this). In any case, we are in a position to apply the standard co-ordinatization theorem of lattice theory; see [23, 54, 14, 32 29]. For this to apply, the dimension of L(P) as a lattice [31] (which is easily seen to coincide with the dimension of P as a transition probability space) must be ≥ 4, so that we must now assume that dim(P) 6= 3; the case dim(P) = 2 is covered directly by Axiom 2. (The fact that dimension 3 is excluded is caused by the existence of so-called non-Desarguesian projective geometries; see [24] for a certain analogue of the co-ordinatization procedure in that case.) Accordingly, for dim(P) 6= 3 there exists a vector space V over a division ring D (both unique up to isomorphism), equipped with an anisotropic Hermitian form θ (defined relative to an involution of D, and unique up to scaling), such that L(P) ' L(V ) as orthocomplemented lattices. Here L(V ) is the lattice of orthoclosed subspaces of V (where the orthoclosure is meant with respect to the orthogonality relation defined by θ). We shall now show that we can use Axiom 2 once again to prove that D = C as division rings. While this may seem obvious from the fact [23, 54] that for any irreducible projection lattice one has D ' (ρ ∨ σ)\σ (for arbitrary atoms ρ 6= σ), which is C by Axiom 2, this argument does not prove that D = C as division rings. The following insight (due to [34], and used in exactly the same way in [60] and [17]) is clear from the explicit construction of addition and mutliplication in D [54, 23]. Let V be 3-dimensional, and let L(V ) carry a topology for which the lattice operations ∨ and ∧ are jointly continuous. Then D (regarded as a subset of the collection of atoms in L(V )), equipped with the topology inherited from L(V ), is a topological division ring (i.e., addition and multiplication are jointly continuous). Let F ∈ L(P) be finite-dimensional. We can define a topology on [∅, F ] (i.e., the set of all Q ∈ L(P) for which Q ⊆ F ) through a specification of convergence. Given a net {Qλ } in F , we say that Qλ → Q when eventually dim(Qλ ) = dim(Q), and if there exists a family of bases {eλi } for {Qλ }, and a basis {ej } P of Q, such that i,j p(eλi , ej ) → dim(Q). This notion is actually independent of P the choice of all bases involved, since j p(ρ, ej ) is independent of the choice of the basis in Q for any ρ ∈ P, and similarly for the bases of Qλ (to see this, extend Pdim(P) dim(Q) dim(P) to a basis {ej }j=1 , and use the property j=1 p(ej , ρ) = 1 for all {ej }j=1 ρ ∈ P). An equivalent definition of this convergence is that Qλ → Q if p(ρλ , σ) → 0 for all σ ∈ F ∧ Q⊥ and all {ρλ } such that ρλ ∈ Qλ .

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Using the criteria in [33], it is easily verified that this defines a topology on F . Moreover, this topology is Hausdorff. For let Qλ → Q and Qλ → R. Then p(ρλ , σ) → 0 for all σ ∈ Q⊥ ∨ R⊥ = (Q ∧ R)⊥ , and {ρλ } as specified above. Choose Pdim(Q∧R) p(ρλ , ej ) = 1, a basis {ej } of Q which extends a basis of Q ∧ R. Then j=1 Pdim(Q) but also j=1 p(ρλ , ej ) = 1 since Qλ → Q. Hence p(ρλ , σ) → 0 for all σ ∈ Q ∧ (Q ∧ R)⊥ . This leads to a contradiction unless Q = R. Lemma 2. The restriction of this topology to any two-sphere ρ ∨ σ ' S 2 in F induces the usual topology on S 2 . Moreover, ∨ and ∧ are jointly continuous on any [∅, F ], where F is a 3-dimensional subspace of L(P). Proof. If we restrict this topology to the atoms in F , then ρλ → ρ if p(ρλ , ρ) → 1. This induces the usual topology on F = σ ∨ τ ' S 2 , since one can easily show that, in F = σ ∨ τ , p(ρλ , ρ) → 1 is equivalent to p(ρλ , ν) → p(ρ, ν) for all ν ∈ σ ∨ τ (cf. [17]). We now take F to be a 3-dimensional subspace. We firstly show that ρλ → ρ and σλ → σ, where ρ and σ are atoms, imply ρλ ∨ σλ → ρ ∨ σ. Let τλ = (ρλ ∨ σλ )⊥ ∧ F , and τ = (ρ ∨ σ)⊥ ∧ F ; these are atoms. Let ρ0λ be the anti-podal point to ρλ in 0 ρλ ∨ σλ (i.e., ρ0λ = ρ⊥ λ ∧ (ρλ ∨ σλ )), and let σλ be the anti-podal to σλ in ρλ ∨ σλ . 0 Then {ρλ , ρλ , τλ } is a basis of F , and so is {σλ , σλ0 , τλ }. The definition of a basis and of ρλ → ρ, σλ → σ imply that p(ρ, τλ ) → 0 and p(σ, τλ ) → 0. Hence p(τ, τλ ) → 1. Now take an arbitrary atom αλ ∈ τλ⊥ ∧ F , and complete to a basis {αλ , α0λ , τλ }, where α0λ ∈ ρλ ∨ σλ . Again, the definition of a basis implies that p(αλ , τ ) → 0. Hence by our second definition of convergence ρλ ∨ σλ → ρ ∨ σ. Secondly, we show that Qλ → Q and Rλ → R, where Q and R are twodimensional subspaces of F , implies Qλ ∧ Rλ → Q ∧ R (we assume Q 6= R, so eventually Qλ 6= Rλ ). Let α = Q⊥ ∧F , β = R⊥ ∧F , γ = Q∧R, and γλ = Qλ ∧Rλ ; as a simple dimension count shows, these are all atoms. By assumption, p(γλ , α) → 0 and p(γλ , β) → 0. Since (α ∪ β)⊥ = (α ∨ β)⊥ by definition of ∨, and (α ∨ β) is two-dimensional, γ is the only point in F which is orthogonal to α and β. Hence p(γλ , γ) → 1; if not, the assumption would be contradicted. But this is precisely the definition of Qλ ∧ Rλ → Q ∧ R. From the classification of locally compact connected division rings [56] we conclude that D = C as division rings; the ring structure is entirely determined by the topology. Moreover, Lemma 2 implies that the orthocomplementation is continuous on 3-dimensional subspaces. If one inspects the way the involution of D is constructed in the proof of the lattice co-ordinatization theorem, one immediately infers that this involution (of C in our case) must then be continuous as well. It can be shown that C only possesses two continuous involutions: complex conjugation and the identity map [54]. The latter cannot define a non-degenerate sesquilinear form (so that, in particular, the lattice L(V ) could not be orthomodular). Hence one is left with complex conjugation, and V must be a complex pre-Hilbert space. The fact that V is actually complete follows from the orthomodularity of L(P) (hence of L(V )). The proof of this statement is due to [10]; see (also cf. [32, Thm.

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11.9], or [14, Thm. 21.4.1]). We will therefore write V = H. We conclude that L(P) is isomorphic to the lattice L(H) of closed subspaces of some complex Hilbert space H. Therefore, their respective collections of atoms P and PH must be isomorphic. Accordingly, we may identify P and PH as sets. Denote the standard transition probabilities (1.16) on PH by pH . With p the transition probabilities in P, we will show that p = pH . 2 isometrically in Refer to the text following Definition 6. We may embed Sref 2 PH; one then simply has p = pH on Sref . Equation (2.1) now reads pH (Tρ∨σ (ρ0 ), Tρ∨σ (σ 0 )) = p(ρ0 , σ 0 );

(4.3)

in particular, pH (Tρ∨σ (ρ0 ), Tρ∨σ (σ 0 )) = 0 iff p(ρ0 , σ 0 ) = 0. On the other hand, we know that p and pH generate isomorphic lattices, which implies that pH (ρ0 , σ 0 ) = 0 iff p(ρ0 , σ 0 ) = 0. Putting this together, we see that pH (Tρ∨σ (ρ0 ), Tρ∨σ (σ 0 )) = 0 iff pH (ρ0 , σ 0 ) = 0. A fairly deep generalization of Wigner’s theorem (see [54, Thm. 4.29]; here the theorem is stated for infinite-dimensional H, but it is valid in finite dimensions as well, for one can isometrically embed any finite-dimensional Hilbert space in an infinite-dimensional separable Hilbert space) states that a bijection T : PH1 → PH2 (where the Hi are separable) which merely preserves orthogonality (i.e., pH2 (T (ρ0 ), T (σ 0 )) = 0 iff pH1 (ρ0 , σ 0 ) = 0) is induced by a unitary or an anti2 , and unitary operator U : H1 → H2 . We use this with H1 = ρ ∨ σ, H2 = Sref T = Tρ∨σ . Since Tρ∨σ is induced by a(n) (anti-) unitary map, which preserves pH , we conclude from (4.3) that pH (ρ0 , σ 0 ) = p(ρ0 , σ 0 ). Since ρ and σ (and ρ0 , σ 0 ∈ ρ ∨ σ) were arbitrary, the proof of Theorem 1 is finished. 4.2. Transition probabilities Our aim is to show that the transition probabilities defined by (1.4) on the pure state space P(A) of the C ∗ -algebra AC (i.e., K = S(A); recall that AC is unital) coincide with those originally defined on P = P(A) = ∂e K (cf. Axiom 5); from Proposition 1 we know that these are given by (1.16). Firstly, A as a Banach space (and as an order-unit space) is isomorphic to the space A(K) of continuous affine functions on K, equipped with the sup-norm. The double dual A∗∗ is isomorphic to Ab (K) (with sup-norm), and the w∗ -topology on Ab (K) as the dual of A(K)∗ is the topology of pointwise convergence, cf. [12, 6]. Since A(K) is w∗ -dense in Ab (K), one may take the infimum in (1.4) over all relevant f in A(K). Since A ⊆ M ⊆ A∗∗ (where M was defined in Proposition 4), by (3.5) one may certainly take the infimum over A(P). But, as we saw in Sec. 3.4, A00 (P) is dense in A(P) when both are seen as subspaces of `∞ (P) with the topology of pointwise convergence. Hence we may take the infimum in (1.4) over all relevant f in A00 (P). Let Q be an orthoclosed subspace of P, and recall that pQ was defined in (3.1). We now show that an equation similar to Eq. (2.19) in [6] holds, viz. pQ = inf {g ∈ A00 (P)| 0 ≤ g ≤ 1, g Q = 1} .

(4.4)

For suppose there exists a 0 ≤ h < pQ for which the infimum is reached. We must

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have h = 1 on Q and h = 0 on Q⊥ , since pQ = 0 on Q⊥ . Then the function pQ − h is ≥ 0, and vanishes on Q and Q⊥ . But such functions must vanish identically: P let pQ − h = i λi pρi . Choose a basis {ej } in Q ∪ Q⊥ . For every point ρ ∈ P, P P P one must have j p(ρ, ei ) = 1. Hence j (pQ − h)(ej ) = i λi = 0. Suppose that pQ − h > 0. Then there will exist another basis {uj } such that f (uj ) > 0 for at P least one j. This implies i λi > 0, which contradicts the previous condition. We conclude that pQ = h, and (4.4) has been proved. The desired result now follows immediately from (4.4) and the observation that by definition pρ (σ) = p(ρ, σ) for atoms Q = ρ. We close this section with a technical comment. If F ⊂ K ⊂ A∗ (again with K = SA) is a w∗ -closed face, then ∂e F ⊆ ∂e K may be equipped with transition probabilities defined by (1.4), in which Ab (K) is replaced by Ab (F ). These coincide with the transition probabilities inherited from ∂e K. For F = K ∩ H for some w∗ -closed hyperplane H ⊂ A∗ (see, e.g., [3, II.5], [6, Sec. 1]), so that Ab (F ) ' H ∗ . By Hahn–Banach, each element of H ∗ can be extended to an element of A∗ , so that any element of Ab (F ) extends to some element of Ab (K). The converse is obvious. The claim then follows from the definition (1.4). This shows, in particular, that Axiom AHS2 is equivalent to our Axiom 2. 4.3. Poisson structure and orientability While not necessary for the main argument in this paper, it is enlightening to see that (given the other axioms) the existence of a Poisson structure on P implies Axiom AHS5, i.e., orientability in the sense of Alfsen et al. [5] (also cf. [50]). We still write K for S(A). These authors define the object B(K) as the space of all affine isomorphisms from B 3 onto a face of K (which in our setting is the state space of A(P) as a JB-algebra), equipped with the topology of pointwise convergence. It follows from Axiom 5 and the argument in [50, p. 499] (or section 3 of [18]) that one can work equally well with the space B(P) of all injective maps from S 2 = PC2 into P which preserve transition probabilities, topologized by pointwise convergence. If ϕ, ψ ∈ B(P) have the same image, then by Axiom 2 and Wigner’s theorem the map ψ −1 ◦ ϕ : S 2 → S 2 lies in O(3) (acting on S 2 ∈ R3 in the obvious way). The maps ψ and ϕ are said to be equivalent if ψ −1 ◦ ϕ ∈ SO(3); the space of such equivalence classes is B(P)/SO(3). The space P is said to be orientable if the Z2 -bundle B(P)/SO(3) → B(P)/O(3) is globally trivial (cf. [5 Sec. 7]). This notion of orientability is equivalent to the one used in [5], cf. [50]. Given ϕ ∈ B(P) and f ∈ A, we form f ◦ ϕ : S 2 → R. We infer from the explicit description of A in Chapter 3 that f ◦ ϕ is smooth. If f, g ∈ A then by (3.7) {f, g} ◦ ϕ(z) = sgn(ϕ)~−1 (ϕ(z)){f ◦ ϕ, g ◦ ϕ}S 2 (z),

(4.5)

where { , }S 2 is the Fubini–Study Poisson bracket on S 2 , and sgn(ϕ) is ±1, depending on the orientation of ϕ. Now suppose that K (hence P) were not orientable. Then there exists a continuous family {ϕt }t∈[0,1] in B(P), for which ϕ0 and ϕ1 have the same image, but

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opposite orientations (cf. the proof of Lemma 7.1 in [5], also for the idea of the present proof). We replace ϕ by ϕt in (4.5). Since {f, g} is continuous, the lefthand side is continuous in t (pointwise in z). On the right-hand side, {f ◦ϕt , g◦ϕt }S 2 is continuous in t, and so is ~−1 ◦ϕt . But sgn(ϕt ) must jump from ±1 to ∓1 between 0 and 1, and we arrive at a contradiction. References [1] M. C. Abbati, R. Cirelli, P. Lanzavecchia, and A. Mania, “Pure states of general quantum-mechanical systems as K¨ ahler bundles”, Nuovo Cim. B83 (1984) 43–59. [2] C. A. Akemann and F. W. Shultz, “Perfect C ∗ -algebras”, Mem. Amer. Math. Soc. 326 (1985) 1–117. [3] E. M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer, 1970. [4] E. M. Alfsen, “On the state spaces of Jordan and C ∗ -algebras”, in Alg`ebres d’op´erateurs et leurs applications en physique math´ematique, ed. A. Connes, Editions CNRS, Paris, 1977. [5] E. M. Alfsen, H. Hanche-Olsen, and F. W. Shultz, “State spaces of C ∗ -algebras”, Acta Math. 144 (1980) 267–305. [6] E.M. Alfsen and F. W. Shultz, “Non-commutative spectral theory for affine function spaces on convex sets”, Mem. Amer. Math. Soc. 172 (1976) 1–120. [7] E.M. Alfsen and F. W. Shultz, “On non-commutative spectral theory and Jordan algebras”, Proc. London Math. Soc. 38 (1979) 497–516. [8] E. M. Alfsen and F. W. Shultz, “State spaces of Jordan algebras”, Acta Math. 140 (1978) 155–190. [9] E. M. Alfsen, F. W. Shultz, and E. Størmer, “A Gelfand-Neumark theorem for Jordan algebras”, Adv. Math. 28 (1978) 11–56. [10] I. Amemiya and H. Araki, “A remark on Piron’s paper”, Publ. RIMS (Kyoto) A2 (1966) 423–427. [11] H. Araki, “On a characterization of the state space of quantum mechanics”, Commun. Math. Phys. 75 (1980) 1–24. [12] L. Asimow and A. J. Ellis, Convexity Theory and its Applications in Functional Analysis, Academic Press, 1980. [13] J. G. F. Belinfante, “Transition probability spaces”, J. Math. Phys. 17 (1976) 285– 290. [14] E. G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics, Cambridge Univ. Press, 1984. [15] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. I: C ∗ - and W ∗ -Algebras, Symmetry Groups, Decomposition of States, 2nd ed., Springer, 1987. [16] L. G. Brown, “Complements to various Stone–Weierstrass theorems for C ∗ -algebras and a theorem of Shultz”, Commun. Math. Phys. 143 (1992) 405–413. [17] R. Cirelli and P. Cotta-Ramusino, “On the isomorphism of a ‘quantum logic’ with the logic of projections in a Hilbert space”, Int. J. Theor. Phys. 8 (1973) 11–29. [18] R. Cirelli, P. Lanzavecchia, and A. Mania, “Normal pure states of the von Neumann algebra of bounded operators as a K¨ ahler manifold”, J. Phys. A16 (1983) 3829–3835. [19] R. Cirelli, A. Mania, and L. Pizzochero, “Quantum mechanics as an infinitedimensional Hamiltonian system with uncertainty structure”, J. Math. Phys. 31 (1990) 2891–2897. [20] R. Cirelli, A. Mania, and L. Pizzochero, “A functional representation for noncommutative C ∗ -algebras”, Rev. Math. Phys. 6 (1994) 675–697. [21] J. Dixmier, C ∗ -Algebras, North-Holland, 1977.

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[22] G. G. Emch, Mathematical and Conceptual Foundations of 20th Century Physics, North-Holland, 1984. [23] K. D. Freyer and I. Halperin, “The von Neumann coordinatization theorem for complemented modular lattices”, Acta Scient. Math. 17 (1956) 203–249. [24] K. D. Freyer and I. Halperin, “On the construction of coordinates for non-Desarguesian complemented modular lattices”, Proc. Kon. Ned. Akad. Wet. A61 (1958) 142–161. [25] J. Glimm and R.V. Kadison, “Unitary operators in C ∗ -algebras”, Pacific J. Math. 10 (1960) 547–556. [26] E. Grgin and A. Petersen, “Duality of observables and generators in classical and quantum mechanics”, J. Math. Phys. 15 (1974) 764–769. [27] H. Hanche-Olsen, “JB-algebras with tensor product are C ∗ -algebras”, Lecture Notes in Math. 1132, 1985, pp. 223–229. [28] H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, Pitman, 1984. [29] S. S. Holland, Jr. “Orthomodularity in infinite dimensions; a theorem of M. Sol` er”, Bull. Amer. Math. Soc. 32 (1995) 205–234. [30] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras I, Academic Press, 1983. [31] G. Kalmbach, Orthomodular Lattices, Academic Press, 1983. [32] G. Kalmbach, Measures and Hilbert Lattices, World Scientific, 1986. [33] J.L. Kelley, General Topology, Van Nostrand, London, 1955. [34] A. Kolmogorov, “Zur Begr¨ undung der projektiven Geometrie”, Ann. Math. 33 (1932) 175–176. [35] H. Kummer, “The foundation of quantum theory and noncommutative spectral theory, I” Found. Phys. 21 (1991) 1021–1069, II: ibid 1183–1236. [36] N. P. Landsman, “Classical behaviour in quantum mechanics: a transition probability approach”, Int. J. Mod. Phys. B10 (1996) 1545–1554. [37] G. Ludwig, An Axiomatic Basis for Quantum Mechanics. Volume 1: Derivation of Hilbert Space Structure, Springer, 1985. [38] J. E. Marsden, Applications of Global Analysis in Mathematical Physics, Publish or Perish, 1974. [39] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 1994. [40] B. Mielnik, “Geometry of quantum states”, Commun. Math. Phys. 9 (1968) 55–80. [41] B. Mielnik, “Theory of filters”, Commun. Math. Phys. 15 (1969) 1–46. [42] B. Mielnik, “Generalized quantum mechanics”, Commun. Math. Phys. 37 (1974) 221– 256. [43] J. von Neumann, “On an algebraic generalization of the quantum mechanical formalism (Part I)”, Math. Sborn. 1 (1936) 415–484; Collected Works Vol. III, ed. A. H. Taub, pp. 492–561. [44] J. von Neumann, “Continuous geometries with a transition probability”, Mem. Amer. Math. Soc. 252 (1981) 1–210 (MS from 1937). [45] S. Pulmannov´ a, “Transition probability spaces”, J. Math. Phys. 27 (1986) 1791–1795. [46] J.E. Roberts and G. Roepstorff, “Some basic concepts of algebraic quantum theory”, Commun. Math. Phys. 11 (1968) 321–338. [47] A. Rodriguez-Palacios, “Jordan axioms for C ∗ -algebras”, Manuscripta Math. 61 (1988) 297–314. [48] J. Schwinger, Quantum Kinematics and Dynamics, W. A. Benjamin, 1970. [49] I. E. Segal, “Postulates for general quantum mechanics”, Ann. Math. 48 (1947) 930– 948. [50] F. W. Shultz, “Pure states as dual objects for C ∗ -algebras”, Commun. Math. Phys. 82 (1982) 497–509.

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RESIDUES AND TOPOLOGICAL YANG MILLS THEORY IN TWO DIMENSIONS KENJI MOHRI National Laboratory for High Physics (KEK), Ibaraki, Japan 305 E-mail : [email protected] Received 5 April 1996 Revised 20 August 1996 A residue formula which evaluates any correlation function of topological SUn Yang– Mills theory with arbitrary magnetic flux insertion in two-dimensions are obtained. Deformations of the system by two-form operators are investigated in some detail. The method of the diagonalization of a matrix-valued field turns out to be useful to compute various physical quantities. As an application we find the operator that contracts a handle of a Riemann surface and a genus recursion relation.

1. Introduction Two-dimensional topological Yang–Mills theory is an example of topological field theories which is precisely the two-dimensional analogue of the Donaldson theory [1]. The physical observables of this theory in the favorable cases are identified with the cohomology classes of the moduli space M of the flat gauge fields on a Riemann surface. The cohomology ring of M [2, 3, 4, 5] recently draws much attention in connection with the Floer cohomology group [6]. The correlation functions of topological Yang–Mills theory, which determine the cohomology ring, have been completely solved in the form of a multiple infinite sum by Witten using the exact solution of two-dimensional physical Yang–Mills theory [7, 8, 9] and the correspondence between physical and topological Yang–Mills theories [10]. Apart from the unsolved problem of the cases of non-compact gauge groups which may be important to analyze topological W gravities, there is still much to do in this theory. First of all it seems difficult to compute the explicit value of a correlation function and to investigate the cohomology ring of topological Yang– Mills theory of gauge groups other than SU2 using the infinite sum formula found in [10]. Thus it would be desirable to find another formula of correlation functions which directly gives their values as rational numbers and is more suitable to study the cohomology ring. Moreover the formula in [10] for the correlators that contain general two-form operators has remained totally implicit because we must perform inversion of matrix field variables to evaluate it. In this paper we give a formula which expresses any correlation function of SUn theory with arbitrary magnetic flux as a residue evaluated at the origin of the Cartan subalgebra generalizing the previous results [2, 11]. 59 Review of Mathematical Physics, Vol. 9, No. 1 (1997) 59–75 c

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In the process we develop systematically the method of inversion of variables using residues and the diagonalization of matrix-valued field. In addition to its practical value, this residue formula sheds light on the structure of the topological Yang–Mills theory. For example, topological Yang–Mills theory may be regarded as a kind of matrix models because the evaluation of correlation functions by the residue can be written as an integration over eigenvalues of Hermite matrixes. We also expect that the residue formula will be useful to consider systematically the quantum deformation of the cohomology ring [4], the coupling of topological Yang–Mills theory to topological gravity and string interpretation of the large N expansion of topological Yang–Mills theory [12]. Jeffrey and Kirwan have studied the non-Abelian localization in [13] and proved the residue formula for SU2 [14]. Blau and Thompson have studied two-dimensional topological Yang–Mills theory as well as other closely related gauge theories based on the path integral approach. They introduced the Abelianization, i.e., the diagonalization of the matrixvalued fields in [15, 16], which naturally reproduces the infinite sum formula mentioned above. In [17], the Abelianization was related to the localization of the gauge field path integral to the reducible gauge fields using the equivariant supersymmetry. The relation between this localization and the non-Abelian localization to the Yang–Mills connections seems still mysterious. This paper is organized as follows. In Sec. 2.1 the magnetic flux for gauge theory on a two-dimensional surface is introduced. Section 2.2 is devoted to the standard construction of the BRST observables. In Sec. 3 we review the result about physical Yang–Mills theory and the infinite sum formula for correlation functions of topological Yang–Mills theory. In Sec. 4, we give the residue formula for the correlation functions with arbitrary magnetic flux which contain no two-form operators but the symplectic form ω. We also describe some concrete examples of the correlation functions. In Sec. 5 we treat the cases in which arbitrary two-form operators are inserted in a correlator. This corresponds to considering deformations of the original topological theory. The most general residue formula for the correlation functions is obtained. Finally in Sec. 6, as an application of the residue formula we find the observable that restricts the path integral to the subspace where the holonomy of any gauge field around a cycle is trivial and does nothing else. 2. Topological Yang Mills on a Riemann Surface 2.1. ’t Hooft magnetic flux Here we describe the geometrical setting for SUn topological Yang–Mills theory on a genus g Riemann surface Σg . We introduce ’t Hooft magnetic flux1 , which is the terminology originally used in four-dimensional gauge theories [10,18,19] as follows. 1 Magnetic flux here represents an element of H 2 (Σ , Z ) ∼ g n = SUn /Zn bundles on Σg .

Zn

which classifies the topology of

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61

Pick a point P on Σg and put the boundary condition on gauge fields that the holonomy around P must be X d , where √ √ 2π −1 2π −1 (2.1) X = diag e n , . . . , e n is a generator of the center Zn of SUn . This is called the SUn gauge theory with d units of magnetic flux. Note that the magnetic flux d is defined only modulo n and the theory with d units of magnetic flux are related to one with (n− d) units of magnetic flux by the charge conjugation. Modulo the degrees of freedom of the ghost zero modes associated with the residual gauge symmetries for the cases of (n, d) 6= 1, path integral of the topological theory can be regarded as an integration over the moduli space of flat gauge fields on Σg −P with the prescribed holonomy around P , which we denote as Mg (n, d). Mg (n, d) is also the absolute minima of the ordinary Yang–Mills action. We list some properties of Mg (n, d) relevant to the later discussion: • Mg (n, d) has the real dimension equal to the ghost number violation 2(n2 − 1)(g − 1). • Mg (n, d) ∼ = Mg (n, n − d) due to the charge conjugation symmetry. • When (n, d) = 1, Mg (n, d) is smooth and has no reducible gauge fields. In the case (n, d) 6= 1, Mg (n, d) always has reducible gauge fields, which make the analysis of the theory difficult. In this paper we define correlation functions of topological Yang–Mills theory as expansion coefficients in coupling constant of the partition function Z() of physical Yang–Mills theory [10]. This procedure indeed gives the correct answer for the case (n, d) = 1. For the other cases (n, d) 6= 1, in addition to a polynomial part, there also exist non-local terms in in the expansion of Z() due to the reducible flat gauge fields, which makes the identification of topological correlation functions with the polynomial part in of Z() somewhat doubtful. However even in these cases the result obtained by this approach is consistent with the Riemann–Roch–Verlinde formula, as we will see later, with the reservation that the existence of the Riemann–Roch formula itself on a moduli space with singularities is also presumed. Thus we will not worry about the singularities associated with reducible flat gauge fields in the cases (n, d) 6= 1 as long as correlation functions are concerned. For the basic facts about the Lagrangian and BRST symmetry see [10, 16, 20]. 2.2. Topological observables Here we review the standard construction of the BRST observables of the topological theory [20]. Mathematical treatments of this subject for general SUn gauge group can be found in [21, 22]. Let (Ai , ψi , φ) be the basic topological multiplet of the topological Yang–Mills theory, with the BRST symmetry δA = ψ, δψ = −Dφ, δφ = 0. √ √ √ −1 −1 −1 φ, 2π ψ, 2π F ). It is useful to redefine the fields as (φ, ψ, F ) = ( 2π

62

K. MOHRI

¯ The There are two ways of component expansions of the matrix-valued field φ. first one is the expansion with respect to an orthonormal basis of Lie algebra, P φ¯ = a φ¯a Ja . The second one, which turns out to be more useful, is defined using a diagonalization φ¯ = diag(z1 , . . . , zn ) =

n−1 X

xi Hi ,

¯ xi = zi − zi+1 = hαi , φi,

(2.2)

i=1

where {Hi } and {αi } are the set of fundamental coweights and the simple roots of SUn respectively, and the Weyl group action is simply the permutations of {zi }. Now the zero form operator of ghost number 2m is defined by Om =

1 Tr(φ¯m ). m!

Next one-form operator of ghost number (2m − 1) is I 1 ¯ m Tr(φ¯(m−1) ψ), Vm (a) = Ca m!

(2.3)

(2.4)

where Ca , 1 ≤ a ≤ 2g are the 1-homology basis of Σg such that Ca · Cb+g = −δab . Note that {Vm (a)} transform among themselves under the mapping class group of Σg [2, 4] and only the modular invariant combination of those; Ξlm =

g X

(Vl (a)Vm (a + g) − Vl (a + g)Vm (a))

(2.5)

a=1

are nonvanishing in the correlation functions. Finally two-form operator of ghost number (2m − 2) is similarly constructed as Z 1 1 (2) =− Tr(mφ¯(m−1) F¯ + m(m − 1)φ¯(m−2) ψ¯2 ). (2.6) Om m! Σ 2 In particular, there always exist observables associated with the degree two Casimir (2) invariant; O2 and ω = O2 which is the standard symplectic two form2 on Mg (n, d). They play the special role in physical/topological Yang–Mills correspondence [10]. The correlation function of the form: + * Y Y Y (2) Oli Vmj (aj ) Onk , Mg (n, d) (2.7) i

j

k

is non-vanishing only if the observables inside the correlator satisfy the ghost number selection rule, X X X 2li + (2mj − 1) + (2nk − 2) = 2(n2 − 1)(g − 1). (2.8) i

j

k

Hereafter we will frequently use the notation g¯ = (g − 1). 2 The ample generator of the Picard group ∼ =

case.

Z is

n ω, (n,d)

and the first Chern class is 2nω in any

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RESIDUES AND TOPOLOGICAL YANG–MILLS THEORY IN TWO DIMENSIONS

3. Physical Yang Mills Theory 3.1. Infinite sum formula In [7, 8, 10], a multiple infinite sum formula was obtained for the partition function of physical Yang–Mills theory. Physical Yang–Mills theory can be described by the same field content as the basic BRST multiplet of the topological Yang–Mills so that the generalized LaX xm x2 with nilpotent δm grangian of the model with a polynomial Q(x) = + 2! m! m≥3

{δm } is given by

¯ + Tr(φF + 1 ψψ). L = −Tr Q(φ) (3.1) 2 Then the partition function is written as the following multiple infinite sum closely related to the multiple zeta value investigated in [23] Z(Q, ω) = (−1)(n−1)d+|∆+|¯g ng

∞ X

···

∞ X

Y

−2¯ g TrQ(Φ) −dhλ1 ,Φi

hα, Φi

e

e

,

ln−1 =1 α∈∆+

l1 =1

(3.2) √ Pn−1 where we set Φ = 2π −1 i=1 Hi li and Hi is the i-th fundamental coweight. The normalization of the partition function is determined so as to produce the corresponding topological correlation function by the expansion in the coupling constant . 3.2. Non-Abelian localization Due to the non-Abelian localization theorem [10, 13], see also [17], the path integral of physical Yang–Mills theory can be localized around the solution of the equation of motion which are the critical points of SY M . Noting that the flat gauge fields are the absolute minima which give the dominant contribution, it is seen that the partition function of the physical Yang–Mills theory has the following connection with the correlation function of the topological theory in the case of (n, d) = 1, D E ¯ Z(Q, ω) = eω eTrQ(φ) , Mg (n, d) + {contributions of non-flat solutions} . (3.3) The first term of the right-hand side of (3.3) is a polynomial in and represents the corresponding correlation function of topological Yang–Mills theory, while the second term refers to the contributions to the path integral from the solutions of the Yang–Mills equation with non-zero action [21] which has the -dependence SY M ' exp(− (2π) 2 ). More precisely it was shown [10] that the infinite sum of the form ∞ X l1 =1

···

∞ X

Y

−2¯ g TrΦ2 /2

hα, Φi

e

TrB(Φ)e−dhλ1 ,Φi

(3.4)

ln−1 =1 α∈∆+

vanishes exponentially for → 0 if (n, d) = 1 and the ghost number of B is greater than 4¯ g|∆+ |. When (n, d) 6= 1, reducible flat gauge fields give additional terms

64

K. MOHRI

which are non-local in the coupling constant. Even in these cases the polynomial part could be given the interpretation as topological correlation functions. Thus we obtain the infinite sum formula for correlation functions of topological Yang–Mills theory which is somewhat conjectural for (n, d) 6= 1 cases;

ω ¯ Mg (n, d) e TrB(φ), = (−1)(n−1)d+|∆+|¯g ng

∞ X l1 =1

···

∞ X

Y

hα, Φi

−2¯ g

TrB(Φ)e−dhλ1 ,Φi . (3.5)

ln−1 =1 α∈∆+

The above formula is well-defined only when the right-hand side converges. This in particular implies that the ghost number of B should be less than 4¯ g |∆+ | in (3.5). 4. Residue Formula 4.1. Residue form with magnetic flux Here we propose the residue formula for correlation functions of topological Yang–Mills theory with arbitrary magnetic flux. The case of (n, d) = (2, 1) and the case of (n, 0) were treated by Thaddeus [2] and Szenes [11] respectively. To present the formula we introduce the following symbol d·i d·i d·i = − . (4.1) θ n n n (k)

Now define the multi-variable residue form Ωg (n, d) for SUn theory with d units of magnetic flux by (n−1)(d−1)+|∆+ |¯ g Ω(k) nk n−1 g (n, d) =(−1)

Y dxi g¯ n−1 i=1

Y i θ(d·i/n)=0

1 kxi (e + 1) 2

Y

xi

ekθ(d·i/n)xi

i θ(d·i/n)=0

kxi − 1)

(ekxi

Y

−2¯g α, φ¯ .

α∈∆+

The residue formula of the correlation function is given by

kω ¯ Mg (n, d) = Res Ω(k) (n, d)TrB(φ) ¯ . e TrB(φ), g {xi =0}

(4.2)

(4.3)

The order of evaluations of residues above is xn−1 < xn−2 < · · · < x1 . The equivalence of the multiple infinite sum formula (3.5) and the residue formula (4.3) can be seen using the localization of infinite residues sum argument as follows. First substituting the partial fraction expansion in the residue form √ X 2π −1 1 eθ(d·i/n)xi m n di √ = , θ(d · i/n) 6= 0 e exi − 1 xi − 2π −1m

(4.4)

m∈Z

X 1 exi + 1 1 √ = , x i 2e −1 x − 2π −1m m∈Z i

(4.5)

RESIDUES AND TOPOLOGICAL YANG–MILLS THEORY IN TWO DIMENSIONS

65

we can see that the residue of the form in the right-hand side of (4.3) evaluated at any dominant integral weight shifted the weight by ρ, {(l1 , . . . , ln−1 )|li ≥ 1}, coincides with a corresponding summand in (3.5) up to a constant ¯ Ω(1) (n, d)TrB(φ) Res √ {xi =2π −1li }

g

= (−1)(n−1)

Y 1 × (−1)(n−1)d+|∆+ |¯g ng hα, Φi−2¯g TrB(Φ)e−dhλ1 ,Φi . n α∈∆+

(4.6)

The set of the dominant integral weights shifted by ρ constitutes the lattice points set of one of the n! Weyl chambers and (4.6) is invariant under the Weyl group action. Thus we can sum over residues over the n! Weyl chambers instead of the single chamber as in (3.5). Let Hα be the subset of the weight lattice perpendicular to the root α, Hα = {(li )|li ∈ Z, hΦ, αi = 0}. Then the union of the lattice points of the n! Weyl chambers coincides with the complement in the weight lattice of the S Hα , and we have union of the hypersurfaces: L = α∈∆+

X (li )∈L

Res √

{x1 =2π −1l1 }

···

Res √

{xn−1 =2π −1ln−1 }

¯ Ω(1) g (n, d)TrB(φ)

¯ Mg (n, d) . =(−1)(n−1) (n − 1)! eω TrB(φ),

(4.7)

Now using the standard residue theorem which tells that the total sum of the residues for one variable with the remaining variables fixed is zero [11] repeatedly from xn−1 to x1 , we can reduce the original residues sum (4.7) to that over sets of T S Hα ∩ Hβ , where each I, beginning with ∅, eventually becomes ∆+ the form: α∈I

β6∈I

in this process. At last (4.7) is expressed by the single residue evaluated at the origin T Hα . Thus the equivalence of the infinite sum formula (3.5) and the residue = α∈∆+

¯ such that (3.5) is convergent. We also conjecture formula (4.3) follows for TrB(φ) ¯ One evidence for this that the residue formula (4.3) is valid for arbitrary TrB(φ). 2¯g Q hα/2,φ¯i ˆ conjecture is the fact that if we insert the A-genus [11] as α∈∆+ ¯i) sinh(hα/2,φ a gauge invariant zero form operator in (4.3),  !2¯g 

¯ Y α/2, φ 

(4.8) Res Ω(k) g (n, d) {xi =0} sinh α/2, φ¯ α∈∆+

we will obtain gives the twisted Verlinde dimension3 [24] of current algebra of level n in accord with the prediction of the (k − n) for any k such that k ≡ 0 mod (n,d) Riemann–Roch formula, the existence of which is also conjectural for the cases of (n, d) 6= 1. 3 If k 6≡ 0 mod n , then the twisted Verlinde dimension is precisely zero, while the residue (4.8) (n,d)

gives a rational number.

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K. MOHRI

4.2. Some examples Here we will give some explicit form of the residue formulas. First for SU2 gauge group, the diagonalization of bosonic ghost becomes 1 1 x1 0 , O2 = x21 . φ¯ = 2 0 −x1 4 The residue forms for d = 0, 1 read as g ¯ Ω(k) g (2, 0) = − (−2k) dx1 (k/2) g ¯

Ω(k) g (2, 1) = (−2k) dx1 k

ekx1 + 1 (x1 )−2¯g . ekx1 − 1

ekx1 /2 (x1 )−2¯g . ekx1 − 1

(4.9) (4.10)

The correlation functions which have been completely solved in [2, 10] can be elegantly expressed as follows; ∞ p X p

1 λg¯ eω eaO2 , Mg (2, 0) = −e− 2 aλ λ/2 cot λ/2 (4.11) g ¯=0 ∞ X

g ¯

λ

ω aO2

e e

1 , Mg (2, 1) = e− 2 aλ

g ¯=0

p λ/2 p . sin λ/2

(4.12)

The generating function of SU2 correlators of all genera is also considered in [5]. Next for SU3 , the diagonalization of φ¯ becomes,   2x1 + x2 0 0 1 , 0 −x1 + x2 φ¯ =  0 3 0 0 −x1 − 2x2 1 2 1 (x + x1 x2 + x22 ), O3 = (2x1 + x2 )(x1 + 2x2 )(x1 − x2 ). 3 1 54 The residue forms for SU3 are given by O2 =

2 Ω(k) g (3, 0) = −3k 2 Ω(k) g (3, 1) = −3k

g¯ g¯

dx1 dx2 (k/2)2 dx1 dx2 k 2

ekx1 + 1 ekx2 + 1 (x1 x2 (x1 + x2 ))−2¯g . ekx1 − 1 ekx2 − 1

ekx1 /3 e2kx2 /3 (x1 x2 (x1 + x2 ))−2¯g . ekx1 − 1 ekx2 − 1

(4.13) (4.14)

We give two simple examples of correlation functions computed by the residue formula

kω aO2 bO3 e e e , M3 (3, 0) =

1 1 19 k 16 − ak 14 + a2 k 12 41513472000 53222400 2419200 1 1 1 1 3 2 10 4 2 a + b k + a − ab k 8 − 120960 4354560 17280 31104 1 7 107 1 5 1 6 2 2 6 3 2 4 a − a b k + a − a b + b k4 + 2592 46656 3888 34992 7558272 (4.15)

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RESIDUES AND TOPOLOGICAL YANG–MILLS THEORY IN TWO DIMENSIONS

3mω aO2 bO3 e e , M3 (3, 1) e 160911 1 15363 2 12 9708939 16 m − am14 + bm13 + a m 512512000 1971200 896 89600 3 1011 3 83 2 21 2 9 abm11 + − a + b m10 + a bm − 320 4480 53760 640 5 7 3 7 2 2 123 4 5 3 a − ab2 m8 − a b− b 3 m 7 + − a5 + a b m6 + 640 384 96 10368 32 192 1 19 7 3 2 107 4 1 6 ab3 m5 + a − a b + b m4 . − − a4 b + (4.16) 48 2592 48 432 93312 Finally for SU4 , the diagonalization of φ¯ becomes   0 0 0 3x1 + 2x2 + x3 1 0 −x1 + 2x2 + x3 0 0  φ¯ =  . 0 0 0 −x1 − 2x2 + x3 4 0 0 0 −x1 − 2x2 − 3x3 =

In this case we have three different theories with magnetic flux units d = 0, 1, 2. 3 Ω(k) g (4, 0) = − 4k

g¯

dx1 dx2 dx3 (k/2)3

ekx1 + 1 ekx2 + 1 ekx3 + 1 ekx1 − 1 ekx2 − 1 ekx3 − 1

(x1 x2 x3 (x1 + x2 )(x2 + x3 )(x1 + x2 + x3 ))−2¯g 3 Ω(k) g (4, 1) = 4k

g¯

dx1 dx2 dx3 k 3

ekx1 /4 e2kx2 /4 e3kx3 /4 ekx1 − 1 ekx2 − 1 ekx3 − 1

(x1 x2 x3 (x1 + x2 )(x2 + x3 )(x1 + x2 + x3 ))−2¯g 3 Ω(k) g (4, 2) = − 4k

g¯

(4.17)

dx1 dx2 dx3 (k/2)k 2

(4.18)

e2kx1 /4 ekx2 + 1 e2kx3 /4 ekx1 − 1 ekx2 − 1 ekx3 − 1

(x1 x2 x3 (x1 + x2 )(x2 + x3 )(x1 + x2 + x3 ))−2¯g

(4.19)

4.3. Bernoulli expansions In principle, by substituting in (4.3) the Fourier expansions [25]: ∞ X 1 ex + 1 x2m = B 2m 2 ex − 1 m=0 (2m)! ∞ X xm xeθ(d·i/n) = , θ(d · i/n) 6= 0 B (θ(d · i/n)) m (ex − 1) m! m=0

(4.20)

(4.21)

we can express any correlation functions by a finite sum of (n − 1) products of Bernoulli polynomials. Here we will present the simplest ones. To this end it is convenient to introduce the following notations; Bm for m 6= 1, and b1 (0) = 0, m! Bm (θ(d · i/n)) . bm (θ(d · i/n)) = m! bm (0) =

68

K. MOHRI

Then the correlation function of SU3 theory can be written as a sum of double products of Bernoulli polynomials

ω a1 a2 e x1 x2 , Mg (3, d) X (−1)m2 2¯g H2¯g −m2 bm1 −a1 (θ(d/3))bm2 −a2 (θ(2d/3)). = (−3)g¯ (4.22)

m1 +m2 =6¯ g

Similarly the correlation function of SU4 theory is given by a sum of triple products of Bernoulli polynomials

ω a1 a2 a3 e x1 x2 x3 , Mg (4, d) X = (−1)d−1 (4)g¯ m1 +m2 +m3 =12¯ g

X

(−1)l1 +l2 +l3 2¯g Hl1 2¯g Hl2 2¯g Hl3

l3 C2¯ g−l2 −m3

l1 ,l2 ,l3 ≥0 l1 +l3 =m1 −6¯ g

bm1 −a1 (θ(d/4))bm2 −a2 (θ(2d/4))bm3 −a3 (θ(3d/4)).

(4.23)

In this way we can express any correlation function of ω and φ as a finite sum of known rational numbers. It would be interesting if we understand the relevance of the arithmetic properties of Bernoulli numbers [23, 25] to two-dimensional gauge theories. 5. Deformations by Two Form Operators 5.1. Witten’s formula So far we have treated correlation functions which contain arbitrary zero operators but do not contain any two-form operator other than the standard symplectic form exp(kω). Now we describe the computation of correlators with arbitrary twoform operator following [10]. Here again the residue method will turn out to be useful. Let O be a gauge invariant polynomial of {φ¯a } of the form, O = O2 +

X

cm Om ,

(5.1)

m≥3

and O(2) be the associated two-form operator. Z O

(2)

=− Σ

∂O ¯ a ∂2O 1 a ¯b ¯ Mab ψ ψ + ¯a F , Mab = ¯a ¯b . 2 ∂φ ∂φ ∂φ

(5.2)

The insertion of exp(O(2) ) in the correlator corresponds to the deformation of the original Lagrangian by the two-form operator. By computing the fermion determinant and the Jacobian of the change of bosonic variables, Witten gave the following formula [10] D

ekO

(2)

E

¯ Mg (n, d) = ekω detM (Q(φ)) ¯ g¯ TrB(Q(φ)), ¯ TrB(φ), Mg (n, d) , (5.3)

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RESIDUES AND TOPOLOGICAL YANG–MILLS THEORY IN TWO DIMENSIONS

¯ is the power series defined by the change of variables; where Q(φ) ba = φba (φ) ¯ ≡ ∂O , φ ∂ φ¯a

b φ¯a = Qa (φ).

(5.4)

5.2. Inversion of variables and residues At first sight it might seem necessary to convert the original field variable φ¯ b in order to evaluate the right-hand side of (5.3). But it into a power series of φ is sufficient to find only the inversion of the gauge invariants {Om }. The gauge ¯ = Om (φ), b b are defined by O bm (φ) 2 ≤ m ≤ n. The invariants constructed by {φ} b two sets of Casimir invariants {Om } and {Om } are related by certain polynomial bm = Fm (O2 , . . . , On ), and we have only to convert them to evaluate equations: O b2 , . . . , O bn ). Now the expansion coefficients the right-hand side of (5.3) Om = Gm (O of Gm defined by X Gm (l2 , . . . , ln )pl22 · · · plnn , (5.5) Gm (p2 , . . . , pn ) = l2 ,...,ln ≥0

can be obtained using the Cauchy formula [26]; Gm (l2 , . . . , ln ) = Res Gm (p)/(p2l2 +1 · · · pnln +1 )dp2 · · · dpn {pi =0}

∂Fi l2 +1 ln +1 /(F2 (q) · · · Fn (q))dq2 · · · dqn . (5.6) = Res qm det {qi =0} ∂qj Thus we get at least formally the following residue formula for the power series Gm of pi ! n dqi ∂Fi Y . (5.7) Gm (p2 , . . . , pn ) = Res qm det ∂qj i=2 (Fi (q) − pi ) {qi =0} 5.3. Diagonalization To get the explicit polynomial relations {Fm } of the previous subsection between the old and new Casimir invariants, it suffices to know only the change of variables for the diagonalization of the fields φ¯ because of gauge invariance; φb =

n−1 X

bl = x bi Hi = diag(b z1 , . . . , zbn ), O

i=1

x bi = yi (x) = Cij

n X 1 (b zi )l , l! i=1

X cm+1 ∂O m (zim − zi+1 = (zi − zi+1 ) + ), ∂xj m!

(5.8)

m≥2

zbi = zi +

X cm+1 1 X zim − cm+1 Om . m! n

m≥2

m≥2

We also have the following determinant formula by the diagonalization !2 2 Y b hα, φi ∂ O ¯ detM (φ) = ndet . ¯ ∂xi ∂xj hα, φi α∈∆

(5.9)

+

Now we can compute any correlation functions (5.3) using (5.7), (5.8) and (5.9).

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K. MOHRI

The consistency of our formalism may be checked by considering deformations of SU2 theory because in SU2 theory any observable can be expressed by ω, O2 and V2 (a). For example the two-form observable associated with O = O2 − (2l − 1)!bl O2l can be written by the observables associated with the Casimir invariant of second degree as ! g X V2 (a)V2 (a + g) . (5.10) O(2) = ω − bl O2l−1 ω − (l − 1)O2l−2 a=1

It can be seen that the use of the Witten’s formula and the direct substitution of (5.10) in the left-hand side of (5.3) give the same answer. Next take, for example, the SU3 theory with the two-form operator associated with O = O2 − 6mO3 . The polynomial relation between old and new Casimir invariants is given by ( b2 = O2 − 18mO3 + 3m2 O22 O (5.11) b3 = O3 − mO22 + 9m2 O2 O3 + m3 (O23 − 54O32 ). O From the residue formula (5.7), we get the inversion of the polynomial relation (5.11) as follows: ( b2 + 18mO b3 + 15m2 O b32 + 270O b23 ) + · · · , b22 + 378m3 O b2 O b3 + m4 (2916O O2 = O b3 + mO b22 + 27m2O b32 + 20O b23 ) + 810m4 O b2 O b3 + m3 (216O b22 O b3 + · · · . (5.12) O3 = O The determinant that appears in Witten’s formula is given by det M = (1 − 12m2 O2 )(1 − 9m2 O2 + 54m3 O3 )2 .

(5.13)

Then by the formula (5.3) we get the results D E (2) 477 8 189 6 2 405 4 4 a + a m − 27a5 m3 − a m e3aO , M2 (3, 1) = 2240 8 4

(5.14)

D E (2) 9708939 16 482733 14 2 27 13 3 a + a m − a m e3aO , M3 (3, 1) = 512512000 98560 28 1244403 12 4 38151 10 6 a m − 243a11 m5 − a m + 4480 16 423549 8 8 a m . − (5.15) 80 5.4. Generalized residue formula One method to compute the correlators with a general two-form operator was bm } perturbatively and to expand the Casimir invariants {Om } into the series of {O substitute them in the right-hand side of (5.3). Here we give another method in the

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RESIDUES AND TOPOLOGICAL YANG–MILLS THEORY IN TWO DIMENSIONS

form of residue formula. First we use the residue formula of the previous section in the right-hand side of (5.3) to obtain D E (2) ¯ Mg (n, d) = Res Ω(k) (n, d)TrB(Q(y))detM g¯(Q(y)) . (5.16) ekO TrB(φ), g {yi =0}

bi to Qi (y) = xi , we obtain Then if we change the integration variables from yi = x the generalized residue formula D E (2) ¯ Mg (n, d) = Res Ω(k) (n, d; O)TrB(x) , (5.17) ekO TrB(φ), g {xi =0}

where (n−1)(d−1)+|∆+ |¯ g nk n−1 Ω(k) g (n, d; O) = (−1)

Y g¯ n−1

dxi

i=1

Y

Jg

i θ(d·i/n)=0

= Jg

1 kyi (e + 1) 2

ekθ(d·i/n)yi

i θ(d·i/n)6=0

n−1 Y kxi i=1

Y

k (ekyi − 1)

e −1 ekyi − 1

Y i θ(d·i/n)=0

Y

−2¯g α, φ¯ α∈∆+

ekyi + 1 ekxi + 1

Y i θ(d·i/n)6=0

ekθ(d·i/n)(yi −xi ) Ω(k) g (n, d) and the Jacobian is

J = det

∂yj ∂xi

(5.18)

= ndet

∂2O ∂xi ∂xj

.

(5.19)

Thus we can say that the deformation of topological Yang–Mills theory by a twoform operator is equivalent to the insertion of a certain zero-form operator. Note that in this section the reduction of the gauge group to the abelian subgroup [15] was the powerful tool to compute explicitly the various physical quantities. 6. Recursion Relation 6.1. Wick contraction of one-form operators Here as an application of the residue formula described above we will consider correlation functions containing one-form observables. It is not difficult to compute them because we can use the physical Yang–Mills Lagrangian (3.1) to integrate all one-form observables in correlators of the topological theory [10] owing to the physical/topological Yang–Mills correspondence. Indeed using the gauge invariance and the diagonalization we get the following contraction formula in the presence of the two-form operator exp(kO(2) ); X 1 −1 ∂Om ∂Ol hVm (a)Vl (b + g)i = − δab Hij , k ∂xi ∂xj ij where Hij is the Hessian Hij =

∂2O ∂xi ∂xj .

(6.1)

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K. MOHRI

6.2. Handle contracting operator; SU2 In [2] Thaddeus rigorously proved that as a cohomology class V2 (a) is the Poincar´e dual of the subspace Ng (a) of Mg (2, 1) where the holonomy around the cycle Ca is trivial. The physical meaning of it is that V2 (a) has only the effect of reducing the path integral to the flat gauge fields that have the trivial holonomy around Ca . Thus V2 (a) may be regarded as a operator which contracts the cycle Ca . The cup product of them V2 (a)V2 (a + g) is the Poincar´e dual to Ng (a) ∩ Ng (a + g) where the holonomies around both the cycles Ca and Ca+g are trivial. Thus we call here H2 (a) ≡ V2 (a)V2 (a + g) the operator that contracts the a-th handle. Noting that Ng (a) ∩ Ng (a + g) is diffeomorphic to Mg−1 (2, 1), we have the relation between correlators of genus g and g − 1, hV2 (a)V2 (a + g)(· · · ), [Mg (2, 1)]i = h(· · · ), [Mg−1 (2, 1)]i ,

(6.2)

where (· · · ) means any operator. H2 (a) is the inverse of the handle operator in the ordinary topological field theories in two-dimensions. The physical derivation follows. Consider a correlation

of this effect [10] is as ¯ kω , [Mg (2, d)] . We can easily infunction of SU2 theory: V2 (a)V2 (a + g)TrB(φ)e tegrate the one-form operators if we use the physical Yang–Mills Lagrangian (3.1) which produces the trivial propagator for the fermions and then return to the topological Lagrangian to get hV2 (a)V2 (a + g)i = − k2 O2 . Then according to the residue formula (4.3) it is clear that the insertion of x2 2 − O2 = − 1 reduces the genus of the surface by one. k 2k Thus we have found the formula which is equivalent to (6.2),

¯ kω , [Mg (2, d)] V2 (a)V2 (a + g)TrB(φ)e

2 kω ¯ ¯ kω , [Mg−1 (2, d)] . = − O2 TrB(φ)e , [Mg (2, d)] = TrB(φ)e (6.3) k 6.3. Handle contracting operator; generalization to SUn The identification of V2 (a) with the a-th cycle contracting operator of SU2 , d = 1 theory was possible [2] because V2 (a) is the only observable that satisfies the two requirements: (1) It should have the ghost number (n2 − 1) = 3. (2) It must be fixed by the modular transformations that fix Ca . It seems impossible to extend this pure topological method to higher rank gauge groups SUn , n > 3. Nevertheless we can identify even for higher rank SUn theories the operator which contracts the a-th cycle by using the generalized residue formula (5.17) and the contraction formula of fermions (6.1). We claim that for general SUn the operator that contracts the a-th handle is the following; Hn (a) =

n−1 Y l=1

l! V2 (a) · · · Vn−1 (a)

n−1 Y l=1

l! V2 (a + g) · · · Vn−1 (a + g).

(6.4)

RESIDUES AND TOPOLOGICAL YANG–MILLS THEORY IN TWO DIMENSIONS

73

Indeed in the presence of the general two-form operator exp(kO(2) ), the Wick contraction of fermions in the physical Yang–Mills theory (3.1) gives, n−1 Y

hHn (a)i =

!2 l!

det (hVn (a)Vm (a + g)i)

l=1

= (−1)

n−1 Y

1 2 (n−2)(n−1)

!2 l!

l=1

= (−1)

1 2 n(n−1)

n−1 Y

1 k n−1

−1 k

n−1

det

!2 detH −1 det

l!

l=1

∂Om −1 ∂Ol H ∂xi ij ∂xj

∂Om ∂xi

2 .

(6.5)

Furthermore we can compute the determinants above as det

∂Om ∂xi

  m n X (zj − znm ) ∂zj ∂Om  ∂zj = det  det = det 1≤i,j≤n−1 ∂xi 1≤j,m≤n−1 ∂xi ∂zj m! j=1 1 1 = (−1) 2 (n−1)(n−2) n

detH =

1 det n

∂yj ∂xi

n−1 Y

!−1 l!

Y (zi − zj ), i<j

l=1

.

Finally we get the Wick contraction formula |∆+ |

hHn (a)i = (−1)

det

∂yj ∂xi

−1

Y 1 ¯ 2. hα, φi n−1 nk

(6.6)

α∈∆+

We see then according to the generalized residue formula (5.17), the insertion of Hn (a) has the effect of reducing the genus of the surface by one. Thus we have the recursion relation D

¯ kO Hn (a)TrB(φ)e

(2)

E D E ¯ kO(2) , [Mg−1 (n, d)] . , [Mg (n, d)] = TrB(φ)e

(6.7)

It is clear that the above equation is also valid even when we insert any one-form operators. Here we describe explicitly the Wick contraction for the case of SU4 and O = O2 . The handle contracting operator for SU4 is H4 (a) = 2 · 3! · V2 (a)V3 (a)V4 (a) · 2 · 3! · V2 (a + g)V3 (a + g)V4 (a + g), and the Wick contraction of this becomes

74

K. MOHRI

−2O2 −3O −4O 3 4 (2!3!)2 2 hH4 (a)i = − −3O 1/4O − 6O −7/12O O 3 4 2 3 2 k3 −4O4 −7/12O2O3 1/36O23 − 1/12O32 + O2 O4 144 1 6 5 4 17 = 3 O2 − O2 O4 − O23 O32 + 16O22 O42 + 6O2 O32 O4 k 72 6 36 3 − 96O43 − O34 4 =

1 2 (x1 x2 x3 (x1 + x2 )(x2 + x3 )(x1 + x2 + x3 )) . 4k 3

(6.8)

References [1] E. Witten, Comm. Math. Phys. 117 (1988) 353; J. Math. Phys. 35 (1994) 5101. [2] M. Thaddeus, J. Diff. Geom. 35 (1992) 131. [3] V. Y. Baranovsky, Izv. Russ. Akad. Nauk. 58 (1994) 204. B. Siebert and G. Tian, “Recursive relations for the cohomology ring of moduli space of stable bundles”, (preprint) alg-geom/9410019. A. D. King and P. E. Newstead, “On the cohomology ring of the moduli space of rank 2 stable bundles on a curve”, (preprint) alg-geom/9502018 . [4] M. Bershadsky, A. Johansen, V. Sadov, and C. Vafa, Nucl. Phys. B448 (1995) 166. [5] D. Zagier, “On the cohomology of moduli spaces of rank two vector bundles over curves”, The Moduli Space of Curves, ed. R. H. Dijkgraaf, C. F. Faber and G. B. M. van der Geer, pp. 533–563, Progress in Mathematics 129, Birkh¨ auser, 1995. [6] S. Dostoglou and D. Salamon, Ann. Math. 139 (1994) 581. S. K. Donaldson, “Floer homology and algebraic geometry”, pp. 119–138 in the book of ref [11]. [7] B. Rusakov, Mod. Phys. Lett. A5 (1990) 693. [8] M. Blau and G. Thompson, Int. J. Mod. Phys. A7 (1991) 3781. [9] E. Witten, Comm. Math. Phys. 141 (1991) 153. [10] E. Witten, J. Geom. Phys. 9 (1992) 303. [11] A. Szenes, “The combinatorics of the Verlinde formulas”, Symposium on Vector Bundles in Algebraic Geometry, Durham 1993, ed. N. J. Hitchin, P. E. Newstead and M. Oxbury, pp. 241–253, London Math. Soc. Lect. Note. Ser. 208, C. U. P., 1994. [12] G. Moore, “2D Yang–Mills theory and topological field theory”, Proceedings of the International Congress of Mathematicians 1994, and (preprint) hep-th/9409044. [13] L. C. Jeffrey and F. C. Kirwan, Topology 34 (1995) 291. [14] L. C. Jeffrey and F. C. Kirwan, Elec. Res. Ann. A. M. S. 01 (1995) 57. [15] M. Blau and G. Thompson, Nucl. Phys. B408 (1993) 345. [16] M. Blau and G. Thompson, “Lectures on 2d gauge theories—topological aspects and path integral techniques”, High Energy Physics and Cosmology 1993, ed. E. Gava, A. Masiero, K. S. Narain, S. Randjbar-Daemi and Q. Shafi, pp. 175–244, ICTP Series in Theoretical Physics 10, World Scientific, 1994. [17] M. Blau and G. Thompson, Nucl. Phys. B439 (1995) 367; “Two aspects of K¨ ahler geometry in the G/G model”, High Energy Physics and Cosmology 1994, ed. E. Gava, A. Masiero, K. S. Narain, S. Randjbar-Daemi and Q. Shafi, pp. 607–620, ICTP Series in Theoretical Physics 11, World Scientific, 1995. [18] G. ’t Hooft, Nucl. Phys. B138 (1978) 1. [19] C. Vafa and E. Witten, Nucl. Phys. B431 (1994) 3. [20] J.-S. Park, Comm. Math. Phys. 161 (1994) 113. [21] M. F. Atiyah and R. Bott, Phil. Trans. Roy. Soc. Lond. A308 (1982) 523.

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[22] R. Earl, “The Mumford relations and the moduli space of rank 3 stable bundles”, (preprint) alg-geom/9503023. [23] D. Zagier, “Values of zeta functions and their applications”, First European Congress of Mathematics II, ed. A. Joseph, F. Migunot, F. Murat, B. Prum and R. Rentschler, pp. 497–512, Progress in Mathematics 120, Birkh¨ auser, 1994. [24] E. Verlinde, Nucl. Phys. B300 (1988) 360. [25] I. Yamaguchi, Sympathetic Number Theory—beautiful cyclotomic fields, a journey into Bernoulli numbers (in Japanese), Sangyo Tosho, 1994. [26] P. M. Morse and H. Feshbach, Methods of Theoretical Physics I, International Series of Pure and Applied Physics, McGraw-Hill, 1953.

THE MODULI SPACE OF YANG MILLS CONNECTIONS OVER A COMPACT SURFACE AMBAR SENGUPTA Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803-4918, USA email: [email protected] Received 3 June 1996 Yang–Mills connections over closed oriented surfaces of genus ≥ 1, for compact connected gauge groups, are constructed explicitly. The resulting formulas for Yang–Mills connections are used to carry out a Marsden–Weinstein type procedure. An explicit formula is obtained for the resulting 2-form on the moduli space. It is shown that this 2-form provides a symplectic structure on appropriate subsets of the moduli space.

1. Introduction In this paper we carry out a Marsden–Weinstein type procedure for Yang–Mills theory over closed oriented surfaces of genus g ≥ 1, with a compact connected Lie group G as gauge group. We obtain an explicit formula for the corresponding symplectic structure. To obtain this description of the symplectic structure, we first present an explicit construction of all solutions of the Yang–Mills equations on compact oriented surfaces. There is a standard natural symplectic structure Ω on the space A of all connections on a principal bundle over a closed oriented surface Σ. The group G of all bundle automorphisms (covering the identity map on Σ) has a natural pull-back action on A, and this action preserves the symplectic structure Ω. Correspondingly there is a moment map J with values in L(G)∗ , the dual of the Lie algebra L(G) of G. We show that Ω, restricted to the set AYM of Yang–Mills connections, induces a symplectic structure on certain subsets of the finite dimensional space (J|AYM )−1 (O)/G, where O is any coadjoint orbit in J(A YM ) ⊂ L(G). In Theorem 6.1, we obtain an explicit description of this symplectic structure in terms of holonomy variations. The expression for this symplectic form is the same as that for the symplectic form on flat connections obtained in [13] (and earlier, in a different framework, in [6]). Some of the motivation for the investigations inR this paper arise from quantum gauge theory. There one considers integrals A/G f dµT where µT is, formally, a probability measure on A/G given by the heuristic formula dµT (ω) = 2 ZT−1 e−||J(ω)|| /(2T ) [Dω] with T a positive parameter, ZT a ‘normalizing constant’, [Dω] the pushforward onto A/G of a formal Lebesgue measure on A, and || · || a natural norm induced on L(G)∗ by the metric on the gauge group and the metric on Σ. 77 Review of Mathematical Physics, Vol. 9, No. 1 (1997) 77–121 c

World Scientific Publishing Company

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computation that when G is semisimple RLetting T ↓ 0R one can see by formal 0 −1 f dµ → f dµ , where A = J (0) is the set of flat connections and µ0 T 0 0 A/G A /G 0 is the normalized volume measure on A /G corresponding to the symplectic strucR ture on J −1 (0)/G. Explicit formulas for a broad class of integrals A/G f dµT are known ([4, 24], and [16, 17]), and using these formulas it has been shown ([5, 18, 19, 24]) in a variety of settings that µT does converge in a suitable sense toRthe symplectic volume measure µ0 . It is reasonable to expect that the behavior of A/G f dµT is also influenced by the critical points of the Yang–Mills functional ||J(·)||2 and that the corresponding moduli spaces and associated symplectic structures are involved. Some support for this comes from the work [25], where Witten has obtained an analog of the Duistermaat–Heckmann formula (which provides an exact expansion of certain partition functions over the critical points of the Hamiltonian). Thus an understanding of the Yang–Mills moduli space and the explicit formula we obtain for the symplectic structure on these spaces may prove to be useful in understanding the influence of the Yang–Mills critical points on the quantum Yang–Mills measure µT . Needless to say, Atiyah and Bott’s standard opus [1] has provided some guidance for the present paper; however, our approach is different and, in a sense, more “hands-on”. We shall now summarize some of our main conclusions. To this end let us first introduce some notation (which will be specified again in later sections more systematically). We work with a principal G-bundle π : P → Σ, with G being a compact connected Lie group, and Σ a compact connected oriented Riemannian 2-manifold of genus g ≥ 1. Let A1 , B1 , . . . , Ag , Bg be loops based at a point o ∈ Σ, generating the fundamental group π1 (Σ, o) subject to the relation that B g Ag Bg Ag · · · B 1 A1 B1 A1 is homotopic to the point loop at o. Pick a basepoint u ∈ π −1 (o), and denote by h(c; ω) the holonomy of a connection ω around a loop c, based at o, with u as initial point. Furthermore, for any ω ∈ A, let ∗Ωω (u) = Ωω (e1 , e2 ) where Ωω is the curvature of ω and e1 , e2 ∈ Tu P project by π∗ to a positively oriented orthonormal basis of To Σ. We shall use the map (with g being the Lie algebra of G) I : A → G2g × g : ω → (h(A1 ; ω), . . . , h(Bg ; ω), ∗Ωω (u))

(1.1)

˜ : x˜ 7→ x be the universal covering of G. to study Yang–Mills connections. Let G The product commutator map −1˜ ˜ ˜g · · · ˜b−1 a ˜ : (a1 , . . . , bg ) 7→ ˜b−1 ˜−1 ˜1 , C : G2g → G g a g bg a 1 ˜ 1 b1 a

˜ covering ai and bi , will also be useful; wherein a ˜i and ˜bi are any elements of G ˜ C makes sense because ker(G → G) is contained in the center of G. For X ∈ g, let GX = {k ∈ G : Ad(k)X = X}, i.e. GX is the isotropy group at X of the ˜ and X ∈ g, we denote by F z the adjoint action of G on g. Next, for z ∈ G X set of all (a1 , . . . , bg ) ∈ G2g X for which C(a1 , . . . , bg ) = z · exp(−|Σ|X), where |Σ| z will denote the subset of G2g × g consisting of all is the area of Σ. Finally, F[X]

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z (ka1 k −1 , . . . , kbg k −1 , Ad(k)X) where (a1 , . . . , bg ) runs over FX and k runs over G. z z The groups GX and G act by conjugation on FX and F[X] , respectively. Let O be the coadjoint orbit through any point µ ∈ J(AYM ) ⊂ L(G)∗ . Denote by G µ the isotropy group at µ of the coadjoint action of G on L(G)∗ . The homomorˆ ˆ ˆ phism G → G : φ 7→ φ(u), where φ(u) ∈ G is specified by φ(u) = uφ(u), restricts to µ a homomorphism G → GX (which is shown in Lemma 5.9.2 to be a submersion). We begin with some results (Proposition 3.12, Theorem 4.1 and Corollary 4.2) z z and F[X] are useful tools for analysing explaining why the map I and the sets FX −1 the moduli spaces (J|AYM ) (O)/G in a concrete way. z , where X = ∗µ(u) (∗µ is the Hodge 1. The map I takes (J|AYM )−1 (µ) into FX ω dual , defined as was explained above for ∗Ω ), and z = [P ] is an element of ˜ → G) which specifies the topology of the bundle P (and is independent ker(G z . of O); I also takes (J|AYM )−1 (O) into F[X] z , for some X ∈ g, 2. If [0, 1] → G2g : t 7→ αt is a C ∞ path lying entirely on FX then there is a family of Yang–Mills connections [0, 1] → AYM : t 7→ ωt , such that:

(i) [0, 1] × P → T ∗ P ⊗ g : (t, p) 7→ ωt (p) is smooth (C ∞ ), (ii) I(ωt ) = αt for all t ∈ [0, 1], (iii) there is a µ ∈ L(G)∗ such that J(ωt ) = µ for all t ∈ [0, 1]. We pause to comment on 2. Taking αt constant, 2 shows that every point z corresponds to a Yang–Mills connection, a result which can be proven by in FX z to smooth other methods (Theorem 6.7 in [1]). The lifts of smooth paths in FX −1 paths in (J|AYM ) (µ) provided by 2 (and the explicit formulas for such a lifting given in the proof of our Theorem 4.1) is crucial to our method of studying the symplectic structure on the corresponding moduli spaces. The holonomy group at u of a Yang–Mills connection ω is contained in GX , where X = ∗Ωω (u). Thus the structure group of the bundle can be reduced to this group and the connection ω to a connection on the corresponding bundle. These ideas are useful in other approaches to studying Yang–Mills connections. For our purposes, it should be noted that this reduction procedure provides a separate reduced bundle for each choice of ω, and thus a family such as ωt would correspond to reduced connections on a family of sub-bundles. In our approach everything takes place on the original fixed bundle P (it may be noted in this context that in the proof of Theorem 4.1 we shall construct local trivializations of P with a transition function taking values in GX , and corresponding connection forms, reflecting the fact that the structure group may be reduced to GX ). Continuing with the map I, the following are from Proposition 5.6 and Theorem 5.9.1. 3. The map I has a linear (directional ) derivative everywhere. 4. If ω ∈ (J|AYM )−1 (µ) and v ∈ Tω (J|AYM )−1 (µ), then I 0 (ω)v is a GX -orbit z if and only if v is a G µ -orbit direction. direction in TI(ω) FX 5. If ω ∈ (J|AYM )−1 (O) and v ∈ Tω (J|AYM )−1 (O) then I 0 (ω)v is a G-orbit z if and only if v is a G-orbit direction. direction in TI(ω) F[X]

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6. The commutative diagram of equivariant maps (J|AYM)−1 (µ)  Iµ y z FX

j

−1 −→ (J|AYM  ) (O)  yI i

−→

(1.2a)

z F[X]

wherein the vertical arrows arise from I and the horizontal ones from inclusions, induces a commutative diagram µ (J|AYM )−1  (µ)/G  I µy z /GX FX

j

−→ (J|AYM)−1 (O)/G  yI i

−→

(1.2b)

z F[X] /G

where the horizontal maps are diffeomorphisms of quotient spaces and the vertical maps are one-to-one local diffeomorphisms of quotients, both in the sense explained in Sec. 5.8.1; we illustrate what this means for I : I is one-to-one, and for any ω ∈ (J|AYM )−1 (O) the (directional ) derivative I 0 (ω) maps G-orbital directions into G-orbit directions in the manner described in 5 and , furthermore, every vector in z is in the image I 0 (ω) Tω (J|AYM )−1 (O) modulo the G-orbit directions. TI(ω) F[X] 7. For every Y ∈ g, FYz is the union of images under I of (J|AYM )−1 (ν) with ν running over some subset of J(AYM ). z z of G2g × g is not, in general, a smooth submanifold, nor is FX The subset F[X] z in general a smooth submanifold of G2g . We may expect the quotient F[X] /G to be made up of pieces (‘strata’) which are themselves smooth connected manifolds. The following result (Theorem 5.7) provides some information about the relationship between such strata and the moduli spaces (J|A YM )−1 (O)/G. Taken together, 1–8 imply that in a sense the connected strata of (J|AYM )−1 (O)/G correspond, by a z /G. The notation is as before. ‘diffeomorphism’ I, to the connected strata of F[X] z /G consisting of projections of smooth paths in 8. If Mα is the subset of F[X] z z and lie entirely on F[X] , then Mα is G2g × g which start at some point α ∈ F[X] −1 contained in the image under I of (J|AYM ) (O)/G for some coadjoint orbit O in L(G)∗ .

Next, in Proposition 5.10, we prove rigorously that the 2-form on the moduli space (J|A YM )−1 (O)/G, induced by the symplectic structure Ω on A, may be z z /GX and F[X] /G (the latter spaces being quotients of transferred to the spaces FX 2g 2g a subset of G and of G × g are more amenable to concrete finite-dimensional analysis). 9. Let Ω be the standard symplectic structure on A (defined in Sec. 5.2 ). Then the restriction of Ω to (J|A YM )−1 (µ) induces a 2-form Ω on (J|AYM )−1 (µ)/G µ . This corresponds to 2-forms on each of the quotient spaces appearing in diagram (1.2b). z /GX We determine (Theorem 6.1) an explicit formula for the 2-form on FX described above. Thus, this formula expresses the 2-form on the moduli space

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(J|AYM )−1 (O)/G in terms of holonomy-variations (such variations describe tangent vectors to the the moduli space): 10. Explicit formula for Ω: Consider the 2-form Ω0 on G4g given, at a point α ∈ G4g , by : X (1) (2) ik hAd(αi−1 · · · α1 )Hi , Ad(αk−1 · · · α1 )Hk ig Ω0 (α · H (1) , α · H (2) ) = 1≤i,k≤4g (1)

(2)

(the vectors H (1) , H (2) ∈ g 4g have components Hi and Hi ) where ik = ±1 according as i < k or i > k, and ii = 0. Consider next the immersion −1 −1 −1 φ : G2g → G4g : (a1 , b1 , . . . , ag , bg ) 7→ (a1 , b1 , a−1 1 , b1 , . . . , ag , bg , ag , bg ) z z is precisely the equivariant 2-form on FX which corThen the restriction φ∗ Ω0 |FX z responds to Ω on FX /GX , described in 9 above.

The expression for Ω, being independent of X, is identical to that in the case of the moduli space of flat connections (i.e. the case X = 0). This is consistent with what one may expect by heuristic reasoning. The case O = {0}, i.e. the case of flat connections, was dealt with by Goldman [6] in terms of representations of π1 (Σ) on G. Karshon [11] provided a proof of the closedness of the symplectic form on the moduli space of flat connections within the algebraic setting of [6]. In [13, 14] the problem was studied directly in terms of variations of connections and direct proofs were given showing both that the 2-form is closed (by simple calculation of the differential) and non-degenerate (by means of a determinant identity). The approach in the present paper is closest to [13, 14] and we use the proof of symplecticity from [13, 14] in the Yang–Mills context. Using the description of Ω given in 10, and results of [13, 14], we show (in Sec. 6.3) that Ω is symplectic on well-behaved subsets of the moduli spaces: z where the isotropy 11. Symplectic nature of Ω: Let UX be the subset of FX group of the GX -action is the center of GX ; then Ω is symplectic on the subset of the moduli space (J|AYM )−1 (O)/G corresponding to UX /GX .

We conclude by working out an example, the case G = SU (n). 2. Notation and Some Basic Results In this section we set up notation and state some basic facts. 2.1. The compact surface Σ Throughout this paper Σ will denote a compact connected oriented twodimensional Riemannian manifold without boundary, of genus g ≥ 1. ˜ Lie algebra g, and the metric h·,·i 2.2. The group G, the cover G, g We shall work with a compact connected Lie group G, with Lie algebra g equipped with an Ad-invariant metric h·, ·ig . In classifying bundles over Σ we shall ˜ of G. use the universal cover G

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2.3. Σ as a quotient of the disk D; orientation for Σ It will be convenient to use a standard procedure for viewing a compact surface as a quotient of a planar disk with suitable identifiications of arcs on the boundary of the disk. It will be necessary for our purposes to spell out this construction in some detail, and introduce some terminology which will be used in Sec. 4. (It is convenient in other approaches, such as in [1], to use the universal cover for Σ. Our approach, however, is to obtain explicit formulas for families of Yang–Mills connections expressed through local trivializations as 1-forms over certain subsets of Σ; this is why we need to spell out a specific quotient in detail.) Let D be the closed unit disk, centered at the origin, in the plane R2 . For t ∈ R, let xt = cos(2πt), sin(2πt) .

(2.1)

We obtain Σ by dividing up the boundary ∂D of D into arcs Ki : [ti−1 , ti ] → ∂D : t 7→ xt

(2.2)

wherein ti =

i , 4g

and pasting together certain of these arcs. More precisely, there is a quotient map q : D → Σ. This map is smooth at all points other than the points xti . However, there is a smooth reparametrization of t 7→ xt near each ti such that the path t 7→ q(xt ) becomes smooth with respect to the new parametrization (see [13]). The quotient map q is defined by requiring the identifications q(xt0 ) = q(xt ) whenever t − tj−1 = tj+2 − t0 , t ∈ [tj−1 , tj ], t0 ∈ [tj+1 , tj+2 ], j ∈ J where def

J = {1, 2, 5, 6, . . . , 4g − 3, 4g − 2} . The point o = q(O) will serve as a convenient basepoint on Σ. The loops Ki0 = q(x0 O).q(Ki ).q(Ox0 )

(2.3)

will be of use to us. We shall write A1 = K10 , B1 = K20 ,

A2 = K50 , B2 = K60 ,

··· .

(2.4)

Thus A1 , B1 , . . . , Ag , Bg generate the fundamental group π1 (Σ; o), subject to the relation that B g Ag Bg Ag · · · B 1 A1 B1 A1 is the identity in homotopy.

(2.5)

For convenience, we will always assume that Σ is equipped with the orientation which makes q orientation-preserving (D having the usual orientation in R2 ).

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83

2.4. The principal bundle P , connections ω, curvature Ωω , and covariant derivative D ω We shall work with a principal G-bundle π : P → Σ; thus there is a smooth map P × G → P : (p, k) 7→ Rk p = pk specifying a right action of G on P . The curvature of a connection ω on P will be denoted Ωω ; thus Ωω = dω + 12 [ω, ω]. 2.5. Parallel transport τω (K), and holonomy h· (K)

Let K : [a, b] → Σ be a piecewise smooth path, and a0 ∈ π −1 K(a) . We shall denote by τω (K)a0 the parallel translate of a0 by ω along K. If K is a loop, then we denote by ha0 (K; ω), the holonomy of ω around K with initial point a0 .

2.6. The set AYM of Yang Mills connections A connection ω on P is a Yang–Mills connection if it satisfies the Yang–Mills equation: (2.6a) d ∗ Ωω + [ω, ∗Ωω ] = 0 i.e. ω is a critical point of the Yang–Mills action Z def ||Ωω ||2g dσ SYM (ω) =

(2.6b)

Σ

with σ being the area-measure on Σ. Here ∗Ωω : P → g : p 7→ Ωω (e1 , e2 ), where e1 , e2 ∈ Tp P project to a positively oriented orthonormal basis (π∗ e1 , π∗ e2 ) of Tπ(p) Σ. We denote by AYM the set of all Yang–Mills connections on P . 2.7. The space A, the groups Go , G and C(P , G) Let A be the infinite dimensional affine space of all connections on P . We denote by G the set of all diffeomorphisms φ : P → P which commute with the right action of G on P and for which π◦φ = π. Then G is a group under composition and acts (on the right) on A by (φ, ω) 7→ φ∗ ω. We will denote by Go the subgroup of G consisting of those φ which pointwise fix the fiber over o. It is sometimes convenient to work with G in a different guise. This is the set C(P, G) of all smooth maps τ : P → G which satisfy τ (pg) = g −1 τ (p)g for all p ∈ P and g ∈ G; thus C(P, G) is a group ˆ specified under pointwise multiplication, and the map G → C(P, G) : φ 7→ φ, ˆ by requiring φ(p) = pφ(p) for all p ∈ P , is an isomorphism. In this notation, ˆ and for the curvatures we have φ∗ Ωω = Ad(φˆ−1 )Ωω (see φ∗ ω = Ad(φˆ−1 )ω + φˆ−1 dφ, Sec. 3 of [2]). 2.8. Holonomies of Yang Mills connections (from [1]) The Yang–Mills equation says that the covariant derivative Dω ∗ Ωω = d ∗ Ω∗ + [ω, ∗Ωω ] vanishes; thus a connection ω is Yang–Mills if and only if the function ∗Ωω is constant along ω-horizontal curves in P . In particular, by considering the values of ∗Ωω at the initial point p and the final point p0 of the horizontal lift of a closed curve C on Σ we see that if ω is a Yang–Mills connection then the holonomy hp (C; ω) commutes with ∗Ωω (p).

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˜ 2.9. Bundle classifier [P ] ∈ G Let S be a simplicial complex triangulating Σ, and s a section of P over the 1-skeleton of S (such a section exists because G is path connected). For each oriented 2-simplex ∆ of S, let s∆ be a section of P over ∆. Then ∂∆ → G : x 7→ s(x)−1 s∆ (x) defines an element γ∆ of π1 (G); here we are writing s(x)−1 s∆ (x) to denote the element g(x) ∈ G, depending continuously on x, for which s∆ (x) = s(x)g(x), and ∂∆ is oriented as the boundary of ∆. We define [P ] to be the product of the γ∆ as ∆ runs over all the positively oriented 2-simplices of S. Identifying π1 (G) with ˜ → G), we shall take [P ] to be an element of ker(G ˜ → G). It may be verified ker(G that [P ] is invariant under subdivisions of S, and is therefore in fact independent of S. We shall use the following result (proved in Sec. 3.17, Proposition 3.18, Theorem 3.1, Theorem 3.9, and Remark 3.12 of [17]; our [P ] here is the inverse of that in [17]) : 2.9.1. Theorem. If π : P → Σ and π 0 : P 0 → Σ are principal G-bundles, then [P ] = [P 0 ] if and only if P and P 0 are equivalent (i.e. there is a G-equivariant diffeomorphism φ : P → P 0 such that π 0 ◦ φ = π). Let ω be a connection on P, and let ai = hu (Ai ; ω), bi = hu (Bi ; ω). Choose ˜ ˜ covering ai , bi ∈ G, respectively. Let γ˜ : [0, 1] → G ˜ be the lift to G, ˜ with a ˜ i , bi ∈ G ˜ γ˜ω (0) = e, (the identity in G), of the path (2.7a) γω : [0, 1] → G : t 7→ hu q(xt O)q(x0 xt )q(Ox0 ); ω where xt = (cos 2πt, sin 2πt) ∈ ∂D as in Sec. 2.3. Then −1˜ ˜ ˜g · · · ˜b−1 a ˜−1 ˜1 γ˜ω (1)−1 . [P ] = ˜b−1 g a g bg a 1 ˜ 1 b1 a

(2.7b)

(It has been noted in Sec. 2.3 that the loop q(xt O)q(x0 xt )q(Ox0 ), consisting of the radial paths q(Ox0 ) and q(xt O), and the path q(x0 xt ) lying on q(∂D), is smoothly parametrizable.) A broader framework for understanding [P ] is available in the general theory in [22]; the bundle P is classified by an element of H 2 (Σ, π1 (G)), and [P ] is the evaluation of this cohomology element on the fundamental 2-cycle for Σ. 3. Representation of Connections by Holonomies and ‘Lassos’ In Sec. 6 we shall derive an explicit formula for the symplectic structure Ω on certain moduli spaces (J|AYM )−1 (O)/G (notation as in Sec. 1) of Yang–Mills z /G, where connections; the formula for Ω will actually be given on a space F[X] z 2g F[X] is a certain subset of G × g introduced below. Thus it will be necessary to understand in some detail the relationship between the space AYM of Yang–Mills z . The purpose of this section is to describe this connections and the spaces F[X] relationship, including at the level of derivatives (since that will be necessary to transfer the symplectic form from one space to another). It will be convenient

85

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

to state most of the results in the setting of the full affine space A rather than the subset AYM . Certain results specific to AYM will be presented in Sec. 5. A broader investigation, not restricted to two dimensions, of questions related to Propositions 3.5 and 3.6 appears in [3] and [7]; in particular, the idea of representing connections by holonomies and lassos (infinitesimal holonomies) is taken from these works. We denote this representation by I in (3.1b) below. The range I(A) has been characterized in [3] and [7]. Note on derivatives: As explained below in Sec. 3.3, we shall often be working with directional derivatives, rather than derivatives in any stronger sense. For the purposes of this paper, directional derivatives suffice. In particular, it is not necessary for our purposes to study any differentiable structures on the infinite dimensional spaces and quotient spaces we work with, although such structures may be introduced and are useful in other contexts. k

3.1. Λ (P,g) k

We shall use the standard notation Λ (P, g) to denote the vector space of all g-valued k-forms η on P for which: (i) η(v1 , . . . , vk ) = 0 whenever some vi is vertical (i.e. π∗ vi = 0); and (ii) Rg∗ η = Ad(g −1 )η for every g ∈ G. The Riemannian metric k

on Σ and the inner-product h·, ·ig on g induce a standard inner-product on Λ (P, g); R 1 for instance, if η, ζ ∈ Λ (P, g) then hη, ζi = Σ hη(e1 ), ζ(e1 )ig + hη(e2 ), ζ(e2 )ig dσ where (e1 , e2 ) are pairs of tangent vectors to P projecting to orthonormal bases on Σ, and dσ is the area measure on Σ. Since Σ is oriented, there is a “Hodge dual” k 2−k 1 1 (P, g); for instance, if ζ ∈ Λ (P, g) then ∗ζ ∈ Λ (P, g) is map ∗ : Λ (P, g) → Λ specified by ∗ζ(e1 ) = ζ(e2 ) and ∗ζ(e2 ) = −ζ(e1 ), where (e1 , e2 ) are as before. If 2 µ ∈ Λ (P, g) then ∗µ(p) = µ(e1 , e2 ) with e1 , e2 ∈ Tp P projecting as before. 3.2. The tangent space Tω A, and other tangent spaces A tangent vector to A (the space of all connections on P ) may be taken to 1 be the difference of two connections, i.e. an element of Λ (P, g). Thus we set 1

Tω A = Λ (P, g), for any ω ∈ A (of course, Tω A does not actually depend on ω). If B is a subset of a manifold M , and m ∈ B, then we shall denote by Tm B the set of all vectors in Tm M which are tangent to C ∞ paths in M which lie entirely on B. We shall use this same definition if B is a subset of A (for instance, B = AYM ), with the understanding that a ‘smooth path’ t 7→ ωt in A is a path for which (t, p) 7→ ωt (p) is smooth. Note that in general Tm B is not a vector space. 3.3. Directional derivatives If t 7→ ωt is a path in A such that (t, p) 7→ ωt (p) is smooth, then by dωt /dt we shall mean the pointwise directional derivative, i.e. it is the g-valued 1-form on ∂ ωt (p); this derivative is an element of P whose value at any p ∈ P is given by ∂t Tωt A. Next suppose f is a map from a subset of A to a subset of a manifold. If

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t 7→ ωt is a path in the domain of f , with (t, p) 7→ ωt (p) smooth, and if A = dωt /dt then by f 0 (ωt )A we shall mean the directional derivative df (ωt )/dt. This notation is meaningful if the derivative exists and does not depend on the choice of the path t 7→ ωt tangent to A. The linearity of the derivative, or even whether its domain is a linear space, will depend on f , and will have to be decided on a case-by-case basis. Differentiation of composite functions need also to be treated thus. We shall use an analogous definition of derivatives for subsets of manifolds, subsets of A, and for the infinite dimensional groups G and G0 . The theme of this section is the representation of a connection ω by the holonomies hu (κi ; ω), for a certain set of loops κi , and by the function which associates to each path starting at o the value of ∗Ωω at the point obtained by parallel translating u along the path by ω. To this end we introduce the following terminology: 3.4. The set P of paths on Σ, and the vector space L of maps P→g Let P be the set of all piecewise smooth paths [0, 1] → Σ based at o. We shall also use the space L of all maps F : P → g. Thus L is a vector space under the pointwise operations. Elements of L are a version of the ‘lassos’ (or lasso functions) introduced in [7]. t If φ : t 7→ φt ∈ L is a map then φ0 (t), or ∂φ ∂t , will denote the element of L whose ∂φt (c) ∂φt t (c) exists for all c ∈ P. We value at c ∈ P is ∂t ; thus ∂t is meaningful if ∂φ∂t take the derivative of a map ψ : M → L, where M is a subset of a manifold (or of A or of G), at m ∈ M , to be the map ψ 0 (m) : Tm M → L whose value on A ∈ Tm M is (ψ ◦ c)0 (0) for any C ∞ path c : [0, 1] → M with c0 (0) = A; thus ψ 0 (m) is meaningful only if (ψ ◦ c)0 (0) exists and depends only on A and not on the choice of c. The group G acts on L by (g, F ) 7→ Ad(g)F , where Ad(g)F is the element of L whose value at any c ∈ L is Ad(g)F (c). For the following, recall that π : P → Σ is a principal G-bundle over the oriented 2-dimensional manifold Σ, g is the Lie algebra of G, o is a basepoint on Σ, u ∈ π −1 (o) a basepoint for P , G is the group of bundle automorphisms of P , and Go the subgroup which fixes u. The curvature of ω ∈ A (the space of all connections on P ), denoted 2 Ωω , is a g-valued 2-form belonging to Λ (P ; g), and its Hodge dual ∗Ωω is the 0

g-valued function on P , belonging to Λ (P ; g), defined in Secs. 2.6 and 3.1. The holonomy of ω ∈ A around a loop κ based at o, with initial point u, is denoted hu (κ; ω). 3.5. Proposition. Let κ1 , . . . , κn be piecewise smooth loops generating the fundamental group π1 (Σ, o). If ω ∈ A, and c : [0, 1] → Σ is a piecewise smooth path on Σ starting at o, let def (3.1a) f ω (c) = ∗Ωω c˜ω (1) where c˜ω is the ω-horizontal lift of c starting at u. Define I : A → Gn × L : ω 7→ (hu (κ1 ; ω), . . . , hu (κn ; ω), f ω ) .

(3.1b)

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

87

Then ˆ −1 I(ω) where φ(u) ˆ ˆ ∈ G is given by φ(u) = uφ(u). (i) I(φ∗ ω) = Adφ(u) (ii) I(ω) = Ad(k −1 )I(ω 0 ), for some k ∈ G, if and only if ω 0 = φ∗ ω for some φ ∈ G; this φ can be chosen with φ(u) = uk. Thus I induces: a one-to-one map I : A/Go → Gn × L, and a one-to-one map I : A/G → (Gn × L)/G. ∗

cω )·k. Moreover, Proof. Suppose ω 0 = φ∗ ω and φ(u) = uk. Then c˜φ ω = (φ−1 ◦˜ ∗ ∗ ∗ ∗ f φ ω (c) = ∗Ωφ ω ≤ ft c˜φ ω (1) = ∗Ωω φ ◦ c˜φ ω (1) cω (1)k) = Ad(k −1 )f ω (c) . = ∗Ωω (˜ This proves (i) and part of (ii). For the converse in (ii), suppose I(ω 0 ) = Ad(k −1 )I(ω). By choosing a ψ ∈ G with ψ(u) = uk (ψ may be constructed by using a local trivialization around u), and considering ψ ∗ ω instead of ω we see that it suffices to assume that k = e, i.e. that I(ω) = I(ω 0 ). We will show below that this implies hu (C; ω) = hu (C; ω 0 ) for every piecewise smooth loop on Σ based at o. Assuming this for the moment, define φ : P → P : p 7→ τω0 (c)τω (c)p where c is any piecewise smooth path on Σ from π(p) to o. The equality of holonomies implies that φ is well-defined, i.e. independent of the choices for the path c. To see that φ is smooth, choose any y ∈ Σ, let uy = τω (cy )u, for some fixed path cy from o to y; consider σω : x 7→ τω (yx)uy , defined for x in some neighborhood of y and yx denoting the radial line (in some fixed chart around y) from y to x. Then σω is smooth (see for instance the theorem in Appendix I of [12]), and we conclude that φ, being the ‘ratio’ of the smooth sections σω and σω0 , is smooth. The definition of φ says that φ maps ω-horizontal curves into ω 0 -horizontal curves, and thus ω 0 and φ∗ ω, being connections with the same horizontal spaces, are equal. Thus it remains only to prove that if I(ω) = I(ω 0 ) then ω and ω 0 have the same holonomies around all piecewise smooth loops based at o. Let C : [0, 1] → Σ be such a loop. Since {κ1 , . . . , κn } generates π1 (Σ, o), and since part of the hypothesis (the first n components of I(ω) = I(ω 0 )) is that ω and ω 0 have equal holonomies around the κi ’s, we see that it suffices to assume that C is contractible as a loop based at o (for, in general, the composite of C with certain of the κi ’s and κi ’s is contractible). There is then (by the lemma in Appendix 7 of [12]) a piecewise smooth homotopy of C into the point curve at o; i.e., there is a continuous map H : [0, 1]2 → Σ : (t, s) 7→ Ht (s) = H(t, s) = H s (t), smooth on each of a finite number of closed rectangles whose union is [0, 1]2 , for which H0 = C, H1 (·) = o, and Ht (0) = Ht (1) = o for every t ∈ [0, 1]. Consider now [0, 1]2 → P : (t, s) 7→ C˜1ω (t, s) = τω (H s |[0, t])τω (H0 |[0, s])u

(3.2a)

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and [0, 1]2 → P : (t, s) 7→ C˜2ω (t, s) = τω (Ht |[0, s])u .

(3.2b)

Then C˜1ω and C˜2ω are piecewise smooth. Define gt,s ∈ G by C˜1ω (t, s) = C˜2ω (t, s)gt,s .

(3.2c)

Thus (t, s) 7→ gt,s is piecewise smooth, and, since t 7→ C˜1ω (t, s) is ω-horizontal, we have ! ∂gt,s −1 ∂ C˜2ω (t, s) + g = 0. ω ∂t ∂t t,s ∂ ∂s

to this, we obtain: ∂ ˜ω ∗ ∂ ∂ ∂gt,s −1 g (C2 ) ω . (3.3) =− ∂s ∂t t,s ∂s ∂t ∂ is 0, and so the right side of Since s 7→ C˜2ω (t, s) is ω-horizontal, (C˜2ω )∗ ω ∂s ∂ ∂ ω ∗ ˜ (3.3) equals (C2 ) dω( ∂t , ∂s ); and, for the same reason, the latter is equal to ∂ ∂ , ∂s ). Thus (C˜2ω )∗ (Dω ω) ( ∂t ! ˜ ω (t, s) ∂ C˜ ω (t, s) ∂ C ∂ ∂gt,s −1 2 2 g , . (3.4) = Ωω ∂s ∂t t,s ∂t ∂s Applying

Thus, noting that the determinant of D(π ◦ C˜2ω )|(t,s) (here [0, 1]2 has its usual orientation as a subset of R2 ) is equal to det H 0 (t, s), we have: Z s ∂gt,s −1 g = f ω (Ht |[0, v]) det H 0 (t, v) dv . (3.5) ∂t t,s 0 Further, g0,s = e. Thus the holonomies gt,1 = hu (Ht ; ω)−1 hu (C; ω) are determined 0 by f ω . Taking t = 1, and recalling that I(ω) = I(ω 0 ), and hence f ω = f ω , we conclude that hu (C; ω) = hu (C; ω 0 ). 3.6. Proposition. The map I in (3.1b) has directional derivatives at all points and (with the first n components moved by left translation to g = Te G) I 0 (ω)Y =

Z −

κ ˜ω 1

!

Z Y, . . . , − κ ˜ω n

Y, Dω YL

(3.6a)

where, as before, c˜ω denotes the ω-horizontal lift, with initial point u, of a path c on Σ based at o, and Z Y (3.6b) Dω YL : c 7→ ∗Dω Y c˜ω (1) − f ω (c), c˜ω

wherein, as usual, for any g-valued 1-form Y on P , Dω Y = dY + [ω, Y ] .

(3.6c)

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

89

Proof. For the first n components of (3.6a), it will suffice to verify the usual differentiation formula Z −1 Y (3.6d) hu (C; ω) δY hu (C; ω) = − ˜ω C

where δY denotes derivative in the Y -direction and C is any piecewise smooth loop on Σ with u being the initial point of the horizontal lift C˜ ω . Define gt,s ∈ G by requiring that s 7→ C˜ ω (s)gt,s be the ωt -horizontal lift of C starting at u. Thus gt,s solves h ∂ C˜ ω (s)g i t,s =0 (3.6e) ωt ∂s with initial conditions gt,0 = e and g0,s = e. Thus (t, s) 7→ gt,s is smooth on [0, 1]2 . ∂ , and using g0,s = e, we obtain ‘Expanding’ the left side of (3.6e) and applying ∂t t=0 ! d ∂ dC˜ ω (s) + (3.6f) Y gt,s = 0 . ds ds ∂t t=0 R Integrating · · · ds, and using gt,0 = e, we obtain Z ∂ Y. (3.6g) gt,s = − ∂t t=0 ˜ ω |[0,s] C Setting s = 1 in (3.6g) gives Eq. (3.6d). It remains to differentiate the last component f ω . Recall that f ω (c) = ω ω c (1)), for c ∈ P. Considering [0, 1] → A : t 7→ ωt , with (t, p) 7→ ωt (p) ∗Ω (˜ ∂ ω = Y , we have (in the second line we have used the notation smooth and ∂t t=0 t d p exp(tH)): p · H, for p ∈ P and H ∈ g, to mean dt t=0 d d ωt ω ω ω ωt c (1)) + d(∗Ω ) c˜ω (1) f (c) = ∗D Y (˜ c˜ (1) dt t=0 dt t=0 Z (3.6g) = ∗Dω Y (˜ cω (1)) + d(∗Ωω ) c˜ω (1) c˜ω (1) · − Y

Z

cω (1)) − ∗Ωω (˜ cω (1)) , = ∗Dω Y (˜

c ˜ω

Y c˜ω

where in the second step we used Eq. (3.6g) and in the last step we used the Ad equivariance of ∗Ωω . For the following, recall that P is the set of piecewise smooth paths on Σ starting at o, L is the vector space of maps P → g, and recall that G acts on L by pointwise conjugation: (Ad(g)f ) (c) = Ad(g)f (c) for every f ∈ L, c ∈ P, and g ∈ G. Combining with the conjugation action (g, x) 7→ gxg −1 of G on Gn , we obtain a left action Ad of G on Gn × L. It will often be convenient to consider the corresponding right-action, given by g 7→ Ad(g −1 ). As noted in Proposition 3.5, the map I : A → Gn × L is then equivariant with respect to the homomorphism ˆ G 7→ G : φ 7→ φ(u), where G is the automorphism group of P , u ∈ π −1 (o) the usual ˆ ˆ basepoint, and φ(u) is specified by φ(u) = uφ(u).

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3.7. Lemma. The orbit map γ˜α : G 7→ Gn × L : k 7→Ad(k −1 )α, where α = (a1 , . . . , an , f ) is any point in Gn × L, has directional derivatives everywhere and in every direction, and this derivative is linear: H, . . . , 1 − Ada−1 H, [f, H] . (3.7a) γ˜α0 (H) = 1 − Ada−1 n 1 The image γα0 (g) corresponds conceptually to the tangent space to the G orbit α · G at α. We shall often write it as α · g: α · g = γ˜α0 (g) . def

(3.7b)

The proof of (3.7a) is obtained readily by a simple calculation. 3.8. The Lie algebras L(G) and L(Go ) We shall review some standard (see, for instance, [2]) notions and terminology concerning the Lie algebra of the automorphism group of the bundle. Recall that ˆ if we associate to each φ ∈ P the map φˆ : P → G given by φ(p) = pφ(p), then ˆ this sets up an isomorphism φ 7→ φ of the group G with the group C(P ; G), the latter being the group of all smooth G-equivariant maps P → G (as in Sec. 2.7). If t 7→ φt ∈ G is smooth, in the sense that (t, p) 7→ φt (p), or equivalently, (t, p) 7→ φˆt (p) ˆ is smooth, then p 7→ φˆt (p)−1 ∂ φt (p) ∈ g is a smooth G-equivariant map P → g; ∂t

conversely, every smooth G-equivariant map H : P → g arises in this way (taking φˆt (p) = exp (tH(p)) ). For this reason, it is standard practice to take the set L(G) of all smooth G-equivariant maps P → g to be the Lie algebra for G (under pointwise operations, L(G) is a Lie algebra). The subalgebra of L(G) consisting of all H ∈ L(G) which vanish at u corresponds to the Lie algebra of Go (the subgroup of G fixing u ∈ π −1 (o)), and will be denoted L(Go ). ˆ in usual notation. From Recall (from Sec. 2.7) that φ∗ ω = Ad(φˆ−1 )ω + φˆ−1 dφ, this it is readily∗ computed that if t 7→ φt ∈ G is tangent at t = 0 to H ∈ L(G), and ∂φt ω ω ∈ A, then ∂t t=0 = dH + [ω, H]. Thus the derivative at ω of the action of G on A can be taken to be the linear map Dω : L(G) → Tω A : H 7→ dH + [ω, H] .

(3.8a)

The image Dω (L(G)) corresponds conceptually to the tangent at ω to the G-orbit ω · G; for this reason we shall often write ω · H = dH + [ω, H]

(3.8b)

and def

def

ω · L(G) = Dω (L(G)) and ω · L(Go ) = Dω (L(Go )) .

(3.8c)

The following result provides a way for viewing the map I : A/G → (Gn × L)/G (of Proposition 3.5) as a ‘diffeomorphism’ onto its image: 3.9. Theorem. Recall, from Proposition 3.5, the map I : A → Gn × L : ω 7→ (hu (K1 ; ω), . . . , hu (Kn ; ω), f ω ) .

(3.9a)

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

Then for any ω ∈ A ,

91

I 0 (ω) (ω · L(G)) = I(ω) · g I 0 (ω)−1 I(ω) · g = ω · L(G)

(3.9b)

ker I 0 (ω) = ω · L(Go ) .

(3.9d)

(3.9c)

Proof. Let H ∈ g, and choose H ∈ L(G) with H(u) = H. Recall the orbit ˆ −1 )I(ω) map γ˜α : G → Gn × L : g 7→ Ad(g −1 ) · α. The relation I(φ∗ ω) = Ad(φ(u) explained in Proposition 3.5 implies I 0 (ω)Dω H = I(ω) · H

(3.9e)

since, taking an appropriate path t 7→ φt ∈ G tangent at t = 0 to H we have ∂φt /∂t|t=0 = Dω H. Thus, I 0 (ω) (ω · L(G)) = I(ω) · g which is (3.9b). To show that I 0 (ω)−1 I(ω) · g = ω · L(G), consider a vector Y ∈ Tω A for which I 0 (ω)Y = I(ω) · H for some H ∈ g. Choosing H ∈ L(G) with H(u) = H, we then have I 0 (ω)Y = I 0 (ω)Dω H. In view of this and (3.9b), it will suffice to I 0 (ω)Y = 0. Define an prove that ker I 0 (ω) = ω · L(Go ), i.e. (3.9d). So suppose R Ad-equivariant map Z : P → g, by requiring Z(p) = c˜ω Y for any piecewise smooth path c : [0, 1] → Σ from o to π(p) for which c˜ω (1) = p; we show that: (a) Z is well-defined (b) Z is smooth (c) Dω Z = Y . Assuming (a) for the moment, (b) follows by writing out the definition of Z in terms of a local trivialization of P over a neighborhood of π(p) on which a chart is defined (with c being made up of a fixed initial segment from o to p followed by ‘radial’ lines, with respect to some chart, from p). Moreover, dZ (d˜ cω (t)/dt) = ω ω cω (t)/dt; Y (d˜ c (t)/dt) shows that D Z and Y agree on the ω-horizontal vector d˜ taking different choices of c|[0, t] we obtain all horizontal directions over c(t), and thus Dω Z = R Y . To prove (a), it will suffice, as in the proof of Proposition 3.5, to prove that c˜ω Y = 0 for every contractible loop c ∈ P. Let H : [0, 1]2 → Σ be, as in the proof of Proposition 3.5, a piecewise smooth homotopy of c into the point loop at o. Let t0 7→ ωt0 ∈ A, for t0 ∈ [0, 1], be such that (t0 , p) 7→ ωt0 (p) is smooth and ∂ωt0 /∂t0 |t0 =0 = Y . Introducing gt,s (ωt0 ) ∈ G as in Proposition 3.5 ([Eqs. (3.2a, b, c)], with respect to the connection ωt0 , we have, as in Eq. (3.5), Z s ∂gt,s (ωt0 ) −1 gt,s (ωt0 ) = f ωt0 Ht [0, v] det H 0 (t, v) dv . (3.9f) ∂t 0 0 ∂ 0 (ω)Y = 0 we have f ωt (Ht |[0, v]) = 0. Thus From the last component of I 0 ∂t t0 =0 applying ∂t∂ 0 t0 =0 to the right side of the above equation, and conjugating by gt,s (ω) we obtain: ∂ −1 ∂gt,s (ωt0 ) gt,s (ω) = 0. (3.9g) 0 ∂t ∂t0 t =0

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Therefore, in particular, g0,1 (ω)−1

∂g0,1 (ωt0 ) −1 ∂g1,1 (ωt0 ) = g (ω) 0 . 1,1 0 t =0 ∂t0 ∂t0 t =0

(3.9h)

As in the discussion following Eq. (3.5), the left side of the above equation equals R 0 and g1,1 (ωt0 ) = hu (c; ωt0 ); then, by (3.6d), the right side of (3.9h) equals − c˜ω Y , thereby proving our claim (a). The preceding result has the following curious consequence, whose significance is that a linear tangent space to A/Go can be defined in terms of Go -equivariant vector fields on fibers of A → A/Go . 3.10. Corollary. Suppose φ∗ ω = ω, for some φ ∈ Go and ω ∈ A. Then for any A ∈ Tω A (3.10) φ∗ A − A ∈ ω · L(Go ) . Proof. Since the mapping I : A → Gn × L is invariant under the action of Go on A it follows, upon taking a smooth path t 7→ ωt with ∂ωt /∂t|t=0 = A, that I 0 (ω)(φ∗ A) = dI(φ∗ ωt )/dt|t=0 = (I ◦ φ∗ )0 (ω)A = I 0 (ω)A = A. Thus φ∗ A − A ∈ ker I 0 (ω), and so, by (3.9d), φ∗ A − A ∈ ω · L(Go ). Since we shall be working mainly with Yang–Mills connections and since, as noted in Sec. 2.8, for such connections the last component f ω of I is constant (this being ∗Ωω (u)), we shall state some of the previous results specialized to a simpler framework. We introduce notation to describe the range I(AYM ). Recall the loops A1 , B1 , . . . , Ag , Bg which generate π1 (Σ, o) subject to the relation that B g Ag Bg Ag · · · B 1 A1 B1 A1 is the identity in π1 (Σ, o). z z and F[X] ) 3.11. Notation (FX

˜ → G), and X ∈ g, and a fixed integer g ≥ 1, we denote by F z For z ∈ ker(G X the set of all (a1 , b1 , . . . , ag , bg ) ∈ G2g satisfying

and

Ad(ai )X = Ad(bi )X = X, for each i ∈ {1, . . . , g}

(3.11a)

−1˜ ˜ ˜1 = z exp(−|Σ|X) ˜b−1 a ˜g · · · ˜b−1 ˜−1 g ˜g bg a 1 a 1 b1 a

(3.11b)

˜ covers x ∈ G. where |Σ| is the area of Σ, and x ˜∈G We also set z = {(ka1 k −1 , kb1 k −1 , . . . , kag k −1 , kbg k −1 , F[X] def

z , k ∈ G} . Ad(k)X) : (a1 , . . . , bg ) ∈ FX

(3.11c)

z z ⊂ G2g , but F[X] ⊂ G2g × g . Note that FX

3.12. Proposition. Let z = [P ], the bundle classifier (Sec. 2.9). For X ∈ g, let def

CX = {ω ∈ AYM : ∗Ωω (u) = X}

(3.12a)

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

def

C[X] = {ω ∈ AYM : ∗Ωω (u) ∈ Ad(G)X} .

93

(3.12b)

Suppose CX 6= ∅. Consider the map I : A → G2g × g : ω 7→ (hu (A1 ; ω), . . . , hu (Bg ; ω), ∗Ωω (u)) .

(3.12c)

Then: z . (i) I maps CX onto FX z . (ii) Go maps CX into itself, and I|CX induces a bijection CX /Go → FX z /G, (iii) G maps C[X] into itself and I|C[X] induces a bijection I : C[X] /G → F[X] z where the action of G on F[X] is by conjugation. Proof. (i) Theorem 2.9.1, describing [P ], along with (3.5) (which implies that z . γ˜ω (1) = exp(−|Σ|X) in the notation of Theorem 2.9.1), implies that I(CX ) ⊂ FX z The fact that I maps CX onto FX is the content of Theorem 4.1(i). (ii) and (iii): In both cases, preservation under Go or G and injectivity follow from Proposition 3.5, while surjectivity is as in (i). 3.13. Proposition. The map I of (3.12c) has directional derivatives at all points and, for any ω ∈ A and Y ∈ Tω A, ! Z Z I 0 (ω)Y =

−

˜ω A 1

Y, . . . , −

˜ω B g

Y, ∗Dω Y (u)

(3.13)

where we use the notation C˜ ω to denote the ω-horizontal lift, with initial point u, of a loop C (note: the first 2g components of (3.13) have been moved by left translation to Te G = g, a convention we shall often use in differentiating G-valued functions). Thus, in particular, I 0 (ω) : Tω A → g 2g+1 is a linear mapping. Proof. This is essentially a special case of the Proposition 3.5. The last ∂ to the defining formula component of (3.13) can be verified directly by applying ∂t ∗Ωωt (u) = dωt (e1 , e2 ) + [ωt (e1 ), ωt (e2 )] where (e1 , e2 ) is a pair of vectors in Tu P which project to a positively oriented t orthonormal basis in To Σ, and ωt ∈ A with (t, p) 7→ ωt (p) smooth and ∂ω ∂t t=0 = Y . 4. Yang Mills Connections In this section we shall construct explicit formulas for families of Yang–Mills connections over the compact surface Σ. Our objectives are: (i) to obtain a smooth family t 7→ ωt of Yang–Mills connections having specified values of hu (Ai ; ωt ), hu (Bi ; ωt ) and ∗Ωωt (u) (these depending smoothly on t); and (ii) to obtain explicit formulas for such ωt in terms of hu (Ai ; ωt ), hu (Bi ; ωt ) and ∗Ωωt (u). We shall show also that when t runs over an interval, the corresponding ∗Ωωt lie in one coadjoint orbit (in a sense to be explained later). These facts will be of essential

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use in later sections in our study of the symplectic structure on Yang–Mills moduli spaces. As has been mentioned in Sec. 1, individual Yang–Mills connections with specified holonomies and ∗Ωω (u) have been constructed by a different method in (Theorem 6.7 of) [1]. 4.1. Theorem. Let Σ be a compact connected oriented 2-dimensional Riemannian manifold of genus g ≥ 1, G a compact connected Lie group whose Lie algebra g has ˜ → G), where G ˜ → G is the universal an Ad-invariant inner-product, and z ∈ ker(G covering of G. (i) Let X ∈ g, and a1 , b1 , . . . , ag , bg ∈ G be such that: −1˜ ˜ ˜1 = z · exp(−|Σ|X) ˜b−1 a ˜g · · · ˜b−1 ˜−1 g ˜ g bg a 1 a 1 b1 a

Ad(ai )X = X

and

Ad(bi )X = X

f or every

i ∈ {1, . . . , g}

(4.1a) (4.1b)

˜ cover bi , ai ∈ G, respectively, for each i ∈ {1, . . . , g}, exp : g → G ˜ ˜i ∈ G where ˜bi , a ˜ → G). Then on a is the exponential map, |Σ| is the area of Σ, and z ∈ ker (G principal G-bundle π : P → Σ whose isomorphism class [P ] is equal to z, there is a Yang–Mills connection ω such that (relative to a basepoint u ∈ π −1 (o)): ∗Ωω (u) = X

(4.2a)

and, for each i ∈ {1, . . . , g}, the holonomies of ω around the basic loops Ai and Bi are given by: (4.2b) hu (Ai ; ω) = ai and hu (Bi ; ω) = bi . (ii) More generally, if 7→ c() = (a1 (), b1 (), . . . , ag (), bg (), X ) is a C ∞ path z (see Sec. 3.11), for some X ∈ g, then on the principal in G2g × g lying in F[X] G-bundle π : P → Σ whose isomorphism class [P ] is z there is a family of Yang– Mills connections 7→ ω , such that (, p) 7→ ω (p) is smooth (here p runs over P ), and with respect to some basepoint u ∈ π −1 (o), ∗Ωω (u) = X

(4.3a)

and, for each i ∈ {1, . . . , g}, hu (Ai ; ω ) = ai ()

and

hu (Bi ; ω ) = bi () .

(4.3b)

Proof. We will prove the more general statement (ii). We shall assume that runs over an interval containing 0. This will be used in the expression (4.11a) below, but entails no loss of generality for this proof. However, this additional hypothesis (actually just that runs over a connected subset of R) will have significance for Corollary 4.2. Step 1. Reduction to constant X We shall write X for X0 . Consider GX = {x ∈ G : Ad(x)X = X}, a closed subgroup of G; the map G → g : k 7→ Ad(k)X has constant rank and so its image,

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

95

the orbit [X] = Ad(G)X ⊂ g, is a smooth submanifold of g, and the induced map G/GX → [X] : k.GX 7→ Ad(k)X is a diffeomorphism. The quotient map G → G/GX , being a principal bundle projection, has smooth local sections; therefore there is a smooth map 7→ k , such that k0 = e

(4.4a)

X = Ad(k−1 )X .

(4.4b)

and Observe that since 7→ ({ai (), bi ()}, X ) is a smooth path in G2g × g lying z ˜ it follows that 7→ ({k ai ()k−1 , , and since z is in the center of G, entirely on F[X] −1 2g z . We shall construct below k bi ()k }) is a smooth path in G lying entirely on FX a smooth family (in the usual sense) of Yang–Mills connections ω0 on P with holonomies hu (Ai ; ω0 ) = k ai ()k−1 ,

hu (Bi ; ω0 ) = k bi ()k−1

(4.4c)

0

for every i ∈ {1, . . . , g}, and ∗Ωω (u) = X. We can choose 7→ Φ ∈ G such that: (i) (, p) 7→ Φ (p) is smooth, (ii) Φ0 = id, and (iii) Φ (u) = uk . Such Φ can be constructed by, for instance, trivializing P in a neighborhood of o and taking Φ over this set to be given by translation by a suitable smooth ‘bump-function’ (see Lemma 5.9.2 for a detailed argument). Then def (4.4d) ω = Φ∗ ω0 provides the family of connections we seek, i.e. each ω is a Yang–Mills connection, ω (p) depends smoothly on (, p) with p running over P , ω has the specified holonomies ai () and bi () around the loops Ai and Bi , and ∗Ωω (u) = X . Therefore we may and will proceed to construct ω , taking X to be the constant X0 = X. Thus the connection ω constructed below is really ω0 . Step 2. Construction of the bundle We shall construct a principal G-bundle P 0 over Σ by dividing Σ into two sets U and V and specifying a transition function s : U ∩V → G. Later we shall specify the connections ω by means of g-valued 1-forms ω,U over U and ω,V over V related in the usual way by s. After that it will be shown that the bundle we construct has isomorphism class equal to z, and that therefore we may take P 0 to be P . We shall use the representation of the surface Σ as a quotient of the planar unit disk D by the map q : D → Σ described in Sec. 2.3. We recall the notation briefly: def

∗ xt = (cos(2πt, sin 2πt), each point of D is thus of the form rxt with r ∈ [0, 1], t∈R def

∗ ti = i/(4g) ∗ the quotient q is specified by q(xt ) = q(xt0 )

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def

whenever t ∈ [tj−1 , tj ], t0 ∈ [tj+1 , tj+2 ], t − tj−1 = tj+2 − t0 , j ∈ J, where J = {1, 2, 5, 6, . . . , 4g − 3, 4g − 2}. We split Σ into two overlapping open subsets: def

def

U = {q(rxt ) : 0 ≤ r < 1, t ∈ [0, 1]} and V = {q(rxt ) : r >

1 , t ∈ [0, 1]} . (4.5) 2

We proceed to construct the transition function s : U ∩ V → G. In fact s will take values in the subgroup GX = {k ∈ G : Ad (k)X = X}. Let aV be a smooth 1-form on V for which daV is the area form dσ on V ; the existence of aV follows by de Rham’s theorem since V can be deformed into the 1-dimensional subspace q(∂D). Let aU be a smooth 1-form on U such that daU = dσ on U . It will be useful to associate to each point (a1 , b1 , . . . , ag , bg ) ∈ G2g the point −1 −1 −1 4g (a1 , b1 , a−1 1 , b1 , . . . , ag , ag , bg , bg ) ∈ G ; thus we set α1 () = a1 (), α2 () = b1 ()−1 , α3 () = a1 ()−1 , α4 () = b1 ()−1 , . . . or, more precisely, α4i−3 () = ai () and α4i−1 () = ai ()−1

α4i−2 () = bi ()

and α4i () = bi ()−1

(4.6a) (4.6b)

with i running over {1, 2, . . . , g}. This procedure and notation will be useful throughout this paper. Consider now a map ξ : R → G satisfying: (ξ1) ξ is smooth (ξ2) ξ(t) has the constant value Z αi (0) · · · α1 (0) · exp

aV X

(4.7)

q(x0 xi )

when t is near ti =

i 4g

(we mean also that ξ(t) = e for t near 0)

(ξ3) ξ(t)ξ(tj−1 )−1 = ξ(t0 )ξ(tj+2 )−1 for j ∈ J = {1, 2, 5, 6, . . . , 4g − 3, 4g − 2}, t ∈ [tj−1 , tj ], t0 ∈ [tj+1 , tj+2 ], t0 − tj+1 = tj − t (ξ4) ξ(t + 1) = ξ(t)ξ(1) for every t ∈ R. (ξ5) ξ(t) ∈ GX for every t ∈ R. The existence of such ξ is apparent upon using the fact that GX is a connected closed subgroup of G and contains each of the points in (4.7). Define (4.8a) s : U ∩ V → G : q(rxt ) 7→ ξ(t)exp{θ(r, t)X} wherein θ : ( 12 , 1) × R → R is any smooth function satisfying dθ = χ∗ (aU − aV )

(4.8b)

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

97

with χ : ( 12 , 1) × R → U ∩ V : (r, t) 7→ q(rxt ). The definition of θ, along with R Stokes’ theorem, implies that θ(r, t + 1) − θ(r, t) = Σ dσ = |Σ|. Then by condition (ξ4), and because, by Eqs. (4.1a) and (4.7), ξ(1) = exp(−|Σ|X), it follows that s is well-defined and smooth. Thus we have a principal G-bundle P 0 with s as transition function between two trivializations. More precisely, there is a principal G-bundle π 0 : P 0 → Σ

(4.9a)

smooth sections ψU : U → P 0 and ψV : V → P

(4.9b)

ψU (x) = ψV (x)s(x) for every x ∈ U ∩ V .

(4.9c)

with satisfying The basepoint over o is now u0 = ψU (o)

(4.9d)

Step 3. Construction of the connections ω We shall construct the connections ω on P 0 by specifying g-valued 1-forms ω,U over U and ω,V over V , related by ω,V = s · ω,U · s−1 + s · ds−1 on U ∩ V . For each i ∈ {1, . . . , 4g}, choose a smooth functionφi : [ti−1 , ti ] → [0, 1], increasing from0 near ti−1 to 1 near ti , (4.10a) and satisfying φj+2 (t0 ) = 1 − φj (t) if j ∈ J, t ∈ [tj−1 , tj ], t0 ∈ [tj+1 , tj+2 ], t0 − tj+1 = tj − t . (4.10b) (Recall that q matches xt and xt0 to the same point on Σ if t and t0 are as above.) Define ξ(0, ) = e, and, inductively, ξ(t, ) = ξ(t)ξ(tj−1 )−1 αj (0)−1 αj φj (t) ξ(tj−1 , ) for t ∈ [tj−1 , tj ], j ∈ J , (4.11a) and, having defined ξ(s, ) for s ∈ [0, tj+1 ], where j, j + 1 ∈ J, we set ξ(t0 , ) = ξ(t, )ξ(tj , )−1 ξ(tj+1 , )

(4.11b)

for t0 ∈ [tj+1 , tj+2 ], j ∈ J, with tj −t = t0 −tj+1 . Note that Eq. (4.11b) is consistent, as it should be, when t0 = tj+1 and t = tj . It is in (4.11a) that we use the added hypothesis that runs over an interval containing 0. The expression for ξ(t, ) in (4.11a) shows that ξ(t, ) = e for t near 0, and also that ξ(t, ) is constant for t near the endpoints of the intervals [tj−1 , tj ], j ∈ J; the ‘nearness’ here, being decided only by the behavior of ξ(t) and φj (and not of the function αj ), is uniform in . This together with the expression for ξ(t0 , ) in (4.11b) shows that ξ(t, ) is constant whenever t ∈ [0, 1] is near (uniformly in ) any of the ti . Thus ξ(t, ) can be extended, by ‘periodicity’ as in condition (ξ4), to a smooth function of (t, ) for all real t.

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Furthermore, since each αi () depends smoothly on , it follows that ξ(t, ) is smooth in (t, ) .

(4.11c)

ξ(t0 , )ξ(tj+2 , )−1 = ξ(t, )ξ(tj−1 , )−1

(4.11d)

Equation (4.11b) implies

which is, for = 0, consistent with condition (ξ3). In particular, ∂ξ(t, ) ∂ξ(t0 , ) 0 −1 ξ(t, )−1 . ξ(t , ) = − ∂t0 ∂t

(4.11e)

From the expression for ξ(tj ) given in (4.7), and from the expression for ξ(t, ) in (4.11a), we have, for j ∈ J, ! Z ξ(tj , ) = exp

aV X

αj ()ξ(tj−1 , ) .

(4.12a)

q(Kj )

Here we have used the notation Kj from Sec. 2.3: Kj is the restriction of the path t 7→ xt ∈ ∂D to t ∈ [tj−1 , tj ]. Recall that J = {1, 2, 5, 6, . . . , 4g − 3, 4g − 2}; we would like to obtain ξ(ti , ) for all i ∈ {1, . . . , 4g} = J ∪ (J + 2). To this end, we set t0 = tj+2 in the expression for ξ(t0 , ) in (4.11b). The resulting expression involves ξ(tj , ), and substituting in the value for this from (4.12a), and using the fact that q(Kj+2 ) = q(K j ), we then have ( !) Z ξ(tj+2 , ) = ξ(tj−1 , ) ξ(tj−1 , )−1 αj+2 () exp

aV X

ξ(tj+1 , ) .

q(Kj+2 )

(4.12b) Combining (4.12a) and (4.12b), we obtain ξ(ti , ) = βi () · · · β1 () wherein

for every i ∈ {1, . . . , 4g}

Z βi () = exp

aV X αi () .

(4.12c)

(4.12d)

q(Ki )

As we have noted before, ξ(t, ) is constant for t near the ti . Setting = 0 in (4.12c,d) and comparing with the expression for ξ(ti ) in (4.7), we see that ξ(ti , 0) = ξ(ti ). Using this in the expressions for ξ(t, ) and ξ(t0 , ) in (4.11a,b), and using condition (ξ3), it follows that ξ(t, 0) = ξ(t) for all t ∈ [0, 1] . (4.12e) From (4.12c,d) we also have, upon using the cancellations arising from q(Kj+2 ) = q(K j ), (4.12f) ξ(1, ) = α4g () · · · α1 () = ξ(1) Let

def γ q(rxt ) = ξ(t)−1 ξ t, f (r)

(4.13a)

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

99

where f : [0, 1] → [0, 1] is a smooth function, zero near r = 0, and equal to 1 for r ≥ 12 . From our previous observations, we conclude that: γ is a well-defined smooth function on U .

(4.13b)

Finally, we can define the g-valued 1-form ω,U on U by: ω,U = −dγ .γ−1 + XaU . def

(4.14a)

Recall the transition function s : U ∩ V → G given in (4.8a). We would like to define ω,V on V by requiring it to be equal to the g-valued 1-form sω,U s−1 −ds·s−1 on U ∩V . To this end it is necessary to verify that the latter extends to a well-defined smooth 1-form on V . So we compute: sω,U s−1 − ds · s−1 = −s · dγ · γ−1 · s−1 + XaU − ds · s−1 = −ξ(t) · dγ · γ−1 · ξ(t)−1 + XaU − {ξ 0 (t)ξ(t)−1 dt + X(aU − aV )} = −d(ξ · γ ) · (ξγ )−1 + XaV (4.13a)

=

−

∂ξ(t, ) ξ(t, )−1 dt + XaV . ∂t

As we have noted before, in (4.11e), ∂ξ(t, ) ∂ξ(t0 , ) 0 −1 ξ(t , ) = − ξ(t, )−1 ∂t ∂t when q matches xt with xt0 onto the same point on Σ. Therefore sω,U s−1 − ds.s−1 extends to a smooth g-valued 1-form ω,V on V : ω,V = −

∂ξ(t, ) ξ(t, )−1 dt + XaV ∂t

on

U ∩V .

(4.14b)

Equivalently: ω,V = Ad (s)ω,U + s · ds−1

on U ∩ V .

(4.14c)

Combining ω,U and ω,V we obtain our connection form ω on the bundle P 0 ; this is the connection on P 0 specified by: ∗ ω = ω,U ψU

and

ψV∗ ω = ω,V .

(4.15)

That (, p) 7→ ω (p) is smooth follows from the smoothness of ω,U and ω,V . That the g-valued 1-forms ω,U and ω,V depend smoothly on (together with the location on Σ) is evident from the expressions for these 1-forms given in (4.14a,b) (the first term on the right of (4.14b) is 0 in a neighborhood of the ‘bad point’ q(xti )), along with the fact that γ (x), as defined in (4.13a), is smooth in (, x). The smoothness of γ (x) in turn follows from the smoothness of ξ(t, ) in (t, ). Finally, the smoothness of ξ(t, ) follows from the hypothesis that the ai () and bi () depend smoothly on .

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Step 4. ω is Yang–Mills and ∗Ωω (u0 ) = X The first term on the right in the expression for ω,U in (4.14a) is (γ−1 )∗ ν where ν is the canonical g-valued left-invariant 1-form on G. Since dν + 12 [ν, ν] = 0, and recalling that daU = dσ on U , we have the following computation for the curvature of ω : 1 1 1 dω,U + [ω,U , ω,U ] = Xdσ − [dγ · γ−1 , XaU ] − [XaU , dγ · γ−1 ] . 2 2 2 Since γ takes values in GX , we conclude that 1 (4.16a) dω,U + [ω,U , ω,U ] = Xdσ . 2 This means ∗ (∗Ωω ) = X . (4.16b) ψU Then, by (2.6a), ω is Yang–Mills. Moreover, evaluating (4.16b) at o gives: ∗Ωω (u0 ) = X

(4.16c)

where u0 is the basepoint ψU (o). Step 5. Holonomies of ω around the loops Ai and Bi If c is a loop on Σ based at o, let us for the moment write h(c; ω ) to mean the holonomy hu0 (c; ω ). We shall show that h(Ai ; ω ) = ai () and h(Bi ; ) = bi () for every i ∈ {1, . . . , g}. To this end it will be convenient to use the loops 0 are just the loops Ki0 in Σ, based at o, from Sec. 2.3: the loops K10 , . . . , K4g A1 , B1 , A1 , B 1 , . . . , Ag , Bg , Ag , B g , in that order. Recall also the element (α1 , . . . , α4g ) ∈ G4g associated to (a1 , . . . , bg ) ∈ G2g , as explained in (4.6a,b). With this notation, our goal is to show that h(Ki0 ; ω ) = αi () for every i ∈ {1, . . . , 4g}. We show first that the path ! Z aV X (4.17) t 7→ ψV (q(xt )) ξ(t, )exp − q(x0 xt )

calculation / {t0 , . . . , is ω -horizontal; this is seen from the following R (valid for t ∈ t4g }), wherein we have written et = exp − q(x0 xt ) aV X : hdn oi dq(xt ) −1 ψV q(xt ) ξ(t, )et = Ad (ξ(t, )et ) ω,V ω dt dt ∂ξ(t, ) et + (ξ(t, )et )−1 ∂t dq(xt ) −ξ(t, )et aV X dt dq(xt ) ∂ξ(t, ) (4.14b) + aV = −ξ(t, )−1 X ∂t dt dq(xt ) −1 ∂ξ(t, ) − aV X + ξ(t, ) ∂t dt = 0.

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In these calculations we have used the fact that ξ(t, ) ∈ GX . allows us to compute the ω The fact that the path (4.17) is ω -horizontal holonomy of the loop q(x0 xti ), with ψV q(xt0 ) as initial point on the fiber. Thus, letting t → ti (see the remarks in Sec. 2.3 concerning smooth reparametrization of t 7→ q(xt )) in (4.17), and recalling the expression for ξ(ti , ) given by (4.12c,d), we have q(x0 xt ); ω = αi () · · · α1 () (4.18) h i ψV q(x0 )

for every i ∈ {1, . . . , 4g}. Since we are concerned ultimately with holonomies of loops based at o, we note the following consequence of (4.18): hu (q(x0 O)q(x0 xti )q(Ox0 ); ω ) = γ −1 αi () · · · α1 ()γ

(4.19a)

where γ ∈ G is specified by ψV q(x0 ) γ = τω q(Ox0 ) u. Now γ can be expressed as (for any r ∈ ( 12 , 1)) Z γ = exp −

q(rx0 ·x0 )

Z aV X s q(rx0 ) exp −

aU X

(4.19b)

q(O·rx0 )

wherein O · rx0 and rx0 · x0 are radial paths in D. Since s q(rx0 ) = exp{θ(r, 0)X} with each αi (). Thus from commutes with each αi (), it follows that γ commutes (4.19a, b) we have h0u (q(x0 O)·q(x0 xti )·q(Ox0 ); ω = αi () · · · α1 (). Consequently: h0u (Ki0 ; ω ) = αi ()

(4.20a)

where Ki0 is the arc q(x0 O)q(xti−1 xti )q(Ox0 ). In view of (4.6a, b), (4.20a) translates into hu0 (Ai ; ω ) = ai () and hu0 (Bi ; ω ) = bi ()

(4.20b)

for every i ∈ {1, . . . , g}. Step 6. The isomorphism class of P 0 is z, i.e. [P 0 ] = z In Proposition 3.12(i) we have shown in particular that if π : P → Σ is a principal G-bundle, and u ∈ π −1 (o) a basepoint, then the map I : ω 7→ ({hu (Ai ; ω) , [P ] hu (Bi ; ω)}) carries any Yang–Mills connection ω into the set FX 0 , with X 0 = [P ] 0 ∗Ωω (u), where FX 0 is the set of (a1 , . . . , bg ) ∈ G2g X 0 satisfying [P ] · exp(−|Σ|X ) = −1 −1 −1 ˜ ˜b−1 a ˜ → G) is the bundle ˜g · · · ˜b1 a ˜1 ˜b1 a ˜1 in usual notation, and [P ] ∈ ker(G g ˜g bg a classifier. Note that the statement of Proposition 3.12(i) is that the map I takes [P ] the set of Yang–Mills connections ω with ∗Ωω (u) = X 0 onto FX 0 (in our present [P ] notation); what we are using here is simply that this map is into FX 0 . (In fact the ‘onto’ part constitutes the first half of the theorem we are currently proving ! The proof of the ‘into’ part was given after the statement of Proposition 3.12, and is a consequence of the bundle-classifier description given in Theorem 2.9.1.)

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Applying the above observations to the connection ω on the bundle P 0 constructed above, and using the basepoint u0 , we have: ˜g ()−1˜bg ()˜ ag () · · · ˜b1 ()−1 a ˜1 ()−1˜b1 ()˜ a1 () exp(|Σ|X) . [P 0 ] = ˜bg ()−1 a

(4.21a)

z , and recalling the definiHowever, recalling our hypothesis (a1 (), . . . , bg ()) ∈ FX z tion of FX , the right side of the above equals z. Thus

[P 0 ] = z .

(4.21b)

Since [P 0 ] specifies the principal G-bundle π 0 : P 0 → Σ up to bundle equivalence, we may take any principal G-bundle π : P → Σ for which [P ] = z and transfer the connections ω all to P by a fixed bundle equivalence which takes u0 to any chosen basepoint u on π −1 (o) in P . Recall the notation I(ω): I(ω) = (hu (A1 ; ω), hu (B1 ; ω), . . . , hu (Ag ; ω), hu (Bg ; ω), ∗Ωω (u)) z , for X ∈ g, consisting of all (ka1 k −1 , kb1 k −1 , . . . , kag k −1 , kbg k −1 , and the set F[X] Ad(k)X) with k ∈ G and (a1 , . . . , bg , X) satisfying (4.1a,b). With this notation, we have the following corollary of the proof of the preceding theorem; this will be useful in Sec. 5.

4.2. Corollary. Let z , f or some X ∈ g . [0, 1] → G2g × g : t 7→ αt be a smooth path lying on F[X]

Then there is a map [0, 1] → AYM : t 7→ ωt such that : (a) (t, p) 7→ ωt (p) is smooth as a map [0, 1] × P → Λ1 P ⊗ g; (b) I(ωt ) = αt for every t ∈ [0, 1]; (c) for every t ∈ [0, 1] there is a φt ∈ G such that φ∗t Ωωt = Ωω0 ; (d) if the last component of αt is the same for every t ∈ [0, 1], then Ωωt = Ωω0 . Proof. We use the notation and construction from the proof of Theorem 4.1. Recall that we constructed Yang–Mills connections ωt satisfying (a) and (b) by first constructing certain Yang–Mills connections ωt0 (in the proof of Theorem 4.1 we wrote instead of t) and then setting ωt = Φ∗t ωt0 , where Φt is an automorphism of the bundle as discussed (4.4d). The connection ωt0 was described using sections ∗ ωt0 Ω = X0 dσ|U , ψU and ψV of P over U and V . We found in (4.16b) that ψU ∗ ωt0 ∗ ω00 Ω = ψU Ω ; by where X0 ∈ g is the last component of α0 . In particular, ψU ωt0 ω00 continuity and equivariance of the curvature, it follows that Ω = Ω . Therefore, 0 0 −1 ∗ ωt ∗ ω0 = Ωωt = Ωω0 = (Φ−1 (Φ−1 t ) Ω 0 ) Ω . Since Φ0 = idP , we can take φt = Φt , completing the proof of (c). For (d) we note that this corresponds to k = e in (4.4b) and hence to Φ = idP .

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In terms of the language and notation to be introduced in Sec. 5.3, the condition def (c) above means that J(ωt ) = Ωωt lies in one coadjoint orbit in the dual Lie-algebra L(G)∗ . 5. Marsden Weinstein Reduction for Yang Mills In this section we shall explore the symplectic structure of the space of connections and will show that a 2-form is induced on certain moduli spaces (J|AYM )−1 (O) /G of Yang–Mills connections. We begin by recording in Secs. 5.1– 5.5 some fairly standard notions and observations which we have drawn largely from [1]. ˜ → G) will be the bundle classifier [P ] Throughout this section, z ∈ ker(G described in Sec. 2.9. 5.1. The product h·∧·i on Tω A Recall that, for any ω ∈ A the tangent space Tω A is the space of 1-forms Λ (P, g); this was explained in Sec. 3.2. If A1 and A2 are two such tangent vectors then we define hA1 ∧ A2 i to be the smooth 2-form on Σ specified by: 1

˜ A2 (Y˜ )ig − hA1 (Y˜ ), A2 (X)i ˜ g hA1 ∧ A2 i(X, Y ) = hA1 (X),

(5.1)

˜ = X and ˜ Y˜ are tangent vectors to P at some point such that π∗ X where X, ˜ π∗ Y = Y . 5.2. The symplectic form on A If A1 and A2 are tangent vectors to A then define Z Ω(A1 , A2 ) = hA1 ∧ A2 i .

(5.2)

Σ

Thus Ω is a constant 2-form on A. Recalling from Sec. 3.1 the inner-product h·, ·i 1 and the Hodge dual ∗ on the space Tω A = Λ (P, g), we have Ω(A1 , A2 ) = hA1 , ∗A2 i, and so Ω is non-degenerate. Thus Ω is a symplectic form on A. 5.3. The Lie algebra L(G), and its dual L(G)∗ In view of the identification (given in Sec. 2.8) ˆ G → C(P, G) : φ 7→ φ, 0

the Lie algebra of G may be taken to be the space Λ (P, g) of all Ad-equivariant smooth functions P → g; we shall set 0

L(G) = Λ (P, g) .

(5.3a)

The metric on g and the Riemannian volume on Σ define an inner-product on L(G) in the obvious way (as noted in Sec. 3.1). If [0, 1]× P → P : (t, p) 7→ φt (p) is smooth

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and each φt ∈ G, with φ0 = id, then the corresponding initial tangent vector Y ∈ ˆt (p) ; on the other hand, if Y ∈ L(G) then Y is tangent L(G) is given by Y (p) = ∂ φ∂t t=0 to the 1-parameter group t 7→ φYt ∈ G specified by p 7→ φYt (p) = p. exp (tY (p)). The Lie algebra corresponding to the subgroup Go is L(Go ) consisting of those Y ∈ L(G) which vanish at u. 2 The dual space L(G)∗ will be taken to be Λ (P, g): 2

L(G)∗ = Λ (P, g) def

(5.3b) R

the dual pairing between µ ∈ L(G)∗ and Y ∈ L(G) is given by Σ π∗ hµ, Yig , where π∗ hµ, Yig is the (real-valued) 2-form on Σ whose lift to P by π ∗ is the 2-form hµ, Yig on P . In terms of the metric on L(G), the pairing of L(G)∗ and L(G) leads to identifying µ ∈ L(G)∗ with the Hodge dual ∗µ ∈ L(G), i.e. the pairing of µ with Y is the same as the inner-product of ∗µ with Y . 5.4. Adjoint and coadjoint actions The adjoint action of G on L(G) is given by (φ, Y) 7→ AdL(G) (φ)Y = Y ◦ φ−1

(5.4a)

since, considering the 1-parameter family φY t (as in Sec. 5.3 above) we have −1 )(p) = φ φ−1 (p) exp tY φ−1 (p) (φφY t φ = p. exp tY φ−1 (p) −1

(p) . = φY◦φ t For any Y ∈ L(G) and µ ∈ L(G)∗ we have (in the notation of Sec. 5.3), π∗ hµ, Y ◦ φi = π∗ h(φ−1 )∗ µ, Yi . Recalling from (5.4a) that Y ◦ φ = (AdL(G) φ−1 )Y, we conclude that the coadjoint action of G on L(G)∗ is given by: ˆ (Ad∗L(G) φ−1 )µ = (φ−1 )∗ µ = Ad(φ)µ

(5.4b)

ˆ φ(p) = pφ(p). ˆ the second equality follows from the relationship between φ and φ: 5.5. Curvature as moment map The action of G on A given by G × A → A : (φ, ω) 7→ φ∗ ω preserves the symplectic structure Ω on A. Differentiation of the map φ 7→ φ∗ ω, for any fixed ω ∈ A, gives a linear map: ∗ def ∂(φt ω) Dω : L(G) → Tω A : Y 7→ Dω Y = ∂t t=0

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wherein {φt } is a path in G tangent at t = 0, to Y . The derivative Dω Y is easily computable (for instance in terms of φˆt ) to give the standard formula Dω Y = dY + [ω, Y]

(5.5a)

(see, for instance, Theorem 5.2.16 in [2]). If Y ∈ L(G), and A is any tangent vector to A, then using (5.5a) and Stokes’ theorem, along with the Ad-invariance of the metric on g, we have: Z

Z hA ∧ Dω Y i =

Ω(A, Dω Y ) = Σ

h dA + [ω, A] , Yi .

Σ

Thus Ω(A, Dω Y ) = dJ|ω (A), Y

(5.5b)

where J : A → L(G)∗ : ω 7→ Ωω

(5.5c)

and dJ|ω (A) is the pointwise directional derivative of J at ω in the direction Y . Equation (5.5b) says that the curvature Ωω plays the role of the moment map of the action of G on A. Moreover, we have Jφ∗ ω = φ∗ Ωω , which, in view of (5.4b), says that J : A → L(G)∗ is G-(anti)equivariant

(5.5d)

wherein L(G)∗ is equipped with the Ad∗ action. These calculations are rigorous, and are formally analogous to the finite dimensional machinery of symplectic mechanics. A more detailed infinite dimensional formulation would entail passing from the directional derivatives we are using to sharper ones, such as Frech´et derivatives; but our interest here is not in such issues. As we shall see, algebraic and topological structural information concerning the symplectic nature of the moduli spaces of Yang–Mills connections can be extracted even by working simply with directional derivatives. Arguing at this level, a Yang–Mills connection ω is a critical point of ||J||2 ; thus ω should satisfy hJ(ω), J 0 (ω)(H)i = 0 for all H ∈ L(G),

(5.5e)

which, by (5.5b) and non-degeneracy of Ω, is equivalent to Dω ∗ J(ω) = 0

(5.5f)

and this is just the Yang–Mills equation of Sec. 2.5. The following result describes coadjoint orbits in L(G)∗ through Yang–Mills connections in terms of the value of the curvature at the basepoint u, i.e. in terms of ∗Ωω (u). Recall that, for X ∈ g, CX denotes the set of all Yang–Mills connections with ∗Ωω (u) = X, and C[X] denotes the union of the sets CAd(k)X as k runs over G.

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5.6. Proposition. (Coadjoint Orbits and C[X] ) −1

(i) For any µ ∈ L(G)∗ , (J|AYM ) (µ) ⊂ CX for some X ∈ g. If O is a coadjoint orbit in L(G)∗ through a point in J(AYM ) then (J|AYM )−1 (O) ⊂ C[X] for some X ∈ g; (ii) J(C[X] ) is a union of coadjoint orbits. (iii) C[X] = (J|AYM )−1 (∪α Oα ), where {Oα }α is a set of coadjoint orbits. Proof. (i) Let ω1 , ω2 ∈ AYM be such that J(ω1 ) and J(ω2 ) lie on the same coadjoint orbit in L(G)∗ , i.e. J(ω1 ) = (Ad∗L(G) φ −1 )J(ω2 ) for some φ ∈ G. In view of the expression for Ad∗L(G) φ−1 given in (5.4b), the relationship between J(ω1 ) and ∗ J(ω2 ) means that Ωω2 = Ωφ ω1 . Then ∗Ωω2 (u) = Ad(k −1 ) ∗ Ωω1 (u)

(5.6)

where k ∈ G is specified by φ(u) = uk. Thus if ω1 ∈ CX then ω2 ∈ CAd(k−1 )X , and so ω1 , ω2 ∈ C[X] . Setting φ = id we obtain the first part of (i). (ii) From (5.6) it is clear that J(C[X] ) is G-invariant. (iii) follows from (i) and (ii).

For our next result, we recall some notation introduced in Sec. 2. We are working with a principal G-bundle π : P → Σ which is classified up to bundle-equivalence ˜ → G). If ω is a connection on P then I(ω) ∈ G2g × g is by an element z ∈ ker(G defined by I(ω) = (hu (A1 ; ω), hu (B1 ; ω), . . . , hu (Ag ; ω), hu (Bg ; ω), ∗Ωω (u)) , z consists of all u ∈ π −1 (o) being the basepoint in P . For X ∈ g, the set FX 2g (a1 , b1 , . . . , ag , bg ) ∈ G satisfying

Ad(ai )X = X = Ad(bi )X for every i and the commutator condition (4.1b): −1 ˜ ˜1 = z · exp(−|Σ|X) , ˜b−1 a g ˜ g · · · b1 a

˜ cover ai , bi ∈ G, and z = [P ] is our usual bundle-classifier. The set where a ˜i , ˜bi ∈ G z F[X] consists of all (ka1 k −1 , kb1 k −1 , . . . , kag k −1 , kbg k −1 , Ad(k)X) with k ∈ G and z . Recall also that, for X ∈ g, C[X] denotes the set of all Yang–Mills (a1 , . . . , bg ) ∈ FX connections ω on P with ∗Ωω (u) ∈ Ad (G)X. As we have seen in Proposition 3.12, z . I(C[X] ) = F[X] 5.7. Theorem. Suppose ω0 , ω1 ∈ AYM are such that I(ω0 ) can be connected to z , for some X ∈ g. Then I(ω1 ) by a smooth path in G2g × g lying entirely on F[X] ∗ ∗ J(ω0 ) and J(ω1 ) lie on the same AdL(G) G-orbit in L(G) . (Note that this does not mean that ω0 and ω1 lie on the same G-orbit.) If ω0 , ω1 ∈ AYM are such that the point (hu (A1 ; ω0 ), . . . , hu (Bg ; ω0 )) can be connected to the point (hu (A1 ; ω1 ), . . . , hu (Bg ; ω1 )) by a smooth path in G2g lying

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107

z entirely on FX , for some X ∈ g, then J(ω0 ) and J(ω1 ) lie on the same Ad∗L(G) (Go )∗ orbit in L(G) . z , joining Proof. Let αt , 0 ≤ t ≤ 1, be a smooth path in G2g × g, lying on F[X] α0 = I(ω0 ) to α1 = I(ω1 ). By Corollary 4.2, there is a map [0, 1] → AYM : t 7→ ωt0 such that:

(a) (t, p) 7→ ωt0 (p) is smooth, (b) I(ωt0 ) = αt for every t ∈ [0, 1], and (c) {J(ωt0 ) : t ∈ [0, 1]} is contained in a single Ad∗L(G) (G)-orbit in L(G)∗ . Since I(ωt ) = I(ωt0 ), for t = 0 and for t = 1, it follows by Proposition 3.5(ii) that ωt and ωt0 , for t ∈ {0, 1}, lie on the same Go -orbit in AYM . Since J is Ad∗L(G) equivariant, it then follows that, for each of the two values for t ∈ {0, 1}, J(ωt ) and J(ωt0 ) lie on the same Ad∗L(G) Go -orbit. Since, as we have already seen, all the J(ωt0 ) lie on a single Ad∗L(G) G-orbit, we conclude that J(ω0 ) and J(ω1 ) lie on one Ad∗L(G) G-orbit. The second part is proved analogously, using Corollary 4.2(d). 5.8.1. k-forms on quotient spaces Let G be a group acting smoothly on a manifold M , by M × G → M : (m, g) 7→ 0 (g), where g is the Lie algebra γm (g). A vector v ∈ Tm M is vertical if it is in γm of G. A k-form over M/G may be taken to be a G-invariant k-form ω on M such that ω(v1 , . . . , vk ) = 0 whenever any of the vi is vertical. To keep the conceptual distinction between k-forms on M and such forms on M/G clear, we shall often write forms over M/G with a bar: ω in place of ω, for instance. The k-form ω on M/G is closed if dω = 0 on M . The above notions make sense even if M is simply a G-invariant subset of a manifold (on which G acts); the tangent spaces now consist of vectors tangent to paths in the appropriate spaces, and thus need not be vector spaces. For example, M z ⊂ G2g ×g introduced in Sec. 3.12, with G acting by conjugation. could be the set FX The notions also make sense if M is a G-invariant subset of the infinite dimensional space A, with the group G acting as usual. Suppose N is an H-space, M a G-space, j : H → G a smooth group homomorphism, if φ : N → M a j-equivariant mapping then a quotient map φ : N/H → M/G is induced. If φ is differentiable (assuming M and N are such that this is meaningful) then the derivative dφ : T N → T M maps vertical vectors to vertical vectors. Now consider a mapping f : N/H → M/G which is of the form φ for some equivariant smooth map φ : M → N . Suppose that: (i) dφ(v) is vertical if and only if v is vertical, and (ii) for any m ∈ M , every vector in Tφ(m) N is in the image of dφm modulo the vertical vectors in Tφ(m) N then we shall say that f is a local diffeomorphism of quotients. If f is a bijection and a local diffeomorphism then we shall say it is a diffeomorphism of quotients. If φ : N → M is smooth, equivariant, and induces a diffeomorphism of quotients φ then the association ω 7→ φ∗ ω induces an isomorphism (in particular, a bijection)

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of k-forms over N/H with k-forms over M/G; here we are concerned with all kforms not only smooth ones. This fact which will be useful for us is the only reason for introducing the terminology concerning diffeomorphisms of quotients. The conveniently suggestive terminology introduced above needs to be used with care. A (local) diffeomorphism of quotients f : N/H → M/G may be equal to ψ for some badly-behaved, but equivariant, ψ. Also f may be diffeomorphism of quotients but f −1 may not be so. However, a composite of smooth maps which induce (local) diffeomorphisms of quotients induces a (local) diffeomorphism of quotients. Suppose now that G acts smoothly on manifolds M and N , and consider an equivariant smooth map φ : M → N . Let y ∈ N and let Gy be the isotropy group at y; thus φ−1 (y) is a Gy -space. Taking j now to be the inclusion Gy → G, the inclusion map iy : φ−1 (y) → φ−1 (y · G) is j-equivariant and thus induces a map iy : φ−1 (y)/Gy → φ−1 (y · G)/G . This map iy is a bijection. Let us verify that iy is a diffeomorphism of quotients. Consider any smooth path [0, 1] → M : t 7→ at lying on φ−1 (yG), with φ(a0 ) = y. Since the orbit map γy : G → N : g 7→ yg is smooth and of constant rank, it follows that y · G is a smooth submanifold of N and the induced map G/Gy → G · y is a diffeomorphism. Combining this with the fact that G → G/Gy is a principal bundle projection, we conclude that there is a smooth path [0, 1] → G : t 7→ kt such that k0 = e and φ(at ) = ykt for every t ∈ [0, 1]. Writing bt = at kt−1 , we have at = bt kt and bt ∈ φ−1 (y). Thus every vector in Ty φ−1 (y · G) is of the form diy (B) + dγy (K) for some B ∈ Ta0 φ−1 (y) and K ∈ g. From this it follows readily that iy is a diffeomorphism of quotients. We shall use this in the context where N = g, with the adjoint action of G on z , and φ : M → N : (a1 , . . . , bg , Y ) 7→ Y . Since φ−1 (X) consists of all it, M = F[X] z z we can identify it, as a GX -space, with FX . (a1 , . . . , bg , X) with (a1 , . . . , bg ) ∈ FX z z Thus we have the diffeomorphism of quotients FX /GX → F[X] /G. 5.8.2. Symplectic structures induced on quotient spaces We shall review here certain well-known observations concerning symplectic structures induced (by the ‘Marsden–Weinstein’ procedure) on quotient spaces. These observations are meant to serve as a guide to our investigations of the Yang– Mills situation, and we will not be concerned here with issues involving smoothness of the quotient spaces. The notation used in this section is meant for use in this section only. Let J : M → L(G)∗ be an Ad∗ -equivariant moment map for a group G acting on a (finite dimensional) symplectic manifold (M, Ω). We shall indicate how Ω induces symplectic structures on the quotient spaces (J|AYM )−1 (O)/G and (J|AYM )−1 (Y )/ GY , wherein Y ∈ L(G)∗ and O is any coadjoint orbit in L(G)∗ . Let O be a coadjoint orbit in L(G)∗ . Let Θ denote the canonical symplectic form on O:

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

˜ Y˜ ] Θp (Xp , Yp ) = p [X,

109

(5.6a)

˜ Y˜ ∈ L(G) are any vectors corresponding, by the where p ∈ O ⊂ L(G)∗ , and X, coadjoint action of G on the orbit O, to the vectors Xp , Yp ∈ Tp O. Consider now the map φ : J −1 (O) → M × O : m 7→ (m, J(m)) . We can view Ω and Θ as being 2-forms on M × O (by lifting by the ‘coordinate projections’). The pullback φ∗ (Ω − Θ) is a closed equivariant 2-form on J −1 (O) which vanishes on the G-orbit directions. This is the 2-form Ω on J −1 (O)/G: Ω on J −1 (O)/G is given by φ∗ (Ω − Θ) .

(5.6b)

(Note that Ω is not, in general, the projection of Ω on J −1 (O)/G; in fact Ω does not, in general, vanish in the G-orbit directions in J −1 (O) and so does not correspond to a form on J −1 (O)/G.) It is clear that Ω is closed; it can be proven to be nondegenerate. Thus Ω is a symplectic structure for J −1 (O)/G. For any Y ∈ O, the inclusion iY : J −1 (Y ) → J −1 (O) induces a bijection iY : J −1 (Y)/GY ' J −1 (O)/G . Since (φ∗ Θ)|J −1 (Y ) = 0 and (φ∗ Ω)|J −1 (Y ) = Ω|J −1 (Y ), we have φ∗ (Ω − Θ)|J −1 (Y ) = Ω|J −1 (Y ) . Thus ∗

iY (Ω) is the same as the projection of Ω|J −1 (Y ) on J −1 (Y )/GY .

(5.6c)

This approach will provide the direction for our explicit determination of the symplectic structure on Yang–Mills moduli spaces in Sec. 6. Returning to the Yang–Mills situation, we show in Proposition 5.10 that the 2-form (in the sense of Sec. 5.8.1) on the quotient (J|AYM )−1 (O)/G obtained by the Marsden–Weinstein procedure (restricted to AYM ) transfers to a 2-form Ω over z /G. Since we have not shown that Ω|AYM is non-degenerate, it is not clear that F[X] z Ω should be non-degenerate. By exploiting the concrete setting of F[X] /G within 2g (G × g)/G, we shall show in Sec. 5 that Ω is indeed symplectic on appropriate smooth subsets of (J|AYM )−1 (O)/G. There are some differences between what we are doing and the usual study of symplectic structures on quotient spaces. We are working with the subset AYM of the full symplectic space A. In the usual setting, one works at the regular values of the moment map, but we are not singling out such points. The setting we are working with is infinite dimensional, though of course the quotient spaces z /G) are finite dimensional. (corresponding to subsets of F[X] z , and 5.9. Relationships between the spaces (J|AYM )−1 (O), F[X] their quotients

We collect together the relationships that exist between the various moduli spaces we have been using.

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Recall that the moment map for the G-action on the symplectic space (A, Ω) is the curvature map J : A → L(G)∗ : ω 7→ J(ω) = Ωω . Let µ ∈ J(AYM ), and denote by O the G-orbit through µ. Let def

def

X(µ) = Xµ = ∗µ(u) , def

i.e. X(µ) = ∗Ωω1 (u) for any ω1 ∈ (J|AYM )−1 (µ). The set [X(µ)] = {Ad(k)X(µ) : k ∈ G} depends only on the coadjoint orbit O in which µ lies. z consists of 2g-tuples (a1 , b1 , . . . , ag , bg ) ∈ G2g such that The set FX(µ) ˜ ˜1 = z · exp(−|Σ|Xµ ), ˜−1 Ad(ai )Xµ = Xµ = Ad(bi )Xµ for each i, and ˜b−1 g a g · · · b1 a ˜ ˜ → G) where y˜ ∈ G is any element projecting to y ∈ G, and z = [P ] ∈ ker(G z consisting classifies the bundle P up to equivalence. We have also the set F[X(µ)] −1 −1 −1 z . The of all (k a1 k, . . . , k bg k, Ad(k )Xµ ) with k ∈ G and (a1 , . . . , bg ) ∈ FX(µ) z z conjugation action makes FX(µ) a GX(µ) -space, and F[X(µ)] a G-space. There is ˆ ˆ the group homomorphism G → G given by φ 7→ φ(u) where φ(u) ∈ G is specified ˆ by φ(u) = uφ(u). The kernel of this homomorphism is Go . Furthermore, the

homomorphism restricts to a homomorphism of isotropy groups G µ → GX(µ) . We have the G-G equivariant mapping z : ω 7→ (hu (A1 ; ω), . . . , hu (Bg ; ω), ∗Ωω (u)) I : (J|AYM )−1 (O) → F[X(µ)]

(5.7a)

which restricts and induces z : ω 7→ (hu (A1 ; ω), . . . , hu (Bg ; ω)) . Iµ : (J|AYM )−1 (µ) → FX(µ)

(5.7b)

5.9.1. Theorem. Let µ, O, X = X(µ), and the map I, be as above, and let G µ be the isotropy group at µ of the G-action on O. Then (i) The map I induces one-to-one local diffeomorphisms of quotients (in the sense of Sec. 5.8.1) I˜µ

I˜

z z and (J|AYM )−1 (µ)/G0 −→ FX . (J|AYM )−1 (O)/G0 −→ F[X]

(5.7c)

The first map is G − G-equivariant (the left side is a G-space, the right side a Gspace), and the second is G µ − GX -equivariant. (ii) We have the commuting diagram of equivariant maps: (J|AYM)−1 (µ)  Iµ y

−→

j

−1 (J|AYM  ) (O)  yI

z FX

i

z F[X]

−→

(5.7d)

wherein the vertical arrows arise from I and the horizontal ones from inclusions (i is given by (a1 , . . . , bg ) 7→ (a1 , . . . , bg , X)). For the quotients we have the commuting diagram: j

µ (J|AYM )−1  (µ)/G  I µy

−→

z /GX FX

−→

i

(J|AYM)−1 (O)/G  yI z F[X] /G

(5.7e)

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111

where the horizontal maps are diffeomorphisms of quotient spaces and the vertical maps are one-to-one local diffeomorphisms of quotients, both in the sense explained in Sec. 5.8.1. Proof. (i) The equivariance and injectivity of the maps in (5.7c) follow directly from Proposition 3.5 and the fact that f ω is constant when ω ∈ AYM . That I is differentiable (as usual, in the sense of directional derivatives) is a special case of Proposition 3.11. Theorem 3.9 (specifically, (3.9d)) implies that I 0 (ω)v is 0 if and only if v is Go -verical. For the surjectivity of the derivative I 0 (ω), we recall Theorem 4.1(ii), where a smooth family of Yang–Mills connections were constructed z ; in Corollary 4.2 we had noted that this corresponding to a smooth path in F[X] family of connections lies within (J|AYM )−1 (O). These facts show that I 0 (ω) is surjective. The results for the second map in (5.7c) are obtained as a consequence, by restriction. (ii) The commutativity of (5.7d) is apparent, while equivariance follows from simple arguments, some of which we have already referred to in (i). In the commuting diagram (5.7e), arguments very similar to those in (i) imply that the map I is one-to-one and a local diffeomorphism of quotients, for instance, Theorem 3.9 (specifically, (3.9c)) implies that I 0 (ω)v is vertical if and only if v is vertical, while (i) implies that I 0 (ω) is surjective. To see that I µ is injective, let ω1 , ω2 ∈ J|(AY M )−1 (µ), and Iµ (ω1 ) = g −1 Iµ (ω1 )g for some g ∈ GX . There is, by (the proof of) Proposition 3.5, a φ ∈ G such that ˆ = g. Applying J, we have µ = φ∗ µ, i.e. φ ∈ G µ . Thus ω1 and ω2 = φ∗ ω1 and φ(u) µ ω2 lie on the same G -orbit. ‘Injectivity’ for Iµ0 (ω) follows from Lemma 5.9.3 below. We have already seen in Secs. 5.8.1 and 5.8.2 that the lower horizontal arrow z z i : FX /GX → F[X] /G is a diffeomorphism of quotients. It is apparent that the top arrow j in (5.7e) is a bijection. Using the properties of the other arrows already proven and chasing the diagram (5.7e) we see that j is a diffeomorphism of quotients. 5.9.2. Lemma. Let µ ∈ J(AYM ), X = ∗µ(u) as usual, GX = {k ∈ G : Ad(k)X = ˆ restricts X}, and gX the Lie algebra of GX . The homomorphism G → G : φ 7→ φ(u) µ µ to a homomorphism G → GX which is surjective, and its derivative L(G ) → g X is also surjective. Proof. For surjectivity of G µ → GX , let g ∈ GX , and pick ω ∈ (J|AYM )−1 (µ). Establish a chart U → R2 on Σ in a neighborhood of o, and consider the smooth section σω : U → P obtained by setting σω (x) to be the parallel translate of u along the radial (in the chart on U ) path from o to x. Since GX is connected, there is a smooth map φ : U → GX such that φ(0) = g, and φ = e near ∂U . Thus there is a unique element ψ ∈ G specified by the requirements that ψ (σω (x)) = σω (x)φ(x) for ˆ x ∈ U and ψ = id outside U ; in particular ψ(u) = g. Since ω ∈ AYM , ∗J(ω) (σω (x)) is the constant X(µ). Thus for any m ∈ Σ, there is a point m ˜ ∈ π −1 (m) such that ˆ m) Adψ( ˜ −1 J(ω)(m) ˜ = J(ω)(m). ˜ By equivariance this holds on all of π −1 (m). µ Thus ψ ∈ G .

112

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Surjectivity of L(G µ ) → gX is seen by applying the above argument to a path in GX . For example, if Y ∈ gX then we can choose a smooth map [0, 1] × U → GX : (t, x) 7→ φt (x) such that φt (o) = exp(tY ) and φt = e near ∂U . The corresponding ψt ∈ G µ trace a smooth path in the sense that (t, p) 7→ ψt (p) is smooth, and corresponds under the homomorphism G µ → GX to the path t 7→ exp (tY ) in GX , thus having derivative Y . 5.9.3. Lemma. Let µ ∈ J(AYM ), X = ∗µ(u), g X the Lie algebra of GX , and z given in (5.7b). ω ∈ (J|AYM )−1 (µ). Recall the map Iµ : (J|AYM )−1 (µ) → FX Then (5.8a) Iµ0 (ω) (ω · L(G µ )) = Iµ0 (ω) · gX (5.8b) Iµ0 (ω)−1 Iµ (ω) · g X = ω · L(G µ ) Proof. The G µ−GX -equivariance of Iµ implies that Iµ0 (ω) (ω · L(G µ )) ⊂ Iµ0 (ω) · g X . Lemma 5.9.2 and the G µ −GX -equivariance of Iµ imply the opposite inclusion, and thus (5.8a) is proven. The proof of (5.8b) will use the analogous result in the setting of A which we proved in (3.9c) of Theorem 3.9. Consider any Y ∈ Tω (J|AYM )−1 (µ) for which Iµ0 (ω)Y = Iµ (ω) · H for some H ∈ g X . By Lemma 5.9.2, we can choose H ∈ L(G µ ) such that H(u) = H. Equivariance of Iµ implies Iµ0 (ω)H = Iµ (ω) · H, and so Iµ0 (ω)Y = Iµ0 (ω)H. Thus it will suffice to prove that ker Iµ0 (ω) ⊂ ω · L(G µ ). In fact, we shall verify that ker Iµ0 (ω) = ω · L(Goµ ), where Goµ = G µ ∩ Go . The equivariance relation Iµ0 (ω)H = Iµ (ω) · H mentioned above implies that ker Iµ0 (ω) ⊃ ω · L(Goµ ). For the opposite inclusion, consider any Y ∈ ker Iµ0 (ω). By (3.9c) of Theorem 3.9, there is a Z ∈ L(Go ) such that Y = ω · Z. The G-equivariance of J then implies that J 0 (ω)Y = J(ω) · Z, where the latter is obtained as the derivative in the Z-direction of the G-coadjoint-orbit through J(ω) ∈ L(G)∗ . Since Y is tangent to the level surface J −1 (µ), it follows that J(ω) · Z = 0, i.e. µ · Z = 0. Thus for every real t, the mapping φˆt : P → G : p 7→ exp (tZ(p)) is an element of C(P ; G) corresponding to some φt ∈ G µ . Noting that (t, p) 7→ φ(t, p) is C ∞ , we conclude that Z ∈ L(G µ ). Since Z(u) = 0, it follows also that φt ∈ Go , and thus Z ∈ L(Goµ ). Note: A version of Lemma 5.9.3 in the general setting, and along the lines of Theorem 3.9 may also be formulated. In the discussions above, we began with a point µ ∈ J(AYM ) (or, equivalently, z z and F[X] . the coadjoint orbit O through µ), and then produced the spaces FX z Conversely, if X ∈ g then, as we noted in Proposition 5.6(ii), FX is the union of images under I of (J|AYM )−1 (µ) for a certain set of points µ ∈ L(G)∗ . Putting together the results and observations made above, we obtain the Marsden–Weinstein reduction Ω of Ω on the various realizations of Yang–Mills moduli spaces: 5.10. Proposition. Let O be a coadjoint orbit in J(AYM ) ⊂ L(G)∗ , µ a point on O, X = Xµ = ∗µ(u), and G µ the isotropy group of the G-action at µ, all as in Sec. 5.9.

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

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Then the 2-form Ω|(J|AYM )−1 (µ) corresponds to a 2-form Ω on (J|AYM )−1 (µ)/G µ (thus, comparing with (5.6c), Ω corresponds to the 2-form produced by the Marsden– Weinstein procedure). This 2-form Ω then corresponds to a 2-form, also denoted Ω, independent of the choice of µ, on (J|AYM )−1 (O)/G by the diffeomorphism of z /GX quotients j in the diagram (5.7e). There are also 2-forms Ω on the quotients FX z and F[X] /G, such that all the 2-forms Ω on these spaces are images of each other via the maps in (5.7e). Proof. Recall from the definition (in Sec. 5.8.1) of 2-forms on quotient spaces, that any G-equivariant 2-form on (J|AYM )−1 (µ) which vanishes in the G µ -orbit directions corresponds to (and actually ‘is’) a 2-form on (J|AYM )−1 (µ)/G µ . The argument that Ω|(J|AYM )−1 (µ) vanishes in the G µ -orbit directions is the same as for finite dimensions: if ω ∈ (J|AYM )−1 (µ), A ∈ Tω (J|AYM )−1 (µ) (i.e. is a vector in Tω A tangent to a ‘pointwise smooth’ path lying on (J|AYM )−1 (µ)), and H ∈ L(G µ ), then using the fact that J is the moment map for the G-action on the symplectic space (A, Ω), we have, with obvious notation, Ωω (A, ω · H) = (dJω (A), H) = 0 the last equality holds because A is tangent to a level surface of J. Thus Ω|(J|AYM )−1 (µ) corresponds to a 2-form Ω on J −1 (µ)/G µ . The diagram (5.7e) now allows us to transfer Ω, by means of the local difz z /GX and F[X] /G; it has feomorphisms of quotients, successively to the spaces FX been noted in Sec. 5.8.1 that diffeomorphisms of quotients allow transfer of k-forms z /GX (and, analogously, from either quotient space to the other. That Ω on FX z z z is on F[X] /G) is indeed specified on all of FX /GX follows from the fact that FX −1 the union of images under I of (J|AYM ) (µ) as µ runs over a subset of J(AYM ) z /G does (Proposition 5.6(ii)). Equivariance of J and I also implies that Ω on F[X] not depend on the choice of µ within its coadjoint orbit O. Finally we can transfer z Ω from F[X] /G to (J|AYM )−1 (O)/G by the diffeomorphism of quotients I (the right vertical arrow in (5.7e)). As noted before in the finite dimensional context, the transferrence of Ω to the space (J|AYM )−1 (O)/G is not implemented by the ‘obvious’ choice of using Ω|(J|AYM )−1 (O). A direct specification of Ω on (J|AYM )−1 (O)/G would require working with Θ as in (5.6a,b). 6. The Symplectic Structure on the Yang Mills Quotients In this section we shall obtain an explicit description of the 2-form Ω on the moduli space (J|AYM )−1 (O) /G induced by Ω, and we shall then show that this form is symplectic on appropriate subsets of the moduli space. z is the set of all (a1 , . . . , bg ) ∈ G2g satisfying Ad(ai )X = X = Recall that FX Ad(bi )X for every i ∈ {1, . . . , g}, and the commutator condition C(a1 , b1 , . . . , ag , bg ) = z · exp(−|Σ|X), where −1˜ ˜ ˜1 ˜ : (a1 , b1 , . . . , ag , bg ) 7→ ˜b−1 a ˜g · · · ˜b−1 ˜−1 C : G2g → G g ˜ g bg a 1 a 1 b1 a

(6.1a)

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AMBAR SENGUPTA

˜ projecting to y ∈ G (since ker(G ˜ → G) ⊂ Z(G), where y˜ denotes any element in G ˜ the choices of the a ˜i and bi do not affect the value of C). If α = (a1 , b1 , . . . , ag , bg ) ∈ G2g it will be convenient to use (as was done earlier in (4.6a,b)) the notation −1 −1 −1 α1 = a1 , α2 = b1 , α3 = a−1 1 , α4 = b1 , . . . , α4g−1 = ag , α4g = bg .

(6.1b)

z z then Tα FX denotes the set of all vectors in Tα G2g which If α = (a1 , . . . , bg ) ∈ FX z . Consistent with the are tangent at α to C ∞ -smooth paths in G2g lying on FX z notation (6.1b), we shall write a typical element of Tα FX in the form

(α1 H1 , α2 H2 , α5 H5 , α6 H6 , . . . , α4g−2 H4g−2 )

(6.1c)

and it will be convenient to set H3 = −Ad(α1 )H1 , H4 = −Ad(α2 )H2 , . . . , H4g = −Ad(α4g−2 )H4g−2 .

(6.1d)

z is contained in C −1 (z · exp Because of the commutator constraint that FX z is in the kernel of dC(α), (−|Σ|X)), any vector (a1 H1 , b1 H2 , . . . , bg H4g−2 ) in Tα FX i.e. satisfies 4g X Ad(αi−1 · · · α1 )−1 Hi = 0 . (6.1e) i=1

With notation and conventions as above, we have the following result. 6.1. Theorem. Let π : P → Σ be a principal G-bundle over a compact connected oriented Riemannian 2-manifold Σ of genus g ≥ 1, with G a compact connected Lie group equipped with an Ad-invariant inner-product h·, ·ig on its Lie algebra g. As explained in Sec. 5, the symplectic structure Ω on the space A induces a 2-form Ω on the space (J|AYM )−1 (O)/G, where O is the coadjoint orbit in L(G)∗ through any z , point in J(AYM ); as in Proposition 5.10, this corresponds to a 2- form Ω on FX z where X = ∗µ(u), with µ ∈ O. Let α ∈ FX , where z = [P ] is the usual bundle z , then: classifier, and H (1) , H (2) ∈ Tα FX 1 X (1) (2) Ωα H (1) , H (2) = ik hAd(αi−1 · · · α1 )−1 Hi , Ad(αk−1 · · · α1 )−1 Hk ig 2 1≤i,k≤4g

(6.2) wherein ik = 1 if i < k, ik = −1 if i > k, and ik = 0 if i = k. Proof. Consider any C ∞ paths 7→ α(1) () and 7→ α(2) ()

(6.3a)

tangent at = 0 to H (1) and H (2) , respectively; i.e. 0

0

(2) (0) = H (2) . α(1) (0)−1 α(1) (0) = H (1) and α−1 2 (0)α (1)

(2)

Let ω and ω Theorem 4.1.

(6.3b)

be the corresponding connections as constructed in the proof of

115

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

(1)

(2)

Recall from Corollary 4.2 that J is constant on the families ω and ω . Therefore (as per the specification of Ω in Proposition 5.10), Ω(H (1) , H (2) ) may be cal(1) (2) culated by evaluating Ω on the tangents, at = 0, to 7→ ω and 7→ ω . We shall use the notation and construction in the proof of Theorem 4.1. Consider for a moment a path 7→ ω of Yang–Mills connections corresponding z as in the proof of Theorem 4.1 (in the language to a smooth path 7→ α() ∈ FX of the proof of Theorem 4.1 we are working with ω0 = ω ). Then by the expression for ω,U given in (4.14a) (in terms of γ which is given by (4.13a)): h i ∂ω,U = −d f (r)ξ(t)−1 ∂2 ξ(t, 0) . ∂ =0

(6.3c)

Moreover, for t ∈ [ti−1 , ti ], calculation using the expression for ξ(t, ) in (4.11a) and the properties of φi shows that (details of this argument are presented below, following the proof): −1 −1 Hi + Adξi−2 Hi−1 + · · · + Ad ξ0−1 H1 ξ(t, 0)−1 ∂2 ξ(t, 0) = φi (t) Ad ξi−1 (6.3d) wherein ξk = ξ(tk ) and Hk = αk (0)−1 α0k (0). (1) (2) Returning to ω and ω , let Θ(1) =

and

(1) ∂ω ∂ =0

Θ(2) =

(2) ∂ω . ∂ =0

Then, using Eq. (6.3c) and Stokes’ theorem for the third equality below, Z D E Θ(1) ∧ Θ(2) Ω(Θ(1) , Θ(2) ) = Σ

Z = U

Z

1

= 0

*

(1) (2) ∂ω,U ∂ω,U ∧ ∂ =0 ∂ =0

d ξ(t)−1 ∂2 ξ (2) (t, 0) dt ξ(t)−1 ∂2 ξ (1) (t, 0), dt

1 X D (1) (2) E + Hi , Hi 2 i=1 g

=

D

X

4g

(6.3d)

+

−1 −1 Hi , fk−1 Hk fi−1 (1)

1≤i
(2)

E g

wherein fi = Ad(αi · · · α1 ). Therefore, by skew-symmetry of Ω, we have: 1 Ω Θ(1) , Θ(2) = 2

X 1≤i,k≤4g

−1 −1 ik hfi−1 Hi , fk−1 Hk i. (1)

(2)

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AMBAR SENGUPTA

Details for (6.3d): Recall (4.11a): ξ(t, ) = ξ(t)ξ(tj−1 )−1 αj (0)−1 αj φj (t) ξ(tj−1 , ) for t ∈ [tj−1 , tj ], j ∈ J (6.4a) wherein J = {1, 2, 5, 6, . . . , 4g − 3, 4g − 2} . This implies, for t ∈ [tj−1 , tj ] and j ∈ J, ξ(t, 0)−1 ∂2 ξ(t, 0) = φj (t)Ad ξ(tj−1 )−1 Hj + ξ(tj−1 )−1 ∂2 ξ(tj−1 , 0) .

(6.4b)

To obtain (6.3d), we now need to extend this to other values of t. Thus consider t0 ∈ [tj+1 , tj+2 ] corresponding, as usual, to t ∈ [tj−1 , tj ] with j ∈ J. Recall from (4.12c) that, for every i ∈ {1, . . . , 4g}, Z aV X αi ()ξ(ti−1 , ) . ξ(ti , ) = exp q(Ki )

This implies ξ(ti , 0)−1 ∂2 ξ(ti , 0) = Ad ξ(ti−1 , 0)−1 Hi + ξ(ti−1 , 0)−1 ∂2 ξ(ti−1 , 0) .

(6.4c)

Now from (4.11d) we have: ξ(t0 , ) = ξ(t, )ξ(tj−1 , )−1 ξ(tj+2 , ) .

(6.4d)

Recalling also from (4.12e) that ξ(t, 0) = ξ(t), and writing ξi for ξ(ti ), we then have from (6.4d): ξ(t0 )−1 ∂2 ξ(t0 , 0)

=

−1 −1 Ad ξj−1 ξj+2 ξ(t)−1 ∂2 ξ(t, 0) −1 −1 −1 −1 ξj+2 ξj−1 ∂2 ξ(tj−1 , 0) + ξj+2 ∂2 ξ(tj+2 , 0) − Ad ξj−1

(6.4b,c)

=

(6.1d)

−1 −1 −1 φj (t)Ad ξj+2 Hj + 0 + Adξj+1 Hj+2 + ξj+1 ∂2 ξ(tj+1 , 0)

=

−1 −1 − φj (t)Ad (ξj+2 αj+2 )Hj+2 + Ad ξj+1 Hj+2

=

−1 + ξj+1 ∂2 ξ(tj+1 , 0) −1 −1 Hj+2 + ξj+1 ∂2 ξ(tj+1 , 0) 1 − φj (t) Ad ξj+1

=

−1 −1 Hj+2 + ξj+1 ∂2 ξ(tj+1 , 0) φj+2 (t0 )Ad ξj+1

This, together with (6.4b), shows that these formulas can be ‘iterated backward’ thereby yielding (6.3d). . Before proceeding to show that Ω is a symplectic 2-form on appropriate subsets z z /GX , we shall make a few simple remarks concerning FX . of FX

THE MODULI SPACE OF YANG–MILLS CONNECTIONS OVER A COMPACT SURFACE

117

z 6.2. Structure of FX

˜ X be the subgroup of G ˜ (the universal cover of G) consisting Let X ∈ g, and let G ˜ ˜ ˜ X is not the universal of all elements k such that Ad(k)X = X (note that in general G ˜ cover of GX ; if G is semisimple then GX is compact but the universal cover of GX ˜ X is a connected Lie group. Every element of G ˜ X can may not be compact). Then G c ˜ be written as a product of an element of GX , the connected component of the center ˜ s = {b−1 a−1 ba : a, b ∈ G ˜ X }, the semisimple part of ˜ X , and an element of G of G X ˜ GX ; this decomposition is unique up to translation by elements of the discrete group ˜s ∩ G ˜ c . Let z = zc zs , with zc ∈ G ˜ c and z = zs ∈ G ˜ s , be such a decomposition G X X X X ˜ → G). of z ∈ ker(G z z 6= ∅. Considering (a1 , . . . , bg ) ∈ FX , we have Suppose FX −1˜ ˜ ˜1 = zs zc exp(−|Σ|X) . ˜b−1 a ˜g · · · ˜b−1 ˜−1 g ˜ g bg a 1 a 1 b1 a

˜ s . Thus z should be decomThe left side, being a product of commutators, is in G X posable as zc zs with (6.5a) zc = exp(|Σ|X) . Conversely, if z and X are such that z can be decomposed as zc zs with zc = ˜ s , then F z is non-empty. exp(|Σ|X) and zs ∈ G X X ˜ X , it follows that so does Note that since both z and zc belong to the center of G zs . z is of the Thus if z and X are such that (6.5a) holds then every element of FX form (6.5b) (a01 a1 , b01 b1 , . . . , a0g ag , b0g bg ) with (a01 , . . . , b0g ) any arbitrary elelement of the torus (GcX )2g (the projection onto G2g ˜ c )2g ), and of (G X

(a1 , . . . , bg ) any element of the projection onto (GsX )2g of the set of elements of ˜ s )2g which satisfy (G X −1˜ ˜ ˜ 1 = zs . ˜b−1 a ˜g · · · ˜b−1 ˜−1 g ˜ g bg a 1 a 1 b1 a

(6.5c)

˜ s is a semisimple compact connected Lie group, solutions to (6.5c) Since G X always exist ([21] contains a proof). However, if X is a regular element of g, i.e. if ˜ X coincides with the maximal torus whose Lie algebra contains X, and if (6.5a) G ˜ s contains only the identity and so in this case F z coincides with the holds, then G X X c 2g ˜ . torus (GX ) = G2g X 6.3. Symplectic nature of Ω In the case G is semisimple and X = 0, it has been proven in [13, 14] that Ω is symplectic on the subset of F0z /G corresponding to the subset of F0z where the isotropy group of the G-conjugation action is the center Z(G). The proof in [13, 14]

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works, with simple modifications, in the present general Yang–Mills case as well. However, we will show how the Yang–Mills case may be related to the flat case and then apply the [13, 14] result. z = (GcX )2g . ExaminaAs we have noted, if (6.5a) holds and X is regular then FX z tion of (6.2) now shows that in this case the 2-form Ω on FX is a simple translationPg invariant 2-form which corresponds to the standard 2-form i=1 (xi dyi − yi dxi ) on z /GX , and since in Rg × Rg , and is thus symplectic. Since (J|AYM )−1 (O)/G ' FX z z the present case FX /GX ' FX , we conclude that Ω is a symplectic form on the Yang–Mills moduli space (J|AYM )−1 (O)/G. ˜ s is non-trivial. As noted above, F z is Now suppose (6.5a) holds but that G X X 2g ˜ c )2g with the essentially the projection onto G of the product of the torus (G X zs ˜ s ˜ s )2g which satisfy GX , by which we mean the set of elements of (G space FX X ˜s → G ˜ X is a covering with the discrete ˜c × G (6.5c). Since the mutliplication map G X X ˜c ∩ G ˜ s }, and since G ˜ → G is also a covering, every C ∞ kernel {(k, k −1 ) : k ∈ G X X ∞ ˜ ˜ s . Noting that the action (as path in GX is a projection of a C path in GcX × G X ‘deck transformations’) of the kernel of this projection is essentially trivial on the z decompose as direct tangent spaces, we conclude that the tangent spaces to FX zs c 2g sums of the corresponding tangent spaces to (GX ) and FX (GsX ), the latter being zs ˜ s (GX ) on G2g . the projection of FX The formula (6.2) for Ω shows that it splits up as a sum of two expressions, one zs (GsX ). Since we have already dealt with Ω for (GcX )2g , for (GcX )2g and one for FX zs ˜ s zs GsX /GX ' we focusnow on the FX GX , or, more exactly, on the quotient FX zs s s FX GX /GX . In other words, we are considering a compact connected semisimple Lie group L = GsX and the 2-form given by (6.2) on the space C −1 (zs )/L where C is the product commutator map −1˜ ˜ ˜g · · · ˜b−1 a ˜−1 ˜1 C(a1 , . . . , bg ) = ˜b−1 g a g bg a 1 ˜ 1 b1 a

˜ s projecting to y ∈ Gs ; thus C here is the same as wherein y˜ is any element of G X X s 2g ˜ the restriction to (GX ) of our usual product commutator map G2g → G. As proven in [13, 14] (Proposition IV.E in [13] and Proposition 3.3 in [14]), this 2-form is symplectic on the ‘smooth part’ of the moduli space C −1 (zs )/L ; by the smooth part we mean the subset corresponding to the points in C −1 (zs ) where the conjugation action of L has the center of L as isotropy group (it was shown in z , this [13] that if g ≥ 2 such a non-empty subset always exists). Returning to FX corresponds to the subset where the isotropy group of the GX -action is Z(GX ). Thus Ω is a symplectic form on the corresponding subsets of the Yang–Mills moduli spaces (J|AYM )−1 (O)/G. 6.4. G = SU (n), an example We shall work out what our conclusions say for the group G = SU (n). The work [1] also provides a description of the Yang–Mills connections in this case. Since SU (n) is simply-connected, there is only one choice for the bundle P (the product bundle) and [P ] = z = e. We may work with a diagonal form for X; thus suppose |Σ|X is a diagonal matrix with diagonal (iλ1 Ik1 , . . . , Iλm Ikm ), where the

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λj are distinct (summing up to 0), and Ikj is the kj × kj -identity matrix,   |Σ|X = 



iλ1 Ik1 ..

 .

. iλm Ikm

Then GX consists of block-diagonal matrices. The center Z(GX ) of GX consists of the matrices   µ1 Ik1   ..   . µm Ikm where each |µj | = 1 and µkmm · · · µk11 = 1. The semisimple part GsX consists of the matrices   A1 ..  , with each Aj ∈ SU (kj ) .  . Am Thus exp(−|Σ|X) belongs to GsX if and only if eik1 λ1 = 1, . . . , eikm λm = 1 . Thus Yang–Mills connections with |Σ|X as ‘total curvature’ exist if and only if each eikj λj = 1. Assume that X satisfies this condition. Let ω be a Yang–Mills connection with ∗Ωω (u) = X. Then the holonomies of ω around the basic loops A1 , B1 , . . . , Ag , Bg can be displayed in the form (a01 a1 , b01 b1 , . . . , a0g , b0g ) with (a01 , b01 , . . . , a0g , b0g ) running over the torus Z(GX )2g and −1 (Ikm ) (a1 , . . . , bg ) running essentially over C1−1 (Ik1 ) × · · · × Cm

where Cj is the product commutator map −1 −1 −1 Cj : SU (kj )2g → SU (kj ) : (a1 , . . . , bg ) 7→ b−1 g ag b g ag · · · b 1 a1 b 1 a1 .

Thus the moduli space of Yang–Mills connections with ∗Ωω (u) = X is identified by means of the map I (see (5.7a, c)) with the product −1 (Ikm )/SU (km ) . Z(GX )2g × C1−1 (Ik1 )/SU (k1 ) × · · · × Cm

(*)

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The 2-form Ω on the moduli space is made up of the corresponding 2-forms, all obtained from the same formula (6.2), on each of the component spaces in the above decomposition. The space Cj−1 (Ikj )/SU (kj ) corresponds to the case X = 0, i.e. it describes the moduli space of flat SU (kj )-connections over Σ. Furthermore, the symplectic structure on this space as obtained above as a component space in (∗) is given by the same formula as (6.2). Thus representation (∗) of the Yang–Mills moduli space for X preserves the symplectic structures as well. The case G = SU (2) is especially simple. Here there are only two possibilities: (i) m = 1, X = 0: flat connections i 0

(ii) m = 2, k1 = k2 = 1, |Σ|X = 2πn 0 −i (n being a non-zero integer), and the moduli space is the 2g-dimensional torus. Since the latter spaces are connected, Theorem 5.7 guarantees that these are exactly the Yang–Mills moduli spaces (J|AYM )−1 (O)/G; furthermore, the symplectic structure is, in all these cases, the P2g ‘constant’ one corresponding to the usual symplectic form i=1 (xi dyi − yi dxi ) on R2g . The symplectic structure on spaces of flat connections have been studied in numerous works [6, 8, 9, 10, 11, 12, 13, 14, 19, 23, 24, 25], to name a few. In particular, methods for computing the volumes of maximal strata of these spaces exist; the above analysis suggests the Yang–Mills moduli space volumes should be obtainable from the volumes for the flat moduli spaces. It is, however, necessary to be careful with regard to the stratifications of the spaces involved. Recall also that the moduli space C[X] /G (where C[X] = {ω ∈ AYM : ∗Ωω (u) ∈ Ad(G)X}), which we have described explicitly above for SU (n), contains but is not necessarily equal to any one of the moduli spaces (J|A YM )−1 (O)/G; as we have noted before, in a certain sense (see Theorem 5.7), every connected stratum of C[X] /G is a stratum of some (J|A YM )−1 (O)/G. Here we are using the word ‘stratum’ in a loose sense. A more detailed study of the spaces (J|AYM )−1 (O)/G, such as determination of their volumes, will require a more detailed delineation of the relationship between the strata of these moduli spaces and those of the moduli spaces C[X] /G. We shall leave these issues for another investigation. As we have seen, the moduli spaces C[X] /G appear quite naturally, and our results above give a fairly detailed picture of these spaces. For instance, in the special case where all the kj equal 2, all the connected strata and their symplectic volumes have been determined in [20] and therefore provide a complete description of the corresponding Yang–Mills moduli spaces C[X] /G and (J|AYM )−1 (O)/G. Acknowledgement At various stages of this work the author has received support from NSF grant DMS 9400961, U.S. Army Research Office #DAAH04-94-G-0249, and from the Alexander von Humboldt Foundation. This paper was written in large part during my stay at the Institut f¨ ur Mathematik, Ruhr-Universit¨at Bochum, and it is a pleasure to thank Professor S. Albeverio and colleagues for their hospitality.

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References [1] M. Atiyah and R. Bott, “The Yang–Mills equations over Riemann surfaces”, Phil. Trans. R. Soc. Lond. A 308 (1982) 523–615. [2] Bleecker, Gauge Theory and Variational Principles, Addison-Wesley, Inc., 1981. [3] B. K. Driver, “Classification of bundle connection pairs by parallel translation and lassos”, J. Funct. Anal. 83 (1989) 185–231. [4] D. Fine, “Quantum Yang–Mills on compact surfaces”, Commun. Math. Phys. 140 (1991) 321–338. [5] R. Forman, “Small volume limits of 2-d Yang–Mills”, Commun. Math. Phys. 151 (1993) 39–52. [6] W. Goldman, “The symplectic nature of fundamental groups of surfaces”, Adv. Math. 54 (1984) 200–225. [7] L. Gross, “A Poincar´e lemma for connection forms”, J. Funct. Anal. 63 (1985) 1–46. [8] J. Huebschmann, “Symplectic and Poisson structures of certain moduli spaces”, preprint (1993). [9] L. C. Jeffrey, “Extended moduli spaces of flat connections on Riemann surfaces”, Math. Annalen 298 (1994) 667–692. [10] L. C. Jeffrey and J. Weitsman, “Toric structures on the moduli space of fat connections on a Riemann surface: Volumes and the moment map”, Adv. Math. 106 (1994) 151– 168. [11] Y. Karshon, “An algebraic proof for the symplectic structure of moduli space”, Proc. Amer. Math. Soc. 116 (1992) 591–605. [12] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. 1, John Wiley and Sons, 1963. [13] C. King and A. Sengupta, “An explicit description of the symplectic structure of moduli spaces of flat connections”, J. Math. Phys. Special Issue on Topology and Physics 35 (1994) 5338–5353. [14] C. King and A. Sengupta, “The semiclassical limit of the two dimensional quantum Yang–Mills model”, J. Math. Phys. Special Issue on Topology and Physics 10 (1994) 5354–5361. [15] C. King and A. Sengupta, “A symplectic structure for connections on surfaces with boundary”, Commun. Math. Phys. 175 (1996) 657–671. [16] A. Sengupta, “Quantum gauge theory on compact surfaces”, Ann. Phys. 221 (1993) 17–52. [17] A. Sengupta, “Gauge theory on compact surfaces”, to appear in Memoirs Amer. Math. Soc. [18] A. Sengupta, “The semiclassical limit for gauge theory on S 2 ”, Commun. Math. Phys. 147 (1992) 191–197. [19] A. Sengupta, “The semiclassical limit of SU (2) and SO(3) gauge theory on the torus”, to appear in Commun. Math. Phys. [20] A. Sengupta, “The semiclassical limit of SU (2) and SO(3) gauge theory on compact surfaces”, Commun. Math. Phys. 169 (1995) 297–314. [21] A. Sengupta, A Yang–Mills Inequality for Compact Surfaces, preprint (1996). [22] N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, 1951. [23] A. Weinstein, “The symplectic structure on moduli space”, in The Floer Memorial Volume, ed. Helmut Hofer et al., Progress in Mathematics, Birkhauser Verlag, 1995. [24] E. Witten, “On quantum gauge theories in two dimensions”, Commun. Math. Phys. 141 (1991) 153–209. [25] E. Witten, “Two dimensional quantum gauge theory revisited”, J. Geom. Phys. 9 (1992) 303–368.

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION A. ABDESSELAM and V. RIVASSEAU Centre de Physique Th´ eorique, Ecole Polytechnique 91128 Palaiseau Cedex, France Received 30 September 1996 We introduce a new type of cluster expansion which generalizes a previous formula of Brydges and Kennedy. The method is especially suited for performing a phase-space multiscale expansion in a just renormalizable theory, and allows the writing of explicit non-perturbative formulas for the Schwinger functions. The procedure is quite model independent, but for simplicity we chose the infrared φ44 model as a testing ground. We used also a large field versus small field expansion. The polymer amplitudes, corresponding to graphs without almost local two and four point functions, are shown to satisfy the polymer bound.

1. Introduction Many of the important results in constructive field theory, for instance the work done on the Yang–Mills theory in 4 dimensions [4, 28], make use of a mathematically rigorous implementation of renormalization group techniques. Albeit well understood at the perturbative level, their application in a constructive framework is still a difficult enterprise. After a period of first successes [20, 27, 17, 22, 15, 4], progress in this branch of mathematical physics was somewhat slowed down. It was partly because new conceptual tools were needed, for instance renormalization group around a surface singularity, involving dynamical N1 expansions and random matrix methods [18, 16, 29]. But another reason is that the heavy technical apparatus needed in the proofs is reaching a critical mass preventing potential readers from being fully convinced by their mathematical rigor, and discouraging actual authors from venturing into the painful task of writing them in detail. Therefore many efforts were devoted since then to the clarification and the improvement of these techniques. One can already observe the growth of two complementary approaches. In the first, a single step of renormalization group applied to algebras of polymer activities is emphasized. One has then to define the proper Banach spaces to support the iteration of the procedure, and to study the corresponding dynamical system. Two versions of this method were developed mainly around Brydges [8, 9, 12, 13] on the one hand, and Pordt [26, 30] on the other. The other approach to rigorous renormalization group methods, is through phase-space expansions. It was initiated by Glimm and Jaffe in [19], and became the main tool of our group at the Ecole Polytechnique [27, 15, 28]. In this approach, the 123 Reviews in Mathematical Physics, Vol. 9, No. 2 (1997) 123–199 c World Scientific Publishing Company

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iteration of many renormalization group steps is treated as a whole. The method looks closer to perturbation theory of which it is a kind of truncated expansion. The idea is to expand enough to put into display the sources of divergences, typically “almost local”, i.e. high frequency, insertions of 2 and 4 point functions in the φ44 model for instance [31]. However we must not expand too much in order to prevent the accumulation of fields at the same spot and with the same frequency. Previous schemes of phase-space expansions used different interpolations for the horizontal expansions (in real space) and the vertical ones (along the frequency axis). The procedure was inductive and the vertical interpolation nonlinear. Furthermore a Mayer expansion step has to be performed at every scale. These features render the previous expansions quite obscure, and the explicit writing of a formula for the output too difficult. In this paper we worked out a new type of phase-space cluster expansion, based on a linear interpolation, which treats horizontal and vertical expansions on the same footing. It generalizes the formula of Brydges and Kennedy for horizontal cluster expansions [11, 8, 2, 1]. This method presents many improvements, since it allows the writing of explicit formulas, somewhat in the spirit of Zimmerman’s forest formula for perturbative renormalization. The method is quite general, and robust enough to bear a large versus small field expansion. We treat here for simplicity, and as a testing ground, the infrared φ44 model. Note that in this special case such a large field versus small field expansion is not necessary, unlike the Yang–Mills theory for instance. We introduce this expansion here for two reasons. One is that we expand a little more than previous treatments of the present model and flirt a little closer with the danger represented by the divergence of the perturbation series. Indeed the polymers that appear in our expansion may have gaps in the vertical direction, i.e. they can couple cubes in the higher frequency slices directly with cubes in the lower ones, with large gaps in between. However we still have the factorization of the functional integrals between polymers that are now more diverse than in old-fashioned expansions. One could compare the difference between our vertical expansion and the previous ones [15] with the difference between the early horizontal cluster expansions of [21], in constructive field theory, and their version by Brydges, Battle and Federbush [5, 6, 10]. The other reason for introducing a large field versus small field expansion is that it was never really written in detail when cast in a multiscale phase-space expansion. Another improvement is also that our method seems to fit better the intrinsic combinatoric structure of functional integrals with interactions. This allows one to shunt the intermediate Mayer expansion steps, which should greatly enhance the clarity of the proofs. This aspect is however postponed to a next paper [3] where the full renormalized model will be constructed. In Sec. 2, we introduce our expansion in full generality. Specializations to multibody interactions in lattice systems, or to the study of p-particle irreducible kernels in field theory as in [23], could be done rather naturally in this formalism. However we refer to [1] for a treatment of these topics. We mention also that a version of these expansions is being featured in the study of the single-slice Anderson model in [29]. Hereupon we specialize our discussion to the example of infrared φ44 .

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An analog of p-particle irreducibility plays a key role here, it corresponds to parts of the expansion that are free from 2 and 4 point “almost local” insertions. In Sec. 3 we prove in detail a polymer bound on the polymers that are free from these insertions. Hence Theorem 2 is the constructive analog of Weinberg’s theorem on convergent graphs. However it already involves summing up all orders of perturbation theory and is the cornerstone of phase-space expansion renormalization group methods. The proof of a similar statement after the extraction of the local part of 2 and 4 point functions is the purpose of our next paper [3]. We intended the present paper to be particularly pedagogical for a non-trivial result in our discipline; this may account for the length and the level of formalization. We painstakingly kept track of numerical constants instead of having stray O(1)’s around, to help the reader check any eventual misunderstanding. The way combinatorics mix with analysis in such a proof makes clarity and unambiguousness very difficult to attain. We hope that one can, with some endeavor, read these technical pages and get, in the end, a sound comprehension of the subject. 2. An All-Purpose Scheme for Cluster Expansions 2.1. Interpolation revisited Generally speaking, cluster expansion techniques in constructive field theory usually stem from a clever application of a Taylor formula with integral remainder. Writing the full Taylor series would amount to completely expanding the perturbation series which most often diverges, and therefore should be avoided. We state in this section, in full generality, the kind of Taylor formula our expansions are based on. Let us suppose we have a finite set of indices L. In applications, L will typically be the set of links between pairs or even larger patches of cubes in the discretization of space which is introduced to prevent local accumulation of vertices. Let H : L (tl )l∈L 7→ H((tl )l∈L ) be a smooth function defined on the cube [0, 1] in RL . As an illustration, one should view H((tl )l∈L ) as the partition function of the system where a coupling corresponding to a link l is weakened by the corresponding parameter L tl , 0 ≤ tl ≤ 1. We define the particular elements 0 and 1 of [0, 1] as the vectors with all entries equal to 0 and 1 respectively. Our expansion which interpolates between H(1) and H(0), inductively generates ordered graphs. The idea is that after the explicit production of a sequence of links, there is a limited possible choice for the next link to be derived. The limitation is the fact that already existing links have connected the cubes into bigger blocks and it would be redundant and even dangerous to produce links inside such a block. Indeed we would then add an arbitrary number of loops and expand too much. So much for the philosophy, now we denote by G the set of finite ordered sequences (l1 , . . . , lk ) made of elements in L, with all possible values of length k. We allow k = 0 hence the empty sequence denoted by ∅. We introduce a partial ordering ≤ on G : if g = (l1 , . . . , lk ) and g0 = (l0 1 , . . . , l0 k0 ) are two sequences, we have g ≤ g0 if and only if k ≤ k 0 and for any a, 1 ≤ a ≤ k, la = l0 a holds, i.e. if g is an initial segment of g0 . Suppose we have a choice map C : G → P(L), where

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P(L) is the power set of L, i.e. a map such that for any two sequences g1 and g2 satisfying g1 ≤ g2 , we have C(g2 ) ⊂ C(g1 ). We define the set of allowed sequences AG, as the set of all sequences g = (l1 , . . . , lk ) such that for any a, 1 ≤ a ≤ k, we have la ∈ C((l1 , . . . , la−1 )). In particular, we always have ∅ ∈ AG. Besides, any initial segment of an allowed sequence is allowed. Finally we suppose that the sequences in AG are of bounded length. We can now state the following interpolation lemma. Lemma 1. Under the previous hypothesis we have X

H(1) =

Z

g=(l1 ,...,lk ) g∈AG

1≥h1 ≥···≥hk ≥0

dh1 . . . dhk

∂kH Tg (h) , ∂tl1 . . . ∂tlk

(1)

where h denotes the vector (h1 , . . . , hk ), and Tg (h) is the (tl )l∈L vector defined in the following way: – if l ∈ / C(∅) then tl = 1, – if l ∈ C(∅)\C((l1 )) then tl = h1 , – if l ∈ C((l1 ))\C((l1 , l2 )) then tl = h2 , .. . – if l ∈ C((l1 , . . . , lk−1 ))\C((l1 , . . . , lk )) then tl = hk , – if l ∈ C((l1 , . . . , lk )) then tl = 0. Summation on k includes all possible values. Proof. This is done by induction. Using the same notations as before, let us define the vector T˜g (h) like we did for Tg (h), except for the last item, where we set tl = hk if l ∈ C((l1 , . . . , lk )), with the convention that h0 = 1. Let n ≥ 0 be an integer, we will prove by induction on n the following formula: H(1) =

Z

X

X

k
g=(l1 ,...,lk ) g∈AG

+

X g=(l1 ,...,ln ) g∈AG

1≥h1 ≥···≥hk ≥0

dh1 . . . dhk

Z 1≥h1 ≥···≥hn ≥0

dh1 . . . dhn

∂kH Tg (h) ∂tl1 . . . ∂tlk

∂nH T˜g (h) . ∂tl1 . . . ∂tln

(2)

Indeed, for n = 0 it is a tautology. Besides, the induction step is a consequence of the fact that, given g = (l1 , . . . , ln ) in AG and h = (h1 , . . . , hn ), 1 ≥ h1 ≥ · · · ≥ hn ≥ 0, we can reexpress ∂nH (3) T˜g (h) ∂tl1 . . . ∂tln as f (hn+1 )|hn+1 =hn . The function f (hn+1 ) of the new interpolation parameter hn+1 , is defined by ∂ nH (tl )l∈L , (4) f (hn+1 ) = ∂tl1 . . . ∂tln

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where (tl )l∈L = T˜g (h), except for the tl ’s with l ∈ C((l1 , . . . , ln )) that are set equal to hn+1 , by definition. We then write Z f (hn ) = f (0) +

df (hn+1 ) dhn+1 Tg (h)

dhn+1 0

=

hn

∂nH ∂tl1 . . . ∂tln Z hn dhn+1 + 0

X ln+1 ∈C((l1 ,...,ln ))

As a result, we get Z 1≥h1 ≥···≥hn ≥0

dh1 . . . dhn

Z =

(5)

∂ n+1 H T˜(g,ln+1 ) (h, hn+1 ) . (6) ∂tl1 . . . ∂tln ∂tln+1

∂nH T˜g (h) ∂tl1 . . . ∂tln

∂nH dh1 . . . dhn Tg (h) ∂tl1 . . . ∂tln 1≥h1 ≥···≥hn ≥0 Z X dh1 . . . dhn dhn+1 + ln+1 |(l1 ,...,ln ,ln+1 )∈AG

×

1≥h1 ≥···≥hn ≥hn+1 ≥0

∂ n+1 H T˜(g,ln+1 ) (h, hn+1 ) , ∂tl1 . . . ∂tln ∂tln+1

(7)

thereby proving (2) at stage n + 1. Now, since sequences in AG have a bounded length, (2) simply reduces to (1) for n large enough. Let us now consider the following specialization and improvement of the previous lemma. We are given a finite nonempty set of objects D, typically the set of cubes in the cell discretization of real space, in single slice models, or of phase-space, in multiscale expansions. P If p is an integer p ≥ 2, we call p-link a map l : D → N such that ∆∈D l(∆) = p. This is an unordered combination of p elements amid D, with possible repetitions. In the φ4 theory, the propagators joining 2 boxes will correspond to 2-links, while vertices, connecting up to 4 boxes in the phase-space decomposition, will be associated to 4-links. In this last example we need to keep record of the multiplicity l(∆) that counts how many fields does the vertex produce in cube ∆. The support of a def p-link l is the set supp l = {∆ ∈ D|l(∆) 6= 0}. The set of p-links is denoted by Lp . In order to handle situations where several types of links are present, we suppose L is the disjoint union of copies of sets Lp . That is we assume we have a partition {L1 , . . . , Lq } of L, a sequence of integers (p1 , . . . , pq ), pν ≥ 2, 1 ≤ ν ≤ q, and a map J : L → ∪p≥2 Lp , such that for every ν, J restricts on Lν to a bijection with Lpν . However we will make a slight abuse of notation by forgetting the distinction between an element of L and the corresponding link J(l). In case there are several sets of links of type Lp for a given p, like in the jungle formulas of [2, 24] one should be more careful.

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The useful version of Lemma 1 we shall need is the one where the smooth L function H is assumed to be defined, may be not on the whole [0, 1] , but on the following subset Ω. A vector (tl )l∈L is called a partition vector if there exists a partition π of D such that tl = 1 if supp l is entirely contained in a block of π and tl = 0 otherwise. The set Ω is by definition the convex hull of all the partition vectors. Furthermore, we impose the following restriction on the choice map C: it is deduced from a connectivity map Π : G → Part(D). Part(D) is the set of partitions of D, and “deduced” means that for any g ∈ G, we have C(g) = Offdiag(Π(g)). π being an arbitrary partition of D, Offdiag(π) denotes the set Offdiag(π) = {l ∈ L| ∀B ∈ π, supp l 6⊂ B} .

(8)

In other words, it is the set of links jumping between two or more blocks of π. Note that Part(D) is equipped with a partial ordering ≤, such that two partitions π1 and π2 satisfy π1 ≤ π2 , if and only if π1 is obtained from π2 by splitting it into smaller pieces. Observe that π1 ≤ π2 is equivalent to Offdiag(π2 ) ⊂ Offdiag(π1 ); we then say that π1 is finer than π2 . Inspecting the proof of Lemma 1, one realizes that its output is still valid, under the present somewhat weaker hypothesis on H. This is because the interpolations involved never take us outside Ω. Indeed, at each time, we modify the argument t ∈ Ω of H or its derivatives, according to the following pattern. We are given g = (l1 , . . . , ln ) in AG and we write t = tdiag (g)+hn toffdiag (g). Here tdiag (g) denotes L the vector in [0, 1] equal to t except for the entries tl with l in C(g) = Offdiag(Π(g)) that are put to zero. By the same token, toffdiag (g) is the vector with entries tl = 1l{l∈C(g)} . We then say t is the value at hn+1 = hn of tdiag (g) + hn+1 toffdiag (g), and interpolate in the variable hn+1 between hn and 0. Hence we can find λ ∈ [0, 1], such that hn+1 = λhn . Therefore we have tdiag (g) + hn+1 toffdiag (g) = λt + (1 − λ)tdiag (g) .

(9)

But Ω is convex, and as a consequence tdiag (g)+ hn+1 toffdiag (g) will be in Ω, as soon as we prove the similar statement for tdiag (g). But this last assertion is true. Indeed, if t is a convex combination of partition vectors t1 , . . . , tn , obtained respectively from the partitions π1 , . . . , πn , then tdiag (g) is a convex combination, with the same weights, of the partition vectors obtained respectively from π1 ∧Π(g), . . . , πn ∧Π(g). The symbol ∧ denotes the greatest lower bound in the lattice Part(D) together with the above-mentioned order relation. In other words, π1 ∧ π2 is the partition whose blocks are the non-empty sets B1 ∩ B2 with B1 ∈ π1 and B2 ∈ π2 . We can restate our result as: Lemma 2. The output of Lemma 1 is still valid if we only assume the definiteness and smoothness of H in Ω. The reason we introduce the partition vectors is the conservation by the interpolation procedure of the positivity of both the covariance and the interaction, when we deal with bosonic models. To apply our formalism to a concrete situation,

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

129

one needs to define the object of interest in terms of a function H depending on coupling parameters tl indexed by finite combinations of degrees of freedom in the system. Then one has to define the connectivity map g 7→ Π(g), for any sequence g. Lemma 2 will produce a polymer expansion, whose blocks are those of Π(g), g being now restricted to allowed sequences. In single-slice cluster expansions with good decay of the propagator, one needs to consider only 2-links and Π(g) is the partition into ordinary connected components of the graph g. One then recovers, modulo symmetrization in the t parameters, the Brydges–Kennedy forest formula [11, 8, 2, 1]. Another easy application is constructive multi-particle analysis in the spirit of [23], albeit more symmetric formulas are obtained. The idea is to take for Π(g) the partition into p-particle irreducible kernels of the graph g. Our method seems also well suited for the treatment of multi-body interactions in Gibbs random fields on lattices. We will not develop these aspects here for which we refer to [1], and tackle the harder generalization of the notion of p-particle irreducibility needed for a phase-space analysis of infrared φ44 for instance. 2.2. Application to infrared φ44 : an explicit small field versus large field multiscale expansion 2.2.1. The model We consider a φ44 theory, with a multiscale decomposition of the bosonic field φ into N + 1 slices. That is we work in a finite cube Λ in R4 , with sides of length M N , where M ≥ 2 is a fixed integer. For each i, 0 ≤ i ≤ N , we fill Λ with cubes whose sides are for instance semi-open intervals of the form [α, β[ and of length (N ) M i . The set of such boxes, with cardinal M N −i , is denoted by Di . We let (N ) D(N ) = ∪0≤i≤N Di , and for any cell ∆ ∈ D(N ) we define its level i(∆) as the (N ) unique i, 0 ≤ i ≤ N , with ∆ ∈ Di . We also denote its volume, here equal to 4i(∆) , by |∆|. If (x, i) ∈ Λ × {0, . . . , N } we define ∆(x, i) as the unique box of M (N ) containing x. The way we picture the set of boxes D(N ) , is by stacking the Di (N ) (N ) (N ) layers Di , putting every Di on top of Di+1 , for any i, 0 ≤ i ≤ N − 1. When we say that some box ∆1 is above another cube ∆2 , this means that i(∆1 ) < i(∆2 ) and ∆1 ⊂ ∆2 . Two elements of D(N ) are said to be vertically neighboring, if we can order them as ∆1 , ∆2 for which i(∆2 ) = i(∆1 ) + 1 and ∆1 ⊂ ∆2 hold. If ∆1 is above ∆2 and these two cubes are vertically neighboring we say that ∆1 is just above ∆2 . Finally, note that two elements of D(N ) are either included one into another or disjoint. Until Sec. 3 we will forget about the N dependence so that D(N ) is now the set D to which we apply the result of Sec. 2.1. We let (φi )0≤i≤N be N + 1 independent scaled Gaussian random fields on Λ. We choose a covariance such that for any x1 and x2 in Λ C(x1 , i1 ; x2 , i2 ) = < φi1 (x1 )φi2 (x2 ) > Z d4 p eip(x1 −x2 ) −M 2i1 p2 −M 2(i1 +1) p2 − e e . = δi1 i2 (2π)4 p2

(10) (11)

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A. ABDESSELAM and V. RIVASSEAU

This gives for the total field φ =

P 0≤i≤N

Z < φ(x1 )φ(x2 ) >=

φi , in the limit N → ∞, the covariance

d4 p eip(x1 −x2 ) −p2 e , (2π)4 p2

(12)

i.e. the covariance of a Gaussian massless field with unit UV cut-off. Note that the fast decrease of the propagator for large momentum, ensures that our measure is supported on smooth fields, see [14, 20]. We write the interaction using a symmetric kernel K(x1 , i1 ; . . . ; x4 , i4 ) which is positive in the sense that, for any test fields φi (x), we have Z

N X

dx1 . . . dx4 K(x1 , i1 ; . . . ; x4 , i4 )φi1 (x1 ) . . . φi4 (x4 ) ≥ 0 .

i1 ,...,i4 =0

(13)

Λ4

Here we choose def

K(x1 , i1 ; . . . ; x4 , i4 ) = gδ 4 (x2 − x1 )δ 4 (x3 − x1 )δ 4 (x4 − x1 ) ,

(14)

where g > 0 is the coupling constant. Therefore the positivity condition (13) simply reduces to !4 Z N X φi (x) dx ≥ 0 . (15) g Λ

i=0

We illustrate our cluster expansion on the partition function Z(Λ) of the system given by  !4  Z Z N X def dµC ((φi )0≤i≤N ) exp −g φi (x) dx , (16) Z(Λ) = Λ

i=0

or more generally on unnormalized Schwinger functions like Z def dµC ((φi )0≤i≤N ) φi1 (x1 ) . . . φin (xn ) S(x1 , i1 ; . . . ; xn , in ) =  exp −g

Z Λ

N X

!4 φi (x)

 dx ,

(17)

i=0

(the normalized objects can be deduced from the expansion of the unnormalized ones by a standard Mayer expansion). Now the way we proceed is by finding a suitable function H((tl )l∈L ) such that H(1) is our quantity of interest. In this concrete example, we apply the result of Lemma 2, with D as a set of objects, def and L = L2 ∪ L4 as a set of links. The parameters (tl )l∈L2 serve to decouple the covariance, whose kernel becomes: def

C[(tl )l∈L2 ](x1 , i1 ; x2 , i2 ) = tl[∆(x1 ,i1 ),∆(x2 ,i2 )] C(x1 , i1 ; x2 , i2 ) ,

(18)

where, for any sequence (∆1 , . . . , ∆p ) of cubes, we denote by l[∆1 , . . . , ∆p ] the p-link def

l defined by l(∆) = #({a| 1 ≤ a ≤ p, ∆a = ∆}), for every ∆ ∈ D.

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

131

Likewise, the parameters (tl )l∈L4 are used to decouple the interaction whose new kernel is: def

K[(tl )l∈L4 ](x1 , i1 ; . . . ; x4 , i4 ) = tl[∆(x1 ,i1 ),...,∆(x4 ,i4 )] K(x1 , i1 ; . . . ; x4 , i4 ) .

(19)

Both kernels are symmetric, and their positivity follows from the next lemma. Lemma 3. Suppose L contains a copy of Lm . Let M(x1 , i1 ; . . . ; xm , im ) be a symmetric kernel, positive in the sense that for any test fields φi (x) we have Z

N X

dx1 . . . dxm M(x1 , i1 ; . . . ; xm , im )φi1 (x1 ) . . . φim (xm ) ≥ 0 ;

(20)

Λm

i1 ,...,im =0

then for any t ∈ Ω the interpolated kernel def

M[(tl )l∈Lm ](x1 , i1 ; . . . ; xm , im ) = tl[∆(x1 ,i1 ),...,∆(xm ,im )] M(x1 , i1 ; . . . ; xm , im ) (21) is also positive. Proof. Since the dependence of M[(tl )l∈Lm ] on t is linear and Ω is the convex hull of partition vectors, it is enough to prove the assertion for such vectors only. Therefore we suppose t is the partition vector associated to π ∈ Part(D). Then by definition N X i1 ,...,im =0

=

X

Z dx1 . . . dxm M[(tl )l∈Lm ](x1 , i1 ; . . . ; xm , im )φi1 (x1 ) . . . φim (xm ) Λm N X

B∈π i1 ,...,im =0

Z Λm

dx1 . . . dxm 1l{supp

l[∆(x1 ,i1 ),...,∆(xm ,im )]⊂B}

M(x1 , i1 ; . . . ; xm , im )φi1 (x1 ) . . . φim (xm ) (22) Z N X X B dx1 . . . dxm M(x1 , i1 ; . . . ; xm , im )φB = i1 (x1 ) . . . φim (xm ), (23) B∈π i1 ,...,im =0

Λm

def

B where φB i (x) = 1l{∆(x,i)∈B} φi (x). The positivity of M applied to φi (x), and summed over the blocks B of π, just proves the assertion.

As a result, the function H((tl )l∈L ) obtained by replacing, in the partition function or the Schwinger function, the kernels C and K by their interpolated version, is well defined. H is also a smooth function. We recall that via an integration by parts one can prove that a derivation with respect to tl , l ∈ L2 , amounts to introducing a functional differential operator Z Z X δ δ 1 (24) C(x1 , i(∆1 ); x2 , i(∆2 )) 2 δφ (x ) δφ i(∆1 ) 1 i(∆2 ) (x2 ) ∆1 ∆2 2 (∆1 ,∆2 )∈D l[∆1 ,∆2 ]=l

in the functional integral, see [14, 20, 7].

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We will be in a position to apply the result of Sec. 2.1, as soon as we define the connectivity map g 7→ Π(g). However, for technical reasons, we postpone this operation after introducing a large field versus small field expansion. 2.2.2. The large field versus small field expansion Following [25], for our large field condition we introduce a C ∞ step function χ  1 if 0 ≤ t ≤ 12    1 − 1 def 1 (25) χ(u) = e2 1 − e t− 2 e− t−1 if 12 ≤ t ≤ 1    0 if t ≥ 1 that interpolates smoothly between 1 and 0 on the interval [0, 1]. We choose also a constant 1 > 0. Given ∆ ∈ Di , and j ≥ i we denote by prj (∆) the unique cube in Dj that contains ∆. Now if Γ is some subset of D, and (φi ) is a configuration of the fields, we pose  4   Z Y X    (1 − χ) g (1+1 ) φi (x) dx χΓ ((φi )) = ∆

∆∈Γ

 ×

Y

 χ g (1+1 )

∆∈D\Γ

Z ∆

i∈IΓ (∆)

 

X

4



 φi (x) dx ,

(26)

i∈IΓ (∆)

where IΓ (∆) is the largest interval of the form {i(∆), i(∆) + 1, . . . , i} such that for any of its elements j, j > i(∆) we have prj (∆) ∈ Γ. Γ is what we call a large field region. The cubes of Γ are large field cubes, those of D\Γ are small field cubes. We now have the identity X χΓ ((φi )) (27) 1= Γ⊂D

which is a consequence of the following elementary algebraic lemma. Lemma 4. Let r ≥ 1 be an integer and let R be a function that associates a real number to any pair (a, τa ), where a is an integer with 1 ≤ a ≤ r, and τa is a map τa : {1, . . . , a − 1} → {0, 1}. Then Y X Y (1 − R)(a, τ |{1,...,a−1} ) × R(a, τ |{1,...,a−1} ) , (28) 1= τ

1≤a≤r τ (a)=1

1≤a≤r τ (a)=0

where the sum is over all maps τ : {1, . . . , r} → {0, 1}. Proof. By induction on r. The case r = 1 is obvious. If the statement holds for r ≥ 1, then the right-hand side for r + 1 may be factorized as: Y Y X (1 − R)(a, τ |{1,...,a−1} ) × R(a, τ |{1,...,a−1} ) τ |{1,...,r}

1≤a≤r τ (a)=1

1≤a≤r τ (a)=0

× (1 − R)(r + 1, τ |{1,...,r} ) + R(r + 1, τ |{1,...,r} ) = 1 ,

(29)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

133

by the induction hypothesis at stage r applied to the sum over the restriction τ |{1,...,r} . To prove (27), we introduce an ordering ∆1 , . . . , ∆r for the elements of D, such that we enumerate first the cubes of DN , then of DN −1 , and up to D0 . We have therefore an identification between subsets Γ of D and their characteristic functions τ : {1, . . . , r} → {0, 1} defined by τ (a) = 1 if ∆a ∈ Γ and 0 otherwise. We take the function R defined by  4   Z X    φi (x) dx . (30) R(a, τ |{1,...,a−1} ) = χ g (1+1 ) ∆a

i∈IΓ (∆a )

This is a consistent definition. Indeed by our numbering, IΓ (∆a ) is the union of {∆a } with a set that depends only on ∆a and Γ ∩ [Di(∆a )+1 ∪ Di(∆a )+2 ∪ . . . ∪ DN ] whose set of labels is included in {1, . . . , a − 1}. Hence IΓ (∆a ) only depends on τ |{1,...,a−1} . Finally (27) follows from Lemma 4. 2.2.3. The expansion Let us study for instance the unnormalized Schwinger function Z S(ξ1 , α1 ; . . . ; ξn , αn ) = dµC[1] ((φi )0≤i≤N ) φα1 (ξ1 ) . . . φαn (ξn )   Z N X dy1 . . . dy4 K[1](y1 , j1 ; . . . ; y4 , j4 ).φj1 (y1 ) . . . φj4 (y4 ). (31) · exp − j1 ,...,j4 =0

Λ4

We begin by inserting the relation (27) in the exponential so that X HΓ (1) , S(ξ1 , α1 ; . . . ; ξn , αn ) =

(32)

Γ⊂D

where HΓ ((tl )l∈L ) is the smooth function on Ω defined by Z def dµC[t] ((φi )0≤i≤N ) φα1 (ξ1 ) . . . φαn (ξn ).χΓ ((φi )) HΓ (t) =   Z N X dy1 . . . dy4 K[t](y1 , j1 ; . . . ; y4 , j4 ).φj1 (y1 ) . . . φj4 (y4 ) . (33) · exp − j1 ,...,j4 =0

Λ4

All we need, to interpolate HΓ (1) along the result of Sec. 2.1, is to specify the connectivity map g 7→ ΠΓ (g). The idea is that blocks of the partition ΠΓ (g) will correspond to convergent polymers, i.e. those without internal “almost local” 2 and 4 point functions that are responsible of the divergences in the bare theory, see [31]. A block of ΠΓ (g) will be a patch of large field regions made of vertically neighboring cubes, and of isolated small field cubes, linked together by propagators, and vertices. Propagators will give a strong connection, whereas vertices will provide a weak

134

A. ABDESSELAM and V. RIVASSEAU

connection. At least five vertices are needed to hook an almost-local insertion to the background or low momentum cubes of the polymer (corresponding to higher indices i(∆)). Therefore an analog for the vertices of 4-particle irreducibility appears in this formalism. To make things more precise, let us first make some combinatorial definitions. We introduce a gluing notion between cubes as follows. Two cubes are said glued if we can order them as ∆1 , ∆2 with i(∆2 ) = i(∆1 ) + 1, ∆2 ∈ Γ, and ∆1 ⊂ ∆2 . Connected components according to gluing are either isolated small field cubes, or a dressed large field block made of vertically neighboring large field cubes together with a layer of small field cubes just on top. Remark that for ∆ ∈ Γ ∩ D0 there are no such small field cubes above ∆; therefore they are not included in the dressed large field block containing ∆. Two elements ∆ and ∆0 of D, such that a 2 or 4-link l occurs in some fixed ordered graph g, and satisfies {∆, ∆0 } ⊂ supp l, are said to be joined. Suppose ∆ and ∆0 are in D, and there exists a sequence of cubes ∆1 = ∆, ∆2 , . . . , ∆u = ∆0 , u ≥ 1, with the property that for any v, 1 ≤ v ≤ u − 1, ∆v and ∆v+1 are either glued or joined; then we declare ∆ and ∆0 to be connected by g and Γ. If V ⊂ D, V is said to be connected by g and Γ, if any distinct cubes ∆ and ∆0 of V are connected by g and Γ. The subsequence of g ∈ G where we delete all p-link l with supp l 6⊂ V and keep the ordering of the remaining links, is denoted by gV . Finally, V is called a 4-vertex irreducible set for some ordered graph g and a large field region Γ (we abreviate by 4-VI), if any graph obtained from gV after deleting at most four 4-links still connects V . Now ΠΓ (g) is by definition the set of maximal 4-VI subsets in D for g and Γ. The blocks of ΠΓ (g) are called the 4-VI components of g and Γ. Beware that 4-vertex irreducibility depends on the shape of Γ. Gluing as well as joining by 2-links provide strong connections, but joining by 4-links is a weak connection: at least 5 of them are needed to make up a strong bond. Lemma 5. ΠΓ (g) thus defined is indeed a partition of D. Proof. Any singleton is 4-VI, therefore ∅ ∈ / ΠΓ (g) and ΠΓ (g) covers D. Let V1 and V2 be two elements of ΠΓ (g) with V1 ∩V2 6= ∅. Let g0 be a subsequence obtained from gV1 ∪V2 by deleting at most four 4-links, then g0V1 is also obtained from gV1 by deleting at most four 4-links. Since V1 is 4-VI, g0V1 connects V1 , and so does g0 of which g0V1 is a subsequence. Likewise g0 connects V2 , and since V1 ∩V2 6= ∅, g0 finally connects V1 ∪ V2 . We have shown that V1 ∪ V2 is 4-VI and by maximality, we get V1 ∪ V2 = V1 = V2 . It is obvious that the map g 7→ C(g) = Offdiag(ΠΓ (g)) is a choice map, i.e. g1 ≤ g2 implies C(g2 ) ⊂ C(g1 ). More generally if g = (l1 , . . . , lk ) and (lj1 , . . . , ljr ) is any of its subsequences, then the partition ΠΓ ((lj1 , . . . , ljr )) is finer than ΠΓ (g) and thus (34) Offdiag(ΠΓ (g)) ⊂ Offdiag(ΠΓ ((lj1 , . . . , ljr ))) . It is easy to see that we cannot have, in an allowed ordered graph, a link appearing more than five times. Since the number of all possible links is finite, this ensures the boundedness of the length of allowed ordered graphs.

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AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

We may now write the full expansion for the Schwinger function, it is an explicit although lenghty formula:

S(ξ1 , α1 ; . . . ; ξn , αn ) =

X

Z

X

Z dh1 . . . dhk

g∈AGΓ

×

Y 1 1≤j≤k lj ∈L2

2

Z

X j j (∆ ,∆ )∈D2 2 1 j j l[∆ ,∆ ]=lj 1 2

Z j

dxj1

∆1

j

dxj2 C(xj1 , i(∆j1 ); xj2 , i(∆j2 ))

∆2

×

Z

X

−

j j (∆ ,...,∆ )∈D4 1 4 j j l[∆ ,...,∆ ]=lj 1 4

1≤j≤k lj ∈L4

δ

δ

δφi(∆j ) (xj1 ) δφi(∆j ) (xj2 ) 1

×φα1 (ξ1 ) . . . φαn (ξn )χΓ ((φi ))

Y

dµC[Tg,Γ (h)] ((φi )0≤i≤N )

1≥h1 ≥···≥hk ≥0

Γ⊂D g=(l1 ,...,lk )

2

Z j

dxj1

...

∆1

j

dxj4 K(xj1 , i(∆j1 ); . . . ;

∆4

xj4 , i(∆j4 )).φi(∆j ) (xj1 ) . . . φi(∆j ) (xj4 ) 1

× exp

4

−

Z

N X

dx1 . . . dx4 K[Tg,Γ (h)](x1 , i1 ; . . . ; x4 , i4 ).φi1 (x1 ) . . . φi4 (x4 ) .

i1 ,...,i4 =0

Λ4

(35)

This is the expression we obtain by applying formula (1) on HΓ for a fixed Γ, then summing over it. Note that our preferred order in which we compute the derivak tives ∂tl ∂...∂tl on H is by applying first the ∂/∂tl for l ∈ L4 , thus producing explicit 1 k vertices, then doing the derivations for l ∈ L2 . In this way, we produce functional differential operators that act on the whole integrand consisting in external fields, explicit vertices, large field conditions and the exponential of the interpolated interaction. However the raw expression of (35) needs further treatment to display its factorizations and to rewrite it as a polymer expansion. If Y ⊂ D is non empty, Y is what we call a polymer. Given (b1 , . . . , bm ), a sequence of not necessarily distinct cubes of Y , and ζ1 , . . . , ζm any points in Λ such that ζq ∈ bq , for every q, 1 ≤ q ≤ m, we define the activity of Y with insertions (bq , ζq )1≤q≤m as:

A(Y ; (bq , ζq )1≤q≤m ) def

=

X

X

ΓY

gY =(l1 ,...,lk ) gY ∈AGΓ ,Y Y Y 4−VI

Z

Z dh1 . . . dhk 1≥h1 ≥···≥hk ≥0

dµC[Tg

Y ,ΓY

(h)]Y

(φY )

136

×

A. ABDESSELAM and V. RIVASSEAU

Y 1 1≤j≤k lj ∈L2

2

Z

X j j (∆ ,∆ )∈Y 2 1 2 j j l[∆ ,∆ ]=lj 1 2

Z j

dxj1

∆1

j

dxj2 C(xj1 , i(∆j1 ); xj2 , i(∆j2 ))

∆2

×

1≤j≤k lj ∈L4

−

Z

X j j (∆ ,...,∆ )∈Y 4 1 4 j j l[∆ ,...,∆ ]=lj 1 4

j

dxj1

j

(xj1 ) δφY

j

i(∆2 )

(xj2 )

Z

... j

∆1

dxj4 K(xj1 , i(∆j1 ); . . . ;

∆4

4

Z

X

exp −

xj4 , i(∆j4 )).φYi(∆j ) (xj1 ) . . . φYi(∆j ) (xj4 ) 1

δφY

δ

i(∆1 )

×φYi(b1 ) (ζ1 ) . . . φYi(bn ) (ζn )χΓY ,Y ((φY ))

Y

δ

Z dx1 . . .

(∆1 ,...,∆4 )∈Y 4

∆1

dx4 ∆4

!

K[TΓY ,gY (h)](x1 , i(∆1 ); . . . ; x4 , i(∆4 )).φYi(∆1 ) (x1 ) . . . φYi(∆4 ) (x4 )

(36)

.

Here we sum ΓY over all subsets of Y with the property that if ∆1 ∈ D, ∆2 ∈ Y , i(∆2 ) = i(∆1 ) + 1 and ∆1 ⊂ ∆2 , then ∆1 ∈ Y also. This means that every cube just on top of ΓY must be in Y . We denote by AGΓY ,Y the set of allowed ordered graphs in Y , defined in the same way as we did in D. This means that we consider GY , the set of ordered graphs gY = (l1 , . . . , lk ) such that for any a, def

1 ≤ a ≤ k, la ∈ LY = {l ∈ L| supp l ⊂ Y }, then we introduce a connectivity map ΠΓY ,Y : GY → Part(Y ) depending on the large field region ΓY . The blocks of ΠΓY ,Y (gY ) are the 4-VI components of Y , with the gluing and joining notions restricted to cubes of Y . Finally we put gY ∈ AGΓY ,Y if and only if for any def

a, 1 ≤ a ≤ k, la ∈ C((l1 , . . . , la−1 )) = Offdiag(ΠΓY ,Y ((l1 , . . . , la−1 ))). The sum in (36) is over every gY ∈ AGΓY ,Y that makes Y into a single 4-VI component, i.e. such that ΠΓY ,Y (gY ) = {Y }. The notation TgY ,ΓY (h) is for the vector (tl )l∈LY defined from (h1 , . . . , hk ) exactly as in Lemma 1 but only for l ∈ LY . The functional integral is on a collection (φY ) of fields supported on Y , that is def

(φY ) = (φYi ) i∈{0,...,N } ,

(37)

Di ∩Y 6=∅

where for any i, φYi (x) is a random field on ∪∆∈Di ∩Y ∆. The measure is Gaussian, with covariance C[TgY ,ΓY (h)]Y defined as before in (18) but restricted to entries (x, i) with ∆(x, i) ∈ Y . Finally we pose:  4   Z Y X    (1 − χ) g (1+1 ) φYi (x) dx χΓY ,Y ((φY )) = ∆

∆∈ΓY

 ×

Y ∆∈Y \ΓY

 χ g (1+1 )

Z ∆

i∈IΓY (∆)

 

X

4



 φYi (x) dx .

(38)

i∈IΓY (∆)

Remark that equality of sequences modulo permutation of the elements is an equivalence relation in G. Any equivalence class is called an unordered graph. If

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g ∈ G, the class of g is denoted by g. We may also, being a little sloppy with notations, characterize g as a map g : L → N, that simply counts the occurrences of a link l ∈ L, in an arbitrary representative g of g. One has naturally an ordering g1 ≤ g2 if and only if g1 (l) ≤ g2 (l) for any l ∈ L. One easily sees that for an ordered graph g, ΠΓ (g) does not depend on the ordering of links in g; however their multiplicity is relevant. As a result, the map g 7→ ΠΓ (g) naturally factorizes def through g 7→ g 7→ ΠΓ (g) = ΠΓ (g). We will hereupon use the same terminology of 4-VI sets and components for ordered and unordered graphs. Given a partition π of D, and a p-link l : D → N, for which we must have P π ∆∈D l(∆) = p, we define the reduced p-link l , on the set π instead of D, as def P follows: lπ is the mapping π → N, such that for all B ∈ π, lπ (B) = ∆∈B l(∆). π Intuitively, l is the link obtained by contracting any block of π to a point. We also define the reduced ordered graph gπ on π from an ordered graph g = (l1 , . . . , lk ) of def

links in D, by gπ = (l1π , . . . , lkπ ). The map g 7→ gπ factorizes naturally through a def

map g 7→ gπ = gπ of unordered graphs. Now consider a partition π of D, for which there are no pairs of glued cubes ∆ and ∆0 falling in two distinct blocks of π. If G π is an unordered graph on π, we can define Π (G) to be the set of 4-VI subsets of π with respect to the graph G. Simply we apply the same definition as for D, using 2 and 4-links on π, but only consider joined elements of π to define connections. The gluing notion for elements of π, is already embodied in them. We denote by 0π the trivial partition {{B}|B ∈ π} of π. def Q Given an unordered graph g, we define its symmetry factor σ(g) = l∈L (g(l))!, recalling that g(l) is the number of occurrences of l in g or its multiplicity. Likewise we define for any l ∈ L the number P l(∆))! def ( . (39) ρ(l) = Q ∆∈D (l(∆))! ∆∈D

Theorem 1. We can write the following polymer expansion: S(ξ1 , α1 ; . . . ; ξn , αn ) =

X

X

π∈Part(D)

gext =(l1 ,...,lk ) π Π (gext π )=0π

Y

Z Z k Y 1 dζ1j . . . dζ14 ρ(lj ) j j σ(gext ) j=1 b1 b4

A(Y ; (bjν , ζνj )Y , (∆(ξq , αq ), ξq )Y ) .

!

(40)

Y ∈π

gext is summed over all unordered graphs on D made of links in Offdiag(π) and π which satisfy Π (gext π ) = 0π . This means that the reduced graph of gext on π has no non-trivial 4-VI components. Remark that there can be no 2-link in gext , since this would create non-trivial 4-VI components in π. Here (l1 , . . . , lk ) is only a choice of ordered representative for gext . For any j, 1 ≤ j ≤ k, (bj1 , . . . , bj4 ) is a choice of boxes such that l[bj1 , . . . , bj4 ] = lj . Finally (bjν , ζνj )Y is the subfamily of (bjν , ζνj ) 1≤j≤k corresponding to the entries ν and j for which bjν ∈ Y . By the 1≤ν≤4

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same token, (∆(ξq , αq ), ξq )Y is the subfamily of (∆(ξq , αq ), ξq )1≤q≤m corresponding to indices q verifying ∆(ξq , αq ) ∈ Y . Proof. Starting from (35), the first thing to do is to organize the sum according to values π of ΠΓ (g). Then if π = {Y1 , . . . , Yr }, we decompose Γ into ΓY1 ∪ . . . ∪ ΓYr def

where ΓYa = Γ ∩ Ya . We see that summing over Γ ⊂ D is the same as summing independently on the ΓYa ’s among subsets of Ya respectively, with the restriction that if ∆1 ∈ D, ∆2 ∈ ΓYa , and ∆1 is just above ∆2 , then ∆1 ∈ Ya . Indeed, ∆1 and ∆2 would then be glued and accordingly would have to fall in the same component Ya . The sum over g = (l1 , . . . , lk ) in AGΓ also splits into independent sums. Denote by gext the subsequence of g that remains after we extract all the disjoint subsequences gY1 , . . . , gYr , that correspond to internal subgraphs of the blocks of π. To show the independence of the sums over gY1 , . . . , gYr and gext we need the following lemmas. The independence of the sums over gY1 , . . . , gYr and gext is intuitively trivial, but the formal proof of such a statement tediously goes along the following lemmas which the trustful reader may safely skip. Lemma 6. Let g = (l1 , . . . , lk ) be in G, not necessarily allowed, and V be a non-empty union of blocks from ΠΓ (g). Denote by ΓV the intersection of Γ with V . ΠΓV ,V (gV ) is by definition a partition of V, namely that of 4-VI components of V, made from the large field region ΓV and the ordered graph gV obtained from g by keeping the internal links of V . We then have the following equalities: ΠΓV ,V (gV ) = ΠΓ (g) ∩ P(V ) = ΠΓ (g)|V ,

(41)

where P(V ) denotes the power set of V, and π|V denotes the partition of V made by the non-empty sets of the form A ∩ V, A ∈ π. Proof. Since V is a non-empty union of blocks ΠΓ (g), it is clear that ΠΓ (g) ∩ P(V ) = ΠΓ (g)|V ; therefore ΠΓ (g) ∩ P(V ) is a partition of V . Now let W be a non-empty subset of V ; we claim that W is 4-VI for gV and ΓV if and only if W is 4-VI for g and Γ. Suppose W is 4-VI in V for the graph gV and the large field region ΓV , and let g0 be any subsequence obtained from gW by deleting at most four 4-links. Since W ⊂ V we have gW = (gV )W , and therefore g0 is obtained from (gV )W by deleting at most four 4-links. Since W is 4-VI for gV and ΓV , we can connect any pair of cubes in W , by a chain of elements in W that are successively glued by ΓV or joined by g0 . But ΓV ⊂ Γ, so that the cubes of the chain are successively glued by Γ or joined by g0 . We have thus proven that W is 4-VI for g and Γ. Suppose W is 4-VI for g and Γ, and let g0 be any subsequence obtained from (gV )W by deleting at most four 4-links. Since (gV )W = gW and W is 4-VI for g and Γ, we conclude that any pair of cubes in W is connected by a chain of cubes in W that are successively glued by Γ or joined by g0 . But if two elements of W are glued by Γ, we can for instance denote them by ∆1 , ∆2 with ∆2 ∈ Γ and ∆1

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just above ∆2 . Since ∆2 ∈ W ⊂ V , we obtain by definition of ΓV , that ∆2 ∈ ΓV . Therefore ∆1 and ∆2 are glued by ΓV . It follows from this remark that W will be 4-VI for gV and ΓV . Now let W ∈ ΠΓ (g)∩P(V ), W is 4-VI for g and Γ, and therefore also for gV and ΓV since W ⊂ V . Let W 0 be a subset of V , containing W , that is 4-VI for gV and ΓV . By the other implication of the above proven equivalence we obtain that W 0 is 4-VI for g and Γ. By maximality of W for this last property due to W ∈ ΠΓ (g), we get W 0 = W . As a result, W is maximal with respect to 4-vertex irreducibility for gV and ΓV , and thus W ∈ ΠΓV ,V (gV ). We have proven ΠΓ (g) ∩ P(V ) ⊂ ΠΓV ,V (gV ), but both are partitions of V . This mere property entails ΠΓ (g) ∩ P(V ) = ΠΓV ,V (gV ). Lemma 7. Let g = (l1 , . . . , lk ) be a sequence in G, Γ be a large field region, and let ΠΓ = {Y1 , . . . , Yr }. Then we have the following equivalence: g ∈ AGΓ ⇔ ∀a, 1 ≤ a ≤ r, gYa ∈ AGΓYa ,Ya .

(42)

Proof. Suppose g ∈ AGΓ and for some a, 1 ≤ a ≤ r, that gYa is the subsequence (lj1 , . . . , lju ) of g. For every v, 1 ≤ v ≤ u, since ΠΓ ((lj1 , lj2 , . . . , lj(u−1) )) ≤ ΠΓ (g), we infer that Ya is a union of blocks in ΠΓ ((lj1 , lj2 , . . . , lj(u−1) )). As a result, Lemma 6 applies and gives ΠΓYa ,Ya ((lj1 , lj2 . . . . , lj(u−1) )) = ΠΓ ((lj1 , lj2 , . . . , lj(u−1) ))|Ya .

(43)

But g ∈ AGΓ entails ljv ∈ Offdiag(ΠΓ ((l1 , l2 , . . . , l(jv )−1 ))) ⊂ Offdiag(ΠΓ ((lj1 , lj2 , . . . , lj(u−1) ))) .

(44)

Now (43) and supp ljv ⊂ Ya , imply ljv ∈ Offdiag(ΠΓYa ,Ya ((lj1 , lj2 . . . . , lj(u−1) ))) .

(45)

By definition, we have just checked that gYa ∈ AGΓYa ,Ya . Conversely, suppose that for any a, 1 ≤ a ≤ r, gYa ∈ AGΓYa ,Ya , and consider j, any integer such that 1 ≤ j ≤ k. If lj is in the subsequence gext of g then lj ∈ Offdiag(ΠΓ (g)) ⊂ Offdiag(ΠΓ ((l1 , l2 , . . . , lj−1 ))) .

(46)

Else, lj must belong to some subsequence gYa = (lµ1 , . . . , lµu ) of g. Suppose j = µv for some v, 1 ≤ v ≤ u. Since ΠΓ ((l1 , l2 , . . . , lj−1 )) ≤ ΠΓ (g), Ya is a union of blocks of ΠΓ ((l1 , l2 , . . . , lj−1 )). Therefore Lemma 6 applies and shows that ΠΓ ((l1 , l2 , . . . , lj−1 ))|Ya = ΠΓYa ,Ya ((l1 , l2 , . . . , lj−1 )Ya ) = ΠΓYa ,Ya ((lµ1 , lµ2 , . . . , lµ(v−1) )) .

(47) (48)

Since gYa ∈ AGΓYa ,Ya , we have lj = lµv ∈ Offdiag(ΠΓYa ,Ya ((lµ1 , lµ2 , . . . , lµ(v−1) ))), and from (48) we conclude that lj ∈ Offdiag(ΠΓ ((l1 , l2 , . . . , lj−1 ))). We have thus checked that g ∈ AGΓ .

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Lemma 8. If g is a sequence in G, π = ΠΓ (g) and gext is the subsequence of g obtained by keeping the links that belong to Offdiag(π), then Ππ (gπext ) = 0π .

(49)

Proof. Let V be a 4-VI subset of π for the reduced graph gπext , and define V = ∪B∈V B. We claim that V is a 4-VI subset of D for g and Γ. To prove this, we consider g0 , any subsequence of gV obtained by deleting at most four 4-links. For any B ∈ V, g0 B is obtained by deleting at most four 4-links from (gV )B = gB . But B is 4-VI for g and Γ, therefore g0B and hence g0 connect B. Remark that (g0 )π is obtained from (gV )π by deleting at most four 4-links, over π this time. Remark also that (gV )π = (gπ )V ; and since V is 4-VI for gπext and consequently for gπ , we deduce that (g0 )π connects V. If ∆ ∈ V , we denote by B(∆) the unique block B ∈ V such that ∆ ∈ B. Now let ∆ and ∆0 be two elements in V . Since (g0 )π connects V, there exists a chain B1 , . . . , Bu , u ≥ 1, in V with B1 = B(∆), Bu = B(∆0 ), and such that for any v, 1 ≤ v ≤ u − 1, there is a link lj in g0 such that Bv and Bv+1 belong to the support → of ljπ . For any v, 1 ≤ v ≤ u − 1, pick a cube ∆← v in Bv and a cube ∆v+1 in Bv+1 , → such that ∆← v and ∆v+1 belong to the support of the above-mentioned link lj . For ← → any v, ∆v and ∆v+1 are joined by g0 . Since g0 connects any B ∈ V, we have that 0 → ← ∆ and ∆← 1 are connected by g , also that for any v, 2 ≤ v ≤ u − 1, ∆v and ∆v 0 → 0 0 are connected by g , and finally that ∆u and ∆ are connected by g . Therefore g0 connects ∆ and ∆0 . This shows that V is 4-VI for g and Γ, and thus cannot be made by more than one block B ∈ π, so that V has to be a singleton. Finally Ππ (gπext ) = 0π , the trivial partition. We also need the converse statement: Lemma 9. Let π be some partition of D. Let for any Y ∈ π, ΓY be a subset of Y satisfying the condition that any cube, just above a cube of ΓY , is in Y . Let also gY be a sequence of links with support in Y , such that ΠΓY ,Y (gY ) = {Y }. Let gext be a sequence made of links in Offdiag(π), such that Ππ (gπext ) = 0π . Finally, let Γ = ∪Y ∈π ΓY . Then for any sequence g obtained by an arbitrary intertwining of the gY , for Y ∈ π, together with gext we have ΠΓ (g) = π .

(50)

Proof. Any Y ∈ π is 4-VI for gY and ΓY , hence it is also 4-VI for g and Γ. From the argument in the proof of Lemma 5, one sees that any subset V of D that is 4-VI for g and Γ, and is maximal for this property, must be a union of Y ’s belonging to a subset V of π. We claim that V is 4-VI for gπext . First note that any subsequence of (gπext )V obtained by deleting at most four 4-links, is of the form (g0 )π where g0 is a subsequence of (gext )V obtained also by deleting at most four 4-links. Now gV is obtained by intertwining of (gext )V and the gY , for Y ∈ V. Therefore if g00 denotes the subsequence obtained from gV by deleting the same 4-links as in the extraction of g0 from (gext )V , then g00 is obtained from gV by deleting at most

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four 4-links and is an intertwining of g0 and the g0Y , Y ∈ V. Now since V is 4-VI for g and Γ, g00 must connect V . If Y and Y 0 are two elements of V, pick a cube ∆ in Y and a cube ∆0 in Y 0 . There exists a chain ∆1 , . . . , ∆u , u ≥ 1, of cubes such that ∆1 = ∆, ∆u = ∆0 and for any v, 1 ≤ v ≤ u − 1, ∆v and ∆v+1 are either glued by Γ or joined by g00 . By assumption on the ΓY , there can be no gluing between cubes belonging to different Y ’s. As a result, there can be no gluing by Γ between a cube in V and a cube in D\V . This, together with the fact that g00 is made of links whose supports lie in V , compels the cubes ∆v , 1 ≤ v ≤ u, to remain in V . Let for every v, 1 ≤ v ≤ u, Y (∆v ) denote the unique Y ∈ V containing ∆v . Now for any v, 1 ≤ v ≤ u − 1, either ∆v and ∆v+1 are glued by Γ and therefore have to belong to the same Y , i.e. Y (∆v ) = Y (∆v+1 ), or ∆v and ∆v+1 are joined by some link lj of g00 . If lj appears in some gY , Y ∈ V, then again we have Y (∆v ) = Y (∆v+1 ). Else, lj must be in (gext )V and therefore (lj )π appears in (g0 )π and joins Y (∆v ) with Y (∆v+1 ). The chain Y (∆1 ) = Y , Y (∆2 ), . . . , Y (∆u ) = Y 0 shows that Y and Y 0 are connected by (g0 )π . To conclude, we have that V is 4-VI for (gext )π , and since Ππ ((gext )π ) = 0π , V must be a singleton. Thus V ∈ π and this completes our proof that ΠΓ (g) = π. This lemma closes the series of verifications that ensure that given a partition π of D, summing over g ∈ AGΓ , such that ΠΓ (g) = π, is the same as summing independently over gY ∈ AGΓY ,Y for any Y ∈ π, with the requirement that ΠΓY ,Y (gY ) = {Y }, together with summing over gext , made of links in Offdiag(π) and such that Ππ (gπext ) = 0π . Finally, one has to sum over all possible intertwinings of gext with the gY , Y ∈ π, to make up g. Remark that this last sum over intertwinings allows the factorization of the integrations in the h parameters. Indeed in (35) each parameter hj is better labeled by the corresponding link lj of g. In that way, the ordering of links inside g imposes the corresponding ordering h1 ≥ . . . ≥ hk over the parameters. Summing over the intertwinings just recomposes a domain of integration on the h parameters where the ordering constraints, between parameters attached to links from different subsequences gY or gext , have been relaxed. Finally remark that the condition on gext does not depend on its ordering. Therefore the sum over gext boils down to a sum over gext , the corresponding unordered graph, then a sum over the choice of ordered representative gext = (l1 , . . . , lkext ). This last sum is performed by summing over the permutations of kext elements, and dividing by the symmetry factor σ(gext ) due to repetition of identical links. The sum over permutations relaxes the ordering constraints over the h parameters corresponding to links of gext , that are now integrated independently from 0 to 1. Besides, it results from the rule in Lemma 1 for defining Tg (h), that nothing in the integrand of (35) depends on the h parameters of gext . Indeed, if g = (l1 , . . . , lk ) and for some j, 1 ≤ j ≤ k, lj is in gext , then ΠΓ ((l1 , . . . , lj−1 )) = ΠΓ ((l1 , . . . , lj )). This results from the fact that those two partitions are finer than ΠΓ (g), so that a subset V of D that is 4-VI for (l1 , . . . , lj−1 ) or (l1 , . . . , lj ), has to be included in a block of π; but then supp lj 6⊂ V and hence (l1 , . . . , lj−1 )V = (l1 , . . . , lj )V . Finally the h parameters of gext are integrated out, yielding a factor 1.

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Now the nice feature of (35), that allows the factorization stated in Theorem 1, is that, by definition, for any l ∈ Offdiag(ΠΓ (g)), (Tg (h))l = 0. Therefore in the covariance as well as in the interaction, there can be no coupling between fields supported by different blocks of π = ΠΓ (g). Hence, the fields (φi )0≤i≤N on Λ can be integrated separately as independent sets (φY ), Y ∈ π, as explained in the definition of the activities. It is also easily seen, by application of Lemma 6, that the part of the interaction and the covariance concerning a block Y is interpolated only with the parameters hj , such that lj ∈ LY , forming the vector hY . In fact, the couplings are weakened by the components of the vector TgY ,ΓY (hY ). A few minor remarks remain in order to complete the proof of the theorem. First, note the factorization Y χΓY ,Y ((φY )) , (51) χΓ ((φi )0≤i≤N ) = Y ∈π

of the large field conditions, since the stack of large field cubes under some cube ∆, whose scales range through the set IΓ (∆), is glued together with ∆. Thus they have to belong to the component Y of ∆, where gluing is done by ΓY only. Second, for any link lj in gext that must be a 4-link, we have chosen a sequence (bj1 , . . . , bj4 ) of boxes such that l[bj1 , . . . , bj4 ] = lj , instead of summing like in (35) over all such sequences. Therefore we had to compensate by the combinatoric factor ρ(lj ). Third, the insertions φαq (ξq ) are gathered together according to the block Y they belong to. For such a block, we have to consider also the insertions φi(bjν ) (ζνj ), with bjν ∈ Y , that come from the links of gext . The proof of the theorem is now complete. To conclude with the algebraic considerations, we state a lemma to control the length of an allowed graph by the size of its supporting polymer. In analogy with trees we get a bound of linear type. Lemma 10. If Y is some polymer, ΓY is a large field region in Y, and gY = (l1 , . . . , lk ) is an allowed graph in Y, with respect to ΓY ; then we have the following bound k ≤ 6#(Y ) − 6 . (52) Proof. We consider gY,1 the subsequence of gY made by the 2-links, and gY,2 made by the 4-links. From the proof of Lemma 7, it is clear that subsequences of allowed graphs are allowed. Being allowed, for a graph of 2-links only, is easily seen from the definition, to imply that it is a forest, i.e. a graph of ordinary links with no loops. Therefore if k1 denotes the length of gY,1 , we must have k1 ≤ #(Y ) − 1. It remains to show that if k2 denotes the length of gY,2 , we have k2 ≤ 5#(Y ) − 5. Remark that we can define the notion of 5-VI subsets with respect to an ordered graph of 2 and 4-links and a large field region in some polymer, exactly as we did for 4-VI subsets. Namely, we ask that it remains connected when we remove up

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to five 4-links from the graph. The lemma will be established as soon as we prove the two following statements. First, an allowed graph gY = (l1 , . . . , lk ), made of 4-links only, is such that the only 5-VI subsets V of Y are contained in the blocks of πY , the partition of connected components in Y when gluing bonds are taken into account only. Second, any graph with the last property and such that each of its links is in Offdiag(πY ), has a length no greater than 5#(Y ) − 5. To prove the first statement, consider (gY )V = (la1 , . . . , lar ) the subsequence of gY , made by the links whose support is entirely contained in V . Suppose that r ≥ 1. Since (gY )V connects V even after removing up to five 4-links, we conclude that the initial segment (la1 , la2 , . . . , la(r−1) ) connects V even after removing up to four 4-links. As a result V is 4-VI with respect to (la1 , la2 , . . . , la(r−1) ), and thus to the larger graph (l1 , l2 , . . . , l(ar )−1 ). But by the allowedness of gY , lar must belong to Offdiag(ΠΓY ((l1 , l2 , . . . , l(ar )−1 ))). However lar has support in V which is included in a component of ΠΓY ((l1 , l2 , . . . , l(ar )−1 )). This leads to a contradiction. Therefore we have r = 0, i.e. the only links that are internal to V are the gluing links. Since V is 5-VI, it is connected, and thus contained in a block of πY , as wanted. Note that a link la of gY is in Offdiag(ΠΓY ((l1 , l2 , . . . , la−1 ))) and thus in Offdiag(πY ), since πY is finer than ΠΓY ((l1 , l2 , . . . , la−1 )). We now prove the second statement by induction on #(Y ) ≥ 1. If #(Y ) = 1, then πY is reduced to {Y }, and since the links of gY must be in Offdiag(πY ), the only possibility is gY = ∅. The bound is satisfied in this case. If #(Y ) ≥ 1 but πY = {Y }, the same conclusion applies. Now suppose #(Y ) ≥ 1 and πY 6= {Y }. Since Y is not contained in a block of πY , Y is not 5-VI. As a result we can remove no more than five 4-links in the graph gY and thus disconnect Y into non-empty disjoint components Y1 , . . . , Yq , with q ≥ 2. Consider for any r, 1 ≤ r ≤ q, the subsequence gYr of gY made of the links that are internal to Yr . It is easy to see that gYr satisfies the hypothesis of the statement we are proving, with respect to the polymer Yr and its partition πYr into connected components for the gluing bonds. By the induction hypothesis, the length kr of gYr , is bounded by 5#(Yr ) − 5. This implies for the length k of gY k ≤5+

q X r=1

kr ≤ 5 +

q X

(5#(Yr ) − 5) ≤ 5

r=1

q X

! #(Yr )

− 5 = 5#(Y ) − 5 , (53)

r=1

since q ≥ 2. This completes the inductive proof of the second statement, and the lemma follows. 3. The Bound on Convergent Polymers 3.1. The main theorem This section will be entirely devoted to the proof of the following theorem. It is a polymer bound in the spirit of [7, 31], that is uniform in the volume and the infrared cut-offs. We consider activities for polymers that are due to 4-VI graphs which do not require the renormalization procedure. The theorem is a constructive analog of Weinberg’s theorem on convergent graphs, with the difference that we

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have to sum all orders of perturbation theory. This result we view as a cornerstone of the constructive approach via phase-space expansions. Theorem 2. There is a constant C > 0 such that, for any η > 0 and K > 0, there exists a g0 > 0, such that for any g satisfying 0 < g ≤ g0 , and for any N ≥ 0 ext we have: for any ∆org ∈ D(N ) , and for any finite family (∆ext s , ζs )s∈S such that ext (N ) ext ext and ζs ∈ ∆s , the following bound: for every s ∈ S, ∆s ∈ D X

ext #(Y ) |A(Y ; (∆ext ≤ η.C #(S) s , ζs )s∈S )|.K

Y |Y ⊂D(N )

Y

(E∆ !)1/2 .

∆∈D (N )

∆org ∈Y {∆ext s |s∈S}⊂Y

Y

M −i(∆s

ext

)

s∈S

(54) = ∆}) that counts the external sources Here E∆ denotes the number #({s ∈ S|∆ext s in ∆. Remark that the case S = ∅, corresponding to vacuum polymers, is included. In the case where S = 6 ∅ we usually do not have a fixed origin ∆org to break translational invariance. To apply the theorem we just have then to choose one of the ∆ext s ’s to be ∆org . Prior to any other consideration, one can rewrite the interaction I inside the exponential in the formula (36) for the activity of a polymer, in a form where the positivity is more patent. Let us denote the contribution of a 4-link l in Y by def

Θl =

Z

Z

X

dx1 . . .

(∆1 ,...,∆4 )∈Y 4 l[∆1 ,...,∆4 ]=l

∆1

dx4 K(x1 , i(∆1 ); . . . ; ∆4

x4 , i(∆4 )).φi(∆1 ) (x1 ) . . . φi(∆4 ) (x4 ) .

(55)

Since we will be working always in some definite polymer Y , without having to distinguish it from other blocks of some partition like in Sec. 2, we will drop the Y subscript from ΓY , gY , TgY ,ΓY , AGΓY ,Y , ΠΓY ,Y and the collection of fields φY . We define for any a, 1 ≤ a ≤ k, X

def

Ξa =

Θl

(56)

l6∈Offdiag(ΠΓ ((l1 ,...,la ))) l∈Offdiag(ΠΓ ((l1 ,...,la−1 )))

and

X

Ξ0 =

Θl ,

(57)

l6∈Offdiag(ΠΓ (∅))

and we introduce the convention h0 = 1 and hk+1 = 0. Then by definition of Tg,Γ we have I=g

k X a=0

ha .Ξa .

(58)

.

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We now perform an Abel transformation I=g

!

k X

k X

a=0

a0 =a

k X

=g

(ha0 − ha0 +1 ) Ξa

(59)

 0  a X (ha0 − ha0 +1 )  Ξa  .

a0 =0

(60)

a=0

But since Offdiag(ΠΓ ((l1 , . . . , la−1 ))) ⊃ Offdiag(ΠΓ ((l1 , . . . , la ))) for every a, 1 ≤ a ≤ k, we have 0

a X

X

Ξa =

Θl

(61)

l6∈Offdiag(ΠΓ ((l1 ,...,la0 )))

a=0

X

=

X

Θl

(62)

B∈ΠΓ ((l1 ,...,la0 )) l|supp l⊂B

For any a0 , 0 ≤ a0 ≤ k, B ∈ ΠΓ ((l1 , . . . , la0 )) and x ∈ Λ we introduce the field def

ΦB (x) =

N X

1l{∆(x,i)∈B} φi (x) .

(63)

i=0

This notation and Eq. (55) allow us to write Z

X

Θl =

4 φB (x) dx .

(64)

Λ

l|supp l⊂B

Finally the interaction becomes I=g

k X

(ha − ha+1 )

a=0

X B∈ΠΓ ((l1 ,...,la ))

Z

4 φB (x) dx .

(65)

Λ

Since 1 ≥ h1 ≥ . . . ≥ hk ≥ 0, the positivity of I, already proven in general in Lemma 3, appears very explicitly here. We see from (54) and the expression (36) that we roughly have the following successive sums to control XXXX Y

Γ

g

“individual contribution” ,

(66)

P

where the last one is over the functional derivation procedures P. The way we bound the sums we perform first, determine what is left to bound the remaining sums. Therefore we start by explaining the last sum namely on Y , because it rules the upstream bounding of the individual contributions.

146

A. ABDESSELAM and V. RIVASSEAU

3.2. The sum over the location of polymers The sum over the position of the polymer Y , in the lattice D(N ) with a number N + 1 of slices, with the restriction to polymers containing a fixed box ∆org , is done thanks to the following proposition that provides uniform bounds in N , that is in the infrared cut-off, and also in the volume cut-off Λ. We formulate this proposition in an arbitrary dimension d, and as a general closed result that can be applied directly in other situations where phase space cluster expansions are needed. The requirements of the lemma seem to be among the very minimal we can demand. Indeed we need a just summable power-like decay of the horizontal links, as well as a factor M −(d+) per connected component, in a very weak sense, of high frequency slices, instead of the stronger hypothesis of a factor M −(d+) per cube. We do not ask M to be large either. If Y is a polymer, i.e. a finite non-empty subset of D, we pose (N ) (67) imax (Y ) = max{i|Y ∩ Di 6= ∅} . We suppose F is an ordinary graph on Y . In the whole section a link means an unordered pair {∆1 , ∆2 } made of two distinct cubes in D(N ) . We do not use here the notion of p-link of Sec. 2. We suppose also that F has no loops. F is then a union of non-overlapping trees, i.e. a forest. We ask that F is made only of horizontal def bonds, i.e. that for any l = {∆1 , ∆2 } in F we have i(∆1 ) = i(∆2 ) = i(l). In that case we pose also def

def

dist2 (l) = dist2 (∆1 , ∆2 ) = inf{d2 (x1 , x2 )|x1 ∈ ∆1 , x2 ∈ ∆2 } .

(68)

The subscript 2 refers to the usual Euclidean distance. (N ) (N ) Given i, 0 ≤ i ≤ N , we let D≤i denote ∪0≤j≤i Dj , and we define a projection (N )

(N )

map pri : D≤i → Di

that associates to any box ∆, with i(∆) ≤ i, the unique box

(N ) Di

(N )

(N )

containing ∆. We denote by Y (i) the subset pri (Y ∩ D≤i ) of Di . pri (∆) of We consider the graph F (i) on Y (i) made by all links {∆1 , ∆2 } for which there (N ) exists ∆01 , ∆02 in Y ∩ D≤i with pri (∆01 ) = ∆1 , pri (∆02 ) = ∆2 and {∆01 , ∆02 } ∈ F . Remark that F (i) need not be a forest. Let Ri (Y, F ) be the set of connected components of Y (i) with respect to the graph F (i) . We say that F is satisfying to Y if #(Rimax (Y ) (Y, F )) = 1. A polymer Y for which there exists a satisfying F , is called admissible. Next we pose def (69) R(Y, F ) = ∪i 0, we introduce T2 (Y, F ) = M −(d+2 )#(R(Y,F )) .

Y

(1 + M −i(l) dist2 (l))−(d+2 ) ,

(70)

l∈F

and T2 (Y ) =

max

F satisfying to Y

T2 (Y, F ) ,

if Y is admissible. We now have the following result.

(71)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

147

Proposition 1. There exists a constant K1 (d, M, 2 ) > 0 such that, for any integer Q ≥ 1 and for any fixed ∆org in D(N ) , we have X

T2 (Y ) ≤ K1 (d, M, 2 )Q .

(72)

Y ⊂D(N ) Y admissible ∆org ∈Y, #(Y )=Q

Proof. We have to organize the bounding term T2 (Y, F ) into a tree decay def allowing one to sum over Y . Having defined Y = ∪i≤imax (Y ) Y (i) ; the first step is to collect a factor less than one per cube of Y . A cube ∆ in Y is called a working cube if it falls into one of the following categories: (a) ∆ is a cube of Y ; (N ) (b) ∆ is a fork of Y that is a cube in Di ∩ Y for which there exist two distinct (N ) cubes ∆1 , ∆2 in Di−1 ∩ Y and contained in ∆. (c) ∆ is in some Y (i) and there exists ∆0 ∈ Y (i) such that dist2 (∆, ∆0 ) > 0 and {∆, ∆0 } ∈ F (i) . W

The set of working cubes is denoted by Y . Let us choose δ an integer, δ ≥ 1, such that M δ ≥ 3d . We say that a cube ∆ of Y is an active cube, if there exists a working cube ∆0 contained in ∆ with i(∆) − i(∆0 ) ≤ δ. The set of active cubes is A denoted by Y . Since such a ∆0 can be associated to at most δ bigger cubes ∆, we have the trivial bound A W (73) #(Y ) ≤ δ.#(Y ) . I def

A

A cube in Y = Y \Y is called an idle cube. Small factors will come from the working cubes and the components in R(Y, F ), so the first thing to show is a bound W on the number of idle cubes, that is linear in #(Y ) and #(R(Y, F )). Let i ≤ imax (Y ); the links of F (i) whose extremities are idle cubes, define conI nected components among the set Y (i) ∩ Y . Those are themselves embedded in larger connected components of the full graph F (i) , namely in some element of Ri (Y, F ). Let X be such a component of idle cubes; we have the following (far from optimal) geometric bound. Lemma 11. #(X) ≤ 3d Proof. Suppose there exist ∆ and ∆0 in X with dist2 (∆, ∆0 ) > 0. Since X is connected there is a sequence ∆1 , . . . , ∆γ , γ ≥ 1 of distinct cubes in X such that ∆1 = ∆, ∆γ = ∆0 and {∆β , ∆β+1 } ∈ F (i) for any β, 1 ≤ β ≤ γ − 1. Since the cubes ∆β , 1 ≤ β ≤ γ, are in X, they are idle and therefore for every β, 1 ≤ β ≤ γ − 1, we have dist2 (∆β , ∆β+1 ) = 0. Since dist2 (∆1 , ∆γ ) > 0 we can consider α the smallest integer, 2 ≤ α ≤ γ, such that dist2 (∆1 , ∆α ) > 0. We will use this to derive a contradiction.

148

A. ABDESSELAM and V. RIVASSEAU

First, notice that if ∆ is an idle cube then i(∆) > δ. Indeed, ∆ ∈ Y implies the W existence of some ∆1 ∈ Y , with ∆1 ⊂ ∆. Since by (a), ∆1 ∈ Y but on the other I hand ∆ ∈ Y we must have i(∆) > i(∆1 ) + δ ≥ δ. By descending induction on j, i(∆) − δ ≤ j ≤ i(∆), we show that there exists a ˆ j has to be ∆ itself. ˆ j in Y (j) and contained in ∆. For j = i(∆), ∆ unique cube ∆ Assume the statement is proven for j, i(∆) − δ < j ≤ i(∆). Since i(∆) − j < δ I ˆ j cannot be in Y , hence there exists a cube b of Y with b ⊂ ∆ ˆ j and and ∆ ∈ Y , ∆ ˆ j−1 as prj−1 (b). We have to show that there is no other cube i(b) < j. Now define ∆

b0 in Y (j−1) and contained in ∆. If there was such a b0 , prj (b0 ) would be in Y (j) ˆ j . Now since and contained in ∆, therefore by the induction hypothesis prj (b0 ) = ∆ 0 j−1 (j−1) j (j) ˆ ˆ ˆ j has to are distinct cubes of Y contained in ∆ ∈ Y , by (b) ∆ b and ∆ I ˆ j ) < δ and ∆ ∈ Y , therefore we have a contradiction. be a fork. But i(∆) − i(∆ This induction argument allows us to define for any idle cube ∆, the unique cube ˆ =∆ ˆ i(∆)−δ with the notations of the ˆ of Y i(∆)−δ contained in ∆. Simply put ∆ ∆ previous proof. We return now to the sequence ∆1 , . . . , ∆α , and remark that for any β, 1 ≤ β ≤ α − 1, {∆β , ∆β+1 } ∈ F (i) , and therefore there exists ∆0β and ∆0β+1 in Y with ∆0β ⊂ ∆β , ∆0β+1 ⊂ ∆β+1 and {∆0β , ∆0β+1 } ⊂ F . By the idleness of ∆β and ∆β+1 , we must have ˆ β ) = i(∆ ˆ β+1 ) . i(∆0β ) = i(∆0β+1 ) < i − δ = i(∆

(74)

ˆ β and by the unicity, they Note that pri−δ (∆0β ) satisfies the defining conditions of ∆ 0 ˆ ˆ β+1 and, as a bymust be equal, and thus ∆β ⊂ ∆β . Likewise we have ∆0β+1 ⊂ ∆ ˆ β+1 } ∈ F (i−δ) . Since ∆β , ∆β+1 are idle, ∆ ˆ β and ∆ ˆ β+1 ˆ β, ∆ product, we obtain {∆ ˆ ˆ cannot be working cubes and by (c) we finally get dist2 (∆β , ∆β+1 ) = 0. This last def

statement is equivalent to dist∞ (x, y) = sup1≤µ≤d |xµ − y µ |. ˆ β, ∆ ˆ β+1 ) = 0 Up to now what we have is that for any β, 1 ≤ β ≤ α − 1, dist∞ (∆ and dist∞ (∆1 , ∆β ) = 0, besides dist∞ (∆1 , ∆α ) > 0. Since ∆1 and ∆α are in the (N ) grid Di of mesh M i , we infer that dist∞ (∆1 , ∆α ) ≥ M i . If diam∞ (A) denotes the diameter for the distance dist∞ of a bounded subset of Rd , we have the elementary inequality dist∞ (A, C) ≤ dist∞ (A, B) + dist∞ (B, C) + diam∞ (B) .

(75)

By iteration we derive ˆ 1, ∆ ˆ α ) ≤ dist∞ (∆ ˆ 1, ∆ ˆ 2 ) + dist∞ (∆ ˆ 2, ∆ ˆ 3 ) + · · · + dist∞ (∆ ˆ α−1 , ∆ ˆ α) dist∞ (∆ ˆ 2 ) + · · · + diam∞ (∆ ˆ α−1 ), (76) + diam∞ (∆ and thus ˆ 1, ∆ ˆ α ) ≤ (α − 2)M (i−δ) , M i ≤ dist∞ (∆1 , ∆α ) ≤ dist∞ (∆

(77)

Mδ ≤ α − 2 .

(78)

implying

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

149

However, since the cubes ∆2 , . . . , ∆α−1 are distinct and verify dist2 (∆1 , ∆β ) = 0 for any β, 2 ≤ β ≤ α − 1, by the geometric constraint that a cube has only 3d − 1 (N ) neighbors in the lattice Di , we conclude that α − 2 ≤ 3d − 1. This together with (78) and the assumption M δ ≥ 3d made at the beginning, leads to a contradiction. As a conclusion for every ∆ and ∆0 in X, there is a connected component of idle cubes swimming in some larger component in Ri (Y, F ), we have dist2 (∆, ∆0 ) = 0. Therefore #(X) ≤ 3d . Note that with further work one can show that #(X) ≤ 2d , besides by refining the previous argument, the assumption M δ ≥ 3d can be weakened to M δ ≥ 2d + 1, however we are not looking here for optimal bounds. Now if X is a connected component of idle cubes in Y (i) , either X is isolated, that is X ∈ Ri (Y, F ), or X is embedded in a strictly bigger element V of Ri (Y, F ). In the second case there must exist an active cube ∆a in V linked by F (i) to some cube ∆b of X. Note that dist2 (∆a , ∆b ) = 0, for if it were not so, ∆a would fall in the category (c) of working cubes, which is forbidden by its assumed idleness. Since distinct components of idle cubes are disjoint and a ∆a can have at most 3d − 1 neighbors, we see that at most 3d − 1 components X can be associated in that way to a single active ∆a . From these facts and (73) we infer that the total number of idle components X is bounded by X A W #(Ri (Y, F )) + (3d − 1).#(Y ) ≤ [#(R(Y, F )) + 1] + δ.(3d − 1).#(Y ) . i≤imax (Y )

(79) From the lemma we deduce I

#(Y ) ≤ 3d (3d − 1)δ#(Y and since #(Y

W

W

) + 3d (#(R(Y, F )) + 1) ,

(80)

) ≥ #(Y ) ≥ 1, and δ ≥ 1, we obtain #(Y ) ≤ δ.#(Y

W

I

) + #(Y )

≤ (9d + 1).δ.#(Y

W

) + 3d #(R(Y, F )) .

(81) (82)

Now define 2

U2 (Y, F ) = M − 2 #(R(Y,F )) . def

Y

− 22 #(R(Y,F )) , 1 + M −i(l) dist2 (l)

(83)

l∈F

so that T2 (Y, F ) = U2 (Y, F ).T 22 (Y, F ) .

(84)

One can easily see that for each l = {∆1 , ∆2 } ∈ F , the set {i|N ≥ i ≥ i(l), dist2 (pri (∆1 ), pri (∆2 )) = 0}

(85)

is nonempty. Let us denote its minimum by i∗ (l). If ∆ ∈ Y is a cube containing either ∆1 or ∆2 , such that i(l) ≤ i(∆) < i∗ (l), since dist2 (pri(∆) (∆1 ), pri(∆) (∆2 )) >

150

A. ABDESSELAM and V. RIVASSEAU

0 and ∆ ∈ {pri(∆) (∆1 ), pri(∆) (∆2 )} ∈ F (i(∆)) , we obtain that ∆ is a working cube of type (c). We then say that ∆ is produced by the link l. Note that many links may produce the same cube, however any type (c) working cube is produced by at least W,b W,c and Y the sets of working cubes of types (b) and one link l. We denote by Y (c) respectively, and by c(l) the number of such cubes produced by the link l. Now if l produces some working cubes, we must have i∗ (l) > i(l) and therefore ∗

dist2 (l) ≥ dist2 (pri∗ (l)−1 (∆1 ), pri∗ (l)−1 (∆2 )) ≥ M i

(l)−1

,

(86)

and thus − 22 2 2 2 ∗ −i(l) dist2 (l) ≤ M − 2 (i (l)−i(l)−1) ≤ M − 4 c(l) .M 2 . 1+M

(87)

As a consequence, we have 2

U2 (Y, F ) ≤ M − 2 #(R(Y,F )) .M

2 2

#(Y )

2

.M − 4 #(Y

W,c

)

,

(88)

W,c

2

. Note that we have thereby extracting a factor M − 4 , less than 1, per cube of Y used the fact that F is a forest on Y , so that #(F ) ≤ #(Y ) − 1. Now consider Y , we can map it into the power set of Y as follows: to each def ∆ ∈ Y , we associate S(∆) = {∆0 ∈ Y |∆0 ⊂ ∆}. The range of S is a set of nonempty subsets of Y that are either included in one another or disjoint; this is what we call a forest of subsets in Y . Lemma 12. If E is a forest of subsets in some finite non-empty set E, then we have the bound #(E) ≤ 2#(E) − 1. Proof. By induction on #(E). If E is a singleton {e}, E must be empty or equal to {{e}} so that the inequality holds. If #(E) > 1, consider the forest of subsets E 0 = E\{E}. If E 0 = ∅ we are done, otherwise let E1 , . . . , Ek be the maximal elements of E 0 , and Ej = E 0 ∩ P(Ej ), 1 ≤ j ≤ k. Since #(Ej ) < #(E), by the induction hypothesis we have #(E) ≤

k X j=1

#(Ej ) + 1 ≤

k X

(2#(Ej ) − 1) + 1.

(89)

j=1

If k ≥ 2, by the disjointness of the Ej , (89) leads to #(E) ≤ 2#(E) − k + 1 ≤ 2#(E) − 1.

(90)

If k = 1, then since #(E1 ) < #(E), (89) becomes #(E) ≤ 2#(E1 ) − 1+ ≤ 2(#(E) − 1) < 2#(E) − 1 .

(91)

W,b

→ P(Y ) is injective. In fact if ∆1 and ∆2 are elements Remark that S : Y W,b , with S(∆1 ) = S(∆2 ), there exists a ∆ ∈ Y with ∆ ⊂ ∆1 ∩ ∆2 . This of Y

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

151

implies for instance ∆1 ⊂ ∆2 . Suppose that the inclusion is strict. Then since (N ) ∆2 is a fork, there exists ∆3 ∈ Y ∩ Di(∆2 )−1 included in ∆2 and distinct from (N )

∆4 = pri(∆2 )−1 (∆1 ) ∈ Y ∩ Di(∆2 )−1 . Therefore ∆3 ∩ ∆4 = ∅, and the non-empty S(∆3 ) verifies S(∆3 ) ⊂ S(∆2 ) but S(∆3 ) ∩ S(∆1 ) ⊂ S(∆3 ) ∩ S(∆4 ) = ∅. This contradicts S(∆1 ) = S(∆2 ). The injectivity allows us to claim that #(Y

W,b

) = #(S(Y

W,b

)).

(92)

W,b

) is a forest of subsets of Y . But, it may be viewed also as a forest of Now S(Y W,b ) itself. Remark that for any subsets of the set M of minimal elements of S(Y fork, ∆ #(S(∆)) ≥ 2, and thus every one of those minimal subsets contain at least two elements. They are also disjoint, and this forces #(M) ≤ 12 #(Y ). By Lemma (12), we obtain W,b )) ≤ 2#(M) − 1 , (93) #(S(Y and finally #(Y

W,b

) ≤ #(Y ) − 1 .

(94)

Consequently #(Y

W

W,c

) ≤ 2#(Y ) + #(Y

),

(95)

and by (82), #(Y ) ≤ 2.δ.(9d + 1)#(Y ) + δ.(9d + 1)#(Y

W,c

) + 3d #(R(Y, F )) .

(96)

This together with (88) proves U2 (Y, F ) ≤ M −3 #(Y ) .C1

#(Y )

def

where 3 =

2 4(9d +1)

,

(97)

def

and C1 = M 2 .

The second step in the proof of the proposition is to organize T 22 (Y, F ) −3 #(Y )

#(Y )

.C1 into a tree decay. Let us introduce G the ordinary graph on .M Y , whose links are those of F , plus all pairs {∆1 , ∆2 } where ∆1 and ∆2 belong to Y , and one of the two is just on top of the other. For any i ≤ imax (Y ) and for any A ∈ Ri (Y, F ), denote A↑ = {∆ ∈ Y |∃∆0 ∈ A, ∆ ⊂ ∆0 } . def

Pose also

(98)

R↑i (Y, F ) = {A↑ |A ∈ Ri (Y, F )} . def

↑

(99) ↑

Since for A ∈ R(Y, F ) we have A ⊂ A , we obtain that the map A 7→ A is injective. Besides it is surjective by definition, therefore R↑ (Y, F ) = ∪i
def

Gi = {{∆1 , ∆2 } ∈ G| i(∆1 ) ≤ i, i(∆2 ) ≤ i} ,

(100)

152

A. ABDESSELAM and V. RIVASSEAU

(N )

the truncation of G above frequency i. Note that every ∆ in Y ∩ D≤i is connected to pri (∆) by Gi since for every j, i(∆) ≤ j < i, {prj (∆), prj+1 (∆)} ∈ Gi . From this property it follows easily that any A↑ ∈ R↑i (Y, F ) is connected by Gi . Besides if ∆1 and ∆2 are connected by Gi , then either pri (∆1 ) = pri (∆2 ) or {pri (∆1 ), pri (∆2 )} ∈ F (i) so that ∆1 and ∆2 fall in the same A↑ of R↑i (Y, F ). Since the A↑ covers (N ) (N ) Y ∩D≤i , we conclude that R↑i (Y, F ) is the set of connected components of Y ∩D≤i with respect to the graph Gi . Considering the fact that Gimax (Y ) connects the unique element Y of R↑ imax (Y ) (Y, F ), we can construct a tree t0 ⊂ Gimax (Y ) connecting Y , with the property that for every i ≤ imax and every A↑ ∈ R↑i (Y, F ), the set of links of t0 that are internal to A↑ form a tree that connects A↑ . The procedure is inductive on i. For def (N ) (N ) 6= ∅} = min{i|Y ∩ Di 6= ∅}, we choose out of i = imin (Y ) = min{i|Y ∩ Di Gi a tree that connects A↑ , for every A↑ ∈ R↑i (Y, F ). The union of such trees is denoted by ti0 . Now suppose imin (Y ) ≤ i < imax (Y ) and we have built ti0 ⊂ Gi , (N ) a forest on D≤i whose connected components are the elements of R↑i (Y, F ). If A↑ ∈ R↑i+1 (Y, F ), we just add to the subforest of ti0 made by the links that are internal to A↑ , any choice of links from Gi+1 to get a connecting tree on A↑ . ti0 plus the newly added links form the set which we denote by ti+1 0 . Finally we take i (Y ) t0 = t0max , and it satisfies the required property. The rule for summation depends on the root ∆org ∈ Y . For any ∆ ∈ Y we define its level lv(∆) as the minimal number of links in t0 to connect it to the root. To any link l = {∆1 , ∆2 } ∈ t0 such that lv(∆1 ) = lv(∆2 ) − 1 we define def

def

f (l) = ∆1 or the father, and s(l) = ∆2 the son. A link l is now oriented from the son to the father. Hence we say that l goes downward if i(f (l)) = i(s(l)) + 1, and it goes upward if i(f (l)) = i(s(l)) − 1. The only other possibility is that l be a horizontal link, i.e. l ∈ F . For such a link we can extract from T2 /2 (Y, F ) a factor (1 + M −i(l) d(l))−(d+2 /2) that is enough to sum s(l) knowing f (l). Note that by construction of G for any ∆ ∈ Y there is at most one link {∆, ∆0 } ∈ G\F with i(∆0 ) > i(∆). Therefore #(G\F ) ≤ #(Y ), and as a result for any upward link we can extract from M −3 #(Y ) a factor M −3 . We now have to show that from T 22 (Y, F ) we can extract also a factor M −(d+2 /2) per downward link. the set of downward links of t0 . To such a link l we associate the Denote by tdown 0 component B(l) ∈ R↑i(s(l)) (Y, F ) that contains s(l). Note that i(s(l)) < imax (Y ); to R↑ (Y, F ). Now we claim that the therefore l 7→ B(l) defines a map from tdown 0 map l 7→ B(l) is injective. This is a consequence of the following lemma about trees. Lemma 13. Suppose t is a tree connecting some finite set E, whose root we denote by eroot . Given two elements e1 and e2 , a minimal subset of t, that contains enough links to connect e1 and e2 , is called a path between e1 and e2 . By definition of a tree such a path exists and is unique, and is denoted by p(e1 , e2 ). We define def

the level of some element e ∈ E as lv(e) = #(p(e, eroot )). Any link l ∈ t is of the def

def

form {e1 , e2 } with lv(e2 ) = lv(e1 ) + 1, and again we pose s(l) = e2 and f (l) = e1 .

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

153

We now have the following result. If F is a subset of E such that the forest def

tF = {l ∈ t|l ⊂ F }

(101)

is a tree connecting F, then there can be at most one link l ∈ t such that s(l) ∈ F and f (l) 6∈ F . Proof. We need first to make the following remarks. Given some e ∈ E, any vertex along p(e, eroot ) has a level at most equal to lv(e). Besides there can be at most one link l ∈ t, with s(l) = e. Indeed if there were two such links l1 and l2 , then f (l1 ) 6= f (l2 ), and from p(f (l1 ), eroot ) ∪ p(f (l1 ), eroot ) we could extract a path between f (l1 ) and f (l2 ). Such a path would have all its vertices of levels at most lv(f (l1 )) = lv(f (l2 )) = lv(e) − 1. But {l1 , l2 } would be a path between f (l1 ) and f (l2 ) having a vertex, namely e, of level lv(e). The existence of these two distinct paths between f (l1 ) and f (l2 ), in the tree t, produces a contradiction. Now let l1 and l2 be two distinct links satisfying the hypothesis of the lemma with respect to some subset F . From the previous remarks we infer that s(l1 ) 6= s(l2 ). Now there exists a sequence e1 , . . . , eα , α ≥ 2, of distinct elements of F , such that e1 = s(l1 ), eα = s(l2 ) and p(s(l1 ), s(l2 )) = {{e1, e2 }, {e2 , e3 }, . . . , {eα−1 , eα }} ⊂ tF .

(102)

Let β, 1 ≤ β ≤ α be such that lv(eβ ) is maximal. Then β cannot satisfy 1 < β < α, for we would have two distinct links {eβ−1 , eβ } and {eβ , eβ+1 } with s({eβ−1 , eβ }) = s({eβ , eβ+1 }) = eβ which is excluded by the previous remarks. Let for instance β = 1, then {e1 , e2 } ⊂ tF and l1 6⊂ tF are two distinct links with s({e1 , e2 }) = s(l1 ) = e1 , this is also impossible. The case β = α is ruled out in the same manner. , with B(l1 ) = B(l2 ) = B ∈ R↑i (Y, F ). We must Now let l1 and l2 be in tdown 0 have i(s(l1 )) = i(s(l2 )) = i; and therefore since l1 and l2 go downward, i(f (l1 )) = i(f (l2 )) = i + 1. Thus f (l1 ) and f (l2 ) are not elements of B. But, by our construction of t0 , the subset (t0 )B of all bonds of t0 that lie in B, is a tree connecting B. The lemma now allows us to conclude that l1 = l2 . The injectivity of l 7→ B(l) implies that ) ≤ #(R↑ (Y, F )) = #(R(Y, F )) . (103) #(tdown 0 hor the set of horizontal links of Let tup 0 denote the set of upward links of t0 , and t0 up t0 . Now #(t0 ) ≤ #(Y ), (88), and (84) show that #(Y )

T2 (Y, F ) ≤ C1

up

2

.M −3 #(t0 ) .M −(d+ 2 )#(t0

down

)

.

Y

−(d+ 22 ) . 1 + M −i(l) dist2 (l)

l∈thor 0

(104) The right-hand side is already a summable tree decay for t0 , however the number of its vertices that are not elements of Y may be too big. In order to control that number we have to prune the tree t0 a little. The result will be a tree ˆt whose support Yˆ verifies Y ⊂ Yˆ ⊂ Y , and has a cardinal bounded linearly by #(Y ).

154

A. ABDESSELAM and V. RIVASSEAU

down Two distinct links in tup are said to be consecutive if they are of the 0 ∪ t0 form l1 = {∆1 , ∆2 } and l2 = {∆2 , ∆3 } with ∆2 6∈ Y , ∆1 just above ∆2 , ∆2 just above ∆3 , and such that no other link of t0 contains ∆2 . ∆2 is then called the down intermediate cube of l1 and l2 . Consecutiveness defines a graph on tup 0 ∪ t0 whose set of connected components we denote by X . The links that belong to some component x ∈ X must all be going upward or all going downward. Otherwise, there would be two consecutive links l1 and l2 in x going in different directions. down is ruled out since Keeping the previous notations, the case l1 ∈ tup 0 , l2 ∈ t 0 ∆2 = s(l1 ) = s(l2 ), and l1 6= l2 contradicts the remark made at the beginning of and l1 ∈ tup the proof of Lemma (13). The remaining case where l1 ∈ tdown 0 0 is also forbidden. Indeed, since the path in t0 going from ∆2 to the root ∆org goes through vertices with levels no greater than lv(∆2 ), and since lv(∆1 ) = lv(∆3 ) = lv(∆2 ) + 1 and there is no link l ∈ t0 apart from l1 and l2 that contains ∆2 , we conclude that ∆2 = ∆org ∈ Y . But this is also excluded in the definition of consecutiveness. We define the subset Y1 of Y obtained by removing all the intermediate cubes of consecutive links. We define the following graph t1 on Y1 def

t1 = (thor 0 ∩ P(Y1 )) ∪ {{∆1 , ∆2 } ∈ D(Y1 )|∆1 6= ∆2 , ∃x ∈ X , ∃l1 ∈ x, ∃l2 ∈ x, ∆1 ∈ l1 , ∆2 ∈ l2 } .

(105)

t1 is simply the graph obtained by replacing maximal linear chains of consecutive vertical links between nearest neighbors by a single link joining their extremities. It is not difficult to see that t1 is a tree connecting Y1 . With the choice of root ∆org , we can define as before for any vertex ∆ of Y1 its level lv(∆) and for any link l ∈ t1 the cubes s(l) and f (l) with respect to the tree structure of t1 . We can also make the distinction between links going upward or backward, namely whether i(s(l)) > i(f (l)) or i(s(l)) < i(f (l)) respectively. Now define the following function on D(N ) × D(N )  0 if ∆1 = ∆2     −(d+ /2) 2  if i(∆1 ) = i(∆2 ), ∆1 6= ∆2 1 + M −i(∆1 ) dist2 (∆1 , ∆2 ) def . F (∆1 , ∆2 ) = −3 [i(∆2 )−i(∆1 )]  if i(∆1 ) < i(∆2 ) M    −(d+2 /2)[i(∆1 )−i(∆2 )] if i(∆1 ) > i(∆2 ) M (106) From (104) we readily deduce Y #(Y ) . F (f (l), s(l)) . (107) T2 (Y, F ) ≤ C1 l∈t1

Indeed, the horizontal links of t0 and t1 are the same so that we still have for them the factors −(d+2 /2) = F (f (l), s(l)) . (108) 1 + M −i(l) dist2 (l) If l is a vertical link of t1 going upward obtained from the component x ∈ X , then since all links in x ⊂ t0 must go upward, we collect from (104) the factor M 3 #(x) = M −3 [i(s(l))−i(f (l))] = F (f (l), s(l)) .

(109)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

155

Likewise if l is a downward link of t1 obtained from some x, we collect from (104) the factor M −(d+2 /2)#(x) = M −(d+2 /2)[i(f (l))−i(s(l))] = F (f (l), s(l)) .

(110)

The last step of pruning is to cut off the leafs of t1 that are in Y1 \Y . A leaf of t1 is a cube ∆ ∈ Y1 such that there is a unique l ∈ t1 with ∆ ∈ l. Now we denote by Yˆ the subset of Y1 obtained by removing the leaves of t1 that are not in Y . We finally def put ˆt = t1 ∩ P(Yˆ ). It is obvious that ˆt is a tree connecting Yˆ , and that Yˆ contains Y . Besides (104) trivially entails Y #(Y ) . F (f (l), s(l)) . (111) T2 (Y, F ) ≤ C1 l∈ˆt

Now the point is that we have the following bound. Lemma 14. #(Yˆ ) ≤ 2#(Y ) − 1 .

(112)

Proof. If ∆ ∈ Yˆ \Y , then ∆ cannot be a leaf of t1 , therefore there exist two distinct links of t1 arriving at ∆. These are vertical links, since horizontal links are between cubes of Y only. They have to be produced by two distinct and thus disjoint components x1 and x2 in X . If l1 ∈ x1 , l2 ∈ x2 and ∆ ∈ l1 ∩ l2 , then l1 6= l2 . We have showed that at least two vertical links in t0 arrive at ∆. Now we claim that ∆ has to be a fork in Y . Indeed, vertical links in t0 are between vertically neighboring cubes. Therefore if l1 = {∆1 , ∆} and l2 = {∆2 , ∆}, then having both ∆1 and ∆2 below ∆ is impossible by ∆1 6= ∆2 . Besides, if we suppose that ∆ is not a fork, the case where ∆1 and ∆2 are above ∆ is also ruled out. The only remaining possibility is that, for instance, ∆1 be above and ∆2 below ∆. But then, since ∆ 6∈ Y and ∆ is supposed not to be a fork, l1 and l2 must be consecutive. This contradicts l1 ∈ χ1 , l2 ∈ χ2 and χ1 6= χ2 . Finally, we have already proven in (94) that the number of forks is bounded by #(Y ) − 1, and the lemma follows. Now it is easy to see that there is a constant K2 (d, M, ) such that X F (∆1 , ∆2 ) ≤ K2 (d, M, ) , sup

(113)

∆1 ∈D (N ) ∆ ∈D (N ) 2

uniformly in N . We now have at our disposal all the needed ingredients to sum over Y . If Y is an admissible set in D(N ) containing ∆org , and such that #(Y ) = Q, we choose a forest F (Y ) that is satisfying to Y and such that T2 (Y ) = T2 (Y, F (Y )). Then starting from Y and F (Y ), we construct, as described previously, a set Yˆ containing Y and a tree ˆt connecting Yˆ . Then the sum over Y is bounded by X X XX Y X T2 (Y ) ≤ C1Q × F (f (l), s(l)) . (114) Y

ˆ Q≤Q ≤2Q−1

ˆ )=Q ˆ ˆ t t Y ⊂Yˆ l∈ˆ Yˆ |#(Y

156

A. ABDESSELAM and V. RIVASSEAU

ˆ is over the cardinal of Yˆ that, according to Lemma (14), has to The sum over Q range through the interval {Q, . . . , 2Q − 1}. The sum over Y , as a subset of Yˆ , gives ˆ a factor 2#(Y ) < 4Q . The sum over ˆt and Yˆ is done in the usual way [19, 20, 31] Q thanks to the tree decay l∈ˆt F (f (l), s(l)). ˆ → D(N ) whose range is Yˆ . For each Yˆ there exist an injection τ : {1, 2, . . . , Q} ˆ Now τ allows Summing over Yˆ is the same as summing over τ and dividing by Q!. ˆ to transport the tree structure ˆt of the varying set Yˆ onto the fixed set {1, 2, . . . , Q}, ˜ yielding a tree t, and a root r˜. We now have X

XY

ˆ ˆt Yˆ |#(Yˆ )=Q

F (f (l), s(l)) =

1 XX ˆ Q! ˜ t

l∈ˆ t

X

Y

F (τ (f (˜l)), τ (s(˜l))) .

r˜ τ |τ (˜ r)=∆org ˜ l∈˜ t

(115) The sum over τ such that τ (˜ r ) = ∆org , is done by summing over τ (j) where j ∈ ˆ is a leaf of ˜t, then for the ancestors all the way up to the root r˜. We then {1, . . . , Q} ˆ ˆ The sum over ˜t obtain a factor K1 (d, M, 2 )Q−1 ; the sum over r˜ costs a factor Q. ˆ Q−2 ˆ . Finally is bounded, with the help of Cayley’s theorem, by Q X

T2 (Y ) ≤ (4C1 )Q ×

Y

X ˆ Q≤Q≤2Q−1

1 ˆ Q−1 ˆ ˆ Q K2 (d, M, 2 )Q−1 . ˆ Q!

Note that for any integer n ≥ 1, nn ≤ en n!, therefore X T2 (Y ) ≤ K1 (d, M, 2 )Q ,

(116)

(117)

Y

with K1 (d, M, 2 ) = 4C1 .e2 .K2 (d, M, 2 )2 .

3.3. A toolkit for the bounds We gather in this section a few lemmas that will be useful in the bounding process. The first two are standard, see [20, 31], but for completeness we prove them in detail. Lemma 15. For any r ≥ 0, there exists a constant K3 (r) ≥ 1 such that, for any x1 and x2 in Λ and i1 and i2 in {0, 1, . . . , N } and for any multi-indices α1 and α2 of length not greater than 2, we have −r −i1 (2+|α1 |+|α2 |) M . |∂xα11 ∂xα22 C(x1 , i1 ; x2 , i2 )| ≤ δi1 ,i2 K3 (r) 1 + M −i1 d2 (x1 , x2 ) (118) As customary, a multi-index α is a quadruplet (α1 , . . . , α4 ) of nonnegative intedef

gers, whose length is denoted by |α| = α1 + · · · + α4 . The notation ∂xα is for 1 4 ∂ |α| /(∂x1 )α . . . (∂x4 )α , where upper indices label the coordinates of x. Proof. We have by (11) ∂xα11 ∂xα22 C(x1 , i1 ; x2 , i2 ) Z 4 d p eip(x1 −x2 ) |α1 |−|α2 | α1 +α2 −M 2i1 p2 −M 2(i1 +1) p2 i p − e e , (119) = δi1 ,i2 (2π)4 p2

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

157

Q4 µ where given a multi-index α = (α1 , . . . , α4 ), pα denotes the product µ−1 (pµ )α , where the pµ ’s are the components of the vector p. We make the change of variables p = M −i1 p˜ so that |∂xα11 ∂xα22 C(x1 , i1 ; x2 , i2 )| ≤ δi1 ,i2 M

−i1 (2+|α1 |+|α2 |)

Z d4 p˜ ei˜pX α1 +α2 −p˜2 −M 2 p˜2 −e e . (120) (2π)4 p˜2 p˜

def

where X = M −i1 (x1 − x2 ). For any values of α1 and α2 ranging over a finite set, the integral in the right-hand side is the Fourier transform of a smooth function in 1 . In particular p˜, therefore it decreases faster than any positive power of (1+|X| 2) there exists a constant K3 (r, α1 , α2 ) ≥ 1, such that Z d4 p˜ ei˜pX α1 +α2 −p˜2 −M 2 p˜2 −r −e e ≤ K3 (r, α1 , α2 ).(1 + |X|2 ) . (2π)4 p˜2 p˜

(121)

Now take K3 (r) = max{K3 (r, α1 , α2 )| |α1 | ≤ 2, |α2 | ≤ 2} and the lemma follows. Note that, with our choice of cut-off, we have even the exponential decrease of the propagator, but we prefer to use only a power law to obtain the bounds on the cluster expansion. The principle of local factorials Lemma 16. The Gaussian bound There exists a constant K4 ≥ 1 such that for any N ≥ 0, any polymer Y, large field region Γ, allowed graph g, and vector of interpolating parameters h = (h1 , . . . , hk ), we have the following bound Z Y Y Y 1 αj −ij (1+|αj |) dµC[T (h)] (φ) ≤ ∂ φ (x ) M (n(∆)!) 2 . (122) K × j 4 xj ij g,Γ j∈J j∈J ∆∈Y Here (xj )j∈J is any family of points in Λ, (ij )j∈J is any corresponding family of scale indices in {0, 1, . . . , N }, and (αj )j∈J is any family of multi-indices such that for every j, |αj | ≤ 2. We require that for each j ∈ J, xj be interior to the cube ∆(xj , ij ). def

Finally we have introduced the notation n(∆) = #({j ∈ J|∆(xj , ij ) = ∆}). Proof. Since the measure is Gaussian, we can integrate thanks to Wick’s theorem. We have a sum over all the contractions c, i.e. over the involutions of J α without fixed points, between the fields. Remark also that a derivative ∂xjj acts αj now on the extremity of the propagator attached to the contracted field ∂xj φij (xj ). Besides the propagators are those of the interpolated covariance C[Tg,Γ (h)] defined in (18). Since we have assumed the points xj to be interior to the corresponding α cubes ∆(xj , ij ), the derivatives ∂xjj do not act on the h dependence that is in general discontinuous across the boundary of a cube ∆. Since the multiplying h parameter is between 0 and 1, we bound the derivated propagator of the interpolated covariance by the corresponding derivated propagator of the original covariance C. Then

158

A. ABDESSELAM and V. RIVASSEAU

Lemma 15 applies and gives with r = 5 Z Y αj dµC[T (h)] (φ) ∂xj φij (xj ) g,Γ j∈J X Y Y p ≤ K3 (5).M −ij (1+|αj |) c

j∈J

−5 δij .ij0 1 + M −ij d2 (xj , xj 0 ) . (123)

{j,j 0 }⊂J j 0 =c(j)

The sum over c is done inductively following a classical argument. Remark that (N ) there exists a constant K5 ≥ 1 such that for any ∆ ∈ D0 , X −5 (1 + d2 (∆, ∆0 )) ≤ K5 . (124) (N )

∆0 ∈D0

Since the boxes of scale i have side M i we have also that for any ∆ ∈ Y −5 X δi(∆),i(∆0 ) 1 + M −i(∆) d2 (∆, ∆0 ) ≤ K5 .

(125)

∆0 ∈Y def

We denote ∆j = ∆(xj , ij ) for every j ∈ J. Suppose we have ordered J as {j1 , . . . , js } such that n(∆j1 ) ≥ n(∆j2 ) ≥ . . . ≥ n(∆js ). To sum over c(j1 ) , we sum first over ∆c(j1 ) , then over c(j1 ) knowing ∆c(j1 ) . The sum over ∆c(j1 ) is done thanks to the following factor in (123) −5 δij1 ,ic(j1 ) 1 + M −ij1 d2 (xj1 , xc(j1 ) ) −5 ≤ δi(∆j1 ),i(∆c(j1 ) ) 1 + M −i(∆j1 ) d2 (∆j1 , ∆c(j1 ) ) , (126) and costs a factor K5 . The sum over c(j1 ) knowing ∆c(j1 ) costs a factor q n(∆c(j1 ) ) ≤ n(∆j1 )n(∆c(j1 ) ) ,

(127)

by our choice of the ordering of J. We now pick the element j with the smallest label in J\{j1 , c(j1 )} and sum over its image by c in the same manner as before thus obtaining a factor q K5 ×

n(∆j )n(∆c(j) ) (128) p and continue the process. Since a square root n(∆j ) will appear exactly once by definition of the contraction scheme c, in the end we have that X Y −5 δij ,ij0 1 + M −ij d2 (xj , xj 0 ) c

{j,j 0 }⊂J j 0 =c(j)

≤

Yq Y q 1 #(J) (K5 n(∆j ) = K52 n(∆)n(∆)

j∈J

≤

p

(129)

∆∈Y n(∆)6=0

#(J) Y p K5 e n(∆)! , ∆∈Y

(130)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

159

n n since for p any integer n, n ≥ 1, we have n ≤ e n!. The lemma follows with K4 = e.K5 .K3 (5).

Lemma 17. The displacement of local factorials Suppose O and E are finite sets, S1 and S2 are two maps from O to E, and G is a function from E × E to [0, +∞[ such that X G(∆1 , ∆2 ) ≤ 1 . (131) sup ∆1 ∈E

∆2 ∈E

We introduce for every ∆ ∈ E, the notation def

n1 (∆) = #({a ∈ O|S1 (a) = ∆}) , def

n2 (∆) = #({a ∈ O|S2 (a) = ∆}) .

(132) (133)

If for every a ∈ O, G(S1 (a), S2 (a)) > 0, then Y

n1 (∆)! ≤ e#(O) ×

∆∈E

Y

Y

n2 (∆)! ×

∆∈E

G(S1 (a), S2 (a))−1 .

(134)

a∈O

Proof. Remark that the left-hand side is the number of permutations σ : O → O such that S1 = S1 ◦ σ. Given such a σ, we define the map ψσ : O → E, by def ψσ = S2 ◦ σ. Now X X 1= #({σ|ψσ = ψ}) , (135) σ

ψ

where the sum over ψ ranges through all the maps ψ : O → E for which there exists a σ such that ψ = ψσ . For a given ψ, looking for σ such that ψσ = ψ is equivalent to looking for σ(a) such that S2 (σ(a)) = ψ(a), for every a ∈ O. There are at most n2 (ψ(a)) possible choices, therefore #({σ|ψσ = ψ}) ≤

Y

n2 (ψ(a)) .

(136)

a∈O

However we supposed that there exists at least one σ0 such that ψ = ψσ0 and thus Y

n2 (ψ(a)) =

a∈O

Y

n2 (S2 (σ0 (a))) =

a∈O

Y

n2 (S2 (a)) ,

(137)

en2 (∆) .n2 (∆)! ,

(138)

a∈O

since σ0 : O → O is a permutation. Finally since Y a∈O

n2 (S2 (a)) =

Y ∆∈E n2 (∆)6=0

n2 (∆)n2 (∆) ≤

Y

∆∈E n2 (∆)6=0

we have #({σ|ψσ = ψ}) ≤ e#(O) ×

Y ∆∈E

n2 (∆)! .

(139)

160

A. ABDESSELAM and V. RIVASSEAU

Let ψ be of the form ψσ , for some permutation σ such that S1 ◦ σ = S1 , we then say that ψ is admissible. This entails Y Y G(S1 (a), ψ(a)) = G(S1 (a), S2 (σ(a))) (140) a∈O

a∈O

Y

=

G(S1 (σ(a)), S2 (σ(a)))

(141)

G(S1 (a), S2 (a)) ,

(142)

a∈O

Y

=

a∈O

since σ is a permutation of O. As a consequence, if for every a ∈ O, G(S1 (a), S2 (a)) > 0, then we have also G(S1 (a), ψ(a)) > 0. Therefore we can write X Y Y X 1= G(S1 (a), ψ(a)) × G(S1 (a), ψ(a))−1 (143) ψ admissible



≤

X

ψ admissible a∈O

Y

a∈O



G(S1 (a), ψ(a)) .

Y

sup ψ admissible

ψ admissible a∈O

! −1

G(S1 (a), ψ(a))

.

a∈O

(144) But by the argument used to prove (142) we conclude, since there exists some admissible ψ (take ψσ with σ the identity), that ! Y Y −1 G(S1 (a), ψ(a)) G(S1 (a), S2 (a))−1 . (145) = sup ψ admissible

a∈O

a∈O

Remark also that by assumption on G, X

Y

G(S1 (a), ψ(a)) ≤

ψ admissible a∈O

Y

X

a∈O

∆∈E

! G(S1 (a), ∆)

≤ 1.

Now putting (139), (145) and (146) together just proves the lemma.

(146)

We now state a lemma that allows to sum small cubes in big ones, with a less than summable vertical decay, typically like M −(4+)|i−j| . This miracle is possible since the small cubes are restricted to an already known polymer Y . The price to pay is a product of local factorials for the big cubes of reference, and a small factor per cube of Y . The idea is to distribute, for each cube of Y , a piece of the associated small factor among the cubes below. This motivates the somewhat poetic name of the lemma. Lemma 18. The rain of small factors Let K6 ≥ 1 and 4 > 0 be two chosen constants. Suppose ∆1 , . . . , ∆p are distinct boxes in D(N ) , and n1 , . . . , np are positive integers. For a given polymer Y in D(N ) , define  nk p Y X def  M −4 (i(∆k )−i(∆k ))  . (147) YY (∆1 , . . . , ∆p ; n1 , . . . , np ) = k=1

∆k ∈Y,∆k ⊂∆k

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

161

Then we have the following bound #(Y )

YY (∆1 , . . . , ∆p ; n1 , . . . , np ) ≤ K6

4

(1 − M − 2 )2 log K6

p −(n1 +···+np ) Y . nk ! . k=1

(148)

P+∞

− 24

. Since (1 − q) i=0 q i = 1 we can write Y −(1−q) P+∞ qi #(Y ) i=0 . K6 1 = K6

Proof. We pose q = M

∆∈Y

#(Y )

≤ K6

.

Y



∆∈Y

#(Y )

≤ K6



#(Y )

≤ K6

p Y



k=1

K6

Y



∆∈D (N )



−(1−q)q

[i(∆)−i(∆)]

∆∈D (N ) ,∆⊂∆

Y

.



Y



(149)

(150)

 −(1−q)q

[i(∆)−i(∆)]

K6

∆∈Y,∆⊂∆

Y





(151)



−(1−q)q

K6

[i(∆k )−i(∆)]

.

(152)

∆∈Y,∆⊂∆k

Therefore we define for any ∆ ∈ D(N ) and n ≥ 1,  Y −(1−q)q[i(∆)−i(∆)] X def K6 × ω(∆, n) = ∆∈Y,∆⊂∆

n M

−4 [i(∆)−i(∆)] 

(153)

∆∈Y,∆⊂∆ 4

for which we can write, recalling that q = M − 2 , X

i(∆)

ω(∆, n) =

n Y

q [i(∆)−ij ] Ai1 ,...,in .

(154)

i1 ,...,,in =0 j=1

Here Ai1 ,...,in denotes def

Ai1 ,...,in =

X

n Y

q [i(∆)−ij ] .

∆1 ,...,∆n j=1

Y

−(1−q)q[i(∆)−i(∆)] K6 ,

(155)

∆∈Y,∆⊂∆

where, for any j, 1 ≤ j ≤ n, the sum over ∆j , 1 ≤ j ≤ n, ranges through all the (N ) cubes ∆j ∈ Y ∩ Dij contained in ∆. Suppose the set {i1 , . . . , in } can be written as {a1 , . . . , , am } with m ≥ 1 and a1 , . . . , am all distinct. Now let νr 1 ≤ r ≤ m denote the number of j’s, 1 ≤ j ≤ n such that ar = ij . Then for every r, 1 ≤ r ≤ m, P we have νr ≥ 1, besides 1≤r≤m νr = n. If for any i, 0 ≤ i ≤ N , ρi denotes #({∆ ∈ Y |i(∆) = i, ∆ ⊂ ∆}), then m Y −(1−q)q[i(∆)−i(∆)] Y . q [i(∆)−ar ]νr (156) K . Ai1 ,...,in = ρνa11 . . . ρνam 6 m r=1

∆∈Y,∆⊂∆

. ≤ ρνa11 . . . ρνam m

ρar Y m m Y −(1−q)q[i(∆)−ar ] . q [i(∆)−ar ]νr . K6 r=1

r=1

(157)

162

A. ABDESSELAM and V. RIVASSEAU

Indeed, we have kept only the powers of K6−1 that come from cubes ∆ ∈ Y , ∆ ∈ ∆ that have a scale among a1 , . . . , am . (157) now reads Ai1 ,...,in ≤

m νr Y exp −(1 − q)(log K6 )q [i(∆)−ar ] ρar q [i(∆)−ar ] ρar (158) r=1

≤ ((1 − q)(log K6 ))−n ×

m Y

(yrνr exp(−yr )) ,

(159)

r=1 def

where yr = (1 − q)(log K6 )q [i(∆)−ar ] ρar , and we have used the bound y ν e−y ≤ ν!, for any y ≥ 0, shows that −n

Ai1 ,...,in ≤ ((1 − q)(log K6 )) By property of the multinomial coefficients, since

and therefore

P

m Y

.

P

1≤r≤m νr

= n. Now

νr ! .

(160)

r=1 1≤r≤m νr

= n, we have

n! ≥ 1. ν1 ! . . . νm !

(161)

Ai1 ,...,in ≤ ((1 − q)(log K6 ))−n .n! .

(162)

This together with (154) implies X

n Y

i(∆)

ω(∆, n) ≤ ((1 − q)(log K6 ))

−n

.n!.

q [i(∆)−ij ]

(163)

i1 ,...,in =0 j=1 −n

≤ ((1 − q)(log K6 ))

.n!.

n Y

+∞ X

j=1

i=0

! q

i

−n .n! . (1 − q)2 (log K6 )

(164) (165)

Finally by (152) and (154), we have YY (∆1 , . . . , ∆p ; n1 , . . . , np ) ≤

#(Y ) K6 .

p Y

ω(∆k , nk ) ,

(166)

k=1

which, together with the bound (165) proves the lemma.

A basic tool that seems unavoidable for just renormalizable bosonic models is what we call domination [15, 31]. It is the use of the positivity of the interaction to bound low momentum fields. An essential ingredient is to compare a field φ(x) to its average on a volume containing x. The average can be bounded by the interaction, or using a small field condition in χΓ , whereas the difference term called fluctuation is expressed with double gradients of the field that are well-behaved regarding bounds. This motivates the following lemma, with a flavor of Sobolev inequalities.

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

163

Lemma 19. There exists a constant K7 ≥ 1 such that for any N ≥ 0, for any pair of cubes ∆1 , ∆2 in D(N ) such that ∆1 ⊂ ∆2 , and for any coordinate indices µ and ν, there exist smooth functions µ,ν : ∆2 × (∆1 × ∆1 × [0, 1] × [0, 1]) → R f∆ 1 ,∆2

(167)

and a smooth function X∆1 ,∆2 with the same domain and range ∆2 , such that the following requirements are satisfied. First we demand that sup

Z 4 X

x∈∆2 µ,ν=1

∆1 ×∆1 ×[0,1]×[0,1]

µ,ν dw|f∆ (x, w)| ≤ 16M 2i(∆2 ) , 1 ,∆2

(168)

and second that for any smooth function φ on ∆2 and for any point x of ∆2 , we have the bound: Z µ,ν dwf∆1 ,∆2 (x, w)∂µ ∂ν φ(X∆1 ,∆2 (x, w)) φ(x) − ∆1 ×∆1 ×[0,1]×[0,1] ≤ K7 .

1 |∆1 |

Z dy φ(y)4

14

M i(∆2 )−i(∆1 ) .

(169)

∆1

Proof. Consider a fixed smooth function δ0 on R4 that takes nonnegative values, has support inside ]0, 1[4 , and verifies Z dy δ0 (y) = 1 . (170) R4

Now if the cube ∆1 is of the form ζ1 + [0, M i(∆1 ) [4 with ζ1 ∈ R4 , we define a smooth function δ∆1 as follows. For any y ∈ R4 we pose def (171) δ∆1 (y) = M −4i(∆1 ) δ0 (y − ζ1 )M −i(∆1 ) . Then δ∆1 takes also nonnegative values, has support in the interior of ∆1 and verifies Z dy δ∆1 (y) = 1 . (172) ∆1

Now let x ∈ ∆2 , then use Taylor’s formula twice and (172) once to write Z 1 du φ(x) (173) φ(x) = |∆1 | ∆1 Z Z 1 1 du φ(u) + ds(x − u)ν ∂ν φ((1 − s)u + sx) (174) = |∆1 | ∆1 0 Z 1 du φ(u) = |∆1 | ∆1 Z Z Z 1 1 du dv ds δ∆1 (v)(x − u)ν ∂ν φ((1 − s)u + sx) + |∆1 | ∆1 ∆1 0 (175)

164

A. ABDESSELAM and V. RIVASSEAU

=

Z Z 1 Z Z 1 1 du φ(u) + du dv ds δ∆1 (v)(x − u)ν |∆1 | ∆1 |∆1 | ∆1 ∆1 0 Z 1 dt ((1 − s)u + sx − v)µ ∂µ ∂ν φ((1 − t)v + t(1 − s)u + tsx) ∂ν φ(v) + 0

=

1 |∆1 | +

Z

Z du φ(u) + ∆1

1 |∆1 |

Z

∆1 ×∆1 ×[0,1]×[0,1]

Z

Z

du ∆1

(176) µ,ν dwf∆1 ,∆2 (x, w)∂µ ∂ν φ(X∆1 ,∆2 (x, w)

1

ds δ∆1 (v)(x − u)ν .

dv ∆1

(177)

0

Here we have introduced def

µ,ν (x, u, v, s, t) = f∆ 1 ,∆2

1 δ∆ (v)(x − u)ν ((1 − s)u + sx − v)µ |∆1 | 1

(178)

and def

X∆1 ,∆2 (x, u, v, s, t) = (1 − t)v + t(1 − s)u + tsx .

(179)

Now remark that by H¨ older’s inequality 14 Z 1 4 du φ(u) ≤ du φ(u) . |∆1 | ∆1 ∆1

Z 1 |∆1 |

(180)

The third expression in Eq. (177) we denote by R and transform by an integration by parts on v. Actually, we have 1 R= |∆1 |

Z

Z

Z

du

1

ds (−∂ν δ∆1 (v))(x − u)ν φ(v) .

dv

∆1

∆1

(181)

0

There is no boundary term, since the support of δ∆1 is in the interior of ∆1 . Since (x − u)ν is bounded by the size M i(∆2 ) of the cube ∆2 , we have Z dv |∂ν δ∆1 (v)||φ(v)| . (182) |R| ≤ M i(∆2 ) ∆1

By applying the Cauchy–Schwarz then the H¨ older inequalities, we obtain Z |R| ≤ M i(∆2 )

dv (∂ν δ∆1 (v))2

12 Z

∆1

12 (183)

∆1

Z ≤M

dv φ(v)2

i(∆2 )+2i(∆1 )

2

dv (∂ν δ∆1 (v)) ∆1

12

1 |∆1 |

Z 4

14

dv φ(v)

Note that, by the change of variable v = ζ1 + M i(∆1 ) v0 , we readily derive Z Z 2 4i(∆1 ) dv (∂ν δ∆1 (v)) = M dv0 M −10i(∆1 ) (∂ν δ0 (v0 ))2 , ∆1

[0,1[4

. (184)

∆1

(185)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

and therefore |R| ≤ M i(∆2 )−i(∆1 )

1 |∆1 |

Z dv φ(v)4

14

! 12

Z dv0 (∂ν δ0 (v0 ))2

.

(186)

[0,1[4

∆1

As a result if we choose as our constant Z def K7 = 1 + max 1≤ν≤4

165

! 12 dv0 (∂ν δ0 (v0 ))2

,

(187)

[0,1[4

the bound (169) follows. Finally, for any coordinate indices µ and ν and any x ∈ ∆2 , we have Z Z 1 Z 1 Z Z µ,ν 2i(∆2 ) du dv ds dt f∆ (x, u, v, s, t) ≤ M dv δ∆1 (v) = M 2i(∆2 ) , 1 ,∆2 ∆1

∆1

0

∆1

0

(188)

so the proof is complete. 3.4. Giving every cube its small factor

Usually, a small factor per cube comes from the coupling constant appearing in explicitly derived vertices whose number is roughly proportional to the cardinal of the polymer. However, this proportionality no longer holds for large field versus small field expansions, since a large field region behaves like a single cube as far as the combinatorics of the expansion are concerned. To obtain a small factor per cube now involves the extraction of a probabilistic estimate on the large field regions that decays exponentially in its volume. We summarize this large deviation argument by saying that “large field regions are typically rare and small”. This is embodied more precisely in the following technical proposition: Proposition 2. Let 5 be some chosen positive constant, then there exists a function U :]0, 1[→]0, 1[ such that limg→0 U (g) = 0 and such that for any N ≥ 0, any polymer Y in D(N ) , any large field region Γ in Y, any allowed graph g making Y 4-vertex irreducible, and for any value of the h parameters, the following inequality holds for 0 < g < 1 : Z Y Y (φ) 1lR 4 1lR 4 dµC 1 −(1+1 ) . φ ≥ 2g φ ≤ g−(1+1 ) [Tg,Γ (h)] ∆ IΓ (∆) ∆ IΓ (∆) ∆∈Γ

·

Y ∆ isolated

g 5 . exp(−I) ≤ U (g)#(Y ) .

∆∈Y \Γ

(189)

P We used here the notations of Sec. 2, and have written φIΓ (∆) instead of i∈IΓ (∆) φi for simplicity. I denotes again the interpolated interaction as reexpressed in (64). Finally a cube ∆ in Y is called isolated if IΓ (∆) = {i(∆)}. This means that the large field block containing ∆ is reduced to ∆, hence no other cube is glued to ∆ by Γ. Note that sharp characteristic functions are used in this bound, and denoted by a 1l, instead of the previously introduced smooth functions χΓ . Indeed this bound will be used after the smooth functions have been bounded by the sharp ones.

166

A. ABDESSELAM and V. RIVASSEAU

Proof. The task of looking for small factors per cube of Y can be reduced to the extraction of such factors for a special category of cubes. This is why we define first the notion of summital cubes of Y . A cube ∆ of Y is said summital if ˜ ∈ D(N ) either i(∆) ≥ 1 and there exists a cube ∆ i(∆)−1 that contains no cube of Y, or ˜ = ∆. i(∆) = 0 in which case we choose ∆ We denote by YS the set of summital cubes of Y . We have Lemma 20. The following inequalities relate the cardinals of Y and YS : #(YS ) ≤ #Y ≤

M4 1 #(YS ) − 4 . M4 − 1 M −1

(190)

Proof. The first inequality is tautological, therefore we concentrate on the second. If ∆ is any cube of D(N ) , we define nY,∆ = #({∆0 ∈ Y |∆0 ⊂ ∆}) and nYS ,∆ = #({∆0 ∈ YS |∆0 ⊂ ∆}). We prove by induction on i(∆), that if nY,∆ > 0 then nYS ,∆ > 0 and M4 1 nY ,∆ − 4 . (191) nY,∆ ≤ 4 M −1 S M −1 This will prove the lemma since Y is non-empty and is contained, together with (N ) YS , in the unique cube of the last slice DN . For i(∆) = 0: if nY,∆ > 0, then ∆ ∈ Y is the only possibility, and by definition ∆ has to be a summital cube of Y . Therefore nY,∆ = nYS ,∆ = 1 and we have equality in (191). Suppose the statement is proven for any ∆ with i(∆) = i, for some fixed i ≥ 0. Let now ∆ be a cube in the following layer, i.e. with i(∆) = i + 1, satisfying nY,∆ > 0. Consider the set J of cubes ∆0 such that ∆0 ⊂ ∆, i(∆0 ) = i and nY,∆0 > 0. By the induction hypothesis, for any ∆0 ∈ J, nYS ,∆0 > 0 and M4 1 nYS ,∆0 − 4 . 4 M −1 M −1 A few cases are to be distinguished: nY,∆0 ≤

(192)

Case 1: ∆ 6∈ Y . We have by (192) X X M4 1 #(J) M4 0 0 nYS ,∆ − 4 nYS ,∆ − 4 . nY,∆ ≤ = 4 nY,∆ = 4 M − 1 M − 1 M − 1 M −1 0 0 ∆ ∈J

∆ ∈J

(193) Since nY,∆ > 0, J must be non-empty and (193) with #(J) ≥ 1 implies (191). Case 2: ∆ ∈ Y − YS . Since ∆ is not summital, all the cubes ∆0 just above ∆ contain a cube of Y , and therefore satisfy nY,∆0 > 0. Therefore #(J) = M 4 , and we can write X X M4 1 0 n nY,∆0 ≤ 1 + − (194) nY,∆ = 1 + Y ,∆ M4 − 1 S M4 − 1 0 0 ∆ ∈J

∆ ∈J

4

= 1+ as wanted.

M4 M4 1 M n = nYS ,∆ − 4 , − Y ,∆ S 4 4 4 M −1 M −1 M −1 M −1

(195)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

167

Case 3: ∆ ∈ YS . Here J can possibly be empty, however we have again: X M4 1 0 nY ,∆ − 4 nY,∆ ≤ 1 + M4 − 1 S M −1 0 ∆ ∈J

= 1+ ≤

#(J) M4 (nYS ,∆ − 1) − 4 M4 − 1 M −1

(196)

M4 M4 1 M4 n = nY ,∆ − 4 , + 1 − Y ,∆ S M4 − 1 M4 − 1 M4 − 1 S M −1

(197)

This completes the proof of the lemma.

Thanks to the lemma, our goal is now limited to extract a small factor per summital non-isolated cube of Y . Let ∆ be such a cube, we consider P (∆) = {∆0 ∈ Y |∆ ⊂ ∆0 } and the partitions p of P (∆) into connected components with respect to gluing by Γ. If X is such a component, let i(X) denote {i(∆0 )|∆0 ∈ X}. Finally when X ranges through p, the collection of the i(X) can be written {I0 (∆), . . . , Iµ(∆) (∆)}, µ(∆) ≥ 0, where for any ν, 0 ≤ ν ≤ µ(∆), Iν (∆) is connected, namely of the form {iν (∆), iν (∆) + 1, . . . , jν (∆)}, with jν (∆) ≥ iν (∆). Besides the ordering has been chosen so that for any ν, jν (∆) < iν+1 (∆). Note def

that i0 (∆) = i(∆) and since ∆ is not isolated, j0 (∆) > i0 (∆). We denote φIν (∆) = Pjν (∆) i=iν (∆) φi . Now let D be some positive constant integer, to be specified later. For any summital non-isolated cube ∆ of Y , we introduce in the left-hand side of (189) the identity Y R R 1l{ φ4 ≤K8 g−(1+1 ) } + 1l{ φ4 >K8 g−(1+1 ) } , (198) 1= ˜ ∆

i0 (∆)≤β≤i0 (∆)+D prβ (∆)∈Y

β

˜ ∆

β

˜ ⊂ ∆ and ∆ ˜ contains no ˜ is the above chosen cube in D(N ) such that ∆ where ∆ cube of Y . Here K8 is some positive constant that will be fixed later. Next we expand the product of such factors. The cubes ∆, such that at least one term of the form 1l{R φ4 >K8 g−(1+1 ) } is selected, are called of type I. Let β(∆) ˜ ∆

β

be a choice of index β, i0 (∆) ≤ β ≤ i0 (∆) + D, prβ (∆) ∈ Y , satisfying the last property. We can bound the characteristic functions of either form, chosen for the other indices β, by one. Remark that one would have to sum over all possible choices in this first expansion, i.e. to pay a factor 2D+1 per non-isolated summital cube. For the cubes such that for any β the selected term is 1l{R φ4 ≤K8 g−(1+1 ) } , which ˜ ∆

we call of type II, we further introduce the factor 1 = 1l{R

˜ ∆

φ4I

0 (∆)

≤K9 g−(1+1 ) }

+ 1l{R

˜ ∆

φ4I

0 (∆)

β

>K9 g−(1+1 ) }

.

(199)

Here again K9 is a constant to be specified later. Now we expand the product of the expressions in the right-hand side of (199) when ∆ ranges over type II cubes.

168

A. ABDESSELAM and V. RIVASSEAU

If the first term of (199) is chosen we call ∆ of type II.1, otherwise ∆ is called of type II.2. For this last category of cubes two cases are possible. In case j0 (∆) ≤ i0 (∆) + D, we say that the cube is of type II.2.1, else we say it is of type II.2.2. Finally for every II.2.2 cube ∆, we introduce a last chopping of the functional integral, using this time a majoration: 1 ≤ 1l

R

aν (∆)16/3

max

˜ ∆

1≤ν≤µ(∆)

X

+

1l

aν (∆)16/3

φ4I

ν (∆)

R ˜ ∆

1≤ν≤µ(∆)

φ4I

≤K10 g−(1+1 )

ν (∆)

>K10 g−(1+1 )

,

(200)

˜

3

where aν (∆) = M 40 (iν (∆)−i(∆)) , and K10 is a constant to be fixed later. We now expand the product of the right-hand side of (200), where ∆ ranges over all cubes of type II.2.2. If the first term of (200) is chosen, ∆ is called of type has been chosen, we say that ∆ R 4 II.2.2a. If a term 1l −(1+1 ) 16/3 aν (∆)

˜ ∆

φI

ν (∆)

>K10 g

is of type II.2.2b.ν. We see that we have a tree of possible choices, on which we have to sum to get a majorant of the whole functional integral in (189). Note however that the distinction between type II.2.1 and type II.2.2 cubes is not to be summed over; one has to take the supremum over the output of both cases. Now let us concentrate on a particular term in this majorant expression. Such a term is of the form: Z Y Y × (φ) 1lR 4 1lR 4 C = dµC 1 [Tg,Γ (h)]

∆

∆∈Γ

Y

× exp(−I) ×

Γ (∆)

Y

g 5 ×

Y

Y

∆ type II.1

i0 (∆)≤β≤i0 (∆)+D prβ (∆)∈Y

Y

∆ type II.2.1

i0 (∆)≤β≤i0 (∆)+D prβ (∆)∈Y

×

∆ type II.2.2a

·1l

max

aν (∆)16/3

˜ ∆

φ4I

ν (∆)

∆ type II.2.2b.ν

φI

Γ (∆)

≤ g−(1+1 )

φ4β(∆) >K8 g−(1+1 ) }

!

.1l{R

φ4β ≤K8 g−(1+1 ) }

φ4I

0 (∆)

˜ ∆

≤K9 g−(1+1 ) }

! ˜ ∆

! .1l{R

φ4β ≤K8 g−(1+1 ) }

˜ ∆

φ4I

0 (∆)

>K9 g−(1+1 ) }

! 1l{R

˜ ∆

φ4β ≤K8 g−(1+1 ) }

.1l{R

φ4I

0 (∆)

˜ ∆

>K9 g−(1+1 ) }

!

≤K10 g−(1+1 )

!

Y

Y

×

R

∆

!

˜ ∆

i0 (∆)≤β≤i0 (∆)+D prβ (∆)∈Y

1≤ν≤µ(∆)

∆∈Y \Γ

˜ ∆

1l{R

Y

Y

−(1+1 )

1l{R

1l{R

Y

×

2g

∆ type I

∆ isolated

×

≥

φI

1l{R

i0 (∆)≤β≤i0 (∆)+D prβ (∆)∈Y

˜ ∆

φ4β ≤K8 g−(1+1 ) }

.1l{R

˜ ∆

φ4I

0 (∆)

>K9 g−(1+1 ) }

!

·1l

aν (∆)16/3

R ˜ ∆

φ4I

ν (∆)

>K10 g−(1+1 )

.

(201)

169

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

R 4 For each ∆ of type I, we use the majoration 1 ≤ K8−1 g (1+1 ) ∆ ˜ φβ(∆) which is valid in the domain of integration defined by the characteristic functions. For ∆ ˆ be the box in Di(∆)+1 containing ∆ (i.e. the box just below). of type II.1, let ∆ ˆ ∈ Γ. Therefore by Since we supposed that the cube ∆ is non-isolated, we have ∆ definition of χΓ , we have the constraint Z 1 −(1+1 ) g ≤ φ4ˆ , (202) 2 ˆ I(∆) ∆ P

def

= where φI(∆) ˆ write

i0 (∆)
ˆ we can use Lemma 19 to φi . But for any x ∈ ∆,

ˆ φ ˆ )(x) + Fluct(∆, ˆ φ ˆ )(x) , (x) = φI(∆) (x) − Fluct(∆, φI(∆) ˆ ˆ I(∆) I(∆)

(203)

where ˆ φ ˆ )(x) def = Fluct(∆, I(∆)

Z ˜ 2 ×[0,1]2 ∆

µ,ν dwf∆, X∆, ˆ (x, w) ˆ ˜ ∆ ˜ ∆ ˆ (x, w)∂µ ∂ν φI(∆)

(204)

ˆ − i(∆) ˜ ≤ 2, satisfies, since i(∆) I(∆) )(x) ≤ K7

φ(x) − Fluct(∆, ˆ φˆ

1 ˜ |∆|

1/4

Z 4

˜ ∆

dy φI(∆) (y) ˆ

M2 .

(205)

As a result we obtain from the elementary inequality (a + b)4 ≤ 8(a4 + b4 ), 1 −(1+1 ) g ≤ 2

Z dx 8 ˆ ∆

4 4 ˆ ˆ φI(∆) (x) − Fluct(∆, φI(∆) )(x) + Fluct(∆, φI(∆) )(x) ˆ ˆ ˆ

4 Z ˆ |∆| 4 8 ˆ K M dx Fluct(∆, φI(∆) )(x) + 8 dy φI(∆) (y)4 ≤8 ˆ ˆ ˜ 7 ˆ ˜ |∆| ∆ ∆ Z

Z ≤8

˜ ∆

Z ≤8

˜ ∆

(206)

(207) 4 Z 4 ˆ φ ˆ )(x) + 8M 16 K74 dx Fluct(∆, dy φI0 (∆) (y) − φi(∆) (y) I(∆) ˜ ∆

(208) 4 Z Z ˆ φ ˆ )(x) 4 + 64M 16 K 4 dx Fluct(∆, φ4I0 (∆) + φ4i(∆) . 7 I(∆) ˜ ∆

˜ ∆

(209)

Z ˜ ∆

R

4 ˜ φi(∆) ∆

≤ K8 g −(1+1 ) and

4 ˆ dx Fluct(∆, φI(∆) )(x) ≥ K11 g −(1+1 ) , ˆ

(210)

But for type II.1 cubes, we have also the constraints 4 −(1+1 ) φ , so as a result ˜ I0 (∆) ≤ K9 g ∆

R

170

A. ABDESSELAM and V. RIVASSEAU

def

with K11 = 18 [(1/2) − 64M 16 K74 (K8 + K9 )]. We suppose we have chosen K8 and K9 such that K11 > 0. Complete fixing of the constants is postponed to the end of the proof. In the integral (201) we can now readily introduce the majoration 1≤

−1 1+1 g K11

Z ˜ ∆

4 ˆ dx Fluct(∆, φI(∆) )(x) ˆ

(211)

that is valid in the domain of integration. Consider now the case where ∆ is of type II.2.1. As we did for type II.1 cubes and with the same notations we can write 4 Z Z ˆ |∆| 1 −(1+1 ) 4 8 4 ˆ g K M ≤ dx Fluct(∆, φI(∆) )(x) +8 dy φI(∆) (y) . (212) ˆ ˆ ˜ 7 2 ˆ ˜ |∆| ∆ ∆ ˜ we have But by H¨ older’s inequality, for any y ∈ ∆  3  4 X (y) ≤  1  φI(∆) ˆ i0 (∆)
and thus Z ˜ ∆

Z

X

dy φI(∆) (y)4 ≤ D3 ˆ

i0 (∆)
Denote now K12

1 = 8

def

˜ ∆



X

φi (y)4 

(213)

i0 (∆)
dy φi (y)4 ≤ D4 K8 g −(1+1 ) .

1 − 8M 16 K74 D4 K8 2

(214)

.

(215)

If we assume we have fixed the constants such that K12 > 0, then we can introduce in the functional integral the majoration 4 Z −1 1+1 ˆ dx Fluct(∆, φI(∆) )(x) . (216) 1 ≤ K12 g ˆ ˜ ∆

Let now ∆ be of type II.2.2a, and a be any integer such that 0 ≤ a ≤ k, where k is the length of the graph g = (l1 , . . . , lk ). Now consider Ba (∆) the block containing ∆ in the partition ΠΓ ((l1 , . . . , la )). Since glued cubes must belong to the same block, P ˜ ⊂ ∆, the field φB (∆) = φi , that appears in the expression (64) of on ∆ a i≥i(∆) pri (∆)∈Ba

the interaction I must have the form:

φBa (∆) = φI0 + φIν1 + · · · + φIνr ,

(217)

˜ we have by H¨older’s with 1 ≤ ν1 < · · · < νr ≤ µ(∆), and r ≥ 0. Now for any x ∈ ∆ inequality −1 φI0 (x) = 1 · φB (∆) (x) + a−1 (218) ν1 (−aν1 φIν1 (x)) + · · · + aνr (−aνr φIνr (x))| a 1/4 3/4 4 + · · ·+ a−4/3 . (219) φBa (∆) (x)+ a4ν1 φ4Iν1 (x)+ · · · + a4νr φ4Iνr (x) ≤ 1 + a−4/3 ν1 νr

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

171

Therefore when we integrate over x we get: Z 3 φ4I0 (x) ≤ 1 + a−4/3 + · · · + a−4/3 ν1 νr ˜ ∆

×

Z

φ4Ba (∆) (x) + a−4/3 a16/3 ν1 ν1

˜ ∆

Z ˜ ∆

φ4Iν1 (x) + · · · + a−4/3 a16/3 νr νr

Z ˜ ∆

φ4Iνr (x) , (220)

and thus Z 3 φ4I0 (x) ≤ 1 + a−4/3 + · · · + a−4/3 ν1 νr ˜ ∆

×

Z ˜ ∆

φ4Ba (∆) (x) + a−4/3 + · · · + a−4/3 ν1 νr

max

1≤ν≤µ(∆)

Z 4 φ (x) . (221) a16/3 ν Iν ˜ ∆

Now, by definition of the aν ’s, µ +∞ X X 1 i ˜ + · · · + a−4/3 M − 10 (iν −i(∆)) ≤ M − 10 ≤ a−4/3 ν1 νr ν=1

(222)

i=0

= (1 − M − 10 )−1 = S . 1

def

(223)

Finally if we define def

K13 =

K9 − K10 S > 0 , (1 + S)3

(224)

R −(1+1 ) we conclude from (221), and the fact that in the present case ∆ ˜ φI0 (x) > K9 g R 4 −(1+1 ) and max1≤ν≤µ(∆)R aν (∆)16/3 ∆ hold in the domain of integra˜ φIν ≤ K10 g 4 −(1+1 ) K13 . This has to be true for any a, 0 ≤ a ≤ k. tion, that so does ∆ ˜ φBa (∆) ≥ g ˜ satisfies As a result the part of the interaction that is integrated in ∆ k X

I∆ ˜ = g

(ha − ha+1 )

a=0 k X

≥g

B∈ΠΓ ((h1 ,...,ha ))

Z (ha − ha+1 )

a=0

≥g

X

X k

Z

4 φB (x) dx

(225)

˜ ∆

4 φBa (∆) (x) dx

˜ ∆

(ha − ha+1 ) g −(1+1 ) K13 = K13 g −1 .

(226)

a=0 −1

Therefore we can extract from the interaction a factor e−I∆˜ ≤ e−K13 g . Indeed ˜ are disjoint, for this is what summital cubes are all about. the ∆ It remains to consider the case of ∆ of type II.2.2b.ν. We have, in the domain of integration on the fields, the inequality: Z 1 φ4Iν (∆) > K10 g −(1+1 ) · M − 10 [iν (∆)−i(∆)] . (227) ˜ ∆

¯ is ¯ is the cube pri (∆) (∆) containing ∆ at scale iν (∆), we have since ∆ But if ∆ ν ¯ > 0, that ∆ ¯ is necessarily a small the top cube of a large field block and i(∆)

172

A. ABDESSELAM and V. RIVASSEAU

R 4 −(1+1 ) field cube. Hence we also have the condition ∆ . This and ¯ φIν (∆) ≤ g (227) entail the existence of a large fluctuation of φIν (∆) at some intermediate ˜ and i(∆). ¯ Define ∆ as the unique cube containing ∆ at scale scale between i(∆) ¯ − i(∆))], ˜ ˜ + E[ 1 (i(∆) where E[x] means the integral part of x. i(∆) 5 ˜ For any x in ∆, we can use Lemma 19 with ∆1 = ∆2 = ∆, to write the decomposition φIν (∆) (x) =

φIν (∆) (x) − Fluct(∆, φIν (∆) )(x) + Fluct(∆, φIν (∆) )(x) ,

where following the notations of the lemma Z µ,ν dwf∆,∆ (x, w)∂µ ∂ν φIν (∆) X∆,∆ (x, w) . Fluct(∆, φIν (∆) )(x) =

(228)

(229)

∆2 ×[0,1]2

The inequality (169) provides us with the bound 1/4 Z 4 φI (∆) (x) − Fluct(∆, φI (∆) )(x) ≤ K7 1 dy φ (y) , Iν ν ν |∆|

(230)

∆

since here i(∆2 ) − i(∆1 ) = 0. We can now perform a similar reasoning to that for type II.1 cubes. We readily obtain Z

Z ˜ ∆

φ4Iν (∆) ≤

˜ ∆

4 dx φIν (∆) (x) − Fluct(∆, φIν (∆) )(x) + Fluct(∆, φIν (∆) )(x)

(231) 4 Z Z 4 dx φIν (∆) (x) − Fluct(∆, φIν (∆) )(x) + 8 dx Fluct(∆, φIν (∆) )(x) ≤8 ˜ ∆

˜ ∆

(232) ˜ · K4 · 1 ≤ 8|∆| 7 |∆|

Z

Z ∆ ˜

dy φ4Iν (y) + 8

≤ 8 · M −4(i(∆)−i(∆)) K74

˜ ∆

4 dx Fluct(∆, φIν (∆) )(x) Z

Z ¯ ∆

dy φ4Iν (y) + 8

˜ ∆

4 dx Fluct(∆, φIν (∆) )(x) .

(233)

(234)

˜ + 1 (i(∆) ¯ − i(∆)) ˜ ¯ entails ∆ ⊂ ∆, ¯ from which we deduced Indeed i(∆) ≤ i(∆) 5 R 4 ≤ i(∆) −(1+ ) 1 , (227) and (234), we infer that the last inequality. Now from ∆ ¯ φIν (∆) ≤ g Z ˜ ∆

4 dx Fluct(∆, φIν (∆) )(x)

g −(1+1 ) 1 ¯ ˜ ˜ − 10 (i(∆)−i( ∆)) 4 −4(i(∆)−i(∆)) ≥ − 8K7 M K10 M (235) 8 1 g −(1+1 ) − 1 (i(∆)−i( ¯ ˜ ˜ ¯ ˜ ∆)) 4 −4(i(∆)−i(∆))+ (i(∆)−i( ∆)) 10 10 M K10 − 8K7 M . (236) = 8

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

173

˜ = E[ 1 (i(∆)−i( ¯ ˜ ≥ 1 (i(∆)−i( ¯ ˜ But i(∆)−i(∆) ∆))] ∆))−1. Besides, since ∆ is a type 5 5 ¯ − i(∆)) ˜ ≥ i1 (∆) − i(∆)) ˜ ≥ II.2.2 cube, we have j0 (∆) ≥ D + 1 + i(∆), so that i(∆) D + 2. Therefore we have ˜

¯

˜

¯

˜

M −4(i(∆)−i(∆))+ 10 (i(∆)−i(∆)) ≤ M 4− 10 (i(∆)−i(∆)) ≤ M 4− 10 (D+2) . As a result Z ˜ ∆

1

7

7

4 1 ¯ ˜ dx Fluct(∆, φIν (∆) )(x) ≥ g −(1+1 ) M − 10 (i(∆)−i(∆)) K14 ,

(237)

(238)

7 def where K14 = 18 K10 − 8K74 M 4− 10 (D+2) . Again we shall see that the constants can be fixed so that K14 > 0. We then introduce in the integral the majoration −1 · 1 ≤ g 1+1 M ( 10 (i(∆)−i(∆)) K14 1

¯

˜

Z ˜ ∆

4 dx Fluct(∆, φIν (∆) )(x)

(239)

that is valid in the domain of integration. The discussion of all possible cases is now complete and we can write a bound on a generic term C of (201). We have: Z Z Y Y (φ) g 5 × φ4β(∆) K8−1 g 1+1 |C| ≤ dµC [Tg,Γ (h)] ˜ ∆ ∆ isolated

×

Y

˜ ∆

∆ type II.1

×

Y

4 Z −1 1+1 ˆ dx Fluct(∆, φI(∆) )(x) K12 g ˆ

∆ type II.2.1

×

Y

∆ type I

4 Z −1 1+1 ˆ dx Fluct(∆, φI(∆) )(x) K11 g ˆ

˜ ∆

−K13 g−1 e

∆ type II.2.2a

×

Y ∆ type II.2.2b.ν

4 Z ¯ ˜ −1 1+1 g M (1/10)(i(∆)−i(∆)) · dx Fluct(∆, φIν (∆) )(x) K14 . ˜ ∆

(240) ˜ as well as the In the right-hand side we can get the spatial integrations on x in ∆, integrals over the parameters w, out from the functional integral, that is performed first. The integrand for this Gaussian functional integral is now a polynomial expression in the fields φi and their double gradients ∂µ ∂ν φi . We bound this functional integral with Lemma 16. Once integrated, every field φi gives a factor K4 M −i , and every double gradient ∂µ ∂ν φi gives a much better factor K4 M −3i . Besides we get Q a product of local factorials ∆∈Y (n(∆)!)1/2 to deal with. We recall that n(∆) denotes the number of fields φi (x) as well as double gradients ∂µ ∂ν φi (x) located in ∆, that is with ∆(x, i) = ∆, that are integrated with respect to dµC[T (h)] . g,Γ

174

A. ABDESSELAM and V. RIVASSEAU

One has also to use the L1 bound (168) on the densities defining the fluctuation fields. Remark that after the bound (122) on the Gaussian integration, nothµ,ν (x, w). ing depends on the parameters w except the corresponding densities f∆ 1 ,∆2 Therefore there is no problem of factorization in using these L1 bounds. Finally ˜ ˜ costs a volume factor M 4i(∆) . Collecting all remark that spatial integration in ∆ these factors we obtain Y Y ˜ g 5 × K8−1 K44 M −4(β(∆)−i(∆)) g 1+1 |C| ≤ ∆ type I

∆ isolated

Y

×

e

−K13 g−1

∆ type II.2.2a

Y

×

∆ type II.1

∆ type II.2.1

Y

×

M

ˆ 2i(∆)−3α 1

... M

ˆ 2i(∆)−3α 4

α1 ,...,α4 =i0 (∆)+1

˜ −1 4 1+1 K4 g M 4i(∆) K12

Y

×

X

j0 (∆) ˜ −1 4 1+1 K4 g M 4i(∆) K11

ˆ ˆ M 2i(∆)−3α1 . . . M 2i(∆)−3α4

X

j0 (∆)

α1 ,...,α4 =i0 (∆)+1

¯ ˜ ˜ −1 4 1+1 K4 g M (1/10)(i(∆)−i(∆)) · M 4i(∆) K14

∆ type II.2.2b.ν

M 2i(∆)−3α1 . . . M 2i(∆)−3α4

X

jν (∆)

·

α1 ,...,α4 =iν (∆)+1

×

Y

(n(∆)!)1/2 .

(241)

∆∈Y

The indices α are the slice labels of the individual fields that have been inteˆ grated and that come in bunches of four. We used the fact that I(∆) = {i0 (∆) + 1, . . . , j0 (∆)} and Iν (∆) = {iν (∆), . . . , jν (∆)}. Needless to say that in (241) the numbers n(∆) depend on the previous choice of the α’s. Note that for a type II.1 ˆ − i(∆) ˜ ≤ 2, we can bound the factor belonging to ∆ in (241) by cube, since i(∆) X

j0 (∆) −1 · K44 · M 16 · K11

˜ ˜ M −3(α1 −i(∆)) . . . M −3(α4 −i(∆)) .

(242)

α1 ,...,α4 =i0 (∆)+1

The analog remark applies to type II.2.1 cubes. For a type II.2.2b.ν cube ∆, in the corresponding factor in (241) we can isolate the contribution of one of the four fields located, say, at slice α ∈ {iν (∆), . . . , jν (∆)}. It gives ˜

˜

˜

˜

M ( 40 (iν (∆)−i(∆))+i(∆)+2i(∆)−3α ≤ M −3(α−i(∆))+ 2 (iν (∆)−i(∆)) , 1

1

(243)

˜ + 1 (iν (∆) − i(∆)) ˜ by definition. since i(∆) ≤ i(∆) 5 Now remark that α ≥ iν (∆). As a result we bound the right-hand side of (243) by M −(5/2)(α−i(∆)) . Therefore, the factor corresponding to ∆ is bounded by X

jν (∆) −1 4 −(1+1 ) K4 g K13

α1 ,...,α4 =iν (∆)

M − 2 (α1 −i(∆)) . . . M − 2 (α4 −i(∆)) . 5

5

(244)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

175

To get rid of the local factorials, we use Lemma 17. Indeed for every pair of boxes def ∆1 , ∆2 in E = D(N ) , define the function 1 9 def (1 − M − 2 )M − 2 (i(∆1 )−i(∆2 )) if ∆1 ⊃ ∆2 . (245) G(∆1 , ∆2 ) = 0 else. For any ∆1 ∈ D(N ) we have, X 1 G(∆1 , ∆2 ) = (1 − M 2 ) ∆2 ∈D (N )

X

M − 2 (i(∆1 ))−i #({∆2 |i(∆2 ) = i, ∆2 ⊂ ∆1 }) 9

0≤i≤i(∆1 )

1 2

= (1 − M )

(246)

X

M

− 12 (i(∆1 )−1)

≤ 1.

(247)

0≤i≤i(∆1 )

Now we take the contracted fields to form the set O in the context of Lemma 17. We define the map S1 as the one that to a field associates its location in phase space, i.e. ˜ We define now the the cube of the corresponding scale α under the concerned ∆. “location of the field after displacement” or its image by S2 as the corresponding cube ∆. Q Q Note that #(O) is bounded by 4#(YS ). Besides, ∆∈E n1 (∆)! = ∆∈Y n(∆)!, where we used the notations of Lemma 17. Now remark that Y Y n2 (∆)! = 4! ≤ 24#(YS ) . (248) ∆∈E

∆ type I or type II.1 or type II.2.1 or type II.2.2b.ν

Indeed a ∆ produces exactly 4 contracted fields if of type I and 4 double gradients if of type II.1, II.2.1 or II.2.2b.ν. Lemma 17 entails Y Y 1 1 9 1/2 2#(YS ) #(Y ) −1/2 (α−i(∆)) S (n(∆)!) ≤e · 24 2 · M4 (1 − M 2 ) . (249) ∆∈Y

field

Remark that for a double gradient field, considering (242), (243) and (244), we see 9 that we can extract at least M − 4 (α−i(∆)) to beat the corresponding factor in (249). However, for a field at scale β(∆) coming from a type I cube ∆, the available factor 5 5 ˜ ˜ M −(β(∆)−i(∆)) in (241) is not sufficient. We have a loss of M 4 (β(∆)−i(∆)) ≤ M 4 D , per such field, to account for. As a result, (241) can be improved to yield Y Y Y −1 g 5 × K8−1 K44 g 1+1 M 5D × e−K13 g |C| ≤ ∆ isolated

×

Y

∆ type I

∆ type II.1

×

Y

∆ type II.2.2a

X

j0 (∆) −1 4 1+1 K4 g M 16 . K11

M

− 34 (α1 −i(∆))

...M

− 34 (α4 −i(∆))

α1 ,...,α4 =i0 (∆)+1

−1 4 1+1 K4 g M 16 . K12

∆ type II.2.1

X

j0 (∆)

α1 ,...,α4 =i0 (∆)+1

M − 4 (α1 −i(∆)) . . . M − 4 (α4 −i(∆)) 3

3

176

A. ABDESSELAM and V. RIVASSEAU

×

−1 4 1+1 K4 g . K14

Y ∆ type II.2.2b.ν

#(YS ) √ 1 . × e2 24(1 − M − 2 )−2

X

jν (∆)

M − 4 (α1 −i(∆)) . . . M − 4 (α4 −i(∆)) 1

1

α1 ,...,α4 =iν (∆)+1

(250)

To sum over the α’s in the case of a type II.1 or type II.2.1 cube, we use the rough bound j0 (∆) ∞ X X 3 1 1 M − 4 (α−i(∆)) ≤ M − 4 i = (1 − M − 4 )−1 . (251) i=0

α=i0 (∆)+1

In the case of a type II.2.2b.ν cube, we use X

jν (∆)

M − 4 (α−iν (∆)) ≤ (1 − M − 4 )−1 , 1

1

(252)

α=iν (∆)

and we keep the factor M − 4 (iν (∆)−i(∆)) per field, in order to sum over the possible cases. Finally: Y Y Y −1 g 5 × K8−1 K44 g 1+1 M 5D × e−K13 g |C| ≤ 1

∆ isolated and summital

Y

×

∆ type I

∆ type II.1

Y

×

1 −1 4 1+1 K4 g M 16 (1 − M − 4 )−4 K12

∆ type II.2.1

Y

×

∆ type II.2.2a

1 −1 4 1+1 K4 g M 16 (1 − M − 4 )−4 K11

1 −1 4 1+1 K4 g (1 − M − 4 )−4 M −(iν (∆)−i(∆)) K14

∆ type II.2.2b.ν

√ 1 × e 24(1 − M − 2 )−2

#(YS )

2

.

(253)

Now to bound the left-hand side of (189), one has to sum over the different possibilities for non-isolated cubes. The sum over the first decomposition (198) of the domain of integration costs a factor 2D+1 per summital non-isolated cube. The dichotomy type II.1/type II.2 costs again a factor 2 per type II cube, and the dichotomy between type II.2.2a and type II.2.2b an other 2 per type II.2.2 cube. Finally the sum over ν for a type II.2.2b cube is done using the factor M −(iν (∆)−i(∆)) P in (253) and its cost is bounded by i M −i = (1−M −1 )−1 . Therefore the left-hand side of (253) is bounded by V (g)#(YS ) where h √ def V (g) = e2 24(1 − M −1/2 )−2 · max g 5 , 2D+1 K8−1 K44 M 5D g 1+1 , −1 4 1+1 −1 4 1+1 K4 g M 16 (1 − M − 4 )−4 , 2D+2 K12 K4 g M 16 (1 − M − 4 )−4 , 2D+2 K11 1

1

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

2D+3 e−K13 g

−1

−1 4 1+1 , 2D+3 K14 K4 g (1 − M − 4 )−4 (1 − M −1 )−1 . 1

Since limg→0 V (g) = 0 and since by Lemma (20) #(YS ) ≤ #(Y ) ≤ M4 M 4 −1

def

177

i

(254)

M4 M 4 −1 #(YS ),

if

), we get the desired bound. we take U (g) = max(V (g), V (g) The only thing to check now in order to complete the proof is the consistency in fixing K8 , K9 , K10 and D. We see that the constraints that we need to satisfy are: 1 1 [ − 64M 16 K74 (K8 + K9 ) > 0 , 8 2 1 1 = [ − 8M 16 K74 D4 K8 ] > 0 , 8 2 K9 = − K10 S > 0 , (1 + S)3

K11 =

(255)

K12

(256)

K13

(257)

and

1 7 [K10 − 8K74 M 4− 10 (D+2) ] > 0 . 8 The first step is to choose K9 small enough such that K14 =

1 − 64M 16 K74 K9 > 0 , 2

(258)

(259)

then pick K10 small enough so that (257) is verified. One has then to take the integer constant D large enough for (258) to be fulfilled. Finally, we choose K8 small enough so that (255) and (256) are satisfied. 3.5. Rearrangement of the expansion Before computing the functional derivatives, once we have chosen a large field region Γ, it is essential to group together the contributions of the graphs g, according to the following notions. First we recall that a dressed large field block is a connected component of Y with respect to gluing by Γ, i.e. an element of ΠΓ (∅). Next we say that two links l and l0 are form-equivalent if they are both 2-links or are both 4-links and satisfy def π the following property. We require that lπ = l0 , where we posed π = ΠΓ (∅) and used the notation of Sec. 2.2.3 for the reduced link with respect to a partition π. This means that for any dressed large field block X ∈ π, def

lπ (X) =

X ∆∈X

l(∆) =

X

l0 (∆) = l0 (X) . def

π

(260)

∆∈X

We now define an equivalence relation among graphs in G. We say that two ordered graphs g = (l1 , . . . , lk ) and g0 = (l10 , . . . , lk0 ) with the same length are form-equivalent, if for any a, 1 ≤ a ≤ k, la and la0 are form-equivalent. An equivalence class of graphs is called a form. The partition ΠΓ (g) into 4-VI components of a graph g depends only on the form of g we denote by hgi. Therefore if g and g0 are form-equivalent, g is allowed if and only if g0 is allowed too.

178

A. ABDESSELAM and V. RIVASSEAU

def

Hereupon we will use the natural and consistent notations ΠΓ (hgi) = ΠΓ (g) def

and Thgi,Γ (h) = Tg,Γ (h). The notion of allowedness naturally extends to forms. It is easy to see that for two allowed form-equivalent graphs g and g0 , for a fixed value of the fields and the h parameters, the exponentiated interactions are the same. We can therefore factorize the sum of functional integrals appearing in (36) that correspond to the graphs in a certain allowed form hgi. Indeed consider Ver(l), the sum of vertices associated to 4-links l0 that are form equivalent to a 4-link l. We suppose that the reduced link lπ has support {X1 , . . . , Xα }, with 1 ≤ α ≤ 4 and X1 , . . . , Xα distinct dressed large field blocks. We can write from (14) and (36), Ver(l) =

Z

X

X

l0 4−link in Y l0 π =lπ

(∆1 ,...,∆4 )∈Y 4 l[∆1 ,...,∆4 ]=l0

(−g) ∆1 ∩···∩∆4

dx φi(∆1 ) (x) . . . φi(∆4 ) (x) . (261)

We define for any x ∈ Λ and any B ⊂ Y , N X

def

B

φ (x) =

1l{∆(x,i)∈B} φi (x)

(262)

i=0

from the collection of fields (φY ). Thus we have Z

X

Ver(l) =

(−g)

dx φ{∆1 } (x) . . . φ{∆4 } (x) .

(263)

Λ

(∆1 ,...,∆4 )∈Y 4 l[∆1 ,...,∆4 ]π =lπ

Here l[∆1 , . . . , ∆4 ]π = lπ means that among the possibly repeated cubes (∆1 , . . . , ∆4 ), lπ (Xβ ) of them must belong to Xβ , for every β, 1 ≤ β ≤ α. The sum over which subset of indices in {1, 2, 3, 4} label the cubes that belong to Xβ , for each β gives a combinatorial factor 4! . π β=1 l (Xβ )!

Qα

(264)

The remaining sum over the positions of the boxes is done thanks to the identity X

φ{∆} (x) = φX (x) ,

(265)

∆∈X def

for every x ∈ Λ. Finally if we pose mβ = lπ (Xβ ), for each β, 1 ≤ β ≤ α, in order that m1 + · · · + mα = 4, we have Ver(l) = (−g)

4! m 1 ! . . . mα !

Z

m1 mα dx φX1 (x) . . . φXα (x) .

(266)

Λ

The above-cited factorization allows us to rewrite the activity (36) of a polymer Y

179

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

in the somewhat more glamorous form ext A(Y, (∆ext s , ζs )s∈S ) Z X X = Γ

×

hgi allowed form Y 4−VI

Y

∆j1

dxj1

Y (−g) × 1≤j≤k lj ∈L4

×

Y

s∈S

dh1 . . . dhk

dµC[Thgi,Γ (h)] (φ)

1>h1 >···>hk >0

Z

Z

1≤j≤k lj ∈L2

Z

∆j2

dxj2

C(xj1 , i(∆j1 ); xj2 , i(∆j2 ))

δ

δφi(∆j ) (xj1 ) δφi(∆j ) (xj2 ) 1

4! mj,1 ! . . . mj,αj !

Z Xj,1

dx φ

!

δ 2

mj,αj mj,1 Xj,αj (x) ... φ (x)

Λ

φi(∆ext (ζsext ) × χΓ ((φ)) s ) 

× exp −g

k X

(ha − ha+1 )

a=0

Z

X B∈ΠΓ ((l1 ,...,la ))

 4 dx φB (x)  .

(267)

Λ

Here g = (l1 , . . . , lk ) is a choice of representative of the form hgi; ∆j1 and ∆j2 are a choice of cubes such that l[∆j1 , ∆j2 ] = lj , for any j, 1 ≤ j ≤ k with lj ∈ L2 . For the 4-links l we have used the same notation as in (266) but with an additional index j to distinguish between the vertices. Before we proceed, we first pose to describe more thoroughly the structure of dressed large field blocks. The links {∆1 , ∆2 }, with ∆1 just on top of ∆2 , naturally endow D(N ) with a tree structure. Once we choose a large field region Γ in Y we select from the above-mentioned links those for which ∆2 ∈ Γ ⊂ Y . These selected links form a subforest of the previous tree, whose connected components in Y are the elements of π. We therefore see that if ∆1 and ∆2 are two elements of X ∈ π, then every cube ∆ ∈ D(N ) satisfying ∆1 ⊂ ∆ ⊂ ∆2 must belong to X. Indeed ∆ is on the unique path in the forest of selected links that connects ∆1 and ∆2 . Besides, there exists a largest cube bg(X) ∈ X that contains all the cubes in X. Briefly, X has to be a stack of cubes lying on bg(X), and at the top of the heap the cubes (N ) must be small field ones except possibly if they are in the highest layer D0 . We now introduce an order relation among dressed large field blocks. If X and X 0 are in π, we say that X is above X 0 if there exist ∆ ∈ X and ∆0 ∈ X 0 such that ∆ ⊂ ∆0 . Remark that this is equivalent to bg(X) ⊂ bg(X 0 ). This order relation is not total but, if X1 , . . . , Xα are the blocks making the support of lπ , for some 4-link l whose contribution Ver(l) does not vanish, then X1 , . . . , Xα must be pairwise comparable for this order relation. Therefore in the formula (267) we suppose Xj,1 , . . . , Xj,αj have been labeled such that Xj,1 is above Xj,β for every β, 2 ≤ β ≤ αj . Xj,1 is the top block of the vertex. If B is a subset of D(N ) and ∆ ∈ D(N ) , we define the shadow of B in ∆ as def ˜. ∆ ˜ sh(∆, B) = ∪ ∆∈B\{∆} ˜ ∆⊂∆

(268)

180

A. ABDESSELAM and V. RIVASSEAU

Therefore if ∆ ∈ B, it is easy to see that ∆ is the disjoint union of the non-empty ˜ ˜ B), where ∆ ˜ ranges through sets ∆\sh( ∆, def ˜ ˜ ⊂ ∆, ∆\sh( ˜ ˜ B) 6= ∅} . ∈ B|∆ ∆, S(∆, B) = {∆

(269)

We now expand in (267), each vertex lj as (−g) ×

4! mj,1 ! . . . mj,αj ! X

Z

˜ j ∈S(bg(Xj,1 ),Xj,1 ) ∆

mj,αj mj,1 dx φXj,1 (x) . . . φXj,αj (x) . (270)

˜ j \sh(∆ ˜ j ,Xj,1 ) ∆

We simply decomposed the domain of integration and used the fact that the integrand vanished out of bg(Xj,1 ). The fields like φXj,1 (x) are called the high momentum fields, whereas φXj,2 (x), . . . , φXj,αj (x) are called the low momentum fields of the considered explicit vertex. After making these decompositions, we finally compute the functional derivatives. We avoid excessive formalization by denoting a generic derivation procedure by P. This contains for every operator δ/δφi (x) in (267) the information about whether it derives an explicit vertex, a large field condition χΓ , an external source φi(∆ext ) (ζ ext ), or the exponentiated interaction. This contains also the information about which particular field appearing in some monomial expression does the contraction δ/δφi (x) hook to. We perform the sum over P through several steps. First we sum over the choices c, for every operator δ/δφ, between deriving an external (ζsext ) or not. Then we sum over the derivation procedures Pext , consource φi(∆ext s ) cerning the δ/δφ operators that act on the external sources. Next we introduce a decomposition for the domain of integration of the δ/δφ operators that do not act on the sources, writing Z Z X dxj1 = dxj1 (271) ∆j1

Z

and

∆j2

˜ j ,Y ) ˜ j \sh(∆ ∆ 1 1

˜ j ∈S(∆j ,Y ) ∆ 1 1

X

dxj2 =

˜ j ∈S(∆j ,Y ) ∆ 2 2

Z ˜ j \sh(∆ ˜ j ,Y ) ∆ 2 2

dxj2 .

(272)

Finally we sum over the derivation procedures Pint of the δ/δφ operators that do not act on the sources. Therefore the ultimate version of the sum we have to bound in (54) looks like XXX X X X X XZ dh C . (273) Y

Γ

hgi

˜ of (∆) vertices

c

Pext

˜ of (∆) derivations

Pint

˜ is over the decompositions (270), (271) and (272) of the spatial The sum over the ∆’s integrations involved. Note also that C denotes a generic individual contribution consisting in a functional integral with a few spatial integrations depending on all

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

181

the previous data. How do we bound such a contribution is the purpose of the next section. 3.6. The bound on individual contributions. The domination of low momentum fields The main ingredient is the bound on the low momentum fields of explicit vertices. The high momentum fields are simply bounded with the Gaussian measure. The low-momentum fields, however must be “dominated” [15, 31]. One already had a flavor of this method in Proposition 2. The idea is that since low-momentum fields are of much lower frequency (i.e. higher index) than the localization cube in which they are integrated, they are statistically almost constant. Therefore we compare them with their average in some cube, whose scale is intermediate between that of the field and that of the localization cube. On the other hand, the bound on the average is of the H¨ older type and uses the small field conditions. Hence it stems indirectly from the positivity of the interaction itself. Finally the bound on the fluctuation part is Gaussian, since the double gradients involved are well-behaved concerning local factorials. The tricky issue is the tuning of the intermediate cube of averaging. The frequency i of this cube is the one appearing in the “effective” power counting factor M −i that we gain for a low momentum field after the domination procedure. If this averaging scale is too high (i too small), the power counting factor is bad and cannot pay all the combinatorial sums we have to perform. If it is too low (i too big), we risk the fluctuation term behaving as badly as would have the initial low momentum field, if bounded by the Gaussian measure only. To bound an individual contribution C, first we bound everything by its modulus. Then we bound every propagator coming from a 2-link lj = l[∆j1 , ∆j2 ], using (118), by: C(xj , i(∆j ); xj , i(∆j )) ≤ K3 (r)M −2i(lj ) 1 + M −i(lj ) d2 (lj ) −r . (274) 1 1 2 2 The notation is that of Proposition 15; r is some large enough fixed integer. We also take out of the functional integral the spatial integrations of the explicit vertices. From (274) the line factors M −i(lj ) are affected by the corresponding contracted fields, as we explain below. 3.6.1. The explicit vertices First we consider the case of an explicit vertex lj . After the functional derivations and taking the modulus, it has the form Z m0 m0 4! dx φX1 (x) 1 . . . φXα (x) α M −i1 . . . M −iγ , (275) g· m1 ! . . . mα ! ∆\sh( ˜ ˜ 1) ∆,X where the m0 integers are the exponents left after derivation, and M −i1 , . . . , M −in are the scaling factors of the derived fields collected from (274). We have 0 ≤ m0β ≤ mβ for any β, 1 ≤ β ≤ α, and m01 + · · · + m0α + γ = 4. From now on, we call field any one of the four factors appearing in (275), including the factors M −i . We treat separately each field, generically denoted by the

182

A. ABDESSELAM and V. RIVASSEAU

symbol φ that we use to label the associated structures, like the averaging cube ∆φ for instance, which is defined independently for each field. We find it simpler in the sequel to define ∆ϕ for any low momentum field, even those that have been derived. However the only ∆ϕ that will be used for averaging are those of the remaining fields. ˜ ϕ the domain First consider a low momentum field |φXϕ (x)|. We denote by ∆ ¯ ϕ the smallest cube of Xϕ of integration of the corresponding vertex lϕ , and by ∆ ˜ ϕ . Remark that ∆ ¯ ϕ cannot contain any other cube of Xϕ . Indeed, by containing ∆ ˜ ϕ is not in Xϕ , therefore ∆ ¯ ϕ has to ¯ ϕ containing ∆ definition, the cube just above ∆ be a small field cube and glues to nothing above it. Note that in this argument, we ˜ ϕ is in the top block of the vertex which we denote by X1,ϕ , which used the fact ∆ ˜ϕ ⊂ ∆ ¯ ϕ is strict. The averaging is distinct from Xϕ , and therefore the inclusion ∆ ˜ ¯ ¯ cube ∆ϕ is chosen such that ∆ϕ ⊂ ∆ϕ ⊂ ∆ϕ ; the rule will be precised later. Now we write: φX (x) ≤ φX (x) − Fluct ∆φ , φXφ (x) + Fluct ∆φ , φXφ (x) , (276) φ φ where using the notation of Lemma 19 with ∆1 = ∆2 = ∆φ , we posed Xφ

Fluct ∆φ , φ

def (x) =

Z ∆2φ ×[0,1]2

µ,ν Xφ dw f∆ (X∆φ ,∆φ (x, w) . (277) ,∆ (x, w)∂µ ∂ν φ φ

φ

Still by Lemma 19 we have the estimate φXφ (x) − Fluct ∆φ , φXφ (x) ≤ K7

1 |∆φ |

Z Xφ

dy φ

4 (y)

1/4 .

(278)

∆φ

Note that we used the fact that φXφ is smooth in ∆φ , which is true since by our ¯ φ , the bunch of frequencies making up φXφ (ξ) does not vary previous remark on ∆ ¯ φ is a small field cube, thanks to the function ¯ φ . Since ∆ when ξ ranges through ∆ χΓ (φ), even modified by the functional derivatives, we have in the domain with non-zero integrand, with respect to the fields, the condition Z 4 dy φXφ (y) ≤ g −(1+1 ) . (279) ∆φ

¯ φ, This entails, since ∆φ ⊂ ∆ 1+ φXφ (x) − Fluct ∆φ , φXφ (x) ≤ K7 M −i(∆φ ) · g − 4 1 .

(280)

The fluctuation part is bounded with the Gaussian measure. In fact we write Z 4 X X Fluct ∆ , φXφ (x) ≤ φ j∈Iφ µ,ν=1

µ,ν f

dw

∆2φ ×[0,1]2

∂µ ∂ν φXφ (X∆ ,∆ (x, w) , φ φ

∆φ ,∆φ (x, w)

(281)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

183

where Iφ is the set of scales making up φXφ . The sums over j, µ and ν, the µν integration over w and the functions f∆ ,∆φ are taken out of the functional inteφ gral. We will later bound the double gradient like in (122) so that the x and w and µ, νR dependences disappear. This enables the factorization of the expressions P4 µν f that are bounded by 16M 2i(∆φ ) from (168). dw (x, w) 2 ∆ ,∆ µ,ν=1 ∆ ×[0,1]2 φ

φ

φ

We consider now the case of a high momentum field φXφ . With the same no P tations, we simply bound it by j∈Iφ φj (x) , and take the sum over j out of the functional integral. 3.6.2. External sources (ζsext ). If no derivation contracts We consider now an external source φi(∆ext s ) ext (ζs ) within the integral, to be controlled by the Gaussian to it, we keep φi(∆ext s ) bound (122). If it is derived, it has to be by a 2-link hooked to ∆ext s , and we simply −i(∆ext ) s . collect from (274) the corresponding factor M 3.6.3. Large field condition vertices We now point our attention to the large field condition χΓ defined in (38). The action of the functional derivatives will produce new vertices of the form Z α dx φIΓ (∆) (x) g 1+1 · M −i1 . . . M −iγ , (282) ∆

with 0 ≤ α ≤ 3, α + γ = 4, the M −iβ , 1 ≤ β ≤ γ, being the scaling factors of the derived fields. These derivations are produced by extremities ∆β of 2-links with i(∆β ) ∈ IΓ (∆) and ∆ ⊂ ∆β . Now remember that any derivative χ(n) of the function (25) used to define the large field conditions has support in [ 12 , 1]. As a result, regardless of the small or large field nature of ∆ we have always at hand the constraint Z φ4Γ(∆) g 1+1 ≤ 1 .

(283)

∆

Therefore using H¨older’s inequality, we bound (282) by Z α

|∆|1− 4 ·

α/4 φ4Γ(∆)

· g 1+1 · M −i1 . . . M −iγ

∆ γ

≤ M −(i1 −i(∆)) . . . M −(iγ −i(∆)) · g (1+1 ) 4 ,

(284)

with 1 ≤ γ ≤ 4. Remark that, in fact, the domain of spatial integration of this ˜ for the involved derivation operators, vertex may be shrunk by the choice of ∆’s but this does not alter this bound. Besides, one can show by explicit computation that the derivatives of the function χ satisfy, for any t ∈ R, |χ(n) (t)| ≤ K15 · (n!)2 , for some constant K15 ≥ 1.

(285)

184

A. ABDESSELAM and V. RIVASSEAU

Note that for each small field cube ∆, we can introduce without changing . We introduce also a factor the functional integral a factor 1l{R φ4 ≤g−(1+1 ) } ∆ IΓ (∆) R per large field cube ∆. This prepares the functional integral 1l 1 4 {

∆

φI

Γ (∆)

≥ 2 g−(1+1 ) }

for applying Proposition 2. 3.6.4. Interaction vertices Finally we have to consider the bound on vertices that are derived from the interaction. Such a vertex has the following form Z

def

V = g .

dx ˜ γ \sh(∆ ˜ γ ,Y )) ∩1≤β≤γ (∆

X

k X

(ha − ha+1 )

a=0

1l{∆1 ,...,∆γ ∈B} |φB (x)|4−γ · M −i1 . . . M −iγ ,

(286)

B∈ΠΓ ((l1 ,...,la )) δ that acted on this vertex. They where γ, 1 ≤ γ ≤ 4, is the number of derivations δφ are labeled by the subscript β, 1 ≤ β ≤ γ. ∆β is the extremity of the 2-link that ˜ β denotes the chosen element of S(∆β , Y ) produced the derivation with label β. ∆ to define the domain of integration; finally iβ is shorthand for i(∆β ). Remark ˜ γ \sh(∆ ˜ γ , Y )) 6= ∅, and this that V gives a non-trivial contribution only if ∩1≤β≤γ (∆

˜ γ def ˜ V. ˜1 = ... = ∆ = ∆ enforces ∆ Note that if there is a B in ΠΓ ((l1 , . . . , la )) such that ∆1 , . . . , ∆γ ∈ B, then B is unique; therefore we denote it by Ba . Now we have by H¨older’s inequality V ≤ gM −i1 . . . M −iγ

k X

(ha − ha+1 )1l{∃Ba ∈ΠΓ ((l1 ,...,la ))|∆1 ,...,∆γ ∈Ba }

a=0

˜ V \sh(∆ ˜ V , Y ))| γ4 |∆

Z Ba

dx φ

4

4−γ 4

(x)

(287)

˜ V \sh(∆ ˜ V ,Y )) ∆

˜

˜

≤ gM −(i1 −i(∆V )) . . . M −(iγ −i(∆V ))

k X

(ha − ha+1 )

a=0

Z . 1l{∃Ba ∈ΠΓ ((l1 ,...,la ))|∆1 ,...,∆γ ∈Ba }

Ba

dx φ

4 (x)

4−γ 4 .

(288)

˜ V −sh(∆ ˜ V ,Y )) ∆

Now since the ha − ha+1 are positive and add up to 1, by concavity of the 4−γ function t 7→ t 4 , we have ˜

˜

V ≤ g.M −(i1 −i(∆V )) . . . M −(iγ −i(∆V )) k X

Z (ha − ha+1 )1l{∃Ba }

a=0 γ

˜

4 dx φBa (x)

! 4−γ 4 (289)

˜ V \sh(∆ ˜ V ,Y ) ∆ ˜

4−γ

4 , ≤ g 4 .M −(i1 −i(∆V )) . . . M −(iγ −i(∆V )) Y∆ ˜ V

(290)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

with def

Y∆ ˜ = g



Z

dx 

˜ ˜ ) ∆\sh( ∆,Y

k X

(ha − ha+1 )

a=0

 4 φB (x)  ,

X

185

(291)

B∈ΠΓ ((l1 ,...,la ))

˜ ∈ D(N ) . for any ∆ (N ) Now Λ being the unique cube in the lowest layer DN , we can write the interaction in the exponential as X Y∆ (292) I= ˜ . ˜ ∆∈S(Λ,Y ) 4−γ

We now use half of if to bound the product of Y∆V4 generated by the derived vertices. Indeed thanks again to the inequality y ν e−y ≤ ν! for y ≥ 0, we have Y

4−γV

V derived vertex

4 Y∆ ˜ V

U 1 × exp(− I) ≤ 2− 4 . 2

Y

14 ˜ , mult1 (∆)!

(293)

˜ ∆∈S(Λ,Y )

where U is the number of underived fields in derived vertices i.e. X def (4 − γV ) . U =

(294)

V derived vertex

˜ counts the number of underived fields localized in ∆\sh( ˜ ˜ Y) ∆, Likewise, mult1 (∆) i.e. X ˜ def = (4 − γV ) . (295) mult1 (∆) V derived ˜ =∆ ˜ vertex | ∆ V

3.6.5. The bound on the functional integral Now that we have explained the bound on field monomials appearing in the functional integral, we explain the bound on the latter. We take out of it all the ˜ scaling factors bounding numerical constants including powers of g, of mult1 (∆)!, −i M , bounds like (274) on the propagators, bounds as in (285) on χ(n) . The sums over scales j of fields, and the spatial integrations are taken out of the functional integral, so that what remains in the end is an expression of the form Z Y Y Y 0 ext |φi(∆ext (ζ )| × |∂µ ∂ν φj (x)| |φj (x)| × C = dµC[Thgi,Γ (h)] ) s s ×

Y

1l{R

∆

Φ4I

Γ

≥ 12 } (∆)

×

Y

1l{R

∆

Φ4I

Γ (∆)

≤1}

1 × exp(− I) . 2

(296)

The product over |φj (x)| is on the remaining high momentum fields, that one over |∂µ ∂ν φj (x)| is on the double gradients generated by the fluctuation terms in (277). 1 Thanks to the Cauchy–Schwarz inequality we now bound C0 by (I1 .I2 ) 2 where Z Y Y def dµC[Thgi,Γ (h)] (297) (φj (x))2 × (∂µ ∂ν φj (x))2 I1 =

186

A. ABDESSELAM and V. RIVASSEAU

and def

I2 =

Z dµC[Thgi,Γ (h)]

Y

1l{R

Φ4I (∆) ≥ 12 } ∆ Γ

×

Y

1l{R

∆

Φ4I

Γ (∆)

≤1}

× exp(−I) .

(298)

The first integral is bounded by Lemma 16. The second is bounded, thanks to Proposition 2, by Y 5 1 g− 2 . (299) I2 ≤ U (g) 2 #(Y ) . ∆ isolated in Y

3.6.6. The choice of averaging cubes We are left with the task of explaining the rule for choosing the averaging cubes ∆ϕ , for low momentum fields. The method consists in two operations. In the first we define a family (∆0ϕ )ϕ of cubes indexed by low momentum fields, ˜ ϕ at scale by letting ∆0ϕ be the unique cube containing ∆ def ˜ ϕ ) + E[ 1 (i(∆ϕ ) − i(∆ ˜ ϕ ))] . i(∆0ϕ ) = i(∆ 4

(300)

˜ ϕ ⊂ ∆0 ⊂ ∆ϕ . Note that necessarily ∆ ϕ The second operation is inductive and goes from small to large scales. Following the notations of Sec. 3.2, we let i be any scale, 0 ≤ i < imax (Y ), we then construct a family (∆iϕ )ϕ of cubes in Y , as follows. Suppose i ≥ 1, and (∆i−1 ϕ )ϕ has been constructed. We consider the ordinary graph F made by the 2-links of hgi. Since hgi is allowed, F must be a forest. Besides, ΠΓ (g) = {Y }. As a consequence, for any component A in Ri (Y, F ), there must be at least one gluing link or five explicit vertices crossing the lower boundary of A. This is to say, their support intersects def (N ) Y>i = Y ∩ (∪j>i Dj ), as well as Y ∩ pr−1 i (A), the set of cubes in Y contained in some cube of A. Indeed if that was not true, we would be able to disconnect Y ∪pr−1 i (A) from the (non-empty since i < imax (Y )) rest of Y , by removing at most four 4-links from g, contradicting the four vertex irreducibility of Y with respect to g and Γ. Suppose there is no gluing link crossing the lower boundary of A, then there are at least five explicit vertices which do. It is not difficult to see that we can pick five low momentum fields ϕ (one in each of the above-mentioned vertices) ˜ ϕ ∈ Y ∩ pr−1 (A) and i(∆ϕ ) > i. If some of these five chosen fields verify such that ∆ i i+1 ˜ ϕ at i(∆ϕ ) ≤ i, we define for such a field ∆iϕ to be the unique cube containing ∆ −1 ˜ ϕ ∈ Y ∩ pr (A) we scale i + 1. For all the other low momentum fields ϕ with ∆ i def

keep ∆iϕ = ∆i−1 ϕ . We do the same for all the components A in Ri (Y, F ), and this completes the definition of (∆iϕ )ϕ . imax (Y ) )ϕ . It is easy to check that the cubes Finally we define (∆ϕ )ϕ to be (∆ϕ ∆ϕ belong to Y , and for any ∆ ∈ Y , the number of fields ϕ such that ∆ϕ = ∆ and ∆ϕ 6= ∆oϕ is at most 5M 4 , i.e. five times the maximal number of components A having an element just above ∆.

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

187

3.6.7. The small factors per cube Section 3.6.1 shows that, in the bound for an explicit vertex, we obtain at least 1−31 def a factor g 4 (g is supposed in the interval ]0, 1[). If now we choose 1 = 16 , we 1+1

1

7

obtain g 8 . From a large field condition vertex, we can extract g 4 = g 24 . Finally from an interaction vertex we can extract at least a g 1/4 . We can thus say that 1 from a vertex of any kind we obtain at least a factor as small as g 8 . Now if we 1 choose 5 = 16 , we collect Y Y 5 1 g8 ≤ g2 . (301) vertices of any kind

isolated cubes of Y

Indeed, any isolated cube ∆ of Y forms a dressed large field block in the support of an explicit vertex, or is an extremity of a 2-link. In the first case we can say that 1 one fourth of the g 8 factor of such an explicit vertex is attributed to this cube. If we cannot find such an explicit vertex, having chosen an order to perform the functional derivatives, we consider the first computed δ/δφi (x) operator coming from a 2-link hooked to ∆. This derivation has to derive a new vertex from the exponentiated interaction or from a large field condition. We can decide that the g 1/8 factor of 5 1 this vertex is attributed to ∆. In any case we get better that g 32 = g 2 . It is easy to see that we have not attributed in this way the same factor twice; and this proves (301). 3.6.8. The handling of Gaussian factorials For any cube ∆ of Y , we define mult2 (∆) to be the number of fields φj (x) or (ζsext ) in the integral C0 of Eq. (296), such that ∆(x, j) = ∆ ∂µ ∂ν φj (x) or φi(∆ext s ) ext or ∆s = ∆, i.e. located in ∆. Note that every initial field appearing in (296) is now counted twice after the Cauchy–Schwarz argument. The Gaussian bound of Q 1 Lemma 16 applied to I1 yields a product of local factorials ∆∈Y ((2.mult2 (∆))!) 2 to deal with. Now we define mult3 (∆) to count the high momentum fields, of the form φj (x), located in ∆, and mult4 (∆) to count the low momentum fields, of the form ∂µ ∂ν φj (x), located in ∆. We recall that E∆ counts the external sources in ∆, and S labels them. By the elementary inequality (p1 + · · · + pm )! ≤ mp1 +···+pm p1 ! . . . pm ! ,

(302)

and since the number of fields in I1 is at most 2.4.k + 2.#(S), we have Y Y 1 1 ((2.mult2 (∆))!) 2 ≤ 34k+#(S) × ((2.mult3 (∆))!) 2 ∆∈Y

×

Y

∆∈Y

1 2

((2.mult4 (∆))!) ×

∆∈Y

Y

1

((2.E∆ )!) 2 .

We now use the elementary multinomial inequality (161) to write Y Y 1 1 ((2.mult3 (∆))!) 2 ≤ ((2.mult5 (X))!) 2 , ∆∈Y

(303)

∆∈Y

X∈π

(304)

188

A. ABDESSELAM and V. RIVASSEAU

def P where mult5 (X) = ∆∈X mult3 (∆), for any X ∈ π. We make another distinction by defining mult6 (∆) to count the low momentum fields ϕ located in ∆ and such that ∆ϕ 6= ∆0ϕ . We define also mult7 (∆) to count the other low momentum fields located in ∆. By the same line of argument and since the total number of occurrences of low momentum fields in I1 is no greater than 2.3.k, we have by (302) Y Y Y 1 1 1 ((2.mult4 (∆))!) 2 ≤ 23k × ((2.mult6 (∆))!) 2 × ((2.mult7 (∆))!) 2 . (305) ∆∈Y

∆∈Y

∆∈Y

We now use the lemma of displacement of local factorials to bound the last two products. First we consider the fields ϕ such that ∆ϕ = ∆0ϕ . Remark that the spatial integration on the corresponding vertex that has been taken out of the functional ˜ ϕ) ˜ ϕ \sh(∆ ˜ ϕ , X1,ϕ )| ≤ M −4i(∆ . This factor can be integral C0 , produces a factor |∆ distributed equally to the four fields composing the vertex. Now if we take the share 1 of a low momentum field ϕ inside the square root I12 , we have such a factor for each of the two copies of the field. But the Gaussian bound (122) produces a constant K4 and a scaling factor M −3jϕ for each of these copies, since it is of double gradient µ,ν type. Recall that we have also L1 bounds on the smearing functions f∆ ,∆ of these ϕ

ϕ

double gradients, that produce a factor 16M 2i(∆ϕ ) . Now the total factor attributed to a copy of a field ϕ is a vertical exponential decay, of varying strength whether we are between the double gradient scale jϕ and the averaging scale i(∆ϕ ), or between ˜ ϕ ). Namely it is the latter and the localization scale i(∆ ˜

16K4 M −3(jϕ −i(∆ϕ ))−3(i(∆ϕ )−i(∆ϕ ))−(i(∆ϕ )−i(∆ϕ )) .

(306)

˜ ϕ ) + E[ 1 (i(∆ϕ ) − i(∆ ˜ ϕ ))] entails 3(i(∆ϕ ) − Now in the present case i(∆ϕ ) = i(∆ 4 9 1 1 ˜ ˜ ˜ ϕ )) − 1 . (i(∆ϕ ) − i(∆ i(∆ϕ )) ≥ 4 (i(∆ϕ ) − i(∆ϕ )), as well as 10 (i(∆ϕ ) − i(∆ϕ )) ≥ 40 10 Therefore we readily bound (306) by ˜

16K4 M − 10 (1 − M − 2 )− 2 .M − 10 (i(∆ϕ )−i(∆ϕ )) .M − 40 (jϕ −i(∆ϕ )) q 1 9 ˜ . (1 − M − 2 )M − 2 (jϕ −i(∆ϕ )) . 1

1

1

9

1

˜

(307)

We recognize in the square root the function G of (245) that allows to compute the ˜ ϕ , at local factorials of the presently considered fields as if they were located in ∆ 1 the cost of an extra factor e 2 . Now we consider fields ϕ such that ∆ϕ 6= ∆0ϕ . By construction we must have ˜ ϕ ) + E 1 (i(∆ϕ ) − i(∆ ˜ ϕ )) . (308) i(∆ϕ ) ≥ i(∆ 4 Since we deal again with double gradients we still have a factor (306), per copy of ϕ, that is grossly bounded by ˜

16K4 M − 10 (1 − M − 2 )− 2 .M − 10 (i(∆ϕ )−i(∆ϕ )) .M − 40 (jϕ −i(∆ϕ )) q 9 1 . (1 − M − 2 )M − 2 (jϕ −i(∆ϕ )) . 1

1

1

9

1

˜

(309)

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

189

Now recall our previous remark that for a given ∆ϕ there can be at most 5M 4 low momentum fields ϕ such that ∆ϕ 6= ∆0ϕ . Therefore the product of local factorials we get after displacement by Lemma 17 is bounded by Y 1 3 ((10M 4 )!) 2 ≤ ((10M 4 )!) 2 k . (310) ∆∈Y ∃ϕ,∆ =∆ ϕ

3.6.9. The bound on an individual contribution C We now have all the elements to write such a bound and it is Y Y ext 1 K3 (r)(1 + M −i(l) d2 (l))−r × K4 M −i(∆s ) |C| ≤ U (g) 2 #(Y ) × s∈S

l∈F

×

Y

1 4

((2.E∆ )!) × 2(1 − M

1 − 40

)−1

4k

× (4!)k × K43k

∆∈Y

×

Y 1 1 1 max[K7 , 16K4e 2 (1 − M − 2 )− 2 ] ϕ

9 1 1 ˜ ˜ ·M − 10 .M − 10 (i(∆ϕ )−i(∆ϕ )) .M − 40 (i(∆ϕ )−i(∆ϕ )) Y Y 3 1 3 ˜ #(Y ) (mult8 (∆)!)2 M − 40 (i(∆δ )−i(∆δ )) × K15 × 32k+ 2 #(S) × 2 2 k × × δ

×

Y

1 2

˜ (mult1 (∆)!) ×

˜ ∆∈S(Λ,Y )

×

Y

Y

∆∈Y

(2.mult5 (X)!)

1 4

X∈π 1

(2.mult9 (∆)!) 4 × ((10M 4 )!)

3k 4

.

(311)

∆∈Y

In this inequality, F is the forest of 2-links of g; s ∈ S labels the sources; ϕ ranges through all the low momentum fields, derivated or not, for which ∆ϕ has always been defined; and δ labels the functional differential operators δ/δφi (x), attached to the extremities of the 2-links, that did not contract to the external sources. ˜ δ is the corresponding chosen integration ∆δ is the concerned extremity, whereas ∆ cube in S(∆δ , Y ) of Eqs. (271) and (272). For every ∆ ∈ Y , mult8 (∆) counts how many times the associated large field condition χ∆ of Eq. (38), is derived. Finally, ˜ ϕ = ∆. mult9 (∆) counts the number of low momentum fields ϕ such that ∆ 4k 1 comes from the choice paid by a factor Note that the factor 2(1 − M − 40 )−1 2, for each low momentum field, between the fluctuation or the average term, in addi1 tion to the choice of scale jϕ , which is summed thanks to the factor M − 40 (jϕ −i(∆ϕ )) . In the case of a high momentum field we sum also on the scale thanks only to a 1 ˜ ˜ mere fraction M − 40 (j−∆) of the available vertical decay M −(j−i(∆)) . Finally the number of high or low momentum fields is bounded by 4k. The (4!)k factor bounds the symmetry factors of the explicit vertices. The K43k term bounds the constants coming with the scaling factor of the high momentum fields, when integrated with the Gaussian bound.

190

A. ABDESSELAM and V. RIVASSEAU

Remark that for a derivated field in an explicit vertex, the volume of integration ˜ δ \sh(∆δ , Y ), where δ is the involved derivation. However of the latter is shrunk to ∆ we use this improvement, only when considering the fourth power of this volume that is attributed to the mentioned derivated field. In any case, for a field derivated ˜ by δ, we collect a factor M −((i(∆δ )−i(∆δ )) . This is more than enough to pay the 3 ˜ M − 40 (i(∆δ )−i(∆δ )) in case the field is in an interaction vertex, or a large field condition vertex, or a source, or a high momentum field. In case it is a low momentum 9 ˜ field ϕ, we need a fraction 37/40 of the initial decay to pay for M − 10 (i(∆ϕ )−i(∆ϕ )) 1 3 ˜ ˜ and M − 40 (i(∆ϕ )−i(∆ϕ )) , and we cannot count on more than M − 40 (i(∆δ )−i(∆δ )) to be allocated to the product over the δ’s. After justifying (311), we can improve it by writing a nicer bound where the Gaussian factorials have undergone a first treatment. By the inequality (302) with m = 2, we have Y Y 1 1 (2.mult5 (X)!) 4 × (2.mult9 (∆)!) 4 ≤ 22k X∈π

×

Y

∆∈Y 1 2

(mult5 (X)!) ×

X∈π

Y

1

(mult9 (∆)!) 2 ,

(312)

∆∈Y

P P since X∈π mult5 (X) + ∆∈Y mult9 (∆) is bounded by the total number of fields in explicit vertices, i.e. by 4k. Now if we define for any X ∈ π, mult10 (X) to count ˜ ϕ ∈ X, by the inequality (161), we the number of low momentum fields ϕ with ∆ have Y Y mult9 (∆)! ≤ mult10 (X)! . (313) ∆∈Y

X∈π

Now let V(X) denote the number of explicit vertices in the form hgi whose top block is X. This implies, for any X ∈ π, mult5 (X) + mult10 (X) = 4V(X). Thus by (161) and (302), we have Y Y Y Y 1 1 1 (mult5 (X)!) 2 × (mult10 (X)!) 2 ≤ [(4V(X))!] 2 ≤ 24k × (V(X)!)2 . X∈π

X∈π

X∈π

X∈π

(314) ˜ counts by definition the derivations δ such that ∆ ˜ δ = ∆, ˜ we have If mult11 (∆) Q Q ˜ ≤ 3.mult11 (∆), ˜ and therefore ˜ 12 ≤ 33k . ∆ ˜ ∈ mult1 (∆) ˜ ∆∈S(Λ,Y ) (mult1 (∆)!) 3 ˜ 2 , since the total number of δ’s is bounded by 2k. We can S(Λ, Y )(mult11 (∆)!) now use Lemma 17 to bound the local factorials in mult8 (∆). Indeed, if δ acts on ˜ δ ⊂ ∆ ⊂ ∆δ , and therefore the large field condition of ∆, we must have ∆ Y Y 1 ˜ (mult8 (∆)!)2 × M − 40 (i(∆δ )−i(∆δ )) ∆∈Y

δ

2 Y Y 1 ˆ   (mult8 (∆)!) × M − 80 (i(∆δ )−i(∆δ ))  , ≤ 

∆∈Y

(315)

δ acting on χ

ˆ δ denotes the cube that labels the large field condition on which δ acts. where ∆ Now define mult12 (∆) to be the number of 2-links hooked to ∆. It must be no less

191

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

than the number of internal derivations δ such that ∆δ = ∆. By Lemma 17, with def ˆ def O equal to the set of δ’s acting on χ, S1 (δ) = ∆ δ and S2 (δ) = ∆δ , we deduce Y 1 Y ˜ (mult8 (∆)!)2 × M 40 (i(∆δ )−i(∆δ )) ∆∈Y

≤

Y

δ

2k 1 (mult12 (∆)!)2 × e2 .(1 − M − 80 )−2 .

(316)

∆∈Y

Finally, using Lemma 10 to bound the exponents k by 6#(Y ), and posing K16 = 287 .327 .e24 .K418 .K15 .(max[K7 , 16K4 e 2 (1 − M − 2 )− 2 ].M − 10 )18 −24 −24 1 1 9 . 1 − M − 40 . 1 − M − 80 .((10M 4 )!) 2 , (317) def

1

1

1

1

the results of this section boil down to Y Y 1 1 ext 1 1 #(Y ) (E∆ !) 2 × 2 2 .3 2 .K4 .M −i(∆s ) |C| ≤ U (g) 2 #(Y ) × K16 × ∆∈Y

Y K3 (r)(1 + M −i(l) d2 (l))−r ×

s∈S

l∈F

Y 9 1 ˜ ˜ M − 10 (i(∆ϕ )−i(∆ϕ )) .M − 40 (i(∆ϕ )−i(∆ϕ )) × ϕ

Y Y 1 ˜ (mult12 (∆)!)2 M − 20 (i(∆δ )−i(∆δ )) × × δ

×

Y

˜ ∆∈S(Λ,Y )

∆∈Y

3 2

˜ (mult11 (∆)!) ×

Y

(V(X)!)2 .

(318)

X∈π

3.7. The sum over the procedures of internal derivations We note that the previous bound does not depend on Pint . Therefore we only have to bound the number of such derivation procedures. We suppose an order has been chosen for the action of the derivation operators generically denoted by δ. For each δ we have a tree of possibilities. Each node of the tree means a new discussion of possible cases. The number we have to bound is the number of procedures as filtered by the successive discussions, i.e. the number of leaves in this tree. At each node we bound the sum over the subordinate procedures, by the number of cases at this stage times the supremum of the analog sums for the chosen case to be discussed at the next stage. This is the standard method of combinatoric factors, see [20, 31]. Now considering a derivation operator δ. By a factor 2, we decide whether it derives a new vertex either from the interaction or a large field condition or the product of explicit vertices, or whether it derives an already derived vertex. In the first case we choose by a factor 3 between the cited possibilities. If δ derives a new vertex from the interaction, we say we have a type I derivation and we only have to choose among the fields of this vertex the one to contract, by a factor 4.

192

A. ABDESSELAM and V. RIVASSEAU

If δ derives an explicit vertex lj , we say that we have a type II derivation. First, ˜δ ⊂ ∆ ˜j ⊂ ˜ j . The latter has to verify ∆ we have to find its localization cube ∆ ˜ δ ) possibilities. Since ∆ ˜ j is in ∆δ , therefore there are at most 1 + i(∆δ ) − i(∆ def

S(bg(Xj,1 ), Xj,1 ) and thus in Xδ = Xj,1 , the top large field block of lj , we have to pay a factor V(Xδ ) to find the concerned vertex. Finally we pay a factor 4, to find the field to contract in the vertex. If δ derives the large field condition χ∆ of some cube ∆, we say we have a type ˜ δ ⊂ ∆ ⊂ ∆δ . As a result we have to pay a III derivation and we must have again ∆ ˜ δ ), then a factor 4 to find the field in the vertex. factor 1 + i(∆δ ) − i(∆ Finally if δ derives an already derived vertex, we say we have a type IV derivation. Now we have to find the previous derivation δ 0 that first derived the vertex on which δ acts. In order to yield a non-zero contribution, we must have the match ˜ δ . As a result we have to pay the factor mult11 (∆ ˜ δ ) to find δ 0 , then a factor ˜ δ0 = ∆ ∆ 3 to choose the field to contract in the vertex, since one has already been derived. To summarize the previous considerations, we write the bound X 1 ≤ max Comb(Pint ) , (319) Pint

Pint

where

!

Y

def

Comb(Pint ) =

24

δ of type I in Pint

×

Y

! ˜ 24.(1 + i(∆δ ) − i(∆δ )).V(Xδ )

Y

×

δ of type II in Pint

! ˜ δ )) 24.(1 + i(∆δ ) − i(∆ ×

δ of type III in Pint

Y

! ˜ δ) 6.mult11 (∆ . (320)

δ of type IV in Pint

Since the number of occurrences of a factor V(X) in (320) is the number of derivations that contract to an explicit vertex with top block X, it is bounded by 4V(X). ˜ is at most the number Similarly, the number of occurrences of a factor mult11 (∆) ˜ ˜ ˜ of δ’s such that ∆δ = ∆, i.e. mult11 (∆). It is now easy to derive Y Y X ˜ δ )) × 1 ≤ 242k × (1 + i(∆δ ) − i(∆ V(X)4V(X) X∈π V(X)6=0

δ

Pint

×

Y

˜ ˜ mult11 (∆) mult11 (∆) ,

(321)

˜ ∆∈S(Y,Λ) ˜ =0 mult11 (∆)6

or X

1 ≤ 242k .e6k ×

Y Y ˜ δ )) × (1 + i(∆δ ) − i(∆ (V(X)!)4 × δ

Pint

X∈π

Y

˜ . (mult11 (∆))!

˜ ∆∈S(Y,Λ)

(322) Note that using the inequality u.e−u ≤ 1, for u ≥ 0, we readily derive ! 1 Y 60.M 60 Y 1 ˜ δ )) (i(∆ )−i( ∆ δ ˜ .M 60 (1 + i(∆δ ) − i(∆δ )) ≤ . log M δ

δ

(323)

193

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION 5

˜ 2 , that Next, we can use Lemma 17 to get rid of the local factorials (mult11 (∆)!) appear after the bound on the sum over Pint . Indeed, we take O to be the set of def def ˜ def internal derivations δ, E = Y , S1 (δ) = ∆ δ , and S2 (δ) = ∆δ , for every δ. Finally, we use the function 1 1 def (1 − M − 150 ).M − 150 (i(∆2 )−i(∆1 )) if ∆1 ⊂ ∆2 (324) G(∆1 , ∆2 ) = 0 else. Now, Lemma 17 yields Y

˜ 52 ) ≤ (mult11 (∆)!)

Y 5 1 5 1 ˜ e 2 (1 − M − 150 )− 2 .M 60 (i(∆δ )−i(∆δ ))

˜ ∆∈S(Λ,Y )

δ

×

Y

5

(mult12 (∆)!) 2 .

(325)

∆∈Y

Besides remark that nothing in the present bounds depend on the h parameters and R therefore the integral 1>h1 >...>hk >0 dh is simply bounded by the powerful factor 1 k! . As a result Y X Z 1 1 1 #(Y ) (E∆ !) 2 dh|C| ≤ .U (g) 2 #(Y ) .K17 . k! Pint ∆∈Y Y 1 1 Y ext −i(∆ ) s × (2 2 .3 2 .K4 .M )× K3 (r)(1 + M −i(l) d2 (l))−r s∈S

×

Y ϕ

×

Y

l∈F

M

9 ˜ ϕ )) − 10 (i(∆ϕ )−i(∆

.M

1 ˜ ϕ )) − 40 (i(∆ϕ )−i(∆

Y Y 1 9 ˜ (mult12 (∆)!) 2 × (V(X)!)6 , (326) M − 60 (i(∆δ )−i(∆δ )) × ∆∈Y

δ

X∈π

where 1

def

K17 = K16 .2412 .e36 .

1 5 60.M 60 5 .e 2 .(1 − M − 150 )− 2 log M

!12 .

(327)

3.8. The final bound We now describe the bounds on the remaining sums in (273). We first bound the ˜ δ of the internal derivations δ, thanks to Lemma sum over the localization cubes ∆ 18, that gives Y X Y 1 1 ˜ mult12 (∆)! . M − 60 (i(∆δ )−i(∆δ )) ≤ e#(Y ) .(1 − M − 120 )−24#(Y ) . ˜ of (∆) derivations

δ

∆∈Y

(328) 1 , and to note that Indeed, we only have to apply (147) with K6 = e, 4 = 60 n1 + · · · + np is bounded by the number of δ’s, thus by 2k ≤ 12#(Y ). To sum over the procedures Pext of external derivations, we have to sum for each such derivation operator δ/δφi (x) located in ∆ = ∆(x, i), over the external

194

A. ABDESSELAM and V. RIVASSEAU

source in ∆ to contract. There are E∆ possibilities, hence we have Y Y X 1≤ (E∆ )mult12 (∆) ≤ eE∆ .mult12 (∆)! . ∆∈Y

Pext

(329)

∆∈Y

The bound over the distinction c between external and internal derivations is given by X 1 ≤ 22k ≤ 212#(Y ) . (330) c

Therefore at this stage we have #(Y ) XX X XZ 1 1 1 dh|C| ≤ U (g) 2 #(Y ) . 212 .e.(1 − M − 120 )−24 .K17 k! ˜ of c (∆) Pext

Pint

derivations

×

Y

1

(E∆ !) 2 ×

×

M

M −i(∆s

ext

×

13

(mult12 (∆)!) 2 ×

∆∈Y def

× C #(S) ×

Y

K3 (r)(1 + M −i(l) d2 (l))−r

l∈F

9 ˜ ϕ )) − 10 (i(∆ϕ )−i(∆

ϕ

Y

)

s∈S

∆∈D (N )

Y

Y

˜

.M − 40 (i(∆ϕ )−i(∆ϕ ))

Y

1

(V(X)!)6 ,

(331)

X∈π 1

1

where C = e.2 2 .3 2 .K4 is the constant appearing in (54). The next step is to get rid of the local factorials thanks to volume effects. First consider the multiplicities V(X). Remark that there exists a numerical constant K18 ≥ 1, such that for any integer p ≥ 1, 6p log p −

(log M )p2 ≤ K18 . 4000

(332)

We then claim that Lemma 21. Y X∈π

(V(X)!)6 ×

Y

˜

M − 80 (i(∆ϕ )−i(∆ϕ )) ≤ K18 1

#(Y )

(333)

ϕ

Proof. We must show that, for any X ∈ π, Y 1 ˜ M − 80 (i(∆ϕ )−i(∆ϕ )) ≤ K18 , (V(X)!)6 ×

(334)

ϕ produced by X

where produced means that ϕ belongs to an explicit vertex whose top block is X. Let us choose a representative g of the given form hgi satisfying the following requirements. If l is any of its 4-links, and Xl is the top block of lπ , then (supp l) ∩ Xl = {bg(Xl )}. Furthermore, we ask that for any other block X in (supp l) ∩ X be reduced to {∆}, where ∆ is the smallest box in X containing bg(Xl ). Such a choice is always possible.

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

195

Now we consider a fixed X ∈ π, and we extract the subsequence gX of g made by the 4-links whose top block is X. As a subsequence of an allowed graph, gX is allowed in Y , with respect to the large field region Γ. It is easy to see that it must also be allowed with respect to the empty large field region, in the subset WX of Y made by the union of the supports of the links in gX . Now from the proof of Lemma 10 we deduce that the length V(X) of gX must satisfy V(X) ≤ 5#(WX ) − 5. As a 0 denotes the set WX \{bg(X)}, we have result if WX 0 )≥ #(WX

V(X) . 5

(335)

But it is easy to check that 0

X ϕ produced by X

#(WX ) X 1 X j 1 ˜ ϕ )) ≥ (i(∆ϕ ) − i(∆ (i(∆) − i(bg(X))) ≥ 80 80 80 0 j=1 ∆∈WX

≥ and thus

X ϕ produced by X

0 ))2 (#(WX , 160

2 1 ˜ ϕ )) ≥ (V(X)) , (i(∆ϕ ) − i(∆ 80 4000

(336)

(337)

from which (334) follows.

Concerning the local factorials in mult12 (∆), we use the following classical volume argument. Lemma 22. There exists a constant K19 ≥ 1 such that for any ∆ ∈ Y, we have Y 13 mult (∆) (1 + M −i(l) d2 (l))−52 ≤ K19 12 . (338) (mult12 (∆)!) 2 . l∈F | ∆∈l

Proof. Remark that since F is a forest, the cubes ∆0 forming the extremities of the links l = {∆, ∆0 } appearing in (338), must be distinct. Remark also that the cubes ∆0 are of the same scale as ∆. If d ≥ 0 is some number, one can (N ) easily see by a volume argument that the number of cubes ∆0 in Di(∆) such that

d2 (∆, ∆0 ) ≤ d.M i(∆) is at most 12 π 2 (d + 4)4 . Thus if mult12 (∆) ≥ 256π 2 , we take ! 1 mult12 (∆) 4 def −4 ≥ 0, (339) d = E π2 and have 12 π 2 (d + 4)4 ≤ 12 mult12 (∆), and therefore at least half of the cubes ∆0 verify d2 (∆, ∆0 ) > d.M i(∆) . As a consequence 13

(mult12 (∆)!) 2 .

Y

mult12 (∆) 13 (1+M −i(l) d2 (l))−52 ≤ mult12 (∆) 2 (1 + d)−26 .

l∈F | ∆∈l

(340)

196

A. ABDESSELAM and V. RIVASSEAU

On the other hand,

1+d≥

mult12 (∆) π2

14

−4,

(341) 13

and thus if we choose K19 large enough so that K19 ≥ (256π 2 ) 2 , then for any p ≥ 256π 2 we have 1 −26 p 4 13 − 4 ≤ K19 , (342) p2. π2

and we easily arrive at (338).

˜ of th evertices is done Now the bound on the sum over the forms hgi and the (∆) by the same move. The method parallels that of the sum over Pint . First we choose the value of k, for which by Lemma 10 there are less than 6#(Y ) possibilities. Next for each index j, 1 ≤ j ≤ k, we decide whether the link la of g is a 2-link or a 4-link. This costs a factor 2k ≤ 26#(Y ) . If lj is chosen to be a 4-link, we pay a ˜ j of the vertex. We use here the huge factor #(Y ) to find the localization cube ∆ ˜ notations of Sec. 3.5. Once we know ∆j , we will also know the top block Xj,1 of ˜ j ∈ X. Next we choose by a factor 3 the vertex, it is the unique X ∈ π such that ∆ π the multiplicity lj (Xj,1 ) i.e. the number of high momentum fields. We choose by a factor 3 the number αj of blocks in the support of ljπ . Suppose we canonically label these disjoint blocks such that Xj,β is above Xj,β+1 for any β, 1 ≤ β ≤ αj − 1. We bound by 33 the number of choices of multiplicities mj,β for the blocks Xj,β , 2 ≤ β ≤ αj . It remains to sum over the location of these blocks. It is enough to ˜ j . But this cube has know for each Xj,β the smallest of its cubes ∆j,β containing ∆ to be the ∆ϕ of at least one low momentum field located in Xj,β . Therefore we can 1 ˜ use the corresponding factor M − 80 (i(∆ϕ )−i(∆ϕ )) especially spared for that purpose in (331). If lj is a 2-link, and if some total ordering was chosen on D(N ) , we sum over the smallest cube of supp lj , with a factor #(Y ). Next we sum on the second cube, thanks for instance to a decay (1 − M −i(lj ) d2 (lj ))−5 that has to be extracted from (331). We summarize these considerations by the following bound X

X

hgi

˜ of (∆) derivations

≤

Y 1 Y − 1 (i(∆ϕ )−i(∆ ˜ ϕ )) M 80 (1 − M −i(l) d2 (l))−5 k! ϕ

X 0≤k≤6#(Y )−6

l∈F

#(Y ) #(Y )k 6 1 2 . max[330 .(1 − M − 80 )−18 , K56 ] , k! k

(343)

) is simply bounded by where K5 is the constant appearing in (124). Now #(Y k! #(Y ) #(Y ) . Besides the sum over Γ ∈ Y , costs a factor 2 . It remains to perform e the sum over Y using Proposition 1. By construction of the averaging cubes ∆ϕ , for any i, 0 ≤ i ≤ imax (Y ) − 1, and for any component A ∈ Ri (Y, F ), there are two possibilities. Either there is a gluing link crossing the lower boundary of A, or ˜ ϕ ), and there are at least five low momentum fields ϕ such that i(∆ϕ ) > i ≥ i(∆

197

AN EXPLICIT LARGE VERSUS SMALL FIELD MULTISCALE CLUSTER EXPANSION

˜ ϕ ) ∈ A. As a result using the notations of Proposition 1 with 2 = 1 , we have pri (∆ 2 Y

M − 2 #(Y ) 9

(1 − M −i(l) d2 (l))− 2 9

Y

˜

M − 10 (i(∆ϕ )−i(∆ϕ )) ≤ T2 (Y ) . 9

(344)

ϕ

l∈F

Therefore we need to choose r at the beginning so as to fulfill r ≥ 52 + 5 + 92 . Now if we define the new constant K20 = 219 .e2 .(1 − M − 120 )−24 .K17 .K3 (r) def

2 .M .K18 .K19

1

9 2

. max[3 (1 − M 30

1 − 80 −18

)

, K56 ].K1

1 4, M, 2

,

(345)

we have by applying Proposition 1, and reverting to the notations of Theorem 2 X ext #(Y ) |A(Y, (∆ext s , ζs )s∈S )|.K Y |Y ⊂∆(N ) ∆ext ∈Y {∆ext s |s∈S}⊂Y



≤

X



6Q.(K.K20 .U (g) 2 )Q  .C #(S) .

Q≥1

1

Y ∆∈D (N )

(E∆ !)1/2 .

Y

M −i(∆s

ext

)

. (346)

s∈S

Now since by Proposition 2 limg→0+ U (g) = 0, for any η and K, we can find a g0 , 0 < g0 < 1, such that 0 < g ≤ g0 implies that the first factor in the right side of (346), is smaller than η. One can easily check that this result is uniform in the IR cut-off N , the volume cut-off Λ, and the collection of external sources. This proves (54), at last. Acknowledgements We thank Jacques Magnen for illuminating discussions, and for the trick in Lemma 19. References [1] A. Abdesselam, Ph.D. Thesis, Ecole Polytechnique, in preparation. [2] A. Abdesselam and V. Rivasseau, “Trees, forests and jungles: a botanical garden for cluster expansions, in Constructive physics”, Proceedings, Palaiseau, France 1994, Lecture Notes in Phys. 446, Springer, 1995. [3] A. Abdesselam and V. Rivasseau, “An explicit phase-space expansion for the renormalized infrared φ44 model”, in preparation. [4] T. Balaban, Commun. Math. Phys. 85 (1982) 603; 86 (1982) 555; 89 (1983) 571; 109 (1987) 249; 116 (1988) 1. [5] G. A. Battle, “A new combinatoric estimate for cluster expansions”, Commun. Math. Phys. 94 (1984) 133. [6] G. A. Battle and P. Federbush, “A phase cell cluster expansion for Euclidean field theories”, Ann. Phys. 142 (1982) 95; “A note on cluster expansions, tree graph identities, extra 1/N ! factors!!!”, Lett. Math. Phys. 8 (1984) 55. [7] D. Brydges, “A short course on cluster expansions, in Critical phenomena, random systems, gauge theories”, Les Houches session XLIII, 1984, Elsevier Science Publishers, 1986.

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[8] D. Brydges, “Functional integrals and their applications”, Cours de Troisi` eme Cycle de la Physique en Suisse Romande, taken by R. Fernandez, Universit´ e de Lausanne or Univ. of Virginia, preprint, (1992). [9] D. Brydges, “Weak perturbations of massless Gaussian measures”, Mathematical Quantum Theory I: Field Theory and Many Body Theory, ed. J. Feldman, R. Froese, and L. Rosen, CRM Proceedings Lecture Notes. [10] D. Brydges and P. Federbush, “A new form of the Mayer expansion in classical statistical mechanics”, Math. Phys. 19 (1978) 2064. [11] D. Brydges and T. Kennedy, “Mayer expansions and the Hamilton-Jacobi equation”, J. Stat. Phys. 48 (1987) 19. [12] D. Brydges and H. T. Yau, “Grad Φ perturbations of massless Gaussian fields”, Commun. Math. Phys. 129 (1990) 351. [13] D. Brydges, J. Dimock and T. Hurd, “The short distance behavior of φ43 ”, Commun. Math. Phys. 172 (1995) 143; “A non-Gaussian fixed point for φ4 in 4-ε dimensions-I”, preprint. [14] “Constructive Quantum field theory”, Proc. 1973 Erice Summer School, ed. G. Velo and A. Wightman, Lecture Notes in Phys. 25, Springer, 1973. [15] J. Feldman, J. Magnen, V. Rivasseau and R. S´ en´eor, “A renormalizable field theory: the massive Gross-Neveu model in two dimensions”, Commun. Math. Phys. 103, (1986) 67; “Construction of infrared φ44 by a phase space expansion”, Commun. Math. Phys. 109, (1987) 473. [16] J. Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, “An infinite volume expansion for many Fermion Green’s functions”, Helvetica Physica Acta, 65 (1992) 679; “Fermionic many-body models”, Mathematical Quantum Theory I: Field Theory and Many Body Theory, ed. J. Feldman, R. Froese and L. Rosen, CRM Proceedings Lecture Notes. [17] J. Feldman and K. Osterwalder, “The Wightman axioms and the mass gap for weakly coupled φ43 quantum field theories”, Ann. Phys. 97 (1976) 80. [18] J. Feldman and E. Trubowitz, “Perturbation theory for many Fermion systems”, Helv. Phys. Acta 63 (1990) 156; “The flow of an electron-phonon system to the superconducting state”, Helvetica Physica Acta 64 (1991) 214. [19] J. Glimm and A. Jaffe, “Positivity of the φ43 hamiltonian”, Fortschr. Phys. 21 (1973) 327. [20] J. Glimm and A. Jaffe, Quantum Physics, A Functional Integral Point of View, Springer, 1987. [21] J. Glimm, A. Jaffe, and T. Spencer, “The particle structure of the weakly coupled P (φ)2 model and other applications of high temperature expansions, Part II: The cluster expansion”, in [14] above. [22] K. Gawedski and A. Kupiainen, “Massless lattice φ44 theory: Rigorous control of a renormalizable asymptotically free model”, Commun. Math. Phys. 99, (1985) 197; “Gross–Neveu model through convergent perturbation expansions”, Commun. Math. Phys. 102 (1985) 1. [23] D. Iagolnitzer and J. Magnen, Commun. Math. Phys. 110 (1987) 511; 111 (1987) 81; 119 (1988) 567; 119 (1988) 609. [24] C. Kopper, J. Magnen, and V. Rivasseau, “Mass generation in the large N Gross– Neveu model”, Commun. Math. Phys. 169 (1995) 121. [25] P. Lemberger, “Large-field versus small-field expansions and Sobolev inequalities”, J. Stat. Phys. 79 (1995) 525. [26] G. Mack and A. Pordt, “Convergent perturbation expansions for Euclidean quantum field theory”, Commun. Math. Phys. 97 (1985) 267. [27] J. Magnen and R. S´en´eor, “Phase space cell expansion and Borel summability of the Euclidean Φ43 -theory”, Commun. Math. Phys. 56 (1977) 237.

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[28] J. Magnen, V. Rivasseau and R.S´ en´eor, “Construction of Y M4 with an infrared cutoff”, Commun. Math. Phys. 155 (1993) 325. [29] G. Poirot, “A single slice 2d Anderson model at weak disorder”, Ecole Polytechnique preprint. [30] A. Pordt, “Renormalization theory for hierarchical models”, Helv. Phys. Acta, 66 (1993) 105; “On renormalization group flows and polymer algebras”, Constructive Physics, Proceedings, Palaiseau, France 1994, Lecture Notes in Phys. 446, Springer, 1995. [31] V. Rivasseau, From Perturbative to Constructive Renormalization, Princeton Univ. Press, 1991. [32] V. Rivasseau, “Cluster expansions with small/large field conditions”, Mathematical Quantum Theory I: Field Theory and Many Body Theory, ed. J. Feldman, R. Froese, and L. Rosen, CRM Proceedings Lecture Notes.

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES INTERACTING WITH A QUANTIZED RADIATION FIELD FUMIO HIROSHIMA Department of Mathematics Faculty of Science Hokkaido University Sapporo 060, Japan Received 10 June 1996 This paper presents a scaling limit of Hamiltonians which describe interactions of N nonrelativistic charged particles in a scalar potential and a quantized radiation field in the Coulomb gauge with the dipole approximation. The scaling limit defines effective potentials. In one-nonrelativistic particle case, the effective potentials have been known to be Gaussian transformations of the scalar potential [J. Math. Phys. 34 (1993) 4478–4518]. However it is shown that the effective potentials in the case of N -nonrelativistic particles are not necessary to be Gaussian transformations of the scalar potential.

1. Introduction The main problem in this paper is to consider a scaling limit of a model in quantum electrodynamics which describes an interaction of many nonrelativistic charged particles and a quantized radiation field in the Coulomb gauge with the dipole approximation. For our discussion we may limit ourselves to the case of a fixed number N of the particles, since N does not change in time. The model we consider is called “the Pauli–Fierz model”, which has been a subject of great interests and by which real physical phenomena of charged particles and a quantized radiation field such as “Lamb shift” can be interpreted. There has been a considerable amount of literature on the Pauli–Fierz model with one-nonrelativistic charged particle, e.g., [1, 2] from the viewpoint of physics and [3, 4, 5, 6, 7, 8] from the mathematical viewpoint. In particular, the authors of [5, 6] have studied a scaling limit of the Pauli–Fierz model with one-nonrelativistic charged particle. We may well extend the scaling limit of one-particle system to N -particle system. The authors of [5, 6] defined Hamiltonians of the Pauli–Fierz model as selfadjoint operators Hρ with an ultraviolet cut-off function ρ acting in the tensor product of the Hilbert space L2 (Rd ) and a Boson Fock space F (W) over W = 2 d ⊕d−1 r=1 L (R ). Introducing scalings with respect to parameters c (the speed of light), m (the mass of the particle) and e (the charge of the particle), the authors have shown the existence of the strong resolvent limits of the scaled self-adjoint operators HρREN (κ)+V ⊗I with an infinite self-energy of the nonrelativistic particle subtracted with a scalar potential V , (we call the limit “the scaling limit of Hρ + V ⊗ I ”):

201 Reviews in Mathematical Physics, Vol. 9, No. 2 (1997) 201–225 c World Scientific Publishing Company

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In [6] we have proved the following: Let V and ρ satisfy some conditions and ∆ be the Laplacian in L2 (Rd ). Then HρREN (κ) + V ⊗ I is self-adjoint and bounded from below uniformly in sufficiently large κ > 0 with ( ) −1 1 − ∆ + Vef f − z ⊗ P0 S −1 , s − lim (HρREN (κ) + V ⊗ I − z)−1 = S κ→∞ 2m∞ where z ∈ C\R, m∞ is a positive constant, S a unitary operator on L2 (Rd )⊗F (W), P0 a projection on F (W) and Veff a multiplication operator defined by Z 2 d dye−|x−y| /2α V (y), Veff (x) = (2πα)− 2 where α is a positive constant. The multiplication operator Veff is called “the effective potential”. One of the strongest methods to analyze the scaling limits in [5, 6] was to find Bogoliubov transformations U, which implements a unitary equivalence between the Pauli–Fierz Hamiltonians Hρ and decoupled Hamiltonians of the form e = − 1 ∆ ⊗ I + I ⊗ Hb + constant, H 2m e where m e is a positive constant and Hb is the free Hamiltonian of the quantized radiation field in F (W); the authors of [5, 6] show equations of the following type: −1 −1 e + U −1 (V ⊗ I)U − z =U H U −1 . (1.1) HρREN + V ⊗ I − z In this paper, the Pauli–Fierz Hamiltonian Hρ~ with N -nonrelativistic charged particles in the Coulomb gauge with the dipole approximation are defined as operators N N acting in the Hilbert space L2 (Rd ) ⊗ · · · ⊗ L2 (Rd ) F (W) ∼ = L2 (RdN ) F (W) by {z } | N

Hρ~ =

d N 2 1 XX −i~Dµj ⊗ I − eI ⊗ Aµ (ρj ) + I ⊗ Hb 2m j=1 µ=1

=−

d N ~2 1 XX ∆ ⊗ I + I ⊗ Hb + 2e~iDµj ⊗ Aµ (ρj ) + e2 I ⊗ A2µ (ρj ) , 2m 2m j=1 µ=1

where Dµj is the differential operator with respect to the j-th variable in the µ-th direction, ∆ the Laplacian in L2 (RdN ), ~ the Planck constant divided by 2π and Aµ (ρj ) the quantized radiation field in the µ-th direction with an ultraviolet cut-off function ρj in the Coulomb gauge. Problems arising in the many-particle systems are as follows: (i) Do there exist any Bogoliubov transformations such as (1.1)? (ii) What kind of scalar potentials V and sets of ultraviolet cut-off functions (ρ1 , . . . , ρN ) does a scaling limit of the Hamiltonian Hρ~ + V ⊗ I exist for? Furthermore, what kind of infinite self-energy should be subtracted from the original Hamiltonian Hρ~ + V ⊗ I?

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

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(iii) If the scaling limit exists, what form does the effective potential have? With this motivation, we continue here to analyze a scaling limit of the Pauli– Fierz model with N -nonrelativistic charged particles. We introduce the same scaling as [6] as follows; c(κ) = cκ, e(κ) = eκ− 2 , m(κ) = mκ−2 . 1

(1.2)

Introducing a pseudo differential operator E REN (D, κ) in L2 (RdN ) with a symbol E REN (p, κ) such that E REN (p, κ) → ∞ as κ → ∞, we define a Hamiltonian Hρ~ REN (κ) by Hρ~ REN (κ) = −E REN (D, κ) ⊗ I + κI ⊗ Hb +

d N 1 XX κ2e~iDµj ⊗ Aµ (ρj ) + e2 I ⊗ A2µ (ρj ) . 2m j=1 µ=1

For sufficiently large κ > 0 and a scalar potential V with some conditions, we shall show that Hρ~ REN (κ) + V ⊗ I is essentially self-adjoint on D(−∆ ⊗ I) ∩ D(I ⊗ Hb ) and bounded from below uniformly in sufficiently large κ > 0, and the existence of Bogoliubov transformations U(κ), which gives a unitary equivalence of Hρ~ REN (κ)+ fρ~ (κ) + Cκ (V ) as follows; V ⊗ I and a self-adjoint operator H fρ~ (κ) + Cκ (V ) − z)−1 U −1 (κ), (Hρ~ REN (κ) + V ⊗ I − z)−1 = U(κ)(H e e fρ~ (κ) = E(D, κ) ⊗ I + κI ⊗ Hb , E(D, κ) is a pseudo differential operator where H 2 dN −1 in L (R ) and Cκ (V ) = U (κ)(V ⊗ I)U(κ) (Theorem 3.5). Then we see that U(κ) → U(∞) as κ → ∞ strongly (Theorem 3.4) and hence we get s − lim (Hρ~ REN (κ) + V ⊗ I − z)−1 κ→∞ = U(∞) (E ∞ (D) + Veff − z)−1 ⊗ P0 U −1 (∞), where E ∞ (D) is a pseudo differential operator in L2 (RdN ) and Veff a multiplication operator. (Theorems 3.6 and 3.7). In the case of one-particle system the effective potential Veff is a Gaussian transformation of a given scalar potential V . However, we shall see that in the N -particle system, Veff is not necessary to be a Gaussian e ∞ )1≤i,j≤N defined e ∞ = (∆ transformation. Actually it is determined by a matrix ∆ ij by 2 2 Z ~ ρˆi (k)ˆ e ρj (k) e∞ = 1 d − 1 dk , (1.3) ∆ ij 2 d mc ~c Rd ω(k)3 e ∞ is non-degenerate, the effective where ω(k) = |k|,k ∈ Rd . In the case where ∆ potential Veff is Gaussian transformations of V . The outline of this paper is as follows. In Sec. 2, we define the Pauli–Fierz Hamiltonian with N -nonrelativistic charged particles in the Coulomb gauge with the dipole approximation and show its self-adjointness. Moreover we construct an exact solution to the Heisenberg equation from the point of view of the operator

204

F. HIROSHIMA

theory (Corollary 2.9). In Sec. 3, when the scaling parameter κ > 0 is sufficiently large, we show that a Bogoliubov transformation can be constructed, and define a renormalized self-adjoint operator Hρ~ REN (κ) which is the original Hamiltonian Hρ~ (κ) with an infinite self-energy of the nonrelativistic charged particles subtracted. We shall show the existence of the scaling limit of Hρ~ + V ⊗ I and give an explicit form of the effective potential. We give typical examples of scalar potentials and sets of ultraviolet cut-off functions. In Sec. 4, we give a physical interpretation of e ∞. the matrix ∆ The author would like to thank Professor A. Arai for helpful discussions. 2. The Pauli Fierz Model and Exact Solution To begin with, let us introduce some preliminary notations. Let H be a Hilbert space over C. We denote the inner product and the associated norm by h∗, ·iH and || · ||H respectively. The inner product is linear in · and antilinear in ∗. The domain of an operator A in H is denoted by D(A). The Fourier transformation (resp. the inverse Fourier transformation) of a function f is denoted by fˆ (resp.fˇ) and f¯ the complex conjugate of f . In this paper, summations over repeated Greek letters are understood. Let W ≡ L2 (Rd ) ⊕ · · · ⊕ L2 (Rd ) . {z } | d−1

We define the Boson Fock space over W by F (W) ≡

∞ M

⊗ns W ≡

n=0

∞ M

Fn (W),

n=0

where ⊗0s W ≡ C and ⊗ns W (n ≥ 1) denotes the n-fold symmetric tensor product. Put F N (W) ≡

N M

M

Fn (W)

n=0

{0}.

n≥N +1

Moreover we define the finite particle subspace of F (W) by F ∞ (W) ≡

∞ [

F N (W).

N =0

The annihilation operator a(f ) and the creation operator a† (f ) (f ∈ W) act on the finite particle subspace and leave it invariant with the canonical commutation relations (CCR): for f, g ∈ W

[a(f ), a† (g)] = f¯, g W , [a] (f ), a] (g)] = 0, where [A, B] = AB − BA, a] denotes either a or a† . Furthermore,

† a (f )Φ, Ψ F (W) = Φ, a(f¯)Ψ F (W) , Φ, Ψ ∈ F ∞ (W).

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We define polarization vectors er (r = 1, . . . , d − 1) as measurable functions er : Rd −→ Rd such that er (k)es (k) = δrs ,

er (k)k = 0,

a.e. k ∈ Rd .

In this paper, we fix polarization vectors er = (er1 , . . . , erd ). The µ-th direction timezero smeared radiation field in the Coulomb gauge with the dipole approximation is defined as operators acting in F (W) by   √ r ! √ r ˜  ˆ ˆ  ~e f 1  † µ d−1 ~eµ f   √ √ + a ⊕ , (2.1) a ⊕d−1 Aµ (f ) = √ r=1 r=1  cω cω 2 and the conjugate momentum o √ √ √ √ i n d−1 r ˆ r ˜ˆ ~ cωe ~ cωe f − a ⊕ f , Πµ (f ) = √ a† ⊕d−1 µ µ r=1 r=1 2

(2.2)

where ge(k) = g(−k). Note that in the case where f is real-valued, Aµ (f ) and Πµ (f ) are symmetric operators. Let Ω = (1, 0, 0, . . .) ∈ F(W). It is well known that L a† (f1 ) . . . a† (fn )Ω, Ω|fj ∈ W, j = 1, . . . , n, n ≥ 1 is dense in F (W). For a nonnegative self-adjoint operator h : W → W, an operator Γ(e−th ) is defined by Γ(e−th )a† (f1 ) . . . a† (fn )Ω = a† (e−th f1 ) . . . a† (e−th fn )Ω, Γ(e−th )Ω = Ω. The operator Γ(e−th ) defines a unique strongly continuous symmetric one-parameter semigroup on F (W). Hence, by Stone’s theorem, there exists a nonnegative selfadjoint operator dΓ(h) in F (W) such that Γ(e−th ) = e−tdΓ(h) . The operator dΓ(h) is called “the second quantization of h”. Put ω e = ω ⊕ · · · ⊕ ω. {z } | The free Hamiltonian Hb in F (W) is defined by ω ). Hb ≡ ~cdΓ(e Let Mn be a Hilbert space defined by Z 2 n |f (k)| ω(k) dk < ∞ , Mn = f Rd

with the inner product Z hf, gin =

f¯(k)g(k)ω(k)n dk. Rd

d−1

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We have the following commutation relations on F ∞ (W), fˆ, gˆ ∈ M−1 , fˆ, gˆ ∈ M1 ,

[Aµ (f ), Aν (g)] = 0, [Πµ (f ), Πν (g)] = 0,

D E [Aµ (f ), Πν (g)] = i~ dµν fˆ¯, gˆ

L2 (Rd )

, fˆ, gˆ ∈ M−1 ∩ M0 ∩ M1 ,

3

and on D(Hb2 ), fˆ ∈ M−1 ∩ M1 , [Hb , Πµ (f )] = i~c2 Aµ (−∆f ), fˆ ∈ M3 ∩ M1 , [Hb , Aµ (f )] = −i~Πµ (f ),

Pd where ∆ is the Laplacian in the L2 -sense in L2 (Rd ) and dµν (k) = r=1 erµ (k)erν (k). The Pauli–Fierz Hamiltonian with N -nonrelativistic charged particles interacting with the quantized radiation field in the Coulomb gauge with the dipole approximation is defined by Hρ~ ≡ Hρ1 ,...,ρN ≡

d N 2 1 XX −i~Dµj ⊗ I − eI ⊗ Aµ (ρj ) + I ⊗ Hb , 2m j=1 µ=1

acting in L2 (Rd ) ⊗ · · · ⊗ L2 (Rd ) {z } |

O

F (W) ∼ = = L2 (RdN ) ⊗ F(W) ∼

N

Z

⊕

RdN

F (W)dx,

where ρ0j s serve as ultraviolet cut-off functions. We introduce a scaling with respect to the parameters c, e, m as in (1.2). Throughout this paper, for objects A = A(c, e, m) containing the parameters c, e, m, we denote the scaled object by A(κ) ≡ A(c(κ), e(κ), m(κ)). We define a class of sets of functions as follows: Definition 2.1. ρ~ = (ρ1 , . . . , ρN ) is in P if and only if (1) ρˆj , j = 1, . . . , N are rotation invariant, ρˆj (k) = ρˆj (|k|), and real-valued, √ √ (2) ρˆj /ω, ρˆj / ω, ρˆj , ωρˆj ∈ L2 (Rd ). Moreover ρ~ is in Pe if and only if in addition to (1) and (2) above. √ (3) For all j = 1, . . . , N , ρˆj /ω ω ∈ L2 (Rd ) and there exist 0 < α < 1 and 1 ≤ √ √ √ d−2 ρj ( s)( s) ∈ Lip(α) ∩ L ([0, ∞)), where Lip(α) is the such that ρˆi ( s)ˆ set of the Lipschitz continuous functions on [0, ∞) with order α, d 3 d 1 ρj (k)ω 2 − 2 (k)| < ∞, supk |ˆ ρj (k)ω 2 − 2 (k)| < ∞, j = 1, . . . , N . (4) supk |ˆ Observe that Definition 2.1 (1) implies that ρj ’s are real-valued functions. Hence Aµ (ρj )’s are symmetric operators. Put H0 = −

1 2 ~ ∆ ⊗ I + I ⊗ Hb , 2m

that H0 is a nonnegative where ∆ is the Laplacian in L2 (RdN ). It is well known 1 2 ~ ∆ ⊗ I ∩ D(I ⊗ Hb ). self-adjoint operator on D(H0 ) = D − 2m Theorem 2.2. [3, 4] For ρ~ ∈ P and κ > 0, the operator Hρ~ (κ) is self-adjoint on D(H0 ) and essentially self-adjoint on any core of H0 and nonnegative.

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207

Let F = F ⊗ I, where F denotes the Fourier transform in L2 (RdN ). It is clear that operators FHρ~ F−1 are decomposable as follows: Z ⊕ −1 Hρ~ (p, κ)dp, FHρ~ (κ)F = RdN

where Hρ~ (p, κ) =

d N 2 1 XX κ~pjµ − eAµ (ρj ) + κHb . 2m j=1 µ=1

Theorem 2.3. [3, 4] For ρ~ ∈ P and κ > 0, the operator Hρ~ (p, κ) is self-adjoint on D(H0 ) and essentially self-adjoint on any core of Hb and nonnegative. Following [3, 4, 6], we shall construct a Heisenberg field concretely. The Heisenberg field Aµ (f, t, κ) with the scaling parameter κ is defined by a solution to the Heisenberg equation: i d Aµ (f, t, κ) = [Hρ~ (p, κ), Aµ (f, t, κ)], dt ~ Aµ (f, 0, κ) = Aµ (f, κ). In order to construct the Heisenberg field in a rigorous way, we shall prepare some technical lemmas. We define an N × N matrix-valued function D(z) = (Dij (z))1≤i,j≤N by Z e2 d − 1 ρˆi (k)ˆ ρj (k) dk, z ∈ C \ [0, ∞). Dij (z) = mδij − 2 2 c d Rd z − |k| Lemma 2.4. Let ( , ) denote the Euclidean inner product. Suppose ρ ~ ∈ Pe. Then the followings hold: (1) The functions Dij (z, κ), 1 ≤ i, j ≤ N, κ > 0 are analytic in C \ [0, ∞). (2) For s ∈ [0, ∞) and κ > 0, the pointwise limit D±ij (s, κ) ≡ limh→0 Dij (s ± ih, κ) exists and has the following form: m 1 e2 Vd d − 1 Hij (s) D±ij (s, κ) = 2 δij − 3 κ κ c2 2 d 2πi e2 Vd d − 1 Kij (s), ± 3 κ c2 2 d √ √ d ρj ( s)s 2 −1 , Kij (s) = ρˆi ( s)ˆ Z Kij (x) dx, Hij (s) = lim →0+ |s−x|> s − x d

where Vd = 2π 2 /Γ( d2 ) (Γ(z) is the gamma function). The convergence is uniform in s ∈ [0, ∞); for any δ > 0, there exists h0 > 0 independent of s, κ, such that for 0 <∀ h ≤ h0 , |Dij (s ± ih, κ) − D±ij (s, κ)| ≤

δ . κ3

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F. HIROSHIMA

Moreover Hij (s) is Lipschitz continuous in s ∈ [0, ∞) with the same order as that of Kij and contained in L (Rd ) with some ≥ 1. (3) Let κ > 0 be sufficiently large. Put D± (s, κ) = (D±ij (s, κ))1≤i,j≤N . Then ~ ∈ CN , there exists a positive constant d1 (κ) such that for (w1 , . . . , wN ) = w inf

s∈[0,∞)

|(D± (s, κ)w, ~ w)| ~ > d1 (κ)|w| ~ 2.

(4) Let κ > 0 be sufficiently large. Then there exists a positive constant d2 (κ) such that for w ~ ∈ CN , inf z∈C\[0,∞)

|(D(z, κ)w, ~ w)| ~ > d2 (κ)|w| ~ 2.

Proof. The statements (1) and (2) are fundamental facts [9]. We shall prove (3). From (2) it follows that ~ w) ~ = (D± (s, κ)w,

m κ2

λ 1 λ (H(s)w, ~ w) ~ ± 2πi 3 (K(s)w, ~ w), ~ |w| ~ 2− κm κ

2 where λ = ec2 V2d d−1 d , H(s) = (Hij (s))1≤i,j≤N , K(s) = (Kij (s))1≤i,j≤N . Since Hij is a Lipschitz continuous function and contained in L ([0, ∞)), it is bounded. Hence we have

|(H(s)w, ~ w)| ~ ≤N×

|Hij (s)| · |w| ~ 2 ≡ α|w| ~ 2.

sup s∈[0,∞),1≤i,j≤N

Thus we can see that for sufficiently large κ > 0 m 1 λ α |w| ~ 2. ~ w)| ~ ≥ 2 1− |(D± (s, κ)w, κ κm Hence we get (3). We shall prove (4). From (2) it follows that for any η > 0, there exists 0 > 0 independent of s ∈ [0, ∞) and κ > 0 such that for 0 <∀ ≤ 0 , ~ w)| ~ − |(D± (s, κ)w,

η |w| ~ 2 ≤ |(D(s ± i, κ)w, ~ w)|. ~ κ3

Hence we have |(D(s ± i, κ)w, ~ w)| ~ ≥

m κ2

η 1 λ α − 3 |w| 1− ~ 2. κm κ

(2.3)

On the other hand, put Π0 = C \ {x + iy|x ≥ 0, |y| ≤ 0 }. Then we see that for x + iy ∈ Π0 ! PN √ Z d 1 λ ∞ (x − s)| i=1 wi ρˆi ( s)|2 s 2 −1 ds |w| ~ − κm 0 (x − s)2 + y 2 √ Z ∞ PN d y| i=1 wi ρˆ( s)|2 s 2 −1 λ ds. +i 3 κ 0 (x − s)2 + y 2

m (D(x + iy, κ)w, ~ w) ~ = 2 κ

2

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

209

Noting that |ab|/a2 + b2 ≤ 1/2, we have 2 Z Z ∞ X N d ∞ (x − s)|PN w ρˆ (√s)|2 s d2 −1 √ 1 2 −1 i i i=1 ≤ ds w ρ ˆ ( s) ds s i i 2 2 0 (x − s) + y 2|y| 0 i=1 ≤ ≡

1 20

Z

N ∞X

0

√ d |ˆ ρi ( s)|2 s 2 −1 ds|w| ~2

i=1

β |w| ~ 2. 0

Since 0 is independent of κ > 0, we see that for sufficiently large κ > 0, 1 λ β m |w| ~ 2 , x + iy ∈ Π0 . |(D(x + iy, κ)w, ~ w)| ~ ≥ 2 1− κ κ m 0

(2.4)

Combining (2.3) and (2.4), we get (4).

From Lemma 2.4 (3) and (4) it follows that for sufficiently large κ > 0, there exist the inverse matrices to D(z, κ) and D± (s, κ), which satisfy ~ 1, w ~2 < sup D−1 ± (s, κ)w

s∈[0,∞)

sup

1 |w ~ 1 ||w ~ 2 |, d1 (κ)

−1 D (z, κ)w ~ 1, w ~2 <

z∈C\[0,∞)

1 |w ~ 1 ||w ~ 2 |. d2 (κ)

(2.5) (2.6)

We set for ρ ~ ∈ Pe and sufficiently large κ > 0   ρˆ1 (k)  ..  2 Q(k, κ) ≡ D−1 + (k , κ)  .  ≡ (Q1 (k, κ), . . . , QN (k, κ)). ρˆN (k) For later use in Appendix, we note that for all s ∈ [0, ∞), √ √ d 1 e2 d−1 V ρj ( s)s 2 −1 . ρˆi ( s)ˆ D+ij (s, κ) − D−ij (s, κ) = 2πi 3 d 2 κ c d

(2.7)

−1 −1 (z))1≤i,j≤N , D−1 Put D−1 (z) = (Dij ± (s) = (D±ij (s))1≤i,j≤N . Then (2.7) implies that N √ √ d d − 1 X −1 2πi e2 −1 Vd D−ik (s, κ)D+jl (s, κ)ˆ ρk ( s)ˆ ρl ( s)s 2 −1 κ 3 c2 d k,l=1

= D−ij (s, κ) − D+ji (s, κ).

(2.8)

Remark 2.5. (1) In [3, 4, 6], the authors define functions D± (s) corresponding to D± (s) defined in this paper. The function 1/D± (s, κ) can be well defined for some ρ and any κ > 0. However, in our case, we do not know whether D± (s, κ) has the inverse or not for all κ > 0. But since, in this paper, we focus on an asymptotic behavior as κ → ∞, it is sufficient to consider the case where κ is sufficiently large.

210

F. HIROSHIMA

(2) For the proof of Lemma 2.4, we do not need Definition 2.1 (4). We define operators Gh (h > 0) by Z f (k 0 ) dk 0 . (Gh f )(k) = d −1 Rd (k 2 − k 02 + ih)(kk 0 ) 2 It is well known and not so hard to see that Gh are bounded linear operators on L2 (Rd ) and the strong limit limh→0 Gh ≡ G exists [4]. Furthermore G is skew symmetric (G∗ = −G). For sufficiently large κ > 0, we can define the following operators: Tµν (κ)f ≡ δµν f +

N d d 1 e2 X Qj (κ)ω 2 −1 Gω 2 −1 dµν ρˆj f, κ3 c2 j=1

1 ≤ µ, ν ≤ d.

Lemma 2.6. Suppose that ρ~ ∈ Pe and κ > 0 is sufficiently large. Then the followings hold. ∗ (κ) are bounded operators on Mα , α = −1, 0, 1 and (Tµν (κ)f ˜) (1) Tµν (κ) and Tµν = Tµν (κ)fe. −1 −1 (0, κ) ≡ D±ij (0, κ) and let f ∈ M−1 . Then (2) Put Dij * + N X ρˆj f Qi (κ) 1 −1 = dµα Dij (0, κ) √ , √ , dνα √ , √ Tµν (κ)f ω ω ω3 ω3 L2 (Rd ) j=1 L2 (Rd )

i = 1, . . . , N. (3) (4) (5) (6)

P 1 e2 ∗ (κ) ] = − N ˆi . [ ω 2 , Tµν i=1 κ3 c2 hQi (κ), ·iL2 (Rd ) dµν ρ ρj = δµν κm2 Qj (κ). Tµν (κ)ˆ ∗ (κ)dνα Tαβ (κ) = dµβ . Tµν r eµ Tµν (κ)dνα Tαβ (κ)esβ = δrs .

Proof. See Appendix.

In the rest of this section, we fix sufficiently large κ > 0 and omit κ in notations b µ (f ) = Πµ (fˆ). We put bµ (f ) = Aµ (fˆ) and Π for simplicity. Define A 1 ∗ r√ 1 ∗ r f 1 (r) b b Aµ √ Tµν eν cωf + iΠµ √ Tµν eν √ B (f, p) = √ cω 2 ~ ~  N  X e Qj erν j √ ~pν + , f ∈ M0 , 3 ,f  ~ (cω) 2 L2 (Rd ) j=1 ! ( 1 1 1 ∗ r√ ∗ r f˜ †(r) ˜ bµ √ T µν e˜ν cωf − iΠ b µ √ T µν e˜ν √ A B (f, p) = √ cω 2 ~ ~  + * N  X e Qj erν ~pjν √ + , 3 ,f  ~ (cω) 2 2 d j=1

L (R )

dN . f ∈ M0 , p = (p , . . . , p ) = (p11 , p12 , . . . , pN d )∈R 1

N

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

211

By the definition of Aµ (f ) and Πµ (f ), for the vector of the form f = f1 ⊕· · ·⊕fd−1 ∈ W, we see that B(f , p) ≡

d−1 X

B (r) (fr , p) = a† (W− f ) + a(W+ f ) +

r=1 †

B (f , p) ≡

N X

Lj pj , f

L2 (Rd )

,

j=1

d−1 X

B

†(r)

N X

(fr , p) = a (W+ f ) + a(W− f ) + Lj pj , f L2 (Rd ) ,

(2.9)

†

r=1

j=1

where (r,s)

W± = (W±

)1≤r,s≤d−1 ,

Lj = (Lrµj )1≤µ≤d, where (r,s)

W+

f =

(r,s) W− f

Lrµj

1 2

1≤r≤d−1



L11j · · · . =  .. ··· Ld−1 1j

 L1dj  ..  , j = 1, . . . , N, . d−1 Ldj

√ √ 1 ∗ s ∗ s 1 √ erµ Tµν eν ω + ωerµ Tµν eν √ ω ω

f,

√ r ∗ s 1 1 r ∗ s√ 1 √ eµ Tµν e˜ν ω − ωeµ Tµν e˜ν √ f˜, = 2 ω ω √ e ~erµ Qj = √ . 2c3 ω 3

We see that, by Lemma 2.6 (1), W± is a bounded operator on W. By virtue of Lemma 2.6 (5) and (6), one can easily see that W± satisfy the following algebraic relations: ∗ ∗ W+ − W− W− = I, W+ ∗

∗

W+ W− − W− W+ = 0,

(2.10)

∗

∗ − W− W− = I, W+ W+ ∗

∗ − W+ W− = 0. W− W+

We put Wα = Mα ⊕ . . . ⊕ Mα , α ∈ R. These relations (2.10) imply that on F ∞ (W) | {z } for f , g ∈ W0 ,

d−1

[B(f , p), B † (g, p)] = hf , giW , [B ] (f , p), B ] (g, p)] = 0, and for Φ, Ψ ∈ F ∞ (W),

† B (f , p)Φ, Ψ F (W) = Φ, B(¯f , p)Ψ F (W) . Lemma 2.7. For f ∈ W0 ∩ W2 , p ∈ RdN and ρ~ ∈ Pe , we have 3

ωf , p), on F ∞ (W) ∩ D(Hb2 ), [ Hρ~ (p), B ] (f , p) ] = ±B ] (~ce where + (resp.-) corresponds to B † (resp.B).

(2.11)

212

F. HIROSHIMA

Proof. Suppose that f ∈ W−2 ∩ W0 ∩ W2 . Then by Lemma 2.6 (3) and (4), one can directly see that (2.11) holds. Next by a limiting argument, one can get (2.11) for f ∈ W0 ∩ W2 . Define 1 A(f , p) ≡ √ B † (f , p) + B(¯f , p) , 2 i Π(f , p) ≡ √ B † (f , p) − B(¯f , p) , 2

f ∈ W0 .

We can easily see that the operators A(f , p)|F ∞ (W) and Π(f , p)|F ∞ (W) are essentially self-adjoint by the Nelson analytic vector theorem [10, Theorem X.39]. We denote the self-adjoint extensions by the same symbols. Theorem 2.8. Suppose ρ ~ ∈ Pe. Then for f ∈ W0 t t ωt exp i Hρ~ (p) A(f , p) exp −i Hρ~ (p) = A(eice f , p), ~ ~ t t ωt f , p). exp i Hρ~ (p) Π(f , p) exp −i Hρ~ (p) = Π(eice ~ ~

(2.12) (2.13)

ωt f , p) = Proof. We only show an outline of the proof. For simplicity, put A(eice ∞ ∞ n (H ) = ∩ D(H ). We can easily see that, by Lemma 2.7, A(f , p, t). Let C b n=1 b E D iA(f ,p,t) ∞ ∞ Ψ, Φ , Ψ, Φ ∈ C (H ) ∩ F (W), f ∈ W ∩ W ∩ W , is differentiable in e −2

b

0

2

t with E d D iA(f ,p,t) e Ψ, Φ dt F (W) i iA(f ,p,t) i e Hρ~ (p)Ψ, e−iA(f ,p,t) Φ = Ψ, Hρ~ (p)Φ − . (2.14) ~ ~ F (W) F (W) From (2.14) it follows that E d D −i t Hρ~ (p) iA(f ,p,t) i t Hρ~ (p) e ~ e e ~ Ψ, Φ = 0. dt F (W) Hence eisA(f ,p,0) = ei ~ Hρ~ (p) eisA(f ,p,t) e−i ~ Hρ~ (p) , t

t

on C ∞ (Hb ) ∩ F ∞ (W).

(2.15)

By a limiting argument, one can see that (2.15) holds for Φ, Ψ ∈ D(Hb ), f ∈ W0 . Since the both sides of (2.15) are one-parameter unitary groups in s ∈ R, Stone’s theorem yields (2.12). (2.13) is quite similar to (2.12). Thus we get the desired result.

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

For t ∈ R, we define operators in F (W) by ! ( √ d−1 1 X ~ icωt r †(r) √ e eν T µν fˆ, p B Aµ (f, t|p) = √ cω 2 r=1 !) √ ~ ˜ + B (r) √ e−icωt erν Tµν fˆ, p cω + * N X ρˆj fˆ −1 i ~pν dµν Dij (0) p ,√ −e cω (cω)3 i,j=1

213

, L2 (Rd )

fˆ ∈ M−1 , µ = 1, . . . , d. Form Lemma 2.5 (2), (5) and (6) it follows that Aµ (f, 0|p) = Aµ (f ).

(2.16)

Corollary 2.9. Suppose ρ ~ ∈ Pe. Then the operator Aµ (f, t|p) is the Heisenberg field with t t (2.17) exp i Hρ~ (p) Aµ (f ) exp −i Hρ~ (p) = Aµ (f, t|p). ~ ~ Proof. It is enough to show (2.17) for a real-valued function f . For a real-valued function f such that fˆ ∈ M−1 , we can see that ! !) ( √ √ d−1 1 X ~ r ~ †(r) (r) √ eν T µν fˆ, p + B √ erν T µν fˆ, p B Aµ (f ) = √ cω cω 2 r=1 + * N X ρˆj fˆ −1 . ~piν dµν Dij (0) p ,√ −e 3 cω (cω) i,j=1 Hence (2.17) follows from Theorem 2.8.

Corollary 2.10. Suppose ρ~ ∈ Pe. Then for Φ ∈ D(Hb ), t t ] ωt f , p)Φ. exp i Hρ~ (p) B (f , p) exp −i Hρ~ (p) Φ = B ] (eice ~ ~ Proof. Note that D(B ] (f , p)) ⊃ D(Hb ) = D(Hρ~ (p)). Thus from (2.12) and (2.13) it follows that on D(Hb ) 1 n t t t t ei ~ Hρ~ (p) A(f , p)e−i ~ Hρ~ (p) = √ ei ~ Hρ~ (p) B † (f , p)e−i ~ Hρ~ (p) 2 o t t + ei ~ Hρ~ (p) B(f , p)e−i ~ Hρ~ (p) , o 1 n ωt ωt f ) + B(e−ice f) , = √ B † (eice 2

214

F. HIROSHIMA

t t t i n t ei ~ Hρ~ (p) Π(f , p)e−i ~ Hρ~ (p) = √ ei ~ Hρ~ (p) B † (f , p)e−i ~ Hρ~ (p) 2 o t t − ei ~ Hρ~ (p) B(f , p)e−i ~ Hρ~ (p) o i n ωt ωt f ) − B(e−ice f) . = √ B † (eice 2 Thus the corollary follows.

3. Bogoliubov Transformations and Scaling Limits In this section, we construct a unitary operator which implements a unitary equivalence of the Pauli–Fierz Hamiltonian and a decoupled Hamiltonian. Moreover we investigate a scaling limit of the Pauli–Fierz Hamiltonian. Unless otherwise stated in this section, we suppose that κ > 0 is sufficiently large. Since the bounded operators W−r,s (κ) have integral kernels PN r s 0 0 1 e2 eµ (k)eµ (k ) j=1 ρˆj (k)Qj (k , κ) (r,s) 0 , W− (k, k , κ) = 3 2 1 κ c 2(|k| + |k 0 |)(|k||k 0 |) 2 (r,s)

such that W− (κ) ∈ L2 (Rd × Rd ), the operator W− (κ) is a Hilbert–Schmidt operator on W. Then from (2.9) and (2.10) it follows that there exist two unitary operators U (κ) (p independent) and S(p, κ) such that [6, Sec. III] U −1 (κ)S(p, κ)−1 B ] (f , p, κ)S(p, κ)U (κ) = a] (f ),

f ∈ W.

(3.1)

Concretely S(p, κ) is given by  ( ! N r −1 X e D (0, κ)ˆ ρ e~ j µ ij d−1 pi a ⊕r=1 √ S(p, κ) = exp  2 µ κ 2~c3 ω 3 i,j=1 − a†

er D−1 (0, κ)ˆ ρj d−1 µ ij ⊕r=1 √ 3 3 2~c ω

!)! .

Theorem 3.1. Suppose ρ ~ ∈ Pe . Then putting S(p, κ)U (κ) = U(p, κ), we see that U(p, κ) maps D(Hb ) onto itself with U(p, κ)Hρ~ (p, κ)U −1 (p, κ) = κHb + E(p, κ),

(3.2)

where E(p, κ) = peiµ (κ) =

d N 2 ~2 X X piµ (κ) + 2(κ), κpiµ + κe 2m i=1 µ=1 N X

pjν ∆ji νµ (κ),

j=1

∆ji νµ (κ)

N d−1 1 e2 X X = 3 2 κ 2c r,s=1 k=1

e ~ 2(κ) = 4mc 2

*

−1 (0, κ)ˆ ρk erν Djk √ , ω3

(r,s) esµ ρˆi −1 √ I + W− (κ)W+ (κ) ω

d−1 r N X X (r,s) esµ ρˆi eµ ρˆi −1 √ , I − W− (κ)W+ √ (κ) . ω ω L2 (Rd ) i=1 r,s=1

, L2 (Rd )

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

215

Proof. For simplicity, we omit the symbol κ. Put U(p)Ω ≡ Ω(p). From [6, Proposition 2.4, Lemma 5.9] it follows that Ω(p) ∈ D(Hb ). Then Ω(p) ∈ D(B(f , p)). By virtue of Corollary 2.10 and (3.1), we can see that for all f ∈ W t (3.3) B(f , p) exp i Hρ~ (p) Ω(p) = 0. ~ Equation (3.3) implies that there exists a positive constant E(p) such that t t exp i Hρ~ (p) Ω(p) = exp i E(p) Ω(p). ~ ~

(3.4)

Hence from Corollary 2.10, (3.1), (3.4) and the denseness of L B † (f1 ) . . . B † (fn )Ω(p), Ω(p) fj ∈ W, j = 1, . . . , n, n ≥ 1 , one can get (3.2)(we refer to [6, Lemma 5.12]). Noting that [6, Lemma 2.2] √ r ! ~eµ ρˆi Ω(p) a ⊕d−1 r=1 √ 2cω ( √ s !) d−1 X ~e ρˆi d−1 −1 (r,s) i † √ µ Ω(p), W− W+ = −e pµ − a ⊕r=1 2cω s=1 one can easily get E(p) by E(p) =

hHρ~ (p)Ω(p), ΩiF (W) . hΩ(p), ΩiF (W)

This completes the proof. The positive constant E(p, κ) can be rewritten as E(p, κ) =

κ2 ~2 2 e κ), p + E REN (p, κ) + E(p, 2m

where d N 2 2 X X e κ) = κ ~ pi bij (κ)pjν , E(p, 2m i,j=1 µ,ν=1 µ µν

bij µν (κ)

=

d N X X

jk ∆jk να (κ) + ∆να (κ) 2

k=1 α=1

!

ik ∆ik µα (κ) + ∆µα (κ) 2

! ,

(3.5)

κ2 ~2 2 e p − E(p, κ). 2m Let M (K) be the set of K × K complex matrices. Note that since ∼ (bij µν (κ)) 1≤i,j≤N,1≤µ,ν≤d ∈ M (N ) ⊗ M (d) = M (dN ) is nonnegative and symmetric, fρ~ (p, κ) by e κ) ≥ 0 for p ∈ RdN . We define Hρ~ REN (p, κ) and H we have E(p, E REN (p, κ) = E(p, κ) −

Hρ~ REN (p, κ) = −E REN (p, κ) + κHb 1 XX −2κe~pjµ Aµ (ρj ) + e2 Aµ (ρj )2 , 2m j=1 µ=1 N

+

d

e κ) + κHb . fρ~ (p, κ) = E(p, H

216

F. HIROSHIMA

Then one can see that Hρ~ REN (κ) ≡ F−1

Z

Hρ~REN (p, κ)dp F

⊕

RdN

= −E REN (D, κ) ⊗ I + κI ⊗ Hb 1 XX −2κe~iDµj ⊗ Aµ (ρj ) + e2 I ⊗ Aµ (ρj )2 , 2m j=1 µ=1 d

N

+

fρ~ (κ) ≡ F−1 H

Z

⊕

e H(p)dp F

RdN

e = E(D, κ) ⊗ I + κI ⊗ Hb , e κ) are pseudo differential operators on L2 (RdN ) with where E REN (D, κ) and E(D, REN e (p, κ) and E(p, κ) respectively. symbols E fρ~ (κ) are essentially Theorem 3.2. Suppose ρ ~ ∈ Pe. Then Hρ~ REN (κ) and H self-adjoint on any core of H0 and bounded from below. e fρ~ (κ) κ), Hρ~ REN (κ) and H Proof. By the definition of E REN (D, κ) and E(D, are symmetric. For f ∈ D(−∆), there exist d1 (κ) and d2 (κ) such that e ||E(D, κ)f ||L2 (RdN ) ≤ d1 (κ)|| − ∆f ||L2 (RdN ) , ||E REN (D, κ)f ||L2 (RdN ) ≤ d2 (κ)|| − ∆f ||L2 (RdN ) . Hence, similar to the proof of Theorem 2.2, the Nelson commutator theorem yields desired results. Remark 3.3. Write E(p, κ) =

N N d d X X ~2 κ2 i ~ 2 κ2 2 X X ~2 κ2 i i p + pµ peµ (κ) + peµ (κ)2 + 2(κ). (3.6) 2m m 2m µ=1 i=1 µ=1 i=1

Then the first and second terms on the right-hand side of (3.6) diverge as κ → ∞ for p 6= 0, but the rest of the terms do not. Actually we see that N d ~2 κ2 X X i peµ (κ)2 κ→∞ 2m µ=1 i=1

lim

1 = 2m

e2 2mc2

d−1 d

2 X N d X α=1 k=1

 

N X j=1

~pjα

ρˆj ρˆk √ ,√ 3 ω ω

2  ,

L2 (Rd )

∞

≡ E (p). Then, by (3.2), concerning an asymptotic behavior of Hρ~ (κ) as κ → ∞, we should subtract the first and second terms in the right-hand side of (3.6) from the original Hamiltonian Hρ~ (κ). However one cannot say that peiµ (κ)2 is real and nonnegative for any p ∈ RdN . To guarantee the nonnegative self-adjointness of the Hamiltonian

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

217

e κ) such as Hρ~ REN (p, κ) with the divergence terms subtracted, we should define E(p, REN (κ) has an interpretation (3.5). In this sense, we may say that the operator Hρ~ of the Hamiltonian Hρ~ (κ) with the infinite self-energy of the nonrelativistic particles subtracted. We define U(κ) = F−1

Z

⊕

RdN

U(κ, p)dp F.

Then we have the following theorem. Theorem 3.4. Suppose that ρ ~ ∈ Pe. Then   N r r X e ρ ˆ e ρ ˆ e~ j j µ µ , Dµj ⊗ a ⊕d−1 − a† ⊕d−1 s − lim U(κ) = exp  r=1 √ r=1 √ 3ω3 3ω3 κ→∞ m 2~c 2~c j=1 ≡ U(∞).

(3.7)

Proof. From [6, Theorem 3.11] it follows (3.7).

We take scalar potentials V to be real-valued measurable functions on RdN and put Cκ (V ) = U −1 (κ)(V ⊗ I)U(κ),

C(V ) = U −1 (∞)(V ⊗ I)U(∞).

(3.8)

We introduce conditions (V-1) and (V-2) as follows. e (V-1) For sufficiently large κ > 0, D(E(D, κ)) ⊂ D(V ) and for λ > 0, −1 e V (E(D, κ) + λ) is bounded with e κ) + λ)−1 || = 0, lim ||V (E(D,

λ→∞

(3.9)

where the convergence is uniform in sufficiently large κ > 0. e (V-2) For λ > 0, V (E(D, κ) + λ)−1 is strongly continuous in κ and e κ) + λ)−1 = V (E ∞ (D) + λ)−1 . s − lim V (E(D, κ→∞

The condition (3.9) yields that, by the Kato–Rellich theorem and commutativity of e e κ)⊗ I + Cκ (V ) are essentially self-adjoint U(κ) and (E(D, κ)+ λ)−1 , operators E(D, e on any core of D(E(D, κ) ⊗ I) and uniformly bounded from below in sufficiently e κ) ⊗ I, large κ > 0. Moreover since I ⊗ Hb is nonnegative and commute with E(D, one can see that e e ρ~ (V, κ) ≡ E(D, κ) ⊗ I + Cκ (V ) + κI ⊗ Hb H e is essentially self-adjoint on any core of D(E(D, κ) ⊗ I + κI ⊗ Hb ) and uniformly bounded from below in sufficiently large κ > 0. In particular, D(H0 ) is a core of fρ~ (V, κ). Put H Hρ~REN (V, κ) ≡ Hρ~REN (κ) + V ⊗ I.

218

F. HIROSHIMA

Theorem 3.5. Let ρ ~ ∈ Pe . Suppose that V satisfies (V-1) and (V-2). Then, for sufficiently large κ > 0, the operator Hρ~REN (V, κ) is essentially self-adjoint on D(H0 ) and bounded from below uniformly in sufficiently large κ > 0. Moreover the unitary operator U(κ) maps D(H0 ) onto itself and for z ∈ C \ R or z < 0 with |z| sufficiently large, −1 −1 fρ~ (V, κ) − z = U(κ) H U −1 (κ). (3.10) Hρ~ REN (V, κ) − z Proof. Since U(κ) maps D(I ⊗ Hb ) onto itself (see Theorem 3.1) and −∆ ⊗ I commutes with U(κ) on D(−∆ ⊗ I), U(κ) maps D(H0 ) onto itself. Put n o S0∞ (RdN ) = f ∈ L2 (RdN )|fˆ ∈ C0∞ (RdN ) . b At first, by Theorem 3.1, we see that for Φ ∈ S0∞ (RdN )⊗D(H b ), fρ~ (V, κ)U −1 (κ)Φ. Hρ~ REN (V, κ)Φ = U(κ)H

(3.11)

By a limiting argument we can extend (3.11) to Φ ∈ D(H0 ). Since D(H0 ) is a fρ~ (V, κ) and U(κ) maps D(H0 ) onto itself, the right-hand side of (3.11) is core of H essentially self-adjoint on D(H0 ). So is the left-hand side of (3.11). Hence (3.10) can be easily shown. We want to consider a scaling limit of Hρ~ REN (V, κ) as κ → ∞. In [5], a general theory of the strong resolvent limit of self-adjoint operators including abstract fρ~ (V, κ) has been established. We shall versions like the self-adjoint operator H apply the theory in [5] with a little modification. Let V satisfy (V-1). Then b since D(C(V )) ⊃ D(−∆)⊗D(H b ), one can define, for Φ ∈ F(W) and Ψ ∈ D(Hb ), a symmetric operator EΦ,Ψ (C(V )) with D(EΦ,Ψ (C(V )) = D(−∆) by hf, EΦ,Ψ (C(V ))giL2 (RdN ) = hf ⊗ Φ, C(V )(g ⊗ Ψ)iF , f ∈ L2 (RdN ), g ∈ D(−∆). In particular, we call EΩ,Ω (C(V )) ≡ EΩ (C(V )) “the partial expectation of C(V ) with respect to Ω” [5, Sec. II]. Theorem 3.6. Let ρ ~ ∈ Pe . Suppose that V satisfies the conditions (V-1) and (V-2). Then for z ∈ C \ R or z < 0 with |z| sufficiently large, s − lim (Hρ~ REN (V, κ) − z)−1 κ→∞

−1

= U(∞) {(E ∞ (D) +EΩ (C(V )) − z)

o ⊗ P0 U −1 (∞),

(3.12)

where P0 is the projection from F (W) to the one-dimensional subspace {αΩ|α ∈ C}. Proof. By (V-1) and (V-2), we see that e κ)) ⊂ D(Cκ (V )) and for λ > 0, (V-1)0 For sufficiently large κ > 0, D(E(D, −1 e Cκ (V )(E(D, κ) + λ) is bounded with e κ) + λ)−1 || = 0, lim ||Cκ (V )(E(D,

λ→∞

where the convergence is uniform in sufficiently large κ > 0.

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

219

e (V-2)0 For λ > 0, Cκ (V )(E(D, κ) + λ)−1 is strongly continuous in κ and e κ) + λ)−1 = C(V )(E ∞ (D) + λ)−1 . s − lim Cκ (V )(E(D, κ→∞

From [5, Sec. II], (V-1)0 and (V-2)0 imply that −1 fρ~ (V, κ) − z = (E ∞ (D) + EΩ (C(V )) − z)−1 ⊗ P0 . s − lim H κ→∞

Thus by Theorems 3.4 and 3.5, we get (3.12). e∞

e ∞ )1≤i,j≤d , (∆ ij ∞

We want to see EΩ (C(V )) more explicitly. For ρ~ ∈ Pe, let ∆ = e ∞ is defined in (1.3). Let Id×d denote d × d-identity matrix. Since ∆ ≡ where ∆ ij e ∞ ⊗ Id×d ∈ M (N ) ⊗ N (d) ∼ ∆ = M (dN ) is a nonnegative symmetric matrix, there exist unitary matrices T ∈ M (dN ) so that   λ1 Id×d   λ2 Id×d   (3.13) T∆∞ T−1 =  , ..   . λN Id×d where λ1 ≥ λ2 · · · ≥ λN ≥ 0. Theorem 3.7. Suppose λ1 ≥ λ2 · · · ≥ λK > 0, λK+1 = · · · = λN = 0 and fix a unitary operator T in (3.13). Let x = (x1 , . . . , xN ), xj ∈ Rd , j = 1, . . . , N and V satisfy Z dy1 · · · dyK |V | ◦ T−1 (y1 , . . . , yK , (Tx)K+1 , . . . , (Tx)N ) RdK

PK

exp −

j=1

|(Tx)j − yj |2

2λ1 · · · λK

! < ∞.

(3.14)

Moreover we suppose that the left-hand side of (3.14) is locally bounded. Then the partial expectation EΩ (C(V )) is given by a multiplication operator Veff ; Z d dy1 . . . dyK V ◦ T−1 (y1 , . . . , yK , (Tx)K+1 , . . . , (Tx)N ) Veff (x) = (2πλ1 . . . λK )− 2 dK R ! PK 2 |(Tx) j − yj | j=1 . × exp − 2λ1 . . . λK e ∞ is non-degenerate, Veff is given by In particular, in the case where ∆ Z |x − y|2 ˜ ∞ )− d2 V (y) exp − Veff (x) = (2π det ∆ dy. ˜∞ 2 det ∆ RdN Proof. Suppose V ∈ S(RdN ), which is the set of the rapidly decreasing infinitely continuously differentiable functions on RdN . Then the direct calculation shows that for f, g ∈ L2 (RdN ) hf, EΩ (C(V ))giL2 (RdN ) = hf, Veff giL2 (RdN ) .

(3.15)

220

F. HIROSHIMA

We next consider the case where V is bounded. In this case we can approximate V dN ), such that by a sequence {Vn }∞ n=1 , Vn ∈ S(R ||V − Vn ||∞ → 0 (n → ∞), where || · ||∞ denotes the sup norm. Then we have EΩ (C(Vn )) → EΩ (C(V )) (n → ∞), strongly. Moreover (Vn )eff (x) → Veff (x) for all x ∈ RdN . Thus for f, g ∈ L2 (RdN ), (3.15) follows for such V . Finally, let V satisfy (3.14). Define V (x) |V (x)| ≤ n, Vn = n |V (x)| > n. Hence for f ∈ L2 (RdN ) and g ∈ D(−∆), we have hf, EΩ (C(Vn ))giL2 (RdN ) → hf, EΩ (C(V ))giL2 (RdN ) (n → ∞). On the other hand, since the left-hand side of (3.14) is locally bounded, we can see that for f ∈ C0∞ (RdN ) and g ∈ D(−∆), hf, (Vn )eff giL2 (RdN ) → hf, Veff giL2 (RdN ) (n → ∞),

which completes the proof.

e ∞ is non-degenerate, since the Remark 3.8. In Theorem 3.7, in the case where ∆ left-hand side of (3.14) is continuous in x ∈ RdN , it is necessarily locally bounded. We call Veff “the effective potential with respect to V ”. We give some examples of scalar potentials V and ultraviolet cut-off functions ρ. Example 3.9. [non-degenerate case] Let 1d−1 e∞ ∆ ij = δij 2 d

~ mc

2

e2 ~c

Z dk Rd

ρˆi (k)2 . ω(k)3

Then there exist positive constants δ1 and δ2 such that for sufficiently large κ > 0 e κ) ≤ δ2 |p|2 . δ1 |p|2 ≤ E(p,

(3.16)

Let d = 3 and V be the Coulomb potential V (x1 , . . . , xN ) = −

N N X X αj βij + , αj ≥ 0, βij ≥ 0. |x | |x − xj | j i j=1 i6=j

Then V is the Kato class potential ([10], Theorem X.16). Namely for any > 0, there exists b ≥ 0 such that D(V ) ⊃ D(−∆) and ||V Φ||L2 (R3N ) ≤ || − ∆Φ||L2 (R3N ) + b||Φ||L2 (R3N ) .

(3.17)

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

221

Together with (3.16) and (3.17), one can see that V satisfies (V-1), (V-2) and for any t > 0 Z 2 |V |(y)e−t|x−y| dy < ∞. R3N

Then the scaling limit of the Pauli–Fierz Hamiltonian with the Coulomb potential exists and has the effective potential given by Z |x−y|2 − 32 V (y)e− 2γ dy, Veff (x) = (2πγ) R3N

( 2 2 )N ~ e 1 ΠN γ= j=1 3 mc ~c

Z

ρˆ2j (k) dk ω(k)3 R3

! .

Moreover E ∞ (D) = −

1 2m

e2 2mc2

2 2 X N

ρˆj 4 2

~2 ∆j , , 3 j=1 ω L2 (R3 )

∆ ⊗I · · · ⊗ I, ∆ is the Laplacian in L2 (R3 ). where ∆j = I ⊗ · · · I ⊗ |{z} the j-th

e ∞ be non-degenerate and V be Example 3.10. [non-degenerate case] Let ∆ the Phillips perturbation with respect to −∆[12]. Then (3.16) holds with some δ1 and δ2 . Hence V satisfies (V-1), (V-2) and for any t > 0 Z 2 |V |(y)e−t|x−y| dy < ∞. RdN

Hence the scaling limit of the Pauli–Fierz Hamiltonian with Phillips perturbation exists and has the effective potential in Theorem 3.7. Example 3.11. [degenerate case] Let V be a real-valued bounded function. Then V satisfies the conditions (V-1) and (V-2). Hence the scaling limit of the Pauli–Fierz Hamiltonian with the scalar potential V exists for all ρ ~ ∈ Pe . Example 3.12. [degenerate case] Let ρi = ρ, i = 1, . . . , N and V satisfy e ∞ = 1 and (V-1), (V-2) and the assumption stated in Theorem 3.7. Then rank ∆ the non-zero eigenvalue C is given by 2 2 Z ~ ρˆ(k)2 e N d−1 dk . C= 2 d mc ~c Rd ω(k)3 Thus the scaling limit of the Pauli–Fierz Hamiltonian with the ultraviolet cut-off function ρ exists and has the following effective potential: Z d |(Tx)1 − y1 |2 dy1 V ◦ T−1 (y1 , (Tx)2 , . . . , (Tx)N ) exp − . Veff (x) = (2πC)− 2 2C Rd Moreover N E ∞ (D) = − 2m

e2 2mc2

2

d−1 d

 2 2 4 d N X X

ρˆ 2

 Dµj  .

ω 2 3 ~ L (R ) µ=1 j=1

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

223

(2) For f ∈ M−1 , D

ω − 2 Qi , ω − 2 dνα Tµν f 3

1

E L2 (Rd )

D E 3 1 = dµα ω − 2 Qi , ω − 2 f

L2 (Rd )

+

N D X

b µν ρˆj f dνα ω − 2 Qi , λω − 2 Qj Gd 3

1

j=1

E L2 (Rd )

= I + II Using (2.8), one can see that N Z X Qi (k)Qj (k)dµν (k 0 )dνα (k)ˆ ρj (k 0 )f (k 0 ) dkdk 0 II = lim λ 2 − k 02 + it)k 2 t→0 (k j=1 √ √ d Z PN N −1 −1 X ρk ( s)ˆ ρl ( s)s 2 −1 Fj (k 0 ) d−1 k,l=1 D−ik (s)D+jl (s)ˆ Vd dsdk 0 = lim λ 02 + it)s t→0 d (s − k j=1

1 X t→0 2πi j=1 N

Z

= lim

1 (s −

k 02

+ it)s

−1 −1 D−ij (s) − D+ji (s) Fj (k 0 )dsdk 0 ,

ρj (k 0 )f (k 0 ). Using the contour integral on the cut plane where Fj (k 0 ) = dµα (k 0 )ˆ CR,δ, (Fig. 1), by (2.5) and (2.6), we have Z

∞

1 −1 −1 D−ij (s) − D+ji (s) ds 02 + it)s (s − k 0 Z ∞ 1 1 −1 −1 D−ij (s) − D+ji (s) ds = lim 02 →0 2πi (s − k + it)s Z −1 −1 Dij Dij (z) (0) 1 dz − lim − = lim →0 R→∞,δ→0 2πi CR,δ, (z − k 02 + it)z −k 02 + it

1 2πi

=−

−1 02 −1 Dij (k − it) (0) Dij − . 02 02 k − it −k + it

Then N Z −1 −1 02 X (0) Dij (k − it)Fj (k 0 ) 0 Fj (k 0 )Dij − dk 02 t→0 k − it k 02 − it j=1

II = lim

D E 3 1 = − dµα ω − 2 Qi , ω − 2 f

L2 (Rd )

*

+

Hence we get (2). (3), (4) They are direct calculations.

dµα ω

− 32

N X j=1

+

1 −1 Dij (0)ˆ ρj , ω − 2 f

. L2 (Rd )

224

F. HIROSHIMA

(5) For f, g ∈ M0 , ∗ dνα Tαβ f, giL2 (Rd ) = hdµβ f, giL2 (Rd ) + λ hTµν

N X b µν ρˆi giL2 (Rd ) hdνβ f, Qi Gd j=1

+λ

N X

b αβ ρˆj f, giL2 (Rd ) hdµα Qj Gd

j=1

+ λ2

N d−1 X b αβ ρˆi f, Qj Gd b µα ρˆj giL2 (Rd ) hQi Gd d i,j=1

= I + II + III + IV. Then, N X

2

IV = lim λ t→0

i,j,k,l=1

d−1 Vd d

Z

√ √ d −1 −1 (s)D+jl (s)ˆ ρk ( s)ˆ ρl ( s)s 2 −1 Fij (k 0 , k 00 ) D−ik (s − k 02 − it)(s − k 002 + it)

dsdk 0 dk 00 ,

≡ lim IVt , t→0

ρj (k 00 )dαβ (k 0 )ˆ ρi (k 0 )f¯(k 0 )g(k 00 ). By using the cut plane where Fij (k 0 , k 00 ) = dµα (k 00 )ˆ integral method as in (2), we have N Z −1 0 X (k + it)Fij (k 0 , k 00 ) −λDij IVt = k 02 − k 002 + 2it i,j=1 −1 00 (k − it)Fij (k 0 , k 00 ) 0 00 −λDij dk dk k 002 − k 02 − 2it N D E X −1 b 2t dµα ρˆj g = −λ (· + it)ˆ ρi G dαβ f, Dji

+

i,j=1

−λ

N D X

−1 b2t dαβ ρˆi f, g (· + it)ˆ ρj G dµα Dij

L2 (Rd )

E

i,j=1

L2 (Rd )

.

By a limiting argument as t → 0, we get lim IVt = −II − III.

t→0

(6) For f, g ∈ M0 ,

r b es f, giL2 (Rd ) eµ Tµν dνα Tαβ esβ f, g L2 (Rd ) = hδrs f, giL2 (Rd ) − λherβ ρj GQ j β −λhf, esµ ρj GQj erµ giL2 (Rd ) b es f, ρj GQ b er giL2 (Rd ) +λ2 hdµβ ρj GQ j β j µ = I − II − III + IV.

A SCALING LIMIT OF A HAMILTONIAN OF MANY NONRELATIVISTIC PARTICLES. . .

225

We see that IV = λ2 lim

N X

→0

Vd

i,j=1

d−1 d

Z

√ √ d ρˆj ( s)ˆ ρi ( s)s 2 −1 Hij (k 0 , k 00 ) dsdk 0 dk 00 (s − k 02 − i)(s − k 002 + i)

N Z X Dij (k 002 − i) Dij (k 02 + i) + 002 = λ lim Hij (k 0 , k 00 )dk 0 dk 00 02 − k 002 + 2i 02 − 2i →0 k k − k i,j=1 = λ lim

→0

+λ

N X

b 2 Qi er giL2 (Rd ) hf, esµ Qj Dij (· + i)G µ

i,j=1

N X

b 2 Qj esµ f, giL2 (Rd ) , herµ Qi Dij (· − i)G

i,j=1

where Hij (k 0 , k 00 ) = Qj (k 0 )Qi (k 00 )esµ (k 0 )erµ (k 00 )f¯(k 0 )g(k 00 ). Note that = ρˆi . Then =λ

N X

b i er giL2 (Rd ) + λ hf, esµ ρˆi GQ µ

i=1

PN j=1

Qj D+ij

N X b j es f, giL2 (Rd ) herµ ρˆj GQ µ j=1

= II + III. Hence we get the desired results.

References [1] H. A. Bethe, “The electromagnetic shift of energy levels”, Phys. Rev. 72 (1947) 339– 342. [2] T. A. Welton, “Some observable effects of the quantum mechanical fluctuations of the electromagnetic field”, Phys. Rev. 74 (1948) 1157–1167. [3] A. Arai, “A note on scattering theory in non-relativistic quantum electrodynamics”, J. Phys. A. Math. Gen. 16 (1983) 49–70. [4] A. Arai, “Rigorous theory of spectra and radiation for a model in a quantum electrodynamics”, J. Math. Phys. 24 (1983) 1896–1910. [5] A. Arai, “An asymptotic analysis and its applications to the nonrelativistic limit of the Pauli–Fierz and a spin-boson model”, J. Math. Phys. 31 (1990) 2653–2663. [6] F. Hiroshima, “Scaling limit of a model in quantum electrodynamics”, J. Math. Phys. 34 (1993) 4478–4578. [7] F. Hiroshima, “Diamagnetic inequalities for a systems of nonrelativistic particles with a quantized radiation field”, Reviews Math. Phys. 8 (1996), 185–204. [8] F. Hiroshima, “Functional integral representation of a model in quantum electrodynamics”, preprint. [9] E. C. Titchmarsh, Theory of Fourier Integrals, 2nd ed., Oxford Univ.Press, 1948. [10] M. Reed and B. Simon, Method of Modern Mathematical Physics II, Fourier Analysis and Self-Adjoint Operator, Academic Press, 1975. [11] E. A. Berezin, Second Quantization, Academic Press, 1966. [12] E. B. Davies, “Properties of the Green’s functions of some Schr¨ odinger operators”, J. London Math. Soc. 7 (1976) 483–493.

RESONANT DECAY NEAR AN ACCUMULATION POINT CHRISTOPHER KING Department of Mathematics and Center for Interdisciplinary Research on Complex Systems Northeastern University, Boston, Mass. 02115, USA

ROGER WAXLER Department of Mathematics S.U.N.Y. at Buffalo, Buffalo, N. Y. 14214, USA Received 22 January 1996 Revised 25 July 1996 We consider the quantum mechanics of a model system in which meta-stable states arise through perturbation of a sequence of embedded simple eigenvalues with an embedded accumulation point. It is shown that the embedded eigenvalues become resonances in the perturbed system. These resonances also accumulate, and the position of the accumulation point is unchanged. The positions of the resonances are estimated uniformly up to the accumulation point. The meta-stable states associated with these resonances have the usual approximately exponential decay with time. Some applications to physical models are discussed.

1. Introduction Hamiltonians of the form

H(λ) =

A λW ∗

λW B

(1.1)

where λ is a real number, and the operators A, B are self-adjoint, have been extensively studied [2, 4, 7, 9, 10, 17, 18] beginning with the work of Friedrichs on what has become known as the Friedrichs model. In the Friedrichs model B is a rank-one operator with a positive eigenvalue E while A has only absolutely continuous spectrum [0, ∞). These Hamiltonians describe simple quantum mechanical models with resonant behavior. In particular, if the discrete spectrum of the operator B has non-empty intersection with the continuous spectrum of A then the Hamiltonian H(0) has eigenvalues embedded in its continuous spectrum. Typically, embedded eigenvalues are unstable under perturbation and disappear for λ 6= 0. Under reasonable conditions on W , H(λ) is expected to have only absolutely continuous spectrum in a neighborhood of an embedded eigenvalue of H(0). Further, if Ψ is an eigenvector of H(0) corresponding to an embedded eigenvalue then, if λ 6= 0 is small enough, Ψ is expected to behave like a quasi-stable state under the dynamics generated by H(λ). 227 Reviews in Mathematical Physics, Vol. 9, No. 2 (1997) 227–241 c World Scientific Publishing Company

228

C. KING and R. WAXLER

To discuss the perturbation of an embedded eigenvalue E, the notion of a resonance associated with E is often introduced ([19], see [15, 16] and [13] for reviews). To simplify the presentation we restrict our attention to the case in which E is a simple embedded eigenvalue. There are various ways to define a resonance. We will use the following formulation, which seems the most elementary. Let Ψ be an −1 Ψi eigenvector of H(0) corresponding to E. The matrix element hΨ, H(λ) − z is an analytic function of z in the upper half-plane. For λ = 0 the matrix element −1 Ψi is meromorphic in an open disk centered at E with a simple hΨ, H(0) − z −1 Ψi has a meromorphic pole at z = E. Suppose that, for λ 6= 0, hΨ, H(λ) − z continuation to an open disk centered at E, whose only singularity is a simple pole below the real axis. Then this pole is called a resonance. Typically, the position of the pole depends on λ and approaches E as λ → 0. For well-behaved perturbations W and isolated eigenvalues E, this position is accurately estimated by lowest order perturbation theory. Our interest here is in the behavior of resonances near accumulation points, an issue which does not arise in the Friedrichs model. Accumulation points for shape resonances have been obtained [14], however, to the best of our knowledge, there have been no results about accumulation points for resonances produced by perturbing embedded eigenvalues. The so-called Auger states [11, 15] in atomic Hamiltonians (excluding hydrogen), the Hamiltonian for an atom in the quantised radiation field [1, 3, 12], and the Hamiltonian for a hydrogen atom (and presumably larger atoms as well) in a sufficiently strong magnetic field [5, 6] all have this structure, namely accumulation points at a threshold embedded in continuous spectrum. Although our model is considerably simpler than these models, it shares some essential features (see the discussion at the end of this section), and we hope that it can point the way toward an analysis of such problems. For the family of models that we consider in this paper, the operator A has purely continuous spectrum [0, ∞) while the operator B has discrete spectrum with an accumulation point at the number 1 and continuous spectrum [1, ∞). Note that when λ = 0 the operator H(0) has embedded eigenvalues with an accumulation point at 1. Our analysis depends on having sufficient control over W ∗ (A − z)−1W and over the resolvent of B. In order to allow a complete and straightforward analysis we d2 1 make simple choices for A, B and W . We choose B = − dx 2 − x + 1, with Dirichlet boundary conditions at x = 0. The operator B is a simple example of a Schr¨odinger operator with an accumulation point. This operator arises in the spectral analysis of hydrogenin a sufficiently strong magnetic field [5]. Its eigenvalues are well known; they are 1 − 4j12 , j = 1, 2, 3, . . . ; and we have detailed estimates for its discrete and continuum eigenfunctions [5]. In choosing A and W we will require that W ∗ (A − z)−1 W have an analytic continuation, as a bounded operator-valued function, to a neighborhood in the complex plane of the real interval [ 34 , 1]. There are many choices for A and W which satisfy this. We will consider two simple cases which mimic the types of situations d2 which arise in practice. For the first case, we choose A = − dx 2 with Dirichlet

RESONANT DECAY NEAR AN ACCUMULATION POINT

229

boundary conditions at x = 0, and we take W to be a multiplication operator with compact support. In the second case we choose A to be a multiplication operator in L2 (R, dk), and we choose W to act as follows: (W ψ)(k) = f (k)(uψ)∧ (k), where u is a multiplication operator with compact support, f is square integrable, and where “∧” denotes the Fourier transform. The first case models the situation in which the decay of the meta-stable atomic state is accompanied by the emission of an electron from the atom. Such is the case with the Auger states in atoms larger than hydrogen [11, 13, 15] as well as with the unstable higher Landau level states for atoms in a strong enough magnetic field [6]. The second case models the situation in which the decay of a single electron atom from an excited state to its ground state is accompanied by the emission of one photon (assume infinite nuclear mass and no spin, and use an ultraviolet cutoff). We can truncate the state space to the direct sum L2 (R3 ) ⊕ F1 , where the first factor L2 (R3 ) is the electron state space with no photons, and the second factor F1 is the one-photon state space with the electron in its ground state. If we use a function with compact support to approximate the ground state, and ignore the degeneracy of L = 1 states, then we get a Hamiltonian H(λ) of the second form described above, with A = |k| and a slightly modified B. Our techniques rely heavily on the explicit form of the operator H(λ). Nevertheless, just as the Friedrichs model has turned out to be a good caricature of what happens when an isolated embedded simple eigenvalue is perturbed, we hope that our model is a good caricature of what happens when embedded thresholds with embedded accumulation points are perturbed. Briefly, our results are as follows. For |λ| > 0 sufficiently small, each embedded eigenvalue of H(0) turns into a resonance of H(λ) in the sense discussed above. The position of each resonance agrees to lowest order with the predictions of perturbation theory. Furthermore, the resonances approach the accumulation point of the embedded spectrum in a regular way. Namely, in our model the embedded eigenval- ues EJ = 1 − 4J1 2 of H(0) approach 1 as J → ∞, in such a way that J 3 EJ+1 − EJ approaches a nonzero limit. The resonances zJ also approach 1 as J → ∞, and in such a way that J 3 zJ − EJ approaches a nonzero limit as J → ∞. The paper is organised as follows. In Sec. 2 we define the model, explain our goals more precisely and state our results. In Sec. 3 we explore the difficulties presented by the embedded accumulation point, and show how to overcome them in our model. In Sec. 4 we consider resonances near the embedded accumulation point, and prove that they behave in the regular way described above. 2. Definition of the Model and Statement of Results The models we consider are defined by Hamiltonians of the form (1.1), where A is self-adjoint, λ is a real number, and W is bounded. We take B = h0 + 1, where h0 is the differential operator h0 = −

d2 1 − , dx2 x

(2.1)

230

C. KING and R. WAXLER

acting in the state space L2 [0, ∞) . H(λ) is defined on the domain appropriate to Dirichlet boundary conditions at 0, namely φ D H(λ) = ψ(0) = 0, φ ∈ D A and ψ ∈ D(h0 ) ψ (here D(A) is the domain of the operator A). That H(λ) is self-adjoint follows immediately from the self-adjointness of the Dirichlet problem for h0 on the halfline, from which it follows that H(0) is self-adjoint, and from the fact that W is bounded, from which it then follows that H(λ) − H(0) is bounded. The Hamiltonian h0 can be related to the 0 angular momentum sector of the Hydrogen atom Hamiltonian, see [11]; its spectrum is o n 1 σ(h0 ) = − 2 j ∈ {1, 2, 3, . . . } ∪ [0, ∞). 4j The eigenvalues − 4j12 are simple. For each j ∈ {1, 2, 3, . . . } we define ψj to be the normalised eigenfunction of h0 with eigenvalue − 4j12 . In Sec. 3 we will write down the explicit expressions for the eigenfunctions ψj , as well as for the continuum eigenfunctions. If we assume that the operator A has spectrum [0, ∞) then it follows that the operator H(0) has spectrum [0, ∞), and has eigenvalues Ej = 1 −

1 4j 2

(2.2)

embedded in the continuous spectrum. Let 0 Ψj = ψj be the corresponding eigenvectors. Note that the spectrum of H(0) has an embedded accumulation point at 1. Our main goal is to analyse resonances of H(λ) associated to the embedded eigenvalues of H(0). Define, for each J ∈ {1, 2, 3, . . . } and for Im z > 0 RJ (λ; z) = hΨJ , H(λ) − z Let

n DJ = z |z − EJ | ≤

−1

ΨJ i.

(2.3)

1 o . (2.4) 16J 3 Note that RJ (0; z) has a meromorphic extension to DJ whose only singularity is a simple pole at z = EJ . We will find a meromorphic continuation of RJ (λ; z) across the positive real half-line for λ 6= 0. The only singularity of this extension is a simple pole whose position we estimate. Explicitly, there is a λ0 > 0, such that for all λ with |λ| < λ0 and for each J, the function RJ (λ; z) has a meromorphic extension to DJ whose only singularity is a simple pole. Let zJ be the position of this pole in DJ . Then Im zJ < 0 and zJ → EJ as λ → 0. We refer to zJ as the perturbative resonance of H(λ) associated to EJ . Our results are enough to easily establish approximate exponential decay in time for hΨJ , e−itH(λ) ΨJ i and

231

RESONANT DECAY NEAR AN ACCUMULATION POINT

to identify the lifetime of the meta-stable state ΨJ with the inverse of the imaginary part of the resonance zJ [8, 19]. We emphasize that our results are uniform in J. Before stating our results we will need some definitions. For Im z 6= 0, we define V (z) = W ∗ (A − z)−1 W acting on L2 [0, ∞) , and we write V (z; x, y) for its integral kernel. the function √ √ ϕ(x) = x J1 (2 x)

(2.5) Further, (2.6)

for x ≥ 0 plays an important role in our analysis. Here J1 is the Bessel function [20]. Note that ϕ is the zero energy continuum eigenfunction for h0 . In particular h0 ϕ = 0 as a differential equation and ϕ(0) = 0. We also define (with abuse of the notation for the inner product in L2 ) Z ∞Z ∞ ϕ(x)V (z; x, y)ϕ(y) dx dy, (2.7) hϕ, V (z)ϕi = 0

0

In all our examples, V (z; x, y) will have compact support, so (2.7) exists. Results. Assume that V (z) has a continuation from the upper half-plane which is an analytic bounded operator-valued function of z in the strip 12 < Re z < 32 , | Im z| < 1. Assume further that its integral kernel V (z; x, y) has compact support. Then there is a λ0 > 0 such that for all λ ∈ (−λ0 , λ0 ) and for all J ∈ {1, 2, 3, . . . } the function RJ (λ; z) has a meromorphic extension to DJ whose only singularity is a simple pole at z = zJ . Define γJ ∈ C by γJ (2.8) zJ = EJ + 3 . J It is immediate that Im γJ ≤ 0, since H(λ) is self adjoint, and that |γJ | ≤ z J ∈ DJ . For all J, γJ satisfies the perturbative estimate

1 16 ,

since

|γJ + λ2 J 3 hψJ , V (EJ )ψJ i| ≤ const λ4 . Further, J 3 hψJ , V (EJ )ψJ i =

1 1 hϕ, V (1)ϕi + O . 2 J2

If we then make the additional assumption Imhϕ, V (1)ϕi = 6 0

(2.9)

then there is a J0 > 0 such that for J > J0 , Im γJ < 0. In addition {γJ } converges as J → ∞. Despite the eigenvalue accumulation, for all λ ∈ (−λ0 , λ0 ) and J > J0 we get the expected approximate exponential decay away from the initial condition ΨJ : |hΨJ , e−itH(λ) ΨJ i − e−itzJ | ≤ const λ2 .

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This follows from Stone’s formula (recall that the ψj are real), Z 1 −itH(λ) ΨJ i = lim e−it Im RJ (λ; + iη) d. hΨJ , e π η↓0

(2.10)

and the existence of the resonant pole, zJ , precisely as in [8]. As discussed in the introduction, we have in mind two classes of examples to which our results apply. In the first case, the state space is L2 [0, ∞) ⊕ L2 [0, ∞) , d2 and A = − dx 2 with Dirichlet boundary conditions at 0. The interaction operator W is multiplication by the real-valued function u(x), which we assume to be continuous √ with compact support in [0, ∞). We write −z for the branch of the function which √ is analytic off the negative real axis, and satisfies −1 = +i. A straightforward calculation shows that for this example, for Im z > 0, √ √ 1 e− −z |x−y| − e− −z (x+y) u(y). V (z; x, y) = u(x) √ 2 −z

(2.11)

It is easy to check that V (z) satisfies the compactness and analyticity assumptions stated above. We note that, since u has compact support, H(λ) − H(0) is not only bounded but is also a relatively compact perturbation of H(0) for all λ. It follows that the essential spectrum of H(λ) is equal to that of H(0), namely σess H(λ) = [0, ∞). In the second case the state space is L2 (−∞, ∞), dk ⊕ L2 [0, ∞) , and A is multiplication by the non-negative real valued function ω(k). The interaction operator W is a bounded map from L2 [0, ∞) to L2 (−∞, ∞), dk , and acts on a state ψ as follows: Z ∞ eikx u(x)ψ(x) dx, (2.12) (W ψ)(k) = f (k) 0

where the function u(x) is real valued and continuous with compact support in [0, ∞). We assume that f is square integrable, which implies that H(λ) is selfadjoint. The integral kernel of the corresponding operator V (z) is Z |f (k)|2 ik(x−y) e dk u(y). (2.13) V (z; x, y) = u(x) ω(k) − z It is easy to see that if both f (k) and ω(k) are analytic in a neighborhood of [ 12 , 32 ], then by deforming the contour of integration in (2.13) we obtain the analytic continuation of V (z) to a strip containing the embedded eigenvalues, and that its kernel has compact support. If this strip is given as 12 < Re z < 32 , | Im z| < b for some b > 0, then our results apply to the embedded eigenvalues EJ with 16J 3 > 1b . 3. Perturbation Theory In this section we recall the standard method for studying resonances using analytic perturbation theory. We point out why the standard method has a difficulty in this model, due to the eigenvalue accumulation, and we show how this

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RESONANT DECAY NEAR AN ACCUMULATION POINT

difficulty is overcome. We will have several occasions to use the following observation from linear algebra. Let V1 and V2 be vector spaces and let M f L= e a be a linear operator on V1 ⊕ V2 . Here M : V1 → V1 , f : V2 → V1 , e : V1 → V2 and a : V2 → V2 . Let M be invertible. Then L is invertible iff α = a − eM −1 f is invertible. If so then L

−1

=

M −1 + M −1 f α−1 eM −1 −α−1 eM −1

−M −1 f α−1 α−1

.

(3.1)

Now let z ∈ DJ . If Im z > 0 then A − z is invertible and, setting M = A − z, we can apply (3.1) to (1.1) and (2.3). Thus RJ (λ; z) can be rewritten −1 ψJ i. RJ (λ; z) = hψJ , h0 + 1 − z − λ2 V (z)

(3.2)

h(z) = h0 + 1 − z − λ2 V (z)

(3.3)

Define of h0 with eigenand introduce the orthogonal projection ΠJ onto the eigenspace 2 ⊥ = 1 − Π we have L [0, ∞) = RanΠ value − 4J1 2 . Writing Π⊥ J J J ⊕ RanΠJ . In 2 ⊥ general, given an operator C in L [0, ∞) for which the restriction Π⊥ J CΠJ is in −n ⊥ 2 vertible on RanΠJ , we will denote by C⊥,J (for n = 1, 2) the operator on L [0, ∞) ⊥ −n on RanΠ⊥ which equals (Π⊥ J CΠJ ) J , equals zero on RanΠJ , and leaves invariant ⊥ the subspaces RanΠJ and RanΠJ . −1 ⊥ ⊥ We will show that Π⊥ J h(z)ΠJ is invertible on RanΠJ and that h(z)⊥,J is an analytic bounded operator-valued function of z, for z ∈ DJ . Thus, another application of (3.1), now with M −1 = h(z)−1 ⊥,J , will show that RJ (λ; z) is analytic in z unless EJ − z − λ2 hψJ , V (z)ψJ i − λ4 hψJ , V (z)h(z)−1 ⊥,J V (z)ψJ i = 0.

(3.4)

If (3.4) is not satisfied then RJ (λ; z) =

1 . hψJ , h(z)ψJ i − λ4 hψJ , V (z)h(z)−1 ⊥,J V (z)ψJ i

(3.5)

If (3.4) is satisfied for some z = zJ then it is immediate that |zJ − EJ + λ2 hψJ , V (z)ψJ i| ≤ const λ4 ,

(3.6)

where const may depend on J. It follows from the self-adjointness of H that (3.4) has no solutions if Imz > 0. Solutions with Imz < 0 are called resonances for H while real solutions are embedded eigenvalues.

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In studying h(z)−1 ⊥,J and (3.4) the chief difficulty to be overcome is that of obtaining estimates which are uniform in J. As J increases the unperturbed eigenvalues EJ accumulate. For large J we have EJ+1 − EJ =

1 1 + O( 4 ). 3 2J J

As a consequence, for z ∈ DJ , 3 k(h0 + 1 − z)−1 ⊥,J k ∼ 2J

(3.7)

2 ⊥ and for a generic bounded perturbation U , Π⊥ J (h0 + 1 − z + λ U )ΠJ is not invertible 2 3 for all z ∈ DJ unless λ kU kJ is small enough. This would require that λ2 ∼ J −3 as J → ∞, which is not the case of interest here. It was noticed in [5] that this difficulty can be overcome if the perturbation U has an integral kernel U (x, y) which approaches zero rapidly enough as either x or y goes to infinity. In the cases at hand U = V (z). As pointed out in the preceding section, because of our assumption that the function u(x) has compact support, V (z; x, y) has compact support in x and y. Let χ be the multiplication operator corresponding to the function 1 ifx ∈ supp(u) (3.8) χ(x) = 0 ifx ∈ / supp(u).

Note that, for each J ∈ {1, 2, 3, . . . }, the operator χ(h0 + 1 − z)−1 ⊥,J χ is an analytic bounded operator-valued function of z, for z ∈ DJ . We will show that χ(h0 + 1 − z)−1 ⊥,J χ is also bounded uniformly in J, thus overcoming the difficulty expressed by (3.7). As in [5] we introduce the eigenfunction expansion for h0 . The discrete eigenfunctions of h0 are [[5], Appendix B] 1 1 3 x ψj (x) = 2− 2 j − 2 xe− 2j x M 1 − j, 2, j while the continuum eigenfunctions of h0 are π i 1 ψp (x) = 1 − e− |p| )− 2 xe−ipx M (1 + , 2, 2ipx . 2p Here M (a, b, z) is the confluent hypergeometric function [20], j ∈ {1, 2, 3, . . . } and the functions ψp (x) are normalized with respect to the measure 2pdp on [0, ∞). It follows from the identity M (a, b, z) = ez M (b − a, b, −z) [20, Eq. 13.1.27] that the functions ψp (x) are real valued. Both ψj and ψp are solutions to the differential equation d2 1 + δ ψ(x) = 0; + dx2 x for ψj we set δ = − 4j12 , for ψp we set δ = p2 . It is easily checked that the following limits hold pointwise: 1

3

lim 2 2 j 2 ψj (x) = lim ψp (x) = ϕ(x).

j→∞

p→0

RESONANT DECAY NEAR AN ACCUMULATION POINT

235

Here ϕ(x) is the zero energy eigenfunction of h0 given by (2.6). We will need some information about the rates of convergence of these limits. The following estimates can be read off from the results of Appendix B of [5]. For x in any bounded subset of [0, ∞) 1 3 7 (3.9) |ψj (x) − 2− 2 j − 2 ϕ(x)| ≤ const j − 2 and |ψp (x) − ϕ(x)| ≤ const p2 .

(3.10)

Lemma 3.1. There is a real number K < ∞ such that for all J ∈ {1, 2, 3, . . . } and for all z ∈ DJ kχ(h0 + 1 − z)−1 (3.11) ⊥,J χk ≤ K and 3 kχ(h0 + 1 − z)−2 ⊥,J χk ≤ K J .

(3.12)

Proof. Let θ1 , θ2 ∈ L2 [0, ∞) , fix J ∈ {1, 2, 3, . . . } and let P1 be the orthogonal projection onto the spectral subspace of h0 contained in [−1, 1]. Then, for all z ∈ DJ , −1 hθ1 , χ(h0 + 1 − z)−1 ⊥,J χθ2 i = hθ1 , χ(h0 + 1 − z)⊥,J P1 χθ2 i

+hθ1 , χ(h0 + 1 − z)−1 ⊥,J (1 − P1 )χθ2 i.

(3.13)

Since k(h0 + 1 − z)−1 ⊥,J (1 − P1 )k ≤ 1 the second term in the expression (3.13) is analytic in z and |hθ1 , χ(h0 + 1 − z)−1 ⊥,J (1 − P1 )χθ2 i| ≤ kθ1 kkθ2 k uniformly for z ∈ DJ . Using the eigenfunction expansion for h0 the first term in (3.13) can be rewritten as follows: hθ1 , χ(h0 + 1 − z)−1 ⊥,J P1 χθ2 i =

X

1 hθ1 , χψj ihχψj , θ2 i Ej − z

j6=J

Z + 0

1

p2

1 hθ1 , χψp ihχψp , θ2 i2pdp. (3.14) +1−z

We now apply the estimates (3.9) and (3.10). We have, for σ ∈ {1, 2}, |hθσ , χψj i − 2− 2 j − 2 hθσ , χϕi| ≤ const j − 2 kθσ k 1

3

7

and |hθσ , χψp i − hθσ , χϕi| ≤ const p2 kθσ k.

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C. KING and R. WAXLER

1 X j −r −2 ln J + O(1) = O(1) 2 Ej − z

The estimates

j6=J

and

Z

1

0

pr 2pdp = p2 + 1 − z

if r = 3 if r > 3

2 ln J + O(1) O(1)

if r = 0 if r > 0

(3.15)

(3.16)

were obtained in Lemma 5.4 of [5] for z ∈ DJ ∩ R and extend easily to all z ∈ DJ (we return to the case r = 3 in the next section where sharper estimates are needed). It follows from these estimates that |hθ1 , χ(h0 + 1 − z)−1 ⊥,J P1 χθ2 i| ≤ const kθ1 kkθ2 k. The bound (3.11) follows. Proving (3.12) is similar. The only difference is that we use the estimates (again as in Lemma 5.4 of [5]) 3−r 1 X j −3−r J if r < 3 ≤ const (3.17) 1 if r ≥ 3 2 |Ej − z|2 j6=J

and

Z 0

1

 2−r J p 2pdp ≤ const ln J  |p2 + 1 − z|2 1 r

if r < 2 if r = 2 if r > 2

(3.18)

for all z ∈ DJ .

⊥ ⊥ An immediate consequence of (3.11) is the invertibility of Π⊥ J h(z)ΠJ on RanΠJ for all z ∈ DJ . If λ is small enough the expansion −1 h(z)−1 ⊥,J = (h0 + 1 − z)⊥,J

∞ N X λ2 V (z)(h0 + 1 − z)−1 ⊥,J

(3.19)

N =0

converges uniformly on DJ , since it can be written as an expansion in powers of V (z) times the operator (3.11). It follows that h(z)−1 ⊥,J is an analytic bounded operator-valued function of z in DJ , that kχh(z)−1 ⊥,J χk ≤ const

(3.20)

3 kχh(z)−2 ⊥,J χk ≤ const J .

(3.21)

and, using (3.12), that Thus (3.5) provides an analytic continuation of RJ (λ; z) across the real line to the set {z ∈ DJ | z is not a solution of (3.4)}. To see that (3.4) has a unique solution and that this solution is a simple zero note that, using (3.9), d hψJ , V (z)ψJ i = |hψJ , V 0 (z)ψJ i| dz ≤ const J −3

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RESONANT DECAY NEAR AN ACCUMULATION POINT

and, using (3.20), (3.21) and (3.9) again, that d hψJ , V (z)h(z)−1 V (z)ψJ i ≤ |hψJ , V 0 (z)h(z)−1 V (z)ψJ i| ⊥,J ⊥,J dz −1 0 + |hψJ , V (z)h(z)−1 ⊥,J h (z)h(z)⊥,J V (z)ψJ i| 0 + |hψJ , V (z)h(z)−1 ⊥,J V (z) ψJ i|

≤ const . Now apply the contraction mapping theorem. Let zJ be the unique solution of (3.4) in DJ . It follows from the comments above that RJ (λ; z) is meromorphic in z for z ∈ DJ with a simple pole at z = zJ . Further, the estimate (3.6) is uniform in J. To estimate the large J behavior of zJ we can use (3.9) and (3.10) in (3.4). Recalling the definitions (2.2) of EJ and (2.8) of γJ we have hψJ , V (EJ +

γJ J 3 )ψJ i

1 γJ 1 hϕ, V (EJ + 3 )ϕi + O( 5 ) 2J 3 J J 1 1 hϕ, V (1)ϕi + O( 5 ). = 3 2J J =

Further, using (3.20), |hψJ , V (EJ + Thus

γJ γJ γJ 1 )h(EJ + 3 )−1 V (EJ + 3 )ψJ i| ≤ const 3 . J3 J ⊥,J J J

1 1 γJ = − λ2 hϕ, V (1)ϕi + O( 2 ) + O(λ4 ). 2 J

(3.22)

4. The Large J Limit In this section we examine the position of the resonant pole zJ as J → ∞ and show that J 3 (zJ − EJ ) has a limit. This pole is given by the unique solution of (3.4) in the disk DJ . Recall the notation introduced in (2.8), namely zJ = EJ + γJ J −3 . Let us write γ zJ (γ) = EJ + 3 J 1 for |γ| ≤ 16 , and define the following function of γ and J: GJ (γ) = −λ2 J 3 hψJ , V (zJ (γ))ψJ i − λ4 J 3 hψJ , V (zJ (γ))h(zJ (γ))−1 ⊥,J V (zJ (γ))ψJ i. (4.1) It follows from (3.4) that γJ is the unique fixed point of the function GJ in the disk n 1o . D0 = γ |γ| ≤ 16 Proposition 4.1. The sequence γJ converges in the disk D0 as J → ∞. Proof. The number γJ is the solution of the equation γJ = GJ (γJ ).

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Recall that GJ is a contraction on D0 for |λ| sufficiently small. Some elementary estimates now show that to prove convergence of the sequence {γJ }, it is sufficient to prove that the sequence {GJ (γ)} converges uniformly for γ ∈ D0 . Using the estimates (3.9) and (3.10), and Lemma 3.1, this reduces to showing convergence −1 of hϕ, V (1)h zJ (γ) ⊥,J V (1)ϕi uniformly in γ. Using the fact that u has compact support, and recalling (3.8), it is sufficient to establish uniform convergence of the se−1 quence of bounded operators χh zJ (γ) ⊥,J χ. Further, by the results of Lemma 3.1, and using the convergent expansion (3.19), it suffices to prove uniform convergence of χ(h0 + 4J1 2 − Jγ3 )−1 ⊥,J χ. As in the proof of Lemma 3.1 we may restrict to the spectral subspace of h0 with spectrum less than 1. Again let P1 be the orthogonal projection on this spectral subspace of h0 . On the orthogonal complement to this subspace the operator (h0 + γ −1 1 4J 2 − J 3 )⊥,J is analytic in γ and uniformly bounded by 1 on D0 . In fact we have the following equality of norm limits: 1 γ −1 = lim (1 − P1 )(h0 + iη)−1 . lim (1 − P1 ) h0 + 2 − 3 J→∞ 4J J ⊥,J η→0

(4.2)

Again using the eigenfunction expansion for h0 we have Z 1 X 1 γ −1 χψj ⊗ ψj χ χψp ⊗ ψp χ χ= + 2pdp. χP1 h0 + 2 − 3 1 γ 1 γ 1 4J J ⊥,J 0 p2 + j6=J − − − 4J 2 4j 2 J3 4J 2 J3 Using the estimates (3.9) and (3.10) leads us back to (3.14). Explicitly, define (1)

ψj

= ψj − 2− 2 j − 2 ϕ, 1

3

ψp(1) = ψp − ϕ and introduce the sum and integral Z 1 1 X −3 1 1 j + 2pdp. 1 γ 1 γ 1 2 0 p2 + j6=J − − − 4J 2 4j 2 J3 4J 2 J3

(0)

GJ (γ) =

(4.3)

Applying (3.9), (3.10), (3.15) and (3.16) we have the existence of the limit (in norm) lim

J→∞

χP1 (h0 +

∞ X = −4 j 2 j=1

Z +2 0

1

1 γ −1 (0) − ) χ − G (γ) χϕ ⊗ ϕχ J 4J 2 J 3 ⊥,J

1 1 (1) (1) (1) (1) p χψj ⊗ ϕχ + p χϕ ⊗ ψj χ + χψj ⊗ ψj χ 3 2j 2j 3

1 χψp(1) ⊗ ϕχ + χϕ ⊗ ψp(1) χ + χψp(1) ⊗ ψp(1) χ dp. p

!

(4.4)

Note that convergence is uniform in γ, and that the limit is independent of γ. The (0) only possible γ dependence in the large J limit is from the term GJ (γ).

239

RESONANT DECAY NEAR AN ACCUMULATION POINT

(0)

We now show that GJ (γ) has a limit as J → ∞. It is easy to show that the integral in (4.3) satisfies the estimate Z

1

0

1 1 . 2pdp = 2 ln J + ln 4 + O 1 γ J p2 + 2 − 3 4J J

To estimate the sum in (4.3) we introduce the number γ − 12 α= 1− J and rewrite the sum as follows: 1 2

1 X 1 γ −1 1 − − = 2α2 J 2 2 2 3 2 4J 4j J j(j − α2 J 2 ) j6=J j6=J X 2 1 1 + − + = j j + αJ j − αJ X

j −3

j6=J

=

1 2 − J (1 + α)J K K X X 1 X 1 1 + + . + lim − 2 K→∞ j j=1 j + αJ j − αJ 1≤j≤K j=1 j6=J

For |z| ≤ 12 , and N ≥ 1 define φz (N ) = Note that lim

N →∞

N X

1 . n + z n=1

∞ X φz (N ) − ln N = C − z

1 n(n + z) n=1

where C is Euler’s constant, and that convergence is uniform in z. Further, let = (α − 1)J; it is easy to see that || ≤ |γ| for all J ≥ 1, and that → then have K X 1 j=1 K X

j

= φ0 (K),

1 = j + αJ j=1

K+J+1 X j=J+1

1 j+

= φ (K + J + 1) − φ (J)

γ 2

as J → ∞. We

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C. KING and R. WAXLER

and (for K > J) X 1 1 = j − αJ (j − J) − 1≤j≤K 1≤j≤K X j6=J

j6=J

=

1 − φ (J) + φ− (K − J). J +

We thus arrive at the following identity: 1 1 1 γ −1 2 1 X −3 1 − − 2φ (J) j − 2− 3 = + 2 2 4J 4j J J J + 2J + j6=J + lim − 2φ0 (K) + φ (J + K + 1) + φ− (K − J) K→∞

∞ X 1 1 1 2 − − 2φ (J) + 22 . = + 2 J J + 2J + n(n − 2 ) n=1

Substituting these expressions for the sum and integral in (4.3), we get the following result: ∞ 1 X 1 (0) − 2 φ . (J) − ln J + ln 4 + O GJ (γ) = 22 n(n2 − 2 ) J n=1 where the remainder O(J −1 ) is uniform in γ. Letting J → ∞ gives (0)

(0)

G∞ (γ) = lim GJ (γ) =

J→∞ ∞ 2X

γ 2

1

n=1 n(n

2

−

γ2 4 )

+γ

∞ X

1 − 2C + ln 4. n(n + γ2 ) n=1

(4.5)

Therefore we conclude that limJ→∞ GJ (γ) exists for every γ ∈ D0 , and that {GJ } converges uniformly in D0 . To approximate γ∞ = lim γJ J→∞

we let

1 γ −1 χ. A(γ) = lim χ h0 + 2 − 3 J→∞ 4J J ⊥,J From (4.2), (4.4) and (4.5) we find that A(γ) exists and is given by the sum of these three terms. Note that ∞ ∞ 2X X 1 1 χϕ ⊗ ϕχ + γ A(γ) − A(0) = γ2 2 γ γ 2 n(n + 2 ) n=1 n(n − 4 ) n=1 =

π2 γ 6

χϕ ⊗ ϕχ + O(γ 3 ).

Recalling (3.19) we have ∞ N X −1 λ2N V (1)A(γ) . χh zJ (γ) ⊥,J χ = A(γ) N =0

RESONANT DECAY NEAR AN ACCUMULATION POINT

241

Substituting in (4.1) it is straightforward to generate an expansion for γ∞ in powers of λ2 . The first few terms are γ∞ = −

λ2 hϕ, V (1)ϕi − λ4 hϕ, V (1)A(0)V (1)ϕi + O(λ6 ). 2

References [1] V. Bach, J. Fr¨ ohlich, and I. M. Sigal, “Mathematical theory of nonrelativistic matter and radiation”, Lett. Math. Phys. 34 (1995) 183–201. [2] E. B. Davies, “Dynamics of a multilevel Wigner–Weisskopf atom”, J. Math. Phy. 15 (1974) 2036–2041. [3] P. A. M. Dirac, Quantum Mechanics, Oxford Univ. Press, 1978. [4] K. O. Friedrichs, Comm. Pure Appl. Math. 1 (1948) 361–406. [5] R. Froese and R. Waxler, “The spectrum of a hydrogen atom in an intense magnetic field”, Reviews in Math. Phys. 6 (1994) 699–832. [6] R. Froese and R. Waxler, “Ground state resonances of a hydrogen atom in an intense magnetic field”, Reviews in Math. Phys. 7 (1995) 311–361. [7] J. Howland, “Spectral concentration and virtual poles, II”, Trans. Amer. Math. Soc. 162 (1971) 141–156; “The Livsic matrix in perturbation theory”, J. Math. Anal. Appl. 50 (1975) 415–437. [8] W. Hunziker, “Resonances, metastable states and exponential decay laws in perturbation theory”, Commun. Math. Phys. 132 (1990) 177–188. [9] C. King, “Exponential decay near resonance, without analyticity”, Lett. Math. Phys. 23 (1991) 215–222. [10] C. King, “Scattering theory for a model of an atom in a quantized field”, Lett. Math. Phys. 25 (1992) 17–28. [11] L. D. Landau and E. M. Lifshitz, Quantum Mechanics 3rd ed., Pergamon, 1977. [12] J. Parker and C. R. Stroud, Jr., “Coherence and decay of Rydberg wave packets”, Phys. Rev. Lett. 56 (1986) 716–719. [13] M. Reed and B. Simon, Methods of Mathematical Physics vol. 2: Fourier Analysis, Self Adjointness, Academic Press, 1975; Methods of Mathematical Physics vol. 4: Analysis of Operators, Academic Press, 1978. [14] M. A. Shubov, “Low-energy chain of resonances for three-dimensional Schr¨ odinger operators with nearly Coulomb potential”, J. Diff. Eq. 114 (1994) 168–198. [15] B. Simon, “Resonances in N-body quantum systems with dilation analytic potentials and the foundations of time-dependent perturbation theory”, Ann. Math. 97 (1973) 247–274. [16] B. Simon, “Resonances and complex scaling: A rigorous overview”, Int. J. Quant. Chem. 14 (1978) 529–542. [17] R. Waxler, “The time evolution of a class of meta-stable states”, Commun. Math. Phys. 172 (1995) 535–549. [18] R. Waxler, “Generalized eigenfunction expansion near resonances”, preprint. [19] V. Weisskopf and E. Wigner, “Berechnung der Nat¨ urlichen Linienbreite auf Grund der Diracshen Lichttheorie”, Z. Phy. 63 (1930) 54–73. [20] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover 1972.

HILBERT SPACE OF SPINOR FIELDS OVER THE FREE LOOP SPACE ´ R. LEANDRE D´ epartement de Math´ ematiques Institut Elie Cartan, Facult´ e des Sciences Universit´ e de Nancy I 54000, Vandoeuvre-les-Nancy France Received 8 January 1996 Revised 4 April 1996

Introduction Let M be a compact oriented manifold of dimension n = 2d. A spin structure over M is a lift of the frame bundle by the spinor group which is a Z/2Z extension of SO(n). There are topological obstruction to construct a spin structure over M [29]. Let us consider the free loop space L(M ): it is the space of smooth applications from the circle S 1 into M . The interest of the free loop space is that infinite dimensional operators over it give, by localization of their index by using the natural circle action over it, topological invariant of the manifold ([59, 63]). In order to work with these operators, we need first of all to define the space of sections of the bundle where they act. These bundles are natural for the de Rham operator over the free loop space or for the Dolbeault operator over the free loop space of a complex manifold. But the situation becomes less simpler if we want to study the Dirac operator: namely topological obstructions to construct the spin bundle over the free loop space appear, which are generally not seen at the level of differential forms but at the level of the Z-cohomology of the free loop space which includes the torsion. Let Q be a principal bundle over M , with compact structure group G. Let us ˜ suppose that G is simply connected and simple. Let L(G) be a central extension of L(G), the free loop space of G [13, 58]. The space of loops of L(Q) appears as a principal bundle over L(M ) with structure group L(G). A spin structure is a lift ˜ of L(Q) by L(G) [40, 63]. The obstruction to construct it is related to the first Pontryaguin class of Q [59, 63, 53, 20] and the most complete way to handle with a construction of this lift is to consider the classifying space of G. But, when G is 2-connected and when M is 2-connected, [20] and [17] present a way to attack this problem by using differential forms over L(M ) and over L(Q). The main ingredient is the notion of transgression of a form over M and over Q, which is a special type of Chen form. The main idea of this differential construction of the string bundle is 243 Reviews in Mathematical Physics, Vol. 9, No. 2 (1997) 243–277 c World Scientific Publishing Company

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the following: there is a 2-closed form over the fiber which allows to construct the central extension. Unfortunately this 2-closed form over the fiber is not a 2-closed form over the total space L(Q). There is a problem to extend it over the total space L(Q) into a 2-closed form with integral values. But when G is 2-connected and Q is 2-connected, we can extend it, modulo a countertem into a 2-integral valued form over L(Q): since L(Q) is simply connected, this allows us to construct a line bundle over L(Q) by hand. There are so many extensions as element of H 2 (LM ; Z) and the obstruction lies in H 3 (L(M ); Z): the construction of [20] or [17] neglects the torsion. Infinite-dimensional operators over the loop space are studied in [39, 52, 45, 47]. They are symmetric, that means there is a measure considered over the free loop space, which is invariant by rotation. It is the B.H.K measure [33, 12]. The stochastic loop are only continuous. The purpose of the infinite-dimensional analysis is to consider an infinite dimensional space with a mesure, to define vector fields suitable in the following sense: there is a divergence associated to them and the involved integration by parts allows us to define by density consistent differential operations. It is in particular well known since [30, 31, 32] that the tangent space is not the natural applicant of the geometry, but a smaller vector space, which is a Hilbert space instead of a Banach space. In our case, the choice of the tangent space is given by [11] or [38]. In particular, [43, 44] and [51] perform the integration by parts over the free loop space which allows one to define by density non-scalar operators which act over the section of infinite-dimensional bundles over the free loop space, when these last ones are defined. For the Dirac operator over the free loop space, the construction of [39] follows closely the choice of [61] of the spin fiber, and is done by hand over a “neighborhood” of the constant loop, which ignores the theory of spin representation of loop groups. The purpose of this paper is to construct some Lp spaces of spinor fields over the whole free loop space. Until now, there is no cohomology theory of the free stochastic loop space, which takes care with the torsion. But there is a beginning of a de Rham theory which works over some Sobolev spaces of forms over the free loop space [38, 46, 47] where the stochastic Chen forms play the key role. The main remark is the following (see [48] and [49]): Chen forms can be integrated over a path in the free loop space, whose derivative is not in the tangent space of the infinite-dimensional analysis of [10] or [38], and allow to consider generalized vector fields, by using the theory of the stochastic integral. This allows in [49], starting from a 2-closed Z valued Chen form over the Brownian bridge, to construct a bundle which is associated: the bundle is defined by its sections; in particular, the bundle has sections, because we construct a line bundle; it is a formal line bundle, because the transition maps are only almost surely defined, because they belong to all the Sobolev spaces, and so are only measurable. In particular, we do not consider the circle bundle associated to this 2-Z-valued Chen form. In the first part of this paper, we study the simplest of the extensions of a fiber which is defined fiberwise to a bigger space. It is the work of [16] over the based

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path space. A given 3-Z-valued closed form gives by transgression a bundle over the set of path Lx,y going from x to y. There is an obstruction to extend it over the whole path space P (M ). When the criterion of the obstruction is satisfied, we construct the set of sections of the formal associated line bundle over P (M ) which belong to all the Sobolev space. The technics of the quasi sure analysis of Getzler– Airault–Malliavin ([27, 3] see [62] for a preliminary work) allow to restrict these sections in sections of a bundle of Lx,y . In particular, we don’t use the positivity theorem of [8] or [2] (see [42] for a version of this theorem adapted to jump processes) in order to show the consistency of these transition functions unlike [49]. We define the circle bundle over P (M ) by defining its space of Lp functionals or some Sobolev spaces attached to it; it is possible because the Haar measure over S 1 is invariant under rotation (see [54]). In the second part, we define over the space of continuous loop, the Lfin (G) bundle of loop in Q over continuous loop in G, Q being a principal bundle over M with structure group G. The choice of the fiber arises from the considerations of Bismut in [10], because Lfin (G) is the set of loops of finite energy in G. We define ˜ fin (G), the basic central over this last one the cocycle which allows to construct L extension of Lfin(G). We extend following [20] this cocycle over Lfin (Q) when the ˜ fin (G) first Pontryaguin class of G is equal to 0. This allows us to define a formal L ˜ bundle over L(M ). The formal fiber is Lfin (M ), but the transition functions are only almost surely defined, because they are constructed from some stochastic integrals. ˜ fin (G), this allows us to define the space of If there is a unitary representation of L p L sections of the associated bundle. The appendix shows, following closely [13], ˜ fin (Spin2d ): this gives the construction that there is a unitary representation of L of the Hilbert space of spinors over the free loop space of a spin manifold. Let us remark that we did not choose the spin representation of [15] or [26], which is not unitary. In the third part, we try to define the space of almost surely bounded functionals ˜ fin (G) over L(M ): in order to do that, over the total space of the formal bundle L we consider the based loop space where the computations are easier (see [49]). Over ˜ fin (G), we construct a measure which is quasi-invariant under the right action of L ˜ ∞ (G) (see [5] and [22] for similar considerations). The transition functions belong L ˜ ∞ (G): unlike the case of a circle bundle, where the Haar measure is invariant to L under rotation, the measure over the fiber is only quasi-invariant under the action ˜ ∞ (G). This shows that only the notion of measurable functional over the total of L space defined almost surely is intrisically given as well as its essential supremum. This part gives an analysis over loop groups with a different measure from the classical one [4, 31]. 1. Line Bundle over the Based Path Space Let M be a Riemannian compact manifold. Let Px (M ) be the space of continuous applications from [0, 1] into M . Let ∆ be the Laplace–Beltrami operator which is associated. P1x is the law of the Brownian motion starting from x and P1,x,y the law of the brownian bridge between x and y: it is a probability measure

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over Lx,y (M ) the space of continuous path starting from x and arriving in y. Let pt (x, y) be the heat kernel associated to the heat semi-group. We have dP1x = p1 (x, y)dy ⊗ dP1,x,y .

(1.1)

Let τt be the parallel transport over a path γt for the Levi–Civita connection: it is almost surely defined. A vector over a path γ. is given by: Xt = τt Ht where Ht lies in Tx (M ) and is such that: Z t Hs0 ds Ht =

(1.2)

(1.3)

0

([11, 38]). Let us give the following hypothesis which is true for the remaining part of this paper: Hypothesis 1.1. M is 2 connected. Let us summarize the theory of [16] which works for smooth path: let ν be a representative at the level of differential forms of an element of H 3 (M, Z). Over each pinned path set Lx,y,∞ (M ), there is a line bundle Λx,y equipped with a connection whose curvature is the transgression of ν modulo a factor 2πi: Z 1 ν(dγs , ., .) . (1.4) τ (ν) = 0

There is an obstruction to patch together all these bundles. It is the cohomology class of ν. If the class of ν is trivial, we can fit together these line bundles by adding a basical form. Let us suppose that dβ = −ν. The global curvature of the global line bundle which modulo a factor 2iπ is Z 1 ν(dγs , ., .) . (1.5) ω = β(γ1 ) + 0

Our goal is to extend the construction of [16] to the case of the brownian path. In [49], we have constructed for the stochastic based loop space a stochastic line bundle by starting from a integral-valued 3-closed form. As a matter of fact, we have defined this line bundle by its Lp sections as its Sobolev sections. ω is closed over Px,fin (M ), the space of path of finite energy. Since Px,fin (M ) is obviously contractible, H2 (Px,fin (M ), Z) = 0, and ω is Z valued. Let γi be a countable dense set of finite energy curves in Px (M ) such that the open balls for the uniform distance B(γi , δ) constitutes a rescovering of Px (M ). Let γref be a reference path. There exists, if γ belongs to B(γi , δ) a distinguish path lt,i (γ) joining γ to γref : lt,i (γ)(s) = expγi (s) (1 − t)(γ(s) − γi (s))

(1.6)

if t ∈ [0, 1] and if t ∈ [1, 2], lt,i (γ) is any curve joining γi to γ. We can normalize the time of this path to be in [0,1].

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If γ ∈ B(γi , δ) ∩ B(γj , δ), there is a surface Si,j (γ) fulfilling lt,i (γ) and lt,j (γ): lt,u,i,j (γ)(s) = explt,i (γ)(s) (1 − t)(lt,j (γ) − lt,i (γ))(s)

(1.7)

and we fulfill the loop lt,i (γ), lt,j (γ) for t ∈ [1, 2] and l1,u,i,j (γ) by any surface: it is possible to do that because Px (M ) is simply connected. We deduce by the theory of stochastic integrals transition maps: " # Z ρi,j (γ) = exp −2πi

ω

(1.8)

Si,j (γ)

The transition map ρi,j belongs modulo a smooth cutoff locally, to all the Sobolev spaces and check: (i) If γ ∈ B(γi , δ) ∩ B(γj , δ), ρi,j (γ)ρj,i (γ) = 1

(1.9)

almost surely. (ii) If γ ∈ B(γi , δ) ∩ B(γj , δ) ∩ B(γk , δ), almost surely: ρi,j (γ)ρj,k (γ)ρk,i (γ) = 1 .

(1.10)

It is true for the following reason: if γ n is the polygonal approximation of γ, ρi,j (γ n ) → ρi,j (γ) almost surely and ρi,j (γ n ) checks (1.9) and (1.10). Let us recall how we construct such smooth cutoff (see [6] for a more systematic exposure). Let g be a function from [0, δ] into [1, ∞] which is infinite if z is larger than 1 δ − α/2, which is smooth over [0, δ − α/2[ and which behaves as (z−δ+α/2) +n when z → (δ − α/2)− . Moreover, z ≤ δ − α is equivalent to g(z) = 1. Let F be an auxiliary function from [1, ∞] into [0, 1], which is equal to 1 only in 1 and with compact support. Let Gi be the functional: Z 1 g(d(γs , γi,s ))ds (1.11) Gi (γ) = F 0

where d is the Riemannian distance. Gi (γ) = 1 if and only if d∞ (γ, γi ) ≤ δ − α and if Gi (γ) > 0, d∞ (γ, γi ) < δ (d∞ denotes the uniform distance between to path over M ). Gi belongs to all the Sobolev spaces ([44, 45, 39]). It is possible to find a sequence of the type Gi which tends to the indicatrice function of B(γi , δ). Definition 1.1. The space of measurable sections of the formal stochastic line bundle Λ over Px (M ) is given by a collection of C-valued random variables over B(γi , δ) such that over B(γi , δ) ∩ B(γj , δ), almost surely: ψj (γ) = ρj,i (γ)ψi (γ) .

(1.12)

Over the fiber, we have a norm which is consistent with the change of local charts and the transition functions. It is given by: kψ(γ)k2 = kψi k2 if γ ∈ B(γi , δ).

(1.13)

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Definition 1.2. The space Lp (Λ) of Lp sections of the stochastic formal line bundle Λ is the set of measurable sections over the line bundle such that: kψkLp (Λ) = k kψk kLp .

(1.14)

In the second expression, we take the Lp norm of the random variable kψk with values in R+ . There is a connection which works over the space of sections of this formal line bundle. The connection form is given over B(γi , δ) by: Z 2 Z 1 dt ω(lt,i (γ)(s))(dlt,i (s), D. li,t (γ)(s)) . (1.15) Ai (γ) = 0

0

We can normalize the path lt,i (γ)(s) such that t ∈ [0, 2]. D. li,t (γ)(s) denotes the H derivative along tangent vectors τs (γ)Hs of li,t (γ)(s). This connection form has the same stochastic structure as the form studied in [38, 46]. Especially, it has a kernel: Z 1

Ai (γ)(τ. H. ) =

ki (s)Hs0 ds

(1.16)

0

and ki (s) belongs to all the Sobolev spaces, modulo a smooth cutoff. Let us remark that Ai (γ) has the same structure as the restriction of the Chen form over the path of Px (M ). Especially dAi has a meaning and modulo the system of charts over B(γi , δ) given by γ → li,. (γ) dAi = ω .

(1.17)

The connection forms are compatible with the transition functionals. By patching together, we get a connection ∇Λ . Moreover the connection is unitary, modulo the factor i. We get namely: 0 Λ 0 hdhψi , ψi0 i, Xi = h∇Λ X ψi , ψi i + hψi , ∇X ψi i

(1.18)

over B(γi , δ). (1.18) has only a formal meaning. The purpose of it is to give a rigorous meaning of (1.18). Let ψ be a section of the formal stochastic line bundle Λ. Let ψi = Gi ψ. Let us suppose that ψi is a random variable over Px (M ) which is smooth in the sense of [44]: it can be seen as a section of Λ over B(γi , δ). Let ∇ be the connection over Px (M ) given by ∇τ. H. ) = τ. DH.

(1.19)

where DHt denotes the H derivative of Ht (see [44] in this context). Lemma 1.3. If ψi is smooth in the sense of [44], ∇k,Λ ψ exists and is an Hilbert–Schmidt cotensor with values in the fiber. Proof. We proceed by induction: ∇Λ ψi = dψi + Ai (γ)ψi

(1.20)

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where Ai is the connection form (we do not specify if we write the normalizing terms). The property is shown in step 1, because the kernels of ki are smooth. Especially ∇Λ ψi is a smooth cotensor in the sense of [44] with values in C. Let us suppose that ∇k,Λ Ψi is a smooth k cotensor with values in C. We get: k,Λ ψi )(Xi , . . . , Xk ) = hd(∇k,Λ (ψi ))(X1 , . . . , Xk ), Xi ∇Λ X (∇ P +Ai (X)(∇k,Λ ψi (X1 , . . . , Xj , . . . , Xk ) − ∇k,Λ ψi (X1 , . . . , ∇X Xj , . . . , Xk ) .

(1.21) By induction hypothesis, the difference between the first and the third terms are smooth as a k + 1 cotensor with values in C. Since Ai is smooth, we deduce the lemma. So if ψi is smooth in the sense of [44], we get ∇Λ∗ = −∇Λ + div

(1.22)

where div is the divergence of a vector field over Px (M ). This allows us to define by density consistently the operation ∇k,Λ , by starting from the finite combination of smooth functionals Gi ψi . Definition 1.4. The space Wk,p (Λ) is the space of sections of the formal line 0 bundle Λ such that ∇k ,Λ ψ belongs to Lp for k 0 ≤ k. The space of sections of k-differentiable Wk,∞− (Λ) is the intersection of the Sobolev spaces Wk,p (Λ). Definition 1.5. The space of smooth sections W∞,∞− (Λ) is equal to the intersection of the spaces Wk,∞− . Remark. W∞,∞− (Λ) is a module over the space of smooth functionals with values in C over Px (M ). Remark. If we change the connection by adding a smooth 1 form, we do not change W∞,∞− . Lemma 1.6. If ψ belongs to W∞,∞− (Λ), ψi = Gi ψ is a scalar functional which belongs to all the Sobolev spaces. Proof. Namely, ∇Λ,k ψi is an algebraic expression between the derivatives of order < k of ψi in the sense of [44], the derivatives of the connection forms which are smooth (see (1.17)). Therefore the highest order differential term in ∇Λ,k ψi is the scalar derivative dk∇ ψi (see [44]). Apparently the functional space associated to the formal stochastic line bundle Λ is defined with respect to a system of charts. Let us take another system of balls B(γj0 , δ) which constitutes an open rescovering of Px (M ). We have a system of transition maps ρj 0 ,i (γ) from the first system of charts to the second system of charts: ψj 0 (γ) = ρj 0 ,i ψi (γ)

(1.23)

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This system allows us to define the space of Lp sections in a consistent way. Moreover the connection forms Ai give, by this system of transition map, connections forms for Aj 0 , which are compatible with the transition map. But we meet a problem: namely we have to reach a smooth section over B(γj0 , δ) as a limit of a sum of a finite number of smooth sections over B(γi , δ). This problem is related to the problem of defining a smooth partition of unity of Px (M ). But in both cases, we have chosen a dense countable set of finite energy curves: this allows us to show that Gj 0 ψj 0 which is smooth in the sense of [44] is a limit of a finite sum of smooth sections over B(γi , δ) in all the Sobolev spaces. It is enough to choose Gin (γ) where γin is a sequence which tends to γj 0 for the uniform norm. So we can add to the original system of charts the open balls B(γ˜i , δ) where γ˜i constitute a countable dense of finite energy curves in the Lx,y (M ), the set of the γi , δ) paths going from x to y endowed with the measure Px, . The set of the balls B(˜ constitutes an open rescovering of Lx,y . We consider over Lx,y the Chen form: Z ωx,y =

1

ν(dγs , ., .) .

(1.24)

0

There is no β contribution. Namely, a vector over Lx,y is written as Xs = τs Hs

(1.25)

with H0 = H1 = 0. We can proceed as before in order to define the space of smooth sections of the formal line bundle Λx,y associated to ωx,y over Lx,y (M ) which is simply connected, because M is 2-connected. We get a functional space W∞,∞− (Λx,y ). We get the space of Lp sections of the formal line bundle Λx,y called Lp (Λx,y ). In order to perform these constructions, we choose the restriction of the connection ∇ over Px (M ): (1.26) ∇Xs = τs DHs where D is the H-derivative. Theorem 1.7. An element ψ of W∞,∞− (Λ) restricts in an element of Lp (Λx,y ). Proof. By the techniques of the quasi-sure analysis (see [3, 43, 44]), the funcγi , δ) restricts into a functional tional gi ψ = ψi with values in C defined over B(˜ over B(˜ γi , δ) ∩ Lx,y (M ). The transition map ρi,j (γ) restricts to the transition map γj , δ) ∩ Lx,y (M ) over Lx,y (M ). Moreover from B(˜ γi , δ) ∩ Lx,y (M ) to B(˜ kψkLp(Λx,y ) ≤ CkψkWk,q (Λ)

(1.27)

for k big enough and q p. In order to see that, let kψkp (γ) be the random variable associated to ψ which takes its values in R+ for p ∈ 2N + . It is a smooth random variable over Px (M ), from the formula (1.18). When we restrict this random variable over Lx,y (M ), we get the norm of the restriction of ψ to the formal line bundle Λx,y . Moreover, by the techniques of the quasi-sure analysis, the Lp (Lx,y (M )) norm

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of this random variable can be estimated in terms of its Sobolev norms over Px (M ). It remains to show that this random variable has derivatives of all orders. We have for instance, Λ (1.28) hdkψk2 , Xi = h∇Λ X ψ, ψi + hψ, ∇X ψi which gives by iteration Λ,2 Λ hd2∇ kψk2 , X, Y i = h∇Λ X ψ, ∇Y ψi + h∇X,Y ψ, ψi + symmetry

(1.29)

This allows us to state the result.

A linear line bundle has always a section. A circle bundle does not always have a section. But over the circle, we can introduce a probability law which is invariant under rotation. This allows us to define the space of Lp functional over the total space of the formal circle bundle associated to ω over Px (M ), which we will call Px (M )(S 1 ). A measurable functional over the formal total space Px (M )(S 1 ) is a set of measurable functionals Fi (γ, ui ), ui ∈ S 1 over B(γi , δ) × S 1 such that over B(γi , δ) ∩ B(γj , δ) (1.30) Fj (γ, uj ) = Fi (γ, ui ) almost surely, where uj = ρj,i ui almost surely. Over the fiber, we put: Z 1/p p |Fi (γ, ui )| dui . kF kp,γ =

(1.31)

S1

Since the measure over the circle is invariant by rotation, (1.31) is consistent with the change of map (1.30). We deduce a space of Lp functionals of the formal circle bundle over Px (M ). It is the space of measurable functionals such that kF kLp (Ps (M)(S 1 ) = (kF k)Lp < ∞ .

(1.32)

In the right-hand side, we take the Lp norm over Px (M ) of the scalar random variable kF kp,γ over Px (M ). Basical functionals are functionals which are constant over the fiber: they belong to Lp (Px (M )(S 1 )) if they belong to the Lp spaces of the path space. Let us go one step further in order to define the space of C 1 functionals over the total space of the formal circle bundle Px (M )(S 1 ) over Px (M ). Let us operate over B(γi , δ). We split the tangent space of our local trivialization of the formal circle bundle in (γ, u) into (X, −Ai (X)u + Y u) where Y ∈ iR, the Lie algebra of the circle in the identity and Ai is the connection form in the system of local char given by B(γi , δ) (we have multiplied the previous 1-form by 2πi). We say that we have split the tangent space of the trivialization into two orthogonal parts: the first one is the horizontal subspace endowed with the projected Hilbert structure; the second one is the vertical space endowed with the natural Hilbert structure over the circle. ˜ over the Let X be a vector over Lx (M ): we define its formal horizontal lift X stochastic circle bundle. It has a rigorous meaning over a local trivialization. Over

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a local trivialization modulo smooth cutoff, we have, since the measure over the circle is invariant by rotation: ˜ i F 0 (γ, ui )]] = −Eγ [Eu [Gi F (γ, ui )hd(Gi F 0 (γ, ui )), Xi]] ˜ Eγ [Eu [hd(Gi F (γ, ui )), XiG 2 0 +Eγ [divX Gi Eu [F (γ, ui )F (γ, ui )]] (1.33) for a big enough class of functionals (we can take the cylindrical functionals and the smooth functionals in ui ). This shows us that in L2 (Px (M )(S 1 ): ∗ DX ˜ + divX . ˜ = −DX

(1.34)

Moreover if we take a canonical vertical vector field Xvert , we have since the vertical measure is invariant by rotation divXvert = 0

(1.35)

This integration by parts allows, by starting from finite combination of functionals Gi Fi (γ, ui ), its H-derivative, which is almost surely defined, and to close this operation. In particular, the space of C 1 functionals over the circle bundle is the following: Definition 1.8. W1,∞− (Px (M )(S 1 )) is the space of functionals F over the stochastic formal circle bundle which have a first order H-derivative dF such that kdF k is a functional over this total space which belong to all the Lp . 2. Chern Simons form and Brownian Bridge Let us recall quickly what is a string structure over the free loop space of a 2connected manifold. Of course, we consider at this stage smooth loop. We introduce a principal bundle over M with compact structure group G. We do in this chapter the supplementary hypothesis: Hypothesis 2.1. G is 2 connected and simple. The total space of the bundle is called Q. Let γs be a loop in M : let qs be a loop in Q over γs . Smooth loop g. in G acts on the loop q. on the left: qs gs is still a loop in Q over the loop γ. The space of smooth loops in Q called L∞ (Q) is a principal bundle with structure group L∞ (G) ([20, 58]). Let us recall what is the basical central extension of L∞ (G). Over G, we introduce a killing form h., .i: they are all proportionals since G is simple. Let L∞ (G) be the 2-form which is invariant under the action of L∞ (G) and which over the Lie algebra of L∞ (G) is defined by: Z 1 1 hX(s), Y 0 (s)ids . (2.1) ω(X, Y ) = 2π 0 ω is a closed 2-form Z valued There exists a smallest Killing product such that 2π over G. Since G is 2-connected, L∞ (G) is simply connected.

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We construct over Lfin (G), the space of finite energy loop in G, or over L∞ (G), the unique circle bundle associated to ω: the unicity of this bundle arises from the fact that L∞ (G) is simply connected. Let e. be the unit loop in L∞ (G). Let lt (g. ) be a smooth path from e. to g. . The bundle is the set of (l. (g. ), α), α ∈ S 1 submitted to the equivalence relation: (l. (g. ), α) = (l.0 (g.0 ), α0 ) if g. = g.0 and if Z ω (2.2) α0 = α exp −i C

where C is a surface with boundary the loop which is built from the path going from e. to g. by lt (g. ) and coming back to e. by lt0 (g. ). We can find such surface because Lfin (G) is simply connected. The values depends on the surface by a multiple of 2iπ, and so (2.2) has a meaning. We define a group law over the bundle by putting: (l. (g. ), α).(l.0 (g.0 ), α0 ) = (l. (g. ) ? l.0 (g.0 ), αα0 )

(2.3)

where l. (g. ) ? l.0 (g.0 ) is the path going from e. to g. g.0 in the following way: first we go from e. to g. by lt (g. ) and secondly from g. to g. g. by g. lt0 (g. ). It is the same as to consider the couple of the paths lt (g. )lt0 (g.0 ) and β where: Z ω αα0 (2.4) β = exp −i C

C is a surface bounded by the triangles lt (g. ), g. lt0 (g.0 ) and lt (g. )lt0 (g.0 ), the last path being runned in the opposite sense. (2.4) defines a group law (it is in particular compatible with the equivalence ˜ ∞ (G) of L∞ (G) relation). This allows us to define the basical central extension L ˜ or Lfin (G) of Lfin (G), which are all isomorphic if G is simply connected (the loop space of G is in this case simply connected too). ˜ ∞ (Q) of L∞ (Q) by The problem to define a string structure is to define a lift L ˜ ∞ (G). L Let AQ be a principal connection over Q. AQ is a Lie G-valued 1-form on the total space Q. If ξ˜q is the fundamental vector at q ∈ Q corresponding to ξ ∈ Lie(G), we have: (2.5) AQ (ξ˜q ) = ξ . The curvature of the 2-form FQ is obtained by 1 FQ = dAQ + [AQ , AQ ] 2

(2.6)

and is basic. Let us recall that 1 [AQ , AQ ](X, Y ) = AQ (X)AQ (Y ) − AQ (Y )AQ (X) 2 in matrix notation. We will follow closely the lines of [20] in order to produce this lift. 1 1 AQ , FQ − [AQ , AQ ] σQ = 8π 2 6

(2.7)

(2.8)

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We have [20], if π denotes the canonical projection from Q to M by using the theory of Chern–Simons ([19]): (2.9) dσQ = −π ∗ p1 (Q) . Moreover, over the fiber, σQ is equal to the 3 invariant form over G. 1 hX, [Y, Z]i . 8π 2

σ(X, Y, Z) =

(2.10)

We say that σQ transgresses σ. We consider the transgression over the free loop space of the total space of σQ . It is given by Z

1

τ (σQ )(XQ , YQ ) =

σQ (dqs , Xs,Q , Ys,Q ) .

(2.11)

0

It is a Chen form over the total space. We have by classical results over Chen forms: Z

1

dτ (σQ )(XQ , YQ , ZQ ) =

p1 (Q)(dγs , Xs , Ys , Zs )

(2.12)

0

where γs is the projection of qs over M and Xs the projection of Xs,Q over T (M ). The last ingredient of [20] is to produce a 2-form ωQ over L∞ (Q) which is equal to ω over the fiber, modulo a normalizing term 2π, which is identified with L∞ (G). [20] produce an explicit formula for the 2-form ωQ ; it is: 0 = ωQ

1 2π

Z 0

1

1 (hAQ , d/dtAQ i − hFQ , AQ (dqs )i 2

(2.13)

d/dtAQ is the form which, to a vector field XQ over L∞ (Q), associates the derivative in time t of AQ (qt )(Xt,Q ). It is the Lie derivative under the natural circle action over the free loop space of Q. We have [20]: 1 0 ω = τ (σQ ) − d 2π Q

1 8π 2

Z

1

hAQ , AQ (dqs )i .

(2.14)

0

We will do the following hypothesis in the remaining part of this paper: Hypothesis 2.2. p1 (Q) represents the trivial cohomology class. By the theory of Chern–Simons form, if p1 (Q) = 0 in cohomology, it is possible to find a representative of ν such that −p1 (Q) = dν and such that σQ − π ∗ ν is a Z-valued closed 3-form over Q ([19] Theorem 3.10]). Let us introduce the form over L∞ (Q): 0 ωQ ωQ = − τ (π ∗ ν) , 2π 2π ωQ 2π

(2.15) ω

is Z-valued over L∞ (Q) and closed. Moreover, 2πQ is invariant under rotation: we use the natural circle action over the free loop space for that. [20] consider the ˜ ∞ (Q) over L∞ (Q) which has form ωQ , they construct the unique circle bundle L

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ωQ as curvature modulo a constant 2πi. They use for that the fact that L∞ (Q) is simply connected, in order to construct this bundle by hand. They construct an ˜ ∞ (Q): this allows them to show that L ˜ ∞ (Q) is in fact an ˜ ∞ (G) over L action of L ˜ ∞ (G) principal bundle over L∞ (M ) with a fiber isomorphic to L ˜ ∞ (G). L Let us come back to the stochastic context. L∞ (M ) becomes the space of continuous loop L(M ) of a Riemannian manifold endowed with the B.H.K. measure: dµ =

p1 (x, x)dx ⊗ dP1,x,x Z . p1 (x, x)dx

(2.16)

M

Over a continuous loop, we consider section of Q qs such that DAQ q. has a bounded energy. In other words, the Bismut bundle Lfin (Q) over L(M ) is the space of loops in Q such that: (2.17) qs = τsQ gs where gs has a bounded energy in G and such that: τ1Q g1 = τ0Q g0

(2.18)

where τsQ is the parallel transport over γs for the connection AQ starting from a given element. This bundle is clearly a Lfin (G) principal bundle (Lfin (G) denotes the space of finite energy loops in G). Moreover, this bundle is clearly invariant under rotation. Let Qt be the bundle over L(M ) pullback by the evaluation map γ → γt of Q. We have a projection map πt : Lfin (Q) → Qt . Let Pfin Q0 be the set of paths over γ0 ∈ Q in the fiber of G. We have an application from Pfin Q0 over Q0 × Q0 which to a path associates its end points. Pfin Q0 is clearly a Le,fin (G) bundle, the based loop space of finite energy loop in G start from the unit element e. It does not, unfortunately, satisfy our requirement to be a Lfin(G) bundle, in order to give a ˜ fin (G), a central treatment similar to [17] for the construction of a lift Lfin (Q) by L extension of Lfin (G), we will state later. This shows us that we will have complications with respect to [49] in order to exhibit a connection over Lfin (Q), ∇∞ . Let us consider an open rescovering of M by a finite number of open subset Oi where Q is trivial, and let us introduce a partition of unity hi associated to Oi . Let LOi (M ) be the set of brownian loop starting from Oi . We can choose over Oi in (2.17) τsQ starting from a given element which depends only on γ0 modulo the trivialization. We will denote LOi ,fin (Q) to be the reciproque image by the projection from Lfin (Q), over L(M ) of LOi (M ). Let us introduce Pfin (G), the set of finite energy path in G. Let fi be the application which associates to a loop γ the couple (γ0 , (τ1Q )−1 ) modulo the trivialization. We have the following commutative diagram: LOi ,fin (Q) → Oi × Pfin (G) ↓ ↓ Oi × G LOi (M ) →

(2.19)

256

´ R. LEANDRE

The vertical maps are the projections. The lower horizontal map is fi and the upper vertical map is fi∗ . More precisely the right vertical map associates to a path g. in Pfin (G) the element of G g1 g0−1 . The second vertical map defines a structure of Lfin (G) bundle over Oi × Pfin (G). Since Oi × G is finite dimensional, there exists a connection ∇∞ i over Oi × Pfin (G). Let us consider the pull-back connection over LOi ,fin (Q) called ∇∞ i . We can now consider the connection over Lfin (Q) defined by: X hi (γ0 )∇∞ (2.20) ∇0,∞ = i . We can rotate the time, and we have another connection called ∇u,∞ . Instead of using Pfin (G) in the commutative diagram (2.19), we can use P∞ (G), the space of smooth paths in G. We get a bundle LOi ,0,∞ (Q) over LOi (M ), which glues together in a bundle L0,∞ (Q) over L(M ). We can choose a connection which we call ∇0,∞ constructed as in the previous way as L0,∞ (Q). Let us remark that L0,∞ (Q) is not invariant under rotation. Since L0,∞ (Q) is a subbundle of Lfin (Q), we can choose the two connections such that they fit together. The connection has the following fundamental property: let us choose a path γ. (u) in L(M ) with the following properties: (i) u → γ0 (u) is smooth. (ii) u → τ1Q (γ. (u)) is smooth over each chart LOi (M ). Let us consider an element over γ. (0) which belongs to LOi ,0,∞ (Q). We can parallel transport it. It remains over each trivialization LOj (M ) in LOj ,0,∞ (Q) and its trajectories are smooth in u. In particular, its trajectories depend in a smooth way of the two data i) and ii) for the C 1 uniform norm over the two data. First Step: Construction of Random Surfaces in Lfin (Q). Let us consider a countable set of finite energy smooth curves γi in L(M ) such that the set of open balls for the uniform distance B(γi , δ) constitutes an open rescovering of L(M ). Let us recall that there exists a distinguished path from γ to a reference path γref . Let us denote this distinguihed curve by li,t (γ). From γ to γi it is defined by: (2.21) li,t (γ)(s) = expγi (s) (1 − t)(γ(s) − γi (s)) and after we choose any curve joining γi to γref , we can normalize the time t to be in [0,1]. If γ ∈ B(γi , δ) ∩ B(γj , δ), there exists a distinguished surface fulfilling the loops li,t (γ) and lj,t (γ). In order to join li,t (γ) to lj,t (γ) in the stochastic part of these paths, we choose: li,j,t,u (γ)(s) = expli,t (γ)(s) (1 − u)(lj,t (γ)(s) − li,t (γ)(s))

(2.22)

We fulfill the triangle constituted by γi , γj and γref by a deterministic surface. We get a random surface in the basis called Si,j (γ). In the sequel, we choose δ small enough such that if γ ∈ B(γi , δ) ∩ B(γj , δ), Q0 is trivial over the set run by γ0 which belongs to one of the Ok , and we can consider the commutative diagramm (2.19).

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In particular, let li,t (γ) be the curve over the basis L(M ) defined as before, and γQ let be an element over γ in L0,∞ (Q). We can parallel transport it over the piece of curve joining γ to γi for the connection ∇0,∞ . We get a curve called li,t (γQ ). We can perform the same manipulation for the path u → li,j,t,u (γ) and we can parallel transport li,t (γQ ) over this curve. The problem is that we do not get a surface with boundary a loop in L0,∞ (Q), because when we parallel transport li,t (γQ ) over u → li,j,t,u (γ), we do not arrive at lt,j (γQ ). We will overcome this difficulty by doing as in [49]. As in [49], we will restrict our open neighborhood in order to produce a nice random surface in L0,∞ (Q), where we can integrate ωQ , by the theory of stochastic integrals. As open rescovering, we will choose the set Wi0 = B(γi , δ) × B(gi,. , δ) where gi,. is in L∞ (G), modulo the previous trivialization which depends only on γ0 and τ1Q (γ). B(gi,. , δ) is the open ball for the C 1 uniform norm topology. Let q. be in Wi0 . First of all, we can choose a distinguished path such that the element in the fiber over γ is applied over gi,. . Secondly, we parallel transport gi,. over li,t (γ) until γi and we introduce a supplementary condition: over γi , we get after parallel transporting an element which is in the fiber in a small tubular neighborhood for the C 1 uniform norm topology of Lfin (G) after trivialization. The condition can be realized if t → li,t (γ)(0) and t → τ1Q (li,t (γ)) remain until arriving in γi in a small neighborhood for the C 1 uniform topology of a given C 2 curve, this last condition having to be seen after choosing suitable trivialization of Q. Let us call Wi the set obtained after this restriction. Let us call qi,. the element of the fiber of L0,∞ over γi which is obtained after parallel transport: this is because there is this restriction over Wi a distinguished curve joining qi,. to a given deterministic element q. (γi ) in the fiber over γi , because L∞ is connected. Let us choose a deterministic path joining q. (γi ) to qref,. . In conclusion, if q. ∈ Wi , we have constructed a distinguished curve joining q. to qref,. . Let us call this distinguished curve li,t (q. ). Let us construct a random surface fulfilling li,t (q. ) and lj,t (q. ) in Wi ∩ Wj . We parallel transport li,t (q. ) over li,j,t,u (γ) when we are over the piece of path which is obtained by parallel transporting of gi,. over li,t (γ). We choose a path joining q. (γi ) to q. (γj ) and we fulfill the deterministic triangle q. (γi ), q. (γj ) and qref,. by a deterministic surface: it is possible because L∞ (Q) is simply connected. In order to get a surface with boundary the loop constituted of the paths li,t (q. ) and lj,t (q. ), there are two holes: the first one is when we push q. over (γ, gi,. ) or q. over (γ, gj,. ). Since δ is small, we can fulfill these two pieces by a vertical distinguished triangle. The second hole, in order to get a surface, is obtained by a loop in Lfin (G). It is in fact a loop in L∞ (G), because the connection ∇0,∞ is in fact a connection for L0,∞ (Q). We fulfill this random loop in L∞ (G) in an arbitrary way. We produce by this procedure over Wi ∩ Wj a surface called Si,j (q. ). A generic element of this surface is called li,j,t,u (q. ). Second Step: Integration of the Curvature over the Random Surface

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There are 3 parts in this integration: (i) The integration of τ (π ∗ ν). It is nothing but the integral of τ (ν) over Si,j (γ) which can be treated R 1 as in the first part (q. is in the fiber of γ). (ii) The integration of 0 hFQ , AQ (dqs )i. But we have: li,j,t,u (q. )(s) = τsQ (li,j,t,u (γ))gs (li,j,t,u (q. )) .

(2.23)

AQ (dli,j,t,u (q. )(s)) = gs−1 (li,j,t,u (q. )dgs (li,j,t,u (q. )) .

(2.24)

Therefore:

Moreover FQ is a basical form. And we have only to integrate over Si,j (γ) the form Z 1 hFQ , gs−1 (li,j,t,u (q. )(s)dgs (li,j,t,u (q. )(s)i (2.25) 0

which leads to a traditional integral. The only point we have to take care is that π(Wi ∩ Wj )Ris not completely B(γi , δ) ∩ B(γj , δ). 1 (iii) The integration of 0 < AQ , d/dsAQ > ds will lead to stochastic integrals. We use the Bismut formula [11] which gives the covariant derivative of the parallel transport. We get: Z s ∂ (τvQ )−1 AQ li,j,t,u (q. )(s) = gs−1 (li,j,t,u (q. )) ∂t 0 ∂ (li,j,t,u (γ))FQ (dli,j,t,u (γ)(v), li,j,t,u (γ)(v))τvQ (2.26) ∂t ∂ (li,j,t,u (γ))gs (li,j,t,u (q. )) + gs (li,j,t,u (q. )) ∂t ∂ li,j,t,u (q. )(s). and there is an analoguous formula for AQ ∂u ∂ ∂ li,j,t,u (γ)(s) or ∂u li,j,t,u (γ)(s). s → li,j,t,u (γ)(s) is a semi-martingale as well as ∂t But s → gs (li,j,t,u (q. )) is not a semi-martingale (we do not precise the localization procedures which appear in this statement). Let us study the critical part where this integration put any problem: s → gs (li,j,t,u )(q. ) is smooth, modulo the trivialization, because ∇0,∞ is in fact a connection over L0,∞ (Q). There are four parts to integrate. First of all, we go from q. to (γ, gi .): we have to integrate over a vertical surface, and this leads to a traditional integral. There is the part which arises from theR lift by the parallel transport: this leads 1 to manipulating expressions of the shape 0 Fs dXs where Xs is a semi-martingale and Fs is a smooth function. We can integrate by parts to conclude. We use, in ∂ gi,j,t,u (q. ) order to understand the stochastic behaviour of this part, the fact that ∂t ∂ gi,j,u,t (q. ). is a smooth function of the time s as well as ∂u The third part arises from the loop in L∞ (G) which is fulfilled by a surface in L∞ (G). The formula (2.26) and the same consideraration as in the previous contribution allow to conclude.

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There remains the integral over the surface fulfilling the triangle q. (γi ), q. (γj ) and qref,. which is deterministic. Let us put if q. ∈ Wi ∩ Wj : " Z # ρi,j (q. ) = exp −i

ωQ .

(2.27)

Si,j (q. )

Let q. ∈ B(γi , δ) × B(gi,. , δ) and πq. be its projection over L(M ). Let (πq. )n be its polygonal approximation by starting from the time t = 0. Let us introduce the corresponding element q.n , with the same element in the fiber as q. , modulo the trivialization. If q. ∈ Wi and if n is big enough, q.n ∈ Wi and we can consider ρi,j (q.n ). Let us remark that li,j,t,u (q.n ) = τ.Q (li,j,t,u (πq.n ))g. (i, j, t, u)(q.n )

(2.28)

and that gi,j,t,u (q.n ) tends when n → ∞ to gi,j,t,u (q. ) for the C 1 uniform norm: namely the parallel transport in time 1 associated to li,j,t,u (πq.n ) over Q tends to the parallel transport in time 1 associated to li,j,t,u (πq. ) over Q. Moreover, πq.n has the same starting point as πq. . This shows us that we can choose, as vertical surface bounded by a loop in L∞ (G) for q.n , a vertical surface which is close from the original vertical surface when n → ∞. This shows us that almost surely in the loop of the basis for all elements in the fiber over πq.n (namely, Wi is a product), we have: ρi,j (q.n ) → ρi,j (q. ) .

(2.29)

ω

Moreover, 2πQ is a Z-valued cohomology class over Lfin (Q): we can fulfill the vertical loop in L∞ (G) by any surface in L∞ (G): the result over ρi,j (q.n ) and therefore over ρi,j (q. ) are not affected. Moreover, ωQ over the space of loop of finite energy of Q determines a bundle. This means that for all q.n ∈ Wi ∩ Wj : ρi,j (q.n )ρj,i (q.n ) = 1

(2.30)

ρi,j (q.n )ρj,k (q.n )ρk,i (q.n ) = 1 .

(2.31)

and for all q.n ∈ Wi ∩ Wj ∩ Wk :

We deduce that almost surely in πq. , for all elements in the fiber, we have over Wi ∩ Wj : (2.32) ρi,j (q. )ρj,i (q. ) = 1 and over Wi ∩ Wj ∩ Wk , almost surely in the curve of the basis for all elements of the fiber: (2.33) ρi,j (q. )ρj,k (q. )ρk,i (q. ) = 1 . ˜ fin (Q) over Lfin (Q). This allows to construct a formal circle bundle L ˜ fin (G) Bundle Over L(m) Third Step: Construction of the Formal L

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˜ fin (G) of Lfin(G). Let us Let us first construct the basical central extension L first remark that Lfin (G) is an Hilbert Lie group which is simply connected. Let us consider the Lfin (G) invariant 2-form ω over it. It is closed and 2πZ-valued. We ˜ ∞ (G). can repeat the same considerations as for the construction of L ˜ ˜ Let us recall that we have an action of L∞ (G) over L∞ (Q), the circle bundle over L∞ (Q) defined by the previous considerations (we have for smooth loop a bundle defined as a topological space). Let l. (q. ) be a path arriving in q. in L∞ (Q) ˜ ∞ (Q). Let l. (g. ) be a and starting from a reference path qref,. and (l. (q. ), α) in L path in L∞ (G) starting from the unit loop and arriving in g. and let us consider ˜ ∞ (G). We choose in order to define the action of a path the element (l. (q. ), β) in L joinning qref,. to q. g. : it is constituted from the path joining qref,. to q. defined by l. (q. ) and after from the path joining q. to q. g. defined by q. l. (g. ). And we choose (αβ) in the circle. Equivalently we choose the path l. (q. )l. (g. ) and the element of the circle R exp[−i S ωQ ]αβ. S is any surface bounded by the path l. (q. ), the path q. l. (g. ) and the path l. (q. )l. (g. ) described in the opposite sense. We can do the same construction for finite energy loop over M and the corre˜ fin (G) bundle over Lfin (M ). ˜ fin (G). We get a L sponding bundle and for L Let q. ∈ Wi ⊂ Lfin(Q): we have constructed a distinguished curve li,t (q. ) joining qref,. to q. . Let us consider the curve li,t (q. )lt (g. ): it is not the distinguished curve corresponding to q. g. belonging to some Wj . But, over the basis we have: π(li,t (q. )lt (g. )) = li,t (πq. ) .

(2.34)

This shows us that li,t (q. )lt (g. ) and lj,t (q. g. ) differ by a loop in Lfin (G). If lt (g. ) has its values in L∞ (G), li,t (q. )lt (g. ) and lj,t (q. g. ) differ by a loop in L∞ (G). We fulfill this loop by a surface C(q) in L∞ (G), and we assimilate the corresponding surface to a vertical surface over li,. (πq. ). By doing as in the second part, we get if q. ∈ Wi and q; g; ∈ Wj " Z # ! (li,t (q. )lt (g. ), α) =

lj,t (q. g. ), exp −i

ωQ α

.

(2.35)

C(q)

It remains to finish our construction to fulfill by any suitable random surface the triangle constituted by li,t (q. ), q. lt (g. ) and li,t (q. )lt (q. ) described in the opposite sense. We put over li,t (q. ) the curve li,t (q. )ls (g. ) with s ≤ t in order to do that, and we get the triangle li,t (q. )ls (g. ), 0 ≤ s ≤ t ≤ 1. We can integrate ωQ over this random surface as it was done in the second part. Let us summarize the previous considerations: if q. ∈ Wi and q. G. ∈ Wj , with the same basical loop γk , we have (li,t (q. ), α).(l. (g. ), β) = (lj,t (q. q. ), ηi,j (q)αβ) .

(2.36)

If we choose the finite energy approximation q.n of q. as before; we have almost surely in the basis for all elements in the fiber: ηi,j (q.n ) → ηi,j (q. ) .

(2.37)

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If Wj is a subset of B(γj , δ) × B(gj,. , δ) with γj 6= γi , we use in addition the change of map given in the second step. In all the cases, we deduce a formula of the shape (2.36) where (2.37) remains true. ˜ fin (G) is compatible with all Since over the finite energy loop, the action of L ˜ ∞ (G) over the the equivalence relations, we deduce from (2.37) that the action of L ˜ formal bundle Lfin (Q) is compatible with the equivalence relation. ˜ fin (Q). Since over Oi the bundle Q Let us now construct some local sections of L 0,∞ over the piece of path li,t (γ) joining γ to is trivial, the parallel transport for ∇ γi can be considered as a random element of L∞ (G). We have restricted B(γi , δ) such that this parallel transport goes into a small neighborhood for the C 1 uniform norm of a path in L∞ (G): in particular, there is a distinguished vertical curve over γi coming from this parallel transport and arriving to a given fixed element of L∞ (G) over γi . We restrict B(γi , δ) such that the parallel transport in Q over a path which belongs to B(γi , δ) remains after trivialization in a small neighborhood of G. We have therefore a smooth local section qi,. (γ) over the restricted set called ˜ fin (Q) the section: Vi of L∞ (Q). We choose as local section of L (li,t (qi,. (γ)), 1) = g˜i,. (γ) .

(2.38)

Let us show that almost surely over Vi ∩ Vj : ρj,i (γ) g˜i,. (γ) = g˜j,. (γ)˜

(2.39)

˜ ∞ (G). where ρ˜j,i (γ) is a random variable in L li,1 (qi,. (γ)) = qi,. (γ) and lj,1 (qj,. (γ)) = qj,. (γ). By restricting Vi and Vj if necessary, we can find an element of L∞ (G) gj,i,. (γ) such that gj,i,. (γ) remains in a small neighborhood of L∞ (G) and depends smoothly on the C 1 uniform topology in L∞ (γ) and such that: (2.40) qi,. (γ) = qj,. (γ)gj,i,. (γ) . As a matter of fact the 3 terms in (2.40) depends on γ only by τ1Q and by γ0 after trivialization of Q. Since G is connected, we get a smooth path in τ1Q and γ0 starting from the unit path in L∞ (G) and arriving in gj,i,. (γ). Let lj,i,. (gj,i, (γ)) be this smooth path. We have (lj,. (qj,i,. (γ)), 1).(lj,i,. (gj,i,. (γ)), 1) = (lj,. (qi,. (γ))lj,i,. (gj,i,. (γ)), αj,i (γ)) .

(2.41)

Let us precise this statement: we have to integrate ωQ over the surface lj,t (qj,. (γ))lj,i,s (gj,i,. (γ)) for 0 ≤ s ≤ t ≤ 1. qj,. (γ) depends smoothly on τ1Q and γ0 as well as gj,i,. (γ); therefore there is a smooth version in the fiber and in gj,i,. (.) if gj,i,. depends on a finite-dimensional parameter and remains in a small neighborhood of a path in L∞ (G) of the integral of ωQ over the surface lj,t (q. )lj,i,s (gj,i, ) for 0 ≤ s ≤ t ≤ 1. (2.41) is checked almost surely in the basis for all elements of the fiber and for all gj,i, in a small neighborhood of a path in l∞ (G) depending smoothly on a finite-dimensional parameter. It remains to choose as element of the fiber the random element qi,. (γ) and the random element gj,i,. (γ). In particular, if

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we choose the polygonal approximation of γ, say γ n , (2.41) is true for γ n , and we see that αj,i (γ n ) → αj,i (γ) almost surely. The last point to check is that: (li,. (qi,. (γ)), 1) = (lj,. (qj,. (γ))lj,i,. (gj,i,. (γ)), βj,i (γ))

(2.42)

and that (2.42) is still true for the polygonal approximation γ n and that βj,i (γ n ) → βj,i (γ) almost surely when n → ∞. The proof is very similar to the first step, the second step and (2.35), but we have to choose q. = qi,. (γ) and g. = gj,i,. (γ). These two random elements depend in fact on a finite number of parameters, τ1Q and γ0 ; the statement can be done as qi,. (γ), qj,. (γ) and gj,i,. (γ) were fixed in the fiber by using the theory of stochastics integrals depending on a finite number of parameters. We deduce that almost surely ρj,i (γ) g˜i (γ) = g˜j (γ)˜

(2.43)

˜ ∞ (G). Moreover ρ˜j,i (γ n ) is defined surely and tends almost and that ρ˜j,i (γ) ∈ L surely to ρ˜j,i (γ) when n → ∞. ˜ fin (G) bundle over Lfin (M ), we have for all γ n the following ˜ fin (Q) is a L Since L relation: (i) If γ n ∈ Vi ∩ Vj : ρi,j (γ n ) = Id . (2.44) ρ˜j,i (γ n )˜ (ii) If γ n ∈ Vi ∩ Vj ∩ Vk ρ˜i,j (γ n )˜ ρj,k (γ n )˜ ρk,i (γ n ) = Id .

(2.45)

Since ρ˜i,j (γ n ) tends almost surely to ρ˜i,j (γ), we deduce that the relations (2.44) and (2.45) are almost surely true for ρ˜i,j (γ). Let us remark that locally, ρ˜i,j (γ) = (li,j,. (gi,j,. (γ)), γi,j (γ))

(2.46)

and that γi,j (γ) belongs to all the Sobolev spaces with one derivative because they use stochastic integrals and because the parallel transport for ∇0,∞ depends smoothly on the paths t → li,t (γ)(0) and t → τ1Q (li,t (γ)) after trivialization of Q. We deduce that the integral which appears from the lift of the basical surface by using ∇0,∞ belong to all the Sobolev spaces with one derivative. It remains to study the contribution of the vertical loop in L∞ (G) which can be fulfilled by any surface in L∞ (G). But this loop depends smoothly on γ for the smooth topology over L∞ (G) (we will precise what it means later), we deduce that the contribution of the vertical surface in γi,j (γ) belongs to all the Sobolev spaces with one derivative. There remain two points to clarify. The first point to clarify is to precise the definition of a smooth loop in L∞ (G) which belongs to all the Sobolev spaces with one stochastic derivative. We can imbed the Lie group in a linear space. Smooth torus with values in G can be seen

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by studying Sobolev–Hilbert spaces over the torus. This allows us to study torus on G (loop over the loop space of G) which depends on γ as random variables with values in suitable Hilbert spaces. The second point it remains to clarify is that we operate over Vi , and so we have to localize the smoothness assumptions. This problem is related to [6]. We would like to introduce functionals with support in Vi which tend almost surely to the indicatrice function of Vi . In order to construct such functionals, we proceed as in [39]. Let g be a function from [0, δ[ into [1, ∞] which is infinite if z is larger than δ − α2 , which is smooth over [0, δ − α2 [ and which behaves as (z − δ + α2 )−n when z → (δ − α2 )− for a big integer n. Moreover, z ≤ δ − α is equivalent to g(z) = 1. Let F be an auxiliary function from [1, ∞] into [0,1] which is equal to 1 only in 1 and which has a bounded support. Let us put: Z 1 g(d(γs , γi,s ))ds . (2.47) Gi (γ) = F 0

Gi (γ) = 1 if and only if d(γ, γi ) ≤ δ − α and gi (γ) is equal to 0 if d(γ, γi ) > δ . Gi belong to all the Sobolev spaces [39]. We can choose Gi with the same shape, except for d(li,t (γ)(0), wt ) and d(d/dtli,t (γ)(0), d/dtwt ) where wt is a curve over M which is C 2 and for τ1Q (li,t (γ)) and d/dtτ1Q (li,t (γ)) after trivialization of Q. We can measure in a smooth way if the curve t → li,t (γ) is closed for the C 1 uniform topology for a given C 2 curve and if the curve in G t → τ1Q (li,y (γ)) is closed for the C 1 uniform norm for a given C 2 curve over G after trivialization of Q0 over B(γi , δ). We can measure in a smooth way in an easier manner if τ1Q (γ) remains in a small neighborhood of G. We have given a smooth version of the 3 conditions which determine Vi : in particular, there exists a sequence Gni which belongs to all the Sobolev spaces over L(M ) with support in Vi almost surely which tends almost surely to the indicatrice function of Vi . We can summarize our discussion: Summary. There exists a countable set of measurable sets Vi which satisfy the following conditions: (i) ∪Vi = L(M ) modulo a set of measure 0. (ii) There exists a sequence of functionals Gni which belong to all the Sobolev spaces with support almost surely in Vi and which tend to the indicatrice function of Vi almost surely. (iii) Over Vi ∩ Vj , there exists an application measurable ρ˜i,j (γ) with values into ˜ ∞ (G) ⊂ L ˜ fin (G) such that almost surely: L ρj,i (γ) = Id ρ˜i,j (γ)˜

(2.48)

and such that over Vi ∩ Vj ∩ Vk almost surely: ρj,k (γ)˜ ρk,i = Id ρ˜i,j (γ)˜

(2.49)

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(iv) Over Vi ∩ Vj , ρ˜i,j (γ) = (gi,j,. (γ), αi,j (γ))

(2.50)

where gi,j (γ) depends smoothly in the ordinary sense from τ1Q (γ) and from γ0 in a trivialization of O0 over Vi and where αi,j (γ) belongs to all the Sobolev spaces with one derivative with values in S 1 : Gni (γ)Gm j (γ)αi,j (γ) belongs to all the Sobolev spaces over the loop space with one derivative. We give now the following hypothesis: ˜ fin (G) called Spin. Hypothesis 2.3. There exists a unitary representation of L Definition 2.1. A measurable section of the associated bundle Spin over L(M ) is a collection of random variables ψi over Vi with values in the representation space of Lfin (G) such that over Vi ∩ Vj almost surely: ψj = ρ˜j,i ψi .

(2.51)

The Hilbertian norm of ψi is compatible with (2.49) since the representation is unitary. Moreover we can associate to a section of Spin the random variable kψk. Definition 2.2. The set of Lp section of Spin called Lp (Spin) is the set of measurable sections ψ such that: 1

kψkLp(Spin) = E[kψkp ] p < ∞ .

(2.52)

˜ fin (Spin2d ) The appendix tells us that there exists a unitary representation of L where Spin2d is the universal cover of SO(2d). ωQ is invariant under the natural circle action over Lfin (Q). So it is natural to get: Theorem 2.3. The natural circle action over L(M ) lifts to a circle action over the section of Spin, which is an isometry of Lp (Spin). Proof. Let q. ∈ Wi . Let (li,. (q. ), α) be an element of the formal circle bundle over Lfin (Q). We rotate γ and q. . We get q. ∈ Wi and we can rotate li,. (q. ). We get u ˜u: (q.u ). We get an application G li,. u (q.u ), α) . (li,. (q. ), α) → (li,.

(2.53)

q.u belongs to some Wj . So there is a distinguished path lj,. (q.u ) from q.u to qref,.. We can suppose that qref,. is invariant under rotation. Let us fulfill the loop constituted u (q.u ) and the path lj,. (q.u ) by a random surface where we can integrate by the path li,. ωQ by using the theory of stochastic integrals. First of all, if we rotate li,. (γ), we still get the distinguished curve joining γ u to γiu . Let us rotate the connection ∇0,∞ into ∇u,∞ . We get a distinguished path going from q u to qref,. called li,.,u (q.u ). It u (q.u ) by a loop in Lfin (G): as a matter of fact, it differs by a loop differs from li,. in L∞,2 (G) where L∞,2 (G) is the subgroup of Lfin (G) constituted of smooth loop of G except for two given times. We can fulfill this loop by a surface, and we can

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265

˜ u is given integrate ωQ as it was done before over this surface. Therefore the map G by: (2.54) (li,. (q. ), α) → (li,j 0 ,.,u (q.u ), αui,j 0 (q. )α) . The path li,j 0 ,.,u (q.u ) is the path constructed from the data q.u , γ u and the element gj 0 ,. obtained after trivialization in time u of Qu . As it was done in the first step and the second step of the discussion given before the definitions, we can fulfill the loop constituted by the paths li,j 0 ,.,u (q.u ) and of lj,. (q.u ) by a suitable surface and we can integrate over it by using the theory of stochastic integral. We can forget the intermediary step given by j 0 and the trivialization of Qu because ωQ is 2π ˜ u becomes: Z-valued. The map G (li,. (q. ), α) → (lj,. (q.u ), ρui,j (q. )α) .

(2.55)

Moreover ρui,j (q.n ) tends almost surely in the basis for all elements in the fiber to ˜ u is compatible with the different type of change of ρui,j (q. ). For q.n , the map G charts, because ωQ is invariant under rotation. ˜ fin (Q) over Lfin (Q) which ˜ u over the formal circle bundle L We deduce a map G gives a periodic group of formal bundle transformations. It is a lift of the natural circle action over Lfin (Q). Let us remark that the 2-cocycle which gives the basical central extension of Lfin (G) is invariant by rotation. We have therefore a circle ˜ fin(G), which is a lift of the natural circle action over Lfin (G). action over L ˜ fin (G), we denote ˜ If q˜ ∈ Lfin (Q), we denote the rotated element by q˜u . If g˜ ∈ L u the rotated element by g˜ . We get: (˜ q g˜)u = q˜u g˜u .

(2.56)

These arguments are formals because we did not define the topological space ˜ fin (Q). We have to operate at the level of sections. L Let Viu be the rotated “neighborhood” of Vi by the circle action. Let ψ be a section of Spin. Over Viu , we put: ψiu (γ) = ψi (γ −u ) .

(2.57)

We get the formula over Viu ∩ Vju ψj (γ) = ρ˜j,i (γ −u )ψiu (γ)

(2.58)

ψj (γ) = ρ˜j,i (γ)ψi (γ) .

(2.59)

since over Vi ∩ Vj If γ ∈ Viu ∩ Vk , we would like to get the formula: ψiu (γ) = ρ˜ui,k (γ)ψk,u (γ) .

(2.60)

˜ fin (Q): to γ u , we can associate the If γ ∈ Vi , there is a section q˜i,. (γ) over Vi of L . u u u u (γ u ) and q˜j,. (γ u ) is the same section q˜i,. (γ ) and the transition map between q˜i,. as the transition map between the non-rotated sections because ωQ is invariant by

266

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u rotation. It remains to compute the transition map between q˜i,. (γ u ) and q˜k,. (γ u ) in u n order to state (2.60). Moreover ρ˜i,j (γ ) tends almost surely to ρ˜ui,k (γ). It remains to show if γ ∈ Vk ∩ Vk0 that:

ψk,u (γ) = ρ˜k,k0 (γ)ψk0 ,u (γ) .

(2.61)

ρui,k (γ))−1 ψiu (γ) ψk,u (γ) = (˜

(2.62)

ρui0 ,k0 )−1 ψiu0 (γ) . ψk0 ,u (γ) = (˜

(2.63)

ρk,k0 (γ)(˜ ρui0 ,k0 (γ))−1 ψiu0 (γ) . ψiu (γ) = ρ˜ui,k (γ)˜

(2.64)

But by (2.60), and So (2.61) gives: But for all γ n , we have surely, because the bundle is invariant under rotation over Lfin (M ): ρk,k0 (γ n )(˜ ρui0 ,k0 (γ n ))−1 = ρ˜i,i0 ((γ n )−u ) (2.65) ρ˜ui,k (γ n )˜ It remains to pass to the limit when n → ∞ in (2.65). It is clear that we have an isometry.

3. Functionals over the String Bundle Let us come back to the based loop space situation in order to simplify the ˜ x,fin(Q) over exposure. We have constructed in [49] a formal principal bundle L Lx (M ): Lx (M ) is the based loop space, that is the space of continuous applications γ from S 1 into M such that γ0 = γ1 = x, endowed with the brownian measure. ˜ x,fin(Q) has as structure group L ˜ e,fin (G), a central extension of Le,fin (G), the based L loop space of loop of finite energy starting from the unit element e in G. This bundle is only formal, because the transition functions are only almost surely defined. If we ˜ e,fin (G), this allows us to construct the space introduce a unitary representation of L p of L sections of the associated bundle. But generally, we have the same problem ˜ x,fin (Q). as in the first part: we do not have sections of L ˜ fin (Q): but the The goal is to define the space of measurable functionals over L problem is more complicated than in the first part, because in the case where the formal fiber is the circle, there is a measure invariant under the group action, the Haar measure. In our situation, we have first of all to construct a measure over the fiber, and to study if this measure is invariant under the group action. ˜ e,fin (G). Let us construct a measure over L We choose for the compact simple 2-connected Lie group the matrix notation. Let Bs be a brownian motion over the Lie algebra of G and C a non-degenerate Gaussian measure independant of B. We consider the following equation: dgs = gs (C + Bs )ds

(3.1)

with g0 = e. Proposition 3.1. g1 has a smooth density q(y) with respect to the Haar measure over G.

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First Proof. We consider the stochastic differential equation: dXs = dBs dgs = gs (C + Bs )ds

(3.2)

By the Hormander’s theorem [34], the couple of (X1 , g1 ) has a smooth density. Second Proof. We will present a version which is closer to the work of [5, 22, 60]. Let us show that there exists integration by parts for g1 . We perform the transformation: C → C + λC 0 Bs → Bs + λHs (3.3) H0 = 0 where Hs is adapted bounded. Let us consider the solution of the differential equation: dgs (λ) = gs (λ)(C + λC 0 + Bs + λHs )ds

(3.3)

for s ≤ 1. We get by the Girsanov formula the following quasi-invariance formula: λ2 hC 0 , C 0 i F (gs1 , . . . , ssk ) exp −λhC, C 0 i − 2 Z 1 Z 1 hHs0 , δBs i − λ2 /2 |Hs0 |2 ds × exp − 0

0

= E[F (gs1 , . . . , gsk )] .

(3.4)

Using the trick of Bismut (see [9]) in order to get formulas for integration by parts, we can take derivative in λ of the first term. We get the integration by parts formula: R1 E[F (gs1 , . . . , gsk )(hC, C 0 i + 0 hHs0 , δBs i)] (3.5) P ∂ gsi > . =E < dgsi F (gs1 , . . . , gsk ), ∂λ Let us compute

∂ ∂λ gsi .

It is the solution of the differential equation: ∂gs (C + Bs )ds +gs (C 0 + Hs )ds ∂λ ∂ g0 = 0 . ∂λ

(3.6)

This can be solved by using the method of the variation of constant. We put: ∂ gs = Ks gs ∂λ

(3.7)

dKs gs + Ks dgs = Ks gs (C + Bs )ds + gs (C 0 + Hs )ds .

(3.8)

dKs = gs (C 0 + Hs )gs−1 ds .

(3.9)

Ks is the solution of

That is

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Z

This shows us that: Ks =

s

gu (C 0 + Hu )gu−1 du .

(3.10)

0

The integration by parts remains true for any Hs which is in all the Lp by density, and which is adapted. In particular any deterministic process Ks which has finite energy and which is deterministic can be written as (3.10). This shows us that: E[hdF (g1 ), K1 g1 i] = E[F (g1 )δK1 ] .

(3.11)

Moreover, by an adaptation of the techniques of [9], we deduce that we have integration by parts of any order: E[hd . . . hdhF (g1 ), K1 g1 iK2 g2 i . . .i] = E[F (g1 )δ(K1 , . . . , Kr )]

(3.12)

Moreover all vector fields over G are locally a finite combination of vector fields of the type Kg with smooth component. Let us use the following formula for the divergence: div(f (g1 )K1 g1 ) = δ(K1 )f (g1 ) − hdf (g1 ), K1 g1 i in order to deduce that g1 has a smooth density (see [9]).

(3.13)

Proposition 3.2. q(e) > 0. Proof. Let us suppose that Bs = 0. The map C → g1 (C) is a submersion in 0. Therefore, for a set of probability > 0 in B. , the map C → g1 (C + B. ) is a submersion and the law image has a density part strictly larger than 0 in e. We get a measure over Pe,fin (G). We use the techniques of the quasi-sure analysis (see [27, 3, 44]) in order to disintegrate this measure. In order to apply these techniques, we have to define a tangent Hilbert space. The tangent space of gs is the set of Ks gs where Ks is twice differentiable. As Hilbert structure, we choose: Z 1 0 2 kKs ”k2 = kKk2 . (3.14) kK0 k + 0

Since we have integration by parts in (3.11), we can define some Sobolev spaces. If Ks gs is a vector fields, we define a connection by putting: ∇(Ks gs ) = (DKs )gs

(3.15)

where D is the H-derivative. This connection is similar to the connection of [43–45, 39, 52] in the Riemannian context. This allows us to define iterated derivatives dr∇ of a functional (see [43, 44] for analoguous considerations in the Riemannian case), and by integrating the Hilbert–Schmidt norm, we get Sobolev norms and the associated Sobolev spaces Wp,r (Pe,fin (G)). Let us consider a functional F which belongs to all the Sobolev spaces. Let us consider the measure over G: f → E[F f (g1 )] .

(3.16)

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We can integrate by parts since F belongs to all the Sobolev spaces. We get a smooth density and which is since q(e) > 0: E[F |g1 = e]q(e) .

(3.17)

This shows us that E[F |g1 = e] is bounded by the Sobolev norms of F over the path group. In particular, if we imbed G into Rd , we get for any integer p > 0: E[|gs − gt |p |g1 = e] ≤ |t − s|p .

(3.18)

We deduce by using classical tightness theorems, that there exists a unique measure over the based loop space of G Le,fin (G) which disintegrates the previous measure over Pe,fin (G). Moreover this measure gives explicit formulas for the expectation of cylindrical functionals F (gs1 , . . . , gsk , g1 ), s1 < s2 < . . . < sk < 1. Namely we can choose deterministic vector fields such that Ksi = 0 for all si except for one. We deduce that the joint distribution of (gs1 , . . . , gsk , g1 ) has a smooth density q(gs1 , . . . , gsk , g1 ). This implies that: Z (F (gs1 , . . . , gsk )q(gs1 , . . . , gsk , e) dπgs1 . . . dπgsk ELe,fin (G) [F (gs1 , . . . , gsk )] = q(e) (3.19) if dπ denotes the Haar measure. (G), we have a two-cocycle with integral values denoted by ω = Over R 1 Le,fin 1 0 4π 2 0 hXs , Ys ids. It gives a central extension of Le,fin (G). We choose the smallest Killing norm over the Lie algebra such that we get a “minimum” extension. We follow for that as in [49] the construction of [17]. We integrate ω over the triangles l0 (g. )lt (g 0 ), lt (g. )l1 (g.0 ) and lt (g. )lt (g.0 ). ˜ e,fin (G). Unlike the We get a central extension of the loop group Le,fin (G) called L ˜ first part the topological space Le,fin (G) is defined, and unlike [54], the projection is always defined. The fiber is a circle. Over the circle, we choose the Haar measure. ˜ e,fin (G). Since the transition maps are of modulus one, we deduce a measure over L The generic element is called g˜. We have the following lemma: ˜ e,fin (G) which projects over π˜ g0 which Lemma 3.3. Let g˜0 be an element of L is a smooth loop of G. Then the law of g˜g˜0 is equivalent to the law of g˜ for the ˜ e,fin (G). probability measure ν over L Proof. Since g˜0 acts fiberwise as a rotation of the circle, and since the Haar measure over the circle is invariant under the group action, it is enough to show that the multiplication by π˜ g0 gives an absolute law with respect to the initial measure over the based loop space. g0,s , if we work over the path space. We Let us compute the differential of gs π˜ have: g0,s = gs π˜ g0,s (π˜ g0 )−1 g0,s ) + gs π˜ g0,s (π g˜s )−1 g0,s . dgs π˜ s (C + Bs )ds(π˜ s dπ˜

(3.20)

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By the Itˆ o formula, g0,s ) = (π˜ g0 )−1 dBs π˜ g0,s + C(s)ds d((π(˜ g0 ))−1 s Bs (π˜

(3.21)

g0 )−1 g0,s ) is still the differential of where C(s) is bounded by C|Bs |. But (π˜ s dBs π(˜ a Brownian motion. Therefore, there is Girsanov density: Z 1 Z 1 hCs0 , δBs i − 1/2 |Cs0 ]2 ds (3.22) I(g. ) = k(g. ) exp 0

0

|Cs0 |

is bounded by C|Bs |. The main problem is that the Girsanov density where does not belong to all the Lp . But, if we take the integral between ti and ti+1 for two close times, I(g. ) belongs in Lp for p closed of 1 but not equals to 1. By conditioning, this shows that E[I(g. )] = 1 and so that the law of the two processes are equivalent. R1 Let us introduce the random variable 0 |Bs |2 ds and Fn a smooth functional from R+ into [0,1], equals to 1 over [1, n] and equals to 0 over [n + 1, ∞[. Let us put: 2 ! X Z 1 Bi,s δBj,s (3.23) χn (g. ) = Fn 0

χn (g. )I(g. ) belongs to all the Sobolev spaces. This shows us that for all functions over the path space which belong to all the Sobolev spaces: g0 )χn (g. π˜ g0 ] = Eloop [F (g. )χn (g. )I(g. )] . Eloop [F (g. π˜

(3.24)

In particular χn (g. )I(g. ) belongs to all the Lp over the based loop space of G. If F g0 )χn (g. π˜ g0 ) is almost is almost surely equal to 0 over the based loop space, F (g. π˜ g0 ) is almost surely equal to 0, and therefore we see that when n → ∞ that F (g. π˜ equal to 0. Therefore the result. ˜ e,fin (Q). Let us introduce We can now define what is a measurable function over L a set Gi of functionals over the based loop space Lx (M ) which are equal to 0 outside the set Vi of the trivialisation, and which are equal to 1 almost surely to 1 over Vi . ˜ e,fin (Q) is trivial, and modulo a choice of a local slice, it is written as Over Vi , L ˜ ˜ e,fin (Q) Vi × Le,fin(G). The transition maps are measurable functionals ρ˜i,j (γ) from L ˜ into Le,fin (Q): therefore γ → ρ˜i,j (γ) is measurable and the loop π ρ˜i,j (γ) is smooth. Therefore the transition function preserves the element of measure 0 over the fiber. ˜ e,fin (Q) is a collection of funcDefinition 3.4. A measurable function over L tionals Fi (γ, g˜i ) over the local trivialization measurable such that over Vi ∩ Vj : Fi (γ, g˜i ) = Fj (γ, g˜j )

(3.25)

almost surely, where g˜i = ρ˜i,j g˜j almost surely. The transition function ρ˜i,j (γ) gives equivalent measures over the fiber. So if F ˜ e,fin (G), its essential supremum essup F is intrisically is a measurable function over L defined.

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271

We have: ˜ e,fin (Q)) is the set of measurable functionals F over the Definition 3.5. L∞ (L total space such that essup |F | is finite. A. Appendix: Projective Representation of Loop Groups This appendix is performed in order to give a brief summary of the spin repre˜ fin (G). It is strongly based upon the work of [58] and [13]. sentation of L Let us consider a complex Hilbert space of infinite dimension which is polarized: H : H+ ⊕ H− .

(a.1)

Let us suppose that H is endowed with a real structure: this is a complex anti-linear map J such that J 2 = 1 and such that: hJx, Jyi = hx, yi .

(a.2)

We introduce a symmetric bilinear form over H: (x, y) = hJx, yi .

(a.3)

The Grassmannian is the set of linear subspaces W such that the first projection over H− is Fredholm and the second projection is Hilbert–Schmidt. We suppose of course that H+ and H− are of infinite dimension. The isotropic Grassmannian GrI (H) is the subset of the Grassmannian such that (a.4) JW = W ⊥ . That is, W is isotropic for (., .). For instance, let us consider an Hilbert basis zn for n ∈ Z of H; zn n < 0 corresponds to a basis of H− and zn n > 0 corresponds to a basis of H+ . The element W which have as Hilbert basis all the zn , n < 0 except for a finite number of zn , n > 0 is an element of the Grassmannian called WS ; S is the set of n which corresponds to the basis of W . If J is the antilinear involution zn → z−n , WS corresponds to an element of the isotropic Grassmannian if JS = S c .

(a.5)

Let A be an Hilbert–Schmidt operator from WS into WS⊥ where WS belongs to the isotropic Grassmannian. The element of the type z + Az, z ∈ WS belongs to the isotropic Grassmannian if and only if: (x, Ay) = −(Ax, y)

(a.6)

for all x, y ∈ WS . It is called an antisymmetric operator for the complex bilinear form (., .). This means that: (a.7) A = −JA∗ J .

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Let us introduce the fermionic Fock space: ΛH+ = ⊕Λn H+

(a.8)

where the sum is taken over the nth antisymmetric power of H+ with n > 0. There is a relation between the isotropic Grassmannian and the Fock space. Without entering into details of the pfaffian line bundle over the isotropic Grassmannian, let us recall how this relation is performed (see [58, 13]). In order to simplify, we will take S = Z− and call the corresponding isotropic space W− . Its orthogonal for the Hermitian form is called W+ and corresponds to zn > 0. Let us introduce a Hilbert–Schmidt operator from W− into W+ . We associate the element of the Fock space: X (z−i , Az−j )zi ∧ zj . (a.9) φ(A) = 1/2 i,j>0

This map is well defined because kφ(A)kΛH+ = 1/2kAk22

(a.10)

where A is the Hilbert–Schmidt norm of the matrix. We consider eφ . We have an explicit form of it by using the Pfaffian. Namely: X X P f (z−νi , Az−νj )zν1 ∧ . . . ∧ zνn . (a.11) eφ(A) = n 0<ν1 <...<νn

The exponential vectors span the Fock space. Let us recall the definition of the relative Pfaffian of two antisymmetric Hilbert–Schmidt operators [37]. Let us consider a real Hilbert space H 0 . A and B are antisymmetric operators over H 0 for the quadratic structure over H 0 . We have if A and B are Hilbert–Schmidt: X P f (AS )P f (BS ) (a.12) P f (A, B) = S

where S describes the set of finite elements of an Hilbert basis of the Hilbert space and AS and BS the finite dimensional matrix deduced from it. This series converges because: X |P f (AS )|2 ≤ exp[1/2kAk22] . (a.13) S

Moreover, we have the following formula: P f (A, B)2 = det(I − AB) ,

(a.14)

det(I − AB) exists because AB is trace class; namely A and B are Hilbert–Schmidt. From (a.11), we deduce that: X X P f ((z−νi , Az−νj ))P f ((z−νi , Bz−νj )) (a.15) heφ(A) , eφ(B) i = n 0<ν1 ···<νn

Moreover, we have: detH− (I + A∗ B) = heφ(A) , eφ(B) i2 . This shows us a similarity with (a.14).

(a.16)

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Let us introduce the restricted orthogonal group Ores (H). It checks several conditions: (i) It belongs to U (H), the unitary group of H. (ii) The matrix element with respect to H in H− ⊕ H+ of g ∈ Ores (H) is a b (a.17) c d b and c are Hilbert–Schmidt. (iii) g commutes with J. In particular, hJg(x), yi = hg(Jx), yi = hJx, gyi .

(a.18)

So g is a rotation for the complex bilinear form (., .) over H. The main remark is the following: g applies an element of GrI (H) over an element of GrI (H). Let w1 be the set of z + Az, z ∈ H− which describes an element of GrI (H). Under the action of g (see (a.17)), it is transformed into the set (a + bA)z + (c + dA)z. Therefore if (a + bA) is invertible, its image is equal to the isotropic subspace associated to Ag where: Ag = (c + dA)(a + bA)−1 .

(a.19)

The set of A such that a+bG is invertible, A Hilbert–Schmidt is big enough because a is Fredholm and b Hilbert–Schmidt. Ores (H) has two connected components: SOres (H) is the connected component of the identity. Moreover, we have the relations [13]: heφ(Ag ) , eφ(Bg ) i2 = det(I + (Ag )∗ B g ) =

det(I + A∗ B) . det(a + bB)(a + bA)∗

(a.20)

We would like to put: g.eφ(A) = µg (A)eφ(Ag )

(a.21)

in order to define an action of SOres (H) over Λ+ H+ . We should have: (µg (A)µg (B))2 = det(a + bB)(a + bA)∗ .

(a.22)

This is related to the problem of defining the square root of the determinant. The main difficulty is to define a square root of the determinant in infinite dimension: namely, only relative pfaffians are existing [37]. Let us introduce µg (A, B) a square root of det(I + (A − B)∗ (a + bA)−1 b) by using a relative pfaffian. ˜ res (H) is the group whose elements consist of the triple Definition A.1. SO (g, A, λ) such that: (i) g ∈ SOres (H), the component of the idendity in Ores (H), A ∈ Ug and λ ∈ C ∗.

´ R. LEANDRE

274

(ii) |λ|2 =

p

det(a + bA)∗ (a + bA)

We identify (g, A, λ) and (g, B, λ0 ) if λ0 = λµg (A, B) .

(a.23)

(g1 , A1 , λ1 ).(g2 , A2 , λ2 ) = (g3 , A3 , λ3 )

(a.24)

The multiplication is given by

where g3 = g1 g2 , A3 ∈ Ug2 ∩ Ug1 such that Ag32 ∈ Ug1 and λ3 = λ1 λ2 µg1 (A1 , Ag32 )µg2 (A2 , A3 ) .

(a.25)

We get a projective representation of SOres (H), that is a representation of ˜ res (H) over Λ+ H + by the following definition: SO Definition A.2. (g, A, λ)eφ(B) = λµg (A, B)eφ(Bg ) .

(a.26)

This representation is unitary over Λ+ H+ (see [13]). Let us complexify Rd : we get Rd ⊗ C. Let ξn (ei ) = exp[2πint − iπt]ei where ei is an orthogonal basis of Rd . The Hilbert space which is spanned is called the space of half-densities over the circle, and is related to the space of the L2 applications from the circle into Rd complexified. More precisely, the space of half densities is the space of elements of the type h(t) exp(−iπt) where h(t) is an element of the L2 spaces of applications from the circle into Rd . H− corresponds to n ≤ 0; H+ corresponds to n > 0. There is a real structure, by taking the conjugation. Let g. be a finite energy loop in Spin2d . Let π be the projection from Spin2d into SO(2d). πg. + which gives a finite energy loop in SO(2d), and therefore an element of SOres (H) called πg. (see [58, 13]). ˜ fin (Spin ) as follows: We construct L 2d ˜ fin (Spin2d) is the set of elements of the triple (g. , A, λ) Definition A.3. L such that: (Spin2d ), A ∈ Uπg. , λ ∈ C ∗ . (i) g ∈ Lfinp (ii) |λ|2 = det(a + bA)∗ (a + bA) if a b (a.27) πg. = c d (g. , A, λ) and (g. , B, λ) are identified if λ0 = λµπg. (B, A) .

(a.28)

The multiplication is given by (g1,. , A1 , λ1 ).(g2,. , A2 , λ2 ) = (g3,. , A3 , λ3 )

(a.29)

where g3,. = g1,. g2,. , A3 ∈ Uπg2,. ∩ Uπg1,. such that A3 ∈ Uπg2,. ∩ Uπg. such g that A32,. ∈ Ug1,. and πg2,.

λ3 = λ1 λ2 µπg1,. (A1 , A3

)µπg2,. (A2 , A3 ) .

(a.30)

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˜ fin (Spin2d ) over Λ+ H+ which is unitary. Therefore, there is a representation of L ˜ [13, Proposition 8.1] shows that Lfin (Spin2d ) is a central extension of Lfin (Spin2d ). Let π the application of the Lie algebra of Spin2d into the Lie algebra of SO(2d). The cocycle of this central extension is represented by (see [13, Proposition 8.2]: Z 1 1 hπXs , (πY )0s ids (a.31) ω(X. , Y. ) = 2π 0 which is proportional to the fundamental cocycle which gives the basical central extension of Lfin (Spin2d ). But over Z,R the image of the Z homology group of second 1 order of Lfin (SO(2d)) by the cocycle 0 hXs , Ys0 ids is Z. Since π is a rescovering, we deduce that the image of the second order Z homology group of Lfin (Spin2d ) is Z modulo a normalizing factor; so (2.31) is the cocycle which gives the basical central ˜ fin (Spin2d ) which are all isomorphic because Spin2d is two-connected. extension of L References [1] S. Aida and D. Elworthy, “Differential calculus on path and loop spaces”, Preprint. [2] S. Aida, S. Kusuoka and D. W. Stroock, “On the support of Wiener functionals”, Asymptotics Problems in Probability Theory: Wiener Functionals and Asymptotics, eds. D. Elworthy. and N. Ikeda, Longman Scientific 284 1993, pp. 3–35. [3] H. Airault and P. Malliavin, “Quasi sure analysis”, Publication Universit´ e Paris VI (1991). [4] H. Airault and P. Malliavin, “Integration on loop groups”, Publication Paris VI (1991). [5] S. Albeverio and R. Hoegh-Krohn, “The energy representation of Sobolev Lie groups”, Compositio Math. 36 (1978) 37–52. [6] S. Albeverio, Z. M. Ma and M. Rockner, “Partition of unity in Sobolev spaces over infinite dimensional state spaces”, Preprint. [7] L. Andersson and G. Peters, “Geometric quantization on Wiener manifolds”, Stochastic analysis, eds. A. Cruzeiro and J. C. Zambrini, Progress in Probability 26, 1990, pp. 29–51. [8] G. Ben-Arous and R. Leandre, “D´ecroissance exponentielle du noyau de la chaleur sur la diagonale II”, P.T.R.F. 90 (1991) 377–402. [9] J. M. Bismut, “Martingales, the Malliavin calculus and hypoellipticity under general Hormander conditions”, Z.W. 56 (1981) 469–505. [10] J. M. Bismut, “M´ecanique al´eatoire, Lecture Notes in Math. 866, Springer, 1981. [11] J. M. Bismut, Large Deviations and the Malliavin Calculus, Progress in Math. 45, Birkhauser, 1984. [12] J. M. Bismut, “Index theorem and equivariant cohomology on the loop space”, Condensed Matter Phys. 98 (1985) 127–166. [13] D. Borthwick, “The Pfaffian line bundle”, Condensed Matter Phys. 149 (1992) 463– 493. [14] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, 1986. [15] J. L. Brylinski Loop spaces, Characteristic Classes and Geometric Quantization, Progress in Math. 107, Birkhauser, 1992. [16] J. L. Brylinski and D. Mac Laughlin, “The geometry of degree-four characteristic classes and of line bundles on loop spaces I”, Duke Math. J. 75 (1994) 603–638. [17] A. L. Carey and M. K Murray, “String structure and the path fibration of a group”, Comm. Math. Phys. 141 (1991) 441–452. [18] K. T. Chen, “Iterated path integrals of differential forms and loop space homology”, Ann. Math. 97 (1973) 213–237.

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[19] S. S. Chern Simons, “Characteristics forms and geometric invariants”, Ann. Maths. 99 (1974) 48–69. [20] R. Coquereaux and K. Pilch, “String structure on loop bundles”, Comm. Math. Phys. 120 (1989) 353–378. [21] A. Cruzeiro and P. Malliavin, “Curvatures of path spaces and stochastic analysis”, Preprint. [22] Y. Daletskii and Y. Shnaiderman, “Diffusions and quasi-invariant measures on infinite dimensional Lie groups”, Funct. Anal. and Appl. 3 (1969) 156–158. [23] B. Driver, “A Cameron-Martin type quasi-invariance for Brownian motion on compact manifolds”, J. Funct. Anal. 110 (1992) 272–376. [24] D. Elworthy, Stochastic Differential Equations on Manifold, London Math. Soc. Lectures Notes Series 20, Cambridge Univ. Press, 1982. [25] O. Enchev and D. W. Stroock, “Towards a Riemannian geometry on the path space over a Riemannian manifold”, Preprint. [26] I. B. Frenkel, “Two constructions of affine Lie algebra representations and BosonFermion correspondence in quantum field theory”, J. Funct. Anal. 44 (1981) 259–327. [27] E. Getzler, “Dirichlet form on a loop space”, Bull. Sci. Math. (2) 113 (1989) 155–174. [28] E. Getzler, J. D. S. Jones and S. Petrack, “Differential forms on loop spaces and the cyclic bar complex”, Topology 30 (1991) 339–371. [29] P. Gilkey, Invariance Theory, the Heat Equation and the Atiyah–Singer Theorem, Math. Lect. Series. 11 Publish and Perish. [30] L. Gross, “Potential theory on Hilbert spaces”, J. Funct. Anal. 1 (1967) 123–181. [31] L. Gross, “Logarithmic Sobolev inequalities on a loop group”, J. Funct. Anal. 102 (1991) 268–312. [32] L. Gross, “Uniqueness of ground states for Schrodinger operators over loop groups”, Preprint. [33] R. Hoegh-Krohn, “Relativistic quantum statistical mechanics in 2 dimensional space time”, Comm. Math. Phys. 38 (1974) 195–224. [34] L. Hormander, “Hypoelliptic second order differential equations”, Acta Math. 119 (1969), 147–171. [35] E. P. Hsu, “Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold”, Preprint. [36] M. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland 1981. [37] A. Jaffe, A. Lesniewski and J. Weitsman, “Pfaffians on Hilbert space”, J. Funct. Anal. 83 (1989) 348–363. [38] J. Jones and R. L´eandre, “Lp Chen forms over loop spaces”, Stochastic Analysis, eds. M. Barlow and N. Bingham, Cambridge Univ. Press, 1991, 104–162. [39] J. D. S. Jones and R. L´eandre, “A stochastic approach to the Dirac operator over the free loop space”, Preprint. [40] T. Killingback, “World-sheet anomalies and loop geometry”, Nucl. Phys. B 288 (1987) 578–588. [41] S. Kobayashi and S. Nomizu, Foundations of Differential Geometry. I, New-York. Intersience, 1969. [42] R. L´eandre, “Strange behaviour of the heat kernel on the diagonal”, Stochastic Processes, Physic and Geometry, ed. S. Albeverio, World Scientific, 1990, 516–528. [43] R. L´eandre, “Integration by parts formulas and rotationally invariant Sobolev Calculus on the free loop space”, XXVII Winter School of Theoretical Physic, eds. R. Gielerak and A. Borowiec, J. Geom. Phy. 11 (1993) 517–528. [44] R. L´eandre, “Invariant Sobolev Calculus on the free loop space”, to be published in Acta Appl. Math. [45] R. L´eandre, “Brownian motion over a Kahler manifold and elliptic genera of level N”,

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HILBERT C∗ -BIMODULES OVER COMMUTATIVE C∗ -ALGEBRAS AND AN ISOMORPHISM CONDITION FOR QUANTUM HEISENBERG MANIFOLDS

Centro de

BEATRIZ ABADIE† Matem´ aticas, Facultad de Ciencias Universidad de La Rep´ ublica Eduardo Acevedo 1139, C.P 11200 Montevideo, Uruguay E-mail: [email protected]

RUY EXEL∗ Departamento de Matem´ atica, Universidade de S˜ ao Paulo Cidade Universit´ aria “Armando de Salles Oliveira” Rua do Mat˜ ao 1010, CEP 05508900 S˜ ao Paulo, Brazil E-mail: [email protected] Received 19 September 1996 A study of Hilbert C ∗ -bimodules over commutative C ∗ -algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.

Introduction In [3], a theory of crossed products of C ∗ -algebras by Hilbert C ∗ -bimodules was introduced and used to describe certain deformations of Heisenberg manifolds constructed by Rieffel (see [12] and [3, Sec. 3.3]). This deformation consists of a c , depending on two real parameters µ and ν, family of C ∗ -algebras, denoted Dµν c turns out to be isomorphic to the and a positive integer c. In case µ = ν = 0, Dµν algebra of continuous functions on the Heisenberg manifold M c . 0 c and Dµc 0 ν 0 cannot be isomorphic unless c = c0 . For K-theoretical reasons [2], Dµν c and Dµc 0 ν 0 are It is the main purpose of this work to show that the C ∗ -algebras Dµν isomorphic when (µ, ν) and (µ0 , ν 0 ) are in the same orbit under the usual action of GL2 (Z) on the torus T 2 (here the parameters are viewed as running in T 2 , since c c and Dµ+n,ν+m are isomorphic for any integers m and n). Dµν c may be described as As indicated above, the quantum Heisenberg manifold Dµν ∗ 2 a crossed product of the commutative C -algebra C(T ) by a Hilbert C ∗ -bimodule. Motivated by this, we are led to study some special features of Hilbert C ∗ -bimodules over commutative C ∗ -algebras, which are relevant to our purposes.

† Partially

∗ Partially

supported by CONICYT, Proyecto 2002, Uruguay. supported by CNPq, Brazil.

411 Reviews in Mathematical Physics, Vol. 9, No. 4 (1997) 411–423 c World Scientific Publishing Company

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In Sec. 1 we consider, for a commutative C ∗ -algebra A, two subgroups of its Picard group Pic(A): the group of automorphisms of A (embedded in Pic(A) as in [5]), and the classical Picard group CPic(A) (see, for instance, [7]) consisting of Hilbert line bundles over the spectrum of A. Namely, we prove that Pic(A) is the semidirect product of CPic(A) by Aut(A). This result carries over a slightly more general setting, and a similar statement (see Proposition 1.1) holds for Hilbert C ∗ bimodules that are not full, partial automorphisms playing then the role of Aut(A). These results provide a tool that enables us to deal with Pic(C(T2 )) in order to prove our isomorphism theorem for quantum Heisenberg manifolds, which is done in Sec. 2. The authors would like to acknowledge the financial support from FAPESP (grant no. 95/4609-0), Brazil, and CONICYT, Uruguay. 1. The Picard Group and the Classical Picard Group Notation. Let A be C ∗ -algebra. If M is a Hilbert C ∗ -bimodule over A (in the M sense of [6, Sec. 1.8]) we denote by h , iM L , and h , iR , respectively, the left and right A-valued inner products, and drop the superscript whenever the context is clear enough. If M is a left (resp. right) Hilbert C ∗ -module over A, we denote by K(AM ) (resp. K(MA )) the C ∗ -algebra of compact operators on M . When M is a Hilbert C ∗ -bimodule over A we will view the elements of hM, M iR (resp. hM, M iL ) as compact operators on the left (resp. right) module M , as well as elements of A, via the well-known identity: hm, niL p = mhn, piR , for m, n, p ∈ M . ˜ is the dual bimodule of M , as defined in The bimodule denoted by M [9, Sec. 6.17]. By an isomorphism of left (resp. right) Hilbert C ∗ -modules we mean an isomorphism of left (resp. right) modules that preserves the left (resp. right) inner product. An isomorphism of Hilbert C ∗ -bimodules is an isomorphism of both left and right Hilbert C ∗ -modules. We recall from [5, Sec. 3] that Pic(A), the Picard group of A, consists of isomorphism classes of full Hilbert C ∗ -bimodules over A (that is, Hilbert C ∗ -bimodules M such that hM, M iL = hM, M iR = A), equipped with the tensor product, as defined in [9, Sec. 5.9]. It was shown in [5, Sec. 3.1] that there is an anti-homomorphism from Aut(A) to Pic(A) such that the sequence 1 −→ Gin(A) −→ Aut(A) −→ Pic(A) is exact, where Gin(A) is the group of generalized inner automorphisms of A. By this correspondence, an automorphism α is mapped to a bimodule that corresponds to the one we denote by Aα−1 (see below), so that α 7→ Aα is a group homomorphism having Gin(A) as its kernel. Given a partial automorphism (I, J, θ) of a C ∗ -algebra A, we denote by Jθ the corresponding [3, Sec. 3.2] Hilbert C ∗ -bimodule over A. That is, Jθ consists of the vector space J endowed with the A-actions:

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C ∗ -BIMODULES OVER

COMMUTATIVE

a · x = ax,

C ∗ -ALGEBRAS AND . . .

413

x · a = θ[θ−1 (x)a] ,

and the inner products hx, yiL = xy ∗ , and hx, yiR = θ−1 (x∗ y) , for x, y ∈ J, and a ∈ A. If M is a Hilbert C ∗ -bimodule over A, we denote by Mθ the Hilbert C ∗ -bimodule obtained by taking the tensor product M ⊗A Jθ . The map m ⊗ j 7→ mj, for m ∈ M, j ∈ J, identifies Mθ with the vector space M J equipped with the A-actions: a · mj = amj,

mj · a = mθ[θ−1 (j)a] ,

and the inner products θ = hx, yiM hx, yiM L L ,

and θ = θ−1 (hx, yiM hx, yiM R R ),

where m ∈ M , j ∈ J, x, y ∈ M J, and a ∈ A. As mentioned above, when M is a C ∗ -algebra A, equipped with its usual structure of Hilbert C ∗ -bimodule over A, and θ ∈ Aut(A) the bimodule Aθ corresponds to the element of Pic(A) denoted by Xθ−1 in [5, Sec. 3], so we have Aθ ⊗ Aσ ∼ = Aθσ fθ ∼ −1 and A for all θ, σ ∈ Aut(A). A = θ In this section we discuss the interdependence between the left and the right structure of a Hilbert C ∗ -bimodule. Proposition 1.1 shows that the right structure is determined, up to a partial isomorphism, by the left one. By specializing this result to the case of full Hilbert C ∗ -bimodules over a commutative C ∗ -algebra, we are able to describe Pic(A) as the semidirect product of the classical Picard group of A by the group of automorphisms of A. Proposition 1.1. Let M and N be Hilbert C ∗ -bimodules over a C ∗ -algebra A. If Φ : M −→ N is an isomorphism of left A-Hilbert C ∗ -modules, then there is a partial automorphism (I, J, θ) of A such that Φ : Mθ −→ N is an isomorphism of A − A Hilbert C ∗ -bimodules. Namely, I = hN, N iR , J = hM, M iR and θ(hΦ(m0 ), Φ(m1 )iR ) = hm0 , m1 iR . Proof. Let Φ : M −→ N be a left A-Hilbert C ∗ -module isomorphism. Notice that, if m ∈ M , and kmk = 1, then, for all mi , m0i ∈ M , and i = 1, ..., n: k

P

P hm, mi iL m0i k P = kΦ( hm, mi iL m0i )k P = k hm, mi iL Φ(m0i )k P = k hΦ(m), Φ(mi )iL Φ(m0i )k P = k Φ(m)hΦ(mi ), Φ(m0i )iR k .

mhmi , m0i iR k = k

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Therefore: k

P P hmi , m0i iR k = sup{m:kmk=1} k mhmi , m0i iR k P = sup{m:kmk=1} k Φ(m)hΦ(mi ), Φ(m0i )iR k P = k hΦ(mi ), Φ(m0i )iR k .

Set I = hN, N iR , and J = hM, M iR , and let θ : I −→ J be the isometry defined by θ(hΦ(m1 ), Φ(m2 )iR ) = hm1 , m2 iR , for m1 , m2 ∈ M . Then, θ(hΦ(m1 ), Φ(m2 )i∗R ) = θ(hΦ(m2 ), Φ(m1 )iR ) = hm2 , m1 iR = hm1 , m2 i∗R = θ(hΦ(m1 ), Φ(m2 )iR )∗ , and θ(hΦ(m1 ), Φ(m2 )iR hΦ(m01 ), Φ(m02 )iR ) = θ(hΦ(m1 ), Φ(m2 )hΦ(m01 ), Φ(m02 )iR iR ) = θ(hΦ(m1 ), hΦ(m2 ), Φ(m01 )iL Φ(m02 )iR ) = hm1 , hΦ(m2 ), Φ(m01 )iL m02 iR = hm1 , hm2 , m01 iL m02 iR = hm1 , m2 hm01 , m02 iR iR = hm1 , m2 iR hm01 , m02 iR = θ(hm1 , m2 iR )θ(hm01 , m02 iR ) , which shows that θ is an isomorphism. Besides, Φ : Mθ −→ N is a Hilbert C ∗ -bimodule isomorphism: Φ(mhm1 , m2 iR · a) = Φ(mθ[θ−1 (hm1 , m2 iR )a] = Φ(mθ(hΦ(m1 ), Φ(m2 )aiR )) = Φ(mhm1 , Φ−1 (Φ(m2 )a)iR ) = Φ(hm, m1 iL Φ−1 (Φ(m2 )a)) = hm, m1 iL Φ(m2 )a = Φ(hm, m1 iL m2 )a = Φ(mhm1 , m2 iR )a , and Mθ . hΦ(m1 ), Φ(m2 )iR = θ−1 (hm1 , m2 iM R ) = hm1 , m2 iR

Finally, Φ is onto because Φ(Mθ ) = Φ(M hM, M iR ) = Φ(M ) = N .

Corollary 1.2. Let M and N be Hilbert C ∗ -bimodules over a C ∗ -algebra A, and let Φ : M −→ N be an isomorphism of left Hilbert C ∗ -modules. Then Φ is

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C ∗ -BIMODULES OVER

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C ∗ -ALGEBRAS AND . . .

415

an isomorphism of Hilbert C ∗ -bimodules if and only if Φ preserves either the right inner product or the right A-action. Proof. Let θ be as in Proposition 1.1, so that Φ : Mθ −→ N is a Hilbert C ∗ -bimodule isomorphism. If Φ preserves the right inner product, then θ is the identity map on hM, M iR and Mθ = M . If Φ preserves the right action of A, then, for m0 , m1 , m2 ∈ M we have: Φ(m0 )hΦ(m1 ), Φ(m2 )iR = hΦ(m0 ), Φ(m1 )iL Φ(m2 ) = hm0 , m1 iL Φ(m2 ) = Φ(m0 hm1 , m2 iR ) = Φ(m0 )hm1 , m2 iR ,

so Φ preserves the right inner product as well.

Proposition 1.3. Let M and N be left Hilbert C ∗ -modules over a C ∗ -algebra A. If M and N are isomorphic as left A-modules, and K(AM ) is unital, then M and N are isomorphic as left Hilbert C ∗ -modules. Proof. First recall that any A-linear map T : M −→ N is adjointable. For if P mi , m0i ∈ M , i = 1, ..., n are such that hmi , m0i iR = 1K(AM) , then for any m ∈ M : T (m) = T

X

X X hm, mi iL T (m0i ) = ξmi ,T m0i (m) , hm, mi iL m0i =

where ξm,n : M −→ N is the compact operator (see, for instance, [8, Sec. 1]) defined by ξm,n (m0 ) = hm0 , miL n, for m ∈ M , and n ∈ N , which is adjointable. Let T : M −→ N be an isomorphism of left modules, and set S : M −→ N , S = T (T ∗ T )−1/2 . Then S is an A-linear map, therefore adjointable. Furthermore, S is a left Hilbert C ∗ -module isomorphism: if m0 , m1 ∈ M , then hS(m0 ), S(m1 )iL = hT (T ∗ T )−1/2 m0 , T (T ∗T )−1/2 m1 iL = hm0 , (T ∗ T )−1/2 T ∗ T (T ∗T )−1/2 m1 iL = hm0 , m1 iL .

We next discuss the Picard group of a C ∗ -algebra A. Proposition 1.1 shows that the left structure of a full Hilbert C ∗ -bimodule over A is determined, up to an isomorphism of A, by its left structure. This suggests describing Pic(A) in terms of the subgroup Aut(A) together with a cross-section of the equivalence classes under left Hilbert C ∗ -modules isomorphisms. When A is commutative there is a natural choice for this cross-section: the family of symmetric Hilbert C ∗ -bimodules (see Definition 1.5). That is the reason why we now concentrate on commutative C ∗ -algebras and their symmetric Hilbert C ∗ bimodules. Proposition 1.4. Let A be a commutative C ∗ algebra and M a Hilbert C ∗ bimodule over A. Then hm, niL p = hp, niL m for all m, n, p ∈ M .

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Proof. We first prove the proposition for m = n, the statement will then follow from polarization identities. Let m, p ∈ M , then: hhm, miL p − hp, miL m, hm, miL p − hp, miL miL = hhm, miL p, hm, miL piL − hhm, miL p, hp, miL miL − hhp, miL m, hm, miL piL + hhp, miL m, hp, miL miL = hmhm, piR hp, miR , miL − hm, miL hp, miL hm, piL − hp, miL hm, piL hm, miL + hp, miL hm, miL hm, piL = hmhp, miR hm, piR , miL − hm, miL hp, miL hm, piL = hmhp, miR , mhp, miR iL − hm, miL hp, miL hm, piL = hhm, piL m, hm, piL miL − hm, miL hp, miL hm, piL = 0. Now, for m, n, p ∈ M , we have: 1X k i hm + ik n, m + ik niL p 4 3

hm, niL p = = =

1 4 1 4

k=0 3 X k=0 3 X

ik hp, m + ik niL (m + ik n) ik phm + ik n, m + ik niR

k=0

= phn, miR = hp, niL m . Definition 1.5. Let A be a commutative C ∗ -algebra. A Hilbert C ∗ -bimodule M over A is said to be symmetric if am = ma for all m ∈ M, and a ∈ A. If M is a Hilbert C ∗ -bimodule over A, the symmetrization of M is the symmetric Hilbert C ∗ bimodule M s , whose underlying vector space is M with its given left Hilbert-module structure, and right structure defined by: m · a = am,

s

= hm1 , m0 iM hm0 , m1 iM R L ,

for a ∈ A, m, m0 , m1 ∈ M s . The commutativity of A guarantees the compatibility of the left and right actions. As for the inner products, we have, in view of Proposition 1.4 : s · m2 = hm0 , m1 iM hm0 , m1 iM L L m2 = hm2 , m1 iM m0 L = m0 · hm2 , m1 iM L s

= m0 · hm1 , m2 iM , R for all m0 , m1 , m2 ∈ M s .

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Remark 1.6. By Corollary 1.2 the bimodule M s is, up to isomorphisms, the only symmetric Hilbert C ∗ -bimodule that is isomorphic to M as a left Hilbert module. Remark 1.7. Let M be a symmetric Hilbert C ∗ -bimodule over a commutative C -algebra A such that K(AM ) is unital. By Remark 1.6 and Proposition 1.3, a symmetric Hilbert C ∗ -bimodule over A is isomorphic to M if and only if it is isomorphic to M as a left module. ∗

Example 1.8. Let A = C(X) be a commutative unital C ∗ -algebra, and let M be a Hilbert C ∗ -bimodule over A that is, as a left Hilbert C ∗ -module, isomorphic to An p, for some p ∈ Proj(Mn (A)). This implies that pMn (A)p ∼ = K(AM ) is isomorphic to a C ∗ -subalgebra of A and is, in particular, commutative. By viewing Mn (A) as C(X, Mn (C)) one gets that p(x)Mn (C)p(x) is a commutative C ∗ -algebra, hence rank p(x) ≤ 1 for all x ∈ X. Conversely, let A = C(X) be as above, and let p : X −→ Proj(Mn (C)) be a continuous map, such that rank p(x) ≤ 1 for all x ∈ X. Then An p is a Hilbert C ∗ -bimodule over A for its usual left structure, the right action of A by pointwise multiplication, and right inner product given by: hm, riL = τ (m∗ r) , for m, r ∈ An p, a ∈ A, and where τ is the usual A-valued trace on Mn (A) (that is, P τ [(aij )] = aii ). To show the compatibility of the inner products, notice that for any T ∈ Mn (A), and x ∈ X we have: (pT p)(x) = p(x)T (x)p(x) = [trace(p(x)T (x)p(x))]p(x) , which implies that pT p = τ (pT p)p. Then, for m, r, s ∈ M : hm, riL s = mpr∗ sp = mτ (pr∗ sp)p = mτ (r∗ s) = m · hr, siR . Besides, An p is symmetric: hm, riR = τ (m∗ r) =

n X

m∗i ri = hr, miL ,

i=1

for m = (m1 , m2 , ..., mn ), r = (r1 , r2 , ...rn ) ∈ M . Therefore, by Remark 1.7, if p, q ∈ Proj(Mn (A)), the Hilbert C ∗ -bimodules An p and An q described above are isomorphic if and only if p and q are Murray-von Neumann equivalent. Notice that the identity of K(AAn p) is τ (p), that is, the characteristic function of the set {x ∈ X : rank p(x) = 1}. Therefore An p is full as a right module if and only if rank p(x) = 1 for all x ∈ X, which happens in particular when X is connected, and p 6= 0. Proposition 1.9. Let A be a commutative C ∗ -algebra. For any Hilbert C ∗ bimodule M over A there is a partial automorphism (hM, M iR , hM, M iL , θ) of A such that the map i : (M s )θ −→ M defined by i(m) = m is an isomorphism of Hilbert C ∗ -bimodules.

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Proof. The map i : M s −→ M is a left Hilbert C ∗ -modules isomorphism. The existence of θ, with I = hM, M iR and J = hM s , M s iR = hM, M iL , follows from Proposition 1.1. We now turn to the discussion of the group Pic(A) for a commutative C ∗ -algebra A. For a full Hilbert C ∗ -bimodule M over A, we denote by [M ] its equivalence class in Pic(A). For a commutative C ∗ -algebra A, the group Gin(A) is trivial, so the map α 7→ Aα is one-to-one. In what follows we identify, via that map, Aut(A) with a subgroup of Pic(A). Symmetric full Hilbert C ∗ -bimodules over a commutative C ∗ -algebra A = C(X) are known to correspond to line bundles over X. The subgroup of Pic(A) consisting of isomorphism classes of symmetric Hilbert C ∗ -bimodules is usually called the classical Picard group of A, and will be denoted by CPic(A). We next specialize the result above to the case of full bimodules. Notation 1.10. For α ∈ Aut(A), and M a Hilbert C ∗ -bimodule over A, we denote by α(M ) the Hilbert C ∗ -bimodule α(M ) = Aα ⊗ M ⊗ Aα−1 . Remark 1.11. The map a ⊗ m ⊗ b 7→ amb identifies Aα ⊗ M ⊗ Aα−1 with M equipped with the actions: a · m = α−1 (a)m,

m · a = mα−1 (a) ,

and inner products ), hm0 , m1 iL = α(hm0 , m1 iM L and hm0 , m1 iR = α(hm0 , m1 iM R ), for a ∈ A, and m, m0 , m1 ∈ M. Theorem 1.12. Let A be a commutative C ∗ -algebra. Then CPic(A) is a normal subgroup of Pic(A) and Pic(A) = CPic(A)×Aut(A) , where the action of Aut(A) is given by conjugation, that is α · M = α(M ). Proof. Given [M ] ∈ Pic(A) write, as in Proposition 1.9, M ∼ = Mθs , θ being an isomorphism from hM, M iR = A onto hM, M iL = A. Therefore M ∼ = M s ⊗ Aθ , where [M s ] ∈ CPic(A) and θ ∈ Aut(A). If [S] ∈ CPic(A) and α ∈ Aut(A) are such that M ∼ = S ⊗ Aα , then S and M s are symmetric bimodules, and they are both isomorphic to M as left Hilbert C ∗ -modules. This implies, by Remark 1.6, that they are isomorphic. Thus we have: gs ⊗ M s ⊗ Aθ ∼ gs ⊗ M s ⊗ Aα ∼ M s ⊗ Aθ ∼ = M s ⊗ Aα ⇒ Aθ ∼ =M =M = Aα , which implies [5, Sec. 3.1] that θα−1 ∈ Gin(A) = {id}, so α = θ, and the decomposition above is unique.

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It only remains to show that CPic(A) is normal in Pic(A), and it suffices to prove that [Aα ⊗ S ⊗ Aα−1 ] ∈ CPic(A) for all [S] ∈ CPic(A), and α ∈ Aut(A), which follows from Remark 1.11. Notation 1.13. If α ∈ Aut(A), then for any positive integers k, l, we still denote by α the automorphism of Mk×l (A) defined by α[(aij )] = (α(aij )). Lemma 1.14. Let A be a commutative unital C ∗ -algebra, and p ∈ Proj(Mn (A)) be such that An p is a symmetric Hilbert C ∗ -bimodule over A, for the structure described in Example 1.8. If α ∈ Aut(A), then α(An p) ∼ = An α(p). Proof. Set J : α(An p) −→ An α(p), r ∈ Aα−1 , and x ∈ An p. Notice that

J(m ⊗ x ⊗ r) = mα(xr), for m ∈ Aα ,

mα(xr) = mα(xpr) = mα(xr)α(p) ∈ An α(p) . Besides, if a ∈ A J(m · a ⊗ x ⊗ r) = J(mα(a) ⊗ x ⊗ r) = mα(axr) = J(m ⊗ a · x ⊗ r) , and J(m ⊗ x · a ⊗ r) = mα(xar) = J(m ⊗ x ⊗ a · r) , so the definition above makes sense. We now show that J is a Hilbert C ∗ -bimodule isomorphism. For m ∈ Aα , n ∈ Aα−1 , x ∈ An p, and a ∈ A, we have: J(a · (m ⊗ x ⊗ r)) = J(am ⊗ x ⊗ r) = amα(xr) = a · J(m ⊗ x ⊗ r) , and J(m ⊗ x ⊗ r · a) = mα(x(rα−1 (a)) = mα(xr)a = J((m ⊗ x ⊗ r) · a) . Finally, hJ(m ⊗ x ⊗ r), J(m0 ⊗ x0 ⊗ r0 )iL = hmα(xr), m0 α(x0 r0 )iL = hm · [(xr)(x0 r0 )∗ ], m0 iL n

p = hm · hx · hr, r00 iA , x0 iA , m0 iL L L An p⊗Aα−1

= hm · hx ⊗ r, x0 ⊗ r0 iL

, m0 iL

= hm ⊗ x ⊗ r, m0 ⊗ x0 ⊗ r0 iL , which shows, by Corollary 1.2, that J is a Hilbert C ∗ -bimodule isomorphism.

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Proposition 1.15. Let A be a commutative unital C ∗ -algebra and M a Hilbert C -bimodule over A. If α ∈ Aut(A) is homotopic to the identity, then ∗

Aα ⊗ M ∼ = M ⊗ Aγ −1 αγ , where γ ∈ Aut(A) is such that M ∼ = (M s )γ . Proof. We then have that K(AM ) is unital so, in view of Proposition 1.3 we can assume that M s = An p with the Hilbert C ∗ -bimodule structure described in Example 1.8, for some positive integer n, and p ∈ Proj(Mn (A)). Since p and α(p) are homotopic, they are Murray-von Neumann equivalent [4, Sec. 4]. Then, by Lemma 1.14 and Example 1.8, we have Aα ⊗ M ∼ = M s ⊗ Aαγ ∼ = M ⊗ Aγ −1 αγ . = Aα ⊗ M s ⊗ Aγ ∼

We turn now to the discussion of crossed products by Hilbert C ∗ -bimodules, as defined in [3]. For a Hilbert C ∗ -bimodule M over a C ∗ -algebra A, we denote by A×M Z the crossed product C ∗ -algebra. We next establish some general results that will be used later. Notation 1.16. In what follows, for A − A Hilbert C ∗ -bimodules M and N we cp write M ∼ = A×N Z. = N to denote A×M Z ∼ Proposition 1.17. Let A be a C ∗ -algebra, M an A − A Hilbert C ∗ -bimodule and α ∈ Aut(A). Then cp ˜, (i) M ∼ =M cp

(ii) M ∼ = α(M ). Proof. Let iA and iM denote the standard embeddings of A and M in A×M Z, respectively. (i) Set ∗ ˜ −→ A×M Z, i ˜ (m) iM˜ : M M ˜ = iM (m) . ˜ ): Then (iA , iM˜ ) is covariant for (A, M g∗ ) = [iM (ma∗ )]∗ = iA (a)iM (m)∗ = iA (a)iM˜ (m) ˜ = iM˜ (ma ˜ , iM˜ (a · m) ˜

M iM˜ (m˜1 )iM˜ (m˜2 )∗ = iM (m1 )∗ iM (m2 ) = iA (hm0 , m1 iM R ) = iA (hm0 , m1 iL ) ,

for a ∈ A and m, m0 , m1 ∈ M . Analogous computations prove covariance on the right. By the universal property of the crossed products there is a homomorphism ˜˜ = M , by reversing the construction above one from A×M˜ Z onto A×M Z. Since M gets the inverse of J. (ii) Set jA : A −→ A×M Z, jα(M) : M −→ A×M Z , defined by jA = iA ◦α−1 , jα(M) (m) = iM (m), where the sets M and α(M ) are identified as in Remark 1.11. Then (jA , jα(M) ) is covariant for (A, α(M )): jα(M) (a · m) = jα(M) (α−1 (a)m) = iA (α−1 (a))iM (m) = jA (a)iα(M) (m) ,

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jα(M) (m0 )jα(M) (m1 )∗ = iM (m0 )iM (m1 )∗ = iA (hm0 , m1 iM ) L α(M)

) = jA (hm0 , m1 iL = jA (αhm0 , m1 iM L

),

for a ∈ A, m, m0 , m1 ∈ M , and analogously on the right. Therefore there is a homomorphism J : A×α(M) Z −→ A×M Z , whose inverse is obtained by applying the construction above to α−1 .

2. An Application: Isomorphism Classes for Quantum Heisenberg Manifolds c For µ, ν ∈ R and a positive integer c, the quantum Heisenberg manifold Dµν 2 [12] is isomorphic [3, Ex. 3.3] to the crossed product C(T )×(Xνc )αµν Z, where Xνc is the vector space of continuous functions on R × T satisfying f (x + 1, y) = e(−c(y − ν))f (x, y). The left and right actions of C(T2 ) are defined by pointwise multiplication, the inner products by hf, giL = f g, and hf, giR = f g, and αµν ∈ Aut(C(T2 )) is given by αµν (x, y) = (x + 2µ, y + 2ν), and, for t ∈ R, e(t) = exp(2πit). c : µ, ν ∈ R, c ∈ Z, c > 0}. Our purpose is to find isomorphisms in the family {Dµν c ∼ We concentrate in fixed values of c, because K0 (Dµν ) = Z3 ⊕ Zc [2]. Besides, since αµν = αµ+m,ν+n for all m, n ∈ Z, we view from now on the parameters µ and ν as running in T. Let M c denote the set of continuous functions on R × T satisfying f (x + 1, y) = e(−cy)f (x, y). Then M c is a Hilbert C ∗ -bimodule over C(T2 ), for pointwise action and inner products given by the same formulas as in X c . The map f 7→ f˜, where f˜(x, y) = f (x, y + ν), is a Hilbert C ∗ -bimodule isomorphism between (Xνc )αµν and C(T2 )σ ⊗ M c ⊗ C(T2 )ρ , where σ(x, y) = (x, y + ν), and ρ(x, y) = (x + 2µ, y + ν). In view of Proposition 1.17 we have: c ∼ Dµν = C(T2 )×C(T2 )σ ⊗M c ⊗C(T2 )ρ Z

∼ = C(T2 )×Mαc µν Z . = C(T2 )×(M c )ρσ Z ∼ As a left module over C(T2 ), M c corresponds to the module denoted by X(1, c) in [11, Sec. 3.7]. It is shown there that M c represents the element (1, c) of K0 (C(T2 )) ∼ = Z2 , where the last correspondence is given by [X] 7→ (a, b), a being the dimension of the vector bundle corresponding to X and −b its twist. It is also proven in [11] that any line bundle over C(T2 ) corresponds to the left module M c , for exactly one value of the integer c, and that M c ⊗ M d and M c+d are isomorphic as left modules. It follows now, by putting these results together, that the map c 7→ [M c ] is a group isomorphism from Z to CPic(C(T2 )). Lemma 2.1.

Pic(C(T2 )) ∼ = Z×δ Aut(C(T2 )) ,

where δα (c) = detα∗ · c, for α ∈ Aut(C(T2 )), and c ∈ Z; α∗ being the usual automorphism of K0 (C(T2 )) ∼ = Z2 , viewed as an element of GL2 (Z).

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Proof. By Theorem 1.12 we have: Pic(C(T2 )) ∼ = CPic(C(T2 ))×δ Aut(C(T2 )) . If we identify CPic(C(T2 )) with Z as above, it only remains to show that α(M c ) ∼ = α∗ ·c det . Let us view α∗ ∈ GL2 (Z) as above. Since α∗ preserves the dimension of M a bundle, and takes C(T2 ) (that is, the element (1, 0) ∈ Z2 ) to itself, we have ! 1 0 . α∗ = 0 detα∗ Now, α∗ (M c ) = α∗ (1, c) = (1, detα∗ · c) = M detα∗ ·c . Since there is cancellation in the positive semigroup of finitely generated projective modules over C(T2 ) [11], the above result implies that α∗ (M c ) and M detα∗ ·c are isomorphic as left modules. Therefore, by Remark 1.7, they are isomorphic as Hilbert C ∗ -bimodules . Theorem 2.2. If (µ, ν) and (µ0 , ν 0 ) belong to the same orbit under the usual c and Dµc 0 ν 0 action of GL(2, Z) on T2 , then the quantum Heisenberg manifolds Dµν are isomorphic. Proof. If (µ, ν) and (µ0 , ν 0 ) belong to the same orbit under the action of GL(2, Z), then αµ0 ν 0 = σαµν σ −1 , for some σ ∈ GL(2, Z). Therefore, by Lemma 2.1 and Proposition 1.17: c 2 ∼ c Mαc µ0 ν 0 ∼ = Mσα −1 = M ⊗ C(T )σασ −1 µν σ cp −1 σ∗ ·c ∼ σ∗ ·c ∼ ) = Mαdet . = C(T2 )σ ⊗ M detσ∗ ·c ⊗ C(T2 )αµν σ−1 ∼ = σ(Mαdet µν µν

In case detσ∗ = −1 we have cp −c ∼ σ∗ ·c ∼ ^ 2 ∼ ⊗ Mc ∼ , Mαdet = Mα−c =M = (M c )α−1 αµν = C(T )α−1 µν µν µν µν

since detα∗ = 1, because αµν is homotopic to the identity.

cp

On the other hand, it was shown in [1, Sec. 0.3] that Mαc −1 ∼ = Mαc µν . µ,ν

cp

Thus, in any case, Mαc µ0 ν 0 ∼ = Mαc µν . Therefore Dµc 0 ν 0 ∼ = C(T2 )×Mαc

µ0 ν 0

c Z∼ . = C(T2 )×Mαc µν Z ∼ = Dµν

References [1] B. Abadie, “Vector bundles over quantum Heisenberg manifolds”, Algebraic Methods in Operator Theory, Birkh¨ auser, (1994) 307–315. [2] B. Abadie, “Generalized fixed-point algebras of certain actions on crossed-products”, Pacific J. Math. 171 (1) (1995) 1–21.

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[3] B. Abadie, S. Eilers and R. Exel, “Morita equivalence for crossed products by Hilbert C ∗ -bimodules”, to appear in the Transactions of the AMS. [4] B. Blackadar, “K-Theory of operator algebras”, MSRI Publ. 5 (1986) Springer-Verlag. [5] G. Brown, P. Green and M. Rieffel, “Stable isomorphism and strong Morita equivalence of C*-algebras”, Pacific J. Math. 71 (2) (1977) pp. 349–363. [6] L. Brown, J. Mingo and N. Shen, “Quasi-multipliers and embeddings of Hilbert C ∗ bimodules”, Canadian J. Math. 46 (6) (1994) 1150–1174. [7] M. J. Dupr´e and R. M. Gillette, “Banach bundles, Banach modules, and automorphisms of C ∗ -algebras”, Research Notes in Math. v. 92, Adv. Publ. Program, Pitman (1983). [8] C. Lance, “Hilbert C ∗ -modules. A toolkit for operator algebraists”, Lecture Notes, University of Leeds (1993). [9] M. Rieffel, “Induced representations of C ∗ -algebras”, Advances in Math. 13 (2) (1974) 176–257. [10] M. Rieffel, “C ∗ -algebras associated with irrational rotations”, Pacific J. Math. 93 (2) (1981) 415–429. [11] M. Rieffel, “The cancellation theorem for projective modules over irrational rotation C ∗ -algebras”, Proc. London Math. Soc. (3), 47 (1983) 285–302. [12] M. Rieffel, “Deformation Quantization of Heisenberg manifolds”, Commun. Math. Phys. 122 (1989) 531–562.

EXPONENTIAL LOCALIZATION FOR MULTI¨ DIMENSIONAL SCHRODINGER OPERATOR WITH RANDOM POINT POTENTIAL ANNE BOUTET DE MONVEL Institut de Math´ ematiques de Jussieu, CNRS UMR 9994 Laboratoire de Physique Math´ ematique et G´ eom´ etrie, case 7012 Universit´ e Paris 7 Denis Diderot 2 place Jussieu F-75251 Paris Cedex 05 and Mathematical Division, B. Verkin Institute 47 Lenin Avenue, Kharkov, 310164 Ukraine

47

VADIM GRINSHPUN Math. Division, B. Verkin Institute Lenin Avenue, Kharkov, 310164 Ukraine Received 10 June 1996 Revised 29 January 1997

We consider a Schr¨ odinger operator −∆α(ω) on L2 (Rd ) (d = 2, 3) whose potential is a sum of point potentials, centered at sites of Zd , with independent and identically distributed random amplitudes. We prove the existence of the pure point spectrum and the exponential decay of the corresponding eigenfunctions at the negative semi-axis for certain regimes of the disorder. In order to prove localization results, we elucidate the structure of the generalized eigenfunctions of −∆α(ω) and the relation between its negative spectrum and the spectra of a family of infinite-order operators on `2 (Zd ). We apply the multiscale analysis scheme to investigate the point spectrum of these operators. We also prove the absolute continuity of the integrated density of states of the operator on the negative part of its spectrum.

0. Introduction Let us consider a random Schr¨ odinger operator on L2 (Rd ), formally defined by − ∆α(ω) = −∆ −

X

αj (ω)δd ( · − j)

(0.1)

j∈Zd

where ∆ is the Laplace operator. We denote by δd ( · − j) the d-dimensional point potential concentrated at j ∈ Zd . We are given identically distributed independent random variables {αj (ω)}j∈Zd , whose probability distributions have compact support and bounded density: Pα {α0 ∈ dα0 } = g(α0 )dα0 , δ −1 = sup g(α0 ) < ∞ . α0

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(0.2)

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The probability-space (Ω, P) is the space of realization of coefficients α = Q (Rj , dPαj ). The operator (0.1) can be rigorously defined {αj }j∈Zd , i.e. Ω = j∈Zd

for d ≤ 3 as the strong resolvent limit of the selfadjoint extensions of the operators −∆|C0∞ (Rd \Λ) , when Λ → Zd (see [1] for a precise definition). The resolvent kernel (−∆α − z)−1 (x, y) is given by (−∆α − z)−1 (x, y) = (−∆ − z)−1 (x, y) +

X

(−∆ − z)−1 (x, j)[Γα (z)]−1 (j, j 0 )(−∆ − z)−1 (j 0 , y) ,

j,j 0 ∈Zd

(0.3) where z ∈ / σ(−∆α ) and the operator Γα (z) : `2 (Zd ) → `2 (Zd ) is defined by the matrix: ( √ if j = j 0 , αj − i4πz , 0 (0.4) Γα (z; j, j ) = −1 0 −(−∆ − z) (j, j ) , otherwise , √ where z denotes the standard determination. In what follows, we consider only the 3 dimensional case; all our proofs and results are also valid in dimension 2, and the case of dimension 1 was already studied [12]. Notice that all our results are valid for the operator with a random point potential which is defined on any lattice Λ = {n1 e1 + n2 e2 + n3 e3 ∈ R3 | (n1 , n2 , n3 ) ∈ Z3 } , where e1 , e2 , e3 is a basis in R3 . The operator −∆α(ω) can be considered as the Hamiltonian describing the motion of a quantum mechanical particle under the influence of the potential created by “point sources” of random strength. The parameter δ can be thought of in some sense as the characteristics of the “degree of disorder” (see Remark following Theorem 1). The point spectrum of a Schr¨ odinger operator with ergodic potential on L2 (Rd ) has been much studied recently (cf. [17, 27, 25, 9, 10], and [7, 28] for a review of results and bibliography). In most models one studies the case of random potentials of the following type: X αj (ω) fj (x − xj (ω)) , Vω (x) = j∈Zd

where the single-site potentials fj corresponding to neighbouring points of the lattice Zd have intersecting supports, or at least supports lying close to each other. The operator (0.1) represents the opposite case of the so-called zero-range potential. Of course, it is in a sense an idealization, because the operator “with the point potential” (or “point interaction Hamiltonian”), defined by (0.1)–(0.4), is not the sum of the Laplace operator and some multiplication operator on L2 (Rd ) for d = 2, 3. The detailed study of the operator (0.1) with potential of general type is the subject of many recent publications. In particular, the case of a random potential

¨ EXPONENTIAL LOCALIZATION FOR MULTI-DIMENSIONAL SCHRODINGER OPERATOR

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was studied in [21, 2] (proof of the non-randomness of the spectrum and of its components, location of the spectrum), [12] (proof of the exponential localization of “eigenstates” for the whole spectrum in dimension 1). A quite complete survey of the results can be found in [1]. In the present work we study the negative part of the spectrum of the operator (0.1)–(0.4), in dimensions d = 2, 3. We prove (Theorem 1) that when the disorder δ increases to infinity, the pure point begins to occupy the negative part of the spectrum of −∆α(ω) , and when δ reaches a certain value δ0 < ∞, then all the negative spectrum becomes pure point. We also prove (Theorem 6) that the density of states of −∆α(ω) on the negative semi-axis is absolutely continuous. The explicit expression (0.3) for the resolvent kernel of −∆α(ω) indicates that the spectral properties of this operator are connected with the spectral properties of the operator Γα(ω) (z) (defined by (0.3)) : for a fixed z, this is an operator of infinite-order on `2 (Z3 ) with random potential, selfadjoint if z = λ < 0. Our approach to study the point spectrum of −∆α(ω) is divided in two parts. First we establish a relation between the spectral properties of −∆α(ω) and Γα(ω) (λ), and secondly we prove the relevant results for Γα(ω) (λ). We prove (Theorem 2) that the spectral measure of −∆α is concentrated on the set of generalized eigenvalues which correspond to the “polynomially bounded” weak solutions (see Definition 1) of the equation − ∆α ψ = λψ .

(0.5)

This theorem is an analogue of the well-known theorem for finite-difference operators of finite order (Sch’noll lemma [32]), and for Schr¨ odinger operators with not “very singular” potentials (so-called Kν -class, [29, 26]). As was shown by [4], this theorem is valid for an arbitrary self-adjoint operator which has the following property: for any bounded Borel set ∆ ⊂ R, Tδ−1 E(∆)Tδ−1 is a trace class operator, (0.6) where Tδ is the multiplication operator by (1 + |x|) on L (R ), with δ > d/2, and E(∆) is the respective spectral projection on the ∆-subspace for A. This implies integral estimates for the corresponding generalized solutions. The pointwise polynomial boundedness depends on the continuity properties of the spectral kernel. Using the explicit expression (0.3) for the resolvent kernel, we prove statements of type (0.6) for −∆α . It follows from the general properties of the point potential that the corresponding weak solutions are not continuous at the points j ∈ Z3 : their behaviour is expected to be of type aj G0λ (x − j) for x → j ∈ Z3 , where Gλ0 (x, y) denotes the kernel of (−∆ − λ)−1 for λ < 0. Theorem 3 shows that any weak solution of (0.5), corresponding to λ < 0, has the form: δ

2

d

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ψλ (x) =

X

ai G0λ (x − i) ,

(0.7)

i∈Zd

where for a.e. λ < 0 (with respect to E(dλ)), the vector a = {ai }i∈Z3 is a (generalized) polynomially bounded solution of the equation Γα (λ)a = 0. Thus there is a one-to-one correspondence between generalized eigenfunctions of the operators −∆α and Γα (λ). This result is deterministic. Thus in order to prove the localization result for −∆α (Theorem 1), it suffices to prove that in the high disorder regime any polynomially bounded generalized eigenfunction of Γα(ω) (λ) decays exponentially as a function of `2 (Z3 ), with probability one (Theorem 5), see precise definitions and notations in Sec. 1. Results of such type appear as one of the ways to establish exponential localization for a discrete Schr¨ odinger operator with random potential (Anderson model). These results rely on the multiscale analysis for estimating the Green function [15, 16, 31, 11, 13, 33, 14]. We use an infinite-order generalization of the multiscale analysis scheme [18, 19] to obtain the required result. We give a schematic proof showing that the estimates of Sec. 4 are uniform in λ. The scheme of [3], which provides another way of proving the Anderson localization for finite-difference operators, is also applicable. However, it requires more restrictive assumptions on the potential distributions, and also some suitable adaptation of the general approach. Notice, that our approach does not require the “multiscale analysis” in R3 (however, this remains possible: localization for −∆α(ω) was first proved by the authors in that way, but turns out to be more technical and less natural than the proof given in this work). We also observe that the integrated density of states of −∆α(ω) at the negative spectrum is absolutely continuous. In order to prove this, we show the relation between the integrated density of states of operators −∆α(ω) and Γα|Λ (ω) (λ) (where α|Λ is the restriction of function α(x) to x ∈ Λ ⊂ Z3 ) using monotonicity in λ of the corresponding operators. The results of this work apply to study the point spectrum of the Schr¨ odinger operator on `2 (Zd ) with a random potential supported by some subspace, and for the related Schr¨odinger operator on a half-space with random boundary conditions [20]. In Sec. 1 we give notations and definitions, state the main results and discuss some applications. In Sec. 2 we prove Theorem 2 on the polynomial boundedness of generalized eigenfunctions. In Sec. 3 we prove Theorem 3 on the structure of weak solutions. In Sec. 4 we outline the proof of Theorem 5: localization of the family of finitedifference infinite-order operators. In Sec. 5 we prove Theorem 6 (result on the density of states). In the Appendix we prove some auxiliary results.

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1. Results Let

(

Gz (x, y) = (−∆α − z)−1 (x, y) , G0z (x, y) = (−∆ − z)−1 (x, y) =

√ i z|x−y|

e 4π|x−y|

z 6∈ σ(−∆α ) , , z 6∈ [0, +∞) ,

be the Green functions (resolvent kernels) of the operators −∆α and −∆ respectively. We denote D(−∆) and D(−∆α ) the domains of the operators −∆ and −∆α respectively (for fixed value of α = α(ω)): D(−∆) = f | f ∈ L2 (R3 ), ∆f ∈ L2 (R3 ) in the weak sense , n ϕ(x) = ϕ (x) + P 0 ∈ R3 \ Z3 , o (1.1) 0 i∈Z3 ai Gz (x P− i), x . D(−∆α ) = ϕ where ϕ0 (x) ∈ D(−∆), ai = j∈Z3 Γ−1 z (i, j)ϕ0 (j) For ϕ ∈ D(−∆α ) we have (−∆α − z) ϕ = (−∆ − z) ϕ0 .

(1.2)

By a we will usually denote the vector {ai }i∈Z3 , and by ϕ0 the vector {ϕ0 (i)}i∈Z3 . It is known [2] that the family of selfadjoint operators {−∆α(ω) }ω∈Ω is ergodic, the spectrum Σ = σ(−∆α(ω) ), as well as the point spectrum (σpp ), the absolutely continuous spectrum (σac ) and the singular continuous spectrum (σsc ) are nonrandom subsets of R (i.e. they coincide with non-random sets with probability one), and the discrete spectrum is empty (with probability one). By σpp we mean the closure of the set of the eigenvalues, then σpp is non-random. We say that a sequence α = {α(i) ∈ supp Pα }i∈Z3 is an admissible potential . We denote by A the set of all admissible potentials. Proposition 1. [2, 21] We have 1. σ (−∆α ) ⊆ Σ, α ∈ A; S 2. Σ = α∈A σ(−∆α ). Proposition 2. [21, 1] Suppose supp Pα ⊂ [µ, ν]. Then σ(−∆α ) has a band structure: [ [am , bm ] , σ(−∆α ) = m∈N

with −∞ < a0 < 0, am < bm , bm ≤ ∞, m ∈ N (the corresponding intervals can intersect). Furthermore, there exists λ0 < ∞ such that if ν < λ0 , and supp Pα = [µ, ν], then σ(−∆α ) = [a0 (µ), b0 (ν)] ∪ [A(µ), ∞) = σ(−∆µ ) ∪ σ(−∆ν ) , where b0 (ν) < 0, A(µ) ≥ 0. We now state our main result: Theorem 1. For any r, 0 < r < 1, there exist δ0 = δ0 (r, Pα ) and δ1 = δ1 (r, Pα ), with δ0 > δ1 > 0, such that :

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(i) For δ1 ≤ δ < δ0 , we can find an increasing function E0 = E0 (δ) ≤ −r < 0 such that the spectrum of −∆α(ω) in the interval (−∞, E0 ) is pure point : σ(−∆α(ω) ) ∩ (−∞, E0 ) ⊂ σpp . (ii) If δ ≥ δ0 (r), then σ(−∆α(ω) ) ∩ (−∞, −r) ⊂ σpp . Moreover, if supp Pα = [µ, ν], and ν < λ0 (with λ0 defined by Proposition 2), then we can choose δ0 such that, for any δ > δ0 : σ(−∆α(ω) ) ∩ (−∞, 0) ⊂ σpp . (iii) The eigenfunctions of −∆α , corresponding to the eigenvalues λ of the point spectrum in cases 1 and 2 decay exponentially at infinity, with probability one: X aλ,ω (i) G0λ (x − i) , ψλ,ω (x) = i∈Z3

where |aλ,ω (i)| ≤ C(λ, ω) e−mλ |i| , 0 < mλ <

√

r.

Remark 1. Theorem 1 remains true if the distribution Pα is H¨older continuous. In this case one considers the potential {λ αi }i∈Z3 and λ measures the strength of the disorder. In order to prove Theorem 1, we study the properties of the spectral kernel of −∆α and show that its spectral measure is concentrated on the set of generalized eigenvalues, corresponding to the solutions of Eq. (0.5) which are “polynomially bounded” in the sense of Definition 1 below. We denote by Eα(ω) (∆) the spectral projection of −∆α(ω) (we will usually omit the ω in our notation when this does not lead to confusion). We say that the statement S(λ) is true for a.e. λ ∈ I ⊂ R with respect to the spectral measure of −∆α , if Eα (∆) = 0 for ∆ = {λ ∈ I | S(λ) is not true}. Let δ > 3/2. We denote L2−δ = L2 (R3 , dµ−δ ), where dµ−δ (x) = (1 + |x|)−2δ dx, and L2δ = L2 (R3 , dµδ ), where dµδ (x) = (1 + |x|)2δ dx. Definition 1. A number λ ∈ R is called a generalized eigenvalue, and a function ψ ∈ L2−δ a generalized eigenfunction of −∆α (relative to λ) if hψ, (−∆α − λ)ϕi = 0

(1.3)

for every ϕ of type (1.1) with ϕ0 ∈ C0∞ , z < inf supp Pα sufficiently small (h · , · i denotes the scalar product of L2 (R3 )). We consider the functions ϕ of the definition above as test functions, and denote by D0 (−∆α ) the set of all such functions, corresponding to some “large negative” z = E0 .

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Remark 2. Lemma A1 of Appendix shows that it is possible to choose E0 in such a way that for any admissible α(ω), ω ∈ Ω, −∆α (D0 (−∆α )) ⊂ L2δ and thus Definition 1 is correct. The following Theorems 2 and 3 are deterministic. Theorem 2. Almost every λ ∈ R with respect to the spectral measure of −∆α is a generalized eigenvalue. The following theorem describes explicitly the structure of the generalized eigenfunctions of −∆α . Let us denote `2−δ = `2 (Z3 , dν−δ ), where ν−δ (i) = (1 + |i|)−2δ , i ∈ Z3 , δ > 3/2, and `2δ = `2 (Z3 , dν+δ ), where ν+δ (i) = (1 + |i|)2δ , i ∈ Z3 . We will say that a function f (x), x ∈ Z3 , decays exponentially at infinity with rate (at least) m > 0, if lim supx→∞ ln |f|x|(x)| ≤ −m. Theorem 3. (i) A function fλ (x) is a generalized eigenfunction of −∆α , relative to a negative generalized eigenvalue λ < 0, if and only if X fi G0λ (x − i) , (1.4) fλ (x) = i∈Z3

where the vector f = {fi }i∈Z3 ∈ `2−δ is a generalized solution of the equation Γα (λ) f = 0 .

(1.5)

(ii) λ < 0 is a (proper) eigenvalue of −∆α if and only if 0 is an eigenvalue of Γα (λ). It is also possible to characterize the generalized eigenfunctions for λ > 0. Theorem 4. A function Fλ (x) is a generalized eigenfunction of −∆α , relative to a positive generalized eigenvalue λ > 0, if and only if X fi G0λ (x − i) ∈ L2−δ , (1.6) Fλ (x) = φ(x) + i∈Z3

where φ(x) is the weak solution of the equation (−∆ − λ)φ(x) = 0, and f = {fj }j∈Z3 ∈ `2−δ satisfy to the equations: √ X Γα (λ; k, j) fj = e−i λk for all k ∈ Z3 , j∈Z3

where Γα (λ; k, j) is given by (0.4). Theorems 2 and 3 imply that there is a strong relation between the spectral properties of the operator −∆α(ω) and the spectral properties of the family Γα (λ). Lemma A2 of Appendix shows that generalized solutions of Eqs. (0.5) and (1.5) exist simultaneously, and also satisfy the same decay or growth conditions at infinity. The point spectrum of finite-difference operators of infinite-order, with random potential was studied recently by [18, 3, 23]. Their results, put in a convenient and improved form for our case, are stated in the following theorem:

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Theorem 5. Let I be an interval, r > 0, I ⊂ (−∞, −r). There exist δ0 = δ0 (r, |I|) and E0 = E0 (δ) (E0 < −r for any δ), such that every generalized solution of (1.5) decays exponentially at infinity with probability 1, for any λ, satisfying one of the following assumptions: a) λ ∈ I if δ ≥ δ0 (r, |I|), or b) λ ∈ (−∞, −E0 (δ)) ∩ I if δ < δ0 (r). Remark 3. Eigenfunction corresponding to an eigenvalue λ < 0, decays at √ infinity with rate at least r. Now we state the result on the density of states for negative energies. Let N (λ) be the integrated density of states of −∆α(ω) (see the discussion in Sec. 5). Theorem 6. Suppose Pα has a density bounded by δ −1 . Given any r > 0, there exists C = C(r) such that for any interval (a, b) ⊂ (−∞, −r) we have |N (b) − N (a)| ≤ 2Cδ −1 |b − a| . Remark 4. If Pα is H¨older continuous of order 0 < γ < 1, then N (λ) is also H¨ older continuous of the same order. Theorem 6 is in fact a corollary of Theorems 1–5, which are proved below. Now we give some consequences of these results. Remark 5. We will prove in a forthcoming paper the theorem on the expansion in generalized eigenfunctions of −∆α (as is done by [4, 29] for a wide class of selfadjoint operators, which however does not include point potentials). Corollary 1. If λ < 0 is a generalized eigenvalue of −∆α , then λ ∈ σ(−∆α ). This follows from Theorem 3, Lemma 3 (Sec. 3) and the fact that the generalized eigenvalues of a finite-difference operator belong to its spectrum [4]. Corollary 2. σ(−∆α ) ∩ (−∞, 0) is the closure of the set of generalized eigenvalues. This follows from Theorem 2, Corollary 1 and the fact that if λ0 ∈ σ(−∆α ) then, for any ε > 0, we have Eα (λ0 − ε, λ0 + ε) > 0. Corollary 3 (Location of the negative spectrum). We have σ(−∆α(ω) ) ∩ (−∞, 0) = λ < 0 | 0 ∈ σ (Γα(ω) (λ)) , σpp (−∆α(ω) ) ∩ (−∞, 0) = λ < 0 | 0 ∈ σpp (Γα(ω) (λ)) .

(1.7) (1.8)

Proof. Suppose λ ∈ σ(−∆α(ω) ). Then Proposition 1 implies λ ∈ σ(−∆α ), where α is some admissible potential. It follows that there exists a sequence Fn ∈ D(−∆α ), such that kFn kL2 = 1, (−∆α − λ)Fn → 0. Lemma 3 implies that there exists a sequence fn ∈ `2 (Z3 ), such that kfn k`2 = 1, Γα fn → 0, so 0 ∈ σ(Γα ) ⊂

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σ(Γα(ω) ) (because Proposition 1 holds for finite-difference ergodic operators). Now assume λ ∈ σpp (−∆α(ω) ). Thus there exists a set Ω0 ⊂ Ω, such that P (Ω0 ) = 1, and λ ∈ σpp (−∆ω0 ) for any ω0 ∈ Ω0 . According to Theorem 3 this implies that 0 ∈ σpp (Γω0 ) for all ω0 ∈ Ω0 , and hence 0 ∈ σpp (Γα(ω) ). The converse is proved in the same manner. Remark 6. It is conjectured that finite-difference ergodic operators of type Γα (λ) on `2 (Zd ) have only pure point spectrum when the dimension d = 2, but might have interval(s) (containing 0) of absolutely continuous spectrum if the dimension d = 3. Corollary 3 shows that in dimension 2 one should expect that all the negative spectrum of −∆α(ω) is pure point, while in dimension 3 there might be some (absolutely?) continuous spectrum in the neighborhood of 0. Corollary 4 (Absence of “rapidly decaying” eigenfunctions in (0, +∞)). If F (x) is the eigenfunction of −∆α , relative to the positive eigenvalue λ > 0, then F has the form (1.6), where X X |fi |2 < ∞ and |fi | = ∞ . i

i

Remark 7. We conjecture that (a, +∞) ⊂ σac (−∆α(ω) ) for some a ≥ 0. 2. Some Properties of the Spectral Kernel of −∆α Proof of Theorem 2. Let H = −∆α , and let E(∆) be the spectral projection of −∆α , GE = (−∆α − E)−1 , E < inf σ(−∆α ), Tδ = (1 + |x|)δ (multiplication operator), δ > 3/2. We follow the strategy of [29, 4]. Lemma 1. For any bounded Borel set ∆ ⊂ R, A(∆) = Tδ−1 E(∆)Tδ−1 is a trace class operator. Proof. Suppose E < inf σ(−∆α ) and consider the function S0 (x) =

1 χΛ (x) , |x| 1/2

where χΛ1/2 (x) is the characteristic function of the cube Λ1/2 (0), centered at 0, with side-length 1/2. Let X S0 (x − i) , S(x) = i∈Z3

then S(x) > 0 and |S −1 (x)| < 1/2. We denote by S the corresponding multiplication operator. Following [4, 29] we call operator on L2 (Rd ), which has an integral kernel a(x, y) obeying Z |a(x, y)|2 dy < ∞ , sup x

a Carleman operator .

Rd

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A. B. de MONVEL and V. GRINSHPUN

The resolvent formula (0.3)–(0.4) implies that S −1 GE is a Carleman operator, since Z S −2 (x) |GE (x, y)|2 d3 y < ∞ sup R3

x

(we apply Lemma A2 of Appendix and the dominated convergence theorem). Hence S −1 GE is a bounded mapping from L2 to L∞ (see [24, 29]). Let us show now that C , x ∈ σ(−∆α ), then the operator if a real function f (x) satisfies f (x) ≤ 1+|x| −1 2 ∞ S f (−∆α ) maps L to L (i.e. is of Carleman type [24]). To this end we consider the function g(x) = (x − E)f (x),

E < inf σ(−∆α ) .

Then f (−∆α ) = (−∆α − E)−1 g(−∆α ), i.e. S −1 f (−∆α ) = [S −1 (−∆α − E)−1 ] g(−∆α ) . Since g(−∆α ) is bounded in L2 , and we have shown that S −1 (−∆α − E)−1 is bounded from L2 to L∞ , then S −1 f (−∆α ) maps L2 to L∞ . Thus S −1 E(∆) is a Carleman operator. Let us denote its kernel by [S −1 E(∆)](x, y), then Z [S −1 E(∆)]2 (x, y) d3 y = B < ∞ . sup x

R3

Now we can show that Tδ−1 E(∆) is a Hilbert–Schmidt operator. Indeed, we have Tδ−1 E(∆) = Tδ−1 S [S −1 E(∆)], and Z (1 + |x|)−2δ S 2 (x)[S −1 E(∆)]2 (x, y) d3 xd3 y 3 3 R ×R Z (1 + |x|)−2δ S 2 (x) d3 x < ∞ . ≤B R3

It follows that Tδ−1 E(∆) and E(∆)Tδ−1 = (Tδ−1 E(∆))∗ are Hilbert–Schmidt opera tors, and A(∆) = Tδ−1 E(∆)Tδ−1 is of trace class. Lemma 1 is proved. Now we follow a standard argument [29]. Clearly, A(∆) is non-negative, and given any bounded set ∆ = ∪∞ n=1 ∆n , such that ∆i ∩ ∆j = ∅ for i = j, we have A(∆) = s-lim

m→∞

m X

A(∆n ) .

n=1

It follows that the operator-valued measure A(∆) is absolutely continuous with respect to the scalar measure ρ(∆) = Tr(Tδ−1 E(∆) Tδ−1 ) , so, by the Radon–Nikodym theorem, for ρ-a.e. λ, there exists a positive measurable trace class operator-valued function a(λ) such that Z a(λ) dρ(λ), Tr(a(λ)) = 1 . A(∆) = ∆

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It is easy to see that ρ is equivalent to E(∆) (in fact, ρ is a spectral measure). Let a(x, y; λ) be the integral kernel of a(λ). Consider the function F (x, y; λ) = (1 + |x|)δ a(x, y; λ) (1 + |y|)δ . One can easily prove that F has the following properties (all the statements are valid for ρ-a.e. λ): Property 1. Z R3 ×R3

|F (x, y; λ)|2 (1 + |x|)−2δ (1 + |y|)−2δ d3 xd3 y ≤ 1 .

(2.1)

Proof. This follows since Tr(a(λ)) = 1.

Property 2. For any finite bounded g(λ), λ ∈ σ(−∆α ), and for any ψ ∈ D0 (−∆α ), ϕ ∈ L2δ : Z Z 3 3 g(λ) F (x, y; λ)ϕ(x)ψ(y) d xd y dρ(λ) . (2.2) hϕ, g(−∆α )ψi = R3 ×R3

σ(−∆α )

Proof. It follows from (2.1) that the integrand in [ · ] is absolutely integrable, and the integral is a bounded function of λ. Thus (2.2) follows from the definition of E(∆). Property 3. F (x, y; λ) is continuous in (x, y) ∈ (R3 × R3 )\(Z3 × Z3 ). Let us show this in detail: we use the fact that −∆α is a self-adjoint extension of −∆|C0∞ (R3 \Z3 ) . We prove the following statement: e0∞ (R3 × f0 = −∆x − ∆y , x, y ∈ R3 , then for any fe0 ∈ C Property 4. Let H 3 3 2 6 R \ Z × Z ) (the set of infinitely differentiable functions on L (R ) with compact support containing no point of Z3 × Z3 ): 3

f0 − 2λ)fe0 i = 0 . hF (λ), (H

(2.3)

Proof. Suppose ϕx , ψy ∈ C0∞ (R3 \ Z3 ), then f0 ϕx ψy ](x, y) = [−∆x ϕ](x) ψ(y) + ϕ(x) [−∆y ψ](y) [H = [Hx ϕ](x) ψ(y) + ϕ(x) [Hy ψ](y) . Using (2.2) we obtain: Z Z g(λ) σ(−∆α )

Z

R3 ×R3

3 3 f F (x, y; λ)H0 ϕ(x)ψ(y)d xd y dρ(λ)

Z

g(λ)

= σ(−∆α )

3

3

2λF (x, y; λ) ϕ(x)ψ(y) d xd y dρ(λ) , R3 ×R3

which implies (2.3), since the last inequality is valid for any bounded g with compact support and any ϕ, ψ ∈ C0∞ (R3 \ Z3 ).

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f0 is a proper Laplacian on C e ∞ (R3 ×R3 \Z3 ×Z3 ). Property 3 follows from 4 since H 0 Property 5. For any fixed y ∈ / Z3 , F ( · , y; λ) is a generalized eigenfunction of −∆α (see Definition 1). Proof. It follows from (2.2) and Fubini’s theorem that for any ψ ∈ D0 (−∆α ), ϕ ∈ C0∞ Z Z F (x, y; λ) (−∆α − λ)ψ(x) d3 x ϕ(y) d3 y = 0 , R3

R3

from which it follows

Z R3

F (x, y; λ) (−∆α − λ)ψ(x) d3 x = 0

(2.4)

in every point y of continuity of F .

Now fix any y 6∈ Z and consider functions fλ (x) = F (x, y; λ). Then for ρ-a.e. λ, f ∈ L2−δ (see (2.1)), and f satisfies (2.4). We have constructed functions which are generalized eigenfunctions for E(∆)-almost all λ ∈ ∆, and this proves Theorem 2. 3

3. Explicit Form of Weak Solutions Proof of Theorem 3. Let fλ (x) be a generalized eigenfunction of −∆α , λ < 0. Thus f ∈ L2−δ , δ > 3/2, and hf, (−∆α − λ) ϕi = 0 for any ϕ of type ϕ(x) = ϕ0 (x) +

X

ai G0E (x − i) ,

(3.1)

(3.2)

i∈Z3

P ∞ for some E < inf supp Pα , ai = j∈Z3 Γ−1 α (E; i, j) ϕ(j), ϕ0 ∈ C0 . It follows from (3.1) and definition of −∆α , that if n o ϕ ∈ H02,2 (R3 \ Z3 ) := ϕ ∈ H02,2 (R3 ) | ϕ(j) = 0, j ∈ Z3 , supp ϕ is finite , then hf, (−∆ − λ)ϕi = 0 .

(3.3)

Let us prove that if g ∈ L2−δ satisfies (3.3) for any ϕ ∈ H02,2 (R3 \ Z3 ), then g ∈ Lin{G0λ ( · − j)}j∈Z3 (i.e. g belongs to the closure of the linear span of {G0λ ( · − j)}j∈Z3 in L2−δ ). First we prove that there exists a sequence {γi }i∈Z3 , depending only on g and λ, such that for any ψ ∈ C0∞ ⊂ D(−∆) with supp ψ ⊂ Λ ⊂ R3 : X γi ψ(i) . hg, (−∆ − λ)ψi = i∈Λ

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For this purpose we consider the function X e ψ(i) θ(x − i) , ψ(x) = ψ(x) − i∈Λ

where the function θ ∈ C0∞ is such that supp θ ⊂ Λ1/2 (0), θ(0) = 1, |(−∆ − λ)θ(x)| ≤ τλ < ∞ . Then ψe ∈ H02,2 (R3 \ Z3 ), and e = 0. hg, (−∆ − λ)ψ)i It follows that hg, (−∆ − λ)ψi =

X

hg, (−∆ − λ) θ(x − i)i ψ(i)

i∈Z3

=

X

γi ψ(i)

i∈Z3

(the r.h.s. is finite since the sum is over a finite number of points i ∈ Z3 ∩ supp ψ). Since g ∈ L2−δ , then g ∈ L2loc , and γi = hg, (−∆ − λ)θ(x − i)i Z ≤ Cλ

!1/2

|g(x)|2 (1 + |x|)−2δ d3 x

!1/2

Z

(1 + |x|)2δ d3 x

Λ1/2 (i)

Λ1/2 (i)

≤ Cλ gi (2 + |i|)δ , (3.4) P where i∈Z3 gi2 < C < ∞. Notice that the γi ’s are uniquely defined by g and λ. Indeed, if we assume that there exist {γi }i∈Z3 and {βi }i∈Z3 such that for any ψ ∈ C0∞ X X hg, (−∆ − λ)ψi = γi ψ(i) = βi ψ(i) , i∈Z3

i∈Z3

and choose ψ such that supp ψ ⊂ Λ1/2 (i), we obtain γi = βi for all i ∈ Z3 . Now let X γi G0λ (x − i) (3.5) h(x) = i∈Z3

(since γi satisfy (3.4) then for λ < 0 the series (3.5) is absolutely convergent for any x 6∈ Z3 and h ∈ L2−δ (Lemma A2 of Appendix); if λ > 0, we can define (3.5) in the weak sense). It follows that hg − h, (−∆ − λ)ψi = 0 for all ψ ∈ C0∞ , so φ = g − h is a weak solution of the equation (−∆ − λ)φ = 0 .

(3.6)

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Now we restrict ourselves to the case λ < 0. Then φ ∈ L2−δ , and (3.6) implies that φ = 0. We have proved, that if f ∈ L2−δ satisfies (3.3), then f=

X

fi G0λ (x − i) ,

(3.7)

i∈Z3

where the series converges in k · kL2−δ and the coefficients {fi }i∈Z3 satisfy (3.4). So far we have only used the fact that −∆α is a self-adjoint extension of −∆|C0∞ (R3 \Z3 ) . Now we consider the explicit form of this extension in order to determine the coefficients of the expansion (3.7). Let ϕ be an arbitrary function of type (3.2), with ϕ0 ∈ C0∞ . We set ϕ0 = {ϕ0 (j)}j∈Z3 , c = {c(j)}j∈Z3 , so that 2 c = Γ−1 E ϕ0 ; E < inf supp Pα , and |E| is chosen large enough, so that ϕ ∈ Lδ and (−∆α − λ)ϕ ∈ L2δ (Lemma A1). We have: X ci GE (x − i) . (3.8) (−∆α − λ)ϕ(x) = (−∆ − λ)ϕ0 (x) + (E − λ) i∈Z3

Substituting (3.7) and (3.8) in (3.1), and taking into account that hG0λ (x − i), (−∆ − λ)ϕ0 i = ϕ0 (i) , hG0λ (x − i), G0E (x − j)i =

1 (Γλ (i, j) − ΓE (i, j)) , E−λ

we obtain X

fi ϕ0 (i) +

X

fi cj [Γλ (i, j) − ΓE (i, j)] = 0 .

i,j∈Z3

i∈supp ϕ0

Since ϕ0 = ΓE c, the last equality can be written X X cj Γλ (j, i) fi = 0 . j∈Z3

(3.9)

i∈Z3

Choosing sequence ϕ0,n = {δ(i − n)}i∈Z3 we determine a sequence cn = {Γ−1 E (j, n)}j∈Z3 , where each cn should satisfy (3.9). This leads to the equation Γ−1 E

X

Γλ (j, i) fi = 0,

j ∈ Z3 .

(3.10)

i∈Z3 2 3 2 3 2 3 Since Γ−1 E : ` (Z ) → ` (Z ) is a bijection, and clearly [ · ] ∈ ` (Z ), we get

Γλ f = 0 . Now assume that for some λ < 0, we have Γλ f = 0,

|fi | ≤ (1 + |i|)δ , i ∈ Z3 ,

(3.11)

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and consider the function f (x) defined by (3.7). It follows from Lemma A2, that f ∈ L2−δ and the above argument shows that f satisfies (3.1) since (3.9) is valid. This proves the first part of Theorem 3. Suppose that 0 is an eigenvalue of Γλ , λ < 0, i.e. we have (3.11) and f ∈ `2 (Z3 ). Lemma A2 shows that f , defined by (3.7), belongs to L2 (R3 ). Let us define X fi G0E (x − i) (3.12) ϕ(x) = ϕ0 (x) + i∈Z3

for some E < 0, E 6= λ, where ϕ0 (x) = (λ − E) (−∆ − E)−1

X

fi G0λ (x − i) .

(3.13)

i∈Z3

Clearly we have ϕ0 (x) = (λ − E) =

X

X

fi G0E G0λ (x − i)

i∈Z3

fi G0λ (x − i) − G0E (x − i) ,

i∈Z3

hence ϕ(x) = f (x) =

X

fi G0λ (x − i) .

i∈Z3

We notice that ϕ ∈ D(−∆α ), since it follows from (3.11) and (3.13) that we have X X fi [Γλ (j, i) − ΓE (j, i)] = fi ΓE (j, i) . ϕ0 (j) = i∈Z3

i∈Z3

It remains to check that F is an eigenfunction: we have −∆α f = (−∆ − E) f + E f = (λ − E) f + E f = λ f .

This proves Theorem 3.

Modifying slightly the last argument, we can prove the following useful statement. Proposition 3. (i) Suppose that a sequence fn ∈ `2 (Z3 ), 0 < kfn k`2 ≤ C < ∞ is such that for some λ < 0, kΓλ fn k`2 → 0 f or n → ∞ ,

(3.14)

then there exists a sequence Fn ∈ D(−∆α ) of functions which satisfy k(−∆α − λ) Fn kL2 → 0 f or n → ∞ and 0 < kFn kL2 ≤ C1 < ∞ .

(3.15)

(ii) Suppose that a sequence Fn ∈ L2 (R3 ) satisfies (3.15). Then there exists a corresponding sequence fn ∈ `2 (Z3 ) such that (3.14) is valid.

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In both cases we can choose the functions of norm 1 for the second sequence if the first one consists of functions of norm 1. Proof. (i) We define Fn (x) = ϕn (x) +

X

fn (i) G0E (x − i) ,

i∈Z3

ϕn (x) = (λ − E) (−∆ − E)−1

X

fn (i) G0λ (x − i) + gn (x) ,

i∈Z3

where gn (x) =

X X

an,i θ(x − i), the function θ ∈ C0∞ is such that supp θ ⊂ Λ1/2 ,

i∈Z3

θ(0) = 1, an,i =

Γλ (i, j)fn (j) = Γλ f n (i), E < 0.

i∈Z3

Then we have Fn =

X

fn (i) G0λ (x − i) + gn (x) ∈ D(−∆α ) ,

i∈Z3

and (−∆α − λ) Fn (x) = (−∆ − λ) gn (x) . We have kan k`2 = kΓλ f n k`2 → 0 as n → ∞, so kgn kL2 → 0 and k(−∆−λ)gn kL2 → 0 as n → ∞. Applying the argument of Lemma A2, we obtain that C10 kfn k`2 ≤ kFn kL2 ≤ C100 kfn k`2 . This shows that we can take normalized functions Fen =

Fn kFn k ,

if fn are normalized.

(ii) Suppose that (−∆α − λ) Fn (x) = Hn (x). Since Fn ∈ D(−∆α ), it has the form X an,i G0En (x − i) , (3.16) Fn (x) = ψn (x) + i∈Z3

for some En < 0, ψn ∈ D(−∆), an = {an,i }i∈Z3 , ψ n = {ψn (i)}i∈Z3 , such that Γ E n an = ψ n . We define f n = an . From (3.16) we obtain (−∆α − λ)Fn (x) = (−∆ − En )ψn (x) + (En − λ)Fn (x) , and it follows that (−∆ − λ)ψn (x) = Hn (x) + (λ − En )

X

an,i G0En (x − i) .

i∈Z3

By using the resolvent identity we pass to the relation X (G0λ − G0En )(x − i) an,i . ψn (x) = G0λ Hn (x) + i∈Z3

(3.17)

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Substituting x = j into (3.17), we get X an,i [ΓEn (i − j) − Γλ (i − j)] . ψn (j) = G0λ Hn (j) + i∈Z3

Since ΓEn an = ψ n , we get that an satisfy the equation: (−∆ − λ)−1 Hn (j) = Γλ an (j),

j ∈ Z3 .

(3.18)

We observe that the r.h.s. of (3.18) is a continuous function of x since it belongs to D(−∆), i.e. it is well defined in the points of Z3 and tends to 0 as n → ∞. It follows also from (3.16) and (3.17) that X an,i G0λ (x − i) + (−∆ − λ)−1 Hn (x) . Fn (x) = i∈Z3

Since Fn are uniformly bounded in L2 (R3 ), k(−∆ − λ)−1 Hn kL2 → 0 as n → ∞, (λ < 0), we obtain, applying the argument of Lemma A2, that C20 kFn kL2 ≤ kan k`2 ≤ C200 kFn kL2 , which shows that we can suppose an normalized if the Fn are normalized.

Remark 8. This lemma implies that there is a geometric relation between the spectral sets of the corresponding operators (see Sec. 1, Corollary 4). 4. Scheme of the Proof for Discrete Operators In this section we will sketch the proof of Theorem 5. Sketch of the proof of Theorem 5. By x, y, z, . . . we will denote integers of Z3 . Let us consider a family of self-adjoint operators on `2 (Z3 ) defined as follows: A(λ) = A0 (λ) + Qλ , where for each λ, A0 (λ) is a non-random convolution operator, i.e. X aλ (x − y)Ψ(y), x ∈ Z3 , Ψ ∈ `2 (Z3 ) , (A0 (λ)Ψ)(x) =

(4.1)

(4.2)

|x−y|≥1

and for each λ the sequence {aλ (x)}x∈Z3 is real and satisfies the conditions √ aλ (x) = aλ (|x|), |aλ (x)| ≤ Ce− |λ||x| , C > 0 . The potential Qλ is a random multiplication operator: p |λ| Ψ(x), (Qλ Ψ)(x) = α(x)Ψ(x) + 4π

(4.3)

(4.4)

where {α(x)}x∈Z3 are real-valued independent random variables with identical distribution and bounded (by δ −1 ) density (see (0.2)).

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Operators A(λ) correspond to the standard probability space and we use the explicit dependence on λ to show that the multiscale analysis scheme can be applied [14, 19]. We denote by BL (x) the cube centered at x ∈ Zd , with sides of length L, and by AB (λ) the restriction of the operator A(λ) to `2 (B) (Dirichlet boundary conditions), by RB (z, λ) = (AB (λ) − z)−1 the resolvent of the operator AB (λ) (z ∈ / σ(AB (λ))), by RB (z, λ; x, y) the corresponding Green’s function, continued by zero on Z3 × Z3 (AB (λ) − z)−1 (x, y), if x, y ∈ B; RB (z, λ; x, y) = 0, otherwise . Let m > 0, E ∈ J ⊂ R, λ ∈ I ⊂ (−∞, −r). Definition 2. The cube BL (x) is (h, m, E, λ)-regular , for a fixed potential, if (i) E 6∈ σ(ABL (x) (λ)) and (ii) |RBL (x) (E, λ; x, y)| ≤ e−m|y−x| for all y such that h/2 ≤ |y − x| ≤ L/2. If this inequality is valid for all E ∈ J ⊂ R, λ ∈ I and for h = 0, we say that BL (x) is (m, E ∈ J, λ ∈ I)-regular . Definition 3. We say that the cube BL is (θ, E, λ)-non-resonant, for a fixed potential, if dist(E, σ(ABL (λ))) ≥ e−l

θ

for some θ, 0 < θ < 1 .

Lemma 2. Let J ⊂ R, I ⊂ (−∞, −r) be intervals. Suppose that for 0 < L0 < L1 < ∞ we have (P1) P the cube BLi is (m0 , E ∈ J, λ ∈ I)-regular ≥ 1 − L−p i , i = 0, 1; for some p > d and 0 < m0 ≤ ρ; (P2) P the cube BL is (θ, E, λ) non-resonant ≥ 1 − L−q for all L ≥ L0 , E ∈ J, λ ∈ I, for some θ (0 < θ ≤ 1/2) and q > q0 (p). Then for any m (0 < m < m0 ) there exist α = α(p) (1 < α <

√ 2) and B0 = B0 (p, θ, α, q, m, m0 ) < ∞ n

such that if L0 > B0 and Ln = (L0 )(α) , then inequality       f or any (E, λ) ∈ J × I P one of two cubes BLn (x)orBLn (y) ≥ 1 − (Ln )−2p     is (Lβn , m, E, λ)-regular holds for all n, for x, y ∈ Zd such that |x − y| ≥ Ln , if β =

2 1+α3 .

(4.5)

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Lemma 3. Suppose that a family of operators Aλ satisfy assumption (P2) of Lemma 2, and that (4.5) is valid for any (E, λ) ∈ J × I, I ⊂ (−∞, −r). Then with probability 1 any generalized solution f ∈ `2−δ of the equation (A(λ) − E)f = 0 decays exponentially at infinity with rate m < m0 . Proofs. The proofs of Lemmas 2 and 3 are quite similar to those of [19], where the corresponding results were obtained for a single operator (4.1)–(4.4) (i.e. for a fixed value of λ). In order to complete the proof of Theorem 5, it is sufficient to verify the validity of assumptions (P1) and (P2) of Lemma 2. (P1) is satisfied in the following two cases. √ Lemma 4. (“Low energy” case). Given 0 < m < r, L1 > L0 > 0, there exists λ0 = λ0 (Pα , m, r, p, L0 , L1 ) ≤ −r < 0 such that for any interval I ⊂ (−∞, −λ0 ) P cubeBLi is(m, E ∈ (−1, 1), λ ∈ I)−regular ≥ 1 − L−p i , i = 0, 1 .

(4.6)

√ Lemma 5. (“High disorder” case). Given l > 0, 0 < m < r, L1 > L0 > 0, there exists δ0 = δ0 (Pα , m, r, p, L0 , L1 , l) < ∞ such that if δ > δ0 , then inequalities (4.6) hold for any interval I ⊂ (−∞, −r) of length |I| = l. Proofs. The proofs of Lemmas 4 and 4.4 are almost the same as in [14, 19]. Assumption (P2) follows from the Wegner bound on the density of states [34] for the finite-difference operators. We formulate the result in the form which will suit us here. Let 1 #{λi (ω) < λ | λi ∈ σ(ΓB (E))} NΓB (E) (λ) = |B| be the normalized counting function of ΓB (E), B ⊂ Z3 , B is finite. Lemma 6. (Wegner lemma). Given any E < 0, I = (a, b) ⊂ R, E NΓB (E) (I) = E NΓB (E) (b) − NΓB (E) (a) ≤ 2 |b − a| δ −1 . This ends the proof of Theorem 5.

Remark 9. Because E < 0, the proof of Lemma 6 can be done following [6] older continuous of without much difficulties or [28]. In the case where Pα0 is H¨ order γ, N (λ) is also H¨older continuous of order γ. Remark 10. Lemmas 2–6 prove more than was stated by in Theorem 5. They imply that all the operators Aλ (with λ in the interval corresponding to the high disorder regime) have pure point spectrum in (−1, 1) with probability 1.

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5. Wegner-Type Estimate for the Density of States In this section we are concerned with the integrated density of states of −∆α(ω) . In particular, we prove Theorem 6. Since the proof of existence of the integrated density of states for the operator −∆α did not appear in the literature as yet, we first briefly discuss the possible way to define it. As before, let Eω (∆) be the spectral projection of −∆α(ω) , χΛ the characteristic function of the cube Λ ∈ R3 , whose sides do not meet the sites of Z3 . We define the measure KΛ,ω (∆) =

1 Tr (χΛ Eω (∆)χΛ ) . |Λ|

A standard argument based on the ergodic theorem (cf. [8]) shows that KΛ,ω converges, weakly for P-almost all ω ∈ Ω, to the non-random limiting measure N (∆) = E Tr (χ1 Eω (∆)χ1 ) ,

(5.1)

where χ1 (x) is the characteristic function of the unit cube in R3 , and E{ · } denotes the expectation value on the probability space (Ω, P). We call function N (λ) = N ((−∞, λ]) the integrated density of states of −∆α(ω) . In what follows we analyse N (λ) for λ < 0. It follows easily from Theorems 2 and 3 and Lemma A2 (cf. (5.1)) that N (λ) is a continuous function for λ < 0. Indeed, we have P λ < 0 is an eigenvalue of − ∆α(ω) = P 0 is an eigenvalue of Γα(ω) (λ) = 0.

(5.2)

The first equality is a consequence of Theorem 3. The second one follows from the continuity of the integrated density of states NΓ(λ) (Lemma 6) and standard arguments [28]. Let us show it in detail. We denote by EΓω (λ) {∆}(x, y) the kernel of the spectral projection of operator Γω (λ), then [28] dNΓ(λ) = E EΓω (λ) {0}(0, 0) . It follows from Lemma 6 (the Wegner lemma) that NΓ(λ) (t) is a continuous function in t = 0, therefore EΓω (λ) {0}(0, 0) = 0 with probability 1, which implies (since EΓω (λ) {0} is ergodic projection) that EΓω (λ) {0}(x, x) = 0 for all x ∈ Z3 with probability 1. Hence EΓω (λ) {0} = 0 with probability 1, and (5.2) follows. Since (5.2) is zero, we have that Eω {λ} = 0 with probability 1, and the statement dN {λ} = 0 follows from (5.1). A more detailed analysis is required to prove the absolute continuity of N (λ). Let B ⊂ R3 be a cube in R3 , Λ = B ∩ Z3 . Consider the operator −∆α,Λ , where α is the restriction of the function α(j) : Z3 → R to the finite subset Λ ⊂ Z3 (i.e. we “switch off” the interaction at the points Z3 \ Λ). The resolvent kernel of

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this operator [1] is given by (−∆α,Λ − z)−1 (x, y) = (−∆ − z)−1 (x, y) +

X

(−∆ − z)−1 (x, j)[Γα(ω) (z)]−1 (j, j 0 )(−∆ − z)−1 (j 0 , y) ,

j,j 0 ∈Λ

(5.3)

where Im z > 0 and operator Γα,Λ (z) : `2 (Λ) → `2 (Λ) is defined as follows: ( √ αj − i4πz , if j = j 0 , 0 Γα,Λ (z; j, j ) = −(−∆ − z)−1 (j, j 0 ) , otherwise (j, j 0 ∈ Λ) .

(5.4)

It is known [1] that for any fixed ω, the negative spectrum of Γα,Λ is discrete and consists of not more than N = |Λ| eigenvalues (number of points in Λ ⊂ Z3 ). It follows that for E < 0, the normalized counting function of Γα,Λ is well defined: NΛ,ω (E) =

1 # {λi (ω) < E} |Λ|

(it can also be well defined for E ≥ 0). If z < 0 and |z| is sufficiently large, then the sequence of resolvents (−∆α,Λ − z)−1 is monotone increasing as Λ → Z3 , and has a strong limit over the filter of cofinite subsets Λ ⊂ Z3 : s-lim (−∆α,Λ − z)−1 = (−∆α − z)−1 .

Λ%Z3

Thus the corresponding sequence of operators −∆α,Λ is monotone decreasing and converges in the strong resolvent sense to −∆α as Λ → Z3 (it is one way to define −∆α , [1]). It follows that the corresponding sequence NΛ,ω (E) is a superadditive stochastic process, that is for any given closed cubes Λ1 , Λ2 , such that Λ1 ∩ Λ2 has empty interior, we have NΛ1 ∪Λ2 ,ω (E) ≥ NΛ1 ,ω (E) + NΛ2 ,ω (E). A standard arguments [22] based on the superadditive ergodic theorem show that the sequence of distribution functions {NΛn ,ω (E)} converges weakly with probability 1 to a none (E), if random limiting distribution function N sup n

1 E {NΛn ,ω (E)} < ∞ . |Λn |

(5.5)

For our purpose it is enough to observe the following: since ∆α is the strong resolvent limit of a decreasing sequence −∆α,Λ , then for any interval Iη ⊂ (−∞, −r) of length η > 0, whose endpoints are not atoms of Eω (∆), we have s-lim EΛ,ω (Iη ) = Eω (Iη ) .

Λ%Z3

(5.6)

Since −∆α(ω) a.s. has no fixed (at least negative) eigenvalues (see (5.2)), we see that (5.6) is valid for any fixed interval, with probability 1. It follows from (5.1) and (5.6) that N (Iη ) ≤ sup Λ

1 E {Tr(EΛ,ω (Iη ))} |Λ|

= sup E {NΛ,ω (Iη )} . Λ

(5.7)

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A. B. de MONVEL and V. GRINSHPUN

Since the set Λ ⊂ Z3 is finite, then for each fixed ω ([1], see also Theorems 2 and 3), we have: NΛ (Iη ) = =

1 # {E ∈ Iη | 0 ∈ σ(ΓΛ (E))} |Λ| 1 # {λ | −η ≤ λ ≤ 0, 0 ∈ σ(ΓΛ (E0 + λ))} , |Λ|

(5.8)

where E0 ≤ −r < 0 is the right endpoint of Iη , and for simplicity we denote ΓΛ = Γα,Λ (see (5.4)). Lemma 7. Assume that 0 ∈ σ(ΓΛ (E + λ)), −η ≤ λ < 0, E ≤ −r < 0. Then dist(0, σ(ΓΛ (E))) < C|λ|, and hence there exists x ∈ σ(ΓΛ (E)) such that |x| < C|λ| (with C = C(r)). Proof. It follows from (5.4) that ΓΛ (E + λ) = ΓΛ (E) + ∆(E, λ) . Using the resolvent identity we get: −1 −1 −1 Γ−1 Λ (E + λ) = ΓΛ (E) − ΓΛ (E) ∆(E, λ) ΓΛ (E + λ) ,

hence

−1 −1 −1 ΓΛ (E) , Γ−1 Λ (E + λ) = (1 + ΓΛ (E) ∆(E, λ))

(5.9)

if 1 + Γ−1 Λ (E) ∆(E, λ) is invertible. Using the Hilbert equation we obtain ∆(E, λ) = G0E − G0E+λ = −λG0E G0E+λ , i.e. if |E| ≤ −r < 0, then k∆(E, λ)k ≤ Suppose that kΓ−1 Λ (E)|| ≤

1 C|λ|

|λ| . r2

(i.e. dist(0, σ(ΓΛ (E))) > Ct) then

kΓ−1 Λ (E) ∆(E, λ)k ≤ from which, choosing C ≥

2 r2 ,

1 , r2 C

(5.10)

we get:

kΓ−1 Λ (E) ∆(E, λ)k ≤

1 . 2

This implies that the operator 1 + Γ−1 Λ (E) ∆(E, λ) is invertible and −1 k ≤ 2. k(1 + Γ−1 Λ (E) ∆(E, λ))

It follows from (5.9) and (5.11) that −1 kΓ−1 Λ (E + λ)k ≤ 2kΓΛ (E)k .

(5.11)

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We have shown that if C ≥

2 r2

...

447

and dist(0, σ(ΓΛ (E))) ≥ C|λ|, then

dist(0, σ(ΓΛ (E + λ))) ≥

C|λ| > 0. 2

This proves Lemma 7.

Lemma 8. Let γ1 (E), . . . , γk (E), 1 ≤ k ≤ |Λ|, be the eigenvalues of matrix ΓΛ (E), E < 0. Then γj (E) are strictly increasing functions of |E|. Proof. This follows from the fact that the matrix |Λ| 1 −√|E||j−j0| ∂ p e ΓΛ = 4π ∂ |E| j,j0=1

is positive-definite [30].

Proof of Theorem 6. We can now finish the proof of Theorem 6. Let us return to (5.8) and suppose that 0 ∈ σ(ΓΛ (E0 + λi )), −η ≤ λi < 0, i = 1, . . . , k . It follows from Lemma 7 that there exist {xi }i=1,...,m such that |xi | ≤ Cη, and xi ∈ σ(ΓΛ (E0 )). Let us denote by xi (λ) the corresponding eigenvalues of ΓΛ (E0 + λ). According to Lemma 8, each xi (λ) increases as λ decreases to −η, and hence cannot cross change sign more than once, i.e. m ≥ k. We have: E {NΛ (Iη )} ≤ E NΓΛ (E0 ) (E0 − Cη, E0 + Cη) , where NΓΛ (E0 ) (λ) is the normalized counting function of ΓΛ (E0 ). The estimate on the density of states for discrete operators ΓΛ (Lemma 6) implies the bound E NΓΛ (E0 ) (E0 − Cη, E0 + Cη) ≤ 2Cηδ −1 .

This proves Theorem 6. Appendix Lemma A1. Let ϕ ∈ D0 (−∆α ), i.e. X ai G0E (x − i) , ϕ(x) = ϕ0 (x) + i∈Zd

P where ϕ0 ∈ C0∞ , E ≤ inf supp Pα − d, ai = j∈Z3 Γ−1 E (i, j)ϕ0 (j). Then there exists d0 such that if d > d0 , then (−∆α − λ)ϕ ∈ L2δ . √ |E| Proof. Let us denote Γ0E = ΓE − 4π . We have (see (0.4)) that kΓ0E k ≤

X i∈Z3 \{0}

√ 1 e− |E| |i| = ε(E) → 0 as E → −∞ . 4π |i|

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It follows from the ergodicity of the potential α that if E < 0, we have p |E| 0 σ(ΓE ) = supp Pα + σ(ΓE ) + 4π (the sum above is the algebraic sum of subsets of R), so it follows that ! p |E| , inf supp Pα − ε(|E|) ≥ d/2 , dist(0, σ(ΓE )) ≥ dist − 4π if |E| is large enough. Using the Combes–Thomas argument, we get the estimate |Γ−1 E (i, j)| ≤

2 e−m(E) |i−j| , d(E)

and it follows that if supp ϕ0 ⊂ Λ, then X −m(|E|)|i| Γ−1 . |ai | ≤ E (i, j) |ϕ0 (j)| ≤ kϕ0 k∞ C1 (E, |Λ|) e j∈Z3 ∩Λ

The next Lemma A2 implies that ϕ ∈ L2δ and (−∆α − λ)ϕ ∈ L2δ . Lemma A2. Let λ < 0 and X fi G0λ (x − i) , F (x) =

f = {fi }i∈Z3 .

(A.1)

i∈Z3

(i) If F ∈ L2 (R3 , dµ), i.e. the series converges in k · kL2 (dµ) , where µ(x) = g 2 (x)d3 x (g(x) = g(|x|) > 0 is a monotone, continuous function of |x|), then for any ε > 0, f ∈ `2 (Z3 , de µ) , where µ e(i) = g 2 (|i| − t), i ∈ Z3 , for some t: |t| = ε.

√ √ µ), where µ e(i) = ge2 (i), i ∈ Z3 (e−ε |λ| |i| ≤ e g (i) ≤ eε |λ| |i| (ii) If f ∈ `2 (Z3 , de for some 0 < ε < 1, ge is a monotone function of |i|, i ∈ Z3 ), then F ∈ L2 (R3 , dµ) ,

g(i). with dµ(x) = g 2 (x)d3 x, and g is a monotone, continuous function of |x|, g(i) = e Proof. (i) We apply once again the argument used in the proof of Theorem 3 (Sec. 3). Clearly, for any ψ ∈ C0∞ X hF, (−∆ − λ)ψi = fi ψ(i) . i∈Z3 ∩ supp ψ

Consider a function θ ∈ C0∞ , supp θ ⊂ Λ1/2 (0), θ(0) = 1, |(−∆ − λ)θ(x)| ≤ τλ < ∞, and denote θi (x) = θ(x − i). Then hF, (−∆ − λ)θi i = fi .

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It follows that fi obey Z

1/2 Z |F (x)|2 g(x)2 d3 x τλ

|fi | ≤ Λ1/2 (i)

1/2 g(x)−2 d3 x .

(A.2)

Λ1/2 (i)

Since g(x) = g(|x|) is monotone and continuous, we have supx∈Λ1/2 |g(x)|−2 ≤ |g(|i| − t)|−2 , for some |t| = 1/2, and from (A.2) we obtain X

|fi g(|i| − t)|2 ≤ kF kL2 (dµ) < ∞ .

i∈Z3

To obtain the result with 0 < |t| < ε, for any ε > 0 we just take the function θε (x) with supp θ ⊂ Λε . Statement (i) is proved. √ (ii) Since ge(i) ≥ eε |λ| |i| , we have that for any fixed x 6∈ Z3 , series (A.1) converges absolutely. Moreover, if function g(x) = g(|x|) is monotone, continuous and g(i) = e g(i), then kF k2L2 (dµ) =

X

X

fi

i

fj hG0λ (x − i), G0λ (x − j)iL2 (dµ) ,

j

where hG0λ (x

−

i), G0λ (x

− j)iL2 (dµ)

1 = 16π 2

Z

e−

R3

≤ C e−m

√

√

√ e |λ||x−j| g(x)2 d3 x |x − i| |x − j| |λ||x−i| −

|λ| |i−j|

g(i)g(j),

0 < m < 1.

Hence kF k2L2 (dµ) ≤ C

X i

=C

X i

fi g(i)

X

fj g(j)e−m

√

|λ| |i−j|

j

fi g(i) φi < C1 kf k2`2 (de , µ)

since it is easy to see that {φi }i∈Z3 ∈ `2 (Z3 ) (and its norm satisfies the corresponding bound). Lemma A2 is proved. Acknowledgments The authors thank L. Pastur for many fruitful discussions. V. G. thanks the Laboratory of Mathematical Physics and Geometry, University Paris 7 for its kind hospitality and Dr. J. Macho. The work was supported in part by the INTAS under Grant INTAS-93-1939 and by State Committee for Science and Technology of Ukraine under grant 3/1/132.

450

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References [1] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New York, Berlin, Heidelberg, 1988. [2] S. Albeverio, R. Hoegh-Krohn, W. Kirsch and F. Martinelli, “The spectrum of the three-dimensional Kronig–Penney model with random point defects”, Adv. Appl. Math. 3 (1982) 435–440. [3] M. Aizenman and S. Molchanov, “Localization at large disorder and at extreme energies: an elementary derivation”, Commun. Math. Phys. 157 (1993) 245–279. [4] J. Berezanskii, Expansion in Eigenfunctions of Self-adjoint Operators, Transl. Math. Monographs 17, American Mathematical Society, Providence, R.I., 1968. [5] A. Boutet de Monvel and V. Grinshpun, “Expansion in generalized eigenfunctions of point interactions hamiltonians”, in preparation. [6] R. Carmona, A. Klein and F. Martinelli, “Anderson localization for Bernoulli and other singular potentials”, Commun. Math. Phys. 108 (1987) 41–66. [7] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨ odinger Operators, Birkh¨ auser, Boston, 1990. [8] H. Cycon, R. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1987. [9] J.-M. Combes and P. D. Hislop, “Localization for some continuous random Hamiltonians in d-dimensions”, J. Funct. Anal. 124 (1994) 149–180. [10] J.-M. Combes, P. D. Hislop and E. Mourre, “Spectral averaging, perturbation of singular spectra, and localization”, Trans. Amer. Math. Soc. 348 (1996) 4883–4894. [11] F. Delyon, Y. Levi and B. Souillard, “Anderson localization for multidimensional systems at large disorder or large energy”, Commun. Math. Phys. 100 (1985) 463– 470. [12] F. Delyon, B. Simon and B. Souillard, “From power pure point to continuous spectrum in disordered systems”, Ann. Inst. H.Poincar´ e , Phys. Th´eor. 42 (1985) 283–309. [13] H. von Dreifus: “On the effects of randomness in ferromagnetic models and Schr¨ odinger operators”, NYU Ph.D. Thesis, 1987. [14] H. von Dreifus and A.Klein, “A new proof of localization in the Anderson tight binding model”, Commun. Math. Phys. 124 (1989) 285–299. [15] J. Fr¨ ohlich and T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy”, Commun. Math. Phys. 88 (1983) 151–184. [16] J. Fr¨ ohlich, F. Martinelli, E. Scoppola and T. Spencer, “Constructive proof of localization in the Anderson tight binding model”, Commun. Math. Phys. 101 (1985) 21–46. [17] I. Goldsheid, S. Molchanov and L. Pastur, “A pure point spectrum of the stochastic one-dimensional Schr¨ odinger operator”, Funct. Anal. Appl. 11 (1977) 1–10. [18] V. Grinshpun, “Point spectrum of random infinite-order operator acting on `2 (Zd )”, Reports Ukrainian Acad. of Sci. (8) (1992) 18–21; (5) (1993) 26–29 (in Russian). [19] V. Grinshpun, “Point spectrum of finite-difference multi-dimensional random operators”, PhD Thesis, University Paris 7, 1994. [20] V. Grinshpun, “Localization for random potentials supported on a subspace”, Lett. Math. Phys. 34 (2) (1995) 103–117. [21] W. Kirsch and F. Martinelli, “On the spectrum of Schr¨ odinger operators with a random potential”, J. Phys. A15 (1982) 2139–2156. [22] W. Kirsch, “Random Schr¨ odinger operators and the density of states”, Stochastic Aspects of Classical and Quantum Systems, eds. S. Albeverio, P. Combe, M. SirugueCollin, Marseille 1983, Lecture Notes in Math. 1109, Springer-Verlag, Berlin, Heidelberg, New York, 1985, 68–102.

¨ EXPONENTIAL LOCALIZATION FOR MULTI-DIMENSIONAL SCHRODINGER OPERATOR

...

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[23] A. Klein, “Localization in the Anderson model with long range hopping”, Braz. J. Phys. 23 (1993) 363–371. [24] V. Korotkov, “Integral operators with Carleman kernels”, Dokl. Akad. Nauk SSSR 165 (1965) 748–757. [25] S. Kotani and B. Simon, “Localization in general one-dimensional random systems: continuum Schr¨ odinger operators. II”, Commun. Math. Phys. 112 (1987) 103–119. [26] V. Kovalenko and Yu. Semenov, “Some problems on expansions in generalized eigenfunctions of the Schr¨ odinger operator with strongly singular potentials”, Russian Math. Surveys 33(4) (1978) 119–157. [27] F. Martinelli and H. Holden, “On absence of diffusion near the bottom of the spectrum for a random Schr¨ odinger operator on L2 (Rν )”, Commun. Math. Phys. 93 (1984) 197– 217. [28] L. Pastur and A. Figotin, Spectra of Random and Almost Periodic Operators, Grundlehren der Math. Wiss., Springer-Verlag, Berlin, Heidelberg, New-York, 1992. [29] B. Simon, “Schr¨ odinger semigroups”, Bull. Amer. Math. Soc. 7 (3) (1982) 447–527. [30] B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, San Francisco, London, 1979. [31] B. Simon and T. Wolff, “Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians”, Commun. Pure Appl. Math. 39 (1986) 75–90. [32] J. Sch’noll, “On the behaviour of the Schr¨ odinger equation”, Mat. Sb. 42 (1957) 273–286 (in Russian). [33] T. Spencer, “Localization for random and quasi-periodic potentials”, J. Stat. Phys. 51 (1988) 1009–1019. [34] F. Wegner, “Bounds on the density of states in disordered systems”, Z. Phys. B44 (1981) 9–15.

QUANTIZATIONS OF FLAG MANIFOLDS AND CONFORMAL SPACE TIME R. FIORESI Department of Mathematics University of California Los Angeles, CA 90095–1555 USA E-mail: [email protected] Received 24 January 1997

In this paper we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2, 4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and euclidean space.

0. Introduction and Summary In the quantum groups setting there has been some attention given to the quantization of the Minkowski space. In their paper [1] Zumino et al. follow an ad hoc approach to this problem: the Minkowski space gets quantized as a real 4-dimensional space with the coaction of the Poincar`e group on it. In the 1980’s following the new ideas introduced by Penrose a few years earlier in [2], Manin introduced the point of view that the Minkowski space is the manifold of the real points of a big cell in the grassmannian of complex two dimensional subspaces of a complex four dimensional space (twistor space) [3]. This is the starting point for the quantizations that are described in this paper. Reshetikhin and Lakshmibai have provided in their paper [4] a quantization of the flag manifold of any complex simple group; however their construction is very abstract and does not seem to provide the deformations of the Pl¨ ucker relations. Taft and Towber [5] on the other hand have a paper on the same subject, though with a very different approach, in which they do have the quantized Pl¨ ucker relations, but their commutation rules resolve in a tautology for the grassmannian coordinates. It is natural to define the quantized coordinate ring of grassmannians and flag manifolds of subspaces of Cn as the graded subring of kq [SLn ] (the Manin deformation of SLn (C), where kq = C[q, q −1 ]) generated by suitable quantum determinants in the matrix entries aij . The essential point is to determine the commutation relations between these generators and show that these relations provide a 453 Reviews in Mathematical Physics, Vol. 9, No. 4 (1997) 453–465 c World Scientific Publishing Company

454

R. FIORESI

presentation of the deformed ring. We do this for the flag spaces kq [F (1, 2; n)], kq [F (1, n−1; n)] and kq [F (2, 3; 4)] as well as the quantized grassmannian kq [G(2, n)]. After the explicit construction of the quantum ring kq [G(2, 4)] of G(2, 4), we introduce the notion of quantum big cell with coordinate ring kq [N ], which is a projective localization of kq [G(2, 4)], in complete analogy with what happens in the classical case. The quantized lower maximal parabolic kq [Pl ] corresponding to the classical maximal parabolic stabilizing he1 , e2 i is then proved to have a coaction on the quantum big cell. We define the quantum complex Minkowski space to have the coordinate ring kq [N ] and the quantum complex conformal group including translations to have the coordinate ring kq [Pl ]. In analogy with the commutative case we define a certain quotient of kq [Pl ], as the quantum complex conformal group kq [C] without the translations. As in the classical case we have that kq [N ] is isomorphic to the 2 by 2 quantum matrix ring. This fact can be used to obtain right away the holomorphic De Rham complex on kq [N ]. We now come to the theory over R. The point of view is that the real spaces and groups, both classical and quantum, are described by the complex objects together with an appropriate involution. In the spirit of Soibelman work [6, 7] it is possible to define 2 involutions ∗M,q , ∗E,q on kq [N ] that allow us to speak of real Minkowski space and real euclidean space. It is also possible to define two involutions ∗C,M , ∗C,E on kq [C] so that we can speak of quantum real conformal group. The quantum real Minkowski space and the quantum real euclidean space will still support a coaction of the real kq [C] with respect to the appropriate involution. With a simple change of coordinates, that generalizes the classical one, the real Minkowski space and the real euclidean space transform into each other. It is easy to construct a real De Rham complex on the real Minkowski space starting from the complex one and using the given involution. Our point of view is group theoretic and not at all ad hoc. One of its advantages is that it provides the quantization of the conformal group naturally. It is capable of generalizing to higher dimensions (some of which are treated here) and provides a basis for the quantization of the Penrose theory. In the special case treated here it leads to a quantization of the complex Minkowski space in tandem with a quantization of the conformal group, then of the real Minkowski and euclidean spaces in tandem with the real conformal group with a compatible coaction that quantizes the classical coaction. 1. The Quantum Homogeneous Spaces kq [G(2, n)], kq [F (1, n − 1; n)], kq [F (1, 2; n)] and kq [F (2, 3; 4)] Let’s define kq [Mn ] as the associative kq -algebra with unit generated by n2 elements aij , i, j ∈ {1 . . . n} subject to the relations (see [8, 9]): aij akl = akl aij i < k, j > l or i > k, j < l aij akj = q −1 akj aij i < k −1 aij akl − akl aij = (q −1 − q)aik ajl i < k, j < l aij ail = q ail aij j < l Let’s define on kq [Mn ] the comultiplication and the counit: X aik ⊗ akj , (aij ) = δij ∆(aij ) =

455

QUANTIZATIONS OF FLAG MANIFOLDS AND CONFORMAL SPACE TIME

kq [Mn ] is a bialgebra with the given ∆ and . Let’s now introduce the notion of quantum determinant: 1...n = D1...n

X

(−q)−l(σ) a1σ(1) . . . anσ(n)

σ∈Sn

where l(σ) denotes the length of the permutation σ. More generally one can define: ...jm = Dij11...i m

X

1 ≤ i1 < · · · < im ≤ n

(−q)−l(σ) ai1 σ(i1 ) . . . aim σ(im )

1 ≤ j1 < · · · < jm ≤ n

σ:(i1 ...im )→(j1 ...jm )

For all the properties of the quantum determinant refer to the papers [8, 9, 10]. This enables us to define the two Hopf algebras [10]: 1...n − 1) kq [SLn ] =def kq [Mn ]/(D1...n 1...n T − 1, aij T − T aij ) kq [GLn ] =def kq [Mn ][T ]/(D1...n

where in both cases the antipode is defined as: ˆ

1...i...n 1...n D S(aij ) = (−q)j−i D1... ˆ j...n 1...n

−1

.

We want now to give a quantization of the grassmannian coordinate ring k[G(2, n)] where G(2, n) is the set of 2-planes in a vector space of dimension n. It will be a generalization of the classical construction (see [11, 12]). Definition-Proposition 1.1. Let kq [G(2, n)] the subalgebra of kq [SLn ] gener, 1 ≤ i1 < i2 ≤ n. Then: ated by Di12 1 i2 kq [G(2, n)] ∼ = kq hλi1 i2 i/IG(2,n) where IG(2,n) is the two sided ideal generated by the relations: (h refers to lexicographic ordering) λi1 i2 λj1 j2 = q −1 λj1 j2 λi1 i2

(i1 i2 ) < (j1 j2 )

with i1 , i2 , j1 , j2 not all distinct. If i1 , i2 , j1 , j2 are all distinct: λi1 i2 λj1 j2 = q −2 λj1 j2 λi1 i2

i1 < i2 < j1 < j2 ,

λi1 i2 λj1 j2 = λj1 j2 λi1 i2 i1 < j1 < j2 < i2 λi1 i2 λj1 j2 = q −2 λj1 j2 λi1 i2 − (q −1 − q)λi1 j1 λi2 j2 i1 < j1 < i2 < j2 λi1 i2 λj1 j2 − q −1 λi1 j1 λi2 j2 + q −2 λi1 j2 λi1 j1 = 0

i1 < i2 < j1 < j2

(p)

We will refer to the relations (p) as the quantum Pl¨ ucker relations, since for q = 1 they reduce to the usual Pl¨ ucker relations for the embedding G(2, 4) ∼ = Q ⊂ P5 where Q is the Klein quadric.

456

R. FIORESI

Proof. Let’s define the map: kq hλi1 i2 i/IG(2,n) λi1 i2

−→ kq [G(2, n)] 7−→ Di12 . 1 i2

By direct computation one can check that this is well defined. The ring kq [G(2, n)] is generated as kq -module by the monomials: DI12 , DI121 . . . DI12r , Ij 6= (i1 , i2 ) and i2 − i1 > 1, I1 < · · · < Ir . kq [G(2, n)] is graded. Consider in fact the grading induced by the degree and assume that there exists an inhomogeneous relation among the generators. Divide all the coefficients of the relation by the highest power of (q − 1) dividing the gcd of the coefficients. Now if q = 1 this will give a relation among the given monomials that in the commutative case are linearly independent. The same argument shows also that the generators are linearly independent (see [4]). This states that there ucker ones, the morphism is 1–1, are no other relations in kq [G(2, n)] besides the Pl¨ hence it is an isomorphism. Proposition 1.2. kq [G(2, n)] is a kq [SL2 ]-comodule and the coaction map is just ∆ restricted to kq [G(2, n)]: kq [G(2, n)] Di12 1 i2

∆

−→

7−→

kq [SLn ] ⊗ kq [G(2, n)] X Dij11ij22 ⊗ Dj121 j2 . 1≤j1 <j2 ≤n

Proof. direct computation

We want now to give similar constructions for the quantized flag rings kq [F (1, n − 1; n)], kq [F (1, 2; n)], kq [F (2, 3; n)] where: F (i1 , . . . , ir ; n) = {(Vi1 , . . . , Vir )|Vi1 ⊂ · · · ⊂ Vir ⊂ V, i1 < . . . ir , dim V = n, dimVi = i}

Definition 1.1. Define kq [F (r, s; n)] as the subalgebra of kq [SLn ] generated by Dj1...s , 1 ≤ i1 < · · · < ir ≤ n, 1 ≤ j1 < · · · < js ≤ n. 1 ...js

, Di1...r 1 ...ir

Theorem 1.3. We have, for n > 2: i) kq [F (1, n − 1; n)] ∼ = kq hλi , λi1 ...in i/IF (1,n−1;n) kq [F (1, 2; n)] ∼ = kq hλi , λi1 i2 i/IF (1,2;n) ∼ kq hλi i , λj j j i/IF (2,3;4) kq [F (2, 3; 4)] = 1 2

1 2 3

where IF (1,n−1;n) , IF (1,2;n) , IF (2,3;4) are the two sided ideals generated by the following sets of relations.

QUANTIZATIONS OF FLAG MANIFOLDS AND CONFORMAL SPACE TIME

457

IF (1,n−1;n) : λi λj = q −1 λj λi

i < j,

λi λi1 ...in−1 = λi1 ...in−1 λi

i ∈ (i1 . . . in−1 )

λi1 ...in−1 λj1 ...jn−1 = q −1 λj1 ...jn−1 λi1 ...in−1 λi λ1...ˆi...n

(i1 . . . in−1 ) < (j1 . . . jn−1 ) X = q −1 λ1...ˆi...n λi + q −1 (q −1 − q) (−q)i−j λj λ1...ˆj...n j
λ1...ˆi...n λi = q X

i−n

(−q)

−1

λi λ1...ˆi...n + (q

λ1...ˆi...n λi =

i

X

−1

− q)

X

(−q)i−j λj λ1...ˆj...n

j>i n−i

(−q)

λi λ1...ˆi...n = 0

(i)

i

IF (1,2;n) : λi λj = q −1 λj λi

IG(2,n) ,

i < j,

λi λj1 j2 = λj1 j2 λi

i ∈ (j1 j2 )

−1

λi λj1 j2 = qλj1 j2 λi i > j1 , j2 λi λj1 j2 = q λj1 j2 λi i < j1 , j2 , −1 −1 λi λj1 j2 = q λj1 j2 λi − (q − q)λj1 λi,j2 i > j1 , i < j2 1 ≤ i1 < i2 < i3 < i4 ≤ n : −2

q λi2 i3 λi1 − q −1 λi1 i3 λi2 + λi1 i2 λi3 = 0 q −2 λi2 i4 λi1 − q −1 λi1 i4 λi2 + λi1 i2 λi4 = 0

(i)

q −2 λi3 i4 λi1 − q −1 λi1 i4 λi3 + λi1 i3 λi4 = 0 q −2 λi3 i4 λi2 − q −1 λi2 i4 λi3 + λi2 i3 λi4 = 0 IF (2,3;4) : λ12 λ134 = q −1 λ134 λ12 ,

IG(2,4) ,

λ12 λ234 = q −1 λ234 λ12

λ13 λ124 = q −1 λ124 λ13 − (q −1 − q)λ12 λ134 , λ13 λ234 = q −1 λ234 λ13 λ14 λ123 = qλ123 λ14 , λ14 λ234 = q −1 λ234 λ14 λ23 λ124 = qλ124 λ23 + (q −1 − q)λ24 λ123 λ23 λ134 = q −1 λ134 λ23 − (q −1 − q)λ12 λ234 λ24 λ123 = qλ123 λ24 , λ24 λ134 = q −1 λ134 λ24 − (q −1 − q)λ12 λ234 λ34 λ123 = qλ123 λ34 , λ34 λ124 = q −1 λ124 λ34 q −2 λ234 λ14 − q −1 λ134 λ24 + λ124 λ34 = 0 q −2 λ234 λ13 − q −1 λ134 λ23 + λ123 λ34 = 0 q −2 λ234 λ12 − q −1 λ134 λ23 + λ123 λ24 = 0

(i)

q −2 λ134 λ12 − q −1 λ124 λ13 + λ123 λ14 = 0 We will refer to the relations (i) in each set as the quantum incidence relations. ii) kq [F (1, n − 1; n)], kq [F (1, 2; n)], kq [F (2, 3; 4)] are kq [SLn ]-comodules and the coaction is given by ∆ restricted to the subrings. kq [F (1, n − 1; n)] λi λi1 ...in−1

∆

−→

7−→ 7−→

kq [SLn ] ⊗ kq [F (1, n − 1; n)] X Dij ⊗ λj X

1≤j≤n

1≤j1 <···<jn−1 ≤n

j ...j

n−1 Di11...in−1 ⊗ λj1 ...jn−1

458

R. FIORESI

kq [F (1, 2; n)]

kq [SLn ] ⊗ kq [F (1, 2; n)] X Dij ⊗ λj

∆

−→

7−→

λi

1≤j≤n

X

7−→

λi1 i2

Dij11ij22 ⊗ λj1 j2

1≤j1 <j2 ≤n

(same for kq [F (2, 3; 4)]). Proof. The formulas in (i) and (ii) are a direct computation. To show the isomorphisms that are stated in (i) one needs an argument similar to the one given in (1.1). 2. The Complex Quantum Minkowski Space and the Complex Quantum Conformal Group The Penrose model of space time introduced by Penrose in the 1970’s is given by the grassmannian G(2, 4) of 2-planes in a complex vector space T of dimension 4 [3]. There is an obvious transitive action of SL4 C on G(2, 4) that gives: G(2, 4) ∼ = SL4 C/Pu , where Pu is the maximal parabolic subgroup of SL4 C leaving the 2-plane spanned by e1 , e2 invariant. The usual complex Minkowski space time MC is embedded in G(2, 4) in a natural 12 6= 0, while G(2, 4) can be obtained from the real way as the big affine cell D12 Minkowski space by complexification followed by compactification [3]. Classically the conformal group can be identified with Pl = Put and its action on the big affine cell N Pu /Pu of SL4 C/Pu is simply given by left multiplication: I2 0 Pl × N Pu /Pu −→ N Pu /Pu N= ν= . p, νPu /Pu 7−→ pνPu /Pu t I2 Since pν =

x

0

tx

y

I2

0

n

I2

=

x

0

tx + yn

y

=

1

0

t + ynx−1

1

x

0

0

y

we actually have an action of Pl on N :

Pl × N x 0 I2 n tx y

0 I2

−→ 7−→

N 1 t + ynx−1

0 1

.

It is our intention to give a quantization of the subgroups Pu , Pl , and most important, of the big affine cell N Pu /Pu viewed as homogeneous space for Pl . This will be the model for the complex Minkowski space. Classically the following equality 12 6= 0) holds for the coordinate rings: (with D12

I2

0

t

I2

x

0

0

y

I2

s

0

I2



a11  .. = . a41

...

...

 a14 ..  . . a44

459

QUANTIZATIONS OF FLAG MANIFOLDS AND CONFORMAL SPACE TIME

Let’s impose the same condition for non commutative coordinates. After some computations one gets: ! 12 12 −1 12 12 −1 D12 D13 D12 a11 a12 −q −1 D23 x= ν=t= 12 12 −1 12 12 −1 a21 a22 −q −1 D24 D12 D14 D12 ! ! (∗) 123 12 −1 123 12 −1 12 −1 23 12 −1 24 D12 D124 D12 D12 −qD12 D12 −qD12 D123 s= y= 124 12 −1 124 12 −1 12 −1 13 12 −1 14 D123 D12 D124 D12 D12 D12 D12 D12 −1

12 Definition 2.1. Let’s define the localization of kq [SL4 ] in D12 ring: 12−1 i =def kq [SL4 ]/(ID12−1 ) kq [SL4 ]hD12

as the following

12

where ID12−1 is given by the set of relations: 12

−1

−1

−1

12 12 aij = aij D12 , D12 12−1 akl D12

=

12−1 akl D12

−1

12 12 D12 akj = qakj D12 ,

+ (q

−1

− q)

−1

−1

12 12 D12 ail = qail D12

i, j = 1, 2 k, l = 3, 4 12 12 12 − q −1 D12 D2k a1l D12 .

12 12 12 D12 Dik a2l D12

−1

−1

12 12 , kq [G(2, 4)]hD12 iproj as Define also projective localization of kq [G(2, 4)] in D12 −1

−1

−1

−1

12 12 12 12 12 12 12 i generated by: D13 D12 , D14 D12 , D23 D12 , the subring of kq [SL4 ]hD12 12 12−1 12 12−1 D24 D12 , D34 D12 .

Proposition 2.1. Let kq [N ] be the subring of kq [SLn ] generated by the elements in t and kq [Pl ] the subring of kq [SLn ] generated by the elements in t, x, y. Then: −1

12 , (i) kq [N ] is the projective localization of kq [G(2, 4)] in D12 k [M ] , (ii) kq [N ] ∼ = q 2 (iii) kq [Pl ] is isomorphic to a quotient of kq [SLn ]/(a13 , a14 , a23 , a24 ) .

Proof. (i) follows directly from the definition of projective localization and (*). (ii) Let’s define a map φ : kq [M2 ] → kq [N ] by: φ(a11 ) = n12 , φ(a12 ) = n11 , φ(a21 ) = n22 , φ(a22 ) = n21 . One can check that φ is well defined. ker φ is trivial. In fact if there were a relation among the tij ’s after multiplying by a suitable power 12 12 12 12 12 it would give rise to a relation among monomials in: D23 , D24 , D13 , D14 , of D12 while by Proposition 1.1 we know that these are linearly independent. kq [SLn ]/(a13 , a14 , a23 , a24 ), as ψ(x) = x, (iii) Let’s kq [Pl ] −→ define themap: ψ : a33 a34 a32 a31 ψ(y) = , ψ(t) = . One can check that this map is well a43 a44 a42 a41 defined and it is surjective. Hence kq [Pl ] is a quotient of kq [SLn ]/(a13 , a14 , a23 , a24 ). Proposition 2.2. kq [N ] is a kq [Pl ]-comodule: kq [Pl ] ⊗ kq [N ] kq [N ] −→ (n) 7−→ (t) ⊗ 1 + (y)S 12 (x) ⊗ (n)

460

R. FIORESI

where 12 a22 D12

12

S (x) =

−1

12 −q −1 a21 D12

12 −qa12 D12 −1

12 a11 D12

−1

! .

−1

Proof. direct computation. 3. Holomorphic De Rham Complex on the Complex Quantum Minkowski Space

We want now to construct for MC,q = kq [N ] the quantum Minkowski space, the quantum analogue of the De Rham complex. Since kq [N ] ∼ = kq [M2 ] we have the following [13]: Definition-Proposition 3.1. Ω∗ (kq [M2 ]) the quantum holomorphic De Rham complex for kq [M2 ] is the associative algebra over kq with generators aij , daij , i, j = 1, 2 and defining relations: aij daij = q 2 daij aij aij akj = q −1 akj aij −1

(daij )2 = 0

aij dakj = qdakj aij −1

daij akj = q akj daij − (q − q)dakj aij daij dakj = −qdakj daij i < k

i
aij ail = q −1 ail aij aij dail = qdail aij j < l daij ail = q −1 ail daij − (q −1 − q)dail aij j < l [ail , akj ] = 0

daij dail = −qdail daij j < l [aij , akl ] = −(q −1 − q)akj ail

i < k, j < l

−1

aij dakl = dakl aij [akj , dail ] = [ail , dakj ] + (q − q)dakl aij i < k, j < l akl daij = daij akl + (q −1 − q)(dakj ail + dail akj ) + (q −1 − q)2 dakl aij i < k, j < l daij dakl = −dakl daij

dajk dali + dali dajk = −(q −1 − q)dakl daij

i < k, j < l

Remark. To get explicitly the relations in the nij coordinates it is necessary the following substitution (see Proposition 2.1 (i)): a11 = n12 , a12 = n11 , a21 = n22 , a22 = n21 . 4. Real Quantum Minkowski and Euclidean Spaces, Real Quantum Conformal Group and Real De Rham Complex It is a general principle that a real algebraic group can be given by the complex algebra of its complexification together with an involution. The function algebra of the real group is recovered as the algebra of the invariant functions with respect to the given involution. Classically the complex Minkowski space MC is given by complex 2 by 2 matrices. Here are two classical involutions [3] of MC :

a

b

c

d

∗M −→

a

c

b

d

a

b

c

d

∗E −→

d

−c

−b

a

.

QUANTIZATIONS OF FLAG MANIFOLDS AND CONFORMAL SPACE TIME

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The set of fixed points of ∗E is the real euclidean space, while the set of fixed points of ∗M is the real Minkowski space [3] M . In order to define a quantum analogue to the involutions ∗M and ∗E we need to consider the quantum conformal group without the translation part. This is because the Soibelman involution [6] of kq [SL4 ] it is not defined on the quotient kq [Pl ]. Definition 4.1. Define quantum conformal group (without translations): kq [C] = kq [Pl ]/(n11 , n12 , n21 , n22 ). 12 ˜ 12 12 ˜ 12 D12 −1), where D12 , D12 Remark. Notice that kq [C] ∼ = kq [GL2 ]×kq [GL2 ]/(D12 are the quantum determinant respectively of the first and second copy of kq [GL2 ]. Moreover, kq [N ] is a kq [C]-comodule i.e. the coaction in (2.2) is well defined modulo the ideal (n11 , n12 , n21 , n22 ).

Definition 4.2. Define on kq [N ] the antilinear involutions:

kq [N ] n11 n21

n12 n22

∗M,q −→

7−→

kq [N ] n22 n21

n12 n11

∗E,q kq [N ] −→ −1 n12 n21 −q n22 7−→ n22 n12 −qn22 q 7−→ q −1

kq [N ] n11 n21

These are well defined automorphism. Define on kq [C] the antilinear involutions: kq [C] xij yij

∗C,M −→ 7−→ 7−→

kq [C] 12

S (yij ) S 12 (xij )

kq [C] xij yij

∗C,M −→ 7−→ 7−→

kq [C] S 12 (xji ) S 12 (yji )

Both maps are well defined and are antiautomorphisms (direct computation). Let’s denote by kq [N ]∗M,q , kq [N ]∗E,q , kq [C]∗C,M , kq [C]∗C,M the set of fixed points for such involutions. Theorem 4.1. kq [N ]∗M,q = kq hM0 , M1 , M2 , M3 i/IM ⊂ kq [N ] ∗E,q

kq [N ]

= kq hE0 , E1 , E2 , E3 i/IM ⊂ kq [N ]

where in analogy with the classical case we set: 1 1 1 ∗ ∗ ∗ (n12 + n12M,q ) M2 = (n11 + n11M,q ) = (n22 + n22M,q ) 2 2 2 −i i 1 ∗ ∗ ∗ (n11 − n11M,q ) = (n22 − n22M,q ) M0 + M1 = (n21 + n21M,q ) M3 = 2 2 2 1 1 −i i ∗ ∗ ∗ ∗ (n12 − n12E,q ) = (n21 − n21E,q ) E0 = (n12 + n12E,q ) = (n21 + n21E,q ) E1 = 2 2 2 2 1 1 −i −i ∗ ∗ ∗ ∗ E2 = (n11 + n11E,q ) = −q (n22 + n22E,q ) E3 = (n11 − n11E,q ) = q (n22 − n22E,q ) 2 2 2 2 M0 − M1 =

462

R. FIORESI

IM : (q −1 − q) 2 q −1 − q (M22 + M32 ) M0 M2= −1 M2 M0 − −1 M1 M2 2 q +q q +q 2 q −1 − q 2 q −1 − q M3 M0 − −1 M1 M3 M1 M2= −1 M2 M1 − −1 M0 M2 M0 M3= −1 q +q q +q q +q q +q 2 q −1 − q M3 M1 − −1 M0 M3 M1 M3= −1 M2 M3=M3 M2 q +q q +q M0 M1=M1 M0 +

IE : (q −1 − q) 2 (E2 + E32 ) 2 2 q −1 − q E3 E0 + i −1 E1 E3 E0 E3 = −1 q +q q +q 2 q −1 − q E3 E1 − i −1 E0 E3 E1 E3 = −1 q +q q +q E0 E1 = E1 E0 − iq

2 q −1 − q E E1 E2 E + i 2 0 q −1 + q q −1 + q 2 q −1 − q E2 E1 − i −1 E0 E2 E1 E2 = −1 q +q q +q E0 E2 =

E2 E3 = E3 E2

Proof. direct computation.

Remark 1. The relations in IM and IE allow us to order the monomials in M0 , M1 , M2 , M3 and E0 , E1 , E2 , E3 respectively. Remark 2. With the chosen coordinates we have that: ∼ M0 − M1 M2 + iM3 E2 + iE3 E0 + iE1 = = . (a ) = (nij ) M2 − iM3 M0 + M1 −q(E2 − iE3 ) E0 − iE1 (2.1)(i) ij This suggests the following definition: Definition 4.3. We call Minkowski (euclidean) length of a vector, the quantum determinant of (nij ) in Minkowski (euclidean) coordinates: detq (n) = M02 − M12 − q −1 (M22 + M32 ) = E02 + E12 + E22 + E32 . Proposition 4.2. kq [N ]∗M,q ∼ = kq [N ]∗E,q Proof. Here is the change of coordinates: 1 i −i 1 M0 =E0 , M1 =−iE1 , M2 = (1−q)E2 + (1+q)E3 , M3 = (1+q)E2 + (1−q)E3 . 2 2 2 2 Theorem 4.3. Let λ be the coaction introduced in Proposition 2.2 for the quantum conformal group kq [C]: kq [N ] −→ kq [C]⊗kq [N ], (n) 7−→ (y)S 12 (x)⊗(n). Then (∗C,M × ∗M,q ) · λ = λ · ∗M,q and (∗C,E × ∗E,q ) · λ = λ · ∗E,q . Proof. direct computation.

Definition-Proposition 4.4. Ω∗ (kq [N ]∗M,q ) quantum De Rham complex for kq [M2 ] is the associative algebra over kq with generators M0 , M1 , M2 , M3 , dM0 , dM1 , dM2 , dM3 and defining relations:

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dM0 dM1 = −dM1 dM0 dM0 dM2 = −

2 q −1 − q dM2 dM0 + −1 dM1 dM2 +q q +q

q −1

dM0 dM3 = −

2 q −1 − q dM dM1 dM3 dM + 3 0 q −1 + q q −1 + q

dM1 dM2 = −

2 q −1 − q dM dM0 dM2 dM + 2 1 q −1 + q q −1 + q

dM1 dM3 = −

2 q −1 − q dM3 dM1 + −1 dM0 dM3 +q q +q

q −1

dM2 dM3 = −dM3 dM2

M0 dM0 =[q − (q 2

−2

− q )]dM0 M0 + (q 2

−1

− q)

1 −1 (q − q)dM1 M0 4

1 1 1 − (q −1 − q)dM0 M1 − (q −1 + q)dM1 M1 + (dM2 M2 + dM2 M3 ) 4 4 2 1 −1 1 2 −1 −1 M0 dM1= q + (q − q)(−q + 3q) dM1 M0 + (q − q) − (q −1 + q)dM0 M0 4 4 1 −1 1 1 −1 + (q − q)dM1 M1 − (−q + 3q)dM0 M1 − (dM2 M2 + dM3 M3 ) 4 4 2 1 M0 dM2 =qdM2 M0 + q(q −1 − q)(dM0 M2 + dM1 M2 ) 2 1 M0 dM3 =qdM3 M0 + q(q −1 − q)(dM0 M3 + dM1 M3 ) 2 1 1 M1 dM0 = q 2 + (q −1 − q)(−q −1 + 3q) dM0 M1 + (q −1 − q) (q −1 − q)dM0 M0 4 4 1 1 1 − (−q −1 + 3q)dM1 M0 + (q −1 − q)dM1 M1 + (dM2 M2 + dM3 M3 ) 4 4 2 1 1 M1 dM1 = q 2 − (q −2 − q 2 ) dM1 M1 + (q −1 − q) (q −1 + q)2 dM0 M0 4 4 1 1 1 − (q −1 − q)dM1 M0 + (q −1 − q)dM0 M1 + (dM2 M2 + dM3 M3 ) 4 4 2 1 M1 dM2 = qdM2 M1 + q(q −1 − q)(dM0 M2 + dM1 M2 ) 2 1 M1 dM3 = qdM3 M1 + q(q −1 − q)(dM0 M3 + dM1 M3 ) 2

464

R. FIORESI

1 M2 dM0 = qdM0 M2 + q(q −1 − q)(dM2 M0 − dM2 M1 ) 2 1 M2 dM1 = qdM1 M2 − q(q −1 − q)(dM2 M0 − dM2 M1 ) 2 1 −1 2 M2 dM2 = q + q(q − q) dM2 M2 2 1 + (q −1 − q)[qdM3 M3 + (dM0 + dM1 )(M0 − M1 )] 2 1 1 M2 dM3 = q 2 + q(q −1 − q) dM3 M2 − q(q −1 − q)dM2 M3 2 2 1 M3 dM0 = qdM0 M3 + q(q −1 − q)(dM3 M0 − dM3 M1 ) 2 1 M3 dM1 = qdM1 M3 − q(q −1 − q)(dM3 M0 − dM3 M1 ) 2 1 −1 1 2 M3 dM2 = q + q(q − q) dM2 M3 − q(q −1 − q)dM3 M2 2 2 1 M3 dM3 = q 2 + q(q −1 − q) dM3 M3 2 1 + (q −1 − q)[qdM2 M2 + (dM0 + dM1 )(M0 − M1 )] 2 Proof. direct computation.

Remark. One can also construct the quantum De Rham complex for the euclidean space. The procedure is the same and the commutation relations in (4.4) get transformed according to (4.2). Acknowledgements I wish to thank Prof. E. Taft for his helpful explanations and especially my advisor Prof. V. S. Varadarajan for his continuous help, encouragement and for the ideas he has given me. References [1] O. Ogievetsky, W. B. Schmidke, J. Wess and B. Zumino, Commun. Math. Phys. 150 (3) (1992) 495. [2] R. Penrose, Rep. Math. Phys. 12 (1977) 65. [3] Y. Manin, Gauge Theory and Holomorphic Geometry, Springer Verlag, 7 (1981). [4] N. Reshetikhin and V. Lakshmibai, Cont. Math. 134 (1992) 145. [5] E. Taft and J. Towber, J. Algebra 142 (1991) 1. [6] Y. S. Soibelman, Amer. Math. Soc. 2 (1991) 161. [7] Y. S. Soibelman, Funct. Anal. Appl. 26 (1992) 225. [8] Y. Manin, Topics in Noncommutative Geometry, Princeton Univ. Press, (1991) 129.

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[9] Y. Manin, Quantum groups and non commutative geometry, Centre de Reserches Mathematiques Montreal (1988) 49. [10] B. Parshall and J. P. Wang, Quantum Linear Groups, Memoirs of the American Mathematical Society 439, American Mathematical Society, Providence RI, (1990) 42. [11] J. Harris, Algebraic Geometry: a First Course, Springer Verlag, 1991. [12] P. Griffiths and J. Harris,Principles of algebraic geometry, Wiley and Sons, New York (1978) 193. [13] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Press (1994) 242.

THE BORN OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST J. HERRIN Center for Clinical Evaluation Sciences Emory University, Atlanta, Georgia 30322, USA E-mail: [email protected]

J. S. HOWLAND∗ Department of Mathematics, University of Virgina Charlottesville, Virginia, USA Received 31 May 1996 Revised 25 November 1996 The problem of calculating asymptotic series for low-lying eigennvalues of Schr¨ odinger operators is solved for two classes of such operators. For both models, a version of the Born–Oppenheimer Approximation is proven. The first model considered is the family d2 2 Hε = −ε4 dx 2 + H(x) in L (R, H) where H(x) : H → H has a simple eigenvalue less than zero. The second model considered is a more specific family Hε = −ε4 ∆ + H(r, ω) in L2 (R3 , C2 ) where the eigenprojection P (ω) of H(r, ω) : C 2 → C 2 is associated with a non-trivial, or “twisted,” fibre bundle. The main tools are a pair of theorems that allow asymptotic series for eigenvalues to be corrected term by term when a family of operators is perturbed.

1. Introduction The Born–Oppenheimer Approximation (BOA), first presented some sixty-five years ago [1], is based on the smallness of the ratio ε4 ≡ me /M of electronic to nucleic mass, in consequence of which the electrons rearrange themselves on a faster time scale than that of nuclear motion. Let the nuclei be frozen into a configuration x, and let E(x) be an eigenvalue of the electronic Hamiltonian H(x), depending smoothly on x — for example, the ground state energy, E0 (x). Also let E(x) have a minimum value of E0 on some manifold M0 of equilibrium configurations. For a diatomic molecule M0 = S 2 , the two-sphere, and BOA asserts the existence of energy levels l(l + 1) 4 ε + O(ε5 ) En,l (ε) = E0 + ε2 (2n + 1)ω + b + R02

(1.1)

where ω, b, and R0 are constants characteristic of E(x). The quantum numbers n and l represent, respectively, the harmonic vibration of the nuclei about the equlibrium configuration and the spatial rotation of the molecule. ∗ Supported by NFS contract DMS-9002357 467 Reviews in Mathematical Physics, Vol. 9, No. 4 (1997) 467–488 c World Scientific Publishing Company

468

J. HERRIN and J. S. HOWLAND

Rigorous proof of (1.1) exists for a number of situations. Combes, Duclos and Seiler [2] offer an exhaustive proof of (1.1) for diatomic molecules and small energies. This work relies on a partitioning technique, called the Livsic (Feshbach) method, which to some extent obscures the content of the proof. Other recent work is that of Hagedorn [3–6] which exploits multiscaling techniques to get asymptotic expansions, to arbitrarily high orders, for particular bound-state eigenvalues and eigenvectors of the diatomic Hamiltonian. Other recent work [7–14], while robust, is likewise difficult. It is the main purpose of this paper to discuss a geometrical correction to (1.1) which arises when the eigenspace bundle is twisted over the manifold M0 = S 2 of equilibrium states. We shall also present a somewhat novel method of derivation of BOA, which we feel illustrates its true nature, by reducing it to (a) a quasi-classical eigenvalue problem and (b) ordinary first order perturbation theory. The method yields the terms up through order ε6 very simply. For clarity, we illustrate this procedure in Sec. 2 for what is probably the simplest Born–Oppenheimer problem. We take Hε = ε4 p2 + H(x) d and H(x) is a family of operators on the fibre space on L2 (R) ⊗ H, where p = −i dx H with negative spectrum consisting of a single non-degenerate eigenvalue E(x). We assume that H(x) is even, i.e., that

H(−x) = H(x) and that E(x) is smooth with a unique minimum value E0 at x = 0. If P (x) is the projection onto the eigenvector ψ(x) of E(x), there exists [15] a unitary adiabatic operator U (x) with U ∗ (x)P (x)U (x) = P (0) . If we now transform Hε by U (x), we obtain 4 2 ε p + E(x) ∗ 4 2 U (x)[ε p + H(x)]U (x) = 0

0 ε4 p2 + T (x)

+ ε4 V

(1.2)

in the decomposition P (0)H ⊕ (I − P (0))H, where T (x) ≥ 0. The term V , given by V = A(x)p + pA(x) + A(x)2 where A(x) = −iU 0 (x)U (x), arises from the commutation of U (x) with p2 , since one has U ∗ (x)p2 U (x) = (U ∗ (x)pU (x))2 = (p − iU 0 (x)U (x))2 . odinger perturbation of the matrix in (1.2). HowWe regard ε4 V as a Rayleigh–Schr¨ 4 2 ever, since ε p + T (x) ≥ 0, the negative spectrum of the matrix operator arises solely from the one-dimensional Schr¨ odinger operator ε4 p2 + E(x) .

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For small ε, this is simply a quasi-classical eigenvalue problem, with eigenvalues 1 2 + O(ε4 ) . En ∼ E0 + ε n + 2 Since the quasi-classical spacing is of order ε2 , while the perturbation is of order odinger ε4 (effectively of order ε3 for technical reasons), first-order Rayleigh–Schr¨ theory gives the eigenvalues up through order ε6 , which is as far as practical BOA calculations are taken. Returning to the general case, if P (x) is the projection onto the eigenvector for E(x), then the map taking the space P (x)H onto x defines a vector bundle over the manifold of nuclear configurations and, in particular, over the equilibrium manifold M0 . For diatomic molecules, M0 = S 2 and the bundle is trivial (or at least assumed so), and this leads to (1.1). However, in Sec. 3 we consider a very simple model with a two-dimensional fibre space H, in which a twisted bundle over S 2 occurs. This leads to the replacement of the usual Laplacian on S 2 by the operator of the Dirac monopole, and hence to the replacement of the spherical eigenvalue l(l + 1) with the Dirac eigenvalue (l + 1)2 . 2. Born Oppenheimer: A Straightforward Case Consider then this example. Let H(x) be self-adjoint in H for each x ∈ R, and assume that σ(H(x)) ∩ (−∞, 0] = {E(x)} where E(x) < 0 is a simple eigenvalue with eigenfunction ψ(x). Let K = L2 (R) ⊗ H and define H as multiplication by d in K with domain D(ˆ p2 ) = W2 (R) ⊗ H (the second H(x) in K. Let pˆ = −i dx ∞ Sobolev space). Assume that E(x), ψ(x) ∈ C (R) and that E(x) has an absolute minimum at 0. Thus E(x) ∈ L∞ (R) and E(x) = E0 + x2 ω02 + x3 ω1 + x4 ω2 + O(x5 ) = E0 + x2 ω02 + x2 W (x) .

(2.1)

Now define H = ε4 pˆ2 + H

on K

p ) ∩ D(H). Then we are prepared to with Hε essentially self-adjoint on D = D(ˆ state the main result of this section. Again, this result is not new, but illustrates a novel method which yields the lower order terms simply and which we will use in the next section to obtain the main result of this paper. 2

Theorem 2.1. Let H, K, H, and Hε be as above. Suppose that H commutes with complex conjugation C, and assume additionally that ψ 0 (x), ψ 00 (x) ∈ L∞ (R, H) ≡ L∞ (R) ⊗ H. Let En (ε), Ψn (ε) be the nth eigenvalues and eigenvectors (counted from the bottom of σ(Hε)) of Hε . Then 6 En (ε) ∼ E0 + ε2 (2n + 1)ω0 + ε4 b(2) n + O(ε )

470

J. HERRIN and J. S. HOWLAND

and Ψn (ε) ∼

X

k Uε−1 φ(k) n ε

k

where (Uε w)(x) = ε

1/2

w(εx). (k)

Note. We will calculate the first few φn ’s and bn ’s. Proof. Define the projection P (x) in H by P (x)φ = hφ, ψ(x)iH ψ(x) . Let P act on K by Pw = P (x)w(x). Writing K= PK ⊕ QK (where Q = I − P) we can write H as E(x) 0 H= 0 T (x) where this defines T (x). Note that T (x) ≥ 0. On PK ⊕ QK, write ε4 pˆ2 as 2 p2 Q Pˆ p P Pˆ 4 ε p2 Q Qˆ p2 P Qˆ 2

d 2 and let p2 denote − dx 2 acting in L (R). For w(x)ψ(x) ∈ PK, we have

Pˆ p2 P(w(x)ψ(x)) = P (x)ˆ p2 {w(x)ψ(x)} = P (x)[−w00 (x)ψ(x) − 2w0 (x)ψ 0 (x) − w(x)ψ 00 (x)] = −w00 (x)ψ(x) − 2w0 (x)hψ 0 (x), ψ(x)iH ψ(x) − w(x)hψ(x)00 , ψ(x)iH ψ(x) = {[p2 + hψ 00 (x), ψ(x)iH ψ(x)]w(x)}ψ(x) . The hψ 0 (x), ψ(x)iH vanishes due to the fact that ψ is real; that is, since H commutes with complex conjugation. (0 = 10 =hψ(x), ψ(x)i0H = 2Rehψ 0 (x), ψ(x)iH .) Writing PK = L2 (R) ⊗ ψ(x) , we find that Pˆ p2 P = (p2 + α(x)) ⊗ 1 where α(x) = hψ 00 (x), ψ(x)iH =

X

βk xk .

Formally then we write 4 2 (ε p + E(x)) ⊗ 1 0 Hε = 0 Qε4 pˆ2 Q + T (x) α(x) ⊗ 1 0 0 Pˆ p2 Q + ε4 + ε4 0 Qˆ p2 P 0 0 hε ⊗ 1 0 ≡ + ε4 V1 + ε4 V2 0 T˜ε ≡ H0ε + ε4 V1 + ε4 V2

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By assumption, T (x) ≥ 0 and hence T˜ε ≥ 0. Thus, the only negative spectrum of H0ε results from hε . Transform hε by ε2 = λ−1 , h(λ) = ε−4 (hε − E0 ) = p2 + λ2 (E(x) − E0 ). Approximate h(λ) by h0 (λ) = p2 + λ2 ω02 x2 , and set k 0 = p2 + ω02 x2 . Define the scaling 1 1 operator Uλ by (Uλ f )(x) = λ 4 f (λ 2 x), so that k 0 = λ−1 Uλ−1 h0 (λ)Uλ . k 0 is just the harmonic oscillator, with eigenvalues en = (2n + 1)ω0 and eigenfunctions φn (x) 1

= Nn e−x ω0 /2 Hn (xω02 ), where Hn (x) is the nth Hermite polynomial. From [16] we know, then, that for h(λ) we have X −l a(l) En0 (λ) ∼ (2n + 1)ω0 λ + a(0) n + n λ 2

l≥1



and

ψn (λ) = Uλ 

X

 −k 2

φ(k) n λ



k≥1 (k)

where φn is just φn times a polynomial. So, for hε , we have X 2l a(l) En0 (ε) ∼ E0 + ε2 (2n + 1)ω0 + n ε

(2.2)

l≥1

and ψn0 (ε) = Uε−1

hX

i k . φ(k) n ε

Thus, as argued earlier, the negative eigenvalues of H0ε are given by (2.2) with eigenvectors Ψ0n (ε) ≡ ψn0 (ε) ⊗ 1. Now we want to “add on” first ε4 V1 and then ε4 V2 . H1ε ≡ H0ε + ε4 V1 is p2 ) ⊆ D(V1 ) so to apply Theorem A.1 essentially self-adjoint on D(ˆ p2 )∩D(H) ⊆ D(ˆ 0 3 (see Appendix) we find that for |z − En | = ε kε4 V1 (H0ε − z)−1 k = kε4 α(x)(hε − z)−1 k ≤ |ε4 | · kαk∞ ·

1 = O(ε) ε3

so that H1ε has asymptotic series for low-level eigenvalues given by (2.2) corrected according to (A.1): En1 (ε) ∼ En0 (ε) + ε4

hΨ0n (ε), V1 P1n Ψ0n (ε)iK hΨ0n (ε), P1n Ψ0n (ε)iK

with eigenvectors given by Ψ1n (ε) = P1n (ε)Ψ0n (ε). Now, to complete the treatment of Hε we must take care of V2 ; in this case Theorem A.2 will apply if we can establish for |z − En1 (ε)| = ε3 that (I) kε8 [V2 (H1ε − z)−1 ]2 k = O(ε) and (II) ε3 · kε4 (H1ε − z)−1 V2 (H1ε − z)−1 k = O(1). We will prove these below. For now, assume (I) and (II) hold. Then Hε has eigenvalues En (ε) with asymptotics given by En (ε) ∼ En0 (ε) + ε4

hΨ0n (ε), V1 P1n Ψ0n (ε)iK hΨ1 (ε), V2 PΨ1n (ε)iK + ε4 n 1 . 0 1 0 hΨn (ε), Pn Ψn (ε)iK hΨn (ε), PΨ1n (ε)iK

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As a first approximation, since hΨ1n (ε), V2 Ψ1n (ε)iK = 0, we have En (ε) ∼ En0 (ε) + ε4 hΨ0n (ε), V1 Ψ0n (ε)iK . We calculate the corrections below and find 6 (6) 7 En (ε) ∼ E0 + ε2 (2n + 1)ω0 + ε4 b(4) n + ε bn + O(ε ) (4)

(4)

(6)

where bn = an + β0 and bn can be calculated.

p2 P. Then Proof of (I) and (II). Let W = Pˆ p2 Q and S = Qˆ 1 −1 1 −1 V2 (Hε − z) V2 (Hε − z) is just W (T˜ε − z)−1 S(hε + ε4 α(x) − z)−1 0 . 0 S(hε + ε4 α(x) − z)−1 W (T˜ε − z)−1 We will show: (a) that ε6 kS(hε + ε4 α(x) − z)−1 k = O(ε) and (b) that ε2 kW (T˜ε − z)−1 k = O(1). For (a), write S = −QP 00 − 2iQP 0 pˆ and note that by our hypotheses, Q, P 0 , and P 00 are all bounded. Thus ε6 kS(hε + ε4 α − z)−1 k ≤ ε6 kP 00 k · k(hε + ε4 α − z)−1 k + ε4 k2P 0 k · kε2 pˆ(hε + ε4 α − z)−1 k ≤ ε6 kP 00 k ·

1 ε3

+ ε4 kP 0 k · kε2 pˆ(ε4 pˆ2 − z)−1 )k · k(ε4 pˆ2 − z)(ε4 pˆ2 + ε4 α + E(x) − z)−1 )k ≤ ε3 kP 00 k + |ε4 |k2P 0 k · B · k(I − (ε4 + E(x))(ε4 pˆ2 + ε4 α + E(x) − z)−1 )k ≤ ε3 kP 00 k + ε4 k2P 00 k · B ·

1 = O(ε) ε3

where we have used Lemma A.3 to bound ε2 pˆ(ε4 pˆ2 − z)−1 and the fact that |z − En1 (ε)| = ε3 to bound (hε + ε4 α − z)−1 . This establishes (a). For (b), we’ll have to work much harder. Let U (x) be the adiabatic operator defined by U 0 (x) = [P 0 (x), P (x)]U (x) with U (0) = I. See [15, p. 99] for a short proof that U (x) exists and has the following properties: U (x)P (0) = P (x)U (x) U ∗ (x) = U −1 (x) . The purpose of introducing U (x) is to allow us to express the differential operators as acting on the same space, namely L2 (R)⊗Q(0)H. Define U on K as multiplication by U (x), so that U is also unitary. Then we have kW (T˜ε − z)−1 k = kU∗ W (T˜ε − z)−1 Uk = kU∗ W UU∗ (T˜ε − z)−1 Uk .

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

473

But p2 Q(x)U (x) U∗ W U = U ∗ (x)P (x)ˆ p2 U (x)U ∗ (x)Q(x)U (x) = U ∗ (x)P (x)U (x)U ∗ (x)ˆ p2 U (x))Q(0) = P (0)(U ∗ (x)ˆ p + B(x))Q(0) = P (0)(ˆ p2 + A(x)ˆ where A(x) and B(x) are bounded operators. Put A1 (x) = P (0)A(x)Q(0) and B1 (x) = P (0)B(x)Q(0). Then, since Q(0) commutes with pˆ2 we get that U∗ W U = (A1 (x)p + B1 (x)) ⊗ 1 on L2 (R) ⊗ Q(0)K. Similarly, we find, putting T0 (x) = U −1 (x)T (x)U (x), p2 Q(x) + T − z)−1 U (x) = (ε4 U ∗ (x)Q(x)ˆ p2 Q(x)U (x) + T0 − z)−1 U ∗ (x)(ε4 Q(x)ˆ = Q(0)(ε4 U ∗ (x)ˆ p2 U (x) + T0 − z)−1 Q(0) = Q(0)(ε4 pˆ2 + ε4 B0 (x) + T0 − z)−1 Q(0) where Q(0)A(x)Q(0) = 0 and B0 (x) = Q(0)B(x)Q(0) is bounded. So we can write U∗ (T˜ε − z)−1 U = (ε4 p2 + ε4 B0 (x) − z)−1 ) ⊗ 1 in L2 (R) ⊗ Q(0)H. Thus we can see that ε2 kW (T˜ε − z)−1 k = ε2 k(A1 (x)p + B1 (x))(ε4 p2 + ε4 B(x) + T0 − z)−1 k ≤ kA1 (x)k · kε2 p(ε4 p2 + ε4 B0 (x) + T0 − z)−1 k + kB1 (x)k · k(ε4 p2 + ε4 B0 (x) + T0 − z)−1 k . The first quantity is bounded by kA1 k(|E0 |/2 + ε3 ) by Lemma A.3. For the second quantity write k(ε4 p2 + ε4 B0 (x) + T0 − z)−1 k ≤ k(ε4 p2 + T0 − z)−1 k · k(ε4 p2 + T0 − z)(ε4 p2 + ε4 B0 (x) + T0 − z)−1 k 1 · k(I − ε4 B0 (ε4 p2 + T0 − z)−1 )−1 k |z| ε4 kB0 k∞ 1 · 1+ ≤ |z| |z|

≤

≤C for ε small. This concludes the proof of (I). Note that these same estimates give us that for z large, kε4 V (Hε0 − z)−1 k < 1. For a proof of (II), we’ve already done most of the work. We need to show that kε7 (H1ε − z)−1 V2 (H1ε − z)−1 k = O(1) .

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We calculate that ε7 (H1ε − z)−1 V2 (H1ε − z)−1 is just ε7 (hε + ε4 α(x) − z)−1 W (T˜ε − z)−1 0

0 ε7 (T˜ε − z)−1 S(hε + ε4 α(x) − z)−1

! .

As above, we have kε2 W (T˜ε − z)−1 k = O(1), and kε3 (hε + ε4 α(x) − z)−1 k = O(1). Similarly, we have as above that kε6 S(hε +ε4 α(x)−z)−1 k = O(1), while k(T˜ε −z)−1 k is bounded for Rez < 0. This establishes (II) and thus completes the proof of Theorem 2.1. The Correction. As a first approximation, we have the correction hΨ0n (ε)| ε V1 |Ψ0n (ε)iK to En0 (ε). Let c0m be the mth order correction to En (ε) contributed by this expression. We have 4

0 Ψn (ε)|ε4 V1 |Ψ0n (ε) = ε4

*

X

εk Uε−1 φ(k) n |V|

=

+ εj Uε−1 φ(j) n

j≥0

k≥0

XX

X

K

−1 (j) εj+k+4 hUε−1 φ(k) n |V|Uε φn iK

j≥0 k≥0

=

X

εl c0l

l≥0

from which c00 = c01 = c02 = 0 (since the term j = k = 0 may contribute corrections of order 3, we don’t know yet that c03 = 0). To calculate the higher order corrections, we need to calculate the contributions of each order from each term in the above (k) expansion. To calculate the c0m ’s, we’ll need to know something about the φn ’s. We know from the exact solution to the harmonic oscillator eigenvalue problem that (0) −x φ(0) n (x) = ψ(x)Φn (x) = ψ(x)Nn e

2

ω0 /2

1

Hn (xω02 )

where Nn is a (real) normalization constant and Hn (x) is the nth Hermite polynomial. Following [16], define ˜ h(λ) = λ−1 Uλ−1 h(λ)Uλ = k 0 + x2 W (xλ− 2 ) 1

where W (x) is as in (2.1). Then formally ˜ h(λ)

X k≥0

−2 Φ(k) = n λ k

X l≥0

−2 e(l) n λ l

X

−2 Φ(k) . n λ 1

k≥0

1 1 ˜ ˜ to an operator unitarily equivalent to h(λ) (under Since λ− 2 −→ −λ− 2 sends h(λ) (l) x −→ −x), we know that en = 0 for l odd. Using these facts we have, formally at least,

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

 k0 

X

475

 −k 2

Φ(k) n λ

   X X l k −2  + wl xl+2 λ− 2   Φ(k) n λ

k≥0

l≥1

   X X − 2l −k 2  2 . e(2l) Φ(k) = n λ n λ l≥0

k≥0

k≥0

By collecting terms of order λ−j/2 we can determine en and Φn . (Technically, we cut the sums off at progressively higher orders, then do the calculation.) Using parity arguments we can calculate that the lowest order corrections to hε = ε4 p2 + Pm E(x) eigenvalues En (ε)=E0 + (2n + 1)ε2 ω0 + l=1 aln ε2l+2 +O(ε2m+4 ) due to the perturbation by ε4 V are given by (j)

(j)

c00 = 0 c01 = 0 c02 = 0 c03 = 0 c04 = β0 c05 = 0 c06 = 2β0 + β1

Z

Z (0) yΦ(1) n Φn dy + β2

2 y 2 [Φ(1) n ] dy .

Beyond order 6, the perturbation V2 makes contributions, and we do not calculate these. Thus, the first few terms in the asymptotic expansion of the low-lying eigenvalues of Hε are just 6 (6) 0 7 En (ε) ∼ E0 + ε2 (2n + 1)ω0 + ε4 (b(4) n + β0 ) + ε (bn + c6 ) + O(ε ) .

3. Born Oppenheimer with a Twist Consider the correspondence Γ(ω): S 2 → CP 1 between the 2-sphere S 2 and the 1-dimensional complex subspaces of C2 . We can think of this correspondence as pairs (z1 (ω), z2 (ω)) of complex numbers with |z1 |2 + |z2 |2 = 1, where Γ(ω) = span{z1 (ω), z2 (ω)}. In spherical coordinates, where φ is the azimuthal angle and θ the longitudinal angle, we have z1 (ω) = cos φ2 and z2 (ω) = sin φ2 eiθ . Thus, Γ(ω) can be taken to be the usual stereograhpic projection of S 2 onto C2 . For ω ∈ S 2 , let P (ω) be the projection in C2 onto the corresponding subspace Γ(ω). Then the projection onto the subspace Γ(ω) in CP 1 is just z1 (ω) z1 (ω)¯ z2 (ω) z1 (ω)¯ . P (ω) = z1 (ω) z2 (ω)¯ z2 (ω) z2 (ω)¯ Note that the vector (z1 (ω), z2 (ω)) changes sign when transported around a circle of constant latitude. This phase change is Berry’s phase; since P is well defined, this

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means that the fibre bundle (R3 , CP 1 , P (ω)) is non-trivial, or “twisted” [17–23]. Set Q(ω) = I − P (ω) and for (r, ω) ∈ [0, +∞) × S 2 = R3 let H(r, ω) act in C2 with H(r, ω) = E(r)P (ω) + T (r, ω)Q(ω) . Let K = L2 (R3 , C2 ) and let H act in K by (Hf )(r, ω) = H(r, ω)f (r, ω) . Let P act in K by (Pf )(r, ω) = P (ω)f (r, ω). Let ∆K be the Laplacian in K and define Hε ≡ −ε4 ∆K + H . This formally defines a Schr¨ odinger operator in K which will be self-adjoint when E(r), T (r, ω) are appropriately restricted; we will assume that E(r), T (r, ω) ∈ L2 (R3 )+ L∞ (R3 ). Then Hε will be self-adjoint with D(Hε ) = D(∆K ) = W2 (R3 , K). We will calculate a version of (1.1) for Hε with: Theorem 3.1. Let H, Hε , and K be as above. Also let E(r) have an absolute minimum E(r0 ) < 0 at r0 > 0 with E 00 (r0 ) = 2ω02 . Take T (r, ω) ≥ 0. Then the eigenvalues of Hε have asymptotic expansions in ε given by (l + 1)2 2 4 (3.1) + O(ε6 ) . En,l (ε) ∼ E(r0 ) + ε (2n + 1)w0 + ε b4 + r02 Proof. The proof is similar in method to that of Theorem 2.1. Begin by writing K = PK ⊕ QK, where Q = I − P, and express Hε formally as P(−ε4 ∆K + E(r))P P(−ε4 ∆K )Q Hε = Q(−ε4 ∆K + T (r, ω))Q Q(−ε4 ∆K )P 4 0 −ε P∆K P + E(r) 0 −P∆K Q 4 +ε = 0 0 −ε4 Q∆K Q + T (r, ω) −Q∆K P ≡ H0ε + ε4 V .

(3.2)

We shall calculate the eigenvalue asymptotics for H0ε and then “perturb” this spectrum by ε4 V. This would be straightforward, except for the degeneracy of our spherical coordinates at the origin. To get around this, we shall need resort to several tricks. Continuing now with (3.2), we note that the negative spectrum of H0ε results exclusively from the formal operator −ε4 P∆K P + E(r) ≡ hε acting in the space PK. We can calculate the compression P∆K P of ∆K to PK: let f ∈ PK be written as 2 3 f = v(r, ω) zz12 (ω) (ω) = v(r, ω)γ(ω), where v ∈ L (R ). Then z1 (ω) P∆K Pf = P∆K Pv(r, ω) z2 (ω) z1 (ω) [¯ z1 ∆z1 + z¯2 ∆z2 ]v(r, ω) = z2 (ω)

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

477

where the symbol ∆ in the last line refers to the Laplacian in L2 (R3 ) with Dirichlet boundary conditions. Thus, z1 ∆z1 + z¯2 ∆z2 ] + E(r)) ⊗ 1 hε = (−ε4 [¯ on L2 (R3 ) ⊗ γ(ω). Continuing, we calculate z1 (∇z1 ) · ∇ + 2¯ z2 (∇z2 ) · ∇ + z¯1 (∆z1 ) + z2 (∆z2 ) z¯1 ∆z1 + z¯2 ∆z2 = ∆ + 2¯ = ∆R −

D2 r2

2

∂ 2 ∂ 1 2 where ∆R = ∂r 2 + r ∂r (again with Dirichlet boundary conditions) and −D + 2 φ is the Dirac Monopole operator [24]. In coordinates, where z1 (ω) = cos 2 , z2 (ω) = sin φ2 eiθ ,

D2 = −

∂ 1 ∂2 1 1 ∂ i φ ∂ φ 1 sin φ − + tan2 + . − sec2 sin φ ∂φ ∂φ sin2 φ ∂θ2 2 2 ∂θ 4 2 2

(3.3)

Thus we can write hε in PK ∼ = L2 (R3 ) ⊗ γ(ω) as hε = (−ε4 ∆R + ε4

D2 + E(r)) ⊗ 1 r2

so to determine the negative spectrum of H0ε it is sufficient to examine h0ε = −ε4 ∆R + ε4

D2 + E(r) r2

in L2 (R3 ) with Dirichlet boundary conditions. Let D(h0ε ) be the domain of this operator. We write L2 (R3 ) as L2 (R+ , r2 dr) ⊗ L2 (S 2 , dω), where R+ = [0, +∞). It is known [25] that D2 has a complete set of eigenvectors δl (ω) in L2 (S 2 , dω) with corresponding eigenvalues (l + 1)2 , l = 0, 1, 2, . . . so we can write L2 (S 2 , dω) = ∞ L Kl where Kl is the eigenspace spanned by δl (ω). Thus l=0

L2 (R+ , r2 dr) ⊗ L2 (S 2 , dω) ∼ =

∞ ∞ M M (L2 (R+ , r2 dr) ⊗ Kl ) = Ll . l=0

l=0

˜ 0ε ) be the subset of D(h0ε ) consisting of finite linear combinations of products Let D(h ˜ 0ε ) ∩ Ll is dense in Ll . ˜ 0ε ) is dense in L2 (R3 ) and thus Dl = D(h f (r)g(ω). Then D(h 0 l 2 + 2 l Decomposing hε according to L = L (R , r dr) ⊗ K we have 2 0 l 4 4 (l + 1) ⊗I hε |Ll ≡ hε ⊗ I = −ε ∆R + E(r) + ε r2 with hlε essentially self-adjoint on Dl . From this series of decompositions we conclude that the set of eigenvalues of hε , and hence of H0ε , is just the union of the

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J. HERRIN and J. S. HOWLAND

sets of the eigenvalues of the hlε ’s acting in Ll . Introduce the unitary operator Sr : L2 (R+ , r2 dr) → L2 (R+ , dr) that takes f (r) → rf (r) and put kεl = Sr−1 hlε Sr ∞ 2 S 0 = ε4 p2 + E(r) + ε4 (l+1) σ(kεl ), so we must only calcur 2 . Thus, σ(Hε ) = l=0

late the eigenvalues of kεl for each l. The operator ε4 p2 + E(r) is the standard (1-dimensional) quasiclassical operator; however, the r12 singularity makes the last term problematic. To get around this we will apply what has been formulated (and referred to) elsewhere as the “twisting trick” (see [26]). The idea is to add an extra degree of freedom so that we can “twist” the r12 singularity out of the way. (This “twisting” is unrelated to the twisting of our fibre bundle — except by the current application, of course!). The Twisting Trick. Choose rb < r0 so that E(rb ) = E(r0 )/2, and let ra < rb . Choose rc > r0 so that E(rc ) = E(rb ) and define E1 (r) by   r ≤ rb    E(r)  E1 (r) = E(rb ) rb ≤ r ≤ rc .     E(r) r ≥ rc On the space L2 (R+ , dr) ⊕ L2 (R+ , dr) define l kε 0 . Kε = 0 ε4 p2 + E1 (r) Note that σ(Kε ) ∩ (−∞, E(rb )] = σ(kεl ) ∩ (−∞, E(rb )]. Define the function ζ(r) ∈ C 2 (R+ ) by   0 r ≥ rb     π . r ≤ r ζ(r) = a 2     smooth otherwise Then put

U (r) =

cos ζ(r)

sin ζ(r)

− sin ζ(r)

cos ζ(r)

≡

C(r)

S(r)

−S(r) C(r)

.

U (r) is unitary on L2 (R+ ) ⊕ L2 (R+ ), and thus σ(Kε ) = σ(U ∗ (r)Kε U (r)). But 2 (using L(r) = ε4 (l+1) r2 ) ε4 p2 + E(r) + ε4 C 2 (r)L(r) 0 U (r)Kε U (r) = 0 ε4 p2 + E1 (r) + ε4 S 2 (r)L(r) −2iζ 0 (r)p − ζ 00 (r) + S(r)C(r)L(r) [ζ 0 (r)]2 . + ε4 [ζ 0 (r)]2 2iζ 0 (r)p + ζ 00 (r) + SCL ∗

Applying quasiclassical analysis [16] to ε4 p2 + E(r) we find that it has eigenvalues ˜n0 (ε) with asymptotics E ˜n0 (ε) ∼ E(r0 ) + ε2 (2n + 1)w0 + ε4 b4 + O(ε6 ) E

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

479

and eigenvectors ψ˜n (ε). The perturbation C 2 (r)L(r) is bounded and positive, so we can apply Theorem A.1 to conclude that ε4 p2 + E(r) + ε4 C 2 (r)L(r) has eigenvalues ˜ 0 (ε) with asymptotics E n,l 2

C (r) ˜ ˜ ˜ ˜ 0 (ε) + ε4 (l + 1)2 hψn (ε), r2 Pn,l (ε)ψn (ε)i ˜ 0 (ε) ∼ E E n,l n hψ˜n (ε), P˜n,l (ε)ψ˜n (ε)i

(3.4)

and eigenvectors ψ˜n,l (ε) = P˜n,l (ε)ψ˜n (ε). Below we calculate the first few corrections and find (l + 1)2 0 2 4 ˜ + O(ε6 ) . En,l (ε) ∼ E(r0 ) + ε (2n + 1)w0 + ε b4 + r02 Now let

V1 (r) = V2 (r) =

−2iζ 0 (r)p

0 2iζ 0 (r)p

0 0

2

[ζ (r)] ζ 00 (r) + S(r)C(r)L(r)

−ζ 00 (r) + S(r)C(r)L(r) [ζ 0 (r)]2

.

ε4 V1 (r) can be added using Theorem A.2 and estimates analogous to those used in the Proof of (I) and (II) in Sec. 2. V2 (r) is bounded, so we can use Theorem A.1 to 1 (ε) of U (r)∗ Kε U (r) have asymptotics add it on and so get that the eigenvalues E˜n,l 1 ˜ 0 (ε) + ε4 (ε) ∼ E E˜n,l n,l

0 0 0 0 hψ˜n,l (ε), V1 P˜n,l (ε)ψ˜n,l (ε)i hψ˜n,l (ε), V2 P˜n,l (ε)ψ˜n,l (ε)i 4 + ε 0 0 0 0 ˜ ˜ ˜ ˜ ˜ ˜ hψ , Pn,l (ε)ψ (ε)i hψ , Pn,l (ε)ψ (ε)i n,l

n,l

n,l

n,l

but V1 (r) = V2 (r) = 0 in a neighborhood of r0 , so there is no contribution of order ε4 . So (l + 1)2 1 (ε) ∼ E(r0 ) + ε2 (2n + 1)w0 + ε4 b4 + (3.5) + O(ε6 ) . E˜n,l r02 Taking the union (over l) of the sets of eigenvalues of the hlε ’s we find that hε and 0 (ε) with asymptotic series given by (3.5). hence H0ε has eigenvalues En,l Returning to (3.2), we see that to complete the treatment of Hε we need to treat the perturbation V. First note that we can write Q(ω) explicitly as z2 (ω) −z2 (ω)¯ z1 (ω) z2 (ω)¯ . Q(ω) = z2 (ω) z1 (ω)¯ z1 (ω) −z1 (ω)¯ and calculate that −Q∆K Q = −∆R +

1 2 D ⊗1 r2

1 ∇ω ⊗ 1 r2 1 −Q∆K P = 2 ∇∗ω ⊗ 1 r −P∆K Q =

on L2 (R3 ) ⊗ γ ⊥ (ω)

from L2 (R3 ) ⊗ γ ⊥ (ω) to L2 (R3 ) ⊗ γ(ω) from L2 (R3 ) ⊗ γ(ω) to L2 (R3 ) ⊗ γ ⊥ (ω)

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J. HERRIN and J. S. HOWLAND

where ∇ω = e−iθ

∂ 1 ∂ − i csc φ − tan φ/2 ∂φ ∂θ 2

and ∇ω ∇∗ω = D2 −

1 . 2

(Note that ∇ω ∇∗ω is the Dirac monopole operator.) From this we can rewrite (3.2) as ! 2 −ε4 ∆R + D 0 0 ∇ω ε4 r 2 + E(r) . + Hε = 2 r2 ∇∗ω 0 0 ε4 ∆R + D2 + T (r, ω) r

The second term here, V, has the same 1/r2 singularity at the origin, so again we will resort to the “twisting trick” to move it out of the way. This time, we define ζ(r), E1 (r) and Sr as before and 1 ra ≤ r ≤ rb r2 g(r) = 0 otherwise 0 g(r)∇ω ≡ g(r)Γ V1 = g(r)∇∗ω 0 cos ζ(r)Sr sin ζ(r)Sr C S 0 0 C= ; S= ; U= −S C 0 cos ζ(r)Sr 0 sin ζ(r)Sr and H1ε =

−ε4 ∆R +

D2 r2

+ E1 (r) −ε4 ∆R +

0

Then Kε ≡

!

0 D2 r2

. + T (r, ω)

H0ε + ε4 V

0

0

H1ε + ε4 V1

has the same negative spectrum as H0ε + ε4 V, hence as does U∗ Kε U. A calculation shows that 0 0 [2iζ 0 p + ζ 00 ]I Hε 0 Hε 0 −[ζ 0 (r)]2 I ∗ 4 . (3.6) U= +ε U [ζ 0 ]2 I 0 H1ε 0 H1ε [−2iζ 0 p − ζ 00 ]I and that U

∗

V 0 0 V1

2

U=

( Cr2 + S 2 g(r))Γ

0

0

( Sr2 + C 2 g(r))Γ

where Γ=

!

2

0

∇ω

∇∗ω

0

.

We will treat the last term in (3.6) as two distinct perturbations, namely −[ζ 0 (r)]2 I ζ 00 (r)I 0 2iζ 0 (r)pI and . 0 −2iζ 0 (r)pI −ζ 00 (r)I [ζ 0 (r)]2 I

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

The perturbation 2iζ 0 (r)p

0

I

481

satisfies the conditions of Theorem A.2, using −I 0 again estimates analogous to those in the proof of (I) and (II) of Sec. 2. 2 We next treat the perturbation ε4 ( Cr2 + S 2 g(r))Γ of H0ε ; this is the most troublesome step, and will require some work. To satisfy the conditions of Theorem A.2, we will need to establish that for |z − En,l (ε)| = ε3 2 (I) kε8 [( Cr2 + S 2 g(r))Γ(H0ε − z)−1 ]2 k = O(ε) and 2 (II) |ε3 | · kε4 (H0ε − z)−1 ( Cr2 + S 2 g(r))Γ(H0ε − z)−1 k = O(1). We will prove these below. For now, assume (I) and (II) hold. The next step is to note that 2 S 2 + C g(r) Γ ≥ E0 /2 . H1ε + ε4 r2 This follows from the construction of g(r) and from ∇∗ω ∇ω = D2 − 12 . Taken with the preceeding assertions, this means that ! 2 0 ( Cr2 + S 2 g(r))Γ 2

( Sr2 + C 2 g(r))Γ

0 contributes only the correction hψ˜n (ε),

C2 + S 2 g(r))Γψn (ε)i r2

to En,l . Finally, we have that

−[ζ 0 (r)]2

−ζ 00 (r)

0 2 −[ζ (r)] ζ 00 (r)

is bounded so we can apply Theorem A.1. This term vanishes in a neighborhood of r0 and so contributes no correction (as a first approximation) to En,l . We know then that Hε has eigenvalues En,l (ε) with asymptotics given by (l + 1)2 2 4 + O(ε6 ) . En,l (ε) ∼ E(r0 ) + ε (2n + 1)ω0 + ε b4 + R2 By the results below (I) and (II) hold. This completes the proof.

Proof of (I) and (II). The proof of both (I) and (II) follows closely the method 2 −1 , used in Sec. 2. First we shall prove (I). Let R0 (z) ≡ (−ε4 ∆R + D r 2 + E(r) − z) 2 D 4 −1 2 1 2 2 1 R1 (z) ≡ (−ε ∆R + r2 + T (r, ω) − z) , and W = (C r2 + S g(r)). Then [(C r2 + S 2 g(r))Γ(H0ε − z)−1 ]2 turns out to be 0 W ∇ω R1 (z)W ∇∗ω R0 (z) . 0 W ∇∗ω R0 (z)W ∇ω R1 (z) We need to show that for |z − En,l (ε)| = ε3 we have ε8 [V (H0ε − z)−1 ]2 = O(ε), so it is sufficient to show that for such z we have

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J. HERRIN and J. S. HOWLAND

(a) kε2 W ∇ω R1 (z)k = O(1) and (b) kε6 W ∇∗ω R0 (z)k = O(ε). Now, since ∇∗ω ∇ω = D2 − 12 ≥ D2 we have (D2 being strictly positive) for u ∈ L2 (S 2 ) that k∇ω D−1 uk2 = h∇ω D−1 u, ∇ω D−1 ui = h∇∗ω ∇ω D−1 u, D−1 ui ≤ hD2 D−1 u, D−1 ui =1 which allows us (using W ≤ κr , some κ) to estimate

2

2 2

ε D

2 W ∇ω 4 4D −1 4 4D −1

+ T − z) ≤ κ · (−ε ∆R + ε + T − z) ε 2 (−ε ∆R + ε r r r r |Imz| 1 1+ ≤κ· |Rez| |z| ≤ C0

for ε near 0

(3.7)

where we have used Lemma A.3. This completes the proof of (a). To complete the proof of (I), we only need to show that ε6 kW ∇ω R0 (z)k = O(ε), and we proceed as above; first, however, we write E(r) as the sum of a positive function V + and a (bounded) negative function V − . Factoring (−ε4 ∆R + ε4 D2 + E(r) − z)−1 = (−ε4 ∆R + ε4 D2 + V + + 1)−1 · (−ε4 ∆R + ε4 D2 + V + + 1) · (−ε4 ∆R + ε4 D2 + E(r) − z)−1 = (−ε4 ∆R + ε4 D2 + V + + 1)−1 (I − (V − − 1 − z) · (−ε4 ∆R + ε4 D2 + E(r) − z)−1 ) we can estimate for |z − En,l (ε)| = ε3 that ε6 kW ∇ω R0 (z)k = k

ε6 W ∇ω (−ε4 ∆R + ε4 D2 + V + + 1)−1 k r2

k(I − (V − − 1 − z)(−ε4 ∆R + ε4 D2 + E(r) − z)−1 )k ε6 W ∇ω (−ε4 ∆R + ε4 D2 + V + + 1)−1 k r2 1 · (1 + (kV − k∞ + |z| + 1) · 3 ) ε ε2 4 4 2 = k 2 W ∇ω (−ε ∆R + ε D + V + + 1)−1 k · (ε4 + Cε) . r

≤k

Using the estimate (3.7) we see that the norm in the last line is bounded and hence that kε6 W ∇ω R0 (z)k = O(ε) for |z − En,l (ε)| = ε3 . This completes the proof of (I).

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

483

As in Sec. 2, we’ve proved en route the fact that kε4 V (H0ε − z)−1 k < 1 for |z| big enough. The proof of (II) will come easily now, using the above results. In the above notation we have (H0ε − z)−1 V(H0ε − z)−1 is

0 R0 (z)W ∇ω R1 (z) 0 R1 (z)W ∇∗ω R0 (z)

.

But ε7 kR0 (z)W ∇∗ω R1 (z)k ≤ ε5 kR0 (z)k · ε2 kW ∇ω R1 (z)k ≤ ε2 · C where we’ve used (3.7). Likewise, ε7 kR1 (z)W ∇ω R0 (z)k ≤ kR1 (z)k · ε7 kW ∇ω R0 (z)k ≤

1 · O(ε2 ) |z|

= O(ε) . This completes the proof of (II). Coefficients. To calculate the coefficients in (3.5) from (3.4) we must refer to [16] for some additional results. The method there (in our notation) is to transform kε ≡ ε4 p2 + E(r) under the unitary operator on L2 [0, +∞) defined by 1

(Dε f )(r) = ε 2 f (εr + r0 ) . −2 E(εr + r0 ). From [16, Corollary 3.2] we know Then kε0 = ε−2 Dε−1 k˜ε Dε = −∂ ∂r 2 + ε that the eigenfunctions φn (ε) of kε0 satisfy 2

|φn (ε)(r)| ≤ Ce−ω0 r

2

/4

(3.8)

for ε small. Translating this information back to our original space, we see that the eigenfunctions of kε are Dε−1 φn (ε) and that the first order correction to En (ε) in (3.4) is (l + 1)2 times ε

4

hDε−1 φn (ε),

C 2 (r) −1 Dε φn (ε)i = ε4 r2

Z

+∞

0

Z

+∞

= ε3 0

Z

+∞

= ε4 −r0 /ε

cos2 ζ(r) −1 |Dε φn (ε)|2 dr r2 2 r − r0 1 ) dr φn (ε)( r2 ε cos2 ζ(εy + r0 ) |φn (ε)(y)|2 dy . (εy + r0 )2

484

J. HERRIN and J. S. HOWLAND

Now, (x + r0 )−2 cos2 ζ(r) is bounded and continuous away from −r0 and can be P∞ written as (x + r0 )−2 cos2 ζ(r) = k=0 ak xk . Note that a0 = r0−2 . Then the above integral can be written as Z +∞ ∞ X 1 ak ε k y k |φn (ε)(y)|2 dy ε4 Dε−1 φn (ε), 2 Dε−1 φn (ε) = ε4 r −r0 /ε k=0

∞

=

ε4 X + ak ε k r02 k=1

Z

+∞

−r0 /ε

y k |φn (ε)(y)|2 dy .

To estimate the remaining terms we use (3.8). The fifth order term integral is estimated by Z Z +∞ +∞ 2 y|φn (ε)(y)| dy ≤ |y||φn (ε)(y)|2 dy −s/ε −s/ε Z

s/ε

= −∞

Z

≤C

|y||φn (ε)(y)|2 dy

s/ε

−∞

= O(e−s

2

|y|e−ω0 y

ω0 /2ε2

2

/2

dy

)

which contributes nothing to the asymptotic series. Higher order terms can be like2 2 wise calculated, being in general of order O(ε5−k e−s ω0 /2ε ) and hence contributing nothing to the series. Thus, the correction is entirely fourth order and equal to (l + 1)2 ε2 /r02 . Appendix. Asymptotic Corrections Consider an analytic family of operators H0 (β) with eigenvalues E0m (β); that is, H0 (β)ψ0m (β) = E0m (β)ψ0m (β). Assume that for each m, E0m (β) is isolated and nondegenerate, and furthermore, that E0m (β) has an asymptotic expansion E0m (β) ∼ a0 P m lk l + am ak β near β = 0. Put H(β) = H0 (β) + β p V . What we’d like is some 1 β + understanding of what happens to E0m (β) under the perturbation V . As long as the distance between the E0m (β)’s doesn’t drop off “too quickly”, and as long as the operator β p V (H0 (β) − z)−1 can be made small enough, we can be sure of finding a unique eigenvalue, with asymptotic series, near E0m (β). This result is precisely stated in: Theorem A.1. Let H0 (β), H(β), E0m (β) and ψ0m (β) be as above, with V closed and symmetric and with am 1 = d(m + c), d 6= 0, and p > l. Suppose that p −1 kβ V (H0 (β) − z) k = O(β) when |z − E0 (β)| = rβ , for some rβ = o(β l ) with β p = o(rβ ). Then H(β) has exactly one eigenvalue E m (β) near E0m (β). The Rayleigh– Schr¨ odinger series for E m (β) is finite term by term and is an asymptotic expansion P n m m m for β small. In particular , writing E m (β) ∼ bm n β , we have bn = an + cn where the correction cm n is given by

485

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

hψ0m (β), V P (β)ψ0m (β)i X m n ∼ cn β . hψ0m (β), P (β)ψ0m (β)i

(A.1)

Proof. Fix m, and let rβ be as described above. Since the separation between the E0m (β)’s is O(β l ) the curve Γβ = {z : |z − E0m (β)| = rβ } contains only the eigenvalue E0m (β). Note that for β small enough, Γβ ⊆ ρ(H(β)), for k(H(β) − z)−1 k = k(I + β p V (H0 (β) − z)−1 )−1 (H0 (β) − z)−1 k ≤ k(I + β p V (H0 (β) − z)−1 )−1 k · k(H0 (β) − z)−1 k which is bounded for z ∈ Γβ and β small. Define I −1 (H0 (β) − z)−1 dz P0 (β) = 2πi Γβ

and P (β) =

−1 2πi

I

(H0 (β) + β p V − z)−1 dz .

Γβ

We want to show that dim{ran P0 (β)} = dim{ran P (β)}, from which it follows that H(β) has exactly one eigenvalue within Γβ . Expand (H(β) − z)−1 as (H(β) − z)−1 = (H0 (β) − z)−1 − β p (H(β) − z)−1 V (H0 (β) − z)−1 so that kP0 (β) − P (β)k

1 I

p −1 −1 β (H(β) − z) V (H0 (β) − z) dz =

2πi

Γβ

1 I

p −1 p −1 −1 −1 β (H0 (β) − z) (I + β V (H0 (β) − z) ) V (H0 (β) − z) dz =

2πi

Γβ ≤ rβ k(H0 (β) − z)−1 k · k(I + β p V (H0 (β) − z)−1 )−1 k · β p kV (H0 (β) − z)−1 k ≤ rβ ·

1 · k(I + β p V (H0 (β) − z)−1 )−1 k · |β|p kV (H0 (β) − z)−1 k rβ

−→ 0 as β goes to 0. This establishes the existence of a unique eigenvalue E m (β) in the interior of Γβ ; all that remains is to show (A.1). But this just follows from the fact that ψ0m (β) has an asymptotic series and from the formula E m (β) =

hψ0m (β), (H0 (β) + β p V )P (β)ψ0m (β)i hψ0m (β), P (β)ψ0m (β)i

= E0m (β) + β p

hψ0m (β), V P (β)ψ0m (β)i hψ0m (β), P (β)ψ0m (β)i

486

J. HERRIN and J. S. HOWLAND

In particular, for n < 2p (that is, as a first approximation) we have (formally) cm n =

dn p m β hψ0 (β)|V |ψ0m (β)i dβ n

from the usual Rayleigh–Schr¨ odinger equations. While the condition kβ p V (H0 (β) − z)−1 k = O(β) is not very strong, it is too strong for some of our purposes. It turns out we can relax this condition if H0 (β) and V are of a special form. Theorem A.2. Let H0 (β), H(β), E0m (β) and V be as above, with am 1 = d(m+c), d 6= 0, and p > l. Additionally, let H0 (β) and V operate on M ⊕ M⊥ by H0 (β) =

A(β)

0

0

B(β)

V =

0

C

D

0

where M is the span of ψnm (β) with A(β)ψnm (β)= E0m (β)ψnm (β). Assume that kβ p V (H0 (β) − z)−1 k < 1 for some z, with β small. Put R0 = R0 (β, z) = (H0 (β) − z)−1 and R = R(β, z) = (H(β) − z)−1 . Assume that for some rβ = o(β l ) with β p = o(rβ ), |z − E0n (β)| = rβ implies that kKβ k = kβ 2p V R0 V R0 k = O(β) and that rβ kβ p R0 V R0 k = O(1). Then H(β) has exactly one eigenvalue E m (β) near odinger series for E m (β) is finite term by term and is E0m (β). The Rayleigh–Schr¨ an asymptotic expansion for β small , where again (A.1) holds. Proof. Define Γβ , P (β) and P0 (β) as in Theorem A.1. Write R = R0 + β p R0 V R0 + β 2p R0 (V R0 )2 + β 4p R0 (V R0 )3 + · · · = (R0 + β p R0 V R0 )(I + β 2p (V R0 )2 + β 4p (V R0 )4 + · · · ) = (R0 + β p R0 V R0 )(I + Kβ + Kβ2 + Kβ3 + · · · ) = (R0 + β p R0 V R0 )(I − Kβ )−1 . Then P (β) − P0 (β) = −

I

1 2πi

(R0 + β p R0 V R0 )(I − Kβ )−1 dz +

Γβ

=−

I R0 dz Γβ

I

1 2πi

1 2πi

β p R0 V R0 dz Γβ

1 + 2πi

I

(R0 + β p R0 V R0 )((I − Kβ )−1 − I) dz .

Γβ

0 W V 0 we integrate the first term around the curve Γβ it vanishes. Thus,

R0 V R0 connects the spaces M and M⊥ (that is, R0 V R0 =

), so when

THE BORN–OPPENHEIMER APPROXIMATION: STRAIGHT-UP AND WITH A TWIST

487

kP (β) − P (β)k ≤ rβ · k(R0 + β p R0 V R0 )k · k(I − (I − Kβ )−1 )k ≤ rβ · kR0 k · kKβ (I − Kβ )−1 k + rβ · kβ p R0 V R0 k · kKβ k · k(I − Kβ )−1 k −→ 0 as β → 0. This establishes the existence of a unique eigenvalue for H(β) near E0m (β), and, as in Theorem A.1, the existence of the asymptotic series follows and (A.1) holds. The following no doubt appears elsewhere, but we state it for reference. Lemma A.3. Let S and T be positive operators, and let S + T be essentially 1 self-adjoint on D ⊆ D(S). Then for z with Rez ≤ −b2 < 0, S 2 (S + T − z)−1 is bounded by b−1 (1 + |Imz|/|z|). Proof. For a2 > 0 we have 0 ≤ S ≤ S + T + a2 on D. Thus for any vector u in the dense set (S + T + a2 )D we have kS 2 (S + T + a2 )−1 uk2 = hS(S + T + a2 )−1 u, (S + T + a2 )−1 ui 1

≤ hu, (S + T + a2 )−1 ui ≤

1 · kuk2 . a2

This proves the lemma for z real. Now let z satisfy the stated conditions, and factor kS 2 (S + T − z)−1 k = kS 2 (S + T − Rez)−1 (S + T − Rez)(S + T − z)−1 k 1

1

1 · k(S + T − Rez)(S + T − z)−1 k ≤√ −Rez 1 ≤ · kI + Imz(S + T − z)−1 k b |Imz| 1 . ≤ · 1+ b |z|

Note. Most of these results first appeared (in less elegant form) in the doctoral dissertation [29] of one of us (Herrin). References [1] M. Born and R. Oppenheimer, Ann. Phys. (Leipzig) 84 (1927) 457–484. [2] J. M. Combes, P. Duclos and R. Seiler, “The Born–Oppenheimer Approximation”, Rigourous Atomic and Molecular Physics, eds. G. Velo and A. Wightman, New York; Plenum, 1981. [3] G. A. Hagedorn, Commun. Math. Phys. 77 (1980) 1.

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[4] G. A. Hagedorn, Ann. Math. 124 (1986) 571. [5] G. A. Hagedorn, Commun. Math. Phys. 116 (1988) 23–44. [6] G. A. Hagedorn, “Multiple scales and the time-independent Born–Oppenheimer approximation”, Differential Equations and Applications, Ohio Univ. Press, 1989. [7] J.-M. Combes, “On the Born–Oppenheimer Approximation”, International Symposium on Mathematical Problems in Theoretical Physics, ed. H. Araki, Berlin, Heidelberg, New York, Springer, 1975. [8] J.-M. Combes, “The Born–Oppenheimer Approximation”, The Schr¨ odinger Equation, eds. W. Thirring, P. Urban and Wien, New York, Springer, 1977. [9] M. Klein, A. Martinez, R. Seiler and X. P. Wang, Commun. Math. Phys. 143 (1992) 607–639. [10] M. Klein, A. Martinez and X. P. Wang, Commun. Math. Phys. 152 (1992) 73–95. [11] A. Martinez, Ann. Inst. H. Poincar´ e Sect. A, 50 (1989) 239–257. [12] A. Martinez, Journ´ees E.P.D. de St. Jean-de-Monts (1988). [13] A. Martinez, J. Differential Equations 91 (1991) 204–234. [14] R. Seiler, Helv. Phys. Acta 46 (1973). [15] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966. [16] B. Simon, Ann. Inst. Henri Poincar´e, 38 (1983) 295. [17] M. G. Benedict, L. Gy. Feher and Z. Horvath, J. Math. Phys. 30 (1989) 1727–1731. [18] G. Herzberg and H. C. Longuet-Higgings, Disc. Farad. Soc. 35 (1963) 77–82. [19] R. Jackiw, Int. J. Mod. Phys. A3 (1988) 285–297. [20] C. A. Mead, Chem. Phys. 49 (1980) 23–38. [21] C. A. Mead, Phys. Rev. Lett. 59 (1987) 161–164. [22] J. Moody, A. Shapere and F. Wilczek, Phy. Rev. Lett. 56 (1986) 893. [23] Wilczek and A. Zee, Phys. Rev. Lett. 52 (1984) 2111. [24] P. A. M. Dirac, Proc. Roy. Soc. A, 127 (1931) 60. [25] I. Tamm, Z. Phys. 71 (1931) 141. [26] E. B. Davies and B. Simon, Commun. Math. Phys. 63 (1978) 277. [27] M. Berry, Proc. Roy. Soc. London A, 392 (1984) 45–57. [28] B. Simon, Phys. Rev. Lett. 51 (1983) 2167–2170. [29] J. Herrin, The Born–Oppenheimer approximation: straight-up and with a twist. Dissertation, University of Virginia, 1990.

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN QUANTUM ELECTRODYNAMICS

Department of

FUMIO HIROSHIMA Mathematics, Faculty of Science

Hokkaido University Sapporo 060, Japan E-mail: [email protected] Received 15 March 1996 This paper presents functional integral representations for heat semigroups with infinitesimal generators given by self-adjoint Hamiltonians (Pauli–Fierz Hamiltonians) describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. By the functional integral representations, some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schr¨ odinger operators with classical vector potentials.

1. Introduction The purpose of this paper is to construct a functional integral representation for the heat semigroup generated by a Hamiltonian of the Pauli–Fierz model in quantum electrodynamics (QED), which describes an interaction of a non-relativistic charged particle in a scalar potential and a quantized radiation field in the Coulomb gauge. Moreover we derive the diamagnetic inequality with respect to the Hamiltonian by using the functional integral representation. This model plays an important role for an interpretation of some physical phenomena, for example the “Lamb shift” [1, 38]. The Pauli–Fierz model has been discussed in both the physics and the mathematics literature [1, 2, 3, 6, 18, 19, 27, 38]. Besides, there are many papers dealing with models describing interactions of non-relativistic particles and a quantized field. For example, the Nelson model [7, 24], persistent model [10, 11] and spin boson model [36, 37], etc. For this kind of models, problems of removal of an infrared cut-off [7, 10, 11, 24], asymptotic behaviors [3, 4, 9, 18], resonances [27], scattering states [1, 2] and dressed one electron states [10, 11] have been discussed. These examples especially are prototype for interaction models of non-relativistic particles with quantized fields. The Wiener path integral method has been studied extensively. In particular, with the help of stochastic integral, path integral representations for the heat semigroups generated by Schr¨ odinger Hamiltonians Hcl =

d 1X (−iDµ − Aµ )2 + V 2 µ=1

489 Reviews in Mathematical Physics, Vol. 9, No. 4 (1997) 489–530 c World Scientific Publishing Company

(1.1)

490

F. HIROSHIMA

with vector potentials Aµ and a scalar potential V have been investigated. The Hamiltonian Hcl has been analyzed by many authors who used the path integral method [15]. These path integral representations are well known as the Feynman– Kac–Itˆ o (FKI) formula. On the other hand, E. Nelson [25, 26] introduced the “generalized path space” in connection with a construction of scalar quantum field models from Markoff fields (the so-called functional integral method). Concretely, the d-dimensional generalized path space is given by Sr0 (Rd ), the set of real tempered distributions on Rd , equipped with a Gaussian measure. As a result, the constructive scalar quantum field models have two representations; one is the original Fock representation, the other is the Schr¨ odinger representation which realizes the Fock space as L2 -space 0 d on Sr (R ). In [14], the authors introduced a natural embedding of L2 (Sr0 (Rd )) into a constant time subspace in L2 (Sr0 (Rd+1 )), by which, the Feynman–Kac–Nelson (FKN) formula relating the relativistic P (φ)1+1 theory to the Euclidean P (φ)2 was obtained. The “generalized path space” was studied more generally and abstracted by [22]. The above classical path integral and the functional integral methods have been applied simultaneously to interaction models of non-relativistic particles and quantized fields. In [4], weak coupling limits for Hamiltonians describing a quantum system of finite number of non-relativistic particles interacting with a massive or massless Bose field were studied, where the FKN formula and the Wiener path integrals were applied. In [12, 13], it was suggested to analyze the Pauli–Fierz models of non-relativistic QED by using the functional integrals and stochastic integrals. The main purpose of this paper is to give a functional integral representation for the Pauli–Fierz model in a rigorous way. The Pauli–Fierz Hamiltonian Hρ,B which we consider is defined as an operator acting in the tensor product MB = L2 (Rd ) ⊗ F(W) by Hρ,B =

d 2 1 X eµ (ρ) + I ⊗ dΓB (e ωB ) . −iDµ ⊗ I − A 2 µ=1

(1.2)

eµ (ρ), Here F (W) denotes the Boson Fock space over W = L2 (Rd ) ⊕ · · · ⊕ L2 (Rd ), A {z } | d−1

a constant fiber direct integral of the µ-th direction time-zero smeared radiation field Aµ (ρ, x) with an ultraviolet cut-off function ρˆ in the Coulomb gauge; ! !) ( r r ˜ i·x ˆe−i·x ˆe 1 d−1 eµ ρ d−1 eµ ρ † + a ⊕r=1 √ , a ⊕r=1 √ Aµ (ρ, x) = √ 2 h h er = (er1 , . . . , erd ), r = 1, . . . , d − 1, polarization vectors, h(k) = |k|, the free photon ωB ), the free Hamiltonian in F (W) (for details, energy with momentum k, and dΓB (e see Sec. 3). Comparing (1.1) with (1.2), functional integral representations for e−tHρ,B (if exist) seem to rely on the FKN and the FKI formulas heavily. Actually, as it will become clear later, these formulas are fundamental in this paper. Until now, for the Hamiltonian Hρ,B , there exist only a few mathematically rigorous

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

491

results [19, 27]. Difficulties for dealing with the Hamiltonian Hρ,B are due to the following facts: Essential self-adjointness. In [1, 2, 3, 6, 18], instead of Hρ,B , Hamiltonians defined by taking the dipole approximation for Hρ,B were studied. This approximation means replacing Aµ (ρ, x) in Hρ,B with Aµ (ρ, 0) ≡ Aµ (ρ);

HD ρ,B

HD ρ,B =

d 1X 2 (−iDµ ⊗ I − I ⊗ Aµ (ρ)) + I ⊗ dΓB (e ωB ) . 2 µ=1

For some ρ’s, it is well known that the Hamiltonians HD ρ,B have a concrete core are not known. As far as we [1, 2]. However, in general, concrete cores of H ρ,B √ know, only in the case where ||ˆ ρ/ h||L2 (Rd ) is sufficiently small, a concrete core of Hρ,B is known [27]. In this paper, we do not discuss the problem of essential self-adjointness of Hρ,B . Instead, we construct a self-adjoint extension of it. Of course, one can construct the Friedrichs extension of Hρ,B , since it is a non-negative symmetric operator [19]. Unfortunately, we do not know whether it coincides with the Friedrichs extension just mentioned. Vector field. For the time-zero smeared radiation field Aµ (f ), f ∈ W, putting the canonical conjugate momentum by Πµ (f ), by the Coulomb gauge, we see that they satisfy the following commutation relations on some suitable domain in F (W): D E , [Aµ (f ), Πν (g)] = i dµν fˆ¯, gˆ L2 (Rd )

[Aµ (f ), Aν (g)] = 0 , [Πµ (f ), Πν (g)] = 0 . Pd−1

r r Here dµν = r=1 eµ eν which is independent of a choice of polarization vectors. These commutation relations suggest that one cannot construct the Schr¨ odinger representation in the same way as in the scalar field theory, i.e., one cannot realize F (W) as L2 -space over Sr0 (Rd ) × . . . . × Sr0 (Rd ). This is due to the fact that {z } | d−1

A1 , . . . , Ad are not independent of each other, since ∂µ Aµ = 0,

(the Coulomb gauge) .

In order to conquer this objection, we have to construct a base space for the Schr¨ odinger representation taking into account the Coulomb gauge [16]. The main idea and technique to achieve our goal will be a certain semigroup idea and to introduce a concept of “Hilbert space-valued stochastic integrals associative with a family of isometries from a Hilbert space to another one”. This is roughly described as follows. In order to construct the Schr¨ odinger representation taking e −1 ] by a quotient into account the Coulomb gauge, we define a real Hilbert space [H space as follows (see Sec. 3): e−1 ] = H−1 ⊕ · · · ⊕ H−1 /N−1 . [H | {z } d

492

F. HIROSHIMA

e −1 ] and the corresponding Constructing a Gaussian random process indexed by [H probability space (Q−1 , dµ−1 ), as in the method used in [14, 15, 25, 26, 30], we can introduce a unitary operator from MB ≡ L2 (Rd ) ⊗ F(W) to M ≡ L2 (Rd ) ⊗ L2 (Q−1 , dµ−1 ). We define Hρ as an operator acting in M by the unitary transform of Hf,B restricted to some domain with some ultraviolet cut-off f (see (3.7)). Concretely it is given by Hρ ≡ Hρ,0 + I ⊗ H0 , Hρ,0 ≡

d 2 1 X −iDµ ⊗ I − φρF ,µ , 2 µ=1

where H0 is the free Hamiltonian in L2 (Q−1 ) and φρF ,µ is a self-adjoint operator odinger representation, we construct a Pauli–Fierz in L2 (Q−1 ). Then, in this Schr¨ Hamiltonian as a self-adjoint extension of Hρ . We can define a family of self-adjoint contraction operators {Qρ,s }s≥0 on M and a subspace M∞ ρ ⊂ M such that the on M at s = 0 exists with weak right-derivative of Qρ,s F for F ∈ M∞ ρ Qρ, − Qρ,0 F = −Hρ,0 F . →0+

w − lim

(1.3)

Moreover we can prove the existence of the strong limit, n

s − lim Q2ρ, n→∞

t 2n

≡ Gρ (t) ,

(1.4)

and give a functional integral representation of Gρ (t). Using the functional integral representations for Gρ (t), we find that Gρ (t) is a strongly continuous self-adjoint one-parameter semigroup, i.e., there exists a unique non-negative self-adjoint operae ρ,0 such that tor H e Gρ (t) = e−tHρ,0 . By (1.3) and (1.4), we see that e ρ,0 H

M∞ ρ

= Hρ,0 |M∞ . ρ

Then we define a mathematically rigorous Pauli–Fierz Hamiltonian by e ρ,0 +I ˙ ⊗ H0 . Hρ = H As a natural embedding from L2 (Sr0 (Rd )) to L2 (Sr0 (Rd+1 )) in the constructive scalar quantum field theory, we can construct the corresponding L2 -space L2 (Q−2 ) such that there exists linear isometries Jt : L2 (Q−1 ) → L2 (Q−2 ) with Jt∗ Js = e−|t−s|H0 . Applying the strong Trotter product formula and (1.5), we have

(1.5)

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

D

...

493

E F, e−tHρ G M t t −n Hρ,0,t− t ∗ −n H ρ,0,t n t t = lim F, Jt Et e Et Et− n · · · Et− n e n→∞

−tH t · · · E nt e n ρ,0, n E nt J0 G

,

(1.6)

M

where Es = Js Js∗ and Hρ,0,s are self-adjoint operators in L2 (Rd ) ⊗ L2 (Q−2 ) defined similarly to Hρ,0 (see Lemma 4.9). Since Hρ,0,s are operators in L2 (Rd ) ⊗ L2 (Q−2 ), it does not immediately follow that one can drop Es ’s in (1.6) by the Markoff property as in the case of Es ’s in the constructive scalar quantum field theory. However for some ρ’s, using functional integral representations for e−tHρ,0,t , we can show that it is possible. To derive the functional integral representations, we introduce the concept of “Hilbert space-valued stochastic integrals associated with the family of isometries {Jt }t≥0 from L2 (Q−1 ) to L2 (Q−2 )”. This D is a key Etool , in our formulation. Then functional integral representations for F, e−tHρ G M

F, G ∈ M, are obtained for some ρ’s. To our knowledge, the functional integral representations presented here are new. The outline of the present paper is as follows. In Sec. 2, following the standard stochastic integral procedure, we extend stochastic integrals to Hilbert space-valued ones and define “Hilbert space-valued stochastic integral associated with a family of isometries from a Hilbert space to another one ” (Theorem 2.5). In Sec. 3, we introduce polarization vectors er , r = 1, . . . , d − 1, and define the two Hilbert spaces L2 (Q−1 , dµ−1 ) and L2 (Q−2 , dµ−2 ). We derive a simple extension of the FKN formula (Proposition 3.4). Moreover it is shown that Hρ is the unitary transform of Hf,B restricted to some domain (3.7). In Sec. 4, we construct the strongly continuous self-adjoint one-parameter semigroup Gρ (t) on M such that its infinitesimal e ρ,0 is a self-adjoint extension of the formally defined Hamiltonian Hρ,0 generator H (Lemmas 4.6, 4.7 and 4.8). We give a rigorous definition of Hρ in terms of the e ρ,0 and I ⊗ H0 . Applying the Trotter product formula [21], ˙ of H form sum + the Hilbert space-valued stochastic integral, and the FKN formula, a functional integral representation for the heat semigroup generated by Hρ with a scalar potential added is derived (Theorem 4.10). Moreover, the functional integral representation is extended to a more general class of potentials in Theorem 4.12. In Sec. 5, we derive some inequalities which are known in the classical case as the diamagnetic inequality [5, 35] and an abstract Kato’s inequality [17, 31, 33] through the functional integral representation. In Sec. 6, we give some remarks, comparing our model with the classical one [35] and scalar field theory [15, 30]. It is a pleasure to thank Prof. A. Arai for raising a problem which led to my consideration of the functional integral representation of a model in QED. 2. Hilbert Space-Valued Stochastic Integrals In this section, we extend the standard stochastic integral to a Hilbert spacevalued one and introduce the “Hilbert space-valued stochastic integral associated with a family of isometries”. [23, 35]

494

F. HIROSHIMA

For a Hilbert space X over C, we denote the inner product and the associated norm by h∗, ·iX and || · ||X , respectively. The inner product is linear in · and antilinear in ∗. The domain of an operator A is denoted by D(A). The notation C(Rd ; X ) denotes the space of strongly continuous functions from Rd to the Hilbert space X . For n = 1, 2, . . ., we denote by C n (Rd ; X ) the subspace of n-times strongly differentiable functions in C(Rd ; X ) and define ( ) ∂ k f (x) < ∞ , Cbn (Rd ; X ) = f ∈ C n (Rd ; X ) sup X |k|≤n,x∈Rd H n (Rd ; X ) = f ∈ C n (Rd ; X ) ∂ k f (·) X ∈ L2 (Rd ), |k| ≤ n , where k = (k1 , k2 , . . . , kd ) is a multi-index ,|k| = k1 +k2 +· · ·+kd , and the derivative ∂ k = ∂1k1 ∂2k2 · · · ∂dkd is taken in the strong topology in X . We fix probabilistic notations. Let (Ω, Db) be a probability space for d -dimensional Brownian motion b(t) = (bµ (t))1≤µ≤d,t≥0 and dµ be the Wiener measure on Rd × Ω defined by dµ = dx ⊗ Db. Let E denote the expectation value with respect to (Ω, Db). Following [28, XIII.16], we use the following identification: L2 (M, dm) ⊗ X ∼ =

Z

⊕

X dm .

M

Let H be a Hilbert space over C. Lemma 2.1. Let f ∈ Cb1 (Rd ; H) and define Jµn (f, b)

2n X k−1 k k−1 = f b t bµ t − bµ t , t ≥ 0, µ = 1, . . . , d . 2n 2n 2n k=1

Then the strong limit Z s − lim Jµn (f, b) ≡ n→∞

t

f (b(s))dbµ 0

exists in L2 (Ω; H). Moreover , for any g ∈ Cb1 (Rd ; H), Z

Z

t

f (b(s))dbµ , 0

Z

t

g(b(s))dbν 0

= δµν E L2 (Ω;H)

0

t

hf (b(s)), g(b(s))iH ds

, (2.1)

where δµν is Kroneker’s delta. Proof. In the same way as in the proof of [35, Theorem 14.2], one can see that {Jµn (f, b)}n≥1 is a Cauchy sequence in L2 (Ω; H). Hence the strong limit of Jµn (f, b) exists in L2 (Ω; H). One can see that hJµn (f, b), Jνn (g, b)iL2 (Ω;H) ! 2n X k−1 k−1 t =E t , g b t δµν . f b 2n 2n 2n H k=1

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

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495

Since hf (b(s)), g(b(s))iH is continuous in s a.e.b ∈ Ω, we have Z t 2n X k−1 k−1 t t ,g b t = ds hf (b(s)), g(b(s))iH , f b lim n→∞ 2n 2n 2n 0 H k=1

a.e.b ∈ Ω . Moreover, n 2 X k−1 k−1 t t ,g b t f b ≤ c0 c00 t , 2n 2n 2n H k=1

where c0 = supx∈Rd ||f (x)||H and c00 = supx∈Rd ||g(x)||H . Hence the Lebesgue dominated convergence theorem yields (2.1). Rt We call 0 f (b(s))dbµ the “H-valued stochastic integral for f ”. Remark 2.2.R (1) As in [35, p. 152], Lemma 2.1 suggests that one can extend t the definition of 0 f (b(s))dbµ from Cb1 (Rd ; H) to arbitrary functions f such that 2 Z t f (b(s))dbµ 2 0

Z

||f (b(s))||2H ds

=E 0

L (Ω;H)

t

Z t Z Rd

0

x2

dx(2πs)− 2 ||f (x)||2H e− 2s d

=

ds < ∞ .

Rs Rt (2) In an obvious way, we can extend 0 f (b(s))dbµ to t f (b(s))dbµ . Then for [t1 , t2 ) ∩ (t3 , t4 ] = φ and f, g ∈ Cb1 (Rd ; H) Z t2 Z t4 f (b(s))dbµ , g(b(s))dbν = 0. (2.2) t1

t3

(3) From (2.1) and (2.2) it follows that L2 (Ω; H).

L2 (Ω,H)

Rt 0

f (b(s))dbµ is strongly continuous in t in

Lemma 2.3. Let f ∈ Cb2 (Rd ; H) and define for t ≥ 0, µ = 1, . . . , d, Sµn (f, b)

2n X k−1 k 1 = t + f b t f b 2 2n 2n k=1 k k−1 t − bµ t . × bµ 2n 2n

Then Z s−

lim Sµ (f, b) n→∞ n

= 0

t

1 f (b(s))dbµ + 2

Z

t

(∂µ f )(b(s))ds

(2.3)

0

Rt in L2 (Ω; H), where 0 (∂µ f )(b(s))ds is the Bochner integral of L2 (Ω; H)-valued function (∂µ f )(b(·)) on Rd .

496

F. HIROSHIMA

Proof. We divide Sµn (f, b) in two parts as follows: Sµn (f, b)

2n X k−1 k k−1 = f b t b t − b t µ µ 2n 2n 2n k=1

2 X k−1 k k k−1 1 t −f b t bµ t −bµ t . f b + 2 2n 2n 2n 2n n

k=1

(2.4) Similarly to Lemma 2.1 [35, p. 160], it is not hard to see that the two terms on the right-hand side (r.h.s.) of (2.4) strongly converges to the two terms on the r.h.s.of (2.3) in L2 (Ω; H), respectively. Remark 2.4. (1) One can easily see that for f ∈ Cb1 (Rd ; H), [2n t]

s − lim

X

n→∞

k=1

Z t k−1 k k−1 f b f (b(s))dbµ . bµ − bµ = 2n 2n 2n 0

Moreover, for f ∈ Cb2 (Rd ; H), X 1 k − 1 k k k−1 f b +f b bµ − bµ s − lim n→∞ 2 2n 2n 2n 2n k=1 Z Z t 1 t f (b(s))dbµ + (∂µ f )(b(s))ds , = 2 0 0 [2n t]

where [ · ] denotes the Gauss symbol. (2) For f ∈ Cb1 (R × Rd ; H), by the same way as the proof of Lemma 2.1, we can show that the following strong limit: 2n X k−1 k−1 k k−1 f t, b t bµ t − bµ t s − lim n→∞ 2n 2n 2n 2n Z ≡

k=1

t

f (s, b(s))dbµ 0

exists in L2 (Ω; H). Let K be a Hilbert space and {It }t≥0 be a family of isometries from H to K. bµ (f, b) by For f ∈ Cb1 (Rd ; H), we define a K-valued stochastic integral J n 2 Z X n

bµ (f, b) J n

=

k=1

k 2n

t

k−1 t 2n

I k−1 f (b(s))dbµ . n t 2

Theorem 2.5. Let f ∈ Cb1 (Rd ; H) such that f˜(s, x) ≡ Is f (x) ∈ Cb1 (R × Rd ; H). Then, in L2 (Ω; K), Z t bµ (f, b) = (2.5) f˜(s, b(s))dbµ . s − lim J n n→∞

0

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

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497

Proof. Fix f ∈ Cb1 (Rd ; H) satisfying the condition in this theorem. Put c0 = sup(s,x)∈R×Rd ||∂s f˜(s, x)||K . At first, we shall show the existence of the bµ (f, b) as n → ∞. By Lemma 2.1 and (2.2), we can see that strong limit of J n 2 b µ bµ (f, b) Jn+1 (f, b) − J n 2 L (Ω;K)

2n Z 2k 2 X 2n+1 t = I 2k−1 t f (b(s)) − I 2k−2 t f (b(s)) dbµ 2k−1 2n+1 2n+1 t n+1 k=1

=

2 X

L2 (Ω;K)

2

Z

n

E

2k−1 2n+1

k=1

Z

n

≤

2 X

E

tc0 2n+1

t

t

2 f (b(s)) − I 2k−2 f (b(s)) ds I 2k−1 n+1 t n+1 t 2

t

2k 2n+1 2k−1 2n+1

k=1

t = 2

2k 2n+1

t

2

tc0 2n+1

!

K

!

2 ds

2

.

Then we have for m > n, b µ bµ (f, b) Jm (f, b) − J n Hence

o n bµn (f, b) J

n≥1

L2 (Ω;K)

3 m t 2 c0 X 1 ≤ √ . 2 k=n+1 2k

is Cauchy in L2 (Ω; K). Then the existence of the strong

bµ limit follows. Secondly we shall show (2.5). We write the strong limit of Jn (f, b) k k−1 µ b (f, b) and put ∆bµ (t, k) ≡ bµ n t − bµ n t . Since It is a contraction by J ∞ 2 2 operator, it can be easily seen that 2 Z t µ ˜(s, b(s))dbµ b (f, b) − J f 2 ∞ 0

L (Ω;K)

2 Z kn t X 2 ˜ k − 1 = lim t, b(s) dbµ f n→∞ k−1 2n n t n

k=1

2

k=1

2

2 k−1 k−1 ˜ −f t, b t ∆bµ (t, k) n n 2 2 L2 (Ω;K) 2 2n Z kn t X k−1 2 ≤ lim f (b(s)) dbµ − f b t ∆bµ (t, k) n n→∞ 2 k−1 2 n t

L (Ω;H)

2 2n Z k 2n X 2n t X k−1 = lim f (b(s)) dbµ − f b t ∆b (t, k) µ k−1 n→∞ 2n n t k=1

2

k=1

L2 (Ω;H)

= 0. Then the proof is complete.

498

F. HIROSHIMA

Rt We call 0 Is f (b(s))dbµ the “K-valued stochastic integral associated with {It }t≥0 ”. We conclude the present section with stochastic integrals over the Wiener paths. Rt Defining 0 f (b(s))dbµ as a strong limit in L2 (Ω; H), for f ∈ H 1 (Rd ; H), we can Rt define 0 f (ω(s))dωµ as a strong limit in L2 (Rd × Ω; H) as follows: Z t 2n X k−1 k k−1 f ω t ωµ t −ωµ t ≡ f (ω(s))dωµ .(2.6) s − lim n→∞ 2n 2n 2n 0 k=1

The existence of this limit can be proven in the same way as in the proof of Lemma 2.1. For f, g ∈ H 1 (Rd ; H), we have Z t Z t Z t e f (ω(s))dωµ , g(ω(s))dων = δµν E hf (ω(s)), g(ω(s))iH ds 0

0

L2 (Rd ×Ω;H)

0

Z = tδµν

Rd

hf (x), g(x)iH dx ,

(2.7)

e denotes the expectation value with respect to (Rd × Ω, dµ). Equation (2.7) where E Rt allows us to extend the definition of 0 f (ω(s))dωµ to f such that the r.h.s. of (2.7) is finite. 3. Probabilistic Description of the Time Zero Radiation Field In this section we define a model which describes a quantum system of a nonrelativistic charged particle interacting with a quantized radiation field with the Coulomb gauge. For mathematical generality, we consider the situation where the charged particle moves in Rd and the quantized radiation field is over Rd . We define polarization vectors er (r = 1, . . . , d − 1) as measurable functions er : Rd → Rd such that er (k) · es (k) = δrs ,

k · er (k) = 0 ,

a.e.k ∈ Rd .

In what follows, fix the polarization vectors er . We introduce two Hilbert spaces e −2 ] as follows. First we define two real Hilbert spaces H−1 and e −1 ] and [H [H H−2 by Z ) ( ˆ(k)|2 | f dk < ∞ , H−1 ≡ f ∈ Sr0 (Rd ) Rd |k| Z ( ) |fˆ(k, k0 )|2 0 d+1 dkdk0 < ∞ , H−2 ≡ f ∈ Sr (R ) Rd+1 |k|2 + |k0 |2 where Sr0 (Rn ) denotes the set of real tempered distributions on Rn (n = d, d + 1) and ˆ denotes the Fourier transformation ( ∨ the inverse Fourier transformation) from S 0 (Rn ) to S 0 (Rn ): Z n f (x)e−ikx dx . fˆ(k) = (2π)− 2 Rn

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

499

Put e−1 = H−1 ⊕ · · · ⊕ H−1 , H | {z } d

e−2 = H−2 ⊕ · · · ⊕ H−2 . H | {z } d

e−1 and H e−2 by We introduce bilinear forms (·, ·)−1 and (·, ·)−2 in H (f, g)−1 =

Z d X µ,ν=1

(f, g)−2 = 2

Rd

¯ dµν (k)fˆµ (k)ˆ gν (k) dk , |k|

Z d X µ,ν=1

Rd+1

¯ dµν (k)fˆµ (k, k0 )ˆ gν (k, k0 ) dkdk0 , 2 |k| + |k0 |2

respectively, where fµ and gµ are the µ-th components of f and g, the symbol ¯ denotes the complex conjugate and dµν (k) ≡

d−1 X

erµ (k)erν (k) ,

r=1

= δµν −

kµ kν . |k|2

We denote the associated semi-norms by | · |−1 and | · |−2 respectively and put n o e−1 |f |−1 = 0 , N−1 = f ∈ H n o e−2 |f |−2 = 0 . N−2 = f ∈ H Then we define pre-Hilbert spaces by the quotient spaces e−1 /N−1 , e−1 ] = H [H e−2 /N−2 , e−2 ] = H [H with inner products h·, ·i−1 and h·, ·i−2 defined by hπ−1 (f ), π−1 (g)i−1 ≡ (f, g)−1 , hπ−2 (f ), π−2 (g)i−2 ≡ (f, g)−2 . e−1 and H e−2 , Here π−1 (f ) and π−2 (f ) denote the equivalence classes of f in H respectively. We denote the norms associated with the inner products h·, ·i−1 and h·, ·i−2 by || · ||−1 and || · ||−2 , respectively. The Hilbert spaces constructed by the e −2 ] with respect to || · ||−1 and || · ||−2 are denoted by e−1 ] and [H completions of [H the same symbols.

500

F. HIROSHIMA

e−1 } and {φ−2 (π−2 (f ))|f ∈ H e−2 } be the Gaussian ranLet {φ−1 (π−1 (f ))|f ∈ H e e dom processes indexed by [H−1 ] and [H−2 ] such that the characteristic functions are given by Z 2 1 eiφj (πj (f )) dµj = e− 4 ||πj (f )||j , j = −1, −2 , Qj

where (Q−1 , dµ−1 ) and (Q−2 , dµ−2 ) denote the underlying measure spaces of these processes, respectively. It is well known that L2 (Qj , dµj ) has the orthogonal decomposition L2 (Qj , dµj ) =

∞ M

ej ]) Γn ([H

n=0

with ej ])=C , Γ0 ([H ej ])=L{: φj (πj (f1 ))φj (πj (f2 )) · · · φj (πj (fn )) : |fk ∈ H ej , k =1, .., n}− , n ≥ 1 , Γn ([H where L{. . .} denotes the linear span of the vectors in {. . .} over C, − the closure in L2 (Qj , dµj ) and : · : the “Wick product” [4]. We denote the complexiej ]C . Suppose that T is a contraction operator from [H e i ]C e j ] by [H fications of [H e j ]C . Corresponding to each such T we can define the contraction operator to [H Γ(T ) : L2 (Qi ; dµi ) −→ L2 (Qj ; dµj ) by Γ(T )Ωi = Ωj , Γ(T ) : φi (πi (f1 )) . . . φi (πi (fn )) : = : φj (T πj (f1 ))φj (T πj (f2 )) . . . φj (T πj (fn )) : . ei ]C (i = −1, −2) we define ei ]C −→ [H For a non-negative self-adjoint operator A : [H dΓ(A) by dΓ(A)Ωi = 0 , dΓ(A) : φi (πi (f1 )) . . . φi (πi (fn )) : = : φi (Aπi (f1 ))φi (πi (f2 )) . . . φi (πi (fn )) : + : φi (πi (f1 ))φi (Aπi (f2 )) . . . φi (πi (fn )) : + · · · + : φi (πi (f1 ))φi (πi (f2 )) . . . φi (Aπi (fn )) : , πi (fk ) ∈ D(A), k = 1, . . . , n , where Ωi denotes the constant function 1 in L2 (Qi , dµi ). It is well known that dΓ(A) has unique self-adjoint extension in L2 (Qi ; dµi ). We denote it by the same symbol dΓ(A). We set L2 (Q−1 , dµ−1 ) = F , L2 (Q−2 , dµ−2 ) = E, φ−1 (·) = φF (·), φ−2 (·) = φE (·) and Ω−1 = ΩF and Ω−2 = ΩE . Put FN =

N M n=0

e−1 ]) Γn ([H

M

{0}

n>N +1

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

501

and define F ∞ by F∞ =

∞ [

FN .

N =0

The standard Boson Fock space [29, X. 7] over W = L2 (Rd ) ⊕ · · · ⊕ L2 (Rd ) is {z } | d−1

defined by [1, 2, 3, 18, 19] F (W) =

∞ M

Fn (W) ,

n=0

Fn (W) = ⊗ns W, n ≥ 1 ,

F0 = C ,

where ⊗ns W denotes the n-fold symmetric tensor product of W. The vacuum vector Ω in F (W) is defined by Ω = {1, 0, 0, . . . .} . The Boson Fock space F (W) describes a Hilbert space of state vectors for the quantized radiation field with the Coulomb gauge. Let F N (W) =

N M

M

Fn (W)

n=0

{0} .

n>N +1

Then a finite particle subspace is defined by F ∞ (W) =

∞ [

F N (W) .

N =0

The annihilation operator a(f ) and the creation operator a† (f ) (f ∈ W) [29] act on the finite particle subspace and leave it invariant with the canonical commutation relations (CCR): for f, g ∈ W

[a(f ), a† (g)] = f¯, g W , [a] (f ), a] (g)] = 0 , where [A, B] = AB − BA, a] denotes either a or a† . Furthermore,

a† (f )Φ, Ψ

F (W)

= Φ, a(f¯)Ψ F (W) ,

Φ, Ψ ∈ F ∞ (W) .

For any contraction operator A : W → W, the“second quantization of A” ΓB (A) : F (W) → F(W) is a bounded operator uniquely determined by ΓB (A)Ω = Ω , ΓB (A)a† (f1 )a† (f2 ) . . . a† (fn )Ω = a† (Af1 )a† (Af2 ) . . . a† (Afn )Ω .

502

F. HIROSHIMA

For a non-negative self-adjoint operator σ in W, the “second quantization of σ”, dΓB (σ), is defined by the infinitesimal generator of the strongly continuous oneparameter semigroup {ΓB (e−tσ )}t≥0 ; ΓB (e−tσ ) = e−tdΓB (σ) . We define the maximal multiplication operator ωB in L2 (Rd ) by (ωB f ) (k) = h(k)f (k) , ωB ) will be the free Hamilwhere h(k) = |k|. Put ω eB = ωB ⊕ · · · ⊕ ωB . Then dΓB (e | {z } d−1

tonian of the quantized radiation field. The second quantization of the identity operator IW on W, dΓ(IW ), is called the number operator. The following inequality is well known 1

||a] (f )Φ||F (W) ≤ ||f ||W × || (dΓ(IW ) + I) 2 Φ||F (W) ,

Φ ∈ F ∞ (W) .

(3.1)

For f ∈ H−1 we define the µ-th direction time-zero radiation field Aµ (f ) (µ = 1, . . . , d) by    !  r ˆ r ˜ˆ  eµ f eµ f 1  , + a ⊕d−1 a† ⊕d−1 (3.2) Aµ (f ) = √ r=1 √ r=1 √ 2 h h  e−1 we put where g˜(k) = g(−k). For g = (g1 , . . . , gd ) ∈ H A(g) ≡

d X

Aµ (gµ ) .

µ=1

We give connection between F and F (W). Here we introduce the subspace D0 in e −1 by H ( √ ∨ r r r e D0 = L f = (f1 , . . . , fd ) ∈ H−1 fµr = erµ hfˆ , ) ∞ d ˆ f ∈ C0 (R \{0}), r = 1, . . . , d − 1 , where C0∞ (Rd \ {0}) denotes the set of infinitely differentiable functions with come −1 with pact support on Rd \ {0}. Then it can be easily seen that D0 is dense in H e−1 ]. Hence respect to the semi-norm |·|−1 , which implies that π−1 (D0 ) is dense in [H L {: φF (π−1 (f1 )) . . . φF (π−1 (fn )) : ΩF , ΩF |fj ∈ D0 , j = 1, . . . , n, n ≥ 1} √ √ ∨ ρ)∨ , . . . , (erd hˆ ρ) ∈ D0 , is dense in F . On the other hand, choosing ρr = (er1 hˆ we see that

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

r

A(ρ )=

d X

Aµ

erµ

√ ∨ hˆ ρ

...

503

µ=1

     d−1 d−1   z }| {   z }| { 1  ˜ˆ ⊕ · · · ⊕ 0 a† 0 ⊕ · · · ⊕ ρˆ ⊕ · · · ⊕ 0+a 0 ⊕ · · · ⊕ ρ =√  . |{z} |{z}   2  the r-th

the r-th

Then we see that L {: A(f1 ) . . . A(fn ) : Ω, Ω|fj ∈ D0 , j = 1, . . . , n, n ≥ 1} is dense in F (W), where : · : denotes the “Wick product” in the Boson Fock space [29, p. 226]. We define the operator ω in H−1 by c (k) = h(k)fˆ(k) , ωf e−1 ] → [H e−1 ] is defined by ω ] : [H and put ω e = ω ⊕ · · · ⊕ ω . Furthermore, [e {z } | d

ωf ), [e ω ]π−1 (f ) = π−1 (e

ej ]|e e−1 } . D([e ω ]) = {π−1 (f ) ∈ [H ωf ∈ H

e −1 ]C as follows: e−1 ]C → [H Extend [e ω ] : [H e1 . ω ]π−1 (f1 ), [e ω ]π−1 (f2 )) , f1 , f2 ∈ H [e ω ] (π−1 (f1 ), π−1 (f2 )) = ([e e−1 ]C , which implies that [e ω] is a Then it is easy to see that Ran ([e ω ] ± i) = [H e self-adjoint operator in [H−1 ]C . Theorem 3.1. There exists a unitary operator U from F (W) to F such that (a) (b) (c) (d) (e)

UΩ = ΩF , e−1 , UA(f )U −1 = φF (π−1 (f )) f ∈ H e UFn (W) = Γn ([H−1 ]), ωB )U −1 = dΓ([e ω ]), UdΓB (e UdΓB (IW )U −1 = dΓ(IF ) ,

e −1 ]. where IF is the identity operator in [H Proof. For fj ∈ D0 , j = 1, . . . , n, we define U : A(f1 ) . . . A(fn ) : Ω = : φF (π−1 (f1 )) . . . φF (π−1 (fn )) : ΩF , UΩ = ΩF . One can easily show that U can be uniquely extended to a unitary operator from F (W) to F with (a), (b) and (c). We shall show (d). Let Xn = L {: φF (π−1 (f1 )) . . . φF (π−1 (fn )) : ΩF |fj ∈ D0 , j = 1, . . . , n} , Yn = L {: A(f1 ) . . . A(fn ) : Ω|fj ∈ D0 , j = 1, . . . , n} .

504

F. HIROSHIMA

Since, as long as ρˆ ∈ C0∞ (Rd \{0}), it follows that exp(−th)ˆ ρ ∈ C0∞ (Rd \{0}), one S∞ S∞ ωB )) leaves n=0 Yn invariant. Hence n=0 Yn is a core can see that exp (−tdΓB (e ωB ) [29, Theorem X. 49]. Moreover, since of dΓB (e ωB )) U −1 = exp −tUdΓB (e ωB )U −1 , U exp (−tdΓB (e S∞ S∞ ωB )U −1 . Noting that on n=0 Xn it follows that n=0 Xn is a core of UdΓB (e UdΓB (e ωB )U −1 = dΓ([e ω ]) . Thus (d) holds. The proof of (e) is similar to that of (d).

ω ]), N = dΓ(IF ). Following [30, Chapter III, 15], we can give We set H0 = dΓ([e connection between F and E. For t ∈ R we define the operator jt by jt : H−1 −→ H−2 , jt f = δt ⊗ f,

f ∈ H−1 ,

where δt is the one-dimensional delta function with mass at {t}. In momentum space, −1 ˆ −itk0 . jc t f (k, k0 ) = (2π) 2 f (k)e We put e jt = jt ⊕ · · · ⊕ jt and define e−1 ] −→ [H e −2 ] , [e jt ] : [H jt f ) . [e jt ]π−1 (f ) = π−2 (e It can be easily seen that [e jt ] is a linear isometry [30, Proposition III.2]. Hence the e −2 ]. We denote the projection onto Ran([e e jt ]) range of [jt ] is a closed subspace of [H by [et ]. Let n o e−2 ] π−2 (f ) ∈ Ran([e jt ]), a ≤ t ≤ b . U[a,b] ≡ L π−2 (f ) ∈ [H We denote the projection onto the closure U[a,b] by [e[a,b] ]. Proposition 3.2. [30, Propositions III.3 and III.4] jt ]∗ = [et ]. (a) [e jt ][e ∗ ω] js ] = e−|t−s|[e . (b) [e jt ] [e (c) Let a ≤ b ≤ c. Then [ea ][eb ][ec ] = [ea ][ec ]. (d) Let a ≤ b ≤ t ≤ c ≤ d. Then [e[a,b] ][et ][e[c,d]] = [e[a,b] ][e[c,d] ]. Proof. (a) is straightforwardly seen. Since we have E D E D js ]π−1 (f ), π−1 (g) = π−2 (e js f ), π−2 (e jt g) [e jt ]∗ [e −1

−2

Z ¯ d gν (k)dµν (k)ei(t−s)k0 fˆµ (k)ˆ 1 X = dkdk0 π µ,ν=1 Rd+1 |k|2 + k02 =

Z d X µ,ν=1

Rd

¯ gν (k)dµν (k)e−|t−s||k| fˆµ (k)ˆ dk , |k|

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

505

(b) holds. Equation (c) follows from (a) and (b). For any π−2 (f ) and π−2 (g), by the definition of [e[a,b] ] and [e[c,d]], they can be presented as follows: [e[c,d] ]π−2 (f ) = lim

n→∞

[e[a,b] ]π−2 (g) = lim

Nn X

fnα ,

fnα ∈ Ran([etnα ]), tnα ∈ [c, d] ,

α=1

m→∞

Mm X

f mβ ,

gmβ ∈ Ran([etmβ ]), tmβ ∈ [a, b] .

β=1

Hence by (c) we have

[e[a,b] ][et ][e[c,d] ]π−2 (f ), π−2 (g) −2 =

=

lim

n,m→∞

lim

n,m→∞

Nn ,Mm X

[et ]fnα , gmβ −2

α,β=1 Nn ,Mm X

fnα , gmβ −2

α,β=1

= [e[a,b] ][e[c,d] ]π−2 (f ), π−2 (g) −2 .

Then (d) follows. We introduce the following notations: Γ([e[a,b] ]) ≡ E[a,b] , Γ([e jt ]) ≡ Jt ,

(3.3)

Γ([et ]) ≡ Et . Proposition 3.3. [30, Theorem III.5] (a) (b) (c) (d)

Jt is a linear isometry from F to E. Jt Jt∗ = Et . Jt∗ Js = e−|t−s|H0 . Let Σ[a,b] denote the σ-algebra generated by L φE (π−2 (f )) π−2 (f ) ∈ U[a,b] and the set of Σ[a,b] -measurable functions in E by E[a,b] . Then Ran E[a,b] = E[a,b] .

(e) (Markoff property) Let a ≤ b ≤ t ≤ c ≤ d. Then E[a,b] Et E[c,d] = E[a,b] E[c,d] . Proof. Equations (a), (b), (c) and (e) follow from Proposition 3.2. Equation (d) follows from [30, Theorem III.8]. As in [30], FKN formula follows from Proposition 3.3.

506

F. HIROSHIMA

Proposition 3.4 ([30, Theorem III.6], FKN formula). Let f1 , . . . , fn ∈ e −1 and G0 , . . . , Gk be bounded measurable functions on Rd . Let t1 , . . . , tk ≥ 0 be H given. Then E D −t1 H0 F −tk H0 F e G . . . e G Ω ΩF , GF 0 1 k F F

=

hΩE , Gs00

. . . Gskk ΩE iE

,

where s0 is arbitrary and sj = s0 +

j X

ti ,

i=1

GF j = Gj (φF (π−1 (f1 )), . . . , φF (π−1 (fn ))) , s jsj f1 )), . . . , φE (π−2 (e jsj fn )) . Gj j = Gj φE (π−2 (e

The Hilbert space of state vectors in the system of a non-relativistic charged particle interacting with a quantized radiation field is given by MB = L2 (Rd ) ⊗ F(W). The unitary operator U given in Theorem 3.1 implements unitary equivalence between MB and Z ⊕ 2 d ∼ F dx . M = L (R ) ⊗ F = Rd

For an H−1 -valued function on Rd , ρ(·) : Rd → H−1 , we put d

z }| { ρeµ (·) = (0, . . . , ρ(·) , . . . , 0) . |{z} the µ-th

Then we define an operator in M by Z ⊕ ρ φF (π−1 (e ρµ (x))) dx . φF ,µ = Rd

Let Dµ , (µ = 1, . . . , d) be the generalized L2 -derivative in the µ-th direction. Then the interaction Hamiltonian of the non-relativistic charged particle with mass 1 and the quantized radiation field is “formally” given as an operator in M by Hρ =

d 2 1 X −iDµ ⊗ I − φρF ,µ + I ⊗ H0 . 2 µ=1

(3.4)

Here “formally” means that we mention nothing about the domain of Hρ . The precise definition will be given in the following section. We set Hρ,0 =

d 2 1 X −iDµ ⊗ I − φρF ,µ . 2 µ=1

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

507

We conclude this section with giving a typical example of the H−1 -valued function ρ(·). One can take ∨ , ρ(x) = fˆ(·)ei·x

(3.5)

where f is a real-valued rapidly decreasing infinitely differentiable function on Rn . In this case, the corresponding standard Boson Fock space element A(e ρµ (x)) is given by    !   r ˆ −i·x r ˜ˆ i·x e e f e f e 1 µ d−1 µ   √ √ ρµ (x)))U = √ + a a† ⊕d−1 ⊕ U −1 φF (π−1 (e r=1 r=1  2 h h ≡ Aµ (f, x) .

(3.6)

Then the function fˆ serves as an ultraviolet cut-off function for photon momenta. Let Z ⊕ Z ⊕ eµ (f ) = Aµ (f, x)dx, Ue = Udx . A Rd

Rd

From (3.6) it follows that for f and ρ in (3.5) e−1

U

d 2 X 1 eµ (f ) + I ⊗ dΓB (e Hρ Ue = ωB ) −iDµ ⊗ I − A 2 µ=1 D

,

(3.7)

D

b ∞ (W) ∩ D(I ⊗ dΓB (e ωB )). Equation (3.7) gives a relation of where D = C0∞ (Rd )⊗F Hamiltonians in the Schr¨ odinger representation and the Boson Fock representation. 4. Functional Integrals In this section we construct a self-adjoint extension of Hρ given formally by (3.4) and derive a functional integral representation for the heat semigroup associated with it. The main idea is to apply the FKN formula and the FKI formula [35, Theorem 15.3]. ρµ (x))) is a self-adjoint operator for For an H−1 -valued function ρ, φF (π−1 (e d each x ∈ R as a multiplication operator in F . Then, for each x, y ∈ Rd , we can define a unitary operator on F by ( !) d X 1 iφF π−1 (e ρµ (x) + ρeµ (y)) (xµ − yµ ) Uρ (x, y) ≡ exp 2 µ=1 ≡ exp (φρ (x, y)) . Let ps (x) be the heat kernel function 1 2 −d 2 , ps (x) = (2πs) exp − |x| 2s

s > 0, x ∈ Rd .

508

F. HIROSHIMA

Then we define a family of the contractive self-adjoint operators {Qρ,s }s≥0 on M by Z ps (x − y)Uρ (x, y)F (y)dy , s > 0 , (Qρ,s F ) (x) = Rd

(Qρ,0 F ) (x) = F (x) , where F (·) ∈ M and the integral is the F -valued Bochner integral. Actually one can easily see that 1 ||Qρ,s F ||M ≤ e−s(− 2 ∆) ||F (·)||F 2 d L (R )

≤ ||F ||M . e −1 ]), µ = 1, . . . , d. ρµ (·)) ∈ Cbn (Rd ; [H Note that if ρ ∈ Cbn (Rd ; H−1 ) then π−1 (e ∞ 1 We define a subspace Mρ in M as follows. For ρ ∈ Cb (Rd ; H−1 ), we say that F ∈ M∞ ρ ⊂ M if and only if the following (i) – (iii) hold: (i) F (·) ∈ H 2 (Rd ; F ). (ii) For each y ∈ Rd , F (y) ∈ F ∞ ,

∂µ F (y) ∈ F ∞ ,

µ = 1, . . . , d .

(iii) (Integration by parts condition) For all G ∈ M, x ∈ Rd (see Lemma 4.3), lim ∂yµ ps (x − y) · hF (y), Uρ (x, y)G(x)iF = 0 ,

y→∞

lim ps (x − y) · ∂yµ hF (y), Uρ (x, y)G(x)iF = 0 ,

y→∞

µ = 1, . . . , d .

Lemma 4.1. Let ρ ∈ Cb2 (Rd ; H−1 ), G ∈ M, and F ∈ M∞ ρ . Then hQρ,s F, GiM is differentiable in s > 0 with lim

s→0+

d hQρ,s F, GiM = h−Hρ,0 F, GiM . ds

(4.1)

For the classical cases [35, Lemma 15.1], analogue of (4.1) for the Schr¨odinger Hamiltonian with vector potentials is important for constructions of path integral representations. In the same way as in the classical case, however, (4.1) cannot be proven directly. To verify (4.1), we need two fundamental lemmas (Lemmas 4.3 and 4.4) as follows. Notice that for F, G ∈ M, Z Z dx ps (x − y) hUρ (x, y)F (y), G(x)i F dy . hQρ,s F, GiM = Rd

Rd

R R Fubini’s lemma allows one to interchange dx and dy. Moreover, we have Z d dxdy ps (x − y) hUρ (x, y)F (y), G(x)i F ds Rd ×Rd Z x2 d + ps (x)dx||F ||M ||G||M < ∞ , ≤ 2s 2s2 Rd

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

so that we can interchange the differential proposition is fundamental.

d ds

and the integral

R

...

509

dxdy. The following

Proposition 4.2. (I) Let f be a Lebesgue measurable bounded function on Rd which is continuous at 0. Then Z ps (x)f (x)dx = f (0) . lim s→0+

Rd

(II) For any α > 0, Z |x|α ps (x)dx = 0 .

lim

s→0+

Proof. Elementary calculations. We introduce notations and estimates. For ρ ∈ Cbr (Rd ; H−1 ), we set ρ,(n)

ρ,(n)

(x, y) ≡ φµ+ (x, y) + φµ− (x, y) , φρ,(n) µ   d X i ρ,(n) ρj (y)) + δn,0 π−1 (e ρj (x)) (xj − yj ) , ∂µn π−1 (e φµ+ (x, y) ≡ φF  2 j=1 ρ,(n)

φµ− (x, y) ≡

i φF −n∂µn−1 π−1 (e ρµ (y)) − δn,1 π−1 (e ρµ (x)) , 2

ρ,(0)

Note that φµ

0 ≤ n ≤ r.

(x, y) = φρ (x, y). For ρ ∈ Cbr (Rd ; H−1 ), put

v u d uX ||∂µn π−1 (e ρj (x))||2−1 , cµ,n (ρ) = sup t x

j=1

ρµ (x))||−1 , dµ,n (ρ) = sup ||∂µn π−1 (e x

0 ≤ n ≤ r,

1 ≤ µ ≤ d.

In the case n = 0, we use notations cµ,0 (ρ) = c0 (ρ) and dµ,0 (ρ) = d0 (ρ). From (3.2) it follows that for ρi ∈ Cbr (Rd ; H−1 ), 0 ≤ ki ≤ r, 1 ≤ µi ≤ d, i = 1, . . . , n, and Φ ∈ F ∞ such that NΦ = N Φ, ρ1 ,(k1 ) (x, y)φµρ22,(k2 ) (x, y) . . . φµρnn,(kn ) (x, y)Φ φµ1 F √ √ √ n 2 N + 1... N + n ≤ ||Φ||F 2n ×Πni=1 {(1 + δ0,ki )cµi ,ki (ρi )|x − y| + (ki + δki ,1 )dµi ,ki −1 (ρi )} . (4.2) Lemma 4.3. Let ρ ∈ Cbr (Rd ; H−1 ), G ∈ F and F ∈ C r (Rd ; F ) such that ∂ F (y) ∈ F ∞ , k = 1, . . . , r − 1. Then hUρ (x, y)F (y), GiF is r-times differentiable in y. In particular, k

510

F. HIROSHIMA

D

E ∂yµhUρ (x, y)F (y), GiF = Uρ (x, y)φρ,(1) (x, y)F (y), G +hUρ (x, y)∂µ F (y), GiF , µ F

ρ∈

Cb1 (Rd ; H−1 ) ,

ρ∈

Cb2 (Rd ; H−1 ) .

(4.3) D E (x, y)∂ F (y), G ∂y2µhUρ (x, y)F, GiF = Uρ (x, y)∂µ2 F (y), G F +2 Uρ (x, y)φρ,(1) µ µ F 2 φρ,(1) + Uρ (x, y) (x, y) + φρ,(2) (x, y) F (y), G , µ µ

F

(4.4)

Proof. Suppose that H ∈ F such that NH = N H, and ρ ∈ Cb1 (Rd ; H). For simplicity we put φρ (x, y) = φ(x, y). Since, by (3.2), F ∞ is the set of the analytic vectors [29, X. 6] of the self-adjoint operator φ(x, y), the following equation follows: hUρ (x, y)H, GiF =

∞ X 1 n hφ (x, y)H, GiF . n! n=0

(4.5)

One can easily derive from (4.2) that each term on the r.h.s.of (4.5) is differentiable with respect to yµ with D E (x, y)H, G ∂yµ hφn (x, y)H, GiF = n φn−1 (x, y)φρ,(1) µ

F

,

from which and (4.2) it follows that k E X 1 D n (x, y)H, G φ (x, y)φρ,(1) µ n! F n=0

√ √ √ N + 1 N + 2... N + n + 1 √ ≤ 2n n! n=0 k X

n

× (cµ,1 (ρ)|x − y| + 2d0 (ρ)) (2c0 (ρ)|x − y|) ||H||F ||G||F .

(4.6)

Then the left-hand side (l.h.s.) of (4.6) converges uniformly in the wider sense with respect to y as k → ∞. Hence the differentiability of hUρ (x, y)H, GiF with respect to yµ follows. From the strong differentiability of F and the fact ∂F (y) ∈ F ∞ , (4.3) follows. Equation (4.4) and the remaining statements can be shown similarly. Lemma 4.4. Let ρ ∈ Cbr (Rd ; H−1 ), G ∈ M, F ∈ M such that F (x) ∈ F ∞ , and 0 ≤ kj ≤ r, j = 1, . . . , n. Then Z

D

lim

X→0

Rd

E ρ,(k ) ρ,(k ) Uρ (x, x − X)φµ1 −1 (x, x − X) . . . φµn −n (x, x − X)F (x − X), G(x) dx

Z

= Rd

D

E ρ,(k ) ρ,(k ) dx . φµ1 −1 (x, x) . . . φµn −n (x, x)F (x), G(x) F

F

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

511

Proof. One can easily see that for all x ∈ Rd s − lim Uρ (x, x − X) = IF X→0

in F . Let K ∈ M be such that NK(x) = N K(x) for all x ∈ Rd . Then Z Rd

D E ρ,(k) ρ,(k) dx φµ− (x, x − X) − φµ− (x, x) K(x), G(x) F

√ N +1 ˜ k dµ,k (ρ)|X|||K||M ||G||M , ≤ √ 2

Pd ρµ (x)) ||−1 . Hence by (4.2) and by where d˜µ,k (ρ) = supx || j=1 ∂j ∂µk−1 π−1 (e lim ||F (· − X) − F (·)||M = 0 ,

X→0

one can directly derive the lemma. Now we can prove Lemma 4.1. Proof of Lemma 4.1. Note that 1 d ps (x − y) = ∆y ps (x − y) . ds 2

Since F ∈ M∞ ρ , one can use the integration by parts formula. Then, by (4.4), we have d Z 1X d hQρ,s F, Gi = dxdyps (x − y)∂y2µ hUρ (x, y)F (y), G(x)iF ds 2 µ=1 Rd ×Rd

( d Z

1X = dxdyps (x − y) Uρ (x, y)∂µ2 F (y), G(x) F 2 µ=1 Rd ×Rd D E +2 Uρ (x, y)φρ,(1) (x, y)∂µ F (y), G(x) µ F

) 2 φρ,(1) + Uρ (x, y) (x, y) φρ,(2) (x, y) F (y), G(x) µ µ F

≡

d Z 1X dxdyps (x − y)Iµ (x, y) 2 µ=1 Rd ×Rd

Z d Z 1X ps (X)dX dxIµ (x, x − X) . = 2 µ=1 Rd Rd

512

F. HIROSHIMA

Here, to apply Proposition 4.1, we divide Iµ (x, y) in two components, Iµ = Iµ+ + Iµ− , as follows: D E ρ,(1) Iµ+ (x, y) = 2 Uρ (x, y)φµ+ (x, y)∂µ F (y), G(x) F * ( 2 ρ,(1) ρ,(1) ρ,(1) φµ+ (x, y) + 2φµ+ (x, y)φµ− (x, y) + Uρ (x, y) ) +

ρ,(2) φµ+ (x, y)

+ F (y), G(x)

, F

D E

ρ,(1) Iµ− (x, y) = Uρ (x, y)∂µ2 F (y), G(x) F + 2 Uρ (x, y)φµ− (x, y)∂µ F (y), G(x) F 2 ρ,(1) ρ,(2) φµ− (x, y) + φµ− (x, y) F (y), G(x) + Uρ (x, y) . F

By (4.2), we have Z dxIµ+ (x, x − X) Rd

√ √ √ N +1 N +1 N +2 (cµ,1 (ρ)|X|) ||∂µ F ||M + ≤ ||G||M 2 √ 2 2 √ o n N +1 2 cµ,2 (ρ)|X|||F ||M × (cµ,1 (ρ)|X|) + 4d0 (ρ)cµ,1 (ρ)|X| ||F ||M + √ 2

≡ 1 |X| + 2 |X|2 , Z

Rd

dxIµ− (x, x − X)

√ N +1 2 ||∂µ F ||M ≤ ||G||M ||∂µ F ||M + 4d0 (ρ) √ 2 √ √ √ N +1 N +2 N +1 2 4d0 (ρ) + √ 2dµ,1 (ρ) ||F ||M + 2 2 ≡ 3 .

(4.7)

By Proposition 4.1 (II), we have Z Z Z dXps (X) 1 |X| + 2 |X|2 = 0 . lim dXps (X) dxIµ+ (x, x − X) ≤ lim s→0+

s→0+

Thus it is enough to analyze the Iµ (·, · − X) component. By Lemma 4.4, it follows that Z Z Iµ− (x, x − X)dx = Iµ− (x, x)dx . (4.8) lim X→0

Rd

Rd

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

513

By (4.7) and (4.8), Proposition 4.1 (I) yields Z Z d X d 1 hQρ,s F, GiM = lim dXps (X) Iµ− (x, x − X)dx lim s→0+ ds s→0+ 2 Rd Rd µ=1 d Z 1X dxIµ− (x, x) = 2 µ=1 Rd

= − hHρ,0 F, GiM .

Lemma 4.5. Let ρ ∈ Cb2 (Rd ; H−1 ), F ∈ M∞ ρ , and G ∈ M. Then hQρ,s F, GiM is right side differentiable at s = 0 with lim

→0+

hQρ, F, GiM − hF, GiM = − hHρ,0 F, GiM .

Proof. We see that Z hQρ,s F, GiM =

Rd

Z ps (X)dX

Z hQρ,0 F, GiM =

Rd

(4.9)

Rd

hUρ (x, x − X)F (x − X), G(x)iF dx

hF (x), G(x)iF dx .

Hence, similar to the proof of Lemma 4.1, it follows that hQρ,s F, GiM is right continuous in s at s = 0. Thus by the Taylor expansion we can see that hQρ, F, GiM − hF, GiM d = lim hQρ,s F, GiM →0+ s→0+ ds lim

= − hHρ,0 F, GiM .

Hence (4.9) follows.

Following [8, 35], we shall construct a strongly continuous one-parameter semigroup from {Qρ,n }n≥1 . For simplicity we put 2n = n∗. Lemma 4.6. Let ρ ∈ Cb2 (Rd ; H−1 ). Then, for all t ≥ 0, the strong limit s − lim Qn∗ ρ, t ≡ Gρ (t) n→∞

n∗

exists. Moreover, Gρ (t) has the following functional integral representation with F, H ∈ M hF, Gρ (t)HiM Z Z = dµ Rd ×Ω

dµ−1 e

iφF

Pd µ=1

Rt 0

π−1 (e ρµ (b(s)+x))dbµ + 12

Rt 0

∂µ π−1 (e ρµ (b(s)+x))ds

Q−1

×F (b(t) + x)H(x) .

(4.10)

514

F. HIROSHIMA

n o Proof. To prove the existence, we show that Qn∗ t ρ, n∗

in M. We see that

is a Cauchy sequence

n≥1

2 n∗ Qρ, t F − Qm∗ t F ρ, n∗

m∗

D

= F, Q2n∗ ρ, t F

E

n∗

M

M

D E + F, Q2m∗ t F ρ, m∗

M

D E m∗ − 2< F, Qn∗ t Q t F ρ, ρ, n∗

m∗

M

.

(4.11)

The last term on the r.h.s.of (4.11) is D

m∗ F, Qn∗ ρ, t Qρ, t F n∗

Z =

E M

m∗

E D m∗ (x) dx F (x), Qn∗ t Q t F n∗

Rd

m∗

F

Z

t (x − x1 )..p t (xn∗−1 − xn∗ )p t (xn∗ − y1 )..p t (ym∗−1 − ym∗ ) p n∗ n∗ m∗ m∗

= Rd ×Rn∗d ×Rm∗d

× hF (x), Uρ (x, x1 )..Uρ (xn∗−1 , xn∗ )Uρ (xn∗ , y1 )..Uρ (ym∗−1 , ym∗ )F (ym∗ )iF dxd~xd~y Z Pd iφ π−1 (2µ,m,n (x)) µ=1 dx F (b(2t) + x), e F , F (x) = Rd

L2 (Ω;F )

where m∗ X t t k + x + ρeµ b (k − 1) + x 2µ,m,n (x) = ρeµ b m∗ m∗ k=1

t t k − bµ (k − 1) × bµ m∗ m∗ +

n∗ X t t k + t + x + ρeµ b (k − 1) + t + x ρeµ b n∗ n∗ k=1

t t k + t − bµ (k − 1) + t . × bµ n∗ n∗ From Lemma 2.3 it follows that for each x ∈ Rd s − lim lim π−1 (2µ,m,n (x)) m→∞ n→∞

Z

2t

π−1 (e ρµ (x + b(s))) dbµ +

= 0

1 2

Z

2t

∂µ π−1 (e ρµ (b(s) + x)) ds 0

e−1 ]). One can easily see that the strong convergence of π−1 (2µ,m,n (x)) in L2 (Ω; [H 2 e−1 ]) implies that for each x ∈ Rd and Φ ∈ F, in L (Ω; [H

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

(

d X

s − lim lim exp iφF m→∞ n→∞

( = exp iφF

π−1 (2µ,m,n (x))

...

515

!) Φ

µ=1 d Z X µ=1

2t

π−1 (e ρµ (x + b(s))) dbµ

0

d Z 1 X 2t + ∂µ π−1 (e ρµ (x + b(s))) ds 2 µ=1 0

!) Φ.

(4.12)

in L2 (Ω; F ). On the other hand we have Pd iφF π−1 (2µ,m,n (·)) µ=1 F (·) F (b(2t) + ·), e L2 (Ω,F ) ≤ ||F (b(2t) + ·)||L2 (Ω;F ) ||F (·)||L2 (Ω;F ) ∈ L1 (Rd ) . Hence, by the Lebesgue dominated convergence theorem, we have E D m∗ lim lim F, Qn∗ t Q t F ρ, ρ, n→∞ m→∞

*

Z =

n∗

m∗

M

dx F (b(2t) + x), Rd

e

iφF

Pd

R 2t

µ=1

0

π−1 (e ρµ (b(s)+x))dbµ + 12

R 2t 0

+

∂µ π−1 (e ρµ (b(s)+x))ds

F (x)

. L2 (Ω;F )

(4.13) E D F Similarly it can be easily seen that F, Q2n∗ ρ, t n∗

M

D E and F, Q2m∗ F ρ, t m∗

M

converge to

the r.h.s. of (4.13) as n, m → ∞, respectively. Then it follows that {Qn∗ t }n≥0 is ρ, n∗ a Cauchy. Equation (4.10) easily follows from (4.13). Lemma 4.7. Let ρ ∈ Cb2 (Rd ; H−1 ). Then the family {Gρ (t)}t≥0 is a strongly continuous one-parameter semigroup on M. Proof. By the definition of Gρ (t) and the proof of Lemma 4.6, it holds that Gρ (t)Gρ (s) = Gρ (t + s) .

(4.14)

We show the strong right continuity at t = 0. Because of (4.14), the weak continuity implies the strong continuity. Hence it is enough to show that lim hGρ (t)F, HiM = hGρ (0)F, HiM ,

t→0+

From (4.10), we have

F ∈ M,

H ∈ M.

516

F. HIROSHIMA

hF, Gρ (t)HiM − hF, Gρ (0)HiM Z = dx F (b(t) + x) − F (x), Rd

e

iφF

Rt

Pd µ=1

0

π−1 (e ρµ (b(s)+x))dbµ + 12

Rt 0

∂µ π−1 (e ρµ (b(s)+x))ds

H(x) L2 (Ω;F )

dx F (x),

Z + Rd

Rt Pd R t iφ π−1 (e ρµ (b(s)+x))dbµ + 12 ∂µ π−1 (e ρµ (b(s)+x)ds) µ=1 0 0 − I H(x) e F

. L2 (Ω;F )

(4.15) The first summand on the r.h.s.of (4.15) can be evaluated as follows: Z d dx F (b(t) + x) − F (x), R

e

Rt

Pd

iφF

µ=1

0

π−1 (e ρµ (b(s)+x))dbµ +

Rt

∂µ π−1 (e ρµ (b(s)+x))ds

0

||F (b(t) + ·) − Ω

2 L (Ω;F )

12

Z ≤ ||H||M

H(x)

F (·)||2M dµ

.

Since lim ||F (b(t) + ·) − F (·)||M = 0 ,

t→0

a.s.b ∈ Ω ,

the first summand on the r.h.s.of (4.15) converges to 0. On the second summand on the r.h.s. of (4.15), one can see that by Remark 2.2 (3) Z s − lim

t→0

0

t

1 π−1 (e ρµ (b(s) + x)) dbµ + 2

Z

t

∂µ π−1 (e ρµ (b(s) + x)) ds

=0

0

e−1 ]). As in the case of (4.12), we see that for Φ ∈ F in L2 (Ω; [H ( s − lim exp iφF t→0

1 + 2

Z

d Z X µ=1

t

π−1 (e ρµ (b(s) + x)) dbµ

0

t

∂µ π−1 (e ρ(b(s) + x)) ds

!) Φ=Φ

0

in L2 (Ω; F ). Hence, by the Lebesgue dominated convergence theorem, we can derive that the second summand converges to 0 as t → 0. The strong continuity in the case t > 0 is proven similarly.

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

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517

By Lemma 4.7 and Stone’s theorem, for each ρ ∈ Cb2 (Rd ; H−1 ), there exists a e ρ,0 in M such that unique positive self-adjoint operator H e Gρ (t) = e−tHρ,0 . e ρ,0 is a Lemma 4.8. Let ρ ∈ Cb2 (Rd ; H−1 ). Then the self-adjoint operator H . self-adjoint extension of Hρ,0 |M∞ ρ e ρ,0 ) and G ∈ M∞ . Then we have Proof. Let F ∈ D(H ρ 1 1 e Qn∗ = lim − I G, F e−tHρ,0 − I G, F t ρ, n∗ n→∞ t t M M n∗−1 X 1 n∗ n∗ j t − I Qρ, n∗ = lim G, Qρ, n∗ t F n→∞ n∗ n∗ t M j=0 Z

1

= lim

n→∞

D n∗ t

0

E [n∗s] t − I Qρ, n∗ G, Qρ, t F n∗

M

ds .

Since, by Lemma 4.5, n∗ t − I Q n∗ G = −Hρ,0 G , n→0 t

w − lim

t the norm || n∗ t (Q n∗ − I)G||M is uniformly bounded in n. By Remark 2.4, we can see that

[n∗s]

s − lim Qρ, t∗ = Gρ (ts) . n→∞

n

Then we have Z 1 1 e e e−tHρ,0 − I G, F = ds −Hρ,0 G, e−tsHρ,0 F . t 0 M M As t → 0 on the both sides, we get E D e ρ,0 F G, H

M

= hHρ,0 G, F iM ,

e ρ,0 ) and which implies that G ∈ D(H e ρ,0 G = Hρ,0 G . H

Thus the proof is complete.

e ρ,0 by the same symbol Hρ,0 . We give a rigorous We denote the extension H ˙ of Hρ,0 and I ⊗ H0 ; definition of Hρ in terms of the form sum + ˙ ⊗ H0 . Hρ = Hρ,0 +I

518

F. HIROSHIMA

Next we study functional integral representations concerning e−tHρ . We introduce a multiplication operator in L2 (Rd ; E) by Z ⊕ ρ,s φE π−2 jes ρeµ (x) dx . φE,µ ≡ Rd

We define an operator acting in L2 (Rd ; E) by Hρ,0,s

d 2 1 X = , −iDµ ⊗ I − φρ,s E,µ 2 µ=1

s ≥ 0.

Since ρ ∈ Cb2 (Rd ; H−1 ), implies that js ρµ (·) ∈ Cb2 (Rd ; H−2 ), one can define a selfadjoint operator Hρ,0,s in the same way as Hρ,0 . Then, the following equation holds for F, H ∈ L2 (Rd ; E) E D F, e−tHρ,0,k H E Z Z R Pd R t iφ π−2 e jk ρ eµ (b(s)+x) dbµ + 12 0t ∂µ π−2 ejk e ρµ (b(s)+x)ds µ=1 0 = dµ dµ−2 e E Rd ×Ω

Q−2

×F (b(t) + x)H(x) . Lemma 4.9. Let ρ ∈ Cb2 (Rd ; H−1 ). Then the following equation holds on L2 (Rd ; E) Js e−tHρ,0, Js∗ = Es e−tHρ,0,s Es , where Js and Es are defined in (3.3). e −1 ], Js eiφF (A) J ∗ = Es eiφE ([jes ]A) Es , and Proof. Note that for any A ∈ [H s ∗ = Js (F (x)). From (4.10) it follows that for F, H ∈ L2 (Rd ; E), E D F, Js e−tHρ,0 Js∗ H

(Js∗ F ) (x)

*

Z = Rd ×Ω

e

iφF

L2 (Rd ;E)

dµ (Js∗ F )(b(t) + x) , Rt

Pd µ=1

0

π−1 (e ρµ (b(s0 )+x))dbµ + 12

Rt 0

∂µ π−1 (e ρµ (b(s0 )+x)ds0 )

+ (Js∗ H)(x) F

*

Z =

dµ F (b(t) + x) , Rd ×Ω

Es e D

iφE

Pd µ=1

jes

Rt 0

π−1 (e ρµ (b(s0 )+x))dbµ + 12

= F, Es e−tHρ,0,s Es H

Rt 0

∂µ π−1 (e ρµ (b(s0 )+x)ds0 )

+ Es H(x) E

E L2 (Rd ;E)

.

Since F, H ∈ L2 (Rd ; E) are arbitrary, the proof is complete.

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

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519

Now we are ready to state the main theorem in this section. Theorem 4.10. Let F, G ∈ M, V ∈ Cb (Rd ) and ρ ∈ Cb2 (Rd ; H−1 ) such that ρµ (x)) ∈ D([e ω ]) , π−1 (e

(4.16)

ω ]π−1 (e ρµ (x))||−1 = sup |e ωρeµ (x)|−1 < ∞ , sup ||[e

x∈Rd

µ = 1, . . . , d .

(4.17)

x∈Rd

Then E D F, e−t(Hρ +V ) G

Z =

M

dµ Rd ×Ω

(

d X

× exp iφE

Z t dµ−2 exp − V (b(s) + x)ds

Z Q−2

0

!)

ρ,µ π−2 (t, x)

Jt F (b(t) + x)J0 G(x) ,

(4.18)

µ=1

where Z ρ,µ (t, x) = π−2

t

0

1 [e js ]π−1 (e ρµ (b(s) + x)) dbµ + 2

Z

t

[e js ]∂µ π−1 (e ρµ (b(s0 ) + x)) ds0 .

0

Proof. By the strong Trotter product formula [21] and Proposition 4.3 (c), we see that E D F, e−t(Hρ +V ) G M

n∗ E t t t G F, e− n∗ Hρ,0 e− n∗ H0 e− n∗ V n→∞ M D t t t t − n∗ Hρ,0 J ∗ t e− n∗ V t e = lim F, Jt∗ Jt e− n∗ Hρ,0 Jt∗ e− n∗ V Jt− n∗ t− n∗ n→∞ E t t − n∗ Hρ,0 J ∗t e− n∗ V t e . . . J n∗ J0 G , D

= lim

n∗

M

from which and Lemma 4.9 it follows that t t t t −n Hρ,0,t− t ∗ − n∗ H − n∗ V ρ,0,t n∗ E t e t Et e Et− n∗ e− n∗ V = lim F, Jt Et e t− n∗ n→∞

...e

t − n∗ V

t t − n∗ Hρ,0, t − n∗ V n∗ t e t E n∗ J0 G E n∗ e

M

≡ lim Sn∗ . n→∞

Let

t n∗

= s. From the definition of Hρ,0,t0 and Lemma 4.6 it follows that D

−sV −sV s s F, Jt∗ Et Qk∗ Et−s Qk∗ ,t Et e ,t−s Et−s e ρ, k∗ ρ, k∗ k→∞ E −sV s . . . e−sV Es Qk∗ E J G e 0 ρ, k∗ ,s s

Sn∗ = lim

M

≡ lim Sn∗,k∗ , k→∞

520

F. HIROSHIMA

where Qρ,t,t0 is defined by operators on L2 (Rd ; E) such that Z (Qρ,t,t0 F )(x) =

Rd

pt (x − y)Uρ,t0 (x, y)F (y)dy,

t > 0,

(Qρ,0,t0 F )(x) = F (x) , and ( Uρ,t0 (x, y) = exp

d X

1 iφE 2

!) [e jt0 ]π−1 (e ρµ (x) + ρeµ (y)) (xµ − yµ )

.

µ=1

One can see that Z Sn∗,k∗ =

Ps (x~1 ) . . . Ps (~xn∗ )e dk∗ Rd × R × · · · × Rdk∗ | {z }

−s

Pn∗ j=1

V (xk∗ j )

n∗

× hF (x), Jt∗ (Et Uρ,t (x~1 )Et ) (Et−s Uρ,t−s (x~2 )Et−s ) · · · (Es Uρ,s (x~n∗ )Es ) J0 G(xkn∗ ) F dxd~x1 · · · d~xn∗ ,

(4.19)

where k∗−1 1 1 2 − xk∗ Ps (x~j ) = ps (xk∗ j−1 − xj )ps (xj − xj ) . . . ps (xj j ), k∗−1 1 1 2 , xk∗ Uρ,α (x~j ) = Uρ,α (xk∗ j−1 , xj )Uρ,α (xj , xj ) . . . Uρ,α (xj j ), !) ( k∗ d X X i−1 1 i−1 i i iφE [e jα ] π−1 ρeµ (xj ) + ρeµ (xj ) (xj,µ − xj,µ ) , = exp 2 µ=1 i=1 k∗ x0j = xk∗ j−1 , x0 ≡ x, j = 1, . . . , n ∗ .

By Proposition 3.3 one can neglect Ej in (4.19), so that Z Sn∗,k∗ = 

* dx F (b(t) + x) ,

Rd

   + d n∗−1 n∗ X X X ∗ [e jjs ]π−1 (2µ,j,k (x))−s V (b(js) + x) J0 G(x) Jt exp iφE  µ=1 j=0

j=1

, L2 (Ω;F )

where k∗ m X m−1 s + js + x + ρeµ b s + js + x ρeµ b k∗ k∗ m=1 m−1 m s + js − bµ s + js , j = 0, . . . , n ∗ −1 . × bµ k∗ k∗

2µ,j,k (x) = 21

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

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521

As in the case of (4.12), we see that for each x ∈ Rd and Φ ∈ E,    d n∗−1   X X [e jjs ]π−1 (2µ,j,k (x)) Φ s − lim exp iφE    k→∞   = exp



µ=1 j=0

 iφE 

d n∗−1 X X µ=1 j=0

  (2µ,j (x) + ∆µ,j (x)) Φ 

in L2 (Ω; E), where Z

2µ,j (x) =

(j+1)s

[e jjs ]π−1 (e ρµ (b(s0 ) + x)) dbµ ,

js

1 2

∆µ,j (x) =

Z

(j+1)s

[e jjs ]∂µ π−1 (e ρµ (b(s0 ) + x)) ds0 .

js

On the other hand, we have Pd Pn∗−1 Pn∗ [e jjs ]π−1 (2µ,j,k (·)) −s V (b(js)+·) Jt F (b(t) + ·), eiφE µ=1 j=0 j=1 J0 G(·) L2 (Ω;E) e ≤ exp −s inf V (x) ||F (b(t) + ·)||L2 (Ω;F ) ||G(·)||L2 (Ω;F ) ∈ L1 (Rd ) . x

Hence, by the Lebesgue dominated convergence theorem, we have * Z lim Sn∗,k∗ =

k→∞

Rd

e

iφE

dx Jt F (b(t) + x) , Pd

Pn∗−1

µ=1

j=0

(

2µ,j (x)+∆µ,j (x))

e

−s

Pn∗ j=1

+ V (b(js)+x)

J0 G(x)

. L2 (Ω;E)

e −2 ]), by Theorem 2.5, ρµ (x)) ∈ Cb1 (R × Rd ; [H Since, by (4.16) and (4.17), [e jt ]π−1 (e d we see that for x ∈ R Z t n∗−1 X s − lim 2µ,j (x) = [ejs ]π−1 (eρµ (x + b(s))) dbµ n→∞

s − lim

n→∞

0

j=0 n∗−1 X

Z ∆µ,j (x) =

t

[e js ]∂µ π−1 (e ρµ (x + b(s))) ds ,

0

j=0

e−2 ]). Then again as in the case of (4.12), for each x ∈ Rd and Φ ∈ E in L2 (Ω; [H    d n∗−1   X X 2 Φ s − lim exp iφE  µ,j (x) + ∆µ,j (x) n→∞   µ=1 j=0

( = exp iφE

d X µ=1

!) ρ,µ π−2 (t, x)

Φ

(4.20)

522

F. HIROSHIMA

in L2 (Ω; E). Passing to the subsequences for n (hereafter denoted again by n), (4.20) holds for each x ∈ Rd and a.e.b ∈ Ω in the strong topology of E. Since V (b(s) + x) is continuous in s for each x ∈ Rd and a.e.b ∈ Ω, we have Z t n∗−1 jt t X V b V (b(s) + x)ds , +x = lim n→∞ n∗ n∗ 0 j=0

x ∈ Rd a.e.b ∈ Ω .

Furthermore, Pd Pn∗−1 −s Pn∗ V (b(js)+·) iφE (2µ,j (·)+∆µ,j (·)) e j=1 µ=1 j=0 J0 G(·) Jt F (b(t) + ·), e

≤ exp −s inf V (x) ||F (b(t) + ·)||F ||F (·)||F ∈ L1 (Rd × Ω; dµ) .

E

x

Hence, again by the Lebesgue dominated convergence theorem we get (4.18).

Note that if we take a typical example of H−1 -valued function ρ(·) as in (3.5) then (4.16) and (4.17) are satisfied and the following equation holds d X

∂µ π−1 (e ρµ (·)) = 0 .

µ=1

The above equation is just the Coulomb gauge condition. Similarly to the classical case [35, Theorem 15.5], we have an interest in extending (4.18) to more general potentials. From (4.18) it follows that for V ∈ Cb (Rd ) and ρ satisfying the conditions stated in Theorem 4.10 D E D E 1 (4.21) ≤ ||F ||F , e−t(− 2 ∆+V ) ||G||F 2 d . F, e−t(Hρ +V ) G M

We define for G ∈ M

L (R )

(

(g sgnG)(x) ≡

G(x) ||G(x)||F ,

||G(x)||F 6= 0,

0,

||G(x)||F = 0 .

Lemma 4.11. Let |V | be a multiplication operator which is − 12 ∆ -form bounded with relative bound . Then for ρ satisfying the condition in Theorem 4.10, |V | is Hρ -form bounded with relative bound ≤ . Proof. Substituting V = 0 and F = g sgn e−tHρ G · ψ, where ψ ∈ C0∞ (Rd ) and ψ ≥ 0, in (4.21), we have E D ψ, ||e−tHρ G||F

L2 (Rd )

D E 1 ≤ ψ, e−t(− 2 ∆) ||G||F

Hence it follows that 1 −tHρ G (x) ≤ e−t(− 2 ∆) ||G(·)||F (x) , e F

L2 (Rd )

.

a.e.x ∈ Rd .

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

...

523

Since Z ∞ 1 1 −1 e−Et t− 2 e−tHρ G (x)dt, a.e.x ∈ Rd , E > 0 , (Hρ + E) 2 G (x)=Γ 2 0 one can see that −1 (Hρ + E) 2 G (x) ≤

! − 12 1 − ∆+E ||G(·)||F (x) , 2

F

a.e.x ∈ Rd .

Then we have Z n o2 1 −1 dx |V (x)| 2 (Hρ + E) 2 G (x) Rd

Z ≤ Thus

Rd

F

(

|V (x)|

1 2

12 −1 |V | (Hρ + E) 2 G ||G||M

M

≤

1 − ∆+E 2

− 12

! ||G(·)||F

)2 dx .

(x)

− 1 12 |V | − 21 ∆ + E 2 ||G(·)||F

L2 (Rd )

||||G(·)||F ||L2 (Rd )

which implies that the following operator norm estimate holds − 12 1 1 1 1 2 − |V | (Hρ + E) 2 ≤ |V | 2 − ∆ + E 2 M

,

.

(4.22)

L2 (Rd )

Since

− 12 1 1 2 − ∆+E lim |V | E→∞ 2

= ,

L2 (Rd )

the lemma follows.

For a multiplication operator V , we set V+ = max{0, V } and V− = max{0, −V }. Let us introduce a class P of potentials. Definition 4.12. A potential V is in the set P if and only if V+ ∈ L1loc(Rd ) and V− is − 12 ∆-form bounded with relative bound < 1. For V ∈ P , we define a quadratic form t by E D 1 E D 1 1 1 + V+2 F, V+2 F t(F, F ) = Hρ2 F, Hρ2 F M

M

D 1 E 1 − V−2 F, V−2 F

M

,

Q(t) = Q(Hρ ) ∩ Q(V+ ) , where Q(A) denotes the form domain of a positive self-adjoint operator A, i.e. 1 Q(A) = D(A 2 ). By Lemma 4.11, t is positive closed form on Q(t). We denote the positive self-adjoint operator associated with t by ˙ + −V ˙ −, Hρ +V ˙ + −V ˙ − ) = Q(t). so that Q(Hρ +V

524

F. HIROSHIMA

Theorem 4.13. Let V ∈ P and ρ ∈ Cb2 (Rd ; H−1 ) satisfy the conditions in ˙ + −V ˙ −. Theorem 4.10. Then (4.18) holds with Hρ + V replaced by Hρ +V Proof. As in [35, Theorem 6.2], one can easily see that by an approximation argument, (4.18) holds for V ∈ L∞ (Rd )(the set of bounded functions). Fix V ∈ P . We set ( ( V+ (x), V+ (x) < n , V− (x), V− (x) < m, , V−m (x) = . V+n (x) = n, V+ (x) ≥ n m, V− (x) ≥ m Then we have E D F, e−t(Hρ +V+n −V−m ) G *

Z M

= Rd ×Ω

(

× Jt F (b(t) + x), exp iφE

Z t dµ exp − (V+n − V−m ) (b(s) + x)ds 0

d Z t X µ=1

+

!)

[e js ]π−1 (e ρµ (b(s) + x)) dbµ

J0 G(x)

0

. E

(4.23) Define closed quadratic forms by D 1 D 1 E E 1 1 2 2 + V+n F, V+n F tn,m (F, F ) = Hρ2 , Hρ2 M

Q(tn,m ) = Q(Hρ ) , D 1 D 1 E E 1 1 2 2 + V+n F, V+n F tn,∞ (F, F ) = Hρ2 , Hρ2 M

Q(tn,∞ ) = Q(Hρ ) , D 1 D 1 E E 1 1 + V+2 F, V+2 F t∞,∞ (F, F ) = Hρ2 , Hρ2 M

M

M

M

D 1 E 1 2 2 − V−m F, V−m F D 1 E 1 − V−2 F, V−2 F

M

D 1 E 1 − V−2 F, V−2 F

M

M

,

,

,

Q(t∞,∞ ) = Q(Hρ ) ∩ Q(V+ ) . ˙ − . We We denote the self-adjoint operator associated with tn,∞ by Hρ + V+n −V have tn,m ≥ tn,m+1 ≥ tn,m+2 ≥ · · · ≥ tn,∞ and tn,m → tn,∞ in the sense of quadratic form on ∪m Q(tn,m ) = Q(Hρ ). Since tn,∞ is closed on Q(Hρ ), by the monotone convergence theorem for forms [20, VIII. Theorem 3.11], the associated positive self-adjoint operators satisfy ˙ − Hρ +V+n −V−m → Hρ +V+n −V in the strong resolvent sense, which implies that for all t ≥ 0, ˙ − exp (−t (Hρ +V+n −V−m )) → exp −t Hρ +V+n −V strongly. Similarly, we have tn,∞ ≤ tn+1,∞ ≤ tn+2,∞ ≤ · · · ≤ t∞,∞

FUNCTIONAL INTEGRAL REPRESENTATION OF A MODEL IN

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525

and tn,∞ → t∞,∞ in the sense of quadratic form on

F ∈ ∩n Q(tn,∞ ) sup tn,∞ (F, F ) < ∞ = Q(Hρ ) ∩ Q(V+ ) . n

Hence the monotone convergence theorem for forms [20, VIII. Theorem 3.13],[32], we get ˙ +n −V ˙ − exp −t Hρ +V

˙ + −V ˙ − → exp −t Hρ +V

t ≥ 0,

,

strongly. On the other hand, the r.h.s.of (4.23) converges to Z Rd ×Ω

Z t dµ exp − (V+n − V− ) (b(s) + x)ds

*

0

(

× Jt F (b(t) + x), exp iφE

d Z X µ=1

t

+

!) [e js ]φ−1 (e ρµ (b(s) + x)) dbµ

J0 G(x)

0

. E

(4.24) as m → ∞ by the monotone convergence theorem for integrals. Also (4.24) converges to the r.h.s of (4.18) as n → ∞ by the Lebesgue dominated convergence theorem. Thus the proof is complete. 5. Inequalities In this section we shall derive some inequalities similar to classical models from the functional integral representation constructed in Sec 4. Let σ(A) be the spectrum of A. For simplicity, we put for V ∈ P ˙ + −V ˙ − ≡ Hρ + V , Hρ +V 1 1 ˙ + −V ˙ − ≡ − ∆ + H0 + V , − ∆ ⊗ I + I ⊗ H0 +V 2 2 1 1 ˙ + −V ˙ − ≡ − ∆+V . − ∆ ⊗ I +V 2 2 Note that − 12 ∆⊗ I + I ⊗ H0 is self-adjoint on D(− 21 ∆⊗ I)∩D(I ⊗ H0 ) and bounded from below. Since Jt is a positivity preserving operator [30]; |Jt Ψ| ≤ Jt |Ψ| ,

Ψ∈F,

from Theorems 4.10 and 4.13 it follows that for V ∈ P D E D E 1 , ≤ |F |, e−t(− 2 ∆+H0 +V ) |G| F, e−t(Hρ+V ) G M M E D 1 ≤ ||F ||F , e−t(− 2 ∆+V ) ||G||F

L2 (Rd )

.

(5.1)

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Theorem 5.1 (Diamagnetic inequality). Let V ∈ P and ρ ∈ Cb2 (Rd ; H−1 ) satisfy the conditions in Theorem 4.10. Then 1 (5.2) inf σ − ∆ + H0 + V ≤ inf σ (Hρ + V ) . 2 Proof. Fix G ∈ M such that EHρ +V ([E0 , E0 + ))G 6= 0, for all 0 < < 0 with some 0 > 0, where EHρ +V denotes the spectral projection of Hρ + V and E0 = inf σ(Hρ + V ). Then by (5.1) we have D E 1 inf σ (Hρ + V ) = lim − log G, e−t(Hρ +V ) G t→∞ t M D E 1 1 ≥ lim − log |G|, e−t(− 2 ∆+H0 +V ) |G| t→∞ t M 1 1 ≥ lim − log |G|2 e−t inf σ(− 2 ∆+H0 +V ) t→∞ t 1 ≥ inf σ − ∆ + H0 + V . 2

Thus (5.2) follows. For G ∈ M define

( sgnG =

G |G| ,

G 6= 0,

0,

G = 0.

Theorem 5.2 (Comparing abstract Kato’s inequality). Let V ∈ P and ρ ∈ Cb2 (Rd ; H−1 ) satisfy the conditions in Theorem 4.10. 1 (1) Suppose that Ψ ∈ D((− 12 ∆ + H0 + V ) 2 ), Ψ ≥ 0 and G ∈ D ((Hρ + V )) . Then 1 |G| ∈ D − 12 ∆ + H0 + V 2 and the following inequality holds < h(sgnG)Ψ, (Hρ + V ) GiM * + 12 12 1 1 ≥ − ∆ + H0 + V Ψ, − ∆ + H0 + V |G| 2 2

.

(5.3)

M

1

(2) Suppose that ψ ∈ D((− 12 ∆ + V ) 2 ), ψ ≥ 0 and G ∈ D ((Hρ + V )). Then 1 ||G(·)||F ∈ D − 12 ∆ + V 2 and the following inequality holds * < h(g sgnG)ψ, (Hρ + V ) GiM ≥

1 − ∆+V 2

12

+ 12 1 ψ, − ∆ + V ||G||F 2

. L2 (Rd )

Proof. The proof is a slight modification of that of [17]. We show (1). By (5.1), we have + + * * 1 I − e−t(Hρ +V ) I − e−t(− 2 ∆+H0 +V ) |F | F ≤ F, . (5.4) |F |, t t M

M

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527

Putting * st (Φ, Φ) ≡

I − e−t(− 2 ∆+H0 +V ) Φ Φ, t 1

+ ,

Φ ∈ M,

L2 (Rd )

one can see that {st }t≥0 is a family of positive closed quadratic forms which increase monotonically as t decreases to zero; st ≤ sk ,

0 < k ≤ t.

Thus the monotone convergence theorem for forms, we see that s∞ (G, G) ≡ supt>0 st (G, G) is a closed quadratic form on R∞ = {F ∈ M| sup(F, F ) < ∞} , t>0

moreover the corresponding positive self-adjoint operators to st converge to 1 − 21 ∆ + H0 + V 2 in the strong resolvent sense. Since, by (5.4), F ∈ D(Hρ + V ) 1 implies that |F | ∈ R∞ , we have |F | ∈ D − 21 ∆ + H0 + V 2 . By (5.1), we have 1 −t(− 12 ∆+H0 +V ) 1 −t(Hρ +V ) e ≤ |F |, e −I G |G| < F, t t M M 1 − < hF, GiM . t

(5.5)

Substituting F = (sgnG)Ψ into (5.5), we have 1 −t(− 12 ∆+H0 +V ) 1 −t(Hρ +V ) e e ≤ Ψ, . −I G − I |G| < (sgnG) Ψ, t t M M Then, taking t → 0+ we see that 1 −t(Hρ +V ) e = −< h(sgnG)Ψ, (Hρ + V ) GiM , −I G lim < (sgnG) Ψ, t→0+ t M and 1 −t(− 12 ∆+H0 +V ) e − I |G| lim Ψ, t→0+ t M * + 12 12 1 1 =− − ∆ + H0 + V Ψ, − ∆ + H0 + V |G| 2 2

. M

Thus (5.3) follows. Using the second inequality in (5.1), one can prove the statement (2) by the same way as that of (1).

528

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6. Remarks (1) In the FKI formula, the Wiener path measure dµ is more useful than the Brownian path measure Db. For the Schr¨ odinger Hamiltonian Hcl =

d 1X (−iDµ − Aµ )2 + V 2 µ=1

the FKI formula is, first, established for an electric vector potential Aµ (·) ∈ C0∞ (Rd ), µ = 1, . . . , d, and after that it is extended to Aµ (·) ∈ L2loc (Rd ) by a limiting argument [35]. In the model which we consider, φρF ,µ corresponds to the classical electric vector potential Aµ (·). But we have no strategy of limiting argument used in the classical model. Then it is necessary to deal with the Hilbert space-valued stochase −1 ]) directly (not using ρµ (·)) ∈ Cb2 (Rd ; [H tic integral for φρF ,µ such that π−1 (e limiting arguments as in the classical case). Then it cannot be assumed that e−1 ]), i.e., one cannot define (see (2.6)) ρµ (·)) ∈ H 2 (Rd ; [H π−1 (e Z

t

φF (π−1 (e ρµ (ω(s)))) dωµ . 0

Therefore we consider the Hilbert space-valued stochastic integral not on the Wiener path but on the Brownian path. (2) In scalar field theory [25, 26, 30], the range of the projection e[a,b] (notations follow [30]) can be characterized by some support properties [30, Proposition III.4], i.e., Ran e[a,b] = f ∈ N suppf ⊂ (a, b) × Rd ¯. In particular, Ran(et ) = f ∈ N suppf ⊂ {t} × Rd ¯. However, the corresponding projection [e[a,b] ], which we introduce in Proposition 3.2, cannot be characterized in such a way. For example, n e−2 ] suppf ⊂ ⊕dµ=1 {t} × Rd ¯. Ran([et ]) 6= π2 (f ) ∈ [H

References [1] A. Arai, “Rigorous theory of spectra and radiation for a model in quantum electrodynamics”, J. Math. Phys. 24 (1983) 1896–1910. [2] A. Arai, “A note on scattering theory in non-relativistic quantum electrodynamics”, J. Phys. A: Math. Gen. 16 (1983) 49–70. [3] A. Arai, “An asymptotic analysis and its application to the nonrelativistic limit of the Pauli–Fiertz and a spin-boson model”, J. Math. Phys. 31 (1990) 2653–2663. [4] A. Arai, “Scaling limit for quantum systems of nonrelativistic particles interacting with a bose field”, Hokkaido Univ. preprint series in math. 59.

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[5] J. Avron, I. Herbst and B. Simon, “Schr¨ odinger operators with magnetic fields I. General interactions”, Duke Math. J. 45 (1978) 847–883. [6] P. Blanchard, “Discussion math´ematique du mod´ele de Pauli et Fierz relatif a ´ la catastrophe intrarouge”, Commun. Math. Phys. 15 (1969) 156–172. [7] J. T. Cannon, “Quantum field theoretic properties of a model of Nelson: domain and eigenvector stability for perturbed linear operators”, J. Funct. Anal. 8 (1971) 102–152. [8] P. R. Chernoff, “Product Formula, Nonlinear Semigroups and Addition of Unbounded Operator”, Amer. Math. Soc. Providence, R. I., 1974. [9] E. B. Davies, “Particle-boson interactions and the weak coupling limit”, J. Math. Phys. 20 (1979), 345–351. [10] J. Fr¨ ohlich, “On the infrared problem in a model of scalar electrons and massless, scalar bosons”, Ann. Inst. Henri Poincar´e 16 (1973) 1–103. [11] J. Fr¨ ohlich, “Existence of dressed one electron states in a class of persistent models”, Fortschritte der Physik 22 (1974) 159–198. [12] J. Fr¨ ohlich and Y. M. Park, “Correlation inequalities and thermodynamic limit for classical and quantum continuous systems”, Commun. Math. Phys. 47 (1974) 271– 317. [13] J. Fr¨ ohlich and Y. M. Park, “Correlation inequalities and thermodynamic limit for classical and quantum continuous systems II. Bose–Einstein and Fermi–Dirac statistics”, J. Stat. Phys. 23 (1980) 701–753. [14] F. Guerra, L. Rosen and B. Simon, “The P (φ)2 Euclidean quantum field theory as classical statistical mechanics”, Ann. of Math. 101 (1975) 111–267. [15] J. Glimm and A. Jaffe, “Quantum Physics”, Springer-Verlag, New York, 1981. [16] L. Gross, “The free Euclidean Proca and electromagnetic fields”, Functional integration and its applications, ed. A. M. Arthurs, Publ. Clarendon Press, 1975. [17] H. Hess, R. Schrader and D. A. Uhlenbrock, “Domination of semigroups and generalization of Kato’s inequality”, Duke Math. J. 44 (1977) 893–904. [18] F. Hiroshima, “Scaling limit of a model in quantum electrodynamics”, J. Math. Phys. 34 (1993) 4478–4518. [19] F. Hiroshima, “Diamagnetic inequalities for systems of nonrelativistic particles with a quantized field”, to appear in Rev. Math. Phys. 8 (2) (1996) 185–203. [20] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heisenberg-New York, 1966. [21] T. Kato and K. Masuda, “Trotter’s product formula for nonlinear semigroups generated by the subdifferentiables of convex functionals”, J. Math. Soc. Japan 30 (1978) 169–178. [22] A. Klein and L. J. Landau, “Singular perturbations of positivity preserving semigroups via path space techniques”, J. Funct. Anal. 20 (1975) 44–82. [23] H. H. Kuo, “Stochastic integrals in abstract Wiener space”, Pacific J. Math. 41 (1972) 469–483. [24] E. Nelson, “Interaction of nonrelativistic particles with a quantized scalar field”, J. Math. Phys. 5 (1964) 1190–1197. [25] E. Nelson, “Construction of quantum fields from Markoff fields”, J. Funct. Anal. 12 (1973) 97–112. [26] E. Nelson, “The free Markov field”, J. Funct. Anal. 12 (1973) 211–227. [27] T. Okamoto and K. Yajima, “Complex scaling technique in non-relativistic massive QED”, Ann. Inst. Henri Poincar´e 42 (1985) 311–327. [28] M. Reed and B. Simon, Method of Modern Mathematical Physics IV, Academic Press, New York, 1975. [29] M. Reed and B. Simon, Method of Modern Mathematical Physics II, Academic Press, New York, 1975.

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[30] B. Simon, “The P (φ)2 Euclidean Field Theory”, Princeton Univ. Press, 1974. [31] B. Simon, “An abstract Kato’s inequality for generator of positivity preserving semigroups”, Indians Univ. Math. J. 26 (1977) 1067–1073. [32] B. Simon, “A canonical decomposition for quadratic forms with applications to monotone convergence theorems”, J. Funct. Anal. 9 (1978) 377–385. [33] B. Simon, “Kato’s inequality and the comparison of semigroups”, J. Funct. Anal. 32 (1979) 97–101. [34] B. Simon, “Maximal and Schr¨ odinger forms”, J. Operator Theory 1 (1979) 37–47. [35] B. Simon, Functional Integral and Quantum Physics, Academic Press, 1979. [36] H. Spohn and R. D¨ umcke, “Quantum tunneling with dissipation and the Ising model over R”, J. Stat. Phys. 41 (1985) 389–423. [37] H. Spohn, “Ground state(s) of the spin-boson Hamiltonian”, Commun. Math. Phys. 123 (1989) 277–304. [38] T. A. Welton, “Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field”, Phys. Rev. 74 (1948) 1157–1167.

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Extended Version ´ DURDEVIC ´ MICO

Instituto de Matematicas, UNAM Area de la Investigacion Cientifica Circuito Exterior, Ciudad Universitaria M´ exico DF, CP 04510, MEXICO Received 24 January 1997 Revised 24 February 1997 A general non-commutative-geometric theory of principal bundles is developed. Quantum groups play the role of structure groups and general quantum spaces play the role of base manifolds. A general conceptual framework for the study of differential structures on quantum principal bundles is presented. Algebras of horizontal, verticalized and “horizontally vertically” decomposed differential forms on the bundle are introduced and investigated. Constructive approaches to differential calculi on quantum principal bundles are discussed. The formalism of connections is developed further. The corresponding operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. In particular the analogs of the basic classical algebraic identities are derived. A quantum generalization of classical Weil’s theory of characteristic classes is sketched. Quantum analogs of infinitesimal gauge transformations are studied. Interesting examples are presented.

Contents 1. 2. 3. 4. 5. 6.

Introduction Preparatory Material Quantum Principal Bundles and the Associated Calculi The Formalism of Connections Characteristic Classes Examples, Remarks and Some Additional Constructions 6.1. Infinitesimal gauge transformations A 6.2. Infinitesimal gauge transformations B 6.3. Internal horizontal forms 6.4. Trivial bundles 6.5. Quantum homogenous spaces 6.6. A constructive approach to differential calculus Appendix A. On Bicovariant Exterior Algebras Appendix B. Multiple Irreducible submodules References

531 534 537 545 563 566

595 602 606

1. Introduction In this study we continue the presentation of the theory of quantum principal bundles. The theory developed in the previous paper [4] was “semiclassical”: structure groups were considered as quantum objects, however base spaces were classical smooth manifolds. Algebraic formalism developed in the previous paper will be 531 Reviews in Mathematical Physics, Vol. 9, No. 5 (1997) 531–607 c World Scientific Publishing Company

532

´ M. DURDEVIC

now generalized and incorporated into a completely quantum framework, following general philosophy of non-commutative differential geometry [2, 3]. Base manifolds, structure groups and corresponding principal bundles will be considered as quantum objects. The paper is organized as follows. Exposition of the theory begins in Sec. 3, with a general definition of quantum principal bundles. This definition will translate into a non-commutative-geometric context classical idea that a principal bundle is a space on which the structure group acts freely on the right, such that the base manifold is diffeomorphic to the corresponding orbit space. After the main definition we go on to questions related to differential calculus on quantum principal bundles. At first, we introduce and analyze a differential ∗ -algebra consisting of “verticalized” differential forms on the bundle. This algebra will be introduced independently of a specification of a complete differential calculus on the bundle. The calculus on the bundle is based on a graded-differential ∗ -algebra (representing differential forms) possessing two important properties. At first, we require that this differential algebra is generated by “functions” on the bundle. This condition ensures uniqueness of various entities naturally appearing in the study of differential calculus. Secondly, we postulate that the group action on “the functions” on the bundle can be extended to an appropriate differential algebra homomorphism (imitating the corresponding “pull back” of differential forms). Quantum counterparts of various important entities associated to differential calculus in the classical theory will be introduced in a constructive manner, starting from the algebra of differential forms on the bundle (and from a given differential calculus on the structure quantum group). In particular, a graded ∗ -algebra representing horizontal forms will be introduced and analyzed. Also, the associated graded-differential ∗ -algebra representing differential forms on the base manifold will be described. It is important to point out a conceptual difference between this approach to differential calculus and the approach presented in the previous paper, where the calculus on the bundle was constructed starting from the standard calculus on the base manifold, and an appropriate calculus on the structure quantum group. The main property of the calculus was a variant of local triviality, in the sense that all local trivializations of the bundle locally trivialize the calculus, too. This property implies certain restrictions on a possible differential calculus on the structure quantum group, as discussed in details in [4]. On the other hand, in this paper we start from a fixed calculus on the group (based on the universal differential envelope of a given first-order differential structure) and the calculus on the base manifold is determined by the calculus on the bundle. However, the calculus on the bundle is not uniquely determined by the previously mentioned two conditions. In Sec. 4 the formalism of connections will be presented. All corresponding basic “global” constructions and results of the previous paper will be translated into the general quantum context. In particular, operators of horizontal projection,

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covariant derivative and curvature will be constructed and investigated. Further, two particularly interesting classes of connections will be introduced and analyzed. The first class consists of connections possessing certain multiplicativity property. This is a trivial generalization of multiplicative connections of the previous paper. The second class consists of connections that are counterparts of classical connections introduced in the previous paper. Here, these connections will be called regular. Intuitively speaking, regular connections are “maximally compatible” with the internal geometrical structure of the bundle. In Sec. 5 a generalization of classical Weil’s theory of characteristic classes will be presented. It turns out that if the bundle admits regular connections, then the Weil classical construction (via the curvature tensor and invariant polynomials over the Lie algebra of the structure group) can be naturally incorporated into the quantum context. The quantum construction gives an intrinsic map between the adjoint-invariant part of the braided-symmetric algebra associated to the leftinvariant part of the first-order calculus over the structure group, and the algebra of cohomology classes represented by differential forms on the base. Finally, in Sec. 6 some examples, remarks and additional constructions are included. In particular, we shall present a general re-construction of differential calculus on the bundle, starting from a given algebra of horizontal forms, and two operators imitating the covariant derivative and the curvature of a regular connection. Further, quantum analogs of infinitesimal gauge transformations will be studied, from two different viewpoints. The formalism that will be developed here possesses several common conceptual points with a theory of quantum principal bundles presented in [1]. However, in formulating the basic structural elements of the formalism, the two approaches significantly differ. At first, higher-order differential calculus on quantum principal bundles does not effectively figure in the formulation of [1], where mainly firstorder quantum differential forms were considered. However, it turns out that the most interesting “purely quantum” phenomenas appear at the level of higer-order differential calculus. Further main differences between two formulations include constructions of horizontal forms and the forms on the base, and the treatment of the operators of horizontal projection, covariant derivative and curvature. On the other hand, both approaches define connection forms essentially in the same way. The interrelations between two formulations will be discussed in more details in Subsec. 6.3. Perhaps the most interesting topic in this area is to analyze under which conditions two concepts of horizontality coincide (which is understandable as a non-trivial condition on the calculus over the bundle). Concerning concrete examples of quantum principal bundles, we shall consider trivial bundles and principal bundles based on quantum homogeneous spaces. The main structural elements of differential calculus and the formalism of connections will be illustrated on these examples. A particular attention will be given to the quantum Hopf fibration. In this example, the total space of the bundle is given by the quantum SU(2) group [12] and the base manifold is a quantum 2-sphere [10].

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We shall consider three different choices for the global calculus on the quantum Hopf fibration — the 3D-calculus [12], the 4D-calculus [14], and finally the infinitedimensional minimal admissible calculus [4] over quantum SU(2). The paper ends with two appendices. The first appendix is devoted to the analysis of the calculus on the bundle in the case when the higher-order calculus on the structure quantum group is described by the corresponding bicovariant exterior algebra [14]. In particular, it will be shown that if the first-order calculus on the group is compatible with all “transition functions” (in the context of the previous paper) then the higher-order calculus based on the exterior algebra possesses this property too. In fact this is equivalent to a possibility of constructing the calculus on the bundle such that all local trivializations of the bundle locally trivialize the calculus. Further, we shall prove that bicovariant exterior algebras describe, in a certain sense, the minimal higher-order calculus on the group such that the corresponding calculus on the bundle possesses the mentioned trivializability property (universal envelopes always describe the maximal higher-order calculus). We shall also analyze similar questions in the context of general theory. In the second appendix the structure of the ∗ -algebra representing “functions” on the bundle is analyzed, in the light of the decomposition of the right action of the structure quantum group into multiple irreducible components. 2. Preparatory Material Before passing to quantum principal bundles we shall fix the notation, and b introduce in the game relevant quantum group entities. We shall use the symbol ⊗ for a graded tensor product of graded (differential ∗ -) algebras. Here, as in the previous paper, we shall deal with compact matrix quantum groups [13] only. However, the compactness assumption is not essential for a large part of the formalism. Let G be a compact matrix quantum group. The algebra of “polynomial functions” on G will be denoted by A. The group structure on G is determined by the comultiplication φ : A → A ⊗ A, the counit : A → C and the antipode κ : A → A. The result of an (n − 1)-fold comultiplication of an element a ∈ A will be symbolically written as a(1) ⊗ . . . ⊗ a(n) . The adjoint action of G on itself will be denoted by ad : A → A ⊗ A. Explicitly, this map is given by ad(a) = a(2) ⊗ κ(a(1) )a(3) .

(2.1)

Let Γ be a first-order differential calculus [14] over G and let X ⊕ ∧k Γ Γ∧ = k≥0

be the universal differential envelope [4, Appendix B] of Γ. Here, the space Γ∧k consists of kth order elements. For each k ≥ 0 let pk : Γ∧ → Γ∧k be the corresponding projection (we shall use the same symbols for projection operators associated to an arbitrary graded algebra built over Γ). Further, let X ⊕ ⊗k Γ Γ⊗ = k≥0

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be the tensor bundle algebra over Γ. Here, Γ⊗k = Γ ⊗A · · ·A ⊗ Γ is the tensor product over A of k-copies of Γ. The algebra Γ∧ can be obtained from Γ⊗ by factorizing through the ideal S ∧ generated by elements Q ∈ Γ⊗2 of the form X dai ⊗A dbi Q= i

where ai , bi ∈ A satisfy

X

ai dbi = 0 .

i

Let us assume that Γ is left-covariant. Let `Γ : Γ → A ⊗ Γ be the corresponding left action of G on Γ. We shall denote by Γinv the space of left-invariant elements of Γ. In other words Γinv = ϑ ∈ Γ: `Γ (ϑ) = 1 ⊗ ϑ . Further, R ⊆ ker() will be the right A-ideal which canonically, in the sense of [14], corresponds to Γ. The map π : A → Γinv given by π(a) = κ(a(1) )da(2)

(2.2)

is surjective and ker(π) = C⊕R. Because of this there exists a natural isomorphism Γinv ↔ ker()/R . The above isomorphism induces a right A-module structure on Γinv , which will be denoted by ◦. Explicitly, π(a) ◦ b = π ab − (a)b , (2.3) for each a, b ∈ A. The maps φ and `Γ admit common extensions to homomorphisms ⊗ ∧ ∧ ⊗ ⊗ ∧ ⊗ `∧ Γ : Γ → A ⊗ Γ and `Γ : Γ → A ⊗ Γ (left actions of G on Γ and Γ ). ⊗k The tensor product of k-copies of Γinv will be denoted by Γinv . The tensor algebra over Γinv will be denoted by Γ⊗ inv . It is naturally isomorphic to the space of left-invariant elements of Γ⊗ . The subalgebra of left-invariant elements of Γ∧ will be denoted by Γ∧ inv . This the space of left-invariant kth algebra is naturally graded. We shall denote by Γ∧k inv be the canonical projection map [14] onto leftorder elements. Let πinv : Γ∧ → Γ∧ inv invariant elements. In the framework of the canonical identification Γ∧ ↔ A ⊗ Γ∧ inv we have πinv ↔ ⊗ id. The following natural isomorphism holds ⊗ ∧ Γ∧ inv = Γinv /Sinv . ⊗2 ∧ is the ideal in Γ⊗ Here Sinv inv , generated by elements q ∈ Γinv of the form

q = π(a(1) ) ⊗ π(a(2) ) , where a ∈ R. This space is in fact the left-invariant part of the ideal S ∧ .

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The right A-module structure ◦ can be uniquely extended from Γinv to Γ∧,⊗ inv , such that 1 ◦ a = (a)1 (ϑη) ◦ a = (ϑ ◦ a(1) )(η ◦ a(2) )

(2.4) (2.5)

for each ϑ, η ∈ Γ⊗,∧ inv and a ∈ A. Explicitly, ◦ is given by ϑ ◦ a = κ(a(1) )ϑa(2) .

(2.6)

∧ The algebra Γ∧ inv ⊆ Γ is d-invariant. The following identities hold

d(ϑ ◦ a) = d(ϑ) ◦ a − π(a(1) )(ϑ ◦ a(2) ) + (−1)∂ϑ (ϑ ◦ a(1) )π(a(2) ) dπ(a) = −π(a(1) )π(a(2) ) .

(2.7) (2.8)

If Γ is ∗ -covariant then the ∗ -involution ∗ : Γ → Γ is naturally extendible to Γ∧ and Γ⊗ (such that (ϑη)∗ = (−1)∂ϑ∂η η ∗ ϑ∗ for each ϑ, η ∈ Γ∧,⊗ ). The maps `∧,⊗ are Γ ∧,⊗ ⊆ Γ are *-invariant. We have hermitian, in a natural manner. Algebras Γ∧,⊗ inv (ϑ ◦ a)∗ = ϑ∗ ◦ κ(a)∗

(2.9)

for each a ∈ A and ϑ ∈ Γ∧,⊗ inv . ∗ Explicitly, the -involution on Γinv is determined by π(a)∗ = −π [κ(a)∗ ] .

(2.10)

Let us now assume that the calculus Γ is bicovariant, and let ℘Γ : Γ → Γ⊗A be the right action of G on Γ. Maps φ and ℘Γ admit common extensions to homomorphisms ⊗ ∧,⊗ → Γ∧,⊗ ⊗ A (right actions of G on corresponding algebras). Let ℘∧ Γ , ℘Γ : Γ $ : Γinv → Γinv ⊗ A be the adjoint action of G on Γinv . The space Γinv is rightinvariant, that is ℘Γ (Γinv ) ⊆ Γinv ⊗ A. We have $ = ℘Γ Γinv . Explicitly, $π = (π ⊗ id)ad .

(2.11)

We shall denote by $⊗ , $⊗k , $∧ and $∧k the adjoint actions of G on the cor∧ responding spaces (coinciding with the corresponding restrictions of ℘⊗ Γ and ℘Γ ). The coproduct map φ : A → A ⊗ A admits the unique extension to the homob Γ∧ of graded-differential algebras. We have morphism φb : Γ∧ → Γ∧ ⊗ b φ(ϑ) = `Γ (ϑ) + ℘Γ (ϑ) ,

(2.12)

∧ b ∧ b ∧ ) ⊆ Γ∧ ⊗ b Γ∧ . Let $: b Γ∧ for each ϑ ∈ Γ. Further, we have φ(Γ inv inv inv → Γinv ⊗ Γ be the corresponding restriction. Explicitly,

$(ϑ) b = 1 ⊗ ϑ + $(ϑ) ,

(2.13)

for each ϑ ∈ Γinv . b $ b and all the introduced adjoint If Γ is a bicovariant ∗ -calculus then the maps φ, and right actions are hermitian, in a natural manner.

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3. Quantum Principal Bundles and the Associated Calculi The aim of this section is twofold. At first, we shall define quantum principal bundles, and briefly describe a geometrical background for this definition. Then, an appropriate differential calculus over quantum principal bundles will be introduced and analyzed. In particular, besides the main algebra consisting of “differential forms” on the bundle, we shall introduce and analyze algebras of “verticalized” and “horizontal” differential forms. Finally, an algebra representing differential forms on the base manifold will be defined. Let us consider a quantum space M , formally represented by a (unital) ∗ -algebra V. At the geometrical level, the elements of V play the role of appropriate “functions” on this space. Definition 3.1. A quantum principal G-bundle over M is a triplet of the form P = (B, i, F ), where B is a (unital) ∗ -algebra while i : V → B and F : B → B ⊗ A are unital ∗ -homomorphisms such that (qpb1) The map i : V → B is injective and b ∈ i(V)

⇔

F (b) = b ⊗ 1

for each b ∈ B. (qpb2) The following identities hold id = (id ⊗ )F (id ⊗ φ)F = (F ⊗ id)F .

(3.1) (3.2)

(qpb3) A linear map X : B ⊗ B → B ⊗ A defined by X(q ⊗ b) = qF (b) is surjective. The elements of B are interpretable as appropriate “functions” on the quantum space P . The map F plays the role of the dualized right action of G on P . Condition (qpb2) justifies this interpretation. The map i: V → B plays the role of the dualized “projection” of P on the base manifold M . Condition (qpb1) says that M is identificable with the corresponding “orbit space” for the right action. Accordingly, the elements of V will be identified with their images in i(V). Finally, condition (qpb3) is an effective quantum counterpart of the classical requirement that the action of G on P is free. It is easy to see that this condition can be equivalently formulated as (qpb4) For each a ∈ A there exist elements bk , qk ∈ B such that 1⊗a=

X k

qk F (bk ) .

(3.3)

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All these conditions explicitly figure (modulo the ∗ -structure) in definition of the concept of a Hopf–Galois extension [11]. However, the Hopf–Galois extensions include in addition the following axiom: (qpb5) The kernel of the map X: B ⊗ B → B ⊗ A coincides with the submodule n o N = gen b ⊗ i(f )q − bi(f ) ⊗ q where b, q ∈ B and f ∈ V. Conditions (qpb3) and (qpb5) say that the projected map X : B ⊗V B → B ⊗ A is bijective. Finally, let us mention that although the formulation of this paper does not assume (qpb5), it is possible to prove [5], using the representation theory [13], that if the structure group is compact then (qpb5) automatically holds. We now pass to questions related to differential calculus on quantum principal bundles. Let P = (B, i, F ) be a quantum principal G-bundle over M . Let us fix a bicovariant first-order ∗ -calculus Γ over G and let us consider a graded vector space ∧ ver(P ) = B ⊗ Γ∧ inv (the grading is induced from Γinv ). Lemma 3.1. (i) The formulas (q ⊗ η)(b ⊗ ϑ) =

X

qbk ⊗ (η ◦ ck )ϑ

(3.4)

b∗k ⊗ (ϑ∗ ◦ c∗k )

(3.5)

k

(b ⊗ ϑ)∗ =

X k

dv (b ⊗ ϑ) = b ⊗ dϑ +

X

bk ⊗ π(ck )ϑ

(3.6)

k

where F (b) = Σk bk ⊗ ck , determine the structure of a graded-differential ∗ -algebra on ver(P ). (ii) As a diferential algebra, ver(P ) is generated by B = ver0 (P ). Proof. Let us first check the associativity of the introduced product. Applying (3.4) and (2.5) and elementary properties of F we obtain X f bk ⊗ (ζ ◦ ck )ϑ (q ⊗ η) (f ⊗ ζ)(b ⊗ ϑ) (q ⊗ η) = k

=

X kl

=

X

f bk ql ⊗

(ζ ◦ ck )ϑ ◦ dl η

(1) (2) f bk ql ⊗ ζ ◦ (ck dl ) (ϑ ◦ dl )η

kl

= (f ⊗ ζ)

X l

bql ⊗ (ϑ ◦ dl )η

= (f ⊗ ζ) (b ⊗ ϑ)(q ⊗ η) , where F (q) =

P

l ql

⊗ dl .

539

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Evidently, ver(P ) is a unital algebra, with the unity 1 ⊗ 1. Now we prove that (3.5) determines a ∗ -algebra structure on ver(P ). We have ∗ ∗ X ∗ bk ⊗ (ϑ∗ ◦ c∗k ) (b ⊗ ϑ)∗ = k

=

X

bk ⊗ (ϑ∗ ◦ ck

(2)∗ ∗

(1)

) ◦ ck

k

=

X

(2) (1) bk ⊗ ϑ ◦ κ−1 (ck ) ◦ ck

k

=

X

(1)

bk ⊗ ϑ ◦ κ−1 (ck )ck (2)

k

= b ⊗ ϑ. Thus, ∗ is involutive. Further, ∗ ∗ X qbk ⊗ (η ◦ ck )ϑ (q ⊗ η)(b ⊗ ϑ) = k

=

X

(2) ∗ (1) (ql bk )∗ ⊗ (η ◦ ck )ϑ ◦ (dl ck )∗

kl

= (−1)∂ϑ∂η

X

(1)∗ (1)∗ ∗ (3) (2)∗ (2)∗ b∗k ql∗ ⊗ ϑ∗ ◦ ck dl η ◦ κ(ck )∗ ck dl

kl

= (−1)∂ϑ∂η

X

b∗k ql∗ ⊗ (ϑ∗ ◦ c∗k dl

(1)∗

)(η ∗ ◦ dl

(2)∗

)

kl

= (−1)∂ϑ∂η

X b∗k ⊗ (ϑ∗ ◦ c∗k ) ql∗ ⊗ (η ∗ ◦ d∗l ) kl

= (−1)∂ϑ∂η (b ⊗ ϑ)∗ (q ⊗ η)∗ . Let us check that (3.6) defines a hermitian differential on the ∗ -algebra ver(P ). We compute ! X bk ⊗ π(ck )ϑ d2v (b ⊗ ϑ) = dv b ⊗ dϑ + k

=

X

bk ⊗ π(ck )dϑ +

k

X

(1)

(2)

bk ⊗ π(ck )π(ck )ϑ +

k

X

bk ⊗ d π(ck )ϑ

k

= 0, according to (2.8). Furthermore, X dv qbk ⊗ (η ◦ ck )ϑ dv (q ⊗ η)(b ⊗ ϑ) = k

=

X k

X (1) (2) qbk ⊗ d (η ◦ ck )ϑ + ql bk ⊗ π(dl ck )(η ◦ ck )ϑ kl

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540

=

X

qbk ⊗ (d(η) ◦ ck )ϑ −

k

+ (−1)∂η +

X

X

X

(1)

(2)

qbk ⊗ π(ck )(η ◦ ck )ϑ

k

(1) (2) qbk ⊗ (η ◦ ck )π(ck )ϑ + (η ◦ ck )dϑ

k

(1)

ql bk ⊗ π(dl ) ◦ ck

(2)

(η ◦ ck )ϑ

kl

+

X

(1)

(2)

qbk ⊗ π(ck )(η ◦ ck )ϑ

k

= (q ⊗ dη)(b ⊗ ϑ) + (−1)∂η (q ⊗ η) + (−1)∂η (q ⊗ η)(b ⊗ dϑ) +

X

X

bk ⊗ π(ck )ϑ

k

ql ⊗ π(dl )η (b ⊗ ϑ)

l

= dv (q ⊗ η) (b ⊗ ϑ) + (−1)∂η (q ⊗ η)dv (b ⊗ ϑ) .

Here, we have used (2.3), (2.5), (2.7) and the main properties of F . Finally, X X ∗ (1)∗ (2)∗ bk ⊗ d(ϑ∗ ◦ c∗k ) + b∗k ⊗ π(ck )(ϑ∗ ◦ ck ) dv (b ⊗ ϑ)∗ = k

=

X

k

b∗k

∗

⊗ (dϑ ) ◦

k

+ (−1)∂ϑ

X

c∗k

−

X

b∗k ⊗ π(ck

(1)∗

)(ϑ∗ ◦ ck

k

b∗k ⊗ (ϑ∗ ◦ ck

(1)∗

k

= (b ⊗ dϑ)∗ + (−1)∂ϑ

X

(2)∗

)π(ck

)+

= (b ⊗ dϑ)∗ − (−1)∂ϑ

X k

= (b ⊗ dϑ)∗ + (−1)∂ϑ = (b ⊗ dϑ) +

X

X

X

)

b∗k ⊗ π(ck

(1)∗

)(ϑ∗ ◦ ck

(2)∗

)

k

b∗k ⊗ (ϑ∗ ◦ ck

)π(ck

b∗k ⊗ (ϑ∗ ◦ ck

(3) (2)∗ ) π κ(ck )∗ ◦ ck

(1)∗

k

∗

(2)∗

(1)∗

(2)∗

)

(2) (1)∗ b∗k ⊗ ϑ∗ π(ck )∗ ◦ ck

k

b∗k

(2) ∗ (1)∗ ⊗ π(ck )ϑ ◦ ck

k

∗ = dv (b ⊗ ϑ) .

Hence, dv is a hermitian differential. To prove (ii) it is sufficient to check that elements of the form qdv (b) linearly generate ver1 (P ). However, this directly follows from property (qpb4) in the definition of quantum principal bundles. In the following it will be assumed that ver(P ) is endowed with the constructed graded-differential ∗ -algebra structure. The elements of ver(P ) are interpretable as verticalized differential forms on the bundle. In classical geometry, these entities are obtained by restricting the domain of differential forms (on the bundle) to the Lie algebra of vertical vector fields.

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

541

Lemma 3.2. There exists the unique homomorphism b Γ∧ Fbv : ver(P ) → ver(P ) ⊗ of (graded) differential algebras extending the map F . We have b Fbv . (Fbv ⊗ id)Fbv = (id ⊗ φ)

(3.7)

The map Fbv is hermitian, in the sense that Fbv ∗ = (∗ ⊗ ∗)Fbv .

(3.8)

Proof. According to (ii) of the previous lemma, the map Fbv is unique, if exists. b Γ∧ by Let us define a linear map Fbv : ver(P ) → ver(P ) ⊗ Fbv (b ⊗ ϑ) =

X

bk ⊗ ϑl ⊗ ck wl

kl

where F (b) = Σk bk ⊗ ck and $(ϑ) b = Σl ϑl ⊗ wl . It is easy to see that such defined b map Fv is a differential algebra homomorphism. Identities (3.7) and (3.8) directly follow form the fact that ver(P ) is generated by B, as well as from property (3.2) and the hermicity of dv respectively. Let us consider a ∗ -homomorphism Fv : ver(P ) → ver(P ) ⊗ A given by Fv = (id ⊗ p0 )Fbv . This map extends the action F . It is interpretable as the (dualized) right action of G on verticalized forms. The following identities hold: (Fv ⊗ id)Fv = (id ⊗ φ)Fv (id ⊗ )Fv = id (dv ⊗ id)Fv = Fv dv .

(3.9) (3.10) (3.11)

The first two identities justify the interpretation of Fv as an action of G. The last identity says that the differential dv is right-covariant. So far about verticalized differential forms. We shall assume that a complete differential calculus over the bundle P is based on a graded-differential ∗ -algebra Ω(P ) possessing the following properties (diff 1) As a differential algebra, Ω(P ) is generated by B = Ω0 (P ). (diff 2) The map F : B → B ⊗ A is extendible to a homomorphism b Γ∧ Fb : Ω(P ) → Ω(P ) ⊗ of graded-differential algebras.

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Let us fix a graded-differential ∗ -algebra Ω(P ) such that the above properties hold. The elements of Ω(P ) will play the role of differential forms on P . It is easy to see that the map Fb is uniquely determined. Lemma 3.3. We have b Fb (Fb ⊗ id)Fb = (id ⊗ φ)

(3.12)

Fb ∗ = (∗ ⊗ ∗)Fb .

(3.13)

Proof. Both identities directly follow from similar properties of F , and from properties (diff 1/2). The formula

F ∧ = (id ⊗ p0 )Fb

(3.14)

defines a ∗ -homomorphism F ∧ : Ω(P ) → Ω(P ) ⊗ A extending the action F . The following identities hold: (F ∧ ⊗ id)F ∧ = (id ⊗ φ)F ∧

(3.15)

(id ⊗ )F ∧ = id

(3.16)

(d ⊗ id)F ∧ = F ∧ d .

(3.17)

The map F ∧ is interpretable as the (dualized) right action of G on differential forms. As a homomorphism between algebras, F ∧ is completely determined by (3.17), and by the fact that it extends F . Now, a very important algebra representing horizontal forms will be introduced in the game. Intuitively speaking, horizontal forms can be characterized as those elements of Ω(P ) possessing “trivial” differential properties along vertical fibers. Definition 3.2. The elements of the graded ∗ -subalgebra hor(P ) = Fb−1 Ω(P ) ⊗ A of Ω(P ) are called horizontal forms. Evidently, B = hor0 (P ). Lemma 3.4. The algebra hor(P ) is F ∧ -invariant. In other words F ∧ hor(P ) ⊆ hor(P ) ⊗ A .

(3.18)

b Fb (ϕ) = (id ⊗ φ)F ∧ (ϕ) Proof. If ϕ ∈ hor(P ) then (Fb ⊗ id)Fb (ϕ) = (id ⊗ φ) ∧ b belongs to Ω(P ) ⊗ A ⊗ A. Hence, F (ϕ) = F (ϕ) ∈ hor(P ) ⊗ A. The following technical lemma will be helpful in various considerations. Lemma 3.5. Let us consider a homogeneous element w ∈ Ωn (P ). Let 0 ≤ k ≤ n be an integer such that (id ⊗ pl )Fb (w) = 0 for each l > k. Then there exist horizontal forms ϕ1 , . . . , ϕm ∈ horn−k (P ) and elements ϑ1 , . . . , ϑm ∈ Γ∧k inv such that (id ⊗ pk )Fb (w) =

m X i=1

F ∧ (ϕi )ϑi .

(3.19)

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Proof. The statement is non-trivial only if (id ⊗ pk )Fb (w) 6= 0. We have X ξij ⊗ aij ϑi , (id ⊗ pk )Fb (w) =

543

(3.20)

ij n−k (P ) and where ϑi ∈ Γ∧k inv are some some linearly independent elements, ξij ∈ Ω aij ∈ A. Applying (2.13), (3.12) and (3.20), and the definition of k we find X Fb (ξij ) ⊗ aij ϑi = (Fb ⊗ pk )Fb (w) ij

= (id2 ⊗ pk ) =

X

X

b ij ϑi ) ξij ⊗ φ(a

ij (1)

(2)

ξij ⊗ aij ⊗ aij ϑi .

ij

Acting by id2 ⊗ πinv on both sides of the above equality we obtain X X Fb (ϕi ) ⊗ ϑi = ξij ⊗ aij ⊗ ϑi , i

(3.21)

ij

where ϕi = Σj ξij (aij ). In other words Fb (ϕi ) =

X

ξij ⊗ aij ,

(3.22)

j

and in particular ϕi ∈ horn−k (P ). Finally, combining (3.20) and (3.22) we conclude that (3.19) holds. We are going to construct a quantum analog for the “verticalizing” homomorphism (in classical geometry, induced by restricting the domain of differential forms on vertical vector fields on the bundle). Proposition 3.6. There exists the unique (graded) differential algebra homomorphism πv : Ω(P ) → ver(P ) reducing to the identity map on B. The map πv is surjective and hermitian. Moreover , Fbv πv = (πv ⊗ id)Fb

(3.23) ∧

Fv πv = (πv ⊗ id)F .

(3.24)

Proof. Let us define a linear (grade-preserving) map πv : Ω(P ) → ver(P ) by requiring πv (w) = (id ⊗ πinv pk )Fb (w) for each w ∈ Ωk (P ). Obviously, πv is reduced to the identity on B. Let us prove that πv is a differential algebra homomorphism. For given forms w ∈ Ωk (P ) and u ∈ Ωl (P ) let us choose elements bi , qj ∈ B and ϑi , ηj ∈ Γ∧k,l inv such that X X F (bi )ϑi (id ⊗ pl )Fb(u) = F (qj )ηj , (id ⊗ pk )Fb (w) = i

in accordance with the previous lemma.

j

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We then have πv (w) =

X

bi ⊗ ϑi

πv (u) =

i

X

qj ⊗ ηj .

j

A direct computation now gives πv (wu) = (id ⊗ πinv pk+l )Fb (wu) X (id ⊗ πinv ) [(F (bi )ϑi ) (F (qj )ηj )] = ij

=

X

(id ⊗ πinv )

i Xh (1) (2) bir qjs ⊗ cir djs (ϑi ◦ djs )ηj rs

ij

=

X

bi qjs ⊗ (ϑi ◦ djs )ηj

ijs

=

" X

 # X bi ⊗ ϑi  qj ⊗ ηj 

i

j

= πv (w)πv (u) , where F (bi ) = Σr bir ⊗ cir and F (qj ) = Σs qjs ⊗ djs . Further, πv d(w) = (id ⊗ πinv pk+1 )dFb (w) " # X F (bi )ϑi = (id ⊗ πinv d) i

= (id ⊗ πinv ) =

X i

= dv

" X

# bir ⊗ cir dϑi + bir ⊗ d(cir )ϑi

ir

bi ⊗ dϑi +

"

X

bir ⊗ π(cir )ϑi

ir

X

# bi ⊗ ϑi

i

= dv πv (w) . Consequently, πv is a homomorphism of differential algebras. The map πv is hermitian, because differentials on Ω(P ) and ver(P ) are hermitian, and the differential algebra Ω(P ) is generated by B. To prove (3.23) it is sufficient to observe that both its sides are differential algebra homomorphisms coinciding with F on B. Finally, (3.24) follows from (3.23), and definitions of F ∧ and Fv . Let us now consider a sequence π

0 → hor1 (P ) ,→ Ω1 (P ) →v ver1 (P ) → 0 of natural homomorphisms of *-B-bimodules.

(3.25)

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

545

Lemma 3.7. The above sequence is exact.

Proof. Clearly, hor1 (P ) ⊆ ker(πv ) ∩ Ω1 (P ) and πv Ω1 (P ) = ver1 (P ). For each w ∈ Ω1 (P ) we have X F (bi )ϑi , (id ⊗ p1 )Fb (w) = i

for some bi ∈ B and ϑi ∈ Γinv , according to Lemma 3.5. This implies X bi ⊗ ϑi . πv (w) = i

Consequently if w ∈ ker(πv ) then (id ⊗ p1 )Fb (w) = 0, and hence w ∈ hor1 (P ).

If a differential calculus on the bundle P is given, then it is possible to construct a natural differential calculus on the base space M . This calculus is based on a graded-differential ∗ -subalgebra Ω(M ) ⊆ Ω(P ) consisting of right-invariant horizontal forms. Equivalently, n o Ω(M ) = w ∈ Ω(P ): Fb(w) = w ⊗ 1 . In a special case when the group G is “connected” in the sense that only scalar elements of A are anihilated by the differential map, the algebra Ω(M ) can be described as Ω(M ) = d−1 hor(P ) ∩ hor(P ) . The differential algebra Ω(M ) is generally not generated by its 0-order subalgebra Ω0 (M ) = V, in contrast to Ω(P ). However, property (diff 1) is not essential for developing a large part of the formalism. Furthermore, it turns out that the algebra hor(P ) is generally not generated by B and hor1 (P ). 4. The Formalism of Connections In this section a general theory of connections on quantum principal bundles will be presented. At first, quantum analogs of pseudotensorial forms will be defined. Let V be a vector space, and let v: V → V ⊗ A be a representation of G in this space. Let ψ(v, P ) be the space of all linear maps ζ : V → Ω(P ) such that the diagram ζ V  )  −−−−−−−−−−−−−−→ Ω(P    ∧  (4.1) v F y y V ⊗ A −−−−−−−−−−−−−−→ Ω(P ) ⊗ A ζ ⊗ id is commutative (intertwiners between v and F ∧ ). The space ψ(v, P ) is naturally graded X ⊕ k ψ (v, P ) , (4.2) ψ(v, P ) = k≥0

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where ψ k (v, P ) consists of maps with values in Ωk (P ). The elements of ψ k (v, P ) can be interpreted as pseudotensorial k-forms on P with values in the dual space V ∗ . Further, ψ(v, P ) is closed with respect to compositions with d: Ω(P ) → Ω(P ). It is also a module, in natural manner, over the subalgebra consisting of right-invariant forms. Let X ⊕ k τ (v, P ) (4.3) τ (v, P ) = k≥0

be the graded subspace of ψ(v, P ) consisting of pseudotensorial forms having the values in hor(P ). The elements of τ (v, P ) are interpretable as tensorial forms on P with values in V ∗ . The space τ (v, P ) is a module over Ω(M ). If the space V is infinite-dimensional and if Ω(P ) possesses infinitely non-zero components then, generally, sums figuring in (4.2) and (4.3) will be “larger” than standard direct sums of spaces (and should be interpreted in the appropriate way). Forms ϕ ∈ τ (v, P ) can be also defined by the following equality Fbϕ(ϑ) = (ϕ ⊗ id)v(ϑ) .

(4.4)

If the space V is endowed with an antilinear involution ∗ : V → V such that v∗ = (∗ ⊗ ∗)v then the formula ϕ∗ (ϑ) = ϕ(ϑ∗ )∗

(4.5)

determines natural *-involutions on ψ(v, P ) and τ (v, P ). For the purposes of this paper, the most important is the case V = Γinv , and v = $. In this case we shall write ψ(P ) = ψ($, P ) and τ (P ) = τ ($, P ). We pass to the definition of connection forms. Definition 4.1. A connection on P is a hermitian map ω ∈ ψ 1 (P ) satisfying πv ω(ϑ) = 1 ⊗ ϑ

(4.6)

for each ϑ ∈ Γinv . The above condition corresponds to the classical requirement that connection forms (understood as lie(G)-valued pseudotensorial ad-type 1-forms) map fundamental vector fields into their generators. Theorem 4.1. Every quantum principal bundle P admits at least one connection. Proof. Let us consider the space W = πv−1 (Γinv ) ∩ Ω1 (P ). By definition this space is invariant under ∗ and F ∧ , and πv (W ) = Γinv . We can write X ⊕ Wα W = α∈T α

where W are corresponding multiple irreducible subspaces (the notation is explained in Appendix B). Similarly X ⊕ α Γinv . Γinv = α∈T

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

547

The following decompositions hold W α ↔ Mor(α, F ∧ ) ⊗ Cn

n Γα inv ↔ Mor(α, $) ⊗ C

where nα is the dimension of α. Further, πv (W α ) = Γα inv for each α ∈ T . In terms of the above identifications the restriction map πv : W α → Γα inv is given by πv µ ⊗ x = πv µ ⊗ x . This map is surjective. Let τ α : Mor(α, $) → Mor(α, F ∧ ) be a left inverse of πv W α . Let τ : Γinv → W be a map defined by X ⊕ τα τ= α∈T α where τα : Γα are given by τα = τ α ⊗ id. By construction, τ intertwines inv → W ∧ $ and F and satisfies πv τ (ϑ) = ϑ for each ϑ ∈ Γinv . Without a lack of generality we can assume that τ is hermitian (if not, we can consider another intertwiner ∗τ ∗ : Γinv → W and redefine τ 7→ (τ + ∗τ ∗)/2). Finally, composing τ and the inclusion map W ,→ Ω(P ) we obtain a connection on P .

Let cn(P ) be the set of all connections on P . This is a real affine subspace of ψ (P ). The corresponding vector space consists of hermitian tensorial 1-forms. Connections can be described in a different, more concise, but equivalent manner, using an algebraic condition which is a symbiosis of the verticalization condition (4.6) and the pseudotensoriality property. 1

Lemma 4.2. A first-order linear map ω : Γinv → Ω(P ) is a connection on P iff ω(ϑ∗ ) = ω(ϑ)∗

(4.7)

Fbω(ϑ) = (ω ⊗ id)$(ϑ) + 1 ⊗ ϑ ,

(4.8)

for each ϑ ∈ Γinv . Proof. It is clear that the above-listed properties imply that ω is a connection on P . Conversely, let us consider an arbitrary ω ∈ cn(P ). Then the pseudotensoriality property and Lemma 3.5 imply that for each ϑ ∈ Γinv we have X F (bk )ϑk , Fbω(ϑ) = (ω ⊗ id)$(ϑ) + k

for some bk ∈ B and ϑk ∈ Γinv . Properties (3.1) and (4.6), and the definition of πv imply 1 ⊗ ϑ = Σk bk ⊗ ϑk . Hence (4.8) holds. Every connection ω canonically gives rise to a splitting of the sequence (3.25), understood as a sequence of left B-modules. Indeed, the map µω : ver1 (P ) → Ω1 (P ) defined by µω (b ⊗ ϑ) = bω(ϑ)

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splits the mentioned sequence. Moreover, the map µω intertwines the corresponding right actions. Conversely, we can define connections via the associated splitting µω . For each ω ∈ cn(P ) let ω ⊗ : Γ⊗ inv → Ω(P ) be the corresponding unital multiplicative extension. Two particularly interesting classes of connections naturally appear in deeper considerations. These classes consist of connections possessing some additional properties that will be called multiplicativity and regularity. Definition 4.2. A connection ω is called multiplicative iff ωπ(a(1) )ωπ(a(2) ) = 0

(4.9)

∧ = 0. for each a ∈ R. Equivalently, ω is multiplicative iff ω ⊗ Sinv

If ω is multiplicative then there exists the unique unital multiplicative extension ⊗ ∧ through Sinv ω ∧ : Γ∧ inv → Ω(P ). This map can be obtained by by factorizing ω ∧ ⊗ (both ω and ω are *-preserving). Condition (4.9) gives a quadratic constraint in the space cn(P ). It is worth noticing that in the classical theory all connections are multiplicative. Definition 4.3. A connection ω is called regular iff X ϕk ω(ϑ ◦ ck ) ω(ϑ)ϕ = (−1)∂ϕ

(4.10)

k

holds for each ϕ ∈ hor(P ) and ϑ ∈ Γinv . Here, F ∧ (ϕ) = Σk ϕk ⊗ ck . Equivalently, X ω ϑ ◦ κ−1 (ck ) ϕk . (4.11) ϕω(ϑ) = (−1)∂ϕ k

In particular, regular connections graded-commute with forms from Ω(M ). The most interesting situations arise when the calculus on the bundle admits connections that are both regular and multiplicative. Relations between regularity and multiplicativity will be treated in detail later — the discussions on this problematics begin after the proof of Proposition 4.9. Regular connections, if exist, form an affine subspace rcn(P ) ⊆ cn(P ). The corresponding vector space consists of hermitian forms ζ ∈ τ 1 (P ) satisfying X ζ ϑ ◦ κ−1 (ck ) ϕk , (4.12) ϕζ(ϑ) = (−1)∂ϕ k

or equivalently ζ(ϑ)ϕ = (−1)∂ϕ

X

ϕk ζ(ϑ ◦ ck )

(4.13)

k

for each ϕ ∈ hor(P ) and ϑ ∈ Γinv . It is important to mention that if ω is regular then the corresponding splitting µω is a splitting of ∗ -bimodules. If the theory is confined to bundles over classical manifolds and if Γ is the minimal admissible calculus then regular connections are precisely classical connections

549

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

(in the terminology of the previous paper). In particular, in the case of classical principal bundles all connections are regular. Let us consider an expression X ϕk ω(ϑ ◦ ck ) (4.14) lω (ϑ, ϕ) = ω(ϑ)ϕ − (−1)∂ϕ k

where ϕ ∈ hor(P ) and ϑ ∈ Γinv . The map lω measures the lack of regularity of ω. Lemma 4.3. We have Fb lω (ϑ, ϕ) =

X

lω (ϑj , ϕk ) ⊗ dj ck ,

(4.15)

jk

where $(ϑ) = Σj ϑj ⊗ dj . In particular , lω is a hor(P )-valued map. Proof. A direct computation gives X X ω(ϑj )ϕk ⊗ dj ck + (−1)∂ϕ ϕk ⊗ ϑck Fblω (ϑ, ϕ) = jk

− (−1)∂ϕ

X

k (3)

(1)

(2)

(4)

ϕk ω(ϑj ◦ ck ) ⊗ ck κ(ck )dj ck

kj

− (−1)∂ϕ =

X

X

(1)

(2)

ϕk ⊗ ck (ϑ ◦ ck )

k

lω (ϑj , ϕk ) ⊗ dj ck .

jk ⊗2 Let σ: Γ⊗2 inv → Γinv be the (left-invariant part of the) canonical braid operator [14]. This map is explicitly [4] given by X ϑk ⊗ (η ◦ ck ) (4.16) σ(η ⊗ ϑ) = k

where $(ϑ) = Σk ϑk ⊗ ck . Let mΩ be the multiplication map in Ω(P ). If ω ∈ rcn(P ) then n o n o mΩ ω ⊗ ϕ = (−1)∂ϕ mΩ ϕ ⊗ ω σ,

(4.17)

for each ϕ ∈ τ (P ). The above equality follows directly from (4.16), the tensoriality of ϕ, and the definition of regular connections. Let us fix a linear map δ: Γinv → Γinv ⊗Γinv such that $⊗2 δ = (δ ⊗ id)$ and such that if δ(ϑ) =

X

ϑ1k ⊗ ϑ2k

k

then d(ϑ) =

X k

ϑ1k ϑ2k ,

−δ(ϑ∗ ) =

X k

1∗ ϑ2∗ k ⊗ ϑk

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for each ϑ ∈ Γinv . Such maps will be called embedded differentials. For each ϑ ∈ Γinv there exists a ∈ ker() such that δ(ϑ) = −π(a(1) ) ⊗ π(a(2) ) π(a) = ϑ .

(4.18)

For given linear maps ϕ, η: Γinv → Ω(P ) let us define (as in [4]) new linear maps hϕ, ηi, [ϕ, η]: Γinv → Ω(P ) by hϕ, ηi = mΩ (ϕ ⊗ η)δ

(4.19)

[ϕ, η] = mΩ (ϕ ⊗ η)c> ,

(4.20)

where c> : Γinv → Γinv ⊗Γinv is the “transposed commutator” map [14], explicitly given by (4.21) c> = (id ⊗ π)$ . The same brackets can be used for linear maps defined on Γinv , with values in an arbitrary algebra. It is easy to see that if ϕ ∈ ψ i (P ) and η ∈ ψ j (P ) then hϕ, ηi, [ϕ, η] ∈ ψ i+j (P ) (the same holds for τ (P )). Further, hϕ, ηi∗ = −(−1)ij hη ∗ , ϕ∗ i ,

(4.22)

as directly follows from (4.5) and the hermicity of δ. For an arbitrary ω ∈ cn(P ) let us consider a map Rω : Γinv → Ω(P ) given by Rω = dω − hω, ωi .

(4.23)

Clearly, this is a pseudotensorial 2-form. Moreover, Lemma 4.4. We have Fb Rω (ϑ) = (Rω ⊗ id)$(ϑ)

(4.24)

for each ϑ ∈ Γinv . Proof. A direct computation gives Fb Rω (ϑ) = dFb ω(ϑ) + Fb ωπ(a(1) )Fbωπ(a(2) ) = d ωπ(a(2) ) ⊗ κ(a(1) )a(3) + 1 ⊗ π(a) + 1 ⊗ π(a(1) )π(a(2) ) + ωπ(a(2) )ωπ(a(5) ) ⊗ κ(a(1) )a(3) κ(a(4) )a(6) + ωπ(a(2) ) ⊗ κ(a(1) )a(3) π(a(4) ) − ωπ(a(3) ) ⊗ π(a(1) ) κ(a(2) )a(4)

= [dω − hω, ωi]π(a(2) ) ⊗ κ(a(1) )a(3) + ωπ(a(2) ) ⊗ κ(a(1) )da(3) − ωπ(a(2) ) ⊗ d κ(a(1) )a(3) − ωπ(a(4) ) ⊗ κ(a(1) )da(2) κ(a(3) )a(5) = (Rω ⊗ id)$(ϑ) .

551

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Here, it is assumed that a ∈ ker() satisfies (4.18) and we have applied (2.2), (2.8), (2.11) and (4.8). Definition 4.4. The constructed map Rω is called the curvature of ω. The curvature Rω implicitly depends on the choice of δ. This dependence disappears if ω is multiplicative. We are going to introduce the operator of covariant derivative. This operator will be first defined on a restricted domain consisting of horizontal forms. Later on, after constructing the horizontal projection operator, we shall extend the domain of the covariant derivative to the whole algebra Ω(P ). For each ω ∈ cn(P ) and ϕ ∈ hor(P ) let us define a new form X ϕk ωπ(ck ) , (4.25) Dω (ϕ) = dϕ − (−1)∂ϕ k

where F ∧ (ϕ) = Σk ϕk ⊗ ck . Lemma 4.5. We have Fb Dω (ϕ) =

X

Dω (ϕk ) ⊗ ck

(4.26)

k

for each ϕ ∈ hor(P ). In particular Dω hor(P ) ⊆ hor(P ) . Proof. We compute Fb Dω (ϕ) = Fb dϕ − (−1)∂ϕ

X

Fb (ϕk )Fbωπ(ck )

k

X X (1) (2) ϕk ⊗ ck − (−1)∂ϕ ϕk ⊗ ck π(ck ) =d k

− (−1)∂ϕ =

X

X

k (3) ϕk ωπ(ck )

k

dϕk ⊗ ck − (−1)∂ϕ

k

=

X

(1)

(2)

(4)

⊗ ck κ(ck )ck

X

(1)

(2)

ϕk ωπ(ck ) ⊗ ck

k

Dω (ϕk ) ⊗ ck .

k

Hence, hor(P ) is Dω -invariant.

Definition 4.5. The constructed map Dω : hor(P ) → hor(P ) is called the covariant derivative associated to ω. Proposition 4.6. (i) The diagram F∧ − − − − − − − −−−−−−−→ hor(P) ⊗ A hor(P )      Dω   Dω ⊗ id y y hor(P ) −−−−−−−−−−−−−−→ hor(P ) ⊗ A F∧ is commutative.

(4.27)

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(ii) We have Dω2 (ϕ) = −

X

i h (1) (2) ϕk dωπ(ck ) + ωπ(ck )ωπ(ck )

(4.28)

k

for each ϕ ∈ hor(P ). In particular , if ω is multiplicative then Dω2 (ϕ) = −

X

ϕk Rω π(ck ) .

(4.29)

k

(iii) If ω is regular then Dω (ϕψ) = Dω (ϕ)ψ + (−1)∂ϕ ϕDω (ψ) ∗

(4.30)

∗

Dω (ϕ ) = Dω (ϕ) ,

(4.31)

for each ϕ, ψ ∈ hor(P ). (iv) We have

(d − Dω ) Ω(M ) = {0}

(4.32)

for each ω ∈ cn(P ). Proof. Diagram (4.27) follows from identity (4.26). Property (iv) follows from definitions of Ω(M ) and Dω . For each ϕ ∈ hor(P ) we have Dω2 (ϕ)

X ∂ϕ = Dω dϕ − (−1) ϕk ωπ(ck ) = − (−1)

∂ϕ

X

k

dϕk ωπ(ck ) −

k

− (−1)∂ϕ =−

X

X

ϕk dωπ(ck )

k

X (3) (1) (2) (4) −dϕk ωπ(ck ) + (−1)∂ϕ ϕk ωπ(ck )ωπ ck κ(ck )ck k

(1) (2) ϕk dωπ(ck ) + ωπ(ck )ωπ(ck ) .

k

If ω is multiplicative then (4.23) and the definition of δ imply that (4.29) holds. Finally, let us assume that ω is regular. Then Dω (ϕψ) = (dϕ)ψ + (−1)∂ϕ ϕdψ X ϕk ψl ω π(ck ) ◦ dl + ϕk ψl (ck )ωπ(dl ) −(−1)∂ϕ+∂ψ kl

X X ∂ϕ ∂ϕ ∂ψ ϕk ωπ(ck ) ψ + (−1) ϕ dψ − (−1) ψl ωπ(dl ) = dϕ − (−1) k ∂ϕ

= Dω (ϕ)ψ + (−1)

l

ϕDω (ψ) ,

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

553

where Σl ψl ⊗ dl = F ∧ (ψ). Further, Dω (ϕ)∗ = d(ϕ∗ ) +

X

ωπ κ(ck )∗ ϕ∗k

k

= d(ϕ∗ ) + (−1)∂ϕ

X

(2) (1)∗ ϕ∗k ω π κ(ck )∗ ◦ ck

k

= d(ϕ∗ ) − (−1)∂ϕ

X

ϕ∗k ωπ(c∗k ) = Dω (ϕ∗ ) .

k

Hence, properties (iii) hold.

Actually, a connection ω is regular if and only if Dω satisfies the graded Leibniz rule. Because of (4.27), the space τ (P ) is closed under taking compositions with Dω . This fact enables us to define the covariant derivative (which will be denoted by the same symbol) as an operator acting in the space τ (P ). Proposition 4.7. We have Dω ϕ = dϕ − (−1)∂ϕ [ϕ, ω]

(4.33)

for each ω ∈ cn(P ) and ϕ ∈ τ (P ). Proof. This directly follows from the tensoriality of ϕ and from definitions of Dω and brackets [, ]. For a given ω ∈ cn(P ) let qω : ψ(P ) → ψ(P ) be a linear map defined by qω (ϕ) = hω, ϕi − (−1)∂ϕ hϕ, ωi − (−1)∂ϕ [ϕ, ω] .

(4.34)

Lemma 4.8. (i) We have Fb qω (ϕ)(ϑ) = (qω ⊗ id)F ∧ ϕ(ϑ)

(4.35)

for each ϕ ∈ τ (P ) and ϑ ∈ Γinv . In particular qω τ (P ) ⊆ τ (P ). (ii) If ω ∈ rcn(P ) then

qω τ (P ) = 0 .

Proof. We compute Fb qω (ϕ)(ϑ) = −Fb ωπ(a(1) )ϕπ(a(2) ) + (−1)∂ϕ Fb ϕπ(a(1) )ωπ(a(2) ) X Fb ϕ(ϑk )ωπ(ck ) − (−1)∂ϕ k

(4.36)

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= − ωπ(a(2) )ϕπ(a(3) ) ⊗ κ(a(1) )a(4) − (−1)∂ϕ ϕπ(a(3) ) ⊗ π(a(1) )κ(a(2) )a(4) + (−1)∂ϕ ϕπ(a(2) )ωπ(a3 ) ⊗ κ(a(1) )a(4) + (−1)∂ϕ ϕπ(a(2) ) ⊗ κ(a(1) )a(3) π(a(4) ) X (3) (1) (2) (4) (1) (2) ϕ(ϑk )ωπ(ck ) ⊗ ck κ(ck )ck + ϕ(ϑk ) ⊗ ck π(ck ) − (−1)∂ϕ k

o Xn hω, ϕi(ϑk ) ⊗ ck − (−1)∂ϕ hϕ, ωi(ϑk ) ⊗ ck = k

− (−1)∂ϕ

X

[ϕ, ω](ϑk ) ⊗ ck − (−1)∂ϕ

k

X

ϕ(ϑk ) ⊗ dck

k

− (−1)∂ϕ ϕπ(a(4) ) ⊗ κ(a(1) )(da(2) )κ(a(3) )a(5) + (−1)∂ϕ ϕπ(a(2) ) ⊗ κ(a(1) )da(3) X = qω (ϕ)(ϑk ) ⊗ ck . k

Here, a ∈ ker() satisfies (4.18), and Σk ϑk ⊗ ck = $(ϑ). Let us assume that ω is regular. Applying (2.3), (2.11) and (4.10), and the definition of brackets h, i and [, ] we obtain hω, ϕi(ϑ) = − ωπ(a(1) )ϕπ(a(2) )

= − (−1)∂ϕ ϕπ(a(3) )ω π(a(1) ) ◦ κ(a(2) )a(4) = − (−1)∂ϕ ϕπ(a(3) )ωπ a(1) − (a(1) )1 κ(a(2) )a(4)

= − (−1)∂ϕ ϕπ(a(1) )ωπ(a(2) ) + (−1)∂ϕ ϕπ(a(2) )ωπ κ(a(1) )a(3)

= (−1)∂ϕ hϕ, ωi(ϑ) + (−1)∂ϕ [ϕ, ω](ϑ) .

Hence, (4.36) holds.

The following proposition gives the quantum counterpart for the classical Bianchi identity. Proposition 4.9. We have (Dω − qω )(Rω ) = hω, hω, ωii − hhω, ωi, ωi

(4.37)

for each ω ∈ cn(P ). Proof. A direct computation gives (Dω − qω )(Rω ) = dRω − hω, Rω i + hRω , ωi = − dhω, ωi − hω, dω − hω, ωii + hdω − hω, ωi, ωi = hω, hω, ωii − hhω, ωi, ωi .

555

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

If the connection ω is multiplicative then the right-hand side of equality (4.37) vanishes. On the other hand, if ω is regular then the second summand on the left-hand side vanishes. It is important to point out that regular connections are not necessarily multiplicative. However, there exists a common obstruction to multiplicativity for all regular connections, so that if one regular connection is multiplicative, then every regular connection possesses the same property. This obstruction will be now analyzed in more details. In general, the lack of multiplicativity of a connection ω is measured by the map rω : R → Ω(P ) given by rω (a) = ωπ(a(1) )ωπ(a(2) ). Lemma 4.10 (i) The following identities hold rω κ(a)∗ = −rω (a)

(4.38)

πv rω (a) = 0

(4.39)

Fbrω (a) = F ∧ rω (a) = (rω ⊗ id)ad(a) .

(4.40)

In particular rω (a) ∈ hor (P ) for each a ∈ R. (ii) Let us assume that P admits regular connections. The map ω 7→ rω is constant on equivalence classes from the space cn(P )/rcn(P ). If ω ∈ rcn(P ) then X ϕk rω (ack ) (4.41) rω (a)ϕ = 2

k

for each a ∈ R and ϕ ∈ hor(P ). Furthermore, drω (a) = hω, ωiπ(a(1) )ωπ(a(2) ) − ωπ(a(1) )hω, ωiπ(a(2) ) .

(4.42)

Proof. A direct computation gives

rω (a)∗ = −ωπ(a(2) )∗ ωπ(a(1) )∗ = −ωπ κ(a(2) )∗ ωπ κ(a(1) )∗ = −rω κ(a)∗

and similarly

h ih i Fb rω (a) = (ω ⊗ id)$π(a(1) ) + 1 ⊗ π(a(1) ) (ω ⊗ id)$π(a(2) ) + 1 ⊗ π(a(2) ) = ωπ(a(2) )ωπ(a(3) ) ⊗ κ(a(1) )a(4) + ωπ(a(2) ) ⊗ κ(a(1) )a(3) π(a(4) ) − ωπ(a(3) ) ⊗ π(a(1) )κ(a(2) )a(4)

= rω (a(2) ) ⊗ κ(a(1) )a(3) + ωπ(a(2) ) ⊗ d κ(a(1) )a(3) = (rω ⊗ id)ad(a) . Let us assume that ω is regular. Then

rω+ϕ (a) = ωπ(a(1) )ωπ(a(2) ) − ϕπ(a(3) )ω π(a(1) ) ◦ κ(a(2) )a(4) + ϕπ(a(1) )ϕπ(a(2) ) + ϕπ(a(1) )ωπ(a(2) ) = ωπ(a(1) )ωπ(a(2) ) + ϕπ(a(1) )ϕπ(a(2) ) + ϕπ(a(2) )ωπ κ(a(1) )a(3) = ωπ(a(1) )ωπ(a(2) ) + ϕπ(a(1) )ϕπ(a(2) )

for each ϕ = ϕ∗ ∈ τ 1 (P ).

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Now if ζ = ζ ∗ ∈ τ 1 (P ) satisfies (4.13) then ϕπ(a(1) )ζπ(a(2) )+ζπ(a(1) )ϕπ(a(2) ) = 0, and in particular ζπ(a(1) )ζπ(a(2) ) = 0 for each a ∈ R. Hence, rω+ϕ+ζ = rω+ϕ . Further, applying (2.3), (4.23) and the tensoriality of the curvature we obtain drω (a) = Rω π(a(1) )ωπ(a(2) ) + hω, ωiπ(a(1) )ωπ(a(2) ) − ωπ(a(1) )Rω π(a(2) ) − ωπ(a(1) )hω, ωiπ(a(2) ) = Rω π(a(1) )ωπ(a(2) ) + hω, ωiπ(a(1) )ωπ(a(2) ) − Rω π(a(3) )ω π(a(1) ) ◦ κ(a(2) )a(4) − ωπ(a(1) )hω, ωiπ(a(2) ) = hω, ωiπ(a(1) )ωπ(a(2) ) − ωπ(a(1) )hω, ωiπ(a(2) ) + Rω π(a(2) )ωπ κ(a(1) )a(3) = hω, ωiπ(a(1) )ωπ(a(2) ) − ωπ(a(1) )hω, ωiπ(a(2) ) . Finally, rω (a)ϕ =

X k

=

X

(1) (2) ϕk ω π(a(1) ) ◦ ck ω π(a(2) ) ◦ ck (1) (2) ϕk ωπ a(1) − (a(1) )1 ck ωπ a(2) − (a(2) )1 ck

k

=

X k

− =

(1)

X

X

(2)

ϕk ωπ(a(1) ck )ωπ(a(2) ck ) −

X

(1)

(2)

(1)

(2)

ϕk ωπ(ack )ωπ(ck )

k

(1) (2) ϕk ωπ(ck )ωπ(ack )

+ (a)

k

X

ϕk ωπ(ck )ωπ(ck )

k

(1)

(2)

ϕk ωπ(a(1) ck )ωπ(a(2) ck ) =

X

k

ϕk rω (ack )

k

for each ϕ ∈ hor(P ). Here, we have used (4.10), and the fact that R ⊆ ker() is a right ideal. Let us assume that P admits regular connections, and let T(P ) be the left ideal in Ω(P ) generated by the space rω (R), for some ω ∈ rcn(P ). Lemma 4.11. (i) The following properties hold: T(P )∗ = T(P )

(4.43)

Fb T(P ) ⊆ T(P ) ⊗ Γ∧

(4.44)

πv T(P ) = {0}

(4.45)

dT(P ) ⊆ T(P ) . (ii) The space T(P ) is a two-sided ideal in Ω(P ).

(4.46)

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

557

Proof. Properties (4.44)–(4.45) directly follow from identities (4.39)–(4.40). Concerning (4.46), it follows from (4.42), and the following observations. At first, ∧2 ∧2 for each a ∈ A. Secondly ω ⊗ (Sinv ) = rω (R). δπ(a)+π(a(1) )⊗π(a(2) ) belongs to Sinv Thirdly, because of (4.11) and (4.40), rω (a)ω(ϑ) ∈ T(P ) for each a ∈ R and ϑ ∈ Γinv . Let us prove (ii). Because of the above inclusion and the fact that ω(Γinv ) and hor(P ) generate Ω(P ) (as follows from the fact that µω splits the sequence (3.25)) it is sufficient to check that rω (a)ϕ ∈ T(P ) for each a ∈ R and ϕ ∈ hor(P ). However, this follows from (4.41) in a straightforward way. Finally, (4.43) follows from (ii) and (4.38). The ideal T(P ) measures the lack of multiplicativity of regular connections. Let ω be a regular connection on P , and let us assume that T(P ) = {0}. Then ω is multiplicative and the map ω ∧ : Γ∧ inv → Ω(P ) possesses the following commutation properties with horizontal forms X ϕk ω ∧ (ϑ ◦ ck ) (4.47) ω ∧ (ϑ)ϕ = (−1)∂ϕ∂ϑ k

ϕω ∧ (ϑ) = (−1)∂ϕ∂ϑ

X

ω ∧ ϑ ◦ κ−1 (ck ) ϕk ,

(4.48)

k

as easily follows from (4.10) and (4.11). The formulas (ψ ⊗ η)(ϕ ⊗ ϑ) = (−1)∂ϕ∂η (ϕ ⊗ ϑ)∗ =

X

X

ψϕk ⊗ (η ◦ ck )ϑ

(4.49)

k

ϕ∗k ⊗ (ϑ∗ ◦ c∗k ) ,

(4.50)

k

determine the structure of a ∗ -algebra in the space vh(P ) = hor(P ) ⊗ Γ∧ inv . The proof is essentially the same as the proof of the ∗ -algebra properties for ver(P ) (actually ver(P ) is a ∗ -subalgebra of vh(P )). Here as usual, F ∧ (ϕ) = Σk ϕk ⊗ ck . The elements of vh(P ) are interpretable as “vertically-horizontally” decomposed differential forms on P . In the following considerations the space vh(P ) will be endowed with this (graded) ∗ -algebra structure. We shall now construct, starting from an arbitrary connection ω, an important isomorphism between the spaces Ω(P ) and vh(P ), extending the splitting µω . This isomorphism will be the base for the construction and the analysis of horizontal projection operators. Also, various interesting questions related to the structure of horizontal forms will be considered. Let us fix a splitting of the form ∧ ∧ Γ⊗ inv = Γinv ⊕ Sinv

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558

∧ in which Γ∧ inv is realized as a complement to the space Sinv , with the help of a ⊗ ∗ ∧ -preserving section ι: Γinv → Γinv (of the factor-projection map) intertwining the adjoint actions. Further, let us assume that the embedded differential δ is given by

δ(ϑ) = ιd(ϑ) . Finally, let us consider a linear map mω : vh(P ) → Ω(P ) given by mω (ϕ ⊗ ϑ) = ϕω ∧ (ϑ) ,

(4.51)

ω∧ = ω⊗ι .

(4.52)

where It is important to mention that the above definition of map ω ∧ reduces, for multiplicative connections, to the previously introduced multiplicative extension. In particular, if ω is multiplicative then mω is independent of the section ι. Further, if ι intertwines ◦-structures then (4.47)–(4.48) hold for each ω ∈ rcn(P ), independently of the multiplicativity property. By construction ω ∧ is hermitian and F ∧ ω ∧ = (ω ∧ ⊗ id)$∧ . Let Fvh : vh(P ) → vh(P ) ⊗ A be the product of actions F ∧ : hor(P ) → hor(P ) ⊗ A and $∧ . This map is a ∗ -homomorphism. Theorem 4.12. (i) The map mω is bijective. (ii) The diagram

is commutative.

mω vh(P  )  ) −−−−−−−−−−−−−−→ Ω(P    ∧  F Fvh  V y y vh(P ) ⊗ A −−−−−−−−−−−−−−→ Ω(P ) ⊗ A mω ⊗ id

(4.53)

(iii) If ω is regular and if T(P ) = {0} then mω is an isomorphism of ∗ -algebras. Proof. We shall first prove that mω is injective. Let us assume that w ∈ ker(mω ), and let w = Σi ϕi ⊗ ϑi where ϑi ∈ Γ∧ inv are homogeneous and linearly independent, and ϕi ∈ hor(P ) are non-zero elements. Let k be the maximum of degrees of elements ϑi . Then X 0 ∧ F (ϕi )ϑi , 0 = (id ⊗ pk )Fb mω (w) = i

according to (4.8), and definitions of F ∧ and hor(P ). The above summation is performed over the indexes i corresponding to kth order elements. This implies Σ0i ϕi ⊗ ϑi = 0, which is a contradiction. Hence, mω is injective. We shall now prove that mω is surjective. For each integer k ≥ 0 let us consider the space Ωk (P ) = (Fb )−1 Ω(P ) ⊗ Γ∧ k ,

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

where Γ∧ k =

X

⊕ ∧i

Γ

559

.

i≤k

In other words, w ∈ Ωk (P ) iff (id ⊗ pl )Fb(w) = 0 for each l > k. Clearly, Ω0 (P ) = hor(P ) and Ωk (P ) ⊆ Ωk+1 (P ) for each k ≥ 0. Further [ Ωk (P ) = Ω(P ) . k

We are going to prove, inductively, that the spaces Ωk (P ) are contained in the image of mω . Evidently, the statement is true for k = 0. Let us assume that it holds for some k. Let us assume that there exists an element w ∈ Ωk+1 (P ) Ωk (P ). Then we have, according to Lemma 3.5 X F ∧ (ϕi )ϑi , Fb (w) = ψ + i ∧ 1+k are linearly where ψ ∈ Ω(P ) ⊗ Γ∧ k and ϕi ∈ hor(P ) \ {0} while ϑi ∈ (Γinv ) independent elements. On the other hand, X X b ϕi ⊗ ϑi = ψ 0 + F ∧ (ϕi )ϑi F mω i 0

i

Γ∧ k.

where ψ ∈ Ω(P ) ⊗ Hence, w − mω (Σi ϕi ⊗ ϑi ) belongs to Ωk (P ) and therefore w ∈ mω vh(P ) . We conclude that mω is surjective. Thus, (i) holds. Intertwining property (4.53) directly follows from the definition of mω and from the right-covariance of ω ∧ . Finally, let us assume that T(P ) = {0} and let ω ∈ rcn(P ). Then ω is multiplicative and ω ∧ is a ∗ -homomorphism. Using (4.47) and (4.49)–(4.50) we obtain X (−1)∂ϕ∂η ψϕk ω ∧ (η ◦ ck )ϑ mω (ψ ⊗ η)(ϕ ⊗ ϑ) = k

X (−1)∂ϕ∂η ψϕk ω ∧ (η ◦ ck )ω ∧ (ϑ) = k

= ψω ∧ (η)ϕω ∧ (ϑ) = mω (ψ ⊗ η)mω (ϕ ⊗ ϑ) , and similarly

X ∗ ∧ ∗ ∗ ϕk ω (ϑ ◦ ck ) mω (ϕ ⊗ ϑ)∗ = k

= (−1)∂ϕ∂ϑ ω ∧ (ϑ∗ )ϕ∗ = (−1)∂ϕ∂ϑ ω ∧ (ϑ)∗ ϕ∗ ∗ = ϕω ∧ (ϑ) ∗ = mω (ϕ ⊗ ϑ) . In other words, mω is a ∗ -algebra isomorphism.

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The spaces Ωk (P ) introduced in the above proof form a filtration of Ω(P ), compatible with the graded-differential *-algebra structure. In particular X j ⊕ Ωk (P ) . Ωk (P ) = j≥0

For each k and ω ∈ cn(P ) the space Ωk (P ) is linearly spanned by elements of the form ϕω ∧ (ϑ), where ϕ ∈ hor(P ) and ϑ ∈ Γ∧i inv , with i ≤ k. Let 0(P ) be the graded-differential ∗ -algebra associated to the filtered algebra Ω(P ). The map q: vh(P ) → 0(P ) given by q(ϕ ⊗ ϑ) = ϕω ∧ (ϑ) + Ωk−1 (P ) , where ϑ ∈ Γ∧k inv , is bijective (and independent of the choice of ω and ι). Moreover, ∗ q is a -isomorphism, as follows from Lemma 4.3 and the horizontality of rω . The introduced filtration is compatible with the map Fb , in the sense that Fb Ωk (P ) ⊆ Ωk (P ) ⊗ Γ∧ k . In other words, Fb is factorizable through the filtration. The diagrams q q vh(P −−−−−−−−−→ 0(P vh(P  )  )  )  ) −−−−−−−−−→ 0(P      b    Fbvh  (4.54) dvh  F d y y y y b Γ∧ −−−−−−−−−→ 0(P ) ⊗ b Γ∧ vh(P ) −−−−−−−−−→ 0(P ) vh(P ) ⊗ q q ⊗ id describe the corresponding factorized maps in terms of vh(P ). The differential dvh : vh(P ) → vh(P ) is given by X ϕk ⊗ π(ck )ϑ + (−1)∂ϕ ϕ ⊗ d(ϑ) . dvh (ϕ ⊗ ϑ) = (−1)∂ϕ k

b Γ∧ is given by Similarly, Fbvh : vh(P ) → vh(P ) ⊗ b . Fbvh (ϕ ⊗ ϑ) = F ∧ (ϕ)$(ϑ) It is worth noticing that dvh ver(P ) = dv and Fbvh ver(P ) = Fbv . The next lemma gives a more detailed description of higher-order horizontal forms. Lemma 4.13. If the bundle admits regular connections then the algebra X ⊕ hork (P ) hor+ (P ) = k≥1

is generated by spaces hor1 (P ) and rω (R) (where ω ∈ rcn(P )). In particular , if T(P ) = {0} then higher-order horizontal forms are algebraically expressible through the first-order ones.

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Proof. The algebra Ω+ (P ) =

X

⊕

561

Ωk (P )

k≥1

is generated by the space Ω (P ) of 1-forms. This fact, together with the ∗ -bimodule splitting Ω1 (P ) = hor1 (P ) ⊕ ver1 (P ) determined by an arbitrary ω ∈ rcn(P ), can be used to prove that each element w ∈ Ωn (P ) is expressible in the form   n X X X  ϕik ω ∧ (ϑik ) + ψjk ω ⊗ (ηjk ) , (4.55) w= 1

k=0

i

j

∧k where ϑik and ηjk are linearly independent elements in the spaces Γ∧k inv and Sinv respectively, while ϕik , ψjk are horizontal (n − k)-forms, expressible as sums of ∧ is generated products of n − k factors from hor1 (P ). Now using the facts that Sinv ∧2 ⊗ ∧2 by Sinv and that ω (Q) is horizontal for each Q ∈ Sinv (because of ω ⊗ (Q) ∈ rω (R)) we can prove, inductively applying (4.10) and using the definition of rω and identity (4.40), that the elements ω ⊗ (ηjk ) are expressible as sums of products of the form rω (a1 ) . . . rω (am )ω ∧ (ϑql ) with l + 2m = k, and a possibly extended set of ϑik . Inserting this in (4.55) we conclude that # " n X X ∧ ϕ˜ik ω (ϑik ) , (4.56) w= k=0

i

where ϕ˜ik are horizontal (n − k)-forms expressible as sums of products of elements from hor1 (P ) and im(rω ). If w ∈ horn (P ) then ϕ˜ik = 0 for each 1 ≤ k ≤ n, according to Theorem 4.12 (i). Hence, higher-order horizontal forms are algebraically expressible via the first-order ones, and the 2-forms from the space rω (R). In the general case (P is an arbitrary bundle and ω is an arbitrary connection) the algebra hor+ (P ) is generated by spaces hor1 (P ), rω (R) and horizontal forms obtained by iteratively acting by lω on the elements from hor1 (P ) and rω (R). It is interesting to observe that in the general case X ϕk rω (ack ) + lω π(a(1) ), lω (π(a(2) ), ϕ) . rω (a)ϕ = k

If the ideal T(P ) is non-trivial (if every ω ∈ rcn(P ) is not multiplicative) then we can “renormalize” the calculus, passing to the factoralgebra Ω? (P ) = Ω(P )/T(P ) and projecting the whole formalism from Ω(P ) on Ω? (P ). It is worth noticing that such a factorization preserves the first-order calculus. In the framework of this projected calculus regular connections become multiplicative. More precisely, let h? (P ) ⊆ Ω? (P ) be the corresponding algebra of horizontal forms, and let Π: Ω(P ) → Ω? (P ) be the projection map. Lemma 4.14. We have h? (P ) = Π hor(P ) .

(4.57)

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In particular , if ω is a regular connection relative to Ω(P ) then Πω is regular in terms of Ω? (P ). Proof. It is evident that Π(hor(P )) ⊆ h? (P ). Let m?ω be the factorization map corresponding to the calculus Ω? (P ) and to the connection Πω. We have then m?ω Πhor(P ) ⊗ id = Πmω .

In particular, (4.57) holds.

With the help of the identification mω the corresponding horizontal projection operator hω : Ω(P ) → hor(P ) can be defined as follows: hω = (id ⊗ p0 )m−1 ω .

(4.58)

Clearly, hω projects Ω(P ) onto hor(P ). The domain of the previously introduced covariant derivative Dω will now be extended from hor(P ) to the whole algebra Ω(P ). Let a map Dω : Ω(P ) → hor(P ) be defined as follows: (4.59) D ω = hω d . This is a straightforward generalization of the corresponding classical definition. The main properties of hω and Dω are collected in the following theorem. Theorem 4.15. (i) The diagrams F∧ F∧ Ω(P )⊗A )⊗A Ω(P  ) −−−−−−−−−→ Ω(P  ) −−−−−−−−−→ Ω(P         Dω ⊗ id  (4.60) hω   hω ⊗ id Dω  y y y y hor(P ) −−−−−−−−−→ hor(P ) ⊗ A hor(P ) −−−−−−−−−→ hor(P ) ⊗ A F∧ F∧ are commutative. (ii) The map Dω extends the previously defined covariant derivative. (iii) If ω is regular and if T(P ) = {0} then hω is a ∗ -homomorphism and Dω (wu) = Dω (w)hω (u) + (−1)∂w hω (w)Dω (u)

(4.61)

Dω (w∗ ) = Dω (w)∗ ,

(4.62)

for each w, u ∈ Ω(P ). Proof. The statement (i) follows from the construction of hω and Dω , and properties (3.17) and (4.53). The statement (iii) is a consequence of property (iii) in Theorem 4.12, and elementary properties of d: Ω(P ) → Ω(P ). Finally, in the first definition (4.25) of the covariant derivative, the differential of a horizontal form ϕ is written as # " X ϕk ⊗ π(ck ) , dϕ = mω Dω (ϕ) ⊗ 1 + (−1)∂ϕ k

which implies that the new definition includes the previous one.

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

563

According to (4.60), compositions of pseudotensorial forms with Dω and hω are tensorial. In particular, it is possible to define, via these compositions, the covariant derivative and the horizontal projection as maps Dω , hω : ψ(P ) → τ (P ). The following proposition gives a more geometrical description of the curvature map, establishing a close analogy with classical geometry. Proposition 4.16. We have Rω = D ω ω

(4.63)

for each ω ∈ cn(P ). Proof. From definitions of mω and Rω we find Rω (ϑ) ⊗ 1 = m−1 ω dω(ϑ) − 1 ⊗ δ(ϑ) , for each ϑ ∈ Γinv . Hence, Dω ω = hω dω = Rω .

Let us assume that ω ∈ rcn(P ) and T(P ) = {0}. The above proposition implies X ϕk Rω (ϑ ◦ ck ) Rω (ϑ)ϕ = k

ϕRω (ϑ) =

X

Rω ϑ ◦ κ−1 (ck ) ϕk

(4.64)

k

for each ϕ ∈ hor(P ) and ϑ ∈ Γinv . Evidently, the above commutation relations are mutually equivalent. To obtain the first it is sufficient to act by Dω on (4.10), and apply (4.61) and (4.63). 5. Characteristic Classes In this section a quantum generalization of classical Weil’s theory of characteristic classes will be presented. Conceptually, we follow the exposition of [9]. As in the classical case, the main result will be a construction of an invariant homomorphism defined on an algebra playing the role of “invariant polynomials” over the “Lie algebra” of G, with values in the algebra of cohomology classes of M . We shall assume that the bundle P admits regular connections, and that T(P ) = {0}. For each k ≥ 0 let Ik⊗ ⊆ Γ⊗k inv be the subspace of $-invariant elements, and let I⊗ be the direct sum of these spaces. Clearly, I⊗ is a unital ∗ -subalgebra of Γ⊗ inv . Let us consider a connection ω. There exists the unique unital homomorphism ⊗ ∗ Rω⊗ : Γ⊗ inv → Ω(P ) extending the curvature Rω . The map Rω is -preserving, horizontally valued, intertwines $⊗ and F ∧ , and multiplies degrees by 2. Here, we are interested for the values of the restriction map Rω⊗ I⊗ . Proposition 5.1. If ϑ ∈ Ik⊗ then the form Rω⊗ (ϑ) belongs to Ω2k (M ). Moreover , if ω ∈ rcn(P ) then Rω⊗ (ϑ) is closed. Proof. Equation (4.24) implies Fb Rω⊗ (ϑ) =

X k

Rω⊗ (ϑk ) ⊗ ck ,

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564

⊗ for each ϑ ∈ Γ⊗ inv , where Σk ϑk ⊗ ck = $ (ϑ). Now, the first statement follows from k the assumption ϑ ∈ I⊗ , and from the definition of Ω(M ). The second statement follows from (4.30), (iv)–Proposition 4.6, and from Bianchi identity Dω Rω = 0, which holds for regular multiplicative connections.

Now we prove that the cohomological class of Rω⊗ (ϑ) in Ω(M ) is independent of ω ∈ rcn(P ). Let τ be another regular connection, and let ωt = ω + tϕ

(5.1)

where ϕ = τ − ω and t ∈ [0, 1], be the segment in the space rcn(P ) determined by ω and τ . Lemma 5.2. We have (d/dt)Rωt = Dωt (ϕ) .

(5.2)

Proof. Applying Lemma 4.8 (ii), Proposition 4.7 and (4.23) we obtain (d/dt)Rωt = (d/dt)[dω + tdϕ − hω + tϕ, ω + tϕi] = dϕ − hϕ, ωi − hω, ϕi − 2thϕ, ϕi = dϕ − hω + tϕ, ϕi − hϕ, ω + tϕi

= Dωt (ϕ) .

i i Let us consider an element ϑ ∈ Γ⊗k inv and let ϑ = Σi ci ϑ1 ⊗ · · · ⊗ ϑk , where ci are i complex numbers and ϑj ∈ Γinv . Applying (5.2) and property (4.30), the definition of Rω⊗ and the Bianchi identity we obtain X ci Dωt (ϕ)(ϑi1 ) . . . Rωt (ϑik ) + · · · + Rωt (ϑi1 ) . . . Dωt (ϕ)(ϑik ) (d/dt)Rω⊗t (ϑ) = i

=

X

ci Dωt ϕ(ϑi1 ) . . . Rωt (ϑik ) + · · · + Rωt (ϑi1 ) . . . ϕ(ϑik ) .

i

Using the tensoriality property of ϕ and Rωt we see that if ϑ ∈ Ik⊗ then the form ψt (ϑ) =

X ci ϕ(ϑi1 ) . . . Rωt (ϑik ) + · · · + Rωt (ϑi1 ) . . . ϕ(ϑik ) i

belongs to Ω2k (M ). Hence (d/dt)Rω⊗t (ϑ) = dψt (ϑ) ,

(5.3)

according to (4.32). Integrating the above equality from 0 to 1, we obtain Z Rτ (ϑ) = Rω (ϑ) + d

1

ψt (ϑ) dt 0

.

(5.4)

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

565

Let H(M ) be the graded ∗ -algebra of cohomology classes associated to Ω(M ). We have proved the following theorem. Theorem 5.3. (i) The cohomological class of Rω⊗ (ϑ) in Ω(M ) is independent of the choice of a regular connection ω, for each ϑ ∈ I⊗ . (ii) The map W : I⊗ → H(M ) given by W (ϑ) = [Rω⊗ (ϑ)] is a unital ∗ -homomorphism.

(5.5)

The homomorphism W plays the role of the Weil homomorphism in classical differential geometry [9]. In fact, in classical geometry the domain of the Weil homomorphism is restricted to the algebra of symmetric invariant elements of the corresponding tensor algebra (invariant polynomials). However, besides simplifying the domain of W such a restriction gives nothing new: the image of the Weil homomorphism will be the same. A similar situation holds in the noncommutative case. Let Σ be the graded ∗ -algebra obtained from Γ⊗ inv by factorizing through the quadratic relations given by the space im(I − σ) ⊆ Γ⊗2 inv . The algebra Σ plays the role of polynoms over the “Lie algebra” of G. The adjoint action $⊗ naturally induces the action $Σ : Σ → Σ ⊗ A. Let I[Σ] ⊆ Σ be the subalgebra consisting of elements invariant under $Σ . Clearly, I[Σ] = I⊗ /(I⊗ ∩ gen{im(I − σ)}) . Lemma 5.4. If ω is regular, then Rω⊗ σ(ϑ) = Rω⊗ (ϑ)

(5.6)

for each ϑ ∈ Γ⊗2 inv . Proof. Applying (4.64), we find o n n o mΩ Rω ⊗ ϕ = mΩ ϕ ⊗ Rω σ for each ϕ ∈ τ (P ). In particular, (5.6) holds.

(5.7)

The above statement implies that both maps W and Rω⊗ are factorizable through the braided-symmetricity relations, so that I[Σ] is the natural domain for the Weil homomorphism. To conclude this section, let us mention that the theory of quantum characteristic classes can be formulated for general quantum principal bundles (without assuming regularity). This requires completely different constructions [6] of universal quantum characteristic classes, incorporating some elements of the classifying

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566

space approach to characteristic classes. It is also worth noticing that two formulations are non-equivalent (for structures admitting regular connections). Finally, it possible to give a full classifying space [7] interpretation of quantum characteristic classes, by introducing quantum analogs of classifying spaces and universal bundles over them. 6. Examples, Remarks and Some Additional Constructions 6.1. Infinitesimal gauge transformations A The *-V-module E = τ 0 (P ) of tensorial 0-forms is definable independently of the choice of a differential calculus on the bundle P . The elements of this space are quantum counterparts of infinitesimal gauge transformations (vertical equivariant vector fields on the bundle). The space E will be here analyzed from this point of view. Explicitly, E is consisting of linear maps ζ: Γinv → B such that the diagram ζ −−−−−−−−−−−−−−→  B Γ inv     $ F y y Γinv ⊗ A −−−−−−−−−−−−−−→ B ⊗ A ζ ⊗ id

(6.1)

is commutative. Let us observe that E is closed under operations h, i and [, ]. In classical geometry, we have [, ] = −2h, i, and [, ]: E × E → E coincides with the standard commutator of vector fields (up to the sign). We are going to construct quantum analogs of contraction operators associated to vector fields representing infinitesimal gauge transformations. For each ζ ∈ E, let us consider a map ιζ : Ω(P ) → Ω(P ) defined by ιζ (w) = −(−1)∂w

X

uk ζ(ηk )

(6.2)

k

where

X k

uk ⊗ ηk = (id ⊗ πinv p1 )Fb(w).

Lemma 6.1. The diagram F∧ )⊗A Ω(P  ) −−−−−−−−−−−−−−→ Ω(P     ιζ   ιζ ⊗ id y y Ω(P ) −−−−−−−−−−−−−−→ Ω(P ) ⊗ A F∧ is commutative.

(6.3)

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

567

Proof. A direct computation gives F ∧ ιζ (w) = −(−1)∂w

X

(id ⊗ p0 )Fb (wi )(ai )F ζp1 (ϑi )

i

= −(−1)∂w

X

(wi ⊗ ai )(ζ ⊗ id)$p1 (ϑi )

i

X b i ϑi ) wi ⊗ φ(a = −(−1)∂w (mΩ ⊗ id)(id ⊗ ζπinv p1 ⊗ p0 ) i

= −(−1)

∂w

(mΩ ⊗ id)(id ⊗ ζπinv p1 ⊗ p0 )(Fb ⊗ id)Fb (w)

= (ιζ ⊗ id)F ∧ (w) , where Fb (w) = Σi wi ⊗ ai ϑi , with wi ∈ Ω(P ), ai ∈ A and ϑi ∈ Γ∧ inv .

The definition of ιζ implies ιζ (ϕw) = (−1)∂ϕ ϕιζ (w) ,

(6.4)

for each ϕ ∈ hor(P ) and w ∈ Ω(P ). In particular, ιζ hor(P ) = {0} ,

(6.5)

for each ζ ∈ E. Let us consider linear maps `ζ : Ω(P ) → Ω(P ) given by `ζ = dιζ + ιζ d .

(6.6)

These maps play the role of the corresponding Lie derivatives. Lemma 6.2. (i) The diagram F∧ − − − − − − − −−−−−−−→ Ω(P )⊗A Ω(P )      `ζ   `ζ ⊗ id y y Ω(P ) −−−−−−−−−−−−−−→ Ω(P ) ⊗ A F∧

(6.7)

is commutative. (ii) The following equality holds `ζ (ϕ) =

X

ϕk ζπ(ck ) ,

(6.8)

k

where ϕ ∈ hor(P ) and Σk ϕk ⊗ ck = F ∧ (ϕ). In particular, `ζ hor(P ) ⊆ hor(P ) .

(6.9)

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Proof. Diagram (6.7) follows from (6.3), (6.6) and (3.17). Identity (6.8) directly follows from definitions of ιζ and `ζ : " # X ∂ϕ ∂ϕ dϕk ⊗ ck + (−1) ϕk ⊗ dck `ζ ϕ = ιζ dϕ = (−1) mΩ (id ⊗ ζπinv p1 ) = mΩ (id ⊗ ζπinv ) =

X

" X

k

#

ϕk ⊗ dck

k

ϕk ζπ(ck ) .

k

Covariance properties (6.3) and (6.7) imply that ψ(P ) is invariant under compositions with ιζ and `ζ . In particular, `ζ τ (P ) ⊆ τ (P ) for each ζ ∈ E. Lemma 6.3. We have `ζ ϕ = [ϕ, ζ] ,

(6.10)

for each ζ ∈ E and ϕ ∈ τ (P ). Proof. It follows from (6.8), definitions of c> and [, ], and the tensoriality of ϕ. Let us compute actions of ιζ and `ζ on connection forms. Lemma 6.4. We have ιζ ω = ζ

(6.11)

`ζ ω = dζ + [ω, ζ] ,

(6.12)

for each ζ ∈ E and ω ∈ cn(P ). Proof. Both identities directly follow from property (4.8) and from definitions of ιζ and `ζ . In the framework of the theory presented in the previous paper, a very important class of infinitesimal gauge transformations naturally appear. These transformations can be described as infinitesimal generators (in the standard sense) of the group of vertical automorphisms of the bundle P . They form a subspace G ⊆ E. A more detailed geometrical analysis shows that the elements from G are naturally identifiable with standard infinitesimal gauge transformations of the classical part Pcl of the bundle P (vertical automorphisms of P are in a natural bijection with standard gauge transformations of Pcl ). Moreover, the space G is closed under brackets [, ] and, in terms of the mentioned identification −[, ] becomes the standard Lie bracket of vector fields. The elements from G naturally act as derivations on Ω(P ). However, the action of the derivation generated by an element ζ ∈ G generally differs from the action of the corresponding Lie derivative `ζ introduced in this subsection. This is visible from (6.12), which can be rewritten in the form `ζ ω = Dω ζ + [ω, ζ] + [ζ, ω] .

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

569

The last two summands generally give a nontrivial non-horizontal contribution, even in the case ζ ∈ G. Only in classical geometry we have [ω, ζ] + [ζ, ω] = 0 (more generally [ϕ, η] = −(−1)∂ϕ∂η [η, ϕ] for each ϕ, η ∈ ψ(P )), due to the antisymmetricity of c> , and the graded-commutativity of Ω(P ). 6.2. Infinitesimal gauge transformations B Motivated by the above remarks, a slightly different approach to defining quantum analogs of the Lie derivative and the contraction operator will be now presented. The main property of this approach is that in the special case of bundles over smooth manifolds, considered in [4], the Lie derivative of an arbitrary element ζ ∈ G coincides with the derivation generated by ζ. We shall also introduce a general quantum counterpart of the space G, and briefly analyze its properties. Lemma 6.5. (i) For each ζ ∈ E there exists the unique ςζ? : vh(P ) → vh(P ) such that ςζ? (hor(P )) = {0} ςζ? (wϑ)

=

ςζ? (w)ϑ

(6.13) ∂w

+ (−1)

wζ(ϑ) ,

(6.14)

for each w ∈ vh(P ) and ϑ ∈ Γinv . (ii) Similarly, for each ζ ∈ E there exists the unique ι?ζ : vh(P ) → vh(P ) such that X ηk ζ(ϑ ◦ ck ) (6.15) ι?ζ (ϑη) = −ϑι?ζ (η) + k

ι?ζ (ϕη) = (−1)∂ϕ ϕι?ζ (η) ,

(6.16)

∧ for each ϕ ∈ hor(P ), η ∈ Γ∧ inv and ϑ ∈ Γinv , where Σk ηk ⊗ck = $ (η). In particular ,

ι?ζ hor(P ) = {0} .

(6.17)

(iii) The following identities hold Fvh ι?ζ = (ι?ζ ⊗ id)Fvh

(6.18)

Fvh ςζ? = (ςζ? ⊗ id)Fvh

(6.19)

ςζ? (wϑ) = ςζ? (w)ϑ + (−1)∂w wςζ? (ϑ) ,

(6.20)

where w ∈ vh(P ) and ϑ ∈ Γ∧ inv . Proof. We shall prove (i) and properties (6.19) and (6.20). The statements about the map ι?ζ follow in a similar way. It is clear that conditions (6.13) and (6.14) uniquely fix the values of ςζ? , if it exists. Also, (6.20) directly follows from (6.14). In order to establish the existence of ςζ? , it is sufficient to check that (6.14) ∧ . is not in contradiction with the quadratic constraint generating the ideal Sinv

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For each a ∈ R, we have o n π(a(1) )π(a(2) ) −→ ζπ(a(1) )π(a(2) ) − π(a(1) )ζπ(a(2) ) = ζπ(a(1) ) π(a(2) ) − ζπ(a(3) ) π(a(1) ) ◦ κ(a(2) )a(4) = ζπ(a(1) ) π(a(2) ) i h − ζπ(a(3) ) π a(1) − (a(1) )1 κ(a(2) )a(4) = ζπ(a(2) )π κ(a(1) )a(3) = 0, and hence ςζ? exists. Finally, let us check (6.19). This equality is satisfied trivially on elements from hor(P ). The definition of ζ implies that it is satisfied on elements from Γinv . Finally, inductively applying (6.14) we conclude that (6.19) holds on the whole algebra vh(P ). Let us assume that the bundle P admits regular connections, as well as that T(P ) = {0}. Lemma 6.6. We have ιζ = mω ι?ζ m−1 ω

(6.21)

for each ζ ∈ E and ω ∈ rcn(P ). Proof. Let us fix a connection ω ∈ rcn(P ). For each ζ ∈ E let ι0ζ : vh(P ) → vh(P ) be a map defined by ι0ζ = m−1 ω ιζ mω . If ϕ ∈ hor(P ) and η ∈ Γ∧ inv then ∧ ∂ϕ −1 ∧ ∂ϕ 0 ι0ζ (ϕη) = m−1 ω ιζ ϕω (η) = (−1) mω ϕιζ ω (η) = (−1) ϕιζ η according to (6.4). Further, if ϑ ∈ Γinv then ∧ ∂η −1 ∧ b ι0ζ (ϑη) = m−1 ω ιζ ω(ϑ)ω (η) = (−1) mω mΩ (id ⊗ ζπinv p1 )F ω(ϑ)ω (η) ! X −1 ∧ ω (ηk ) ⊗ ϑ ◦ ck = mω mΩ (id ⊗ ζ) k

io h ω(ϑ)mΩ (id ⊗ ζπinv p1 )Fbω ∧ (η) + (−1) X ∧ ηk ζ(ϑ ◦ ck ) − m−1 = ω (ω(ϑ)ιζ ω (η)) k X ηk ζ(ϑ ◦ ck ) − ϑι0ζ (η) . = ∂η

k

m−1 ω

n

571

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Here, Σk ηk ⊗ ck = $∧ (η) and we have used elementary properties of entities figuring in the game. Applying (ii) Lemma 6.5 we find that ι0ζ = ι?ζ . Hence (6.21) holds. For each ω ∈ rcn(P ) and ζ ∈ E let ςζ,ω : Ω(P ) → Ω(P ) be a map introduced via the diagram ςζ? − − − − − − − − −−−−−−→ vh(P vh(P )  )      (6.22) mω   mω y y Ω(P ) −−−−−−−−−−−−−−→ Ω(P ) ςζ,ω and let `ζ,ω : Ω(P ) → Ω(P ) be a map given by `ζ,ω = dςζ,ω + ςζ,ω d .

(6.23)

It is evident that `ζ,ω and ςζ,ω are right-covariant maps, in the sense that F ∧ `ζ,ω = (`ζ,ω ⊗ id)F ∧ ∧

∧

F ςζ,ω = (ςζ,ω ⊗ id)F .

(6.24) (6.25)

The maps `ζ,ω and ςζ,ω , are also interpretable as quantum counterparts of the Lie derivative and the contraction operator respectively. In contrast to the classical case, these maps are generally connection-dependent. However, Lemma 6.7. If w ∈ Ω1 (P ), then ςζ,ω (w) = ιζ (w) ,

(6.26)

for each ζ ∈ E and ω ∈ rcn(P ). In particular , operators `ζ and `ζ,ω possess the same restrictions on hor(P ). Proof. It follows from the fact that ι?ζ and ςζ? coincide on the spaces hor(P ) and hor(P ) ⊗ Γinv . Covariance properties (6.24)–(6.25) enable us to define actions of `ζ,ω and ςζ,ω in the space ψ(P ). The ω-dependence of constructed operators becomes explicitly visible if we consider the action of `ζ,ω on connection forms. Lemma 6.8. We have `ζ,ω τ = Dτ ζ + [τ − ω, ζ] + [ζ, τ − ω]

(6.27)

for each ζ ∈ E, ω ∈ rcn(P ) and τ ∈ cn(P ). In particular, `ζ,ω τ is always tensorial. Proof. Using Lemmas 6.3 and 6.7, and properties (6.11) and (6.23), we obtain `ζ,ω τ = `ζ,ω ω + [τ − ω, ζ] = dζ + [τ − ω, ζ] + ςζ,ω dω .

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On the other hand, (4.23) together with the regularity of ω, tensoriality of ζ and the definition of ςζ,ω gives ςζ,ω dω(ϑ) = ςζ,ω (hω, ωi + Rω )(ϑ) = ςζ,ω hω, ωi(ϑ) = −ζπ(a(1) )ωπ(a(2) ) + ωπ(a(1) )ζπ(a(2) ) h i = −ζπ(a(1) )ωπ(a(2) ) + ζπ(a(3) )ω π(a(1) ) ◦ κ(a(2) )a(4) = −ζπ(a(2) )ωπ κ(a(1) )a(3)

= −[ζ, ω](ϑ) , where a ∈ A satisfies (4.18). Consequently, `ζ,ω τ = Dω ζ + [τ − ω, ζ] = Dτ ζ + [τ − ω, ζ] + [ζ, τ − ω] .

Finally, let G ⊆ E be the space of elements ζ satisfying ζ(ϑ)ϕ =

X

ϕk ζ(ϑ ◦ ck )

(6.28)

k

for each ϕ ∈ hor(P ) and ϑ ∈ Γinv . Let us assume that G is nontrivial. Proposition 6.9. (i) The space G is closed under the action of brackets [, ]. We have ζπ(a(1) )ξπ(a(2) ) − ξπ(a(1) )ζπ(a(2) ) = [ζ, ξ]π(a)

(6.29)

for each ζ, ξ ∈ G and a ∈ A. In particular, brackets [, ] determine a Lie algebra structure on G. (ii) Operators ςζ,ω and `ζ,ω are ω-independent, if ζ ∈ G. (iii) The following identities hold `?ζ (wu) = `?ζ (w)u + w`?ζ (u)

(6.30)

ςζ (wu) = ςζ (w)u + (−1)∂w wςζ (u)

(6.31)

ςζ ςξ + ςξ ςζ = 0

(6.32)

`?ζ `?ξ − `?ξ `?ζ = −`?[ζ,ξ]

(6.33)

`?ζ ςξ − ςξ `?ζ = −ς[ζ,ξ] .

(6.34)

Here ζ, ξ ∈ G and w, u ∈ Ω(P ) while ςζ = ςζ,ω and `?ζ = `ζ,ω . Proof. We compute

573

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

ξπ(a(1) )ζπ(a(2) ) = ζπ(a(3) )ξ π(a(1) ) ◦ κ(a(2) )a(4) = ζπ(a(1) )ξπ(a(2) ) − [ζ, ξ]π(a) . Let us check that G is closed under the brackets [, ]. Using properties (6.7)–(6.8) and (6.28)–(6.29), we find ([ζ, ξ]π(a))ϕ =

X k

−

(1)

ϕk ζ π(a(1) ) ◦ ck

X

(2)

ξ π(a(2) ) ◦ ck

(1)

ϕk ξ π(a(1) ) ◦ ck

(2)

ζ π(a(2) ) ◦ ck

k

X (1) (2) (1) (2) ϕk ζπ(a(1) ck )ξπ(a(2) ck ) − ϕk ξπ(a(1) ck )ζπ(a(2) ck ) = k

−

X

`ζ (ϕk )ξπ(ack ) −

k

+ =

X k

+ =

X

(1)

(2)

(1)

(2)

ϕk ζπ(ack )ξπ(ck )

k

`ξ (ϕk )ζπ(ack ) +

k

ϕk [ζ, ξ]π(ack ) −

X

X k

=

X

X X

X

ϕk ξπ(ack )ζπ(ck )

k

ξπ(a)`ζ (ϕ) −

X

k

ζπ(a)`ξ (ϕ) +

k

X

ζπ(a)ϕk ξπ(ck )

k

ξπ(a)ϕk ζπ(ck )

k

ϕk [ζ, ξ]π(ack ) ϕk [ζ, ξ] π(a) ◦ ck ,

k

where a ∈ ker() and ϕ ∈ hor(P ). Now we shall prove identities (6.30)–(6.34). Let us observe that (6.30) directly follows from (6.31) and (6.23). On the other hand the fact that ςζ,ω is an antiderivation together with Lemma 6.7 shows that ςζ,ω (and therefore `ζ,ω ) is ω-independent. Evidently, (6.31) is equivalent to the fact that ςζ? is an antiderivation on vh(P ). Having in mind identity (6.20) and property (6.13) it is sufficient to check that ςζ? (ϑϕ) = ςζ? (ϑ)ϕ , for each ϑ ∈ Γ∧ inv and ϕ ∈ hor(P ). However, this easily follows from property (6.28) and the definition of ςζ? . For each ζ, ξ ∈ G the anticommutator of ςζ? and ςξ? is an antiderivation on vh(P ). This anticommutator vanishes on hor(P ) and Γinv . Therefore it vanishes identically. Having in mind that `?ζ are derivations on Ω(P ) commuting with the differential, and that derivations form a Lie algebra, it is sufficient to check that (6.33) holds on elements b ∈ B. We have

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(`?ζ `?ξ − `?ξ `?ζ )(b) = `?ζ

X

bk ξπ(ak ) − `?ξ

k

=

X

bk ζπ(ak )

k

i Xh (1) (2) (1) (2) bk ξπ(ak )ζπ(ak ) − bk ζπ(ak )ξπ(ak ) k

=

X

bk [ξ, ζ]π(ak )

k

= `?[ξ,ζ] (b) , where F (b) = Σk bk ⊗ ak . Similarly, it is sufficient to check that (6.34) holds on elements of the form ω(ϑ). We have ? ? ? `ζ ςξ − ςξ `ζ ω = `ζ ξ − ςξ Dω ζ = [ξ, ζ] = −ς[ζ,ξ] ω , which completes the proof.

6.3. Internal horizontal forms In this subsection we shall consider in more details interrelations between [1] and the theory developed here. At the level of spaces the formulation of [1] contains (modulo the ∗ -structure) an additional condition which is essentially the same as (qpb5), such that the concept of a quantum principal bundle turns out to be equivalent to the concept of a Hopf–Galois extension [11]. In our formulation this additional condition generally will not be satisfied, however in the most interesting case of compact structure groups it turns out [5] that (qpb5) holds automatically. The main difference between two approaches appears at the level of differential calculus. At first, it is assumed in [1] that the higher-order differential calculus on the bundle uniquely follows from the first-order one, being based on (an appropriate) universal envelope of the first-order differential structure. In contrast to this, the theory presented here allows to change the higher-order calculus without modifying the first-order calculus. Such a flexibility allows us to pass from one calculus to another when necessary, without changing the space of connection forms. As one example of this type of modification, let us mention the “renormalization” procedure considered in Sec. 4. Another important example is the modification of the calculus on the bundle, naturally appearing if we pass from the universal envelope Γ∧ to the braided exterior algebra Γ∨ , at the structure group level. This is considered in details in Appendix A. Secondly, the total “pull back” of the right action does not figure in [1] and horizontal forms are defined in a completely different “internal” way, independently of any covariance property of the calculus on P . We shall discuss horizontal forms of [1] in the framework of general differential structures introduced in Sec. 3. Let P = (B, i, F ) be a quantum principal Gbundle over M and let Ω(P ) be an arbitrary graded-differential ∗ -algebra satisfying properties (diff 1/2). Let Ωhor ⊆ hor(P ) be a (∗ -) subalgebra generated by B and d i(V) . This algebra is a counterpart of horizontal forms introduced in [1]. One of the main conditions postulated in [1] (in the definition of differential calculus) is that the sequence

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

575

π

0 → Ω1hor ,→ Ω1 (P ) →v ver1 (P ) → 0 is exact. According to Lemma 3.7 this is equivalent to Ω1hor = hor1 (P ) .

(6.35)

This condition can be understood as a (relatively strong) condition on the bundle, if applied to the universal case, for example. Namely, a trivial differential calculus on the bundle can always be constructed by taking the universal differential envelope of B (conditions (diff 1/2) hold). In this case, the condition (6.35) will not hold if we assume that the first-order calculus over G is non-universal. However, if we assume that the calculus over G is also universal, then it turns out that (6.35) is equivalent to the injectivity assumption (qpb5). Furthermore, it is worth noticing that if (6.35) holds and if the bundle admits regular and multiplicative connections then Ωhor = hor(P ). Also, two horizontal algebras coincide if Ω(M ) is generated (as a differential algebra) by i(V). This follows from the fact that each ϕ ∈ hor(P ) can be written in the form X wi bi ϕ= i

where bi ∈ B and wi ∈ Ω(M ), as explained in Appendix B. The definition of connection forms given in [1] is (modulo the ∗ -structure and differences between differential calculi) equivalent to the definition proposed in this work. On the other hand, the horizontal projection, covariant derivative and the curvature map described in [1] can be consistently defined only for very restricted differential structures, and are generally different from hω , Dω and Rω constructed here, which are essentially based on the horizontal-vertical decomposition map mω and the embedded differential δ respectively. 6.4. Trivial bundles For given quantum space M and compact matrix quantum group G let us define a -algebra B and maps F : B → B ⊗ A and i: V → B by ∗

B =V ⊗A id ⊗ φ = F i(f ) = f ⊗ 1 . The triplet P = (B, i, F ) is a trivial quantum principal bundle over M . Geometrically P = M × G. The algebra of verticalized forms is isomorphic to the tensor product ver(P ) = V ⊗ Γ∧ , with dv ↔ id ⊗ d.

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Let Ω(M ) be an arbitrary graded-differential ∗ -algebra generated by V = Ω0 (M ), representing a differential calculus on M . Then it is natural to define the algebra Ω(P ) (representing a differential calculus on the bundle P ) as the graded tensor product b Γ∧ . Ω(P ) = Ω(M ) ⊗ We then have

Fb = id ⊗ φb .

Horizontal forms constitute a ∗ -subalgebra hor(P ) = Ω(M ) ⊗ A . We shall now analyze in more details the structure of tensorial forms, connection forms, and operators of covariant derivative and curvature, in the special case of trivial bundles. Let X be the graded ∗ -Ω(M ) module of linear maps L: Γinv → Ω(M ). Lemma 6.10. For each ϕ ∈ τ (P ), there exists the unique Lϕ ∈ X such that ϕ(ϑ) = (Lϕ ⊗ id)$(ϑ)

(6.36)

for each ϑ ∈ Γinv . The above formula establishes an isomorphism between graded ∗ -Ω(M )-modules. Proof. For a given L ∈ X let ϕL : Γinv → Ω(P ) be a map determined by equality ϕL (ϑ) = (L ⊗ id)$(ϑ). Evidently, the image of ϕL is contained in hor(P ) and F ∧ ϕL = (L ⊗ φ)$ = (L ⊗ id)$ ⊗ id $ = (ϕL ⊗ id)$ . Hence ϕL ∈ τ (P ). It is clear that the map L 7→ ϕL is a monomorphism of ∗ -Ω(M )modules (because L = (id ⊗ )ϕL ). Let us consider an arbitrary tensorial form ϕ. Acting by id ⊗ ⊗ id on the tensoriality identity for ϕ we find that (6.36) holds, with Lϕ = (id ⊗ )ϕ. The following lemma gives a similar description of connection forms. Lemma 6.11. (i) The formula ω(ϑ) = (Aω ⊗ id)$(ϑ) + 1 ⊗ ϑ

(6.37)

establishes a bijective affine correspondence between connections on P and hermitian elements of X 1 . (ii) A connection ω is regular iff Aω (ϑ)ζ = (−1)∂ζ ζAω (ϑ)

(6.38)

Aω (ϑ ◦ a) = (a)Aω (ϑ)

(6.39)

for each a ∈ A, ϑ ∈ Γinv and ζ ∈ Ω(M ).

577

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Proof. The formula ω0 (ϑ) = 1 ⊗ ϑ determines a canonical “flat” connection on P . The statement (i) follows from the previous lemma and the fact that hermitian elements of τ 1 (P ) form the vector space associated to cn(P ). Let us assume that ω ∈ rcn(P ). In other words ω(ϑ)(ζ ⊗ a) = (−1)∂ζ (ζ ⊗ a(1) )ω(ϑ ◦ a(2) ) for each ζ ∈ Ω(M ), a ∈ A and ϑ ∈ Γinv . This is equivalent to X

Aω (ϑk )ζ ⊗ ck a = (−1)∂ζ

k

X

ζAω (ϑk ◦ a(1) ) ⊗ ck a(2) ,

(6.40)

k

where Σk ϑk ⊗ ck = $(ϑ). Acting by id ⊗ on both sides on the above equality we obtain (6.41) Aω (ϑ)ζ(a) = (−1)∂ζ ζAω (ϑ ◦ a) , which is equivalent to conditions listed in (ii). Conversely (6.41) imply (6.40), evidently. The bijection ω ↔ Aω generalizes the classical correspondence between connections and their gauge potentials. In the previous paper, a similar correspondence was established, at the local level. This was possible because of the local triviality of considered bundles. However, in the general quantum context it is not possible to speak about local domains on the base space, and hence it is not possible to speak about locally trivial bundles. Lemma 6.12. We have Rω (ϑ) = (F ω ⊗ id)$(ϑ)

(6.42)

where F ω ∈ X 2 is a hermitian element given by F ω = dAω − hAω , Aω i .

(6.43)

Further , if ϕ ∈ hor(P ) then Dω (ϕ) ↔ q ω,ϕ

(6.44)

q ω,ϕ = dLϕ − (−1)∂ϕ [Lϕ , Aω ] .

(6.45)

where Proof. Inserting (6.36) and (6.37) in (4.33), we obtain (Dω ϕ)(ϑ) =

X

dLϕ (ϑk ) ⊗ ck + (−1)∂ϕ

k

X

Lϕ (ϑk ) ⊗ dck

k

i Xh (1) (2) (1) (2) − (−1)∂ϕ Lϕ (ϑk ) ⊗ ck π(ck ) + Lϕ (ϑk )Aω π(ck ) ⊗ ck =

k

dL − (−1)∂ϕ [Lϕ , Aω ] ⊗ id $(ϑ) . ϕ

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Similarly, inserting (6.37) in (4.23) we obtain X X dAω (ϑk ) ⊗ ck − Aω (ϑk ) ⊗ dck + 1 ⊗ dϑ Rω (ϑ) = k

k

+ 1 ⊗ π(a

(1)

)π(a

(2)

) + Aω π(a(2) ) ⊗ κ(a(1) )a(3) π(a(4) )

− Aω π(a(3) ) ⊗ π(a(1) )κ(a(2) )a(4) + Aω π(a(2) )Aω π(a(3) ) ⊗ κ(a(1) )a(4) = dAω − hAω , Aω i ⊗ id $(ϑ) . Here, a ∈ ker() is such that (4.18) holds.

Infinitesimal gauge transformations are in a natural bijection with linear maps γ: Γinv → V. The elements of G correspond to functions γ satisfying γ(ϑ ◦ a) = (a)γ(ϑ) wγ(ϑ) = γ(ϑ)w for each ϑ ∈ Γinv , a ∈ A and w ∈ Ω(M ). Let us assume that Γ is the minimal admissible (bicovariant ∗ -) calculus over G (in the sense of [4]). The following natural identifications hold G ↔ Z 0 (M ) ⊗ lie(Gcl ) rcn(P ) ↔ Z 1 (M ) ⊗ lie(Gcl ) , where Gcl is the classical part of G and Z(M ) is the (graded) centre of Ω(M ). The example of trivial bundles can be further straightforwardly generalized by relaxing the assumption that Ω(P ) is the graded tensor product. It is sufficient to ask that we have a decomposition Ω(P ) ↔ Ω(M ) ⊗ Γ∧ , which is Ω(M )-linear and b This is a graded-differential variant of cleft [11] which intertwines Fb and id ⊗ φ. Hopf–Galois extensions. 6.5. Quantum homogeneous spaces Let H be a compact matrix quantum group and let G be a (compact) subgroup of H. At the formal level, this presumes a specification of a ∗ -epimorphism (the corresponding “restriction map”) j: B → A such that (j ⊗ j)φ0 = φj j = 0 κj = jκ0 . Here B is the functional Hopf ∗ -algebra for H. In what follows entities endowed with the prime will refer to H. The ∗ -homomorphism F : B → B ⊗ A given by F = (id ⊗ j)φ0

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

579

is interpretable as the right action of G on H. Let M be the corresponding “orbit space”. At the formal level, M is represented by the fixed-point ∗ -subalgebra V ⊆ B. Let i: V ,→ B be the inclusion map. Lemma 6.13. The triplet P = (B, i, F ) is a quantum principal G-bundle over M . Proof. It is evident that conditions (qpb1/2) of Definition 3.1 are satisfied. We have 1 ⊗ j(b) = κ(b(1) )F (b(2) ) , for each b ∈ B. Hence (qpb4) holds.

Because of the inclusion φ0 (V) ⊆ B ⊗ V there exists a natural left action of H on M , defined by φ0 i: V → B ⊗ V. This action is “transitive” in the sense that only scalar elements of V are invariant. In this sense M is understandable as a quantum homogeneous H-space. Now a construction of a differential calculus on H will be presented, which explicitly takes care of the “fibered” geometrical framework. Let Ψ be a left-covariant first-order ∗ -calculus over H and let R0 ⊆ ker(0 ) be the corresponding right B-ideal. Let us assume that j(R0 ) ⊆ R 0

0

0

(id ⊗ j)ad (R ) ⊆ R ⊗ A

(6.46) (6.47)

where R ⊆ ker() is the right A-ideal which determines the calculus Γ over G. Condition (6.46) ensures the existence of the projection map ρ: Ψinv → Γinv , which is determined by the formula (6.48) ρπ 0 = πj . The meaning of condition (6.47) is that the calculus Ψ is right-covariant, relative to G. Consequently, there exists the corresponding adjoint action χ: Ψinv → Ψinv ⊗ A. This map is explicitly given by χπ 0 = (π 0 ⊗ j)ad0 .

(6.49)

Maps ρ and χ are hermitian and ρ(ϑ ◦ a) = ρ(ϑ) ◦ j(a)

(6.50)

$ρ = (ρ ⊗ id)χ .

(6.51)

In particular, the space L = ker(ρ) is a ∗ - and χ-invariant submodule of Ψinv . Let us now assume that the full calculus on the bundle P (⇔ the fibered H) is described by a graded-differential ∗ -algebra Ω(P ) built over Ψ which is such that the map F (and therefore χ) can be extended to a differential algebra homomorphism b ∧ (that is, property (diff 2) holds). Let us consider a ∗ -invariant Fb : Ω(P ) → Ω(P ) ⊗Γ and χ-invariant complement L⊥ of L in Ψinv .

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Lemma 6.14. A linear map ω: Γinv → Ω(P ) given by ω(ϑ) = ρL⊥

−1

(ϑ)

(6.52)

is a connection on P . Proof. By construction, it follows that ω is a hermitian pseudotensorial 1-form. Condition (4.6) directly follows from the observation that πv (ϑ) = 1 ⊗ ρ(ϑ)

for each ϑ ∈ Ψinv .

Let us assume that the subspace L⊥ is also a submodule of Ψinv . In other words ρ⊥ (ϑ ◦ b) = ρ⊥ (ϑ) ◦ b ,

(6.53)

where ρ⊥ : Ψinv → L is the projection map, corresponding to the splitting Ψinv = L ⊕ L⊥ . In what follows we shall identify the spaces L⊥ and Γinv , via the map ρ. The right B-module structure on L⊥ can be naturally “projected” to the right A-module structure on this space, so that ρ : Γinv ↔ L⊥ becomes a right A-module isomorphism, because of L⊥ ◦ ker(j) = {0} . Further, let us assume that Ω(P ) is left-covariant and let l(P ) ⊆ Ω(P ) be a -subalgebra consisting of left-invariant elements. Finally, let us assume that ω constructed in the above lemma is a regular and multiplicative connection. This assumption implies certain specific algebraic relations between elements of l(P ). It is clear that elements η ∈ L are horizontal. Hence the following relations hold X ηk (ϑ ◦ ak ) = 0 (6.54) ϑη + ∗

k

X

ηk ⊗ ak = χ(η). where ϑ ∈ L⊥ and k The action of the covariant derivative on elements from B and L is described by Lemma 6.15. The following identities hold κ(b) = κ0 (b(1) )Dω (b(2) )

(6.55)

Dω (b) = b(1) κ(b(2) ) Dω κ(b) = −Rω πj(b) − κ(b

(1)

(6.56) )κ(b

(2)

),

(6.57)

where κ = ρ⊥ π 0 . Proof. Evidently, (6.55) and (6.56) are mutually equivalent. Equation (6.56) directly follows from (2.2), (4.25) and from the definition of ω. Acting by Dω on both sides of (6.55) and applying (4.29)–(4.30) we obtain

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

581

Dω κ(b) = Dω κ0 (b(1) )Dω (b(2) ) + κ0 (b(1) )Dω2 (b(2) ) = Dω κ0 (b(1) ) b(2) κ(b(3) ) − κ0 (b(1) )b(2) Rω πj(b(3) ) = −κ(b(1) )κ(b(2) ) − Rω πj(b) .

In fact, formula (6.57) defines Dω and Rω . If π 0 (b) ∈ L then Dω π 0 (b) = −κ(b(1) )κ(b(2) ) and similarly

Rω π 0 (b) = −κ(b(1) )κ(b(2) ) ,

if κ(b) = 0. The above two formulas are in fact equivalent to (6.57). Both give the same consistency condition for l(P ). If b ∈ R0 then κ(b(1) )κ(b(2) ) = 0 .

(6.58)

The above constraint generates a further constraint in the third-order level, because it must be compatible with Dω satisfying the graded Leibniz rule. Explicitly, Rω πj(b(1) )κ(b(2) ) − κ(b(1) )Rω πj(b(2) ) = 0 for each b ∈ R0 . More generally, it follows that X ηk Rω (ϑ ◦ ck ) Rω (ϑ)η =

(6.59)

k

X ηk ⊗ ck = χ(η). where ϑ ∈ Γinv and η ∈ L, while k It is worth noticing that ρ is extendible to a homomorphism ρ∧ : l(P ) → Γ∧ inv of graded-differential algebras. In terms of the canonical identification Ω(P ) ↔ B ⊗ l(P ) of spaces, the verticalization homomorphism is given by πv ↔ id ⊗ ρ∧ . Motivated by the derived expressions and constraints we shall now construct “the universal” higher-order calculus on the bundle, admitting regular and multiplicative connections of the described geometrical nature. The starting point will be a leftcovariant ∗ -calculus Ψ over H, endowed with a splitting of the form Ψinv ∼ = L⊕ Γinv . We shall assume that this splitting possesses all the properties introduced above. Let K1 be the ideal in the tensor algebra L⊗ generated by elements of the form w = κ(b(1) ) ⊗ κ(b(2) )

(6.60)

where b ∈ R0 . The formulas D π 0 (b) = −κ(b(1) )κ(b(2) ) R π 0 (q) = −κ(q (1) )κ(q (2) ) consistently define linear maps D: L → L⊗ /K1 and R: Γinv → L⊗ /K1 . Here, πj(b) = 0 and κ(q) = 0 respectively. We have Dκ(b) = −Rπj(b) − κ(b(1) )κ(b(2) ) .

´ M. DURDEVIC

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Let K2 be the ideal in L⊗ /K1 generated by relations of the form X ηk R(ϑ ◦ ck ) R(ϑ)η = k

where χ(η) = Σk ηk ⊗ ck . The map D can be uniquely extended to a first-order derivation D: L? → L? , ? ⊗ where L = (L /K1 )/K2 . Indeed, it is sufficient to check that the graded Leibniz rule for D is not in contradiction with relations generating K1 and K2 . This follows from the above-derived equations. Both ideals K1 and K2 are right and ∗ -invariant, in a natural manner. In other words L? is a ∗ -algebra, endowed with the right action χ: L? → L? ⊗ A. Let us assume that R is factorized through the ideal K2 . By construction, D and R are hermitian and right-covariant maps. In particular, it follows that DR = 0. The ◦-structure on L⊗ can be naturally “projected” to L? , through ideals K1 and K2 . The following identities hold D(η ◦ b) = D(η) ◦ b − κ(b(1) )(η ◦ b(2) ) + (−1)∂η (η ◦ b(1) )κ(b(2) ) R ϑ ◦ j(b) = R(ϑ) ◦ b .

(6.61) (6.62)

Finally, let us consider a graded ∗ -algebra defined as horP = B ⊗ L? at the level of graded vector spaces, while the product and the ∗ -structure are given by (q ⊗ η)(b ⊗ ϑ) = qb(1) ⊗ (η ◦ b(2) )ϑ (b ⊗ ϑ)∗ = b(1)∗ ⊗ (ϑ∗ ◦ b(2)∗ ) . Evidently, B and L? are *-subalgebras of horP . The formulas D(b ⊗ ϑ) = b(1) ⊗ κ(b(2) )ϑ + b ⊗ D(ϑ) F ? (b ⊗ ϑ) = F (b)χ(ϑ) define extensions D: horP → horP and F ? : horP → horP ⊗ A of the previously introduced maps. By construction, F ? defines the action of G by “automorphisms” of horP . Further, Lemma 6.16. The map D is a hermitian right-covariant first-order antiderivation on horP . The following identities hold X ϕk Rπ(ck ) D2 (ϕ) = − R(ϑ)ϕ =

X k

where F ? (ϕ) =

X k

ϕk ⊗ ck .

k

ϕk R(ϑ ◦ ck ) ,

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

583

Proof. We compute D(ϑb) = D(b(1) )ϑ ◦ b(2) + b(1) D(ϑ ◦ b(2) ) = b(1) κ(b(2) )ϑ ◦ b(3) + b(1) D(ϑ) ◦ b(2) −b(1) κ(b(2) )ϑ ◦ b(3) + (−1)∂ϑ b(1) (ϑ ◦ b(2) )κ(b(3) ) = D(ϑ)b + (−1)∂ϑ ϑD(b) . This implies that D is a (first-order) antiderivation on horP . Furthermore, it is sufficient to check that the relation between D2 and R holds on elements from L and B. We have D2 (b) = D b(1) κ(b(2) ) = b(1) κ(b(2) )κ(b(3) ) − b(1) Rπj(b(2) ) − b(1) κ(b(2) )κ(b(3) ) = −b(1) Rπj(b(2) ) and similarly D2 κ(b) = −D Rπj(b) + κ(b(1) )κ(b(2) ) = Rπj(b(1) )κ(b(2) ) − κ(b(1) )Rπj(b(2) ) = κ(b(3) )R πj(b(1) ) ◦ j κ(b(2) )b(4) − κ(b(1) )Rπj(b(2) ) = −κ(b(2) )Rπj κ(b(1) )b(3) . Finally, we have to check the commutation relation between R and B. Direct transformations give Rπj(q)b = −κ(q (1) )κ(q (2) )b = −b(1) κ(q (1) ) ◦ b(2) κ(q (2) ) ◦ b(3) = −b(1) κ(q (1) b(2) )κ(q (2) b(3) ) = −b(1) Rπj(qb(2) ) , where κ(q) = 0. This completes the proof.

Now the construction of the full differential calculus on the bundle P can be completed applying ideas of Subsec. 6. The initial splitting is naturally understandable as a regular and multiplicative connection ω on P . Maps D and R are interpretable as the corresponding operators of covariant derivative and curvature. The full algebra Ω(P ) is left-covariant (over H). The associated firstorder calculus coincides with Ψ. The corresponding differential ∗ -subalgebra l(P ) of left-invariant elements can be independently described as follows. At the level of (graded) vector spaces l(P ) = L? ⊗ Γ∧ inv .

´ M. DURDEVIC

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The differential ∗ -algebra structure is specified by X ηϑk ⊗ (ξ ◦ ck )ζ (η ⊗ ξ)(ϑ ⊗ ζ) = (−1)∂ξ∂ϑ k

X

(ϑ ⊗ ζ)∗ =

k ∧

d (ϑ ⊗ ζ) = D(ϑ) ⊗ ζ + (−1)

∂ϑ

ϑ∗k ⊗ (ζ ∗ ◦ c∗k )

X

ϑk ⊗ π(ck )ζ + (−1)∂ϑ ϑd∧ (ζ) .

k

Here d∧ : l(P ) → l(P ) is the corresponding differential and Finally, d∧ (ζ) = d(ζ) + R(ζ)

X k

ϑk ⊗ ck = χ(ϑ).

for ζ ∈ Γinv . The map ρ∧ is given by ρ∧ ↔ L ⊗ id, where L is a character on L? specified by L (L) = {0}. As a concrete example, let us consider the quantum Hopf fibering [10]. This bundle is described by the quantum group H = Sµ U (2), and its subgroup G = Hcl = U (1). The base manifold is the quantum 2-sphere [10]. By definition [12], B is the ∗ -algebra generated by elements α and γ and the following relations αα∗ + µ2 γγ ∗ = 1 αγ = µγα

α∗ α + γ ∗ γ = 1

αγ ∗ = µγ ∗ α

γγ ∗ = γ ∗ γ .

The fundamental representation is given by α −µγ ∗ u = (u† )−1 = γ α∗ where µ ∈ (−1, 1) \ {0}. Let us first assume that the first-order differential structure Ψ over H coincides with the 3D-calculus [12], based on a right B-ideal n o R0 = gen γ 2 , γγ ∗ , γ ∗2 , αγ − γ, αγ ∗ − γ ∗ , µ2 α + α∗ − (1 + µ2 )1 . The space Ψinv is 3-dimensional and spanned by elements η3 = π 0 (α − α∗ )

η+ = π 0 (γ)

η− = π 0 (γ ∗ ) .

The corresponding right B-module structure ◦ is specified by µ2 η3 ◦ α = η3

η3 ◦ α∗ = µ2 η3

µη± ◦ α = η±

η± ◦ α∗ = µη±

with Ψinv ◦ γ = Ψinv ◦ γ ∗ = {0}. The *-algebra A of polynomial functions on G is generated by the canonical unitary element z = j(α) (and we have j(γ) = j(γ ∗ ) = 0). Let us assume that

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

585

Γ is a left-covariant calculus over G based on the right A-ideal R = j(R0 ). The space Γinv is 1-dimensional, and spanned by ζ = ρ(η3 ) = π(z − z ∗ ). We have ρ(η+ ) = ρ(η− ) = 0 and Γ∧k = {0} for k ≥ 2. However, the calculus based on Γ∧ differs from the classical differential calculus, because the right A-module structure on Γinv is given by µ2 ζ ◦ z = ζ ζ ◦ z ∗ = µ2 ζ . The space L is spanned by η+ and η− . Let us define a splitting Ψinv ∼ = L ⊕ Γinv (⇔ the space L⊥ ) by identifying η3 and ζ. The corresponding relations determining the algebra L? are 2 2 = η− =0 η+

η+ η− = −µ2 η− η+ while the third-order relations are trivialized. Additional relations determining the algebra l(P ) are µ4 η3 η+ = −η+ η3 η3 η− = −µ4 η− η3

η2 = 0 .

It is worth noticing that Ω(P ) = Ψ∧ and hence the higher-order calculus on the bundle coincides with the calculus constructed in [12]. Modulo differences between general formulations, the canonical regular connection ω (associated to the fixed splitting of Ψinv ) coincides with a connection constructed in [1] using different methods. The curvature of ω is given by Rω (ζ) = dη3 = µ(1 + µ2 )η− η+ .

(6.63)

Next, let us consider the case of a 4D-calculus over Sµ U (2). This calculus Ψ is bicovariant and ∗ -covariant. By definition [14] the corresponding R0 is generated (as a right ideal) by multiplets n o n o 3 = aγ, a(α − α∗ ), aγ ∗ 1 = a µ2 α + α∗ − (1 + µ2 )1 o n 5 = γ 2 , γ(α − α∗ ), µ2 α∗2 − (1 + µ2 )(αα∗ − γγ ∗ ) + α2 , γ ∗ (α − α∗ ), γ ∗2 where a = µ2 α + α∗ − (µ3 + 1/µ)1. The space Ψinv is 4-dimensional. A natural basis is given by elements τ = π 0 (µ2 α + α∗ ) η+ = π 0 (γ)

η3 = π 0 (α − α∗ )

η− = π 0 (γ ∗ ) .

Elements η+ , η− and η3 form a triplet (relative to the adjoint action). The element τ is $0 -invariant. We have [4, Sec. 6] η+ ◦ γ ∗ = η− ◦ γ = −

1 − µ2 (1 + µ)(1 − µ2 ) τ − η3 µ(1 + µ2 )(1 − µ3 ) µ(1 + µ2 )

´ M. DURDEVIC

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η+ ◦ γ = η− ◦ γ ∗ = 0 η+ ◦ α = η+ = η+ ◦ α∗ η3 ◦ γ = −

1 − µ2 η+ µ

η3 ◦ γ ∗ = −

1 − µ2 η− µ

2µ (1 + µ)(1 − µ2 ) τ− η3 2 3 µ(1 + µ )(1 − µ ) 1 + µ2

−η3 ◦ α∗ = η3 ◦ α =

η− ◦ α = η− = η− ◦ α∗

2µ µ(1 + µ)(1 − µ2 ) τ+ η3 2 3 (1 + µ )(1 − µ ) 1 + µ2

τ ◦γ =

(1 − µ)(1 − µ3 ) η+ µ

τ ◦ α∗ =

µ(1 − µ)(1 − µ3 ) 1 + µ4 τ − η3 µ(1 + µ2 ) 1 + µ2

τ ◦ γ∗ =

(1 − µ)(1 − µ3 ) η− µ

τ ◦α =

(1 − µ)(1 − µ3 ) 1 + µ4 τ + η3 . µ(1 + µ2 ) µ(1 + µ2 )

It turns out that the ideal R = j(R0 ) is generated by the element z + µz ∗ −(1 + µ)1. The space L is spanned by elements η+ , η− and ξ = τ + (1 − µ3 )/(1 + µ) η3 . It is worth noticing that ξ is (the unique) common characteristic vector for all operators of the form ◦b (where b ∈ B). Explicitly, ξ◦α=

1 ξ µ

ξ ◦ α∗ = µξ

ξ ◦ γ = ξ ◦ γ∗ = 0 .

However, the space L does not possess a ◦-invariant complement. A natural choice of the complement L⊥ is to consider the subspace spanned by η3 (following a weak analogy with the previous example). The calculus on G = U (1) is nonstandard. The higher-order calculus is trivial. The corresponding right A-module structure is given by 1 ζ ◦ z = µζ ζ ◦ z∗ = ζ µ where ζ = ρ(η3 ). Let us assume that the higher-order calculus on the bundle is based on the bicovariant exterior algebra Ψ∨ [14]. Let ω be the connection corresponding to the above splitting. This connection is not multiplicative. The most general form of a connection (coming from a complement L⊥ ) is ω(ζ) = η3 + tξ

(6.64)

where t ∈ <. All these connections are non-regular. A connection ω is multiplicative iff (η3 + tξ)2 = 0, which is equivalent to t = −(1 + µ)/(1 − µ3 ), in other words the corresponding complement is spanned by τ . However it is worth noticing that if the higher-order calculus is described by Ψ∧ then ω is not multiplicative, because τ 2 6= 0 in this case. The curvature is given by µt µ + (6.65) (τ η3 + η3 τ ) . Rω = d(η3 + tξ) = 1 − µ2 (1 − µ)(1 − µ3 )

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

587

We see that for the above-mentioned special value of t the curvature vanishes. Finally, let us assume that the calculus on G is classical (based on standard differential forms). Let us assume that Ψ is a bicovariant ∗ -calculus. This implies (X ⊗ id)ad0 (R0 ) = {0}, where X is the canonical generator of lie(G). In other words, Ψ is admissible (in the sense of the previous paper). Let us assume that Ψ is the minimal admissible (bicovariant ∗ −) calculus. The space Ψinv can be naturally identified with the algebra D consisting of all elements b ∈ B invariant under the left action of G on H (polynomial functions on a quantum 2-sphere). In terms of this identification $0 ↔ (φ0 D): D → D ⊗ B

( ) ◦ b ↔ κ(b(1) )( )b(2) .

Let τ be the element corresponding to 1 ∈ D and let us assume that L⊥ is spanned by τ . Explicitly, 2 π 0 (µ2 α + α∗ ) . (6.66) τ= 2 µ −1 The element τ is right-invariant (L⊥ consists precisely of right-invariant elements of Ψinv and each integer-valued irreducible multiplet appears without degeneracy in the decomposition of $0 into irreducible components), and τ ◦ b = 0 (b)τ .

(6.67)

In particular σ(τ ⊗ ϑ) = ϑ ⊗ τ σ(ϑ ⊗ τ ) = τ ⊗ ϑ .

(6.68)

Let us assume that the higher-order calculus on the bundle is described by the exterior algebra Ψ∨ . We have τ2 = 0 dτ = 0. Identities (6.67) and (6.68) imply τ w = (−1)∂w wτ for each w ∈ Ψ∨ . The connection ω corresponding to L⊥ is regular and multiplicative. Moreover, ω is flat in the sense that Rω = 0. 6.6. A constructive approach to differential calculus Every regular connection ω induces the isomorphism mω : vh(P ) ↔ Ω(P ) (if T(P ) = {0}). Moreover, if two algebras are identified with the help of mω then it is possible to express the differential structure on Ω(P ) in terms of the algebra structure on vh(P ), and the following maps: Dω : hor(P ) → hor(P ) ∧ d: Γ∧ inv → Γinv

Rω : Γinv → hor(P )

F ∧ : hor(P ) → hor(P ) ⊗ A .

´ M. DURDEVIC

588

Explicitly, d∧ (ϕ ⊗ ϑ) = Dω (ϕ) ⊗ ϑ + (−1)∂ϕ

X

ϕk ⊗ π(ck )ϑ + (−1)∂ϕ ϕd∧ (ϑ) ,

(6.69)

k

where d∧ is the corresponding differential on vh(P ) and d∧ Γ∧ inv is fixed by d∧ (ϑ) = d(ϑ) + Rω (ϑ) , for ϑ ∈ Γinv , and extended on Γ∧ inv by the graded Leibniz rule. It is worth noticing that the curvature Rω is completely determined by the covariant derivative, as easily follows from (4.29) and property (qpb4) from Sec. 3. Indeed, we have X qk Dω2 (bk ) (6.70) Rω π(a) = − k

where qk , bk ∈ B are such that (3.3) holds. In this subsection an “opposite” construction will be presented, which builds the algebra Ω(P ) and a regular multiplicative connection ω starting from a ∗ -algebra playing the role of horizontal forms, and three operators imitating the covariant derivative, curvature and the right action. Let P = (B, i, F ) be a quantum principal G-bundle over M . Let X ⊕ horkP horP = k≥0

be a graded ∗ -algebra such that hor0P = B. Further, let us assume that a gradepreserving ∗ -homomorphism F ? : horP → horP ⊗ A extending the map F is given such that ? ? ? (6.71) (F ⊗ id)F = (id ⊗ φ)F (id ⊗ )F ? = id .

(6.72)

Let us assume that a linear first-order map D: horP → horP is given such that the following properties hold D(ϕψ) = D(ϕ)ψ + (−1)∂ϕ ϕD(ψ) D∗ = ∗D

(6.73) (6.74)

F ? D = (D ⊗ id)F ? .

(6.75)

Finally, let us assume that there exists a linear map R: Γinv → horP such that X ϕk Rπ(ck ) , (6.76) D2 (ϕ) = − k

for each ϕ ∈ horP , where F ? (ϕ) = Σk ϕk ⊗ ck . Evidently, D plays the role of the covariant derivative. The map R is determined uniquely by the above condition. It plays the role of the curvature map. Explicitly

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Rπ(a) = −

X

qk D2 (bk )

589

(6.77)

k

where qk , bk ∈ B are such that (3.3) holds. We pass to a “reconstruction” of the calculus Ω(P ). Let us first analyze the map R in more details. Proposition 6.17. The following identities hold ?

F R = (R ⊗ id)$

(6.78)

DR = 0

(6.79)

R∗ = ∗R X ϕk R(ϑ ◦ ck ) . R(ϑ)ϕ =

(6.80) (6.81)

k

Proof. Acting by D ⊗ id on equality (3.3) and using (6.73) we obtain 0=

X

D(qk )F (bk ) +

X

k

qk (D ⊗ id)F (bk ) .

k

This, together with (6.75) and (6.76), implies 0=

X k

D(qk )D2 (bk ) +

X

qk D3 (bk ) = D

X

k

qk D2 (bk ) = −DRπ(a) .

k

Hence, (6.79) holds. Further, (3.3) implies 1⊗1⊗a=

X

F (qk )F (bkj ) ⊗ akj

kj

and hence 1 ⊗ κ(a(1) ) ⊗ a(2) =

X

qki bkj ⊗ zki ⊗ akj ,

(6.82)

ijk

where F (bk ) = Σj bkj ⊗ akj and F (qk ) = Σi qki ⊗ zki . From (6.82) we find 1 ⊗ κ(a) =

X

F (qk )bk

(6.83)

qk bk

(6.84)

qki F (bkj ) ⊗ zki akj .

(6.85)

k

(a)1 =

X k

1 ⊗ a(2) ⊗ κ(a(1) )a(3) =

X ijk

´ M. DURDEVIC

590

Identity (6.84) also directly follows from (3.3). A direct computation now gives X ∗ X ∗ 2 ∗ bk D (qk ) = − D2 (b∗k )qk∗ = Rπ(a) , R π(a)∗ = −Rπ κ(a)∗ = k

k

and similarly (R ⊗ id)$π(a) = Rπ(a(2) ) ⊗ κ(a(1) )a(2) X =− qki D2 (bkj ) ⊗ zki akj ijk

= −F ?

X

qk D2 (bk )

k ?

= F Rπ(a) . This proves (6.78) and (6.80). Finally, let us prove (6.81). We have X X qk bkj ϕn ⊗ akj cn = ϕn ⊗ acn , n

nkj

for each ϕ ∈ horP . Hence X X qk bkj ϕn ⊗ π(akj cn ) = ϕn ⊗ π(a) ◦ cn , n

nkj

if a ∈ ker(). The above equality, together with (6.76) implies X X X qk bkj ϕn Rπ(akj cn ) = − qk D2 (bk ϕ) = ϕn R π(a) ◦ cn . nkj

n

k

On the other hand, (6.77) and (6.84) imply X X qk D2 (bk )ϕ = − qk D2 (bk ϕ) . Rπ(a) ϕ = − k

k

Hence, property (6.81) holds. ∗

Let us consider the graded space Ω(P ) = horP ⊗Γ∧ inv , endowed with the following -algebra structure X ψϕk ⊗ (η ◦ ck )ϑ (6.86) (ψ ⊗ η)(ϕ ⊗ ϑ) = (−1)∂ϕ∂η ∗

(ϕ ⊗ ϑ) =

X

k

ϕ∗k

⊗ (ϑ∗ ◦ c∗k ) ,

(6.87)

k ∗ where F ? (ϕ) = Σk ϕk ⊗ ck . Algebras horP and Γ∧ inv are understandable as -subalgebras of Ω(P ), in a natural manner.

Lemma 6.18. satisfying

There exists the unique antiderivation d∧ : Γ∧ inv → Ω(P ) d∧ (ϑ) = R(ϑ) + d(ϑ)

for each ϑ ∈ Γinv .

(6.88)

591

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

Proof. The graded Leibniz rule implies that the values of d∧ on higher-order forms are completely determined by the restriction d∧ Γinv (and we have d∧ 1 = 0.) Hence, d∧ is unique, if exists. Let us prove that d∧ can be consistently constructed by extending, with the help of the graded Leibniz rule, a linear map (acting on Γinv ) given by (6.88). The extension exists iff contradictions do not appear at the level of second-order constraints defining the algebra Γ∧ inv . Simple transformations give o n π(a(1) )π(a(2) ) → Rπ(a(1) )π(a(2) ) − π(a(1) )π(a(2) )π(a(3) ) + π(a(1) )π(a(2) )π(a(3) ) − π(a(1) )Rπ(a(2) ) = Rπ(a(1) )π(a(2) ) − Rπ(a(3) ) π(a(1) ) ◦ κ(a(2) )a(4) = Rπ(a(2) )π κ(a(1) )a(3) = 0, for each a ∈ R. Hence, d∧ exists.

The formula d∧ (ϕ ⊗ ϑ) = D(ϕ) ⊗ ϑ + (−1)∂ϕ ϕd∧ (ϑ) + (−1)∂ϕ

X

ϕk ⊗ π(ck )ϑ

(6.89)

k

defines a linear first-order map d∧ : Ω(P ) → Ω(P ) extending d∧ introduced in the previous lemma. Proposition 6.19. The following identities hold d∧ ∗ = ∗d∧

(6.90)

∧ 2

(d ) = 0

(6.91)

d∧ (wu) = d∧ (w)u + (−1)∂w wd∧ (u) .

(6.92)

Proof. A direct calculation gives for w ∈ Γinv and u = ϕ ⊗ ϑ, X ϕk ⊗ (w ◦ ck )ϑ d∧ (wu) = (−1)∂ϕ d∧ =

X k

−

k

ϕk ⊗ d(w ◦ ck )ϑ +

X

X

X ϕk ⊗ (w ◦ ck ) d∧ (ϑ) + (−1)∂ϕ D(ϕk ) ⊗ (w ◦ ck )ϑ

k

+

X

ϕk R(w ◦ ck ) ⊗ ϑ

k

ϕk ⊗

(1) π(ck )(w

k

◦

(2) ck )ϑ

k

= −w D(ϕ) ⊗ ϑ − (−1)∂ϕ wϕd∧ (ϑ) + d∧ (w)u X w ϕk ⊗ π(ck )ϑ − (−1)∂ϕ k ∧

= d (w)u − wd∧ (u) .

´ M. DURDEVIC

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A similar computation shows that (6.92) holds for w ∈ horP , X wl ϕk ⊗ π(dl ck )ϑ + (−1)∂ϕ+∂w wϕd∧ (ϑ) + D(wϕ) ⊗ ϑ d∧ (wu) = (−1)∂ϕ+∂w lk

= D(w)ϕ ⊗ ϑ + (−1)∂w wD(ϕ) ⊗ ϑ + (−1)∂w+∂ϕ wϕd∧ (ϑ) X X wϕk ⊗ π(ck )ϑ + (−1)∂ϕ+∂w wl ϕk ⊗ π(dl ) ◦ ck ϑ + (−1)∂ϕ+∂w k ∧

∂w

= d (w)u + (−1)

kl ∧

wd (u) ,

where Σl wl ⊗ dl = F ? (w). It follows that (6.92) holds for arbitrary w, u ∈ Ω(P ). Let us prove that the square of d∧ vanishes. We have (d∧ )2 π(a) = d∧ −π(a(1) )π(a(2) ) + Rπ(a) = DRπ(a) + π(a(1) )π(a(2) )π(a(3) ) − π(a(1) )π(a(2) )π(a(3) ) − Rπ(a(1) ) π(a(2) ) + π(a(1) )Rπ(a(2) ) + Rπ(a(2) )π κ(a(1) )a(3) = Rπ(a(3) ) π(a(1) ) ◦ κ(a(2) )a(4) − Rπ(a(1) ) π(a(2) ) + Rπ(a(2) )π κ(a(1) )a(3) =0 for each a ∈ A. Further, X ϕk π(ck ) (d∧ )2 ϕ = d∧ Dϕ + (−1)∂ϕ = D2 ϕ − (−1)∂ϕ

X

k

D(ϕk )π(ck ) + (−1)∂ϕ

k

+

X

D(ϕk )π(ck )

k

o Xn (1) (2) ϕk π(ck )π(ck ) + ϕk Rπ(ck ) + ϕk dπ(ck ) = 0 , k

for each ϕ ∈ horP . Having in mind that spaces horP and Γinv generate Ω(P ), and using the fact that the square of an antiderivation is a derivation we conclude that (d∧ )2 = 0. Finally, X ϕ∗k π(c∗k ) d∧ (ϕ∗ ) = D(ϕ∗ ) + (−1)∂ϕ k

= (Dϕ)∗ + (−1)∂ϕ

X k

∂ϕ

= Dϕ + (−1)

X k

∧

∗

= d (ϕ)

ϕ∗k π(ck )∗ ◦ ck (2)

∗ ϕk π(ck )

(1)∗

593

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

for each ϕ ∈ horP . It follows that d∧ is a hermitian map.

We are going to prove that Ω(P ) satisfies condition (diff 2). The formula b Fb (ϕ ⊗ ϑ) = F ? (ϕ)$(ϑ)

(6.93)

b Γ∧ extending F ? and $. b determines a linear map Fb : Ω(P ) → Ω(P ) ⊗ Proposition 6.20. The map Fb is a homomorphism of differential ∗ -algebras. Proof. For each ϕ ∈ horP and ϑ ∈ Γinv we have X Fb ϕk (ϑ ◦ ck ) Fb (ϑϕ) = (−1)∂ϕ k

∂ϕ

= (−1)

X k

+ (−1)∂ϕ ∂ϕ

= (−1)

(1) (2) ϕk ⊗ ck (ϑ ◦ ck )

X

X

(1)

(3)

(2)

(4)

(ϕk ⊗ ck )(ϑl ◦ ck ⊗ κ(ck )al ck )

kl

ϕk ⊗ (ϑck ) + (−1)∂ϕ

X

k

(1)

(2)

ϕk (ϑl ◦ ck ) ⊗ al ck

kl

X = 1 ⊗ ϑ + $(ϑ) ϕk ⊗ ck k

= Fb (ϑ)Fb (ϕ) . Here,

X

?

k

ϕk ⊗ ck = F (ϕ) and $(ϑ) =

X l

ϑl ⊗ al . Using the facts that F ? and generate Ω(P ), we conclude that Fb is

$ b are multiplicative, and that horP and Γinv multiplicative, too. Let us prove that Fb intertwines differentials. We have X ∧ ∂ϕ b b ϕk π(ck ) F d (ϕ) = F D(ϕ) + (−1) =

X

k

Dϕk ⊗ ck + (−1)∂ϕ

k

+ (−1)∂ϕ =

X

=

k

X

(2)

(3)

(1)

(2)

(4)

ϕk π(ck ) ⊗ ck κ(ck )ck

Dϕk ⊗ ck + (−1)∂ϕ

+ (−1)∂ϕ

(1)

ϕk ⊗ ck π(ck )

k

k

k

X

X

X

X

ϕk ⊗ dck

k (1)

(2)

ϕk π(ck ) ⊗ ck

k

d∧ (ϕk ) ⊗ ck + (−1)∂ϕ

X k

ϕk ⊗ dck

= d∧ ⊗ id + (−1)∂ϕ id ⊗ d Fb(ϕ) .

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Further, if ϑ ∈ Γinv then b Fbd∧ (ϑ) = F ? R(ϑ) + $(dϑ) = (R ⊗ id)$(ϑ) + 1 ⊗ dϑ + (d ⊗ id)$(ϑ) − (id ⊗ d)$(ϑ) = (d∧ ⊗ id)$(ϑ) − (id ⊗ d)$(ϑ) + 1 ⊗ dϑ = d∧ ⊗ id + (−1)∂∗ id ⊗ d 1 ⊗ ϑ + $(ϑ) b . = d∧ ⊗ id + (−1)∂∗ id ⊗ d $(ϑ) Hence, Fb preserves differential structures. Finally, X Fb ϕ∗k ⊗ (ϑ∗ ◦ c∗k ) Fb (ϕ ⊗ ϑ)∗ = k

=

X

ϕ∗k (ϑ∗i ◦ ck

(1)∗

) ⊗ wi∗ ck

(2)∗

ki

=

X

(ϕk ϑi )∗ ⊗ wi∗ c∗k

ki

∗ = Fb (ϕ ⊗ ϑ) for each ϕ ∈ horP and ϑ ∈ Γ∧ b = Σi ϑi ⊗ wi . Thus, Fb is a hermitian inv , where $(ϑ) map. Clearly, B = Ω0 (P ). The algebra of horizontal forms corresponding to Ω(P ) coincides with the initial one. In other words, we can write horP = hor(P ). Further, F ? coincides with (the restriction of) the corresponding right action F ∧ . Let us consider a map ω: Γinv → Ω(P ) given by ω(ϑ) = 1 ⊗ ϑ. Proposition 6.21. (i) The map ω is a regular multiplicative connection on P . In particular , T(P ) = {0}. (ii) We have R = Rω and D = Dω . Proof. It is evident that ω is a hermitian map. According to the definition of Fb , we have Fb ω(ϑ) = $(ϑ) b = $(ϑ) + 1 ⊗ ϑ = (ω ⊗ id)$(ϑ) + 1 ⊗ ϑ . Hence, ω is a connection on P . Multiplicativity of ω directly follows from the fact ∧ ∧ that Γ∧ inv is a subalgebra of Ω(P ). In particular, ω (ϑ) = 1 ⊗ ϑ for each ϑ ∈ Γinv . Regularity follows from the definition of the product in Ω(P ). Finally, (ii) follows from definitions of Rω , Dω and d∧ . The corresponding factorization map mω reduces to the identity. It is worth noticing that the construction of the algebra ver(P ) of verticalized forms can be understood as a trivial special case of the construction presented in this subsection. Indeed, if we define horP = B (with horkP = {0} for k ≥ 1), D = 0 (and hence R = 0) and F ? = F then Ω(P ) = ver(P ) and Fb = Fbv . The algebra (vh(P ), dvh ) can be viewed in a similar way.

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

595

Appendix A. On Bicovariant Exterior Algebras In the presented theory we have assumed that the higher-order calculus on the structure quantum group is described by the corresponding universal envelope. This assumption is conceptually the most natural. However all the formalism can be repeated (straightforwardly, or with natural modifications) if the higher-order calculus on G is described by an appropriate non-universal differential structure. Here, it will be assumed that the higher-order calculus on G is based on the corresponding bicovariant exterior algebra [14]. The appendix is devoted to the analysis of some aspects of these structures, interesting from the point of view of differential calculus on quantum principal bundles. Let Γ be a bicovariant first-order differential calculus over G and let us consider the canonical flip-over automorphism σ: Γ ⊗A Γ → Γ ⊗A Γ (its “left-invariant” restriction is given by (4.16)). The corresponding exterior algebra [14] Γ∨ can be constructed by factorizing Γ⊗ through the ideal S ∨ = ker(A) . X⊕ An is the “total antisymmetrizer” map, with An : Γ⊗n → Γ⊗n given Here A = n by X (−1)π σπ An = π∈Sn

where σπ is the operator obtained by replacing transpositions i ↔ i + 1 figuring in a minimal decomposition of π, by the corresponding σ-twists (this definition is consistent, due to the braid equation for σ). By definition, A acts as the identity transformation on A and Γ. The following decomposition holds Ak+l = (Ak ⊗ Al )Akl where Akl =

X

(A.1)

(−1)π σπ−1

π∈Skl

and Skl ⊆ Sk+l is the subset consisting of permutations preserving the order of the first k and the last l factors. The differential map d: A → Γ can be naturally extended to the differential on the whole Γ∨ . By universality, there exists the unique graded-differential homomorphism I∧∨ : Γ∧ → Γ∨ reducing to identity maps on A and Γ. If Γ is ∗ -covariant then the ∗ -involutions on Γ and A can be extended to the ∗ -structure on Γ∨ , so that I∧∨ is a hermitian map. Proposition A.1. (i) There exists the unique differential algebra homomorb Γ∨ extending the map φ. phism φ∨ : Γ∨ → Γ∨ ⊗ (ii) There exists the unique graded-antimultiplicative extension κ∨ : Γ∨ → Γ∨ of the antipode κ, satisfying (A.2) κ ∨ d = d κ∨ .

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(iii) The following identities hold (φ∨ ⊗ id)φ∨ = (id ⊗ φ∨ )φ∨

(A.3)

m∨ (κ∨ ⊗ id)φ∨ = m∨ (id ⊗ κ∨ )φ∨ = 1∨

(A.4)

(∨ ⊗ id)φ∨ = (id ⊗ ∨ )φ∨ = id

(A.5)

∨

∨

∨

where m is the product map in Γ and is a linear functional specified by ∨ (ϑ) = p0 (ϑ) . (iv) If Γ is ∗ -covariant then (∗κ∨ )2 (ϑ) = ϑ

(A.6)

φ∨ ∗ = (∗ ⊗ ∗)φ∨ .

(A.7)

Proof. Uniqueness of maps φ∨ and κ∨ is a consequence of the fact that Γ∨ is generated, as a differential algebra, by A. Let κ!: Γ → Γ be the canonical extension of the antipode map [4, Appendix B]. This map is explicitly given by the diagram A⊗ Ψ ⊗ A ←−−−−−−−−−−−−−−   κ ⊗ id ⊗ κ  y A ⊗ Ψ ⊗ A −−−−−−−−−−−−−−→

Ψ     −κ! y Ψ

where the horizontal arrows denote the twofold multiplication and coaction map respectively. There exists the unique graded-antimultiplicative extension κ⊗ : Γ⊗ → Γ⊗ of κ and κ! We have (A.8) σκ⊗ (ϑ) = κ⊗ σ(ϑ) for each ϑ ∈ Γ ⊗A Γ. This directly follows from the fact that κ! maps left-invariant to right-invariant elements, and conversely. Hence, κ⊗ σπ = σjπj κ⊗

(A.9)

for each π ∈ Sn , where j is the “total inverse” permutation. This implies that κ⊗ commutes with operators An . Therefore κ⊗ can be factorized through the ideal S ∨ . In such a way we obtain a graded-antimultiplicative map κ∨ : Γ∨ → Γ∨ satisfying (A.2). Let us now consider a A ⊗ A-module Ψ given by Ψ = (A ⊗ Γ) ⊕ (Γ ⊗ A) . It is easy to see that Ψ is a bicovariant bimodule over the group G×G. In particular, the corresponding right and left actions of G × G on Ψ are given by ℘Ψ (a ⊗ ϑ) ⊕ (η ⊗ b) = φ(a)℘Γ (ϑ) ⊕ ℘Γ (η)φ(b) `Ψ (a ⊗ ϑ) ⊕ (η ⊗ b) = φ(a)`Γ (ϑ) ⊕ `Γ (η)φ(b) ,

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

597

where on the right-hand side the appropriate tensor multiplication is assumed. The following natural isomorphism holds Ψinv ∼ = Γinv ⊕ Γinv . Further, Ψ is a first-order calculus over G × G, in a natural manner. The corresponding differential map D: A ⊗ A → Ψ is given by D = d ⊗ id + id ⊗ d. In terms of the above identification the corresponding right A ⊗ A-module structure on Ψinv is given by (ϑ ⊕ η) ◦ (a ⊗ b) = (a)(ϑ ◦ b) ⊕ (η ◦ a)(b) (A.10) ⊗2 and the action of the corresponding flip-over operator Σ: Ψ⊗2 inv → Ψinv is determined by the block-matrix   σ 0 0 0 0 0 τ 0 Σ= (A.11)  0 τ 0 0 0 0 0 σ ⊗2 where τ : Γ⊗2 inv → Γinv is the standard transposition. This implies

b Γ∨ . Ψ∨ ∼ = Γ∨ ⊗ Let φ⊕ : Γ → Ψ be a map given by φ⊕ = `Γ ⊕ ℘Γ . The following identities hold Dφ = φ⊕ d

(A.12)

φ⊕ (ϑa) = φ⊕ (ϑ)φ(a)

(A.13)

φ⊕ (aϑ) = φ(a)φ⊕ (ϑ).

(A.14)

Equalities (A.13)–(A.14) imply that φ and φ⊕ can be consistently extended to a homomorphism φ⊗ : Γ⊗ → Ψ⊗ . The following inclusion holds φ⊗ ker(A) ⊆ ker(AΣ ) , where AΣ is the antisymmetrizer corresponding to Σ. Hence φ⊗ can be projected through ideals ker(A) and ker(AΣ ). In such a way we obtain the homomorphism b Γ∨ . Equality (A.12) implies that φ∨ intertwines the corresponding φ∨ : Γ∨ → Γ∨ ⊗ differentials. Properties (A.3)–(A.5) as well as (A.6)–(A.7) simply follow from analogous properties for φ and κ. It is worth noticing that κ∨ and φ∨ can be obtained by projecting κ∧ and φb from Γ∧ to Γ∨ . Let us now turn to the conceptual framework of the previous paper, and assume that M is a classical compact smooth manifold, and P a locally trivial principal G-bundle over M . Further, let us assume that Γ is the minimal admissible leftcovariant calculus over G (this calculus is bicovariant and ∗ -covariant, too). Let

´ M. DURDEVIC

598

τ = (πU )U∈U be an arbitrary trivialization system for P , and let Cτ be the corresponding G-cocycle, consisting of “transition functions” ψUV : S(U ∩ V ) ⊗ A → S(U ∩ V ) ⊗ A. The restrictions ϕUV = ψUV A are (∗ -homomorphisms) explicitly given by ϕUV = (gVU ⊗ id)φ where gVU : U ∩ V → Gcl represent the classical Gcl -cocycle corresponding to Cτ (understood here as ∗ -homomorphisms gVU : A S(U ∩ V )). → 1 There exists the unique map ]UV : Γ → Ω (U ∩ V ) ⊗ A ⊕ S(U ∩ V ) ⊗ Γ satisfying ]UV (aξ) = ϕUV (a)]UV (ξ) ]UV (da) = [d ⊗ id + id ⊗ d] ϕUV (a) , for each a ∈ A and ξ ∈ Γ. This also implies ]UV (ξa) = ]UV (ξ)ϕUV (a) ⊗ b Γ⊗ → Ω(U ∩ V ) ⊗ and hence there exists (the unique) homomorphism ϕ⊗ UV : Γ extending both ϕUV and ]UV . All antisymmetrizing operators are left and right covariant in a natural manner. In particular they are reduced in the corresponding spaces of left-invariant elements. In what follows we shall denote by the same symbols their restrictions in Γ⊗ inv (if there is no ambiguity from the context).

Proposition A.2. We have ∨ b ∨ ϕ⊗ UV [S ] ⊆ Ω(U ∩ V ) ⊗ S

(A.15)

for each (U, V ) ∈ N 2 (U). Proof. The ideal S ∨ is bicovariant. In particular, it has the form ∨ S∨ ∼ = A ⊗ Sinv ∨ where Sinv is the left-invariant part of S ∨ . The following equality holds X ∂ UV (ϑk ) ⊗ ck ]UV (ϑ) = 1 ⊗ ϑ +

(A.16)

k

for each ϑ ∈ Γinv . Here, Σk ϑk ⊗ ck = $(ϑ) and ∂ UV : Γinv → Ω(U ∩ V ) is the map specified by (A.17) ∂ UV π(a) = gVU (a(1) )d gUV (a(2) ) . Let us observe that ∂ UV (η)∂ UV (ζ) = −

X

∂ UV (ζk )∂ UV (ηk )

(A.18)

k

for each η, ζ ∈ Γinv , where Σk ζk ⊗ ηk = σ(η ⊗ ζ). Indeed this follows from (4.16), and from the identity (A.19) ∂ UV (ϑ ◦ a) = (a)∂ UV (ϑ) .

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

599

⊗ Now for an arbitrary ϑ ∈ Γ⊗n inv let us consider the elements ϑk = (id ⊗ pk )ϕUV (ϑ) where 0 ≤ k ≤ n. It follows from (4.16) and (A.16) that these elements have the form ⊗ idk )Alk (ϑ) (A.20) ϑk = (∇UV l UV are components with l = n − k. Here we have identified Γ⊗ and A ⊗ Γ⊗ inv , and ∇l UV UV of the unital multiplicative extension of the map ∇ = (∂ ⊗ id)$. According to (A.18) the above expression can be rewritten as

ϑk =

1 UV (∇ Al ⊗ idk )Alk (ϑ) . l! l

In particular (id ⊗ Ak )(ϑk ) =

1 UV (∇ ⊗ idk )An (ϑ) , l! l

according to (A.1). Hence, (A.15) holds. ϕ⊗ UV

(A.21)

∨

can be factorized through the ideal S . In such a way we obtain The map ∨ ∗ b ∨ a homomorphism ϕ∨ UV : Γ → Ω(U ∩ V )⊗Γ of graded-differential -algebras. Now the formula ∨ (α ⊗ ϑ) = αϕ∨ (A.22) ψUV UV (ϑ) b Γ∨ (extending both defines a graded-differential ∗ -automorphism of Ω(U ∩ V ) ⊗ ∨ ψUV and ϕUV ). These maps are Ω(U ∪ V )-linear and satisfy the following cocycle conditions ∨ ∨ ∨ ψVW (ϕ) = ψUW (ϕ) ψUV b Γ∨ . for each (U, V, W ) ∈ N 3 (U) and ϕ ∈ Ωc (U ∩ V ∩ W ) ⊗ Applying the above results, and using similar constructions as in the previous paper, it is possible to construct a graded-differential ∗ -algebra Λ(P ), representing the corresponding “differential forms” on the bundle. This algebra is locally trivial, in the sense that any local trivialization (U, πU ) of the bundle can be “extended” b Γ∨ ↔ Λ(P |U ). to a local representation of the form Ω(U ) ⊗ By construction, the right action F can be (uniquely) extended to the homob Γ∨ of graded-differential ∗ -algebras. Moreover, all morphism F ∨ : Λ(P ) → Λ(P ) ⊗ entities naturally appearing in the differential calculus on P constructed from the universal envelope Γ∧ have counterparts in the calculus based on Λ(P ), and all algebraic relations are preserved (because the formalism can be obtained by “projecting” the first calculus on Λ(P )). In a certain sense, Γ∨ is the minimal bicovariant graded algebra (built over Γ) compatible with all possible “transition functions” for the bundle P . Namely, let us assume that N ⊆ Γ⊗ is a bicovariant graded-ideal satisfying N k = {0} for k ∈ {0, 1} and b (A.23) ϕ⊗ UV (N ) ⊆ Ω(U ∩ V ) ⊗ N for each trivialization system τ = (πU )U∈U and each (U, V ) ∈ N 2 (U). This ensures the possibility to construct the corresponding global algebra for the bundle P , which b Γ⊗ /N . locally will be of the form Ω(U ) ⊗

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600

Lemma A.3. Under the above assumptions we have N ⊆ S∨ .

(A.24)

Proof. We shall prove inductively that k ∨k ⊆ Sinv , Ninv

(A.25)

2 . Applying (A.23) and for each k ≥ 2. Let us assume that Σi ϑi ⊗ ηi = ψ ∈ Ninv (A.16) we obtain X X X ∂ UV (ϑik )∂ UV (ηil ) ⊗ cik dil + ∂ UV (ϑik ) ⊗ cik ηi − ∂ UV (ηil ) ⊗ ϑi dil 0= ikl

ik

il

where $(ϑi ) = Σk ϑik ⊗ cik and $(ηi ) = Σl ηil ⊗ dil . In particular, X X ∂ UV (ϑik ) ⊗ cik ηi − ∂ UV (ηil ) ⊗ ϑi dil . 0= ik

il

In other words, modulo the identification Γ ↔ A ⊗ Γinv , we have (∇UV ⊗ id)(I − σ)(ψ) = 0 . Having in mind that the family of maps ∇UV distinguishes elements of Γinv (a consequence of the minimality of Γ) we conclude that ψ = σ(ψ). In other 2 ∨2 ⊆ ker(I − σ) = Sinv . words Ninv Let us assume that (A.25) holds for some k ≥ 2. Then UV ⊗ Ak )A1k (ψ) 0 = (id ⊗ Ak pk )ϕ⊗ UV (ψ) = (∇ k+1 . Because of the arbitrariness of τ it follows that for each ψ ∈ Ninv

Ak+1 (ψ) = (id ⊗ Ak )A1k (ψ) = 0 , k+1 ∨ k+1 ⊆ (Sinv ) . in other words Ninv

We pass to the study of the problem of passing from Γ∧ to Γ∨ , in the framework of the general theory. Let P = (B, i, F ) be a quantum principal G-bundle over a quantum space M . Lemma A.4. Modulo the natural identification Γ∧ ↔ A ⊗ Γ∧ inv we have (id ⊗ pk )Fbω ⊗n = (F ∧ ω ⊗l ⊗ [ ]∧ k )Alk

(A.26)

where n = k + l, and ω is an arbitrary connection on P . Proof. Essentially the same reasoning as in the proof of Proposition A.2, applying identity (4.8) instead of (A.16). The above equation implies ∨ ∨ b Γ∧ + Ω(P ) ⊗ b [S ∨ ]∧ , ) ⊆ ω ⊗ (Sinv )⊗ Fb ω ⊗ (Sinv

(A.27)

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

601

for each ω ∈ cn(P ). Let us assume that the calculus (described by Ω(P )) admits regular connections, and that T(P ) = {0} (multiplicativity of regular connections). Let Υ(P ) ⊆ Ω(P ) be the space linearly generated by elements of the form ∨ and ϕ ∈ hor(P ), while ω ∈ rcn(P ). ϕω ⊗ (ϑ) where ϑ ∈ Sinv Lemma A.5. (i) The space Υ(P ) is a (two-sided) graded-differential ∗ -ideal in Ω(P ), independent of the choice of a regular connection ω. (ii) We have

b [S ∨ ]∧ . b Γ∧ + Ω(P ) ⊗ (A.28) Fb Υ(P ) ⊆ Υ(P ) ⊗ −1 ∨ ∧ ∗ Proof. The space mω Υ(P ) = hor(P ) ⊗ [Sinv ] is a graded two-sided -ideal in vh(P ). Hence the space Υ(P ) is the graded ∗ -ideal in Ω(P ). Inclusion (A.28) directly follows from (A.27). Let us prove that d Υ(P ) ⊆ Υ(P ). It is sufficient to check that dω ⊗ (ϑ) ∈ Υ(P ) ∨ . The following equality holds for each ϑ ∈ Sinv (A.29) dω ⊗ (ϑ) = ω ⊗ δ ∗ (ϑ) + mΩ Rω ⊗ ω ⊗ A1k−1 (ϑ) ⊗ ⊗ ∗ for each k ≥ 2 and ϑ ∈ Γ⊗k inv . Here, δ : Γinv → Γinv is the unique (hermitian) antiderivation extending a given embedded differential map δ. In particular, if ∨k then both summands on the right-hand side of the above equality belong ϑ ∈ Sinv ∨k ∨ k+1 ∨k ∨ k−1 ) ⊆ (Sinv ) and A1k−1 (Sinv ) ⊆ Γinv ⊗ (Sinv ) . to Υ(P ), because of δ ∗ (Sinv Finally, let ζ be an arbitrary (hermitian) tensorial 1-form satisfying (4.13). We then have X 1 mΩ (ζ ⊗k Ak ⊗ ω ⊗l )Akl (ϑ) (ω + ζ)⊗ (ϑ) = k! k+l=n

for each ϑ ∈

Γ⊗n inv .

This shows that Υ(P ) is ω-independent.

Hence, it is possible to pass jointly to the factoralgebra Λ(P ) = Ω(P )/S ∨ (as a representative of the calculus on the bundle), and to the exterior bicovariant algebra Γ∨ (representing the calculus on G). This factorization does not change the first-order differential structure. For this reason the spaces of connection forms associated to both calculi on P are the same. Further, the spaces hor(P ) and Ω1 (P ) are preserved. This implies that regular connections relative to Ω(P ) and Λ(P ) are the same. By construction, regular connections are multiplicative, relative to Λ(P ), too. Lemma A.3 establishing the “minimality” of the exterior algebra has a general quantum counterpart. Let us assume that the calculus on the bundle is such that tensorial 1-forms ζ satisfying (4.13) distinguish elements of Γinv . In the case of bundles over classical compact manifolds [4] the algebra Ω(P ) built from the minimal admissible calculus Γ possesses this property. Let us consider a bicovariant graded-ideal N ⊆ Γ⊗ satisfying N k = {0} for k ∈ {0, 1}. The space Nω = mω hor(P ) ⊗ [Ninv ]∧ is a two-sided ideal in Ω(P ). Let us assume that Nω is independent of ω ∈ rcn(P ). Lemma A.6. Under the above assumptions N ⊆ S ∨ .

602

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The construction of “global” differential calculus on the bundle given in Subsec. 6 can be applied to the exterior algebra case, too. Starting from algebras horP and Γ∨ inv it is possible to construct a graded ∗ ∨ ∨ -algebra structure on Λ(P ) = horP ⊗ Γinv . Using maps R, D, and d: Γ∨ inv → Γinv ∨ it is possible to construct a natural differential d on Λ(P ). All constructions are the same as in the universal envelope case. The only nontrivial point is to prove the analog of Lemma 6.18. Let R? : Γ⊗ inv → Λ(P ) be the unique antiderivation extending R. The following equality holds (A.30) R? (η) = R ⊗ [ ]∨n A1n (η) n+1 ∨ . This implies that R? can be factorized through Sinv . In for each η ∈ (Γ⊗ inv ) ∨ ∨ ∨ ∨ such a way we obtain the map R : Γinv → Λ(P ). Finally, the map d : Γinv → Λ(P ) is defined by d∨ (ϑ) = d(ϑ) + R∨ (ϑ) .

Appendix B. Multiple Irreducible Submodules In this appendix the structure of the ∗ -algebra B of functions on a quantum principal bundle P = (B, i, F ) will be analyzed from the viewpoint of the representation theory [13] of the structure quantum group. For every finite-dimensional unitary representation u of G, let us denote by Hu the corresponding carrier Hilbert space. In other words, u is a comodule structure map u: Hu → Hu ⊗ A. Furthermore, let us denote by Eu = Mor(u, F ) the space of all intertwiners ϕ: Hu → B between representations u and F . The spaces Eu are V-bimodules, in a natural manner. Moreover, a detailed analysis [8] shows that the structure of the bundle P is completely expressible in terms of these bimodules. Geometrically, V-bimodules Eα are interpretable as (spaces of smooth sections of) the associated vector bundles. Let T be the complete set of mutually inequivalent irreducible unitary representations of G. Let us assume that every α ∈ T is acting in a standard unitary space Hα = Cnα , where nα is the dimension of α. Let (e1 , . . . , enα ) be the absolute basis in Cnα . We have α(ei ) =

nα X

ej ⊗ αji ,

j=1

where αij ∈ A are the corresponding matrix elements. The representation F : B → B ⊗ A of G in B is highly reducible. For each α ∈ T let B α ⊆ B be the multiple irreducible subspace corresponding to α. Evidently, B α is a bimodule over V. We have X ⊕ α B . B= α∈T

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

603

Bimodules Eα ⊗ Cnα and B α are naturally isomorphic, via the correspondence ϕ ⊗ x ↔ ϕ(x) .

(B.1)

This identification intertwines F B α and id ⊗ α. Irreducible G-multiplets in B α are of the form ϕ(e1 ), . . . , ϕ(enα ) , for some intertwiner ϕ. Let us fix i, j ∈ {1, . . . , nα }. There exist intertwiners µ1 , . . . , µd ∈ Eα and numbers cklpq ∈ C, where k, l ∈ {1, . . . , d} and p, q ∈ {1, . . . , nα }, such that X

α cklpq bα∗ kp F (blq ) = 1 ⊗ αij ,

(B.2)

klpq

where bα kp = µk (ep ). Equivalently X

α ∗ cklpq F (bα∗ kp )blq = 1 ⊗ αji .

(B.3)

klpq

The above equalities imply that the summation over indexes satisfying (p, q) 6= (i, j) can be dropped. In other words, X

α ckl bα∗ ki F (blj ) = 1 ⊗ αij

(B.4)

α ∗ ckl F (bα∗ ki )blj = 1 ⊗ αji

(B.5)

kl

X kl

α where ckl = cklij . From the fact that bα k1 , . . . , bkn form a G-multiplet it follows that the above equalities hold for arbitrary i, j ∈ {1, . . . , nα }. Equalities (B.4)–(B.5) imply that the numbers ckl can be always choosen such that the matrix (ckl ) is hermitian. Further, without a lack of generality we can assume that the matrix (ckl ) is nonsingular. In what follows, it will be assumed that the matrix (ckl ) is positive. Diagonalizing this matrix, and redefining in the appropriate way intertwiners µk we obtain X α bα∗ (B.6) ki F (bkj ) = 1 ⊗ αij k

X

α ∗ F (bα∗ ki )bkj = 1 ⊗ αji .

(B.7)

k

Equivalently, the following identities hold X α bα∗ ki bkj = δij .

(B.8)

k

In particular, it follows that all irreducible representations appear in the decomposition of F into irreducible components (all V-bimodules B α are non-trivial).

´ M. DURDEVIC

604

Let us consider, for each f ∈ V, the elements X α∗ bα %kl (f ) = ki f bli .

(B.9)

i

These elements are F -invariant and the following identities hold X

%kl (f )∗ = %lk (f ∗ )

(B.10)

%kn (f )%nl (g) = %kl (f g) .

(B.11)

n

Indeed, we have F (bα ki ) =

X

bα j kj

⊗ αji , and a direct computation gives

X α∗ F (bα F %kl (f ) = ki f bli ) i

=

X

α∗ ∗ bα km f bln ⊗ αmi αni

imn

=

X

α∗ bα km f bln ⊗ δmn

mn

= %kl (f ) ⊗ 1 , and similarly X X X X α∗ α α∗ α∗ α∗ %kn (f )%nl (g) = bα bα bα ki f bni bnj gblj = ki f gblj δij = ki f gbli = %kl (f g) . n

nij

ij

i

In other words, the maps %kl : V → V realize a ∗ -homomorphism %: V → Md (V), where Md (V) is the ∗ -algebra of d × d-matrices over V. In particular, pr1 = %(1) is a projector in Md (V). Let us consider the free left V-module V d , with the absolute basis (ε1 , . . . , εd ). The elements of the algebra Md (V) are understandable as endomorphisms (acting on the right) of V d , in a natural manner. Explicitly, this realization is given by X Akl εl . (B.12) (εk )A = l

Let E ⊆ V d be the left V-submodule determined by the projector pr1 . Evidently, E is %-invariant and %, together with the left multiplication, determine a unital V-bimodule structure on E. Let H1 : V d → Eα be the left V-module homomorphism given by H1 (εk ) = µk .

(B.13)

H1 ψ%(f ) = H1 (ψ)f ,

(B.14)

The following identity holds

in particular H1 (ψpr1 ) = H1 (ψ).

GEOMETRY OF QUANTUM PRINCIPAL BUNDLES II

605

This implies that the restriction (H1 E): E → Eα is a homomorphism of unital V-bimodules. Now we shall prove that H1 E is bijective. Let us assume that ψ ∈ ker(H1 ). α∗ This implies Σki ψk bα ki bli = 0, for each l ∈ {1, . . . , d} where ψ = Σk ψk εk . In other words, ψpr1 = 0, which means that H1 E is injective. We have µ = Σk qk µk for each µ ∈ Eα , where qk ∈ V are elements given by qk = Σi µ(ei )bα∗ ki . In other words, elements µk span the left V-module Eα . This implies that H1 is surjective. Hence, Eα are finite and projective, as left V-modules. This implies that left V-modules B α are finite and projective, too. Relations (B.8) can be rewritten in the form B † B = In ,

(B.15)

where B is a d × n matrix with coefficients bα ki and In is the unit matrix in Mn (B). Let us assume that the following additional relations hold (ZBC −1 )> B ∗ = In ,

(B.16)

where C ∈ Mn (C) is the canonical intertwiner [13] between α and its second contragradiant αcc , and Z ∈ Md (C) is a strictly positive matrix. Relations of this type naturally appear in a C ∗ -algebraic version of the theory of quantum principal bundles. The matrix Z is connected with modular properties of an appropriate invariant integral on the bundle. Let us consider a map λ: V → Md (V) given by X −1 bα∗ ]li . (B.17) λkl (f ) = ki f [ZBC i

We then have λ(f )† = Z ∗ λ(f ∗ )(Z ∗ )−1

(B.18)

λ(f )λ(g) = λ(f g) .

(B.19)

Further, the elements of Md (V) are naturally identificable with endomorphisms of the right V-module V d (acting on the left). Let F ⊆ V d be a right V-submodule determined by a projection pr2 = λ(1). The map λ induces a unital left V-module structure on F , so that F becomes a V-bimodule. Let H2 : V d → Eα be a right V-module homomorphism given by H2 (εk ) = νk , where νk = Σl Zkl µl . Then the restriction (H2 F ): F → Eα is a bimodule isomorphism. It is important to mention that the established finite projectivity property extends [8] to all bimodules Eu . A similar consideration can be applied to all covariant algebras figuring in the game. In particular, X ⊕ α H hor(P ) = α∈T

where H are corresponding α-multiple irreducible subspaces (relative to the decompositions of F ∧ ). These spaces are Ω(M )-bimodules. The following natural α

606

´ M. DURDEVIC

decompositions hold Hα = Fα ⊗ Cnα , where now Fu = Mor(u, F ∧ ) are graded Ω(M )-bimodules representing vector-bundle valued forms on the base space. Every ϕ ∈ hor(P ) can be written in the form ϕ = Σk wk bk , where wk ∈ Ω(M ) and bk ∈ B. Indeed, it is sufficient check the statement for elements of some irreducible multiplet. Moreover, we have natural decompositions Fu ↔ Eu ⊗V Ω(M ) ↔ Ω(M ) ⊗V Eu

(B.20)

hor(P ) ↔ B ⊗V Ω(M ) ↔ Ω(M ) ⊗V B .

(B.21)

Let us assume that ϕ1 , . . . , ϕnα ∈ Hα is an irreducible α-multiplet. We then have X α ϕj bα∗ ϕi = kj bki jk

for each i ∈ {1, . . . , nα }. On the other hand all Σj ϕj bα∗ kj belong to Ω(M ). + 1 In particular, if hor (P ) is generated by hor (P ) and if every first-order horizontal form ϕ can be written as ϕ = Σk bk d(gk ), where bk ∈ B and gk ∈ V, then the spaces Hα are linearly spanned by elements of the form hα = bα d(f1 ) . . . d(fn ), where fk ∈ V and bα ∈ B α . References [1] T. Brzezinski and S. Majid, Quantum group gauge theory on quantum spaces, Commun. Math. Phys. 157 (1993) 591–638. [2] A. Connes, Non-commutative differential geometry, Extrait des Publications Math´ematiques–IHES 62 (1986). [3] A. Connes, Noncommutative Geometry, Academic Press, 1994. [4] M. Durdevi´c, Geometry of quantum principal bundles I, Commun. Math. Phys. 175 (1996) 457–521. [5] M. Durdevi´c, Quantum principal bundles and Hopf–Galois extensions, preprint, Inst. of Math. UNAM, M´exico, 1995. [6] M. Durdevi´c, Characteristic Classes of quantum principal bundles, preprint, Inst. of Math. UNAM, M´exico, 1995. [7] M. Durdevi´c, Quantum Classifying Spaces and Universal Quantum Characteristic Classes, Lectures, QGroups and QSpaces Minisemester, Stefan Banach International Mathematical Center, November ‘95, BC Publications, 40 (1997) 315–327. [8] M. Durdevi´c, Quantum principal bundles and Tannaka–Krein duality theory, Rep. Math. Phys. 38(3) (1996) 313–324. [9] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publ., New York, London, 1963. [10] P. Podles, Quantum spheres, Lett. Math. Phys. 14 (1987) 193–202. [11] H. J. Schneider, Hopf–Galois Extensions, Cross Products, and Clifford Theory Advances in Hopf Algebras, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 158 (1994). [12] S. L. Woronowicz, Twisted SU(2) group. An example of a non-commutative differential calculus, RIMS Kyoto Univ. 23 (1987) 117–181. [13] S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987) 613–665. [14] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups/quantum groups, Commun. Math. Phys. 122 (1989) 125–170.

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Note added in proof The first version of this paper was written at the Faculty of Physics, Belgrade University, Serbia. The extended version was written at the Institute of Mathematics, UNAM, Mexico. This work has been partially supported by the Investigation Project IN106897 of DGAPA/UNAM.

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY HAGEN NEIDHARDT

E-mail :

Universit¨ at Potsdam Institut f¨ ur Mathematik P. O. Box 601553 14415 Postdam Deutschland [email protected]

VALENTIN ZAGREBNOV Universit´ e de la M´ editerran´ ee D´ epartement de Physique CPT-Luminy, Case 907 13288 Marseille Cedex 9 France E-mail : [email protected] Received 15 April 1996 Let the pair of self-adjoint operators {A ≥ 0, W ≤ 0} be such that: (a) there is a dense domain D ⊆ dom(A) ∩ dom(W ) such that H˙ = (A + W )|D is semibounded from below (stability domain), (b) the symmetric operator H˙ is not essentially self-adjoint (singularity of the perturbation), ˆ of A˙ = A|D is maximal with respect to W , i.e., (c) the Friedrichs extension A √ dom( −W ) ∩ ker(A˙ ∗ − ηI) = {0}. η < 0. Let {Wn }∞ n=1 be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W . The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians ˜n = A ˜ + Wn exists and is unique for any self-adjoint extension A ˜ of A, ˙ (ii) to H describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence {Wn }∞ n=1 such ˜ + Wn tends in the strong resolvent sense to unique (right Hamiltonian) ˜n = A that H . ˆ =A ˆ + W , otherwise the limit is not unique. H

1. Introduction History of the definition of the “right Hamiltonian” is rather long (see e.g. Simon [12]). But we would like to mention that the “physical approach” to definition of the Hamiltonian in case of singular interactions is always based on a paradigm of removing cut-offs or regularization. The papers [6, 7, 11, 12, 13, 15] scrutinize this approach in the framework of the abstract scheme of functional analysis. From this point of view all singular perturbations could be divided into two complementary classes: positive and non-positive. 609 Reviews in Mathematical Physics, Vol. 9, No. 5 (1997) 609–633 c World Scientific Publishing Company

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H. NEIDHARDT and V. ZAGREBNOV

ˆ proposed in [11, 15] for positive perturbations is a The “right Hamiltonian” H straightforward application of the property of semiboundedness of the perturbation W and the monotonicity of the regularizing sequence of cut-off potentials. In this . ˆ = A + W as the right Hamiltonian case one gets automatically the form sum H which is expected. The situation drastically changes when we come to non-positive perturbations which are singular enough that the operator A + W on dom(A) is not semibounded from below. The first abstract result in this direction is due to [7]. The important (for the mathematical and physical comprehension of the situation) notion of the “stability” domain D ⊆ dom(A) ∩ dom(W ) with the property (1.1) (see below) was introduced there. Then, in the paper [8] the notion of an admissible regularizing sequence was added. These two notions are crucial for the understanding how the regularizing sequence chooses itself the “right” (i.e. unique) regularized Schr¨ odinger operator. In [8] the phenomenon was clarified on an abstract operator-theoretical level. Moreover, the conditions formulated in [7] were essentially relaxed which improves the main result of [7] for finite deficiency indices case and gives a natural sufficient condition for the case of infinite deficiency indices. More precisely the situation can be described as follows. Let A ≥ 0 be a selfadjoint operator on some separable Hilbert space H which can be regarded as the Hamiltonian of some quantum system. Let this operator be perturbed by selfadjoint operator W . With respect to quantum mechanics the operator A can be thought as the Laplacian −∆ on L2 (Rn ) and W as a multiplication operator which arises from some potential. We assume that the perturbation W is form bounded with respect to A on some dense subset D (called stability domain) of dom(A) ∩ dom(W ), i.e. |(W f, f )| ≤ a(Af, f ) + b(f, f ),

0 < a < 1,

0 < b,

(1.1)

f ∈ D ⊆ dom(A) ∩ dom(W ) . ˙ Let us introduce the naturally defined symmetric operator H, ˙ = Af + W f, Hf

˙ = D, f ∈ dom(H)

(1.2)

which is semibounded from below by (1.1). We are interested in the case that H˙ is not essentially self-adjoint, i.e., W is a singular perturbation for A. Since for singular perturbations H˙ is not essentially self-adjoint, it is unclear which self-adjoint operator is a right Hamiltonian for the perturbed quantum system. The only requirement is that the right Hamiltonian has to be a semibounded ˙ Since it is a priori unclear which of them is the right self-adjoint extension of H. one we have to apply additional arguments to find out this extension [8]. A standard procedure goes as follows [6, 7, 8, 9, 11, 15]. We choose a sequence of bounded self-adjoint operators {Wn }∞ n=1 which tends in the strong resolvent sense to W as n −→ ∞. Since the sequence {Wn }∞ n=1 , which is called a regularizing sequence for the singular perturbation W , consists of bounded operators we have no problems to associate with it a sequence of self-adjoint operators {Hn }∞ n=1

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

Hn f = Af + Wn f,

f ∈ dom(Hn ) = dom(A) .

611

(1.3)

{Hn }∞ n=1

is called the approximating sequence. If the approximatThe sequence ing sequence tends in the strong resolvent sense to some semibounded self-adjoint ˙ then one believes that this H is the right Hamiltonian for the extension H of H, singular perturbed system. However, if A˙ = A|D is not essentially self-adjoint, then there are a lot of self-adjoint extensions A˜ of A˙ = A|D which represent different unperturbed systems. Moreover, replacing A by A˜ one again satisfies condition (1.1) Since ˙ = Af + W f = Af ˜ + W f, Hf f ∈D (1.4) ˜ Hence, the the operator W is singular not only for A but for all extensions A. ˜ Therefore, problem of finding out the right Hamiltonian arises for all extensions A. using the same method as before one has to investigate the convergence of the ˜ n }∞ approximating sequences {H n=1 , ˜ + Wn f, ˜ n f = Af H

˜ n ) = dom(A) ˜ f ∈ dom(H

(1.5)

˜ for different A. ˜ and to identify their limits H As it follows from above the perturbation W is singular as well for the Friedrichs extension Aˆ of A˙ = A|D. Using (1.1) one can prove [8] that the approximating sequence ˆ + Wn f, ˆ n ) = dom(A), ˆ ˆ n f = Af f ∈ dom(H (1.6) H converges in the strong resolvent sense to the so-called form extension .

ˆ = Aˆ + W , H

(1.7)

which means the extension given by the KLMN-Theorem [10]. Natural question which arises is to describe all extensions of H˙ which can be obtained by the above procedure. Surprisingly it turns out that under some natural ˆ is the only one which is available in this way. conditions the form sum H In the present paper this result is proven for non-positive singular perturbations, i.e., for W ≤ 0, (1.8) under the assumption that the Friedrichs extension Aˆ is maximal with respect to W . This means that if for some semibounded self-adjoint extension A˜ we have √ (1.9) dom(˜ ν ) ⊆ dom −W , ˜ then A˜ coincides with the Friedrichs where ν˜ is the quadratic form associated with A, √ extension Aˆ and, obviously, ν˜ = νˆ. Note that one always has dom(ˆ ν ) ⊆ dom( −W ) by (1.1). Under these assumptions we shall show that if the approximating sequence ˜ n }∞ {H n=1 , such that

p p

˜ n − z −1 |Wn | =m(z) 6= 0 , (1.10) sup |Wn | H

< +∞, n

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H. NEIDHARDT and V. ZAGREBNOV

and if it tends in the strong resolvent sense to some semibounded self-adjoint ex˜ of H, ˙ then H ˜ coincides with the form sum H ˆ (1.7). Conversely, for tension H ˙ there is a regularizing sequence any semibounded self-adjoint extension A˜ of A, ˜ n }∞ such that the corresponding approximating sequence {H {Wn }∞ n=1 n=1 tends in the strong resolvent sense to the form sum (1.7) and the condition (1.10) is satisfied. In other words, if one assumes that for a regularizing sequence and for a semibounded self-adjoint extension A˜ of A˙ the corresponding approximating se˜ n }∞ quence {H n=1 satisfies the condition (1.10), then either this sequence converges in the strong resolvent sense to the form sum (1.7) (if it happens the regularizing sequence is called well-chosen) or there is no convergence at all. It turns out, that for each semibounded self-adjoint extension of A˙ a well-chosen regularizing sequence always exists. Therefore, for any self-adjoint extension A˜ the well-chosen regularization “picks out” the right Hamiltonian which is in this case (1.7). It remains open the question whether there always exists a non-increasing (see ˜ n }∞ tends in the strong resolvent sense Sec. 4) regularizing sequence such that {H n=1 ˆ and the condition (1.10) is satisfied. We would like to note that if the maxto H imality condition is omitted then there are semibounded self-adjoint extensions of A˙ such that for a given regularizing sequence the corresponding approximating sequences satisfy the condition (1.10) but converge to different self-adjoint extensions ˙ Therefore, we lose the uniquenes of the limit with pleasant property of the of H. regularization to choose the right Hamiltonian. The paper is organized as follows. In Sec. 2 we recall some necessary facts from the Krein’s extension theory. The aim of Sec. 3 is to find an explicit formula for the resolvent of the perturbed operator which links the Friedrichs extension ˆ the extension parameter B which coresponds to A˜ and a bounded perturbation A, W . The problem of convergence of approximating sequences is solved in the whole generality in Sec. 4. In Sec. 5 we discuss the consequences of Sec. 4, in particular, we prove the above-mentioned results. 2. Extensions Let A0 ≥ 0 be some non-negative closed symmetric operator which can be regarded as the closure of A˙ = A|D, where A is a self-adjoint operator and D ⊆ dom(A). To describe all semibounded self-adjoint extensions A˜ of A0 obeying A˜ ≥ ηI, η < 0, we introduce the deficiency subspace Nη = ker(A∗0 − ηI). By Theorem 2.9 of [2] there is an one-to-one correspondence between semibounded self-adjoint extensions A˜ obeying A˜ ≥ ηI and non-negative closed quadratic forms q˜ on Nη . This correspondence is given in terms of quadratic forms. Denoting by ν˜ ˜ i.e. the quadratic form which is associated with A, q q q A˜ − ηf, A˜ − ηf + η(f, f ), f ∈ dom A˜ − η , (2.1) ν˜(f, f ) = and by νˆ the form which corresponds to Friedrichs extension Aˆ of A0 we have .

q) dom(˜ ν ) = dom(ˆ ν ) + dom(˜

(2.2)

613

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

.

(where M + N means that M ∩ N = {0}) and for g ∈ dom(ˆ ν ) and h ∈ dom(˜ q) ν˜(g + h, g + h) = νˆ(g, g) + 2η<e(g, h) + η(h, h) + q˜(h, h) .

(2.3)

Note that it is not necessary that the domain of the quadratic form q˜ is dense in Nη or is closed subspace of Nη . The Friedrichs extension Aˆ of A0 corresponds to the quadratic form q˜ = qˆ = 0 with dom(ˆ q ) = {0} (ˆ q = +∞ in the terminology of [2]). The smallest self-adjoint extension Aˇ of A0 obeying Aˇ ≥ ηI is called the Krein extension with respect to η and is given by the trivial quadratic form q˜ = qˇ = 0 with dom(ˇ q ) = Nη . All other extensions obeying A˜ ≥ ηI are between these two, ˇ ˜ ˆ i.e. A ≤ A ≤ A. ˜ η = dom(˜ q) ⊆ Since q˜ is a non-negative closed quadratic form on the subspace Q ˜ Nη , there exist a non-negative self-adjoint operator B ≥ 0 on Qη such that the representation q˜(h, h) =

√

√ Bh, Bh ,

h ∈ dom(˜ q ) = dom

√ B

(2.4)

holds. As far as relation (2.3) is equivalent to

2

q

q √

2

2

A˜ − η g + h = Aˆ − ηg +

Bh

(2.5)

we see that ker(A˜ − η) = {0} is equivalent to ker(B) = {0}. Assuming ker(B) = {0} which implies the existence of B −1 we find the representation A˜ − η

−1

f = Aˆ − η

−1

f + B −1 P f,

f ∈ H,

P f ∈ ran(B) ,

(2.6)

˜η. where P denotes the orthogonal projection from H onto Q It turns out, however, that the relation (2.6) defines a self-adjoint extension A˜ of A0 even in the case when one only assumes the invertibility of B. In this case the formula (2.6) defines a self-adjoint extension of A0 which is not necessarily semibounded. In particular, if B ≤ 0 and invertible we get an extension for which ˜ where ρ(·) denotes the resolvent the interval (η, 0) is a spectral gap, i.e. (η, 0) ⊆ ρ(A) set. The relation (2.6) can be used to compute the resolvent (A˜ − η − z)−1 in terms of the Friedrichs extension Aˆ and the operator B. To this end we introduce the analytic operator-valued function M (z) = z 2 P Aˆ − η − z

−1

˜ η −→ Q ˜η, P + zI : Q

=m(z) 6= 0 .

(2.7)

which is usually called the abstract Weyl function [3, 4]. Notice that the imaginary part of the Weyl function has a fixed sign for z in the upper or lower half plane, i.e. −2 Aˆ − η P . =m(M (z)) = =m(z) P Aˆ − η Aˆ − η − z

(2.8)

Moreover, since =m(M (z)) has bounded inverse we immediately find that the operator-valued function (B − M (z)) has the same property for each =m(z) 6= 0. If

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H. NEIDHARDT and V. ZAGREBNOV

B is bounded and has a bounded inverse from (2.6) we derive the Krein representation [1] −1 = Aˆ − η − z −1 −1 + Aˆ − η Aˆ − η − z P (B − M (z))−1 P Aˆ − η Aˆ − η − z (2.9)

A˜ − η − z

−1

for =m(z) 6= 0. Note that the right-hand side of (2.9) makes sense if B is any self-adjoint operator. A straightforward computation shows that for any self-adjoint operator B the relation (2.9) defines a self-adjoint extension A˜ of A0 . Conversely, if A˜ is any ˜ η of the deficiency self-adjoint extension of A0 , then there is a unique subspace Q ˜ subspace Nη and a self-adjoint operator B acting on Qη such that the Krein representation (2.9) holds for any non-real z. In the following the self-adjoint operator ˜ η = dom(B) ⊆ Nη is called the extension parameter B defined on the subspace Q ˜ of the extension A. 3. Perturbations Let W be a bounded self-adjoint operator on H. We consider the self-adjoint ˜ operator H, ˜ = A˜ + W , H (3.1) where A˜ is a self-adjoint extension of the non-negative closed symmetric operator ˜ − z)−1 , A0 with extension parameter B. Our goal is to compute the resolvent (H ˆ the extension parameter B =m(z) 6= 0, in terms of the Friedrichs extension A, and the perturbation W . It turns out that for this purpose the so-called perturbed abstract Weyl function L(z), cf. (2.7), L(z) = P (W − z) Aˆ + W − η − z

−1

˜ η −→ Q ˜η , (W − z)P + zI : Q

=m(z) 6= 0 , (3.2) plays a very important rˆ ole. A straightforward computation shows that −2 ˆ − η − z) Aˆ − η P . =m(L(z)) = =m(z)P Aˆ − η H

(3.3)

Hence the functions L(z) is a so-called operator-valued Nevanlinna function. ˆ − η − z|−1 has a bounded inverse it is easy to see Since the operator (Aˆ − η)|H that the operator-valued function Γ(z), ˜η ˜ η −→ Q Γ(z) = B + P W P − L(z) : Q

(3.4)

has the same property for each non-real z. Further, we introduce the operator-valued function Ω(z), Ω(z) = Aˆ + W − (W − z)P (B + P W P − z)−1 P (W − z) : dom Aˆ −→ H , (3.5)

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

615

ˆ = Aˆ + W . =m(z) 6= 0, which can be regarded as a non-selfadjoint perturbation of H Introducing the operator-valued function K(z), K(z) = (W − z)P (B + P W P − z)−1 P (W − z) + zI : Nη −→ Nη ,

(3.6)

=m(z) 6= 0, one gets that Ω(z) − (η + z)I = Aˆ + W − K(z) − ηI .

(3.7)

However, K(z) is as well an operator-valued Nevanlinna. This follows from the relation =m(K(z)) = =m(z) [W P − (B + P W P )]|B + P W P − z|−2 × [P W − (B + P W P )] + (I − P )} ,

(3.8)

=m(z) 6= 0. Hence Ω(z) − z − η is a dissipative operator for each non-real z. Let us prove that the operator Ω(z) − η − z has a bounded inverse for each non-real z. By (3.7) and (3.8) we obtain that ker(Ω(z) − η − z) = {0} for each non-real z. Moreover, the relation n ˆ − η − z −1 (W − z) ˆ − η − z −1 + H f = (Ω(z) − η − z) H ˆ −η−z × P (B + P W P − L(z))−1 P (W − z) H

−1

}f ,

(3.9)

which holds for each f ∈ H and any non-real z, implies ran(Ω(z) − η − z) = H. This yields that (Ω(z) − η − z) has a bounded inverse for each non-real z. The operator-valued functions Γ(z) and Ω(z) are related by ˆ − η − z −1 (Ω(z) − η − z)−1 = H ˆ − η − z −1 (W − z) P Γ(z)−1 P (W − z) H ˆ − η − z −1 + H

(3.10)

and Γ(z)−1 = (B + P W P − z)−1 + (B + P W P − z)−1 P (W − z) × (Ω(z) − η − z)−1 (W − z)P (B + P W P − z)−1 ,

(3.11)

=m(z) 6= 0. We need both of these relations in the following. Proposition 3.1. Let A˜ be a self-adjoint extension of A0 given by the extension ˜ η = dom(B) ⊆ Nη . Further , let W be a bounded operator parameter B defined on Q ˜ η . Then the resolvent on H and let P be the orthogonal projection from H onto Q ˜ − η − z)−1 admits the representations (H ˆ − η − z −1 ˜ − η − z −1 = H H h i h i ˆ − η − z −1 (W − z) P Γ(z)−1 P I − (W − z) H ˆ − η − z −1 , + I− H (3.12)

616

H. NEIDHARDT and V. ZAGREBNOV

and ˜ −η−z H

−1

= P (B + P W P − z)−1 P + [I − P (B + P W P − z)−1 P (W − z)](Ω(z) − η − z)−1 × [I − (W − z)P (B + P W P − z)−1 P ] ,

(3.13)

for =m(z) 6= 0. Proof. Let =m(z) 6= 0. Using (3.1) we get ˜ − η − z −1 f = I + W A˜ − η − z −1 A˜ − η − z f, H

f ∈ dom A˜ . (3.14)

Taking into account (2.9) we find n ˜ −η−z f = H ˆ − η − z Aˆ − η − z −1 H −1 + W Aˆ − η Aˆ − η − z P (B − M (z))−1 −1 o × P Aˆ − η Aˆ − η − z A˜ − η − z f

(3.15)

ˆ − η − z)−1 , we get Multiplying on the left by the operator (Aˆ − η)(H ˜ −η−z f ˆ − η − z −1 H Aˆ − η H o n ˆ − η − z −1 W Aˆ − η Aˆ − η − z −1 P (B − M (z))−1 P = I + Aˆ − η H −1 A˜ − η − z f . × Aˆ − η Aˆ − η − z

(3.16)

A straightforward computation shows that ˆ − η − z −1 W Aˆ − η Aˆ − η − z −1 P (B − M (z))P −1 I + Aˆ − η H ˆ − η − z −1 W Aˆ − η Aˆ − η − z −1 P Γ(z)−1 P . = I − Aˆ − η H (3.17) ˆ − η − z)f = g, one gets Setting (H o −1 n ˜ − η − z −1 g g−W H Aˆ − η Aˆ − η − z o n ˆ − η − z −1 W Aˆ − η Aˆ − η − z −1 P Γ(z)−1 P = I − Aˆ − η H ˆ − η − z −1 g . × Aˆ − η H

(3.18)

ˆ − η − z)−1 one ˆ − η − z)−1 = I − (W − z)(H for each g ∈ H. Since (Aˆ − η)(H simplifies (3.18) to o −1 n ˜ − η − z −1 g g−W H Aˆ − η − z ˆ − η − z W Aˆ − η Aˆ − η − z −1 g− H ˆ − η − z −1 g . × P Γ(z)−1 P Aˆ − η H

ˆ −η−z = H

−1

(3.19)

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

Using the resolvent identity, −1 ˆ − η − z −1 = H ˆ − η − z −1 W − H Aˆ − η − z −1 = Aˆ − η − z W we find Aˆ − η − z

Aˆ − η − z ˆ −η−z H

617

−1 −1

o −1 n ˜ − η − z −1 g g−W H

ˆ − η − z −1 ˆ − η − z −1 g − Aˆ − η − z −1 W H = H ˆ − η − z −1 g . × Aˆ − η P Γ(z)−1 P Aˆ − η H

(3.20)

Multiplying on the left by (Aˆ − η − z), we obtain n ˆ − η − z −1 g ˜ − η − z −1 g = W H W H ˆ −η−z + H

−1

o ˆ − η − z −1 g . Aˆ − η P Γ(z)−1 P Aˆ − η H

(3.21)

If ker(W ) = {0}, then the relation (3.12) is proved. If ker(W ) 6= {0}, then we choose a τ ∈ R such that ker(W − τ I) = {0} and set W 0 = W − τ I and z 0 = z − τ . Using the identities ˜ − (η + z)I = A˜ + W 0 − z 0 I H we find ˜ −η−z W0 H

−1

ˆ −η−z + H

g = W0

−1

n

ˆ − (η + z)I = Aˆ + W 0 − z 0 I and H

ˆ −η−z H

−1

g

o ˆ − η − z −1 g Aˆ − η P Γ0 (z 0 )−1 P Aˆ − η H

(3.22)

with Γ0 (z 0 ) = B + P W 0 P − L0 (z 0 ) ˆ −η−z = B + P W 0 P − P (W 0 − z 0 ) H

−1

(W 0 − z 0 )P − z 0 I

= Γ(z)

(3.23)

which yields ˜ −η−z W0 H

−1

ˆ −η−z + H

= W0

−1

n

ˆ −η−z H

−1

g

o ˆ − η − z −1 g . (3.24) Aˆ − η × P Γ(z)−1 P Aˆ − η H

Since ker(W 0 ) = {0} we get (3.12). To prove (3.13) we insert (3.11) into (3.12) and use the identity ˆ − η − z −1 (W − z) ˆ − η − z −1 = (Ω(z) − η − z)−1 − H H × P (B + P W P − z)−1 P (W − z)(Ω(z) − η − z)−1 , =m(z) 6= 0.

(3.25)

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H. NEIDHARDT and V. ZAGREBNOV

At the end of this section let us note that formulas (3.12) and (3.13) have a converse. Straightforward computations prove −1 −1 −1 − Aˆ − η (W − z) Aˆ − η P Γ(z)−1 P = − Aˆ − η h i −1 ˜ − η − z −1 + I + Aˆ − η (W − z) H h −1 i × I + (W − z) Aˆ − η − z

(3.26)

and −1 (W − z) Aˆ − η −1 ˜ − η − z −1 (W − z) Aˆ − η . (3.27) + Aˆ − η (W − z) H

(Ω(z) − η − z)−1 = Aˆ − η

−1

− Aˆ − η

−1

In the next sections we use the above representations to resolve the problem of convergence for regularizing sequences of bounded operators {Wn }∞ n=1 . 4. Convergence Let us start with definition of regularizing sequence. Definition 4.1. A sequence {Wn }∞ n=1 of bounded self-adjoint operators is called regularizing sequence for the self-adjoint operator W if s − lim (Wn − z)−1 = (W − z)−1 ,

=m(z) 6= 0 .

n→∞

(4.1)

If W ≤ 0, then the sequence {Wn }∞ n=1 is called non-increasing regularizing sequence for W if (4.2) 0 ≤ −W1 ≤ −W2 ≤ · · · ≤ −Wn ≤ · · · ≤ −W . In the following we restrict ourselves to non-positive perturbations. Notice that if {Wn }∞ n=1 is a non-increasing regularizing sequence for the non-positve perturbation W then {−Wn }∞ n=1 is a non-decreasing sequence and by Theorem 3.3 of [5] there is always an operator W∞ such that s − lim (Wn − z)−1 = (W∞ − z)−1 ,

=m(z) 6= 0 .

n→∞

(4.3)

Definition 4.1 means that W = W∞ . By [14] we find dom

√ −W = {f ∈ H : sup(−Wn f, f ) < +∞}

(4.4)

n

and

√

√ −W f, −W f = lim (−Wn f, f ), n→∞

Then by the Theorem 3.13 of [5] one has p √ −Wn + If = −W + If, lim n→∞

f ∈ dom

f ∈ dom

√ −W ,

√ −W .

(4.5)

(4.6)

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

619

In accordance with (1.1) we assume that for non-positive perturbations W one has a stability domain D 0 ≤ (−W f, f ) ≤ a(A0 f, f ) + b(f, f ),

f ∈ D,

0 < a < 1, b > 0 ,

(4.7)

where A0 is a non-negative closed symmetric operator for which D is a core, i.e. A0 |D = A0 . Denoting by Aˆ the Friedrichs extension of A0 and by νˆ the quadratic form which is associated with Aˆ we immediately get from (4.7) that √ (4.8) dom(ˆ ν ) ⊆ dom −W and 0≤

√ √ −W f, −W f ≤ aˆ ν (f, f ) + b(f, f ),

f ∈ dom(ˆ ν ),

0 < a < 1, b > 0 , (4.9) which shows that −W is relatively bounded with respect to Aˆ in the form sense. . ˆ = Aˆ + W is well defined and the Then by the KLMN-theorem [10] the form sum H ˆ is semibounded from below. Using (4.9) we directly find operator H s − lim

n→∞

p −1 −1 √ −Wn Aˆ − z = −W Aˆ − z

(4.10)

−1 √ −1 p −Wn = Aˆ − z −W

(4.11)

s − lim Aˆ − z n→∞

p √ −1 √ −1 p −Wn Aˆ − z −Wn = −W Aˆ − z −W , s − lim n→∞

(4.12)

ˆ n }∞ , for =m(z) 6= 0. As it is shown in [8] the approximating sequence {H n=1 ˆ n = Aˆ + Wn , H

(4.13)

ˆ i.e. converges in the strong sense to H, ˆn − z s − lim H

−1

n→∞

ˆ −z = H

−1

,

=m(z) 6= 0 .

(4.14)

Using (4.10)–(4.12) one proves in addition to (4.14) the following limits: s − lim

n→∞

p √ ˆ − z −1 ˆ n − z −1 = −W H −Wn H

(4.15)

−1 p √ ˆ − z −1 −W −Wn = H

(4.16)

ˆn − z s − lim H n→∞

p √ √ p ˆ − z −1 −W . ˆ n − z −1 −Wn = −W H −Wn H s − lim n→∞

(4.17)

The goal of this section is to find necessary and sufficient conditions ensuring ˜ n }∞ defined in (1.3) converges to the form that the approximating sequence {H n=1 ˆ in the generalized strong resolvent sense which means that in addition to sum H the strong resolvent convergence we have p √ ˆ − z −1 ˜ n − z −1 = −W H −Wn H s − lim n→∞

(4.18)

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H. NEIDHARDT and V. ZAGREBNOV

˜n − z s − lim H n→∞

s − lim

n→∞

√ −1 p ˆ − z −1 −W −Wn = H

p √ p √ ˆ − z −1 −W ˜ n − z −1 −Wn = −W H −Wn H

(4.19) (4.20)

for some non-real z. According to this definition, the limits (4.15) – (4.17) imply that ˆ n }∞ the approximating sequence {H n=1 converges in the generalized strong resolvent ˆ sense to the form sum H, see (1.7) ˆ by H ˆ n one gets the representation Replacing in (3.12) Γ by Γn , W by Wn and H ˜n − η − z H

−1

ˆ n − η − z −1 = H h i ˆ n − η − z −1 (Wn − z) P Γn (z)−1 + I− H h i ˆ n − η − z −1 , × P I − (Wn − z) H

(4.21)

with Γn (z) = B + P Wn P − Ln (z) and ˆn − η − z Ln (z) = P (Wn − z) H

−1

(4.22)

(Wn − z)−1 P + zI .

(4.23)

Proposition 4.2. Let W be a non-positive self-adjoint operator and let A0 be a non-negative closed symmetric operator . Assume that there is a core D of A0 such that (4.7) is satisfied (stability domain). Further , let {Wn }∞ n=1 be a non-increasing regularizing sequence for W and A˜ be a self-adjoint extension of A0 given by the ˜ n }∞ extension parameter B. Then the approximating sequence {H n=1 converges in ˆ if and only if one has the the generalized strong resolvent sense to the form sum H following limits s − lim Γn (z)−1 n→∞ p −Wn P Γn (z)−1 s − lim n→∞ p s − lim Γn (z)−1 P −Wn n→∞ p p −Wn P Γn (z)−1 P −Wn s − lim n→∞

=0

(4.24)

=0

(4.25)

=0

(4.26)

=0

(4.27)

for some non-real z. Proof. Due to (3.26) one gets representation −1 −1 −1 − Aˆ − η (Wn − z) Aˆ − η P Γn (z)−1 P = − Aˆ − η h i −1 i h ˜ n − η − z −1 I + (Wn − z) Aˆ − η −1 . + I + Aˆ − η (Wn − z) H (4.28) ˜ n }∞ converges in the generalized strong resolvent sense to Assume that {H n=1 ˆ Then taking into account (4.10)–(4.12) we immediately find the the form sum H.

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ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

relations (4.24)–(4.27). Conversely, if the relations (4.24)–(4.27) are satisfied, then ˜ n }∞ converges in the generalized by (4.21) and (4.15)–(4.17) we obtain that {H n=1 strong resolvent sense to the form sum. Since the conditions (4.24)–(4.27) are not very practicable it is necessary to look for more comfortable conditions. To this end we use the representation (3.13): ˜ n − η − z −1 = P (B + P Wn P − z)−1 P H +[I − P (B + P Wn P − z)−1 P (Wn − z)] (Ωn (z) − η − z)−1 × [I − (Wn − z)P (B + P Wn P − z)−1 P ]

(4.29)

with Ωn (z) = Aˆ + Wn − (Wn − z)P (B + P Wn P − z)−1 P (Wn − z),

n = 1, 2, . . . (4.30)

which follows from (3.5) by replacing Ω by Ωn and W by Wn . Propostion 4.3. Let W be non-positive self-adjoint operator and let A0 be nonnegative closed symmetric operator . Assume that there is a core D of A0 such that (4.7) is satisfied (stability domain). Further , let {Wn }∞ n=1 be non-increasing regu˜ larizing sequence for W and let A be self-adjoint extension of A0 which corresponds ˜ n }∞ converges in to the extension parameter B. If the approximating sequence {H n=1 ˆ the generalized strong resolvent sense to the form sum H, then ˆ − η − z −1 , (4.31) s − lim (Ωn (z) − η − z)−1 = H n→∞

p √ ˆ − η − z −1 , −Wn (Ωn (z) − η − z)−1 = −W H s − lim

(4.32)

p √ ˆ − η − z −1 −W , −Wn = H

(4.33)

n→∞

s − lim (Ωn (z) − η − z)−1 n→∞

s − lim

n→∞

p p √ √ ˆ − η − z −1 −W −Wn (Ωn (z) − η − z)−1 −Wn = −W H

for some non-real z. If in addition the condition

p

p

sup −Wn P (B + P Wn P − z)−1 P −Wn < +∞

(4.34)

(4.35)

n

is satisfied , then s − lim (B + P Wn P − z)−1 n→∞ p −Wn P (B + P Wn P − z)−1 s − lim n→∞ p s − lim (B + P Wn P − z)−1 P −Wn n→∞ p p −Wn P (B + P Wn P − z)−1 P −Wn s − lim n→∞

= 0,

(4.36)

= 0,

(4.37)

= 0,

(4.38)

= 0,

(4.39)

for some non-real z. Conversely, if the conditions (4.31)–(4.34) and (4.36)–(4.39) ˜ n }∞ converges in the generalized strong resolvent sense to are satisfied , then {H n=1 ˆ the form sum H.

622

H. NEIDHARDT and V. ZAGREBNOV

Proof. From (3.27) we find the representation −1 −1 −1 − Aˆ − η (Wn − z) Aˆ − η (Ωn (z) − η − z)−1 = Aˆ − η −1 ˜ n − η − z −1 (Wn − z) Aˆ − η −1 . + Aˆ − η (Wn − z) H

(4.40)

˜ n }∞ converges in the generalized strong resolvent sense to Assume now that {H n=1 ˆ Then using (4.10)–(4.12) one gets directly the existence of limits (4.31)–(4.34). H. Taking into account the representation ˆ − η − z −1 = Aˆ − η −1 − Aˆ − η −1 (W − z) Aˆ − η −1 H + Aˆ − η

−1

ˆ −η−z (W − z) H

−1

(W − z) Aˆ − η

−1

(4.41)

ˆ − η − z)−1 . one proves that the right-hand side of (4.40) converges to (H To show (4.36)–(4.39) we note that (4.35) yields the estimate

p

p

sup −Wn P (B + P Wn P − z)−1 = sup (B + P Wn P − z)−1 P −Wn < +∞ . n

n

(4.42) This comes from the representation p p −Wn P {(B + P Wn P − z)−1 − (B + P Wn P − z)−1 }P −Wn p p = 2i=m(z) −Wn P (B + P Wn P − z)−1 (B + P Wn P − z)−1 P −Wn (4.43) which implies the estimate

2

p

|=m(z)| (B + P Wn P − z)−1 P −Wn

p

p

≤ −Wn P (B + P Wn P − z)−1 P −Wn .

(4.44)

Therefore, the condition (4.35) guarantees the condition (4.42). Since (B + P Wn P − z)−1 = Γn (z)−1 ˆ n − η − z)−1 (Wn − z)P Γn (z)−1 , −(B + P Wn P − z)−1 P (Wn − z)(H (4.45) from Proposition 4.2 and relations (4.15)–(4.17), together with the estimates (4.35) and (4.42), we get limits (4.36)–(4.39). Conversely, if (4.31)–(4.34) and (4.36)–(4.39) are valid, then the representation ˜ n }∞ converges in the generalized strong resolvent sense to (4.29) shows that {H n=1 ˆ the form sum H. 5. Consequences Here we study some consequences of the results obtained in the previous section.

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

623

Proposition 5.1. Let W be a non-positive self-adjoint operator and let A0 be a non-negative closed symmetric operator . Assume that the pair {A0 , W } has a stability domain such that (4.7) is satisfied . Further , let {Wn }∞ n=1 be a nonincreasing regularizing sequence for W and let A˜ be self-adjoint extension of A0 given ˜ n }∞ by extension parameter B. If the corresponding approximating sequence {H n=1 ˆ converges in the generalized strong resolvent sense to the form sum H, then √ (5.1) dom −W ∩ dom(B) = {0} . If the extension parameter B is non-negative, then √ √ dom −W ∩ dom( B) = {0} .

(5.2)

√ Proof. Assume there is a non-trivial h ∈ dom( −W ) ∩ dom(B). Then by (4.6) √ √ we have limn→∞ −Wn + Ih = −W + Ih. Taking into account (4.22) and (4.23) as well as (4.14)–(4.17) and (4.24)–(4.27) one finds lim Γn (z)−1 (Γn (z) + z)h = 0 .

(5.3)

n→∞

On the other hand, we get Γn (z)−1 (Γn (z) + z)h = h + z lim Γn (z)−1 h = h, n→∞

=m(z) 6= 0 ,

(5.4)

which contradicts (5.3). Hence, √ h = 0.−1 If B ≥ 0, then operator BΓn (z) , =m(z) 6= 0, is well-defined for each n = 1, 2, . . .. Furthermore, we have

2

p

2

√

BΓn (z)−1 h = (h, Γn (z)−1 h) + −Wn P Γn (z)−1 h ˆn − η − z +(P (Wn − z) H

−1

(Wn − z)P Γn (z)−1 h, Γn (z)−1 h) .

Applying again (4.14)–(4.17) and (4.24)–(4.27) we find √ =m(z) 6= 0 . s − lim BΓn (z)−1 = 0, n→∞

(5.5)

(5.6)

Using representation (Γn (z)−1 (Γn (z) + z)h, h0 ) =

√ √ Bh, BΓn (z)−1 h0 p p −Wn h, −Wn Γn (z)−1 h0 − ˆ n − η − z −1 − (Wn − z) H ×(Wn − z)h, Γn (z)−1 h0 ,

(5.7)

√ √ h ∈ dom( −W ) ∩ dom( B), h0 ∈ Nη , =m(z) 6= 0, as well as (4.14)–(5.17) and (4.24)–(4.27) one gets √ √ h ∈ dom −W ∩ dom B . (5.8) w − lim Γn (z)−1 (Γn (z) + z)h = 0, n→∞

624

H. NEIDHARDT and V. ZAGREBNOV

However, this contradicts (5.4). The conditions (5.1) and (5.2) imply that sup(−Wn h, h) = ∞ ,

(5.9)

n

√ for non-trivial elements h ∈ dom(B) or h ∈ dom( B). According to Definition 2.6 of [8] the property (5.9) for B ≥ 0 means that the regularizing sequence {Wn }∞ n=1 is νˆ-admissible. Hence, Proposition 5.1 shows that admissibility is necessary in order ˆ ˜ n }∞ to the form sum H. to get convergence of the approximating sequence {H n=1 √ Next we are interested in the question what is going on if dom( −W ) ∩ Nη 6= ˜ n }∞ converge {0}. It turns out that in this case not all approximating sequences {H n=1 ˆ to the form sum H. Proposition 5.2. Let W be a non-positive self-adjoint operator and let A0 be a non-negative closed symmetric operator . Assume that the pair {A0 , W } has a √ stability domain D such that condition (4.7) is satisfied . If dom( −W ) ∩ Nη 6= {0}, Nη = ker(A∗0 − ηI), for some η < 0, then there is a continuum of different semibounded self-adjoint extensions A˜ of A0 such that √ (5.10) dom(˜ ν ) ⊆ dom −W , dom(˜ ν ) ∩ Nη = dom

√

−W ∩ Nη ,

∀η < 0 ,

(5.11)

and √ √ −W f, −W f ≤ a ˜ν˜(f, f ) + ˜b(f, f ), f ∈ dom(˜ ν ), 0 < a ˜ < 1, ˜b > 0

(5.12)

˜ where ν˜ is the closed quadratic form associated with A. √ Proof. First we remark that if dom( −W ) ∩ Nη = {0} for some η < 0, then this relation holds for any η < 0. To see this we have to use the representation [2] . dom(A∗0 ) = dom Aˆ + Nη

(5.13)

√ which holds√for each η < 0. Assume √ that dom( −W ) ∩ Nη = {0} for √ some η < 0. ˆ we have dom( −W )∩Nη0 = Since dom( −W )∩dom(A∗0 ) = dom( −W )∩dom(A) {0} for any η 0 < 0. Let η < − ab where the constants a and b are taken from (4.7). We set √ √ 0 ≤ ρ < ∞, q˜ρ (h, h) = ρ −W h, −W h , √ (5.14) h ∈ dom(˜ qρ ) = dom −W ∩ Nη . Since W is a non-positive self-adjoint operator the quadratic form q˜ρ is non-negative and closed. In accordance with (2.4) we associate with q˜ρ a semi-bounded selfadjoint extension A˜ρ of A0 . By (2.3) and (2.4), we have √ . qρ ) ⊆ dom −W (5.15) dom(˜ νρ ) = dom νˆ + dom(˜

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

625

and ν˜ρ (g + h, g + h) − ηkg + hk2 = νˆ(g, g) − ηkgk2 + q˜ρ (h, h), g ∈ dom(ˆ ν ), h ∈ dom(˜ q) .

(5.16)

Using the estimate √ √ −W (g + h), −W (g + h) √

≤ (1 + )

√ √ 1 √ −W g, −W g + 1 + −W h, −W h ,

> 0,

(5.17)

and (4.9) we find √ √ −W (g + h), −W (g + h) √ 1+ √ −W h, −W h ≤ (1 + ) aˆ ν (g, g) + bkgk2 + which implies √ √ −W (g + h), −W (g + h) √ 1 √ b kgk2 + −W h, −W h ≤ (1 + )a νˆ(g, g) + . (1 + ) a a Since

b (1+)a

≤

b a

1 a .

(5.19)

< −η, one gets the estimate

√ √ −W (g + h), −W (g + h) ≤ (1 + ) a νˆ(g, g) − ηkgk2 + q˜ρ (h, h) for each ρ ≥

(5.18)

(5.20)

Therefore, we have √ √ −W (g + h), −W (g + h) ≤ (1 + )a˜ νρ (g + h, g + h) − (1 + ) aηkg + hk2 .

(5.21)

Choosing small enough we satisfy the condition a ˜ = (1 + )a < 1. Setting ˜b = −(1 + )η we prove (5.12). We are going to show (5.11). For the chosen η < − ab the condition (5.11) is satisfied by construction. In order to show the condition (5.11) for each η < 0 we note that .

ν ) + {dom(˜ qρ ) ∩ Nη } dom(˜ νρ ) ∩ dom(A∗0 ) = dom(ˆ n o √ . = dom(ˆ ν ) + dom −W ∩ Nη

(5.22)

√ which yields dom(˜ νρ ) ∩ Nη = dom( −W ) ∩ Nη for each η < 0. 1 quadratic forms q˜ρ are different, the corresponding selfSince for different ρ ≥ a ˜ adjoint extensions Aρ are different too. Hence, there is a continuum of self-adjoint extensions of A0 satisfying (5.10)–(5.12).

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H. NEIDHARDT and V. ZAGREBNOV

Replacing Aˆ by A˜ one easily gets that (5.10) and (5.12) are equivalent to (4.8) . ˜ = A˜ + W is well defined and (4.9). Again by the KLMN-Theorem the form sum H ˆ ˜ n = A˜ + Wn and replacing A, and semibounded from below. Further, setting H ˆ and H ˆ n by A, ˜ H ˜ and H ˜ n , respectively, we obtain relations similar to (4.10)– H (4.12) and (4.14)–(4.17). Hence, the expectation that each approximating sequence √ ˆ ˜ n }∞ converges to H is false if dom( −W ) ∩ Nη 6= {0}. {H n=1 √ ν ) has to be Condition dom( −W ) ∩ Nη = {0} means that the domain dom(ˆ maximal with respect to W , i.e., if A˜ is any semibounded self-adjoint extension of A0 with corresponding quadratic form ν˜ such that √ (5.23) dom(ˆ ν ) ⊆ dom(˜ ν ) ⊆ dom −W , ˆ then dom(˜ ν ) = dom(ˆ ν ) which yields A˜ = A. Obviously the last remark implies the following question. Assume that νˆ is ˜ n }∞ maximal with respect to W and there is approximating sequence {H n=1 which ˜ ˙ ˜ converges to some self-adjoint extension H of H. Does H coincide necessarily with ˆ the form sum H? Proposition 5.3. Let W be a non-positive self-adjoint operator and let A0 be a non-negative closed symmetric operator such that dom(ˆ ν ) is maximal with respect to W. Assume that (4.7) is satisfied for a core D of A0 and A˜ is a self-adjoint extension of A0 . Further, let {Wn }∞ n=1 be a regularizing sequence for W such that p √ √ |Wn | + If = −W + If, f ∈ dom −W . (5.24) lim n→∞

˜ n }∞ If the corresponding approximating sequence {H n=1 converges in the strong ˜ of H˙ and resolvent sense to a semibounded self-adjoint extension H

p p

˜ n − z −1 |Wn | (5.25) sup |Wn | H

< +∞ n

˜ coincides with the form sum H. ˆ for some non-real z, then H Proof. Let us introduce the sequence {Sn (z)}∞ n=1 of bounded operators p p ˜ n − z −1 |Wn | + I (5.26) Sn (z) = |Wn | + I H which is uniformly bounded by (5.25), i.e. supn kSn (z)k < +∞. This yields the existence of a bounded operator S(z) such that for a subsequence {nk }∞ k=1 we have weak limit (5.27) w − lim Snk (z) = S(z) . k→∞ √ Let f, h ∈ dom( −W ). Then taking into account (5.24) we find p p ˜ n − z −1 |Wn | + If, |Wn | + Ih lim (Snk (z)f, h) = H k k k nk →∞

=

˜ −z H

√ −1 √ −W + If, −W + Ih

= (S(z)f, h) ,

(5.28)

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

which shows that

√ ˜ ⊆ dom −W dom H

627

(5.29)

and S(z)f =

√

˜ −z −W + I H

−1 √ −W + If,

f ∈ dom

√ −W .

(5.30)

A straightforward computation shows that if S(z) makes sense and is bounded for ˜ in particular, for real points some non-real z, then the same holds for each z ∈ ρ(H), ˜ They exist by the assumption that H ˜ is semibounded. of the resolvent set ρ(H). −1 ˜ ˜ Hence there is a τ ≤ 0 such that H −√τ I ≥ 0 and the inverse √ operator (H − τ I) −1 ˜ exists and is bounded. Since S(τ ) = −W + I(H − τ ) −W + I is bounded we √ ˜ − τ )1/2 ) ⊆ dom( −W ). Let µ ˜ be the quadratic form which is obtain that dom((H ˜ Then we have associated with H. dom(˜ µ) ⊆ dom Let us introduce closed quadratic form ν˜ √ √ −W f, −W f , ν˜(f, f ) = µ ˜(f, f ) +

√ −W .

f ∈ dom(˜ ν ) = dom(˜ µ) .

(5.31)

(5.32)

Obviously the form ν˜ is semibounded. Moreover, if f ∈ D ⊆ dom(A0 ) ∩ dom(W ), then ˙ f − (W f, f ) = (A0 f, f ) . (5.33) ν˜(f, f ) = Hf, Hence, the closed quadratic form ν˜ defines a semibounded self-adjoint extension A˜ ν ) is maximal with respect to W and of A0 . Since dom(ˆ dom(ˆ ν ) ⊆ dom(˜ ν ) ⊆ dom

√ −W

(5.34)

ˆ Therefore dom(˜ we get dom(˜ ν ) = dom(ˆ ν ) which yields ν˜ = νˆ and A˜ = A. µ) = dom(ˆ ν ) and √ −W f, −W f √ √ −W f, −W f = νˆ(f, f ) −

µ ˜(f, f ) = ν˜(f, f ) −

√

=µ ˆ(f, f ) f ∈ dom(˜ µ) = dom(ˆ µ) = dom(ˆ ν ), where µ ˆ is the quadratic form which is associated ˆ Consequently we obtain H ˜ = H. ˆ with H. If the dom(ˆ ν ) is maximal with respect to the perturbation W and the additional condition (5.25) holds, then Proposition 5.3 tells us that the approximating sequence ˆ or it does not converge at all. ˜ n }∞ either converges to the form sum H {H n=1 Finally let us show that for each semibounded self-adjoint extension A˜ of A0 there is always a regularizing sequence {Wn }∞ n=1 for W ≤ 0 such that the corˆ in the ˜ n }∞ tends to the form sum H responding approximating sequence {H n=1 generalized strong resolvent sense.

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H. NEIDHARDT and V. ZAGREBNOV

Proposition 5.4. Let W be a non-positive self-adjoint operator and let A0 be a non-negative closed symmetric operator such that dom(ˆ ν ) is maximal with respect to W, that the pair {A0 , W } has a stability domain and condition (4.7) is satisfied for a core D of A0 . Then for each semibounded self-adjoint extension A˜ of A0 there is a regularizing sequence {Wn }∞ n=1 for W such that |(Wn f, f )| ≤ a(A0 f, f ) + (b + 2)(f, f ),

f ∈ D,

(5.35)

where a and b are taken from (4.7), and the corresponding approximating sequence ˜ n }∞ given by (1.3) obeys {H n=1 ˆ − z −1 ˜ n − z −1 = H (5.36) s − lim H n→∞

p √ ˜ n − z −1 = −W H ˆ − z −1 |Wn | H s − lim

(5.37)

p √ ˆ − z −1 −W ˜ n − z −1 |Wn | = H s − lim H

(5.38)

n→∞

n→∞

s − lim

n→∞

p √ p √ ˜ − z −1 |Wn | = −W H ˆ − z −1 −W |Wn | H

(5.39)

for non-real z. Proof. Let {Vn }∞ n=1 be a non-increasing regularizing sequence for W ≤ 0. ˜ Choosing η Further, let B be the extension parameter which corresponds to A. ˜ smaller than inf σ(A) we can assume that B ≥ 0. We set Xn = P Vn P ≤ 0, ˜ η −→ H such that n = 1, 2, . . .. There is an isometry Γn : Q p p −Vn + IP = Γn −Xn + IP, n = 1, 2, . . . (5.40) Introducing the operator Jn = I − 2Γn Γ∗n ,

n = 1, 2, . . .

we define a sequence {Wn }∞ n=1 of bounded operators by p p Wn = − −Vn + IJn −Vn + I − I .

(5.41)

(5.42)

∞ for W √ . By monotonicity for Let us show that √ {Wn }n=1 is a regularizing sequence √ each f ∈ dom( −W ) we have s − limn→∞ −Vn + If = −W + If , see (4.6). Using (5.40) we get representation

Γ∗n f = √

p 1 P −Vn + If, −Xn + I

f ∈ dom

√ −W , n = 1, 2 . . .

(5.43)

Since dom(ˆ ν ) is maximal with respect to W we have s− limn→∞ (−Xn + I)−1/2 = 0. See Lemma 2.10 of [8]. This immediately implies √ f ∈ dom −W . (5.44) lim Γ∗n f = 0, n→∞

∗ Since {Γ∗n=1 }∞ n=1 is a sequence of partial isometries this yields s − limn→∞ Γn = 0. ∗ Hence one gets s − limn→∞ Γn Γn = 0. Therefore s − limn→∞ Jn = I. If τ ∈ R is

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ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

sufficiently small, then the operator (I + τ (I − Vn )−1 Jn )−1 has a bounded inverse such that 1 . (5.45) k(I + τ (I − Vn )−1 )−1 Jn k ≤ 1 − |τ | Notify that kJn k = 1. Therefore, one gets the following representation p p −1 −Vn + IJn −Vn + I + τ = √

1 Jn (I + τ (I − Vn )−1 Jn )−1 −Vn + I

1 × √ . −Vn + I

(5.46)

Since the sequence {Vn }∞ n=1 tends in the strong resolvent sense to W and s − limn→∞ Jn = I we find p p −1 −Vn + IJn −Vn + I + τ = (−W + I + τ )−1 . (5.47) s − lim n→∞

√ √ Hence, { −Vn + IJn −Vn + I}∞ n=1 tends in the strong resolvent sense to −W + I which shows, see (5.42), that {Wn }∞ n=1 tends in the strong resolvent sense to W . ∞ Hence {Wn }n=1 is a regularizing sequence for W . However, the sequence {Wn }∞ n=1 is not, in general, a non-increasing one. To prove (5.35) we note that by (5.41) and (5.42) one has p p Wn = − −Vn + I{I − 2Γn Γ∗n } −Vn + I − I p p (5.48) = Vn + 2 −Vn + IΓn Γ∗n −Vn + I which yields Vn ≤ Wn ,

n = 1, 2, . . .

(5.49)

Moreover, since kΓn k ≤ 1 we find Wn ≤ −Vn + 2I,

n = 1, 2, . . .

(5.50)

which proves the estimate |(Wn f, f )| ≤ (−Vn f, f ) + 2(f, f ),

n = 1, 2, . . .

f ∈ H.

(5.51)

Since −Vn ≤ −W , n = 1, 2, . . ., we obtain (5.35) by (4.7). ˜ n }∞ To verify the convergence of the approximating sequence {H n=1 to the form ˆ sum H we use the representation (4.29). Then by (5.40) one finds p p P Wn P = −P −Vn + IJn −Vn + IP + P p p (5.52) = − −Xn + IΓ∗n Jn Γn −Xn + I + P . Since Γ∗n Jn Γn = −I one finally gets P Wn P = −Xn + 2P,

n = 1, 2 . . . .

(5.53)

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H. NEIDHARDT and V. ZAGREBNOV

This immediately implies B + P Wn P − I = B − Xn + I ≥ I,

n = 1, 2 . . . .

(5.54)

Now we calculate operator-valued function Ωn (z) for z = 1. Using (4.30) and (5.54) we find p p Ωn (1) = Aˆ + I − −Vn + IJn −Vn + I p p p p − −Vn + IJn −Vn + I P (B − Xn + I)−1 P −Vn + IJn −Vn + I . (5.55) Taking into account (5.41) one gets Ωn (1) = Aˆ + I n op p p p −Vn + I. − −Vn + I Jn + Γn −Xn + I(B − Xn + I)−1 −Xn + IΓ∗n (5.56) Inserting (5.41) into (5.56), we finally obtain o n p p p Ωn (1) = Aˆ + Vn + −Vn + IΓn 2I − −Xn + I(B − Xn + I)−1 −Xn + I p (5.57) ×Γ∗n −Vn + I . Note that

n o p p 0 ≤ Γn 2I − −Xn + I(B − Xn + I)−1 −Xn + I)−1 Γ∗n ≤ 2I .

(5.58)

Therefore, by (5.57) one gets ˆ. Ωn (1) ≥ Aˆ + Vn ≥ H

(5.59)

ˆ inf σ(A)}, ˜ then If η + 2 < min{inf σ(H), ˆ n − (η + 1)I ≥ H ˆ − (η + 1)I ≥ I , Ωn (1) − (η + 1)I ≥ H

(5.60)

ˆ n = Aˆ + Vn , n = 1, 2, . . .. Hence, the inverse operator (Ωn (1) − η − 1)−1 where H exists and we have (5.61) 0 ≤ (Ωn (1) − η − 1)−1 ≤ I . ˆ − η − 1)−1 . By (5.57) the Let us show that s − limn→∞ (Ωn (1) − η − 1)−1 = (H

operator (Ωn (1) − η − 1)−1 admits the representation

1 1 (I + Zn )−1 q , (Ωn (1) − η − 1)−1 = q ˆn − η − 1 ˆn − η − 1 H H

(5.62)

where

o n p p p 1 −Vn + IΓn 2I − −Xn + I(B − Xn + I)−1 −Xn + I Zn = q ˆn − η − 1 H × Γ∗n

p 1 . −Vn + I q ˆn − η − 1 H

(5.63)

ON THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS: GENERAL THEORY

631

ˆ n }∞ to H ˆ and (5.44) one By the generalized strong resolvent convergence of {H n=1 has (5.64) I = s − lim (I + Zn )−1 n→∞

which yields ˆ −η−1 s − lim (Ωn (1) − η − 1)−1 = H

−1

n→∞

.

(5.65)

Since dom(ˆ ν ) is maximal with respect to W we have s − lim (B − Xn + I)−1 = 0 . n→∞

(5.66)

Since (5.66) yields the existence of the limit s − limn→∞ (B − Xn + I)−1/2 and √ the sequence of operators { −Vn + IP (B − Xn + I)−1/2 }∞ n=1 is uniformly bounded we get p −Vn + I P (B − Xn + I)−1 = 0 . (5.67) s − lim n→∞

Using (5.40) and (5.44), we obtain s − lim (B − Xn + I)−1 P n→∞

p −Vn + I = 0 .

(5.68)

The estimate (5.50) gives |Wn | ≤ −Vn + 2I , which implies s − lim

n→∞

p |Wn |P (B − Xn + I)−1 = 0 .

(5.69) (5.70)

Since {Wn }∞ n=1 converges in the strong resolvent sense to W , one has s − lim

n→∞

p √ −1 −1 |Wn | + I = −W + I .

(5.71)

Hence, we obtain p 1 =0 (5.72) |Wn | p |Wn | + I p However, the sequence {(B−Xn +I)−1/2 P |Wn |}∞ n=1 is uniformly bounded. Therefore, we get p (5.73) s − lim (B − Xn + I)−1/2 P |Wn | = 0 , s − lim (B − Xn + I)−1 P n→∞

n→∞

which yields s − lim (B − Xn + I)−1 P n→∞

and s − lim

n→∞

p |Wn | = 0

p p |Wn |P (B − Xn + I)−1 P |Wn | = 0 .

(5.74) (5.75)

Taking into account (4.15)–(4.17), (5.42), the representation (5.62) as well as the convergences (5.64), (5.70), (5.74), (5.75) and s − limn→∞ Jn = I we finally obtain p |Wn |P (B − Xn + I)−1 P Wn (Ωn (1) − η − 1)−1 = 0 , (5.76) s − lim n→∞

632

H. NEIDHARDT and V. ZAGREBNOV

s − lim (Ωn (1) − η − 1)−1 Wn P (B − Xn + I)−1 P n→∞

p |Wn | = 0 ,

p |Wn |P (B − Xn + 1)−1 P Wn (Ωn (1) − η − 1)−1 n→∞ p × Wn P (B − Xn + 1)−1 P |Wn | = 0 .

(5.77)

s − lim

(5.78)

Using representation (4.29) and convergences (5.65), (5.66), (5.70), (5.74), (5.75), (5.76)–(5.78) we prove (5.36)–(5.39). Remark 5.5. We note that the sequence {Wn }∞ n=1 defined by (5.42) dependents ˜ η but not on the chosen semibounded self-adjoint extension A˜ of on the subspace Q A0 . Moreover, the construction holds for any non-increasing regularizing sequence {Vn }∞ n=1 of W . Acknowledgements The first author (H.N.) would like to acknowledge Centre de Physique Th´eorique–Luminy as well as Universit´e de la M´editerran´ee and Universit´e de Toulon et du Var for hospitality and support during his visit to Marseille. The paper was accomplished during the visit of the second author (V.Z.) to Potsdam University. He thanks the Sonderforschungsbereich 288 in Berlin for financial support. References [1] N. I. Akhiezer and I. M. Glasmann, Theory of Linear Operators in Hilbert Space, Vol. II. Frderick Ungar Publ. Co., New York, 1963. [2] A. Alonso and B. Simon, “The Birman–Krein–Visik theory of self-adjoint extensions of semibounded operators”, J. Operator Theory 4 (1980) 251–270. [3] V. A. Derkach and M. M. Malamud, “On the Weyl function and lacunary Hermitian operators”, Dokl. Akad. Nauk SSSR 293 (5) (1980) 1041–1046. [4] M. M. Malamud, “On the formula of generalized resolvents for non-densely defined Hermitian operators”, Ukr. Math. J. 44 (12) (1992) 1658–1688. [5] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin– Heidelberg–New York, 1966. [6] M. Klaus, Removing cut-offs from one-dimensional Schr¨ odinger operators, J. Phys. A: Math. Gen. 13 (1980) L295–L298. [7] G. Nenciu, “Removing cut-offs from singular perturbations: an abstract result”, Lett. Math. Phys. 7 (1983) 301–306. [8] H. Neidhardt and V. A. Zagrebnov, Regularization and convergence for singular perturbations, Commun. Math. Phys. 149 (1992) 573–586. [9] H. Neidhardt and V. A. Zagrebnov, “Singular perturbations, regularization and extension theory”, Operator Theory: Advances and Applications Vol. 70 Birkh¨ auser Verlag, Basel, 1994, pp. 299–305. [10] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness, Academic Press, New York–San Francisco–London, 1975. [11] M. Schechter, “Cut-off potentials and form extensions”, Lett. Math. Phys. 1 (1976) 265–273. [12] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton Univ. Press, Princeton, NJ, 1971.

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[13] B. Simon, “Quadratic forms and Klauder’s phenomenon: a remark on very singular perturbations”, J. Funct. Anal. 14 (1973) 295–298. [14] B. Simon, “A canonical decomposition for quadratic forms with applications to monotone convergence theorems”, J. Funct. Analysis 28 (1978) 377–385. [15] V. A. Zagrebnov, “On singular potential interactions in quantum statistical mechanics”, Trans. Moscow Math. Soc. 41 (1980) 101–120.

CONTINUITY OF SYMPLECTICALLY ADJOINT MAPS AND THE ALGEBRAIC STRUCTURE OF HADAMARD VACUUM REPRESENTATIONS FOR QUANTUM FIELDS ON CURVED SPACETIME RAINER VERCH∗ Dipartimento di Matematica Universit` a di Roma II “Tor Vergata” I-00133 Roma, Italy E-mail : [email protected] Received 8 October 1996 Revised 26 February 1997 We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its “purification”. As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its “purification” induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1 -properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

1. Introduction In the first part of this paper we shall investigate a special case of relative continuity of symplectically adjoint maps of a symplectic space. By this, we mean the following. Suppose that (S, σ) is a symplectic space, i.e. S is a real-linear vector space with an anti-symmetric, non-degenerate bilinear form σ (the symplectic form). A pair V, W of linear maps of S will be called symplectically adjoint if σ(V φ, ψ) = σ(φ, W ψ) for all φ, ψ ∈ S. Let µ and µ0 be two scalar products on S and assume that, for each pair V, W of symplectically adjoint linear maps of (S, σ), the boundedness of both V and W with respect to µ implies their boundedness with respect to µ0 . ∗ Supported

by a Von Neumann Fellowship of the Operator Algebras Network, EC Human Capital and Mobility Programme. Address after 1 April 1997: Inst. f. Theor. Physics, Universit¨ at G¨ ottingen, Bunsenstr. 9, D-37073 G¨ ottingen, Germany. 635 Reviews in Mathematical Physics, Vol. 9, No. 5 (1997) 635–674 c

World Scientific Publishing Company

636

R. VERCH

We refer to such a situation as relative µ − µ0 continuity of symplectically adjoint maps (of (S, σ)). A particular example of symplectically adjoint maps is provided by the pair T, T −1 whenever T is a symplectomorphism of (S, σ). (Recall that a symplectomorphism of (S, σ) is a bijective linear map T : S → S which preserves the symplectic form, σ(T φ, T ψ) = σ(φ, ψ) for all φ, ψ ∈ S.) In the more specialized case to be considered in the present work, which will soon be indicated to be relevant in applications, we show that a certain distinguished relation between a scalar product µ on S and a second one, µ0 , is sufficient for the relative µ − µ0 continuity of symplectically adjoint maps. (We give further details in Sec. 2, and in the next paragraph.) The result will be applied in Sec. 3 to answer a couple of open questions concerning the algebraic structure of the quantum theory of the free scalar field in arbitrary globally hyperbolic spacetimes: the local definiteness, local primarity and Haag-duality in representations of the local observable algebras induced by quasifree Hadamard states, as well as the determination of the type of the local von Neumann algebras in such representations. Technically, what needs to be proved in our approach to this problem is the continuity of the temporal evolution of the Cauchy-data of solutions of the scalar Klein–Gordon equation (∇a ∇a + r)ϕ = 0

(1.1)

in a globally hyperbolic spacetime with respect to a certain topology on the Cauchydata space. (Here, ∇ is the covariant derivative induced by the metric g on the spacetime, and r an arbitrary real-valued, smooth function.) The Cauchy-data space is a symplectic space on which the said temporal evolution is realized by symplectomorphisms. It turns out that the classical “energy-norm” of solutions of (1.1), which is given by a scalar product µ0 on the Cauchy-data space, and the topology relevant for the required continuity statement (the “Hadamard oneparticle space norm”), induced by a scalar product µ1 on the Cauchy-data space, are precisely in the relation for which our result on relative µ0 − µ1 continuity of symplectically adjoint maps applies. Since the continuity of the Cauchy-data evolution in the classical energy norm, i.e. µ0 , is well known, the desired continuity in the µ1 -topology follows. The argument just described may be viewed as the prime example of application of the relative continuity result. In fact, the relation between µ0 and µ1 is abstracted from the relation between the classical energy-norm and the one-particle space norms arising from “frequency-splitting” procedures in the canonical quantization of (linear) fields. This relation has been made precise in a recent paper by Chmielowski [11]. It provides the starting point for our investigation in Sec. 2, where we shall see that one can associate with a dominating scalar product µ ≡ µ0 on S in a canonical way a positive, symmetric operator |Rµ | on the µ-completion of S, and a family of scalar products µs , s > 0, on S, defined as µ with |Rµ |s as an operator kernel. Using abstract interpolation, it will be shown that then relative µ0 − µs continuity of symplectically adjoint maps holds for all 0 ≤ s ≤ 2. The relative µ0 − µ1 continuity arises as a special case. In fact, it turns out that

CONTINUITY OF SYMPLECTICALLY ADJOINT MAPS AND HADAMARD VACUA

637

the indicated interpolation argument may even be extended to an apparently more general situation from which the relative µ0 −µs continuity of symplectically adjoint maps derives as a corollary, see Theorem 2.2. Section 3 will be concerned with the application of the result of Theorem 2.2 as indicated above. In the preparatory Sec. 3.1, some notions of general relativity will be summarized, along with the introduction of some notation. Section 3.2 contains a brief synopsis of the notions of local definiteness, local primarity and Haag-duality in the the context of quantum field theory in curved spacetime. In Sec. 3.3 we present the C ∗ -algebraic quantization of the KG-field obeying (1.1) on a globally hyperbolic spacetime, following [16]. Quasifree Hadamard states will be described in Sec. 3.4 according to the definition given in [45]. In the same section we briefly summarize some properties of Hadamard two-point functions, and derive, in Proposition 3.5, the result concerning the continuity of the Cauchy-data evolution maps in the topology of the Hadamard two-point functions which was mentioned above. It will be seen in Sec. 3.5 that this leads, in combination with results obtained earlier [64, 65, 66], to Theorem 3.6 establishing detailed properties of the algebraic structure of the local von Neumann observable algebras in representations induced by quasifree Hadamard states of the Klein–Gordon field over an arbitrary globally hyperbolic spacetime. 2. Relative Continuity of Symplectically Adjoint Maps Let (S, σ) be a symplectic space. A (real-linear) scalar product µ on S is said to dominate σ if the estimate |σ(φ, ψ)|2 ≤ 4 · µ(φ, φ) µ(ψ, ψ) ,

φ, ψ ∈ S ,

(2.1)

holds; the set of all scalar products on S which dominate σ will be denoted by µ q(S, σ). Given µ ∈ q(S, σ), we write Hµ ≡ S for the completion of S with respect to the topology induced by µ, and denote by σµ the µ-continuous extension, guaranteed to uniquely exist by (2.1), of σ to Hµ . The estimate (2.1) then extends to σµ and all φ, ψ ∈ Hµ . This entails that there is a uniquely determined, µ-bounded linear operator Rµ : Hµ → Hµ with the property σµ (x, y) = 2 µ(x, Rµ y) ,

x, y ∈ Hµ .

(2.2)

The antisymmetry of σµ entails for the µ-adjoint Rµ∗ of Rµ Rµ∗ = −Rµ ,

(2.3)

and by (2.1) one finds that the operator norm of Rµ is bounded by 1, || Rµ || ≤ 1. The operator Rµ will be called the polarizator of µ. In passing, two things should be noticed here: (1) Rµ |S is injective since σ is a non-degenerate bilinear form on S, but Rµ need not be injective on all of Hµ , as σµ may be degenerate. (2) In general, it is not the case that Rµ (S) ⊂ S.

638

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Further properties of Rµ will be explored below. Let us first focus on two significant subsets of q(S, σ) which are intrinsically characterized by properties of the corresponding σµ or, equivalently, the Rµ . The first is pr(S, σ), called the set of primary scalar products on (S, σ), where µ ∈ q(S, σ) is in pr(S, σ) if σµ is a symplectic form (i.e. non-degenerate) on Hµ . In view of (2.2) and (2.3), one can see that this is equivalent to either (and hence, both) of the following conditions: (i) Rµ is injective, (ii) Rµ (Hµ ) is dense in Hµ . The second important subset of q(S, σ) is denoted by pu(S, σ) and defined as consisting of those µ ∈ q(S, σ) which satisfy the saturation property µ(φ, φ) =

sup ψ∈S\{0}

|σ(φ, ψ)|2 , ψ ∈ S. 4µ(ψ, ψ)

(2.4)

The set pu(S, σ) will be called the set of pure scalar products on (S, σ). It is straightforward to check that µ ∈ pu(S, σ) if and only if Rµ is a unitary antiinvolution, or complex structure, i.e. Rµ−1 = Rµ∗ , Rµ2 = −1. Hence pu(S, σ) ⊂ pr(S, σ). Our terminology reflects well-known relations between properties of quasifree states on the (CCR-) Weyl-algebra of a symplectic space (S, σ) and properties of σ-dominating scalar products on S, which we shall briefly recapitulate. We refer to [1, 3, 5, 45, 49] and also references quoted therein for proofs and further discussion of the following statements. Given a symplectic space (S, σ), one can associate with it uniquely (up to C ∗ -algebraic equivalence) a C ∗ -algebra A[S, σ], which is generated by a family of unitary elements W (φ), φ ∈ S, satisfying the canonical commutation relations (CCR) in exponentiated form, W (φ)W (ψ) = e−iσ(φ,ψ)/2 W (φ + ψ) ,

φ, ψ ∈ S .

(2.5)

The algebra A[S, σ] is called the Weyl-algebra, or CCR-algebra, of (S, σ). It is not difficult to see that if µ ∈ q(S, σ), then one can define a state (i.e., a positive, normalized linear functional) ωµ on A[S, σ] by setting ωµ (W (φ)) := e−µ(φ,φ)/2 ,

φ∈S.

(2.6)

Any state on the Weyl-algebra A[S, σ] which can be realized in this way is called a quasifree state. Conversely, given any quasifree state ωµ on A[S, σ], one can recover its µ ∈ q(S, σ) as ∂ ∂ ωµ (W (tφ)W (τ ψ)) , φ, ψ ∈ S . (2.7) µ(φ, ψ) = 2Re ∂t ∂τ t=τ =0 So there is a one-to-one correspondence between quasifree states on A[S, σ] and dominating scalar products on (S, σ).

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639

Let us now recall the subsequent terminology. To a state ω on a C ∗ -algebra B there corresponds (uniquely up to unitary equivalence) a triple (Hω , πω , Ωω ), called the GNS-representation of ω (see e.g. [5]), characterized by the following properties: Hω is a complex Hilbert space, πω is a representation of B by bounded linear operators on Hω with cyclic vector Ωω , and ω(B) = hΩω , πω (B)Ωω i for all B ∈ B. Hence one is led to associate with ω and B naturally the ω-induced von Neumann algebra πω (B)− , where the bar means taking the closure with respect to the weak operator topology in the set of bounded linear operators on Hω . One refers to ω (respectively, πω ) as primary if πω (B)− ∩ πω (B)0 = C · 1 (so the center of πω (B)− is trivial), where the prime denotes taking the commutant, and as pure if πω (B)0 = C · 1 (i.e. πω is irreducible — this is equivalent to the statement that ω is not a (non-trivial) convex sum of different states). In the case where ωµ is a quasifree state on a Weyl-algebra A[S, σ], it is known that (cf. [1, 49]) (I) ωµ is primary if and only if µ ∈ pr(S, σ), (II) ωµ is pure if and only if µ ∈ pu(S, σ). We return to the investigation of the properties of the polarizator Rµ for a dominating scalar product µ on a symplectic space (S, σ). It possesses a polar decomposition (2.8) Rµ = Uµ |Rµ | on the Hilbert space (Hµ , µ), where Uµ is an isometry and |Rµ | is symmetric and has non-negative spectrum. Since Rµ∗ = −Rµ , Rµ is normal and thus |Rµ | and Uµ commute. Moreover, one has |Rµ |Uµ∗ = −Uµ |Rµ |, and hence |Rµ | and Uµ∗ commute as well. One readily observes that (Uµ∗ + Uµ )|Rµ | = 0. The commutativity can by the spectral calculus be generalized to the statement that, whenever f is a real valued, continuous function on the real line, then [f (|Rµ |), Uµ ] = 0 = [f (|Rµ |), Uµ∗ ] ,

(2.9)

where the brackets denote the commutator. In a recent work [11], Chmielowski noticed that if one defines for µ ∈ q(S, σ) the bilinear form (2.10) µ ˜(φ, ψ) := µ(φ, |Rµ |ψ) , φ, ψ ∈ S, then it holds that µ ˜ ∈ pu(S, σ). The proof of this is straightforward. That µ ˜ dominates σ will be seen in Proposition 2.1. To check the saturation property (2.4) for µ ˜, it suffices to observe that for given φ ∈ Hµ , the inequality in the following chain of expressions: 1 |σµ (φ, ψ)|2 = |µ(φ, Uµ |Rµ |ψ)|2 4 = |µ(φ, −Uµ∗ |Rµ |ψ)|2 = |µ(|Rµ |1/2 Uµ φ, |Rµ |1/2 ψ)|2 ≤ µ(|Rµ |1/2 Uµ φ, |Rµ |1/2 Uµ φ) · µ(|Rµ |1/2 ψ, |Rµ |1/2 ψ)

640

R. VERCH

is saturated and becomes an equality upon choosing ψ ∈ Hµ so that |Rµ |1/2 ψ is parallel to |Rµ |1/2 Uµ φ. Therefore one obtains for all φ ∈ S, sup ψ∈S\{0}

|σ(φ, ψ)|2 = µ(|Rµ |1/2 Uµ φ, |Rµ |1/2 Uµ φ) 4µ(ψ, |Rµ |ψ) = µ(Uµ |Rµ |1/2 φ, Uµ |Rµ |1/2 φ) =µ ˜(φ, φ) ,

which is the required saturation property. Following Chmielowski, the scalar product µ ˜ on S associated with µ ∈ q(S, σ) will be called the purification of µ. It appears natural to associate with µ ∈ q(S, σ) the family µs , s > 0, of symmetric bilinear forms on S given by µs (φ, ψ) := µ(φ, |Rµ |s ψ) ,

φ, ψ ∈ S .

(2.11)

˜ = µ1 . The subsequent We will use the convention that µ0 = µ. Observe that µ proposition ensues. Proposition 2.1. (a) µs is a scalar product on S for each s ≥ 0. (b) µs dominates σ for 0 ≤ s ≤ 1. (c) Suppose that there is some s ∈ (0, 1) such that µs ∈ pu(S, σ). Then µr = µ1 for all r > 0. If it is in addition assumed that µ ∈ pr(S, σ), then it follows ˜. that µr = µ1 for all r ≥ 0, i.e. in particular µ = µ (d) If µs ∈ q(S, σ) for some s > 1, then µr = µ1 for all r > 0. Assuming ˜. additionally µ ∈ pr(S, σ), one obtains µr = µ1 for all r ≥ 0, entailing µ = µ ˜: We have µ fs = µ ˜ = µ1 (e) The purifications of the µs , 0 < s < 1, are equal to µ for all 0 < s < 1. Proof. (a) According to (b), µs dominates σ for each 0 ≤ s ≤ 1, thus it is a scalar product whenever s is in that range. However, it is known that µ(φ, |Rµ |s φ) ≥ µ(φ, |Rµ |φ)s for all vectors φ ∈ Hµ of unit length (µ(φ, φ) = 1) and 1 ≤ s < ∞, cf. [60, p. 20]. This shows that µs (φ, φ) 6= 0 for all nonzero φ in S, s ≥ 0. (b) For s in the indicated range there holds the following estimate: 1 |σ(φ, ψ)|2 = |µ(φ, Uµ |Rµ |ψ)|2 4 = |µ(φ, −Uµ∗ |Rµ |ψ)|2 = |µ(|Rµ |s/2 Uµ φ, |Rµ |1−s/2 ψ)|2 ≤ µ(Uµ |Rµ |s/2 φ, Uµ |Rµ |s/2 φ) · µ(|Rµ |s/2 ψ, |Rµ |2(1−s) |Rµ |s/2 ψ) ≤ µs (φ, φ) · µs (ψ, ψ) ,

φ, ψ ∈ S .

Here, we have used that |Rµ |2(1−s) ≤ 1. (c) If (φn ) is a µ-Cauchy-sequence in Hµ , then it is, by continuity of |Rµ |s/2 , also a µs -Cauchy-sequence in Hs , the µs -completion of S. Via this identification, we

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obtain an embedding j : Hµ → Hs . Notice that j(ψ) = ψ for all ψ ∈ S, so j has dense range; however, one has µs (j(φ), j(ψ)) = µ(φ, |Rµ |s ψ)

(2.12)

for all φ, ψ ∈ Hµ . Therefore j need not be injective. Now let Rs be the polarizator of µs . Then we have 2µs (j(φ), Rs j(ψ)) = σµ (φ, ψ) = 2µ(φ, Rµ ψ) = 2µ(φ, |Rµ |s Uµ |Rµ |1−s ψ) = 2µs (j(φ), j(Uµ |Rµ |1−s )ψ) ,

φ, ψ ∈ Hµ .

This yields Rs ◦j = j ◦Uµ |Rµ |1−s

(2.13)

on Hµ . Since by assumption µs is pure, we have Rs2 = −1 on Hs , and thus j = −Rs jUµ |Rµ |1−s = −j(Uµ |Rµ |1−s )2 . By (2.12) we may conclude |Rµ |2s = −Uµ |Rµ |Uµ |Rµ | = Uµ∗ Uµ |Rµ |2 = |Rµ |2 , which entails |Rµ |s = |Rµ |. Since |Rµ | ≤ 1, we see that for s ≤ r ≤ 1 we have |Rµ | = |Rµ |s ≥ |Rµ |r ≥ |Rµ | , hence |Rµ |r = |Rµ | for s ≥ r ≥ 1. Whence |Rµ |r = |Rµ | for all r > 0. This proves the first part of the statement. For the second part we observe that µ ∈ pr(S, σ) implies that |Rµ |, and hence also |Rµ |s for 0 < s < 1, is injective. Then the equation |Rµ |s = |Rµ | implies that |Rµ |s (|Rµ |1−s −1) = 0, and by the injectivity of |Rµ |s we may conclude |Rµ |1−s = 1. Since s was assumed to be strictly less than 1, it follows that |Rµ |r = 1 for all r ≥ 0; in particular, |Rµ | = 1. (d) Assume that µs dominates σ for some s > 1, i.e. it holds that 4|µ(φ, Uµ |Rµ |ψ)|2 = |σµ (φ, ψ)|2 ≤ 4 · µ(φ, |Rµ |s φ) · µ(ψ, |Rµ |s ψ) ,

φ, ψ ∈ Hµ ,

which implies, choosing φ = Uµ ψ, the estimate µ(ψ, |Rµ |ψ) ≤ µ(ψ, |Rµ |s ψ) ,

ψ ∈ Hµ ,

i.e. |Rµ | ≤ |Rµ |s . On the other hand, |Rµ | ≥ |Rµ |r ≥ |Rµ |s holds for all 1 ≤ r ≤ s since |Rµ | ≤ 1. This implies |Rµ |r = |Rµ | for all r > 0. For the second part of the statement one uses the same argument as given in (c).

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R. VERCH

(e) In view of (2.13) it holds that |Rs |2 j = −Rs2 j = −Rs jUµ |Rµ |1−s = −jUµ |Rµ |1−s Uµ |Rµ |1−s = −jUµ2 (|Rµ |1−s )2 . Iterating this one has for all n ∈ N |Rs |2n j = (−1)n jUµ2n (|Rµ |1−s )2n . Inserting this into relation (2.12) yields for all n ∈ N µs (j(φ), |Rs |2n j(ψ)) = µ(φ, |Rµ |s (−1)n Uµ2n (|Rµ |1−s )2n ψ) = µ(φ, |Rµ |s (|Rµ |1−s )2n ψ) ,

φ, ψ ∈ Hµ .

(2.14)

For the last equality we used that Uµ commutes with |Rs |s and Uµ2 |Rµ | = −|Rµ |. Now let (Pn ) be a sequence of polynomials on the intervall [0, 1] converging uniformly to the square root function on [0, 1]. From (2.14) we infer that µs (j(φ), Pn (|Rs |2 )j(ψ)) = µ(φ, |Rµ |s Pn ((|Rµ |1−s )2 )ψ) ,

φ, ψ ∈ Hµ

for all n ∈ N, which in the limit n → ∞ gives µs (j(φ), |Rs |j(ψ)) = µ(φ, |Rµ |ψ) , as desired.

φ, ψ ∈ Hµ ,

˜ = µ iff Proposition 2.1 underlines the special role of µ ˜ = µ1 . Clearly, one has µ µ ∈ pu(S, σ). Chmielowski has proved another interesting connection between µ and µ ˜ which we briefly mention here. Suppose that {Tt } is a one-parametric group of symplectomorphisms of (S, σ), and let {αt } be the automorphism group on A[S, σ] induced by it via αt (W (φ)) = W (Tt φ), φ ∈ S, t ∈ R. An {αt }-invariant quasifree state ωµ on A[S, σ] is called regular if the unitary group which implements {αt } in the GNS-representation (Hµ , πµ , Ωµ ) of ωµ is strongly continuous and leaves no nonzero vector in the one-particle space of Hµ invariant. Here, the one-particle space is d π (W (tφ))Ωµ , φ ∈ S. It is proved in [11] spanned by all vectors of the form dt t=0 µ that, if ωµ is a regular quasifree KMS-state for {αt }, then ωµ˜ is the unique regular ˜ can be quasifree groundstate for {αt }. As explained in [11], the passage from µ to µ seen as a rigorous form of “frequency-splitting” methods employed in the canonical quantization of classical fields for which µ is induced by the classical energy norm. We shall come back to this in the concrete example of the Klein–Gordon field in Sec. 3.4. It should be noted that the purification map ˜· : q(S, σ) → pu(S, σ), µ 7→ µ ˜, assigns to a quasifree state ωµ on A[S, σ] the pure quasifree state ωµ˜ which is again a state on A[S, σ]. This is different from the well-known procedure of assigning to

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643

a state ω on a C ∗ -algebra A, whose GNS representation is primary, a pure state ω0 on A◦ ⊗ A. (A◦ denotes the opposite algebra of A, cf. [75].) That procedure was introduced by Woronowicz and is an abstract version of similar constructions for quasifree states on CCR- or CAR-algebras [45, 54, 75]. Whether the purification map ωµ 7→ ωµ˜ can be generalized from quasifree states on CCR-algebras to a procedure of assigning to (a suitable class of) states on a generic C ∗ -algebra pure states on that same algebra, is in principle an interesting question, which however we shall not investigate here. Theorem 2.2. (a) Let H be a (real or complex) Hilbert space with scalar product µ( . , . ), R a (not necessarily bounded) normal operator in H, and V, W two µbounded linear operators on H which are R-adjoint, i.e. they satisfy W dom(R) ⊂ dom(R)

V ∗ R = RW

and

on dom(R) .

(2.15)

Denote by µs the Hermitean form on dom(|R|s/2 ) given by µs (x, y) := µ(|R|s/2 x, |R|s/2 y) ,

x, y ∈ dom(|R|s/2 ), 0 ≤ s ≤ 2 .

We write || . ||0 := || . ||µ := µ( . , . )1/2 and || . ||s := µs ( . , . )1/2 for the corresponding semi-norms. Then it holds for all 0 ≤ s ≤ 2 that V dom(|R|s/2 ) ⊂ dom(|R|s/2 )

and

W dom(|R|s/2 ) ⊂ dom(|R|s/2 ) ,

and V and W are µs -bounded for 0 ≤ s ≤ 2. More precisely, the estimates || V x ||0 ≤ v || x ||0

and

|| W x ||0 ≤ w || x ||0 ,

x∈H,

(2.16)

with suitable constants v, w > 0, imply that || V x ||s ≤ ws/2 v 1−s/2 || x ||s

and

|| W x ||s ≤ v s/2 w1−s/2 || x ||s ,

(2.17)

for all x ∈ dom(|R|s/2 ) and 0 ≤ s ≤ 2. (b) (Corollary of (a)) Let (S, σ) be a symplectic space, µ ∈ q(S, σ) a dominating scalar product on (S, σ), and µs , 0 < s ≤ 2, the scalar products on S defined in (2.11). Then we have relative µ − µs continuity of each pair V, W of symplectically adjoint linear maps of (S, σ) for all 0 < s ≤ 2. More precisely, for each pair V, W of symplectically adjoint linear maps of (S, σ), the estimates (2.16) for all x ∈ S imply (2.17) for all x ∈ S. Remarks. (i) In view of the fact that the operator R of part (a) of the Theorem may be unbounded, part (b) can be extended to situations where it is not assumed that the scalar product µ on S dominates the symplectic form σ. (ii) When it is additionally assumed that V = T and W = T −1 with symplectomorphisms T of (S, σ), we refer in that case to the situation of relative continuity of the pairs V, W as relative continuity of symplectomorphisms. In Example 2.3 after

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R. VERCH

the proof of Theorem 2.2 we show that relative µ ˜ − µ continuity of symplectomorphisms fails in general. Also, it is not the case that relative µ − µ0 continuity of symplectomorphisms holds if µ0 is an arbitrary element in pu(S, σ) which is dominated by µ (|| φ ||µ0 ≤ const.|| φ ||µ , φ ∈ S), see Example 2.4 below. This shows that the special relation between µ and µ ˜ (respectively, µ and the µs ) expressed in (2.11, 2.15) is important for the derivation of the Theorem. Proof of Theorem 2.2. (a) In a first step, let it be supposed that R is bounded. From the assumed relation (2.15) and its adjoint relation R∗ V = W ∗ R∗ we obtain, for 0 > 0 arbitrarily chosen, V ∗ (|R|2 + 0 1)V = V ∗ RR∗ V + 0 V ∗ V = RW W ∗ R∗ + 0 V ∗ V ≤ w2 |R|2 + 0 v 2 1 ≤ w2 (|R|2 + 1) with := 0 v 2 /w2 . This entails for the operator norms || (|R|2 + 0 1)1/2 V || ≤ w || (|R|2 + 1)1/2 || , and since (|R|2 + 1)1/2 has a bounded inverse, || (|R|2 + 0 1)1/2 V (|R|2 + 1)−1/2 || ≤ w . On the other hand, one clearly has || (|R|2 + 0 1)0 V (|R|2 + 1)0 || = || V || ≤ v . Now these estimates are preserved if R and V are replaced by their complexified versions on the complexified Hilbert space H ⊕ iH = C ⊗ H. Thus, identifying if necessary R and V with their complexifications, a standard interpolation argument (see Appendix A) can be applied to yield || (|R|2 + 0 1)α V (|R|2 + 1)−α || ≤ w2α v 1−2α for all 0 ≤ α ≤ 1/2. Notice that this inequality holds uniformly in 0 > 0. Therefore we may conclude that || |R|2α V x ||0 ≤ w2α v 1−2α || |R|2α x ||0 ,

x ∈ H , 0 ≤ α ≤ 1/2 ,

which is the required estimate for V . The analogous bound for W is obtained through replacing V by W in the given arguments. Now we have to extend the argument to the case that R is unbounded. Without restriction of generality we may assume that the Hilbert space H is complex, otherwise we complexify it and with it all the operators R, V , W , as above, thereby preserving their assumed properties. Then let E be the spectral measure of R, and denote by Rr the operator E(Br )RE(Br ) where Br := {z ∈ C : |z| ≤ r}, r > 0.

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Similarly define Vr and Wr . From the assumptions it is seen that Vr∗ Rr = Rr Wr holds for all r > 0. Applying the reasoning of the first step we arrive, for each 0 ≤ s ≤ 2, at the bounds || Vr x ||s ≤ ws/2 v 1−s/2 || x ||s

and || Wr x ||s ≤ v s/2 w1−s/2 || x ||s ,

which hold uniformly in r > 0 for all x ∈ dom(|R|s/2 ). From this, the statement of the Proposition follows. (b) This is just an application of (a), identifying Hµ with H, Rµ with R and V, W with their bounded extensions to Hµ . Example 2.3. We exhibit a symplectic space (S, σ) with µ ∈ pr(S, σ) and a ˜, symplectomorphism T of (S, σ) where T and T −1 are continuous with respect to µ but not with respect to µ. Let S := S(R, C), the Schwartz spaceRof rapidly decreasing testfunctions on R, viewed as real-linear space. By hφ, ψi := φψ dx we denote the standard L2 scalar product. As a symplectic form on S we choose σ(φ, ψ) := 2 Imhφ, ψi ,

φ, ψ ∈ S . 2

d Now define on S the strictly positive, essentially selfadjoint operator Aφ := − dx 2 φ+ 2 φ, φ ∈ S, in L (R). Its closure will again be denoted by A; it is bounded below by 1. A real-linear scalar product µ will be defined on S by

µ(φ, ψ) := RehAφ, ψi ,

φ, ψ ∈ S .

Since A has lower bound 1, clearly µ dominates σ, and one easily obtains Rµ = −iA−1 , |Rµ | = A−1 . Hence µ ∈ pr(S, σ) and µ ˜ (φ, ψ) = Rehφ, ψi ,

φ, ψ ∈ S .

Now consider the operator T :S →S,

(T φ)(x) := e−ix φ(x) , 2

x ∈ R, φ ∈ S .

˜. The Obviously T leaves the L2 scalar product invariant, and hence also σ and µ 2 ˜ invariant inverse of T is just (T −1 φ)(x) = eix φ(x), which of course leaves σ and µ as well. However, T is not continuous with respect to µ. To see this, let φ ∈ S be some non-vanishing smooth function with compact support, and define φn (x) := φ(x − n) ,

x ∈ R, n ∈ N .

Then µ(φn , φn ) = const. > 0 for all n ∈ N. We will show that µ(T φn , T φn ) diverges for n → ∞. We have µ(T φn , T φn ) = hAT φn , T φn i Z ≥ (T φn )0 (T φn )0 dx Z ≥

(4x2 |φn (x)|2 + |φ0n (x)|2 ) dx −

Z

4|xφ0n (x)φn (x)| dx , (2.18)

646

R. VERCH

where the primes indicate derivatives and we have used that |(T φn )0 (x)|2 = 4x2 |φn (x)|2 + |φ0n (x)|2 + 4 · Im(ixφn (x)φ0n (x)) . Using a substitution of variables, one can see that in the last term of (2.18) the positive integral grows like n2 for large n, thus dominating eventually the negative integral which grows only like n. So µ(T φn , T φn ) → ∞ for n → ∞, showing that T is not µ-bounded. Example 2.4. We give an example of a symplectic space (S, σ), a µ ∈ pr(S, σ) and a µ0 ∈ pu(S, σ), where µ dominates µ0 and where there is a symplectomorphism T of (S, σ) which together with its inverse is µ-bounded, but not µ0 -bounded. We take (S, σ) as in the previous example and write for each φ ∈ S, φ0 := Reφ and φ1 := Imφ. The real scalar product µ will be defined by µ(φ, ψ) := hφ0 , Aψ0 i + hφ1 , ψ1 i ,

φ, ψ ∈ S ,

where the operator A is the same as in the example before. Since its lower bound is 1, µ dominates σ, and it is not difficult to see that µ is even primary. The real-linear scalar product µ0 will be taken to be µ0 (φ, ψ) = Rehφ, ψi ,

φ, ψ ∈ S .

We know from the example above that µ0 ∈ pu(S, σ). Also, it is clear that µ0 is dominated by µ. Now consider the real-linear map T : S → S given by T (φ0 + iφ1 ) := A−1/2 φ1 − iA1/2 φ0 ,

φ∈S.

One checks easily that this map is bijective with T −1 = −T , and that T preserves the symplectic form σ. Also, || . ||µ is preserved by T since µ(T φ, T φ) = hφ1 , φ1 i + hA1/2 φ0 , A1/2 φ0 i = µ(φ, φ) ,

φ∈S.

On the other hand, we have for each φ ∈ S µ0 (T φ, T φ) = hφ1 , Aφ1 i + hφ0 , A−1 φ0 i , and this expression is not bounded by a (φ-independent) constant times µ0 (φ, φ), since A is unbounded with respect to the L2 -norm. 3. The Algebraic Structure of Hadamard Vacuum Representations 3.1. Summary of notions from spacetime-geometry We recall that a spacetime manifold consists of a pair (M, g), where M is a smooth, paracompact, four-dimensional manifold without boundaries, and g is a Lorentzian metric for M with signature (+ − − −). (Cf. [33, 52, 70], see these references also for further discussion of the notions to follow.) It will be assumed that (M, g) is time-orientable, and moreover, globally hyperbolic. The latter means

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that (M, g) possesses Cauchy-surfaces, where by a Cauchy-surface we always mean a smooth, spacelike hypersurface which is intersected exactly once by each inextendable causal curve in M . It can be shown [15, 28] that this is equivalent to the statement that M can be smoothly foliated in Cauchy-surfaces. Here, a foliation of M in Cauchy-surfaces is a diffeomorphism F : R × Σ → M , where Σ is a smooth 3-manifold so that F ({t} × Σ) is, for each t ∈ R, a Cauchy-surface, and the curves t 7→ F (t, q) are timelike for all q ∈ Σ. (One can even show that, if global hyperbolicity had been defined by requiring only the existence of a non necessarily smooth or spacelike Cauchy-surface (i.e. a topological hypersurface which is intersected exactly once by each inextendable causal curve), then it is still true that a globally hyperbolic spacetime can be smoothly foliated in Cauchy-surfaces, see [15, 28].) We shall also be interested in ultrastatic globally hyperbolic spacetimes. A globally hyperbolic spacetime is said to be ultrastatic if a foliation F : R × Σ → M in Cauchy-surfaces can be found so that F∗ g has the form dt2 ⊕(−γ) with a complete (t-independent) Riemannian metric γ on Σ. This particular foliation will then be called a natural foliation of the ultrastatic spacetime. (An ultrastatic spacetime may posses more than one natural foliation, think e.g. of Minkowski-spacetime.) The notation for the causal sets and domains of dependence will be recalled: Given a spacetime (M, g) and O ⊂ M , the set J ± (O) (causal future/past of O) consists of all points p ∈ M which can be reached by future/past directed causal curves emanating from O. The set D± (O) (future/past domain of dependence of O) is defined as consisting of all p ∈ J ± (O) such that every past/future inextendible causal curve starting at p intersects O. One writes J(O) := J + (O) ∪ J − (O) and D(O) := D+ (O) ∪ D− (O). They are called the causal set, and the domain of dependence, respectively, of O. For O ⊂ M , we denote by O⊥ := int(M \ J(O)) the causal complement of O, i.e. the largest open set of points which cannot be connected to O by any causal curve. A set of the form OG := int D(G), where G is a subset of some Cauchy-surface Σ in (M, g), will be referred to as the diamond based on G; we shall also say that G ⊥ is again a diamond, is the base of OG . We note that if OG is a diamond, then OG based on Σ \ G. A diamond will be called regular if G is an open, relatively compact subset of Σ and if the boundary ∂G of G is contained in the union of finitely many smooth, closed, two-dimensional submanifolds of Σ. Following [45], we say that an open neighbourhood N of a Cauchy-surface Σ in (M, g) is a causal normal neighbourhood of Σ if (1) Σ is a Cauchy-surface for N , and (2) for each pair of points p, q ∈ N with p ∈ J + (q), there is a convex normal neighbourhood O ⊂ M such that J − (p) ∩ J + (q) ⊂ O. Lemma 2.2 of [45] asserts the existence of causal normal neighbourhoods for any Cauchy-surface Σ. 3.2. Some structural aspects of quantum field theory in curved spacetime In the present subsection, we shall address some of the problems one faces in the formulation of quantum field theory in curved spacetime, and explain the notions

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of local definiteness, local primarity, and Haag-duality. In doing so, we follow our presentation in [67] quite closely. Standard general references related to the subsequent discussion are [26, 31, 45, 71]. Quantum field theory in curved spacetime (QFT in CST, for short) means that one considers quantum fields propagating in a (classical) curved background spacetime manifold (M, g). In general, such a spacetime need not possess any symmetries, and so one cannot tie the notion of “particles” or “vacuum” to spacetime symmetries, as one does in quantum field theory in Minkowski spacetime. Therefore, the problem of how to characterize the physical states arises. For the discussion of this problem, the setting of algebraic quantum field theory is particularly well suited. Let us thus summarize some of the relevant concepts of algebraic QFT in CST. Let a spacetime manifold (M, g) be given. The observables of a quantum system (e.g. a quantum field) situated in (M, g) then have the basic structure of a map O → A(O), which assigns to each open, relatively compact subset O of M a C ∗ -algebra A(O),1 with the properties:2 Isotony : Locality :

O1 ⊂ O2 ⇒ A(O1 ) ⊂ A(O2 )

(3.1)

O1 ⊂ O2⊥ ⇒ [A(O1 ), A(O2 )] = {0} .

(3.2)

A map O → A(O) having these properties is called a net of local observable algebras over (M, g). We recall that the conditions of locality and isotony are motivated by the idea that each A(O) is the C ∗ -algebra formed by the observables which can be measured within the spacetime region O on the system. We refer to [31] and references given there for further discussion. The collection of all open, relatively compact subsets of M is directed with respect to set-inclusion, and so we can, in view of (3.1), form the smallest C ∗ -algebra S || . || which contains all local algebras A(O). For the description of a A := O A(O) of all positive, system we need not only observables but also states. The set A∗+ 1 normalized linear functionals on A is mathematically referred to as the set of states on A, but not all elements of A∗+ 1 represent physically realizable states of the system. Therefore, given a local net of observable algebras O → A(O) for a physical system over (M, g), one must specify the set of physically relevant states S, which is a suitable subset of A∗+ 1 . determines We have already mentioned in Sec. 2 that every state ω ∈ A∗+ 1 canonically its GNS representation (Hω , πω , Ωω ) and thereby induces a net of von Neumann algebras (operator algebras on Hω ) O → Rω (O) := πω (A(O))− . Some of the mathematical properties of the GNS representations, and of the induced nets of von Neumann algebras, of states ω on A can naturally be interpreted physically. Thus one obtains constraints on the states ω which are to be viewed 1 Throughout the paper, C ∗ -algebras are assumed to be unital, i.e. to possess a unit element, denoted by 1. It is further assumed that the unit element is the same for all the A(O). 2 where [A(O ), A(O )] = {A A − A A : A ∈ A(O ), j = 1, 2}. 1 2 1 2 2 1 j j

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as physical states. Following this line of thought, Haag, Narnhofer and Stein [32] formulated what they called the “principle of local definiteness”, consisting of the following three conditions to be obeyed by any collection S of physical states. T Local Definiteness: O3p Rω (O) = C · 1 for all ω ∈ S and all p ∈ M . Local Primarity: For each ω ∈ S, Rω (O) is a factor. Local Quasiequivalence: For each pair ω1 , ω2 ∈ S and each relatively compact, open O ⊂ M , the representations πω1 |A(O) and πω2 |A(O) of A(O) are quasiequivalent. Remarks. (i) We recall (cf. the first Remark in Sec. 2) that Rω (O) is a factor if Rω (O) ∩ Rω (O)0 = C · 1, where the prime means taking the commutant. We have not stated in the formulation of local primarity for which regions O the algebra Rω (O) is required to be a factor. The regions O should be taken from a class of subsets of M which forms a base for the topology. (ii) Quasiequivalence of representations means unitary equivalence up to multiplicity. Another characterization of quasiequivalence is to say that the folia of the representations coincide, where the folium of a representation π is defined as the which can be represented as ω(A) = tr(ρ π(A)) with a density set of all ω ∈ A∗+ 1 matrix ρ on the representation Hilbert space of π. (iii) Local definiteness and quasiequivalence together express that physical states have finite (spatio-temporal) energy-density with respect to each other, and local primarity and quasiequivalence rule out local macroscopic observables and local superselection rules. We refer to [31] for further discussion and background material. A further, important property which one expects to be satisfied for physical states ω ∈ S whose GNS representations are irreducible 3 is Haag-Duality: Rω (O⊥ )0 = Rω (O), which should hold for the causally complete regions O, i.e. those satisfying (O⊥ )⊥ = O, where Rω (O⊥ ) is defined as the von Neumann algebra generated by all the Rω (O1 ) so that O1 ⊂ O⊥ . We comment that Haag-duality means that the von Neumann algebra Rω (O) of local observables is maximal in the sense that no further observables can be added without violating the condition of locality. It is worth mentioning here that the condition of Haag-duality plays an important role in the theory of superselection sectors in algebraic quantum field theory in Minkowski spacetime [31, 59]. For local nets of observables generated by Wightman fields on Minkowski spacetime it follows from the results of Bisognano and Wichmann [4] that a weaker condition of “wedge-duality” is always fulfilled, which allows one to pass to a new, potentially larger local net (the “dual net”) which satisfies Haag-duality. In quantum field theory in Minkowski-spacetime where one is given a vacuum state ω0 , one can define the set of physical states S simply as the set of all states 3 It is easy to see that, in the presence of local primarity, Haag-duality will be violated if π is not ω irreducible.

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on A which are locally quasiequivalent (i.e., the GNS representations of the states are locally quasiequivalent to the vacuum-representation) to ω0 . It is obvious that local quasiequivalence then holds for S. Also, local definiteness holds in this case, as was proved by Wightman [72]. If Haag-duality holds in the vacuum representation (which, as indicated above, can be assumed to hold quite generally), then it does not follow automatically that all pure states locally quasiequivalent to ω0 will also have GNS representations fulfilling Haag-duality; however, it follows once some regularity conditions are satisfied which have been checked in certain quantum field models [19, 61]. So far there seems to be no general physically motivated criterion enforcing local primarity of a quantum field theory in algebraic formulation in Minkowski spacetime. But it is known that many quantum field theoretical models satisfy local primarity. For QFT in CST we do not know in general what a vacuum state is and so S cannot be defined in the same way as just described. Yet in some cases (for some quantum field models) there may be a set S0 ⊂ A∗+ 1 of distinguished states, and if this class of states satisfies the four conditions listed above, then the set S, defined as consisting of all states ω1 ∈ A∗+ 1 which are locally quasiequivalent to any (and hence all) ω ∈ S0 , is a good candidate for the set of physical states. For the free scalar Klein–Gordon field (KG-field) on a globally hyperbolic spacetime, the following classes of states have been suggested as distinguished, physically reasonable states4 (1) (quasifree) states fulfilling local stability [3, 22, 31, 32] (2) (quasifree) states fulfilling the wave front set (or microlocal) spectrum condition [6, 47, 55] (3) quasifree Hadamard states [12, 68, 45] (4) adiabatic vacua [38, 48, 53] The list is ordered in such a way that the less restrictive condition preceeds the stronger one. There are a couple of comments to be made here. First of all, the specifications (3) and (4) make use of the information that one deals with the KG-field (or at any rate, a free field obeying a linear equation of motion of hyperbolic character), while the conditions (1) and (2) do not require such input and are applicable to general — possibly interacting — quantum fields over curved spacetimes. (It should however be mentioned that only for the KG-field (2) is known to be stronger than (1). The relation between (1) and (2) for more general theories is not settled.) The conditions imposed on the classes of states (1), (2) and (3) are related in that they are ultralocal remnants of the spectrum condition requiring a certain regularity of the short distance behaviour of the respective states which can be formulated in generic spacetimes. The class of states (4) is more special and can only be defined for the KG-field (or other linear fields) propagating in Robertson– Walker-type spacetimes. Here a distinguished choice of a time-variable can be made, and the restriction imposed on adiabatic vacua is a regularity condition on their 4 The following list is not meant to be complete, it comprises some prominent families of states of the KG-field over a generic class of spacetimes for which mathematically sound results are known. Likewise, the indicated references are by no means exhaustive.

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spectral behaviour with respect to that special choice of time. (A somewhat stronger formulation of local stability has been proposed in [34].) It has been found by Radzikowski [55] that for quasifree states of the KG-field over generic globally hyperbolic spacetimes the classes (2) and (3) coincide. The microlocal spectrum condition is further refined and applied in [6, 47]. Recently it was proved by Junker [38] that adiabatic vacua of the KG-field in Robertson– Walker spacetimes fulfill the microlocal spectrum condition and thus are, in fact, quasifree Hadamard states. The notion of the microlocal spectrum condition and the just-mentioned results related to it draw on pseudodifferential operator techniques, particularly the notion of the wave front set, see [20, 36, 37]. Quasifree Hadamard states of the KG-field (see definition in Sec. 3.4) have been investigated for quite some time. One of the early studies of Hadamard forms, mainly in the context of classical fields on manifolds, is [12]. The importance of quasifree Hadamard states, especially in the context of the semiclassical Einstein equation, is stressed in [68]. Other significant references include [24, 25] and, in particular, [45] where, apparently for the first time, a satisfactory definition of the notion of a globally Hadamard state is given, cf. Sec. 3.4 for more details. In [66] it is proved that the class of quasifree Hadamard states of the KG-field fulfills local quasiequivalence in generic globally hyperbolic spacetimes and local definiteness, local primarity and Haag-duality for the case of ultrastatic globally hyperbolic spacetimes. As was outlined in the beginning, the purpose of the present chapter is to obtain these latter results also for arbitrary globally hyperbolic spacetimes which are not necessarily ultrastatic. It turns out that some of our previous results can be sharpened, e.g. the local quasiequivalence specializes in most cases to local unitary equivalence, cf. Theorem 3.6. For a couple of other results about the algebraic structure of the KG-field as well as other fields over curved spacetimes we refer to [2, 6, 16, 17, 18, 40, 41, 46, 63, 64, 65, 66, 73, 74]. 3.3. The Klein Gordon field In the present section we summarize the quantization of the classical KG-field over a globally hyperbolic spacetime in the C ∗ -algebraic formalism. This follows in major parts the the work of Dimock [16], cf. also references given there. Let (M, g) be a globally hyperbolic spacetime. The KG-equation with potential term r is (∇a ∇a + r)ϕ = 0

(3.3)

where ∇ is the Levi–Civita derivative induced by the metric g, the potential function r ∈ C ∞ (M, R) is arbitrary but fixed, and the sought for solutions ϕ are smooth and real-valued. Making use of the fact that (M, g) is globally hyperbolic and drawing on earlier results by Leray, it is shown in [16] that there are two uniquely determined, continuous5 linear maps E ± : C0∞ (M, R) → C ∞ (M, R) with the properties (∇a ∇a + r)E ± f = f = E ± (∇a ∇a + r)f ,

f ∈ C0∞ (M, R) ,

5 With respect to the usual locally convex topologies on C ∞ (M, R) and C ∞ (M, R), cf. [13]. 0

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and supp(E ± f ) ⊂ J ± (supp(f )) ,

f ∈ C0∞ (M, R) .

The maps E ± are called the advanced(+)/retarded(–) fundamental solutions of the KG-equation with potential term r in (M, g), and their difference E := E + − E − is referred to as the propagator of the KG-equation. One can moreover show that the Cauchy-problem for the KG-equation is wellposed. That is to say, if Σ is any Cauchy-surface in (M, g), and u0 ⊕ u1 ∈ C0∞ (M, R) ⊕ C0∞ (M, R) is any pair of Cauchy-data on Σ, then there exists precisely one smooth solution ϕ of the KG-equation (3.3) having the property that PΣ (ϕ) := ϕ|Σ ⊕ na ∇a ϕ|Σ = u0 ⊕ u1 .

(3.4)

The vectorfield na in (3.4) is the future-pointing unit normalfield of Σ. Furthermore, one has “finite propagation speed”, i.e. when the supports of u0 and u1 are contained in a subset G of Σ, then supp(ϕ) ⊂ J(G). Notice that compactness of G implies that J(G) ∩ Σ0 is compact for any Cauchy-surface Σ0 . The well-posedness of the Cauchy-problem is a consequence of the classical energy-estimate for solutions of second order hyperbolic partial differential equations, cf. e.g. [33]. To formulate it, we introduce further notation. Let Σ be a Cauchy-surface for (M, g), and γΣ the Riemannian metric, induced by the ambient Lorentzian metric, on Σ. Then denote the Laplacian operator on C0∞ (Σ, R) corresponding to γΣ by ∆γΣ , and define the classical energy scalar product on C0∞ (Σ, R) ⊕ C0∞ (Σ, R) by Z (u ⊕ u , v ⊕ v ) := (u0 (−∆γΣ + 1)v0 + u1 v1 ) dηΣ , (3.5) µE 0 1 0 1 Σ Σ

where dηΣ is the metric-induced volume measure on Σ. As a special case of the energy estimate presented in [33] one then obtains Lemma 3.1. (Classical energy estimate for the KG-field.) Let Σ1 and Σ2 be a pair of Cauchy-surfaces in (M, g) and G a compact subset of Σ1 . Then there are two positive constants c1 and c2 so that there holds the estimate E E c1 µE Σ1 (PΣ1 (ϕ), PΣ1 (ϕ)) ≤ µΣ2 (PΣ2 (ϕ), PΣ2 (ϕ)) ≤ c2 µΣ1 (PΣ1 (ϕ), PΣ1 (ϕ))

(3.6)

for all solutions ϕ of the KG-equation (3.3) which have the property that the supports of the Cauchy-data PΣ1 (ϕ) are contained in G.6 We shall now indicate that the space of smooth solutions of the KG-equation (3.3) has the structure of a symplectic space, locally as well as globally, which comes in several equivalent versions. To be more specific, observe first that the Cauchy-data space 6 The formulation given here is to some extent more general than the one appearing in [33] where it is assumed that Σ1 and Σ2 are members of a foliation. However, the more general formulation can be reduced to that case.

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DΣ := C0∞ (Σ, R) ⊕ C0∞ (Σ, R) of an arbitrary given Cauchy-surface Σ in (M, g) carries a symplectic form Z δΣ (u0 ⊕ u1 , v0 ⊕ v1 ) := (u0 v1 − v0 u1 ) dηΣ . Σ

It will also be observed that this symplectic form is dominated by the classical energy scalar product µE Σ. Another symplectic space is S, the set of all real-valued C ∞ -solutions ϕ of the KG-equation (3.3) with the property that, given any Cauchy-surface Σ in (M, g), their Cauchy-data PΣ (ϕ) have compact support on Σ. The symplectic form on S is given by Z (ϕna ∇a ψ − ψna ∇a ϕ) dηΣ

σ(ϕ, ψ) := Σ

which is independent of the choice of the Cauchy-surface Σ on the right-hand side over which the integral is formed; na is again the future-pointing unit normalfield of Σ. One clearly finds that for each Cauchy-surface Σ the map PΣ : S → DΣ establishes a symplectomorphism between the symplectic spaces (S, σ) and (DΣ , δΣ ). A third symplectic space equivalent to the previous ones is obtained as the quotient K := C0∞ (M, R)/ker(E) with symplectic form Z f (Eh) dη , f, h ∈ C0∞ (M, R) , κ([f ], [h]) := M

C0∞ (M, R)

→ K and dη is the metric-induced volume where [ . ] is the quotient map measure on M . Then define for any open subset O ⊂ M with compact closure the set K(O) := [C0∞ (O, R)]. One can see that the space K has naturally the structure of an isotonous, local net O → K(O) of subspaces, where locality means that the symplectic form κ([f ], [h]) vanishes for [f ] ∈ K(O) and [h] ∈ K(O1 ) whenever O1 ⊂ O⊥ . Dimock has proved in [16, Lemma A.3] that moreover there holds K(OG ) ⊂ K(N )

(3.7)

for all open neighbourhoods N (in M ) of G, whenever OG is a diamond. Using this, one obtains that the map (K, κ) → (S, σ) given by [f ] 7→ Ef is surjective, and by Lemma A.1 in [16], it is even a symplectomorphism. Clearly, (K(OG ), κ|K(OG )) is a symplectic subspace of (K, κ) for each diamond OG in (M, g). For any such diamond one then obtains, upon viewing it (or its connected components separately), equipped with the appropriate restriction of the spacetime metric g, as a globally hyperbolic spacetime in its own right, local versions of the just introduced symplectic spaces and the symplectomorphisms between them. More precisely, if we denote by S(OG ) the set of all smooth solutions of the KG-equation (3.3) with the property that their Cauchy-data on Σ are compactly supported in G, then the map PΣ restricts to a symplectomorphism (S(OG ), σ|S(OG )) → (DG , δG ), ϕ 7→ PΣ (ϕ). Likewise, the symplectomorphism [f ] 7→ Ef restricts to a symplectomorphism (K(OG ), κ|K(OG )) → (S(OG ), σ|S(OG )).

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To the symplectic space (K, κ) we can now associate its Weyl-algebra A[K, κ], cf. Sec. 2. Using the aforementioned local net-structure of the symplectic space (K, κ), one arrives at the following result. Proposition 3.2 [16]. Let (M, g) be a globally hyperbolic spacetime, and (K, κ) the symplectic space, constructed as above, for the KG-equation with smooth potential term r on (M, g). Its Weyl-algebra A[K, κ] will be called the Weyl-algebra of the KG-field with potential term r over (M, g). Define for each open, relatively compact O ⊂ M, the set A(O) as the C ∗ -subalgebra of A[K, κ] generated by all the Weyl-operators W ([f ]), [f ] ∈ K(O). Then O → A(O) is a net of C ∗ -algebras fulfilling isotony (3.1) and locality (3.2), and moreover primitive causality, i.e. A(OG ) ⊂ A(N )

(3.8)

for all neighbourhoods N (in M ) of G, whenever OG is a (relatively compact) diamond. It is worth recalling (cf. [5]) that the Weyl-algebras corresponding to symplectically equivalent spaces are canonically isomorphic in the following way: Let W (x), x ∈ K denote the Weyl-generators of A[K, κ] and WS (ϕ), ϕ ∈ S, the Weyl-generators of A[S, σ]. Furthermore, let T be a symplectomorphism between (K, κ) and (S, σ). Then there is a uniquely determined C ∗ -algebraic isomorphism αT : A[K, κ] → A[S, σ] given by αT (W (x)) = WS (T x), x ∈ K. This shows that if we had associated e.g. with (S, σ) the Weyl-algebra A[S, σ] as the algebra of quantum observables of the KG-field over (M, g), we would have obtained an equivalent net of observable algebras (connected to the previous one by a net isomorphism, see [3, 16]), rendering the same physical information. 3.4. Hadamard states We have indicated above that quasifree Hadamard states are distinguished by their short-distance behaviour which allows the definition of expectation values of energy-momentum observables with reasonable properties [26, 68, 69, 71]. If ωµ is a quasifree state on the Weyl-algebra A[K, κ], then we call i λ(x, y) := µ(x, y) + κ(x, y) , 2

x, y ∈ K ,

its two-point function and Λ(f, h) := λ([f ], [h]) ,

f, h ∈ C0∞ (M, R) ,

its spatio-temporal two-point function. In Sec. 2 we have seen that a quasifree state is entirely determined through specifying µ ∈ q(K, κ), which is equivalent to the specification of the two-point function λ. Sometimes the notation λω or λµ will be used to indicate the quasifree state ω or the dominating scalar product µ which is determined by λ. For a quasifree Hadamard state, the spatio-temporal two-point function is of a special form, called Hadamard form. The definition of Hadamard form which

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655

we give here follows that due to Kay and Wald [45]. Let N be a causal normal neighbourhood of a Cauchy-surface Σ in (M, g). Then a smooth function χ : N × N → [0, 1] is called N -regularizing if it has the following property: There is an open neighbourhood, Ω∗ , in N × N of the set of pairs of causally related points in N such that Ω∗ is contained in a set Ω to be described presently, and χ ≡ 1 on Ω∗ while χ ≡ 0 outside of Ω. Here, Ω is an open neighbourhood in M × M of the set of those (p, q) ∈ M × M which are causally related and have the property that (1) J + (p) ∩ J − (q) and J + (q) ∩ J − (p) are contained within a convex normal neighbourhood, and (2) s(p, q), the square of the geodesic distance between p and q, is a well-defined, smooth function on Ω. (One observes that there are always sets Ω of this type which contain a neighbourhood of the diagonal in M × M , and that an N -regularizing function depends on the choice of the pair of sets Ω∗ , Ω with the stated properties.) It is not difficult to check that N -regularizing functions always exist for any causal normal neighbourhood; a proof of that is e.g. given in [55]. Then denote by U the square root of the VanVleck–Morette determinant, and by vm , m ∈ N0 the sequence determined by the Hadamard recursion relations for the KG-equation (3.3), see [12, 23, 27] and also [30] for their definition. They are all smooth functions on Ω.7 Now set for n ∈ N, V (n) (p, q) :=

n X

vm (p, q)(s(p, q))m ,

(p, q) ∈ Ω ,

m=0

and, given a smooth time-function T : M → R increasing towards the future, define for all > 0 and (p, q) ∈ Ω, QT (p, q; ) := s(p, q) − 2i(T (p) − T (q)) − 2 , and GT,n (p, q) :=

1 4π 2

U (p, q) + V (n) (p, q)ln(QT (p, q; )) , QT (p, q; )

where ln is the principal branch of the logarithm. With this notation, one can give the following: Definition 3.3 [45]. A C-valued bilinear form Λ on C0∞ (M, R) is called a Hadamard form if, for a suitable choice of a causal normal neighbourhood N of some Cauchy-surface Σ, and for suitable choices of an N -regularizing function χ and a future-increasing time-function T on M, there exists a sequence H (n) ∈ C n (N × N ), so that Z ΛT,n (3.9) Λ(f, h) = lim (p, q)f (p)h(q) dη(p) dη(q) →0+

for all f, h ∈

C0∞ (N, R),

M×M 8

where

T,n (n) ΛT,n (p, q) , (p, q) := χ(p, q)G (p, q) + H

(3.10)

7 For any choice of Ω with the properties just described. 8 The set Ω on which the functions forming GT,n are defined and smooth is here to coincide with

the Ω with respect to which χ is defined.

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and if, moreover, Λ is a global bi-parametrix of the KG-equation (3.3), i.e. it satisfies Λ((∇a ∇a + r)f, h) = B1 (f, h)

and

Λ(f, (∇a ∇a + r)h) = B2 (f, h)

for all f, h ∈ C0∞ (M ), where B1 and B2 are given by smooth integral kernels on M × M .9 Based on results of [24, 25], it is shown in [45] that this is a reasonable definition. The findings of these works will be collected in the following: Proposition 3.4. (a) If Λ is of Hadamard form on a causal normal neighbourhood N of a Cauchy-surface Σ for some choice of a time-function T and some N -regularizing function χ (i.e. that (3.9), (3.10) hold with suitable H (n) ∈ C n (N × N )), then so it is for any other time-function T 0 and N -regularizing χ0 . (This means that these changes can be compensated by choosing another sequence H 0(n) ∈ C n (N × N ).) (b) (Causal Propagation Property of the Hadamard Form) If Λ is of Hadamard form on a causal normal neighbourhood N of some Cauchy-surface Σ, then it is of Hadamard form in any causal normal neighbourhood N 0 of any other Cauchysurface Σ0 . (c) Any Λ of Hadamard form is a regular kernel distribution on C0∞ (M × M ). (d) There exist pure, quasifree Hadamard states (these will be referred to as Hadamard vacua) on the Weyl-algebra A[K, κ] of the KG-field in any globally hyperbolic spacetime. The family of quasifree Hadamard states on A[K, κ] spans an infinite-dimensional subspace of the continuous dual space of A[K, κ]. (e) The dominating scalar products µ on K arising from quasifree Hadamard states ωµ induce locally the same topology, i.e. if µ and µ0 are arbitrary such scalar products and O ⊂ M is open and relatively compact, then there are two positive constants a, a0 such that a µ([f ], [f ]) ≤ µ0 ([f ], [f ]) ≤ a0 µ([f ], [f ]) ,

[f ] ∈ K(O) .

Remark. Observe that this definition of Hadamard form rules out the occurence of spacelike singularities, meaning that the Hadamard form Λ is, when tested on functions f, h in (3.9) whose supports are acausally separated, given by a C ∞ kernel. For that reason, the definition of Hadamard form as stated above is also called global Hadamard form (cf. [45]). A weaker definition of Hadamard form would be to prescribe (3.9), (3.10) only for sets N which, e.g., are members of an open covering of M by convex normal neighbourhoods, and thereby to require the Hadamard form locally. It is known from examples that such a local notion of 9 We point out that statement (b) of Proposition 3.4 is wrong if the assumption that Λ is a global bi-parametrix is not made. In this respect, Definition C.1 of [66] is imprecisely formulated as the said assumption is not stated. There, like in several other references, it has been implicitly assumed that Λ is a two-point function and thus a bi-solution of (3.3), i.e. a bi-parametrix with B1 = B2 ≡ 0.

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Hadamard form is in general properly weaker than that of the global Hadamard form [29, 51]. However, Radzikowski [56] recently proved a conjecture due to Kay [29] saying that, if Λ is locally of Hadamard form and in addition the spatio-temporal two-point function of a state on A[K, κ] and thus dominates the symplectic form κ (|κ([f ], [h])|2 ≤ 4 Λ(f, f )Λ(h, h)), then it is already globally of Hadamard form. Radzikowski’s proof makes use of a characterization of Hadamard forms in terms of their wave front sets which was mentioned above. A different, weaker definition of Hadamard form has recently been given in [44]. We should add that the usual Minkowski-vacuum of the free scalar field with constant, non-negative potential term is, of course, an Hadamard vacuum. This holds, more generally, also for ultrastatic spacetimes, see below. Notes on the proof of Proposition 3.4. The property (a) is proved in [45]. The argument for (b) is essentially contained in [25] and in the generality stated here it is completed in [45]. An alternative proof using the “propagation of singularities theorem” for hyperbolic differential equations is presented in [55]. Also property (c) is proved in [45, Appendix B] (cf. [66, Proposition C.2]). The existence of Hadamard vacua (d) is proved in [24] (cf. also [45]); the stated Corollary has been observed in [66] (and, in slightly different formulation, already in [24]). Statement (e) has been shown to hold in [66, Proposition 3.8]. In order to prepare the formulation of the next result, in which we will apply our result of Sec. 2, we need to collect some more notation. Suppose that we are given a quasifree state ωµ on the Weyl-algebra A[K, κ] of the KG-field over some globally hyperbolic spacetime (M, g), and that Σ is a Cauchy-surface in that spacetime. Then we denote by µΣ the dominating scalar product on (DΣ , δΣ ) which is, using the symplectomorphism between (K, κ) and (DΣ , δΣ ), induced by the dominating scalar product µ on (K, κ), i.e. µΣ (PΣ Ef, PΣ Eh) = µ([f ], [h]) ,

[f ], [h] ∈ K .

(3.11)

Conversely, to any µΣ ∈ q(DΣ , δΣ ) there corresponds via (3.11) a µ ∈ q(K, κ). Next, consider a complete Riemannian manifold (Σ, γ), with corresponding Laplacian ∆γ , and as before, consider the operator −∆γ + 1 on C0∞ (Σ, R). Owing to the completeness of (Σ, γ) this operator is, together with all its powers, essentially selfadjoint in L2R (Σ, dηγ ) [10], and we denote its selfadjoint extension by Aγ . Then one can introduce the Sobolev scalar products of mth order, hu, viγ,m := hu, Am γ vi ,

u, v ∈ C0∞ (Σ, R), m ∈ R ,

where on the right-hand side is the scalar product of L2R (Σ, dηγ ). The completion of C0∞ (Σ, R) in the topology of h . , . iγ,m will be denoted by Hm (Σ, γ). It turns out that the topology of Hm (Σ, γ) is locally independent of the complete Riemannian metric γ, and that composition with diffeomorphisms and multiplication with smooth, compactly supported functions are continuous operations on these Sobolev spaces. (See Appendix B for precise formulations of these statements.) Therefore,

658

R. VERCH

whenever G ⊂ Σ is open and relatively compact, the topology which h . , . im,γ induces on C0∞ (G, R) is independent of the particular complete Riemannian metric γ, and we shall refer to the topology which is thus locally induced on C0∞ (Σ, R) simply as the (local) Hm -topology. ˜ , γ˜ ), given in a Let us now suppose that we have an ultrastatic spacetime (M 2 ˜ ˜ natural foliation as (R × Σ, dt ⊗ (−γ)), where (Σ, γ) is a complete Riemannian ˜ and {0} × Σ. ˜ Consider again Aγ = selfadjoint manifold. We shall identify Σ ∞ ˜ 2 ˜ ˜ γ), extension of −∆γ + 1 on C0 (Σ, R) in LR (Σ, dηγ ) with ∆γ = Laplacian of (Σ, ◦ and the scalar product µΣ˜ on DΣ˜ given by 1 −1/2 hu0 , A1/2 v i + hu , A v i 0 1 1 γ γ 2 1 hu0 , v0 iγ,1/2 + hu1 , v1 iγ,−1/2 = 2

µ◦Σ˜ (u0 ⊕ u1 , v0 ⊕ v1 ) :=

(3.12)

for all u0 ⊕u1 , v0 ⊕v1 ∈ DΣ˜ . It is now straightforward to check that µ◦Σ˜ ∈ pu(DΣ˜ , δΣ˜ ), in fact, µ◦Σ˜ is the purification of the classical energy scalar product µE ˜ defined in Σ Eq. (3.5). (We refer to [11] for discussion, and also the treatment of more general situations along similar lines.) What is furthermore central for the derivation of the next result is that µ◦Σ˜ corresponds (via (3.11)) to a Hadamard vacuum ω ◦ on the Weyl-algebra of the KG-field with potential term r ≡ 1 over the ultrastatic ˜ dt2 ⊕ (−γ)). This has been proved in [24]. The state ω ◦ is spacetime (R × Σ, ˜ dt2 ⊕ (−γ)); it called the ultrastatic vacuum for the said KG-field over (R × Σ, is the unique pure, quasifree ground state on the corresponding Weyl-algebra for the time-translations (t, q) 7→ (t + t0 , q) on that ultrastatic spacetime with respect to the chosen natural foliation (cf. [40, 42]). ◦ ◦ Remark. The passage from µE ˜ to µΣ ˜ , where µΣ ˜ is the purification of the classiΣ cal energy scalar product, may be viewed as a refined form of “frequency-splitting” procedures (or Hamiltonian diagonalization), in order to obtain pure dominating scalar products and hence, pure states of the KG-field in curved spacetimes, see ˜ is not a Cauchy-surface lying in the natural [11]. However, in the case that Σ foliation of an ultrastatic spacetime, but an arbitrary Cauchy-surface in an arbitrary globally hyperbolic spacetime, the µ◦Σ˜ may fail to correspond to a quasifree Hadamard state — even though, as the following Proposition demonstrates, µ◦Σ˜ gives locally on the Cauchy-data space DΣ˜ the same topology as the dominating scalar products induced on it by any quasifree Hadamard state. More seriously, µ◦Σ˜ may even correspond to a state which is no longer locally quasiequivalent to any quasifree Hadamard state. For an explicit example demonstrating this in a closed Robertson–Walker universe, and for additional discussion, we refer to Sec. 3.6 in [38].

We shall say that a map T : DΣ → DΣ0 , with Σ, Σ0 Cauchy-surfaces, is locally continuous if, for any open, locally compact G ⊂ Σ, the restriction of T to C0∞ (G, R) ⊕ C0∞ (G, R) is continuous (with respect to the topologies under consideration).

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Proposition 3.5. Let ωµ be a quasifree Hadamard state on the Weyl-algebra A[K, κ] of the KG-field with smooth potential term r over the globally hyperbolic spacetime (M, g), and Σ, Σ0 two Cauchy-surfaces in (M, g). Then the Cauchy-data evolution map TΣ0 ,Σ := PΣ0 ◦PΣ−1 : DΣ → DΣ0

(3.13)

is locally continuous in the Hτ ⊕ Hτ −1 -topology, 0 ≤ τ ≤ 1, on the Cauchydata spaces, and the topology induced by µΣ on DΣ coincides locally (i.e. on each C0∞ (G, R) ⊕ C0∞ (G, R) for G ⊂ Σ open and relatively compact) with the H1/2 ⊕ H−1/2 -topology. Remarks. (i) Observe that the continuity statement is reasonably formulated since, as a consequence of the support properties of solutions of the KG-equation with Cauchy-data of compact support (“finite propagation speed”) it holds that for each open, relatively compact G ⊂ Σ there is an open, relatively compact G0 ⊂ Σ0 with TΣ0 ,Σ (C0∞ (G, R) ⊕ C0∞ (G, R)) ⊂ C0∞ (G0 , R) ⊕ C0∞ (G0 , R). (ii) For τ = 1, the continuity statement is just the classical energy estimate. It should be mentioned here that the claimed continuity can also be obtained by other methods. For instance, Moreno [50] proves, under more restrictive assumptions on Σ and Σ0 (among which is their compactness), the continuity of TΣ0 ,Σ in the topology of Hτ ⊕ Hτ −1 for all τ ∈ R, by employing an abstract energy estimate for first order hyperbolic equations (under suitable circumstances, the KG-equation can be brought into this form). We feel, however, that our method, using the results of Sec. 2, is physically more appealing and emphasizes much better the “invariant” structures involved, quite in keeping with the general approach to quantum field theory. Proof of Proposition 3.5. We note that there is a diffeomorphism Ψ : Σ → Σ0 . ˜ → M of M in CauchyTo see this, observe that we may pick a foliation F : R × Σ ˜ surfaces. Then for each q ∈ Σ, the curves t 7→ F (t, q) are inextendible, timelike curves in (M, g). Each such curve intersects Σ exactly once, at the parameter value ˜ As F is a diffeomorphism and t = τ (q). Hence Σ is the set {F (τ (q), q) : q ∈ Σ}. ∞ ˜ τ : Σ → R must be C since, by assumption, Σ is a smooth hypersurface in M , ˜ are diffeomorphic. The same argument shows that Σ0 and one can see that Σ and Σ ˜ and therefore, Σ and Σ0 , are diffeomorphic. Σ Now let us first assume that the g-induced Riemannian metrics γΣ and γΣ0 on Σ, respectively Σ0 , are complete. Let dη and dη 0 be the induced volume measures on Σ and Σ0 , respectively. The Ψ-transformed measure of dη on Σ0 , Ψ∗ dη, is given through Z Z (u◦Ψ) dη = Σ

u (Ψ∗ dη) ,

Σ0

u ∈ C0∞ (Σ0 ) .

(3.14)

Then the Radon–Nikodym derivative (ρ(q))2 := (Ψ∗ dη/dη 0 )(q), q ∈ Σ0 , is a smooth, strictly positive function on Σ0 , and it is now easy to check that the linear map ϑ : (DΣ , δΣ ) → (DΣ0 , δΣ0 ) ,

u0 ⊕ u1 7→ ρ · (u0 ◦Ψ−1 ) ⊕ ρ · (u1 ◦Ψ−1 ) ,

660

R. VERCH

is a symplectomorphism. Moreover, by the result given in Appendix B, ϑ and its inverse are locally continuous maps in the Hs ⊕ Ht -topologies on both Cauchy-data spaces, for all s, t ∈ R. By the energy estimate, TΣ0 ,Σ is locally continuous with respect to the H1 ⊕ H0 topology on the Cauchy-data spaces, and the same holds for the inverse (TΣ0 ,Σ )−1 = TΣ,Σ0 . Hence, the map Θ := ϑ−1 ◦TΣ0 ,Σ is a symplectomorphism of (DΣ , δΣ ), and Θ together with its inverse is locally continuous in the H1 ⊕ H0 -topology on DΣ . Here we made use of Remark (i) above. Now pick two sets G and G0 as in Remark (i), ˜ of Ψ−1 (G0 ) ∪ G in then there is some open, relatively compact neighbourhood G Σ. We can choose a smooth, real-valued function χ compactly supported on Σ ˜ It is then straightforward to check that the maps χ◦Θ◦χ and with χ ≡ 1 on G. χ◦Θ−1 ◦χ (χ to be interpreted as multiplication by χ) are a pair of symplectically adjoint maps on (DΣ , δΣ ) which are bounded with respect to the H1 ⊕ H0 -topology, i.e. with respect to the norm of µE Σ . At this point we use Theorem 2.2(b) and consequently χ◦Θ◦χ and χ◦Θ−1 ◦χ are continuous with respect to the norms of the (µE Σ )s , 0 ≤ s ≤ 2. Inspection shows that (µE Σ )s (u0 ⊕ u1 , v0 ⊕ v1 ) =

1 hu0 , A1−s/2 v0 i + hu1 , A−s/2 γΣ γΣ v1 i 2

for 0 ≤ s ≤ 2. From this it is now easy to see that Θ restricted to C0∞ (G, R) ⊕ C0∞ (G, R) is continuous in the topology of Hτ ⊕ Hτ −1 , 0 ≤ τ ≤ 1, since χ◦Θ◦ χ(u0 ⊕ u1 ) = Θ(u0 ⊕ u1 ) for all u0 ⊕ u1 ∈ C0∞ (G, R) ⊕ C0∞ (G, R) by the choice of χ. Using that Θ = ϑ−1 ◦TΣ0 ,Σ and that ϑ is locally continuous with respect to all the Hs ⊕ Ht -topologies, s, t ∈ R, on the Cauchy-data spaces, we deduce that TΣ0 ,Σ is locally continuous in the Hτ ⊕ Hτ −1 -topology, 0 ≤ τ ≤ 1, as claimed. If the g-induced Riemannian metrics γΣ , γΣ0 are not complete, one can make them into complete ones γˆΣ := f · γΣ , γˆΣ0 := h · γΣ0 by multiplying them with η and suitable smooth, strictly positive functions f on Σ and h on Σ0 [14]. Let dˆ dˆ η 0 be the volume measures corresponding to the new metrics. Consider then the η ), (φ2 )2 := (dˆ η 0 /dη 0 ), which are C ∞ and strictly density functions (φ1 )2 := (dη/dˆ ˆ ˆ ˆ positive, and define (DΣ , δΣ ), (DΣ0 , δΣ0 ) and ϑ like their unhatted counterparts ˆE but with dˆ η and dˆ η 0 in place of dη and dη 0 . Likewise define µ Σ with respect to γˆΣ . Then TˆΣ0 ,Σ := φ2 ◦TΣ0 ,Σ ◦φ1 (understanding that φ1 , φ2 act as multiplication operators) and its inverse are symplectomorphisms between (DΣ , δˆΣ ) and (DΣ0 , δˆΣ0 ) which are locally continuous in the H1 ⊕ H0 -topology. Now we can apply the above ˆ = ϑˆ−1 ◦TˆΣ0 ,Σ and, hence, TˆΣ0 ,Σ is locally continuous argument showing that Θ in the Hτ ⊕ Hτ −1 -topology for 0 ≤ τ ≤ 1. The same follows then for TΣ0 ,Σ = −1 ˆ φ−1 2 ◦TΣ0 ,Σ ◦φ1 . For the proof of the second part of the statement, we note first that in [24] it is ˆ , gˆ) of the form shown that there exists another globally hyperbolic spacetime (M ˆ = R × Σ with the following properties: M ˆ , gˆ), and a causal normal neigh(1) Σ0 := {0} × Σ is a Cauchy-surface in (M ˆ of bourhood N of Σ in M coincides with a causal normal neighbourhood N ˆ , in such a way that Σ = Σ0 and g = gˆ on N . Σ0 in M

CONTINUITY OF SYMPLECTICALLY ADJOINT MAPS AND HADAMARD VACUA

661

ˆ lies properly to the past of (2) For some t0 < 0, the (−∞, t0 ) × Σ-part of M 2 ˆ N, and on that part, gˆ takes the form dt ⊕ (−γ) where γ is a complete Riemannian metric on Σ. ˆ , gˆ) is a globally hyperbolic spacetime which equals (M, g) on This means that (M a causal normal neighbourhood of Σ and becomes ultrastatic to the past of it. ˆ κ Then consider the Weyl-algebra A[K, ˆ ] of the KG-field with potential term rˆ ˆ , R) agrees with r on the neighbourhood N ˆ = N ˆ , gˆ), where rˆ ∈ C0∞ (M over (M ˆ and is identically equal to 1 on the (−∞, t0 ) × Σ-part of M . Now observe that the ˆ of the respective KG-equations on (M, g) and (M ˆ , gˆ) coincide propagators E and E ∞ when restricted to C0 (N, R). Therefore one obtains an identification map ˆ , [f ] = f + ker(E) 7→ [f ]ˆ= f + ker(E)

f ∈ C0∞ (N, R) ,

ˆ N ˆ ) which preserves the symplectic forms κ and κ between K(N ) and K( ˆ. Without ˆ N ˆ ). This identidanger we may write this identification as an equality, K(N ) = K( ˆ N ˆ ), κ ˆ N ˆ )) lifts to a C ∗ -algebraic fication map between (K(N ), κ|K(N )) and (K( ˆ |K( isomorphism between the corresponding Weyl-algebras ˆ N ˆ ), κ ˆ N ˆ )] , A[K(N ), κ|K(N )] = A[K( ˆ |K( (3.15) ˆ ([f ]ˆ) , W ([f ]) = W

f ∈ C0∞ (N, R) .

Here we followed our just indicated convention to abbreviate this identification as ˆ) = M ˆ in (M ˆ , gˆ), implying an equality. Now we have D(N ) = M in (M, g) and D(N ˆ ˆ ˆ that K(N ) = K and K(N ) = K. Hence A[K(N ), κ|K(N )] = A[K, κ] and the same for the “hatted” objects. Thus (3.15) gives rise to an identification between A[K, κ] ˆ κ and A[K, ˆ ], and so the quasifree Hadamard state ωµ induces a quasifree state ωµˆ ˆ κ on A[K, ˆ ] with µ ˆ([f ]ˆ, [h]ˆ) = µ([f ], [h]) ,

f, h ∈ C0∞ (N, R) .

(3.16)

This state is also a Hadamard state since we have i Λ(f, h) = µ([f ], [h]) + κ([f ], [h]) 2 i ˆ ([f ]ˆ, [h]ˆ) , =µ ˆ([f ]ˆ, [h]ˆ) + κ 2

f, h ∈ C0∞ (N, R) ,

and Λ is, by assumption, of Hadamard form. However, due to the causal propagation property of the Hadamard form this means that µ ˆ is the dominating scalar product ˆ κ ˆ κ on (K, ˆ ) of a quasifree Hadamard state on A[K, ˆ ]. Now choose some t < t0 , and let ˆ , gˆ) corresponding Σt = {t} × Σ be the Cauchy-surface in the ultrastatic part of (M to this value of the time-parameter of the natural foliation. As remarked above, the scalar product µ◦Σt (u0 ⊕ u1 , v0 ⊕ v1 ) =

1 hu0 , v0 iγ,1/2 + hu1 , v1 iγ,−1/2 , 2

u0 ⊕ u1 , v0 ⊕ v1 ∈ DΣt , (3.17)

662

R. VERCH

is the dominating scalar product on (DΣt , δΣt ) corresponding to the ultrastatic ˆ , gˆ), which is an Hadamard vacuum. vacuum state ω ◦ over the ultrastatic part of (M Since the dominating scalar products of all quasifree Hadamard states yield locally the same topology (Proposition 3.4(e)), it follows that the dominating scalar product µ ˆΣt on (DΣt , δΣt ), which is induced (cf. (3.11)) by the the dominating scalar product of µ ˆ of the quasifree Hadamard state ωµˆ , endows DΣt locally with the same topology as does µ◦Σt . As can be read off from (3.17), this is the local H1/2 ⊕ H−1/2 topology. To complete the argument, we note that (cf. (3.11, 3.13)) ˆΣt (TΣt ,Σ0 (u0 ⊕u1 ), TΣt ,Σ0 (v0 ⊕v1 )) , µ ˆ Σ0 (u0 ⊕u1 , v0 ⊕v1 ) = µ

u0 ⊕u1 , v0 ⊕v1 ∈ DΣ0 .

But since µ ˆΣt induces locally the H1/2 ⊕ H−1/2 -topology and since the symplectomorphism TΣt ,Σ0 as well as its inverse are locally continuous on the Cauchy-data ˆΣ0 induces the spaces in the H1/2 ⊕ H−1/2 -topology, the last equality entails that µ local H1/2 ⊕ H−1/2 -topology on DΣ0 . In view of (3.16), the Proposition is now proved. 3.5. Local definiteness, local primarity, Haag-duality, etc. In this section we prove Theorem 3.6 below on the algebraic structure of the GNS-representations associated with quasifree Hadamard states on the CCRalgebra of the KG-field on an arbitrary globally hyperbolic spacetime (M, g). The results appearing therein extend our previous work [64, 65, 66]. Let (M, g) be a globally hyperbolic spacetime. We recall that a subset O of M is called a regular diamond if it is of the form O = OG = int D(G) where G is an open, relatively compact subset of some Cauchy-surface Σ in (M, g) having the property that the boundary ∂G of G is contained in the union of finitely many smooth, closed, two-dimensional submanifolds of Σ. We also recall the notation Rω (O) = πω (A(O))− for the local von Neumann algebras in the GNS-representation of a state ω. The C ∗ -algebraic net of observable algebras O → A(O) will be understood as being that associated with the KG-field in Proposition 3.2. Theorem 3.6. Let (M, g) be a globally hyperbolic spacetime and A[K, κ] the Weyl-algebra of the KG-field with smooth, real-valued potential function r over (M, g). Suppose that ω and ω1 are two quasifree Hadamard states on A[K, κ]. Then the following statements hold. (a) The GNS-Hilbert space Hω of ω is infinite dimensional and separable. (b) The restrictions of the GNS-representations πω |A(O) and πω1 |A(O) of any open, relatively compact O ⊂ M are quasiequivalent. They are even unitarily equivalent when O⊥ is non-void. (c) For each p ∈ M, we have local definiteness, \ Rω (O) = C · 1 . O3p

More generally, whenever C ⊂ M is the subset of a compact set which

CONTINUITY OF SYMPLECTICALLY ADJOINT MAPS AND HADAMARD VACUA

663

is contained in the union of finitely many smooth, closed, two-dimensional submanifolds of an arbitrary Cauchy-surface Σ in M, then \

Rω (O) = C · 1 .

(3.18)

O⊃C

(d) Let O and O1 be two relatively compact diamonds, based on Cauchy-surfaces Σ and Σ1 , respectively, such that O ⊂ O1 . Then the split-property holds for the pair Rω (O) and Rω (O1 ), i.e. there exists a type I∞ factor N such that one has the inclusion Rω (O) ⊂ N ⊂ Rω (O1 ) . (e) Inner and outer regularity  Rω (O) = 

[

OI ⊂O



\

Rω (OI ) 00 =

Rω (O1 )

(3.19)

O1 ⊃O

holds for all regular diamonds O. (f) If ω is pure (a Hadamard vacuum), then we have Haag-Duality Rω (O)0 = Rω (O⊥ ) for all regular diamonds O. (By the same arguments as in [65, P roposition 6], Haag-Duality extends to all pure (but not necessarily quasifree or Hadamard) states ω which are locally normal (hence, by (d), locally quasiequivalent) to any Hadamard vacuum.) (g) Local primarity holds for all regular diamonds, that is, for each regular diamond O, Rω (O) is a factor. Moreover, Rω (O) is isomorphic to the unique hyperfinite type III1 factor if O⊥ is non-void. In this case, Rω (O⊥ ) is also hyperfinite and of type III1 , and if ω is pure, Rω (O⊥ ) is again a factor. Otherwise, if O⊥ = ∅, then Rω (O) is a type I∞ factor. Proof. The key point in the proof is that, by results which for the cases relevant here are to large extent due to Araki [1], the above statement can be equivalently translated into statements about the structure of the one-particle space, i.e. essentially the symplectic space (K, κ) equipped with the scalar product λω . We shall use, however, the formalism of [40, 45]. Following that, given a symplectic space (K, κ) and µ ∈ q(K, κ) one calls a real linear map k : K → H a one-particle Hilbert space structure for µ if (1) H is a complex Hilbert space, (2) the complex linear span of k(K) is dense in H and (3) i hk(x), k(y)i = λµ (x, y) = µ(x, y) + κ(x, y) 2 for all x, y ∈ K. It can then be shown (cf. [45, Appendix A]) that the GNSrepresentation of the quasifree state ωµ on A[K, κ] may be realized in the following

664

R. VERCH

way: Hωµ = Fs (H), the Bosonic Fock-space over the one-particle space H, Ωωµ = the Fock-vacuum, and +

πωµ (W (x)) = ei(a(k(x))+a

(k(x)))−

,

x∈K,

where a( . ) and a+ ( . ) are the Bosonic annihilation and creation operators, respectively. Now it is useful to define the symplectic complement F v := {χ ∈ H : Im hχ, φi = 0 ∀φ ∈ F } for F ⊂ H, since it is known that (i) Rωµ (O) is a factor iff k(K(O))− ∩ k(K(O))v = {0}, (ii) Rωµ (O)0 = Rωµ (O⊥ ) iff k(K(O))v = k(K(O⊥ ))− , T T − (iii) O⊃C Rωµ (O) = C · 1 iff O⊃C k(K(O)) = {0} , cf. [1, 21, 35, 49, 58]. After these preparations we can commence with the proof of the various statements of our Theorem. (a) Let k : K → H be the one-particle Hilbert space structure of ω. The local one-particle spaces k(K(OG ))− of regular diamonds OG based on G ⊂ Σ are topologically isomorphic to the completions of C0∞ (G, R) ⊕ C0∞ (G, R) in the H1/2 ⊕ H−1/2 -topology and these are separable. Hence k(K)− , which is generated by a countable set k(K(OGn )), for Gn a sequence of locally compact subsets of Σ eventually exhausting Σ, is also separable. The same holds then for the one-particle Hilbert space H in which the complex span of k(K) is dense, and thus separability is implied for Hω = Fs (H). The infinite-dimensionality is clear. (b) The local quasiequivalence has been proved in [66] and we refer to that reference for further details. We just indicate that the proof makes use of the fact that the difference Λ − Λ1 of the spatio-temporal two-point functions of any pair of quasifree Hadamard states is on each causal normal neighbourhood of any Cauchysurface given by a smooth integral kernel — as can be directly read off from the Hadamard form — and this turns out to be sufficient for local quasiequivalence. The statement about the unitary equivalence can be inferred from (g) below, since it is known that every ∗-preserving isomorphism between von Neumann algebras of type III acting on separable Hilbert spaces is given by the adjoint action of a unitary operator which maps the Hilbert spaces onto each other. See e.g. Theorem 7.2.9 and Proposition 9.1.6 in [39]. (c) Here one uses that there exist Hadamard vacua, i.e. pure quasifree Hadamard states ωµ . Since by Proposition 3.4 the topology of µΣ in DΣ is locally that of H1/2 ⊕ H−1/2 , one can show as in [66, Chap. 4 and Appendix] that under the T stated hypotheses about C it holds that O⊃C k(K(O))− = {0} for the one-particle Hilbert space structures of Hadamard vacua. From the local equivalence of the topologies induced by the dominating scalar products of all quasifree Hadamard states (Proposition 3.4(e)), this extends to the one-particle structures of all quasifree Hadamard states. By (iii), this yields the statement (c).

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(d) This is proved in [65] under the additional assumption that the potential term r is a positive constant. (The result was formulated in [65] under the hypothesis that Σ = Σ1 , but it is clear that the present statement without this hypothesis is an immediate generalization.) To obtain the general case one needs in the spacetime deformation argument of [65] the modification that the potential term rˆ of the KGˆ , gˆ) is equal to a positive constant on its ultrastatic field on the new spacetime (M part while being equal to r in a neighbourhood of Σ. We have used that procedure already in the proof of Proposition 3.5, see also the proof of (f) below where precisely the said modification will be carried out in more detail. (e) Inner regularity follows simply from the definition of the A(O); one deduces that for each A ∈ A(O) and each > 0 there exists some OI ⊂ O and A ∈ A(OI ) so that || A − A || < . It is easy to see that inner regularity is a consequence of this property. So we focus now on the outer regularity. Let O = OG be based on the subset G of the Cauchy-surface Σ. Consider the symplectic space (DΣ , δΣ ) and the dominating scalar product µΣ induced by µ ∈ q(DΣ , δΣ ), where ωµ = ω; the corresponding one-particle Hilbert space structure we denote by kΣ : DΣ → HΣ . Then we denote by W(kΣ (DG )) the von Neumann algebra in B(Fs (HΣ )) generated by the unitary groups of the operators (a(kΣ (u0 ⊕ u1 )) + a+ (kΣ (u0 ⊕ u1 )))− where u0 ⊕ u1 ranges over DG := C0∞ (G, R) ⊕ C0∞ (G, R). So W(kΣ (DG )) = Rω (OG ). It holds generally T T that G1 ⊃G W(kΣ (DG1 )) = W( G1 ⊃G kΣ (DG1 )− ) [1], hence, to establish outer regularity, we must show that \ kΣ (DG1 )− = kΣ (DG )− . (3.20) G1 ⊃G

In [65] we have proved that the ultrastatic vacuum ω ◦ of the KG-field with potential term ≡ 1 over the ultrastatic spacetime (M ◦ , g ◦ ) = (R × Σ, dt2 ⊕ (−γ)) (where γ is any complete Riemannian metric on Σ) satisfies Haag-duality. That means, we have (3.21) R◦ω◦ (O◦ )0 = R◦ω◦ (O◦⊥ ) for any regular diamond O◦ in (M ◦ , g ◦ ) which is based on any of the Cauchy-surfaces {t} × Σ in the natural foliation, and we have put a “◦” on the local von Neumann algebras to indicate that they refer to a KG-field over (M ◦ , g ◦ ). But since we have inner regularity for R◦ω◦ (O◦⊥ ) — by the very definition — the outer regularity of R◦ω◦ (O◦ ) follows from the Haag-duality (3.21). Translated into conditions on the one-particle Hilbert space structure k◦Σ : DΣ → HΣ◦ of ω ◦ , this means that the equality \ k◦Σ (DG1 )− = k◦Σ (DG )− (3.22) G1 ⊃G

holds. Now we know from Proposition 3.5 that µΣ induces locally the H1/2 ⊕H−1/2 topology on DΣ . However, this coincides with the topology locally induced by µ◦Σ on DΣ (cf. (3.11)) — even though µ◦Σ may, in general, not be viewed as corresponding to a Hadamard vacuum of the KG-field over (M, g). Thus the required relation (3.20) is implied by (3.22).

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(f) In view of outer regularity it is enough to show that, given any O1 ⊃ O, it holds that (3.23) Rω (O⊥ )0 ⊂ Rω (O1 ) . The demonstration of this property relies on a spacetime deformation argument similar to that used in the proof of Proposition 3.5. Let G be the base of O on the Cauchy-surface Σ in (M, g). Then, given any other open, relatively compact subset G1 of Σ with G ⊂ G1 , we have shown in [65] that there exists a globally hyperbolic ˆ , gˆ) with the properties (1) and (2) in the proof of Proposition 3.5, spacetime (M and with the additional property that there is some t < t0 such that − ˆ 1 ) ∩ Σt . ˆ ⊂ int D(G int J(G) ∩ Σt Here, Σt = {t} × Σ are the Cauchy-surfaces in the natural foliation of the ultrastatic ˆ , gˆ). The hats indicate that the causal set and the domain of dependence part of (M ˆ , gˆ). This implies that we can find some regular diamond are to be taken in (M t t ˆ ˆ O := intD(S ) in (M , gˆ) based on a subset S t of Σt which satisfies

ˆ int J(G) ∩ Σt

−

ˆ 1 ) ∩ Σt . ⊂ S t ⊂ int D(G

(3.24)

ˆ 1 ), one derives from (3.24) the relations ˆ := int D(G) ˆ ˆ1 := int D(G Setting O and O ˆ1 . ˆ ⊂ Ot ⊂ O O

(3.25)

ˆ⊥ ˆ ⊥ ⊂ (Ot )⊥ ⊂ O O 1

(3.26)

These are equivalent to ˆ , gˆ). where ⊥ is the causal complementation in (M Now as in the proof of Proposition 3.5, the given Hadamard vacuum ω on the Weyl-algebra A[K, κ] of the KG-field over (M, g) induces a Hadamard vacuum ω ˆ on ˆ ˆ the Weyl-algebra A[K, κ ˆ ] of the KG-field over (M , gˆ) whose potential term rˆ is 1 on ˆ , gˆ). Then by Proposition 6 in [65] we have Haag-duality the ultrastatic part of (M ˆ ωˆ (Oˆt ) ˆ ωˆ (Oˆt ⊥ )0 = R R

(3.27)

for all regular diamonds Oˆt with base on Σt ; we have put hats on the von Neumann ˆ κ algebras to indicate that they refer to A[K, ˆ ]. (This was proved in [65] assuming ˆ , gˆ) is globally ultrastatic. However, with the same argument, based on that (M primitive causality, as we use it next to pass from (3.28) to (3.30), one can easily establish that (3.27) holds if only Σt is, as here, a member in the natural foliation ˆ , gˆ).) Since Ot is a regular diamond based on Σt , we of the ultrastatic part of (M obtain ˆ ωˆ (Ot ) ˆ ωˆ ((Ot )⊥ )0 = R R and thus, in view of (3.25) and (3.26), ˆ ⊥ )0 ⊂ R ˆ ωˆ ((Ot )⊥ )0 = R ˆ ωˆ (Ot ) ⊂ R ˆ ωˆ (O ˆ1 ) . ˆ ωˆ (O R

(3.28)

CONTINUITY OF SYMPLECTICALLY ADJOINT MAPS AND HADAMARD VACUA

667

ˆ , gˆ) coincides with (M, g) on a Now recall (see proof of Proposition 3.5) that (M causal normal neighbourhood N of Σ. Primitive causality (Proposition 3.2) then entails ˆ ⊥ ∩ N )0 ⊂ R ˆ ωˆ (O ˆ1 ∩ N ) . ˆ ωˆ (O (3.29) R ˆ \ G) and O ˆ1 are diamonds in (M ˆ , gˆ) based on Σ. ˆ ⊥ = intD(Σ On the other hand, O ˆ , gˆ) coincide on the causal normal neighbourhood N of Σ, one Since (M, g) and (M ˜ ˆ G)∩N ˜ ˜ ∈ Σ. Hence, with O = int D(G), obtains that int D(G)∩N = int D( for all G O1 = int D(G1 ) (in (M, g)), we have that (3.23) entails Rω (O⊥ ∩ N )0 ⊂ Rω (O1 ∩ N ) (cf. the proof of Proposition 3.5) where the causal complement ⊥ is now taken in (M, g). Using primitive causality once more, we deduce that Rω (O⊥ )0 ⊂ Rω (O1 ) .

(3.30)

The open, relatively compact subset G1 of Σ was arbitrary up to the constraint G ⊂ G1 . Therefore, we arrive at the conclusion that the required inclusion (3.23) holds for all O1 ⊃ O. (g) Let Σ be the Cauchy-surface on which O is based. For the local primarity one uses, as in (c), the existence of Hadamard vacua ωµ and the fact (Proposition 3.5) that µΣ induces locally the H1/2 ⊕H−1/2 -topology; then one may use the arguments of [66, Chap. 4 and Appendix] to show that due to the regularity of the boundary ∂G of the base G of O there holds k(K(O))− ∩ k(K(O))v = {0} for the one-particle Hilbert space structures of Hadamard vacua. As in the proof of (c), this can be carried over to the one-particle structures of all quasifree Hadamard states since they induce locally on the one-particle spaces the same topology, see [66, Chap. 4]. We note that for Hadamard vacua the local primarity can also be established using (3.18) together with Haag-duality and primitive causality purely at the algebraic level, without having to appeal to the one-particle structures. The type III1 -property of Rω (O) is then derived using Theorem 16.2.18 in [3] (see also [73]). We note that for some points p in the boundary ∂G of G, O admits domains which are what is in Sec. 16.2.4 of [3] called “βp -causal sets”, as a consequence of the regularity of ∂G and the assumption O⊥ 6= ∅. We further note that it is straightforward to prove that the quasifree Hadamard states of the KGfield over (M, g) possess at each point in M scaling limits (in the sense of Sec. 16.2.4 in [3], see also [22, 32]) which are equal to the theory of the massless KG-field in Minkowski-spacetime. Together with (a) and (c) of the present Theorem this shows that the the assumptions of Theorem 16.2.18 in [3] are fulfilled, and the Rω (O) are type III1 -factors for all regular diamonds O with O⊥ 6= ∅. The hyperfiniteness follows from the split-property (d) and the regularity (e), cf. Proposition 17.2.1 in [3]. The same arguments may be applied to Rω (O⊥ ), yielding its type III1 property (meaning that in its central decomposition only type III1 -factors occur)

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and hyperfiniteness. If ω is a Hadamard vacuum, then Rω (O⊥ ) = Rω (O)0 is a factor unitarily equivalent to Rω (O). For the last statement note that O⊥ = ∅ implies that the spacetime has a compact Cauchy-surface on which O is based. In this case Rω (O) = πω (A[K, κ])00 (use the regularity of ∂G, and (c), (e) and primitive causality). But since ω is quasiequivalent to any Hadamard vacuum by the relative compactness of O, Rω (O) = πω (A[K, κ])00 is a type I∞ -factor. We end this section and therefore, this work, with a few concluding remarks. First we note that the split-property signifies a strong notion of statistical independence. It can be deduced from constraints on the phase-space behaviour (“nuclearity”) of the considered quantum field theory. We refer to [9, 31] for further information and also to [62] for a review, as a discussion of these issues lies beyond the scope of this article. The same applies to a discussion of the property of the local von Neumann algebras Rω (O) to be hyperfinite and of type III1 . We only mention that for quantum field theories on Minkowski spacetime it can be established under very general (model-independent) conditions that the local (von Neumann) observable algebras are hyperfinite and of type III1 , and refer the reader to [7] and references cited therein. However, the property of the local von Neumann algebras to be of type III1 , together with the separability of the GNS-Hilbert space Hω , has an important consequence which we would like to point out (we have used it implicitly already in the proof of Theorem 3.6(b)): Hω contains a dense subset ts(Hω ) of vectors which are cyclic and separating for all Rω (O) whenever O is a diamond with O⊥ 6= ∅. But so far it has only been established in special cases that Ωω ∈ ts(Hω ), see [64]. At any rate, when Ω ∈ ts(Hω ) one may consider for a pair of regular diamonds O1 , O2 with O1 ⊂ O2 and O2⊥ nonvoid the modular operator ∆2 of Rω (O2 ), Ω (cf. [39]). The split property and the factoriality of Rω (O1 ) and Rω (O2 ) imply the that the map 1/4

Ξ1,2 : A 7→ ∆2 AΩ ,

A ∈ Rω (O1 ) ,

(3.31)

is compact [8]. As explained in [8], “modular compactness” or “modular nuclearity” may be viewed as suitable generalizations of “energy compactness” or “energy nuclearity” to curved spacetimes as notions to measure the phase-space behaviour of a quantum field theory (see also [65]). Thus an interesting question would be if the maps (3.31) are even nuclear. Summarizing it can be said that Theorem 3.6 shows that the nets of von Neumann observable algebras of the KG-field over a globally hyperbolic spacetime in the representations of quasifree Hadamard states have all the properties one would expect for physically reasonable representations. This supports the point of view that quasifree Hadamard states appear to be a good choice for physical states of the KG-field over a globally hyperbolic spacetime. Similar results are expected to hold also for other linear fields. Finally, the reader will have noticed that we have been considering exclusively the quantum theory of a KG-field on a globally hyperbolic spacetime. For recent developments concerning quantum fields in the background of non-globally hyperbolic spacetimes, we refer to [44] and references cited there.

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Acknowledgements I would like to thank D. Buchholz for valuable comments on a very early draft of Sec. 2. Moreover, I would like to thank C. D’Antoni, R. Longo, J. Roberts and L. Zsido for their hospitiality, and their interest in quantum field theory in curved spacetimes. I also appreciated conversations with R. Conti, D. Guido and L. Tuset on various parts of the material of the present work. Appendix Appendix A For the sake of completeness, we include here the interpolation argument in the form we use it in the proof of Theorem 2.2 and in Appendix B below. It is a standard argument based on Hadamard’s three-line-theorem, cf. Chap. IX in [57]. Lemma A.1. Let F , H be complex Hilbert spaces, X and Y two non-negative, injective, selfadjoint operators in F and H, respectively, and Q a bounded linear operator H → F such that Q Ran(Y ) ⊂ dom(X). Suppose that the operator XQY admits a bounded extension T : H → F . Then for all 0 ≤ τ ≤ 1, it holds that Q Ran(Y τ ) ⊂ dom(X τ ), and the operators X τ QY τ are bounded by || T ||τ || Q ||1−τ . Proof. The operators ln(X) and ln(Y ) are (densely defined) selfadjoint operators. Let the vectors x and y belong to the spectral subspaces of ln(X) and ln(Y ), respectively, corresponding to an arbitrary finite interval. Then the functions C 3 z 7→ ez ln(X) x and C 3 z 7→ ez ln(Y ) y are holomorphic. Moreover, eτ ln(X) x = X τ x and eτ ln(Y ) y = Y τ y for all real τ . Consider the function F (z) := hez ln(X) x, Qez ln(Y ) yiF . It is easy to see that this function is holomorphic on C, and also that the function is uniformly bounded for z in the strip {z : 0 ≤ Re z ≤ 1}. For z = 1 + it, t ∈ R, one has |F (z)| = |he−it ln(X) x, XQY eit ln(Y ) yiF | ≤ || T || || x ||F || y ||H , and for z = it, t ∈ R, |F (z)| = |he−it ln(X) x, Qeit ln(Y ) yiF | ≤ || Q || || x ||F || y ||H . By Hadamard’s three-line-theorem, it follows that for all z = τ + it in the said strip there holds the bound |F (τ + it)| ≤ || T ||τ || Q ||1−τ || x ||F || y ||H . As x and y were arbitrary members of the finite spectral interval subspaces, the last estimate extends to all x and y lying in cores for the operators X τ and Y τ , from which the the claimed statement follows. Appendix B For the convenience of the reader we collect here two well-known results about Sobolev norms on manifolds which are used in the proof of Proposition 3.5. The

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notation is as follows. Σ and Σ0 will denote smooth, finite dimensional manifolds (connected, paracompact, Hausdorff); γ and γ 0 are complete Riemannian metrics on Σ and Σ0 , respectively. Their induced volume measures are denoted by dη and dη 0 . We abbreviate by A the selfadjoint extension in L2 (Σ, dη) of the operator −∆γ + 1 on C0∞ (Σ), where ∆γ is the Laplace–Beltrami operator on (Σ, γ); note that [10] contains a proof that (−∆γ + 1)k is essentially selfadjoint on C0∞ (Σ) for all k ∈ N. A0 will be defined similarly with respect to the corresponding objects of (Σ0 , γ 0 ). As in the main text, the mth Sobolev scalar product is hu, viγ,m = hu, Am vi for u, v ∈ C0∞ (Σ) and m ∈ R, where h . , . i is the scalar product of L2 (Σ, dη). Anagolously we define h . , . iγ 0 ,m . For the corresponding norms we write || . ||γ,m , respectively, || . ||γ 0 ,m . Lemma B.1. (a) Let χ ∈ C0∞ (Σ). Then there is for each m ∈ R a constant cm so that || χu ||γ,m ≤ cm || u ||γ,m , u ∈ C0∞ (Σ) . (b) Let φ ∈ C ∞ (Σ) be strictly positive and G ⊂ Σ open and relatively compact. Then there are for each m ∈ R two positive constants β1 , β2 so that β1 || φu ||γ,m ≤ || u ||γ,m ≤ β2 || φu ||γ,m ,

u ∈ C0∞ (G) .

Proof. (a) We may suppose that χ is real-valued (otherwise we treat real and imaginary parts separately). A tedious but straightforward calculation shows that the claimed estimate is fulfilled for all m = 2k, k ∈ N0 . Hence Ak χA−k extends to a bounded operator on L2 (Σ, dη), and the same is true of the adjoint A−k χAk . Thus by the interpolation argument, cf. Lemma A.1, Aτ k χA−τ k is bounded for all −1 ≤ τ ≤ 1. This yields the stated estimate. (b) This is a simple corollary of (a). For the first estimate, note that we may replace φ by a smooth function with compact support. Then note that the second estimate is equivalent to || φ−1 v ||γ,m ≤ β2 || v ||γ,m , v ∈ C0∞ (G), and again we use that instead of φ−1 we may take a smooth function of compact support. Lemma B.2. Let (Σ, γ) and (Σ0 , γ 0 ) be two complete Riemannian manifolds, N and N 0 two open subsets of Σ and Σ0 , respectively, and Ψ : N → N 0 a diffeomorphism. Given m ∈ R and some open, relatively compact subset G of Σ with G ⊂ N, there are two positive constants b1 , b2 such that b1 || u ||γ,m ≤ || Ψ∗ u ||γ 0 ,m ≤ b2 || u ||γ,m ,

u ∈ C0∞ (G) ,

where Ψ∗ u := u◦Ψ−1 . Proof. Again it is elementary to check that such a result is true for m = 2k with k ∈ N0 . One infers that, choosing χ ∈ C0∞ (N ) with χ|G ≡ 1 and setting χ0 := Ψ∗ χ, there is for each k ∈ N0 a positive constant b fulfilling || Ak χΨ∗ χ0 v ||γ,0 ≤ b || (A0 )k v ||γ 0 ,0 ,

v ∈ C0∞ (Σ0 ) ;

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671

here Ψ∗ v := v ◦Ψ. Therefore, Ak ◦χ◦Ψ∗ ◦χ0 ◦(A0 )−k extends to a bounded operator L2 (Σ0 , dη 0 ) → L2 (Σ, dη) for each k ∈ N0 . Interchanging the roles of A and A0 , one obtains that (A0 )k ◦χ0 ◦Ψ∗ ◦χ◦A−k also extends, for each k ∈ N0 , to a bounded operator L2 (Σ, dη) → L2 (Σ0 , dη 0 ). The boundedness transfers to the adjoints of these two operators. Observe then that for (Ψ∗ )† , the adjoint of Ψ∗ , we have (Ψ∗ )† = ρ2 ◦(Ψ∗ ) on C0∞ (N ), and similarly, for the adjoint (Ψ∗ )† of Ψ∗ we have (Ψ∗ )† = Ψ∗ ◦ρ−2 on C0∞ (N 0 ), where ρ2 = Ψ∗ dη/dη 0 is a smooth density function on N 0 , cf. Eq. (3.14). It can now easily be worked out that the interpolation argument of Lemma A.1 yields again the claimed result. References [1] H. Araki, “A lattice of von Neumann algebras associated with the quantum theory of a free Bose field”, J. Math. Phys. 4 (1963) 1343. [2] U. Bannier, “On generally covariant quantum field theory and generalized causal and dynamical structures”, Commun. Math. Phys. 118 (1988) 163. [3] H. Baumg¨ artel and M. Wollenberg, Causal Nets of Operator Algebras, Akademie Verlag, Berlin, 1992. [4] J. J. Bisognano and E. Wichmann, “On the duality condition for a hermitean scalar field”, J. Math. Phys. 16 (1975) 985. , “On the duality condition for quantum fields”, J. Math. Phys. 17 (1976) 303. [5] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vols. 1 and 2, Springer-Verlag, New York, Berlin, Heidelberg, 1986. [6] R. Brunetti, K. Fredenhagen and M. K¨ ohler, “The microlocal spectrum condition and Wick polynomials of free fields in curved spacetimes”, Commun. Math. Phys. 180 (1996) 633. [7] D. Buchholz, C. D’Antoni and K. Fredenhagen, “The universal structure of local algebras”, Commun. Math. Phys. 111 (1987) 123. [8] D. Buchholz, C. D’Antoni and R. Longo, “Nuclear maps and modular structures”; I, J. Funct. Anal. 88 (1990) 223; II, Commun. Math. Phys. 129 (1990) 115. [9] D. Buchholz and E. Wichmann, “Causal independence and the energy-level density of states in quantum field theory”, Commun. Math. Phys. 106 (1986) 321. [10] P. R. Chernoff, “Essential self-adjointness of powers of generators of hyperbolic equations”, J. Funct. Anal. 12 (1973) 401. [11] P. Chmielowski, “States of scalar field on spacetimes with two isometries with timelike orbits”, Class. Quantum Grav. 11 (1994) 41. [12] B. S. DeWitt and R. W. Brehme, “Radiation damping in a gravitational field”, Ann. Phys. (N.Y.) 9 (1960) 220. [13] J. Dieudonn´e, Treatise on Analysis, Vol. 3, Academic Press, New York, 1972. [14] J. Dieudonn´e, Treatise on Analysis, Vol. 4 (Chap. XX.18, Probl. 6), Academic Press, New York, 1972. [15] J. Dieckmann, “Cauchy-surfaces in globally hyperbolic spacetime”, J. Math. Phys. 29 (1988) 578. [16] J. Dimock, “Algebras of local observables on a manifold”, Commun. Math. Phys. 77 (1980) 219.

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[69] R. M. Wald, “Trace anomaly of a conformally invariant quantum field in curved spacetime”, Phys. Rev. D17 (1978) 1477. [70] R. M. Wald, General Relativity, Univ. of Chicago Press, 1984. [71] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, Univer. of Chicago Press, 1994. [72] A. S. Wightman, “La th´eorie quantique locale et la th´eorie quantique des champs”, Ann. Inst. H. Poincar´ e 1 (1964) 403. [73] M. Wollenberg, “Scaling limits and type of local algebras over curved spacetime”, in Operator Algebras and Topology, eds. W. B. Arveson et al., Pitman Research Notes in Mathematics Series 270, Longman, Harlow, 1992. [74] M. Wollenberg, “On the relation between conformal structure in space-time and nets of local algebras of observables”, Lett. Math. Phys. 31 (1994) 195. [75] S. L. Woronowicz, “On the purification of factor states”, Commun. Math. Phys. 28 (1972) 221. , “On the purification map”, Commun. Math. Phys. 30 (1973) 55.

ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN YONG-SHENG LI∗ Institute of Applied Physics and Computational Mathematics P.O. Box 8009, Beijing 100088 People’s Republic of China and Department of Mathematics, Huazhong University of Science and Technology Wuhan, Hubei 430074 People’s Republic of China

BO-LING GUO Institute of Applied Physics and Computational Mathematics P.O. Box 8009, Beijing 100088 People’s Republic of China Received 10 November 1996 1991 Mathematics Subject Classification: 35Q55, 58F39 In this paper the authors consider the Cauchy problem of dissipative Zakharov equations in R and prove the existence of the maximal attractor. Keywords: dissipative Zakharov equations, bounded absorbing set, asymptotically smooth, maximal attractor.

1. Introduction In the laser plasma interaction, the most general useful model for the strong Langmuir turbulence is described by Zakharov equations (Za). These are a pair of coupled equations with one for the envelope of the Langmuir wave of high-frequency electric field and with another for the ion-acoustic wave density perturbation. By linearizing Maxwell’s equations and using the ion and electric hydrodynamical approximations, Zakharov [1] derived the following model of Langmuir turbulence in plasma: iEt + ∆E − nE = 0 ,

(1.1)

ntt − ∆n = ∆|E|2 .

(1.2)

These equations can also be obtained less rigorously but in a simpler way by using the dispersion relation for Langmuir wave and the ponderomotive force ∗ Supported

by the Postdoctoral Foundation of China and Natural Science Foundation of Hubei

Province. 675 Reviews in Mathematical Physics, Vol. 9, No. 6 (1997) 675–687 c World Scientific Publishing Company

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Y.-S. LI and B.-L. GUO

created by the high-frequency wave propagation in plasma [2]. Zakharov equations play important roles in the strong turbulence theory for plasma waves. An important effect in this theory is the so-called Langmuir collapse. Zakharov equations (1.1) and (1.2) have one-dimensional soliton solutions. Under the assumption of slow varying of the perturbation of ion density, one may neglect the term ntt in (1.2) and obtain n = −|E|2 , thus one gets the cubic nonlinear Schr¨ odinger equation (NLS), which is the quasi-static limit of (1.1) and (1.2) iEt + ∆E + |E|2 E = 0 . Due to its importance, Zakharov equations draw much attention of many mathematicians and physicists. In recent years, Zakharov equations have been quite extensively studied theoretically and numerically. The existence of solutions can be seen in [3–10]. We must also note that some numerical experiments show that the solutions of two and three dimensional Zakharov equations may develop singular in finite time [11–14]. These are consistent with the physical phenomenon of Langmuir collapse. Later, Glangetas and Merle proved theoretically that Zakharov equations have exact self-similar singular solutions in two space dimensions [15]. However, smooth global solutions to one-dimensional problems always exist [4, 10]. When the dissipativity is taken into account, we have Zakharov equations of the following form: iEt + ∆E + iαE − nE = f ,

(1.3)

ntt − ∆n + βnt − ∆|E|2 = g .

(1.4)

The long time behavior of (1.3) and (1.4) on a bounded interval Ω = (0, l) has been studied in [16], where I. Flahaut proved the existence of a finite dimensional maximal attractor in the weak topology of phase space H 2 × H 1 × L2 (Ω). In [17] the author considered the strong dissipative case, found a decomposition for the semigroup and then proved that the maximal attractor exists in the norm topology of H 2 × H 1 × L2 (Ω). We note that, although Za (1.1) and (1.2) closely resembles the dissipative NLS, the long time behavior of (1.3) and (1.4) is very different from the dissipative NLS (see [16, 18]). The latter has a norm topological global attractor in an unbounded domain case [18], but the former has one only in the weak topology even in the case of a bounded domain [16]. In the present paper we consider the one-dimensional dissipative Zakharov equations (1.3) and (1.4) on the whole spatial space complemented with initial data E(0, x) = E0 (x), n(0, x) = n0 (x), nt (0, x) = n1 (x),

x∈ R .

(1.5)

We are going to prove that (1.3) and (1.5) has a maximal attractor in the phase space H 2 × H 1 × L2 ( R ) which attracts bounded sets in the topology of H 3 × H 2 × H 1 ( R ).

ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN

677

We note that, in the unbounded domain case of our present paper, the imbedding 0 of H s (R) into H s (R) (s > s0 ) is not compact. To overcome this difficulty we adapt the methods in [18] and in [19] and utilize the Kuratowskii α-measure of noncompactness to show the asymptotic smoothness of the semigroup S(t). Then we can apply the theory of [20] to prove the existence of the maximal attractor. We introduce the following standard notations. We denote the spaces of complex valued functions and real valued functions by the same symbols. For s ≥ 0, 1 ≤ p ≤ ∞, H s, p (R) is the usual Sobolev spaces of orders s. H s (R) = H s, 2 (R). (·, ·) denotes the inner product in L2 (R). We denote by k · kp the norm of Lp (R) and by k · ks,p the norm of H s, p (R). Especially k · k = k · k2 . For any Banach space E, Cb (I; E) denotes the space of continuous and bounded functions on an interval I ⊂ R with values in E. C is a generic constant and may assume various values from line to line. In the subsequent sections we shall repeatedly use the Gagliardo–Nirenberg inequality [21] kDm ukλr , kDj ukp ≤ Ckuk1−λ q where j 1 = +λ p n

1 m − r n

u ∈ Lq ∩ H m,r (Rn ) , +

1−λ , q

j ≤ λ ≤ 1. If m−j − nr is a nonnegative 1 ≤ q, r ≤ ∞, j is an integer, 0 ≤ j ≤ m, m j integer, then the inequality holds for m ≤ λ < 1.

2. Global Solutions and Bounded Absorbing Sets We introduce a transformation m = nt + δn, where δ is a positive constant. Then the problem (1.3)–(1.5) is equivalent to the following: iEt + ∆E + iαE + nE = f , nt + δn = m ,

t ∈ R+ ,

(2.1)

x∈R

mt + (β − δ)m + (1 − δ(β − δ))n − ∆n = ∆|E|2 + g , (E, n, m)(0, x) = (E0 , n0 , m0 )(x) ,

x∈R

(2.2) (2.3) (2.4)

1 }, then A = 1 − δ(β − δ) − ∆ is a where m0 = δn0 + n1 . Choose δ ≤ min{ β2 , 2β positive, self-adjoint and elliptic operator of order 2, and is a homeomorphism from H s (R) into H s−2 (R). We set

V = H 2 × H 1 × L2 (R),

X = H 3 × H 2 × H 1 (R) .

Then X ,→ V with continuous (but not compact) imbedding. In this section we establish time-uniform a priori estimates of solutions (E, n, m) in V and then in X which guarantee the existence of the global solutions and bounded absorbing sets.

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Lemma 2.1. Let f ∈ H 2 (R), g ∈ H 1 (R). Then (1) for any (E0 , n0 , m0 ) ∈ V, the solution of (2.1)–(2.4) belongs to L∞ (R+ ; V ); (2) for any (E0 , n0 , m0 ) ∈ X, the solution of (2.1)–(2.4) belongs to L∞ (R+ ; X). Proof. following:

Taking the analogous procedure as in [16, 22] we can obtain the

Z d kEk2 + 2αkEk2 = 2Im f E dx; dt (2) Let Z Z H0 (t) = 2k∇Ek2 + 2 n|E|2 dx + 4Re f E dx

(2.5)

(1)

+ k(−∆)− 2 mk2 + (1 − δ(β − δ))k(−∆)− 2 nk2 + knk2 , 1

1

Z I0 (t) = 4αk∇Ek2 + 2(2α + δ)

Z n|E|2 dx + 4αRe

f E dx

+ 2(β − δ)k(−∆)− 2 mk2 + 2δ(1 − δ(β − δ))k(−∆)− 2 nk2 + 2δknk2 1

Z +

1

(−∆)− 2 m(−∆)− 2 g dx , 1

1

then dH0 (t) + I0 (t) = 0 . dt

(2.6)

(3) Let Z H1 (t) = 2k∆Ek − 4Re 2

Z nE∆E dx − 4Re

f ∆E dx

+ kmk2 + (1 − δ(β − δ))knk2 + k∇nk2 , Z I1 (t) = 4αk∆Ek2 − 4(2α + δ)

Z nE∆E dx − 4αRe

f ∆E dx

+ 2(β − δ)kmk2 + 2δ(1 − (β − δ))knk2 + 2δk∇nk2 Z + 2Im

Z nf ∆E dx + 2Im Z

Z − 4Re

n2 E∆E dx

m|∇E|2 dx −

gm dx ,

then dH1 (t) + I1 (t) = 0 . dt

(2.7)

ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN

(4) Let

Z

H2 (t) = 2k∇∆Ek − 4Re 2

679

Z ∇(nE)∇∆E dx − 4Re

∇f ∇∆E dx

+ k∇mk2 + (1 − δ(β − δ))k∇nk2 + k∆nk2 , Z Z I2 (t) = 4αk∇∆Ek2 − 4(2α + δ) ∇(nE)∇∆E dx − 4αRe ∇f ∇∆E dx + 2(β − δ)k∇mk2 + 2δ(1 − (β − δ))k∇nk2 + 2δk∆nk2 Z Z + 2Re m∇E∇∆E dx + 2Im ∇(nf )∇∆E dx Z

Z ∇(n2 E)∇∆E dx − 2Re

+ 2Im

Z

Z − 2Re

∇m∇E∆E dx

∇m∇(|∇E| ) dx − 2

∇m∇g dx ,

then

dH1 (t) + I2 (t) = 0 . (2.8) dt From (2.5) we can easily get the estimate kEk ≤ C. 1 β , 2 }, by estimating the indefinite sign terms one by one Since δ ≤ min{ 2β

and we see that both H0 (t) and I0 (t) are equivalent to k∇Ek2 + k(−∆)− 2 mk2 + 1 k(−∆)− 2 nk2 + knk2 , up to a constant C(kf k, kgk, kEk). Thus from (2.6) we can find a β0 > 0 such that 1

dH0 (t) + β0 H0 (t) ≤ K0 dt where K0 = K0 (kf k, kgk, kEk). By Gronwall inequality we get H0 (t) ≤ H0 (0)e−β0 t +

K0 (1 − e−β0 t ) , β0

and obtain the estimates for k∇Ek, knk, and k(−∆)− 2 mk. Since now k∇Ek, kEk∞ and knk are bounded in R+ , we infer that both H1 (t) and I1 (t) are equivalent to the square of the norm of (E, n, m) in V , up to a constant C(kf kH 1 , kgk, kEkH 1 , knk). So from (2.7) we can take a suitable β1 > 0 such that 1

dH1 (t) + β1 H1 (t) ≤ K1 , dt where K1 = K1 (kEkH 1 , knk, kf kH 1 , kgk). By Gronwall inequality we obtain H1 (t) ≤ H1 (0)e−β1 t +

K1 (1 − e−β1 t ) , β1

and the desired estimates for k∆Ek, k∇nk and kmk. Similarly we get the estimates on k∇∆Ek, k∆nk and k∇mk. From the above estimates we can prove, without any difficulty, the existence of solutions and bounded absorbing sets.

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Lemma 2.2. Let f ∈ 2(R), g ∈ H 1 (R). Then (1) for every (E0 , n0 , m0 ) ∈ V, problem (2.1)–(2.4) has a unique solution (E, n, m) ∈ Cb R+ ; V . Let S(t) be the solution operator, i.e. (E(t), n(t), m(t)) = S(t)(E0 , n0 , m0 ) is the solution of (2.1)–(2.4) with initial data (E0 , n0 , m0 ) ∈ V. Then S(t) is a semigroup in V, uniformly continuous on any compact interval [0, T ], and has a bounded absorbing set B1 ⊂ V. (2) S(t) is a continuous semigroup in X and also has a bounded absorbing set B2 ⊂ X. 3. The Decomposition In this section we shall decompose the semigroup S(t) generated by (2.1)–(2.4). Let B ⊂ X = H 3 × H 2 × H 1 (R) be contained in a ball of radius R centered at 0. Then there is a constant C(R) such that for any (E0 , n0 , m0 ) ∈ B, the corresponding solution (E(t), n(t), m(t)) = S(t)(E0 , n0 , m0 ) satisfies kE(t)kH 3 + kn(t)kH 2 + km(t)kH 1 ≤ C(R), Let χL (x) ∈

C0∞ (R)

∀t ≥ 0 .

with 0 ≤ χL ≤ 1 satisfy 1, |x| ≤ L, χL (x) = 0, |x| ≥ 1 + L .

Then for any σ ∈ (0, 1], there exists an L(σ) > 0 (sufficiently large) such that kf − fσ k2H 2 (R) ≤ σ,

where fσ = f χL(σ)

kg − gσ k2H 1 (R) ≤ σ,

where gσ = gχL(σ)

k∆|E|2 (1 − χL(σ) )k2H 1 (R) ≤ σ . We define (Eσ , nσ , mσ )(t) = S1σ (t)(E0 , n0 , m0 ) to be the solution of the following problem: iEσt + ∆Eσ + iαEσ − iσ∆Eσ + nEσ = f − fσ − iσ∆E ,

(3.1)

nσt + δnσ = mσ ,

(3.2)

mσt + (β − δ)mσ + (1 − δ(β − δ))nσ − ∆nσ = (∆|E|2 + g)(1 − χL (σ) ) ,

(3.3)

x ∈ R.

(3.4)

Eσ (0, x) = E0 (x), nσ (0, x) = n0 (x), mσ (x) = m0 (x),

Then (uσ , vσ , wσ ) = S2σ (t)(E0 , n0 , m0 ) = S(t)(E0 , n0 , m0 ) − S1σ (t)(E0 , n0 , m0 ) = (E − Eσ , n − nσ , m − mσ ) satisfies iuσt + ∆uσ + iαuσ − iσ∆uσ + nuσ = fσ (x) ,

(3.5)

wσ = vσt + δvσ ,

(3.6)

wσt + (β − δ)wσ + (1 − δ(β − δ))vσ − ∆vσ = (∆|E|2 + g)χL(σ) ,

(3.7)

uσ (0, x) = 0,

vσ (0, x) = wσ (0, x) = 0 ,

x ∈ R.

(3.8)

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ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN

For S1σ (t) we have the following lemma. Lemma 3.1. There is a constant C > 0 and an increasing function ω(σ) with ω(0) = 0 such that the solution of (3.1)–(3.4) satisfies kEσ kH 2 , knσ kH 1 , kmσ kL2 ≤ C, f or all 0 < σ ≤ 1 and t ≥ 0 , kEσ kH 2 , knσ kH 1 , kmσ kL2 ≤ ω(σ), f or all 0 < σ ≤ 1 and t ≥ t∗ (∃ t∗ > 0) . Proof. Multiplying (3.1) by 2E σ , integrating over R, and then taking imaginary parts we get d kEσ k2 + 2αkEσ k2 + 2σk∇Eσ k2 dt Z Z = 2Im (f − fσ )E σ dx + 2σIm ∇E∇E σ dx ≤ Ckf − fσ k2 + αkEσ k2 + σk∇Eσ k2 + σk∇Ek2 , therefore d kEσ k2 + αkEσ k2 + σk∇Eσ k2 ≤ Cσ . dt By Gronwall inequality kEσ k2 ≤ kE0 k2 e−αt +

Cσ (1 − e−αt ) . α

Therefore kEσ k2 ≤ C,

for all t ≥ 0 and 0 < σ ≤ 1 .

(3.9)

Taking t1 = t1 (R) > 0 such that e−αt1 kE0 k2 ≤ e−αt1 R2 < σ, then kEσ k2 ≤ Cσ

for all t ≥ t1 ,

(3.10)

√ thus kEσ k ≤ C σ, t ≥ t1 . Multiplying (3.1) by ∆2 E σ , integrating over R and taking imaginary parts we get 1 d k∆Eσ k2 + αk∆Eσ k2 + σk∇∆Eσ k2 2 dt Z Z = −Im ∆nEσ ∆E σ dx − 2Im ∇n∇Eσ ∆E σ dx Z

Z +Im

∆(f − fσ )∆E σ dx + σIm

∇∆E∇∆E σ dx

≤ Ck∆nk kEσ k∞ k∆Eσ k + 2k∇nk4 k∇Eσ k4 k∆Eσ k

682

Y.-S. LI and B.-L. GUO

+ k∆(f − fσ )k k∆Eσ k + σk∇∆Ek k∇∆Eσ k 1

7

1

15

≤ Ck∆nk kEσ k 4 k∆Eσ k 4 + CknkH 2 kEσ k 8 k∆Eσ k 8

α 3 1 k∆Eσ k2 + k∆(f − fσ )k2 + σk∇∆Eσ k2 + σk∇∆Ek2 6 2α 4 α σ ≤ k∆Eσ k2 + k∇∆Eσ k2 + Ck∆nk8 kEσ k2 2 2 +

2 2 2 + Cknk16 H 2 kEσ k + Ckf − fσ kH 2 + σk∇∆Ek ,

thus, d 2 k∆Eσ k2 + αk∆Eσ k2 + σk∇∆Eσ k2 ≤ Cσ + C(knk16 H 2 + 1)kEσ k . dt By Gronwall inequality we obtain k∆Eσ k2 ≤ k∆E0 k2 e−αt +

2 C(knk16 H 2 + 1)kEσ k (1 − e−αt ) . 2

(3.11)

By (3.9) and (3.11), k∆Eσ k2 ≤ C,

for all t ≥ 0 and 0 < σ ≤ 1 .

Taking t2 = t2 (R) ≥ t1 such that e−αt2 k∆E0 k2 ≤ e−αt2 R2 < σ, then by (3.11) and (3.10) we have k∆Eσ k2 ≤ Cσ

for all t ≥ t2 ,

√ thus k∆Eσ k ≤ C σ, t ≥ t2 . Multiplying (3.3) by 2mσ , integrating by parts, using (3.2) we get d (kmσ k2 + (1 − δ(β − δ))knσ k2 + k∇nσ k2 ) dt +2(β − δ)kmσ k2 + 2δ(1 − δ(β − δ))knσ k2 + 2δk∇nσ k2 ) Z = 2 (∆|E|2 + g) (1 − χL(σ) )mσ dx ≤ (β − δ)kmσ k2 + C(k∆|E|2 (1 − χL(σ) )k2 + kg − gσ k2 ) ≤ Cσ . Let Hσ (t) = kmσ k2 + (1 − δ(β − δ))knσ k2 + k∇nσ k2 , recall that δ ≤ 12 β, we get d Hσ (t) + 2δHσ (t) ≤ Cσ , dt by Gronwall inequality we get Hσ (t) ≤ Hσ (0)e−2δt +

Cσ (1 − e−2δt ) . 2δ

Thus kmσ k2 + knσ k2 + k∇nσ k2 ≤ C,

∀t ≥ 0,

0 < σ ≤ 1.

(3.12)

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ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN

Take t3 = t3 (R) ≥ t2 such that Hσ (0)e−δt3 ≤ CR2 e−δt3 < σ, then kmσ k2 + knσ k2 + k∇nσ k2 ≤ Cσ . Thus we have Lemma 3.1. Lemma 3.2. There are constants C1 (σ), C2 (σ) such that kxuσ k, kx∇uσ k, kxDx2 uσ k ≤ C1 (σ) , kxvσ k + kx∇vσ k + kxwσ k ≤ C2 (σ) . Proof. We consider uσ first. Multiplying (3.5) by 2x2 uσ , taking imaginary parts we get Z Z Z d x2 |uσ |2 dx + 2α x2 |uσ |2 dx + 2σ x2 |∇uσ |2 dx dt Z Z Z = −2Im fσ x2 uσ dx + 4σRe x∇uσ uσ dx + 4Im x∇uσ uσ dx , (3.13) then the right-hand side of (3.13) is less than or equal to (4 + 4σ)kx∇uσ k kuσ k + 2kxfσ k kxuσ k 4 1 + 4 kuσ k2 + αkxuσ k2 + kxfσ k2 , ≤ σkx∇uσ k2 + σ α thus d kxuσ k2 + αkxuσ k2 + σkx∇uσ k2 dt 4 1 kuσ k2 + kxfσ k2 . ≤ 4+ σ 4α

(3.14)

Because fσ has a compact support, kxfσ k is finite. By Gronwall inequality we have kxuσ k2 ≤ C1 (σ),

∀t ≥ 0 .

Differentiating (3.5) twice with respect to x we have an equation analogous to (3.5) i∂x2 uσt + ∆∂x2 uσ + iα∂x2 uσ − iσ∆∂x2 uσ + n∂x2 uσ = Fσ (x)

4

=

∂x2 fσ (x) − ∂x2 nuσ − 2∂x n∂x uσ .

Thus we have (3.13) with uσ replaced by ∂x2 uσ , fσ replaced by Fσ . Noting that Z ∂x2 nuσ x2 ∂x2 uσ dx Z =−

h i ∂x n ∂x uσ x2 ∂x2 uσ + 2uσ xk ∂x2 uσ + uσ x2 ∂x3 uσ dx ,

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Y.-S. LI and B.-L. GUO

we obtain d dt

Z

Z 2

x

|∂x2 uσ |2

dx + 2α

Z 2

x

|∂x2 uσ |2

dx + 2σ

x2 |∂x3 uσ |2 dx

Z h i ∂x2 fσ − 2∂x n∂x uσ x2 ∂x2 uσ dx = −2Im Z

h i ∂x n ∂x uσ x2 ∂x2 uσ + 2uσ xk ∂x2 uσ + uσ x2 ∂x3 uσ dx

− 2Im

Z

Z + 4σRe

x∂x3 uσ ∂x2 uσ dx + 4Im

x∂x3 uσ ∂x2 uσ dx .

(3.15)

Since (E, n, m)(t) is bounded in X = H 3 ×H 2 ×H 1 (R), and since (Eσ , nσ , mσ )(t) is bounded in V = H 2 ×H 1 ×L2 (R), uσ = E −Eσ and vσ = n−nσ are also bounded in H 2 (R) and H 1 (R) respectively. Therefore, by kuσ kH 2 , kxuσ k, kfσ k, kxfσ k, kx∂x2 fσ k ≤ C(σ), and k∇nk∞ ≤ CknkH 2 ≤ C, we infer that the right-hand side of (3.15) is controlled 2kx∂x2 fσ k kx∂x2 uσ k + 6k∇nk∞ kx∇uσ k kx∂x2 uσ k + 4k∂x nk∞ ; kxuσ k k∂x2 uσ k + 2k∂x nk∞ kxuσ k kx∂x3 uσ k + (4 + 4σ)kx∇∂x2 uσ k k∂x2 uσ k ≤ σkx∂x3 uσ k2 + αkx∂x2 uσ k2 + C(σ) kx∂x2 fσ k2 + k∇nk2H 1 kx∂x uσ k2 + k∇nk2H 2 kxuσ k2 + kuσ k2H 2 Z ≤σ

x2 |∂x3 uσ |2 dx + C(σ) + C(σ)kx∂x uσ k2 .

Therefore we get Z Z Z d x2 |∂x2 uσ |2 dx + α x2 |∂x2 uσ |2 dx + σ x2 |∂x3 uσ |2 dx dt ≤ C(σ) + C(σ)kx∇uσ k2 . From (3.14), kx∇uσ k2 ≤ C(σ) −

1 d σ dt

(3.16) Z x2 |uσ |2 dx ,

by Gronwall inequality we have (noting that uσ (0, x) = 0) Z t Z 2 2 2 e−2α(t−s) C(σ) + C(σ)kx∇uσ k2 (s) ds x |∂x uσ | dx ≤ 0

Z d e−2α(t−s) C(σ) − x2 |uσ (s)|2 dx ds ds 0 Z t d e−2α(t−s) kxuσ (s)k2 ds ≤ C(σ) − C(σ) ds 0 ≤ C(σ)

Z

t

ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN

685

s=t ≤ C(σ) − C(σ)e−2α(t−s) kxuσ k2

s=0

Z

t

+ C(σ)

2αkxuσ (s)k2 e−2α(t−s) ds

0

≤ C(σ) − C(σ)kxuσ k2 + C(σ)C1 (σ)(1 − e−2αt ) ≤ C2 (σ) . By integrating by parts we have Z Z Z x2 |∇uσ |2 dx = − 2x∇uσ uσ dx − x2 ∆uσ uσ dx ≤ 2kxuσ k k∇uσ k + kx∆uσ k kxuσ k ≤ C(σ) . Now we turn to prove the inequality for vσ . Taking the inner product of (3.7) with 2x2 wσ , we get Z Z Z d x2 |wσ |2 dx + (1 − (β − δ)) x2 |vσ |2 dx + x2 |∇vσ |2 dx dt Z Z + 2(β − δ) x2 |wσ |2 dx + 2δ(1 − (β − δ)) x2 |vσ |2 dx Z + 2δ

x2 |∇vσ |2 dx Z

Z (∆|E| + g)χL(σ) x wσ dx − 4 2

=2

2

x∇wσ vσ dx .

Applying Gronwall inequality we get kxwσ k2 + (1 + δβ)kxvσ k2 + kx∇vσ k2 ≤ C3 (σ) . Hence the lemma. 4. The Existence of Attractor Assume that f ∈ H 2 (R), g ∈ H 1 (R). Then S(t) forms a semigroup. In the previous sections we have shown that S(t) has a bounded absorbing set in V and in X respectively, and splitted S(t) so that we can make use of the so-called Kuratowskii α-measure of noncompactness to prove the asymptotical smoothness of S(t) and thus to construct the maximal attractor. We recall that the α-measure of a set A in a Banach space E is defined by α(A)

4

=

inf{d | there is a finite covering of A of diameter < d} ,

which have the property [20, 23] α(A ∪ B) ≤ α(A) + α(B), α(A) = 0,

(subaddictivity)

A is compact in E .

We are going to prove the main theorem.

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Y.-S. LI and B.-L. GUO

Theorem 4.1. Let f ∈ H 2 (R), g ∈ H 1 (R), S(t) be the semigroup generated by (2.1)–(2.4). Then there exists a set A ⊂ V satisfying (1) S(t)A = A, ∀t ≥ 0, (2) lim dist V (S(t)B, A) = lim sup dist V (S(t)y, A) = 0, ∀ B ⊂ X bounded, t→∞

(3) A is compact in V.

t→∞ y∈B

That is, A is a maximal compact attractor in V which attracts bounded sets of X in the topology of V. To prove the theorem, we need the following compact imbedding lemma. Lemma 4.2 (see [24, 25]). Let s > s1 be integers. Then the imbedding of H s (Rn ) ∩ H s1 (Rn , (1 + x2 ) dx) into H s1 (Rn ) is compact. From Lemmas 3.2 and 4.2, we see that S2σ (t) defined by (3.5)–(3.8) is compact from X into V . Therefore for any B ⊂ X bounded, α(S2σ (t)B) = 0,

∀t ≥ 0 .

From Lemma 3.1, we find that ∀ε > 0, there exist a σ and thus a t0 > 0 such that kS1σ (t)(E0 , n0 , m0 )k < ε, ∀ t ≥ t0 and (E0 , n0 , m0 ) ∈ B, B ⊂ X bounded , that is, α(S1σ (t)B) ≤ 2ε,

∀t ≥ t0 .

So we have α(S(t)B) ≤ α(S1σ (t)B) + α(S2σ (t)B) = α(S1σ (t)B) ≤ 2ε ∀t ≥ t0 and thus lim α(S(t)B) = 0,

t→∞

∀B ⊂ X bounded .

Therefore S(t) is asymptotically smooth. By Theorem 3.4.6 in [20] we complete the proof of the theorem. References [1] V. E. Zakharov, Sov. Phys. JEPT 35 (1972) 908–914. [2] V. G. Makhankov, “Dynamics of classical solitons (in nonintegrable systems)”, Phys. Rep. Phys. Lett. C 35 (1978) 1. [3] H. Added and S. Added, C. R. Acad. Sci. Paris 229 (1984) 551–554. [4] Bo-ling Guo and Long-jun Sheng, Acta Math. Appl. Sinica 5 (2) (1982) 310–324 (in Chinese). [5] C. E. Kenig, G. Ponce and L. Vega, J. Funct. Anal. 127 (1995) 204–234. [6] Yong-sheng Li, J. Partial Diff. Eq. 5 (2) (1992) 81–93. [7] T. Ozawa and Y. Tsutsumi, Diff. Integr. Eqs. 5 (1992) 721–745.

ATTRACTOR FOR DISSIPATIVE ZAKHAROV EQUATIONS IN AN UNBOUNDED DOMAIN

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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T. Ozawa and Y. Tsutsumi, Publ. Res. Inst. Math. Sci. 28 (1992) 329–361. S. H. Schochet and M. I. Weinstein, Commun. Math. Phys. 106 (1986) 569–580. C. Sulem and P. L. Sulem, C. R. Acad. Sci. Paris 289 (1979) 173–176. M. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem and X. P. Wang, Phys. Rev. A 46 (1992) 7869–7876. D. L. Newman, P. A. Robinson and M. V. Goldman, Phys. Rev. Lett. 62 (18) (1989) 2132–2135. P. A. Robinson, D. L. Newman and M. V. Goldman, Phys. Rev. Lett. 61 (6 and 7) (1988) 702–705. G. C. Papanicolaou, C. Sulem, P. L. Sulem and X. P. Wang, Phys. Fluids B 3 (4) (1991) 969–980. L. Glangetas and F. Merle, Commun. Math. Phys. 160 (1994) 173–215. I. Flahaut, Nonlinear Anal. 16, (1991) 599–633. Yong-sheng Li and Qing-yi Chen, “Strong topological global attractors for Zakharov equations”, preprint. P. Lauren¸cot, Nonlinear Diff. Eq. Appl. 2 (3) (1995) 357–369. E. Feireisl, Nonlinear Anal. 23 (1994) 187–195. J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, Vol. 25, Amer. Math. Soc., Providence, RI, 1988. A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, 1969. Yong-sheng Li and Qing-yi Chen, J. Math. 16 (3) (1996), 242–254. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, London, New York, Tokyo, North-Holland, 1992. A. V. Babin and M. I. Vishik, Proc. Roy. Soc. Edinburgh, 116A (1990) 221–243. Bo-ling Guo and Yong-sheng Li, “Attractor for dissipative Klein–Gordon–Schr¨ odinger Equations in R3 ”, J. Diff. Eq., to appear.

THE CONNES LOTT PROGRAM ON THE SPHERE J. A. MIGNACO∗ , C. SIGAUD and F. J. VANHECKE Instituto de F´ısica ao, Rio de Janeiro, Brasil UFRJ, Ilha do Fund˜ E-mail : [email protected] E-mail : [email protected] E-mail : [email protected]

A. R. DA SILVA Instituto de Matem´ atica ao, Rio de Janeiro, Brasil UFRJ, Ilha do Fund˜ E-mail : [email protected] Received 7 November 1996 We describe a classical Schwinger-type model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π2 (S 2 ), we construct hermitian connections with values in the universal differential envelope. Instead of describing matter by the usual Dirac spinors yielding the standard Schwinger model on the sphere, we apply the Connes–Lott program to the Hilbert space of complexified inhomogeneous forms with its Atiyah–K¨ ahler structure. This Hilbert space splits in two minimal left ideals of the Clifford algebra preserved by the Dirac–K¨ ahler operator D = i(d−δ). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra with Clifford action on the “spinors” of the Hilbert space. The subsequent steps of the Connes–Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared. Keywords: Non-commutative geometry, Schwinger model, K¨ ahler fermions.

1. Introduction Since Dirac’s seminal paper [10], the static magnetic monopole has been of considerable interest, not as much as a physically realised system but mainly as an example of a theory with non-trivial topological features. It is a curious coincidence that Dirac’s paper appeared in the same year as Hopf’s paper [13] on circle bundles over the sphere. The geometrical fibre bundle approach appeared in the seventies by Wu and Yang [19] in a language more accessible to physicists and by Greub and Petry [12] in a more formal way. Another approach using the differential characters of Cheeger and Simons [2] was made by Coquereaux [7]. The monopole is described by its magnetic field defined on R3 \{O}, homeomorphic to S 2 ×R+ , of which the sphere S 2 is a deformation retract. The non-trivial topological features are thus common to both R3 \{O} and S 2 and for simplicity reasons we ∗ Partially

supported by CNPq, Brasil

689 Reviews in Mathematical Physics, Vol. 9, No. 6 (1997) 689–717 c World Scientific Publishing Company

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J. A. MIGNACO et al.

restrict ourselves to the case of the sphere on which we include spinor fields. This actually means that we switched from the Dirac monopole in three dimensions to a Schwinger-type model on the sphere. Here we examine this model in the light of Serre–Swan’s theorem on the correspondence between the sections of (complex) vector bundles over the sphere S 2 and the projective modules over the algebra of smooth complex-valued functions on the sphere A = C ∞ (S 2 , R) ⊗ C. We then apply the Connes–Lott programa with a suitable Hilbert space and Dirac operator and obtain equations analogous to those of the standard (classical) Schwinger model on the sphere [14]. The main interest of our work consists in an explicitly worked out example of the Connes–Lott scheme applied to a relatively simple system. Since the approach is entirely algebraic, it is feasible to generalise to the tensor product C ∞ (M, R) ⊗ AF , where AF is a finite, involutive algebra, not necessarily commutative, e.g. the “standard model” choice C ⊕ H ⊕ M3 (C). In Sec. 2 we define our notation for the relevant differential geometry on the sphere. The classification in topological sectors of the hermitian finite projective modules MP over A is given in Sec. 3. The hermitian connections ∇P with values in the universal differential envelope Ω• A are also constructed. In Sec. 4 we make our choice of a Hilbert space H and a Dirac operator D. Instead of the usual square integrable spinors on the sphere and the usual Dirac operator associated to the Levi– Civita connection, we choose the space of sections of the Atiyah–K¨ahler bundle with its Clifford module structure and with D = i(d − δ) as Dirac operator. We do not discuss here the possible physical interpretation of these “K¨ ahler spinors” which remains controversial and for which we refer to [11, 1 and 9]. The representation of the algebra in H, together with the Dirac operator D induces a representation of the universal envelope Ω• A in H. However, this representation is not a differential one and an unwanted ideal has to be quotiented out. The elimination of this socalled “junk” is done in the standard manner [3] and yields a representation in H of the connection with values in the de Rham exterior algebra. Finally, in Sec. 5 the projective modules MP are tensored over A with the Hilbert space H allowing to construct a covariant Dirac operator D(∇P ) in MP ⊗a H, used in the matter action. The Yang–Mills functional, in each topological sector, is obtained by the Dixmier trace of the square of the corresponding curvature operator. Some conclusions are drawn and further prospects are presented in Sec. 6. The appendix contains a miscelanea of formulae useful in explicit calculations. 2. Differential Geometry on the Sphere S 2 The sphere of radius r is defined as S 2 = p = (x, y, z) ∈ R3 | x2 + y 2 + z 2 = r2 and the stereographic projection on the equatorial plane is given in the austral chart HA = p ∈ S 2 |z < r by φA : HA → R2 : (x, y, z) → (ξ, η) , where ξ =

y x , η= , r−z r−z

(2.1)

a We refer to the original Connes–Lott paper [6], to the review article [17] and to Connes’ book [3].

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THE CONNES–LOTT PROGRAM ON THE SPHERE

with inverse

φA −1 : R2 → HA : (ξ, η) → (x, y, z) ,

2ξ 2η y x = = , , 2 2 r 1+ξ +η r 1 + ξ 2 + η2 In the boreal chart HB = p ∈ S 2 | − r < z , one has

ξ 2 + η2 − 1 z = . r 1 + ξ 2 + η2

φB : HB → R2 : (x, y, z) → (ξ 0 , η 0 ) , where ξ 0 =

x y , η0 = , r+z r+z

where

(2.2)

with inverse φB −1 : R2 → HB : (ξ 0 , η 0 ) → (x, y, z) , where

2ξ 0 x = , r 1 + ξ 02 + η 02

2η 0 y = , r 1 + ξ 02 + η 02

1−ξ0 −η0 z = . r 1 + ξ 02 + η 02 2

2

It is useful to introduce the complex coordinates ζA = ξ + iη with its complex c c = ξ− iη in HA and ζB = ξ 0 − iη 0 , ζB = ξ 0 + iη 0 in HB . In the conjugate ζA T 2 overlap HA HB = (x, y, z) ∈ S | − r < z < r , both coordinates are related by ζA ζB = 1, displaying the complex structure of S 2 ' CP1 . Spherical coordinates are defined on this overlap by x = r sin θ cos ϕ ,

y = r sin θ sin ϕ ,

z = r cos θ ,

related to the coordinates above by ζA = cotg(θ/2) exp(+iϕ) , ζB = tg(θ/2) exp(−iϕ) .

(2.3)

Note that the signs are chosen such that the orientation defined by (ξ, η) corresponds to a positive inward oriented normal on S 2 as imbedded in R3 and is opposed to that of (ξ 0 , η 0 ) and (θ, ϕ). The metric on the sphere S 2 is obtained as the pull-back of the Euclidean metric on R3 : gE = dx ⊗ dx + dy ⊗ dy + dz ⊗ dz . In HA it is written as g|A = θξ ⊗ θξ + θη ⊗ θ η ,

(2.4)

with the Zweibein θξ =

2 2 dξ and θη = dη , where fA = 1 + |ζA |2 . fA fA

In terms of θ A = θ ξ + iθ η =

2 fA dζA

g|A =

and its conjugate θcA , the metric reads

1 c θ ⊗ θA + θA ⊗ θcA . 2 A

Similarly in HB , the Zweibein is given by θξ0 =

2 2 dξ 0 and θη0 = dη 0 , where fB = 1 + |ζB |2 , fB fB

(2.5)

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J. A. MIGNACO et al.

or by θB = θξ0 − iθ η0 =

2 fB dζB

and θcB , so that the metric reads

g|B = θξ0 ⊗ θ ξ0 + θ η0 ⊗ θη0 = In the overlap HA

T

1 c θB ⊗ θB + θB ⊗ θcB . 2

(2.6)

HB , one has the relation:

θA = −

2 ζA θ , |ζA |2 B

and θcA = −

c2 ζA θc . |ζA |2 B

(2.7)

In this overlap, spherical coordinates can also be used and one has fA−1 = sin2 (θ/2) , fB−1 = cos2 (θ/2) ; θA = − exp(iϕ) dθ − i sin θ dϕ , θ B = exp(−iϕ) dθ − i sin θ dϕ . In HA the structure functions of the Zweibein field are given by dθ ξ = η θξ ∧ θη and dθη = ξ θη ∧ θξ or by dθA =

1 ζA θ A ∧ θcA , 2

and the Levi–Civita connection reads: ∇θ ξ = − ξθη − ηθ ξ ⊗ θη and ∇θ η = − ηθ ξ − ξθ η ⊗ θ ξ or by 1 c θA ⊗ θA . ∇θ A = − ζA θ cA − ζA 2

(2.8)

(2.9)

Similar expressions hold in HB and the relation with (2.8) and (2.9), in the interT section HA HB , is easily worked out. The oriented area element ω on the sphere is given, respectively in HA and HB by ω|A = τ := θξ ∧ θη = 2i θ A ∧ θcA , 0

0

ω|B = −τ 0 := −θξ ∧ θη = 2i θB ∧ θ cB T and in the overlap HA HB , one has

(2.10)

ω|A = ω|B = −dθ ∧ sin θdϕ . On the sphere, a local basis of the space of differential forms F • (M ), is given in HA by ξ η 1, θ , θ , τ . The only non-zero C ∞ (S 2 , R)-valued scalar products of these basis vectors, as defined in (A.10) are g −1 1, 1 = g −1 θξ , θ ξ = g −1 θ η , θη = g −1 τ, τ = 1 .

THE CONNES–LOTT PROGRAM ON THE SPHERE

693

The multiplication table of the real Clifford algebra C`(S 2 ), defined in (A.26) of the appendix, is given in Table 1. Table 1. Clifford multiplication. ∨

1

θξ

θη

1

1

θξ

θη

τ

ξ

ξ

1

τ

θη

θη

θη

−τ

1

−θξ

τ

τ

−θ η

θξ

−1

θ

θ

τ

Besides the trivial idempotent 1, the primitive idempotents are of the form P = 12 1 + bξ θξ + bη θη + cτ , with b2ξ + b2η = 1 + c2 , bξ , bη , c ∈ C ∞ (S 2 , R). The Dirac–K˝ ahler operators, D = i(d − δ) and DD = i(dD − δ D ), defined in (A.32) and (A.33), conserve the left-, IPE = C`(S 2 ) ∨ P , and right-ideals, IPD = P ∨ C`(S 2 ), if and only if ∇X P = 0 for any vector field X on the sphere. This happens if bξ = bη = 0 and 1 + c2 = 0, so that there is no real solution. Thus, although the real representation of C`(S 2 ) is reducible, the invariant subspaces are not conserved by the Dirac–K˝ ahler operators. However, in the complexified Clifford algebra C`C (S 2 ), E = there is a solution, namely P± = 12 (1 ± iω), so that the minimal left-, I± C C 2 D 2 C` (S ) ∨ P , and right-ideals, I = P ∨ C` (S ), are conserved by D, respec±

±

±

tively DD . The complexified Clifford algebra C`C (S 2 ) can then be decomposed in a sum E E + I− , providing two equivalent represenof minimal left ideals: C`C (S 2 ) = I+ C 2 E tations of C` (S ) on these ideals I . To write down explicit matrices for these ±

E in each chart, e.g. in HA , one takes representations, one chooses a local basis of I± ξ A A A A P± Q± , where Q± = θ ∨ P± .

P+A =

1 (1 + iτ ) , 2

QA + =

1 ξ 1 (θ + iθ η ) = dζA , 2 fA

1 1 ξ 1 η (1 − iτ ) , QA dζ c . − = (θ − iθ ) = 2 2 fA A ξ0 B QB ∨ P±B and ± , where Q± = θ

P−A = In HB , one has P±B P+B =

1 (1 − iτ 0 ) , 2

QB + =

0 1 ξ0 1 (θ − iθ η ) = dζB , 2 fB

0 1 1 ξ0 1 (1 + iτ 0 ) , QB + iθ η ) = dζ c , − = (θ 2 2 fB B T with the relation in HA HB given by BA B B , P±A QA ± = P± Q± T±

P−B =

(2.11) (2.12)

(2.13)

(2.14)

(2.15)

694

where

J. A. MIGNACO et al.

  T+BA = 

1 0

−





0 ζA |ζA |

2  ,

 T−BA = 

1 0

−

0 c ζA |ζA |

 2  .

A general inhomogeneous form is now written as Ψ = Ψ(+) + Ψ(−) , with ! A,(0) ψ± (±) A A , Ψ |A = P± Q± A,(1) ψ± ! B,(0) ψ± (±) B B , Ψ |B = P± Q± B,(1) ψ± T and, in HA HB , one has the gauge transformation: ! ! B,(0) A,(0) ψ± ψ± BA = T± . B,(1) A,(1) ψ± ψ±

(2.16)

(2.17)

E , locally An element Φ of the Clifford algebra is then represented in each ideal I± A B by the matrices R± (Φ), R± (Φ): A A A (2.18) Φ ∨ P±A QA ± = P± Q± R± (Φ) , B B B (2.19) Φ ∨ P±B QB ± = P± Q± R± (Φ) .

Explicit matrices are obtained from Table 2. Table 2. Representation of the Clifford algebra. ∨

A P+

QA +

A P−

QA −

1

A P+

QA +

A P−

QA −

θξ

QA +

A P+

QA −

A P−

η

−iQA + A −iP+

A iP+ iQA +

iQA − A iP−

A −iP−

θ

τ

RA ± (1)

=

ξ RA ± (θ ) =

In the overlap region HA

T

10 01 01 10

! ,

RA ± (τ )

=

! η , RA ± (θ ) =

−iQA −

!

∓i 0 0 ±i

,

0 ±i ∓i 0

!

(2.20) .

HB , one has

BA RA RB ± (Φ) = T± ± (Φ)

T±BA

−1

.

(2.21)

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A c A A c A In C`C (S 2 ), the complex conjugation yields (P± ) = P∓ , (Q± ) = Q∓ so that the A A basis P± Q± is orthonormal (see Table 3) with respect to the sesquilinear form h−1 (Ψ, Φ) = g −1 (Ψc , Φ), given in (A.21).

Table 3. Orthonormality relations. h−1

A P+

QA +

A P−

QA −

A P+

1/2

0

0

0

QA + A P− QA −

0

1/2

0

0

0

0

1/2

0

0

0

0

1/2

This hermitian scalar product with values in C ∞ (S 2 , R)C is given by h−1 (Ψ, Φ) = h−1 (Ψ(+) , Φ(+) ) + h−1 (Ψ(−) , Φ(−) ) , where, in chart HA say, (±)

h−1 Ψ|A , Φ|A (±)

=

(2.22)

1 A,(0) c A,(0) A,(1) A,(1) (ψ± ) φ± + (ψ± )c φ± . 2

(2.23)

E , a hermitian scalar product, as in Integrating over the sphere yields, in each I± (A.22), with complex values: Z h−1 Ψ(±) , Φ(±) ω . (2.24) hΨ(±) | Φ(±) i = S2

is then completed with this product to form a Hilbert space H(P± ) Each and the total Hilbert space is the direct orthogonal sum H(P+ ) ⊕ H(P− ). The Dirac–K˝ ahler operator conserves this splitting and reads: E I±

DΨ = D(+) Ψ(+) + D(−) Ψ(−) with, on each H(P± ): (±) A Ψ|A D(±)

=

P±A

A



Q± 

0 A,(0)

D(±)

A,(1)

D(±)

 

0

(2.25)

A,(0)

ψ±

A,(1)

! .

(2.26)

ψ±

Each Hilbert space H(P± ) is Z2 graded by the parity of the corresponding differential form with grading operator ±iω represented by the matrix ! 1 0 . (2.27) 0 −1 With respect to this grading, H(P± ) is further decomposed as H(P± ) = H(0) (P± )⊕ (0) ahler operator is odd. The operators D(±) : H(0) (P± ) → H(1) (P± ) and the Dirac–K˝ (1)

H(1) (P± ) and D(±) : H(1) (P± ) → H(0) (P± ) are formally adjoint in the sense that (1)

(0)

(0)

(1)

(1)

(0)

hΨ± |D(±) Φ± i − hD(±) Ψ± |Φ± i = 0 .

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c It is easier to use the complex coordinates ζA and ζA to write explicit expressions A for D(±) , represented by the matrices:

  A =i  D(+) 

0 fA

fA

∂ ∂ζA

  A =i  D(−)

0 fA

∂ c ∂ζA

fA

 ∂ − ζ A c  ∂ζA ,  0  ∂ c − ζA  ∂ζA . 0

(2.28)

(2.29)

The index of the Dirac–K˝ ahler operator, restricted to the even forms, is (0) (0) (0) Index D(±) = dim Ker D(±) − dim Coker D(±) (0) (1) = dim Ker D(±) − dim Ker D(±) .

(2.30)

(0)

Now, Ker D(+) is given by the functions ψ such that f ∂ζ∂A ψ = 0, i.e. the anti-holomorphic functions on the sphere, and these are the constants, so (0) (1) that dim(Ker D(+) ) = 1. On the other hand, Ker D(+) is given by φ such that f ∂ζ∂c − ζA φ = 0. The substitution φ = f γ yields f 2 ∂ζ∂c γ = 0, so that the A A solutions are φ = f × constant. These, however, are not integrable over the sphere, (1) so that dim(Ker D(+) ) = 0 and Index (D(+,0) ) = 1 − 0. The same argument applies (0)

to D(−) and the total index is (0) (0) Index D(+) + D(−) = 2 − 0 ,

(2.31)

which equals the Euler number of the sphere. The index could also be obtained as the difference between the even and the odd zero modes of D. The spectrum of the Dirac operator is obtained solving the eigenvalue equation: ∂ A,(1) A,(0) = λψ+ , i fA c − ζA ψ+ ∂ζA ∂ A,(0) A,(1) = λψ+ , i fA ψ+ ∂ζA

(2.32)

which yields −fA2

∂2 A,(0) A,(0) = λ2 ψ+ . c ψ+ ∂ζA ∂ζA

(2.33)

To make a link with well-known facts, it is useful to introduce the Killing vector fields generating the SO(3) action on S 2 :

697

THE CONNES–LOTT PROGRAM ON THE SPHERE

`b+ = − `b− = +

∂ ∂ + ζA 2 c ∂ζA ∂ζA ∂ c2 ∂ + ζA c ∂ζA ∂ζA

, ,

(2.34)

∂ c ∂ − ζA , `bz = + ζA c ∂ζA ∂ζA with the Casimir: ∂2 b + `b2 = −f 2 b 2 = 1 `b+ `b− + `b− `+ (2.35) L z A c . 2 ∂ζA ∂ζA p It follows that the eigenvalues are λ = ± `(` + 1) with multiplicity (2` + 1) and eigenvectors given by ! A,(0) ψ+ (±, `, m) A , (2.36) ψ(+) (±, `, m) = A,(1) ψ+ (±, `, m) with A,(0)

ψ+

A,(1)

ψ+

(±, `, m) = Y`,m , ±i c ˆ `ˆ− + ζA (±, `, m) = p `z Y`,m `(` + 1) p ±i c = p (` + m)(` − m + 1)Y`,m−1 + m ζA Y`,m . `(` + 1)

3. The Projective Modules Over C ∞ (S 2 , R) ⊗ C Let M be a (right) module over the algebrab A = C ∞ (S 2 , R)⊗C with involution a → a+ given here by complex conjugation a+ = ac . Assume M to be endowed with a sesquilinear form h, i.e. a mapping h : M × M → A : X, Y → h(X, Y ) ,

(3.1)

which is bi-additive and obeys h(Xa, Y b) = a+ h(X, Y ) b . It is nondegenerate if h(X, Y ) = 0, ∀X ⇒ Y = 0 and h(X, Y ) = 0, ∀Y ⇒ X = 0. It is hermitian + and positive definite if it is hermitian and if h(X, X) is if h(X, Y ) = h(Y, X) a positive element of A for all X 6= 0. The adjoint, S† , with respect to h of an endomorphism S ∈ ENDA (M) is defined by h(X, S† Y ) = h(SX, Y ) . The universal graded differential envelope of A (see e.g. [8, 17]) is the graded differential algebra ) ( ∞ M • (k) Ωu (A) , du . (3.2) Ωu (A) = k=0 b Throughout this section, elements of the algebra A are denoted by {a, b, . . .}, vectors of the module M by {X, Y, . . .}, and complex numbers by {κ, λ, . . .}.

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Its most pragmatic definition goes as follows. Let du A be a copy of A as a set and consider the free algebra over C generated by the elements of A and du A, then Ω•u (A) can be defined as this free algebra modulo the relations κdu a + λdu b = du (κa + λb) , du (a b) = (du a) b + a (du b) , du 1 = 0 . The involution in A can be extended to Ω•u (A) such thatc (du a)+ = du a+ ,

(3.3)

and, with αu being the main automorphism (see (A.1)) of the graded algebra Ω•u (A), one obtains ∀ψu ∈ Ω•u (A) , (du ψu )+ = −αu du (ψu+ ) .

(3.4)

In fact, Ω•u (A) is an A-bimodule so that its tensor product over A with M, Ω•u (M) = M ⊗a Ω•u (A), is well defined. The hermitian structure h of (3.1) can be extended to Ω•u (M) by h : Ω•u (M) × Ω•u (M) → Ω•u (A) : (X ⊗a ψu , Y ⊗a φu ) → h(X ⊗a ψu , Y ⊗a φu ) = ψu+ h(X, Y ) φu .

(3.5)

(1)

A connection in M is a maping ∇ : M → Ωu (M) : X → ∇X , such that: ∇(X + Y ) = ∇X + ∇Y , ∇(X a) = (∇X)a + X ⊗a du a .

(3.6)

The connection also can be extended to Ω•u (M): (k+1) (M) : ∇ : Ω(k) u (M) → Ωu

X ⊗a ψu → ∇(X ⊗a ψu ) = (∇X)ψu + X ⊗a du ψu .

(3.7)

The square of the connection is its curvature which is a right module homomorphism: (k+2) (M) , i.e. ∇2 (X ⊗a ψu a) = ∇2 (X ⊗a ψu ) a . (3.8) ∇2 : Ω(k) u (M) → Ωu The connection is compatible with the hermitian structure if: du h(X, Y ) = h(∇X, Y ) + h(X, ∇Y ) . This is easily extended to Ω•u (M): cu ∈ Ω• (M), du h(α (ψ cu ), φ cu ) = h(∇ψ cu , φ cu ) + h(ψ cu , ∇φ cu ) . cu , φ ∀ψ u u c Several authors, e.g. Connes in [3], use a different convention: (d a)+ = −d a+ . u u

(3.9)

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699

The curvature is then always a hermitian operator: h(∇2 X, Y ) − h(X, ∇2 Y ) = 0 .

(3.10)

N A free module of finite rank is a module isomorphic to A and has ai basis Ei ; i = 1, . . . , N so that each element of M can be written as X = Ei x . The j hermitian structure is then given by h(X, Y ) = (xi )+ h( ¯i j y . 1 if i = j , In the basis By definition, in a standard unitary basis: h¯i j = 0 otherwise. (1) Ei , the connection ∇ is given by an N × N matrix with entries in Ωu (A):

∇Ei = Ej ⊗a ω j i ,

(3.11)

∇X = Ei ⊗a (du xi + ω i j xj ) .

(3.12)

(1)

where ω j i ∈ Ωu (A) , so that

The curvature of the connection is then given by ∇2 Ei = Ej ⊗a ρj i ;

ρj i = du ω j i + ω j k ω k i

∈ Ω(2) u (A) .

(3.13)

The compatibility with the hermitian structure reads: ` du h¯ij = (ω k i )+ hkj ¯ + h¯i` ω j ,

(3.14)

` (ρk i )+ hkj ¯ − h¯i` (ρ j ) = 0 .

(3.15)

An endomorphism S ∈ ENDA (M), given by SEi = Ej sji , has adjoint S† given by S† Ei = Ej (s† )ji such that h¯i k (s† )kj = (s` i )+ h`¯j . A hermitian projective module of finite rank MP over A is obtained from a free module M as the image of a hermitian projection operator P ∈ ENDA (M) such that P2 = P and P† = P. An element X ∈ M belongs to MP = P M iff PX = X. The hermitian structure h in M defines an hermitian structure hP in MP by restricting X, Y to MP : hP (X, Y ) = h(X, Y ) = h(PX, Y ) = h(X, PY ) = h(PX, PY ) .

(3.16)

Such a projection operator can be expanded and the Pauli– in terms of the 2identity Gell-Mann hermitian traceless matrices λα ; α = 1, . . . , N − 1 as P=

1 a 1 + bα λα , 2

where a and the bα are real-valued functions on the manifold (here S 2 ). Using the multiplication of the λ matrices in standard notation: λα λβ = γ λγ dαβ + γ fαβ , leads to a2 +

2 α β N δαβ b b

= 2a,

2 a bα + γ dαβ bα bβ = 2 bγ .

(3.17)

2 N δαβ

+

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It can be shown that, in the case of the sphere, it is enough to consider only N = 2 so that, besides the trivial solutions P = 1 and P = 0, the general solution is obtained as P(~n) =

1 (1 + nα σα ) , 2

(3.18)

where nα are real functions on the sphere such that δαβ nα nβ = 1. The projection operators are thus given by mappings S 2 → S 2 and it can be shown that homotopic mappings define isomorphic projective modules. These are thus classified by the second group π2 (S 2 ) ' Z. homotopy In each homotopy class ~n , a representative can be chosen in various ways. Our choice is the following. First we choose coordinates in the domain sphere and in the target sphere such that the point (1,0,0) of the domain is mapped on the fixed base point in the target sphere and, in this target sphere, coordinates are chosen such that the fixed base point is also given by (1,0,0). Let HAim and HBim be the austral and boreal charts of the image sphere with complex coordinates νA , νB , then in each homotopy class ~n we choose , if ~n = +n ; c n c n , νB = ζB , if ~n = −n , νA = ζA νA = ζA

n

, νB = ζB

n

(3.19) (3.20)

where n is a non-negative integer. In this way, HAim and HBim are the images of HA and HB . In a standard unitary basis Ei ; i = 1, 2 of A2 the projection operator PEi = Ej pj i , with the usual representation of the Pauli matrices, is given by the matrix: ! 1 + nz nx − iny 1 . (3.21) P= 2 nx + iny 1 − nz It is given in HAim , respectively in HBim , byd ! c |νA |2 νA 1 1 ; PB = PA = 1 + |νA |2 1 + |νB |2 νA 1

1

!

νB

c νB |νB |2

An element X = Ei xi ∈ A2 , given by the column matrix X =

x1 x2

.

(3.22)

, belongs to

MP if PX = X . Here x1 , x2 are functions on S 2 , given in the chart HA by functions c c 2 ), related by x1A = νA xA . In HB they are given by x1B , x2B , with x1A , x2A of (ζA , ζA 2 c 1 the relation xB = νB xB . A local, normalized basis is defined in HA , by ! c νA 1 p , (3.23) EA = (E1 E2 ) EA ; EA = 1 1 + |νA |2 d Here we obtain, in a quite natural way, the so-called Bott projection, used in algebraic K-theory [18]. We thank J. M. Gracia-Bond´ıa for pointing this out.

THE CONNES–LOTT PROGRAM ON THE SPHERE

and, in HB by EB = (E1 E2 ) EB ;

EB =

1

!

c νB

1 p . 1 + |νB |2

701

(3.24)

In HA ∩ HB , they are related by the (passive) gauge transformation: B , EA = EB gA

B with gA =

c νA . |νA |

(3.25)

In these local bases an element X ∈ MP is written as X = EA xA in HA and X = EB xB in HB with B A x . xB = gA

(3.26)

The hermitian product of two elements X and Y of MP reads hP (X, Y ) = (xA )+ y A in HA = (xB )+ y B in HB .

(3.27)

A connection ∇ in M induces a connection ∇P in MP . ∇P = P ◦ ∇ : MP → MP ⊗a Ω(1) u (A) ,

(3.28)

∇P X = Ei ⊗a pi j du xj + (pik ω`k p`j ) xj ,

(3.29)

such that

In matrix notation, this reads ∇P X = Pdu X + P(ω)P X .

(3.30)

∇P 2 X = P(du (κ))P + (κ)2 + P(du P)(du P)P X .

(3.31)

With (κ) = P(ω)P , the curvature ∇P 2 reads

A general hermitian connection in A2 is given by ! 1 ω1 σ (ω) = , i σ + ω2 (1)

with ω1 , ω2 and σ in Ωu (A) and ω1 + = ω1 , ω2 + = ω2 . The matrix (κ) is given in HA by c c 1 1 κA νA κA νA νA p , (3.32) (κ) = p 2 κ ν κ A A A 1 + |νA | 1 + |νA |2 where 1 1 1 c c νA ω1 νA + νA σ + σ + νA + ω2 p ; κA = p 1 + |νA |2 i 1 + |νA |2

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and in HB by 1

(κ) = p 1 + |νB |2

κB νB κB

κB νB c νB κB νB

1 p , 1 + |νB |2

(3.33)

where 1 1 1 c c (ω1 + νB σ + + σ νB + νB ω2 νB )p . κB = p 1 + |νB |2 i 1 + |νB |2 In HA , on X = EA xA , one obtains ∇P X |A = EA du xA + γA xA ,

(3.34)

with the (universal) total potential γA = κA + µA , where p p 1 1 c − 1 + |νA |2 du 1 + |νA |2 p νA du νA µA = p 2 1 + |νA | 1 + |νA |2

(3.35)

will be called the (universal) monopole potential. In the same way, in HB , on X = EB xB , one obtains ∇P X |B = EB (du xB + γB xB ) ,

(3.36)

with γB = κB + µB , where p p 1 1 c − 1 + |νB |2 du 1 + |νB |2 p . (3.37) νB du νB µB = p 2 1 + |νB | 1 + |νB |2 The curvature operator reads ∇P 2 X |A = EA RA xA , with RA = du γA + γA γA , ∇P 2 X |B = EB RB xB , with RB = du γB + γB γB .

(3.38)

The gauge transformations in the overlap region HA ∩ HB are B −1 B ) κB g A , κA = (gA B −1 B B −1 B ) µB gA + (gA ) du gA , µA = (gA B −1 B B −1 B ) γB gA + (gA ) du gA , γA = (gA B −1 B ) RB gA . RA = (gA

(3.39) (3.40)

4. The Spectral Triple {A, H, D} The spectral triple {A, H, D}, as defined by Connes [4, 5], is given here by the algebra A = C ∞ (S 2 , R) ⊗ C with complex conjugation as involution and kf kA = supx∈S 2 |f (x)| as C∗ -algebra norm. The Hilbert space H is given by the

THE CONNES–LOTT PROGRAM ON THE SPHERE

703

E completion H(P+ ) of the left ideal I+ of sections in the Clifford algebra bundle with inner product: Z Ψc ∧ ? Φ . (4.1) hΨ, Φi = S2

On this Hilbert space, there is a faithful ∗-representation π0 of A given here by pointwise multiplication: π0 : A → L(H) : f → fb = π0 (f ) (fbΨ)(x) = f (x) Ψ(x)

(4.2)

Faithfulness implies that operator norm and C∗ -algebra norm coincide: kfˆk = kf kA . The unbounded, essentially Dirac operator iso D = i(d−δ), restricted n selfadjoint, p to H(P+ ). Its spectrum is ± `(` + 1) ; ` = 0, 1, 2, . . . , so that the resolvent (D − z)−1 , z 6∈ R , is compact. Furthermore, its commutator with fˆ is calculated using formula (A.30): i h1 D, fˆ Ψ = (d − δ)f Ψ − f (d − δ)Ψ i = θ µ ∨ ∇~eµ f Ψ − f θµ ∨ ∇~eµ Ψ = θ µ ∨ ~eµ (f ) Ψ = df ∨ Ψ It is bounded, with norm squared:

h 1 i 2

D, fˆ = supx∈S 2 h−1 (df, df ) . i o np `(` + 1) ; ` = 0, 1, 2, . . . , with multiplicities The eigenvalues of |D| are ( 1 if ` = 0 . µ` = 2(2` + 1) if ` 6= 0

(4.3)

(4.4)

These eigenvalues are ordered p in an increasing sequence so that the order number nL of the eigenvalue λnL = L(L + 1), counting the multiplicity, is nL = 1 +

L−1 X

2(2` + 1) = 2L2 − 1 .

`=1

It follows then that the order of λnL as nL → ∞ is (nL )1/2 . In Connes’ terminology, the spectral triple is called (d, ∞) summable with d = 2. It can then be shown that e the Dixmier trace TrDix fˆ|D|−d exists and, in our case, is given by Z 1 fω. TrDix fˆ|D|−2 = 2π S 2 e We refer to [3, 17] for the definition and properties of the Dixmier trace.

(4.5)

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The Dirac operator extends π0 to a ∗-representation π of Ω•u (A) in H: π : Ω•u (A) → L(H) : ψu → π(ψu ) , 1 1 b b b D, f1 . . . D, fk π(f0 du f1 . . . du fk ) = f0 i i (given here by) = f0 df1 ∨ . . . ∨ dfk ∨ .

(4.6)

However, this representation is not differential since ψu ∈ Ker(π) does not imply π(du ψu ) = 0, which is easily seen by the standard example: π 2f du f − du (f 2 ) = (2f df − d(f 2 )) = 0 , while π du (2f du f − du (f 2 )) = π(2du f du f ) = 2df ∨ df = 2g −1 (df, df ) Id . To obtain a graded differential algebra of operators in H one has to take the quotient of Ω•u (A) by the graded differential ideal, often called “junk”, J = J0 + du (J0 ) =

∞ M (J0 (k) + du J0 (k−1) ) ,

(4.7)

k=0

where J0 = Ker(π). In this way, one obtains the graded differential algebra: Ω•D (A) =

∞

Ω•u (A) M (k) = ΩD (A) , J

(4.8)

k=0

with canonical projection πD : Ω•u (A) → Ω•D (A)

(4.9)

The classical homomorphism theorem, applied to the representation π, yields the isomorphism: (k) π Ωu (A) (k) . (4.10) ΩD (A) ∼ = π du J0 (k−1) (k) (k) In π Ωu (A) , the scalar product of R and S, belonging to π Ωu (A) , is defined by hR, Si(k) = TrDix (R† S|D|−d ) .

(4.11)

(k) (k) In the corresponding Hilbert space completion Hu of π Ωu (A) , let P (k) be the projector on the orthogonal complement of π(du J0 (k−1) ). Then, hP (k) R, P (k) Si(k)

THE CONNES–LOTT PROGRAM ON THE SPHERE

705

(k)

depends only on the equivalence classes of R and S in ΩD (A) and defines a scalar ⊥ (k) (k) (k) which contains ΩD (A) as a dense product in HD = P (k) Hu = π(du J0 (k−1) ) (k)

subspace. Indeed, let RD and SD belong to ΩD (A), which means that RD and SD (k) are equivalence classes of elements R and S in Ωu (A) modulo π(du J0 (k−1) ), then hRD , SD i(k),D = hP (k) R, P (k) Si(k) .

(4.12)

(k)

Furthermore, with this scalar product, each HD is endowed with a left- and a right representation of the unitary group u+ u = uu+ = 1 of the algebra A: bSD i(k),D = hRD , SD i(k),D = hRD u b, SD u bi(k),D . hb u RD , u

(4.13)

u, S u ˆi(k) , consequence of the This follows from hˆ uR, u ˆSi(k) = hR, Si(k) = hRˆ assumed “tameness” [17] of the Dixmier trace: TrDix (fˆR† |D|−d ) = TrDix (R† fˆ|D|−d ) ; f ∈ A ,

(4.14)

and from the fact that P (k) is a bimodule homomorphism: P (k) fb1 Rfb2 = fb1 P (k) (R) fb2 .

(4.15)

It was shown by Connes [3, 17] that, in the commutative Riemannian case, this quotient amounts to identify Ω•D (A) with the usual de Rham algebra of differential forms. This can be seen from formulae (A.27) and (A.28), which yield f0 df1 ∨ df2 = f0 df1 ∧ df2 + f0 g −1 (df1 , df2 ) f0 −1 g (d(f1 + f2 ), d(f1 + f2 )) = f0 df1 ∧ df2 + 2 − g −1 (df1 , df1 ) − g −1 (df2 , df2 ) . 2 Now, f g −1 (dh, dh) = f dh∨dh = d(f h)∨dh+ d( −f 2 )∨d(h ) and, since (f h)dh+ (1) −f 2 −1 ( 2 )d(h ) = 0, f g (dh, dh) belongs to π(du J0 ), so that

P (2) (f0 df1 ∨ df2 ) = f0 df1 ∧ df2 .

(4.16)

This is generalised to arbitrary k: P (k) (f0 df1 ∨ df2 ∨ . . . ∨ dfk ) = f0 df1 ∧ df2 ∧ . . . ∧ df .

(4.17)

Besides, the “Wick theorem” relating Clifford products with exterior products (see, e.g. [15]), resulting from (A.27) and (A.28), yields an explicit expression for the

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“junk”: f0 df1 ∨ df2 ∨ . . . ∨ dfk − f0 df1 ∧ df2 ∧ . . . ∧ dfk = +f0 g −1 (df1 , df2 ) df3 ∧ df4 ∧ . . . ∧ dfk − f0 g −1 (df1 , df3 ) df2 ∧ df4 ∧ . . . ∧ dfk + ··· + f0 g −1 (df1 , df2 ) g −1 (df3 , df4 ) df5 ∧ . . . ∧ dfk − ···

(4.18)

The trace theorem [3, 17] yields: hRD , SD i(k),D

1 Z (k) † (k) −2 = TrDix P R P S|D| ρc ∧ ∗σ , = 2π S 2

(4.19)

where ρ, σ are the de Rham forms corresponding to RD and SD . 5. The Action To construct an action functional for the potential and the matter field, the projective modules MP = PM are tensored over A with the Hilbert space H, yielding a new Hilbert space: HF = MP ⊗a H ,

(5.20)

whose generic elements, denoted by |Fi, are linear combinations of the factorisable states |X ⊗a Ψi, with X ∈ MP and Ψ ∈ H. The scalar product, on these states, is defined by (5.21) hhX ⊗a Ψ k Y ⊗a Φii = hΨ, π0 hP (X, Y ) Φi . The representation π of Ωu (A) (4.6) induces a right A-module homomorphism: Rπ : Ω•u (MP ) = MP ⊗a Ω•u (A) → L(H, HF ) ,

(5.22)

defined by Rπ (X ⊗a φu )Ψ = |X ⊗a π(φu )Ψi . This homomorphism, in turn, induces a linear map: Oπ : HOMA MP , Ω•u (MP ) → L(HF ) : T → Oπ (T) , defined by Oπ (T)|X ⊗a Ψi = |Rπ (TX)Ψi .

(5.23)

707

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The Dirac operator in H and the connection ∇P in MP allow for the construction of a covariant Dirac operator D(∇P ) in HF : D(∇P )|X ⊗a Ψi = |X ⊗a DΨi + iRπ (∇P X)Ψ ,

(5.24)

which has a well-defined action on |X f ⊗a Ψi = |X ⊗a fbΨi. Furthermore, D(∇P ) is formally self adjoint in HF and is covariant under the unitary automorphisms U of MP D(U ∇P )U|Fi = UD(∇P )|Fi ,

(5.25)

where U ∇P = U ◦ ∇P ◦ U† . a self-adjoint operator RF in HF to the The linear map (5.23) Oπ associates (2) curvature ∇P 2 ∈ HOMA MP , Ωu (MP ) so that the Dixmier trace in HF defines a functional: I[∇P ] = TrDix (RF † RF |D(∇P )|−d ) .

(5.26)

D On the other hand, the projection of ∇P with n oId ⊗ πD yields a connection ∇P on (1) MP with values in ΩD (A). In the basis Ei of M, this connection is given by (3.28): j ∇D P (PEi ) = PEj (γ) i ,

(5.27)

(1)

with (γ)i j ∈ ΩD (A), obeying (γ)i j = P i k (γ)k ` P ` j .

(2) 2 ) is a homomorphism of HOM , Ω (M ) , and is M The curvature (∇D A P P P D

given by (3.31): j 2 (∇D P ) P Ei = PEj (RD ) i ,

withf (RD )

j i

(5.28)

m = P i k dD (γ)k ` P ` j + (γ)i k (γ)k i + P i k dD P k `dD P ` m P j. (2)

Let (R)j i be one of the operators in H belonging to π Ωu (A) , whose equij

valence class is (RD ) i . The Dixmier trace in H: i

j † (2) j (2) i −d SYM [∇D P ] = h(RD ) j (RD ) i i(2),D = TrDix (P (R) i P (R) j |D| ) (5.29) n o is well defined and, due to tameness (4.14), does not depend on the basis Ei used in M. According to Connes [3]: D SYM [∇D P ] = Inf{I[∇P ] ; (Id ⊗ πD )∇P = ∇P } .

(5.30)

The action for the matter field, living in HF , is given by D SF [|Fi, ∇D P ] = hF|D(∇P )Fi , fd

D

is the differential in Ω•D (A).

(5.31)

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D where D(∇D P ) is the Dirac operator with connection ∇P . Finally, the total action D for the matter field |Fi and the connection ∇P , is D D S[|Fi, ∇D P ] = SF [|Fi, ∇P ] + λ SYM [∇P ] ,

(5.32)

where λ is a coupling constant. This action and the resulting Euler–Lagrange equations will be written down explicitly in our case study. The connection ∇D P in MP is obtained from (3.34) and (3.36) as D ∇D P EA = EA γA , in HA ;

D ∇D P EB = EB γB , in HB ,

(5.33)

where the total potentials are locally given by D = κ + µD γA A ;

D γB = κ + µD B .

(5.34)

Here, κ = πD (κA ) = πD (κB )

(5.35)

is a globally defined one-form on S 2 , while the monopole potentials: 1/2 c c (νA dνA − νA dνA ) , 1 + |νA |2 1/2 c c = (νB dνB − νB dνB ) , 1 + |νB |2

µD A = µD B

(5.36)

are local one-forms, related by a gauge transformation in the overlap region: D B −1 B dgA . µD A = µB + (gA )

(5.37)

The curvature two-form is also globally defined: ρ = dκ + ρm ,

(5.38)

D with the monopole field ρm = dµD A = dµB given by

ρm =

1 1 c c dνA ∧ dνA = dνB ∧ dνB . (1 + |νA |2 )2 (1 + |νB |2 )2

(5.39)

The integral of the curvature yields the Chern character of the projective module: Z Z 1 1 ρ= ρm . (5.40) ch(MP ) = 2iπ S 2 2iπ S 2 For example, when the homotopy class of π2 (S 2 ) is [~n] = ±n and, provided we choose the representatives in (3.19) and (3.20), then Z n2 |ζA |2n−2 1 c dζA ∧ dζA = ∓n . (5.41) ch(MP ) = ± 2iπ S 2 (1 + |ζA |2n )2

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The monopole potential naturally depends on the choice of the representative in the relevant homotopy class. With our choice in (3.19), they are given in spherical coordinates in HA ∩ HB by µD A = ∓i n

(cotgθ/2)2n dϕ ; 1 + (cotgθ/2)2n

µD B = ±i n

(tgθ/2)2n dϕ . 1 + (tgθ/2)2n

(5.42)

Both are related by the gauge transformation: D µD A − µB = ∓i n dϕ .

(5.43)

It should be noted that the usual potentials, which are given by µD A = ∓in

(cotgθ/2)2 dϕ ; 1 + (cotgθ/2)2

µD B = ±in

(tgθ/2)2 dϕ , 1 + (tgθ/2)2

(5.44)

differ from ours by a global differential form. The action for the Yang–Mills potential is, again, obtained from the trace theorem: Z 1 D ρc ∧ ∗ρ . (5.45) SYM [∇P ] = 2π S 2 The elements |Fi of HF are given, locally, by |Fi|A = EA ⊗a ΨA and |Fi|B = EB ⊗a ΨB , with EA and EB (or ΨA and ΨB ) related by the passive gauge transformation (3.25). Since hP (EA , EA ) = 1, the scalar product of |Fi|A with |Gi|A = EA ⊗a ΦA is Z g −1 (ΦcA , ΨA )ω . (5.46) hh |Gi k |Fi ii = hΦA , ΨA i = S2

Here g −1 (ΦcA , ΨA )ω may be replaced by ΦcA ∧ (?ΨA ) = (?0 ΦcA ) ∧ ΨA , since forms of degree less than the highest, vanish under the integration symbol. The covariant Dirac operator acts on |Fi as D D(∇D P )|Fi = EA ⊗a DA (γA )ΨA ,

with D D )ΨA = i(d − δ)ΨA + iγA ∨ ΨA . DA (γA

Using the selfadjointness of the Dirac operator, the action becomes Z 1 g −1 (ΨcA , DA ΨA ) + g −1 ((DA ΨA )c , ΨA ) ω . SF = 2 S2

(5.47)

(5.48)

The Euler–Lagrange equations for the connection are obtained from the total action D = κ+µD (5.32) by the variation ϑ(κ) of κ in the connection γA A , so that ϑ(ρ) = dϑ(κ) and ϑ(DA ΨA ) = iϑ(κ) ∨ ΨA .

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Using the proper formulae of the appendix, it is easy to see that Z 1 ϑ(κ)c ∧ ?δρ + ?0 δρc ∧ ϑ(κ) ϑ(SYM ) = 2π S 2 Z 1 g −1 (ϑ(κ)c , δρ) + g −1 (δρc , ϑ(κ)) ω . = 2π S 2

(5.49)

Under this variation of κ, the matter action varies as Z 1 g −1 (ΨcA , iϑ(κ) ∨ ΨA ) + g −1 (−iϑ(κ)c ∨ ΨcA , ΨA ω . ϑ(SF ) = 2 S2 Introducing the current of Eq. (A.38): c , θ ν ∨ ΨA ) θ µ = j c , j = gµν g −1 (ψA

the variation of the matter action becomes Z i g −1 (j c , ϑ(κ)) − g −1 (ϑ(κ)c , j ) ω . ϑ(SF ) = 2 2

(5.50)

(5.51)

S

Combining Eq. (5.49) and (5.51) yield the Euler–Lagrange equations: i λ δρ − j = 0 2π 2 i c λ c δρ + j = 0 . 2π 2

(5.52)

The matter equation results from the variation ϑ(ΨA ) in (5.48), which yields the covariant Dirac equation of Benn and Tucker [1]: i(d − δ)ΨA + i(κ + µD A ) ∨ ΨA = 0 .

(5.53)

The system of coupled equations (5.52) and (5.53) is consistent provided δj = 0 is satisfied and this results from (A.39) which holds also for the covariant Dirac equation. In the absence of matter, the absolute minimum of SYM is reached when ρ = 0, i.e. dκ + ρm = 0 .

(5.54)

Using the Hodge decomposition and H1deRham(S 2 ) = 0, we may write κ = dχ0 + δφ2 ,

(5.55)

where χ0 is a zero-form and φ2 is a two-form. Substitution in (5.54) yields dδφ2 + ρm = 0 . If we put f = ?φ2 , this becomes d ? α(df ) + ρm = 0 or ∆f + ?−1 ρm = 0 ,

(5.56)

where ∆ is the Laplacian on the sphere. This result is due to Jayewardena [14] for the (classical) Schwinger model on S 2 .

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6. Conclusions In this article, we made explicit the procedure of Connes and Lott [6] for the algebra of complex valued functions on the spere S 2 , describing a classical Schwinger-type model on the sphere. Here, the non-trivial topological features of the theory show up in the projective modules over this algebra with their connections. For each projective module (i.e. each “sector” of the theory), the Lagrangian appears as that of a constrained theory in the sense that the monopole field is fixed and the Euler–Lagrange equations look accordingly. The Connes–Lott program, restricted here to a commutative algebra, provides a systematic and consistent way of dealing with topologically nontrivial aspects of gauge theories. A similar treatment of the, still commutative, algebra C ∞ (S 2 , R) ⊗ (C ⊕ C) will lead to a generalised Schwinger model on the sphere and includes also “Higgs” type of phenomena. Finally, the genuine non-commutative algebra of quaternionic-valued functions on S 4 describes instantons in the Connes–Lott framework. Ackowledgements We thank Dr. Jos´e M. Gracia-Bond´ıa for discussions during his visit at UFRJ, Rio de Janeiro, made possible by the financial support of CLAF (Centro Latino Americano de F´ısica). Appendix Let M be a Riemannian manifold of dimension N with cotangent bundle τ (M ) : T ∗ (M ) → M and let Λ• τ ∗ (M ) be the exterior product bundle. Its space PN of sections consists of the differential forms F • (M ) = k=0 F (k) (M ) which, with the exterior product ∧ and the exterior differential d, becomes a graded differential algebra.g The main automorphism α of the graded algebra F • (M ), ∧ is defined by ∗

α(ψ ∧ φ) = α(ψ) ∧ α(φ) , α(f ) = f , and

α(ξ) = −ξ ,

f ∈ F (0) (M )

(A.1)

ξ ∈ F (1) (M )

and the main antiautomorphism β by β(ψ ∧ φ) = β(φ) ∧ β(ψ) ,

and

β(f ) = f ,

f ∈ F (0) (M ) ,

β(ξ) = ξ ,

ξ ∈ F (1) (M ) .

g We follow the conventions of Kobayashi–Nomizu [16] for the Cartan exterior calculus.

(A.2)

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Besides the usual exterior differential d and interior product ι(X) which are antiderivations on F • (M ) acting from the left, i.e. d ψ ∧ φ = (dψ) ∧ φ + α(ψ) ∧ (dφ) and ι(X) ψ ∧ φ = (ι(X)ψ) ∧ φ + α(ψ) ∧ (ι(X)φ) ,

(A.3) (A.4)

it appears useful to also define a “right” exterior differential and interior product by dD ψ ∧ φ = ψ ∧ (dD φ) + (dD ψ) ∧ α(φ) and ιD (X) ψ ∧ φ = ψ ∧ (ιD (X)φ) + (ιD (X)ψ) ∧ α(φ) .

(A.5) (A.6)

They are related to the usual d and ι(X) byh dD = d ◦ α = −α ◦ d , and ιD (X) = −ι(X) ◦ α = α ◦ ι(X) .

(A.7)

The sections of the tangent bundle τ (M ) : T (M ) → M are the vector fields X (M ) on A vector field X ∈ X (M ) acts on the differential graded the manifold. algebra F • (M ), ∧, d through the interior product ι(X) and the Lie derivative: (A.8) L(X) = d ◦ ι(X) + ι(X) ◦ d = dD ◦ ιD (X) + ιD (X) ◦ dD . o n Let {~eµ }, {θµ } be a pair of dual local bases of the tangent, respectively cotangent, bundle with structure functions given by [~eµ , ~eν ] = ~eκ κ Cµν

or dθ κ = −

1 κ Cµν θ µ ∧ θ ν . 2

(A.9)

The metric on the manifold defines a scalar product on F • (M ) with values in C ∞ (M ): g −1 : F • (M ) × F • (M ) → C ∞ (M ) : (ψ, φ) → g −1 ψ, φ , which is C ∞ (M )-bilinear, symmetric and such that forms of different order are orthogonal. Let ψ = ψα1 ...αk θα1 ∧ . . . ∧ θ αk , φ = φβ1 ...βk θβ1 ∧ . . . ∧ θβk , then g −1 ψ, φ = k! ψα1 ...αk φβ1 ...βk g α1 β1 . . . g αk βk

(A.10)

This product has the following properties: g −1 ξ ∧ χ, φ = g −1 χ, ˜ξc φ ,

g −1 χ ∧ ξ, φ = g −1 χ, φ c˜ξ ,

(A.11)

where ˜ξ is the vector field defined by ˜ξc η = η c˜ξ = g −1 η, ξ . h We will also write ι(X)ψ = Xcψ and ιD (X)ψ = ψcX.

(A.12)

713

THE CONNES–LOTT PROGRAM ON THE SPHERE

The Hodge duals ? and ? 0 are the C ∞ (M )-linear mappings: ? k : F (k) (M ) → F (N −k) (M ) : φ → ? k φ , ? 0k : F (k) (M ) → F (N −k) (M ) : ψ → ? 0k ψ given by ψ ∧ (? k φ) = (? 0k ψ) ∧ φ = g −1 ψ, φ ω .

(A.13)

? k = (−1)k(N −k) ? k0

(A.14)

They are related by

and on inhomogeneous forms the action of ? and ? 0 is straightforward. The above properties (A.11) of g −1 imply that η. (A.15) ? φ ∧ ξ = ˜ξc (?φ) , ? 0 η ∧ ψ = (? 0 ψ) c˜ Since ?1 = ? 0 1 = ω , the repeated application of (A.15) leads to k 1 ? ξ 1 ∧ · · · ξ k = ˜ξ c · · · ˜ξ c ω , 1 k ? 0 ξ k ∧ · · · ξ 1 = ω c˜ξ · · · c˜ξ .

(A.16)

The inverse of the Hodge star-operators are given by ? 0N −k ◦ ? k = ? N −k ◦ ? k0 = Sign(det g) Id k .

(A.17)

Other useful properties are ? ◦ α = (−1)N α ◦ ? , ? 0 ◦ α = (−1)N α ◦ ? 0 , g −1 ? ψ, φ = g −1 ψ, ? 0 φ , ? 0 ψ ∧ η = ? 0 η˜c ψ . ξ ∧ ?φ = ? φ c˜ξ ,

(A.18) (A.19) (A.20)

The complexification of the exterior bundle yields complex-valued differential C forms F • (M ) with a naturally defined operationof complex conjugation ψ → ψ c C and an involution ψ → ψ † related by ψ c = β ψ † . On F • (M ) a C ∞ (S 2 , R)C sesquilinear form is given by (A.21) h−1 ψ, φ = g −1 ψ c , φ . When M is compact,i it defines a Hermitian scalar product in F • (M ) : Z Z Z −1 c h ψ ∧ (?φ) = (? 0 ψ c ) ∧ φ . ψ, φ ω = hψ | φi = C

M

M

(A.22)

M

i If M were not compact, the scalar product can be defined for differential forms of compact support or for square integrable forms with the Riemannian invariant measure.

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The completion of F • (M ) with this product (A.22) yields a Hilbert space H(M ). The adjoint operators of d and dD with respect to this scalar product are the codifferentials δ and δ D obtained as follows: Z (dψ c ) ∧ (?φ) hdψ|φi = C

M

Z

d ψ c ∧ (?φ) −

= M

Z

Z α(ψ c ) ∧ d(?φ) M

ψ c ∧ ?(?−1 (d(?(α(φ)))))

= M

= hψ|δφi , so that δ = ?−1 ◦ d ◦ ? ◦ α = (−1)N +1 ? 0

−1

◦ d ◦ ?0 ◦ α.

(A.23)

In the same way, Z

(? 0 ψ c ) ∧ (dD φ) = · · · = hδ D ψ|φi ,

hψ|d φi = D

M

and δD = ? 0

−1

◦ dD ◦ ? 0 ◦ α = (−1)N +1 ?−1 ◦dD ◦ ? ◦ α .

(A.24)

Both codifferentials are related by δ D = α ◦ δ = −δ ◦ α .

(A.25)

Just as the exterior algebra Λ• (τ ∗ ) is obtained as the quotient of the tensor N• ∗ (τ ) by the two-sided ideal Iext generated by elements of the form {ξ ⊗ algebra η + η ⊗ ξ}: ) ( • O • ∗ ∗ (τ )/Iext , ∧ , Λ (τ ) = the Clifford algebraj is obtained as the quotient by the ideal ICliff generated by elements of the form {ξ ⊗ η + η ⊗ ξ − 2g −1 (ξ, η) 1}, ) ( • O (±) ∗ ∗ (τ )/ICliff , ∨ . C` (τ ) = N• ∗ (τ ) persists Since Iext is generated by homogeneous elements, the Z-grading of • ∗ in Λ (τ ), while ICliff being generated by inhomogeneous but even elements of N• ∗ (τ ), only a Z2 grading survives in C`(±) (τ ∗ ). j A detailed account on Clifford algebra and the usual Dirac operator with special emphasis to applications in noncommutative geometry can be found in the lecture notes “CLIFFORD GEOMETRY: A Seminar” by J. C. V´ arilly and J. M. Gracia–Bond´ıa of the University of Costa Rica.

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As vector spaces both Λ• (τ ∗ ) and C`(±) (τ ∗ ) are isomorphic so that they can be considered as a single vector space with two different products ∧ and ∨, yielding the so-called Atiyah-K˝ ahler algebra. This algebraic construction can be done in the cotangent space of each point of the manifold M , yielding an Atiyah–K˝ ahler • algebra-bundle with its space of sections {F (M ), ∧, ∨}. The relation between the two products was given by K˝ahler (see [11]) and, in our notation, reads ψ∨φ=

N X 1 α1 β1 g . . . g αk βk k!

. . . ψ ce~α1 . . . ce~αk ∧ e~βk c . . . e~β1 cφ . . . .

k=1

(A.26) In particular, ξ ∨ φ = ξ ∧ φ + ˜ξc φ ,

(A.27)

η. ψ ∨ η = ψ ∧ η + ψ c˜

(A.28)

and

Further useful formulae are g −1 ξ ∨ ψ, φ = g −1 ψ, ξ ∨ φ , Also

? ψ ∨ η = η ∨ ?ψ ,

g −1 ψ, φ ∨ η = g −1 ψ ∨ η, φ . ? 0 ξ ∨ φ = ? 0φ ∨ ξ ,

which imply ? η 1 ∨ . . . ∨ ηk = ηk ∨ . . . ∨ η 1 ∨ ω , ? 0 ξ1 ∨ . . . ∨ ξk = ω ∨ ξk ∨ . . . ∨ ξ1 , or, more generally ?φ = β φ ∨ ω ,

? 0ψ = ω ∨ β ψ .

(A.29)

The exterior differentials and codifferentials can be written in terms of the Levi– Civita covariant derivative as follows: dψ = θ µ ∧ ∇~eµ ψ

µ and δψ = −˜θ c ∇~eµ ψ ;

µ dD ψ = ∇~eµ ψ ∧ θµ , and δ D ψ = − ∇~eµ ψ c˜θ .

(A.30) (A.31)

The Hermitian K˝ ahler–Dirac operators are defined by Dψ = i d − δ ψ = iθ µ ∨ ∇~eµ ψ ,

(A.32)

DD ψ = i dD − δ D ψ = i ∇~eµ ψ ∨ θµ .

(A.33)

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A local current can be constructed as follows : g −1

1 Dψ c , φ = g −1 θµ ∨ (∇~eµ ψ c ), φ i = g −1 ∇~eµ ψ c , θµ ∨ φ = ~eµ g −1 ψ c , θµ ∨ φ − g −1 ψ c , ∇~eµ (θ µ ∨ φ) = ~eµ g −1 ψ c , θµ ∨ φ − g −1 ψ c , (∇~eµ θµ ) ∨ φ) − g −1 ψ c , θµ ∨ (∇~eµ φ) .

So that the current Jµ = g −1 ψ c , θµ ∨ φ

(A.34)

obeys µ

~eµ (J ) +

Γµµ ν Jν

=g

−1

1 Dψ i

c

−1 c 1 ,φ + g ψ , Dφ i

(A.35)

and is covariantly conserved if ψ and φ obey the Dirac–K˝ahler equation Dψ = 0 = Dφ. The current density: p det g (A.36) J µ = g −1 ψ c , θµ ∨ φ is then divergence free: divJ = ~eµ (J µ ) + Cµµ ν J ν = 0 .

(A.37)

j = gµν g −1 (ψ c , θ ν ∨ ψ)θ µ ,

(A.38)

δj = 0

(A.39)

The current one-form:

obeys

and is dual to the (N-1)-form of Benn and Tucker [1] which is closed when the Dirac–K˝ ahler equation is satisfied. References [1] [2] [3] [4] [5] [6] [7]

I. M. Benn and R. W. Tucker, Commun. Math. Phys. 89 (1983) 341. J. Cheeger and J. Simons, Lect. Notes in Math. 1167 (1985) 50. A. Connes, Noncommutative Geometry, Acad. Press, London, 1994. A. Connes, J. Math. Phys. 36 (1995) 6194. A. Connes, preprint hep-th 9603053. A. Connes and J. Lott, Nucl. Phys. (Proc. Suppl.) 18B (1990) 29. R. Coquereaux, J. Math. Phys. 26 (1985) 3176.

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[8] R. Coquereaux, J. Geom. Phys. 6 (1989) 425. [9] A. Crumeyrolle, Orthogonal and Symplectic Clifford Algebras, Kluwer Acad. Publ., Dordrecht, 1990. [10] P. A. M. Dirac, Proc. Roy. Soc. A133 (1931) 60. [11] W. Graf, Ann. Inst. H. Poincar´ e 29 (1978) 85. [12] W. Greub and H.-R. Petry, J. Math. Phys. 16 (1975) 1347. [13] H. Hopf, Math. Ann. 104 (1931) 637. [14] C. Jayewardena, Helv. Phys. Acta 61 (1988) 636. ´ [15] D. Kastler, Introduction ` a l’Electrodynamique Quantique, Dunod, Paris, 1961. [16] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, J. Wiley, New York, Vol. I (1963), Vol. II (1969). [17] J. C. V´ arilly and J. M. Gracia-Bond´ıa, J. Geom. and Phys. 12 (1993) 223. [18] N. E. Wegge-Olsen, K-theory and C∗ -algebras, Oxford Univ. Press, Oxford, 1994. [19] T. T. Wu and C. N. Yang, Phys. Rev. D12 (1975) 3845; Nucl. Phys. B107 (1976) 365.

SOME MATHEMATICAL PROBLEMS RELATED TO CLASSICAL-QUANTUM INTERACTIONS R. OLKIEWICZ∗ Institute of Theoretical Physics, University of Wroclaw Pl. Maxa Borna 9, PL-50-204 Wroclaw, Poland E-mail : [email protected] Received 20 June 1994 Revised 26 April 1996 A mathematical model describing the interaction between classical and quantum systems is proposed. The discrete case of a counter as well as the continuous case of the SQUID-tank model are discussed.

Introduction Quantum mechanics, whose basic laws were formulated in the twenties, still remains the most fundamental theory we know. It successfully investigates, among other things, the behaviour of electrons in atoms, molecules and solids. But the quantum dynamics based on Schr¨ odinger equation makes it difficult to describe irreversible processes like the decay of unstable particles and measurement processes. It follows from the fact that it is not easy to introduce a consistent definition of an observation and an event in the framework of the quantum theory [1–4]. In the Machida–Namiki model of the measurement [5] and its reformulation given by Araki [6, 7], which provides a new formalism for description of the process of measurement in which the measured object is microscopic but the measuring apparatus is described macroscopically, an infinite time is needed for an event to occur [8]. Moreover it seems to be impossible to apply quantum mechanics to explain the occurrence of macroscopic effects. Macroscopic systems are usually described either by classical physics of a few classical parameters or by quantum statistical mechanics if the quantum nature is essential. However, there are examples of macroscopic quantum phenomena, where a large number of particles can be described by a few degrees of freedom. In these cases the evolution of the quantum object reflects the presence of the classical environment on one hand and, on the other hand, the modification of the dynamics of the classical system through some appropriate expectation value appears. So it would be useful to possess a model in which classical parameters would be coupled with quantum states in such a way that

∗ Supported

by KBN Grant no 1236/P3/93/04.

719 Reviews in Mathematical Physics, Vol. 9, No. 6 (1997) 719–747 c World Scientific Publishing Company

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a flow of information from the quantum system to the classical one and the influence of the classics on the quantum process would be ensured. Recently a mathematically consistent description of such an interaction between classical and quantum systems was proposed [9–16]. The two most essential ingredients of Blanchard’s and Jadczyk’s model are: central quantities and dissipative dynamics. Central quantities represent the classical degrees of freedom and label the different Hilbert spaces that are needed for a complete description of a given system. The term “central” means that they are measurable with all other quantities, i.e. the operators representing them commute with all other observables and thus belong to the centre of the algebra. Dissipation means that the time evolution is described not by a unitary group but by the more general concept of a completely positive semigroup. Because automorphisms of an algebra leave its centre invariant this step is necessary to enable the transfer of information between the classical system and the quantum one. The fact that the classical and the quantum system must be coupled by a dissipative rather than reversible dynamics follows from no-go theorem [8, 17], where it was shown in a general algebraic framework that the information about the measured object cannot be transmitted to values of macroscopic observable as long as the dynamics of the total system is reversible in time. Let us stress that although the dissipative dynamics overcomes the difficulty with the coupling of classical and quantum degrees of freedom in a system described by an algebra with a non-trivial centre, it is given in a special way, neither quantum (by a commutator) nor classical (by a Poisson bracket). It is also worth emphasizing that, although this joint system can be thought as an open one and thus put into a unitarily evolving larger quantum system, such a dilation is generally non-unique and neglects the role of the central quantities. Let us recall that a dilation of a semigroup of completely positive maps on a C ∗ -algebra means that the action of the semigroup is given by a composition of a conditional expectation and a group of ∗ -automorphisms on a larger C ∗ -algebra [18, 19]. Thus it is proposed to deal directly with the coupling of classical and quantum systems which are both given by a physical experiment and to consider the Liouville equation as exact. Putting the central quantities and the dissipative dynamics into action simultaneously a new perspective is obtained: central quantities evolve with time, with their evolution depending on the actual state of a quantum subsystem. Some examples such as the model of a counter [9, 11], the Stern–Gerlach experiment [11], the quantum Zeno effect [10, 11], the cloud chamber model (with GRW spontaneous localization) [14, 15] and the coupling between a SQUID and a damped classical oscillating circuit [11] have been considered. But all of them were treated separately. The purpose of this paper is to develop this theory and to build a mathematical apparatus which both unifies all the examples mentioned above and permits the construction of some new ones. Although the proposed scheme seems to be very promising it is only a phenomenological model not a fundamental approach. Nevertheless we believe that this model is indeed a minimal extension of the quantum theory that accounts for events and it can be successfully applied to a large class of physical phenomena.

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In Sec. 1 we give a review of the basic facts concerning the general theory of semigroups on Banach spaces, completely positive maps on operator algebras and, finally, the generators of positive and completely positive semigroups. We would like to notice that some references in this section do not refer to original papers but to more attainable textbooks. In Sec. 2 a mathematical model describing the interaction between a classical system with a finite number of degrees of freedom represented by an algebra of continuous functions on a symplectic manifold and a quantum system described by an algebra of all bounded linear operators acting on a separable Hilbert space is proposed. In Sec. 3 two examples describing both a discrete and continuous case are considered. 1. Mathematical Background 1.1. Why completely positive semigroups? The concept of completely positive semigroups comes from the theory of open systems. The time evolution of an open quantum system S interacting with an external system R may be described in the following way. Let us consider the joint isolated system S + R. Its time evolution is governed by a one-parameter group of unitary operators Ut ∈ L(HS ⊗ HR ). For a reference state ωR of the reservoir R the reduced dynamics of S in the Schr¨ odinger picture is defined by Λt ρ = TrR (Ut ρ ⊗ ωR Ut∗ ) , where ρ is a state of S and TrR denotes the partial trace. The dual Heisenberg dynamics is given by Λ∗t A = E (Ut∗ A ⊗ 1R Ut ) , where E is the conditional expectation from L(HS ⊗ HR ) onto L(HS ) with respect to ωR . Thus Λ∗t as the composition of a ∗ -automorphism with a conditional expectation, is a family of completely positive and normal maps. Generally the function t → Λ∗t does not possess simple properties and satisfies a complicated integrodifferential equation. However for a large class of interesting physical phenomena we can derive, using certain limiting procedures, such as the weak coupling limit, the low density limit or the singular coupling limit [20], an approximate Markovian master equation for the reduced dynamics of S. This means that we may restrict ourselves to those Λ∗t which fulfil the semigroup condition. For the application of positive semigroups in dynamical description of non-equilibrium statistical mechanics see [21]. 1.2. Basic theory of one-parameter semigroups Semigroups are known mainly from applications into differential equations and to probability theory. Linear differential equations in Banach spaces are connected with the theory of one-parameter semigroups by the following theorem. For a linear and densely defined operator A the initial value problem

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d u(t) = Au(t), dt u(0) = f

t≥0

has a unique solution for every f ∈ D(A) if A generates a strongly continuous semigroup. Up to some technical details the converse theorem is also true. The second main area of applications of semigroups is the probability theory, particularly Markov processes. The hypothesis that a particle has no memory of the past implies that the transition probability p(t, x, s, E) satisfies the Chapman–Kolmogorov equation Z p(t, x, s, E) = p(t, x, u, dy)p(u, y, s, E) for any t < u R< s. For a temporally homogeneous process we have that p(t + s, x, E) = p(t, x, dy)p(s, y, E), which leads to the semigroup property for the evolution operator Z (Tt f )(x) = p(t, x, dy)f (y) acting in a suitable function space. Since there are several textbooks concerning one-parameter semigroups on Banach spaces, e.g. Davies (1980) [22] and Goldstein (1985) [23] we restrict ourselves to present briefly some definitions and results. Definition 2.1. A one-parameter semigroup on a Banach space E is a map T : R+ → L(E) such that (a) Tt+s = Tt ◦ Ts , (b) T0 = id, (c) T is continuous in one of the following topologies on L(E) : uniform, strong or weak. It is evident that uniform ⇒ strong ⇒ weak. Proposition 2.2 [23]. strongly continuous.

A weakly continuous one-parameter semigroup is

Strongly continuous semigroups will be called C0 -semigroups. Definition 2.3. To every C0 -semigroup Tt there belongs an operator A, called the generator and defined on D(A) = {x ∈ E : lim

t→0

by Ax = lim

t→0

Tt x−x t

Tt x − x t

exists in E}

for x ∈D(A).

The generator A is uniquely determined by Tt . The converse statement is also true: a semigroup is uniquely determined by its generator.

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Proposition 2.4. Let A be a generator of a C0 -semigroup. Then (a) D(A) is dense in E, (b) A is a closed operator, (c) every Tt -invariant and dense subspace of D(A) is a core for A, T n (d) D(A∞ ) := ∞ n=1 D(A ) is a core for A. Using the property (d) we may write Tt = etA . Given a C0 -semigroup it is rather difficult to determine the domain of its generator. So it is important to know which subspaces of D(A) determines the semigroup uniquely. Proposition 2.5. Let A be the generator of a C0 -semigroup Tt , and let D0 be a core for A. Then A |D0 determines the unique C0 -semigroup (Tt ). Proof. Let B be the generator of another C0 -semigroup St and B |D0 = A |D0 . Because B is closed we have that A ⊂ B. But A and B are generators so there exist w1 , w2 ∈ R such that (w1 , ∞) ⊂ ρ(A) , (w2 , ∞) ⊂ ρ(B). So ρ(A) ∩ ρ(B) 6= ∅ and hence A = B. The following theorem shows that cores are the only domains of uniqueness. Theorem 2.6 [24]. Let A be the generator of a C0 -semigroup and D0 a subspace of D(A). If D0 is not a core of A, then there exists an infinite number of extensions of A |D0 which are generators. The interesting class of semigroups of bounded operators is that of contractions. Now we characterize their generators. For x ∈ E, let Tx := {φ ∈ E ∗ : kφk = 1 and φ(x) = kxk}. Definition 2.7 [25]. A densely defined operator A on a Banach space E is called dissipative if one of the following equivalent conditions is satisfied: (a) ∀ x ∈ D(A) ∃ φ ∈ Tx E Re φ (Ax) ≤ 0, (b) ∀ x ∈ D(A) ∀ φ ∈ Tx E Re φ (Ax) ≤ 0, (c) ∀ x ∈ D(A) ∀ α > 0 k(1 − αA)xk ≥ kxk. Definition 2.8. A dissipative operator A is called m-dissipative if ρ(A) ∩ (0, ∞) 6= ∅. It turns out that if A is m-dissipative then (0, ∞) ⊂ ρ(A). Theorem 2.9 [23]. An operator A generates a contraction C0 -semigroup iff A is m-dissipative. Remark 2.10. A dissipative operator is closable and its closure is also dissipative. We call A maximal dissipative if it has no proper dissipative extensions. If A is m-dissipative then it is maximal dissipative, but the converse statement is true only when E is a Hilbert space.

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Definition 2.11. A semigroup Tt∗ ∈ L(E ∗ ) defined by (Tt∗ φ)(x) = φ(Tt x) is called the adjoint semigroup. It is clear that Tt∗ is ∗ -weak continuous. It follows from Proposition 2.2 that if the Banach space E is reflexive it is also strongly continuous. However, for a general Banach space Tt∗ may not be strongly continuous. To get this kind of continuity we have to restrict Tt∗ to a norm closed subspace E0∗ of E ∗ defined by E0∗ = {φ ∈ E ∗ : lim Tt∗ (φ) = φ in a norm}. The C0 -semigroup Tt∗ |E0∗ is called t→0

the semigroup dual. Definition 2.12. Let E and F be Banach spaces. By E⊗F we mean the completion of the algebraic tensor product E ⊗F with respect to one of the following norms: ! i=m i=m X X kxi k kyi k : u = xi ⊗ yi , xi ∈ E, yi ∈ F kuk1 = inf i=1

i=1 ∗

kuk2 = sup (hu, φ ⊗ ψi : φ ∈ E , ψ ∈ F ∗ , kφk = kψk = 1). Theorem 2.13 [24]. Let Tt and St be C0 -semigroups on Banach spaces E and F respectively, and let A and B be their generators. Then Tt ⊗ St is a C0 -semigroup on E⊗F and the operator closure of A ⊗ id + id ⊗ B |D(A)⊗D(B) is its generator. The last point we discuss is perturbation theory. Proofs of the following theorems may be found in [23]. Theorem 2.14. Let A generate a C0 -semigroup on E and let B ∈ L(E). Then A + B defined on D(A) generates a C0 -semigroup. Theorem 2.15. Let A generate a contraction C0 -semigroup and let B be dissipative operator with D(A) ⊂ D(B). If there are constants 0 ≤ a < 1 and b ≥ 0 such that kB xk ≤ akA xk + bkxk for all x ∈ D(A), then A + B, with the domain D(A), generates a contraction C0 -semigroup. Remark 2.16. If the Banach space E is reflexive we may allow a from the above theorem to be equal 1. 1.3. Completely positive maps There are several papers, e.g. [26–29] which deal with positive and completely positive maps on C ∗ -algebras, so we present here only some basic facts. Let A and B be unital C ∗ -algebras. Let Mn (A) denote the C ∗ -algebra A ⊗ Mn , where Mn is the algebra of n × n complex matrices.

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Definition 3.1. We say Φ : A → B is n-positive if Φ ⊗ 1n : Mn (A) → Mn (B) is positive, i.e. maps positive elements into positive elements. The set of all such Φ will be denoted Pn [A, B]. Definition 3.2. Φ is called completely positive if Φ ∈ Pn [A, B] for all natural n. We denote the set of all such Φ by P∞ [A, B]. Theorem 3.3 [28]. If A or B is commutative then P1 [A, B] = P∞ [A, B]. The converse is also true. Theorem 3.4 [27]. If P1 [A, B] = P2 [A, B] then either A or B is commutative. This implies that if both A and B are noncommutative then P1 6= P2 . Generally, we have that Pn−1 6= Pn for any n. For example, it was shown in [27] that for n ≥ 2, Φ(A) = [(n − 1) TrA] 1n − A, A ∈ Mn is (n − 1)-positive but not n-positive. However, if one of the algebras is equal to Mn we have the following result: Theorem 3.5 [27]. Pn [Mn , B] = P∞ [Mn , B] Pn [A, Mn ] = P∞ [A, Mn ] Now we state the Stinespring theorem [29] but in a slightly extended form. Theorem 3.6 [28]. If Φ ∈ P∞ [A, L(H)] then there exists a representation (π, K) of A, a normal representation (ρ, K) of the von Neumann algebra Φ(A)0 and a bounded linear operator V from H into K such that Φ(A) = V ∗ π(A)V,

A∈A

ρ(x) ∈ π(A)0 and ρ(x)V = V x,

x ∈ Φ(A)0

K = [π(A)V H] The triple {(π, K), (ρ, K), V } satisfying the above conditions is unique. Remark 3.7. If Φ(e) = 1, e is the unit in A, then V is an isometry of H into K. Let L[A, L(H)] denote the Banach space of linear and bounded maps from A into L(H). It is evident that P∞ [A, L(H)] is a convex cone in L[A, L(H)]. Now we discuss topological properties of P∞ . For r > 0 let Lr [A, L(H)] denote the ball of radius r in L[A, L(H)]. Let us topologize Lr at first. By definition, a net Φν ∈ Lr converges to Φ ∈ Lr if Φν (A) → Φ(A) in the weak operator topology, for every A ∈ A. A convex subset O ⊂ L[A, L(H)] is open if O ∩ Lr is open in Lr for

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every r > 0. The convex open sets form a base for a locally convex Hausdorff topology on L[A, L(H)], called BW-topology. Theorem 3.8 [26]. P∞ [ A, L(H) ] is P∞ [A, L(H)] ∩ Lr [A, L(H)] is BW-compact.

BW-closed

in

L[ A, L(H) ] and

The above result is used to prove the following extension theorem. Theorem 3.9 [26]. Let A0 be a C ∗ -subalgebra of A and contain the identity of A. For every Φ0 ∈ P∞ [A0 , L(H)] there exists Φ ∈ P∞ [A, L(H)] such that Φ |A0 = Φ0 . Thus, if M is a von Neumann algebra acting on a Hilbert space H then every Φ0 ∈ P∞ [M, M] can be extended to Φ ∈ P∞ [L(H), L(H)]. So it would be convenient to possess a description of such a Φ. Unfortunately this is done with the additional assumption that Φ is normal. Using the Stinespring theorem Kraus [30] has proved the following: Theorem 3.10. Let H be a separable Hilbert space. If Φ ∈ P∞ [L(H), L(H)] and Φ is normal, then Φ(A) =

∞ X

Vi∗ A Vi ,

Vi ∈ L(H) and

i=1

∞ X

Vi∗ Vi < ∞ .

i=1

The above result was generalized by Evans [31] who proved that every normal Φ ∈ P∞ [M, M], where M is a von Neumann algebra acting on a separable Hilbert P space H has the form Φ(A) = Vi∗ A Vi but with Vi ∈ L(H). The simplicity of the above expression leads to the following definition. Definition 3.11. A normal and completely positive map Φ on a von Neumann algebra M is called inner if there exists a sequence {Vi }, Vi ∈ M such that for P every x ∈ M , Φ(x) = Vi∗ x Vi in the ultraweak topology. It is evident that innerness is not a trivial notion. For example: Theorem 3.12 [32]. If E is a conditional expectation from a von Neumann algebra M onto a von Neumann subalgebra N = 6 M with N 0 ∩ M = M0 ∩ M, then E cannot be inner. The sufficient condition for a normal and completely positive map defined on a σ-finite von Neumann algebra to be inner was obtained in [33]. 1.4. Generators of positive semigroups In this subsection we connect the notions of positivity and complete positivity with the general theory of semigroups and discuss positive semigroups on unital operator algebras.

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Theorem 4.1 [34]. Let L be a bounded ∗ -linear operator on a C ∗ -algebra A. Then the following conditions are equivalent : (a) (b) (c) (d)

etL is positive for all t ≥ 0, (λ − L)−1 is positive for all large positive λ, L(A2 ) + AL(1)A ≥ L(A)A + AL(A) for all self-adjoint A ∈ A, L(1) + u∗ L(1)u ≥ L(u∗ )u + u∗ L(u) for all unitaries u ∈ A.

The above theorem gives a characterization of the generators of positive and uniformly continuous semigroups. Now we state the result for strongly positive semigroups, i.e. those which fulfil Tt (A∗ A) ≥ Tt (A)∗ Tt (A). Theorem 4.2 [34]. Let L be a bounded ∗ -linear operator on a C ∗ -algebra A. Then the following conditions are equivalent : (a) etL is strongly positive, (b) L(A∗ A) ≥ L(A∗ )A + A∗ L(A) for all A ∈ A. It is evident that strongly positive semigroups form a proper subclass of positive semigroups. For example, let A = M2 (C) and L(A) = AT − A. Because for all self-adjoint A, (AT − A)2 ≥ 0, so L(A2 ) ≥ L(A)A + AL(A) and thus etL is positive. On the other hand, for 0 1 A= 2 0 we have that L(A∗ A) = 0 and L(A∗ )A + A∗ L(A) =

2 0 0 −4

,

so the condition (b) in Theorem 4.2 is not satisfied. Definition 4.3. A ∗ -linear operator L, which satisfies the condition L(A∗ A) ≥ L(A∗ )A + A∗ L(A) is called a dissipation. If the matrix inequality [L(A∗i Aj )] ≥ [L(A∗i )Aj + A∗i L(Aj )] holds for all finite sequences A1 , A2 , . . . , An ∈ A then L is called a complete dissipation. Theorem 4.4 [25]. An everywhere-defined dissipation is bounded and dissipative. It follows that etL generated by an everywhere-defined dissipation L is not only strongly positive, but also contractive. Theorem 4.5 [25, 35]. Let L be a bounded ∗ -linear map on a C ∗ -algebra. Then the following conditions are equivalent : (a) etL is a semigroup of completely positive contractions, (b) L is a complete dissipation.

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The notion of dissipation and complete dissipation may be generalized to the unbounded case, when the operator L is defined on a dense ∗ -subalgebra D(L) ⊂ A. However such a generalization has no direct implications for semigroups. Some results in this direction were obtained in [36]. For example, they proved the following theorems. Theorem 4.6. Let L be a densely defined ∗ -linear operator on a C ∗ -algebra A, whose domain is closed under the star operation. Then the following conditions are equivalent : (a) L is the generator of a C0 -semigroup of positive contractions, (b) R(1 − αL) = A ∀α > 0 and φ(L(A)) ≤ 0 for any self-adjoint A ∈ D(L) such that A = A+ − A− with A+ 6= 0, and for some φ ∈ TA+ A. TA+ A denotes the set of normalized tangent functionals at the point A+ ∈ A introduced in Sec. 1.2. Theorem 4.7. Let Tt be a C0 -semigroup on a C ∗ -algebra A with the ∗ -linear generator L. Then the following conditions are equivalent : (a) Tt is strongly positive, (b) (1 − L)−1 (A∗ A) + A∗ A ≥ (1 − L)−1 (A∗ )A + A∗ (1 − L)−1 (A) for all A ∈ A and all small > 0. It is evident that the general form of a complete dissipation is of great importance for studying dynamical semigroups, at least those ones which are uniformly continuous. The so-called canonical decomposition of the generator of a uniformly continuous semigroup of completely positive and normal maps on a von Neumann algebra was obtained by Gorini, Kossakowski and Sudarshan for finite dimensional matrix algebras [37], and by Lindblad for hyperfinite factors acting on a separable Hilbert space [35]. Subsequently Christiensen generalized this result to arbitrary von Neumann algebras [38]. The last step was made by Christiensen and Evans [39] who proved the following. Theorem 4.8. Let etL , t ≥ 0 be a uniformly continuous semigroup of completely positive maps on a C ∗ -algebra A acting on a Hilbert space. Then there exists a completely positive map ψ from A into the ultraweak closure A of A and an operator k ∈ A such that the generator L is equal to L(A) = ψ(A) + k ∗ A + Ak . There has also been considerable interest in obtaining the canonical decomposition for unbounded generators. Only partial results are known. The following theorem was obtained by Davies [40]. Theorem 4.9. Let Tt , t ≥ 0 be an ultraweakly continuous semigroup of completely positive and normal maps on L(H), leaving the compact operators invariant and having a pure invariant state Ω. Let L be the generator of the predual semigroup

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acting on Tr(H). Then Lρ = Y ρ + ρY ∗ +

∞ X

Bn ρBn∗

n=1

for all ρ from a dense subspace D ⊂ Tr(H). Here Y denotes the generator of the contraction semigroup on H given by AΩ → Tt (A)Ω, A ∈ L(H), and Bn are linear maps from Dom(Y ) into H such that Bn Ω = 0 for all n. Moreover hY ψ, ψi + hψ, Y ψi +

∞ X

kBn ψk2 ≤ 0

n=1

for all ψ ∈ Dom(Y ). A sufficient condition for an operator L, L(A) = Y ∗ A + AY + B ∗ AB to be the generator of a dynamical semigroup on L(H) exploiting the classical theory of Markov processes was obtained by Chebotariev [41]. However, the problem to find a general form for the unbounded generators is still open. And the answer will be not so simple, as one may suppose, since Holevo [42] has shown that there is a dynamical semigroup on L(H) with an unbounded generator which cannot be written in the canonical form. 2. The Model 2.1. The classical system Let us consider a nonrelativistic classical system with a finite number of degrees of freedom. The system of first order differential equations of Hamilton reflect the presence of a mathematical structure, namely a symplectic structure. In many applications it appears as the cotangent bundle of a configuration space. It is endowed with the natural symplectic form and with the position mapping given by the projection. Sometimes, however, we need to consider a more general situation when there are no natural functions of position or momentum as in the Lobachevski space case. So we assume a phase space is a symplectic manifold (M, ω). What we observe are smooth and real-valued functions on M . They form an abelian algebra C ∞ (M, R). Because, in general, they are unbounded, they do not possess good topological properties. So for this mathematical reason we assume a classical algebra Ac is the C ∗ -algebra C0 (M ). Apart from observables we are interested in states which reflect our knowledge about all possible initial conditions. A set of states Sc = {φ ∈ C0 (M )∗ : φ ≥ 0, kφk = 1}. Pure states are represented by point measures on M . The time evolution is governed by an appropriate function H on M called the Hamiltonian of the system. The Hamiltonian flow on M has the nice property of leaving the symplectic form invariant. It turns out that a flow which retains ω need not come from a Hamiltonian vector field but from a local Hamiltonian vector field given by a closed 1-form on M via the canonical isomorphism Tx∗ M → Tx M [43]. However, quite often, such an evolution is not sufficient for our purposes. Thus we assume the time evolution is just a flow on M ,

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i.e. a mapping g : (t, x) → gt (x) such that: (a) g : R × M → M is smooth, (b) for any t, gt is a diffeomorphism of M , (c) t → gt is a group homomorphism. Its generator is a complete vector field X on M . Proposition 1.1. Let gt be a flow on M. Then αt : C0 (M ) → C0 (M ), (αt f )(x) = f (gt−1 x), x ∈ M is a strongly continuous one parameter group of automorphisms of Ac . Proof. M is a metrizable space with a metric d. Let f0 ∈ Cc (M ) with suppf0 = K0 . Let K = g([−1, 1] × K0 ). Because g is continuous, K is compact. Let h(t) = supx∈K d(gt−1 x, x). From continuity of f0 and h we get ∀ > 0 ∃δ < 1 |t| < δ ⇒ ∀x ∈ M |f0 (gt−1 x) − f0 (x)| < . Continuity of αt for f ∈ C0 (M ) follows from /3 argument.

Sometimes we need to consider a more general situation when X = X(t). In this case we get a propagator. Proposition 1.2. If Zct exp(X(u)du)

T (s, t) = s

R exists, where b denotes the multiplicative integral, then (a) T (s, t) is a diffeomorphism of M , (b) (s, t) → T (s, t)x is smooth for any x ∈ M , (c) T (t, t) =identity, T (s, t) = T (t, s)−1 , T (u, s) = T (u, t) · T (t, s). Evolution on C0 (M ) starting at epoch t0 is a strongly continuous one parameter family of automorphisms (αt f )(x) = f (T (t, t0 )−1 x) such that αt0 = identity. For technical reasons R we use the von Neumann algebra formalism. Let ϕ : Cc (M ) → C, ϕ(f ) = M f ω n , n = 12 dim M. By the representation theorem we get a positive measure µ on some σ-algebra E on M [44]. It has the following properties: (a) µ is a regular measure, (b) µ(K) < ∞ if K is compact, (c) E contains R the Borel σ-algebra B, (d) ϕ(f ) = f dµ for f ∈ Cc (M ). Any other measure ν defined on a σ-algebra F with these properties agrees with µ on E ∩ F. So we get the unique measure µ defined on the σ–algebra B of Borel subsets of M . Let H1 = L2 (M, B, µ). We have a natural representation

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731

π of Ac in H1 , given by (π(f )Ψ)(x) = f (x)Ψ(x). Let us take A¯c = π(Ac )00 = L∞ (M, B, µ). Now a set of states Sc consists of normed elements from L1 (M, B, µ)+ . Moreover αt extends to an ultraweakly continuous one parameter group (family) of automorphisms of A¯c whose generator is a densely (in ultraweak topology) defined derivation δc (or a family of derivations δc (t) with one common domain). It is worth emphasizing that physically interesting functions like Hamiltonians now have the status of operators affiliated to A¯c . 2.2. The quantum system A physical system is characterized by: its measureable attributes called observables, its modes of preparation called states and a prediction rule which gives for every state the expected value of every observable. Let H be a separable Hilbert space associated with a quantum system. It permits us to make the following identifications. As a quantum algebra Aq we take L(H). Self-adjoint, generally unbounded, operators on H play the role of observables of the quantum system. They are obviously affiliated to Aq . A set of states Sq = {ρ ∈ Tr(H)+ : Trρ = 1}. Pure states are represented by one dimensional projectors in Aq , i.e. rays in H. The expectation value of an observable A in a state ρ is given by TrρA. The time evolution governed by a self-adjoint operator H is an ultraweakly continuous one parameter group of automorphisms of Aq given by A → eiHt Ae−iHt . Its generator is a densely defined (in the ultraweak topology) derivation on L(H) δq = ad(iH) = i[H, ·]. Remark 2.1. If H = H(t) then under some assumptions [45] we get a unitary propagator Zcs exp(H(u)du) . U (t, s) = t

2.3. The total system Now let us consider the joint system. From the structural point of view the world is divided into Q × C, the quantum and the classical part. They are put together by tensoring a non-commutative quantum algebra with a commutative algebra of functions and coupled via a specific dynamics that can be encoded in an irreversible Liouville evolution equation for statistical states of the total system. Algebra. The total algebra AT is given by AT = A¯c ⊗ Aq as von Neumann ˜ = H1 ⊗ H. This means that AT is the algebra of weakly measurable algebras on H ˜ < ∞. We have a functions A˜ : (M, B, µ) → L(H) such that ess supx∈M kA(x)k natural inclusions A¯c → AT , Aq → AT . Moreover for each pure state ρ on Aq there onto is a conditional expectation (in the sense of Umegaki) Eρ : AT −→ A¯c . Similarly, for each state ϕ on A¯c , which may be always written as ϕ(f ) = hϕ1/2 , : f ϕ1/2 iL2 , ϕ1/2 ∈ L2 (M, B, µ), kϕ1/2 kL2 = : 1, f ∈ L∞ (M, B, µ), there is a conditional onto expectation Eϕ : AT −→ Aq [46].

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States. States are normal and positive functionals on AT with norm 1. Because AT ∗ is equal to a norm closed (in A∗T ) space generated by product states, so AT ∗ = L1 (M, B, µ; Tr(H)) is a space of weakly measurable functions M → Tr(H) such that Z kρ˜(x)kTr dµ(x) < ∞ . M

Thus a set of states equal to Z ST = {ρ˜ ∈ AT ∗ : ρ˜(x) ∈ Tr(H)+ a.e. and

Tr(˜ ρ(x))dµ(x) = 1} . M

The mean value of A˜ ∈ AT in a state ρ˜ ∈ ST is given by Z ˜ ρ(x)] . ˜ ρ˜ = dµ(x) Tr[A(x)˜ hAi M

We have natural mappings Sc × Sq 3 (ϕ, ρ) → ϕ ⊗ ρ ∈ ST and

ST 3 ρ˜

: X XXXXz

ρ=

R

ρ˜(x)dµ(x) ∈ Sq

ϕ ∈ Sc , ϕ(x) = Tr˜ ρ(x)

Evolution. The coupling between a classical and a quantum system reveals itself in two different ways. Firstly, the quantum generator becomes dependent on the classical parameters and, secondly, the dissipative generator responsible for the transmission of information from the quantum to the classical system appears. Let ˜ = Wc (A) ˜ + Wq (A) ˜ + L(A), ˜ where Wc = δc ⊗ id, Wq = δq (x) A˜ ∈ AT and let WT (A) and L describes the interaction between the classical and the quantum system. It is worth stressing that in this formalism the evolution of the classical parameters does not depend on the quantum Hamiltonian itself. It depends only on the classical generator and the dissipative coupling. Remark 3.1. Let us consider the reversible part of the dynamics with Wq = id ⊗ δq . By passing to the preduals of A¯c and Aq we obtain from Theorem 2.13 in Sec. 1 a well-defined C0 -group on L1 (M, B, µ) ⊗ Tr(H). This means that the operator closure of (Wc + Wq ) |D(δc )⊗D(δq ) is a generator of an ultraweakly continuous one-parameter group of automorphisms of AT . So it remains to check that Wc + Wq + L is the generator of a completely positive semigroup. Proposition 3.2. Let L be a bounded complete dissipation. Then Wc + Wq + L generates a completely positive ultraweakly continuous semigroup of contractions.

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

733

Proof. Let us write Wcq for Wc + Wq . Let α > 0. Then Range[1 − α(Wcq + L)] = Range[α−1 − (Wcq + L)] ⊃ Range[α−1 − (Wcq + L)](α−1 − Wcq )−1 = Range[1 − L(α−1 − Wcq )−1 ] Because Wcq is dissipative we have k(α−1 − Wcq )−1 k ≤ α and kL(α−1 − Wcq )−1 k ≤ αkLk. Thus 1 − L(α−1 − Wcq )−1 is invertible for small α and so Range[1 − α(Wcq + L)] = AT for small α. Moreover, 1 − α(Wcq + L) is injective for all α > 0 because Wcq + L is a dissipative operator. So (0, ∞) ∩ ρ(Wcq + L) 6= ∅ and Wcq + L is m-dissipative. It follows that Range[1 − α(Wcq + L)] = AT for all α > 0. So by Theorem 4.6 in Sec. 1 we have that Wcq + L generates a positive semigroup of contractions. Because Wcq is a derivation and L is a complete dissipation the same calculations may be applied to Wcq ⊗ 1n + L ⊗ 1n , where 1n is the unit in Mn×n . Construction of L. We assume the following: (a) Let P = P ∗ be an observable on H with a spectral measure dE(λ), (b) Let to any α ∈ R corresponds a shift on the phase space. By the shift we mean a morphism of (M, B, µ), i.e. a bijective map hα : M → M such that hα and h−1 α are measurable and leave the measure µ invariant. Moreover, we demand that for any f ∈ Cc (M ) and any x ∈ M a mapping α → f (h−1 α x) is x) is a unitary Borel measurable. Then Uα : H1 → H1 , (Uα Ψ)(x) = Ψ(h−1 α operator. Moreover α → hf1 , Uα f2 i is measurable for any f1 , f2 ∈ Cc (M ) and so α → Uα is weakly measurable, (c) Let f : M × R → R be a uniformly bounded and Borel measurable function such that Z∞ | f (x, α) |2 dα < ∞ .

sup x∈M

−∞

From the above data we construct the operator L in the form: ˜ , ˜ = W (A) ˜ − 1 {W (1), ˜ A} L(A) 2 ˜ is the unit in AT and {·, ·} stands for the anticommutator. The operator where 1 W is formally given by Z∞ ˜ α x)dE(λ2 ) . ˜ dαf (x, λ1 − α)f (x, λ2 − α)dE(λ1 )A(h W (A)(x) = −∞

At first we define an operator V˜α : V˜α (x) =

R

f (x, λ − α)dE(λ), x ∈ M, α ∈ R.

Proposition 3.3. V˜α ∈ AT , supα kV˜α k < ∞ and α → V˜α is weakly measurable. Proof. V˜α (x) ∈ Aq for any x ∈ M . Moreover Z |f (x, λ − α)|2 dhE(λ)Ψ, Ψi ≤ sup |f (x, λ)|2 . sup kV˜α (x)k2 = sup sup x

x kΨk=1

x,λ

734

R. OLKIEWICZ

Because for every Ψ1 , Ψ2 ∈ H and every α ∈ R the function Z x → hΨ1 , V˜α (x)Ψ2 i = f (x, λ − α)dhE(λ)Ψ1 , Ψ2 i is Borel measurable, we have that V˜α ∈ AT . In the same way we may prove that kV˜α k2 =

sup ˜ H,k ˜ Ψk=1 ˜ Ψ∈

˜ 2 kV˜α Ψk

Z = sup

Z dµ(x)

˜ kΨk=1

˜ ˜ , Ψ(x)i |f (x, λ − α)|2 dhE(λ)Ψ(x)

≤ sup |f (x, λ)|2 . x,λ

˜ i = ψi ⊗ Ψi , where ψi ∈ H1 and Ψi ∈ H i = 1, 2. Then Finally, let Ψ Z Z ˜ 2i = ˜ 1 , V˜α Ψ ψ¯1 (x)ψ2 (x)f (x, λ − α) dµ(x)dhE(λ)Ψ1 , Ψ2 i α → hΨ is Borel measurable. But linear combinations of simple tensors ψ ⊗ Ψ form a dense ˜ so α → V˜α is weakly measurable. subspace of H ˜α = U ˜α · V˜α . Although U ˜α ∈ L(H1 ) ⊗ Aq but for any ˜α = Uα ⊗ 1 and let W Let U ∗ ˜˜ ∗ ˜˜ ˜ ˜ ˜ A ∈ AT we have that Uα AUα ∈ AT and so Wα AWα ∈ AT . ˜ α is weakly measurable. ˜ α∗ A˜W Proposition 3.4. For any A˜ ∈ AT , α → W ˜ 1, Ψ ˜2 ∈ ˜ α is weakly measurable. Let Ψ Proof. It is enough to show that α → W ˜ Then H. ∞ X ˜ 2i = ˜ ˜n ih˜ ˜ 2i , ˜α · V˜α Ψ ˜ ∗Ψ ˜ 1, U hU en , V˜α Ψ hΨ α 1, e n=1

˜ Because U ˜α and V˜α are weakly measurable, where {˜ en } is an orthonormal basis in H. ˜ α. so is W ˜ be defined by Now let Wn (A) Zn ˜Ψ ˜ 2i = ˜ 1 , Wn (A) hΨ

˜ αΨ ˜ 2i , ˜ 1, W ˜ ∗ A˜W dαhΨ α

˜ 1, Ψ ˜2 ∈ H ˜. Ψ

−n

˜ αΨ ˜ 2 i is Borel measurable and bounded the ˜ ∗ A˜W ˜ 1, W Because the function α → hΨ α integral on the right-hand side exists and defines a linear and bounded operator on ˜ also belongs to AT . ˜ As the weak limit of operators from AT , Wn (A) H. ˜ exists and belongs to AT . ˜ = limn→∞ Wn (A) Proposition 3.5. W (A) Proof. Let A˜ be a positive operator. Then ˜ α ≤ kAk ˜ α = kAk ˜ W ˜ α∗ W ˜ V˜α2 ˜ α∗ A˜W W

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

Z∞

˜ α Ψ, ˜ Ψi ˜ ≤ kAk ˜ ˜ α∗ A˜W dαhW

−∞

735

Z∞ ˜ 2 dαkV˜α Ψk

−∞

Z

Z∞ ˜ = kAk

dα

−∞

Z

˜ ≤ kAk(sup x

Z∞ dµ(x)

M

˜ ˜ |f (x, λ − α)|2 dhE(λ)Ψ(x), Ψ(x)i

−∞

˜ 2. |f (x, α)|2 dα)kΨk

˜2 ∈ H ˜ and ∀ A˜ ∈ AT , ˜ 1, Ψ formula we have that ∀ Ψ RSo∞ from the polarization ∗ ˜˜ ˜ ˜ ˜ ˜ −∞ dαhΨ1 , Wα AWα Ψ2 i exists. This means that Wn (A) is weakly convergent to ˜ and thus W (A) ˜ ∈ AT . W (A) Now let us consider a ∗ -linear map W : AT → AT . Proposition 3.6. W is a completely positive map. Proof. Wn is the weak limit of completely positive maps and W is the weak limit of Wn . Because ∀ k ∈ N W ⊗ 1k , 1k ∈ Mk×k , is the weak limit of positive ˆ is positive. maps, so for any positive operator Aˆ from AT ⊗ Mk×k W ⊗ 1k (A) Proposition 3.7. W is normal. Proof. We have to show that W is an adjoint map of some bounded T : ˜ such that ρ˜ = ˜ → Tr(H). ˜ Let ρ˜ ∈ Tr(H) ˜ + . There exists a basis {em } in H Tr(H) P∞ m=1 cm Pm , Pm is a projector onto em . At first we show that Zn ρ) = Tn (˜

˜ α ρ˜W ˜ α∗ dαW

−n

˜ exists in Tr(H). ˜ + and Because it exists in L(H) Sk =

m=k X

hem , Tn (˜ ρ)em i

m=1

=

m=k X

Zn

˜ α ρ˜W ˜ α∗ em i dαhem , W

m=1−n

Zn ≤

˜ α ρ˜W ˜ ∗ kTr ≤ 2n(sup kW ˜ α k2 )kρ˜kTr , dαkW α

−n

˜ + . Moreover ρ) ∈ Tr(H) so Tn (˜

α

736

R. OLKIEWICZ

Z∞

Z∞

˜ α ρ˜W ˜ α∗ kTr = dαkW

−∞

dα Z∞ dα

∞ X

˜ ∗ em k 2 kρ˜2 W α 1

m=1

−∞

Z∞ =

˜ α ρ˜W ˜ α∗ em i hem , W

m=1

−∞

=

∞ X

dα

∞ ∞ X X

1 ˜ ∗ em , en i|2 . |h˜ ρ2 W α

m=1 n=1

−∞ 1 ˜ α∗ ∈ HS(H). ˜ So we get But ρ˜ 2 W

Z∞ dα −∞

∞ ∞ X X

Z∞ ˜ α en i| = cn |hem , W 2

n=1 m=1

dα ∞ X n=1

=

∞ X n=1

Z∞ dαkV˜α en k2

cn −∞

cn

dα

−∞

x

R∞ −∞

Z

Z∞

·dhE(λ)en (x), en (x)i ≤ kρ˜kTr sup Thus T (˜ ρ) =

˜ α en k 2 cn k W

n=1

−∞

=

∞ X

Z

Z∞ dµ(x)

M

|f (x, λ − α|2

−∞

|f (x, α)|2 dα .

˜ α ρ˜W ˜ α∗ ∈ Tr(H) ˜ + and Tr(T (˜ ˜ = Tr(˜ ˜ for any dαW ρ)A) ρW (A))

A˜ ∈ AT .

˜ where F ∈ A¯c ⊗ 1 and (F Ψ)(x) ˜ ˜ = W (A) ˜ − 1 {F, A}, = 3.8. Let L(A) 2 R Theorem 2 ˜ ˜ ( |f (x, α| dα)Ψ(x). Then L is a bounded complete dissipation and L(1) = 0, where ˜ is the unit in AT . 1 ρ) = T (˜ ρ) − 12 {F, ρ˜} is a generator of a Remark 3.9. The adjoint of L, i.e. L∗ (˜ one parameter semigroup of completely positive and conservative maps of ST . ⊥ ˜ Proof. Because AT ∗ = Tr(H)/A T , where

˜ : Tr(˜ ˜ = 0 ∀ A˜ ∈ AT } ˜ ∈ Tr(H) ρA) A⊥ T = {ρ ⊥ and L∗ (A⊥ T ) ⊂ AT we have that L∗ : ST → ST .

Remark 3.10. It is evident that nowhere did we use the fact the quantum algebra is the type I factor. All results will still be valid if we replace L(H) by an arbitrary von Neumann algebra acting on a separable Hilbert space.

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

737

3. Examples 3.1. The discrete case A classical system is supposed to possess three distinct pure states. One of them can be viewed as the initial state of a counter while the other two correspond to the two possible results of a measurement. So the classical phase space M = {1, 2, 3} is a three point space. The classical algebra Ac is in this case equal to C(M ) = C3 . The set of states Sc is the space of probabilistic measures on M , i.e. a classical state Pi=3 is a tuple p = (p1 , p2 , p3 ) such that pi ≥ 0 and i=1 pi = 1. Because there is no nonzero derivation on C(M ) [25] we have no classical evolution. Let the two-dimensional Hilbert space H = C2 corresponds to the quantum system. Thus Aq = M2×2 and the states are described by positive matrices with the sum of their eigenvalues equal to 1. For any H = H ∗ ∈ Aq , δq = i[H, ·] is the generator of a uniformly continuous one-parameter group of automorphisms of Aq . The algebra of the total system is given by AT = C3 ⊗ M2×2 . It is convenient to note that it is the algebra of block diagonal matrices on the space C3 ⊗C2 . It means that every A˜ ∈ AT has the form A˜ = diag(A1 , A2 , A3 ) with Ai ∈ M2×2 . States are represented by block diagonal matrices ρ˜ = (ρ1 , ρ2 , ρ3 ), where ρi are positive Pi=3 2 × 2 matrices such that i=1 Trρi = 1. Each state ρ˜ projects on a quantum state P ρ= ρi and on a classical one p = (Trρ1 , Trρ2 , Trρ3 ). Let us now describe the operator L. Let P = P ∗ ∈ M2×2 , P = λ1 P1 +λ2 P2 , where P1 and P2 are orthogonal one-dimensional projectors. We may assume that λ1 < λ2 . Let us define shifts of M = {1, 2, 3} by    U1 for λ1 − λ2 ≤ α < 0 hα = U2 for 0 ≤ α < λ2 − λ1   id otherwise       010 001 100       U1 =  1 0 0  U2 =  0 1 0  id =  0 1 0  . 001 100 001 Finally, assume that f (i, α) does not depend on i and is equal to ( f (α) =

(λ2 − λ1 )− 2 1

for

λ1 ≤ α ≤ λ2

0 otherwise.

Then assumptions (a), (b) and (c) of the construction of the operator L in the previous section are fulfilled and we get V˜α (i) = f (λ1 − α)P1 + f (λ2 − α)P2 ˜ α∗ A˜W ˜ α (i) = V˜α A(h ˜ α (i))V˜α W

738

Z∞

R. OLKIEWICZ

˜ α (i) ˜ α∗ A˜W dαW

−∞

Z∞ ˜ α (i)) · [f (λ1 − α)P1 + f (λ2 − α)P2 ] dα[f (λ1 − α)P1 + f (λ2 − α)P2 ]A(h

= −∞

Z∞

Z∞ ˜ α (i))P1 + dα|f (λ1 − α)| P1 A(h

˜ α (i))P2 dα|f (λ2 − α)|2 P2 A(h

2

= −∞

−∞

˜ 1 (i))P1 + P2 A(U ˜ 2 (i))P2 . = P1 A(U So for 

0

A1

 A˜ =  0 0

A2 0

0



 0  A3

˜ equal to A1 , A2 , A3 ∈ M2×2 we have that L(A) 

P1 A2 P1 +P2 A3 P2 −A1  0 

0 P1 A1 P1 +P2 A2 P2 −A2

0 0

0

P1 A3 P1 +P2 A1 P2 −A3

0

  .

Thus L + 1 ⊗ δq is the symmetric generator of a counter [9, 11]. If we put δq = 0 then Tt = etL is given by Tt (diag(A1 , A2 , A3 )) 1 = diag e−t A1 + (1 − e−t )2 (P1 A1 P1 + P2 A1 P2 ) 2 +

1 (1 − e−2t )(P1 A2 P1 + P2 A3 P2 ) , 2

1 1 e−t A2 + (1−e−t)2 (P1 A2 P1 +P2 A2 P2 )+ (1 − e−2t )(P1 A1 P1 + P2 A2 P2 ) , 2 2 1 1 e−t A3 + (1−e−t )2 (P1 A3 P1 +P2 A3 P2 )+ (1−e−2t )(P1 A3 P1 +P2 A1 P2 ) . 2 2 The time evolution of a quantum observable A given by Tt (diag(A, A, A)) remains quantum and equal to e−t A + (1 − e−t )P (A),

P (A) = P1 AP1 + P2 AP2 .

This implies that T∞ (A) = P (A) which coincides with the quantum measurement projection postulate.

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

739

On the contrary to the reversible evolution which retains the centre of the algebra dissipative dynamics maps the classical algebra Ac ⊗ 1, i.e. the centre of AT into the whole algebra AT . ˜ ρ] = At last let us consider the time evolution of states, which is given by Tr[(Tt A)˜ ˜ Tr[A(Tt∗ ρ˜)]. Let us specify the initial state as ρ˜ = (ρ, 0, 0), where ρ ∈ Sq . Then 1 Tt∗ ρ˜ = diag e−t ρ + (1 − e−t )2 (P1 ρP1 + P2 ρP2 ) , 2 1 1 −2t −2t (1 − e )P1 ρP1 , (1 − e )P2 ρP2 . 2 2 Its projection onto classical states gives p1 (t) =

1 (1 + e−2t ) , 2

p2 (t) =

1 (1 − e−2t )TrP1 ρ , 2

p3 (t) =

1 (1 − e−2t )TrP2 ρ . 2

Because p2 (∞) = 12 TrP1 ρ and p3 (∞) = 12 TrP2 ρ so by analysing the states of classical subsystem for large t we obtain information about the expectation values of projectors P1 and P2 in the initial quantum state ρ. 3.2. The continuous case In this example we will consider the SQUID coupled to a damped classical harmonic oscilator. The radio frequency SQUID-tank magnetometer consists of an LC oscillatory circuit called the tank circuit which is coupled via a mutual inductance to a thick (compared to the magnetic penetration depth) superconducting ring, containing a weak link constriction. In this coupled system, the tank circuit acts as an external flux source for the SQUID ring, which induces a screening current in the ring. This screening current is coupled back to the tank circuit due to the mutual inductance [47, 48]. The equation ωR ˙ 1 ∂Ek 2 ¨ Xcl + ωR Xcl = F −µ Xcl + QR M ∂ftot describing the modification of the classical equation of motion due to the quantum system was obtained in [49]. Here the classical coordinate Xcl is the average of the collective coordinate Xcl = hXT i. For a superconducting weak link ring the above equation takes the simple one-dimensional form C φ¨ +

1 ∂Ek 1 ˙ φ + φ = Iin (t) − µ , R L ∂Φtot

where φ is the classical flux variable and Φtot is the total external flux. In the limit, the expectation value of the superconducting screening current flowing around the

740

R. OLKIEWICZ

ˆ − Φtot )/Λ is given by [50] ring Is = (Φ hk|Is |ki = −

∂Ek ∂Φtot

and the flux equation for the tank circuit becames C φ¨ +

1 1 ˙ ˆ − Φtot )/Λ|ki . φ + φ = Iin (t) + µhk|(Φ R L

We will try to obtain this equation within the mathematical framework introduced in Sec. 2. Let us begin by a description of the classical system. The phase space is M = R2 with the canonical symplectic form ω = dx ∧ dp. Let X(t) = p

∂ ∂ + (I(t) − p − x) , ∂x ∂p

where I(t) is a continuous function, be a vector field on M responsible for the time evolution. Instead of looking at the convergence of the multiplicative integral it is better to consider this case from the point of view of differential equations. The integral curves for X(t) are given by: x˙ = p, p˙ = I(t) − p − x which leads to x ¨ + x˙ + x = I(t). Equivalently it may be written as x˙ x 0 0 1 = + . −1 −1 p˙ p I(t) For each t0 ∈ R and (x0 , p0 ) ∈ R2 we have the unique global solution Zt x0 0 x0 ds , = R(t, t0 ) + R(t, s) gt I(s) p0 p0 t0

where R(t, s) is the resolvent of the corresponding homogenous differential equation [51]. Thus for any epoch t0 we have a strongly continuous one parameter family αt of automorphisms of C0 (R2 ) with αt0 =identity. Now let us consider the quantum object called the SQUID. Its Hamiltonian defined on H = L2 (R, dΦ) is given by ! ˆ ˆ − Φx )2 ˆ2 Φ (Φ Q + − ~ν cos 2π , H= 2C 2Λ Φ0 ˆ is the position operator, Φ0 = ˆ = −i d , Φ where Q dΦ magnetic flux. The evolution generator δq = i[H, ·]. The algebra of the total system is described by

h 2e

L∞ (R2 , dxdp) ⊗ L(L2 (R, dΦ)) . We construct the operator L in the following way: Let ˆ − Φtot , Pˆ = Φ

and Φx is the external

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

741

where Φtot = Φx + µx. Let the shift of (R2 , B, dxdp) be defined by hα (x, p) = (x, p−α). Then (Uα Ψ)(x, p) = Ψ(x, p+α). Finally, let f : R2 ×R → R, f (x, p, α) = f0 (α − x), where f0 is a continuous, bounded and square integrable function. Again the assumptions (a), (b) and (c) are fulfilled and we get Z∞ V˜α (x, p) =

f (x, p, λ − α)dE(λ) −∞

Z∞ f0 (λ − α − x)dE(λ)

= −∞

= f0 (Pˆ − x − α) , where dE(λ) is the spectral measure of the observable Pˆ . So Z∞ ˜ p − α)f0 (Pˆ − x − α) dαf0 (Pˆ − x − α)A(x,

˜ L(A)(x, p) = −∞



−

Z∞

 ˜ p) f02 (α)dα · A(x,

−∞

ρ). Moreover the total generator and similarly for L∗ (˜ ρ) = δc (t) ⊗ 1(˜ ρ) − δq (x)(˜ ρ) + L∗ (˜ ρ) , W∗ (˜ where δq (x) = i[H(x), ·] and H(x) is the SQUID Hamiltonian with Φx replaced by Φtot , is the SQUID-tank generator [11]. It is worth to emphasize that δc (t) acts on the space of states and so it is adjoint to X(t) i.e. δc (t) ϕ = −

∂ ∂ (p ϕ) − [(I(t) − p − x) ϕ] ∂x ∂p

= −p

∂ϕ ∂ϕ − (I(t) − p − x) + ϕ, ∂x ∂p

ϕ ∈ Sc .

R 1 Let us assume in addition that |α| 2 f0 is square integrable and αf02 (α)dα = 0. It is clear that Cc∞ (R2 )+ with L1 -norm equal 1 is dense in Sc and belongs to the domain of δc (t) for every t. Similarly we have Proposition 2.1. For every self-adjoint operator H on H there is a set D+ ⊂ Sq , which is dense in trace norm in Sq and such that δq (D+ ) = i[H, D+ ] ⊂ Tr(H)SA . Proof. Let H =

R

λdE(λ) and let H0 =

( D+ =

ρ: ρ=

∞ S

En H, En = E[−n, n]. Let

n=1 k X i=1

ci Pvi ,

k ∈ N,

k X i=1

) ci = 1

,

742

R. OLKIEWICZ

where Pvi is a one dimensional projector onto vi ∈ H0 . Because H0 is dense in H, D+ is dense in the set of all finite dimensional states and thus in Sq . Moreover Pk ∀ ρ ∈ D+ , ρ = i=1 ci Pvi there exists n ∈ N such that vi = En vi . So ρH =

k X

ci Pvi En H .

i=1

Becuse En H is bounded we have that ρH and Hρ are bounded too and belong to Tr(H). ˜ + is dense in ST (by D ˜ + we understand the ˜ + = Cc∞ (R2 )+ ⊗ D+ . Then D Let D set of finite convex combinations of ϕi ⊗ ρi with total trace equal to unity. Let us ˜ + and define choose ρ˜ ∈ D R ˜ ρ(x ⊗ 1)) = xTr(˜ x¯ = Tr(˜ ρ(x, p))dxdp R ˜ p¯ = Tr(˜ ρ(p ⊗ 1)) = pTr(˜ ρ(x, p))dxdp . From the evolution equation t ≥ 0 and so

d ˜t dt ρ

˜ H) ˜ SA for every = W∗ (˜ ρt ) we have that ρ˜˙ t ∈ Tr( Z xTr(ρ˜˙ (x, p))dxdp

x ¯˙ = Z p¯˙ =

pTr(ρ˜˙ (x, p))dxdp

are well defined. Moreover we may calculate that Z ρ)(x, p))dxdp x¯˙ = xTr(W∗ (˜ Z ρ))(x, p)]dxdp xTr[(δc (t) ⊗ 1(˜

=+

Z

Z −

ρ))(x, p)]dxdp + xTr[(1 ⊗ δq (˜

∂ ρ˜ ∂ ρ˜ (x, p) + x(p + x − I(t))Tr (x, p) ∂x ∂p Z + xTr˜ ρ(x, p)]dxdp − i xTr[H(x), ρ˜(x, p)]dxdp

Z =

xTr[L∗ (˜ ρ)(x, p)]dxdp

[(−xp)Tr

Z +

ρ(x, p − α)] dαdxdp xTr[f02 (Pˆ − x − α)˜ Z

− kf0 k2L2

xTr˜ ρ(x, p)dxdp .

Integrating by parts and using the fact that ρ˜ vanishes at infinity we can rewrite the first part as Z pTr[˜ ρ(x, p)]dxdp .

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

743

The second part vanishes because Tr[H(x), ρ˜(x, p)] = 0. By making the change of variables p → r + α we get the last term equal to 0 too. Thus x¯˙ = p¯. In the similar way we may obtain that Z ρ)(x, p)dxdp p¯˙ = pTr(W∗ (˜ Z = −¯ p−x ¯ + I(t) +

kf0 k2L2

¯ Tr[Pˆ ρ˜(x, p)]dxdp − kf0 k2L2 x

¯) , = −¯ p−x ¯ − I(t) + kf0 k2L2 (hPˆ iρ − x R where hPˆ iρ = TrPˆ ρ and ρ = ρ˜(x, p)dxdp. It may be written as one equation ˆ − Φtot )iρ . ¨¯ + x x = I(t) + kf0 k2L2 h(Φ x ¯˙ + (1 + kf0 k2L2 )¯ 3.3. Concluding remarks The purpose of this paper was to introduce a possible approach to the coupling between classical and quantum systems. We also discussed, using two examples, some properties of the proposed scheme. A quantum process is modified by the presence of a classical environment and modifies the classical evolution of its environment through some appropriate expectation value. It is obtained by an evolution of Liouville type of an algebra with a non-trivial centre. The elements of the centre stand for classical parameters. The construction of the dissipative generator, although it seems to be a bit artificial, possesses a straightforward justification. Let us assume that the function f (x, α) is replaced by the multiplication of a function on the classical phase space and the Dirac’s measure on R, i.e. f (x, α) = f0 (x)δ(α). Then the operator W : AT → AT acts in the following way: Z Z Z ˜ α x)dE(λ2 ) ˜ dαδ(λ1 − α)δ(λ2 − α)dE(λ1 )A(h W (A)(x) = f02 (x) Z = f02 (x)

˜ λ x)dE(λ) , dE(λ)A(h

which, of course, makes no sense. So introducing the function f and performing R double integration (firstly, to obtain the operator V˜α (x) = f (x, λ − α)dE(λ) and, secondly, to smear out the action along all α by Z∞ ˜ α x)V˜α (x)dα V˜α (x)A(h

˜ W (A)(x) = −∞

P ˜ i x)Pi to gives the meaning to it. It is a generalization of the expression i Pi A(h the continuous case. It is also clear that our scheme may be applied to the following, more general, situation. Instead of a quantum observable P let us take a locally compact and abelian group G with a strongly continuous unitary representation on H. Then by the N.A.G. theorem there exists a spectral measure dE(λ) on the group of ˆ [52]. Let the shifts on the phase space be labeled by elements of G ˆ characters G

744

R. OLKIEWICZ

ˆ Then for α ∈ G ˆ we have and the function f be defined on M × G. Z V˜α (x) = f (x, λ − α)dE(λ) ˆ G

Z

˜ α x)V˜α (x)dν(α) , V˜α (x)A(h

˜ W (A(x)) = ˆ G

ˆ Such a generalization may where dν(α) is the unique invariant Haar measure on G. be useful when a finite set of pairwise commuting quantum observables is considered. Finally, let us sketch some other applications of the presented formalism. At first let us consider the cloud chamber model [14, 15]. The quantum system is just a quantum particle moving in a space E = Rn according to the free Hamiltonian H. The classical system is given by a continuous medium such that every point of it can be in one of the two states: black and white. For simplicity let us restrict ourselves only to the class of all finite subsets of E, i.e. to those states of the medium which are everywhere white except in a finite number of points. Let us denote this class by C. As AT , the algebra of the total system, we take bounded functions defined on C with values in L(H), where H is the Hilbert space used for description of the quantum particle. The statistical states of the total system are described by a family P {ρX }X∈C (ρX is a positive, trace class operator on H) such that X TrρX = 1. For any a ∈ E let us take a nonnegative function ga : E → R+ , which describes a sensitivity of the detector located at the point a ∈ E. We assume moreover that for R any b ∈ E there is daga2 (b) = k. Let us also define a flip of X by a(X) = X ÷ a, where ÷ is the symmetric difference of sets. Then with suitable chosen operators V˜α we obtain the following Liouville evolution equation for statistical states Z daˆ ga ρa(X) gˆa − kρX . ρ˙ X = −i[H, ρX ] + Here gˆa denotes the operator on H given by the function ga . For a quantum state P ρ = X ρX we arrive at Z ga − kρ , ρ˙ = −i[H, ρ] + daˆ ga ρˆ what is exactly of the type discussed by Ghirardi, Rimini and Weber [53]. At last let us indicate a practical advantage of the proposed scheme. It is possible [10, 13, 14] to interpret the continuous time evolution of statistical states of the total system in terms of a piecewise deterministic random process with values in pure states. The process consists of random jumps accompanied by changes of a classical state interspersed by random periods of Schr¨odinger-type deterministic evolution. For the definition and basic properties of PD processes see [54]. For simplicity let us assume that the classical system is discrete. The time evolution of statistical states of the total system can be written in a component way as ρ˙ n = −i[Hn , ρn ] +

m X j=1

∗ Vnj ρj Vnj −

1 {Λn , ρn } , 2

SOME MATHEMATICAL PROBLEMS RELATED TO. . .

745

P ∗ where n = 1, 2, . . . , m describes the classical system and Λn = j Vjn Vjn . Such an equation encodes in a unique way the algorithm for generating admissible histories of individual systems, what generalizes that of Quantum Monte Carlo method [16]. Suppose that at time t0 the system is described by a quantum state vector ψ0 and a classical state n. Then choose a uniform random number p ∈ [0, 1], and proceed with the continuous time evolution by solving the modified Schr¨ odinger equation 1 ψ˙t = −iHn − Λn ψt 2 with the initial vector ψ0 until t = t1 , where t1 is determined by Z

t1

hψt , Λn ψt idt = p .

t0

Then jump. When jumping, change n → j with probability pn→j =

kVjn ψt1 k2 hψt1 , Λn ψt1 i

and change ψt1 → ψ1 =

Vjn ψt1 . kVjn ψt1 k

Repeat the steps replacing t0 , ψ0 , n with t1 , ψ1 , j. Using that algorithm we may calculate numbers that are needed in experiments and cannot be obtained in another way [55]. Let us also point out that contrary to the pure quantum case that process is unique. The proof of the uniqueness based on the analysis of the corresponding Markov–Feller generator is given in [56]. Acknowledgments I would like to thank Prof. A. Jadczyk for helpful discussions. I am also grateful to W. Hebisch for useful comments. Finally, I would like to thank the referees for comments improving the clarity of the paper. References [1] J. Bell, “Against measurement”, in Proc. NATO Advanced Study Institute, ed. A. I. Miller, NATO ASI Series B vol. 226, Plenum Press, New York, 1990. [2] J. Bell, “Towards an exact quantum mechanics”, in Themes in Contemporary Physics II. Essays in honour of Julian Schwinger’s 70th birthday, eds. S. Deser and R. J. Finkelstein, World Scientific, Singapore, 1989. [3] R. Haag, “Fundamental irreversibility and the concept of events”, Commun. Math. Phys. 132 (1990) 245–251. [4] R. Haag, “Events, histories, irreversibility”, in Quantum Control and Measurement, Proc. ISQM Satellite Workshop, eds. H. Ezawa and Y. Murayama, North Holland, Amsterdam, 1993.

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[5] S. Machida and M. Namiki, “Theory of measurement in quantum mechanics”, Progr. Theor. Phys. 63 (1980) 1457–1473 and 1833–1847. [6] H. Araki, “A Remark on Machida–Namiki theory of measurement”, Progr. Theor. Phys. 64 (1980) 719–730. [7] H. Araki, “A continuous superselection rule as a model of classical measuring apparatus in quantum mechanics”, in Fundamental Aspects of Quantum Theory, Proc. NATO Adv. Res. Workshop, eds. V. Gorini and A. Frigerio, NATO ASI Series B 144, Plenum Press, New York, 1986. [8] M. Ozawa, “Cat Paradox for C ∗ -dynamical systems”, Progr. Theor. Phys. 88 (1992) 1051–1064. [9] Ph. Blanchard and A. Jadczyk, “On the interaction between classical and quantum systems”, Phys. Lett. A 175 (1993) 157–164. [10] Ph. Blanchard and A. Jadczyk, “Strongly coupled quantum and classical systems and Zeno’s effect”, Phys. Lett. A 183 (1993) 272–276. [11] Ph. Blanchard and A. Jadczyk, “Classical and quantum intertwine”, in Proc. Symposium on Foundations of Modern Physics, ed. P. Bush, World Scientific, 1993. [12] Ph. Blanchard and A. Jadczyk, “From quantum probabilities to classical facts”, in Advances in Dynamical Systems and Quantum Physics, ed. R. Figario, World Scientific, 1994. [13] A. Jadczyk, “Topics in quantum dynamics”, in Proc. First Caribb. School of Math. and Theor. Phys., ed. R. Coquereaux, World Scientific, Singapore, 1995. [14] A. Jadczyk, “Particle tracks, events and quantum theory”, Progr. Theor. Phys. 93 (1995) 631–646. [15] A. Jadczyk, “On quantum jumps, events and spontaneous localization models”, Found. Phys. 25 (1995) 743–762. [16] Ph. Blanchard and A. Jadczyk, “Event-enhanced quantum theory and piecewise deterministic dynamics”, Ann. Phys. 4 (1995) 583–599. [17] N. P. Landsman, “Algebraic theory of superselection sectors and the measurement problem in quantum mechanics”, Int. J. Mod. Phys. A 6 (1991) 5349–5371. [18] D. E. Evans and J. T. Lewis, “Dilations of dynamical semigroups”, Commun. Math. Phys. 50 (1976) 219–227. [19] E. B. Davies, “Dilations of completely positive maps”, J. London Math. Soc 17 (2) (1978) 330–338. [20] R. Alicki and K. Lendi, “Quantum dynamical semigroup and applications”, Lect. Notes Phys. 286 (1987). [21] W. A. Majewski, “Dynamical semigroups in the algebraic formulation of statistical mechanics”, Fortschr. Phys. 32 (1984) 89–133. [22] E. B. Davies, One-parameter Semigroups, London-New York-San Francisco, Academic Press, 1980. [23] J. A. Goldstein, Semigroups of Operators and Applications, Oxford Univ. Press, 1985. [24] R. Nagel and U. Schlotterbeck (part A-1), W. Arenolt (part A-2), “One-parameter semigroups of positive operators”, ed. by R. Nagel (1986), Lect. Notes Math. 1184. [25] O. Bratteli, “Derivations, dissipations and groups actions on C ∗ -algebras”, Lect. Notes Math. 1229 (1986). [26] W. B. Arveson, “Subalgebras of C ∗ -algebras”, Acta Math. 123 (1969) 141–224. [27] M. D. Choi, “Positive linear maps on C ∗ -algebras”, Can. J. Math. 24 (1972) 520–529. [28] M. Takesaki, Theory of Operator Algebras 1, Springer-Verlag, New York Inc., 1979. [29] W. F. Stinespring, “Positive functions on C ∗ -algebras”, Proc. Amer. Math. Soc. 6 (1955) 211–216. [30] K. Kraus, “General state changes in quantum theory”, Ann. Phys. 64 (1971) 311–335. [31] “D. E. Evans, “Quantum dynamical semigroups, symmetry groups and locality”, Acta Appl. Math. 2 (1984) 333–352.

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[32] M. Choda, N. Moritani, T. Sano and H. Takehana, “Inner completely positive maps on von Neumann algebras”, Math. Japonica 35 (1990) 331–334. [33] J. A. Mingo, “The correspondence associated to an inner completely positive map”, Math. Ann. 284 (1989) 121–135. [34] D. E. Evans and H. Hanche-Olsen, “The generators of positive semigroups”, J. Funct. Anal. 32 (1979) 207–212. [35] G. Lindblad, “On the generators of quantum dynamical semigroups”, Comm. Math. Phys. 48 (1976) 119–130. [36] O. Bratteli and D. W. Robinson, “Positive C0 -semigroups on C ∗ -algebras”, Math. Scand. 48 (1981) 259–274. [37] V. Gorini, A. Kossakowski and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems”, J. Math. Phys. 57 (1976) 821–825. [38] E. Christensen, “Generators of semigroups of completely positive maps”, Commun. Math. Phys. 62 (1978) 167–171. [39] E. Christensen and D. E. Evans, “Cohomology of operator algebras and quantum dynamical semigroups”, J. London Math. Soc. 20 (1979) 358–368. [40] E. B. Davies, “Generators of dynamical semigroups”, J. Funct. Anal. 34 (1979) 421– 431. [41] A. Chebotariev, “Sufficient conditions for dissipative dynamical semigroups to be conservative”, Theor. Math. Phys. 80 (1989) 192–211 (in Russian). [42] A. C. Holevo, “There is non-standard dynamical semigroup on L(H)”, UMN 51 (1996) 225–226 (in Russian). [43] W. I. Arnold, Mathematical Methods in Classical Mechanics, Moscow, 1974 (in Russian). [44] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis Vol. 1, Springer-Verlag, Berlin, 1963. [45] K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965. [46] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras Vol. 2, Academic Press, Inc., 1986. [47] F. London, Superfluids Vol. 1, Macroscopic Theory of Superconductivity, Dover, New York, 1961, Section E. [48] B. D. Josephson, Phys. Lett. 1 (1962) 251. [49] T. P. Spiller, T. D. Clark, R. J. Prance and H. Prance, “The adiabatic monitoring of quantum objects”, Phys. Lett. A 170 (1992) 273–279. [50] J. F. Ralph, T. P. Spiller, T. D. Clark, H. Prance, R. J. Prance and A. J. Clippingdale, “Non-linear behaviour in the rf-SQUID magnetometer”, Physica D 63 (1993) 191–201. [51] L. Schwartz, Analyse Math´ematique, Hermann, Paris, 1967. [52] K. Maurin, General Eigenfunction Expansions and Unitary Representation of Topological Groups, PWN Warszawa, 1968. [53] G. C. Ghirardi, A. Rimini and T. Weber, “An attempt to a unified description of microscopic and macroscopic systems”, in Fundamental Aspects of Quantum Theory, Proc. NATO Adv. res. Workshop, eds. V. Gorini and A. Frigerio, NATO ASI Series B 144, Plenum Press, New York, 1986. [54] M. H. A. Davies, “Piecewise deterministic markov processes. A general class of nondiffusion stochastic models”, J. R. Statis. Soc. B 46 (1984) 353–388. [55] K. Møller, “Density matrices and the quantum Monte Carlo method in quantum optics”, preprint, Aarhus, IFA-94/08. [56] A. Jadczyk, G. Kondrat and R. Olkiewicz, “On Uniqueness of the Jump Process in Event Enhanced Quantum Theory”, J. Phys. A: Math. Gen. 30 (1997) 1863–1880.

QUASIVECTORS AND TOMITA TAKESAKI THEORY FOR OPERATOR ALGEBRAS ON Π1 -SPACES VICTOR S. SHULMAN∗ Dept. of Math., Vologda Polytechnic Institute 15 Lenina Street Vologda 160000 Russia E-mail : [email protected] Received 4 April 1996 Mathematics Subject Classification: 46K15, 47B50, 46L10 We consider operator algebras, which are symmetric with respect to an indefinite scalar product. It is shown, that in the case when the rank of indefiniteness is equal to 1 there exists a working modular theory, and in particular a precise analogue of the Fundamental Tomita’s Theorem holds: For any weakly closed J-symmetric operator algebra J with identity on a Π1 -space H which has a cyclic and separating vector, there is an antilinear J-involution j : H → H such that jJ j = J 0 . The paper also contains a full proof of the Double Commutant Theorem for J-symmetric operator algebras on Π1 -spaces. Keywords and phrases :J -symmetric operator algebras on Pontryagin Π1 -spaces, quasivectors on operator ∗ -algebras, Tomita–Takesaki theory for quasivectors, Double Commutant Theorem.

1. Introduction The problem of existence of modular automorphisms and antiinvolutions for operator algebras, symmetric with respect to an indefinite scalar product, arises in connection with some models of the quantum field theory (see [1, 2] and references therein). However attempts to develop the modular theory for scalar products with finite rank of indefiniteness by analogy with the classical approach in [12] met with obstacles, which originated in the complicated spectral behaviour of modular operators. As a result, in order to work indefinite-selfadjoint observables one should impose some strong restrictions [2]. The aim of the present work is to show that, at least in the case of rank-one indefinite forms, the obstructions can be taken over and the modular theory can be produced. In particular, in this case the exact analogue of the fundamental theorem of Tomita holds: ∗ The

author is grateful to the University of North London for inviting him to Britain for a threemonth research visit.

749 Reviews in Mathematical Physics, Vol. 9, No. 6 (1997) 749–783 c World Scientific Publishing Company

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Theorem 1.1. For any weakly closed J-symmetric operator algebra J with identity on a Π1 space H which has a cyclic and separating vector, there is an antilinear J-involution j : H → H such that jJ j = J 0 . An indefinite scalar product with a finite rank K of indefiniteness turns a linear space into a Pontrjagin’s space ΠK . In other words, ΠK -spaces are indefinite metric spaces which decompose into the direct sum of a Hilbert space and a K-dimensional “negative” space. For operator algebraists, ΠK -spaces are attractive primarily because of the fact that the theory of J-symmetric operator algebras on these spaces combines, in a natural way, two well-developed theories: C ∗ -algebras and operator algebras on finite-dimensional spaces. From the first works of Naimark (see, for example, [8]) and Ismagilov [4] it became clear that quite apart from deep intrinsic problems this theory sheds new light on and raises interesting new questions about self-adjoint operator algebras on Hilbert spaces. To see the importance of ΠK -spaces for physical applications it suffices to take into account, that all irreducible representations of Lorentz group can be realised as J-unitary (preserving indefinite metrics) representations on these spaces. Our approach to modular theory is based on the classification (division into several types and construction of canonical models for every type) of J-symmetric operator algebras on Π1 -spaces, obtained in [11]. For this purpose, essential development and refinement were done of the main technical tool in [11] — the theory of quasivectors of operator ∗ -algebras on Hilbert spaces. In this way we obtain the precise description of commutants of canonical models and give the full proof of the Double Commutant Theorem for J-symmetric operator agebras on Π1 -spaces, outlined in [11]. In what follows only generic algebras are considered, that is, the algebras which have at least one neutral invariant subspace. The case of algebras without such invariant subspaces is quite easy — it was shown in [6], that all of them (if weakly closed) are similar to W ∗ -algebras and, therefore Theorem 1.1 clearly holds for them. I am deeply indebted to Edward Kissin for inspiration to write this paper and help in preparing it. 2. Weakly ∗ -Closed Quasivectors on Operator ∗ -Algebras Let H be a Hilbert space and B(H) be the algebra of all bounded operators on H. For A ∈ B(H), M ⊂ B(H) and K ⊆ H, we write Ker A = {x ∈ H : Ax = 0} ,

Ker M =

\

Ker A ,

A∈M

AK = linear hull of {Ax : x ∈ K} and MK = linear hull of

[

AK .

A∈M

By M0 we denote the commutant of M : M0 = {T ∈ B(H) : T A = AT , for A ∈ M}. The closure of a set is denoted by the bar; if the topology is not evident,

QUASIVECTORS AND TOMITA–TAKESAKI THEORY FOR

...

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W

we add the corresponding symbol (for example, M means the closure of M in the weak operator topology). We consider the uniform (u) and the weak (w) topologies on Hilbert spaces and the uniform (u), weak (w) and strong (s) operator topologies on operator subspaces. The fact that, for topologies with the same dual space, the closures of a convex set coincide is used systematically and without further comments. ˜ denotes the anti-Hilbert space — antipod of H, that is, the Hilbert The symbol H space with the same elements as H in which the scalar product and multiplication by a scalar are defined as follows: ˜. (x, y)H˜ = (y, x) and λ · x = λx, for λ ∈ C and x ∈ H ˜ ⊕ H is a Hilbert space with the usual scalar product. In the The orthogonal sum H ˜ ⊕ H) we consider the product direct products B(H) × H and D(H) = B(H) × (H topologies u × u, u × w, w × w and so on. We write σ for the topology w × w and τ for the topology u × w. Let U be a subalgebra in B(H). For a linear map q : U → H, one can consider its graph G(q) = {(A, q(A)) : A ∈ U} in B(H) × H and, if U is a ∗ -algebra, its ∗ -graph Γ(q) = {(A, q(A∗ ) ⊕ q(A) : A ∈ U} in D(H). if G(q) is closed in the u × u or σ topology, the map q is called closed or weakly closed, respectively. If U is a ∗ -algebra and Γ(q) is closed in the u × u or σ topology, we say that q is ∗ -closed or, respectively, weakly ∗ -closed. Since the topologies w and s on B(H) determine the same dual and the topologies ˜ ⊕ H determine the same dual and since Γ(q) is convex, one can u and w on H formulate the ∗ -closedness of q in several other ways. Definition 2.1. A linear mapping q : U → H is a quasivector if q(AB) = Aq(B), for A, B ∈ U . It is bounded if there is C > 0 such that kq(A)k ≤ CkAk, for A ∈ U. We are interested in the structure of ∗ -closed and weakly ∗ -closed quasivectors of -algebras. Clearly, a quasivector is ∗ -closed if and only if U is complete with respect to the norm kAkq = kAk + kq(A)k + kq(A∗ )k . ∗

Every element x ∈ H defines a bounded quasivector qx by the formula: qx (A) = Ax, for A ∈ U . If U has identity 1, any quasivector on U is bounded, since q(A) = Aq(1). If q is a bounded ∗ -closed quasivector then the norms k · k and k · kq are equivalent and U is an operator C ∗ -algebra. The name “quasivector” is justified by the following result.

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Proposition 2.2. If q is a bounded quasivector from U into H then there is x ∈ H such that q = qx . Proof. Let {Eλ }λ∈∧ be a bounded approximate identity in U. The net {q(Eλ )}λ∈∧ is bounded and has a cluster point x in H in the weak topology. Since q is bounded, q(A) is a cluster point of the net {q(AEλ )}λ∈∧ in the norm topology, for any A ∈ U. Therefore, for y ∈ H, (q(A), y) = lim(q(AEλ ), y) = lim(q(Eλ ), A∗ y) = (x, A∗ y) = (Ax, Y ) . λ

λ

Thus q(A) = Ax.

Lemma 2.3. If q is a ∗ -closed quasivector from U into H then q(A) ∈ AH, for every A = A∗ ∈ U, and there is a bounded sequence {En }∞ n=1 for selfadjoint operators in U commuting with A such that En q(A) uniformly converge to q(A). Proof. Let A be the operator ∗ -algebra of all polynomials of A with constant term zero. Then A has a bounded selfadjoint approximate identity {En }. Hence kq(En A)k = kEn q(A)k ≤ kEn k kq(A)k is bounded and the sequence {q(En A)}∞ n=1 has a cluster point x in H in the weak topology. Since A and En commute, w q(En A) = q(AEn ) = Aq(En ) ∈ AH. Hence x ∈ AH and, thus x ∈ AH. We also have (En A)∗ = AEn = En A, so that (A, x ⊕ x) is a cluster point of the set {(En A, q((En A)∗ ) ⊕ q(En A))}∞ n=1 in the topology τ on DH). Since Γ(q) is closed with respect to τ , (A, x ⊕ x) ∈ Γ(q). Thus q(A) = x ∈ AH and, therefore, En q(A) uniformly converge to q(A). From Lemma 2.3 it follows that q(U) ⊆ UH . Lemma 2.4. Let q be a ∗ -closed (i) If U has an operator with a bounded and U is an operator (ii) If Q is the orthoprojection on

(2.1)

quasivector from U into H. bounded inverse then 1H ∈ U, so that q is C ∗ -algebra. UH and q is unbounded then Q ∈ / U.

Proof. Let an operator B in U have a bounded inverse. The operator T = B ∗ B is selfadjoint and has a bounded inverse. Hence there are real polynomials Pn (t), with Pn (0) = 0, such that Tn = Pn (T ) converge to T −1 with respect to the norm in B(H). Then T Tn2 T converge to 1H and q(T Tn2 T ) = T Tn2q(T ) converge to T −1 q(T ). Thus (T Tn2 T, q((T Tn2T )∗ ) ⊕ q(T Tn2 T )) converge to (1H , T −1 q(T ) ⊕ T −1 q(T )) in D(H). Since Γ(q) is closed, 1H ∈ U. Therefore q is bounded and U is an operator C ∗ -algebra. Part (i) is proved. If 1H ∈ U, there is B ∈ U such that k1H − Bk < 1. Hence B has a bounded inverse and, by (i), 1H ∈ U and q is bounded. Since q(U) ⊆ UH, we consider U as an operator ∗ -algebra on UH, q as a quasivector from U into UH and Q as the / U. identity operator on UH. Therefore it follows that Q ∈

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We shall turn now to the study of the structure of weakly ∗ -closed quasivectors. They are, clearly, ∗ -closed. We consider some examples of weakly ∗ -closed quasivectors and, as we shall see later, these examples are, in fact, universal. Example 2.5. 1) Let R be a positive bounded operator on a Hilbert space H such that Ker R = {0} and let v ∈ H. Let A be a W ∗ -algebra on H. Set U = {A ∈ A : AR = RA, Av ∈ RH and A∗ v ∈ RH} and q(A) = R−1 Av, for A ∈ U . Then U is an operator ∗ -algebra and q is a quasivector from U into H. Let a net {Aλ }λ∈Λ of operators from U converge to A in the weak operator topology, let q(A∗λ ) = R−1 A∗λ v converge to x and q(Aλ ) = R−1 Aλ v converge to y with respect to the norm on H. Then A commutes with R and, for any z ∈ H, (Av, z) = lim(Aλ v, z) = lim(Rq(Aλ ), z) = (Ry, z) . Hence Av = Ry. Similarly, A∗ v = Rz. Thus A ∈ U and q is weakly ∗ -closed. P 2) Let H = ∞ k=1 ⊕Hk be the orthogonal sum of Hilbert spaces Hk , let Pk be the orthoprojections on Hk and let vk ∈ Hk . Let A be a W ∗ -algebra on H. Set ) ( ∞ ∞ X X 2 ∗ 2 kAvk k < ∞ and kA vk k < ∞ U = A ∈ A : APk = Pk A, for all k, k=1

and q(A) =

∞ X

k=1

⊕Avk , for A ∈ U .

k=1

Then U is an operator ∗ -algebra and q is a quasivector from U into H. If P∞ P∞ 2 k=1 kvk k < ∞ then v = k=1 ⊕vk ∈ H and q is bounded. If q is not bounded, it is ∗ -closed and even weakly ∗ -closed. Indeed, let a net {Aλ }λ∈Λ of operators from U converge to A in the weak operator topology, q(A∗λ ) converge to x and q(Aλ ) converge to y with respect to the norm on H. Then for any k, APk = Pk A and the elements Pk q(A∗λ ) and Pk q(Aλ ) converge uniformly to Pk x and Pk y, respectively. Since Pk q(A∗λ ) = A∗λ vk and Pk q(Aλ ) = Aλ vk , and since A∗λ converge to A∗ in the weak operator topology, we obtain that A∗ vk = Pk x and Avk = Pk y. Therefore ∞ X

kAvk k2 =

k=1 ∞ X k=1

kA∗ vk k2 =

∞ X

kpk yk2 = ky||2 < ∞ and

k=1 ∞ X

kPk xk2 = kx||2 < ∞ .

k=1

Thus A ∈ U, x = q(A∗ ) and y = q(A), so that q is weakly ∗ -closed.

The first goal of this section is to prove a “double commutant” theorem for weakly ∗ -closed quasivectors. To formulate it we need to introduce some additional notations. Let q be a ∗ -closed quasivector from U into H. Set K = UH. Then

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K ⊥ = Ker U. We define now a quasivector from a ∗ -subalgebra of the commutant U’ of U into H. For T ∈ U 0 , qT (A) = T q(A), for A ∈ U , is a quasivector from U into H, since, for A, B ∈ U, qT (AB) = Tq (AB) = T Aq(B) = AT q(B) = AqT (B) . If qT is bounded then, by Proposition 2.2, there is y ∈ H such that qT (A) = Ay, for A ∈ U . If K ⊥ 6= {0}, the element y is not unique and defined up to a summand from k ⊥ . However, there is a unique such element in K which we denote by q 0 (T ). By Uq0 we denote the set of all operators T ∈ U 0 such that both quasivectors qT and qT ∗ are bounded. It is clear that q 0 is a linear mapping from Uq0 into K and that (2.2) T q(A) = qT (A) = Aq 0 (T ), for A ∈ U and T ∈ Uq0 . Lemma 2.6. If q is ∗ -closed then Uq0 is a ∗ -subalgebra of U 0 and the mapping q is a weakly ∗ -closed quasivector from Uq0 into K. 0

Proof. Clearly, Uq0 is a linear operator ∗ -family. Let S, T ∈ Uq0 . The subspace K is invariant for all operators from U 0 . Hence Sq 0 (T ) ∈ K. For A ∈ U, qST (A) = ST q(A) = SAq 0 (T ) = ASq 0 (T ) . Therefore qST is a bounded quasivector. Similarly, q(ST )∗ is bounded, so that ST ∈ Uq0 . Hence Uq0 a ∗ -subalgebra of U 0 and q 0 (ST ) = Sq 0 (T ). Thus q 0 is a quasivector from Uq0 into K. Let a net {Tλ }λ∈Λ of operators from Uq0 converge to a bounded operator T in the weak operator topology and let q 0 (Tλ ) and q 0 (Tλ∗ ) converge, respectively, to x and y in K in the weak topology. Since U 0 is a W ∗ -algebra, T ∈ U 0 . For every z ∈ K and A ∈ U, it follows from (2.2) that (T q(A), z) = lim(Tλ q(A), z) = lim(Aq 0 (Tλ ), z) = (Ax, z) , (T ∗ q(A), z) = lim(Tλ∗ q(A), z) = lim(Aq 0 (Tλ∗ ), z) = (Ay, z) . Therefore T q(A) = Ax and T ∗ q(A) = Ay, for A ∈ U, so that the quasivectors qT and qT ∗ are bounded. Thus T ∈ Uq0 . Repeating the construction before Lemma 2.6, we denote by Uq00 = (Uq0 )0q0 the set of all operators S in U 00 for which the quasivectors qS0 = Sq 0 (·) and qS0 ∗ = S ∗ q 0 (·) are bounded and by q 00 (S) we denote a unique element in Uq0 H such

QUASIVECTORS AND TOMITA–TAKESAKI THEORY FOR

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that Sq 0 (T ) = T q 00 (S), for all T ∈ Uq0 . By Lemma 2.6, Uq00 is a ∗ -subalgebra of U 00 , q 00 is a weakly ∗ -closed quasivector from Uq00 into H and Sq 0 (T ) = T q 00 (S), for T ∈ Uq0 and S ∈ Uq00 .

(2.3)

We shall now state the “double commutant” theorem for quasivectors. Theorem 2.7. Let q be a weakly ∗ -closed quasivector from an operator -algebra U into H and let Q be the orthoprojection on Ker U. Then Uq0 is weakly dense in U 0 , Uq00 = U + {CQ}, q 00 (Q) = 0 and q 00 |U = q. ∗

From (2.2) and (2.3) we have that U + {CQ} ⊆ Uq00 and q 00 |U = q. Thus to finish the proof of Theorem 2.7 it suffices to show that Uq0 is weakly dense in U 0 and Uq00 ⊆ U + {CQ} .

(2.4)

Let U1 and U2 be operator ∗ -algebras on Hilbert spaces H1 and H2 and let U1 ⊕U2 be their orthogonal sum on H1 ⊕ H2 . Let qi , i = 1, 2, be quasivectors from Ui into Hi and q = q1 ⊕ q2 be the quasivector from U1 ⊕ U2 into H1 ⊕ H2 q(A1 ⊕ A2 ) = q1 (A1 ) ⊕ q2 (A2 ), for A1 ∈ U1 and A2 ∈ U2 . Then q is ∗ -closed (resp. weakly ∗ -closed) if and only if q1 and q2 are ∗ -closed (resp. weakly ∗ -closed). The proof of the next lemma is evident. Lemma 2.8. If U1 H1 = H1 then (U1 ⊕ U2 )0q = (U1 )0q1 ⊕ (U2 )0q2 and q 0 = q10 ⊕ q20 . If, in particular, q2 = 0 then (U1 ⊕ U2 )0q = (U1 )0q1 ⊕ (U2 )0 and q 0 = q10 ⊕ 0. Let U be the restriction of U to K = UH and q be the restriction of q to U considered as a quasivector from U into K. Then U = U ⊕ {0} is the orthogonal sum of the restrictions of U to K and K ⊥ and q = q ⊕ 0. Hence, by Lemma 2.8, Uq0 = U0q ⊕ B(K ⊥ ), Uq00 = U00q ⊕ {C1K ⊥ }, q 0 = q0 ⊕ 0 and q 00 = q00 ⊕ 0 . From this and from (2.4) we conclude that to prove Theorem 2.7 it suffices to consider the case when UH = H. Before proceeding further with the proof of Theorem 2.7, we have to obtain some auxillary results. We shall start with the “double commutant” theorem for A-submodules of the direct product A × H, where A is a W ∗ -algebra on H. Let L be a left A-submodule of A × H, that is, B(A, x) = (BA, Bx) ∈ L, for (A, x) ∈ L and B ∈ A. By L0 we denote the left A0 -submodule of A0 × H which consists of all pairs (B, y), where B ∈ A0 and y ∈ H, such that Bx = Ay, for all (A, x) ∈ L .

(2.5)

It is clear that L0 is closed in the σ-topology. Similarly, L00 = (L0 )0 is a left A00 -submodule of A00 × H closed in the σ-topology.

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Theorem 2.9. Let A be a W ∗ -algebra. If a left A-submodule L of A × H is closed in the σ-topology then L00 = L. Proof. Let H(n+1) be the orthogonal sum of n + 1 copies of H and let zi ∈ H for i = 1, . . . , n. Since L is a left A-submodule of A × H, L = {x ⊕ Az1 ⊕ . . . ⊕ Azn : (A, x) ∈ L} is a linear manifold of H(n+1) invariant for the W ∗ -algebra A(n+1) . Let Q be the orthoprojection on the subspace L⊥ and Q = (Qij ), 0 ≤ i, j ≤ n, be its block-matrix form. Then Q commutes with algebra A(n+1) , so that every element Qij commutes with A. Hence Qij ∈ A0 . Since Q(x ⊕ Az1 ⊕ . . . ⊕ Azn ) = 0, for (A, x) ∈ L, we have that 0 = Qi0 x +

n X

Qij Azj = Qi0 x + A

j=1

n X

Qij zj , for i = 0, . . . , n .

j=1

It follows that the pairs (Qi0 , yi ) ∈ L0 , where yi = − Let (B, u) ∈ (L0 )0 . Then B ∈ A00 and 0 = Qi0 u − Byi = Qi0 u +

n X

BQij zj = Qi0 u +

j=1

n X

Pn j=1

Qij zj , i = 0, . . . , n .

Qij Bzj , for all i = 0, . . . , n .

j=1

Hence Q(u ⊕ Bz1 ⊕ . . . ⊕ Bzn ) = 0, so that u ⊕ Bz1 ⊕ . . . ⊕ Bzn belongs to the closure of L. Therefore, for every ε > 0, there is (A, x) ∈ L such that ku − xk, < ε and kBzi − Azi k < ε, for i = 1, . . . , n . Hence (B, u) belongs to the closure of L in the s-topology. Since L is closed in the σ-topology, it is closed in the s-topology, so that (B, u) ∈ L. We go back now to a weakly ∗ -closed quasivector q. Its graph G(q) does not need to be closed in the σ-topology in B(H) × H. In other words, q is not, generally speaking, a closed map with respect to the weak operator topology on B(H) and σ the weak topology on H. It appears that q is weakly closable: the closure G(q) of G(q) in the σ-topology is the graph of a linear map. Lemma 2.10. (i) The quasivector q is weakly closable. W (ii) The weak closure p of q is a quasivector from a left ideal J of U into W K = UH and p(AB) = Ap(B), for A ∈ U and B ∈ J. (iii) The operator ∗ -algebra U is a right ideal of J. (iv) J ∗ ∩ J = U. σ

Proof. To prove that G(q) is the graph of a linear map it suffices to show that σ (0, x) ∈ G(q) implies x = 0. σ Let (B, x) ∈ G(q) . There is a net {(Bλ , q(Bλ ))}λ∈Λ in G(q) which converge to (B, x) in the σ-topology. Then, for any A ∈ U, the operators ABλ converge

QUASIVECTORS AND TOMITA–TAKESAKI THEORY FOR

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to AB in the weak operator topology, q(ABλ ) = Aq(Bλ ) converge to Ax and q((ABλ )∗ ) = Bλ∗ q(A∗ ) converge to B ∗ q(A∗ ) in the weak topology on H. Since q is weakly ∗ -closed, (AB, B ∗ q(A∗ ) ⊕ Ax) ∈ Γ(q), so that AB ∈ U and Ax = q(AB) .

(2.6)

σ

If (0, x) ∈ G(q) then, by (2.6), Ax = 0 for any A ∈ U, so that x ∈ Ker U. On the other hand, it follows from (2.1) that x ∈ UH. Since UH∩ Ker U = {0}, we obtain that x = 0. Part (i) is proved. Let p be the closure of q and J be the domain of p. From (2.6) it follows that U is a right ideal of J and part (iii) holds. We also have Ap(B) = q(AB), for A ∈ U and B ∈ J . W

Let {Aλ }λ∈Λ be a directed set of elements from U which converge to A ∈ U in the weak operator topology. For B ∈ J, it follows from the above argument that Aλ B ∈ U and q(Aλ B) = Aλ p(B) converge to Ap(B). Since the operators Aλ B converge to AB in the weak operator topology, we have that AB ∈ J and that p(AB) = lim q(Aλ B) = Ap(B) . Part (ii) is proved. The inclusion U ⊆ J ∩ J ∗ is evident. To prove part (iv) it suffices to show that A ∈ U, for any A = A∗ ∈ J. There is a net {Aλ }λ∈Λ of elements from U which converges to A in the strong operator topology and q(Aλ ) converge to p(A) with respect to the norm. Set C = max(kyk, kp(A)k, kAxk, kAyk, 1), for x, y ∈ H. For 1 > ε > 0, let λε ∈ Λ be such that, for λε ≤ λ, ε ε ε , k(A − Aλ )yk ≤ and kp(A) − q(Aλ )k ≤ . k(A − Aλ )xk ≤ 3C 3C 3C Then kAλ xk ≤ 2C, kAλ yk ≤ 2C and |(A2 − A∗λ Aλ )x, y)| = |(Ax, Ay) − (Aλ x, Aλ y)| ≤ |((A − Aλ )x, Ay)| + |(Aλ x, (A − Aλ )y)| ≤ k(A − Aλ )xk kAyk + kAλ xk k(A − Aλ )yk ≤

ε ε + 2C 3 3C

=ε and |(Ap(A) − q(A∗λ Aλ ), y)| ≤ |(Ap(A) − A∗λ p(A), y| + |(A∗λ p(A) − A∗λ q(Aλ ), y)| ≤ kq(A)k k(A − Aλ )yk + kp(A) − q(Aλ )k kAλ yk ε ε + 2C = ε. 3 3C Hence A∗λ Aλ converge to A2 in the weak operator topology and q(A∗λ Aλ ) converge to AP (A) in the weak topology in H. Since q is weakly ∗ -closed and A∗λ Aλ are selfadjoint, ≤

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A2 ∈ U and q(A2 ) = Ap(A) . Combining this result with parts (ii) and (iii), we obtain that An ∈ U and q(An ) = An−1 p(A), for all n ≥ 2. Hence R(A)A ∈ U and q(R(A)A) = R(A)p(A) for any polynomial R with constant term 0. The algebra P(A) of all polynomials of A with constant term zero has a bounded countable approximate identity {Rk (A)}∞ k=1 of selfadjoint elements. Hence the operators Rk (A)A in U uniformly converge to A and the set R = {Rk (A)p(A)}∞ k=1 in H is bounded. Since every ball in H is compact in the weak topology, the set R has a ¯ ⊕ H) is a cluster point of the cluster point x. Hence (A, x ⊕ x) in B(H) × (H ∞ subset {(Rk (A)A, q(Rk (A)A) ⊕ q(Rk (A)A))}k=1 of Γ(q) with respect to the topology generated by the weak operator topology on B(H) and the weak topology on H. Since Γ(q) is closed in this topology, (A, x ⊕ x) ∈ Γ(q). Thus A ∈ U. Having proved that q is closable, we can bow apply to its closure p the same procedure as was used above for q, to construct quasivectors p0 and p00 from some subalgebras of U 0 and U 00 , respectively, into K. For any T ∈ U 0 , pT (A) = T p(A), for A ∈ J , is a quasivector from J into K. We denote by Jp0 the set of all operators T in U 0 such that T p(A) = Ax, for some x ∈ K and all A ∈ J . (2.7) Since U ⊆ J, we have that, for any T ∈ Jp0 , there is a unique x in K which satisfies (2.7) and which we denote by p0 (T ). It is evident that Jp0 is a left ideal of U 0 , that p0 is a quasivector from Jp0 into K and p0 (ST ) = Sp0 (T ), for S ∈ U 0 and T ∈ Jp0 .

(2.8)

From (2.7) it follows that T p(A) = Ap0 (T ), for A ∈ J and T ∈ Jp0 .

(2.9)

Lemma 2.11. (i) Ker J 0 p = {0} . (ii) q 0 = p0 |Uq0 , Uq0 = Jp0 ∩ (Jp0 )∗ and Ker Uq0 = {0}. W

Proof. The graph G(p) of p in U × H is closed in σ-topology. By Lemma 2.6, W W W it is a left U -submodule of U × H, since, for A ∈ U and (B, p(B)) ∈ G(p), AB ∈ J and A(B, p(B)) = (AB, Ap(B)) = (AB, p(AB)) . Hence, by Theorem 2.9, G(p)00 = G(p). Using (2.5) and (2.9), we obtain G(p)0 = {(T, p0 (T ) + z) : T ∈ Jp0 andz ∈ Ker U} .

(2.10)

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Suppose that x ∈ Ker Jp0 . Then the pair (0H , x), where 0H is the zero operator on H, belongs to G(p)00 , since 0H (p0 (T ) + z) = T x = 0, for all T ∈ Jp0 . Hence (0H , x) ∈ G(p) so that x = 0. Thus Ker Jp0 = {0} and part (i) is proved. We observe now that Uq0 ⊆ Jp0 . To prove this we assume that B ∈ Uq0 . Then B ∈ U 0 and Bq(A) = Aq 0 (B), for all A ∈ U . Let C ∈ J. Then there is a net {(Cλ , q(Cλ )}λ∈Λ of elements of U × H which converges to (C, p(C)) in the σ-topology. For every z ∈ H, (Bp(C), z) = lim(Bq(Cλ ), z) = lim(Cλ q 0 (B), z) = (Cq 0 (B), z) . Therefore Bp(C) = Cq 0 (B). Since q 0 (B) ∈ K, it follows from (2.7) and (2.9) that B ∈ Jp0 and p0 (B) = q 0 (B). Hence Uq0 ⊆ Jp0 and q 0 = p0 |Uq0 . Since Uq0 is a ∗ -algebra, Uq0 ⊆ Jp0 ∩ (Jp0 )∗ . On the other hand, if A ∈ U and T = T ∗ ∈ J p then p(A) = q(A) and, by (2.9), T q(A) = T p(A) = Ap0 (T ) . Since T is selfadjoint, T ∗ q(A) = T q(A) = Ap0 (T ). Hence T ∈ Uq0 . Thus Uq0 = Jp0 ∩ (Jp0 )∗ . Since Jp0 is a left ideal of U 0 and since Uq0 contains all selfadjoint elements of we have A∗ A ∈ Uq0 , for any A ∈ Jp0 . If 0 6= x ∈ Ker Uq0 then A∗ Ax = 0, so that kAxk2 = (x, A∗ Ax) = 0 and x ∈ Ker Jp0 . Form (i) we conclude that Ker Uq0 = {0}. Jp0 ,

We denote by Jp00 = (Jp0 )0p0 the set of all operators S in U 00 such that Sp0 (T ) = T x, for some x ∈ H and all T ∈ Jp0 .

(2.11)

Since Ker Jp0 = {0}, there is a unique x, for any S ∈ Jp00 , which satisfies (2.11). We denote it by p00 (S). In fact, p00 (S) ∈ K and it is evident that Jp00 is a left ideal of U 00 , that p00 is a quasivector from Jp00 into K and Sp0 (T ) = T p00 (S), for T ∈ Jp0 and S ∈ Jp00 .

(2.12)

Corollary 2.12. Jp00 = J + {CQ}, where Q is the orthoprojection on Ker U.

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V. S. SHULMAN

Proof. By (2.9), J ⊆ Jp00 . Since p0 (Jp0 ) ⊆ K, it follows from (2.11) that Q ∈ Jp00 , so that J + {CQ} ⊆ Jp00 . Conversely, let S ∈ Jp00 and let SQ = 0. It follows from (2.5), (2.10) and (2.12) that (S, p00 (S)) ∈ G(p)00 . By Theorem 2.9, G(p)00 = G(p), so that S ∈ J. For arbitrary S ∈ J. For arbitrary S ∈ Jp00 , we obtain from (2.4) that there is t ∈ C such that (S − tQ)Q = 0. Hence S ∈ J + {CQ}. We are now ready to proceed with the proof of Theorem 2.7. Proof of Theorem 2.7. Since Uq0 is an operator ∗ -algebra, it has a bounded approximate identity {Eλ }λ∈Λ of selfadjoint elements. Since Ker Uq0 = {0}, we have Uq0 H = H. Therefore Eλ converge to 1H in the strong operator topology. Since Uq0 = Jp0 ∩ (Jp0 )∗ , all Eλ belong to Jp0 . Since Jp0 is a left ideal of U 0 , all Eµ AEλ belong to Jp0 , for any A ∈ U 0 . All Eλ A∗ Eµ also belong to Jp0 , so that Eµ AEλ ∈ Uq0 . Consequently, U 0 is the closure of Uq0 in the weak operator topology. Let A ∈ Jp0 . Since Jp0 is a left ideal of U 0 , we have that Eλ A and A∗ Eλ belong to Jp0 . Therefore Eλ A ∈ Jp0 ∩ (Jp0 )∗ = Uq0 . By (2.8) and Lemma 2.11, q 0 (Eλ A) = p0 (Eλ A) = Eλ p0 (A) converge to p0 (A) in H .

(2.13)

Let S = S ∗ ∈ Uq00 . Replacing T by Eλ A in (2.3) and making use of (2.13), we obtain Sp0 (A) = lim Sq 0 (Eλ A) = lim Eλ Aq 00 (S) = Aq 00 (S) . It follows from (2.12) that S ∈ Jp00 . By Corollary 2.12, S ∈ J + {CQ} and, since S is selfadjoint, we obtain from Lemma 2.10 (iv) that S ∈ U + {CQ}. Thus Uq00 ⊆ U + {CQ}. Combining this with (2.4), we conclude the proof. We shall now apply Theorem 2.7 and Lemma 2.11 to obtain a description of the structure of weakly ∗ -closed quasivectors on operator ∗ -algebras on separable Hilbert spaces. Theorem 2.13. Let U be an operator ∗ -algebra on a separable Hilbert space H and q be a weakly ∗ -closed quasivector from U into H. There are a positive operator S on H with the dense domain D(S) and an element v ∈ UH such that AD(S) ⊆ D(S) and SA|D(S) = AS|D(S) , for A ∈ U , and (i) Av ∈ D(S) and q(A) = SAv, for A ∈ U; W (ii) U = {A ∈ U : Av ∈ D(S) and A∗ v ∈ D(S)}. Proof. Let B1 = {A ∈ B(H) : kAk ≤ 1} be the unit ball of B(H) and let M = {B ∈ Uq0 : kBk ≤ 1, kq 0 (B)k ≤ 1 and kq 0 (B ∗ )k ≤ 1} be a subset of B1 . Since H is separable, there is a metric ρ(A, B) on B1 which defines a topology equivalent to the weak topology on B1 [3, Theorem V.5.1]. From

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this and from the fact that B1 is compact it follows that B1 is separable and, hence, M is also separable [3, Theorems I.6.11 and I.6.15]. Let L = {Ri }∞ i=1 be a dense subset of M . The bounded operator R=

∞ X

!1/2 2

−1

Ri∗ Ri

i=1

is positive, kRk ≤ 1 and, by Lemma 2.11, 2

Ker R = Ker R =

∞ \

Ker Ri = Ker M = Ker Uq0 = {0} .

i=1

Therefore R maps H into a dense linear manifold D. For A ∈ U, it follows from (2.2) that kqR (A)k2 = kRq(A)k2 = (R2 q(A), q(A) =

∞ X

2−i kRi q(A)k2

i=1

=

∞ X i=1

2−i kAq 0 (Ri )k2 ≤ kAk2

∞ X

2−i kq 0 (Ri )k2 ≤ kAk2 ,

i=1

since all kq 0 (Ri )k ≤ 1. Hence qR and qR∗ = qR are bounded quasivectors. Thus R ∈ Uq0 and RA = AR, for A ∈ U. Therefore D is invariant for all operators from U. By (2.2), Rq(A) = Aq 0 (R), for A ∈ U, where q 0 (R) ∈ UH. Set S = R−1 and v = q 0 (R) . Then S is a positive operator, D(S) = D is dense in H and AD(S) ⊆ D(S), SA|D(S) = AS|D(S) and q(A) = SAq 0 (R) = SAv, for A ∈ U . W

Let A = A∗ ∈ U and Av ∈ D(S). Then there is z ∈ H such that Rz = Av. It follows from Theorem 2.7 that to prove (ii) it suffices to show that A ∈ Uq00 , that is, Aq 0 (B) = Bz, for all B ∈ Uq0 ,

(2.14)

For every B in Uq0 which can be factorized as B = F R, with F ∈ U 0 , it follows from (2.8) and Lemma 2.11 that Aq 0 (B) = Aq 0 (F R) = Ap0 (F R) = AF p0 (R) = F Aq 0 (R) = F Av = F Rz = Bz (2.15) Now we prove that all Ri in L admit this factorization. Indeed, Ri∗ Ri ≤ 2i R2 , so that kRi xk2 = (Ri∗ Ri x, x) ≤ 2i (R2 x, x) = 2i kRxk2 , for any x ∈ H. Let Ui be linear mappings from D(S) into H such that Ui Rx = Ri x, for x ∈ H .

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Then Ui are bounded on D(S) and extend to bounded operators on H which we will also denote by Ui . Since R and all Ri belong to U 0 for any C ∈ U, CUi Rx = CRi x = Ri Cx = Ui RCx = Ui CRx . Hence CUi |D(S) = Ui C|D(S) . Since C and Ui are bounded, they commute. Thus Ui ∈ U 0 . Since Ri = Ui R ∈ Uq0 , it follows from (2.15) that Aq 0 (Ri ) = Ri z, for all Ri ∈ L . Let B ∈ M . By the definition of M , all elements q 0 (Ri ) and q 0 (Ri∗ ) belong to the unit ball H1 = {x ∈ H : kxk ≤ 1}. Since H is separable, the weak topology on H1 is equivalent to the topology defined by a metric τ on H1 [3, Theorem V.5.1], so that H1 is a compact metric space. Since the set L is dense in M , there are elements {Ri(n) }∞ n=1 in L which converge to B with respect to the metric ρ on B1 and such 0 ∗ ) converge to an element that q (Ri(n) ) converge to an element u ∈ H1 and q 0 (Ri(n) 0 w ∈ H1 with respect to the metric τ . Since q is weakly ∗ -closed, u = q 0 (B). Hence, for any x ∈ H, (Aq 0 (B), x) = lim(Aq 0 (Ri(n) ), x) = lim(Ri(n) z, x) = (Bz, x) . Thus Aq 0 (B) = Bz and (2.14) holds for all operators from M . Since, for every B ∈ Uq0 , there is t > 0 such that tB ∈ M , we obtain that (2.14) holds for all operators from Uq0 . If the element v in Theorem 2.13 belongs to D(S) then q(A) = SAv = ASv = Az, for A ∈ U , where z = Sv, so that q is a bounded quasivector. Decomposing the operator S in Theorem 2.13 in the orthogonal sum of bounded operators, we obtain the representation of a weakly ∗ -closed quasivector as the orthogonal sum of bounded quasivectors. Corollary 2.14. Let q be a weakly ∗ -closed quasivector on an operator -algebra U on a separable Hilbert space H. There are subspaces Hk invariant for U P∞ and elements vk ∈ Hk such that H = k=1 ⊕Hk and P∞ (i) q(A) n= k=1 ⊕Avk , where Avk ∈ Hk ; o P∞ W P∞ ∗ 2 kA v k < ∞ . (ii) U = A ∈ U : k=1 kAvk k2 < ∞ and k k=1 ∗

Proof. Let E(λ), λ ∈ [0, ∞), be the spectral function of the positive operator S in Theorem 2.13. The projections Ek = E(k)−E(k −1) commute with all operators P∞ W W A ∈ U , so that the subspace Hk = Ek H are invariant for U , H = k=1 ⊕Hk and Ek v ∈ D(S). Set vk = SEk v ∈ Hk . By Theorem 2.13, Avk = ASEk v = SAEk v and

QUASIVECTORS AND TOMITA–TAKESAKI THEORY FOR

q(A) = SAv = lim E(n)SAv = lim SA n→∞

n→∞

We also have that

(

D(S) =

∞ X

n X

⊕Ek v = lim

⊕xk : xk ∈ Hk and

k=1

Therefore the condition Av = ∞ X

Similarly, A∗ v =

k=1

∞ X

763

⊕Avk =

k=1

∞ X

⊕Avk .

k=1

) kSxk k2 < ∞

.

k=1

P∞ k=1

⊕AEk v ∈ D(S) is equivalent to the condition

kSAEk vk2 =

k=1

P∞

n X

n→∞

k=1

...

∞ X

kAvk k2 < ∞ .

k=1

⊕A∗ Ek v ∈ D(S) implies

P∞ k=1

kA∗ vk k2 < ∞.

Let q, U and H be the same as in Corollary 2.14. Let R ∈ Uq0 be the positive operator constructed in Theorem 2.13 such that Ker R = {0}, let J be a maximal commutative subalgebra of U 0 which contains R and let T be the compact space of all maximal ideals of J . Then there is an isometric ∗ -isomorphism B → {B(t)} of the algebra J onto the algebra C(T ) of all continuous functions on T . There is also a Borel measure σ and a measurable family {H(t)} of Hilbert spaces on T such that H is isometrically isomorphic to the direct integral of these spaces. We identify them and write Z H(t)dσ(t) . H= T

Every operator B ∈ J is the multiplication operator by B(t): Bh = {B(t)h(t)}, where h = {h(t)} ∈ H . W

Every operator A ∈ U is decomposable, that is, there is a measurable operator function {A(t)} such that A(t) ∈ B(H(t)), that kAk = |kA(t)k|∞ and (Ah)(t) = A(t)h(t), for almost all t. Corollary 2.15. There is a measurable vector function ζ = {ζ(t)}, with W ζ(t) ∈ H(t), such that the operator ∗ -algebra U consists of all operators A ∈ U for which Z Z kA(t)ζ(t)k2 dσ(t) < ∞ and kA(t)∗ ζ(t)k2 dσ(t) < ∞ . T

T

Furthermore, q(A) = {A(t)ζ(t)} , f or A ∈ U If q is unbounded, {ζ(t)} does not belong to H, that is,

W

R

.

T

kζ(t)k2 dσ(t) = ∞.

Proof. The operator R acts on H as the operator of multiplication by the function R(t). Set ΩR = {t ∈ T : R(t) = 0}. Since Ker R = {0}, we have σ(ΩR ) = 0 and the operator S = R−1 acts on H as the multiplication operator by the function R(t)−1 for all t ∈ T \ΩR . Let v = v(t) be the element of H such that

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q(A) = SAv, for A ∈ U. The condition Av ∈ D(S) means that Z kR(t)−1 A(t)v(t)k2 dσ(t) < ∞ . T −1

Setting ζ(t) = R(t) v(t), for t ∈ T \ΩR , and making use of Theorem 2.13, we complete the proof of the corollary. 3. Structure of ∗ -Closed Quasivectors We consider now ∗ -closed quasivectors on operator ∗ -algebras and, making use of the structure of weakly ∗ -closed quasivectors established in Theorem 2.13 and Corollaries 2.14 and 2.15, we obtain a full description of their structure. The main result in this direction is the following theorem. Theorem 3.1. Let q be a ∗ -closed quasivector on an operator ∗ -algebra U on W a Hilbert space H. Then there are an operator ∗ -algebra U˜ in U and a weakly ∗ -closed quasivector q˜ on U˜ such that U = U˜ ∩ U and q = q˜|U , where U is the uniform closure of U. Since q(U) ⊆ UH , it follows that to prove Theorem 3.1 it suffices to consider the case when UH = H, so that Ker U = {0}. Therefore we assume this up to the end of the proof of Theorem 3.1. Let q be a ∗ -closed quasivector on an operator ∗ -algebra U. The graph G(q) of q in B(H) × H is not necessarily uniformly closed. In other words, q is not, generally speaking, a closed mapping with respect to the norms on B(H) and H. However, it is closable and, repeating the argument of Lemma 2.10, we obtain that the uniform closure G(q) of G(q) is the graph of a linear mapping γ, that is, G(γ) = G(q). Denoting by I the domain of γ, we establish the following lemma. Lemma 3.2. (i) I is a uniformly dense left ideal of U and U = I ∩ I ∗ . (ii) The quasivector q extends to a quasivector γ from I into UH such that γ(T A) = T γ(A), f or T ∈ U and A ∈ I . We denote by N the linear manifold of H which consists of all x such that |(q(A), x)| ≤ Cx kAk, A ∈ U , where Cx is a positive constant which depends on x. Lemma 3.3. (i) The linear manifold N is dense in H. (ii) For any x ∈ N, there are y, z ∈ H such that (γ(B), x) = (By, z) for B ∈ I. Proof. Let f be a bounded functional on U × H. Then there are a bounded functional g on U and x ∈ H such that

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f ((A, y)) = g(A) + (y, x), for (A, y) ∈ U × H .

765

(3.1)

If f vanishes on the graph G(γ) = {(A, γ(A)) : A ∈ I} of γ, 0 = f ((A, γ(A))) = g(A) + (γ(A), x), for A ∈ I . Hence x ∈ N . Suppose that N is not dense in H and let z ∈ N ⊥ . The element (0H , z) lies in U × H and does not belong to G(γ) (0H is the zero operator on H). By Hahn–Banach Theorem there is a bounded functional f on U × H which vanishes on G(γ) and f ((0H , z)) = 1. By the above argument, f has form (3.1), where x ∈ N so that 1 = f ((0H , z)) = g(0H ) + (z, x) = 0 . This contradiction shows that N is dense in H. Part (i) is proved. Let x ∈ N . The functional fx on I defined by the formula fx (B) = (γ(B), x), for B ∈ I , is bounded. Indeed, for any B ∈ I, there are An ∈ U which converge to B with respect to the norm and such that q(An ) converge to γ(B) in H. Then |(γ(B), x)| = lim |(q(An ), x)| ≤ Cx lim kAn k = Cx kBk . Hence fx extends to a bounded functional on U which we also denote by fx . Making use of the polar decomposition of the functional fx , we obtain that there is a representation πf of U on a Hilbert space Hf with a scalar product <, > and cyclic vectors ξ and η for πf such that fx (B) = (γ(B), x) = hπf (B)ξ, ηi, for B ∈ I . For every R ∈ U, it follows from Lemma 15.14 that (γ(B), Rx) = (γ(R∗ B), x) = hπf (R∗ B)ξ, ηi = hπf (B)ξ, πf (R)ni .

(3.2)

The linear manifold L = {Rx : R ∈ U} in H and its closure L are invariant for U. Let ρ be the restriction of the identity representation of U on H to L and let T be the mapping from L into Hf such that T (Rx) = πf (R)η . If Rx = 0 for some R ∈ U then, by (3.2), hπf (B)ξ, πf (R)ηi = 0, for all B ∈ I. Since ξ and η are cyclic for πf and since I is dense in U, the linear manifolds {πf (B)ξ : B ∈ I} and {πf (B)η : B ∈ I} are dense in Hf . Hence πf (R)η = 0, so that T is well defined, Ker T = {0} and Im T is dense in Hf , since η is cyclic. We have that T ρ|L = πf T |L , since, for A ∈ U and Rx ∈ L, T ρ(A)(Rx) = T ARx = πf (AR)η = πf (A)πf (R)η = πf (A)T (Rx) . We shall show now that the mapping T is closable. Let Rn x converge to 0 and πf (Rn )η converge to ϑ ∈ Hf . By (3.2), for any B ∈ I, hπf (B)ξ, ϑi = limhπf (B)ξ, πf (Rn )ηi = lim(γ(B), Rn x) = 0 .

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Since {πf (B)ξ : B ∈ I} is dense in Hf , we have ϑ = 0 and T is closable. Let T be the closure of T . Then T ρ(A)|D(T ) = πf (A)T |D(T ) , for A ∈ U. If y ∈ Ker T , there are Rn x ∈ L which converge to y and such that T Rn x = πf (Rn )η converge to 0. For every B ∈ I, it follows from (3.2) that 0 = limhπf (B)ξ, πf (Rn )ηi = lim(γ(B), Rn x) = (γ(B), y) . Thus y = 0 and Ker T = {0}. Applying now Theorem 1 [9, Sec. 21], we obtain that there is an isometry operator U from L onto Hf such that U ρ(A) = πf (A)U for A ∈ U. Set y = U −1 ξ and z = U −1 η. Then y, z ∈ L ⊆ H and (γ(B), x) = hπf (B)ξ, ηi = hπf (B)U y, U zi = (ρ(B)y, z) = (By, z) .

σ

The corollary below shows that q is weakly closable, that is, G(q) is the graph of a linear mapping. Corollary 3.4. Every ∗ -closed quasivector q on an operator ∗ -algebra U is weakly closable. σ

Proof. If (0H , u) ∈ G(q) , there is a net {Aλ }λ∈Λ of elements of U which converges to 0H in the weak operator topology such that q(Aλ ) converge to u int the weak topology on H. By Lemma 3.3, for x ∈ N , there are y, z ∈ H such that (u, x) = lim(q(Aλ ), x) = lim(Aλ y, z) = 0 .

Since N is dense in H, we have u = 0.

We denote by r the weak closure of the ∗ -closed quasivector q and by I the W domain of r. Then U ⊆ I ⊆ I ⊆ U and, repeating the proof of Lemma 2.10, we W obtain that I is a weakly dense left ideal of U and r(SA) = Sr(A), for S ∈ U

W

and A ∈ I .

Set U˜ = I ∩ I ∗ and q˜ = r|U˜ . Then U˜ is an operator ∗ -algebra and it is easy to check that q˜ is a weakly ∗ -closed quasivector from U˜ into H. Proof of Theorem 3.1. Clearly, U ⊆ U˜ ∩ U and q˜|U = r|U = q. To conclude the proof we only have to show that U˜ ∩ U ⊆ U. From Lemma 3.2 it follows that to prove this inclusion it suffices to show that I ∩ U ⊆ I. The graph G(r) is the σ-closure of G(q) and G(γ) ⊆ G(r) ⊆ B(H) × H. Let / I then (B, r(B)) ∈ / G(γ) B ∈ I ∩ U. Then (B, r(B)) ∈ G(r) ∩ (U × H). If B ∈ and, by the Hanh–Banach Theorem, there is a bounded functional f on B(H) × H which vanishes on G(γ) but does not equal zero on (B, r(B)). By (3.1), there are a bounded functional g on B(H) and an element x ∈ H such that 0 = f ((A, γ(A))) = g(A) + (γ(A), x), for A ∈ I .

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Therefore x ∈ N and, by Lemma 3.3, there are y, z ∈ H such that (γ(A), x) = (Ay, z) .

(3.3)

g(A) + (Ay, z) = 0, for A ∈ I .

(3.4)

Hence Since g is bounded, (3.4) holds for all A ∈ U . Since B ∈ U, it follows from (3.1) that f ((B, r(B))) = g(B) + (r(B), x) = −(By, z) + (r(B), x) . Since B ∈ I, there is a net {Aλ }λ∈Λ of elements from U which converge to B in the weak operator topology and such that q(Aλ ) converge to r(B) in the weak topology on H. Therefore, by (3.3), (By, z) = lim(Aλ y, z) = lim(q(Aλ ), x) = (r(B), x) , so that f ((B, r(B))) = 0. This contradiction shows that (B, r(B)) ∈ G(γ), so that B ∈ I. Corollary 3.5. The results of Theorem 2.13 and Corollaries 2.14 and 2.15 W remain valid for a ∗ -closed quasivector if the W ∗ -algebra U is replaced by the C ∗ -algebra U. 4. Tomita Takesaki Theory for Biinjective Quasivectors We now consider a particular type of quasivectors: biinjective and establish further links between the algebras U and Uq0 and between the quasivectors q and q 0 . Definition 4.1. A ∗ -closed quasivector q form U ⊆ B(H) into H is biinjective if both quasivector q and q 0 are injective: Ker q = Ker q 0 = {0}. For the subsequent use we need the following two auxillary results. Lemma 4.2. If q is a ∗ -closed quasivector from U into H then (q(A), q 0 (T ∗ )) = (q 0 (T ), q(A∗ )), f or all A ∈ U and T ∈ Uq0 . Proof. Since q is linear, it suffices to prove the lemma for A = A∗ . By Lemma 2.3, there are selfadjoint operators En in U which commute with A such that En q(A) uniformly converge to q(A). By (2.2), (q(A), q 0 (T ∗ )) = lim(En q(A), q 0 (T ∗ )) = lim(q(A), En q 0 (T ∗ )) = lim(q(A), T ∗ q(En )) = lim(T q(A), q(En )) = lim(Aq 0 (T ), q(En )) = lim(q 0 (T ), Aq(En )) = lim(q 0 (T ), En q(A)) = (q 0 (T ), q(A)) .

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Lemma 4.3. A weakly ∗ -closed quasivector q from U into H is injective if and only if q 0 (Uq0 ) = UH. Proof. Let q be injective. By Lemma 2.6, q 0 (Uq0 ) ⊆ UH. Denote by P the projection on UH q 0 (Uq0 ). The linear manifold q 0 (Uq0 ) is invariant for Uq0 , since q 0 is a quasivector on Uq0 . By Theorem 2.7, Uq0 is weakly dense in U 0 . Therefore the subspace q 0 (Uq0 ) is invariant for U 0 . Since the subspace UH is also invariant for U 0 , the projection P belongs to U 00 . Moreover, for any T ∈ Uq0 , we have P q 0 (T ) = 0, so that P ∈ Uq00 . It follows from Theorem 2.7 and (2.2) that P ∈ U and that T q(P ) = 0 for all T ∈ Uq0 . By Lemma 2.11, Ker Uq0 = {0}, so that q(P ) = 0. Since q is injective, P = 0 and q 0 (Uq0 ) = UH. Conversely, if q 0 (Uq0 ) = UH and q(A) = 0 for some A ∈ U, then, by (2.2), 0 Aq (T ) = T q(A) = 0, for all T ∈ Uq0 . Hence UH ⊆ Ker A, so that A = 0. The corollary below establishes some equivalent conditions for a quasivector to be biinjective. Corollary 4.4. For a weakly ∗ -closed quasivector q the following conditions are equivalent. (i) q is injective and q(U) = H. (ii) q is biinjective. (iii) q 0 is injective and q 0 (Uq0 ) = H. Proof. By Lemma. 4.3, the quasivector q 0 is injective if and only if q 00 (Uq00 ) = Uq0 H . It follows from Lemma 2.11 that Ker Uq0 = {0}, so that Uq0 H = H. From Theorem 2.7 we obtain that q 00 (Uq00 ) = q(U). Combining this with the above formula, we conclude that q 0 is injective if and only if q(U) = H. (i) ⇒ (ii) follows from the above argument. (ii) ⇒ (iii). By Lemma 4.3, q 0 (Uq0 ) = UH. Since q 0 is injective, q(U) = H. By (2.1), q(U) ⊆ UH, so that UH = H. Hence q 0 (Uq0 ) = H. (iii) ⇒ (i). Since q 0 is injective, q(U) = H. Therefore UH = H, so that q 0 (Uq0 ) = UH. By Lemma 4.3, q is injective. We shall now define an algebraic structure and an involution in q(U) which turn it into a left Hilbert algebra (see [12]). Let q be a biinjective weakly ∗ -closed quasivector from an operator ∗ -algebra U ⊆ B(H) into H. Set A = q(U) and endow A with the algebraic structure and the involution from U: q(A) ◦ q(B) = q(AB) = Aq(B) and V0 q(A) = q(A)# = q(A∗ ) and with the scalar product from H. Then V0 is an antilinear operator on A. Since q is biinjective, H is the closure of A with respect to the norm associated with (,).

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Proposition 4.5. A is a left Hilbert algebra. Proof. Clearly, A is an algebra. For any A ∈ U, the linear mapping Lq(A) is defined on A by the formula: Lq(A) q(B) = q(A) ◦ q(B) = Aq(B) . Therefore it is bounded and extends to the operator A on H. For A, B, C ∈ U, (q(A) ◦ q(B), q(C)) = (Aq(B), q(C) = (q(B), A∗ q(C)) = (q(B), q(A)# ◦ q(C)) . By Lemma 2.3, for any A ∈ U, there are En ∈ U such that En q(A) uniformly converge to q(A). Thus A2 is dense in A, since En q(A) = q(En ) ◦ q(A) ∈ A2 . Finally, Let q(An ) converge to 0 and V0 q(An ) = q(A)# = q(A∗n ) converge to x ∈ H. By Lemma 4.2, for any T ∈ Uq0 , 0 = lim(q(An ), q 0 (T ∗ )) = lim(q 0 (T ), q(A∗n )) = (q 0 (T ), x) . Since q is biinjective, q 0 (Uq0 ) is dense in H. Therefore x = 0, so that V0 is closable and A is a left Hilbert algebra. As usual in the theory of left Hilbert algebras, we denote by V the closure of the antilinear operator V0 : q(A) → q(A)# . Let V ∗ be the adjoint of V . For x ∈ H, we denote by Rx the linear mapping from A into H such that Rx q(A) = Ax, A ∈ U , and set A0 = {x ∈ D(V ∗ ) : Rx is bounded}. Then A0 is a right Hilbert algebra with respect to multiplication x ◦ y = Ry x, with the involution x = V ∗ x and the scalar product inherited from H (see [12]). Proposition 4.6. (i) A0 = q 0 (Uq0 ), V ∗ q 0 (T ) = q 0 (T ∗ ) and Rq0 (T ) extends to T, for any T ∈ Uq0 . (ii) q 0 (T1 ) ◦ q 0 (T2 ) = q 0 (T2 T1 ) and q 0 (T ) = q 0 (T ∗ ). Proof. If x = q 0 (T ), where T ∈ Uq0 , then, by (2.2), Rx q(A) = Ax = Aq 0 (T ) = T q(A) . Therefore Rx is bounded and extends to T . Moreover, q 0 (T ) ∈ D(V ∗ ) and V ∗ q 0 (T ) = q 0 (T ∗ ) , because, by Lemma 4.2, (V0 q(A), q 0 (T )) = (q(A∗ ), q 0 (T )) = (q 0 (T ∗ ), q(A)) , for any A ∈ U. Thus q 0 (Uq0 ) ⊆ A0 .

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Conversely, let x ∈ A0 , so that x ∈ D(V ∗ ) and Rx is bounded. Then the extension T of Rx to H belongs to U 0 . Indeed, for any A, B ∈ U, T Aq(B) = T q(AB) = Rx q(AB) = A(Bx) = ARx q(B) = AT q(B) . Since q(U) is dense in H, we obtain that T ∈ U 0 . We also see that qT (A) = T q(A) = Rx q(A) = Ax, for A ∈ U so that qT is a bounded quasivector. Moreover, the operator T ∗ extends RV ∗ x , since, for A, B ∈ U, (RV ∗ x q(A), q(B)) = (AV ∗ x, q(B)) = (V ∗ x, A∗ q(B)) = (V ∗ x, q(A∗ B)) = (V q(A∗ B), x) = (q(B ∗ A), x) = (q(A), Bx) = (q(A), Rx q(B)) = (q(A), T q(B)) = (T ∗ q(A), q(B)) . Therefore qT ∗ (A) = T ∗ q(A) = RV ∗ x q(A) = A(V ∗ x), so that qT ∗ is a bounded quasivector. Hence T ∈ Uq0 and x = q 0 (T ). Part (i) is proved and part (ii) follows immediately. Theorem 4.7. Let q be a biinjective weakly ∗ -closed quasivector from U ⊆ B(H) into H. Then there exists an antilinear involution operator J on H such that JUJ = Uq0 and q 0 (JAJ) = Jp(A), for any A ∈ U. Proof. For ξ ∈ H, we denote by Lξ the linear mapping from A0 into H : Lξ = Rx ξ and define A00 = {ξ ∈ D(V ) : Lξ is bounded}. Then A00 is a left Hilbert algebra and A ⊆ A00 . Repeating the argument of the Proposition 4.6, we obtain that A00 = q 00 (Uq00 ). Since q is biinjective, q(U) = H and it follows from Theorem 2.7 that Uq00 = U and q 00 = q. Therefore, A = A00 , so that A is a perfect left Hilbert algebra. Let (4.1) V = J∆1/2 be the polar decomposition of V , where ∆ = V ∗ V . It follows from the Tomita Theorem (see [12]) that JA = A0 . In other words, Jq(U) = q 0 (Uq0 ) and JLq(A) J = RJq(A) , for every A ∈ U. Then Jp(A) = q 0 (T ), where T ∈ Uq0 . Since Lq(A) = A and Rq0 (T ) = T , we obtain that JAJ = T , so that JUJ = Uq0 and Jq(A) = q 0 (JAJ), for A ∈ U. 5. Double Commutant Theorem for J-Symmetric Operator Algebras on Π1 -Spaces First we provide some information about J-symmetric operator algebras on ΠK -spaces. Let H = H− ⊕ H+ be an orthogonal decomposition of a Hilbert space H with a scalar product (x, y),let K± = dim H± and K = min(K− , K+ ) < ∞. The 0 defines indefinite form involution operator J = −1 0 1

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[x, y] = (Jx, y) on H and, with this form, H is called a Πk -space. We always assume that K− ≤ K+ , so that K = K− . A subspace L in H is positive (resp. negative) if [x, x] > 0 (resp. [x, x] < 0), for x 6= 0, and neutral if [x, x] = 0, for all x ∈ L. The subspace L[⊥] = {y ∈ H : [x, y] = 0 for all x ∈ L} is the J-orthogonal complement of L. If L ∩ L[⊥] = {0}, then H decomposes in the direct and J-orthogonal sum H = L[+]L[⊥] . In particular, if L is positive, then decomposition (5.1) holds. For any A ∈ B(H), the operator A# = JA∗ J is called the J-adjoint of A and [Ax, y] = [x, A# y], for x, y ∈ H . An operator algebra J on H is J-symmetric if A ∈ J implies A# ∈ J . It is non-degenerate if it has no neutral invariant subspaces. Otherwise it is called generic. J-symmetric operator algebras J and J1 on H and H1 , respectively, are J-equivalent, if there is a bounded operator S from H onto H1 which preserves the indefinite scalar product ([Sx, Sy] = [x, y]) and such that the mapping A → SAS −1 is an isomorphism of J onto J1 . As usual, we denote by J 0 the commutant of J : J 0 = {A ∈ B(H) : AB = BA, for B ∈ J }. It was shown in [6] that any weakly closed non-degenerate J-symmetric operator algebra J is similar to a W ∗ -algebra, that is, there is a bounded operator T from H onto a Hilbert space H with a bounded inverse such that {T AT −1 : A ∈ J } is a W ∗ -algebra on H. Therefore J = J 00 for all such algebras. If, however, J is generic then, generally speaking, J = 6 J 00 . This section considers generic algebras on Π1 -spaces and establishes conditions under which J = J 00 . In particular, it shows that J = J 00 if J is commutative. In order to do this we have to describe the canonical models of generic, uniformly closed algebras on Π1 -spaces obtained in [10,11]. Let now H = H− ⊕ H+ be a Π1 -space, that is, dim H− = 1, and let x ∈ H− , y ∈ H+ and kxk = kyk = 1. Set ξ = (x + y)/21/2 , η = (−x + y)/21/2 and H = H+ {Cy} . Then ξ and η are neutral elements in H, kξk = kηk = 1, [ξ, η] = 1 and H = {Cξ} ⊕ H ⊕ {Cη} . We denote H by Π1 (H) and 0

[αξ + z + βη, α0 ξ + z 0 + β 0 η] = αβ + βα0 + (z, z 0 ) , where α, α0 , β, β 0 ∈ C .

(5.1)

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For u, v ∈ H, we denote by u ⊗ v the rank one operator on H: (u ⊗ v)z = (z, u)v, for z ∈ H . Then A(u⊗v) = u⊗Av, (u⊗v)A = (A∗ u)⊗v, for A ∈ B(H), and (u⊗v)∗ = v⊗u . (5.2) Let J1 = η ⊗ ξ and J3 = ξ ⊗ η. We can consider J1 as an operator from {Cη} onto {Cξ} and J3 as the inverse of J1 . Witt respect to decomposition (5.1) the operator J has the following block-matrix form: 

0

 J = 0 J3

0 1H 0

J1



 0 . 0

Let U be an operator C ∗ -algebra on H. 1) Class M 0 . Let 1H 6= U. By M 0 we denote the class of all operator algebras    0 0     λ   0 M (U) =  0 λ1H + B 0  : λ ∈ C and B ∈ U     0 0 λ on Π1 (H). They are uniformly closed and J-symmetric. 2) Classes M 2a , M 2b , M 3a , M 3b . Let 1H ∈ U and let R be a subspace of H invariant for U. The operator algebras    λ  2a M (U) =  0   0    λ  2b M (U, R) =  0   0    λ  3a M (U) =  0   0    λ  3b M (U, R) =  0   0

  0    0  : B ∈ U and λ ∈ C ,   0 λ   y⊗ξ tJ1    B η ⊗ z  : B ∈ U, y, z, ∈ R, and λ, t ∈ C ,   0 λ   0 0    B 0  : B ∈ U and λ, µ ∈ C ,   0 µ   y⊗ξ tJ1    B η ⊗ z  : B ∈ U, y, z ∈ R, and λ, µ, t ∈ C   0 µ 0 B

on Πq (H) are uniformly closed and J-symmetric and they form, respectively, the classes M 2a , M 2b , M 3a and M 3b .

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3) Class M 1 . Let U be an operator ∗ -algebra on H, let 1H ∈ / U and let q be a -closed quasivector from U into H. Let R be a subspace of H invariant for U and orthogonal to q(U), let D be a linear manifold in Ker(U) orthogonal to R and let V be a closed antilinear operator on D such that V 2 = 1D . By (2.1), q(U) ⊆ UH, so that q(U) and D are orthogonal. Since R is invariant for U and U has an approximate identity which converges to the projection operator on UH in the strong operator topology, R = R1 ⊕ R2 where the subspaces R1 = R ∩ UH and R2 = R ∩ Ker U are invariant for U. Since q(U) is invariant for U, H has the following decomposition in the orthogonal sum of invariant subspace ∗

R p p H = q(U) ⊕ N1 ⊕ R1 ⊕ R2 ⊕ N2 ⊕ D , p p p p UH Ker U

(5.3)

where N1 = UH (q(U) ⊕ R1 ) and N2 = (Ker U) R2 D. By M 1 (U, q, R, D, V ) we denote the operator algebra on Π1 (H) which consists of operator   tJ1 λ (q(B ∗ ) + y + V u) ⊗ ξ   λ1H + B η ⊗ (q(B) + z + u)  (5.4) 0 0

0

λ

where B ∈ U, y, z ∈ R, u ∈ D and λ, t ∈ C. They are uniformly closed and Jsymmetric and these algebras form the class M 1 . If J = M 1 (U, q, R, D, V ), we will often write R(J ) for R, N1 (J ) for N1 and so on. Theorem 5.1. [11]. The algebras from different classes are not J-equivalent. Any generic, uniformly closed J-symmetric operator algebra on a Π1 -space is J-equivalent to one of the model algebras on Π1 (H) from one of the classes M 0 , M 1 , M 2a , M 2b , M 3a , M 3b . The algebras M 0 (U), M 2a (U), M 2b (U, R), M 3a (U) and M 3b (U, R) are weakly closed if and only if U is a W ∗ -algebra. The algebra M 1 (U, R, D, V ) is weakly closed if and only if the quasivector q is weakly ∗ -closed. The canonical models of algebras allow one to solve with comparative ease many natural problems which arise in the theory of operator algebras on Π1 -spaces such as the description of invariant subspaces, the commutant and the double commutant of algebras. The commutants of model algebras from all classes, apart from M 1 , can be obtained by straightforward calculations. The description of the commutant of algebras from the class M 1 and, consequently, the double commutant theorem for the algebras is more complicated than for other classes. This is no surprising in view of the fact that while the model algebras from the classes M 0 , M 2 and M 3 are constructed in a simple way from much studied and, at present, quite habitual operator C ∗ -algebras, the model algebras from the class M 1 are based on a less usual notion of quasivector.

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Let J = M 1 (U, R, D, V ). We denote by QR and QD the orthoprojections on R and D and by Q2 the orthoprojection N2 (cf. decomposition (5.3)). We also set Q = QR + QD . Let Uq0 and q 0 be, respectively, the subalgebra of B(H) and the weakly ∗ -closed quasivector form Uq0 into H constructed in Sec. 2. By V ∗ we denote the adjoint operator of V on D and by D(V ∗ ) the domain of V ∗ . Then V ∗ is an antilinear operator on D(V ∗ ) and (V ∗ )2 = 1D(V ∗ ) . Since R and D are orthogonal to q(U), we have qQ (A) = Qq(A) = 0, for A ∈ U, so that Q ∈ Uq0 (see (2.2)). Hence A = (1H − Q)Uq0 (1H − Q) is a subalgebra of Uq0 . Proposition 5.2. (M 1 (U, q, R, D, V ))0 = M 1 (A, q 0 |A , N2 , D(V ∗ ), V ∗ ). Proof. The operator algebra J contains the following subspaces     0 0 tJ1       Γ = Γ(t) =  0 0 0  : t ∈ C ,     0 0 0     0 0 q(A∗ ) ⊗ ξ       A η ⊗ q(A)  : A ∈ U , U(q) = Aq =  0     0 0 0     0 y⊗ξ 0       0 η ⊗ z  : y, z ∈ R , JR = Φ(y ⊕ z) =  0     0 0 0     0 Vu⊗ξ 0       0 η ⊗ u : u ∈ D JV = Φ(V u ⊕ u) =  0     0 0 0

(5.5)

and J = {C1H } + U(q) + JR + JV + Γ. Using simple block-matrix calculations, we obtain that Γ ⊂ J 0 and that J 0 ⊆ {C1H } + Γ + B, where B is a linear subspace of bounded operators on Π1 (H) which have the block-matrix form   0 w⊗ξ 0   F (T, x, w) =  0 T η ⊗ x  , where T ∈ B(H) and x, w ∈ H . 0

0

The operators from the subspaces U(q), JR and JV also have this form. Making use of (5.2), we obtain that operators F (T, x, w) and F (T1 , x1 , w1 ) commute if and only if T ∗ W1 = T1∗ w, T x1 = T1 x, T T1 = T1 T and (x, w1 ) = (x1 , w) . Choosing as F (T1 , x1 , w1 ) operators from JR , JV and U(q), we obtain that F (T, x, w) ∈ J 0 if and only if

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R ⊆ Ker T ∩ Ker T ∗ ,

(5.6)

x, w ∈ R⊥ ,

(5.7)

D ⊆ Ker T ∩ Ker T ∗ ,

(5.8)

(x, V u) = (u, w), for any u ∈ D ,

(5.9)

T ∗ q(A∗ ) = A∗ w, T q(A) = Ax, T A = AT ,

(5.10)

(x, q(A∗ )) = (q(A), w), for any A ∈ U .

(5.11)

It follows from (5.6) and (5.8) that T QR = QR T = T QD = QD T = 0. Hence QT = T Q = 0, so that T = (1H −Q)T (1H −Q). From (5.10) we obtain that T ∈ Uq0 . Thus T ∈ U. From (5.10) and from the discussion before Lemma 2.6 we have that x = q 0 (T ) ⊕ x0 and w = q 0 (T ∗ ) ⊕ w0 , where x0 and w0 belong to Ker U and q 0 (T ) and q 0 (T ∗ ) belong to UH. From (5.7) we conclude that q 0 (T ) and q 0 (T ∗ ) lie in q(U) ⊕ N1 (J ) and that x0 and w0 lie in N2 (J ) ⊕ D. Decomposing x0 = x2 ⊕ xD and w0 = w2 ⊕ wD , where x2 = Q2 x0 , xD = QD x0 , w2 = Q2 w0 and wD = QD w0 , we can see that conditions (5.6)–(5.11) do not make any restrictions on x2 and w2 and that the only restriction on xD and wD follows from (5.8): (xD , V u) = (u, wD ), for all u ∈ D . Therefore we conclude that xD can be any element from D(V ∗ ) providing that wD = V ∗ xD . Thus x = q 0 (T ) ⊕ x2 ⊕ xD and w = q 0 (T ∗ ) ⊕ w2 ⊕ V ∗ xD , where x2 , w2 ∈ N2 and xD ∈ D(V ∗ ). The only condition we have not considered is (5.11). Substituting the expressions for x and w in it, we obtain (q 0 (T ), q(A∗ )) = (q(A), q 0 (T ∗ )) . However, by Lemma 2.12, this condition is valid automatically.

For any subspace L in H, let BL = QL B(H)QL , where QL is the orthoprojection on L in H. Theorem 5.3. Let J be a weakly closed, generic J-symmetric operator algebra on a Π1 -space H and let 1H ∈ J . (i) (ii) (iii) (iv)

If J belongs to one of the classes M 0 , M 2a or M 3a then J = J 00 . If J ∈ M 3b then J 6= J 00 Let J = M 2b (U, R). Then J = J 00 if and only if BR ⊆ U. Let J = M 1 (U, q, R, D, V ). Then J 00 = J if and only if BR ⊆ U.

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Proof. Part (i) follows from the fact that operator algebras from the classes M 0 , M 2a and M 3a are similar to W ∗ -algebras. Part (ii) can be obtained by direct block-matrix calculations. Part (iii) is a simple version of part (iv) and also can be obtained by direct block-matrix calculations. We proceed now with the proof of (iv). Recall that J is weakly closed if and only if q is a weakly ∗ -closed quasivector. Let, as in Proposition 5.2, A = (1H − Q)Uq0 (1H − Q). Set p = q 0 |A . Then p is a quasivector from A into H. By Theorem 2.7, Uq0 is weakly dense in U 0 . Since 1H ∈ U 0 , we have Uq0 H = U 0 H = H. Therefore Uq0 has a two-sided approximate identity {Eλ }λ∈Λ which converges to 1H in the strong operator topology. Hence the operators Fλ = (1H − Q)Eλ (1H − Q) belong to A and, for any x ∈ H R(J ) D, the elements Fλ x converge to x. Thus AH = (1H − Q)H = q(U) ⊕ N1 (J ) ⊕ N2 (J ) and Ker A = QH = R(J ) ⊕ D. Since V is closed, D(V ∗ ) = D. From the proof of Proposition 5.2 it follows that p(A) = q 0 (A) ⊆ q(U) ⊕ N1 (J ) and R(J 0 ) = N2 (J ). Taking all this into account, we have the following decomposition of H into the orthogonal sum of subspaces invariant for J 0 q(U) ⊕ N1 (J ) N2 (J ) R(J ) p p p p p p H = p(A) ⊕ N1 (J 0 ) ⊕ R(J 0 ) ⊕ N2 (J 0 ) ⊕ D . p p p p AH Ker A

(5.12)

Let PR be the orthoprojection on R(J 0 ) and set P = PR + QD . Let A0p and p0 be, respectively, the subalgebra of B(H) and the weakly ∗ -closed quasivector from A0p into H constructed according to the procedure described before Lemma 2.6. Set L = (1H − P )A0p (1H − P ) . By Proposition 5.2, the commutant J 0 = M 1 (A, p, R(J 0 ), D(V ∗ ), V ∗ ). The orthoprojection P plays the same role for the algebra J 0 as Q plays for J . Hence, using the same argument as before Proposition 5.2, we have that P ∈ A0p , so that L is a ∗ -subalgebra of A0p . Applying Proposition 5.2 again, we obtain J 00 = (M 1 (A, p, R(J 0 ), D(V ∗ ), V ∗ ))0 = M 1 (L, p0 |L , N2 (J 0 ), D(V ∗∗ ), V ∗∗ ) . Since V is closed, V ∗∗ = V and D(V ∗∗ ) = D. We can also see from (5.12) that N2 (J 0 ) = R(J ), so that J 00 = M 1 (L, p0 |L , R(J ), D, V ). Thus J = J 00 if and only if U = L and q = p0 |L . Let K = AH and A˜ be the restriction of A to K. Then A = A˜ ⊕ {0} and p generates a quasivector p from A˜ into K such that p = p ⊕ 0. By ˜ 0p ⊕ B(K ⊥ ). Taking into account (5.12) and the fact that Lemma 2.8, A0p = (A) 1H − P = (1K − PR ) ⊕ QR , we obtain that L = (1H − P )A0p (1H − P ) = (1K − PR )(A1 )0p (1K − PR ) ⊕ QR B(K ⊥ )QR .

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Thus BR = QR B(H)QR = {0} ⊕ QR B(K ⊥ )QR ⊆ L. From this it follows that if J = J 00 then U = L, so that BR ⊆ U. Conversely, let QR B(H)QR ⊆ U. Then QR ∈ U, so that R(J ) = R1 (J ) ⊆ UH and R2 (J ) = {0}. Hence P = PR + QD = Q2 + QD . Set H1 = q(U) ⊕ N1 (J ) and H2 = R(J ) ⊕ N2 (J ) ⊕ D . Thus decomposition (5.3) becomes H2 p p p H = q(U) ⊕ N1 (J ) ⊕ R(J ) ⊕ N2 (J ) ⊕ D p p p p UH Ker U p

H1

(5.13)

Since all subspaces in (5.13) are invariant for U, H1 and H2 are invariant for U. Let U1 and U2 be the restrictions of U to H1 and H2 . We have that U2 = BR = QR B(H)QR = QR B(H2 )QR and U = U1 ⊕ U2 . The quasivector q generates quasivectors q1 and q2 on U1 and U2 such that q = q1 ⊕ q2 . We have that q2 = 0, since q(QR ) = QR q(QR ) = 0, so that q = q1 ⊕ 0. By Lemma 2.8, U˜q0 = (U1 )0q1 ⊕ (U2 )0 = (U1 )0q1 ⊕ [{C1R } ⊕ B(N2 (J ) ⊕ D)] and q 0 = q10 ⊕ 0 . We have that 1H − Q = 1H1 ⊕ Q2 . Therefore A = (1H − Q)Uq0 (1H − Q) = (1H1 ⊕ Q2 )Uq0 (1H1 ⊕ Q2 ) = (U1 )0q1 ⊕ Q2 B(H2 )Q2 and p = q 0 |A = q10 ⊕ 0. By Lemma 2.10(ii), (U1 )0q1 H1 = H1 . Hence, applying Lemma 2.8 again, we obtain that A0p = (U1 )00q1 ⊕ (Q2 B(H2 )Q2 )0 = (U1 )00q1 ⊕ [{C1N2 } ⊕ B(R(J ) ⊕ D)] and p0 = q100 ⊕ 0. Since 1H − P = 1H1 ⊕ QR , it follows that L = (1H − P )A0p (1H − P ) = (1H1 ⊕ QR )((U1 )00q1 ⊕ [{C1N2 } ⊕ B(R(J ) ⊕ D)])(1H1 ⊕ QR ) (U1 )00q1 ⊕ QR B(H2 )QR = (U1 )00q1 ⊕ U2 . By Theorem 2.7, (U1 )00q1 = U1 and q100 = q1 . Thus L = U1 ⊕ U2 = U and p0 |L = q. Corollary 5.4. Any commutative, weakly closed, J-symmetric operator algebra with identity J on a Π1 -space coincides with its double commutant. Proof. By Theorem 5.3, J = J 00 if J belongs to the classes M 0 , M 2a or M 3a . Clearly, J cannot belong to M 3b . It follows from (5.2) and (5.5) that Φ(y ⊕ 0)Φ(0 ⊕ y) = Γ(t) and Φ(0 ⊕ y)Φ(y ⊕ 0) = 0, for y ∈ H , where t = kyk2 . Hence if J = M 1 (U, q, R, D, V ) or J = M 2b (U, R) then R = {0} and, by Theorem 5.3, J = J 00 .

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6. Tomita Takesaki Theory for J-symmetric Operator Algebras on Π1 -Spaces We list now some problems which will be answered in this section. Let J be a weakly closed J-symmetric operator algebra on a Π1 -space H and 1H ∈ J . A vector x ∈ H is called bicyclic for J if it is cyclic for both S and J 0 . Problems. 1) If J has a bicyclic vector, does J coincide with its double commutant? 2) If a vector x in H is cyclic and separating for J = (Ax = 0, for A ∈ J , implies A = 0), is x bicylic? 3) If J has a bicyclic vector, does there exist an antilinear J-involution j : H → H such that J 0 = jJ j? First of all we will find conditions under which a vector is separating. Lemma 6.1. (i) A vector x = αξ ⊕ v ⊕ βη ∈ Π1 (H) is separating for an algebra J = M 1 (U, q, R, D, V ) if and only if β 6= 0, R = {0} and Bv + βq(B) = 0 implies B = 0 . In particular, if v = 0 then q must be injective. (ii) A vector x = αξ⊕v⊕βη ∈ Π1 (H) is separating for an algebra J = M 2b (U, R) if and only if β 6= 0, R = {0} and v is a separating vector for U. (iii) Algebras from the class M 3b do not have separating vectors. Proof. Let 0 6= y ∈ R and B ∈ U. It follows from (5.4) that the operators     0 y ⊗ ξ tJ1 0 0 J1     0 0  A = Γ(1) =  0 0 0  , A(y, t) =  0 0 0 0 0 0 0 

and

0  βˆ =  0 0

q(B ∗ ) ⊗ ξ B 0

 sJ1  η ⊗ q(B)  0

belong to J = M (U, q, R, D, V ). Let x be separating. If β = 0 then Ax = 0. Thus β 6= 0. Set t = −(v, y)/β. Then A(y, t)x = ((v, y) + tβ)ξ = 0 which contradicts the assumption that x is separating. Thus R = {0}. ˆ = [(v, q(B ∗ ))+βs]ξ ⊕ [Bv+βq(B)] = Bv+βq(B). Set s = −(v, q(B ∗ ))/β. Then Bx Since x is separating, Bv + βq(B) = 0 implies B = 0. Conversely, let all the conditions of (i) hold. Then all operators in J have block-matrix form (5.4), where y = x = 0. Therefore (see (5.4)) 1

T (B, q(B ∗ ) + V u, q(B) + u, λ, t)x = 0 implies λ + (v, q(B ∗ ) + V u) + tβ = 0, λv + Bv + βq(B) + βu = 0 and λβ = 0 .

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Since β 6= 0, we have λ = 0D. Since u ∈ D and D is orthogonal to UH, the above conditions become (v, q(B ∗ ) + V u) + tβ = 0 and (Bv + βq(B)) ⊕ βu = 0 and we conclude that u = 0, Bv + βq(B) = 0 and (v, q(B ∗ )) + tβ = 0. Hence B = 0 and t = 0, so that x is separating. Part (i) is proved. Parts (ii) and (iii) can be proved in the same way. Let x = αξ ⊕ v ⊕ η be a separating vector for a weakly closed operator algebra J = M 1 (U, q, R, D, V ). It is easy to check that the operator   1 v⊗ξ tJ1   −η ⊗ v  , where t = −kvk2 /2 , T =  0 1H 0

0

1

is J-unitary: T # T = T T # = 1H . it follows from (5.4) by direct calculations that T M 1 (U, q, R, D, V )T # = M 1 (U, qv , R, D, V ), where qv : B → q(B) + Bv is a weakly ∗ -closed quasivector from U into H. Thus the algebras M 1 (U, q, R, D, V ) and M 1 (U, qv , R, D, V ) are J-equivalent. It also follows that the vector T x = γξ ⊕η, where γ = α + kvk2 /2, is separating for M 1 (U, qv , R, D, V ). Therefore, replacing if necessary an algebra J from the class M 1 by a J-equivalent algebra, we can always assume that a separating vector for J has the form x = αξ ⊕ η. Theorem 6.2. A vector is separating for a weakly closed J-symmetric operator algebra J on a Π1 -space H if and only if it is cyclic for J 0 . Proof. If Ax = 0, for an operator A ∈ J , then ABx = BAx = 0, for all B ∈ J 0 . Therefore if x is cyclic for J 0 , it is separating for J . It is well known that separating vectors for a W ∗ -algebra J with identity are cyclic for J 0 . Weakly closed non-degenerate algebras and algebras from the classes M 0 , M 2a and M 3a are similar to W ∗ -algebras, so that the theorem holds for them. By Lemma 6.1, algebras from the class M 3b do not have separating vectors. For algebras from the class M 2b the proof of the theorem is straightforward. We now prove the theorem for algebras from the class M 1 . Let H = Π1 (H) = {Cξ} ⊕ H ⊕ {Cη}, J = M 1 (U, q, R, D, V ) and let x be separating for J . By Lemma 6.1, R(J ) = {0} and it follows from (5.3) that H = q(U) ⊕ N1 (J ) ⊕ N2 (J ) ⊕ D . p p p p UH Ker U

(6.1)

We obtain from Proposition 5.2 that J 0 = M 1 (A, q 0 |A , N2 (J ), D(V ∗ ), V ∗ ), where A = (1H − QD )Uq0 (1H − QD ) and (R(J 0 ) = N2 (J ). Following the discussion before the theorem, we assume that x = αξ ⊕ η. Set H = J 0 x. We shall now see formulae (5.5) to show that H = H.

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Since Γ(1) ∈ J 0 , we have Γ(1)x = ξ ∈ H. Since R(J 0 ) = N2 (J ), we have that Φ(0 ⊕ z) ∈ J 0 , for z ∈ N2 (J ). Hence Φ(0 ⊕ z)x = z ∈ H. Thus N2 (J ) ⊆ H. Since Φ(V u ⊕ u) ∈ J 0 , for u ∈ D(V ∗ ), it follows that Φ(V u ⊕ u)x = u ∈ H. Therefore D(V ∗ ) = D ⊆ H. We also have Aq0 x = q 0 (A)) ∈ H, for A ∈ A. Hence to prove that H = H we have to show that the linear manifold {q 0 (A) : A ∈ A} is dense in UH .

(6.2)

Since x = αξ ⊕ η is separating, it follows form Lemma 6.1 that the quasivector q is injective. Making use of Lemma 4.3, we obtain that q 0 (Uq0 ) = UH. Combining this with (6.2), we conclude that to complete the proof that H = H it suffices to show that q 0 (Uq0 ) = q 0 (A) . Set K = UH. By U˜ we denote the restriction of U to k. Then U = U˜ ⊕ {0} and q generates a quasivector q from U˜ into K such that q = q ⊕ 0. By Lemma 2.8, ˜ 0 ⊕ B(K ⊥ ) and q 0 = q0 ⊕ 0. Taking into account (6.1), we have that Uq0 = (U) q 1H − QD = 1K ⊕ Q2 , where Q2 is the orthoprojection on N2 (J ). Therefore A = (1H − QD )Uq0 (1H − QD ) = (U˜)0q ⊕ B(N2 (J )) and q 0 (Uq0 ) = q0 (U˜)0q = q 0 (A) . Thus H = H and x is a cyclic vector for J 0 .

Corollary 6.3. (Solution of Problem 2). If a vector is cyclic and separating for a weakly closed J-symmetric operator algebra on a Π1 -space then it is bicyclic. Problem 1) has a positive solution for Π1 -space subject to a more general condition. Theorem 6.4. (Solution of Problem 1). Any weakly closed J-symmetric operator algebra with identity on a Π1 -space which has a separating vector coincides with its double commutant. Proof. Non-degenerate operator algebras and algebras from the classes M 0 , M and M 3a always coincide with their double commutant (see Theorem 5.3). Algebras from the class M 3b do not have separating vectors. If J belongs to the class M 1 or M 2b and has a separating vector then, by Lemma 6.1, R = {0}. Hence, by Theorem 5.3, J coincides with its double commutant. 2a

For an antilinear operator S on H, the J-adjoint operator S # is defined by [S # x, y] = [Sy, x], for x, y ∈ H . An antilinear operator S is called a J-involution if S # = S = S −1 . The solution of problem 3) is given by Theorem 1.1.

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Proof of Theorem 1.1. By Theorem 6.2, J has a separating and cyclic vector x. Let J = M 1 (U, q, R, D, V ). Following the discussion before Theorem 6.2, we assume that x = αξ ⊕ η. By Lemma 6.1, R = R(J ) = {0}. By Theorem 6.4, J = J 00 . Hence it follows from Theorem 6.2 that the vector x is also cyclic and separating for J 0 . By Proposition 5.2, J 0 = M 1 (A, p, N2 (J ), D(V ∗ ), V ∗ ), where A = (1H − Q)Uq0 (1H − Q) and p = q 0 |A . By Lemma 6.1, N2 (J ) = R(J 0 ) = {0}. Therefore decomposition (5.12) turns into H = UH ⊕ D = AH ⊕ D, where UH = AH and Ker U = Ker A = D . Since x is separating for both J and J 0 , it follows from Lemma 6.1 that the quasivector q from U into H and the quasivector p from A into H are injective. Set K = UH = AH and by U˜ denote the restriction of U to K. Then U = U˜ ⊕{0} and q generates a quasivector q from U˜ into K such that q = q ⊕ 0. By Lemma 2.8, 0 ⊕ B(K ⊥ ) and q 0 = q0 ⊕ 0. Since R(J ) = {0}, Q = QR + QD = QD , so that Uq0 = U˜q 0 ⊕ {0}. Since q and p are injective quasivectors, we A = (1H − Q)Uq0 (1H − Q) = U˜q have that the quasivector q is biinjective. Applying Theorem 4.7, we obtain that there exists an antilinear involution operator j0 on K such that j0 U˜ j0 = U˜q0 and q0 (j0 Bj0 ) = j0 q(B), for B ∈ U˜ .

(6.3)

To extend j0 to an antilinear involution operator on Π1 (H), we consider the polar decomposition of the closed antilinear operator V on D = H K: V = j1 |V | , where |V | is a positive operator and J1 is an antilinear unitary operator on D. We have that j1 j1∗ = 1D and, since V 2 x = x, for x ∈ D, V −1 = V . Hence |V |−1 j1∗ = j1 |V |, so that |V |−1 = j1 |V |j1 = j12 (j1∗ |V |j1 ) . The operator j12 is unitary and the operator J1∗ |V |j1 is positive, so that the above formula is the polar decomposition of the positive operator |V |−1 . Uniqueness of the polar decomposition gives j12 = 1D . Therefore j1∗ = j1 and j1 is an antilinear involution on D. We also have that j1 D(V ∗ ) = D and j1 V = j12 |V | = |V | = |V |j12 = V ∗ j1 .

(6.4)

The operator ˆj = j0 ⊕ j1 is an antilinear involution on H and, it follows from (6.3) and (6.4) that ˆjU ˆj = A, ˆjD(V ∗ ) = D, ˆjV = V ∗ ˆj and p(ˆjB ˆj) = ˆjq(B), for B ∈ U .

(6.5)

Finally, setting ¯ , ¯ ξ ⊕ βη j2 (αξ ⊕ βη) = α we obtain an antilinear J-involution on {Cξ} ⊕ {Cη} and the operator j = ˆj ⊕ j2 is an antilinear J-involution on Π1 (H), since, [jy, z] = αδ + βγ + (ˆjyH , zH ) = αδ + βγ + (ˆjzH , yH ) = [jz, y]

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for y = αξ ⊕ yH ⊕ βη and z = γξ ⊕ zH ⊕ δη, where yH , zH ∈ H. Taking into account that R(J ) = R(J 0 ) = {0}, we obtain from (5.4) that J consists of operators   tJ1 λ (q(B ∗ ) + V u) ⊗ ξ   λ1H + B η ⊗ (q(B) + u)  , T (B, u, λ, t) =  0 0 0 λ where B ∈ U, u ∈ D and λ, t ∈ C, and that J 0 consists of operators   sJ1 µ (p(A∗ ) + V ∗ v) ⊗ ξ   µ1H + A η ⊗ (p(A) + v)  , S(A, v, µ, s) = =  0 0

0

µ

where A ∈ A, v ∈ D(V ∗ ) and µ, s ∈ C. To finish the proof of the theorem we shall show that (6.6) jT (B, u, λ, t)j = S(ˆjB ˆj, ˆju, λ, t) , where, by (6.5), ˆjB ˆj ∈ A and ˆju ∈ D(V ∗ ). Set A = ˆjB ˆj and v = ˆju. Let y = αξ ⊕ z ⊕ βη, where z ∈ H. Then jy = ¯ Making use of (6.5), we obtain that T (B, u, λ, t)jy = α0 ξ ⊕ z 0 ⊕ β 0 η, α ¯ ξ ⊕ ˆjz ⊕ βη. where α + (ˆjz, q(B ∗ ) + V u) + tβ¯ = λ¯ α + (ˆjq(B ∗ ) + ˆjV u, z) + tβ¯ α0 = λ¯ α + (p(A∗ ) + V ∗ v, z) + tβ¯ , λ¯ α + (p(ˆjβ ∗ ˆj) + V ∗ ˆju, z) + tβ¯ = λ¯ ¯ + u) and β 0 = λβ¯ . z 0 = λˆjz + B ˆjz + β(q(B) Using (6.5) and the fact that ˆj is an antilinear involution, we have jT (B, u, λ, t)jy = α ¯ 0 ξ ⊕ ˆjz 0 ⊕ β¯0 η , where ¯ + (z, p(A∗ ) + V ∗ v) + t¯β, α ¯ 0 = λα

¯ β¯0 = λβ

ˆ ¯ =λ ¯ ˆj 2 z + ˆjB ˆjz + β ˆjq(B) + β ˆju ˆjz 0 = ˆj(λˆjz + B ˆjz + βq(B) + βu) ¯ + Az + βp(A) + βv . = λz ¯ t¯)y, we conclude that (6.6) holds. Thus the Comparing this with S(ˆjB ˆj, ˆju, λ, theorem is proved for algebras from the class M 1 . Let now J belong to another class, for example, J = M 2b (U, R) and let x = αξ ⊕ v ⊕ βη be a bicycle vector. By Theorem 6.2, x is separating and cyclic for J and J 0 . By Lemma 6.1, R(J ) = {0}, β 6= 0 and v is a separating vector for U. By direct calculations we obtain that J 0 = M 2b (U 0 , {0}). Since x is cyclic for J , the vector v is cyclic for U. It follows from Tomita–Takesaki theory (see [12]) that there is an antilinear involution operator ˆj on H such that ˆjU ˆj = U 0 . Let

QUASIVECTORS AND TOMITA–TAKESAKI THEORY FOR

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j2 be the antilinear J-involution operator on {Cξ} ⊕ {Cη} considered above. The operator j = ˆj ⊕ j2 is antilinear J-involution operator on H. With respect to the decomposition Π1 (H) = ({Cξ} ⊕ {Cη}) ⊕ H, J = L ⊕ U and J 0 = L ⊕ U 0

n o λ t are the orthogonal sums of the operator algebra L = : λ, t ∈ C on 0 λ {Cξ} ⊕ {Cη} and the W ∗ -algebras U and U 0 respectively. Therefore jJ j = j2 Lj2 ⊕ ˆjU ˆj = L ⊕ U 0 = J 0 .

References [1] K. Yu. Dadashyan and S. S. Horygii, “On field algebras in quantum theory with indefinite metric, I”, Theoretical and Math. Phys. 54 (1) (1983) 57–77. [2] K. Yu Dadashyan and S. S. Horygii, “On field algebras in quantum theory with indefinite metric, II”, Theoretical and Math. Phys. 62 (1) (1985) 30–44. [3] N. Dunford and J. T. Schwartz, “Linear Operators”, vol. I, Interscience Inc., New York, 1988 (new edn.). [4] R. S. Ismagilov, “Rings of operators in a space with an indefinite metric”, Dokl. Akad. Nauk SSSR 171 (2) (1966) 269–271. [5] R. Larsen, An Introduction to the Theory of Multipliers, Berlin, Springer–Verlag, 1971. [6] V. I. Liberzon and V. S. Shulman, “Nondegenerate operator algebras in spaces with indefinite metric”, Izv. Akad. Nauk SSSR 37 (3) (1973), 533–38; Math. USSR Izvestija 7 (1973) 529–534. [7] A. I. Loginov and V. S. Shulman, “Vector-valued duality for modules over Banach algebras”, Izvestiia Rosiiskoi Akad. Nauk 57 (4) (1993) 3–35. [8] M. A. Naimark, “Commutative algebras of operators in the space Π1 ”, Rev. Roumaine Math. Pures Appl. 9 (1964) 499–528. [9] M. A. Naimark, Normed Algebras, Nauka, Moscow, 1968. [10] V. S. Shulman, “On operator algebras in spaces with indefinite metric of Π1 type”, Dokl. Akad. Nauk SSSR 201 (1) (1971) 44–47. [11] V. S. Shulman, “Symmetric Banach algebras of operators in a type Π1 space”, Mat. Sb. 89 (2) (1972) 264–79; Math. USSR Sbornik 18 (2) (1972) 267–283. [12] M. Takesaki, “Tomita’s theory of modular Hilbert algebras and its application”, Lecture Notes on Mathematics, 128, Springer–Verlag, 1972.

SUPERSELECTION STRUCTURES FOR C ∗ -ALGEBRAS WITH NONTRIVIAL CENTER ¨ ´ HELLMUT BAUMGARTEL and FERNANDO LLEDO Mathematisches Institut der Universit¨ at Potsdam Am Neuen Palais 10, Postfach 601 553 D-14415 Potsdam, Germany E-mail: [email protected] E-mail: [email protected] Received 18 December 1996 We present and prove some results within the framework of Hilbert C ∗ -systems {F , G} with a compact group G. We assume that the fixed point algebra A ⊂ F of G has a nontrivial center Z and its relative commutant w.r.t. F coincides with Z, i.e. we have A0 ∩F = Z ⊃ C1l. In this context we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. Z. Finally, we give several characterizations of the stabilizer of A.

1. Introduction The Doplicher–Roberts superselection theory [1–8] starts with a C ∗ -algebra A with trivial center, i.e. Z(A) := Z = C1l. A is interpreted as the algebra of quasilocal observables. The field algebra F ⊃ A, together with the gauge group G is then constructed as a special C ∗ -dynamical system {F , G} (cf. [9]), namely as a crossed product [6], also called a Hilbert C ∗ -system in [10, Chap. 10]. It satisfies the condition that the relative commutant is trivial, i.e. A0 ∩ F = C1l. The paper by Fredenhagen, Rehren and Schroer [12], where conformal theories in 1+1 dimensions are studied, suggests that a nontrivial center (containing for example (global) “Casimir operators”) of the universal algebra may also appear in physically relevant examples, and this situation is related to the superselection theory of the model (see also [13]). Furthermore, there are good mathematical reasons for considering the center of A to be nontrivial, and indeed this case has been treated in the past. For example, in the framework of strict symmetric monoidal C ∗ -categories with conjugates Doplicher and Roberts [7, Secs. 2 and 3] present some results where (ι, ι) is not necessarily trivial. Further, Longo and Roberts [11, Sec. 2] also study the notion of conjugation in the more general setting of strict monoidal C ∗ -categories without assuming that (ι, ι) is trivial. They also present a result for the case that (ι, ι) is finite dimensional. One of the problems of dealing with a nontrivial center of A is mentioned in [7, Introduction]: “There is, however, no known analogue of Theorem 4.1 of [6] for a C ∗ -algebra with a non-trivial center and hence nothing resembling a ‘duality’ in this more general setting.” The theorem mentioned before guarantees 785 Reviews in Mathematical Physics, Vol. 9, No. 7 (1997) 785–819 c World Scientific Publishing Company

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the existence of a C ∗ -algebra containing an algebraic Hilbert space that satisfies the usual nice properties, once A with Z(A) = C1l and a suitable endomorphism are given. Contrarily, the present paper deals with Hilbert C ∗ -systems {F , G} for compact groups G, where the fixed point algebra A has nontrivial center Z and satisfies the condition A0 ∩ F = Z ⊃ C1l .

(1)

We adopt a pure mathematical point of view and we will use the words field algebra F and observable algebra A only in a “metaphorical” sense, not claiming any relation to QFT. We remind that the condition A0 ∩ F = C1l (which implies Z = C1l) leads to the property π(A)00 = U (G)0 , where π denotes a so-called regular representation of {F , G}, i.e. π is the GNS-representation of the state ω(F ) := ω0 (Πι F ) and the GNS-representation of A w.r.t. ω0 is faithful (see, for example, [13, p. 18 ff.]) In the general case given by (1) it can be shown that for a regular representation π the equation π(A) ∩ U (G)00 = C1l holds. In this case we have that the condition π(A)00 = U (G)0 implies Z = C1l and, therefore, if we assume Z ⊃ C1l, then the proper inclusion π(A)00 ⊂ U (G)0 must hold (note that π(A)00 ⊆ U (G)0 is always true). Roughly speaking we can say that the group G does not determine the “observables” completely. We hope that this (mathematical) model will serve to get familiar with certain structures (e.g. Hilbert Z-modules) that may possibly appear when trying to construct G and F starting from the Doplicher–Roberts analysis in [7], for example in the special case that the “statistical dimensions” d(ρ) are scalars. However, at the present we have no convincing argument that this could be even possible. The paper is structured in 8 sections: In Sec. 2 we collect standard results concerning Hilbert C ∗ -systems that will be used later on. In Sec. 3 the notion of a Hilbert Z-module is introduced. It is a natural generalization of the usual notion of an algebraic Hilbert space [1, Sec. 2] when the center of the observable algebra Z is nontrivial and Eq. (1) is satisfied. In Sec. 4 the bijection between the set of right Hilbert Z-modules and the set of canonical endomorphism is extended to a functor between the corresponding categories. From very general arguments it is easy to see that the original notion of irreducible endomorphism ρ, i.e. (ρ, ρ) = C1l, is not meaningful anymore when Z is nontrivial. Section 5 proposes a generalization of this concept. A first justification of this new notion is given by the observation that the action of the inverse of an “irreducible” endomorphism restricted to the center can be described by certain continuous function acting on spec Z (cf. with Remark 5.5). Section 6 deals with the decomposition theory of a general Hilbert Z-module H = HZ, with H a group b The invariant algebraic Hilbert space, in terms of HD = HD Z, where D ∈ G. main results of the article are presented in Sec. 7, where different statements and

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characterizations concerning the stabilizer of A, stab A, are proved. For example, stab A is identified as a certain subgroup of the group of all continuous functions from spec Z into G, and the description of this subgroup uses the functions associated to irreducibles mentioned above. 2. Basic Material on Hilbert C ∗ -System We start introducing some notation and results concerning Hilbert C ∗ -systems. General references are [10, Chap. 10] [14, 13]. A Hilbert C ∗ -system is denoted by {F , G}, where G ⊂ aut F is a compact group w.r.t. the pointwise norm topology. ΠD , D ∈ Gb (the dual of G), are the spectral projections that satisfy the orthonormality relation ΠD1 ◦ ΠD2 = δD1 D2 ΠD1 ,

D1 , D2 ∈ Gb .

b we put For the trivial representation class ι ∈ G, A := Πι F = {F ∈ F: g(F ) = F,

g ∈ G} ,

i.e. A is the fixed point algebra in F w.r.t. G. Further, the spectrum of G, b ΠD 6= 0} , spec G := {D ∈ G: can be defined equivalently as the “Arveson spectrum” (cf. [13]). According to the definition of a Hilbert C ∗ -system we have spec G = Gb and to each D ∈ Gb there corresponds an algebraic Hilbert space HD ⊂ ΠD F , dim HD = dim D = d, such that supp HD = 1 and G acts irreducibly on HD . Further, the unitary representation G HD is an element of of the equivalence class D. Recall that if {ΦD, i }di=1 is an orthonormal in HD , i.e. the basis elements satisfy Φ∗D, i ΦD, i0 = δii0 1l , then supp HD :=

d X

ΦD, i Φ∗D, i = 1l .

i=1

In terms of the matrix elements we have g (ΦD, i ) =

d X

ΦD, i0 UD, i0 i (g) ,

g ∈G.

i0 =1

b the conjugated representation of D ∈ G. b If D is related We denote by D ∈ G, to the matrix (Ui0 i (g))i0 ,i ∈ Matd (C) as above, then D is realized by the complex ∈ Matd (C) w.r.t. the conjugated orthonormal conjugated matrix Ui0 i (g) i0 ,i

{ΦD, i }di=1 of HD . The following equations will be useful for later on [15, 16], [17, p. 182]: o n F = clok·k span ΠD F : D ∈ Gb ΠD F = span{A · HD }, ∗

where k · k = k · kF is the C -norm in F .

D ∈ Gb ,

(2) (3)

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By end A we denote the set of all unital endomorphisms of A. Further for λ, µ ∈ end A we consider, as usual, the intertwiner space (λ, µ) := {A ∈ A: A λ(X) = µ(X) A ,

X ∈ A} .

If H denotes an arbitrary G-invariant algebraic Hilbert space with support 1l in F , then the canonical endomorphism associated to H is λH (F ) :=

h X

Ψi F Ψ∗i ,

F ∈F,

(4)

i=1

where {Ψi }hi=1 is an orthonormal basis of H. λH is unital and since G leaves H invariant we have λH (A) ⊆ A, i.e. λH ∈ end A. If H := HD , then ρHD is briefly denoted by ρD . From Eqs. (2) and (3) (see also [10, Subsec. 10.1.3] we obtain the relation   X 0 (ρD , ι) · HD  . (5) A ∩ F = clok·k  b D∈G This implies that A0 ∩ F = Z iff (ρD , ι) = {0} for all D 6= ι and therefore, from our fundamental assumption, A0 ∩ F = Z ⊃ C1l ,

(6)

we get the following disjointness relation between the canonical endomorphisms ρD , b D ∈ G: (7) (ρD1 , ρD2 ) = {0} , D1 6= D2 , D1 , D2 ∈ Gb . Note that Eq. (7) implies also Eq. (6). For a general λ ∈ end A and, in particular, for λ := ρH we get Z ⊆ (λ, λ) .

(8)

On the other hand we also obtain the inclusions, C1l ⊆ λ(Z) ⊆ (λ, λ) ,

(9)

because from the relation Z A = A Z, A ∈ A, Z ∈ Z, it follows that λ(Z) λ(A) = λ(A) λ(Z) , and therefore, λ(Z) ∈ (λ, λ). Note that since λ is not surjective in general we cannot assure that λ(Z) ∈ Z for all Z ∈ Z. Therefore a typical feature of the present case is expressed by the fact that, in general, λ(Z) 6⊂ Z .

(10)

Equation (8) implies that the usual “intrinsic” (i.e. group independent) notion of irreducible endomorphism λ, namely (λ, λ) = C1l, is meaningless in our

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situation (cf. nevertheless with Sec. 5 and with [10, Subsec. 10.1.3] for further details concerning this point). Next we introduce the *-subalgebra F0 ⊂ F and the A-valued scalar product h·, ·iA on F . We select first, a family of algebraic Hilbert spaces {HD }D∈Gb b of corresponding to {F, G} and, second, a corresponding family {ΦD, i }d , D ∈ G, i=1

orthonormal bases. Then,    X F0 :=   D

d X

! AD, i ΦD, i

: AD, i

i=1

finite sum

   ∈A ⊂F.  

(11)

F0 is a dense *-subalgebra of F . On F we can define the following A-valued scalar product, hF1 , F2 iA := Πι (F1 F2∗ ) , F1 , F2 ∈ F . It satisfies the equations: 1 δD D δi i 1l , D1 , D2 ∈ Gb , i1 = 1, . . . , d1 , i2 = 1, . . . , d2 . hΦD1 ,i1 , ΦD2 , i2 iA = d1 1 2 1 2 hA1 F1 , A2 F2 iA = A1 hF1 , F2 iA A∗2 , A1 , A2 ∈ A, F1 , F2 ∈ F . hF , F iA ≥ 0

and

From this we obtain for F1 :=

P

hF , F iA = 0 iff

F = 0,

AD, i ΦD, i ∈ F0 and F2 :=

i,D

P

F ∈ F0 . BD, i ΦD, i ∈ F0 ,

i,D

the equation hF1 , F2 iA =

X1 ∗ AD, i BD, i. d i,D

{ΨH, i }hi=1

as specified in (4). Denote by H a conjugated Consider H, ρH and algebraic Hilbert space (carrying the conjugated representation with orthonormal basis given by {ΨH,i }hi=1 . Putting RH :=

h X

ΨH, i ΨH, i ∈ A

and

(12)

i=1

ε(H1 , H2 ) :=

X

ΨH2 , i ΨH1 , j Ψ∗H2 , i Ψ∗H1 , j ,

(13)

i,j

we get the following relations: RH ∈ (ι, ρH ◦ ρH ) .

(14)

ε(H1 , H2 ) ∈ (ρH1 ◦ ρH2 , ρH2 ◦ ρH1 ) .

(15)

∗ ΨH, i , i = 1, . . . , h Ψ∗H, i = RH

(16)

RH = ε(H, H)RH .

(17)

∗ ∗ RH = RH RH RH

=

h1l .

1l = ε(H1 , H2 ) ε(H2 , H1 ) .

(18) (19)

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The so-called standard left inverse is given by φH (A) :=

1 ∗ R ρ (A) RH , h H H

A ∈ A.

(20)

3. Hilbert Z-modules The stabilizer, stab A, is a subgroup of aut F defined by stab A := {β ∈ aut F : β(A) = A for all A ∈ A} . The study of stab A in the present situation leads in a natural way to the notion of a right Hilbert Z-module [18, Chap. 15] (see the following two propositions). Some of the results of this and the next section may be compared by putting Z = C1l with standard results in e.g. [1]. Note that we work with finite-dimensional algebraic Hilbert spaces. Let H be an G-invariant algebraic Hilbert space in F of finite dimension d. Then we define the free right Z-module H by extension ) ( d X Φi Z i : Z i ∈ Z , H := HZ = i=1

where {Φi }di=1 is an orthonormal basis in H. In other words, the set {Φi }di=1 becomes a module basis of H and dim H = d. For H1 , H2 ∈ H put hH1 , H2 iH := H1∗ H2 ∈ Z . Then, {H, h· , ·iH } is a Hilbert (right) Z-module or a Hilbert Z-module, for short. A system of d elements {Ψi }di=1 ⊂ H with Ψi =

d X

Φi0 Zi0 i ,

Zi0 i ∈ Z , i0 , i = 1, . . . , d ,

(21)

i0 =1 d

is an orthonormal basis of H, i.e. hΨi , Ψk i = δik , if the matrix Z := (Zi0 i )i0 ,i=1 ∈ Matd (Z) satisfies (22) Z∗ Z = 1ld . Using Gelfand’s Theorem we denote the values of the corresponding matrix-valued function on spec Z by Z(ϕ), ϕ ∈ spec Z: spec Z 3 ϕ 7−→ Z(ϕ) := (Zi0 i (ϕ))i0 ,i . Then Eq. (22) is equivalent to Z(ϕ)∗ Z(ϕ) = 1ld ,

ϕ ∈ spec Z .

Recall that in the finite-dimensional case, Eq. (23) implies Z(ϕ)Z(ϕ)∗ = 1ld ,

ϕ ∈ spec Z ,

(23)

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and again by Gelfand’s Theorem we obtain ZZ∗ = 1ld .

(24)

The last equation implies that the canonical endomorphism ρH can now be associated to H = HZ, because ρH (A) =

d X

Ψi A Ψ∗i ,

A ∈ A,

i=1

where now {Ψi }di=1 can be any orthonormal basis in H, is independent of the choice of the orthonormal basis in H: d X

Ψi A Ψ∗i =

X

Φi0 Zi0 i A Zi∗00 i Φ∗i00

i,i0 ,i00

i=1

=

=

X i0 ,i00

i

X

X

i0 ,i00

=

Φi0

X

d X

Φi0

! Zi0 i A Zi∗00 i

Φ∗i00

! Zi0 i Zi∗00 i

A Φ∗i00

i

Φi A Φ∗i .

i=1

For this reason we use the notation ρH = ρH , H = HZ. We emphasize that ρH does not characterize anymore the algebraic Hilbert space H (as in the case where Z = C1l). However we have: Proposition 3.1. Let H = HZ be a Hilbert Z-module as above and let ρH be the corresponding canonical endomorphism. Then the relation H∈H

iff

A ∈ A,

H A = ρH (A) H ,

holds. With other words, ρH characterizes the Hilbert Z-module H uniquely. Proof. Let H ∈ H. Then we can write H =

d P

Φi Zi for certain Zi ∈ Z and

i=1

{Φi }di=1 an orthonormal basis of H. We compute directly HA =

d X i=1

Φi Z i A =

d X i=1

Φi A Zi = ρH (A)

d X

Φi Zi = ρH (A) H .

i=1

Conversely, suppose that H ∈ F satisfies the equation HA = ρH (A) H, A ∈ A. Then, since Φi A = ρH (A) Φi , and A Φ∗i = Φ∗i ρH (A) for all A ∈ A, i = 1, . . . , d,

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we have Φ∗i H A = A Φ∗i H, A ∈ A, i.e. we get Φ∗i H ∈ A0 ∩ F = Z. Putting now d P Φi Zi ∈ H. Zi := Φ∗i H ∈ Z, we finally obtain, H = i=1

Proposition 3.2. Let H = HZ be a Hilbert Z-module as above. Then, H is stab A-invariant, i.e. β(H) ⊂ H for all β ∈ stab A. Proof. Let β ∈ stab A and Φ, Ψ ∈ H. Since ρH (A) ∈ A for A ∈ A, we have Φ∗ β(Ψ) A = Φ∗ β(ΨA) = Φ∗ β(ρH (A)Ψ) = Φ∗ ρH (A) β(Ψ) = A Φ∗ β(Ψ) , for all A ∈ A, and therefore Φ∗ β(Ψ) ∈ A0 ∩ F = Z. In particular, putting as in the preceding proposition Zi := Φ∗i β(Ψ) ∈ Z, where {Φi }di=1 is an orthonormal basis d P Φi Zi ∈ H. of H, we obtain β(Ψ) = i=1

Next we ask the question of how to characterize G-invariant algebraic Hilbert spaces that are contained in a given H = HZ, with H itself a G-invariant algebraic Hilbert space. By UH (g) ∈ Matd (C), g ∈ G, we denote the unitary matrix representation of G given on H, w.r.t. an orthonormal basis {Φi }di=1 specified in H. If we choose another orthonormal basis {Ψi }di=1 in H, related to {Φi }di=1 by means of the unitary matrix Z ∈ Matd (Z) of Eq. (21), then the representation of G w.r.t. the new basis is given by the matrices V (g) ∈ Matd (Z), defined by V (g) := Z∗ UH (g) Z ,

g ∈G.

(25)

In contrast to UH (g), the matrix V (g) cannot in general be associated to a constant matrix-valued function on spec Z. The condition “V (g), g ∈ G, is a constant matrix-valued function on spec Z” reads Z(ϕ1 )∗ UH (g) Z(ϕ1 ) = Z(ϕ2 )∗ UH (g) Z(ϕ2 ) ,

ϕ1 , ϕ2 ∈ spec Z , g ∈ G ,

or Z(ϕ2 )Z(ϕ1 )∗ UH (g) = UH (g) Z(ϕ2 )Z(ϕ1 )∗ ,

ϕ1 , ϕ2 ∈ spec Z , g ∈ G ,

(26)

i.e. Z(ϕ2 )Z(ϕ1 )∗ is an intertwiner of UH (G). Now consider the special case that G acts irreducibly on H. Then Eq. (26) is equivalent to Z(ϕ2 ) = µ(ϕ1 , ϕ2 ) Z(ϕ1 ) , where |µ(ϕ1 , ϕ2 )| = 1, ϕ1 , ϕ2 ∈ spec Z. Let ϕ1 := ϕ0 be a fixed point of spec Z and put W := Z(ϕ0 ). Then we get the condition |µ(ϕ)| = 1 , ϕ ∈ spec Z , d where µ(·) is a continuous scalar function. Let W := Wi0 i 0 Z(ϕ) = µ(ϕ) W ,

∗

∗

U ∈ Z with U (ϕ) = µ(ϕ). Then U U = U U = 1l and d

Z = (Wi0 i U )i0 ,i=1 .

i ,i=1

∈ Matd (C) and

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In other words, we obtain that V (g), g ∈ G, is a constant matrix-valued function w.r.t. {Ψi }di=1 iff d X Φi0 Wi0 i U , U ∈ Z unitary . Ψi = i0 =1

e i U and we obtain from Eq. (25): e i = P Φi0 Wi0 i , then Ψi = Φ Putting Φ i0

d ∗ e V (g) = Z∗ UH (g) Z = diag U ∗ · Uf H (g)·diag U = U Ui0 i U 0

i ,i=1

= Uf H (g) ,

g ∈G,

e d where the matrix Uf H (g) corresponds to the orthonormal basis {Φi }i=1 of H. We have obtained from the preceding considerations the following result: Lemma 3.3. If G acts irreducibly on H and if H = HZ, then H0 ⊂ H is a G-invariant algebraic Hilbert space iff H0 = HZ, where Z ∈ Z is unitary. Remark 3.4. Using the right Hilbert Z-module {H, h·, ·iH } one can construct canonically a continuous field of Hilbert spaces, a so-called “Dixmier field” [19, Chap. 10]. Recall that according to Gelfand’s Theorem Z ∼ = C(spec Z). For ϕ ∈ spec Z we put N (ϕ) := {H ∈ H: hH, HiH (ϕ) = 0} . e Denoting by H(ϕ) the coset in H/N (ϕ) associated to H ∈ H, we consider for a fixed ϕ ∈ spec Z the space n o e e H(ϕ) := H(ϕ) ∈ H/N (ϕ) : H ∈ H , as a preHilbert space with scalar product given by f2 (ϕ)iϕ := hH1 , H2 iH (ϕ) , f1 (ϕ), H hH

H1 , H2 ∈ H .

e e . Then it can be Denote by H(ϕ) the completion of H(ϕ), i.e. H(ϕ) := clok·kϕ H(ϕ) ! Q e Q H(ϕ) , is a continuous field of Hilbert H(ϕ), shown that the pair spaces.

ϕ∈spec Z

ϕ∈spec Z

4. The Canonical Functor In this section we will show that the bijection between H = HZ, H being G-invariant, and ρH established in Proposition 3.1 can be extended to a functor from the category of the right Hilbert Z-modules into the category of unital endomorphisms of A. The first part of this section is concerned with Hilbert Z-modules, H = HZ, where H is a finite-dimensional algebraic Hilbert space, but not necessarily G-invariant.

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Let H1 , H2 be two such modules. By LZ (H1 → H2 ) we denote the set of all Z-module morphisms form H1 into H2 , i.e. if T ∈ LZ (H1 → H2 ), then T is linear and satisfies T (H1 Z) = T (H1 ) Z ,

H1 ∈ H1 , Z ∈ Z .

For T ∈ LZ (H1 → H2 ) there is always an adjoint T ∗ ∈ LZ (H2 → H1 ) such that hH2 , T H1 iH2 = hT ∗ H2 , H1 iH1 ,

Hi ∈ Hi , i = 1, 2 .

1 2 ⊂ H1 and {Ψj }dj=1 ⊂ H2 , then T is charIndeed, given orthonormal bases {Φi }di=1 d2 P Ψj Zji , acterized by Z = (Zji )j,i ∈ Matd2 ×d1 (Z), via the equation T (Φi ) =

j=1

i = 1, . . . , d1 . In this case, T ∗ is given by T ∗ (Ψj ) :=

d1 P i=1

∗ ∗ Φi Zji and Z∗ := Zji

i,j

∈

Matd1 ×d2 (Z). Definition 4.1. Let T ∈ LZ (H1 → H2 ) be characterized by Z = (Zji )j,i ∈ Matd2 ×d1 (Z) as above. Then we define Tb :=

d2 d1 X X

Ψj Zji Φ∗i ∈ F .

i=1 j=1

Proposition 4.2. The assignment LZ (H1 → H2 ) 3 T 7−→ Tb ∈ F, with Tb given in the preceding definition, satisfies the following properties: (i) T (H1 ) = TbH1 , H1 ∈ H1 . (ii) Tb = 0implies ∗ T = 0 (injectivity). ∗ b c (iii) T = T . (iv) If T12 ∈ LZ (H2 → H1 ) and T23 ∈ LZ (H3 → H2 ), then we have T12 ◦ T23 ∈ c \ ◦ T23 = Tc LZ (H3 → H1 ) and T12 12 · T23 . b (v) kT kLZ (H1→H2 ) = kT kF . 1 2 and {Ψj }dj=1 orthonormal bases of H1 and H2 , Proof. (i) Denote by {Φi }di=1 respectively. Then the equation

Tb · Φi0 =

d2 d1 X X

Ψj Zji Φ∗i Φi0 = T (Φi0 ) ,

i=1 j=1

holds for all i0 = 1, . . . , d1 .

d2 d1 P P Ψj Zji Φ∗i = 0. Then, multiplying from the left (ii) Suppose that Tb := i=1 j=1

with Ψ∗j0 and form the right with Φi0 we get Zj0 i0 = 0 for all j0 = 1, . . . , d2 , i0 = 1, . . . , d1 , and therefore T = 0.

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(iii) Recall that if T is realized by the matrix Z = (Zj,i )j,i , then T ∗ is given by ∗ , so that the matrix Z∗ = Zji i,j  ∗ ∗ X X ∗ Ψj Zji Φ∗i  = Φi Zji Ψ∗j = Tc∗ . Tb =  i,j

i,j

d3 (iv) Add to the orthomal bases introduced in (i) the orthonormal bases {Ωk }k=1 (12)

of H3 . Denoting the matrices of T12 and T23 by Z12 = Zij and Z23 = i,j (23) Zjk , respectively, we get that the matrix of T12 ◦ T23 is given by j,k

 Z12 Z23 = 

d2 X

 (12) (23) Zij Zjk 

j=1

∈ Matd1 ×d3 (Z) . i,k

Then we calculate

    X X (12) (23) c  Φi Zij Ψ∗j  ·  Ψj 0 Zj 0 k Ω∗k  Tc 12 · T23 = j 0 ,k

i,j

=

X i,k

 Φi 

X

 (12) (23) Zij Zjk  Ω∗k

j

\ = T12 ◦ T23 . (v) Put H1 = H2 =: H. Then from (ii)–(iv) it follows that LZ (H) and L\ Z (H) ⊂ F are *-isomorphic *-algebras. Both algebras are C ∗ -algebras with C ∗ -norms k·kLZ (H) and k·kF , respectively. Therefore the isomorphy implies that the C ∗ -norms coincide, i.e. kT kLZ (H) = kTbkF , T ∈ LZ (H). In the general case we have that if T ∈ LZ (H1 → ∗ ∗T = T b Tb. Therefore, H2 ), then T ∗ T ∈ LZ (H1 ) and Td kT k2LZ (H1→H2 ) = kT ∗ T kLZ (H1 ) = kTb∗TbkF = kTbk2F

and the proof is concluded.

In the following we restrict again to the case where H = HZ, with H an G-invariant algebraic Hilbert space. From Proposition 3.2 we know that, in this case, H is stab A-invariant. Recall that g ∈ G acts on H as a unitary operator UH (g), i.e. g H = UH (g) ∈ L(H) and if an orthonormal basis {Φi }di=1 is given, then the representation UH of G in H is specified by a scalar unitary d × d-matrix, (Ui0 i (g))i0 ,i ∈ Matd (C): g(Φi ) =

d X

Φi0 Ui0 i (g) .

i0 =1

In analogy we consider next those Z-module morphisms which are G-invariant.

¨ ´ H. BAUMGARTEL and F. LLEDO

796

Definition 4.3. Let Hi = Hi Z, i = 1, 2, be Hilbert Z-modules, where the associated algebraic Hibert spaces Hi are G-invariant. Denote by UHi (g), g ∈ G, the corresponding unitary representations on Hi . Then the subset LZ (H1 → H2 ; G) ⊂ LZ (H1 → H2 ) is defined as the set of all intertwining operators: LZ (H1 → H2 ; G) := {T ∈ LZ (H1 → H2 ): UH2 (g) ◦ T = T ◦ UH1 (g) , Lemma 4.4. Let T ∈ LZ (H1 → H2 ). Then T ∈ LZ (H1 → H2 ; G) iff g T (H1 ) = T g(H1 )

g ∈ G} .

g ∈ G , H1 ∈ H 1 .

Proof. Obvious, since the following equations, = UH2 (g) ◦ T (H1 ) g T (H1 ) = UH2 (g) T (H1 ) and

T (g(H1 )) = T UH1 (g)(H1 )

=

T ◦ UH1 (g) (H1 ) ,

hold for all H1 ∈ H1 .

Proposition 4.5. Let T ∈ LZ (H1 → H2 ). Then, T ∈ LZ (H1 → H2 ; G) Even more we have

iff

Tb ∈ A .

o n Tb: T ∈ LZ (H1 → H2 ; G) = (ρH1 , ρH2 ) ,

(27)

(28)

i.e. the mapping T 7−→ Tb exhausts the whole intertwiner space (ρH1 , ρH2 ). First, let T ∈ LZ (H1 → H2 ; G). Then, according to Lemma 4.4 we have Proof. T g(H1 ) = g T (H1 ) for all g ∈ G, H1 ∈ H1 . From Proposition 4.2 (i) we get further Tb · g(H1 ) = T g(H1 ) = g T (H1 ) = g Tb · H1 = g Tb · g(H1 ) , so that g Tb = Tb and, therefore, Tb ∈ A. Second, if g Tb = Tb for all g ∈ G we have g T (H1 ) = g Tb · H1 = g Tb · g(H1 ) = Tb · g(H1 ) = T g(H1 ) , which by Lemma 4.4 implies that T ∈ LZ (H1 → H2 ; G). 1 2 and {Ψj }dj=1 orthonormal basis of H1 and H2 , To prove Eq. (28) let {Φi }di=1 respectively. Then d2 d1 X X Ψj Zji Φ∗i ∈ F , Tb := i=1 j=1

where Z = (Zji )j,i ∈ Matd2 ×d1 (Z) denotes the matrix corresponding to T ∈ LZ (H1 → H2 ; G). But from the definition of the canonical endomorphisms we have for A ∈ A:

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  ! X X ∗ ∗ Tb ρH1 (A) =  Ψj Zji Φi  Φi0 A Φi0 i0

i,j

  X Ψj Zji A Φ∗i  = i,j

    X X Ψj A  Ψ∗j Ψj 0  Zj 0 i Φ∗i  = j0

i,j

   X X Ψj A Ψ∗j   Ψj 0 Zj 0 i Φ∗i  = i,j 0

j

= ρH2 (A) Tb . On the other hand if B ∈ (ρH1 , ρH2 ), for all A ∈ A, i = 1, . . . , d1 , j = 1, . . . , d2 we have that Ψ∗j B Φi A = Ψ∗j B ρH1 (A) Φi = Ψ∗j ρH2 (A) B Φi = A Ψ∗j B Φi , and Ψ∗j B Φi ∈ A0 ∩ F = Z. Putting Zji := Ψ∗j B Φi ∈ Z and using the support 1 2 and {Ψj }dj=1 we obtain properties of the spaces generated by {Φi }di=1 B :=

X

Ψj Zji Φ∗i .

i,j

Therefore, denoting by T the operator in T ∈ LZ (H1 → H2 ) which is characterized by the matrix Z = (Zji )j,i ∈ Matd2 ×d1 (Z) we have from the last equation that Tb = B. Since B ∈ A we even know from the first part of the proposition that T ∈ LZ (H1 → H2 ; G). We have the following direct consequence of the preceding result: Corollary 4.6. If T ∈ LZ (H), then T ◦ UH (g) = UH (g) ◦ T , g ∈ G

iff

Tb ∈ (ρH , ρH ) ,

i.e. T is an intertwiner of the representation UH (g) on H iff Tb is an intertwiner of the canonical endomorphism, ρH . As it was announced at the beginning of this section we can now extend the mapping ρH 7−→ H, F(ρH ) := H to a functor by means of the assignment (ρH1 , ρH2 ) 3 A 7−→ F(A) ∈ LZ (H1 → H2 ; G) ,

798

¨ ´ H. BAUMGARTEL and F. LLEDO

where

F(A) (H1 ) := A · H1 .

In other words, F(A) can be characterized, once the orthonormal bases are chosen in H1 and H2 , by the matrix Z = (Zji )j,i ∈ Matd2 ×d1 (Z) , that satisfies the equation Z ◦ UH1 (g) = UH2 (g) ◦ Z ,

g ∈G.

In particular, since Z ⊆ (ρH , ρH ) (cf. with Eq. (8)), for each H = HZ, where H is e∈Z G-invariant, we have that ZH ∈H for all H ∈ H. This means that to each Z e= Z ei0 i ∈ Matd (Z), such that there corresponds a matrix Z i0 ,i

e Φi = Z

d X

Φi0 Zei0 i .

(29)

i0 =1

Therefore, the tensor product of two Hilbert Z-modules H1 and H2 , H1 H2 := spanZ (H1 · H2 ) = spanZ {H1 H2 : Hk ∈ Hk , k = 1, 2} , is again a Hilbert Z-modules. Indeed, this follows from the computation: (H1 H2 )∗ H10 H20 = H2∗ hH1 , H10 iH1 H20 = hH2 , hH1 , H10 iH1 H20 iH2 ∈ Z , where Hk , Hk0 ∈ Hk , k = 1, 2. In other words H1 H2 is the inner tensor product of the Hilbert Z-modules H1 and H2 w.r.t. *-homomorphism Z → LZ (H2 ) defined in Eq. (29) (see also [20]). Obviously, we have H1 H2 = (H1 H2 )Z, where H1 H2 denotes the C-tensor product, spanC (H1 · H2 ), of H1 and H2 . 5. Irreducible Endomorphisms In the present section we will determine the intertwiner space (ρH , ρH ), where H = HZ and the algebraic Hilbert space H is invariant and irreducible w.r.t. G. Theorem 5.1. Let H be as described above. Then the equation (ρH , ρH ) = ρH (Z) ,

(30)

holds. Proof. The inclusion ρH (Z) ⊆ (ρH , ρH ) follows from Eq. (9). To prove the other inclusion suppose that A ∈ (ρH , ρH ). Then from the relation (28), there exists T ∈ LZ (H → H ; G), such that A = Tb. According to Corollary 4.6 this means that T ◦ UH (g) = UH (g) ◦ T ,

g ∈G,

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NONTRIVIAL CENTER

where UH is the unitary representation of G on H (cf. with the paragraph before Definition 4.3). Choosing an orthonormal basis {Φi }di=1 of H, we can rewrite the preceding equation as d X

Zii0 UH, i0 i00 (g) =

i0 =1

d X

UH, ii0 (g) Zi0 i00 ,

g ∈ G , i, i00 = 1, . . . , d ,

(31)

i0 =1

where (Zii0 )i,i0 ∈ Matd (Z) is the matrix characterizing T and UH, ii0 (g)

i,i0

cor-

responds to UH (g). Therefore, by Gelfand’s Theorem, we can associate to each Zii0 ∈ Z a continuous function Zii0 (·) ∈ C(spec Z), satisfying Zii0 (ϕ) = ϕ(Zii0 ), ϕ ∈ spec Z. From Eq. (31) we obtain d X

Zii0 (ϕ) UH, i0 i00 (g) =

i0 =1

d X

UH, ii0 (g) Zi0 i00 (ϕ) , ϕ ∈ spec Z , g ∈ G , i, i00 = 1, . . . , d .

i0 =1

Denoting by T (ϕ) ∈ L(H) the operator whose scalar matrix w.r.t. the orthonormal basis {Φi }di=1 is (Zii0 (ϕ))i,i0 , we get from the preceding equation T (ϕ) ◦ UH (g) = UH (g) ◦ T (ϕ) ,

ϕ ∈ spec Z , g ∈ G .

But UH (G) is irreducible, hence T (ϕ) = c(ϕ)1lH , follows, where c(·) ∈ C(spec Z). Again, by Gelfand’s Theorem, the function c(·) can be associated to an element Z0 ∈ Z, such that c(ϕ) = Z0 (ϕ). We obtain from this Zii0 = Z0 δii0 or T = Z0 1lH . But, since A = Tb we get from the last equation A=

d X

Φi Zii0 Φ∗i0 =

i,i0 =1

d X

Φi Z0 δii0 Φ∗i0 =

i,i0 =1

d X

Φi Z0 Φ∗i = ρH (Z0 )

i=1

and the proof is concluded. Corollary 5.2. If H is irreducible, then the inclusion Z ⊆ ρH (Z) , holds.

Proof. Use Eq. (8) and the preceding theorem.

We have therefore the following relations for the canonical endomorphism, ρD ≡ b ρHD , with HD = HD Z, D ∈ G: (ρD , ρD0 ) = {0} ,

D 6= D0 ,

(ρD , ρD ) = ρD (Z) .

D, D0 ∈ Gb

¨ ´ H. BAUMGARTEL and F. LLEDO

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Motivated by Theorem 5.1 we introduce the notion of an irreducible endomorphism, which is independent of the group G. Definition 5.3. An arbitrary endomorphism λ ∈ end A is said to be irreducible if (λ, λ) = λ(Z). Note that this definition coincides with the usual notion of irreducibility in the case where Z = C1l (see for instance [17]). Proposition 5.4. If ρH ∈ end A is irreducible, then ρ−1 H can be considered on Z and the inclusion ρ−1 H (Z) ⊆ Z , holds, i.e. ρ−1 H Z ∈ end Z. Proof. From the existence of a left inverse (cf. Eq. (20)) it follows that ρH is injective. Then, Corollary 5.2 ends the proof. Remark 5.5. According to the preceding Proposition we have that if the endomorphism λ := ρH is irreducible, then λ−1 Z is a unital injective endomorphism of Z. But according to Gelfand’s Theorem the category of unital abelian C ∗ -algebras and their ∗ -homomorphisms and the category that is opposite to the category of compact topological spaces and their continuous maps are isomorphic (see e.g. [21, Chap. IV]. This means that to any unital endomorphism λ−1 Z there corresponds a continuous mapping fλ : spec Z −→ spec Z , such that

λ−1 (Z) (ϕ) = Z fλ (ϕ) ,

Z ∈ Z , ϕ ∈ spec Z ,

where fλ is surjective in our case. λ−1

−−−−−−→

Z

Z

↓

↓

C(spec Z)

C(spec Z)

↑

↑

spec Z

fλ

←−−−−−−

spec Z

From the preceding comments we can divide the irreducible endomorphisms λ := ρH into two different families: the first one is characterized by the fact that λ−1 Z is also a surjective mapping. In this case the equations Z = λ(Z) = (λ, λ)

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hold. For the second family λ−1 Z is not surjective and the following chain of proper inclusions can be easily established . . . ⊂ λ−1

n

(Z) ⊂ λ−1

(n−1)

(Z) ⊂ . . . ⊂ λ−1 (Z) ⊂ Z .

If Z is finite-dimensional only the first type of endomorphisms will appear. The continuous mappings fλ ≡ fρH , H irreducible, are essential ingredients of the Hilbert C ∗ -system {F, G}. Roughly, the function fλ reflects, at the level of the spectrum, the action of an irreducible endomorphism λ on Z. Example 5.6. A simple example that illustrates the present situation is constructed as follows: let Ω be a compact topological space and B := CAR(h, Γ) the C ∗ -algebra of the canonical anticommutation relations over an infinite-dimensional Hilbert space h with an antiunitary involution Γ [22]. Define the C ∗ -algebra A := C(Ω, B) = {f : Ω −→ B: f is continuous} with the natural operations and C ∗ -norm. From Z(B) = C1l we obtain that Z(A) = C(Ω, C) ⊃ C1l . Define next the automorphism γ ∈ aut A as (γf ) (p): = β−1 (f (p)) , p ∈ Ω, f ∈ A, where β−1 denotes the Bogoljubov automorphism of B corresponding to the unitarity U : = −1l [10, p. 130]. The field algebra F is constructed using the automorphism γ as in [17, Sec. 3.6] and since (id, γ) = {0} we obtain A0 ∩ F = Z(A). The automorphism γ is irreducible and satisfies γ Z(A) = Z(A) = (γ, γ) . The group in this example is G = Z2 = {id, α} (α ∈ aut F satisfies α2 = id) and Gb = {ι, χ} ∼ = Z2 . 6. Decomposition of H in Terms of HD , D ∈ Gb As at the beginning of Sec. 3, we consider a Hilbert Z-module H = HZ, where the algebraic space H is G-invariant. Denote the associated canonical endomorphism by λ := ρH . Further, we will need the quantities HD = HD Z and ρD associated to irreducible D ∈ Gb and defined in Sec. 2. As before, UH (G) and UD (G) ≡ UHD (G) are the unitary representation of G on H and on HD , respectively. UD is irreducible. Proposition 6.1. With the notation introduced above the following properties are true: b (i) The intertwiner space (ρD , λ) is a Hilbert ρD (Z)-module, D ∈ G. 0 0 b (ii) For D, D ∈ G and D 6= D the Hilbert modules (ρD , λ) and (ρD0 , λ) are mutually orthogonal.

¨ ´ H. BAUMGARTEL and F. LLEDO

802

(iii) The Hilbert ρD (Z)-module (ρD , λ) is a free module. There exists an orm(D) thonormal basis {CD, l }l=1 , where m(D) denotes the multiplicity of D ∈ Gb in the decomposition of UH as a direct sum of irreducible representations. Further, the following equation holds: X

m(D)

supp (ρD , λ) =

∗ CD, l CD, l = PD ,

l=1

where PD is the uniquely determined isotypic projection belonging to D in the mentioned decomposition. Proof. (i) Let A, B ∈ (ρD , λ). Since the endomorphisms ρD are irreducible, we have that A∗ B ∈ (ρD , ρD ) = ρD (Z) . (ii) Let A ∈ (ρD , λ) and B ∈ (ρD0 , λ) with D, D0 ∈ Gb and D 6= D0 . From Eq. (7) we get A∗ B ∈ (ρD0 , ρD ) = {0} . (iii) First we decompose UH on H into a direct sum of irreducible components. For D ∈ Gb we write explicitly X

m(D)

PD =

ED, l ,

(32)

l=1 m(D)

where the orthonormal family of projections {ED, l }l=1 satisfy ED, l ◦ UH (g) = UH (g) ◦ ED, l , g ∈ G, and the subspaces {ED, l H}l are irreducible. Then, according [ to Corollary 4.6 we have that E D, l ∈ (λ, λ). We denote by b i = 1, . . . , d, l = 1, . . . , m(D)} {ΨD, i, l : D ∈ G, an adapted orthonormal basis of H w.r.t the decomposition specified in (32). Further, choose an orthormal basis {ΦD, i }di=1 of HD and put CD, l :=

d X

ΨD, i, l Φ∗D, i .

(33)

i=1

b l = 1, . . . , m(D). Indeed, note first that We will show that CD, l ∈ (ρD , λ), D ∈ G, g(CD, l ) = CD, l , g ∈ G, so that CD, l ∈ A. Moreover, we obtain for all A ∈ A CD, l ρD (A) =

d X

ΨD, i, l Φ∗D, i ρD (A)

i=1

=

d X

ΨD, i, l A Φ∗D, i

i=1

= λ(A)

d X

ΨD, i, l Φ∗D, i

i=1

= λ(A) CD, l ,

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where for the third equation we have used Proposition 3.1 and the fact that ΨD, i, l ∈ H. ∗ 0 Next, we obtain by a direct computation that CD, l CD, l0 = 0, for l 6= l : ! ! d d X X ∗ ∗ ∗ ΦD, i ΨD, i, l · ΨD, i0 , l0 ΦD, i0 CD, l CD, l0 = i0 =1

i=1

X

=

ΦD, i Ψ∗D, i, l ΨD, i0 , l0 Φ∗D, i0

i,i0

= 0, because the family of projections {ED, l }l are mutually orthogonal and, therefore, the equation Ψ∗D, i, l ΨD, i0 , l0 = 0 holds for l 6= l0 . Note that from Eq. (33) we get ΨD, i, l = CD, l ΦD, i . m(D)

Now we have to prove that {CD, l }l=1 b Then we have (ρD , λ), D ∈ G.

(34)

is a module basis of (ρD , λ). Let B ∈

∗ ∗ ∗ CD, l B ρD (A) = CD, l λ(A) B = ρD (A) CD, l B , ∗ i.e. CD, l B ∈ (ρD , ρD ) = ρD (Z). We put ∗ CD, l B =: ρD (ZD, l ) ,

(35)

for some ZD, l ∈ Z. Furthermore, we have ∗ CD 0, l B = 0 ,

D0 6= D ,

(36)

∗ because CD 0 , l B ∈ (ρD , ρD 0 ) = {0}. Moreover, we calculate using Eq. (33)

X

m(D) ∗ CD, l CD, l =

l=1

d X X

XX l

=

X

ΨD, i, l Φ∗D, i

·

d X

! ΦD, i0 Ψ∗D, i0 , l

i0 =1

i=1

l

=

!

ΨD, i, l Φ∗D, i ΦD, i0 Ψ∗D, i0 , l

i,i0

ΨD, i, l Ψ∗D, i, l

l,i

= PD , where we have used the fact that {ΨD, i, l }D, i, l is an adapted basis w.r.t. the deP ΨD, i, l Ψ∗D, i, l = 1. From composition specified in Eq. (32). Note that supp H = D,l,i

Eq. (35) we get 

X

m(D)

PD B = 

l=1

 ∗  CD, l CD, B= l

X l

CD, l ρD (ZD, l )

¨ ´ H. BAUMGARTEL and F. LLEDO

804

and from Eq. (36) we obtain



PD0 B = 



m(D0 )

X

∗ B = 0, CD0 , l CD 0, l

l=1

for all D0 6= D. From this we finally have X

m(D)

B=

CD, l ρD (ZD, l )

l=1

and the proof is concluded.

b Theorem 6.2. With the notation of the beginning of this section, let D ∈ G. Then (ρD , λ) HD is a Hilbert Z-module. Further, H can be decomposed into the following orthogonal direct sum: H = ⊕ (ρD , λ) HD . D

Proof. Let Ak ∈ (ρD , λ), Xk ∈ HD , Zk ∈ Z, k = 1, 2, so that Ak Xk Zk ∈ (ρD , λ) HD . According to Proposition 6.1 (i) we have that A∗1 A2 = ρD (Z) for some Z ∈ Z and, therefore, ∗

(A1 X1 Z1 ) (A2 X2 Z2 ) = Z1∗ X1∗ A∗1 A2 X2 Z2 = Z1∗ X1∗ ρD (Z) X2 Z2 = Z1∗ X1∗ X2 Z Z2 = Z1∗ hX1 , X2 iHD Z Z2 ∈ Z . b follows from The mutual orthogonality of the Hilbert Z-modules (ρD , λ) HD , D ∈ G, the mutual orthogonality of the ρD (Z)-module (ρD , λ) (cf. Proposition 6.1 (ii)). It remains to show that Pc D · (HZ) = (ρD , λ) HD Z where, as before, G H acts by the unitary representation UH (G) and PD ∈ L(H) denotes the isotypical b Recall that PD ◦ UH (g) = UH (g) ◦ PD , g ∈ G, implies projection w.r.t. D ∈ G. c PD ∈ (λ, λ) ⊂ A (cf. Eq. (28)). The family {ΦD, i }di=1 denotes an orthonormal basis of HD . First, we prove Pc D · (HZ) ⊆ (ρD , λ) HD Z. For H ∈ HZ we have Pc D H ∈ span{A HD } = ΠD F and therefore Pc DH =

d X

Ai ΦD, i ,

Ai ∈ A .

i=1

To prove that Ai ∈ (ρD , λ), i = 1, . . . , d, we take B ∈ A and put Pc DHB =

d X i=1

Ai ΦD, i B =

d X i=1

Ai ρD (B) ΦD, i .

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On the other hand we get c c Pc D H B = PD λ(B) H = λ(B) PD H =

d X

λ(B) Ai ΦD, i .

i=1

Recall that {ΦD, i }di=1 is also an A-module basis of F0 . Therefore λ(B) Ai = Ai ρD (B) for all B ∈ A, i.e. Ai ∈ (ρD , λ). Second, to prove the other inclusion take A ∈ (ρD , λ). Then the inclusion A HD ⊆ HZ follows from (A ΦD, i ) B = A ρD (B) ΦD, i = λ(B) (A ΦD, i ) ,

B ∈ A,

so that A ΦD, i ∈ HZ according to Proposition 3.1. Finally, for g ∈ G we have g(A ΦD, i ) = A g(ΦD, i ) = A UHD (ΦD, i ) =

d X

(A ΦD, i0 ) UD, i0 i (g) ,

i0 =1

so that A ΦD, i transforms according to D ∈ Gb and therefore A HD ⊆ Pc D HZ.

Remark 6.3. According to Proposition 3.2 the Hilbert Z-module H = HZ is stab A-invariant. This means that for each β ∈ stab A and for an orthonormal basis {Ψi }di=1 of H we obtain β(Ψi ) =

d X

Ψi0 Zi0 i (β) ,

Zi0 i (β) ∈ Z ,

i0 =1

i.e. to each β ∈ stab A there corresponds a matrix Z(β) = Zi0 i (β) {Ψi }di=1

i0 ,i

∈ Matd (Z).

we get immediately that (cf. also with From the orthonormality of the basis Lemma 7.2) Z(β)∗ Z(β) = 1ld , , 1ld ∈ Matd (Z). We also have where Z(β)∗ = Zi∗0 i (β) i,i0

Z(β1 ◦ β2 ) = Z(β1 )Z(β2 )

and Z(ι) = 1ld ,

where β1 , β2 ∈ stab A and ι denotes the identical automorphism. In particlular, choosing an orthonormal basis {ΦD, i }di=1 of HD we denote the corresponding matrices by ZD = (ZD, i0 i )i0 ,i ∈ Matd (Z) . Choose D1 , D2 ∈ Gb and consider HD1 HD2 as well as the Hilbert Z-module HρD1 ◦ρD2 = HD1 HD2 , HDk = HDk Z, k = 1, 2. Recall that the representation UHD1 HD2 (G) on HD1 HD2 belongs to the class D1 D2 . Let ZD1 D2 = ZD1 D2 , i01 i02 i1 i2 i0 ,i0 ,i ,i 1

2

1

2

¨ ´ H. BAUMGARTEL and F. LLEDO

806

be the matrix associated to β HD1 HD2 w.r.t. orthonormal basis {ΦD1 , i1 ΦD2 , i2 }i1 ,i2 . According to Theorem 6.2 we have the following decomposition of HD1 HD2 : HρD1 ◦ρD2 = HD1 HD2 = ⊕ (ρD , ρD1 ◦ ρD2 ) HD . D

This means that there is an orthonormal basis in HD1 HD2 of the form {CD, l ΦD, i }l,i , l = 1, . . . , m(D), i = 1, . . . , d, m(D) being the multiplicity of D ∈ Gb in the decomposition of UHD1 HD2 (G) (cf. with Proposition 6.1 and with Eq. (34)), P 1 D2 d · m(D) = d1 d2 . Denote by ΓD1 D2 = ΓD ∈ Matd1 d2 (C) where i1 i2 ,D l i i1 i2 ,D l i

D

the corresponding scalar unitary transformation matrix (Clebsch-Gordan matrix)

ΦD1 , i1 ΦD2 , i2 =

d X m(D) X X

1 D2 ΓD i1 i2 ,D l i CD, l ΦD, i .

l=1 i=1

D

Then we have ZD1 D2 (β) = Γ

D1 D2

diag m(D) ZD

ΓD1 D2

−1

,

(37)

D

where diag m(D) ZD := diag (ZD , . . . , ZD ). {z } | m(D)-times

Remark 6.4. Note that the expression (13) is not independent of the choice of the orthonormal module basis in H. There is, however, the possibility to define an ε associated to the Hilbert Z-module: let H = HZ with H irreducible, i.e. ρH is irreducible. Then, according to Lemma 3.3, the set of all G-invariant algebraic Hilbert space H0 ⊂ H is given by H0 = HZ with Z ∈ Z unitary. The first step is to select from each class {HZ}Z∈U (Z) exactly one representative H ⊂ H. Now, according to Lemma 6.2 we have for an arbitrary Hλ with λ = ρH the unique decomposition Hλ = ⊕ (ρD , λ) HD . b D∈G Thus, by Hλ := ⊕ (ρD , λ) HD , b D∈G where HD now denotes the representative in the corresponding class {HD Z}Z∈U (Z) , we obtain a unique G-invariant algebraic Hilbert space and on the basis of this choice we define ε(H1 , H2 ) := ε(H1 , H2 ) , where ε(H1 , H2 ) is given by the expression (13). Unfortunately, this definition of ε(H1 , H2 ) depends on the initial choice of the representatives from each class {HZ}Z∈U (Z).

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807

7. The Stabilizer, stab A We start with a first characterization of the elements of stab A. Recall the b and A-scalar product h·, ·iA introduced notions of spectral projection, ΠD , D ∈ G, in Sec. 2. Lemma 7.1. If β ∈ stab A, then the equation β ◦ ΠD = ΠD ◦ β ,

D ∈ Gb ,

(38)

holds and, further, we have β ∈ stab A

iff hβ(F1 ), β(F2 )iA = hF1 , F2 iA ,

F1 , F2 ∈ F .

(39)

Proof. We prove first Eq. (38). Let β ∈ stab A and take F ∈ ΠD F . Then from Proposition 3.1, Proposition 3.2 and Eq. (3) we have β(F ) ∈ A HD Z = A ρD (Z) HD ⊆ A HD = ΠD F and, therefore, (ΠD ◦ β)(F ) = β(F ) for all F ∈ ΠD F . This implies ΠD ◦ β ◦ ΠD = β ◦ ΠD and

D0 6= D, D0 , D ∈ Gb .

ΠD0 ◦ β ◦ ΠD = 0 , For F ∈ F0 we have F =

X

ΠD0 F ,

b D0∈G (formal infinite sum) and using this expression we obtain !! X ΠD0 (F ) = (β ◦ ΠD )(F ) , (ΠD ◦ β)(F ) = ΠD ◦ β ◦ D0

so that Eq. (38) holds. Now, if β ∈ stab A, then β(A) = A for all A ∈ A or β ◦ Πι = Πι . But from Eq. (38) we have also the relation Πι = Πι ◦ β and since ∗ = (Πι ◦ β)(F1 F2∗ ) , hβ(F1 ), β(F2 )iA = Πι β(F1 ) β(F2 ) we obtain hβ(F1 ), β(F2 )iA = hF1 , F2 iA , for all F1 , F2 ∈ F. To prove the other implication suppose that hβ(F1 ), β(F2 )iA = hF1 , F2 iA , F1 , F2 ∈ F, i.e. Πι ◦ β = Πι . Since Πι ◦ Πι = Πι we get Πι ◦ β ◦ Πι = Πι

(40)

¨ ´ H. BAUMGARTEL and F. LLEDO

808

and

D 6= ι , D ∈ Gb .

Πι ◦ β ◦ ΠD = 0 ,

(41)

{ΦD, i }di=1

an orthonormal basis of HD From this we have for D 6= ι, A ∈ A and = hβ(A), ΦD, i iA = hA, β −1 (ΦD, i )iA = A β −1 (ΦD, i ) = 0 , β(A) D, i

because

ι

= Πι ◦ β −1 ◦ ΠD (ΦD, i ) = 0 . β −1 (ΦD, i ) = Πι β −1 (ΦD, i ) ι

This implies

β(A) = β(A)

(β ◦ Πι )(F ) = (Πι ◦ β ◦ Πι )(F ) ,

or ι

F ∈F.

Using Eq. (40) we obtain finally β ◦ Πι = Πι ◦ β ◦ Πι = Πι and β(A) = A for all A ∈ A = Πι F , i.e. β ∈ stab A.

Lemma 7.2. Let Z = (Zi0 i )i0 ,i ∈ Matd (Z) be the matrix corresponding to β ∈ stab A restricted to the Hilbert Z-module H = HZ, where the orthonormal basis {Ψi }di=1 is fixed. Then Z is unitary, i.e. the equations Z∗ Z = ZZ∗ = 1ld hold. In terms of the entries we can equivalently write d X

Zi∗0 i Zi0 j = δij =

i0 =1

d X

∗ Zii0 Zji 0 .

i0 =1

Proof. Write H, H 0 ∈ H as H =

d P

Ψ i Xi , H 0 =

i=1

d P

Ψj Yj , where Xi , Yi ∈ Z,

j=1

i = 1, . . . , d. Then we have on the one hand ∗

0

H H =

d X

Xi∗ Yi ∈ Z

i=1

and on the other hand H ∗ H 0 = β(H ∗ H 0 ) = β(H)∗ β(H 0 )=

X

Xi∗ Zi∗0 i Ψ∗i0 Ψj 0 Zj 0 j Yj =

i,j,i0 ,j 0

Therefore the equation,

d P i0 =1

X

Xi∗ Zi∗0 i Zi0 j Yj .

i,j,i0

Zi∗0 i Zi0 j = δij , holds. The second equation, ZZ∗ =

1ld , follows from Gelfand’s Theorem and from the fact that the scalar matrices (Zi0 i (ϕ))i0 ,i ∈ Matd (C), ϕ ∈ spec Z, are finite-dimensional (cf. with Sec. 3). The preceding lemma says that we can associate to each β ∈ stab A and each Hilbert Z-module H, a unitary module morphism stab A 3 β 7−→ UH (β) ∈ LZ (H) , because β(HZ) = β(H) Z, H ∈ H, Z ∈ Z.

(42)

SUPERSELECTION STRUCTURES FOR

C ∗ -ALGEBRAS WITH

809

NONTRIVIAL CENTER

From the definition of the functor F, given after Corollary 4.6, and from Eq. (28) we know that the elements A ∈ (ρH1 , ρH2 ) determine via F(A), the set of intertwining operators between UH1 (G) and UH2 (G). Next we prove that this intertwining property is still valid for UH (β). Proposition 7.3. Let β ∈ stab A, H1 , H2 be Hilbert Z-module and UH1 (β), UH2 (β) the corresponding unitary module morphisms given in Eq. (42). If A ∈ (ρH1 , ρH2 ), then the following intertwining relation holds: UH2 (β) ◦ F(A) = F(A) ◦ UH1 (β) ,

β ∈ stab A ,

(43)

where F is the functor defined after Corollary (4.6). Proof. First note that Eq. (43) can be rewritten as UH2 (β) (AH1 ) = A UH1 (β) (H1 ) , A ∈ (ρH1 , ρH2 ), H1 ∈ H1 . According to Eq. (42), β Hk acts via UHk (β) ∈ LZ (Hk ) as β(Hk ) = UHk (β)(Hk ), k = 1, 2. Further we have β(AF ) = A β(F ), F ∈ F. Therefore we obtain UH2 (β) (AH1 ) = β(AH1 ) = A β(H1 ) = A UH1 (β) (H1 ) ,

and the proof is concluded.

From Definition 4.1 we can associate to a unitary module morphism UH (β) ∈ LZ (H) an element of the field algebra U\ H (β) ∈ F. Obviously the assignment stab A 3 β 7−→ U\ H (β) , is a unitary representation of stab A in F . Lemma 7.4. The representation stab A 3 β 7−→ U\ H (β) , is continuous, where in stab A we use the topology of pointwise norm convergence and in F the topology given by the C ∗ -norm. Proof. Suppose that (βn )n −→ β, i.e. kβn (F ) − β(F )kF −→ 0 for all F ∈ F. Now if {Ψi }di=1 is an orthonormal basis in H, and using the support property, supp H = 1l, we obtain

! ! d d

X X

\ \ Ψi Ψ∗i − U\ Ψi Ψ∗i

UH (βn ) − U\ H (β) = UH (βn ) H (β)

F i=1

≤

d

X

\

UH (βn )Ψi − U\ H (β)Ψi i=1

=

d X

i=1

F

F

kβn (Ψi ) − β(Ψi )kF ,

i=1

which proves the assertion.

¨ ´ H. BAUMGARTEL and F. LLEDO

810

Next we will show that the unitary operators UH (β) are “generated” by the elements UHD (β), D ∈ Gb (recall Theorem 6.2). Proposition 7.5. Let β ∈ stab A. Then each unitary module morphism UH (β) , where HD = HD Z and HD is is uniquely determined by the family UHD (β) b D∈G b Precisely, if H is the the algebraic Hilbert space corresponding the irreducible D ∈ G. Hilbert Z-module associated to the endomorphism λ ≡ ρH ∈ end A, if {ΦD, i }di=1 is an orthonormal basis of HD and if H3H=

d XX

AD, i ΦD, i ,

AD, i ∈ (ρD , λ) ,

D i=1

is the orthogonal decomposition of H according to Theorem 6.2, then we get XX AD, i UHD (β) (ΦD, i ) . UH (β) (H) = D

i

Proof. Since β ∈ stab A, we have XX AD, i β(ΦD, i ) , β(H) = i

D

and according to Proposition 7.3 the equation XX AD, i UHD (β) (ΦD, i ) , UH (β) (H) = D

i

finishes the proof.

b which Proposition 7.5 justifies that we restrict to the study of UHD (β), D ∈ G, determine completely the morphisms UH (β) for a general Hilbert Z-module H. Denote by UZ (HD ) the set of all unitary module morphisms in LZ (HD ). Proposition 7.6. The mapping, stab A 3 β 7−→ V (β) :=

Y

UHD (β) ∈

D

Y

UZ (HD ) ,

D

is a group monomorphism and a homeomorphism, where in stab A we take the Q UZ (HD ) the Tychonoff product topology same topology as in Lemma 7.4 and in D

generated by the operator norm topology in LZ (HD ). b and, Proof. First note that if V (β1 ) = V (β2 ), then UHD (β1 ) = UHD (β2 ), D ∈ G, therefore, β1 (ΦD, i ) = β2 (ΦD, i ) for all elements of the orthonormal basis ΦD, i ∈ HD , b hence β1 (F ) = β2 (F ), for all F ∈ F0 . Since F = clok·k F0 we get β1 = β2 . D ∈ G, Further, from UHD (β1 ◦ β2 ) = UHD (β1 ) ◦ UHD (β2 ), β1 , β2 ∈ stab A, we obtain Q V (β1 ◦ β2 ) = V (β1 ) ◦ V (β2 ) and V (ι) = 11HD = 1l. D

SUPERSELECTION STRUCTURES FOR

C ∗ -ALGEBRAS WITH

NONTRIVIAL CENTER

811

Second, to prove the homeomorphism property, note that the continuity already follows from Proposition 4.2 (v) and Lemma 7.4. For the rest of the proof we follow arguments given in [5, Lemma 3.2]. Suppose that we have a sequence {βn }n ⊂ stab A such that Y Y UZ (HD ) 3 V (βn ) −→ V ∈ UZ (HD ) . D

D

From the equation βn (ΨD Z) = βn (ΨD ) Z = UHD (βn ) (ΨD ) Z, ΨD ∈ HD , Z ∈ Z, we get that UZ (HD ) 3 UHD (βn ) =: Un −→ UHD =: U ∈ UZ (HD ) w.r.t. the operator norm topology. But from Un−1 − U −1 = Un−1 (U − Un )U −1 , we also have UZ (HD ) 3 Un−1 −→ U −1 ∈ UZ (HD ) . In other words, we have βn (ΨD Z) −→ (UHD )(ΨD ) Z and (ΨD ) Z . βn−1 (ΨD Z) −→ UH−1 D Therefore, we can define for F ∈ F0 β(F ) := limk·kF βn (F ) , γ(F ) := limk·kF βn−1 (F ) ,

F ∈ F0 F ∈ F0 .

Since βn and βn−1 are automorphisms of F the limits β and γ can be extended by continuity to F = clok·kF F0 , respectively. So we have F = (γ ◦ β)(F ) = (β ◦ γ)(F ) ,

F ∈F,

i.e. β ∈ aut F , γ = β −1 and even more β ∈ stab A, so that U = UHD = UHD (β) and V = V (β). Thus V (βn ) −→ V implies βn −→ β, i.e. V (stab A) is closed and finally we have that the assignment β 7−→ V (β) is a homeomorphism. From the preceding result it follows also that if Z is finite-dimensional, then stab A is compact as in the case where Z = C1l. b Recall that if we consider a fixed orthonormal basis {ΦD, i }di=1 of HD , D ∈ G, then UHD (β) corresponds to a matrix ZD (β) = ZD, i0 i (β) 0 ∈ Matd (Z) , HD = HD Z , i ,i

by means of

d X ΦD, i0 ZD, i0 i (β) . UHD (β) (ΦD, i ) = i0 =1

¨ ´ H. BAUMGARTEL and F. LLEDO

812

Using Gelfand’s Theorem we can also interpret ZD (β) as a continuous matrixvalued function on spec Z, i.e. for each ϕ ∈ spec Z we get a unitary scalar matrix ZD (β) (ϕ) ∈ Matd (C). In the next theorem we will characterize the subgroup G ⊂ stab A. Theorem 7.7. Let β ∈ stab A and ZD (β) be the corresponding matrix from b are fixed. Then, Matd (Z), where the orthonormal basis {ΦD, i }di=1 of HD , D ∈ G, β∈G

iff ZD (β) ∈ Matd (C) ,

for all D ∈ Gb .

In other words β ∈ G iff the corresponding functions (ZD (β))(·) are constant unitary matrix functions on spec Z. Proof. Define first the set b S := {β ∈ stab A: ZD (β) ∈ Matd (C) , D ∈ G} and note that S is a subgroup of stab A. Further we have that G ⊆ S (cf. with the remark before Proposition 3.1). We prove the other inclusion G ⊇ S. First note that for β ∈ S ⊆ stab A we have β(H) ⊆ H , for all G-invariant algebraic Hilbert H ⊂ F. Now we consider the set C of all functions G 3 g 7−→ fH2 ,H1 (g) := hH1 , g(H2 )iH = H1∗ g(H2 ) ∈ C1l ,

H1 , H2 ∈ H ,

where H runs through all finite-dimensional and G-invariant algebraic Hilbert spaces in F . Obviously, these functions are continuous on G, i.e. C ⊆ C(G). Further (i) C is closed w.r.t. multiplication, because (H1 H10 )∗ g(H2 H20 ) = H10 ∗ H1∗ g(H2 ) g(H20 ) = H1∗ g(H2 ) H10 ∗ g(H20 ) . (ii) C is closed w.r.t. linear combinations. Let H1 , H2 be given. Then H := W1 H1 +W2 H2 , with W1 , W2 ∈ A, W1∗ W1 = W2∗ W2 = 1l, W1∗ W2 = W2∗ W1 = 0, W1 W1∗ + W2 W2∗ = 1l, is also of the required type, and (W1 H1 + W2 H2 )∗ g(W1 H1 + W2 H2 ) = H1∗ W1∗ g(W1 )g(H1 ) + H2∗ W2∗ g(W1 )g(H1 ) + H1∗ W1∗ g(W2 )g(H2 ) + H2∗ W2∗ g(W2 )g(H2 ) = H1∗ g(H1 ) + H2∗ g(H2 ) , where we have used that g(Wk ) = Wk , since Wk ∈ A, k = 1, 2. (ii) The function c(g) ≡ 1 belongs to C, because C1l is an (irreducible) invariant subspace.

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(iii) G is separated by C, because from g1 H = g2 H for all admissible H we obtain immediately g1 = g2 . (v) The complex-conjugated functions of elements in C belong also to C. Namely, let {Ψi }di=1 be an orthonormal basis of H and {Ψi }di=1 an orthonormal basis of a conjugated space H (w.r.t. G). Then according to d P ∗ Ψi and g(Ψi ) = Ψi0 Ui0 i (g), where (Ui0 i (g))i0 i Sec. 2 we have Ψ∗i = RH i0 =1

is a scalar matrix. Then (g(Ψi ))∗ =

d X

Ψ∗i0 Ui0 i (g)

i0 =1

= g(Ψ∗i )

∗ g RH Ψi

=

∗ g(Ψi ) = RH

∗ RH

=

d X

Ψi0 Vi0 i (g)

i0 =1

=

d X

Ψ∗i0 Vi0 i (g) ,

i0 =1

so that Vi0 i (g) = Ui0 i (g) . Therefore, according to the Stone–Weierstraß Theorem the *-algebra C is dense in C(G) and therefore also dense in L2 (G). Now let β ∈ S, then β(X) ∈ H for X ∈ H. As in [23, pp. 206–207] we define an operator U on C by Uβ (fX,Y ) := fX, β(Y ) . We calculate for Xk ∈ Hk , k = 1, 2: Z Z Uβ (fX1 ,Y1 ) (g) Uβ (fX2 ,Y2 ) (g) dg = fX1 , β(Y1 ) (g) fX2 , β(Y2 ) (g) dg G

G

Z =

(βY1 )∗ g(X1 ) ((βY2 )∗ g(X2 ))∗ dg

G



= (βY1 )∗ 

G

 = β Y1∗  = β Z

Z

Z

 g(X1 X2∗ ) dg  (βY2 ) 

g(X1 X2∗ ) dg Y2 

G

Z

fX1 ,Y1 (g) fX2 , Y2 (g) dg 

G

fX1 ,Y1 (g) fX2 , Y2 (g) dg ,

= G



¨ ´ H. BAUMGARTEL and F. LLEDO

814

where for the fourth equation we have used the relation

R G

g(X1 X2∗ ) dg ∈ A. This

equation expresses the uniqueness of the definition of U as an operator on C and, simultaneously, its isometry property w.r.t. the scalar product in L2 (G). By continuous extension of Uβ to the whole L2 (G) we obtain a unitary operator on L2 (G), which is also denoted by Uβ . Moreover, we have Uβ (fX1 ,Y1 fX2 ,Y2 ) = Uβ (fX1 X2 ,Y1 Y2 ) = fX1 X2 , β(Y1 )β(Y2 )

=

fX1 X2 , β(Y1 Y2 )

=

fX1 , β(Y1 ) fX2 , β(Y2 )

= Uβ (fX1 ,Y1 ) · Uβ (fX2 ,Y2 ) , i.e. Uβ ∈ aut C, hence Uβ ∈ aut C(G). But according to Gelfand’s Theorem the automorphisms of C(G) correspond bijectively to the homeomorphisms of G. Therefore, there is a gβ ∈ G such that if e is the unit element in G the equation (Uβ f )(e) = f (gβ−1 ) ,

f ∈ C(G) , gβ ∈ G .

Hence we obtain for X, Y ∈ H hβ(Y ), XiH = fX, β(Y ) (e) =

Uβ (fX,Y ) (e) = fX,Y gβ−1 = hY, gβ−1 (X)iH

= hY, U (gβ−1 )(X)iH = hY, U (gβ )∗ (X)iH = hU (gβ )Y , XiH = hgβ (Y ), XiH , where g(X) = UH (g)X and UH (g) is unitary for g ∈ G. Recall that F = C ∗(A, {H}), which implies β = gβ . For the next theorem recall Proposition 5.4, Remark 5.5, Remark 6.3, Proposition 7.5 and Proposition 7.6. Theorem 7.8. Let stab A 3 β 7−→ V (β) :=

Y

UHD (β) ∈

D

Y

UZ (HD ) ,

D

be the assignment specified in Proposition 7.6 and choose a fixed orthonormal basis in HD , {ΦD, i }di=1 . The d×d-matrices in Matd (Z) associated to UHD (β) are denoted by ZD = (ZD, i0 i )i0 ,i . Then the following conditions hold ZD1 D2 = ρ−1 D2 (ZD1 ) ⊗ ZD2 , (ZD ) ZtD , 1l = ρ−1 D

D1 , D2 ∈ Gb

D, D ∈ Gb ,

(44) (45)

where ZD1 D2 is given in Remark 6.3 by formula (37) and ρ−1 D2 (ZD1 ) i0 i = 0 ρ−1 D2 (ZD1 , i0 i ) , with i, i = 1, . . . , d. The superindex t denotes the transposed matrix.

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NONTRIVIAL CENTER

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Proof. Since β ∈ stab A is an automorphism, we have on the one hand for i1 = 1, . . . , d1 , i2 = 1, . . . , d2 , XX ΦD1 , i01 ΦD2 , i02 ZD1 D2 , i01 i02 i1 i2 β(ΦD1 , i1 ΦD2 , i2 ) = i01

=

i02

XX i01

i02

(ρD1 ◦ ρD2 )(ZD1 D2 , i01 i02 i1 i2 ) ΦD1 , i01 ΦD2 , i02

and on the other hand β(ΦD1 , i1 ) β(ΦD2 , i2 ) =

XX i01

=

i02

XX i01

i02

ΦD1 , i01 ZD1 , i01 i1 ΦD2 , i02 ZD2 , i02 i2 ρD1 (ZD1 , i01 i1 ) (ρD1 ◦ ρD2 )(ZD2 , i02 i2 ) ΦD1 , i01 ΦD2 , i02 ,

so that we obtain ρD2 ZD1 D2 , i01 i02 i1 i2 = ZD1 , i01 i1 ρD2 ZD2 , i02 i2 . This equation implies (44). Further, from Eqs. (12) and (16) we have X X X ∗ ∗ ∗ ∗ ∗ ∗ ZD, ZD, ΦD, i0 ρ−1 ZD, β(ΦD, i )∗ = i0 i ΦD, i0 = i0 i RD ΦD, i0 = RD i0 i D i0

and also

i0

i0

X ∗ ∗ ΦD, i = RD ΦD, i0 ZD, i0 i . β Φ∗D, i = β RD i0

From these equations we obtain ∗ ZD, ZD, i0 i = ρ−1 i0 i , D

which implies (44).

Taking again into account Remark 6.3 we can formulate the following counterpart of Theorem 7.8. Theorem 7.9. Suppose that the matrices ZD = (ZD, i0 i )i0 ,i ∈ Matd (Z) ,

D ∈ Gb ,

satisfy the properties (44) and (44) of the preceding theorem, where ZD1 D2 is defined in Eq. (37). Then the linear mapping γ: F0 −→ F0 defined by γ(A) := A , A ∈ A , X ΦD, i0 ZD, i0 i , γ(ΦD, i ) := i0

¨ ´ H. BAUMGARTEL and F. LLEDO

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is an automorphism of the *-algebra F0 which can be uniquely extended to an automorphism γ of F , with γ ∈ stab A. Proof. The properties γ(ΦD1 , i1 ΦD2 , i2 ) = γ(ΦD1 , i1 )γ(ΦD2 , i2 ) and γ(ΦD, i )∗ = b ik = 1, . . . , dk , k = 1, 2, i = 1, . . . , d, follow directly from γ(Φ∗D, i ), D1 , D2 , D ∈ G, Eqs. (44) and (44) (cf. the proof of the preceding theorem). Furthermore, γ −1 exists on F0 and is defined by (cf. Lemma 7.2) γ −1 (A) := A , A ∈ A , X ∗ ΦD, i0 ZD, γ −1 (ΦD, i ) := ii0 , i0

so that γ is an automorphism of F0 . It remains to show that γ is isometric on F0 and, therefore, it can be uniquely and isometrically extended to an automorphism of F , also denoted by γ. In this case γ ∈ stab A. In order to prove the isometry of γ we use the A-valued scalar product h·, ·iA and its properties stated in Sec. 2. First we prove the equation

Indeed, put F1

hγ(F1 ), γ(F2 )iA = hF1 , F2 iA , F1 , F2 ∈ F0 . P P = AD1 , i1 ΦD1 , i1 and F1 = BD2 , i2 ΦD2 , i2 . D1 , i1

Then we

D2 , i2

compute hγ(F1 ), γ(F2 )iA X X ∗ = AD1 , i1 hγ(ΦD1 , i1 ), γ(ΦD2 , i2 )iA BD 2 , i2 D1 , i1 D2 , i2

=

X X X D1 , i1 D2 , i2 i01 ,i02

=

X XX 1 ∗ ∗ AD1 , i1 ρD1 ZD BD 0 i00 ZD1 , i0 i1 00 , i 1 1 1 1 , i1 1 d1 0 00

D1 , i1 i1

=

∗ ∗ AD1 , i1 ρD1 ZD1 , i01 i1 hΦD1 , i01 , ΦD2 , i02 iA ρD2 ZD2 , i02 i2 BD 2 , i2

i1

X 1 ∗ AD1 , i1 BD 1 , i1 d1

D1 , i1

= hF1 , F2 iA . Next recall (see e.g. [10, pp. 201–203] that the C ∗ -norm k · kF in the Hilbert C -system {F, G} can be written as ∗

kF kF = kπ(F )kop ,

F ∈F,

12

where k ·kop is the operator norm w.r.t. the norm |F | := hF, F iA of the operator π(F ) on F , defined by π(F ) (X) := XF ∗ , F, X ∈ F .

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Note that we have |γ(F )| = |F | for all F ∈ F0 . Further, we get ∗ π(γ(F ))(X) = X (γ(F )) = γ γ −1 (X) F ∗ = γ −1 (X) F ∗ ,

F ∈ F0 .

This equation implies

π(γ(F ))

op

or, equivalently,

= kπ(F )kop ,

γ(F ) = kF kF , F

F ∈ F0 ,

F ∈ F0 ,

and the proof is concluded.

Finally, we are able to give a characterization of stab A in terms of spec Z, G b more precisely in terms of the and the irreducible endomorphisms, ρD , D ∈ G, continuous mappings fD = fρD : spec Z −→ spec Z which correspond to ρ−1 D (see Remark 5.5). First we consider an element of the group C(spec Z → G). Then to each ϕ ∈ spec Z there corresponds g(ϕ) ∈ G, such that the assignment ϕ 7→ g(ϕ) is continuous. Recall that g(ϕ) acts unitarily on HD , i.e. g(ϕ)(ΦD, i ) =

d X

ΦD, i0 UD, i0 i g(ϕ) ,

i0 =1

where UD, i0 i (g(ϕ))

i0 ,i

is a continuous unitary matrix-valued function on spec Z

which in its turn determine the elements ZD, i0 i ∈ Z, i, i0 = 1, . . . , d via the relation ZD, i0 i (ϕ) := UD, i0 i (g(ϕ)) ,

ϕ ∈ spec Z .

The matrix ZD = (ZD, i0 i )i0 ,i ∈ Matd (Z) is unitary. Next we define a closed subgroup T ⊂ C(spec Z → G). Definition 7.10. The continuous function spec Z 3 ϕ 7−→ g(ϕ) ∈ G is an element of T ⊂ C(spec Z → G) if the following two conditions are satisfied for all ϕ ∈ spec Z: ZD1 D2 (ϕ) = (ZD1 ◦ fD2 )(ϕ) ⊗ ZD2 (ϕ) , D1 , D2 ∈ Gb 1l = ZD ◦ fD (ϕ) ZD (ϕ)t , D, D ∈ Gb ,

(46) (47)

where ZD1 D2 (ϕ) is the matrix-valued function on spec Z associated to ZD1 D2 (see Eq. (37)), the superindex t means the transposed matrix and the functions fD , D ∈ b are given in Remark 5.5. G,

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Note that the fact that g(ϕ) ∈ G already implies that the scalar unitarities g(ϕ) HD1 HD2 and g(ϕ) HD satisfy the Eq. (37). Since Γ is a scalar (hence a constant) unitarity, the matrix ZH corresponding to a given continuous function ϕ 7→ g(ϕ) and to a Hilbert Z-module H, satisfies also Eq. (37). Theorem 7.11. The automorphism β ∈ aut F satisfies β ∈ stab A iff there is a continuous function spec Z 3 ϕ 7−→ g(ϕ) ∈ G , such that the corresponding matrices ZD = (ZD, i0 i )i0 ,i ∈ Matd (Z) satisfy the conditions (46) and (47). Moreover, stab A is isomorphic and homeomorphic to the subgroup T ⊂ C(spec Z → G). Proof. (i) Let β ∈ stab A, so that, according to Theorem 7.8, the associated matrices satisfy the conditions (44) and (44). Then the matrices ZD (ϕ), with ϕ ∈ spec Z fixed, are constant matrix functions satisfying (46) and (47). Therefore, from b Theorem 7.7 there is an automorphism g(ϕ) ∈ G associated to {ZD (ϕ)}D , D ∈ G, and ϕ 7→ g(ϕ) is continuous (note Proposition 7.6)). Using the function fD defined in Remark 5.5 the conditions (44) and (44) can be rewritten in the form of Eqs. (46) and (47) of Definition 7.10. So the function ϕ 7→ g(ϕ) is an element of T . (ii) Conversely, if ϕ 7→ g(ϕ) is an element of T , then, according to the remarks before Definition 7.10, we have the corresponding unitary matrices ZD , that satisfy by assumption the conditions (46) and (47), which can be rewritten in the form (44) and (44) of Theorem 7.8. Then, by Theorem 7.9, they define an automorphism β ∈ stab A. The bijection between stab A and T , stab A ↔ T , is an isomorphism and an homeomorphism by Theorem 7.7. References [1] J. E. Roberts, Symp. Math. 20 (1976) 335. [2] S. Doplicher and J. E. Roberts, Bull. Amer. Math. Soc. 11 (1984) 333. [3] S. Doplicher and J. E. Roberts, “C ∗ -algebras and duality for compact groups: Why there is a compact group of internal gauge symmetries in particle physics”, in Proceedings of the International Conference of Mathematical Physics (Marseille 1986), eds. M. Mebkhout and R. S´eneor, World Scientific, Singapore, 1987. [4] S. Doplicher and J. E. Roberts, J. Funct. Anal. 74 (1987) 96. [5] S. Doplicher and J. E. Roberts, J. Operator Theory 19 (1988) 283. [6] S. Doplicher and J. E. Roberts, Ann. Math. 130 (1989) 75. [7] S. Doplicher and J. E. Roberts, Invent. Math. 98 (1989) 157. [8] S. Doplicher and J. E. Roberts, Commun. Math. Phys. 131 (1990) 51. [9] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Springer Verlag, Berlin, 1987. [10] H. Baumg¨ artel and M. Wollenberg, Causal Nets of Operator Algebras. Mathematical Aspects of Algebraic Quantum Field Theory, Akademie Verlag, Berlin, 1992. [11] R. Longo and J. E. Roberts, K-Theory 11 (1997) 103. [12] K. Fredenhagen, K.-H. Rehren and B. Schroer, Rev. Math. Phys. Special Issue, 113 (1992). [13] H. Baumg¨ artel, Math. Nachr. 172 (1995) 15. [14] H. Baumg¨ artel, Math. Nachr. 161 (1993) 361.

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[15] K. Shiga, J. Math. Soc. Japan 7 (1955) 224. [16] D. E. Evans and T. Sund, Rep. Math. Phys. 17 (1980) 299. [17] H. Baumg¨ artel, Operatoralgebraic Methods in Quantum Field Theory. A series of Lectures, Akademie Verlag, Berlin, 1995. [18] N. E. Wegge-Olsen, K-Theory and C ∗ -Algebras, Oxford Science Publ., Oxford Univ. Press, Oxford, 1994. [19] J. Dixmier, C ∗ -algebras, North Holland Publishing Co., Amsterdam, 1977. [20] E. C. Lance, Hilbert C ∗ -modules. A Toolkit for Operator Algebrists, Cambridge Univ. Press, Cambridge, 1995. [21] A. Y. Helemskii, Banach and Locally Convex Algebras, Claredon Press, Oxford, 1993. [22] H. Araki, “Bogoliubov automorphisms and Fock representations of the canonical anticommutation relations” in Operator Algebras and Mathematical Physics (Proceedings of the summer conference held at the University of Iowa, 1985), eds. P. E. T. Jorgensen and P. S. Muhly, Amer. Math. Soc., Rhode Island, 1987. [23] M. Takesaki, “Operator algebras and their automorphism group”, in Operator algebras and group representations (Proceedings of the international conference held in Neptune, Romania, 1980), eds. G. Arsene et al., Pitman Monographs and Studies in Mathematics Vol. 18, Boston, 1984.

GEOMETRIC REALIZATIONS OF REPRESENTATIONS OF FINITE LENGTH CHARLES H. CONLEY Mathematics Department University of North Texas Denton TX 76203, USA E-mail : [email protected] Received 25 February 1997 1991 Mathematics Subject Classification: 22E45 Let G = H × Rn be a semidirect product Lie group. We reduce the problem of deciding which indecomposable representations of G may be realized in subquotients of spaces of sections of vector bundles over infinitesimal neighborhoods of orbits of H in the dual of Rn to a problem involving only representations of the H-stabilizers of the orbits.

0. Introduction Let P be the Poincare group, SL2 C ×s R1,3 , and let X0+ be the forward light cone, a massless orbit of SL2 C in the dual of R1,3 . The Gupta–Bleuler theory of quantum electrodynamics is based on the natural representation, W0 , of P in the space of R1,3 -valued functions on X0+ . It is neither unitary nor irreducible; it is an indecomposable representation that is induced from the inhomogeneous stabilizer, or little group, of X0+ . In [9], G. Rideau considered the problem of constructing alternate theories of quantum electrodynamics based on other indecomposable representations W having the same composition series as W0 . It is a well known consequence of the Mackey machine that all irreducible representaions of semidirect product groups such as P are induced, but this project led Rideau to discover that there are non-induced indecomposable representations of P. He found that W0 admits deformations whose moduli space is CP1 , none of which is induced, and at the end of [9] he proposed a general study of non-induced indecomposable representations of semidirect products, which is the framework into which this paper fits. Let G = H ×s A be the semidirect product Lie group formed by a real Lie group H acting linearly on a real vector group A, and assume that the orbits of the dual action of H on A∗ are locally closed. Fix an orbit O of H in A∗ and a point p0 in O, and let S be the subgroup of H stabilizing p0 . Given any representation σ of S in a finite dimensional complex vector space V , let IndH S σ be the representation of H induced by σ in the space Cc∞ (O : H ×S V ) of smooth compactly supported sections of the H-vector bundle over O associated to V . For any element 821 Reviews in Mathematical Physics, Vol. 9, No. 7. (1997) 821–851 c World Scientific Publishing Company

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p of A∗ , let eip denote the character a 7→ exp iha, pi of A, and note that σ ⊗ eip0 is a representation of the inhomogeneous stabilizer S ×s A in the space V . Here we consider smooth representations of finite length of G, having a topologically split ip0 ). We will view composition series of representations of the form IndG SA (σ ⊗ e such composition series representations as acting in the space Cc∞ (O : H ×S V ), where the action of H is the induced one and the action of a ∈ A is multiplication by the character function ξa : p 7→ exp iha, pi. The first general results on this category of representations were obtained by A. Guichardet. In [6], he proved (among many other things) that it contains no indecomposable representations having in their composition series representations associated to more than one orbit O, and so in this paper we restrict our study to the category ExtG O of smooth topologically split representations of G whose composition series elements are all induced from the stabilizer of O. This category is defined more precisely in Sec. 2. In [7], he proved that when the tangent bundle T O admits an H-covariant complementary bundle in the trivial bundle O × A∗ over O, all representations in ExtG O are induced and ExtG O is categorically isomorphic to the category of representations of finite length of S ×s (Tp0 O)⊥ , such that Tp0 O⊥ acts by the character eip0 on each element of the composition series. This is a very strong result which completes the study proposed by Rideau for such orbits, which are generic for most G. For example, if G = P then Guichardet’s theorem applies to all orbits except the two light cones. Our approach stems from the work of Cassinelli, Truini, and Varadarajan, who discovered that the non-induced deformations of W0 found by Rideau may be realized in subquotients of the space of R1,3 -valued functions on the first-order infinitesimal neighborhood of the light cone in its ambient space (R1,3 )∗ [2]. This suggests that there is a generalization of the Mackey machine which realizes the representations from ExtG O in H-vector bundles over infinitesimal neighborhoods of O in its ambient space A∗ . In order to decide this question, it will be necessary to define the appropriate notion of homogeneous vector bundles over infinitesimal neighborhoods. Using the results of the present paper, we have proven in [5] the following preliminary version. Let Cc∞ O(n) be the the functions on the nth-order infinitesimal neighborhood of O, which carry a natural representation V (n) of G. Under the assumption that H is an algebraic group whose finite dimensional representations are all rational, any element of ExtG O of length n + 1 may be realized as a subquotient of the representation V (n) ⊗ V , where V is some finite dimensional representation of H, extended trivially to G. The proof of this depends on the results of [3], [4], and as mentioned, this paper. The main result of [4] is the construction of a category equivalence F from ExtG O to a certain subcategory C of the finite dimensional representations of SA. The definition of C is complicated: if U is an object of ExtG O, then as an object of C, the representation FU of SA is given with several additional structures. Our results here are the proof that the representation V (n) ⊗ V above lies in ExtG O, the calculation of F (V (n) ⊗ V ), and our main theorem, which gives a condition for U to be a subquotient of V (n) ⊗ V in terms of F U and F (V (n) ⊗ V ).

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In order to handle the extra structures attached to F U, we have found it convenient to define a notion of “representations of the infinitesimal neighborhood S ∞ of a Lie subgroup S in its ambient group H” (we remark that we do not feel that these objects are worth study for their own sake). It turns out that F U is a representation of (SA)∞ , where the ambient group is G, and that it can be restricted to a representation of S ∞ , where the ambient group is H. Any representation V of H can also be restricted to a representation of S ∞ , and our main theorem is that U is a subquotient of V (n) ⊗ V if, as a representation of S ∞ , F U is a subquotient of V . It is well known that if S is an algebraic subgroup of an algebraic group H, then any rational representation of S is a subquotient of the restriction of a rational representation of H to S; see for example [1]. In [5] we prove the analog of this with S ∞ replacing S. Coupled with the main theorem of this paper, this gives the subquotient theorem of [5] stated above. Our results are organized as follows. In Sec. 1 we define representations of S ∞ , and in Sec. 2 we review our prior results in [3] and [4]. In Sec. 3 we link subquotients in ExtG O to subquotients in C, and in Sec. 4 we associate a representation of S ∞ to each object in C. In Sec. 5 we give a precise definition of Cc∞ O(n) , and Sec. 6 we show that V (n) is in ExtG O. In Sec. 7 we give explicit formulae for F (V (n) ⊗ V ), in Sec. 8 we prove the main theorem described above, and in Sec. 9 we apply it to Rideau’s original example. We thank the UC Berkeley Mathematics Department, and in particular J. A. Wolf, for their support while these results were obtained. 1. Representations of Neighborhoods of Subgroups We begin with some definitions which are motivated by our main results. Throughout this paper, we will use gothic letters for complexified Lie algebras of real Lie groups, and we will write U(h) and Ur (h) for the universal enveloping algebra of a Lie algebra h and its standard filtration, respectively. Wherever it is convenient, we will write 1 for the identity map of any set with itself, and if π is a representation of some group K 3 k we will write πk for π(k) whenever it abbreviates the notation. Definition. Let H be a real Lie group, and let S be a Lie subgroup. A representation of the infinitesimal neighborhood of S in H, or more briefly a representation of S ∞ , is a complex finite dimensional vector space W0 , a representation σ of S on W0 , an S-invariant subspace W1 of W0 , and for each X ∈ h a linear map σ(X) : W1 → W0 such that 1. Let dσ : s → End(W0 ) be the differential of σ|S . Then for all X ∈ s, dσ(X)|W1 = σ(X). 2. For all s ∈ S and X ∈ h, σ(s) ◦ σ(X) ◦ σ(s−1 ) = σ(Ads X) on W1 . 3. For all X, Y ∈ h, [σ(X), σ(Y )] = σ([X, Y ]) wherever the left hand side is defined, i.e., on the intersection of the inverse images of W1 under the maps σ(X) and σ(Y ). We will often refer to such a representation simply as σ, W , where it is understood that W denotes the flag W0 ⊃ W1 and σ denotes the actions of both S and h.

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Associated to any representation σ, W of S ∞ is a flag W0 ⊃ W1 ⊃ W2 ⊃ . . . which extends the given flag W0 ⊃ W1 , defined inductively by \ σ(X)−1 (Wk−1 ) , Wk = X∈h

the intersection of the inverse images of Wk−1 under σ(X) as X runs over h. It follows from an inductive argument that Wk is indeed a subspace of Wk−1 for all k, and another inductive argument using condition (2) in the definition of representations of S ∞ shows that Wk is S-invariant for all k ≥ 0. The flag {Wk } has the property that σ(X)Wk ⊂ Wk−1 for all X ∈ h, and in fact its definition is the same as defining it to be the finest such flag. It follows from conditions (1) and (3) above that for each Z in U(s)Ur (h), there is a map σ(Z) : Wr → W0 such that σ(Z)(Wk ) ⊂ Wk−r for all k, and σ(Z)σ(Z 0 ) = σ(ZZ 0 ) wherever both sides are defined, for all Z and Z 0 in U(h). The point is that for X1 , . . . , Xp in h, we may define, σ(X1 . . . Xp ) : Wp → W0 to be σ(X1 ) . . . σ(Xp ); this definition is unambiguous by condition (3) and the Poincare–Birkhoff–Witt (PBW) theorem. Furthermore, if all but r of the Xi ’s lie in s, σ(X1 . . . Xp ) is defined from Wr to W0 . The representations of S ∞ form a category, for which we write RS ∞ . A morphism T from an object σ, W to an object σ 0 , W 0 is a linear map T : W0 → W00 such that T (W1 ) ⊂ W10 and T intertwines both the S and h actions. An inductive argument shows that T respects the flags associated to W and W 0 , i.e., T (Wk ) ⊂ Wk0 for all k ≥ 0. We now generalize some of the standard constructions for representations of S to RS ∞ . There is an obvious notion of subobjects in RS ∞ . If σ, W is as above and V = {V0 ⊃ V1 } is a subflag of W0 ⊃ W1 , such that Vk is an S-invariant subspace of Wk for k = 0, 1 and σ(X)(V1 ) ⊂ V0 for all X ∈ h, then V is a subobject of σ, W . In this case one checks that for all k ≥ 0 the associated flag space Vk is an S-invariant subspace of Wk . There is also a notion of quotient objects, but only certain subobjects are permitted as factors. Specifically, if σ, W and V are as above and V1 = V0 ∩ W1 , then we define the quotient representation σ, W/V of S ∞ by (W/V )k = Wk /Vk for k = 0, 1 and σ the obvious quotient map. The property V1 = V0 ∩ W1 is necessary and sufficient to insure that (W/V )1 is canonically a subspace of (W/V )0 . We will refer to this property as “the quotient property”. An inductive argument shows that if V has this property then Vk = Wk ∩ V0 for all k, and also that Wk /Vk is canonically a subspace of (W/V )k for all k. Note that in general (W/V )k may not be equal to Wk /Vk . Our first lemma shows that with these definitions morphisms have the usual property. The proof is left as an exercise. Lemma 1.1. Let σ, W and σ 0 , W 0 be representations of S ∞ , and let T : W → W 0 be a morphism. Then the subflag T (W ) of W 0 is σ 0 -invariant, and so it is a subobject of σ 0 , W 0 . The kernel subflag Kk = T −1 (0) ∩ Wk (k = 0, 1) of W is a subobject of σ, W having the quotient property, and T factors through to an isomorphism T : W/K → T (W ).

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Retain σ, W and σ 0 , W 0 as representations of S ∞ . We define the direct sum object σ ⊕ σ 0 , W ⊕ W 0 to be the obvious direct sum action on the flag (W ⊕ W 0 )k = object σ ⊗ σ 0 acting in the flag Wk ⊕ Wk0 . We also define the tensor product 0 0 0 0 (W ⊗ W )k = Wk ⊗ Wk , k =0 0, 1, by σ ⊗ σ (s) = σ(s) ⊗ σ (s) for s ∈ S, and 0 σ ⊗σ (X) = σ(X)⊗1⊕1⊗σ (X) for X ∈ h. It is easy to check that this definition makes σ ⊗ σ 0 a representation of S ∞ . Here (W ⊗ W 0 )k ⊃ Wk ⊗ Wk0 for all k ≥ 0, but in general the containment may be proper. This tensor product has the usual properties of associativity and commutativity, and the trivial object W0triv = W1triv = C, σ triv |S = 1, σ triv |h = 0 is a unit for it. For us, its most important property is that W ⊗ W 0 contains a subobject V defined by X Wi ⊗ Wj0 , k = 0, 1 . Vk = i+j=n−k

P If k ≥ 2, the associated flag space Vk is easily seen to contain the space i+j=n−k Wi ⊗ Wj0 , although the containment may be proper. A few comments are in order regarding the category RS ∞ . First, we do not have dual objects, because for an object σ, W of RS ∞ , W1∗ is not a subspace of W0∗ . Also, it is not possible to divide by subrepresentations unless they have the quotient property, and so RS ∞ is not an abelian category. Finally, it is not clear which objects should be thought of as irreducible. Objects with no subrepresentations of any kind take the form that W0 is an irreducible representation of S and W1 = 0, but objects with no subrepresentations that have the quotient property are more difficult to describe. Thus we suspect that the definition of RS ∞ can be improved upon, but it will do for our applications. We will conclude this section with some examples of objects of S ∞ that will be useful later on. Examples. The (complex finite dimensional) representations of S sit inside RS ∞ as the full subcategory of objects σ, W such that W1 = 0. At the opposite extreme, the ordinary (S, h)-representations sit inside RS ∞ as the subcategory of objects σ, W such that W1 = W0 ; these representations of S ∞ are called restrictions of representations of S, h. We will need the following lemma in Sec. 9. Lemma 1.2. If σ ˜ is a representation of S on a space W0 and W1 = W0S is the subspace of invariant vectors, then there is a representation σ of S ∞ on W0 ⊃ W1 defined by σ|S = σ ˜ and σ|h = 0. ˜ = π|S , let V1 = V0S , and let If π is a representation of H on a space V0 , let σ ∞ σ be the representation of S on V0 ⊃ V1 defined as in the last paragraph. Let V˜1 be a linearly isomorphic copy of V1 , and define a representation π ˜ of H in V0 ⊕ V˜1 ∞ by π ˜ = π ⊕ 1. Then σ is isomorphic to an S -subquotient of the restriction of π ˜ ˜ |S ∞ . to the S ∞ representation π Proof. The first sentence is easy. For the second, let j : V1 → V˜1 be an ˜ |S ∞ acts in isomorphism, and let Y0 = {(v, jv) : v ∈ V1 } ⊂ V0 ⊕ V˜1 . By definition, π the flag V0 ⊕ V˜1 ⊃ V0 ⊕ V˜1 , and the flag V0 ⊕ V˜1 ⊃ V˜1 is an S ∞ -subrepresentation

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of π ˜ |S ∞ on which h acts trivially (and which does not have the quotient property). Furthermore, the flag Y0 ⊃ 0 is an S ∞ -subrepresentation of V0 ⊕ V˜1 ⊃ V˜1 which does have the quotient property, and one checks that the quotient is isomorphic to σ. Note that σ is usually not a subrepresentation of π|S ∞ , because V1 is usually not V0H . If N, π is any representation of S and n is any non-negative integer, then we may define a representation W, σ of S ∞ by W0 = Un (h) ⊗s N , W1 = Un−1 (h) ⊗s N , σ|S = Ad ⊗ π, and σ|h the usual left action on the first factor. In this case it is an exercise to prove that Wk = Un−k (h) ⊗s N for k ≤ n. An interesting example for us is the following. Let A be a real representation of H, let O be the orbit under H of a point p0 in the dual A∗ of A, and let S be the stabilizer of p0 . Assume that O is locally closed in A. Let ∆np0 A∗ be the space of differential operators of order ≤ n on A∗ supported at p0 , so that elements of ∆np0 A∗ are linear maps δ : Cc∞ (A∗ ) → C such that δf depends only on the n-jet ∗ n ∗ of f at p0 . Then there is a natural (S, h)-action on ∆∞ p0 A such that S leaves ∆p0 A n+1 ∗ invariant and h maps it into ∆p0 A , and so we get for each n a representation W ∗ of S ∞ such that Wk = ∆n−k p0 A for k = 0, 1. Later we will develop a more explicit ∗ description of W , and prove that in fact Wk = ∆n−k p0 A for all k. 2. Review of Prior Results Throughout this paper fix G = H ×s A and O, p0 , and S as in the introduction. In this section we define the category of smooth representations of G of finite length with a topologically split composition series as described in the introduction, and we recall the main results of [3] and [4]. We will use Schwartz’ notations D and E in place of Cc∞ and C ∞ , and if B is a vector bundle over a space O then D(B) and E(B) will always denote the sections of B, equipped with their usual topologies of uniform convergence of all derivatives on all compact sets [10]. We will write DB instead of D(B) whenever it simplifies the notation, and similarly for E. We will write B(p) or Bp for the fiber of B at p ∈ O. Definition. Let ExtG O be the category of smooth representations of G in topological vector spaces V that admit a G-invariant flag V = V0 ⊃ . . . ⊃ Vn ⊃ Vn+1 = 0 with the following properties. First, for each i there is a closed subspace of Vi complementary to Vi+1 , and second, each subquotient Vi /Vi+1 is topologically equivip0 ) for some finite dimensional complex alent to the representation IndG SA (σ ⊗ e representation σ of S. Morphisms are continuous linear intertwining maps. Henceforth fix a real subbundle C of the vector bundle O × A∗ over O that is complementary to the tangent bundle T O. As we said in the introduction, when C can be chosen to be an H-bundle Guichardet has completely described ExtG O [7], and so our previous work was concerned mainly with the exceptional cases where this cannot be done. However, it applies in all cases, and so what follows is general.

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Associated to C there is a full subcategory ExtC G O of ExtG O such that the inclusion functor is an equivalence of categories, and which is much easier to study. Definition. An object of the category ExtC G O is a representation U in ExtG O, together with an integer n and an ordered set σ 0 , . . . , σn of finite dimensional representations of S in spaces Vi , which are related to U as follows. Let Fi be the i ip0 ) H-vector bundle H ×S Vi over O, and let U i be the representation IndG SA (σ ⊗ e acting in DFi as in the introduction. L 1. The representation U acts in the topological vector space n0 DFi , and so for all g ∈ G we may view Ug as a matrix with entries Ugij : DFj → DFi . Ln 2. The flag j DFi , 0 ≤ j ≤ n, is G-invariant, and U defines the representations U j in the subquotients DFj . In other words, for all g in G the matrix Ugij is lower triangular, with diagonal entries Ugii = Ugi . 3. For all a ∈ A, Uaij is a smooth section of Hom(Fj , Fi ), i.e., Ua acts fiberwise. It has the form Ua = ξa exp la , where ξa is defined in the introduction and la is Ln a continuous endomorphism of 0 DFi depending linearly on a such that laij is 0 for i ≤ j and is a smooth section of Hom(Fj , Fi ) for i > j. Furthermore, for all p ∈ O, la (p) = 0 for all a in the subspace Cp⊥ of A that annihilates Cp . Note that exp la is polynomial in la , as la is nilpotent. Let U, F0 , . . . , Fn and U 0 , F00 , . . . , Fn0 0 be two objects of ExtC G O. A morphism between them is a continous linear map T :

n M 0

0

DFi →

n M

DFi0

0 0

intertwining the actions of U and U , such that the matrix entries T ij : DFj → DFi0 are smooth sections of Hom(Fj , Fi0 ), i.e., such that T acts fiberwise. Theorem 2.1. The forgetful functor from to ExtC G O to ExtG O which maps an object U, F0 , . . . , Fn to U is an equivalence of categories. If U, F0 , . . . , Fn is any ij object of ExtC G O, then for all h ∈ H and 0 ≤ j ≤ i ≤ n the matrix entry Uh is an order ≤ i − j differential operator from Fj to Fi above the diffeomorphism ij is an order ≤ 1 differential h : O → O, and for all X ∈ h the matrix entry UX operator above the identity map from O to O. Proof. A proof of this theorem is given mainly in [3], but first see Sec. 2 of [4]. By a differential operator of order ≤ r from Fj to Fi above the diffeomorphism h : O → O, we mean a continuous linear map D : DFj → DFi such that for any s ∈ DFj and p ∈ O, Ds(p) depends only on the rth jet of s at h−1 p. For a more detailed definition, see Sec. 1 of [4]. This theorem is the starting point of [4]. It shows that any object U, F0 , . . . , Fn of ExtC G O acts locally, and this may be used to associate to U a representation of

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SA. This is complicated because H can act by differential operators, and so U may not be induced, but Theorem 2.1 allows us to proceed as in Secs. 3 and 4 of [4], which we will now review. For any vector bundle B over O, let ∆r B be the vector bundle over O whose sections are differential operators of order ≤ r from DB to DO. Let λ be the quasiregular representation of H on DO, and extend it to G by making A act so that λa is multiplication by the character function ξa . Fix an object U, F0 , . . . , Fn of ExtC G O, and for 0 ≤ l ≤ k < ∞ define vector bundles ˜kl (U) = E

l M

∆k−i Fi .

i=0

˜kl (U), and similarly for all the other maps We will usually write simply E˜kl for E ˜kl ⊂ E˜k0 l0 if k ≤ k 0 and l ≤ l0 , so that and spaces we will associate to U. Note that E ˜ ˜ {Ekl } is a biflag. A section D of Ekl is a differential operator from ⊕ni=0 DFi to DO in an obvious way, and its order is graded in a way compatible with the grading of the order of the differential operator Uh given by Theorem 2.1. Consequently, ˜kl , and we define a representation ∆U of G on λh ◦ D ◦ Uh−1 is again a section of E ˜ the biflag {DEkl } by ∆Ug (D) = λg ◦ D ◦ Ug−1 . It is easy to see that ∆U is induced from SA, and we will write σ(U) or simply σ ˜kl (p0 )}. This is for the inducing representation, which acts in the biflag of fibers {E the representstion of SA which is associated to U in [4]. 0 0 0 There is a similar construction for morphisms in ExtC G O. Let U , F0 , . . . , Fn0 be 0 0 ˜0 ˜ a second object of ExtC G O, and write Ekl for Ekl (U ), and so on. If T : U → U 0 is a morphism, then there is an induced morphism ∆T : ∆U → ∆U defined by ∆T (D) = D ◦ T , i.e., ∆T is the transpose T T of T . Since morphisms in ExtC GO 0 into DE˜k+n,n for all 0 ≤ l ≤ k. We act fiberwise, we find that ∆T maps DE˜kl will write µ(T ) or simply µ for the inducing morphism, and we will see later in this section that it is closely related to T (p0 )T . When n = n0 and T maps the invariant subspace ⊕ni=j DFi of U into the invariant subspace ⊕ni=j DFi0 of U 0 for 0 ≤ j ≤ n, ˜ 0 ) ⊂ DE˜kl . we say that T is a flag morphism. In this case we find that ∆T (DE kl The maps U 7→ σ(U) and T 7→ µ(T ) make up a functor from ExtC G O to the finite dimensional representations of SA, which is easily seen to be faithful. However, in order to have a useful “little group method” for ExtC G O it is necessary to determine the range of this functor. This is done in [4]; the essential idea is that the repre˜nn (p0 ) = ⊕n ∆n−i Fi (p0 ) is determined by its restriction sentation σ of SA on E i=0 to the non-invariant subspace ⊕ni=0 Fi∗ (p0 ). To be more precise, we need several definitions. Let S 0 = SA, s0 = s⊕ a, and Ur = Ur (g)U(s0 ) for 0 ≤ r. Choose a subspace r of h complementary to s and an ordered basis r1 , . . . , rq of r, and let Rr be the subspace Pq I of Ur (g) spanned by the monomials rI = r1I1 · · · rqq such that |I| = 1 Ii ≤ r. Note that Ur = U(s0 )Ur (g) = U(s0 )Rr .

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˜kl (p0 ), and let Nk (U) = ⊕k Li . Define biflags Let Li (U) = Fi∗ (p0 ), let Ekl (U) = E 0 Jkl (U) =

l M 0

Li ⊗ Uk−i ,

Mkl (U) =

l M

Li ⊗ Rk−i .

0

An element ω ∈ Nn may be viewed as an order 0 differential operator at p0 on ⊕n0 DFi , and one checks that if ω ∈ Nl and Z ∈ Uk are such that ω ⊗ Z is in Jkl , then ω ◦ UZ is in Ekl . Let ρ(U) : Jkl → Ekl be the resulting map of biflags ω ⊗ Z 7→ ω ◦ UZ . It turns out that ρ(Jkl ) = Ekl and that ρ maps Mkl isomorphically onto Ekl , and so we may let q(U) : Ekl → Mkl be the right inverse of ρ. In order to understand the way in which σ is determined by its restriction to Nn , we will use some rather cumbersome constructions. Define a map η(U) : S 0 → HomC (Nn , Mnn ) by ηs = q ◦ σs |Nn for s ∈ S 0 . Note that Nn is a subspace of both Enn and Jnn , and that ρ|Nn = 1, so ηs may also be written as q ◦ σs ◦ ρ|Nn . Let dη : s0 → HomC (Nn , Mnn ) be the differential, so that for X ∈ s0 , dηX = q ◦ σX |Nn = q ◦ σX ◦ ρ|Nn . We will usually write simply η for dη. For δ ∈ Jkl and Z ∈ Ur , let δ · Z ∈ Jk+r,l be the obvious product: when P δ = ω ⊗ Z 0 , δ · Z = ω ⊗ Z 0 Z. Define a map N (U) : S 0 → EndC ( kl Jkl ) by Ns (ω ⊗ Z) = (ηs ω) · Ads Z . We will write N for the differential dN , so that if X ∈ s0 then NX (ω ⊗ Z) = ηX ω · Z + ω ⊗ adX Z. Roughly, we may think of N as η ⊗ Ad, but this is imprecise because η ⊗ 1 and 1 ⊗ Ad do not commute. Let r be the right regular representation of U(s0 ) on the biflag {Jkl }, so that if Z ∈ U(s0 ) and δ ∈ Jkl , then rZ δ = δ · Z T , where Z 7→ Z T is the usual antiautomorphism of U. It is easy to check that η maps Nk into Mkk for 0 ≤ k ≤ n, that N and r both leave Jkl and also the space Kkl defined below invariant for 0 ≤ l ≤ k, and that for s ∈ S 0 and X ∈ s0 we have ρ ◦ Ns = σs ◦ ρ and ρ ◦ rX = σX ◦ ρ. However, N is not usually a representation of S 0 on Jkl . Let Kkl be the subspace of Jkl defined by Kkl = {ηX ω · Z + ω ⊗ XZ : ω ∈ Ni , Z ∈ Uk−i , X ∈ s0 , 0 ≤ i ≤ l} . It is not too hard to prove that Kkl = kernel ρ|Jkl , and hence that Jkl = Mkl ⊕ Kkl . It follows from this that for any r, s ∈ S 0 and X, Y ∈ s0 , Nr Ns − Nrs and [NX , NY ] − N ([X, Y ]) both map Nk into Kk−1,k−1 for 0 ≤ k ≤ n. We have seen that starting out knowing only the spaces Li and the map η, we can construct the biflags Jkl , Mkl , and Kkl , and the map N . Since ρ is essentially the quotient map which divides by Kkl , and ρ intertwines N and σ, we can reconstruct the representation σ from η and the Li . This will serve as motivation for the definition of the category C of representations of SA to which ExtC G O is isomorphic. Definition. An object σ of the category C is a representation of S 0 = SA, along with the following additional structures. First, we are given an integer n(σ) ≥ 0

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and n(σ) + 1 complex finite dimensional vector spaces L0 (σ), . . . , Ln(σ) (σ). From these we form biflags Jkl (σ) and Mkl (σ) and a flag Nk (σ), exactly as above. We will usually suppress the argument σ on all the maps and spaces associated to σ. Second, we are given asmooth map η(σ) : S 0 → HomC (Nn , Mnn ), such that ηs (Nk ) ⊂ Mkk P for all s ∈ S 0 . From η we form a map N (σ) : S 0 → EndC kl Jkl and another biflag Kkl (σ), again exactly as above, and we also write η and N for the differentials of η and N at e ∈ S 0 . It is easy to check that N leaves both Jkl and Kkl invariant. The map η is required to have the following properties. 1. For all r, t in SA, Nr ◦ Nt − Nrt maps Nk into Kk−1,k−1 for 1 ≤ k ≤ n, and it maps N0 to 0. 2. There is a linear map L(σ) : A → End(Nn ) such that for all a ∈ A, ηa = e−iha,p0 i exp L−a . Furthermore, for all a ∈ A, La (Nk ) ⊂ Nk−1 for 1 ≤ k ≤ n, La (N0 ) = 0, and if ha, Cp0 i = 0, La = 0. Note that La is nilpotent, and that ηa leaves Nk invariant. Property 2 derives from Property 3 in the definition of ExtC G O. In [4] we prove (with considerable difficulty) that Property 1 is equivalent to requiring that Jkl = Mkl ⊕ Kkl . It follows that Jkl /Kkl is naturally a subspace of Jk0 l0 /Kk0 l0 for k ≤ k 0 and l ≤ l0 , and so we may define a biflag Ekl (σ) = Jkl /Kkl . As before, let ρ(σ) : Jkl → Ekl be the quotient map, and let q(σ) : Ekl → Mkl be its right inverse. Since P N leaves Kkl invariant, it factors through ρ to a map from S 0 to End Ekl . It is immediate from Property 1 that this map is a representation leaving Ekl invariant, and we define σ itself by making it equal to this representation. Note that if we are given only η and the Li , we can reconstruct σ. In order to define morphisms in C, let σ 0 be another object and denote all the associated maps and spaces by primes. A morphism µ : σ → σ 0 is a linear P 0 P Ekl intertwining σ and σ 0 , given along with a linear map map µ : Ekl → 0 τ (µ) : Nn → Nn0 such that ρ ◦ (τ ⊗ 1) = µ ◦ ρ. Here τ ⊗ 1 is the obvious P endomorphism of Jkl , and we have suppressed the argument µ of τ as usual. The condition that µ be an S 0 -map is equivalent to (τ ⊗ 1)◦ηs = ηs0 ◦τ for all s ∈ 0 0 S , and it is a consequence of the relation between µ and τ that µ(Ekl ) ⊂ Ek+n 0 ,n0 . 0 0 0 and If n = n and τ (Nk ) ⊂ Nk for 0 ≤ k ≤ n, then one checks that µ(Ekl ) ⊂ Ekl we say that µ is a flag morphism. That concludes the definition of C. It is clumsy, but it is useful for computations. The reader may ask why we do not replace the strange Property 1 with the requirement that Jkl = Mkl ⊕ Kkl ; the reason is that Property 1 is more useful for treating the types of examples in [4]. We remark that Properties 1 and 2 together lead to the fact that for any X ∈ s0 , ηX maps Nk into Nk + (Nk−1 ⊗ R1 ), which is expected from the form of UX given in Theorem 2.1. The following theorem is the main result of [4]. Theorem 2.2. There is a functor F : ExtC G O → C defined by F (U) = {σ(U), η(U), L0 (U), . . . , Ln (U)}

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and F (T ) = {µ(T ), τ (T )}, where τ (T ) is the transpose T (p0 )T of T (p0 ). This functor is a contravariant equivalence of categories. Definition. We conclude our review by defining an extension of any object σ of C to a representation of S 0∞ (see Sec. 1), where S 0 is viewed as a subgroup of G. We will write also σ for this extension. The flag for σ as a representation of S 0∞ is Enn ⊃ En−1,n−1 , and for X ∈ g, σX is obtained by factoring the right P regular representation r of g on Jkl = Nn ⊗ U(g) through ρ. The point is that rX (Jkl ) ⊂ Jk+1,l and rX (Kkl ) ⊂ Kk+1,l , so we may define σX : Ekl → Ek+1,l by σX ◦ ρ = ρ ◦ rX . The compatibility conditions σ must meet to be a representation of S 0∞ follow from ρ ◦ rX = ρ ◦ NX = σX ◦ ρ for all X ∈ s0 . It is not hard to use the PBW Theorem to check that the flag associated to σ is Enn ⊃ En−1,n−1 ⊃ . . . ⊃ E1,1 ⊃ E0,0 ⊃ 0. If µ : σ → σ 0 is a flag morphism between two objects of C, then it is also an S 0∞ -morphism. In Sec. 4 we will describe those representations of S 0∞ which arise from objects of C in this way. In Secs. 7 and 8 we will need the following facts. First, (σα + ihα, p0 i)(Ekk ) ⊂ Ek−1,k−1 for any σ ∈ C and α ∈ a. This will be clear if we show that Nα + ihα, p0 i maps Jkk into Jk−1,k−1 . Since Nα = ηα ⊗1+1⊗adα , and ηα +ihα, p0 i = −Lα maps Nk into Nk−1 by the definition of C while adα maps Ur into Ur−1 because a is an ideal in g, the result follows. Second, if σ = F (U) for some U in ExtC G O, then for X ∈ g the action of σX on En−1,n−1 is σX δ = −δ ◦UX , as here ρ is defined by ρ(δ ⊗Z) = δ ◦UZ . Third, if σ ∈ C and we write σ ˜ for the restriction of the representation σ of S 0∞ to a representation of S ∞ with the same flag Enn ⊃ En−1,n−1 , then it follows again from the PBW theorem that the flag associated to σ ˜ is Enn ⊃ En−1,n−1 ⊃ . . . ⊃ E0,0 ⊃ 0, just as it is for σ. 3. Subquotients Recall from the introduction that this paper is a step towards deciding which objects of ExtG O are subquotients of certain geometrically natural representations of G. By Theorem 2.1 it is enough to do this for objects of ExtC G O. To do so we will use the functor F to move the problem to the category C, and so we must study the relation between subquotients in ExtC G O and in C. Throughout this section fix an O, let σ, η, L0 , . . . , Ln be the object F (U) of C, and object U, F0 , . . . , Fn of ExtC G let Bj = ⊕nj Fi so that U acts in the flag DB0 ⊃ . . . ⊃ DBn . By refining the flag associated to U if necessary, we may and do assume in this section that the H-bundles Fi are irreducible. Our first lemma shows that ExtC G O is closed under taking the natural type of subquotient. Lemma 3.1. Suppose that B00 is a subbundle of B0 such that DB00 is a G-invariant subspace of DB0 , and let U 0 = U|DB00 . Let Bj0 = B00 ∩ Bj , and choose subbundles Fi0 of B00 such that Bj0 = ⊕nj Fi0 for 0 ≤ j ≤ n. Then Fi0 is either 0 or isomorphic to Fi , and U 0 , F00 , . . . , Fn0 is an object of ExtC G O. Define a flag of bundles Bj00 = Bj /Bj0 , let U 00 be the quotient representation on the flag DB000 ⊃ . . . ⊃ DBn00 , and choose subbundles Fi00 of B000 such that Bj00 = ⊕nj Fi00

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for 0 ≤ j ≤ n. Then Fi00 is either 0 or isomorphic to Fi , and U 00 , F000 , . . . , Fn00 is an object of ExtC G O. Proof. First note that it is indeed possible to choose the Fi0 as required, for 0 ) is a closed Gexample by putting a Hermitian metric on B0 . Since D(Bj0 /Bj+1 subspace of the irreducible induced representation D(Bj /Bj+1 ), it is 0 or isomorphic to DFj , and so Fj0 is 0 or isomorphic to Fj . Both U 0 and U 00 will inherit property 3 in the definition of ExtC G O from U; the rest is easy. The reader should keep in mind that B0 is not naturally an H-bundle unless U is induced. Recall the elementary fact that if F is any H-bundle and V is the induced representation of G on DF , then any closed G-invariant subspace of DF is of the form DF 0 for some H-subbundle F 0 of F . Lemma 3.1 does not generalize this fact to say that any closed G-invariant subspace of DB0 is of the form DB00 for some B00 as in the lemma. Such a generalization would be of interest to the author. We remark that Fj0 need not be a subbundle of Fj . This points out a flaw in our exposition: in the definition of objects of ExtC G O we should specify only the flag {Bj } and not the splitting {Fi }. We gave the definition as it stands in [4] so that we could write U as a matrix, and we will not change it here so that it will be easier to refer to that paper. Similarly, in the definition of objects of C it would be better to specify only the flag {Nj } rather than the splitting {Li }. We now define subquotients in C. Suppose that Nn0 is a subspace of Nn such that ηs (Nn0 ) ⊂ Nn0 ⊗ U(g) for all s ∈ S 0 . Let Nj0 = Nn0 ∩ Nj , and choose subspaces L0i of Nn0 such that Nj0 = ⊕nj L0i for 0 ≤ j ≤ n (some L0i may be 0). Let η 0 = η|Nn0 0 0 0 and define Mkl , Jkl , and Kkl as in the definition of C, but using η 0 and the L0i in place of η and the Li . 0 and η 0 has Lemma 3.2. For all s ∈ S 0 and 0 ≤ j ≤ n, ηs0 (Nj0 ) ⊂ Mjj 0 properties 1 and 2 in the definition of C, so there is an object σ of C such that η(σ 0 ) = η 0 and Li (σ 0 ) = L0i . Such objects will be called subobjects of σ. The 0 0 are naturally subspaces of Ekl , and the inclusion µ : Ekl → Ekl associated spaces Ekl 0 is a C-morphism such that τ (µ) is the inclusion Nn → Nn . 0 = Proof. Since Mjj

Pj 0

Ni0 ⊗ Rj−i , we have

j X 0 Ni ⊗ Rj−i = Mjj . ηs (Nj0 ) ⊂ ηs (Nn0 ) ∩ ηs (Nj ) ⊂ Nn0 ⊗ U(g) ∩ 0 0 0 0 = Kkl + Mkl . Clearly η 0 inherits property 2 from η. Now by Lemma 4.2 of [4], Jkl 0 0 0 0 0 On the other hand, Kkl ⊂ Kkl , Mkl ⊂ Mkl , and Kkl ∩ Mkl = 0, so Jkl = Kkl ⊕ Mkl . We have mentioned that this is equivalent to η 0 having property 1; the proof is 0 0 0 = Jkl /Kkl is a subspace of Ekl essentially that of Lemma 3.8 in [4]. Last, Ekl 0 0 0 0 because Jkl ∩ Kkl = (Kkl ⊕ Mkl ) ∩ Kkl = Kkl .

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Let σ 0 be a subobject of σ as above, and define spaces Nj00 = Nj /Nj0 . Then ⊂ . . . ⊂ Nn00 is a flag, although some of these inclusions may not be proper, and Pl Pl 00 00 00 00 and Mkl as usual: Jkl = 0 Ni00 ⊗Uk−i and Mkl = 0 Ni00 ⊗Rk−i . we may define Jkl Let π : Nn ⊗ U(g) → Nn00 ⊗ U(g) be the canonical projection, and note that π(Mkl ) = P 0 00 00 0 0 0 , π(Jkl ) = Jkl , kernel(π) = Jkl , and Jkl ∩Mkl = Mkl . Since ηs (Nk0 ) ⊂ Mkk for Mkl 0 00 00 00 00 00 00 all s ∈ S , ηs factors through π to a map ηs : Nn → Mnn such that ηs (Nk ) ⊂ Mkk for 0 ≤ k ≤ n. N000

Lemma 3.3. The map η 00 has properties 1 and 2 in the definition of C, and so by choosing spaces L00i such that Nk00 = ⊕k0 L00i for 0 ≤ k ≤ n we get an object σ 00 of C such that η(σ 00 ) = η 00 and Li (σ 00 ) = L00i (some of the L00i may be 0). The biflag 00 0 00 is the quotient of Ekl by Ekl , and the canonical projection µ0 : Ekl → Ekl is Ekl 00 0 00 a C-morphism from σ to σ such that τ (µ ) is the projection π : Nn → Nn . The object σ 00 will be called the quotient of σ by σ 0 , written σ/σ 0 . Proof. Clearly η 00 inherits property 2 from η. To check property 1, note that π ◦ Ns = Ns00 ◦ π and Nk00 = π(Nk ), so 00 )(Nk00 ) = π ◦ (Ns ◦ Nt − Nst )(Nk ) ⊂ π(Kk−1,k−1 ) . (Ns00 ◦ Nt00 − Nst 00 . Everything else follows from the easy fact that π(Kkl ) = Kkl

The next lemma says that the definitions of subobjects and quotient objects in C give the usual homomorphism theorem. Its proof follows from the fact that if σ ), µ:σ→σ ˜ is a C-morphism, then (τ ⊗ 1) ◦ ηs = η˜s ◦ τ , where we write η˜ = η(˜ τ = τ (µ), and so on. Lemma 3.4. If µ : σ → σ ˜ is a C-morphism, then the subspace Nn0 = kernel(τ ) 0 ˜ 0 = image(τ ) of N ˜n˜ defines a of Nn defines a subobject σ of σ. The subspace N n ˜ 0 0 ˜ , and µ factors through to an isomorphism from σ/σ to σ ˜0. subobject σ ˜ of σ Here by an isomorphism α : ν → ν 0 between two objects of C we mean a morphism such that τ (α) is a linear isomorphism; the two objects may be of different lengths. We will leave it to the reader to check that if α is an arbitrary morphism, P P then α : kl Ekl (ν) → kl Ekl (ν 0 ) is injective (respectively, surjective) if and only if τ (α) is injective (respectively, surjective). Next we must check that the functor F defines a contravariant bijection from the subquotients of U of the type discussed at the beginning of this section to the subquotients of σ = F (U). Lemma 3.5. Let all notation be as in Lemma 3.1, and let σ 0 = F (U 0 ) and σ 00 = F (U 00 ). Then σ 00 is a subobject of σ and σ 0 = σ/σ 00 . Conversely, suppose that σ 00 is any subobject of σ and let σ 0 = σ/σ 00 . Then there is a subbundle B00 of B0 such that DB00 is G-invariant and F (U 0 ) = σ 0 , where U 0 = U|DB00 . Furthermore, if U 00 is the quotient on D(B0 /B00 ) then F (U 00 ) = σ 00 . Proof. We will leave the first statement to the reader and prove only its converse. Let π : σ → σ 0 be the projection morphism. Since F is essentially surjective

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there is an object V of ExtC G O acting in a flag of bundles DC0 ⊃ . . . ⊃ DCm such that F (V) = σ 0 , and since F is fully faithful there is a morphism T : V → U such that F (T ) = π. The map T : C0 → B0 acts fiberwise, and so if we can show that it is injective on every fiber then its image T (C0 ) will be the required subbundle B00 ˜ of B0 . Recall the bundles of graded differential operators E˜kl (U), and write E(U) P P ˜ ˜ for kl Ekl (U) and similarly E(U) for the fiber kl Ekl (U) of E(U) at p0 . In Sec. 2 ˜ ˜ we defined a fiberwise map ∆T : E(U) → E(V) by ∆T (D) = D ◦ T , such that ˜ ˜ and E(V) are G-bundles and ∆T ∆T (p0 ) = F (T ) = π : E(U) → E(V). Now E(U) is a G-map, so since π is surjective, ∆T is surjective on all fibers. It follows that T is injective; we leave the rest to the reader. 4. Relations Between C, RS 0∞ , and RS ∞ Recall that at the end of Sec. 2 we defined a map from objects of C to objects of RS 0∞ , and from flag morphisms in C to RS 0∞ -morphisms. Here we will describe the image of this map. Whenever we refer to S 0∞ , S 0 is being regarded as a subgroup of G, while whenever we refer to S ∞ , S is being regarded as a subgroup of H. Let π be a representation of S 0∞ acting in V0 ⊃ V1 , and let {Vi } be the associated flag defined in Sec. 1. Assume that we are given an integer n and subspaces Nn ⊂ V0 , Nn−1 ⊂ V1 , . . . , N0 ⊂ Vn , such that N0 ⊂ . . . ⊂ Nn is a flag with the following Pl Pl properties. First, if we let Jkl = 0 Ni ⊗ Uk−i and Mkl = 0 Ni ⊗ Rk−i and define ρ : Jnn → V0 by ρ(ω ⊗ Z) = π(Z T )ω, then ρ restricts to an isomorphism from Mnn to V0 . Second, π|A leaves Nn invariant, and π(a) − exp iha, p0 i maps Nk to Nk−1 for 0 ≤ k ≤ n and is 0 for a ∈ Cp⊥0 . We will see that such π comprise the image of our object map from C to RS 0∞ . Define Ekl to be ρ(Jkl ), and note that Ekl ⊂ Vn−k because ρ(Ni ⊗ Uk−i ) = π(Uk−i )Ni ⊂ π(Uk−i )Vn−i ⊂ Vn−k . Let q : V0 → Mnn be the right inverse of ρ, and for s ∈ S 0 define ηs : Nn → Mnn to be q ◦ πs . As usual, define Kkl = {ηX ω · Z + ω ⊗ XZ : ω ∈ Ni , Z ∈ Uk−i , X ∈ s0 , 0 ≤ i ≤ l} . Lemma 4.1. In the setting just described , let L0 , . . . , Ln be any subspaces of Nn such that Nk = ⊕k0 Li . Then η, L0 , . . . , Ln define an object σ of C such that η(σ) = η, Li (σ) = Li , ρ(σ) = ρ, q(σ) = q, Ekl (σ) = Ekl , and the extension of σ to a representation of S 0∞ on Enn ⊃ En−1,n−1 is equal to π. In particular , ρ(Mkk ) = Ekk = Vn−k for 0 ≤ k ≤ n and Vn+1 = 0. Conversely, the extension of any object σ of C to an object of RS 0∞ has all of the above additional structures and properties in the way indicated by the notation. Proof. We start by checking that ρ(Mkk ) = Ekk = Vn−k for 0 ≤ k ≤ n; the same argument also shows that Vn+1 = 0. This is true for k = n, and so we may induct downward on k. We know that ρ(Mkk ) ⊂ Ekk ⊂ Vn−k , and so it will do to suppose that there exists v ∈ Vn−k − ρ(Mkk ) and derive a contradiction. Recall that h = s ⊕ r, r1 , . . . , rq is an ordered basis of r, and the subspace Rr of Ur used to define Mkl has the monomials rI with |I| ≤ r as a basis. Therefore any element

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P of Mnn may be written uniquely as I ωI ⊗ rI , where ωI ∈ Nn−|I| . In particular, P since q(v) ∈ Mnn − Mkk we have q(v) = I ωI ⊗ rI , where ωI 6∈ Ni and |I| ≥ k − i for some I. It follows that q(v) · rq ∈ Mnn − Mk+1,k+1 , but −ρ(q(v) · rq ) = π(rq )v is in Vn−k−1 because v ∈ Vn−k . By induction, Vn−k−1 = ρ(Mk+1,k+1 ), and so q(π(rq )v) ∈ Mk+1,k+1 is not equal to −q(v) · rq . This is a contradiction, because both of them lie in Mnn and have the same image under ρ. Next we will verify that η defines an object of C. To check that ηs (Nk ) ⊂ Mkk , note that ηs (Nk ) ⊂ q◦πs (Vn−k ) = q(Vn−k ) = Mkk . It follows that Kkl ⊂ Jkl , and so as we saw in the proof of Lemma 3.2, Jkl = Mkl + Mkl . By construction ρ(Kkl ) = 0, and so in fact Jkl = Mkl ⊕ Kkl . We also saw in the proof of Lemma 3.2 that this means that η has property 1. The condition on π|A implies that η|A = π|A and hence that η has property 2, so the object σ of C exists. We find from Jkl = Mkl ⊕ Kkl that Knn = kernel(ρ) and Kkl = Knn ∩Jkl , from which it follows that Ekl (σ) = Ekl . We leave the rest to the reader. Lemma 4.2. Let σ and σ 0 be two objects of C such that n = n0 . If µ : En → En0 is an S 0∞ -morphism between the S 0∞ -actions of σ and σ 0 such that µ(Nk ) ⊂ Nk0 0 for all k, then µ is a flag morphism in the category C. In particular , µ(Ekl ) ⊂ Ekl . Proof. Let τ : Nn → Nn0 be µ|Nn . Then for ω ⊗ Z ∈ Jkl we have 0 ρ0 ◦ (τ ⊗ 1)(ω ⊗ Z) = ρ0 ◦ rZ T ◦ τ (ω) = σZ T ◦ µ(ω)

= µ ◦ σZ T (ω) = µ ◦ ρ(ω ⊗ Z) . Hence ρ0 ◦ τ ⊗ 1 = µ ◦ ρ, the condition for µ to be a C-morphism. It is a flag morphism by definition, and as we remarked in the definition of C the last sentence is easy. We will finish this section by developing a technique for combining an object σ, η, N0 ⊂ . . . ⊂ Nn of C with an object π of RS ∞ (not RS 0∞ ) to form a new object of C, which we refer to as σ ⊗ π. Write Ekl for Ekl (σ) and so on, as usual, and let π act in V0 ⊃ V1 with associated flag {Vi }. For convenience we will write Ek , Mk , and Jk for Ekk , Mkk , and Jkk , respectively, and we write Wk for Vn−k . Define spaces ˜k = E

k X 0

Ei ⊗ Wk−i

˜k = and N

k X

Ni ⊗ Wk−i

0

for 0 ≤ k ≤ n, view σ as a representation of S 0∞ as in Lemma 4.1, extend π to a ˜ be the tensor product representation of S 0∞ by making A act trivially, and let σ 0∞ ˜ ˜ representation σ ⊗ π of S acting in the flag En ⊃ En−1 , as defined in Sec. 1. ˜n−1 ⊃ . . . ⊃ E˜0 ⊃ 0, and the ˜n ⊃ E Lemma 4.3. The flag associated to σ ˜ is E ˜ ˜ ˜ into an object of C as in Lemma 4.1. subflag Nn ⊃ . . . ⊃ N0 of this flag makes σ We define the object σ ⊗ π of C to be σ ˜.

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Suppose that π 0 , V00 ⊃ V10 is another representation of S ∞ and that T : π → π 0 is a S ∞ -morphism. Then µ = 1 ⊗ T is a C-morphism from σ ⊗ π to σ ⊗ π 0 , and in fact it is a flag morphism. The map τ (µ) is µ|N˜n . ˜ kl from the flag {N ˜k } as usual, define ρ˜ : J˜nn → E ˜n Proof. Define J˜kl and M T ˜ ˜ ˜ ˜ by ρ˜(ω ⊗ Z) = σ ˜ (Z )ω, and for convenience define Jk = Jkk and Mk = Mkk . It is easy to check that ρ˜(J˜k ) ⊂ E˜k , and so by Lemma 4.1 we need only prove that ρ˜ ˜ k isomorphically onto Ek for 0 ≤ k ≤ n, and that σ ˜ |A has the special form maps M given in that lemma. This last statement is easy, because σ|A has this special form and π|A = 1. ˜k is an isomorphism, we first check that M ˜ and ˜k → E To prove that ρ˜ : M Pkk ˜ ˜ Ek have the same dimension. Since dim(Mi ) = dim(Ei ), dim(Ek ) = dim( 0 Mi ⊗ Pk P ˜ k are naturally isomorphic to Wk−i ). But both 0 Mi ⊗Wk−i and M i+j+l=k Ni ⊗ ˜ ˜ Rj ⊗ Wl , and so we are done. Next we will check that ρ˜(Mk ) = Ek by induction on k, which will complete the proof of the first paragraph of the lemma. Since ˜0 = N ˜0 , the induction starts. We will proceed in two steps, where the first ˜0 = E M ˜k = ρ˜(M ˜ k ). To show the forward containment step is a proof that σ ˜ (R1 )E˜k−1 + N ˜ ˜ ˜ k−1 ) = E ˜k−1 by induction. For this, ρ(Mk−1 ) ⊂ ρ˜(Mk ), as ρ˜(M we only need σ ˜ (R1 )˜ note that ˜ k−1 ) = ρ(M σ ˜ (R1 )˜

k−1 X 0

˜i ⊂ σ ˜ (R1 )˜ σ (Rk−i−1 )N

k−1 X

˜i σ ˜ (Rk−i + Uk−i−1 )N

0

˜ k + J˜k−1 ) = ρ˜(M ˜ k) + because R1 Rk−i−1 ⊂ Rk−i + Uk−i−1 . But this last sum is ρ˜(M ˜ k ), as needed. The reverse containment is easy. ˜k−1 ⊂ ρ˜(M E ˜k = E ˜k . The forward For the second step we must prove that σ ˜ (R1 )E˜k−1 + N Pk ˜ containment is clear; for the reverse containment, recall that Ek = 0 Ei ⊗ Wk−i . ˜k by induction on i. Since E0 ⊗ ˜ (R1 )E˜k−1 + N We will show that Ei ⊗ Wk−i ⊂ σ ˜k , the induction begins and we may assume Ei−1 ⊗ Wk−i+1 ⊂ Wk = N0 ⊗ Wk ⊂ N ˜k−1 + N ˜k . Then we have that Ei−1 ⊗ Wk−i+1 + σ ˜ (R1 )(Ei−1 ⊗ Wk−i ) + Ni ⊗ σ ˜ (R1 )E ˜k−1 + N ˜k , and so since σ ˜ (R1 )E ˜ = σ ⊗ π as a representation of S 0∞ , Wk−i is in σ σ(R1 )Ei−1 + Ni = Ei , and π(R1 )Wk−i ⊂ Wk−i+1 , the result follows. We leave the second paragraph to the reader. 5. Functions on Neighborhoods of Submanifolds In Sec. 6 we will define certain objects of ExtG O which arise geometrically as spaces of vector-valued functions on infinitesimal neighborhoods of the orbit O in its ambient space A∗ , and so in this section we must define functions on neighborhoods of O. Heuristically we think of the nth order infinitesimal neighborhood of O in A∗ as a space, O(n) , but in fact we will not define O(n) but only the functions on it. Roughly, a function on O(n) assigns smoothly to each point of O the n-jet of a function on A∗ . Similar constructions may be found in the beginning of Griffiths’ paper [8]. We remark that the functional analysis we will use in this section is all easy; most of the work is in setting up notation.

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Suppose that M is any smooth manifold and N is any regularly imbedded submanifold. Let ∆M be the vector bundle of differential operators on M . This is a complex vector bundle of countably infinite rank with the usual filtration of bundles ∆r M of differential operators of order ≤ r, which are of finite rank. A section of E∆M or D∆M is a smooth or smooth compactly supported differential operator mapping DM to itself, respectively. An element of the fiber ∆M (p) at a point p ∈ M is a differential operator supported at p mapping DM to C. For each integer r ≥ 0, we define the rth jet bundle J r M above M to be the vector bundle dual to ∆r M . If F and D are sections of J r M and ∆r M , respectively, we have the function hD, F i on M whose value hD(p), F (p)i at p is given by the duality. Let jr : DM → DJ r M be the map assigning to a function f its rth jet. In other words, jr f is defined by the equation hD, jr f i = Df for all D ∈ D∆r M . The map jr is not a vector bundle map; it is easily seen to be a differential operator of order r. The fibers of J r M have an algebra structure induced by multiplication of functions, and so DJ r M is an algebra under fiberwise multiplication and jr is an algebra homomorphism. Since ∆r M is a subbundle of ∆s M for r ≤ s, there is a quotient map of bundles πrs : J s M → J r M . One checks that πrs is an algebra map, that πqr ◦ πrs = πqs for q ≤ r, and that jr and πrs ◦ js are equal as maps from DM to DJ r M . Note that D(J r M |N ) is an algebra under fiberwise multiplication, and that we may regard ∆r N as a subbundle of ∆r M |N by restriction in the obvious way. Definition. We define the space DN (n) of smooth compactly supported functions on N (n) to be the subspace of D(J n M |N ) consisting of sections F such that for each p ∈ N , there is an open neighborhood V of p in M (not N ) and a smooth function fV on V such that for all q ∈ V ∩ N we have jn fV (q) = F (q) . So locally, functions on N (n) are restrictions to N of n-jets of functions on M . We give DN (n) the subspace topology it inherits from D(J n M |N ), and we define the support supp(F ) ⊂ N of F ∈ D(N (n) ) to be its support as a section of J n M |N . Lemma 5.1. The space DN (n) is a closed subalgebra of D(J n M |N ), and so it is an inductive limit of Frechet spaces and is complete. Proof. It is clear that DN (n) is a subalgebra from the definitions. To prove that it is closed, let F be any section in D(J n M |N ), and choose D and E in D∆n M such that the composition E ◦ D is still of order ≤ n, and E|N lies in the subspace D∆n N of D(∆n M |N ). Since ∆n M |N and J n M |N are dual to each other, there is a fiberwise pairing which gives a function hD|N , F i in DN . A local calculation shows that if F ∈ DN (n) , then the function obtained by applying the differential operator E|N ∈ D∆n N to hD|N , F i is equal to h(E ◦ D)|N , F i, and furthermore that F is in DN (n) if and only if this equality holds for all E and D as above. For each choice of D and E this is a closed condition, and so DN (n) is closed.

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Recall the following standard notation: if L is any manifold, B is any vector bundle over L, and K is any subset of L, let DK B = {s ∈ DB : supp(s) ⊂ K}. When K is compact, DK B is a closed Frechet subspace of DB, and DB is the inductive limit of the spaces DK B as K runs over a compact exhaustion of L. Similarly, for any subset K of N let DK N (n) = DK (J n M |N ) ∩ DN (n) . Then DN (n) is the inductive limit of the closed Frechet subspaces DK N (n) as K runs over a compact exhaustion of N . Remark. Perhaps the first definition of DN (n) one thinks of is the following. Since N is regularly imbedded in M it is locally closed in M , and so we may choose V open in M such that N is a closed subset of V . Let IN (V ) be the ideal of functions in DV that vanish on N . It would be natural to define DN (n) to be the space DV /IN (V )n+1 with the quotient topology, but this adds unnecessary difficulty to our results. Note that for the definition of DN (n) we are using, there is a natural continuous linear map πn : DV → DN (n) defined by πn f = (jn f )|N , and it is clear that πn is an algebra homomorphism whose kernel is IN (V )n+1 . By Lemma 5.2, πn is surjective, and we expect that it factors through to a linear homeomorphism from DV /IN (V )n+1 with the quotient topology to DN (n) , but we do not have and will not need a proof of this. The next two lemmas give some basic properties of DN (n) . We will use the obvious fact that for any V as above, DN (n) 3 F is a module for the algebra EV 3 g, where gF = (jn g)|N F . Lemma 5.2. First , if {Uα } is any open cover of N and F ∈ DN (n) , then F is a finite sum of elements of DN (n) each having support in one of the Uα . Second , the map πn defined above is surjective. Proof. First, let us replace the given open cover by a locally finite refinement {U1 , U2 , . . .} such that for each open set Ui in N there is an open set Vi in M such that Ui = Vi ∩ N and Ui is closed in Vi . Let V 0 = ∪i Vi , and let {θi } be a partition P of unity for the open cover {Vi } of V 0 . One checks that F = i θi F is the desired finite sum. Second, choose a locally finite open cover U1 , U2 , . . . of N such that for each i, there is an open set Vi in V with Ui = Vi ∩ N and Ui closed in Vi , and a function fi in EVi such that (jn fi )|Ui = F |Ui . Let V 0 = ∪i Vi , and let {θi } again be a partition P of unity for V 0 subordinate to {Vi }. One checks that F = πn ( i θi fi ). Recall from above the surjective bundle maps πrs : J s M → J r M for r ≤ s, and write also πrs for the associated surjective fiberwise maps from D(J s M |N ) to D(J r M |N ). Let Irs = {F ∈ DN (s) : πrs F = 0}; then Irs is the closed subspace of DN (s) of functions that vanish to order r. Let Ω1 M be the 1-form bundle over M . The tangent bundle T N is canonically a subbundle of T M |N , and we define T ⊥ N ⊂ Ω1 M |N to be the annihilating bundle of T N . For any vector bundle B, let S n B be its nth symmetric power bundle. When B is a real bundle, we will adopt the convention that S n B is complexified.

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Lemma 5.3. First , the maps πrs ◦ πs and πr defined above are equal , πrs (DN (s) ) = DN (r) , and Irs is an ideal in the algebra DN (s) . Second , πrs factors through to a linear homeomorphism from DN (s) /Irs with the quotient topology to DN (r) . Third , Irs admits a closed complementary subspace in DN (s) . Fourth, the product Irs Iqs is contained in Ir+q+1,s , and so the action of DN (s) on Irs factors through to an action of DN (s−r−1) . Fifth, the DN = DN (0) -module Ir−1,r is DN -isomorphic and linearly homeomorphic to D(S r T ⊥ N ). Proof. Let V be as above Lemma 5.2. The first statement follows from the facts that πrs ◦ js and jr are equal as maps from DV to DJ r V , that πr (DV ) = DN (r) , and that πrs is an algebra map. Second, to see that πrs factors through to a linear homeomorphism it is enough to prove that πrs is an open map. For any compact subset K of N it is clear that πrs (DK N (s) ) ⊂ DK N (r) . If we can prove that this containment is an equality, then since DK N (s) is Frechet the open mapping theorem implies that the restriction of πrs to it is open, and so πrs is open because DN (r) is the inductive limit of spaces DK N (r) . Thus we come down to proving that given F ∈ DN (r) , there exists F 0 ∈ DN (s) such that πrs F 0 = F and suppF 0 = suppF . Choose a locally finite open cover {Uα } of N with the following properties. For each Uα there is an open set Vα in M such that Uα = Vα ∩ N , Uα is closed in Vα , and Vα is a coordinate chart on M with coordinates (x, y) = (x1 , . . . , xp , y1 , . . . , yq ) such that Vα is the cube {(x, y) : |xi | < 1, |yj | < 1}, and Uα is the subset of Vα cut out by y = 0. Note that x gives coordinates for Uα . Let V 0 = ∪α Vα , and let {θα } be a partition of unity for V 0 subordinate to {Vα } (the Vα may not be locally finite, so to construct the θα , choose a locally finite refinement). A local calculation shows that we may choose P Fα0 in DN (s) such that suppFα0 = supp(θα F ) and πrs Fα0 = θα F , and so F 0 = α Fα0 is the required element of DN (s) . For the third statement, retain the open cover Uα . We will need products of the J coordinates xi and yj , and we will use the multinomial notations y J = y1J1 . . . yq q P (s) and |J| = j Jj . Given F ∈ DUα N , a local calculation shows that there are P unique functions cJ (x) ∈ DUα N for all |J| ≤ s such that F = js ( |J|≤s cJ y J ) on Uα . Since suppF ⊂ Uα , F lies in Irs if and only if cJ ≡ 0 whenever |J| ≤ r. We define a continuous linear projection pα : DUα N (s) → Irs by   X cJ y J  pα (F ) = js  r<|J|≤s

P on Uα and pα (F ) ≡ 0 off Uα . Define p : DN (s) → Irs by p(F ) = α pα (θα F ); then since Irs is an ideal and the pα are projections, p is a continuous linear projection. Therefore DN (s) = kernel (p) ⊕ Irs , which proves the third statement. Fourth, recall that πrs : J s M → J r M is a bundle map and an algebra map on fibers. One checks locally that the kernels of πrs on the fibers form a subbundle Krs of J s M whose fibers are ideals, and that the product of the fibers Krs (p)Kqs (p) is contained in Kr+q+1,s (p) for all p ∈ M . Since J r M = J s M/Krs and Kr+q+1,s = 0

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if q ≥ s − r − 1, the J s M action on Krs factors through to an action of J s−r−1 M . The statement now follows from Irs = D(Krs |N ) ∩ DN (s) . For the fifth statement we must define the transversal order of elements of the fiber ∆r M (p) for p ∈ N . One of the sets Uα from above contains p, and the coordinates (x, y) on Vα give us the basis ∂xi |p and ∂yj |p of T M (p). If we write J ∂yJ for ∂yJ11 . . . ∂yqq , then ∆r M (p) has the basis {∂xI ∂yJ |p : |I + J| ≤ r}. For k ≤ r, we define the subbundle ∆r (M, N, k) of differential operators of order ≤ r and transversal order ≤ k by letting the fiber ∆r (M, N, k)(p) have the basis {∂xI ∂yJ |p : |J| ≤ k, |I + J| ≤ r}. This definition is coordinate free, and one sees that the quotient bundle ∆r M |N /∆r (M, N, r − 1) is naturally isomorphic to S r (T ⊥ N )∗ in much the same way that one sees that ∆r M/∆r−1 M ∼ = SrT M . r (r) we have the pairing hD, F i ∈ Recall that for D ∈ D(∆ M |N ) and F ∈ DN DN . One checks that if F ∈ Ir−1,r and f ∈ DN = DN (0) , then hD, f F i = hf D, F i = f hD, F i, and that hD, F i ≡ 0 for all F ∈ Ir−1,r if and only if D ∈ D∆r (M, N, r − 1). Since (∆r M |N /∆r (M, N, r − 1))∗ and S r T ⊥ N are isomorphic, it follows that there is a DN -isomorphism from Ir−1,r to D(S r T ⊥ N ), and using the open mapping theorem as above shows that it is a homeomorphism. 6. Representations in Functions on Neighborhoods of Orbits Recall from the introduction that O is a locally closed orbit of H in A∗ , and so O is regularly imbedded in A∗ . In this section we will prove that there is a canonical representation of G in DO(n) which is an object of ExtG O, and exhibit an equivalence from this object to an object of ExtC G O which will be useful for (n) is not canonically a vector bundle over O, while computations. Note that DO objects in ExtC G O act in vector bundles. The H-action on A∗ defines an H-bundle structure on J n A∗ such that jn intertwines the representations of H on DA∗ and DJ n A∗ . Since O is H-invariant, there is a smooth representation of H on D(J n A∗ |O ), and one finds that DO(n) is an invariant subspace. We write V (n) for the subrepresentation on DO(n) , and we (n) extend V (n) to a representation of G by letting Va be multiplication by (jn ξa )|O for all a ∈ A, where ξa is the character function from the introduction. It may be of interest to note that if we let V = A∗ \∂O, where ∂O is the boundary of O, then V is a H-invariant open set in A∗ containing O as a closed subset, and πn : DV → DO(n) is a G-map. Clearly the maps πrs are G-maps, and so the ideals Ikn of DO(n) are G-invariant. Thus V (n) admits the G-flag DO(n) ⊃ I0,n ⊃ . . . ⊃ In−1,n . Using Lemma 5.3 one checks that the isomorphism from DO(n) /Ikn to DO(k) restricts to an isomorphism from Ik−1,n /Ik,n to Ik−1,k , and also that the isomorphism from In−1,n to D(S n T ⊥ O) is a G-map, where D(S n T ⊥ O) is a G-space via the Hbundle structure on S n T ⊥ O coming from the natural H-bundle structure on T ⊥ O, and the character action of A. Lemma 5.3 now yields the following proposition.

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Proposition 6.1. The representation V (n) of G on DO(n) is an object of ExtG O whose composition series is isomorphic to D(S k T ⊥ O), 0 ≤ k ≤ n. Our plan is to apply the results of [4] to reduce the study of DO(n) to the study of certain representations of SA, and to use [4] we must find an isomorphism of DO(n) with an object of ExtC G O. To this aim let e be the endpoint map from the total space of the real subbundle C of O × A∗ to A∗ : for p ∈ O and c ∈ Cp ⊂ A∗ , e(p, c) = p + c. We will write Ct for the total space of the bundle C, so that for example ECt denotes the smooth functions from Ct to C, while EC denotes the smooth sections of C. Given any vector space V , recall that S k V is its kth symmetric tensor power (complexified if V is real), and let Sn V = ⊕n0 S i V . We will view Sn V as an algebra via the projection from the algebra SV = S∞ V to Sn V whose kernel is the ideal hV in+1 = ⊕i>n S i V . Similarly, for any vector bundle B we have the vector bundle Sn B, whose fibers Sn (Bp ) are algebras. We will use the endpoint map e to define an isomorphism n : DSn C ∗ → DO(n) . Elements of the fiber Sn Cp∗ are polynomials of degree ≤ n on Cp , and so a section s ∈ DSn C ∗ defines a function s˜ ∈ ECt that is polynomial on fibers: for (p, c) ∈ Cp , s˜(p, c) is s(p)(c), the value of the polynomial s(p) at c. For future reference, for ˜ kn = k ≤ n let π ˜kn : Sn C ∗ → Sk C ∗ be the projection along the bundle of ideals K i ∗ ∗ ˜kn for the corresponding map DSn C → DSk C ∗ ⊕k
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for any q ∈ U 0 , (x(q), y) are linear coordinates on the plane q + Cq = e(Cq ) ⊂ A∗ defined near (q, 0). In other words, for q ∈ U 0 fixed and c ∈ Cq varying c 7→ y(q + c) are linear coordinates on Cq defined near 0 (of course, linear coordinates are defined everywhere, but V 0 ∩ (q + Cq ) is a neighborhood of (q, 0)), and x(q + c) = x(q) is independent of c. Given F ∈ O such that suppF ⊂ U 0 , there are unique functions cJ ∈ DU 0 O P such that F = jn ( |J|≤n cJ y J ) on U 0 . For x fixed at the value x(q) for any q ∈ U 0 , P y J is a polynomial on Cq , and so |J|≤n cJ y J may be viewed as a section s in DU 0 Sn C ∗ such that n s = F . This shows that at least when s and F have support in U 0 , suppn s = supps and F is in the image of n . It also shows that F is in ˜ kn ) if and only if cJ ≡ 0 for |J| ≤ k, which is the same as F ∈ Ikn . Now n (DK the usual partition of unity arguments imply that suppn s = supps in general, n ˜ kn ) = Ikn . is surjective, and n (DK ˜kn s = πkn n s, it is enough to prove it on U 0 . Since Sn C ∗ = To prove that k π ∗ ˜ ˜ kn such that s = Sk C ⊕ Kkn , there are unique sections s0 of Sk C ∗ and s00 of K 0 00 −1 0 s ◦ e˜ ) on U , we find that we come down to proving s + s . Now using n s = jn (˜ πkn n s00 = 0, and this follows from our choice of coordinates. Second, n is injective because it preserves supports, and we have already seen that it is surjective. It is a homeomorphism by the same type of open mapping theorem argument used in the proof of Lemma 5.3. To check that it is an algebra map, it will do to prove n (ss0 ) = n (s)n (s0 ) for both s and s0 having support in U 0 . This comes down to proving that the difference t = s˜s˜0 − (ss0 )˜ is a function in Et C such that jn (t ◦ e˜−1 ) is 0 on U 0 . Now s˜ and s˜0 are polynomial of degree ≤ n on the fibers Cp , and a review of the definition of multiplication in Sn C ∗ shows that (ss0 )˜ is the part of s˜s˜0 of degree ≤ n. A coordinate calculation finishes the proof. We will leave the third statement to the reader; it may be proven by using the differential operators ∂yj , which are sections of C|U 0 ⊂ T A∗ |U 0 , and the definition of the isomorphism in Lemma 5.3. The isomorphism n allows us to transport V (n) to an equivalent representation (n) ˜ on DSn C ∗ , and we will prove that V˜ (n) is an object of ExtC V G O. To simplify notation let us write , V, and V˜ in place of n , V (n) , and V˜ (n) when n is specified by the context. We begin with some isomorphisms, which will be viewed as identifications when it is convenient. Any δ ∈ Sn A∗ defines a constant coefficient differential operator Dδ on A∗ , and there is an isomorphism between the bundles A∗ × Sn A∗ and ∆n A∗ defined by (p, δ) 7→ Dδ (p). Any Ω ∈ Sn A defines a polynomial PΩ on A∗ , and this defines a duality h , iA between Sn A∗ and Sn A: hδ, ΩiA = Dδ (0)PΩ . Note that this procedure defines a duality h , iV between Sn V and Sn V ∗ for any vector space V , such that h , iV = h , iV ∗ . For p ∈ A∗ , let PΩ,p be the shifted polynomial PΩ,p (q) = PΩ (q − p). Since J n A∗ is dual to ∆n A∗ and Sn A is dual to Sn A∗ , we get an isomorphism between J n A∗ and A∗ × Sn A. One checks that it maps the n-jet of PΩ,p at p to (p, Ω), and hence that it is an algebra isomorphism on each fiber which identifies the ideals Kkn (p) and hAik+1 .

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Under these identifications, DO(n) sits inside D(J n A∗ |O ) = D(O : Sn A), the functions on O with values in Sn A, and the ideal Ikn of DO(n) sits inside the ideal D(O : hAik+1 ). Let r : O × Sn A → Sn C ∗ be the fiberwise map r(p, Ω) = PΩ |Cp , and write also r for the resulting algebra map from D(O : Sn A) to DSn C ∗ . An easy calculation using the definition of gives the following lemma. Lemma 6.3. The isomorphism −1 : DO(n) → DSn C ∗ is the restriction of r to DO(n) . Consequently, for p ∈ O, s ∈ DSn C ∗ , and δ ∈ Sn Cp ⊂ Sn A∗ we have hδ, s(p)iCp = hδ, s(p)iA∗ . We remark that as a map from DSn C ∗ to D(O : Sn A), is not a fiberwise map but a differential operator of order n. We are now ready to compute the action of ˜ Let ζ = dξ, so that for the Lie algebra a on DSn C ∗ under the representation V. ∗ α ∈ a, ζα : A → C is the function ζα (p) = ihα, pi. Proposition 6.4. The representation V˜ together with the splitting Sn C ∗ = ∗ ˜ is an object of ExtC G O. If α ∈ a, s ∈ DSn C , and p ∈ O, then (Vα s)(p) = ∗ r(p, ζα (p) + iα)s(p). Here ζα (p) + iα ∈ S1 A is a polynomial on A with constant term ζα (p), r(p, ζα (p) + iα) ∈ S1 Cp∗ is its restriction to Cp , and the product is in the algebra Sn Cp∗ .

⊕n0 S i C ∗

Proof. Let F = (s). We must prove that −1 (Vα F )(p) = r(p, ζα (p) + iα)s(p). Since −1 (Vα F ) = r(Vα F ) and r is an algebra map, it is enough to prove (Vα F )(p) = (ζα (p) + iα)F (p), where the product is in the algebra Sn A. But (Vα F )(p) = jn ζα (p)F (p), and since ζα is just the linear polynomial iα on A∗ , our identifications give jn ζα (p) = ζα (p) + iα. In light of Proposition 6.1 and the fact that (⊕nk+1 DS i C ∗ ) = Ikn , V˜ will be an object of ExtC G O provided it satisfies the third property in the definition of that category. We have just seen that V˜α acts fiberwise, and if α lies in the subspace C ⊥ ⊗ C of a then r(p, α) = 0, so V˜α F (p) = ζα (p)F (p). Lifting these statements from a to A completes the proof. Note that now Theorem 2.1 applies to describe the actions of H and h under ˜ V, although in fact it is easy to prove this description directly for this particular representation. ˜ (n) 7. Computation of F V Let π be any representation of H in a finite dimensional complex space V , and extend it to a representation of G by making A act trivially. Then it follows from Proposition 6.1 that the representation V (n) ⊗ π of G is in ExtG O, and our goal is to reduce the problem of deciding which objects U of ExtG O are subquotients of V (n) ⊗ π, for some n and π, to a problem involving only the representation of S ∞ associated to U at the end of Sec. 2. The map ⊗ 1 is an equivalence from V (n) ⊗ π to V˜ (n) ⊗ π, and it follows from Proposition 6.4 that V˜ (n) ⊗π together with the splitting Sn C ∗ ⊗V = ⊕n0 S i C ∗ ⊗V is

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C in ExtC G O. In this section we will compute the action of the functor F : ExtG O → C (n) ˜ from Sec. 2 on V ⊗ π, which will allow us to reduce our subquotient question to a problem in the category C. As before we write V˜ and V for V˜ (n) and V (n) . We begin by proving that ˜ of graded differential operators on Sn C ∗ are ˜kk (V) for 0 ≤ k ≤ n the bundles E k ∗ isomorphic to the bundles ∆ A |O of differential operators of order ≤ k on A∗ , restricted to O. This isomorphism is the transpose T of the map of Sec. 6, and it will allow us to move F V˜ to an equivalent object of C acting in the flag ∆n A∗ (p0 ) = Sn A∗ . This object may be thought of roughly as F V, although V is not in ExtC G O. We saw in the discussion around Lemma 6.2 that for any p ∈ O, we may choose neighborhoods V ⊃ U 3 p as follows. First, U is open in O and closed in V , V is open in A∗ , and U = V ∩ O. Second, U is a coordinate chart on O with ˜i , y˜j on the coordinates xi and C|U is trivial, and so we may define coordinates x ˜i (q, c) = xi (q) and c 7→ y˜j (q, c) total space of C|U such that for any (q, c) ∈ C|U , x are linear coordinates on Cq . Third, there is an open set V˜ ⊂ C|U such that the endpoint map e is a diffeomorphism from V˜ to V . We will write e˜ : V˜ → V for this ˜i and diffeomorphism, which we use to define coordinates xi , yj on V by xi ◦ e˜ = x yj ◦ e˜ = y˜j . Recall that : DSn C ∗ → DO(n) is defined on U as follows. Given s ∈ DSn C ∗ , we view s as a function s˜ : Ct → C that is polynomial on fibers, and we let s ◦ e˜−1 )|U . So roughly speaking, is the transpose of e−1 . Now recall from s|U = jn (˜ the beginning of Sec. 5 that elements D ∈ D(∆n A∗ |O ) act on elements F ∈ DO(n) to give functions DF ∈ DO, and so we have the map T D = D ◦ : DSn C ∗ → DO. Note that with the identifications at the end of Sec. 6, D(p) ∈ Sn A∗ , F (p) ∈ Sn A, and DF (p) = hD(p), F (p)iA .

˜ and T is a fiberLemma 7.1. For any D in D(∆n A∗ |O ), T D is in DE˜nn (V), ˜nn (V). ˜ Therefore it arises from a vector wise isomorphism from D(∆n A∗ |O ) to DE T n ∗ ˜ ˜ bundle isomorphism : ∆ A |O → Enn (V), and T maps ∆k A∗ isomorphically to ˜kk (V) ˜ for 0 ≤ k ≤ n. E ˜kl (V). ˜ First we check that if D is in D(D∆k A∗ |O ) Proof. Let us write E˜kl for E T ˜ then D is in DEkk . It follows from our choice of coordinates that there are P unique functions dIJ (x) on U such that D|U = |I+J|≤k dIJ ∂xI ∂yJ , where we use the multinomial notation from the proof of Lemma 5.3. Now the y˜j are nothing but a section basis of the bundle C ∗ |U , and so for any s in DSn C ∗ there are unique P ˜ = x ◦ e˜, y˜ = y ◦ e˜, functions sJ (x) on U such that s|U = |J|≤n sJ y˜J . Since x P J J0 T I J!dIJ ∂x sJ (x). Since dIJ = 0 if and ∂y y = J!δJJ 0 , we find that D(s)|U = |I + J| > k, we know at least that T D is a differential operator of order ≤ k. To check that it is graded so as to be in DE˜kk , recall the flag of vector bundles ˜ 0n ⊃ . . . ⊃ DK ˜ n−1,n ⊃ 0 defined below Proposition 6.1. This is the flag Sn C ∗ ⊃ K ˜ ˜ kn if and only if sJ = 0 for |J| ≤ k for all choices of left invariant by V, where s ∈ K ˜kl shows that it consists of differential U as above. A review of the definition of DE ∗ ˜ ln , and to operators of order ≤ k from DSn C → DO that restrict to 0 on DK

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˜ in for 0 ≤ i ≤ l. In particular, T D differential operators of order ≤ k − i − 1 on DK ˜ is in DEkk if and only if dIJ = 0 whenever |I + J| > k, and so T maps D(∆k A∗ |O ) surjectively to DE˜kk . For any f ∈ DO, T (f D) = f T D, and so T is a fiberwise map. We leave it to the reader to check that it is injective. Recall the fiberwise restriction map r : DO ⊗ Sn A → DSn C ∗ , which restricts to ˜nn , let rT (D) be D ◦ r : DO(n) → DO. the inverse of on DO(n) . For any D ∈ DE T ˜nn to ∆n A∗ |O It follows from Lemma 7.1 that r is a bundle isomorphism from E k ∗ T ˜kk to ∆ A |O , and that it is the inverse of . which takes E ˜nn . Let µ : E ˜nn (p0 ) → Definition. Let φ˜(n) = F V˜ (n) , the object of C acting in E n ∗ ∗ T ∆ A (p0 ) = Sn A be the isomorphism r (p0 ), and use µ to transport φ˜(n) to an object φ(n) of C acting in Sn A∗ . We will usually write simply φ˜ and φ for these objects. Instead of finding the action of φ˜ we will find the action of φ, which should be thought of as FV. It may seem that we could have avoided V˜ altogether, but since ˜ V is not canonically an object of ExtC G O we defined V in order to use the machinery of [4]. To understand the sense in which φ = F V, let us define a representation ∆V of G on D(∆n A∗ |O ) by ∆Vg (D) = λg ◦ D ◦ Vg−1 . Here ∆Vg D is well defined because sections of ∆n A∗ are determined by their action on DO(n) . By the above remarks rT intertwines ∆V˜ and ∆V, and so since µ = rT (p0 ) is a C-isomorphism by definition, we see that the S 0 -action of φ is simply ∆V|S 0 (p0 ), the little group representation of ∆V at p0 . Using the last paragraph of Sec. 2, we see that the g-action φX : En−1,n−1 (φ) → Enn (φ) is φX (δ) = −δ ◦ VX . We will now compute Ekk (φ) and Nk (φ), and from now on we will write Ek ˜ = ∆k A∗ (p0 ) = Sk A∗ . In computing instead of Ekk . By definition Ek (φ) = µ(Ek (φ)) ˜ we will use the fact that if Y ⊂ Z is any subspace of any vector Nk (φ) = µ(Nk (φ)) space and r : SZ ∗ → SY ∗ is restriction of polynomials, then rT : SY → SZ is ˜ is the space of order 0 differential operators at p0 on DSn C ∗ inclusion. Since Nk (φ) ˜ =K ˜ kn (p0 ), we find that Nk (φ) ˜ kn (p0 )⊥ ⊂ (Sn C ∗ p0 )∗ . that annihilate the ideal DK Henceforth we will write C0 for Cp0 and identify the dual of Sn C0∗ with Sn C0 , which ˜ = Sk C0 . Now on order 0 differential operators µ = rT (p0 ) is r(p0 )T , gives Nk (φ) and so the fact noted above gives Nk (φ) = Sk C0 . Next we will use the fact that φ = F V to compute the S 0∞ -action of φ. Let us write λ for the canonical action of H on DA∗ , extended to an action of G by the character action of A. We will also continue to use λ for the same action on DO, as we did in Sec. 2. Let us write ` for the graded action of H on SA, and `∗ for the graded action on SA∗ . Then ` and `∗ are dual actions, and we make A∗ × SA∗ and A∗ × SA into H-bundles by the product action. Now λ makes ∆n A∗ and J n A∗ into H-bundles, and we leave it to the reader to check that our identification of ∆n A∗ with A∗ × Sn A∗ is an H-isomorphism, and so the identification of J n A∗ with A∗ × Sn A is also, by duality. The bundle O ×Sn A∗ = ∆n A∗ |O is also an H-bundle by the product action, and a review of the definition of V shows that the restriction ∆V|H of the representation

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∆V defined above is the representation λ ⊗ ` on DO ⊗ Sn A∗ associated to this H-bundle structure. If Z is any vector space and z ∈ Z, let mz : SZ → SZ be multiplication by z, and for α ∈ Z ∗ let ∂α : SZ → SZ be the derivative along α of elements of SZ, viewed as polynomials on Z ∗ . Then ∂α is a derivation of the algebra SZ such that ∂α (Sn Z) ⊂ Sn−1 Z, and on Z we have ∂α z = hα, zi. It is elementary that the transpose ∂αT : SZ ∗ → SZ ∗ is mα . We are now in a position to compute φ. Proposition 7.2. The object φ of C acts in Sn A∗ , with Ek (φ) = Sk A∗ and Nk (φ) = Sk C0 . For α ∈ a, s ∈ S, and X ∈ h, we have φα = −ζα (p0 ) − i∂α , φs = `s , and φX = `X + mXp0 : Sn−1 A∗ → Sn A∗ . Proof. We have already seen the first sentence. The formula for φα follows from the proof of Proposition 6.4 and the fact that mTα = ∂α , and φs = `s follows from ∆V = λ⊗`, since φ|S is the little group representation of ∆V|H at p0 . For δ ∈ Sn A∗ , we compute φX δ = −δ ◦ VX using the identifications from the end of Sec. 6. As an element of ∆n A∗ (p0 ), δ is Dδ |p0 , and if Ω ∈ Sn A then as an element of J n A∗ (p0 ), Ω is the n-jet jn PΩ,p0 (p0 ) of the polynomial PΩ,p0 at p0 . Now VX jn PΩ,p0 = jn λX PΩ,p0 , and one finds that λX PΩ,p0 = PΩ0 ,p0 , where Ω0 = `∗X Ω − ∂Xp0 Ω. Thus hφX δ, Ωi = . −hδ, (`∗X − ∂Xp0 )Ωi = h(`X + mXp0 )δ, Ωi, completing the proof. We will now consider the representation V˜ ⊗ π defined at the beginning of this section. In fact, it will be more convenient to study V˜ ⊗ π ∗ , and we will write V π and V˜ π for V ⊗ π ∗ and V˜ ⊗ π ∗ . Then r ⊗ 1 : V π → V˜ π is a G-isomorphism, and ˜ ⊗V ˜ ⊗ π, so the object φ˜π = F V˜ π of C acts in Ek (φ˜π ) = Ek (φ) clearly ∆V˜ π = (∆V) π π π ∗ ˜ ⊗ V . Define an object φ of C with Ek (φ ) = Sk A ⊗ V and with Nk (φ˜ ) = Nk (φ) Nk (φπ ) = Sk C0 ⊗ V by making µ ⊗ 1 : φ˜π → φπ an equivalence, where µ : φ˜ → φ is the equivalence from above. Proposition 7.3. The object φπ of C is φ ⊗ π, where the representation π of H is restricted to a representation of S ∞ with flag V = V0 = V1 , and the tensor product is the one between C and RS ∞ defined in Sec. 4. Proof. If we define a representation ∆V π of G on D(∆n A∗ |O )⊗V by ∆Vgπ (D) = λg ◦ D ◦ Vgπ−1 , then rT ⊗ 1 is a G-isomorphism and so ∆V π is actually (∆V) ⊗ π. Furthermore, φπ = FV π in the same sense as before, and so φπ |S 0 is the inhomogeneous little group representation of ∆V ⊗ π at p0 , which is φ|S 0 ⊗ π. To check that φπX = φX ⊗ 1 + 1 ⊗ πX , choose δ ∈ Sn A∗ and v ∈ V and compute φπX (δ ⊗ v) = −(δ ⊗ v) ◦ (V ⊗ π ∗ )X , using the fact that V ⊗ π ∗ is an ordinary representation of H. In Sec. 8 we will need the fact that the restriction of φ to a representation of S ∞ is in fact the restriction of a representation φ˜ of H on Sn A∗ , i.e., φ|S ∞ = ˜ S ∞ . To understand why this is the case, note that φ is an S 0∞ -subrepresentation φ| of an ordinary S 0 , g-representation on the entire symmetric algebra SA∗ , and the

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restriction of this representation to S, h leaves the ideal hA∗ in+1 invariant. Hence it defines a quotient representation on Sn A∗ , and this representation extends φ|S ∞ to an ordinary representation of S, h. To extend φ|S ∞ further to a representation of H, recall that φ is defined using the identification δ 7→ Dδ |p0 of Sn A∗ with ∆n A∗ (p0 ). The action of φ on ∆n A∗ (p0 ) is dual to the natural action on J n A∗ (p0 ), and this space is naturally isomorphic to the polynomials Pn A∗ on A∗ of degree ≤ n. Now Pn A∗ carries a natural action of H, the dual of which extends φ|S ∞ to H. Furthermore, translation by p0 in Pn A∗ intertwines the H action on J n A∗ (p0 ) with that on J n A∗ (0), which is just `∗ . Working this out yields the following results. For any p ∈ A∗ , define an endomorphism mp : δ 7→ pδ of S n A∗ such that mp maps the ideal hA∗ in to 0. Let φ˜ be the representation of H on Sn A∗ defined by ˜ φ˜h = exp(mhp0 − mp0 ) ◦ `h ; since `h ◦ mp ◦ `−1 h = mhp , φ is indeed a representation. Differentiating at e yields φ˜X = `X + mXp0 , which gives the next proposition. Proposition 7.4. The restriction of φ˜ to a representation of S ∞ on the flag Sn A∗ ⊃ Sn−1 A∗ is φ|S ∞ . Furthermore, exp(mp0 ) intertwines the representations φ˜ and ` of H. 8. The Main Theorem We may now prove our main theorem, which gives a condition for an object π U of ExtC G O to be isomorphic to a subquotient of V for some π, in terms of the 0∞ restriction of the representation of S associated to F U to S ∞ . The main strength of the theorem is that in deciding whether or not U is a subquotient of V π , it is not necessary to consider the A-action of F U. Theorem 8.1. Let U be an object of ExtC G O, and write σ for F U. Then U π is isomorphic to a subquotient of V for some representation π of H on a finite dimensional complex vector space V if the representation σ ˜ = σ|S ∞ of S ∞ on the flag En ⊃ En−1 is a subquotient of the restriction of π to a representation of S ∞ on the flag V = V0 = V1 . Conversely, if U is isomorphic to a subquotient of V π for some π, then σ ˜ is a subquotient of the restriction of a representation of H to ∞ S (this representation of H is not necessarily π). Remark. It is well known that if H is a real linear algebraic group and S ⊂ H is an algebraic subgroup, then all rational representations of S are subquotients of restrictions of rational representations of H, see e.g. [1]. If there is an analog of this fact showing that for some wide class of subgroups S of algebraic groups H, a wide class of representations of S ∞ are subquotients of restrictions of representations of H, we will be able to say that “most” of the representations in π ExtC G O are isomorphic to subquotients of representations of the form V . In light of Theorem 2.1, this would mean that generically, representations in ExtG O are isomorphic to subquotients of representations of the form V π .

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Proof. If U is isomorphic to a subquotient of V π , then σ is a subquotient of FV π = φ⊗π in both the categories C and RS 0∞ by Lemma 3.5. By Proposition 7.4, ˜ of H, and so the converse (φ⊗π)|S ∞ is itself the restriction of the representation φ⊗π statement of the theorem is proven. Suppose now that σ ˜ is isomorphic to a subquotient of π|S ∞ in the category ˜ is isomorphic to a subquotient of φ ⊗ π RS ∞ . Then Lemma 4.3 shows that φ ⊗ σ in the category C, where the tensor product is the one defined in Sec. 4 mapping C × RS ∞ to C. Therefore if we can show that σ is isomorphic to a subquotient of φ ⊗ σ ˜ in C, we will have that σ is isomorphic to a subquotient of φ ⊗ π in C. Lemma 3.5 will then show that U is isomorphic to a subuotient of V π , completing the proof of the theorem. We will in fact show that σ is isomorphic to a subobject of φ ⊗ σ ˜ in the category C, by defining a C-monomorphism z : σ → φ ⊗ σ ˜ . Write Ek and Nk for Ek (σ) and Nk (σ), and recall from the end of Sec. 2 that the flag associated to the representation ˜ ) and N˙ k = σ ˜ of S ∞ on En ⊃ En−1 is En ⊃ . . . ⊃ E0 ⊃ 0. Let E˙ k = Ek (φ ⊗ σ ˜ ); by definition Nk (φ ⊗ σ E˙ k =

M

Sp A∗ ⊗ Eq

M

and N˙ k =

p+q=k

Sp C0 ⊗ Eq .

p+q=k

In order to define z, we must define an algebra representation Σ of Sn A on En . For α ∈ a = AC , let Σα = i(σα + ζα (p0 )). Since σ is a representation of the abelian Lie algebra a, this defines a representation Σ of the U(a) = SA. Using the last paragraph of Sec. 2 we find that Σα (Ek ) ⊂ Ek−1 , so the ideal hAin+1 acts as 0 under Σ and hence Σ factors through to a representation of Sn A. For any Ω ∈ Sn A, let ΣTΩ : En∗ → En∗ be the transpose of ΣΩ . Then ΣT is an algebra representation of Sn A on En∗ , and we define the map z : En → Sn A∗ ⊗ En by defining its transpose z T : Sn A ⊗ En∗ → En∗ to be z T (Ω ⊗ ) = ΣTΩ . We remark that if σ = φ, ΣTα = mα and z T : Sn A ⊗ Sn A → Sn A is just multiplication; this case led us to discover z. Since z T is surjective, z is injective, and we will show that z(Nk ) ⊂ N˙ k and that z is a S 0∞ -map. Then it will follow from Lemma 4.2 that z(Ek ) ⊂ E˙ k and that z is a flag monomorphism in C, which means that σ is isomorphic to a subobject of φ ⊗ σ ˜ and Theorem 8.1 is proven. In order to proceed we need a lemma, whose proof will be omitted. Lemma 8.2. Let A, B, Y , and Z be finite dimensional vector spaces, α : A → Z and β : B → Y linear maps, Z0 and Z1 subspaces of Z, A0 ⊂ . . . ⊂ An ⊂ A and B0 ⊂ . . . ⊂ Bn ⊂ B flags, and denote transposes with a superscript T . First , T α(A0 ) ⊂ Z0 if and only if αT (Z0⊥ ) ⊂ A⊥ 0 . Second , if j : Z0 → Z is inclusion, j is ⊥ ⊥ ∗ ∗ ⊥ ⊥ ⊥ projection. Third , (A0 ⊗B0 ) = A0 ⊗B +A ⊗B0 . Fourth, (Z0 +Z1 ) = Z0 ∩Z1⊥ . Fifth, (α ⊗ β)T = αT ⊗ β T . Last ,  

X

p+q=k

⊥ Ap ⊗ Bq  =

X r+s=k−1

⊥ A⊥ r ⊗ Bs .

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P In proving that z(Nk ) ⊂ N˙ k , we will actually prove that z(Nk ) ⊂ p+q=k Sp C0 ⊗ Nq . Writing hC0⊥ i for the ideal in Sn A generated by C0⊥ , we have (Sp C0 )⊥ = hC0⊥ i + hAip+1 , and so by Lemma 8.2 it will do to prove that * + X T ⊥ r+1 ⊥ Σ (hC0 i + hA i)Ns = 0 . Nk , r+s=k−1

Since ΣhC0⊥ iNk = 0, the terms containing C0⊥ are 0, and since ΣT hAr+1 iNs⊥ ⊂ Nr+s+1 = Nk , the equation holds. We will check that z is an S 0∞ -map by checking separately that it is an S-map, ˜s ) ◦ z, or σsT ◦ z T = an a-map, and a h-map. For s ∈ S we need z ◦ σs = (φs ⊗ σ z T ◦ (φTs ⊗ σsT ). Applying both sides to some Ω ⊗ ∈ Sn A ⊗ En∗ , we see that we need σsT ◦ ΣT (Ω) = ΣT (φTs Ω) ◦ σsT . This comes down to Σ(Ω) = σs ◦ Σ(φTs Ω) ◦ σs−1 , and it is enough to check this for Ω = α ∈ a. By Proposition 7.2, φs = `s , and so φTs = `∗ (s−1 ) acts on α to give Ad−1 s α. Now the definition of Σ and the fact that sp0 = p0 give the desired equality. For α ∈ a, (φ ⊗ σ ˜ )α = φα ⊗ 1 because σ ˜ is trivial on A, and so we need to check that z ◦ σα = (φα ⊗ 1) ◦ z. Taking the transpose and applying both sides to Ω ⊗ , we need σαT ◦ ΣT (Ω) = ΣT (φTα Ω), or Σ(Ω) ◦ σα = Σ(φTα Ω). Now φTα = −ζα (p0 ) − imα , so Σ(φTα Ω) = Σ(−ζα (p0 ) − iα) ◦ Σ(Ω) = σα ◦ Σ(Ω) = Σ(Ω) ◦ σα . ˜ )X ◦ z for X ∈ h, because σX It is a little tricky to check that z ◦ σX = (φ ⊗ σ and φX are not endomorphisms of En and Sn A∗ , but rather are maps from En−1 to En and Sn−1 A∗ to Sn A∗ , respectively. First, a direct argument like the one used to prove that z(Nk ) ⊂ N˙ k shows that z(En−1 ) ⊂ Sn−1 A∗ ⊗ En−1 , and so both sides ∗ map En−1 to Sn A∗ ⊗ En . Let us write ρ for both of the projections En∗ → En−1 T ∗ ∗ and Sn A → Sn−1 A. Then we find that (z ◦ σX ) : Sn A ⊗ En → En−1 acts on Ω ⊗ T T ◦ ΣT (Ω), while z T ◦ (φ ⊗ σ ˜ )TX = z T ◦ (φTX ◦ ρ + ρ ⊗ σX ) acts on it to to give σX T T T T give Σ (φX Ω) ◦ ρ + Σ (ρΩ) ◦ σX . Thus we need to check that Σ(Ω) ◦ σX and Σ(φTX Ω) + σX ◦ Σ(ρΩ) are equal as maps from En−1 to En . Here φX = `X + mXp0 gives φTX Ω = −ρ`∗X Ω + ∂Xp0 Ω, and note that since ΣhAn iEn−1 = 0, Ω and ρΩ have the same action on En−1 under Σ. Hence we come down to proving that [σX , ΣΩ ] = Σ(−`∗X Ω + ∂Xp0 Ω). Since both ad(σX ) ◦ Σ and Σ ◦ (∂Xp0 − `∗X ) are Σderivations from Sn A to End(En ), it is enough to check the equality for Ω = α ∈ a, where it is easy. The following variation on Theorem 8.1 can be more practical than the theorem for computations. For example, if U = V π then σ = F U = φ⊗π, and so Theorem 8.1 ˜ exhibits U as a subquotient of V φ⊗π , where φ˜ is the representation of H from Proposition 7.4. Here U is actually equal to V π , a much simpler representation ˜ than V φ⊗π , and it would be nice to have an improvement of Theorem 8.1 which would sense this and exhibit U as a subquotient of V π . Corollary 8.3 does this, as the restriction of φ ⊗ π to a representation of S ∞ on Sn A∗ ⊗ V has π|S ∞ as

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a quotient. To see this, divide by the S ∞ -subrepresentation hA∗ i ⊗ V , and use Proposition 7.4 to check that the quotient is π|S ∞ and that it has the property required by Corollary 8.3. Corollary 8.3. Given U in ExtC G O, let σ = F U. Suppose that the restriction of σ to a representation σ ˜ of S ∞ on En ⊃ En−1 has a quotient S ∞ -representation σ on a flag Xn ⊃ Xn−1 , such that the subspace Xn∗ of En∗ generates En∗ under the action of the representation σ ∗ |A . If there is a representation π of H on a space V such that σ is a subquotient of π|S ∞ , then U is a subquotient of V π . Proof. We must show that σ is a C-subquotient of φ ⊗ π. Since σ is a S ∞ subquotient of π, φ ⊗ σ is a C-subquotient of φ ⊗ π by Lemma 4.3, and so we need only show that σ is a C-subquotient of φ ⊗ σ. Let ρ : En → Xn be the quotient map intertwining σ ˜ = σ|S ∞ with σ, let z : En → Sn A∗ ⊗ En be the C-map from the proof of Theorem 8.1, and let z : En → Sn A∗ × Xn be (1 ⊗ ρ) ◦ z. By Lemma 4.3, 1 ⊗ ρ is a C-map from φ ⊗ σ ˜ to φ ⊗ σ, and so z is a C-map from σ to φ ⊗ σ. Now z T : Sn A ⊗ Xn∗ → En∗ is just z T , and so z T (Sn A ⊗ Xn∗ ) = ΣT (Sn A)Xn∗ . The assumption that σ ∗ (U(a))Xn∗ = En∗ therefore gives that z T is surjective, and so z is injective and σ is actually a C-subobject of φ ⊗ σ. 9. An Example We will now apply our main theorem to the example of Rideau’s mentioned in the introduction. Let A = R1,3 , and let H = SL2 C act on A via the double cover of the identity component of SO(1, 3) by H. Then G is the Poincare group, and we take O to be the forward light cone. Let F0 be the trivial bundle O × C, let F1 be the complexified tangent bundle Tc O, and let U i be the canonical representation of G in DFi for i = 0, 1. Let σi be the little representation of S in Fi∗ (p0 ), so that U i is induced from the representation σ ∗ ⊗ eip0 of S 0 . In [9], Rideau studied indecomposable representations U of G acting in DF0 ⊕DF1 that leave DF1 invariant and define U 0 and U 1 in the subquotients. He found that there is a bijection from the set of equivalence classes of these representations to the projective complex plane P1 , which is described in the notation of this paper on p. 663 of [4]. There we classified the objects σ (actually, η(σ)) of C such that σ = F U for some such U, indexing them by elements (p, x) of C2 . Write σp,x for the object of C with index (p, x), and let Up,x be a representation in ExtC G O with F Up,x = σp,x . We found 0 0 ∼ that Up,x = Up0 ,x0 if and only if [p, x] = [p , x ] in P1 or both indices are (0, 0), and that Up,x decomposes if and only if (p, x) = (0, 0). Furthermore, we saw that U0,x is induced and is isomorphic to the natural representation on DO ⊗ A if x 6= 0, and that Up,x is not induced if p 6= 0. Here we will prove that for all (p, x), Up,x has the type of geometric realization sought after in this paper. This result was first obtained by Cassinelli, Truini, and Varadarajan [2]. Proposition 9.1. For all (p, x) in C2 , Up,x is isomorphic to a subquotient of V (1) ⊗ (π ⊕ 1), where 1 is the trivial representation of H on C and π is the representation of H on A.

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Proof. We will use Corollary 8.3, and for brevity we will use the notation from Sec. 6 of [4]. Fix (p, x), and write σ for σp,x and η for η(σp,x ). Since E1,1 is linearly isomorphic to M1,1 and σ acts in E1,1 (see Sec. 2 of this paper), we may transport σ to M1,1 . Here M1,1 = L0 ⊕ (L0 ⊗ r) ⊕ L1 , where L0 has basis f00 , L1 has basis f1−2 , f10 , f12 , and r has basis M+ , M− , M3 . In order to describe the S 0∞ -action of σ, we must give the actions of s0 and r. The space E0,0 = M0,0 is just L0 , and it is only here that σ|r is defined. A review of the definition of the S-structure of objects of C shows that M ∈ r acts by right multiplication: σM f00 = −f00 ⊗ M . The s0 -action follows from the formulae for the action of η given in [4]; we will simply state the relevant facts. The flag L0 ⊗ r ⊃ 0 is an S ∞ -subrepresentation of the representation σ|S ∞ on M1,1 ⊃ L0 which has the quotient property. Keep in mind that when we speak of S 0∞ , S 0 is being viewed as a subgroup of G, while when we speak of S ∞ , S is being viewed as subgroup of H. Writing X1 ⊃ X0 for the quotient flag, we find that the quotient S ∞ -representation is isomorphic to one of two representations of S ∞ , depending on whether x is 0 or not. In both cases, h acts by 0 on X0 . If x 6= 0, the S-action on X1 is isomorphic to π, and if x = 0 it is isomorphic to σ0 ⊕ σ1 . In either case, it follows from Lemma 1.2 that X1 ⊂ X0 is isomorphic to a subquotient of the restriction of π ⊕ 1 to a representation of S ∞ . We leave the reader to use the formulae for η|a from [4] to check that X1∗ generates E1∗ under the action of A, and then apply Corollary 8.3 to complete the proof. References [1] G. Cassinelli, G. Olivieri, P. Truini and V. S. Varadarajan, “On some nonunitary representations of the Poincar´e group and their use for the construction of free quantum fields”, J. Math. Phys. 30 (1989) 2692–2707. [2] G. Casinelli, P. Truini and V. S. Varadarajan, “Hilbert space representations of the Poincar´e group for the Landau gauge”, J. Math. Phys. 32 (1991) 1076–1090. [3] C. Conley, “Representations of finite length of semidirect product Lie groups”, J. Funct. Anal. 114 (1993) 421–457. [4] C. Conley, “Little group method for smooth representations of finite length”, Duke Math. J. 79 (3) (1995) 619–666. [5] C. Conley, “Geometric realizations of representations of finite length II”, to appear in Pacific J. Math. [6] A. Guichardet, “Extensions des repr´esentations induites des produits semidirects”, J. f¨ ur reine angew. Math. 310 (1979) 7–32. [7] A. Guichardet, “Repr´esentations de longeur finie des groupes de Lie inhomog` enes”, Ast´erisque 124–125 (1985) 212–252. [8] P. Griffiths, “The extension problem in complex analysis II”, Amer. J. Math. 88 (1966) 366–446. [9] G. Rideau, “Noncompletely reducible representations of the Poincar´ e group associated with the Lorentz gauge”, J. Math. Phys. 19 (1978) 1627–1634. [10] L. Schwartz, Th´eorie des distributions 3rd ed., Hermann, Paris, 1966. [11] V. S. Varadarajan, “Geometry of Quantum Theory,” 2nd ed., Springer-Verlag, New York, 1985.

ENERGY DEPENDENT BOUNDARY CONDITIONS AND THE FEW-BODY SCATTERING PROBLEM P. KURASOV Dept. of Mathematics, Stockholm University 10691 Stockholm, Sweden; Dept. of Mathematics, Ruhr Uni.-Bochum 44780 Bochum, Germany; Dept. of Mathematics, Lule˚ aUniversity 97187 Lule˚ a, Sweden; Dept. of Mathematical and Computational Physics St. Petersburg Univ, 198904 St. Petersburg, Russia Received 5 February 1997 An exactly solvable problem with energy dependent interaction is investigated in the present paper. The selfadjoint model operator describes the scattering problem for three one-dimensional particles. It is shown that this problem is equivalent to the diffraction problem in the sector with energy dependent boundary conditions. The problem is solved with the help of the Sommerfeld–Maluzhinetz representation, which transforms the partial differential equation for the eigenfunctions to a functional equation on the integral densities. The solution of the functional equation can be constructed explicitly in the case of identical particles. The three-body scattering matrix describing rearrangement and excitation processes is represented in terms of analytic functions.

1. Introduction Energy dependent interactions play an important role in the modern mathematical physics. Such interactions allow to model complicated physical phenomena and solve the problem exactly at the same time. A disadvantage of these problems is that they are usually described by non-selfadjoint operators or even by operator bundles. The corresponding eigenfunctions do not satisfy orthogonality and completeness properties. This produces additional difficulties during the investigation of these problems and limits the number of the phenomena which can be described in habitual terms. We show that some of these problems can be solved by considering operators in certain extended Hilbert spaces. In this approach operator bundles with energy dependent interactions appear as restrictions of selfadjoint operators. Resolvents of the operators with energy dependent interactions can be calculated with the help of the M. G. Krein formula [43]. A wide class of such operators is well known under the name of operators with zero-range (or delta functional) potentials. The interaction in such problems is described by boundary conditions on some low dimensional manifolds. The most complete set of these problems has been collected in the monographs by S. Albeverio et al. [5] and Yu. N. Demkov and V. N. Ostrovsky [10]. Similar problems were 853 Reviews in Mathematical Physics, Vol. 9, No. 7 (1997) 853–906 c World Scientific Publishing Company

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studied with the help of the method of point interactions with internal structure. This method leads to a new class of exactly solvable Schr¨ odinger operators with a richer spectral structure [40, 41]. Selfadjoint operators describing physical phenomena are defined in the orthogonal sum of standard Hilbert spaces and certain internal spaces, describing the interaction. Applications of the discussed methods to the two-body problem were considered by V. M. Adamyan and B. S. Pavlov [1]. A similar scattering problem for three particles in the three-dimensional space has been studied by B. S. Pavlov, Yu. A. Kuperin, K. A. Makarov, S. P. Merkuriev and A. K. Motovilov [19, 20, 21, 29, 30, 42]. This investigation was inspired by the papers [47, 36, 37, 49, 3, 4], where the system of three particles in three-dimensional space interacted via delta potential has been studied. The present paper is devoted to the three-body problem in one space dimension. The few-body scattering problems form a wide class of complicated quantum mechanical problems [35, 39, 44]. Some of the difficulties already appear at the level of the three-body operator. Such operators describe the following processes: rearrangement (12)+ 3 → 1 + (23), breakup (12)+ 3 → 1 + 2 + 3, capture 1 + 2 + 3 → (12) + 3 and excitation (12) + 3 → (12)∗ + 3. The corresponding scattering matrix describes the interaction between several asymptotic channels. As the standard scattering problem leads to complicated calculations, exactly solvable models should play an important role in the investigation of these phenomena. An application of the discussed method of boundary conditions to the case of few-body problems leads to a wide class of operators which can be studied exactly. Unfortunately the simplest models do not give a possibility to describe complicated phenomena. The complexity of the model problem increases with the number of phenomena which can be described by the model. Several systems of one-dimensional particles with interaction of this type have been analyzed. The investigation of this three-body problem was started from the simplest problems such as the system of identical particles [13, 50], the system of impenetrable particles [14], the system of two particles interacting with a wall [2, 28]. Some of the solutions were expressed in terms of the elementary functions [8], for example in the case of identical particles. The scattering solution can be constructed with the help of the Bethe Ansatz [12] in this case. More realistic problems describing nonidentical particles lead to complicated equations for the eigenfunctions. These equations can be solved using certain integral transformations. Using the Sommerfeld–Maluzhinetz integral representation [31, 32] one transfers the partial differential equation into a difference equation for some analytic function. The solution of this equation can be constructed with the help of special functions [2, 14, 15, 25–28], and it can be expressed in terms of elementary functions when the problem has some symmetry properties. Usually this case coincides with the one for which the solution can be presented by the Bethe Ansatz. The standard delta functional interaction defines the unique two-body bound state or resonance. The corresponding model cannot be used to describe collisional deexcitation processes. The two-cluster and three-cluster channels are orthogonal in the case of equal particles. Breakup and capture processes are forbidden in this model.

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The case of identical particles is studied in the present paper. The model operator is constructed following the general scheme suggested by P. S. Pavlov. This is the first application of this scheme to the soluble problem of three-body scattering in dimension one with nontrivial two-body interaction (see [18] where a nontrivial three-body interaction has been introduced). The model constructed has a more realistic scattering matrix and richer structure of the spectrum than the model in [8]. In particular it permits to describe rearrangement processes. The model operator is a selfadjoint perturbation of the “asymptotic”Hamiltonian, describing the free motion in the system of three particles. The model operator describes a system of three arbitrary particles, but later we confine our consideration to the case of identical particles in order to be able to express the solution of the scattering problem in terms of elementary functions. The solution of the twobody scattering problem can be presented by a combination of plane waves. The problem has an arbitrary number of bound states and resonances. The solution of the three-body scattering problem cannot be presented in terms of a Bethe Ansatz and it is calculated using the Sommerfeld–Maluzhinetz transformation. The equation for the eigenfunctions is equivalent to the Helmholtz equation in the sector with energy dependent boundary conditions. The diffraction problem is transformed to a functional equation, which is solved exactly. Analytical properties of the solutions of the functional equation are investigated. Singularities of these solutions are determined by the two-body bound states and resonances. The analytical solution of the functional equation yields the analytical two- and three-body scattering matrices. Simple formulae for these scattering matrices give us the possibility to investigate the relations between the two- and three-body spectral characteristics. Some of the results presented here were discussed by the author in [22]. Our model can be used in statistical physics calculations in order to investigate the relations between the spectral characteristics of two-body operators and the thermodynamic parameters. Such investigations have been started in [24]. Developed methods can be applied to the study of diffraction problems in the domains with singularities. The model operator is presented by certain block operators acting in the orthogonal sum of Hilbert spaces. The interaction between the components is determined by boundary conditions, which can be considered as antidiagonal singular operators. Thus the operator constructed is close to the set of matrix selfadjoint operators studied recently by V. M. Adamyan, F. V. Atkinson, H. Langer, R. Mennicken and A. Shkalikov [6]. The paper is organized as follows. The two-body Schr¨odinger operator is constructed in Sec. 2. The two-body scattering matrix is calculated. Relations with the standard Schr¨ odinger operator are discussed here following the paper [23]. In Sec. 3 a generalization of the model to the three-body case is considered. The symmetric three-body operator is constructed. A selfadjoint extension of this operator is calculated with the help of the von Neumann theory. The equation for the deficiency elements is transformed into the vector difference equation for an analytic function in Sec. 3. The symmetries of the corresponding equations are discussed. Investigation of a special invariant basis leads to a system of independent

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two-dimensional difference equations. It is shown that these equations can be decoupled in the case of indistinguishable particles. The solution of the difference equation is obtained in Sec. 4 for this system. Properties of the deficiency elements are studied in Sec. 5. A selfadjoint extension of the symmetric three-body operator is constructed in Sec. 6. The three-body scattering matrix is calculated, it is expressed in terms of elementary functions. The scattering solution is presented by a combination of plane waves constructed in the form of the Bethe Ansatz plus a certain outgoing wave. The outgoing wave is equal to the limit of the deficiency element calculated earlier, when the spectral parameter λ approaches the real line. Relations between the spectral properties of the two- and three-body operators are discussed. 2. The Two-Body Hamiltonian This section is devoted to the construction of the model operator describing the two-body problem on the line. We first recall some standard facts concerning the two-body Schr¨ odinger operator with the interaction introduced by usual potential. We concentrate our attention to the properties of the corresponding scattering data. The main part of this section is devoted to the construction of the model for a twobody operator. The two-body quantum mechanical problem contains the following asymptotic channels: two noninteracting particles; two particles in a bound state. The model operator is defined as a selfadjoint perturbation of the orthogonal sum of the Hamiltonians describing each asymptotic channel. These operators are the two-dimensional Laplace operator and the one-dimensional matrix second derivative operator with a diagonal thres hold matrix. The entries of the latter matrix coincide with the energies of the two-body bound states. The eigenfunctions of the model operator corresponding to the discrete and continuous spectra are calculated explicitly. The scattering matrix is expressed in terms of elementary functions. We discuss how to select model operators with the standard properties of the scattering data. 2.1. Interaction determined by a potential The Schr¨ odinger operator describing two one-dimensional quantum particles with equal masses has the following form: AV = −

1 2

d2 d2 + 2 2 dr1 dr2

+ V (| r1 − r2 |) ,

(2.1)

where r1 , r2 denote the coordinates of the particles. Here the interaction is determined by a potential V which depends only on the distance between the particles. The center of mass motion can be separated using Jacobi coordinates x12 = r1 − r2 , y12 =

r1 + r2 , 2

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the Schr¨ odinger operator can be decomposed as follows: AV = −

1 ∂2 2 × I + I × AV , 4 ∂y12

AV = −

d2 + V (| x12 |) . dx212

(2.2)

The operator AV has been studied as a selfadjoint operator in the Hilbert space L2 (R) for potentials with the finite first momentum [11, 33, 9, 34]: Z +∞ |x V (x)|dx < ∞ . (2.3) −∞

The scattering problem for the two-body Schr¨ odinger operator is formulated with the unperturbed operator equal to the second derivative operator A0 = −

d2 , dx212

defined on the standard domain Dom(A0 ) = W22 (R). The unperturbed and perturbed operators have the same branch of absolutely continuous spectrum [0, ∞). The perturbed operator AV with the interaction V can have some additional negative eigenvalues — two-body bound states. Let f− (x, k), f+ (x, k), k ∈ R \ {0} be the solutions of the equation AV f = k 2 f in the generalized sense with the following asymptotics: f− (x, k) ∼ eikx , f+ (x, k) ∼ e−ikx ,

x → +∞ , (2.4a)

x → −∞ .

The solutions fj (k, x) are asymptotic to sums of exponentials as x → ∓∞ 1 R− (k) −ikx eikx + e , T− (k) T− (k)

x → −∞,

1 R+ (k) ikx e−ikx + e , f+ (x, k) ∼ T+ (k) T+ (k)

x → +∞ .

f− (x, k) ∼

(2.4b)

T+ (k) R− (k) S(k) = R+ (k) T− (k) is called the scattering matrix. This matrix is unitary The matrix

(2.5)

|T− |2 + |R− |2 = 1 = |T+ |2 + |R+ |2 , T− (k)R+ (−k) + R− (k)T+ (−k) = 0 . One can prove that the transition coefficients coincide and that the following asymptotics for the coefficients of the scattering matrix are valid [11] when k → ∞: 1 ; T+ (k) = T− (k) = 1 + O |k| (2.6) 1 1 ; R+ (k) = O |k| . R− (k) = O |k|

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The following estimates are valid in the low-energies domain: T± (k) = O(k), R± (k) = −1 + O(k), k → 0 .

(2.7)

These asymptotics will be called “standard” ones in what follows. The model operators which will be constructed in the next section define unitary scattering matrices. But the coefficients of these matrices do not necessarily have the standard asymptotics. In order to make the model realistic we confine our consideration to the model operators with the standard asymptotics of the scattering matrix (see Sec. 2.5). 2.2. The model operator The model two-body operator is constructed as a perturbation of the operator describing possible asymptotic channels for the two-body problem. 2 The 2oper ∂ ∂ ⊕ ator A20 is equal to the orthogonal sum of two operators A20 = − 12 ∂r 2 + ∂r 2

A20

1

2

2

− 41 dyd 2 + A12 . The first operator in this orthogonal sum acts in the Hilbert space 12

L2 (R2 ) and describes two noninteracting particles. The second operator describes two coupled particles moving together. The energies of the bound states are equal to the eigenvalues of the finite dimensional selfadjoint matrix A12 acting in the finite dimensional space H12 . We suppose that the eigenvalues of A12 are negative. The second operator acts in the Hilbert space L2 (R, H12 ). The standard separation of the center of mass motion gives the following operator: A20 = A1,2 ⊕ A12 ,

A1,2 = −

d2 dx21,2

(2.8)

which acts in the orthogonal sum of the Hilbert spaces H 2 = L2 (R) ⊕ H12 . The unperturbed operator for the scattering problem can be chosen equal to the A20 . The perturbed operator can be constructed by restricting first the operator A20 to a certain symmetric operator and then extending it to another selfadjoint operator. The interaction between the channels will be introduced using some generalized boundary conditions. The restriction of the operator A1,2 → A1,20 to the domain Dom(A1,20 ) = u ∈ W22 (R), u(0) = 0, u0 (0) = 0 is a symmetric operator with the deficiency indices (2, 2). The adjoint operator is defined by the same differential expression Dom(A∗1,20 ) = u ∈ W22 (R\{0}) . The boundary form of the adjoint operator is equal to u, v ∈ Dom(A∗1,20 ) hA∗1,20 u, viL2 − hu, A∗1,20 viL2 du du d¯ v d¯ v = , h¯ vi + [¯ v ] − hui − [u] dx dx dx dx x=0

(2.9)

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where [∗] and h∗i denote the jump and the mean value of function at the origin [f (x)] ≡ f (x + 0) − f (x − 0) , hf (x)i ≡

f (x + 0) + f (x − 0) . 2

(2.10)

We restrict the operator A12 to the operator A120 defined on the domain {u12 ∈ H12 : hu12 , θi = 0}. The restricted operator is a symmetric but not densely defined operator in H12 . Thus one cannot use directly the von Neumann theory to construct the selfadjoint extensions of the operator A120 . The restricted total operator A200 is defined by the orthogonal sum of the symmetric operators A200 = A1,20 ⊕ A120 on the domain Dom(A200 )={(u1,2 , u12 ) ∈ H 2 : u1,2 ∈ W22 (R), u1,2 (0)=0, u01,2 (0)=0; hu12 , θi=0} . We define the perturbed operator describing the interacting particles as a certain selfadjoint extension of the operator A200 . Theorem 2.1. Let the real parameters a, b, c, d satisfy the following equality: a b = −1 , det (2.11) c d θ ∈ H12 . The operator A2 U = A2

u1,2 u12

 A∗1,20 u1,2 , = du1,2 + bhu1,2 i |x=0 θ A12 u12 + a dx 

(2.12)

defined on the domain of functions from Dom(A∗1,20 ) ⊕ H12 satisfying the boundary conditions du1,2 + dhu1,2 i |x=0 = hu12 , θi c dx (2.13) [u1,2 ] |x=0 = 0 is a selfadjoint extension of the operator A200 . Remark. The operator A2 will be called the perturbed model two-body operator in the sequel. Similar two-body operators has been suggested first by K. Makarov [29, 30]. Proof. Consider first any element U from the domain of the operator A200 . The operators A2 and A200 map this element to one and the same element of the Hilbert space H 2 = L2 (R) ⊕ H12 . It follows that the operator A2 is an extension of the operator A200 . We are going to prove that the operator A2 is densely defined. Let U be a given element from the Hilbert space H 2 3 U = (u1,2 , u12 ). Consider an arbitrary element (˜ u1,2 , u12 ) from the domain of the operator A2 . The difference

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(u1,2 , u12 ) − (˜ u1,2 , u12 ) = (u1,2 − u˜1,2 , 0) belongs to the space L2 (R) ⊂ H 2 . The restricted operator A1,20 is densely defined, thus for every given > 0 there exists u1,2 −ˆ u1,2 kL2 < function u ˆ1,2 from the domain of the operator A1,20 such that k u1,2 −˜ u1,2 + u ˆ1,2 , u12 ) k< and (˜ u1,2 + uˆ1,2 , u12 ) . This implies that k (u1,2 , u12 ) − (˜ belongs to the domain of the operator A2 . Thus the operator A2 is densely defined. We calculate now the boundary form of the operator (11) on the functions from Dom (A∗1,20 ) ⊕ H12

hA2 U, V iH − hU, A2 V iH =

du1,2 du1,2 hv1,2 i + [v1,2 ] dx dx ! dv1,2 dv1,2 − [u1,2 ] |x=0 −hu1,2 i dx dx du1,2 + a + bhu1,2 i |x=0 hv12 , θi dx dv1,2 + bhv1,2 i |x=0 −hu12 , θi a dx

(2.14)

This boundary form vanishes on the domain of the operator A2 hA2 U, V i − hU, A2 V i =

du1,2 dv1,2 hv1,2 i − hu1,2 i dx dx

dv1,2 dv1,2 du1,2 + bchu1,2 i + ac dx dx dx du1,2 hv1,2 i + bdhu1,2 ihv1,2 i + ad dx dv1,2 dv1,2 du1,2 − adhu1,2 i − ac dx dx dx du1,2 hv1,2 i − bdhu1,2 ihv1,2 i |x=0 −cb dx ! a dv1,2 du1,2 |x=0 1 + det = hv1,2 i − hu1,2 i dx dx c

b d

=0

(2.15)

Thus the operator A2 is a symmetric extension of the operator A200 . The adjoint operator is defined by the same formula (2.12) and its domain is a subset of Dom(A∗1,20 ) ⊕ H12 . If an element U = (u1,2 , u12 ) ∈ Dom(A∗1,20 ) ⊕ H12 belongs to the domain of the adjoint operator then the boundary form (1.14)

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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861

2 should be equal h toi zero for any V ∈ Dom(A ). Consider elements V such that dv1,2 hv1,2 i|x=0 = dx |x=0 = hv12 , θi = 0. The boundary form for such V is equal to dv

1,2 i and it follows that every function u1,2 must be continuous at the origin −[u1,2 ]h dx [u1,2 ] = 0. Consider now elements V such that hv12 , θi = 0. Similar calculations show that the boundary values of U should satisfy the first condition (2.13). It follows that the adjoint operator A2∗ has the same domain as the operator A2 and thus it is selfadjoint.

The operators A2 and A20 are in general two different selfadjoint extensions of the symmetric operator A200 . The set of constructed operators A2 does not coincide with the set of all selfadjoint extensions of the operator A200 . The advantage of the method presented is that the operator A2 is defined explicitly. One can use the fact that the operators A20 and A2 are two selfadjoint extensions of the same symmetric operator A200 to calculate the resolvent of A2 . 2.3. The resolvent The resolvent of the perturbed operator for all λ, =λ 6= 0 can be calculated using the modified Kreins formula and the resolvent of the unperturbed operator RA20 (λ) = RA1,2 (λ) ⊕ RA12 (λ) . The resolvent of √the operator A1,2 is the integral operator with the kernel ei

λ|x−y|

√ . The branch of the square root is fixed by the condi2i λ √ tion =λ > 0 ⇒ = λ > 0. The resolvent of the operator A12 coincides with the following matrix RA12 (λ) = (A12 − λ)−1 . The following function will play an important role in the sequel:

rA1,2 (λ, x, y) =

D(λ) =

bhRA12 (λ)θ, θi + d . ahRA12 (λ)θ, θi + c

(2.16)

The function R(λ) = hRA12 (λ) θ, θi is analytic in the upper half-plane =λ > 0 and has positive imaginary part there. The real constants a, b, c, d define a conformal map of the upper half-plane onto itself due to the conditions (2.11). It follows that the function D(λ) is analytic in the upper half-plane and has positive imaginary part there. Lemma 2.1. The resolvent of the perturbed operator RA2 (λ) = (A2 − λ)−1 is the matrix operator of the form RA2 (λ) = RA20 (λ) +

∆R(1,2)(1,2) (λ)

∆R(1,2)(12) (λ)

∆R(12)(1,2) (λ)

∆R(12)(12) (λ)

! .

(2.17)

862

P. KURASOV

The operators ∆R(1,2)(1,2) (λ), ∆R(12)(1,2) (λ) are the integral operators with the following kernels: √

√ D(λ) ei λ|x| √ ei λ|y| √ , ∆r(1,2)(1,2) (λ, x, y) = − D(λ) + 2i λ 2i λ

∆r(12)(1,2) (λ, y) =

√ aD(λ) − b √ ei λ|y| (A12 − λ)−1 θ . D(λ) + 2i λ

(2.18) (2.19)

The operators ∆R(1,2)(12) (λ), ∆R(12)(12) (λ) are equal to √

∆R(1,2)(12) (λ) = ei

λ|x|

h(A12 − λ)−1 ∗, θi √ √ , (2ai λ + b)R(λ) + 2ci λ + d

√ h(A12 − λ)−1 ∗, θi(A12 − λ)−1 θ √ √ . ∆R(12)(12) (λ) = −(2ai λ + b) (2ai λ + b)R(λ) + 2ci λ + d

(2.20)

(2.21)

Proof. Consider an arbitrary F ∈ H 2 . Let RA2 (λ)F = G. This implies that G ∈ Dom(A2 ) and F = (A2 − λ)G. The last equation can be written for the components as follows: d2 − 2 − λ g1,2 (x) = f1,2 (x) ; dx (2.22) dg1,2 + bhg1,2 i |x=0 θ − λg12 = f12 . A12 g12 + a dx We apply the operator RA12 (λ) to the left- and right-hand sides of the second equation dg1,2 (2.23) + bhg1,2 i |x=0 RA12 (λ) θ . g12 = RA12 (λ)f12 − a dx The projection on the element θ gives the following relation: dg1,2 + bhg1,2 i |x=0 hRA12 (λ)θ, θi . hg12 , θi = hRA12 (λ)f12 , θi − a dx

(2.24)

Every solution to (2.22) which is continuous at the origin is given by √ λ|x|

g1,2 = RA1,2 (λ)f1,2 + qei

,

where q is a parameter which will be fixed later. The boundary values of the function g1,2 at the origin are equal to Z hg1,2 i|x=0 =

∞

−∞

√

ei λ|y| √ f1,2 (y)dy + q; 2i λ

√ dg1,2 |x=0 = 2i λq . dx

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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The element G belongs to the domain of the operator A2 and satisfies the boundary conditions (2.13). It follows from (2.24) that the boundary values of g1,2 should satisfy the following equation:

dg1,2 |x=0 + (d + hRA12 (λ)θ, θib) hg1,2 i|x=0 (c + hRA12 (λ)θ, θia) dx = hRA12 (λ)f12 , θi . The parameter q can now be calculated q=

1 √ 2i λ + D(λ) 1 hRA12 (λ)f12 , θi − D(λ) c + hRA12 (λ)θ, θia

×

Z

∞ −∞

√

ei λ|y| √ f1,2 (y)dy 2i λ

! .

It follows that g12

√ √ Z 2i λ(aD(λ) − b) ∞ ei λ|y| √ √ f1,2 (y)dyRA12 (λ) θ = RA12 (λ)f12 + 2i λ + D(λ) −∞ 2i λ √ a2i λ + b hRA12 (λ)f12 , θiRA12 (λ) θ , − √ 2i λ(c + R(λ)a) + bR(λ) + d √

g1,2

ei λ|x| = RA1,2 (λ)f1,2 + √ 2i λ + D(λ) ×

1 hRA12 (λ)f12 , θi − D(λ) c + hRA12 (λ)θ, θia

Formulas (2.18–2.21) follow from the last two equations.

Z

∞ −∞

√

ei λ|y| √ f1,2 (y)dy 2i λ

! .

2.4. Spectrum, eigenfunctions, scattering matrix The singularities of the resolvent RA2 (λ) are situated at the points which √ satisfy the equation D(λ) + 2i λ = 0. They correspond to the eigenvalues of the operator A2 . The absolutely continuous spectrum of the operator is determined by the discontinuity of the resolvent on the positive part of the real axis due to the √ discontinuity of the function λ there. The discrete spectrum eigenfunctions are solutions of the equation A2 Ψs = 2 , are the negative real solutions of the λs Ψs , where λs > 0, s = 1, 2, . . . , Nbs equation 2i

p λs = −D(λs ) .

(2.25)

864

P. KURASOV

The eigenfunctions can be explicitly calculated: Ψ s = cs s (x) = e−χs |x| , ψ1,2

s ψ1,2 (x)

!

s ψ12

,

√ χs = −i λs > 0 ,

(2.26)

s = −(−2aχs + b)(A12 + χ2s )−1 θ . ψ12

The constant cs can be determined from the normalizing condition k Ψs k= 1: −1/2 1 + (−2aχs + b)2 k (A12 + χ2 )−1 θ k2 . (2.27) cs = χs The continuous spectrum eigenfunctions Ψ = (ψ1,2 , ψ12 ) are generalized solutions of the following equation:   d2 − 2 ψ1,2   dx ψ1,2   (2.28)  = λ ψ12 ,  dψ1,2 A12 ψ12 + a + bhψ1,2 i |x=0 θ dx satisfying the boundary conditions (2.13). Equation (2.28) can be reduced to the usual one-dimensional Schr¨ odinger equation on the axis with energy dependent boundary conditions at the origin. This reduction is similar to the one carried out in the proof of Lemma 2.2. The second of Eq. (2.28) dψ1,2 + bhψ1,2 i |x=0 θ + A12 ψ12 = λψ12 a dx can be solved as follows: dψ1,2 + bhψ1,2 i |x=0 (A12 − λ)−1 θ . ψ12 = − a dx Substitution into the boundary conditions (2.13) gives the following energy dependent boundary conditions for the component ψ1,2 : i h dψ1,2 dx bR(λ) + d |x=0 = − ≡ −D(λ) , (2.29) hψ1,2 i aR(λ) + c [ψ1,2 ] |x=0 = 0 . The multiplicity of the continuous spectrum is equal to 2. As in (2.4a) and (2.4b) the following representations for the eigenfunctions can be used: ! ψ±1,2 (x) 1 , λ = k2 Ψ± (λ) = √ 2 πk ψ±12 ( ikx e + R− (k)e−ikx ; x < 0 (2.30) ψ−1,2 (λ, x) = x>0 T− (k)eikx ; ( T+ (k)e−ikx ; x<0 ψ+1,2 (λ, x) = −ikx ikx + R+ (k)e ; x > 0 e

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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865

The left and right reflection and transition coefficients are identical due to the symmetry of the problem R− (k) = R+ (k) ≡ R(k),

T− (k) = T+ (k) ≡ T (k) .

The transition and reflection coefficients are calculated from the energy dependent boundary conditions (2.29): T (k) =

2ik D(λ) + 2ik

−D(λ) . R(k) = D(λ) + 2ik The components ψ±12 (λ) of the eigenfunctions are identical: √ 4 λ aD(λ) − b √ (A12 − λ)−1 θ . ψ±12 (λ) = ψ12 (λ) = i √ π D(λ) + 2i λ

(2.31)

(2.32)

The reflection and transition coefficients form the unitary scattering matrix: ! T (k) R(k) S(k) = . (2.33) R(k) T (k) The unitarity of the scattering matrix we calculated can be proven directly using the fact that the function D(k 2 ) is real for the real values of the parameter k. The discrete spectrum eigenfunctions Ψs and continuous spectrum eigenfunctions Ψ± (λ) define the spectral decomposition of the operator A2 : Theorem 2.2. Let F = (f1,2 , f12 ), G = (g1,2 , g12 ) ∈ H 2 have infinitely differentiable outside the origin components f1,2 , g1,2 with compact support, then the following formula is valid : 2

hF, GiH 2 =

Nbs X

hF, Ψs iH 2 hΨs , GiH 2

s=1

+

XZ

Z

+∞

−∞

+∞

dλ −∞

0

α=±

×

Z

∞

f1,2 (x)ψα1,2 (x)dx + hf12 , ψα12 i

ψα1,2 (x)g1,2 (x)dx + hψα12 , g12 i .

(2.34)

Moreover if F ∈ Dom(A2 ) then 2

hA F, GiH 2 = 2

Nbs X

λs hF, Ψs iH 2 hΨs , GiH 2

s=1

+

XZ α=±

Z ×

Z

∞

+∞

λdλ 0

+∞

−∞

−∞

f1,2 (x)ψα1,2 (x)dx + hf12 , ψα12 i

ψα1,2 (x)g1,2 (x)dx + hψα12 , g12 i .

(2.35)

866

P. KURASOV

The theorem can be proven integrating the resolvent of the operator A2 over the contour surrounding the discrete and continuous spectra. 2.5. Restrictions on the model Only the model operators with the standard behaviour of the scattering matrix (2.4) will be considered in what follows. The function D(λ) is a rational function. It is analytic in the upper half-plane and has positive imaginary part there. It is real on the real axis. Every such function has the following asymptotics at infinity: D(λ) = c1 λ + c0 + O( λ1 ), c1 , c0 ∈ R, c1 ≥ 0. The transition coefficient T (k) tends to one at infinity only if the linear term in the asymptotics is absent (c1 = 0). Only the model operators with the perturbation determined by the zero parameter d possess such property. The reflection coefficient tends to zero at infinity in this case and the scattering matrix has the standard behaviour at infinity (2.6). The scattering matrix (2.7) has standard zero energy behaviour if no zero energy bound state is present D(0) 6= 0 .

(2.36)

In the sequel we are going to consider only the model operators with standard behaviour of the scattering matrix. The singularities of the scattering matrix are situated on the positive √ part of the imaginary axis on the k-plane and in the lower half-plane (k = λ). These singularities correspond to the bound states and resonances respectively. We are going to consider the case a = d = 0.a . The number of the eigenvalues of the perturbed and unperturbed operators coincide in this case. Lemma 2.2. Let a = d = 0 and all the eigenvalues of A12 be negative. The equation D(k 2 ) + 2ik = 0

(2.37)

has exactly N12 = dimH12 solutions in the upper half-plane =k > 0. All these solutions are situated on the imaginary axis. Proof. The solutions of the equation on the physical sheet are situated on the √ imaginary axis because the functions D(λ) and 2i λ have imaginary parts with the same sign on the λ-plane outside the real axis, where these functions are real. The function D(λ) = b2 R(λ) considered on the real axis is a continuous increasing function on each interval not containing the singularities which coincide with the eigenvalues of the operator A12 . The number of the singularities on the negative halfaxis coincides √with N12 , since all the eigenvalues of the operator A12 are negative. The function 2i λ is a negative increasing function. It follows that the equation a We shall present formulas generally for nonzero a and d, but the final result will be proven for a and d equal to zero

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

...

867

has exactly dimH12 solutions in the upper half-plane =k > 0 and all these solutions are situated on the imaginary axis. The constant b can be considered as a perturbation parameter. The eigenvalues of the operator A2 tend to the eigenvalues of the operator A12 in the limit b → 0. Equation (2.37) has exactly 2N12 + 1 solutions since all these solutions are roots of a polynomial in k of degree 2N12 + 1. The number of the solutions on the nonphysical sheet =k < 0 is equal to N12 + 1. We suppose that all these solutions are situated on the imaginary axis. This is true if the parameter b is small. In the general situation only N12 − 1 solutions are situated on the imaginary axis in the lower half-plane and two solutions can have nontrivial real part. The discrete part of the operator A2 will be denoted by A2d = A2 Pd , where Pd is the discrete spectrum projection for the operator A2 . The discrete spectrum eigenfunctions Ψs , s = 1, 2, . . . , N12 , form an orthogonal basis in the finite dimensional subspace Hd2 = Pd H 2 . The operator A2d is a diagonal operator in this basis X X (−χ2s )gs Ψs . gs Ψs = A2d We note that the operator A2d is unitary equivalent to an operator in H12 since the spaces H12 and Hd2 have the same dimension. The constructed model two-body problem is exactly solvable in the following sense: the eigenfunctions and scattering data can be expressed in terms of elementary functions. Different scattering channels are orthogonal as in the usual two-body scattering problem. The channel Hamiltonians which were used in the construction of the problem are not the channel Hamiltonians for the perturbed operator because the two-body discrete spectrum changes during the perturbation. The channel Hamiltonian corresponding to the cluster decomposition (two particles in a bound state) can be defined in the space L2 (R, H12 ) by the following expression: d2 A˜12 = − 2 + A2d . dy12 The space L2 (R, H12 ) can be embedded into the space H2 with the help of the discrete spectrum eigenfunctions of the operator A2 as it has been described in [44]. 3. Three-Body Hamiltonian This section is devoted to the construction of the three-body symmetric operator with nontrivial two-body interactions. A rich symmetry group of the operator will be described. Selfadjoint extensions of the symmetric operator will be constructed using the von Neumann theory in Sec. 6. Sommerfeld–Maluzhinetz integral transformation will be used in this section to transform the differential equation on the

868

P. KURASOV

deficiency elements into the difference functional equation. We restrict our consideration later on to the case of indistinguishable particles. The corresponding simplified functional equation will also be derived. 3.1. Symmetric three-body operator The model three-body operator will be defined as a perturbation of the orthogonal sum of operators describing possible asymptotic channels for the system of three particles. These asymptotic channels are: three noninteracting particles (operator A1,2,3 ); two particles in a bound state and the third particle free (operators A12,3 , A23,1 , A31,2 ); three particles in a bound state (operator A123 ). We restrict our consideration to the case of particles with equal masses. Let (α, β, γ) be a cyclic permutation of the numbers (1, 2, 3) then the three-body Jacobi coordinates can be written as follows: y123 =

1 (r1 + r2 + r3 ) , 3

xαβ = (rα − rβ ) , r 4 1 (rα + rβ ) − rγ . xαβ,γ = 3 2

(3.1)

Then we define the channel Hamiltonians as follows: 1 ∂2 ∂2 ∂2 + + in H1,2,3 = L2 (R3 ) , A1,2,3 = − 2 ∂r12 ∂r22 ∂r32 ! 1 ∂2 ∂2 + 2 + Aαβ in Hαβ,γ = L2 (R2 , Hαβ ) , (3.2) Aαβ,γ = − 2 6 ∂y123 ∂xαβ,γ A123 = −

1 ∂2 + A123 2 6 ∂y123

in

H123 = L2 (R, H123 ) .

Here the operators Aαβ , A123 are certain selfadjoint matrices in the finite dimensional Hilbert spaces Hαβ , H123 respectively. The unperturbed operator A30 is the orthogonal sum of the asymptotic channel Hamiltonians: A30 = A1,2,3 ⊕ A12,3 ⊕ A23,1 ⊕ A31,2 ⊕ A123

(3.3)

acting in the Hilbert space H3 = H1,2,3 ⊕ H12,3 ⊕ H23,1 ⊕ H31,2 ⊕ H123 .

(3.4)

The operator A30 can be decomposed into the tensor sum A30 = −

1 ∂2 × I + I × A30 , 2 6 ∂y123

which corresponds to the separation of the center of mass motion. The threebody operator with the separated center of mass motion will be investigated. The

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

...

869

corresponding unperturbed operator is equal to the orthogonal sum of the channel operators: A30 = A1,2,3 ⊕ A12,3 ⊕ A23,1 ⊕ A31,2 ⊕ A123 .

(3.5)

It acts in the orthogonal sum of the Hilbert spaces H 3 = H1,2,3 ⊕ H12,3 ⊕ H23,1 ⊕ H31,2 ⊕ H123 .

(3.6)

The operators used in the decomposition of the operator A30 are equal to ! ∂2 ∂2 + 2 A1,2,3 = − ∂x212 ∂x12,3 ! ∂2 ∂2 + 2 =− ∂x223 ∂x23,1 ! ∂2 ∂2 + 2 in H1,2,3 = L2 (R2 ) ; =− ∂x231 ∂x31,2 Aαβ,γ = −

∂2 + Aαβ in Hαβ,γ = L2 (R, Hαβ ) . ∂x2αβ,γ

(3.7)

The scalar product in the Hilbert space H 2 will be denoted by hh·, ·ii. We are going to restrict our consideration to the case of the trivial space H123 for simplicity. The interaction between the channels will be introduced by restricting first the operator A30 to a certain symmetric operator A300 and constructing its different selfadjoint extension. The operator A1,2,3 describes the free motion of particles on the plane Λ = {r ∈ R3 | r1 + r2 + r3 = 0}. In analogy to Sec. 2 the interaction with the cluster operator Aαβ,γ should be introduced on the line `γ , where the coordinates of the particles α and β coincide rα = rβ . These lines `1 , `2 , `3 divide the plane Λ onto six equal sectors. The point of the intersection of these lines needs very careful consideration. Thus on the first step only the functions with the support separated from the origin will be considered. The operator defined on such functions will be symmetric only but not selfadjoint. In order to introduce the two-body interaction we restrict the operator A1,2,3 → A1,2,30 to the set of the smooth functions, vanishing in a neighborhood of the lines `γ , γ = 1, 2, 3. The adjoint operator is defined on the domain W22 (Λ\{`γ }). Functions from this domain can have singularities on the lines `γ . Let us denote by Λ1 , Λ2 , . . . , Λ6 the six sectors on the Λ-plane. Thus we introduce the following subspace of bounded functions: Definition 3.1. The subspace C0∞ ⊂ W22 (Λ \ {`γ }) consists of all infinitely differentiable outside the lines `γ bounded functions with compact support separated from the origin. The support of a function from the defined subspace is not necessarily separated from the screens `γ . The functions from C0∞ can be discontinuous on the lines

870

P. KURASOV

`γ but the boundary values of the functions and their normal derivatives from the both sides of the lines exist and are absolutely continuous functions with compact support. Lemma 3.1. Let u1,2,3 , v1,2,3 ∈ C0∞ . Then the boundary form of the operator A3∗ 00 is equal to hhA∗1,2,30 u1,2,3 , v1,2,3 ii − hhu1,2,3 , A∗1,2,30 v1,2,3 ii =

3 Z X γ=1

dxαβ,γ

∂u1,2,3 ∂u1,2,3 hv1,2,3 [v1,2,3 ¯ i+ ¯ ] ∂xαβ ∂xαβ

∂¯ v1,2,3 ∂¯ v1,2,3 . − [u1,2,3 ] −hu1,2,3 i ∂xαβ ∂nγ xαβ =0

(3.8)

where the sum is taken over all cyclic permutations (α, β, γ) of the numbers (1, 2, 3) parameterized by the number γ. Proof. The lemma can be proven by integrating by parts in the domain Λ\{`γ } which is possible because the functions u1,2,3 , v1,2,3 are twice continuously ∂u ∂v , u1,2,3 , ∂x1,2,3 , v1,2,3 differentiable outside the lines `γ . The boundary values ∂x1,2,3 α,β α,β exist and they are continuous functions with compact support on every line `γ . Thus all integrals in formula (3.8) converge. Theorem 3.1. The operator u1,2,3 A3 U 3 = A3 uαβ,γ   A∗1,2,30 u1,2,3  (3.9) = ∂u1,2,3 Aαβ,γ uαβ,γ + aαβ,γ + bαβ,γ hu1,2,3 i |`γ θαβ,γ ∂xαβ P3 ∞ ∞ defined on the domain of functions from C∞ 0 = C0 ⊕ γ=1 C0 (R \ {0}, Hα,β ) satisfying the boundary conditions ∂u1,2,3 + dαβ,γ hu1,2,3 i |`γ = huαβ,γ , θαβ,γ i, cαβ,γ ∂xαβ [u1,2,3 ] |`γ = 0 is symmetric if the real parameters a, b, c, d satisfy the following condition: a b = −1 det c d and θαβ,γ ∈ Hαβ .

(3.10)

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

...

871

Proof. Consider two arbitrary elements U, V ∈ C∞ 0 . The boundary form of the operator A3 is equal to hA3 U, V iH 3 − hU, A3 V iH 3 = hhA∗1,2,30 u1,2,3 , v1,2,3 ii − hhu1,2,3 , A∗1,2,30 v1,2,3 ii +

3 Z X

dxαβ,γ

γ=1

∂u1,2,3 aαβ,γ + bαβ,γ hu1,2,3 i |xαβ =0 hθαβ,γ , vαβ,γ i ∂xαβ

) ∂v1,2,3 + bαβ,γ hv1,2,3 i |xαβ =0 − huαβ,γ , θαβ,γ i aαβ,γ ∂xαβ =

3 Z X γ=1

dxαβ,γ

∂u1,2,3 ∂u1,2,3 hv1,2,3 i |xαβ =0 + [v1,2,3 ] |xαβ =0 ∂xαβ ∂xαβ

∂v1,2,3 ∂v1,2,3 − hu1,2,3 i |xαβ =0 −[u1,2,3 ] |xαβ =0 ∂xαβ ∂nγ ∂u1,2,3 + aαβ,γ + bαβ,γ hu1,2,3 i |xαβ =0 hθαβ,γ , vαβ,γ i ∂xαβ ) ∂v1,2,3 + bαβ,γ hv1,2,3 i |xαβ =0 . − huαβ,γ , θαβ,γ i aαβ,γ ∂xαβ The integrated expression vanishes at every point on the lines `γ due to the condi tions (3.10). It follows that the operator A3 on C∞ 0 is symmetric. The operator A3 with the domain C∞ 0 is not selfadjoint and its selfadjoint extensions can be described in terms of the deficiency elements. The deficiency elements will be calculated in the following sections. We discuss first the symmetries of the constructed operator. 3.2. The symmetry group We consider in the sequel the system of identical particles. In terms of the constructed model it means that the operators Aαβ,γ , the constants aαβ,γ , bαβ,γ , cαβ,γ , dαβ,γ and the vectors θαβ,γ are equal. The function u1,2,3 can be considered in three different coordinate systems related to the three cluster decompositions of the three particles. We denote the corresponding functions by the index 1, 2 or 3 in such a way that u31,2,3 (x12 , x12,3 ) = u11,2,3 (x23 , x23,1 ) = u21,2,3 (x31 , x31,2 ) . The symmetries of the system of three identical particles interacting via even potential are described by the dihedral group D12 [16] generated by two elements s and t such that s6 = 1, t2 = 1, tst = s−1 .

872

P. KURASOV

The constructed model symmetric operator has the same symmetry group. The element s of order 6 corresponds to the rotation of the plane Λ on the angle π/3   α  γ u1,2,3 (−xαβ , −xαβ,γ ) (xαβ , xαβ,γ ) v1,2,3     v uβγ,α(−xαβ,γ ) αβ,γ (xαβ,γ )     sU = V ⇒  = .   vβγ,α (xβγ,α )   uγα,β (−xβγ,α ) 

vγα,β (xγα,β )

(3.11)

uαβ,γ (−xγα,β )

The element t can be chosen equal to the operator Zγ of the transposition of the particles α and β   γ   γ u1,2,3 (−xαβ , xαβ,γ ) v1,2,3 (xαβ , xαβ,γ )     v uαβ,γ (xαβ,γ ) αβ,γ (xαβ,γ )     tU = Zγ U = V ⇒  =  . (3.12)   vβγ,α (xβγ,α )   uγα,β (xβγ,α ) vγα,β (xγα,β )

uβγ,α (xγ,α,β )

The transpositions Zγ generate the subgroup of permutations P3 , which consists of 6 elements [16]. The element s generates important cyclic subgroup, namely the group of the central rotations on the plane Λ by the angles nπ/3. As if the operator A3 commutes with the rotations sn A3 sn = sn A3 the Hilbert space is decomposable into the orthogonal sum of Hilbert spaces of functions, which are quasi invariant with respect to the rotations s: sU = e−imπ/3 U, m = 0, 1, 2, 3, 4, 5 .

(3.13)

Let us denote by Pm the projector on these quasi invariant elements. Every such element is defined by its values in one of the sectors Λ0 = {x12 < 0, x31 < 0} on the plane Λ and values of the functions u12,3 on the positive halfaxis. The transformation m 0 Tm : U → (um o , u1 ) ∈ L2 (Λ ) ⊕ L2 (R+ , H12 )

Tm U = ((Pm u1,2,3 )|Λ0 , (Pm u12,3 )|R+ ) is invertible on such functions. The operator 6Tm is norm preserving. The Hilbert space H 3 and the operator A3 can be decomposed as follows: H3 = ⊕

5 X

−1 Tm Hm ,

Hm = L2 (Λ0 ) ⊕ L2 (R+ , H12 ) ;

m=0

A3 = ⊕

5 X

−1 Tm Am Tm .

m=0

Let us introduce the polar coordinates in such a way that Λ0 = {(r, ϕ)|0 ≤ r ≤ ∞, 0 ≤ ϕ ≤ π/3}.

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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Lemma 3.2. The operator Am is defined by the following formula:  !  −∆r,ϕ um 0 um 0 (r, ϕ) , = Am ∂2 m m (r) um + A + `(u )θ − u 12 1 1 0 ∂r2 ! m a ∂um 0 m −imπ/3 ∂u0 `(u0 ) = ϕ=0 − e r ∂ϕ ∂ϕ ϕ=π/3 +

873

(3.14)

b m u0 |ϕ=0 +e−imπ/3 um 0 |ϕ=π/3 2

on the domain of functions from Tm C∞ 0 satisfying the boundary conditions: ! m c ∂um d m 0 m −imπ/3 ∂u0 u |ϕ=0 + hu1 , θi = ϕ=0 − e r ∂ϕ ∂ϕ ϕ=π/3 2 0 + e−imπ/3 um 0 |ϕ=π/3

,

−imπ/3 m u0 |ϕ=π/3 . um 0 |ϕ=0 = e

(3.15) (3.16)

3 −1 Proof. Consider any U ∈ Tm C∞ 0 . Then Am = Tm A Tm . The domain of the operator coincides with the set Tm Dom(A3 ). The boundary conditions (3.15) and (3.16) follow from the boundary conditions (3.10) and the fact that every element −1 U satisfies (3.13). Similarly formula (3.14) for the operator Am follows from Tm (3.9) and (3.13).

3.3. Deficiency elements and Sommerfeld Maluzhinetz transformation The deficiency elements for the operator Am are solutions of the equation A∗m Gm = λGm , =λ > 0 , √ k = λ, =k ≥ 0 ,

(3.17)

Gm = (g0m , g1m ) . The operator A∗m is defined by the same differential expression (3.14) on the set of functions from W22 (Λ0 )⊕W22 (R+ , H12 ) satisfying the boundary conditions (3.15) and (3.16). We consider first only bounded functions continuously differentiable outside the lines `γ which are from the domain of A∗m . These functions are not necessarily equal to zero at the origin. The set of deficiency elements from this class is not trivial. The Sommerfeld–Maluzhinetz integral representation [38, 31, 32] will be used to solve the system of Eq. (3.17). We consider here the limit case where k is real and positive. This limit of the deficiency element will be used to calculate the eigenfunctions of the extended three-body operator.

874

P. KURASOV

We suppose that the components of the function Gm can be presented by the following integrals over the plane waves: Z m 1 m m eikr cos α g˜+ (α + ϕ) + g˜− (α + π/3 − ϕ) dα , (3.18) g0 (k) = 2πi Γ Z 1 m eikr cos α g˜1m (α)dα , g1 (k) = 2πi Γ where Γ is a contour in the complex plane α. The contour goes to infinity for real positive k in the upper half-plane in the strips: (2m − 1)π < <α < 2πm, m = 0, 1 . The contour Γ is not closed and has two infinite branches. We chose first the contour Γ in such a way, that no singularity of the density function is situated inside the contour. It is possible because the singularities are situated at a finite distance from the real axis. This assumption will be justified later when the integral density will be calculated. The integral densities are supposed to be analytic functions in α in the region of deformation of the contour Γ. Moreover we are going to carry out integration by parts during these calculations. We assume that the contribution of the boundary terms is equal to zero. These calculations will be justified later for the calculated solution only. The contour Γ will be chosen in a special way later, but at this time we fix some contour Γ1 , which is situated in the upper half-plane =α > Q, Q > 0. The positive real number Q will be determined after the calculation of the solution. The component g0m satisfies the first equation (3.17) due to the special dependence of the integral density on the angle ϕ. The proof of this fact can be carried out by integration by parts which is possible due to our assumption. Integration by parts in the second equation (3.17) for the component g1m gives the following equations for the integral densities g˜1m : g1m (A12 − λ sin2 α)˜ m m m m = − {aik sin α g˜+ (α) − g˜− (α + π/3) − e−imπ/3 (˜ g+ (α + π/3) − g˜− (α)) m m m m (α) + g˜− (α + π/3) + e−imπ/3 (˜ g+ (α + π/3) + g˜− (α)) θ . (3.19) + b/2 g˜+ Solving the matrix equation (3.19) we get one of the solutions of the equation. But the general solution contains an additional decreasing exponent which belongs to the kernel of the Sommerfeld–Maluzhinetz transformation p (3.20) exp i λ − A12 r h . Here h is a vector from H12 . It parametrizes the solution. The matrix exponential function can be presented by the Sommerfeld integral with the following integral density (see [31, 32]): −

k sin α √ h. k cos α − λ − A12

(3.21)

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The general solution of the second equation (3.17) in the Sommerfeld representation is equal to k sin α √ h k cos α − λ − A12 i n h m m m m (α) − g˜− (α + π/3) − e−imπ/3 (˜ g+ (α + π/3) − g˜− (α)) − aik sin α g˜+

g˜1m (α) = −

io h m m m m (α) + g˜− (α + π/3) + e−imπ/3 (˜ g+ (α + π/3) + g˜− (α)) + b/2 g˜+ × (A12 − λ sin2 α)−1 θ .

(3.22)

We exclude the component g1 by projecting the solution onto the element θ. The m (α) is obtained: following difference equation on the integral densities g˜± m m (α + π/3) + g˜− (α + π/3) e−imπ/3 g˜+ m m (α) + e−imπ/3 g˜− (α) + 2f (α) , = Π(α) g˜+

(3.23)

where the following notations are used: 2ik sin α + D(λ sin2 α) −1 = (T (k sin α) + R(k sin α)) , 2ik sin α − D(λ sin2 α) k sin α √ h, θ k cos α − λ − A12 . f (h, α) = (2aik sin α − b)R(λ sin2 α) + 2cik sin α − d Π(α) =

(3.24)

The two-body transition and reflection coefficients appear in the latter formula.b We get the second difference equation from the second boundary condition: m m m m (α + π/3) − g˜− (α + π/3) = g˜+ (α) − e−imπ/3 g˜− (α) . e−imπ/3 g˜+

(3.25)

These equations can be written in the vector form for the two component functions m m g+ (α), g˜− (α)): g˜m (α) = (˜ 1 g˜ (α + π/3) = 2 m

eimπ/3 (Π(α) + 1)

Π(α) − 1 imπ/3 e . + f (α) 1

Π(α) − 1 e−imπ/3 (Π(α) + 1)

g˜m (α) (3.26)

We are able to obtain the solution for these difference equations in terms of elementary functions only for m = 0, 3. The solution to the difference equation for general m will be presented in one of the future publications. The cases m = 0, 3 b The vector h can be considered as a parameter in the latter formula. We are going to use the simplified notation f (α) in the following sections except Sec. 7.

876

P. KURASOV

correspond to the system of particles with the wave function symmetric or antisymmetric with respect to the transpositions of the particles (boson or fermion systems correspondingly). The eigenbasis of the matrix Π(α) − 1 1 eimπ/3 (Π(α) + 1) 2 Π(α) − 1 e−imπ/3 (Π(α) + 1) is independent of α in this case and the matrix system of the difference equations can be reduced to two independent ordinary difference equations. 3.4. Functional equation for identical particles We consider in the sequel the case where the wave function of three particles has boson symmetry. The case of fermions leads to the trivial interaction between the particles because every continuous antisymmetric function is equal to zero at the origin. As a result of this the three-body and two-body channels for the fermion problem would be separated in our model. Thus we restrict our consideration to the system of bosons. To obtain the three-body boson Hamiltonian we restrict the operator A3 to the set Hb3 of functions invariant with respect to the subgroup of ∞ transpositions P3 . The corresponding subset of C∞ 0 will be denoted by C0b . Every element U symmetric with respect to the permutation of the particles is determined by its values in the sector Λ0 and values of the function u12,3 and u23,1 on the positive and negative halfaxes respectively. Thus the transformation P : Hb3 → L2 (Λ0 ) ⊕ (R+ , H12 ) ⊕ (R+ , H12 )   u1,2,3 (x12 , x12,3 )|Λ0   1   u (x )| 12,3 12,3 R+  PU =    2   1 u23,1 (−x23,1 )|R+ 2 is invertible on bosonic elements. The operator 6P preserves the norm of the element. Lemma 3.3. The operator Ab = P A3 P −1 is defined by the following differential expression:   −∆r,ϕ u0    2  2a ∂u0 u0 (r, ϕ)   − ∂ +A | + + bu | u θ 12 1 ϕ=0 0   ϕ=0 ∂r2 r ∂ϕ Ab  u1 (r)  =  (3.27)   2   u2 (r) 2a ∂u0 ∂ − 2 + A12 u2 + − ϕ=π/3 + bu0 ϕ=π/3 θ ∂r r ∂ϕ on the domain of functions from P C∞ 0b satisfying the boundary conditions c ∂u0 d |ϕ=0 + u0 |ϕ=0 , r ∂ϕ 2 c ∂u0 d |ϕ=π/3 + u0 |ϕ=π/3 . hu2 , θi = − r ∂ϕ 2 hu1 , θi =

(3.28)

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Proof. The proof of the lemma is quite similar to the proof of Lemma 3.4. The domain of Ab consists of all U such that P −1 U ∈ Dom(A3 ). The boundary conditions (3.28) follow from this inclusion. The operator Ab is symmetric and it commutes with the reflection operator with respect to the bisector:     u0 (r, π/3 − ϕ) u0 (r, ϕ)     u2 (r) Y  u1 (r)  =  . u2 (r) u1 (r) Consider the transformations Ps (Pa ) of all symmetric (antisymmetric) elements 00 00 from P C∞ 0b to L2 (Λ ) ⊕ L2 (R+ , H12 ) (Λ denotes the sector on the plane Λ with 00 the angle π/6 : Λ = {(r, ϕ) | 0 ≤ r < ∞, 0 ≤ ϕ ≤ π/6}). These transformations are invertible and the operator Ab can be decomposed as follows: Ab = Ps−1 As Ps ⊕ Pa−1 Aa Pa . Lemma 3.4. The operators As and Aa are defined by the following formula:  !  −∆r,ϕ u0 u0 (r, ϕ) (3.29) = As,a 2a ∂u0 ∂2 |ϕ=0 +bu0 |ϕ=0 θ − 2 + A12 u1 + u1 (r) ∂r r ∂ϕ ∞ on the domain of functions from Ps C∞ 0b and Pa C0b satisfying the following boundary conditions:

As : hu1 , θi =

c ∂u0 d |ϕ=0 + u0 |ϕ=0 r ∂ϕ 2

∂u0 |ϕ=π/6 = 0 ; ∂ϕ c ∂u0 d |ϕ=0 + u0 |ϕ=0 Aa : hu1 , θ i = r ∂ϕ 2 u0 |ϕ=π/6 = 0 .

(3.30)

The proof is quite similar to the proof of Lemmas 3.4 and 3.5. The boundary conditions on the line ϕ = 0 for the functions from the domain of the operators As and Aa coincide. The operators Aa and As are symmetric but not selfadjoint. The bounded deficiency elements for the operators As and Aa can be presented using the Sommerfeld– Maluzhinetz transformation by the following integrals: Z 1 eikr cos α {˜ g0s (α + ϕ) + g˜0s (α + π/3 − ϕ)} dα g0s = 2πi Γ Z 1 eikr cos α {˜ g0a (α + ϕ) − g˜0a (α + π/3 − ϕ)} dα g0a = (3.31) 2πi Γ Z 1 eikr cos α g˜1s,a dα . g1s,a = 2πi Γ

878

P. KURASOV

The functions g0s (g0a ) satisfy the Helmholtz equation and Neumann (Dirichlet) boundary condition on the line ϕ = π/6 respectively. Functional equations for the meromorphic functions g˜0s , g˜0a can be derived from the difference equations (3.23) by the following substitution: As :

m=6 6 g˜+

Aa :

6 = g˜− = g˜0s

m=3 3 g˜+

3 = −˜ g− = g˜0a

We get the following functional difference equations: g0s (α) + f (α) g˜0s (α + π/3) = Π(α)˜ g0a (α) − f (α) . g˜0a (α + π/3) = −Π(α)˜

(3.32)

Here the function Π(α) is defined in (3.24). The solutions of these equations will be derived in the next section. 4. Solution of the Functional Equations The difference equations (3.32) are investigated in the present section. These two equations have similar structure and can be solved using the same method. The solution of the first equation will be discussed in detail. The final formula for the solution of the second equation will be presented. 4.1. Method of iterations for the functional equation The coefficients of the functional equation g0s (α) + f (α) g˜0s (α + π/3) = Π(α)˜

(4.1)

possess the following properties: Π(α + π) = Π(−α) = Π−1 (α) ,

(4.2)

Π(α + 2π) = Π(α) ,

(4.3)

f (α + 2π) = f (α) .

(4.4)

The singularities and zeroes of the equation coefficients are situated at finite distance from the real axis. The function Π(α) = (T (k sin α) + R(k sin α))−1 is an unimodular rational function of k sin α. The singularities of the function T (k) + 2 ) R(k) = ik−D(k ik+D(k2 ) coincide with the two-body bound states and the resonances. We made assumption in Sec. 2.5 that all the two-body resonances are pure imaginary. The zeroes and singularities of Π(α) are symmetric with respect to each other in accordance with the property (4.2). Then the singularities and zeroes of Π(α) are situated on the lines <α = πs, s ∈ Z in the complex plane α. The zeroes of the function Π(α) on the line <α = 0 will be denoted by iγm ,

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m = −N12 − 1, . . . , −2, −1, 0, 1, 2, . . . , N12 in such a way that positive m correspond to the two-body bound states. Then the zeroes of the function Π(α) are situated at the points (−1)s+1 iγm + sπ,

s ∈ Z,

m = −N12 − 1, . . . , −2, −1, 0, 1, . . . , N12 .

The function Π(α) possesses a remarkable Blaschke representation. Let us denote by iχn the two-body resonances and bound states on the k-plane in such a way, that positive m correspond to the bound states. Then the following representation holds: Π(α) = Πn

ik sin α + χn . ik sin α − χn

We are going to use this representation in the next section during the calculation of the residues of the functions Π(α) at the singular points. We are going to look for the solutions of the equation which are analytic functions in a neighborhood of infinity, i.e. in the region =α ≥ max |γm |. The parameter Q introduced in the previous section can now be chosen equal to Q = max |γm |. The solution of the functional equation can be expressed in terms of the elementary functions using the properties of the coefficients outlined above. The solution has to be a meromorphic function on the plane α. The general solution of the functional equation is formed as a sum of particular solution of the inhomogeneous equation and general solution of the homogeneous equation. The general solution of the homogeneous equation y(α + π/3) = Π(α)y(α)

(4.5)

is represented by the product of one particular solution and arbitrary π/3 periodic function. To derive the particular solution of the inhomogeneous equation we iterate this equation 5 times. All solutions of the inhomogeneous equation satisfy this new equation y(α + 2π) = y(α) + σ(α) , where σ(α) = f (α + 5π/3) +Π(α + 5π/3)f (α + 4π/3) +Π(α + 5π/3)Π(α + 4π/3)f (α + π) +Π(α + 5π/3)Π(α + 4π/3)Π(α + π)f (α + 2π/3) +Π(α + 4π/3)Π(α + π)f (α + π/3) +Π(α + π)f (α) .

(4.6)

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P. KURASOV

The function σ(α) is a 2π periodic function. Consequently one of the solutions y ∗ (α) of Eq. (4.6) is equal to y ∗ (α) =

α σ(α) . 2π

(4.7)

The general solution of the homogeneous Eq. (4.6) is a 2π periodic function. Hence we arrive at the following Ansatz for the solution of the functional Eq. (4.1): g˜0s (α) = y ∗ (α) + y0 (α) ,

(4.8)

where y0 (α) is a 2π periodic function. Substitution of this Ansatz (4.8) into Eq. (4.1) gives the following equation for the periodic function y0 (α) 1 y0 (α + π/3) = Π(α)y0 (α) + f (α) − σ(α + π/3) . 6

(4.9)

Here we have used the fact that the function σ(α) satisfies the homogeneous equation σ(α + π/3) = Π(α)σ(α) .

(4.10)

The solution of (4.9) can be calculated using the following Ansatz: y0 (α) = a1 f (α + 5π/3) +a2 Π(α + 5π/3)f (α + 4π/3) +a3 Π(α + 5π/3)Π(α + 4π/3)f (α + π) +a4 Π(α + 5π/3)Π(α + 4π/3)Π(α + π)f (α + 2π/3) +a5 Π(α + 4π/3)Π(α + π)f (α + π/3) +a6 Π(α + π)f (α) . Substitution of this representation into (4.9) gives the following relations for the constants {aj }6j=1 : a1 = a6 + 5/6 , aj = aj+1 + 1/6 . So we have got a one parameter set of solutions of the functional equation. Thus the following Theorem has been proven.

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Theorem 4.1. The function α + t f (α + 5π/3) g˜0s (α) = 2π α + t − 1/6 Π(α + 5π/3)f (α + 4π/3) + 2π α + t − 2/6 Π(α + 5π/3)Π(α + 4π/3)f (α + π) + 2π α + t − 3/6 Π(α + 5π/3)Π(α + 4π/3)Π(α + π)f (α + 2π/3) + 2π α + t − 4/6 Π(α + 4π/3)Π(α + π)f (α + π/3) + 2π α + t − 5/6 Π(α + π)f (α) (4.11) + 2π for every value of the parameter t is a solution of the difference functional equation (4.1), in the region =α > max |γm |. The set of functions g˜0s (α) does not coincide with the set of all solutions of the functional equation. One can easily write down the complete set of meromorphic solutions of the equation but we are not going to do that here. The one parameter family derived contains the solution we are searching for. It is necessary to calculate the density g˜1s (α) in order to reconstruct all components of the deficiency element. Lemma 4.1. Let Gs = (g0s , g1s ) be a solution of the differential equation for the deficiency elements satisfying the boundary conditions (3.30). Let Gs be presented by the Sommerfeld integrals (3.31) with the density g˜0s given by (4.11). Then the integral density g˜1s is equal to g˜1s (α) = −

k sin α h √ k cos α − λ − A12 2 4ik sin α(aD(λ sin2 α) − nb) 1 s + f (α)(2aik sin α − b) g˜0 (α) + 2 2ik sin α − D(λ sin2 α) × (A12 − λ sin2 α)−1 θ .

(4.12)

Proof. The integral density can be reconstructed with the help of Eq. (3.22) if s s (α) = g˜− (α) = g˜0s (α) and g˜1s (α) = 12 g˜1m (α): we put m = 0 and g˜+ g˜1s (α) = −

k sin α h √ k cos α − λ − A12 2 1 g0s (α) − g˜0s (α + π/3)) + b (˜ g0s (α) + g˜0s (α + π/3))} − {2aik sin α (˜ 2 × (A12 − λ sin2 α)−1 θ .

882

P. KURASOV

The latter formula can be modified using the fact that g˜0s (α) is a solution of Eq. (4.1) g˜1s (α) = −

k sin α √ h/2 k cos α − λ − A12 1 g0s (α) − f (α)) + b ((1 + Π(α))˜ g0s (α) + f (α))} − {2aik sin α ((1 − Π(α))˜ 2 ×(A12 − λ sin2 α)−1 θ .

Formula (4.12) follows now from (3.14). 4.2. Analytical properties of the solution

The analytical properties of the solution derived are described by the following: Theorem 4.2. The integral densities g˜0s (α + ϕ) + g˜0s (α + π/3 − ϕ), g˜1s (α) are meromorphic on the whole complex α-plane. The singularities of the function g˜0s (α + ϕ) + g˜0s (α + π/3 − ϕ) are poles of finite multiplicity at the lattice of points −ϕ + (−1)s+1 iγm − nπ/3 + sπ,

s = 0, ±1, ±2, . . . ;

−π/3 + ϕ + (−1)s+1 iγm − nπ/3 + sπ,

n = 0, 1, 2,

s = 0, ±1, ±2, . . . ;

n = 0, 1, 2 .

The same is true for the density g˜1s with the lattice of points (−1)s+1 iγm − nπ/3 + sπ,

s = 0, ±1, ±2, . . . ;

π/3 + (−1)s+1 iγm − nπ/3 + sπ,

n = 0, 1, 2,

s = 0, ±1, ±2, . . . ;

n = 0, 1, 2 .

Proof. The singularities of the function g˜0s (α) are caused by the singularities of the functions Π(α) and f (α). The singularities of the function f (α) are situated at the points, where the denominator is equal to zero (2aik sin α − b)R(λ sin2 α) + 2cik sin α − d = 0 ⇒ 2ik sin α = D(λ sin2 α) . These singularities coincide with the singularities of the function Π(α). Some additional singularities can be caused by the singularities of the numerator k sin √α h, θi but these singularities cancel with the singularities of the deh k cos α− λ−A12 nominator. As the result the function f (α) is analytic in a neighborhood of these points. Hence the function g˜0s (α) has singularities at the points . . . , iγm − 2π/3,

iγm − π/3,

iγm , −iγm + π/3,

−iγm + 2π/3, . . .

The integral density of the Sommerfeld integral for the component g0s (r, ϕ) has singularities at the points iγm − 2π/3 − ϕ, iγm − π/3 − ϕ, iγm − ϕ, −iγm + π/3 − ϕ, −iγm + 2π/3 − ϕ, . . . iγm − π + ϕ, iγm − 2π/3 + ϕ, iγm − π/3 + ϕ, −iγm + ϕ, −iγm + π/3 + ϕ, . . . .

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The singularities of the function g˜1s (α) are situated at the same points as the singularities of the functions g˜0s (α), g˜0s (α + π/3). Additional singularities can appear at the points corresponding to the eigenvalues λj of the operator A12 : λ sin2 αj = λj . The function D(λ sin2 α) is equal to b/a at these points: D(λ sin2 αj ) =

b , a

if a 6= 0 and it has pole of the second order there if a = d = 0. The first term in (4.12) has the following singularity near the point αj : −

k sin α h hj k sin α p √ ∼α→αj − ej + O(1) , k cos α − λ − A12 2 k cos α − λ − λj 2

where ej is the eigenvector of A12 corresponding to the eigenvalue λj . The first term in the square brackets in (4.12) has second-order zero at the points α = αj . Thus the second term of (4.12) possesses the following representation: 4ik sin α(aD(λ sin2 α) − b) 1 s +f (α)(2aik sin α−b) (A12 −λ sin2 α)−1 θ g˜ (α) 2 0 2ik sin α − D(λ sin2 α) ∼α→αj

k sin α 1 λj − λ sin2 α 1 p hj θ¯j θj ej + O(1) . 2 k cos α − λ − λj |θj |2 λj − λ sin2 α

Thus the function g˜1s (α) is bounded in the neighborhood of the point αj .

Thus the singularities of all integral densities are situated on finite distance from the real axis. The assumption formulated in Sec. 3.3 holds for the densities we calculated. The integral densities are analytic everywhere in the region =α > max |γm |, thus the corresponding integrals are solutions of the system of the differential equations for the deficiency elements. However the functions are exponentially increasing ones for large r and consequently do not belong to the Hilbert space even for =λ > 0. The asymptotic behaviour of the integrals for r → ∞ will be discussed in the next section. The integration contour should be changed in order to make the functions square integrable. The solution of the second Eq. (64) can be derived in the same way: g˜0a (α) = −

α + t f (α + 5π/3) 2π α + t − 1/6 Π(α + 5π/3)f (α + 4π/3) + 2π α + t − 2/6 Π(α + 5π/3)Π(α + 4π/3)f (α + π) − 2π α + t − 3/6 Π(α + 5π/3)Π(α + 4π/3)Π(α + π)f (α + 2π/3) + 2π

884

P. KURASOV

α + t − 4/6 Π(α + 4π/3)Π(α + π)f (α + π/3) − 2π α + t − 5/6 Π(α + π)f (α) . + 2π

(4.13)

It contains also arbitrary parameter t. The zeroes and the singularities of the function g˜0a are situated at the same points as those of g˜0s . 5. Properties of the Deficiency Elements We discuss here the properties of the solutions of the difference equation derived in the previous section. It will be shown that the Sommerfeld integrals with such densities over the contour Γ1 are not square integrable functions. Another contour of integration will be chosen. The corresponding integrals will be elements of the Hilbert space for λ : =λ > 0. The asymptotic behaviour for large r → ∞ and for small r → 0 will be investigated. 5.1. Asymptotics for large r The asymptotic behaviour at infinity will be studied with the help of the steepest descent method. The saddle points for the Sommerfeld integral are α = 0 and α = π. These two critical points define outgoing and incoming spherical waves in the asymptotics of the component g0s : r 2π −iπ/4 ikr s 1 e e (˜ g0 (ϕ) + g˜0s (π/3 − ϕ)); α=0: 2πi kr (5.1) r 2π iπ/4 −ikr s 1 s e e (˜ g0 (π + ϕ) + g˜0 (4π/3 − ϕ)) . α=π: 2πi kr The second point defines an exponentially increasing function for k with positive imaginary part. The solution of the difference equation contains free parameter t. This parameter can be chosen in a special way to make the amplitude of the incoming spherical wave equal to zero. Let the parameter t be equal to −1/6. Then the amplitude of the incoming spherical wave vanishes. Lemma 5.1. If t = −1/6 then the solution g˜0s of the difference equation (4.1) satisfies the equation g˜0s (π + ϕ) + g˜0s (4π/3 − ϕ) = 0 for every ϕ. Proof. The coefficients of the difference equation possess the following properties: f (−α) = Π(α + π)f (α) , Π(−α) = Π(α + π) .

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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Then the following calculations can be performed: ϕ + 2/6 f (ϕ + 2π/3) g˜0s (π + ϕ) = 2π ϕ + 1/6 Π(ϕ + 2π/3)f (ϕ + π/3) + 2π ϕ Π(ϕ + 2π/3)Π(ϕ + π/3)f (ϕ) + 2π ϕ − 1/6 Π(ϕ + 2π/3)Π(ϕ + π/3)Π(ϕ)f (ϕ − π/3) + 2π ϕ − 2/6 Π(α + π/3)Π(ϕ)f (ϕ − 2π/3) + 2π ϕ − 3/6 Π(ϕ)f (ϕ − π) + 2π ϕ + 2/6 Π(−ϕ + π/3)f (−ϕ − 2π/3) = 2π ϕ + 1/6 Π(−ϕ + 2π/3)Π(−ϕ + π/3)f (−ϕ − π/3) + 2π ϕ Π(−ϕ + 3π/3)Π(−ϕ + 2π/3)Π(−ϕ + π/3)f (−ϕ) + 2π ϕ − 1/6 Π(−ϕ + π/3)Π(−ϕ + 2π/3)f (−ϕ + π/3) + 2π ϕ − 2/6 Π(−α + π)f (−ϕ + 2π/3) + 2π ϕ − 3/6 f (−ϕ + π) + 2π = −˜ g0s (−ϕ + 4π/3) . The proof of the lemma is accomplished.

(5.2)

The amplitude of the incoming spherical wave for t = −1/6 is equal to zero. However the contour Γ1 cannot be deformed to the steepest descent one in the region of the analyticity of the solution g˜0s (α + ϕ) + g˜0s (α + π/3 − ϕ). The residues at the poles of the integral density would add exponentially increasing terms into the asymptotics. It means that the Sommerfeld integral over the contour Γ1 is not an element of the Hilbert space. Hence a new contour of the integration must be chosen. The new contour Γ2 goes to infinity in the same strips as contour Γ1 and passes the saddle points α = 0 and α = π. It surrounds all corresponding to the resonances singularities in the region =α > 0, π ≥ <α ≥ 0 and all corresponding to the bound states singularities in the region =α > 0, 2π ≥ <α ≥ π. No other singularities are situated inside the contour.

886

P. KURASOV

Lemma 5.2. The asymptotics of the integral g0 (r, ϕ) =

1 2πi

Z eikr cos α (˜ g0s (α + ϕ) + g˜0s (α + π/3 − ϕ))dα Γ2

is given by 1 2πi

r

2π −iπ/4 ikr s e e (˜ g0 (α + ϕ) + g˜0s (α + π/3 − ϕ)) kr o Xn eikr cos(iγm −ϕ) + eikr cos(iγm +ϕ−π/3) +

g0 (r, ϕ) ∼r→∞

m>0

× (f (α) + Π(α + 4π/3)f (α + π/3)

(5.3)

+Π(α + 5π/3)Π(α + 4π/3)f (α + 2π/3) +Π(α)Π(α + 4π/3)Π(α + 5π/3)f (α + π)) |α=iγm 2 tan iγm Πm , where Πm = Πn6=m

χm + χn . χm − χn

Proof. If ϕ 6= 0, π/3 then the asymptotics of the integral is given by the steepest descent method in accordance with Lemma 5.1 and formulas (5.1). If ϕ = 0 or ϕ = π/3 then the contour Γ2 cannot be transformed to the steepest descent one without passing through the singularities of the integral density. The asymptotics of the integral for ϕ = 0 contains in addition to the spherical outgoing wave (5.1) the outgoing surface waves which are determined by the residues at the points iγm , iγm + 2π, m > 0 for ϕ = 0, X

Res (˜ g0s (α + ϕ) + g˜0s (α + π/3 − ϕ)) |α=iγm +2π−ϕ

m>0

−

!

X

Res (˜ g0s (α

+ ϕ) +

g˜0s (α

+ π/3 − ϕ)) |α=iγm −ϕ

eikr cos(iγm −ϕ)

m>0

=

X m>0

=

X

Res (˜ g0s (α)) |α=iγm +2π

−

X

! Res (˜ g0s (α)) |α=iγm

eikr cos(iγm −ϕ)

m>0

(f (α) + Π(α + 4π/3)f (α + π/3)

m>0

+ Π(α + 5π/3)Π(α + 4π/3)f (α + 2π/3) + Π(α)Π(α + 4π/3)Π(α + 5π/3)f (α + π)) |α=iγm Res (Π(α + π)) |α=iγm eikr cos(iγm −ϕ) .

(5.4)

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The residue of the function Π(α) can be calculated using the Blaschke representation Y ik sin α − χn ik sin α − χm |α=iγm Res |α=iγm Res Π(α + π) |α=iγm = ik sin α + χn ik sin α + χm n6=m

= 2 tan iγm

Y χm + χn = 2 tan iγm Πm . χm − χn

n6=m

The calculated residues determine the surface waves in the asymptotics. These functions decrease exponentially inside the sector. Really | eikr cos(iγm −ϕ) |∼ e−kr(exp(−γm )−exp(γm )) sin(ϕ)/2 is an exponentially decreasing function for γm > 0, π/3 > ϕ > 0. But this function does not decrease exponentially for ϕ = 0 and real λ. The residues at the points iγm − π/3, iγm + 5π/3 can be analyzed in the same way. Thus the asymptotics of the integral is given by (5.3). A similar method can be applied to investigate the properties of the element g1s (r). The difference is that the saddle points do not give the main contribution to the asymptotics in this case. Lemma 5.3. The asymptotics of the integral Z 1 g˜s (α)eikr cos α dα g1s (r) = 2πi Γ2 1 is given by the formula X s eikr cos(iγm ) ψ12 × [f (α) + Π(α + 4π/3)f (α + π/3) g1s (r) ∼r→∞ m>0

+Π(α + 5π/3)Π(α + 4π/3)f (α + 2π/3) +Π(α)Π(α + 4π/3)Π(α + 5π/3)f (α + π)] |α=iγm tan iγm Πm ,

(5.5)

s is defined by (2.26). where the element ψ12

Proof. The proof of this Lemma is quite similar to the proof of Lemma 5.2. Theorem 5.1. The Sommerfeld integrals g0s (r, ϕ), g1s (r) of the densities + ϕ) + g˜0s (α + π/3 − ϕ) and g˜1s (α) over the contour Γ2 form the deficiency elements for the operator As corresponding for λ = k 2 , =λ > 0.

g˜0s (α

Proof. The Sommerfeld integrals over the contour Γ2 are bounded functions on every compact subset of Λ00 and R+ respectively. These functions decrease exponentially at infinity for λ with positive imaginary part. It follows that the corresponding functions are elements from the Hilbert space L2 (Λ00 )⊕ L2 (R+ , H12 ).

888

P. KURASOV

The integrals over the contour Γ1 are solutions of the differential equations and satisfy the boundary conditions (3.30). This is true because the integration by parts gives no boundary terms, since the function eik cos α decreases exponentially in the strips where the contour Γ1 tends to ∞. The integral over the contour Γ2 differs from the integral over the original contour Γ1 by the residues at the points iγm + 2π − ϕ, iγm + 5π/3 − ϕ, iγm + 4π/3 − ϕ,

m > 0;

iγm + 5π/3 + ϕ, iγm + 4π/3 + ϕ, iγm + π + ϕ, −iγm + π − ϕ, −iγm + 2π/3 − ϕ, −iγm + π/3 − ϕ,

m ≤ 0.

(5.6)

−iγm + 2π/3 + ϕ, −iγm + π/3 + ϕ, −iγm + ϕ . The residues at the points corresponding to m > 0 give the set of the surface waves coming along the boundary of the sector from infinity and going away after two reflections. This set of functions is similar to the set of surface waves which will be obtained in Sec. 7. The residues for m ≤ 0 correspond to the analogous set of the resonance functions. Both sets of functions satisfy the differential equations and the boundary conditions. The corresponding residues for the component g1s must be taken into account also. It follows that the integrals over the contour Γ2 form the deficiency elements for the operator As . The calculated deficiency elements depend on the parameter h ∈ H12 . It will be shown in the next section that there exist N12 linearly independent deficiency elements. Thus we are going to use the following notation for the deficiency element we constructed Gsλ (h) ∈ L2 (Λ00 ) ⊕ (R+ , H12 ). 5.2. Boundary values of the deficiency elements We are going to study the behaviour of the calculated functions in a neighborhood of the point zero. The zero and the first components are bounded continuous function there. The boundary values of the first component at the origin will be calculated. Lemma 5.4. The boundary values of the component g1s at the origin are given by the following formulas: g1s (0) =

h 2 +

3 −1 1 XX θ A12 − λ sin2 (iγm + π + nπ/3) 2 m>0 n=0

4ik sin α(aD(λ sin2 α) − b) ×Res g˜0s (α) 2ik sin α − D(λ sin2 α) + f (α)(2aik sin α − b) |α=iγm +π+nπ/3

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

+

...

889

3 −1 1 XX θ A12 − λ sin2 (−iγm + nπ/3) 2 n=0 m≤0

4ik sin α(aD(λ sin2 α) − b) ×Res g˜0s (α) 2ik sin α − D(λ sin2 α) +f (α)(2aik sin α − b) |α=−iγm +nπ/3 .

(5.7)

p h ∂g1s (0) = i λ − A12 ∂r 2 +

3 1 XX ik cos(iγm + π + nπ/3) 2 m>0 n=0

−1 θ × A12 − λ sin2 (iγm + π + nπ/3) 4ik sin α(aD(λ sin2 α) − b) × Res g˜0s (α) 2ik sin α − D(λ sin2 α) +f (α)(2aik sin α − b) |α=iγm 2+π+nπ/3

+

3 1 XX ik cos(−iγm + nπ/3) 2 n=0 m≤0

−1 θ × A12 − λ sin2 (−iγm + nπ/3) 4ik sin α(aD(λ sin2 α) − b) ×Res g˜0s (α) 2ik sin α − D(λ sin2 α) +f (α)(2aik sin α − b) |α=−iγm +nπ/3 .

(5.8)

Proof. The value u(0) of a Sommerfeld integral at the point zero is determined by behaviour of the integral density u ˜(α) at infinity. Particularly, the following equation is valid for every even function analytic inside the contour of the integration [31, 32]: 1 lim u ˜(α) . (5.9) i α→i∞ Thus the boundary values of the integral over the initial contour Γ1 with the density u(0) =

k sin α h √ k cos α − λ − A12 2 4ik sin α(aD(λ sin2 α) − b) 1 + f (α)(2aik sin α − b) g˜0s (α) + 2 2ik sin α − D(λ sin2 α)

g˜1s (α) = −

×(A12 −λ sin2 α)−1 θ

890

P. KURASOV

√ are equal to h2 and i λ − A12 h2 correspondingly. To calculate the boundary values of the first component of the deficiency element g1s it is necessary to add the residues at the singular points situated between the contour Γ1 and Γ2 . We get formulas (5.7) and (5.8). All the residues which appeared in (5.7) and (5.8) can be calculated explicitly. The corresponding calculations are presented in the Appendix A. The boundary values of the first component are linear functions of the vector h. ˜s can be introduced: Therefore the following matrices Bs and B  h   g1s (0) = (1 + Bs (k)) 2 (5.10) s p  h ∂g  1 (0) = (i λ − A + B ˜ 12 s (k)) . ∂r 2 Lemma 5.5. The matrix 1 + Bs (k) is invertible. Proof. The deficiency elements Gsλ (h) corresponding to different h ∈ H12 have different asymptotics at infinity. Suppose that there exists h ∈ H12 , h 6= 0 such that g1s (0) = 0 .

(5.11)

Consider the following expression: hA∗s Gs (h), Gs (h)i = k 2 hGs (h), Gs (h)i . Integrating by parts two times both components we get the following formula: hA∗s Gs (h), Gs (h)i = hGs (h), A∗s Gs (h)i = k¯2 hGs (h), Gs (h)i .

(5.12)

The boundary terms from the second component vanish due to the condition (5.11). Similar terms produced by the boundary of Λ00 vanish due to the conditions (3.10). The proof is similar to the proof of the symmetry of the operator A3 (Theorem 3.3). It follows from (5.12) that Gsλ (h) = 0 and therefore h = 0. We got the contradiction which proves the Lemma. Corollary 5.1. The boundary values g1s (0) span the finite dimensional space H12 . The matrix Cs (λ) connecting the boundary values of the first component can be introduced: ∂g1 (0) = Cs (λ)g1 (0) ∂r √ ˜s (1 + Bs )−1 . ⇒ Cs (λ) = i λ − A + B

(5.13)

A similar procedure can be also carried out for the antisymmetric deficiency element. ˜a and Ca (λ). The corresponding matrices will be denoted by Ba , B

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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891

6. Selfadjoint Three-Body Operator Two N12 -dimensional families of deficiency elements for the operators As and Aa define 2N12 × 2N12 family of symmetric extensions of the operator As ⊕ Aa . We describe this family by the boundary conditions at the origin. We prove that the extended symmetric operator is bounded from below. The selfadjoint operator will be defined by the Friedrichs procedure. Consider the operator A∗b = A∗s ⊕ A∗a restricted to the domain: s a s a D = C∞ ¯ (h), Gλ ¯ (h)}h∈H12 . 0b + L{Gλ (h), Gλ (h), Gλ

Every symmetric extension of the operator Ab to a subset of D coincides with the restriction of the operator A∗b to this subset. Lemma 6.1. The boundary form of the operator A∗b for every two elements U, V ∈ D is equal to hA∗b U, V i − hU, A∗b V i =

∂ s ∂ u1 (0), v1s (0) − us1 (0), v1s (0) ∂r ∂r ∂ a ∂ u1 (0), v1a (0) − ua1 (0), v1a (0) . + ∂r ∂r

(6.1)

Proof. The Lemma can be proven by integrations by parts if one takes into account that the functions u0 , u1 and v0 , v1 are bounded and twice differentiable. ∂ s ∂ a u1 (0), ∂r u1 (0)) Lemma 6.2. Let =λ > 0. The boundary values (us1 (0), ua1 (0), ∂r of the deficiency elements U ∈ L{Gsλ (h), Gaλ (h), Gsλ¯ (h), Gaλ¯ (h)}h∈H12 span the finite dimensional vector space H12 ⊕ H12 ⊕ H12 ⊕ H12 .

Proof. The subspace L{Gsλ (h), Gaλ (h), Gsλ¯ (h), Gaλ¯ (h)}h∈H12 has dimension 4(dim H12 ) which coincides with the dimension of the orthogonal sum H12 ⊕ H12 ⊕ H12 ⊕ H12 . Thus it is sufficient to show that the linear map between the two vector spaces of equal dimension: η : L{Gsλ (h), Gaλ (h), Gsλ¯ (h), Gaλ¯ (h)}h∈H12 → H12 ⊕ H12 ⊕ H12 ⊕ H12 , ∂ s ∂ a u1 (0), ∂r u1 (0)) η(U ) = (us1 (0), ua1 (0), ∂r

has zero kernel. Let U be an element from the kernel of the map η. Then it has the following representation: U = Uλ + Uλ¯ , Uλ ∈ L{Gsλ (h), Gaλ (h)}h∈H12 , Uλ¯ ∈ L{Gsλ¯ (h), Gaλ¯ (h)}h∈H12 . Consider the following scalar product: ¯ ¯ , Uλ i . hA∗b (Uλ + Uλ¯ ), Uλ i = λhUλ , Uλ i + λhU λ

892

P. KURASOV

Integrating by parts in the first scalar product we get the following equation ¯ A∗ Uλ i = λhU ¯ λ , Uλ i + λhU ¯ ¯ , Uλ i . hA∗b (Uλ + Uλ¯ ), Uλ i = hUλ + U λ, λ b It follows that hUλ , Uλ i = 0 because λ has nontrivial imaginary part.

Lemma 6.3. The restriction AbL of the operator A∗b to the subspace of functions from D which boundary values span Lagrangian subspace L of H12 ⊕H12 ⊕H12 ⊕H12 with respect to the boundary form (6.1) is a symmetric extension of the operator Ab . Proof. The domain of the operator Ab is in D and their boundary values are trivial. It follows that every restriction of A∗b to the domain defined by the Lagrangian subspace is an extension of the operator Ab . This restriction defines symmetric operator because the boundary form vanishes on every domain determined by the Lagrangian subspace. Lemma 6.4. The operator AbL is bounded from below. Proof. The operator AbL is a finite dimensional extension of the operator Ab . Hence it is enough to prove that both operators As and Aa are bounded from below. We consider the case where a = d = 0 for simplicity. Let U ∈ Ps C∞ 0b . Then Z (−∆u0 (r, ϕ))¯ u0 (r, ϕ)rdrdϕ hAs U, U i = Λ00

Z ∞ ∂2 bu0 (r, 0)hθ, u1 (r)idr − 2 + A12 u1 (r), u1 (r) dr + ∂r 0 0 2 Z ∞ Z ∞ Z ∂ 1 ∂ 2 u0 (r, 0)¯ u1 (r) dr |∇u0 | rdrdϕ + u(r, 0)dr + = r ∂ϕ ∂r Λ00 0 0 Z ∞ Z ∞ hA12 u1 (r), u1 (r)idr + bu0 (r, 0)hθ, u1 (r)idr + Z

∞

+

0

Z =

Λ00

Z

|∇u0 |2 rdrdϕ +

Z

0 ∞

Z

hA12 u1 (r), u1 (r)idr +

0

0

∞

2 ∂ u1 (r) dr ∂r

∞

2<(hu1 , θi¯ u0 (r, 0))dr .

+b 0

The latter integral can be estimated as follows: Z ∞ 1 <(hu1 (r), θiu0 (r, 0)dr| ≤ |b| k u1 k2 + k u0 k2L2 + k ∇u0 k2L2 |2b 0 with arbitrary positive . The following estimate is valid for the quadratic form of the operator Z ∞ |b| k u0 k2L2 , hA12 u1 (r), u1 (r)idr − |b| k u1 k2 − hAs U, U i ≥ 0

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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893

provided |b| < 1. The operator As is bounded from below since every term in the latter formula can be estimated by the norm of the element U . Now the selfadjoint operator describing the system of three one-dimensional particles can be defined as follows. First consider the symmetric extension AbL of the operator Ab to the set of functions from D satisfying the boundary conditions: ! ! us,1 (0) u0s,1 (0) =Q , (6.2) u0a,1 (0) ua,1 (0) where Q is a selfadjoint matrix. These boundary conditions determine some Lagrangian subspace in the space of boundary values. Hence the operator LbL is a symmetric and bounded from below operator (Lemmas 6.3 and 6.4). The operator AbL commutes with the symmetry operator with respect to the bisector if the boundary conditions have the following form: u0s,1 (0) = Qs us,1 (0) ,

(6.3)

u0a,1 (0) = Qa ua,1 (0) .

The boundary conditions (6.3) will be used in the sequel. These boundary conditions describe the symmetric extensions AaL and AsL of the operators Aa and As and AsL = AaL ⊕ AsL . Every symmetric operator which is bounded from below can be extended to a selfadjoint operator using Friedrichs procedure. We are going to keep the same notation Ab = Aa ⊕ As for the selfadjoint operator describing the system of three-one dimensional particles. The spectral properties and scattering matrix of this operator will be discussed in the next section. 7. Spectrum, Scattering Matrix The spectral properties of the operators As and Aa are quite similar. We are going to study the spectrum and scattering matrix for the operator As . The spectrum of the operator As consists of the continuous spectrum [0, ∞) corresponding to the processes with three free particles, branches of the continuous spectrum [−χ2m , ∞), m = 1, 2, . . . , N12 , corresponding to the two-body bound states, and probably some eigenvalues. The scattering matrix will be calculated from the asymptotics of the continuous spectrum eigenfunctions of the operator As . 7.1. Definition of the scattering matrix We are going to calculate the continuous spectrum eigenfunctions of the operator As . These functions are generalized solutions of the following equation:  

−∆r,ϕ u0 (r, ϕ) −

∂ 2 u1 (r) + A12 u1 (r) + bu0 (r, 0)θ ∂r2

 =λ

u0 (r, ϕ) u1 (r)

! (7.1)

894

P. KURASOV

satisfying the boundary conditions hu1 (r), θi =

c ∂u0 (r, 0) ; r ∂ϕ

∂u0 (r, π/6) = 0; ∂ϕ

(7.2)

∂u1 (0) = Qs u1 (0) . ∂r The function u0 (r, ϕ) is defined in the sector 0 ≤ r < ∞, 0 ≤ ϕ ≤ π/6. The function u1 (r) is a vector valued function on R+ taking values in H12 . Both functions u0 (r, ϕ) and u1 (r) are continuous and bounded. The asymptotics for large r of the zero component of every solution of the problem (7.1), (7.2) are equal to the sum of incoming and outgoing plane waves, spherical wave, and surface waves. The continuous spectrum eigenfunctions can be separated onto two classes in accordance to the type of the incoming channel. The eigenfunctions of the first type correspond to the three-body incoming channel. Such eigenfunctions contain only one incoming plane wave in the asymptotics of the zero component. The eigenfunctions of the second type contain only incoming surface wave in the asymptotics of the zero component. In addition to these incoming waves the asymptotics of the zero component contains a set of outgoing waves: plane, spherical and surface ones. This set of eigenfunctions will be called incoming. The second complete set of the eigenfunctions, so called outgoing set, is determined by the different outgoing waves. The eigenfunctions contain in their asymptotics only one outgoing wave and a set of incoming waves. The scattering matrix can be defined as an operator connecting the spectral representations with respect to these two sets of the eigenfunctions. We are going to define the scattering matrix from the asymptotics of the incoming set of eigenfunctions. The scattering matrix is an integral operator of the form: ( S(λ) =

S33

S32

S23

S22

) ,

(7.3)

acting in the space L2 (0, π/6)⊕K12 , dim K12 = N12 . The operator S33 is an integral operator with the kernel s33 (λ, ϕ, ϕ0 ). The operator S23 is a matrix integral operator with the kernel s23 (λ, m, ϕ0 ), m = 1, 2, . . . , N12 . The operators S32 and S22 are operators of multiplication by the matrices s32 (λ, ϕ, m), s22 (λ, m, n). Consider first the incoming eigenfunction determined by the incoming plane wave: uin 0 (λ, ϕ0 , r, ϕ) =

exp(−ikr cos(ϕ − ϕ0 )) 2 √ , k = λ, 2π 2k

0 < ϕ0 < π/6 .

(7.4)

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

...

895

Then the asymptotics at infinity r → ∞ of the zero component of this function contains the following outgoing waves: R33 (λ, ϕ0 )

+

exp(−ikr cos(ϕ − ϕ0 − π)) 1 e−iπ/4 ikr √ √ + a33 (λ, ϕ, ϕ0 ) e 2k πr 2π 2k

N12 X

cm s23 (λ, m, ϕ0 ) √ p eikr cos(ϕ−iγm ) . 4 2 2π λ + χ m m=1

(7.5)

The normalizing constant for the surface wave is determined by Eq. (2.27), −χ2m is the energy of the two-body bound state. The real number γm is given by the formula iγm = arctan √iχm 2 . The scattering amplitude a33 and three-body reflection λ+χm

coefficient R33 form the kernel s33 of the scattering matrix: s33 (λ, ϕ, ϕ0 ) = R33 (λ, ϕ0 )δ(ϕ − ϕ0 ) + a33 (λ, ϕ0 , ϕ) .

(7.6)

The scattering amplitude s23 coincides with the kernel of the operator S23 . Similarly the eigenfunction determined by incoming surface wave: cn p e−ikr cos(ϕ−iγn ) , n = 1, 2, . . . , N12 , λ > −χ2n (7.7) uin 0 (λ, n, r, ϕ) = √ 4 2π λ + χ2n has asymptotics at infinity, which contains the following outgoing waves: s32 (λ, ϕ, n)

+

1 e−iπ/4 ikr √ e 2k πr

N12 X

cm s22 (λ, m, n) √ p eikr cos(ϕ−iγm ) . 4 2 2π λ + χ m m=1

(7.8)

The scattering amplitudes s32 , s22 are equal to the coefficients of the matrix operators S32 , S22 . 7.2. Calculation of the scattering matrix The incoming eigenfunctions defined by the three-body plane wave can be presented by the sum of plane waves and the limit of the deficiency element on the real axis u(λ, ϕ0 ) = uplane (λ, ϕ0 ) + Gsλ (h(λ, ϕ0 )) ,

(7.9)

where uplane denotes the set of plane waves and Gsλ (h(λ, ϕ0 )) is the limit of the deficiency element on the real axis. The set of plane waves is the result of multiple reflections of the incoming plane wave from the boundaries of the sector in accordance to the laws of the geometrical optics. The total number of the reflected waves is equal to 11. The reflection coefficient from the boundary ϕ = π/6 is equal to 1 for the symmetric functions (to −1 for Aa ). The reflection coefficients from the boundary ϕ = 0 are determined by the two-body scattering matrix only

896

P. KURASOV

P (k⊥ ) = T (k⊥ ) + R(k⊥ ). Here k⊥ denotes the perpendicular component of the wave vector. We note that the reflection coefficient P (k) is related to the coefficient Π(α) in the difference functional equation as Π(α) = P (k sin α)−1 . The incoming eigenfunctions determined by the three-body incoming plane wave can be parameterized by the energy λ = k 2 , 0 ≤ λ < ∞ and the angle ϕ0 , 0 ≤ ϕ0 ≤ π/6 of the incoming wave (7.4). The set of the induced plane waves consists of the twelve waves with the following zero component: up0 (λ, ϕ0 , r, ϕ) =

1 √ {exp (−ikr cos(ϕ − ϕ0 )) 2π 2k +Π−1 (ϕ0 ) exp (−ikr cos(ϕ + ϕ0 )) +Π−1 (ϕ0 ) exp (−ikr cos(ϕ − ϕ0 − π/3)) +Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 ) exp (−ikr cos(ϕ + ϕ0 + π/3)) +Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 ) exp (−ikr cos(ϕ − ϕ0 − 2π/3)) +Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 )Π−1 (2π/3 + ϕ0 ) exp (−ikr cos(ϕ + ϕ0 + 2π/3)) + exp (−ikr cos(ϕ + ϕ0 − π/3)) +Π−1 (π/3 − ϕ0 ) exp (−ikr cos(ϕ − ϕ0 + π/3)) +Π−1 (π/3 − ϕ0 ) exp (−ikr cos(ϕ + ϕ0 − 2π/3)) +Π−1 (π/3 − ϕ0 )Π−1 (2π/3 − ϕ0 ) exp (−ikr cos(ϕ − ϕ0 + 2π/3)) +Π−1 (π/3 − ϕ0 )Π−1 (2π/3 − ϕ0 ) exp (−ikr cos(ϕ + ϕ0 + π))

+Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 )Π−1 (2π/3 + ϕ0 ) exp (−ikr cos(ϕ − ϕ0 − π)) .(7.10)

The first component for the set of plane waves is given by up1 (λ, ϕ0 , r) =

1 √ {exp (−ikr cos(ϕ0 )) ψ12 (λ sin2 (ϕ0 )) 2π 2k +Π−1 (ϕ0 ) exp (−ikr cos(ϕ0 + π/3)) ψ12 (λ sin2 (ϕ0 + π/3)) +Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 ) exp (−ikr cos(ϕ0 + 2π/3)) ψ12 (λ sin2 (ϕ0 + 2π/3)) + exp (−ikr cos(ϕ0 − π/3)) ψ12 (λ sin2 (−ϕ0 + π/3)) +Π−1 (π/3 − ϕ0 ) exp (−ikr cos(ϕ0 − 2π/3)) ψ12 (λ sin2 (−ϕ0 + 2π/3))

+Π−1 (π/3 − ϕ0 )Π−1 (2π/3 − ϕ0 ) exp (−ikr cos(ϕ0 + π)) ψ12 (λ sin2 (ϕ0 )) (7.11)

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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897

where the function ψ12 (λ) is defined in (2.32). Thus the boundary values of the first component at the point zero are equal to up1 (λ, ϕ0 , 0) =

1 √ { ψ12 (λ sin2 (ϕ0 )) 2π 2k +Π−1 (ϕ0 )ψ12 (λ sin2 (ϕ0 + π/3)) +Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 )ψ12 (λ sin2 (ϕ0 + 2π/3)) +ψ12 (λ sin2 (−ϕ0 + π/3)) +Π−1 (π/3 − ϕ0 )ψ12 (λ sin2 (−ϕ0 + 2π/3))

+Π−1 (π/3 − ϕ0 )Π−1 (2π/3 − ϕ0 ) ψ12 (λ sin2 (ϕ0 )) ,

(7.12)

−ik ∂up1 (λ, ϕ0 , r) √ {cos(ϕ0 ) ψ12 (λ sin2 (ϕ0 )) = ∂r 2π 2k r=0 +Π−1 (ϕ0 ) cos(ϕ0 + π/3)ψ12 (λ sin2 (ϕ0 + π/3)) +Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 ) cos(ϕ0 +2π/3)ψ12 (λ sin2 (ϕ0 +2π/3)) + cos(ϕ0 − π/3)ψ12 (λ sin2 (−ϕ0 + π/3)) +Π−1 (π/3 − ϕ0 ) cos(ϕ0 − 2π/3)ψ12 (λ sin2 (−ϕ0 + 2π/3))

+Π−1 (π/3 − ϕ0 )Π−1 (2π/3 − ϕ0 ) cos(ϕ0 + π) ψ12 (λ sin2 (ϕ0 )) . (7.13) The boundary values of the first component of the set of plane waves do not satisfy in general the boundary conditions (7.2) at the origin. Hence it is necessary to add some outgoing wave, which can be presented as a limit of the deficiency element on the real axis. Every deficiency element can be parameterized by the vector h(λ, ϕ0 ). Substitution of the representation (7.9) into the boundary conditions (7.2) gives the following inhomogeneous linear equation on the vector h(λ, ϕ0 ): ∂up1 (λ, ϕ0 , r) ∂r

p ˜s h(λ, ϕ0 ) + i λ − A12 + B 2 r=0 h(λ, ϕ0 ) . = Qs up1 (e, ϕ0 , 0) + (1 + Bs ) 2

(7.14)

The vector h(λ, ϕ0 ) can be calculated as p −1 ∂up p 1 ˜ + Qs u1 |r=0 . (7.15) − h(λ, ϕ0 ) = 2 i λ − A12 + Bs − Qs (1 + Bs ) ∂r

898

P. KURASOV

The first two components of the scattering matrix can be calculated now from the asymptotics of the solution ss33 (λ, ϕ, ϕ0 ) = δ(ϕ − ϕ0 )Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 )Π−1 (2π/3 + ϕ0 ) √ g0s (h(λ, ϕ0 ), ϕ) + g˜0s (h(λ, ϕ0 ), π/3 − ϕ)) ; +(−i 2k) (˜

(7.16)

ss23 (λ, m, ϕ0 ) √ p 2π 4 λ + χ2m (f (h(λ, ϕ0 ), α) + Π(α + 4π/3)f (h(λ, ϕ0 ), α + π/3) = cm +Π(α + 5π/3)Π(α + 4π/3)f (h(λ, ϕ0 ), α + 2π/3) +Π(α)Π(α + 5π/3)Π(α + 4π/3)f (h(λ, ϕ0 ), α + π)) |α=iγm 2 tan iγm Πm . (7.17) The eigenfunctions corresponding to the surface waves can be considered in the same way. These eigenfunction can be presented by the following sum: u(λ, m) = usurf (λ, m) + G(λ, h(λ, m)) .

(7.18)

The eigenfunctions are parameterized by the energy of the incoming surface wave (7.4) λ ∈ [−χ2m , ∞) and the two-body bound state m, which can be represented formally as a plane wave with the complex wave vector cn p e−ikr cos(ϕ−iγn ) . uin 0 (λ, n, r, ϕ) = √ 2π 4 λ + χ2n The set of surface waves can be constructed using the two-body scattering data: (λ, m, r, ϕ) usurf 0

( cn =√ p e−ikr cos(ϕ−iγm ) + e−ikr cos(ϕ−π/3+iγm ) 2π 4 λ + χ2n

+Π−1 (π/3 − iγm )e−ikr cos(ϕ+π/3−iγm ) +Π−1 (π/3 − iγm )e−ikr cos(ϕ−2π/3+iγm ) +Π−1 (2π/3 − iγm )Π−1 (π/3 − iγm )e−ikr cos(ϕ+2π/3−iγm ) ) +Π−1 (2π/3 − iγm )Π−1 (π/3 − iγm )e−ikr cos(ϕ−π+iγm )

.

(7.19)

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

...

899

The first component of the set of plane waves can also be calculated: n s cn −ikr cos(iγm ) ψ12 p (λ, m, r) = √ e usurf 1 2 2π 4 λ + χ2n + e−ikr cos(−π/3+iγm ) ψ12 (λ sin2 (π/3 − iγm )) + Π−1 (π/3 − iγm )e−ikr cos(−2π/3+iγm ) ψ12 (λ sin2 (2π/3 − iγm )) ψs + Π−1 (2π/3 − iγm )Π−1 (π/3 − iγm )e−ikr cos(ϕ−π+iγm )) 12 , 2 (7.20) s is defined by (2.26). The boundary values of the first component at the where ψ12 origin are given by

cn p usurf 1 (λ, m, 0) = √ 2π 4 λ + χ2n

s ψ12 + ψ12 (λ sin2 (π/3 − iγm )) 2

+Π−1 (π/3 − iγm )ψ12 (λ sin2 (2π/3 − iγm )) s ψ12 −1 −1 , +Π (2π/3 − iγm )Π (π/3 − iγm ) 2

(7.21)

−ikcn ∂usurf 1 (λ, m, r) s = √ p {cos(−iγm ) ψ12 ∂r 2 2π 4 λ + χ2n r=0 + cos(−π/3 + iγm )ψ12 (λ sin2 (π/3 − iγm )) +Π−1 (π/3 − iγm ) cos(−2π/3 + iγm )ψ12 (λ sin2 (2π/3 − iγm )) s }. +Π−1 (2π/3 − iγm )Π−1 (π/3 − iγm ) cos(−π + iγm ) ψ12

(7.22) The vector h(λ, m) can be calculated from the boundary conditions: p ∂usurf 1 + i λ − A12 +B ˜s h(λ, m) ∂r r=0 2 h(λ, m) surf = Qs u1 |r=0 +(1 + Bs ) 2 −1 p ˜s − Qs (1 + Bs ) ⇒ h(λ, m) = 2 i λ − A12 + B ∂usurf + Qs usurf − 1 1 ∂r

!

r=0

.

(7.23)

(7.24)

900

P. KURASOV

The components S22 , S32 of the scattering matrix can be calculated from the asymptotics of the constructed eigenfunction: ss22 (λ, m, n) = δnm Π−1 (2π/3 − iγm )Π−1 (π/3 − iγm ) √ p 2π 4 λ + χ2m (f (h(λ, m), α) + Π(α + 4π/3)f (h(λ, m), α + π/3) + cm +Π(α + 5π/3)Π(α + 4π/3)f (h(λ, m), α + 2π/3) +Π(α)Π(α + 5π/3)Π(α + 4π/3)f (h(λ, m), α + π)) |α=iγn ×2 tan iγn Πn (−iγn )Θ(λ + χ2m ) ,

(7.25)

√ g0s (h(λ, m), ϕ) + g˜0s (h(λ, m), π/3 − ϕ)} Θ(λ) . ss32 (λ, ϕ, m) = −i 2k {˜

(7.26)

Here Θ(λ) is the Heaviside function. We succeeded in precise analytical calculation of the whole three-body scattering matrix for the case of identical particles. The following theorem has been proven. s s s s , S23 , S32 , S22 with the kernels Theorem 7.1. The matrix integral operators S33

ss33 (λ, ϕ, ϕ0 ) = δ(ϕ − ϕ0 )Π−1 (ϕ0 )Π−1 (π/3 + ϕ0 )Π−1 (2π/3 + ϕ0 ) √ g0s (h(λ, ϕ0 ), ϕ) + g˜0s (h(λ, ϕ0 ), π/3 − ϕ)) ; +(−i 2k) (˜ ss23 (λ, m, ϕ0 )

√ p 2π 4 λ + χ2m = (f (h(λ, ϕ0 ), α) + Π(α + 4π/3)f (h(λ, ϕ0 ), α + π/3) cm +Π(α + 5π/3)Π(α + 4π/3)f (h(λ, ϕ0 ), α + 2π/3) +Π(α)Π(α + 5π/3)Π(α + 4π/3) × f (h(λ, ϕ0 ), α + π)) |α=iγm 2 tan iγm Πm ;

√ g0s (h(λ, m), ϕ) + g˜0s (h(λ, m), π/3 − ϕ)} Θ(λ) ; ss32 (λ, ϕ, m) = −i 2k {˜ ss22 (λ, m, n) = δnm Π−1 (2π/3 − iγm )Π−1 (π/3 − iγm ) √ p 2π 4 λ + χ2m (f (h(λ, m), α) + Π(α + 4π/3)f (h(λ, m), α + π/3) + cm +Π(α + 5π/3)Π(α + 4π/3)f (h(λ, m), α + 2π/3) +Π(α)Π(α + 5π/3)Π(α + 4π/3)f (h(λ, m), α + π)) |α=iγn ×2 tan iγn Πn (−iγn )Θ(λ + χ2m ) ; form the scattering matrix

Ss =

for the operator As .

s s S33 S32 s s S23 S22

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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901

The energies of the three-body bound states can be calculated as follows. The wave function of the three-body bound state is equal to the limit of some deficiency element on the real axis. Substitution of the boundary values of the deficiency element into the boundary conditions (7.2) gives the dispersion equation for the energy of the three particles bound state: p ˜s − Qs (I + Bs ) = 0 . det i λ − A12 + B

(7.27)

The solutions of the dispersion equation are situated on the negative real axis λ. The solutions define the singularities of scattering amplitudes. Equations (7.14) and (7.23) for the vector h cannot be solved at these points. It should be underlined that additional singularities in the scattering amplitudes are produced by the two-body bound states and resonances. A more rich structure of the three-body bound states can be obtained by adding the three-body space of interaction. The bound state eigenfunctions are orthogonal to the continuous spectrum eigenfunctions. Thus the investigation of the three-body model scattering problem is accomplished. All eigenfunctions were presented by Sommerfeld integrals, however the scattering matrix was calculated in terms of elementary functions. This was possible due to the simple geometry of the problem. But the scattering matrix for nonidentical particles should contain some special functions. The components s23 , s32 of the scattering matrix are continuous functions of the angles for all ϕ. The component s33 contains a singularity corresponding to the back scattering. In the case of the two-body zero energy resonance the function g˜0s (α) has a singularity at the origin. As the result, the asymptotics of the Sommerfeld integral cannot be calculated by the saddle point method directly for ϕ = 0. Hence the scattering amplitude is discontinuous at ϕ = 0 in this case. The analytical continuation of the scattering matrix has singularities at the points of discrete spectrum. All these properties of the scattering amplitudes are similar to the properties of amplitudes for the standard three-body Hamiltonians with the two-body potentials. Our model can be used for the investigation of the influence of the presence of zero energy eigenstate or resonance on the analytical properties of the corresponding scattering amplitudes [45, 17]. It is important to discuss the propagator estimates for the model constructed [46]. These questions will be studied in one of the future publications. 8. Acknowledgements The author owes special thanks to Professor B. S. Pavlov for continuous discussions and many fruitful remarks. The author wants to thank all the members of B. S. Pavlov scientific group for support and collaboration, especially Yu. A. Kuperin and K. A. Makarov. It is a pleasure for him to thank Professor S. Albeverio for the scientific support and helpful suggestions. He thanks Professors V. M. Adamyan, J. Boman, V. S. Buslaev, L. Thomas, H. Holden, A. Holst, P. Exner, N. Elander, E. Skibsted for their interest to his work.

902

P. KURASOV

The author thanks St. Petersburg University, where the main part of this work has been done. He is also indebted to the Manne Siegbahn Institute and Stockholm University for the financial support and hospitality. He thanks Department of Mathematics, Lule˚ a University, where this work was accomplished for the excellent working opportunities. This work was supported by the Alexander von Humboldt foundation. Appendix A Analytical formulas for the boundary values of the deficiency elements We introduce the following linear operators ρm : H12 → C m > 0 : ρm : h → Res g˜0s (α) |α=iγm +2π ( 1 iγm m + = 2 tan iγm Π Π(iγm + 5π/3)Π(iγm + 4π/3)Π(iγm ) 2π 2 k sin iγm √ ∗, θ k cos iγm + λ − A12 × (2aik sin iγm + b)R(λ sin2 iγm ) + 2cik sin iγm + d 1 iγm + Π(iγm + 5π/3)Π(iγm + 4π/3) + 2π 3 k sin(iγm + 2π/3) √ ∗, θ k cos(iγm + 2π/3) + λ − A12 × (2aik sin(iγm + 2π/3) − b)R(λ sin2 (iγm + 2π/3)) + 2cik sin(iγm + 2π/3) − d 1 iγm + Π(iγm + 4π/3) + 2π 6 k sin(iγm + π/3) √ ∗, θ k cos(iγm + π/3) + λ − A12 × (2aik sin(iγm + π/3) − b)R(λ sin2 (iγm + π/3)) + 2cik sin(iγm + π/3) − d k sin(iγm ) ) √ ∗, θ iγm k cos(iγm ) + λ − A12 , + 2π (2aik sin(iγm ) − b)R(λ sin2 (iγm )) + 2cik sin(iγm ) − d (−1)

m ≤ 0 : ρm : h → Res g˜0s (α) |α=−iγm +π . Then the boundary values of the deficiency element at the origin are: −1 h 1 Xn θ (−2aik sin(iγm ) + b) A12 − λ sin2 (iγm ) g1s (0) = + 2 2 m>0 +(−2aik sin(iγm + 5π/3) + b)Π(iγm + 2π/3) −1 θ × A12 − λ sin2 (iγm + 5π/3)

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

...

903

+(−2aik sin(iγm + 4π/3) + b)Π(iγm + π/3) −1 θ ×Π(iγm + 2π/3) A12 − λ sin2 (iγm + 4π/3) −1 θ +(2aik sin(iγm + 5π/3) + b) A12 − λ sin2 (iγm + 5π/3) +(2aik sin(iγm + 4π/3) + b)Π(iγm + 2π/3) −1 θ × A12 − λ sin2 (iγm + 4π/3) +(2aik sin(iγm + π) + b)Π(iγm + π/3)

−1 o θ ρm h ×Π(iγm + 2π/3) A12 − λ sin2 (iγm + π)

+

−1 1 Xn θ (−2aik sin(iγm ) + b) A12 − λ sin2 (iγm ) 2 m≤0

+(−2aik sin(−iγm + 2π/3) + b)Π(−iγm + 5π/3) −1 × A12 − λ sin2 (−iγm + 2π/3) +(−2aik sin(−iγm + π/3) + b)Π(−iγm + 4π/3) −1 θ ×Π(−iγm + 5π/3) A12 − λ sin2 (−iγm + π/3) −1 θ +(2aik sin(−iγm + 2π/3) + b) A12 − λ sin2 (−iγm + 2π/3) +(2aik sin(−iγm + π/3) + b)Π(−iγm + 5π/3) −1 θ × A12 − λ sin2 (−iγm + π/3) +(2aik sin(−iγm ) + b)Π(−iγm + 4π/3)

−1 o θ ρm h . ×Π(−iγm + 5π/3) A12 − λ sin2 (iγm )

(A.1)

p h ∂g1s (r) |r=0 = i λ − A12 ∂r 2 −1 1 Xn θ ik cos(iγm )(−2aik sin(iγm ) + b) A12 − λ sin2 (iγm ) + 2 m>0 +(−2aik sin(iγm + 5π/3) + b)Π(iγm + 2π/3) −1 θ × ik cos(iγm + 5π/3) A12 − λ sin2 (iγm + 5π/3) +(−2aik sin(iγm + 4π/3) + b)Π(iγm + π/3)Π(iγm + 2π/3) −1 θ ×ik cos(iγm + 4π/3) A12 − λ sin2 (iγm + 4π/3)

904

P. KURASOV

+(2aik sin(iγm + 5π/3) + b) −1 θ ×ik cos(iγm + 5π/3) A12 − λ sin2 (iγm + 5π/3) +(2aik sin(iγm + 4π/3) + b)Π(iγm + 2π/3) −1 θ ×ik cos(iγm + 4π/3) A12 − λ sin2 (iγm + 4π/3) +(2aik sin(iγm + π) + b)Π(iγm + π/3)Π(iγm + 2π/3) −1 o θ ρm h ×ik cos(iγm + π) A12 − λ sin2 (iγm + π) +

1 X {(−2aik sin(iγm ) + b)ik cos(−iγm + π) 2 m≤0

−1 θ × A12 − λ sin2 (iγm ) +(−2aik sin(−iγm + 2π/3) + b)Π(−iγm + 5π/3) −1 θ ×ik cos(−iγm + 2π/3) A12 − λ sin2 (−iγm + 2π/3) +(−2aik sin(−iγm + π/3) + b)Π(−iγm + 4π/3)Π(−iγm + 5π/3) −1 θ ×ik cos(−iγm + π/3) A12 − λ sin2 (−iγm + π/3) +(2aik sin(−iγm + 2π/3) + b) −1 θ ×ik cos(−iγm + 2π/3) A12 − λ sin2 (−iγm + 2π/3) +(2aik sin(−iγm + π/3) + b)Π(−iγm + 5π/3)

−1 θ ×ik cos(−iγm + π/3) A12 − λ sin2 (−iγm + π/3)

+(2aik sin(−iγm ) + b)Π(−iγm + 4π/3)Π(−iγm + 5π/3) −1 o θ ρm h . ×ik cos(iγm ) A12 − λ sin2 (iγm )

(A.2)

References [1] V. M. Adamyan and B. S. Pavlov, “Zero-radius potentials and M.G.Krein’s formula for generalized resolvents”, Zap. Nauchn. Sem. LOMI 149 (1986), Issled. Linein. Oper. Teor. Funktsii XY, 7–23, 186. [2] S. Albeverio, “Analytische L¨ osung eines idealisierten Stripping- oder Beugungsproblems”, Helv. Phys. Acta 40 (1967) 135–184. [3] S. Albeverio, R. Høegh-Krohn and L. Streit, “Energy forms, Hamiltonians, and distorted Brownian paths”, J. Math. Physics 18 (1977) 907–917. [4] S. Albeverio, R. Høegh-Krohn and T. T. Wu, “A class of exactly solvable three-body quantum mechanical problems and the universal low energy behaviour”, Phys. Lett. 83A (1981) 105–109. [5] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988. [6] F. V. Atkinson, H. Langer, R. Mennicken and A. Shkalikov, “The essential spectrum of some matrix operators”, Math. Nachr. 167 (1994) 5–20.

ENERGY DEPENDENT BOUNDARY CONDITIONS AND

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[7] M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Dordrecht, Reidel, 1987. [8] V. S. Buslaev, S. P. Merkuriev and S. P. Salikov, “On the diffractional character of the scattering problem for three one-dimensional particles”, Problemu Mat. Fiz., vup.9, Leningrad Univ. Press (1979) (in Russian). [9] P. Deift and E. Trubowitz, “Inverse scattering on the line”, Commun. Pure Appl. Math. 32 (1979) 121–251. [10] Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and their Applications in Atomic Physics, Plenum Press, New York, 1988. [11] L. D. Faddeev, “Inverse problem of quantum scattering theory”, II, Sovrem. Probl. Matem. 3 (1976) 93. [12] M. Gaudin, La Fonction d’Onde de Bethe, Masson, Paris, 1983. [13] J. B. McGuire, “Study of exactly soluble one-dimensional N -body problems”, J. Math. Phys. 5 (1964) 622–636. [14] J. B. McGuire and C. A. Hurst, “The scattering of three impenetrable particles in one dimension”, J. Math. Phys. 13 (1972) 1595–1607. [15] J. B. McGuire and C. A. Hurst, “Three interacting particles in one dimension: An algebraic approach”, J. Math. Phys. 29 (1988) 155–168. [16] G. G. Hall, Applied Group Theory, Longman, 1967. [17] H.Isozaki, “Structure of S-matrix for three body Schr¨ odinger operators”, Comm. Math. Phys. 146 (1992) 241–258. [18] Yu. A. Kuperin, K. A. Makarov and B. S. Pavlov, “One-dimensional model of threeparticle resonances”, Theoret. Math. Phys. 63 (1985) 376–382. [19] Yu. A. Kuperin, K. A. Makarov, S. P. Merkuriev, A. K. Motovilov and B. S. Pavlov, “Quantum few-body problem with internal structure I: Two-body problem”, Teor. i Mat. Fiz. 75 (1988) 630–639. [20] Yu. A. Kuperin, K A. Makarov, S. P. Merkuriev, A. K. Motovilov and B. S. Pavlov, “Quantum few-body problem with internal structure II: Three-body problem”, Teor. i Mat. Fiz. 76 (1988) 834–847. [21] Yu. A. Kuperin, K. A. Makarov, S. P. Merkuriev, A. K. Motovilov and B. S. Pavlov, “Extended Hilbert space approach to few-body problems”, JMF 31 (1990) 199–201. [22] P. B. Kurasov, “Three one-dimensional bosons with an internal structure”, in ˇ Schr¨ odinger Operators, Standard and Nonstandard, eds. P. Exner, and P. Seba, Proc. Dubna, USSR, 1988, World Scientific, 1989. [23] P. B. Kurasov, “Zero-range potentials with internal structures and the inverse scattering problem”, Lett. Math. Phys. 25 (1992) 287–297. [24] P. Kurasov, V. Kurasov and B. Pavlov, “Second virial coefficient for one-dimesnional system”, in “On three levels”, eds M. Fannes et al., Plenum Press, New York, 1994, 423–428. [25] K. Lipszyc, Acta Phys. Polon. A42 (1972) 571. [26] K. Lipszyc, Acta Phys. Polon. A44 (1973) 115. [27] K. Lipszyc, “One-dimensional model of the rearrangement process and the Faddeev equations”, J. Math. Phys. 15 (1974), 133–138. [28] K. Lipszyc, “On the application of the Sommerfeld–Maluzhinetz transformation to some one-dimensional three-particle problems”, J. Math. Phys. 21 (1980) 1092–1102. [29] K. A. Makarov, “On delta-like interactions with internal structure and semibounded from below three-body Hamiltonian”, preprint FUB/HEP- 88–13, Berlin 1988, 16p. [30] K. A. Makarov, “Semiboundedness of the energy operator of a three-particle system with pair interaction”, St. Petersburg Math. J. 4 (1993) 967–980. [31] G. D. Maluzhinetz, “Inversion formula for Sommerfeld integral”, Dokl. Akad. Nauk SSSR 118 (1958) 1099.

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[32] G. D. Maluzhinetz, “Connections between inverse formulas for Sommerfeld’s integral and formulas of Kantorovich–Lebedev”, Dokl. Akad. Nauk SSSR 119 (1958) 49–51. [33] V. A. Marchenko, Spectral theory of Sturm–Liouville operators, Naukova Dumka, Kiev, 1972 (Russian). [34] A. Melin, “Operator methods for inverse scattering on the real line”, Commun. Partial Diff. Eq. 10 (1985), 677–766. [35] S. P. Merkuriev and L. D. Faddeev, Quantum Scattering Theory for Few Body Systems, Nauka, Moscow, 1985 (in Russian). [36] R. A. Minlos and L. D. Faddeev, “On the point interaction for a three-particle system in quantum mechanics”, Sov. Phys. -Doklady 6 (1962) 1072–1074. [37] R. A. Minlos and L. D. Faddeev, “Comment on the problem of three particles with point interactions”, Sov. Phys. JETP 14 (1962) 1315–1316. [38] P. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company, New York, 1953. [39] R. Newton, Scattering Theory of Waves and Particles, Second edition, SpringerVerlag, Berlin, 1989. [40] B. S. Pavlov, “A model of zero-radius potential with internal structure”, Teor. i Mat. Fiz. 59 (1984) 345–353. [41] B. S. Pavlov, The theory of extensions and explicitly-soluble models, Russian Math. Surveys 42 (1987) 127–168. [42] B. S. Pavlov, “Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schr¨ odinger operator with pointwise potential”, Math. USSR Sbornik, 64 (1989) 161–175. [43] B. S. Pavlov, “Construction of soluble models based on few-body modification of M. Krein formula” (in preparation). [44] M. Reed and B. Simon, Methods of modern mathematical physics, vol. III, Scattering Theory, Academic Press, New York, 1979, pp. 75–79. [45] E. Skibsted, “Propagation estimates for N -body Schr¨ odinger operators”, Commun. Math. Phys. 142 (1991) 67–98. [46] E. Skibsted, “Smoothness of N -body scattering amplitudes”, Rev. Math. Phys. 4 (1992) 619–658. [47] G. V. Skorniakov and K. A. Ter-Martirosian, “Three body problem for short range forces. I. Scattering of low energy neutrons by deuterons”, Soviet Phys. JETP, 4 (1957) 648–661. [48] S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, AMS, Providence, 1963. [49] L. Thomas, “Multiparticle Schr¨ odinger Hamiltonians with point interactions”, Phys. Rev. D 30 (1984) 1233–1237. [50] C. N. Yang, “S-matrix for the one-dimensional N -body problem with repulsive or attractive δ-function interaction”, Phys. Rev. 168 (1968) 1920–1923.

CONTINUOUS QUANTUM MEASUREMENT: LOCAL AND GLOBAL APPROACHES S. ALBEVERIO Fakult¨ at f¨ ur Mathematik Ruhr Universit¨ at Bochum D 44780 Bochum Germany

V. N. KOLOKOL’TSOV Department of Mathematics, Statistics and O.R. Nottingham Trent University Burton Street Nottingham, NG1 4BU UK

O. G. SMOLYANOV Mechanics-Mathematics Department Moscow State University 119899, Moscow Russia Received 11 December 1996 In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schr¨ odinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menski’s formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schr¨ odinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator.

1. Introduction One can distinguish roughly two approaches in an attempt to give an adequate description of the continuously observed quantum system. In the first (which we shall call local approach), one looks for an evolutionary equation, which generalises the Schr¨ odinger unitary evolution in such a way as to include the interaction of a system with a measuring apparatus and the influence of the latter on the state vector of the system. The following nonlinear (but almost surely norm preserving) aposterior Schr¨ odinger equation was obtained in 1988 by V. P. Belavkin [8]: √ λ (1) dφ + iH + (R − hRi)2 φ dt = λ(R − hRi)φ dW (t) . 2 907 Reviews in Mathematical Physics, Vol. 9, No. 8 (1997) 907–920 c World Scientific Publishing Company

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Here the selfadjoint operators H and R = (R1 , . . . , Rn ) in a Hilbert space H stand for the Hamiltonian of a free dynamics and for the measured physical values respectively, λ ≥ 0 characterises the accuracy of measurement, W = (W 1 , . . . , W n ) is the standard Wiener vector-valued process, and hRi = hRiφ = (φ, Rφ)/(φ, φ) denotes the mean value of an operator with respect to the state vector φ. The particular cases of this stochastic equation corresponding to a projection R = P and a position R = x operators were considered also in [24, 20] respectively. As was noted by Belavkin in [9] and proved by the Ito formula, if one makes the shift √ (2) dB(t) = dW (t) + 2 λhRidt in the Wiener process, then Eq. (1) written in terms of the process B can be obtained by the normalisation φ = ψ/kψk from the following Belavkin quantum filtering equation √ λ 2 (3) dψ + iH + R ψ dt = λRψ dB(t) , 2 which is linear but not norm preserving. In quantum measurement theory, the process B is called the output process. Equation (2) shows that observing √ the trajectories of B means measuring in a continuous way R (more exactly, of 2 λhRiφ ˙ . disturbed by the white noise W The most interesting example of Eq. (3) is given by the equation √ λ i 2 ∆ − iV (x) − kxk ψ dt + λxψ dB(t) , dψ = (4) 2 2 where H = L2 (Rn ), R is the multiplication by the coordinate x, and H = V (x) − ∆/2

(5)

is the standard quantum mechanical Hamiltonian. Equation (4) stands for the quantum filtering of the wave packet under continuous measurement of the position of the quantum particle in the potential field V . It corresponds to the aposterior Schr¨ odinger Eq. (1) of the form √ λ i 2 ∆ − iV (x) − (x − hxi) φ dt + λ(x − hxi)φ dB(t) , (6) dφ = 2 2 which was derived independently by V. Belavkin [8] and L. Diosi [20] for this case. Let us note that it is sometimes convenient to use the equivalent form √ i ∆ − iV (x) − λkxk2 ψ dt + λxψ dS B(t) , dψ = (7) 2 of Eq. (4) written in terms of the so-called symmetric (or Stratonovich) differential dS , which is connected with the more usual Ito differential by the well-known law ψ dS B = ψ dB + dψ dB/2 .

(8)

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The second approach (which we can call a global one) generalises directly the Feynman path integral (alternative to the Schr¨ odinger and Heisenberg) description of the quantum evolution. This approach to the continuous position observation problem of a quantum system described by the Hamiltonian (5) was initiated by B. Menski [34, 35] in 1979 (see also [36]), who postulated the heuristic propagator ! ) ( Z Z t ξ˙2 (τ ) 2 − V (ξ(τ ) + iλ|q(τ ) − ξ(τ )| dτ , K(t, x, y, [q(.)]) = Dξ exp i 2 0 (9) where the formal integral is taken over the set of all continuous paths ξ(τ ) joining y and x in time t and λ is again some constant. In other words, if the initial state of the system is given by the function φ(x) and the result of continuous measurement of its position is given by the curve q(τ ), τ ∈ [0, t], then the finite state is given by the function Z φ(t, x) = K(t, x, y, [q(.)])φ(y) dy . (10) To be more precise, in Menski’s formula the constant λ has the form λ = (2δ 2 t)−1 with some other constant δ, but since in Menski’s formula the time is fixed (he does not consider any evolution equation), this dependence of λ on t seems not to be essential. The aim of the present paper is to bring together (both in the sense of physics and mathematics) two indicated approaches to the description of the continuous position observation of a quantum particle. More precisely, starting only from standard quantum mechanical concepts, we intend to give a deduction (or physical mathematical motivation) of Eqs. (4) and (9), and state mathematically rigorous connections between them. In particular, we shall show that the local evolution that leads to (9) is given by the Belavkin filtering Eq. (4). Let us stress that the discovery of Eq. (1) (and its more general version [8]) was a result of a long development of the mathematical physical (and philosophical) concept of continuous quantum measurement. As important previous steps let us note (i) the discovery of the so-called master equation (describing continuously observed quantum system in terms of an averaged density matrix, see e.g. [19]) that gave rise to the solution in this framework of the quantum Zeno paradox (see [15] and also [16], where the Feynman path represetation for the solutions of the master equation was proposed) and to the general theory of completely positive semigroups [33, 2, 3], (ii) the development of mathematical models of quantum filtering and nonideal nondemolition measurement [10, 11, 12], (iii) the phenomenological models [37, 24], of the continuous state reduction, the operational approach to quantum continuous measurements [15, 16] and based on them GRW model [23] (see also [7]). A more complete bibliography as well as different points of view on Eq. (1) and its possible applications in theoretical and experimental physics, are presented in the Proceedings [13]. One can distinguish two main approaches to this equation. One of them, most consequently carried through by V. Belavkin, considers (1) as a consequence of unitary evolution after some process of filtering with respect to observed data. From another point of view. (see e.g. [26, 27, 37, 17] and references

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therein), one should postulate Eq. (1) as a more general (more fundamental) law of nature (state diffusion model), which describes open systems. We note here only that, on the one hand, the standard Schr¨ odinger equation is surely a particular case of (1) (when λ vanishes), as any closed system can be considered as an (idealised) particular case of an open system, but on the other hand, the desire to deduce a new equation as a consequence of already well known and commonly adopted laws is very natural and the possibility of this deduction is of importance. In the next section we shall try to present the simplest possible unified deduction of both (1) and (9). The use of a (very simple) model of unitary interaction of a quantum system with a measuring apparatus proves to be very useful for justifying the statistical law 2 of the observed process. The discussion of Sec. 2 is given on a heuristic level. On the other hand, the last Sec. 3 presents a mathematical treatment of Eq. (4). Namely, we state the mathematical relation between (4) and (9), presenting the solutions of (4) in terms of the mathematically well-defined Feynman path integral (in fact, even in two forms: as an infinite dimensional oscillator integral and as an integral over a suitable Wiener space). For other mathematical properties of Eqs. (1) and (4) see, for instance, [28, 29, 25, 14] (well posedness theorems and the construction of WKB-type asymptotic solution), and [30, 31, 32] (long time asymptotics of its solutions). The main results of this paper have been announced in [4]. 2. Local and Global Description of Continuous Quantum Observations For simplicity, we confine ourselves to the consideration of the continuous measurement of the position of a one-dimensional quantum particle described by the standard Hamiltonian (5), the evolution of the wave function without observation being then given by ψt = exp{−itH}ψ0. To deal with a more general case describing the continuous observation of a physical value corresponding to a selfadjoint operator R in a Hibert space H (or a set of commuting operators) one should simply use the L2 (X, dµ) representation of H, where the operator R is the multiplication operator and the points of X present the spectrum of R (possible results of measurement of the corresponding physical value). Let us note also that the Postulates 1 and 2 given below are largely used in the literature on quantum continuous measurement (see e.g. [20] and references therein). Let a measurement at time t of the position of the particle yields the value q. Postulate 1. Nonideal measurement principle. After such a measurement, the wave function ψt (x) transforms into a new function which up to normalisation has the form (11) ψt (x) exp{−α(x − q)2 } , where the coefficient α characterises the accuracy of our measurement. Note that the ideal measurement would give a reduction ψt (x) 7→ ψt (x)δ(x − q) and thus nonideal measurement means only that we take instead of δ-function its Gaussian approximation, which is natural when taking into account that the sum of (infinitly) many small independent errors has a normal distribution.

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Let us explain how this postulate can be deduced from a unitary evolution combined with the standard von Neumann (ideal) measurement postulate. We shall measure the position x of our particle X indirectly by reading out (sharp and direct) the value of “analogous” physical variable y of a measuring apparatus Y (the position of a quantum meter), which is directly connected with (or directly influenced by) the considered quantum system X. More precisely, we consider the state of the quantum meter to be described by vectors in a Hilbert space L2 (R), variable y stands also for the operator of multiplication by the argument y in it (we systematically denote the position of a system and the corresponding multiplication operator by the same symbol), and the direct influence of X on Y means that the (unitary) evolution of the compound system in L2 (R) ⊗ L2 (R) = L2 (R2 ) (the interaction between the quantum particle and the quantum meter in the process of measurement) reduces to the shift of y by value of x, i.e. it is described by the law f (x, y) 7→ f (x, y − x) . Furthermore, we suppose that we can always prepare a fixed initial state of the quantum meter, say a Gaussian one, r 4 2α α exp{−αy 2 } Φ (y) = π (which is the simplest approximation to the δ-function giving an ideal element, where the value of the operator y is exactly 0). Thus the process of measurement is described in the following way. We have a state φ of our quantum particle and prepare an initial state Φα of the quantum meter Y . Then we switch on the interaction and get as a result the function φ(x)Φα (y − x) in L2 (R2 ). At last we read out (directly and sharply) the value of the second variable, say y = q, which gives us (due to the standard von Neumann reduction postulate) the state 11 of the considered system X. It is important to point out that from this unitary model we get even more than just Postulate 1. Namely, since the square of the magnitude of the state function φ(x)Φα (y − x) defines the density of the joint distribution of x and y = q, the distribution of the measured value q can be obtained by integration in x and thus the probability density of q is equal to r Z 2α q φ2 (x) exp{−2α(z − x)2 } dx . (12) Pα (z) = π Now we fix a time t and make nonideal measurements of the position of the quantum particle at moments tk = kδ, where δ = t/n, n ∈ N . Postulate 2. Continuous limit of discrete observations. As n → ∞, the accuracy of each measurement will be proportional to the time between successive measurements, i.e. α = δλ, where the constant λ reflects the properties of the measuring apparatus.

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This postulate is very natural. It means that the accuracy of each measurement decreases, when the time between them decreases (one cannot make in a fixed period of time infinitely many equally accurate measurements). In the unitary picture (presented above) of the interaction of X with the quantum meter Y , this postulate means that if the time between measurements that we use to bring the quantum meter back to the initial state decreases, then the “quality” of this initial state also decreases, i.e. it gives worse approximations to the idealised δ(y) state of the meter, where its position is strictly zero. Postulates 1 and 2 give everything one needs to get Menski’s formulas (9) and (10). In fact, supposing that between measurements the evolution of the quantum system is described by the law of free Hamiltonian, one concludes that after n measurements the resulting wave function will have the form ψtn

=

n Y

exp{−δλ(x − qj )2 } exp{−iδH} .

j=1

From the Trotter formula exp{A + B} = limn→∞ (exp{A/n} exp{B/n})n for noncommuting operators A, B (more precisely, its nonhomogeneous version) one sees (we shall give below also an alternative deduction without the use of the Trotter formula) that ψtn , n → ∞, tends to the solution of the equation ψ˙ = −(iH + λ(x − q(t))2 )ψ ,

(13)

where q(τ ) is the function taking values qj at times tj . But this is the standard Schr¨ odinger equation with the complex (and time depending) potential V (x) − iλ(x − q(t))2 . Writing down the propagator (or the Green function of the Cauchy problem) for Eq. (13) in terms of the formal Feynman path integral, one gets directly formulas (9) and (10). Let us now modify Eq. (13). Observing that the term λq(t)2 ψ on the r.h.s. can be dropped, because the solutions of the equation with and without this term are proportional and this difference is irrelevant for the purposes of quantum mechanics (where we are interested only in normalised states), one gets the equation (14) ψ˙ + iH + λx2 ψ = 2λxq(t)ψ . Rt Introducing the function Q(t) = 0 q(τ ) dτ , one can rewrite (14) in equivalent differential form (15) dψ + iH + λx2 ψ dt = 2λxψ dQ(t) , √ which written in terms of the function B(t) = 2 λQ(t) is formally the same as the Belavkin linear equation in the Stratonovich form (7). But in fact, the Belavkin result states more, namely the statistical properties (2) of the trajectories of B. It implies, in particular, that the function q(t) in (9) should be a rather singular object, which almost surely makes sense only as a distribution and not as a continuous

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913

curve. In order to see this singularity, one should take into account the probability density 12 of the measured value q. It implies that if the state function of our quantum particle X was φ(τ, x) at the instant τ (in fact, τ = tk for some k), then the expectation of the value q at the instant τ + δ is given by the formula r Z Z 2λδ q zφ2 (τ, x) exp{−2λt(x − z)2 } dxdz Eqτ +δ = zPλδ (z) dz = π and is thus obviously equal to the mean value hxiτ of the position of the particle in the state φ(τ, x). Moreover, the dispersion of this random variable r Z 2λδ 2 (z − hxiτ )2 φ2 (, τ, x) exp{−2λδ(x − z)2 } dxdz E((qτ +δ − Eqτ +δ ) ) = π r Z 2λδ (x − hxiτ − z)2 φ2 (τ, x) exp{−2λδz 2 } dxdz = π = h(x − hxiτ )2 iτ +

1 4λδ

has the form (4λδ)−1 (1 + O(δ)) (at least for wave functions φ(τ, x) with a finite second moment with respect to the corresponding probability distribution). Thus the random variable qτ +δ − hxiτ has vanishing expectation and a dispersion (or the second moment) of the order (4λδ)−1 . Therefore, one cannot consider q(t) as a continuous curve. This object is very singular. The simplest model for the process of errors in the continuous observation (and also the most natural and commonly ˙ , which used in the classical stochastic theory of measurement) is the white noise W can be defined as a formal derivative of the standard Wiener process W (t). √ −1 Taking −1 into account that the dispersion (4λδ) has the random value (2 λδ) W (δ), one arrives at the following: Postulate 3. White noise model of errors. The difference between qt+δ √ −1 and hxit is approximately equal to (2 λδ) W (δ) for small δ, or more precisely, 1 ˙ q(t) = hxit + √ W (t) , 2 λ equivalently, in terms of the Ito differential of the stochastic process Q(t) + Ror, t q(τ ) dτ 0 1 (16) dQ(t) = hxit dt + √ dW (t) . 2 λ √ In terms of the process B(t) = 2 λQ(t) Eqs. (15) and (16) coincide with (1) and (7) respectively, if one understands the differential in (15) as a Stratonovich differential. The latter is natural, because in order to derive (15), we have made some manipulations considering q(t) as a usual function and such manipulations for ˙ can be justified only for Stratonovich differentials, which satisfy the standard W rules of calculus. Nevertherless, to make the situation clearer, we give now (using

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S. ALBEVERIO, V. N. KOLOKOL’TSOV and O. G. SMOLYANOV

Postulates 1–3 only) a simple alternative deduction of (3) and (4) using neither the Stratonovich differentials nor the Trotter product formula. Due to the Postulates, the change dψ(t) of the wave function ψt = ψ(t, x) of our system X in a small time δ = dt is given approximately by the formula ( 2 ) dW (t) exp{−iH dt} ψt − ψt . (17) dψ(t) = exp −λdt x − hxit − √ 2 λdt We can rewrite the first exponent as exp{−λx2 dt + 2λxhxidt +

√ λx dW (t)} ,

because we can drop the multiplier which only depends on the time (as we have already mentioned we are interested in the evolution of the wave function up to normalisation). Developing this exponent in a Taylor series, using the Ito rule dW (τ ) dW (τ ) = dτ for the stochastic differentials, and omitting the remainders of order O((dt)3/2 ) we can write it approximately as √ 1 − 1/2λx2 dt + 2λxhxidt + λx dW (t) . Inserting this in (17) and representing also the second exponent in (17) approximately as 1 − iH dt, we get the formula (4) for the differential dψ with √ √ dB(t) = 2 λ dQ(t) = dW (t) + 2 λhxi dt . As we have mentioned, the Belavkin aposterior Eq. (1) can be obtained from the quantum filtering Eq. (3) by normalisation (with the help of Ito’s formula). However, in the case under consideration, we can get the normalised Eq. (6) also directly from (17) by another recombinations of terms in the first exponent. Namely, up to a multiplier which depends only on time (and which can therefore again be omitted), we can write the first exponent in (17) also as √ exp{−λ(x − hxit )2 dt + λ(x − hxit ) dW (t)} and then the same manipulations as above will bring us directly to Eq. (6). To conclude this section, let us mention, that if the state vector evolves in time according to Eq. (1), then for the expectation (with respect to the Wiener measure ¯ one gets of the process W ) of the corresponding density matrix ρ = E(φ ⊗ φ), directly (using Ito’s formula) the equation ρ˙ = −i[H, ρ] −

λ [R, [R, ρ]] , 2

(18)

which is called the master (or Lindblad) equation. In fact, one of the popular ways to deduce (1) in physics (see e.g. [20, 25, 26, 27]) is by postulating Eq. (18) as the fundamental evolution law for open systems and then choosing the coefficients of a stochastic Ito equation for the wave function in such a way that it would fit in (18) after the transform to the density matrix and the following averaging. This can be

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done in a non unique way. For instance, as it follows from [8], the same master equation corresponds to a completely different type of aposteriori equation for the counting measurements. In this approach, Eq. (1) is no longer connected directly with the measurement process, however at the same time the nice interpretation of dB as the output process of an indirect observation is lost. 3. Rigorous Path Integral Representation for the Solutions of the Linear Quantum Filtering Equation We give here two different (and valid for different assumptions on the potential and the initial function) representations of the solutions of Eq. (4) in terms of functional integrals, first being an integral over a suitable Wiener space, another being an infinite dimensional oscillatory integral. Theorem 1. Let V (resp. η) be an entire analytic function on C n such √ that for any compact K ∈ Rn√there exists a constant C > 0 such that ImV (x + iy) < C(1 (resp. |η(x + iy)| < exp{C(1 + |y|)} for all y ∈ Rn and x ∈ K, where √ + |y|) iπ/4 i=e . Then there exists a (strong) solution of the Cauchy problem for Eq. (4) with initial data ψ0 = η, which can be represented in the form √ Z t √ √ λ (x + iξ(τ )) dB(τ ) ψ(t, x) = E0t η(x + iξ(t)) exp Z × exp −

0 t

√ λ(x + iξ(τ )) + iV (x + iξ(τ )) dτ

2

,

(19)

0

where Est denotes the expectation with respect to the Wiener measure for the Wiener process ξ (independent on the process B) on the space C0 [s, t] of all continuous functions ξ : [s, t] 7→ Rn such that ξ(s) = 0. The integral with respect to dB(τ ) is understood in Ito’s sense. Proof. That the initial condition is satisfied can be checked in a straightforward manner, once the existence of integral in (19) is established. The latter is implied by the Fernique theorem [22]. To prove that (19) is a solution of (4), we shall use the machinery based on Ito’s formula. Let V1 (x) = −(λkxk2 + iV (x)) , Z t √ Z t √ √ V1 (x + iξ(τ )) dτ + λ (x + iξ(τ ))dB(τ ) , fs (x; ξ, B, t) = exp s

s

where ξ, B are two independent standard Wiener processes on [s, t] (and ξ(s) = B(s) = 0). Then by the Ito formula √ √ √ √ λ dfs (., t) = fs (., t) V1 (x + iξ(t)) dt + λ(x + iξ(t)) dB(t) + (x + iξ(t))2 dt . 2 Again by Ito’s formula √ √ √ √ i dη(x + iξ(t)) = η 0 (x + iξ(t)) i dξ + η 00 (x + iξ(t)) dt . 2

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Hence, using the independence of the processes ξ, B and once more by Ito’s formula one gets √ √ √ λ i d(fs (., t)η(x + iξ(t)) = fs (., t) − (x + iξ(t))2 − iV (x + iξ(t)))η + η 00 dt 2 2 √ √ √ + λ(x + iξ(t))η dB(t) + iη 0 dξ(t) , √ where η, η 0 , η 00 on the r.h.s. are taken at the point x+ iξ(t). Taking the expectation Est of both sides of this equation and introducing the notation √ (Tst η)(x) = Est fs (., t)η(x + iξ(t)) we get (Tst η)(x) = η(x) +

Z t√ λ(Tsτ (xη))(x) dB(τ ) s

Z t i λ kxk2 + iV (x) − ∆ η (x) dτ , Tsτ − 2 2 s which implies that the function (19) satisfies Eq. (4). We turn now to the second representation formula, which we discuss rather briefly. Using any of the natural definitions of oscillatory integrals in a Hilbert space, for instance by means of the Parcival equality [1] or by means of a discrete approximation [21, 5] (see also [6, 18, 38, 39] and references therein), one easily gets the following result. Proposition. Let L1 , L2 be commuting symmetric trace class operators in a real Hilbert space H such that L2 is nonnegative and L = L1 + iL2 is invertible. Furthermore, let l ∈ H and let f be the Fourier transform of a finite complex Borel measure µ on H: Z ei(h,y) dµ(y) .

f (h) = µ ˆ(h) = H

Then the oscillator intergral FH (or Feynman integral ) of the function i f (h, Lh) + (l, h) f (h) (h) = exp gL,l 2 is well defined and is given by the formula Z i f −1/2 −1 FH (gL,l ) = det(1 + L) exp − ((y − il), (1 + L) (y − il)) dµ(y) . (20) 2 H We need the following particular case of this construction. Let Ht be the space of continuous curves γ : [0, t] 7→ Rn such that γ(t) = 0 and the derivative γ˙ of γ (in the sense of distribution) belongs to L2 ([0, t]). The scalar product in Ht is defined as Z t

(γ1 , γ2 ) =

γ˙ 1 (s)γ˙ 2 (s) ds . 0

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This Ht is sometimes called the Cameron–Martin space. If V, η are Fourier transforms of finite complex measures in Rn , then the function Z t V (γ(s) + x) ds η(γ(t) + x) f (γ) = exp i 0

is the Fourier transform of a finite Borel measure on Ht (for any x). Furthermore, if Rt B(t) denotes the trajectory of the Wiener process, then the curve lB (τ ) = τ B(s) ds belongs to Ht with probability one, and for γ ∈ Ht Z t Z t (l, γ) = B(s)γ(s) ˙ ds = γ(s) dB(s) , 0

0

where the latter integral is understood inR the Ito sense. Finally, if we define the t operator L by the formula (γ1 , Lγ2 ) = ω 0 γ1 (s)γ2 (s) ds, where ω is any complex constant with positive imaginary part, then all conditions of the Proposition are f ) is given by formula (20). satisfied and the integral FHt (gL,l Theorem 2. If V and η are Fourier transforms of finite complex measures on Rn , then there exists a (strong) solution of the Cauchy problem for Eq. (4) with initial data ψ0 = η, which can be represented in the form Z t √ λ(γ(τ ) + x) dB(τ ) ψ(t, x) = FHt η(γ(t) + x) exp 0

Z t 2 (λ(γ(τ ) + x) + iV (γ(τ ) + x)) dτ . × exp − 0

Sketch of the proof. The simplest way is to use the Stratonovich form (7) of Eq. (4). Then we approximate the trajectory B of the Wiener process by a sequence of smooth curves Bn tending to B as n → ∞. For smooth curves the proof of the theorem can be given in a quite similar way as in the case of the usual Schr¨ odinger equation (see e.g. [1, 39]). We can finish the proof exploiting the fact that the sequence of solutions of the Stratonovoch equation with Bn placed instead of B tends to the solution of (7). We should note only that the latter assertion is proved in the literature for many classes of equations, which do not seem to include directly Eq. (7). But the methods of these proofs, for instance that of [40], can be applied to get the convergence in our case as well. Remark 1. The integrals giving the solution of (4) in Theorems 1 and 2 can be represented as the limit of a sequence of finite-dimensional integrals that one normally uses to introduce the heuristic Feynman integral. Our results above show, in particular, that the latter sequence of finite-dimensional integrals converges, i.e. in the present case the heuristic Feynman path integral has a rigorous mathematical meaning. Using the standard heuristic notation for this Feynman integral and √ ˙ putting formally B(t) = 2 λq(t) one gets a heuristic representation of solutions of the Eq. (4) by means of (9) and (10).

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Remark 2. Representations analogous to Theorems 1 and 2 can be proved also for the case of the continuous measurement of the position of an anharmonic oscillator, i.e. when one puts the additional term −iωkxk2ψ dt (with some positive ω) into the r.h.s. of (4). Acknowledgements We are gratefull for the financial support of SFB 237 which gave us the opportunity to work together at Ruhr Universit¨ at Bochum in February 1996, where this work was started. References [1] S. Albeverio and R. Hoegh-Krohn, Mathematical Theory of Feyman Path Integrals, LNM 523, Springer, 1976. [2] S. Albeverio, R. Hoegh-Krohn and G. Olsen, “Dynamical semigroups and Markov processes on C ∗ -algebras”, J. Reine Ang. Math. 319 (1980) 29–37. [3] S. Albeverio and R. Hoegh-Krohn, “Dirichlet forms and Markov semigroups on C ∗ -algebras”, Commun. Math. Phys. 56 (1977) 173–187. [4] S. Albeverio, V. N. Kolokoltsov and O. G. Smolyanov, “Repr´ esentation des solution de l’´equation de Belavkin pour la mesure quantique par une version rigoureuse de la formule d’integration fonctionelle de Menski”, to appear in Comptes Rendu de l’Acad. Sci. [5] S. Albeverio and Zd. Brzezniak, “Finite dimensional approximations approach to oscillatory integrals in infinite dimensions”, J. Funct. Anal. 113 (1993) 177–244. [6] S. Albeverio, Ph. Combe, R. Hoegh-Krohn, G. Rideau, M. Sirugue-Collin and R. Stora, eds., Feynman Path Integrals, LNP 106, Springer, 1979. [7] J. Bell, “Speakable and Unspeakable in Quantum Mechanics”, Cambridge Univ. Press, 1987. [8] V. P. Belavkin, “Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes”, in Proc. Bellman Continuous Workshop, Sophia-Antipolis 1988, LNCIS 121 (1988) 245–265. [9] V. P. Belavkin, “A new wave equation for a continuous nondemolition measurement”, Phys. Lett. A 140 (1989) 355–358. [10] V. P. Belavkin, “Optimal linear randomized filtration of quantum Boson signals”, Problems of Control and Information Theory, 3 (1974) 47–62. [11] V. P. Belavkin, “Optimal quantum filtering of Markovian signals”, Problems of Control and Information Theory, 7 (1978) 345–360. [12] V. P. Belavkin, “Optimal filtering of Markovian signals in white quantum noise”, Radio Eng. Electron. Phys. 18 (1980) 1445–1453. [13] V. P. Belavkin, R. Hirota and R. Hudson, eds., Quantum Communications and Measurement. Proc. of Intern. Workshop held in Nottingham, July 1994. Plenum Press, N.Y., 1996. [14] V. P. Belavkin and V. N. Kolokoltsov, “Quasi-classical asymptotics of quantum stochastic equations”, Teor. Matem. Fizika 89 (1991) 163–178. Engl. Transl. in Theor. Math. Phys. [15] A Barchielli, L. Lanz and G. M. Prosperi, “Statistics of continuous trajectories in quantum mechanics: Operator valued stochastic processes”, Found. Phys. 13 (1983) 779–812. [16] A. Barchielli, L. Lanz and G. M. Prosperi, “A model for the macroscopic description and continual observation in quantum mechanics”, Nuovo Cimento 72 B:1 (1982) 79–121.

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[17] Ph. Blanchard and A. Jadczyk, “Event-Enhanced Formalism of Quantum Theory or Columbus Solution to the Quantum Measurement Problem”, BiBoS Preprint 655/7/94. Published in [13]. [18] A. M. Chebotarev and V. P. Maslov, “Processus a ` sauts et leur application dans la m´ecanique quantique”, pp. 58–72 in [6]. [19] E. B. Davies, Quantum Theory of Open System, London, Acad. Press, 1976. [20] L. Diosi, “Continuous quantum measurement and Ito formalism”, Phys. Lett. A 129 (1988) 419–423. [21] D. Elworthy and A. Truman, “Feynman maps, Cameron–Martin formulae and anharmonic ascillator”, Ann. Inst. Henri Poincar´ e, Physique Th´ eorique, 41 (2) (1984) 115–142. ´ [22] X. Fernique, Regularit´e des trajectoires des fonctions al´eatoires Gaussiennes, Ecole d’Et´e de Probabilit´es de Saint-Flour 4-1974, LNM 480 (1975), ed. P. Hennequin, pp. 2– 96. [23] G. C. Ghirardi, A. Rimini and T. Weber, “A model for unified quantum description of macroscopic and microscopic systems”, in Quantum Probability and Applications II , eds. L. Accardi, W. von Waldenfels, LNM 1136 (1985), Berlin, Springer, 223–233. [24] N. Gisin, “Quantum measurement and stochastic processes”, Phys. Rev. Lett. 52 (19) (1984) 1657–1660. [25] D. Gatarek and N. Gisin, “Continuous quantum jumps and infinite dimensional stochastic equations”, J. Math. Phys. 32 (1991) 2152–2156. [26] N. Gisin and I. Percival, “Quantum state diffusion, localization and quantum dispersion entropy”, J. Phys. A: Math. Gen. 26 (1993) 2233–2243. [27] N. Gisin and I. Percival, “Quantum state diffusion picture of physical processes”, J. Phys. A: Math. Gen. 26 (1993) 2245–2260. [28] V. N. Kolokoltsov, “Application of the quasi-classical methods to the investigation of the Belavkin quantum filtering equation”, Mat. Zametki 50 (1991) 153–156. Engl. Transl. in Mathem. Notes. [29] V. N. Kolokoltsov, “Stochastic Hamilton–Jacobi–Bellman equation and stochastic method WKB”, Inst. Math. Ruhr-Uni Bochum, preprint 236 (1994). To appear in Proc. Int. Workshop “Idempotency”, Bristol, 3–7 October 1994, Cambridge Univ. Press, 1995. [30] V. N. Kolokoltsov, “Long time behavior of the solutions of the Belavkin quantum filtering equation”, Proc. Int. Workshop “Quantum Communications and Measurement”, Nottingham, July 1994, Plenum Press, N.Y., 1995, 429–439. [31] V. N. Kolokoltsov, “Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate”, J. Math. Phys. 36 (6) (1995) 2741–2760. [32] V. N. Kolokoltsov, “Localization and analytic properties of the simplest quantum Langevin equation”, Inst. Math. Ruhr-Uni Bochum, preprint 275 (1995). [33] G. Lindblad, “On the generators of quantum dynamical semigroups”, Commun. Math. Phys. 48 (1976) 119–130. [34] B. M. Menski, “Quantum restrictions for continuous observation of an oscillator”, Phys. Rev. D 20 (2) (1979) 384–387. [35] B. M. Menski, “Quantum restrictions on the measurement of the parameters of motion of a macroscopic oscillator”, Sov. Phys. JETP 50 (1979) 667–674. [36] B. M. Menski, Path Groups, Moscow, Nauka, 1983 (in Russian). [37] P. Pearle, “Reduction of the state vector by a nonlinear Schr¨ odinger equation”, Phys. Rev. D 143 (1976) 857–868. [38] O. G. Smolyanov and M. O. Smolyanova, “Transformations of Feynman integrals under nonlinear transformations of the phase space”, Teor.i Matem Fizika 100 (1) (1994) 3–13, Engl. Transl. in Theor. Mathem. Phys.

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[39] O. G. Smolyanov and E. Shavgulidze, Continual Integrals, Moscow Univ. Publisher, 1990 (in Russian). [40] H. Sussman, “On the gap between deterministic and stochastic ordinary differential equations”, Ann. Prob. 6 (1) (1978) 19–41.

SYMMETRY GROUPS IN QUANTUM MECHANICS AND THE THEOREM OF WIGNER ON THE SYMMETRY TRANSFORMATIONS GIANNI CASSINELLI Department of Physics and I.N.F.N., University of Genoa via Dodecaneso 33, 16146 Genoa, Italy E-mail: [email protected]

ERNESTO DE VITO Department of Mathemathics, University of Modena via Campi 213/B, 41100 Modena, Italy E-mail: [email protected]

PEKKA J. LAHTI Department of Physics, University of Turku FIN-20014 Turku, Finland E-mail: [email protected]

ALBERTO LEVRERO Department of Physics and I.N.F.N., University of Genoa via Dodecaneso 33, 16146 Genoa, Italy E-mail: [email protected] Received 21 April 1997 Various mathematical formulations of the symmetry group in quantum mechanics are investigated and shown to be mutually equivalent. A new proof of the theorem of Wigner on the symmetry transformations is worked out.

1. Introduction The Hilbert space formulation of quantum mechanics points out several mathematical objects whose physical meaning is connected with the probabilistic structure of the theory. Among them there are: (1) (2) (3) (4) (5) (6)

the set of pure states P with the notion of transition probability, the convex set S of states, the orthomodular lattice L of the closed subspaces, the partial algebra E of the positive operators bounded by the unit operator, the Jordan algebra Br of the self-adjoint bounded operators, the C ∗ -algebra B of the bounded operators.

The automorphisms of these sets, that is, the one to one maps from a given set onto itself preserving the corresponding relevant structure (transition 921 Reviews in Mathematical Physics, Vol. 9, No. 8 (1997) 921–941 c World Scientific Publishing Company

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probability, convexity,. . .) are natural candidates to represent the symmetries of quantum mechanics. Moreover, the set of the automorphisms of any of the previous mathematical objects forms a group, denoted by Aut (·), under the natural composition of mappings. So there are several groups which can be used to represent the symmetries of quantum mechanics. This poses the question on the equivalence of these groups, that is, if they are isomorphic in some natural way. It is well known that the answer to the above question is positive (at least if the dimension of the Hilbert space is greater than two). Moreover, the Wigner theorem shows that one can associate to each element of Aut (P) a unitary or antiunitary operator which is unique up to a phase factor. In this way the previous groups Aut (·) are shown to be isomorphic to the symmetry group Σ(H) of the Hilbert space, that is, the group of unitary or antiunitary operators modulo the phase group. This allows using the theory of unitary representations of groups in order to implement symmetry in quantum mechanics. Much of these facts are well known since a long time ago and there is a rich literature on this topic. However, up to our knowledge there is no complete concise and simple treatment of the relations among these different symmetry groups. The present paper is devoted to fill this gap with special care on the following aspects: (1) to point out the two dimensional case where some of the previous identifications fail to be true; (2) to discuss the topological properties of these isomorphisms that are indispensable in applying the theory of unitary representations; (3) to give a new simple proof of the Wigner theorem, whose need has been emphasised by Weinberg in his recent book [1]. We base our treatment on the following works which we believe are the essential contributions to the problems we are concerned with: (1) Wigner’s book [2]: it contains the original idea on the isomorphism between Aut (P) and Σ(H). (2) Uhlhorn’s paper [3]: it proves Wigner theorem with a weaker assumption, but assuming that the dimension of the Hilbert space is greater than two. It also studies some relations between Aut (P) and Aut (B). (3) Bargmann’s paper [4]: it gives the first complete proof of the Wigner theorem without any assumption on the dimension of H. (4) Varadarajan’s book [5]: it discusses the isomorphisms of some of the automorphism groups and the Wigner theorem using the fundamental theorem of projective geometry. This makes these results less accessible and they are also subject to the dimension limitation dim(H) ≥ 3. (5) Simon’s review [6]: it makes a survey of the relations among Aut (P), Aut (S), Aut (Br ) and Σ(H) and it points out the case of the time evolution as a one parameter group of symmetries. (6) Ludwig’s book [7]: it discusses in great detail the properties of the various automorphisms on E, assuming again that dim(H) ≥ 3.

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The paper is organised in the following way. In Sec. 2 we define the previous groups of automorphisms and we endow them with some natural topologies arising from the probabilistic structure of quantum mechanics. In the following section we give a new proof of the Wigner theorem based on the idea of positive cones in the Hilbert space. In Sec. 4 we show that the groups Aut (P), Aut (S), Aut (L), Aut (E), Aut (Br ), Aut (B) and Σ(H) are isomorphic. In the last section we discuss the topological properties of these groups and we prove that they are second countable metrisable topological groups and that the previous isomorphisms are in fact homeomorphisms. As a final introductory comment we notice that when the Hilbert space is one dimensional all the above automorphism groups have only one element whereas the group Σ(H) has two elements. From now onwards we assume that dim H ≥ 2. 2. The Automorphism Groups We first introduce some notations. Let H be a complex separable Hilbert space associated with a quantum system. The inner product h | i is taken to be linear with respect to the second argument. We let tr[·] denote the trace functional and P [ϕ] the projection on the one-dimensional subspace [ϕ] generated by the nonzero vector ϕ ∈ H (so that for any ψ ∈ H, P [ϕ]ψ = hϕ|ψi hϕ|ϕi ϕ). We denote by B1 the set of trace class operators on H and by B1,r the set of self-adjoint trace class operators on H. We introduce next the various mathematical objects with their specific structures and the corresponding groups of automorphisms. 2.1. Let P be the set of one-dimensional projections on H. From physical point of view P is the set of pure states of a quantum system. We endow P with the notion of transition probability: P × P 3 (P1 , P2 ) 7→ tr[P1 P2 ] ∈ [0, 1] . The corresponding automorphisms are the bijective maps α : P → P that satisfy the condition tr[α(P1 )α(P2 )] = tr[P1 P2 ] for all P1 , P2 ∈ P. We call them P-automorphisms. The set of such maps is denoted by Aut (P) and it forms a group. If α satisfies only the weaker condition tr[α(P1 )α(P2 )] = 0 ⇐⇒ tr[P1 P2 ] = 0 P1 , P2 ∈ P , we call it a weak P-automorphism. In this way, a weak P-automorphism is a bijective map preserving only the zero probabilities. The group of weak Pautomorphisms is denoted by Aut w (P). Clearly Aut (P) is a subgroup of Aut w (P). We endow both Aut (P) and Aut w (P) with the initial topology defined by the following set of functions: fPP1 ,P2 : α 7→ tr[P1 α(P2 )],

P1 , P2 ∈ P .

This is the natural topology with respect to transition probabilities.

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2.2. Let S be the set of positive trace class operators of trace one, S = {T ∈ B1 | T ≥ O, tr[T ] = 1} . The elements of S represent the states of the quantum system. The set S is a convex subset of the vector space B1 , that is, if T1 , T2 ∈ S and 0 ≤ w ≤ 1, then wT1 + (1 − w)T2 ∈ S. We observe that P is a subset of S, in fact it is the set of extremal points of S. The relevant automorphisms are the bijective maps V : S → S such that V (wT1 + (1 − w)T2 ) = wV (T1 ) + (1 − w)V (T2 ) for all T1 , T2 ∈ S, and for all 0 ≤ w ≤ 1. These are the S-automorphisms and they form a group denoted by Aut (S). We endow Aut (S) with the initial topology defined by the following set of functions: S : V 7→ tr[AV (T )], fA,T

A ∈ Br , T ∈ S ,

that are related to the probabilistic interpretation of the elements of S. We have the following properties of the S-automorphisms: Lemma 2.1. Let V ∈ Aut (S). (1) V is the restriction of a trace-norm preserving linear operator on B1,r ; (2) if P ∈ P, then V (P ) ∈ P; (3) if V (P ) = P for all P ∈ P, then V is the identity. Proof. (1) We recall that V has a unique extension to a positive trace-preserving bijective linear map Vˆ on B1,r . For any T ∈ B1,r , write T = T + − T − , where T ± = 12 (|T | ± T ). Then kVˆ (T )k1 = kVˆ (T + − T − )k1 = kVˆ (T + ) − Vˆ (T − )k1 ≤ kVˆ (T + )k1 + kVˆ (T − )k1 = kT + k1 + kT − k1 = kT k1 . Since V −1 has the same properties than V we conclude that kVˆ (T )k1 = kT k1 . (2) Let P ∈ P and assume that V (P ) = wT1 + (1 − w)T2 for some 0 < w < 1, T1 , T2 ∈ S. Then P = wV −1 (T1 )+(1−w)V −1 (T2 ), so that P = V −1 (T1 ) = V −1 (T2 ) and thus V (P ) = T1 = T2 showing that V (P ) ∈ P. P (3) Any T ∈ S can be expressed as T = i wi Pi for some sequence (wi ) of P weights [0 ≤ wi ≤ 1, wi = 1] and for some sequence of elements (Pi ) ⊂ P with the series converging in the trace norm. The continuity of V thus gives V (T ) = T for all T ∈ S whenever V (P ) = P for all P ∈ P.

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2.3. Let L be the set of the closed subspaces of H. This set can also be described as the set of projections on H, identifying the closed subspace M ∈ L with the corresponding projection (denoted by the same symbol) onto M . L is a complete orthocomplemented (orthomodular) lattice with respect to the subspace inclusion as the order relation and the orthogonal complement M 7→ M ⊥ as the orthocomplementation. By the previous identification, P is also a subset of L. In fact, it is precisely the set of the minimal elements of L. From a physical point of view the elements of L can be interpreted as the propositions on the system. A function τ : L → L is an L-automorphism if it is bijective and preserves the orthogonality and the order on L, that is, for all M, M1 , M2 ∈ L, τ (M ⊥ ) = τ (M )⊥ , M1 ⊂ M2 ⇐⇒ τ (M1 ) ⊂ τ (M2 ). The group Aut (L) of the L-automorphisms is a topological space with respect to the initial topology defined by the functions L : τ 7→ tr[T τ (M )], fT,M

T ∈ S, M ∈ L .

We recall that tr[M T ] can be interpreted as the probability that the proposition M is true in the state T . We list some properties of the L-automorphisms in the following lemma. Lemma 2.2. Let τ ∈ Aut (L). (1) Let (Mi )i∈I be any family in L, then τ sup Mi = sup τ (Mi ) i∈I

τ

inf Mi

i∈I

i∈I

= inf τ (Mi ) . i∈I

(2) If P ∈ P, then τ (P ) ∈ P. (3) If τ (P ) = P for all P ∈ P, then τ is the identity map. Proof. (1) It suffices to show only the first relation. Since τ preserves the order we have Mk ⊂ sup Mi i∈I

τ (Mk ) ⊂ τ sup Mi i∈I

sup τ (Mi ) ⊂ τ sup Mi . i∈I

i∈I

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Since τ −1 shares the properties of τ we conclude that τ (supi∈I Mi ) = supi∈I τ (Mi ). (2) As τ preserves the order it maps one dimensional projections into one dimensional projections. (3) This statement follows from the first one by observing that any element M of L is the supremum of the one dimensional subspaces contained in it. 2.4. Let E be the set of operators E on H such that O ≤ E ≤ I. E is a partial algebra with respect to the sum. We define an E-automorphism as a bijective map f from E onto E preserving the partially defined sum, that is, satisfying E + F ≤ I ⇐⇒ f (E) + f (F ) ≤ I and, in this case, f (E + F ) = f (E) + f (F ) . We denote by Aut (E) the group of the E-automorphisms and we endow it with the initial topology defined by the following functions E : f 7→ tr[T f (E)], fE,T

E ∈ E, T ∈ S .

Lemma 2.3. Let f ∈ Aut (E). Then (1) f is order preserving, that is, E ≤ F ⇐⇒ f (E) ≤ f (F ) ; (2) if (Ei )i∈I is any family of elements of E such that sup Ei ∈ E

and sup f (Ei ) ∈ E

i∈I

i∈I

sup f (Ei ) = f sup Ei ;

then

i∈I

i∈I

(3) f (O) = O and f (I) = I; (4) if (Ei )i∈I is an increasing net of elements of E, then sup Ei ∈ E

and sup f (Ei ) ∈ E

i∈I

i∈I

and sup f (Ei ) = f i∈I

sup Ei . i∈I

Proof. (1) If E ≤ F then F = (F − E) + E, with F − E ∈ E, hence f (F ) = f (F − E) + f (E), so that f (E) ≤ f (F ). Since f −1 shares the properties of f , the converse is also true.

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(2) See the proof of point (1) of Lemma 2.2. (3) This follows from the bijectivity of f and from the fact that O = inf E and I = sup E. (4) Since (Ei )i∈I is an increasing net and it is norm bounded, it is a standard result that supi∈I Ei exists in E. Due to (1) the same holds for (f (Ei ))i∈I so that by (2) the proof is complete. 2.5. Let Br be the set of self-adjoint operators on H. Br is a commutative algebra with respect to the Jordan product Br × Br 3 (A, B) 7→

AB + BA ∈ Br . 2

A function S : Br → Br is a Br -automorphism if it is a linear bijection and preserves the Jordan product, that is, for any A, B ∈ Br , 1 1 (AB + BA) = (S(A)S(B) + S(B)S(A)) . S 2 2 We denote the group of Br -automorphisms by Aut (Br ). We put on Aut (Br ) the initial topology defined by Br : S 7→ tr[T S(A)] , fT,A

T ∈ S , A ∈ Br .

One readily observes that a linear bijection S : Br → Br is a Br -automorphism if and only if satisfies S(A2 ) = S(A)2 ,

A ∈ Br .

In fact, if S ∈ Aut (Br ), then S(A2 ) = S(A)2 for all A ∈ Br . Conversely, S((A + B)2 ) = S(A2 ) + S(AB) + S(BA) + S(B 2 ) = (S(A + B))2 = S(A)2 + S(A)S(B) + S(B)S(A) + S(B)2 gives S(A)S(B) + S(B)S(A) = S(AB + BA). Remark 2.1. In his paper [6] Simon defines also a weak Br -automorphism as a linear bijection preserving the Jordan product for pairs of commuting (in the algebra B) bounded self-adjoint operators. The previous observation shows immediately that weak Br -automorphisms are in fact Br -automorphisms. The following lemma collects the basic properties of the Br -automorphisms. Lemma 2.4. Let S ∈ Aut (Br ). Then (1) for any A, B ∈ Br , A ≤ B if and only if S(A) ≤ S(B), (2) for any E ∈ E, S(E) ∈ E.

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Proof. (1) Since S is a linear bijection it suffices to show that S preserves the positivity. But if A ≥ O, then A = (A1/2 )2 and S(A) = S(A1/2 )2 ≥ O. (2) Taking into account point (1) it is sufficient to prove that S(I) = I. Since S(M 2 ) = S(M )2 for all M ∈ L and S is a bijective order preserving map, it sends the greatest projection to the greatest projection, that is, S(I) = I. 2.6. The set B of bounded operators is a unital C ∗ -algebra. A function Φ : B → B is a B-automorphism if it is a linear or antilinear bijection and satisfies for all A, B ∈ B the conditions Φ(A∗ ) = Φ(A)∗ Φ(AB) = Φ(A)Φ(B) . In the linear case the notion of a B-automorphism is the usual notion of a C ∗ -isomorphism. Let Aut (B) denote the group of B-automorphism with the initial topology defined by the functions B : Φ 7→ tr[T Φ(A)] , fT,A

T ∈ B1 , A ∈ B .

This topology is the natural one since B is the dual of B1 . 3. The Wigner Theorem We go on to prove the Wigner theorem. We emphasise that the proof does not depend on the dimension of the Hilbert space. Theorem 3.1. Let α ∈ Aut(P). There is a unitary or an antiunitary operator U on H such that for any P ∈ P, α(P ) = U P U ∗ . U is unique up to a phase factor . Proof. Fix α ∈ Aut P. Let ω ∈ H, ω 6= 0, be a fixed vector and define Oω := {ϕ ∈ H | hω|ϕi > 0} . We observe that Oω is a cone, that is, Oω + Oω ⊂ Oω and λOω ⊂ Oω , λ > 0. Let ω 0 be a vector in the range of the projection α(P [ω]) such that kω 0 k = kωk and define the cone Oω0 . The proof of the theorem will now be split in five parts. Part 1. We show that there is a function Tω : Oω → Oω0 such that for all ϕ, ϕ1 , ϕ2 ∈ Oω , λ > 0,

SYMMETRY GROUPS IN QUANTUM MECHANICS AND . . .

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kTω ϕk = kϕk , Tω (λϕ) = λTω ϕ , Tω (ϕ1 + ϕ2 ) = Tω ϕ1 + Tω ϕ2 , P [Tω ϕ] = α(P [ϕ]) .

To define Tω we observe first that for any vector ϕ ∈ Oω , there is a unique vector ψ ∈ Oω0 , kψk = kϕk, such that α(P [ϕ]) = P [ψ]. We denote ψ = Tω ϕ. This defines a function Tω : Oω → Oω0 . Observe that Tω ω = ω 0 . By definition, Tω is norm preserving, positively homogeneous, and α(P [ϕ]) = P [Tω ϕ]. Also for any ϕ1 , ϕ2 ∈ Oω , (+) |hTω ϕ1 |Tω ϕ2 i| = |hϕ1 |ϕ2 i| . We prove next the additivity of Tω . Let ϕ1 , ϕ2 ∈ Oω . By the definition of Oω , ϕ1 and ϕ2 are linearly dependent (over C) if and only if ϕ1 = λϕ2 for some λ > 0. If ϕ1 = λϕ2 then Tω (ϕ1 + ϕ2 ) = Tω ((λ + 1)ϕ2 ) = (λ + 1)Tω ϕ2 = λTω ϕ2 + T ϕ2 = Tω ϕ1 + Tω ϕ2 . Assume now that ϕ1 , ϕ2 are linearly independent. We observe first that for any ψ ∈ H, if hTω ϕi |ψi = 0, i = 1, 2, then hϕi |γi = 0, i = 1, 2, for any γ ∈ α−1 (P [ψ]), and thus hTω (ϕ1 + ϕ2 )|ψi = 0. Hence Tω (ϕ1 + ϕ2 ) = z1 Tω ϕ1 + z2 Tω ϕ2 for some z1 , z2 ∈ C. Since ϕ1 , ϕ2 are linearly independent there are two uniquely defined vectors θ1 , θ2 in [ϕ1 , ϕ2 ], the subspace generated by the vectors ϕ1 , ϕ2 , such that hθi |ϕj i = δij , i, j = 1, 2. In fact, they are θi = (hϕj |ϕj iϕi − hϕj |ϕi iϕj ) / (hϕj |ϕj ihϕi |ϕi i − hϕi |ϕj ihϕj |ϕi i) , i = 1, 2, i 6= j. Let θi0 ∈ H be such that kθi0 k = kθi k and P [θi0 ] = α(P [θi ]). Writing ϕ = ϕ1 + ϕ2 , 1 = hϕ|θi i = |hϕ|θi i|2 = |hTω ϕ|θi0 i|2 = |zi |2 so that that |zi | = 1. Since ϕ1 , ϕ2 , ϕ ∈ Oω and Tω ϕ1 , Tω ϕ2 , Tω ϕ ∈ Oω0 one has hω|ϕi = |hω|ϕi| = |hω 0 |Tω ϕi| = hω 0 |Tω ϕi, which gives hω|ϕ1 i + hω|ϕ2 i = z1 hω|ϕ1 i + z2 hω|ϕ2 i .

(*)

But then hω|ϕ1 i + hω|ϕ2 i = |hω|ϕ1 i + hω|ϕ2 i| = |z1 hω|ϕ1 i + z2 hω|ϕ2 i| ≤ |z1 hω|ϕ1 i| + |z2 hω|ϕ2 i| = hω|ϕ1 i + hω|ϕ2 i , which shows that z1 hω|ϕ1 i = λz2 hω|ϕ2 i for some λ ∈ R. Therefore, 0 < z1 hω|ϕ1 i + z2 hω|ϕ2 i = (1 + λ)z2 hω|ϕ2 i, which shows that the imaginary part of z2 equals 0

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and one thus has z2 = ±1. Similarly, one gets z1 = ±1. From Eq. (*), where hω|ϕ1 i, hω|ϕ2 i > 0, one finally gets z1 = z2 = 1. This completes the proof of the additivity of Tω . Part 2. Let ψ ∈ H, ψ 6= 0, and assume that T is any function Oψ → H, having the properties (a)–(d). Then for any ϕ ∈ Oω ∩ Oψ , T (ϕ) = zTω (ϕ) ,

(e)

for some z ∈ T. Indeed, by the property (d), it holds that for any ϕ ∈ Oω ∩ Oψ , T ϕ = f (ϕ)Tω ϕ, with f (ϕ) ∈ T, and it remains to be shown that f (ϕ) is constant on Oω ∩ Oψ . For any λ > 0 and ϕ ∈ Oω ∩ Oψ , T (λϕ) = f (λϕ)Tω (λϕ) = λf (λϕ)Tω ϕ and T (λϕ) = λT ϕ = λf (ϕ)Tω ϕ. Hence λf (λϕ)Tω ϕ = λf (ϕ)Tω ϕ. Since Tω ϕ 6= 0 for ϕ 6= 0, this gives f (ϕ) = f (λϕ). Consider next vectors ϕ1 , ϕ2 ∈ Oω ∩ Oψ such that ϕ1 6= λϕ2 for any λ > 0 (so that ϕ1 , ϕ2 are linearly independent over C). Then T (ϕ1 + ϕ2 ) = f (ϕ1 + ϕ2 )Tω (ϕ1 + ϕ2 ) = f (ϕ1 )Tω ϕ1 + f (ϕ2 )Tω ϕ2 . Using again the above vectors θ1 , θ2 , associated with ϕ1 , ϕ2 one easily gets, e.g., f (ϕ1 + ϕ2 ) = f (ϕ1 ) for any ϕ2 ∈ Oω ∩ Oψ . Hence f (ϕ) is constant on Oω ∩ Oψ and thus Tω is unique modulo a phase on the cone Oω . Part 3. Let ω ∈ H, ω 6= 0, and let Tω : Oω → Oω0 be defined as in part 1. We show next that Tω has one of the following two properties, either hTω ϕ1 |Tω ϕ2 i = hϕ1 |ϕ2 i

(f)

hTω ϕ1 |Tω ϕ2 i = hϕ2 |ϕ1 i

(g)

for all ϕ1 , ϕ2 ∈ Oω , or for all ϕ1 , ϕ2 ∈ Oω . First of all, let ϕ1 , ϕ2 ∈ Oω . Then hTω (ϕ1 + ϕ2 )|Tω (ϕ1 + ϕ2 )i = hϕ1 + ϕ2 |ϕ1 + ϕ2 i. Using the additivity of Tω and the inner product this shows, in view of (+), that either hTω ϕ1 |Tω ϕ2 i = hϕ1 |ϕ2 i or hTω ϕ1 |Tω ϕ2 i = hϕ2 |ϕ1 i. We show next that for a fixed ϕ ∈ Oω , either hTω ϕ|Tω ψi = hϕ|ψi or hTω ϕ|Tω ψi = hψ|ϕi for all ψ ∈ Oω . To prove this assume on the contrary that there are vectors ϕ1 , ϕ2 ∈ Oω such that hTω ϕ|Tω ϕ1 i = hϕ|ϕ1 i(6= hϕ1 |ϕi) and hTω ϕ|Tω ϕ2 i = hϕ2 |ϕi (6= hϕ|ϕ2 i). By a direct computation of hTω ϕ|Tω (ϕ1 + ϕ2 )i one observes that this leads to a contradiction. By a similar counter argument one shows finally that either hTω ϕ|Tω ψi = hϕ|ψi for all ϕ, ψ ∈ Oω or hTω ϕ|Tω ψi = hψ|ϕi for all ψ ∈ Oω . Part 4. We construct next a unitary or antiunitary operator U of H for which α(P ) = U P U ∗ for all P ∈ P. Let ω ∈ H and Tω : Oω → Oω0 be given as in part one. Let M = [ω]⊥ and M = [ω 0 ]⊥ and define a function S : M → M 0 by 0

Sϕ := Tω+ϕ ϕ, Sϕ := 0,

ϕ 6= 0

ϕ=0

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where Tω+ϕ is the operator on the cone Oω+ϕ with the choice of the phase given by Tω+ϕ ω = ω 0 . S is well defined since for any ϕ ∈ M , ϕ 6= 0, we have ϕ ∈ Oω+ϕ . Moreover, for any two ϕ, ψ ∈ M , Tω+ϕ = Tω+ψ on the cone Oω+ϕ ∩ Oω+ψ , which contains at least the vector ω for which Tω+ϕ ω = Tω+ψ ω. According to part 3 any Tω+ϕ , ϕ ∈ M , has either the property (f) or the property (g). Due to the fact that for all ϕ, ψ ∈ M , Tω+ϕ = Tω+ψ on the intersection of their defining cones, all the operators Tω+ϕ , ϕ ∈ M , are of the type (f) or they all are of the type (g). We proceed to show that S is in the first case a unitary operator and in the second case an antiunitary operator. In fact the proofs of the two different cases are similar and we treat only the case that all Tω+ϕ , ϕ ∈ M , are of the type (f). We show first that for any ϕ ∈ M, λ ∈ C, S(λϕ) = λSϕ. In fact, if λϕ = 0, the result is obvious, otherwise we have hTω (ω + λϕ)|Tω (ω + ϕ)i = hω + λϕ|ω + ϕi ¯ = kωk2 + λhϕ|ϕi hTω (ω + λϕ)|Tω (ω + ϕ)i = hTω+λϕ (ω + λϕ)|Tω+ϕ (ω + ϕ)i = hTω+λϕ ω + Tω+λϕ (λϕ)|Tω+ϕ ω + Tω+ϕ ϕi = hω 0 + S(λϕ)|ω 0 + Sϕi = kω 0 k2 + hS(λϕ)|Sϕi . ¯ But S(λϕ) = Tω+λϕ (λϕ) ∈ Since kωk = kω 0 k this gives hS(λϕ)|Sϕi = λhϕ|ϕi. α(P [λϕ]) and Sϕ ∈ α(P [ϕ]), which shows that S(λϕ) = zSϕ for some z ∈ C. ¯ ¯ and Therefore, λhϕ|ϕi = hS(λϕ)|Sϕi = z¯hSϕ|Sϕi = z¯hϕ|ϕi, which gives z¯ = λ, thus S(λϕ) = λSϕ. To show the additivity of S on M , let ϕ1 , ϕ2 ∈ M . If ϕ1 = λϕ2 , λ ∈ C, then the homogeneity of S gives the additivity. Therefore, assume that ϕ1 , ϕ2 are linearly independent. Let θ1 , θ2 be the unique vectors in [ϕ1 , ϕ2 ] such that hθi |ϕj i = δij . Then S(ϕ1 + ϕ2 ) = Tω+ϕ1 +ϕ2 (ϕ1 + ϕ2 ) = Tω+θ1 +θ2 (ϕ1 + ϕ2 ) = Tω+θ1 +θ2 ϕ1 + Tω+θ1 +θ2 ϕ2 = Tω+ϕ1 ϕ1 + Tω+ϕ2 ϕ2 = Sϕ1 + Sϕ2 . Hence S : M → M 0 is a linear map. It is also isometric since for any ϕ ∈ M , ϕ 6= 0, hSϕ|Sϕi = hTω+ϕ ϕ|Tω+ϕ ϕi = hTϕ ϕ|Tϕ ϕi = hϕ|ϕi. Moreover, for any unit vector ϕ ∈ M one has P [Sϕ] = α(P [ϕ]). To show the surjectivity of S, let ψ ∈ M 0 , ψ 6= 0. Since α is surjective there is a unit vector ϕ ∈ M such that α(P [ϕ]) = P [ψ]. Hence Sϕ = λψ for some λ ∈ C. Since kϕk = 1, also kSϕk = 1 so that λ 6= 0 and thus S( ϕ λ ) = ψ. This concludes the proof of the unitarity of S. We now have H = [ω] ⊕ M = [ω 0 ] ⊕ M 0 and we define U : H → H such that U (λω + ϕ) = λω 0 + Sϕ for all λ ∈ C, ϕ ∈ M . If S is antiunitary we define U instead

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¯ 0 + Sϕ. Clearly, the operator U is unitary (antiunitary) and it by U (λω + ϕ) = λω is related to the function α according to α(P ) = U P U ∗ for any P ∈ P. Part 5. Let V : H → H be related to α according to α(P ) = V P V ∗ , P ∈ P. By change of phase we may assume that V ω = ω 0 . Let ϕ ∈ M . The operator V has, in particular, the properties (a)–(d) on Oω+ϕ so that V , when restricted on Oω+ϕ , equals with zTω+ϕ for some z ∈ T. But since V ω = ω 0 = zTω+ϕ ω = zω 0 , one has that for any ϕ ∈ M , V |Oω+ϕ = Tω+ϕ , that is, V ϕ = Sϕ on M . Therefore, V equals with U on M , showing that V = U whenever M 6= {0}. In other words, U is unique modulo a phase factor and the unitary or the antiunitary nature of U is completely determined by α ∈ Aut (P) (apart from the trivial case of H being one-dimensional). Moreover, the operator U does not depend on the choice of the vector ω. This ends the proof of the theorem. The content of the Wigner theorem suggests to introduce another group of quantum symmetries. ¯ denote the group of unitary and antiunitary operators on H. It is a Let U ∪ U metrisable second countable topological group with respect to the induced strong operator topology. Let T = {zI|z ∈ T} be the phase group which is the closed ¯ centre of U ∪ U. ¯ Let Σ(H) denote the quotient group U∪ U/T, endowed with the quotient topology. We call it the symmetry group on H and denote its elements by [U ], with ¯ Σ(H) is a metrisable topological group satisfying the second axiom of U ∈ U ∪ U. countability. 4. The Group Isomorphisms We now proceed to show that the groups of automorphisms introduced in Sec. 2 are isomorphic. To work out this plan, we are going to prove that the arrows in the following diagram: Aut x (L)  8

9

−→ 7

Autw (P) ←−

5

4

Aut  (S)  y6

←− Aut (E) ←−

Aut (P)

−→

1

Σ(H)

2

−→

Autx(Br )  3 Aut (B)

are injective group homomorphisms. The diagram contains two loops, one on the right-hand side and one on the left-hand side. We prove that the maps obtained by composing the arrows along both loops are the identity. From this it follows that all the maps are isomorphisms. The arrows between the automorphism groups are natural in the sense that they are defined in terms of some natural relations between the sets which the automorphisms act on. In particular, the arrows 3, 4, 6 and 8 are induced by the inclusions P⊂S P ⊂ L. E ⊂ Br ⊂ B Since a P-automorphism is a weak P-automorphism the arrow 7 is the natural inclusion. The arrows 5 and 9 are based on the duality between B and B1 . The

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arrows 1 and 2 reflect the natural action of the unitary (or antiunitary) group on P and B, respectively. We notice that there are other natural relations giving rise to (a priori) different homomorphisms, for example the inclusion of L in E; however, our choice is motivated by the aim of presenting as simple a proof as possible for the various isomorphisms. We come back to this issue at the end of the section. A particular care is needed to define the arrow 9 since we have to assume that the dimension of H is at least three. This will be clarified by the discussion after Corollary 4.2. We consider first the right-hand side of the diagram. The proofs that the arrows 1, 2 and 3 are injective homomorphisms are immediate. We summarise these results in the following propositions. Proposition 4.1. Any α ∈ Aut (P) defines (via the Wigner theorem) an equivalence class of unitary or antiunitary operators [Uα ] such that α(P ) = Uα P Uα∗ ,

P ∈ P.

The map Aut (P) 3 α 7→ [Uα ] ∈ Σ(H) is an injective group homomorphism. Proposition 4.2. Any [U ] ∈ Σ(H) defines a ΦU ∈ Aut (B) by ΦU (A) = U AU ∗ , A ∈ B. The map Σ(H) 3 [U ] 7→ ΦU ∈ Aut (B) is an injective group homomorphism. Proposition 4.3. Any Φ ∈ Aut (B), when restricted on Br , is a Br -automorphism SΦ . The map Aut (B) 3 Φ 7→ SΦ ∈ Aut (Br ) is an injective group homomorphism. Now we turn to the fourth arrow. Proposition 4.4. Any S ∈ Aut (Br ), when restricted on E, is an E-automorphism fS . The map Aut (Br ) 3 S 7→ fS ∈ Aut (E) is an injective group homomorphism. Proof. By Lemma 2.4 fS is a well-defined bijective map from E onto E. Since S is linear it follows that fS is an E-automorphism. The map S 7→ fS is obviously a group homomorphism. To show that it is injective suppose that fS (E) = E for all E ∈ E. Let A ∈ Br , then S(A) = S(A+ − A− ) = S(A+ ) − S(A− ) A+ A− − kA− kfS = kA+ kfS kA+ k kA− k = A+ − A− , so that S is the identity.

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To define mapping 5 we need a lemma. Lemma 4.1. Let p : E → [0, 1] be a function with the following properties: (1) if E + F ≤ I, then p(E + F ) = p(E) + p(F ), (2) if (Ei )i∈I is an increasing net in E, then p sup Ei = sup p(Ei ) . i∈I

i∈I

Then there is a unique positive trace class operator T such that for all E ∈ E p(E) = tr[T E] . Proof. We notice first that p(E) = p(E + O) = p(E) + p(O), so that p(O) = 0. We prove next that for all E ∈ E and 0 < λ < 1, p(λE) = λp(E) .

(**)

If λ is rational this follows from the additivity of p. Let 0 < λ < 1 and let (rn ) be an increasing sequence of positive rationals converging to λ. Then sup (rn E) = λE n

and this implies that

p(λE) = p sup {rn E} n

= sup p(rn E) n

= λp(E) . We now extend p first to the set of positive operators B+ , defining A + , A ∈ B+ , p (A) = kAkp kAk and then to the set of self-adjoint operators Br , letting 1 + p (A + |A|) − p+ (|A| − A) , A ∈ Br . pˆ(A) = 2 From the additivity of p and from property (**) it follows that pˆ is linear. Moreover, by construction pˆ is positive and it is the unique linear extension of p to Br . The linear map pˆ is, in fact, normal. If (Ai )i∈I is an increasing norm bounded positive net in Br , then, letting c = supi kAi k, ( Aci )i∈I is an increasing net in E and we have Ai pˆ sup Ai = cp sup c i i Ai = c sup p c i = sup pˆ(Ai ) . i

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Hence pˆ is a linear positive normal function on Br . It is well known that pˆ defines a unique positive trace class operator T such that pˆ(A) = tr[T A] ,

A ∈ Br .

Since pˆ is uniquely defined by its restriction p on E the proof is complete. Proposition 4.5. Let f ∈ Aut (E). There is a unique Vf ∈ Aut (S) such that f (P ) = Vf (P ) for all P ∈ P. Moreover , the correspondence Aut (E) 3 f 7→ Vf ∈ Aut (S) is an injective group homomorphism. Proof. Let f ∈ Aut (E). For all T ∈ S define the map from E to [0, 1] by E 7→ tr[T f −1 (E)] . Using now the statement (4) of Lemma 2.3 and Lemma 4.1 there is a positive trace class operator T 0 such that tr[T f −1 (E)] = tr[T 0 E] ,

E ∈ E.

Taking E = I we have tr[T 0 ] = 1, hence T 0 ∈ S. We define Vf from S to S as Vf (T ) = T 0 so that tr[Vf (T )E] = tr[T f −1(E)] ,

E ∈ E.

Using this formula it is straightforward to prove that Vf ∈ Aut (S) and that f 7→ Vf is a group homomorphism. Moreover, suppose that Vf (T ) = T for all T ∈ S, then tr[T (E − f −1 (E))] = 0 ,

E ∈ E, T ∈ S.

Hence E = f −1 (E) for all E ∈ E, that is, f is the identity. This shows the injectivity of the map f 7→ Vf and ends the proof. Finally we have: Proposition 4.6. Any V ∈ Aut (S) restricted to P defines an element αV ∈ Aut (P). The function Aut (S) 3 V 7→ αV ∈ Aut (P) is an injective group homomorphism. Proof. Let V ∈ Aut (S). By Lemma 2.1 its restriction αV on P is well defined and bijective. Let Vˆ be the trace-norm preserving linear extension of V given by Lemma 2.1. Let P1 , P2 ∈ P. A simple calculation shows that p 2 1 − tr[P1 P2 ] = kP1 − P2 k1 = kVˆ (P1 − P2 ) k1 = kV (P1 ) − V (P2 )k1 p = 2 1 − tr[V (P1 )V (P2 )] ,

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so that αV preserves the transition probabilities. The map V 7→ αV is clearly a group homomorphism and its injectivity follows from Lemma 2.1. Let G denote any of the six groups on the right-hand side of the diagram. Starting from G and composing the injective group homomorphisms one obtains an injective group homomorphism φG of G into G. Corollary 4.1. The map φG is the identity on G. Proof. It is sufficient to prove the statement for a particular choice of G. Choosing G = Σ(H) the proof is immediate. In fact, let [U ] ∈ Σ(H); a simple computation shows that the image of [U ] with respect to the composition of the first three homomorphisms is the E-automorphism E 3 E 7→ U EU ∗ ∈ E . Using the properties of the trace this is mapped to the S-automorphism S 3 T 7→ U T U ∗ ∈ S and then to the P-automorphism P 3 P 7→ U P U ∗ ∈ P . The statement follows now from the Wigner theorem. The previous proposition implies that the six injections on the right-hand side of the diagram are all isomorphisms. We stress that this holds without any assumption on the dimension of the Hilbert space. Now we consider the left-hand side. The homomorphism 6 is defined in Proposition 4.6 while the homomorphism 7 is trivial. In fact we have the following statement. Proposition 4.7. The natural immersion Aut (P) ,→ Autw (P) is an injective group homomorphism. The following proposition describes the homomorphism 8. Proposition 4.8. Let α ∈ Autw (P). There is a unique τα ∈ Aut (L) such that τα (P ) = α(P ) for all P ∈ P. Moreover, the map Autw (P) 3 α 7→ τα ∈ Aut (L) is an injective group homomorphism . Proof. Let α ∈ Autw (P). For all M ⊂ H, M 6= {0}, let τα (M ) = {ψ ∈ α([φ]) : φ ∈ M, φ 6= 0} ,

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and put τα ({0}) = {0}. We observe that τα−1 (τα (M )) = Φ ∈ α−1 ([ψ]) : ψ ∈ τα ([φ]), φ ∈ M, φ 6= 0 = Φ ∈ α−1 (α[φ]) : φ ∈ M, φ 6= 0 = CM . In the same way we have that τα (τα−1 (M )) = CM . Let now M ∈ L. We then have τα (M ⊥ ) = τα (M )⊥ . In fact, if φ ∈ M and ψ ∈ M ⊥ are nonzero vectors, then α(P [φ]) ⊥ α([ψ]). Hence τα (M ) ⊥ τα (M ⊥ ) τα (M ⊥ ) ⊂ τα (M )⊥ and, since M = τα−1 (τα (M )), one concludes that τα (M ⊥ ) = τα (M )⊥ . Moreover, since M is a closed subspace, τα (M ) = τα ((M ⊥ )⊥ ) = (τα (M ⊥ ))⊥ , proving that τα (M ) is a closed subspace. We denote by τα the map from L to L sending M to τα (M ). Obviously τα is bijective and preserves the order and the orthogonality, that is, τα ∈ Aut (L). Finally, by construction, τα (P ) = α(P ) for all P ∈ P. A standard calculation shows that the map α 7→ τα is a group homomorphism. The statement 3 of Lemma 2.2 shows that it is also injective. This concludes the proof. We end with the following proposition where the assumption on the dimension of the Hilbert space is essential. Proposition 4.9. Let dim(H) ≥ 3. Given τ ∈ Aut (L) there is a unique Vτ ∈ Aut (S) such that Vτ (P ) = τ (P ) for all P ∈ P. Moreover, the map Aut (L) 3 τ 7→ Vτ ∈ Aut (S) is an injective group homomorphism. Proof. Let τ ∈ Aut (L). Since τ is a lattice orthoisomorphism on L the mapping L 3 M 7→ tr[T τ −1 (M )] ∈ [0, 1] is a generalised probability measure on L for all T ∈ S. According to a theorem of Gleason [8] (which holds if the dimension of H is greater than 2) there is a unique T 0 ∈ S such that tr[T 0 M ] = tr[T τ −1 (M )] for all M ∈ L. The induced function T 7→ T 0 =: Vτ (T ) is one-to-one onto and it preserves the convex structure of S, that is, Vτ ∈ Aut (S). Clearly the map τ 7→ Vτ is a group homomorphism.

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We show now that Vτ (P ) = τ (P ) for all P ∈ P. It is sufficient to prove that tr[Vτ (P1 )P2 ] = tr[τ (P1 )P2 ] ,

P1 , P2 ∈ P .

Since Vτ , restricted to P, is a P-automorphism we have tr[Vτ (P1 )P2 ] = tr[P1 Vτ−1 (P2 )] = tr[P1 Vτ −1 (P2 )] = tr[τ (P1 )P2 ] . Suppose now that Vτ (T ) = T for all T ∈ S. Then, in particular, τ (P ) = P for all P ∈ P so that by Lemma 2.2 τ is the identity. This shows the injectivity of the map τ 7→ Vτ . Similarly to Corollary 4.1 we have the following statement. Corollary 4.2. Let dim H ≥ 3. The composition map of the arrows 6 to 9 is the identity on each group of automorphisms. Proof. We compose the maps starting from Aut (S). Let V ∈ Aut (S). Its restriction αV to P is a (weak) P-automorphism. Hence, by Proposition 4.8, αV defines an L-automorphism ταV such that ταV (P ) = V (P ) for all P ∈ P. Hence the corresponding S-automorphism given by Proposition 4.9 is V on P. From Corollaries 4.1 and 4.2 we conclude that if the dimension of the Hilbert space is greater that two all the injections of the diagram are isomorphisms and all the groups are isomorphic. On the other hand, if the dimension of H is 2, the groups on the right-hand side of the diagram are still isomorphic, while for the left-hand side we will prove that in the diagram Aut (S)

6

−→

Aut (P)

7

−→

Autw (P)

8

−→

Aut (L) ,

the maps 6 and 8 are still surjective while the range of the injection 7 is a proper subset of Autw (P). As a consequence one obtains that the assumption on the dimension of H in Proposition 4.9 cannot be avoided. The fact that the injection 6 is surjective follows directly from Corollary 4.1. The surjectivity of the arrow 8 is the content of the following proposition. Corollary 4.3. The homomorphism α 7→ τα defined in Proposition 4.8 is surjective (without any assumption on the dimension of H). Proof. Let τ ∈ Aut (L). Its restriction to P is a weak P-automorphism since τ preserves orthogonality, hence Proposition 4.8 gives the result.

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The fact that Aut (P) is a proper subset of Autw (P) was first shown by Uhlhorn [3]. The following is a simplified version of his example. Example 4.1. Consider the two dimensional Hilbert space H = C2 . The set P of one-dimensional projections on C2 consists exactly of the operators of the form 12 (I + ~a · ~σ ), where ~a ∈ R3 , k~ak = 1, and ~σ = (σ1 , σ2 , σ3 ) are the Pauli matrices. Therefore, any α : P → P is of the form 12 (I + ~a · ~σ ) 7→ 12 (I + ~a0 · ~σ ) so that α is bijective if and only if ~a 7→ ~a0 =: f (~a) is a bijection on the unit sphere of R3 . Writing ~a = (1, θ, φ), θ ∈ [0, π], φ ∈ [0, 2π] we define a function f such that f (1, θ, φ) = (1, θ, φ) whenever θ 6= π2 and we write f (1, π2 , φ) = (1, π2 , g(φ)), with g(φ) = φ2 /π for 0 ≤ φ ≤ π and g(φ) = (φ − π)2 /π + π for π ≤ φ ≤ 2π. The function α : P → P defined by f is clearly bijective. Using the fact that tr[ 12 (I + ~a · ~σ ) 12 (I + ~b · ~σ )] = 12 (1 + ~a · ~b) one immediately observes that α preserves transition probability zero but not, in general, other transition probabilities. Hence / aut (P). α ∈ Autw (P), but α ∈ We noticed at the beginning of the section that there exist some other natural ways to define isomorphisms between the various groups. However, they lead to the same isomorphism we obtained composing the arrows of the diagram. Consider for instance the following examples. (1) Composing the homomorphisms 5, 6, 7, 8 we obtain an injective group homomorphism from Aut (E) to Aut (L). This is exactly the map induced by the inclusion L ⊂ E. This homomorphism is surjective if and only if the dimension of H is greater than two. (2) Starting from any group Aut(X) one can obtain an isomorphism onto Aut (P) (if X = L we assume dim H ≥ 3). This is the map induced by the inclusion P ⊂ X. 6. Conclusion Using the results of the previous two sections we shall describe the various groups of automorphisms in terms of unitary or antiunitary operators, taking into account also the topological properties. Let X denote one of the sets S, P, E, Br , B or L. In the case X = L we suppose that the dimension of H is greater than 2. We denote by Aut (X) the group of automorphisms of X, endowed with the topology defined in Sec. 2. By the results of Sec. 4 Aut (X) is isomorphic to Σ(H) and for any element χ ∈ Aut (X) there is a unitary or anti-unitary operator, uniquely defined by χ up to a phase factor, such that A ∈ X. χ(A) = U AU ∗ := χU (A), Proposition 6.1. The map jX : Σ(H) 7→ Aut (X) defined as jX ([U ]) = χU ,

¯, U ∈ U∪U

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is a group homeomorphism and Aut (X) is a second countable, metrisable, topological group. Proof. Taking into account that the topological group Σ(H) is second countable and metrisable, the only fact to be proven is that the map jX is a homeomorphism. ¯ → Aut (X), U 7→ JX (U ) := We demonstrate first that the function JX : U ∪ U ¯ χU is continuous. Since U∪ U is second countable, it suffice to show that if (Un )n≥1 ¯ then (JX (Un ))n≥1 is convergent in is a (strongly) convergent sequence in U ∪ U, ¯ we have, for instance for X = S, Aut (X). As U 7→ U −1 is continuous in U ∪ U, S (JS (Un )) = lim tr[AJS (Un )(T )] lim fA,T

n→∞

n→∞

= lim tr[AUn T Un−1 ] n→∞

S (U ) , = tr[AU T U −1 ] = fA,T

for all A ∈ Br , T ∈ S, which shows the continuity of JS . The other cases are shown as well. By definition of quotient topology, this proves also that jX is continuous. −1 is continuous. Consider It remains to be shown that the inverse mapping jX of unit vectors in H. Since P the group Aut (X) and let (ϕi )i≥1 be a dense sequence is contained in X, then the sequence of functions fPX[ϕi ],P [ϕj ]

gives Aut (X) i,j≥1

a metrisable topology, which a priori is weaker than the one defined above for −1 is continuous in this weaker topology. It suffices Aut (X). We shall show that jX again to consider only sequences. Let (γn ) be a convergent sequence in Aut (X), with −1 −1 (γn ) → jX (γ) in Σ(H). To proceed assume on the γn → γ. We will show that jX −1 contrary that jX is not continuous so that there is an open set O ⊂ Σ(H) such that ¯ / O for a subsequence (γn ) of (γn ). Let Uk , U ∈ U ∪ U j −1 (γ) ∈ O but j −1 (γn ) ∈ X

X

k

k

such that jX ([Uk ]) = γnk and jX ([U ]) = γ. The sequence (Uk ) is bounded, so that ¯ with Uk → V . But then it has a weakly convergent subsequence (Ukh ) in U ∪ U, h 2 2 tr[P [ϕi ]γnkh (P [ϕj ])] = |hϕi |Unkh ϕj i| → |hϕi |V ϕj i| and tr[P [ϕi ]γnkh (P [ϕj ])] → tr[P [ϕi ]γ(P [ϕj ])] = |hϕi |U ϕj i|2 , which shows that [V ] = [U ]. Since Unkh → V also strongly we thus have [Unkh ] → [V ] = [U ] which is a contradiction. This shows that −1 : Aut (X) → Σ(H) is continuous. This ends the proof. jX References [1] S. Weinberg, The Quantum Theory of Fields, Vol I, Cambridge Univ. Press, Cambridge, USA, 1995, Appendix A, pp. 91–96. [2] E. P. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum, Fredrick Vieweg und Sohn, Braunschweig, Germany, 1931, pp. 251– 254; Group Theory and Its Application to the Quantum Theory of Atomic Spectra, Academic Press Inc., New York, 1959, pp. 233–236. [3] U. Uhlhorn, Arkiv Fysik 23 (1962) 307. [4] V. Bargmann, J. Math. Phys. 5 (1964) 862. [5] V. S. Varadarajan, Geometry of Quantum Theory, Vol. I, D. Van Nostrand Co. Inc., New York, 1968, Geometry of Quantum Theory, second edition, Springer-Verlag, Berlin, 1985.

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[6] B. Simon, “Quantum dynamics: from automorphism to hamiltonian”, in Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann, eds. E. H. Lieb, B. Simon, A. S. Wightman, Princeton Series in Physics, Princeton University Press, Princeton, New Jersey 1976, pp. 327–349. [7] G. Ludwig, Foundations of Quantum Mechanics, Vol. I, Springer Verlag, New York, 1983. [8] A. M. Gleason, J. Math. Mech. 6 (1957) 567.

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR: WHAT IS A CLASSICAL CONFIGURATION?∗ T. SCHMITT Ackerstr. 11, 10115 Berlin Germany E-mail: [email protected] Received 14 October 1996 We discuss the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the “space” of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M . We discuss the two main approaches to supermanifolds, and we examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin–Kostant–Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the superfunctionals considered in [44] are nothing but superfunctions on M . We propose a programme for future mathematical work, which applies to any classical field model with fermion fields. A part of this programme will be implemented in the successor paper [45].

Contents Introduction 1. Supergeometry and Its Relationship to Quantum Field Theory 1.1. Geometric models of quantum field theory and classical configurations 1.2. Remarks on the history of supergeometry 1.3. A short account of the Berezin–Leites–Kostant approach 1.4. Supergeometry and physicists 1.5. Supergeometry and hermitian conjugation 1.6. Sketch of the deWitt–Rogers approach 1.7. From Berezin to deWitt–Rogers 1.8. Molotkov’s approach and the role of the algebra of constants 1.9. Comparison of the approaches 1.10. B-valued configurations 1.11. Families of configurations 1.12. Classical configurations, functionals of classical fields, and supermanifolds 1.13. Example: Fermions on a lattice 2. Infinite-Dimensional Supermanifolds 2.1. Formal power series 2.2. Analytic power series 2.3. Insertions 2.4. Superfunctions 2.5. Superfunctions and ordinary functions 2.6. Superdomains 2.7. Supermanifolds 2.8. Morphisms of smf’s 2.9. Supermanifolds and manifolds

∗ Special

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thanks to the late German Democratic Republic who made this research possible by continuous financial support over ten years. 993

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2.10. Cocycle description; products 2.11. Comparison with finite-dimensional Berezin smfs 2.12. Sub-supermanifolds 2.13. Example: the unit sphere of a super Hilbert space References

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Introduction Although this paper logically continues [44], it can be read independently. We begin with a discussion of the conceptual difficulties connected with the anticommutativity of classical (=unquantized) fermion fields, and we argue that only a supergeometric approach allows a convincing description of these fields; indeed, the “space” of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold. Before explicating this, we discuss the two main approaches to supermanifolds, starting with the approach of Berezin, Leites, Kostant, et al. together with the hermitian modification proposed in [39]. We then review the second, alternative approach proposed by B. deWitt and A. Rogers and followed by quite a few authors and its connection to the first one as well as the “interpolating” approach of Molotkov ([29]), and we examine the reasons why many physicists tend to prefer it although the Berezin–Kostant–Leites approach is the more fundamental one. In particular, we show that the deWitt–Rogers supermanifolds suffer from the same conceptual shortcomings as the naive configuration notion in classical field theory with anticommuting fields. The deWitt–Rogers approach is intimately connected with an “interim solution” of the conceptual difficulties mentioned: one considers “B-valued configurations”, i.e. instead of the usual real or complex numbers, one uses a Grassmann algebra B (or some relative of it) as target for the classical fields, providing in this way the apparently needed “anticommuting values”. However, this approach is suspicious for the very same reasons as the deWitt–Rogers supermanifold approach; in particular, in view of the arbitrariness in the choice of B. Therefore we go one step beyond and re-interpret B-valued configurations as families of configurations parametrized by supermanifolds. Once this is done, it is natural to state the question for a universal family and a corresponding moduli space — an idea which directly leads to the infinite-dimensional configuration supermanifold mentioned. Hence, we have to develop the infinite-dimensional variant of the Berezin– Kostant–Leites approach, basing on previous work of the author. In fact, we will show that the superfunctionals considered in [44] are nothing but superfunctions on an infinite-dimensional supermanifold. If the model is purely bosonic then a superfunctional is an ordinary functional, i.e. a function the domain of definition of which is a function space. Generalizing the philosophy of the paper [5] from quantum mechanics onto quantum field theory, we propose a programme for future mathematical work, which applies to any classical field model with fermion fields, and which aims at understanding its mathematical structure. A part of this programme is implemented in the successor paper [45].

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1. Supergeometry and Its Relationship to Quantum Field Theory 1.1. Geometric models of quantum field theory and classical configurations The modern realistic quantum field theoretic models, like quantum electrodynamics, Salam–Weinberg electroweak theory, quantum chromodynamics, and all these unified in the standard model, have achieved remarkable success in predicting experimental results, in one case even with an accuracy up to 11 digits. Even for a mathematician, which turns up his nose about the mathematical and logical status of these models, this should be strong evidence that these models have something to do with nature, and that they should be taken seriously. Even if it should turn out that they do not produce rigorous quantum theories in Wightman’s sense, they have proven to give at least a kind of semiclassical limit plus quite a few quantum corrections of a more fundamental theory (provided the latter exists), and therefore it remains a promising task to clear up the mathematical structure, and to put it into a consistent framework as far as possible. Moreover, there exist phenomenological models in hadron and nuclear physics which are rather unlikely to be implementable as rigorous quantum theories, due to non-renormalizability; nevertheless, they give information, and thus should be taken serious. The models mentioned start with formulating a classical field theory of fields on Minkowski space R4 , and the hypothetical final theory is thought to arise by quantization of the classical theory in a similar sense as quantum mechanics arises by quantization of point mechanics. Of course, the concept of quantization is problematic even in the latter situation; but Kostant–Souriau geometric quantization gives at least one mathematically well-defined procedure. In the field theoretic situation, where we have infinitely many degrees of freedom, geometric quantization in its usual form does no longer apply, and there arises a gap between mathematics and physics: The models which have been constructed rigorously (like the P (Φ)d models, the Thirring model, the Gross–Neveu model (cf. [31, 19] and references therein)), are only toy models in small space-time dimensions, while the realistic models mentioned above are treated up to now only heuristically (this statement is not affected by the fact that some special features have been successfully treated by rigourous mathematical methods, like e.g. the instanton solutions of euclidian pure Yang–Mills theory). A popular device to tackle the field quantization problem heuristically is Feynman’s path integral. Inspite of the fact that only some aspects are mathematically understood (cf. e.g. Osterwalder–Schrader axiomatics a la Glimm–Jaffe [17]), and that only the bosonic part can be understood as integration over a measure space, it has been very successful in deriving computation rules by formal manipulation like variable changing, integration by parts, standard Gaussian integrals. Now the path integral is supposed to live on the space M of classical configurations Ξ of the model: under some restrictions onto the Lagrangian density L[Ξ], R “each one” of them enters formally with the weight exp(i d4 x L[Ξ](x)/~) (or, in the euclidian variant, minus instead of the imaginary unit).

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Now, if fermion fields enter the game then a complication arises (and this applies in the standard models both to the matter fields and to the ghost fields in the Faddeev–Popov scheme): Any classical fermion field Ψα has to be treated as anticommuting, (1.1.1) [Ψα (x), Ψβ (y)]+ = 0 (x, y ∈ R4 are space-time points; physicists use to say that Ψα (x) is an “a-number”), and this makes the concept of a “classical configuration” for fermion fields problematic. (“Fermion” refers always to the statistics, not to the spin. The difference matters for ghost fields, which do not obey the Pauli Theorem.) In [3], Berezin developed a recipe to compute Gaussian functional integrals over fermions in a heuristic way (actually, his recipe is mathematically precise but applies only to rather “tame” Gaussian kernels; however, the practical application is heuristic because the kernels are far from being “tame”). Thus, in the standard models, one can often circumvent the problem mentioned above by “throwing out” the fermions by Gaussian integration, so that only bosonic configurations remain. (However, the Faddeev–Popov scheme for the quantization of gauge theories does the opposite: after some formal manipulations with the bosonic functional integral, a determinant factor appears which is represented by a fermionic Gaussian integral. The rest is Feynman perturbation theory.) On the other hand, the problem of understanding the notion of a classical configuration becomes urgent in supersymmetric models, because — at least in component field formulation — supersymmetry simply does not work with the naive notion of configuration with commuting fields, and the computations of physicists would not make sense. Only anticommuting fields behave formally correct. Also, in the mathematical analysis of the BRST approach to the quantization of constrained systems (cf. e.g. [24]), the fermionic, anticommuting nature of the ghost degrees of freedom (ghost fields in the context of Yang–Mills theory) has to be taken into account. Contrary to naive expectations, the law (1.1.1) is not automatically implemented by considering superfield models on supermanifolds: The components fµ (x) of a superfunction X f (x, ξ) = fµ (x)ξ µ are still R- (or C-) valued and therefore commuting quantities while e.g. for the chiral superfield (cf. e.g. [51, Ch. V], therein called “scalar superfield”) Φ(y, θ) = A(y) + 2θψ(y) + θθF (y) ,

(1.1.2)

the Weyl spinor ψ(x), which describes a left-handed fermion, has to satisfy the law (1.1.1) — at least if non-linear expressions in ψ have to be considered. Thus, in the framework of superfield models, it still remains as obscure as in component field models how to implement (1.1.1) mathematically. Nevertheless, we will see that it is just supergeometry which is the key to an appropriate solution of this problem. But before establishing the link, we review the history and the main approaches to supergeometry.

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Unfortunately, this problem has been mostly neglected in the existing literature both on super differential geometry and on mathematical modelling of field theories with fermions. In the worst case, fermions were modelled by ordinary sections in spinor bundles, like e.g. in [12]. (Of course, this is okay as long as all things one does are linear in the fermion fields. For instance, this applies to the vast literature on the Dirac equation in a gauge field background. On the other hand, if e.g. the first Yang–Mills equation with spinor current source or supersymmetry transformations have to be considered then this approach becomes certainly inconsistent.) 1.2. Remarks on the history of supergeometry “Supermathematics”, and, in particular, super differential geometry, has its roots in the fact (which is familiar and unchallenged among physicists, although not seldom ignored by mathematicians) that in the framework of modern quantum field theory, which uses geometrically formulated field theories, fermions are described on a classical level by anticommuting fields. Indeed, practical experience in heuristic computations showed that if keeping track of the signs one can handle anticommuting fields formally “in the same way” as commuting (bosonic) ones. The most striking example for this is the Feynman path integral, or its relative, the functional integral, for fermion fields; cf. [3]. This experience led F. A. Berezin to the conclusion that “there exists a non-trivial analogue of analysis in which anticommuting variables appear on equal footing with the usual variables” (quotation from [4]), and he became the pioneer in constructing this theory. The book [4] which summarizes Berezin’s work in this field, reflects supergeometry “in statu nascendi”. The papers [23, 25], and the book [26] give systematic, methodologically closed presentations of the foundations of supergeometry; cf. [25] also for more information on the history of the subject. In the meantime, supergeometry turned to more special questions, in particular, to the investigation of super-analogues of important classical structures. As examples from the vast literature, let us mention Serre and Mekhbout duality on complex supermanifolds [30], Lie superalgebras and their representations (see e.g. [21, 35]), integration theory on supermanifolds [6, 7, 33], deformations of complex supermanifolds [49], integrability of CR structures [37, 38], the investigation of twistor geometry and Penrose transform in the super context ([27] and references therein). (Of course, the quotations are by no means complete.) Manin’s book [27] seems to be one of the most far-reaching efforts to apply supergeometry to classical field models in a mathematical rigorous way. Also, one should mention the “Manin programme” [28] which calls for treating the “odd dimensions” of super on equal footing with the ordinary “even dimensions” as well as with the “arithmetic” or “discrete dimensions” of number theory, and for considering all three types of dimensions in their dialectic relations to each other. Now if speaking on the history of supergeometry one should also mention the alternative approach of B. deWitt and A. Rogers to supermanifolds, which was followed (and often technically modified) by several authors (cf. e.g. [20, 50, 14]),

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and the theory of Molotkov [29] which in some sense unifies both approaches in an elegant way and allows also infinite-dimensional supermanifolds. We will comment this below. 1.3. A short account of the Berezin Leites Kostant approach Roughly speaking, a finite-dimensional supermanifold (smf ) X is a “geometrical object” on which there exist locally, together with the usual even coordinates x1 , . . . , xm , odd, anticommuting coordinates ξ1 , . . . , ξn . In order to implement this mathematically, one defines a (smooth) superdomain U of dimension m|n as a ringed space U = (space(U ), O∞ ) where space(U ) ⊆ Rm is an open subset, and the structure sheaf is given by ∞ O∞ (·) := C (·) ⊗R R[ξ1 , . . . , ξn ]

(1.3.1)

where ξ1 , . . . , ξn is a sequence of Grassmann variables. Thus, O∞ (V ) for open V ⊆ U is the algebra of all formal sums X X fµ (x)ξ µ (1.3.2) f (x|ξ) = fµ1 ···µn (x)ξ µ1 · · · ξ µn = where the sum runs over all 2n tuples µ = (µ1 , . . . , µn ) ∈ Zn2 , and the fµ (x) are smooth functions V → R. One interprets (1.3.2) as the Taylor expansion of f (x|ξ) w. r. to the Grassmann variables; due to their anticommmutativity, it terminates after 2n terms. Here and in the following, we use the subscript ∞ at the structure sheaf in order to emphasize that we are considering the smooth variant of the theory; the symbol O without subscript is reserved to complex- and real-analytic structure sheaves, as already considered in [44, 3.5]. Note that in the finite-dimensional situation, the definitions of the complex- and real-analytic variants of (1.3.1) are straightforward. The “globality theorems” like [23, Props. 2.4.1, 2.4.2, etc.], or the quirk of [23] of defining morphisms by homomorphisms of the algebra of global superfunctions, do no longer work in these situations. However, if one instead follows the standard framework of ringed spaces there is no genuine obstacle to a satisfactory calculus. Although the use of a real Grassmann algebra in (1.3.1) looks quite natural, it is in fact highly problematic; we will discuss this in 1.5 below. Now one defines a supermanifold (smf ) as a ringed space X = (space(X), O∞ ) with the underlying space paracompact and Hausdorff which is locally isomorphic to a superdomain. (Kostants definition is easily seen to be equivalent to this one.) Morphisms of smfs are just morphisms of ringed spaces. We will not mention here the various differential-geometric ramifications; in Sec. 2, we will develop the infinite-dimensional variant of some of them. We only note that every smf determines an underlying C ∞ manifold, and that, conversely, every C ∞ manifold can be viewed as smf. For later use we note also that to every finite-dimensional Z2 -graded vector space V = V0 ⊕ V1 there belongs a linear superspace (often called also “affine superspace”, in analogy with the habits of algebraic geometry) L(V ) which has the even part V0

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as underlying space and C ∞ (·) ⊗ ΛV1∗ as structure sheaf. Here ΛV1∗ is the exterior algebra over the dual of the odd part; due to the dualization, one has a natural embedding of the whole dual into the global superfunctions, V ∗ ⊆ O∞ (L(V )) ,

(1.3.3)

and V ∗ is just the space of linear superfunctions. This procedure is just a superization of the fairly trivial fact that every finite-dimensional vector space can be viewed as smooth manifold. It is of outstanding importance that a morphism from an arbitrary smf X to a linear smf L(V ) is known by knowing the coordinate pullbacks: Fix a basis u1 , . . . , uk ∈ V0∗ , ξ1 , . . . , ξl ∈ V1∗ of the dual V ∗ ; because of (1.3.3), these elements become superfunctions u1 , . . . , uk , ξ1 , . . . , ξl ∈ O∞ (L(V )). It is reasonable to call this tuple a coordinate system on L(V ), and we have (cf. [25, Theorem 2.1.7], [36, Theorem 3.1], [39, Corollary 3.3.2]): Theorem 1.3.1. Given superfunctions u01 , . . . , u0k ∈ O∞ (X)0 , ξ10 , . . . , ξl0 ∈ O∞ (X)1 there exists a unique morphism µ : X → L(V ) such that µ∗ (ui ) = u0i , µ∗ (ξj ) = ξj0 for all i, j. This Theorem makes the morphisms between the local models controllable. One can give it a more invariant form: Corollary 1.3.2. Given an smf X, a finite-dimensional Z2 -graded vector space V, and an element µ ˆ ∈ (O∞ (X)⊗V )0 there exists a unique morphism µ : X → L(V ) ˆ i = µ∗ (v) within O∞ (X). such that for any v ∈ V ∗ we have hv, µ In this form, it will also generalize to the infinite-dimensional situation (cf. Theorem 2.8.1 below); however, the real-analytic analogon O (·)⊗V of the sheaf O∞ (·)⊗V will have to be replaced by a bigger sheaf O V (·) of “V -valued superfunctions”. Remark. Several authors, like e.g. [29], consider generalized frameworks based have to be apon a ZN 2 grading, so that the parity rule as well as the first sign rule L -graded algebra R = plied to each degree | · |j separately. Thus, a ZN 2 (i1 ,...,iN )∈ZN 2 N R(i1 ,...,iN ) is Z2 -commutative iff ba = (−1)|a|1 |b|1 +...|a|N |b|N ab for homogeneous a, b ∈ R. Some people claim that the use of such a structure is appropriate in order to separate physical fermion field d. o. f.’s from ghost field d. o. f.’s. However, as observed e.g. in [18, Ex. 6.15], one can work without such a generalization, thus saving at least some notations: One may construct from R a Z2 -graded algebra Rs by taking the total degree, M (Rs )k := R(i1 ,...,iN ) (i1 ,...,iN )∈ZN 2 : i1 +...+iN ≡k mod 2

1000

for k = 0, 1; thus |a|s =

T. SCHMITT

P

|a|i . Also, one modifies the multiplication law by P |a| |b| a ·s b := (−1) i>j i j ab

i

for homogeneous elements. This new product is again associative; if R was ZN 2 -commutative then Rs is Z2 -commutative. Analogous redefinitions can be made for modules and other algebraic structures, so that there is no real necessity to consider such generalized gradings. In particular, instead of constructing the exterior differential on an smf as even operator (cf. e.g. [23]), which leads to a Z2 × Z+ -graded commutative algebra of differential forms, it is more appropriate to use an odd exterior differential. This leads to an algebra of differential forms which is still Z2 × Z+ -graded but where only the Z2 degree produces sign factors. Such an approach does not only simplify computations but it opens the way for a natural supergeometric interpretation of exterior, interior, and Lie derivatives as vector fields on a bigger smf, and it leads straightforwardly to the Bernstein–Leites pseudodifferential forms. Cf. [6, 36, 42]. 1.4. Supergeometry and physicists Supergeometry has both its sources and its justification mainly in the classical field models used in quantum field theory. Nowadays it is most used for the formulation and analysis of supersymmetric field models in terms of superfields which live on a supermanifold (cf. e.g. [51, 27]). However, most of the physicists who work in this field either do not make use of the mathematical theory at all, or they use only fragments of it (mainly the calculus of differential forms and connections, and the Berezin integral over volume forms). The first and most obvious reason for this is that one can do formulation and perturbative analysis of superfield theories without knowing anything of ringed spaces α ˙ (one has to know how to handle superfunctions f (xµ , θα , θ ) which depend on comα ˙ muting coordinates xµ and anticommuting coordinates θα , θ ; for most purposes it is inessential how this is mathematically implemented). The second reason is fairly obvious, too: Despite the invasion of manifolds, fibre bundles, cohomology, Chern classes etc. into mathematical physics — the work with e.g. morphisms (instead of “genuine maps” of sets) or group objects (instead of “genuine groups”) is not easy even for the mathematically well-educated physicist. However, the difficulties are of purely psychological nature; the algebraic geometer has to deal with quite similar (and sometimes still far more abstract) structures since Grothendieck’s revolution in this area (one should mention here that also C ∞ supergeometry has learned a lot from algebraic geometry). But there are also two further, less obvious reasons. One of them is a certain conceptual shortcoming of Berezin’s theory concerning the treatment of complex conjugation, which made it almost inapplicable to physical models. In [39], the present author made a proposal to modify this theory in order to adapt it to the needs of quantum physics; cf. 1.5 below for a short account. Finally, the fourth, and perhaps most severe reason is the problem of modelling classical fermion fields mentioned above. In particular, the Berezin–Kostant–Leites

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approach does not provide anticommuting constants, which seem to be indispensable (but are, as we will see in 1.11, in fact not). Although supergeometry is usually considered to be merely a tool for the formulation of supersymmetric models in terms of superfields on supermanifolds, its conceptual importance is much wider: while the xi describe usual, “bosonic” degrees of freedom the ξj describe new, “fermionic” degrees of freedom which are not encountered in classical mechanics. In fact, supergeometry is the adequate tool to describe the classical limit of fermionic degrees of freedom. This point of view is pursued also in the paper [5] of Berezin and Marinov. However, there seems to be a minor conceptual misunderstanding in that paper which we want to comment on: The authors argue that one can describe the algebra of classical observables of a spinning electron in an external field as a Grassmann algebra R = C[ξ1 , ξ2 , ξ3 ] (or, more precisely, as the even, real part of R). Time evolution of observables f ∈ R is governed by df /dt = {H, f } where {·, ·} is a Poisson bracket making R a Lie superalgebra, and H ∈ R0 is the Hamiltonian. In supergeometric language, the phase space is the 0|3-dimensional hermitian linear supermanifold (cf. 1.5) L(R0|3 ) equipped with the symplectic structure ω = (dξ1 )2 + (dξ2 )2 + (dξ3 )2 which induces the Poisson bracket in R = O∞ (L(R0|3 )). Now the notion of “phase space trajectory” introduced in [5, 2.1] as a smooth map R → R1 , t 7→ ξ(t), leads to confusion. Indeed, t 7→ ξ(t) describes the time evolution of the observable ξ(0) rather than that of a state (that is, we have a “Heisenberg picture” at the unquantized level). In accordance with this, for the “phase space distribution” ρ of [5, 2.2], there is no sensible notion of a δ-distribution which would indicate a “pure state”. Thus, it should be stressed that in this approach — as well as in our one to be developed below — there do not exist “individual configurations” on the unquantized level. With this point of view on [5], one may say that the programme presented in Sec. 1.12 below is the logical extension of the philosophy of [5] from classical mechanics onto classical field theory. For a systematic (but rather formal) treatment of supermanifolds as configuration spaces of mechanical systems, cf. the book [18] and references therein. In 1.13, we will see how supermanifolds emerge naturally as configuration spaces of lattice theories; the advantage of those is that the finite-dimensional calculus is sufficient. Now if we consider a field (instead of, say, a mass point) then we will have infinitely many degrees of freedom. Roughly speaking, the bosonic and fermionic field strengthes Φi (x), Ψj (x) for all space-time points x are just the coordinates of the configuration space. The appearance of anticommuting functions on the configuration space indicates that the latter should be understood as an infinite-dimensional supermanifold. We will elaborate this philosophy from 1.11 on. However, the use of anticommuting variables is not restricted to the unquantized theory: “Generating functionals” of states and operators in Fock space were invented by F. A. Berezin [3] thirty years ago. While in the purely bosonic case

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they are analytic functions on the one-particle state space H, their geometric interpretation remained up to now obscure on the fermionic side: they are just elements of an “infinite-dimensional Grassmann algebra” the origin of which was not clear. Infinite-dimensional supergeometry provides a satisfactory solution of the riddle: the Grassmann algebra mentioned above is nothing but the algebra of superfunctions on the 0|∞-dimensional supermanifold L(H). More generally, supergeometry allows to unify bosonic and fermionic case, instead of considering them separately, as Berezin did. Cf. [43] for a sketch; a detailed presentation will appear elsewhere. Another natural appearance of supergeometry is in quantized euclidian theory, which is given by its Schwinger functions, i.e. the vacuum expectation values of the quantized fields: S I|J (X|Y ) = S i1 ,...,ik |j1 ,...,jl (x1 , . . . , xk |y1 , . . . , yl ) ˆ i (xk )Ψ ˆ j1 (y1 ) · · · Ψ ˆ j (yl )ivac ˆ i1 (x1 ) · · · Φ : = hΦ k l Z 1 [DΦ][DΨ]Φi1 (x1 ) · · · Φik (xk ) · Ψj1 (y1 ) · · · Ψjl (yl ) = N −1 Seuc [Φ|Ψ] . × exp ~ Here Seuc Φ|Ψ is the euclidian action which depends on bosonic fields Φi and fermionic ones Ψj (for notational simplicity, we use real field components). Usually, one assumes the Schwinger functions to be tempered distributions defined on the whole space, S I|J ∈ S 0 (Rd(k+l) ), and satisfying the Osterwalder– Schrader axioms (cf. [8, 9.5.B]). The Schwinger functions now are the coefficient functions (cf. [44, 2.3]) of the Euclidian generating functional Z X 1 X dXdY S I|J (X|Y ) Zeuc [JΦ |JΨ ] = k!l! i ,...,i ,j ,...,j k,l≥0

× =

1 N

1

1

l

Φ Ψ Ψ JΦ i1 (x1 ) · · · Jik (xk )Jj1 (y1 ) · · · Jjl (yl )

Z [DΦ][DΨ] exp Z

+

k

dx

X i

−1 Seuc [Φ|Ψ] ~

JΦ i (x)Φi (x) +

X

!! JΨ j (x)Ψj (x)

,

j

where the functional variables JΦ , JΨ are (formal) “external sources”; note that JΦ , JΨ are commuting and anticommuting, respectively. In the purely bosonic case, Zeuc [JΦ ] : S(Rd ) ⊗ V ∗ → C is the characteristic function of a measure on the euclidian configuration space S 0 (Rd ) ⊗ V of the theory; here V is the field target space as in [44, 2.2]. However, if fermionic fields are present then, due to the antisymmetry of the Schwinger functions in the fermionic sector, Zeuc [JΦ |JΨ ] cannot be interpreted as a map any longer. Instead of this, it is at least a formal power series in the sense of [44, 2.3], and thanks to the

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usual regularity condition on the Schwinger functions, it becomes a superfunction Zeuc[JΦ |JΨ ] ∈ O (L(S(Rd ) ⊗ V ∗ )). (It is a challenging task to construct some superanalogon of measures, so that Zeuc[JΦ |JΨ ] becomes the characteristic superfunction of a supermeasure on the euclidian configuration supermanifold L(S 0 (Rd ) ⊗ V ) of the theory.) Apart from the last remark, analogous remarks apply also to the (time-ordered) Green functions in Minkowski theory. Note, however, that, although their existence is generally assumed, it does not follow from the Wightman axioms; cf. also [8, 13.1.A]. 1.5. Supergeometry and hermitian conjugation Here, we give a short account of [39]. Usually, one uses the sign rule in the following form: Sign Rule. Whenever in a multilinear expression standing on the r.h.s. of an equation two adjacent terms A, B are interchanged (with respect to their position on the r.h.s.) a sign (−1)|A||B| occurs. This rule has is origin in the commutation rule of Grassmann algebras, and it is well known wherever these are applied, e.g. in supergeometry as well as in homological algebra, differential geometry (in treating exterior forms), and algebraic topology. However, the law of operator conjugation in the quantized theory, (AB)∗ = ∗ ∗ B A yields in the classical limit the rule f g = gf for superfunctions f, g on a supermanifold. This seems to contradict the sign rule, and the best way out of the trouble is the following: One applies the sign rule only to complex multilinear expressions. In fact, one avoids the use of merely real-linear terms; that is, all vector spaces the elements of which appear in multilinear terms should be complex. With this first step, the contradiction is resolved, but there remains uncertainty. Thus, we have to do more. Skew-linearity should appear only in the form of explicit hermitian conjugation, and the latter is treated by: Second Sign Rule. If conjugation is applied to a bilinear expression containing the adjacent terms a, b (i.e. if conjugation is resolved into termwise conjugation), either a, b have to be rearranged backwards, or the expression acquires the sign factor (−1)|a||b| . Multilinear terms have to be treated iteratively. These rules have consequences for the very definition of supermanifolds: In a natural way, one defines a hermitian ringed space as a pair X = (space(X), O∞ ) where space(X) is a topological space, the structure sheaf O∞ is now a sheaf of complex Z2 -graded algebras which are equipped with a hermitian conjugation, i.e. an antilinear involution − : O∞ → O∞ which satisfies f g = gf

for f, g ∈ O∞ .

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Now one defines a hermitian superdomain U of dimension m|n as a hermitian ringed space U = (space(U ), O∞ ) where space(U ) ⊆ Rn is open again but ∞ O∞ (·) = CC (·) ⊗C C[ξ1 , . . . , ξn ]

where ξ1 , . . . , ξn is again a sequence of Grassmann variables. Thus, O∞ (V ) for open V ⊆ U is the algebra of all formal sums (1.3.2) where, however, the smooth functions fµ (x) are now complex-valued. The hermitian conjugation is defined by the properties f = f for f ∈ CR∞ , ξ i = ξi for all i. Thus f (x|ξ) =

X

(−1)|µ|(|µ|−1)/2 fµ (x)ξ µ .

(1.5.1)

Now one defines a hermitian supermanifold as a hermitian ringed space with the underlying space paracompact and Hausdorff which is locally isomorphic to a hermitian superdomain. Note that the definition of the linear superspace belonging to a finite-dimensional Z2 -graded vector space V = V0 ⊕ V1 also changes: ∗ L(V ) = (V0 , CC∞ (·) ⊗C ΛV1,C ).

Theorem 1.3.1 slightly changes because the coordinate pullbacks now have to be required real: u01 , . . . , u0k ∈ O∞ (X)0,R , ξ10 , . . . , ξl0 ∈ O∞ (X)1,R . Also, in Corollary 1.3.2, the element µ ˆ has to be required real: µ ˆ ∈ (O∞ (X) ⊗ V )0,R . Cf. [39] for the rest of the story. Remark. It is interesting to note that among the people who worked on supercalculus questions, DeWitt was one of the few who did not walk into the trap of “mathematical simplicity” concerning real structures: Although in his book [14] he does not formulate the second sign rule, he actually works in the hermitian framework instead of the traditional one from the beginning. In particular, his “algebra of supernumbers” Λ∞ is a hermitian algebra (cf. [44, 2.1] or [39]), and the “supervector space” introduced in [14, 1.4] is the same as a free hermitian module over Λ∞ . Also, the book [18], being concerned with both classical and quantum aspects of ghosts and physical fermions, uses in fact a hermitian approach. 1.6. Sketch of the deWitt Rogers approach The difficulties mentioned motivated physicists to look for an alternative approach to the mathematical implementation of supermanifolds. This approach was pioneered by B. deWitt [14] and A. Rogers [32], and it was followed (and often technically modified) by several authors (cf. e.g. [20, 50]). Cf. the book [9] for more on the history. The basic idea consists in realizing m|n-dimensional superspace as the topological space B0m × B1n , where B = B0 ⊕ B1 is a suitable topological Z2 -commutative algebra. Suitability is defined differently in each version of the theory, but for the present discussion, the differences do not matter much. Here we follow the original

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paper of Rogers [32], that is, B = Λ[β1 , . . . βk ] is a finite-dimensional Grassmann P P |cµ |, it becomes a Banach algebra. Equipping B with the norm k cµ β µ k := algebra. (We note that, from the point of view of the Berezin approach, it looks suspicious that odd things appear as geometrical points. One should really work better with the purely even object B0m × (ΠB1 )n where Π is the parity shift symbol. In this form, m|n-dimensional superspace will naturally appear also in 1.7 below.) Now a map f : U → B where U ⊆ B0m × B1n is open is called superdifferentiable iff there exist maps Di f : U → B such that P kf (x + y) − f (x) − yi Di f (x)k =0 lim kyk for x = (x1 , . . . , xm+n ) ∈ U and y → 0 in B0m × B1n . The problem with this definition is that, while the even partial derivatives D1 f, . . . , Dk f are uniquely determined, the odd ones, Dm+1 f, . . . , Dm+n f are determined only up to a summand of the form g(x)β1 · · · βk where the map g : U → R is arbitrary (!). This is the first appearance of the “truncation effects” which are typical for finite-dimensional B. They can be avoided by using infinite-dimensional Grassmann algebras, but only at the price of other technical and conceptual complications; so we stick here to k < ∞. Let G∞ (U ) be the algebra of all infinitely often superdifferentiable functions on U . More exactly, one defines inductively Gl (U ) by letting G0 (U ) be just the set of all continuous maps U → B, and Gl+1 (U ) be the set of all f : U → B which are superdifferentiable such that all Di f can be choosen in Gl (U ); finally, T G∞ (U ) := l>0 Gl (U ). In an obvious way, G∞ (U ) is a Z2 -graded commutative algebra over B. Note that since B is a 2k -dimensional vector space, f encodes 2k real-valued functions on RN with N := 2k−1 (m + n); and superdifferentiability requires not only that these functions are differentiable but that their derivatives satisfy a certain linear system of relations, somewhat analogous to the Cauchy–Riemann relations in function theory. The expansion (1.6.1) below is a consequence of these relations. The most obvious example for a superdifferentiable function is the projection ole of the ith xi : B0m × B1n → B from the ith factor; xi ∈ G∞ (U ) will play the rˆ coordinate. Sometimes we will have to treat the even and the odd ones separately; so we will also write (u1 , . . . , um , ξ1 , . . . , ξn ) := (x1 , . . . , xm+n ). Now let : B → R be the unique algebra homomorphism; thus, for every a ∈ B, its “soul” a − (a) is nilpotent, and a is invertible iff its “body”, the number (a), is non-zero. Remark. In the various variants of the theory, one has always such a unique body projection; however, if B is an infinite-dimensional algebra, the soul may be only topologically nilpotent. We get a “body projection” from to superspace to ordinary space: : B0m × B1n → Rm ,

(u1 , . . . , um , ξ1 , . . . , ξn ) 7→ ((u1 ), . . . , (um )) .

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Let U ⊆ B0m × B1n be open. Given f ∈ C ∞ ((U )) ⊗ B, i.e. a B-valued smooth function on (U ), we get by “Grassmann analytic continuation” an element z(f ) ∈ G∞ (U ), z(f )(u) :=

X

νm ∂1ν1 · · · ∂m f ((u))

ν1 ,...,νm ≥0

(u1 − (u1 ))ν1 · · · (um − (um ))ν1 ; ν1 ! · · · νm !

the sum is actually finite. One gets an injective algebra homomorphism z : C ∞ ((U )) → G∞ (U ), and it turns out that G∞ (U ) is generated as algebra by the union of the image of this homomorphism with the set of coordinates {x1 , . . . , xm+n }. Indeed, one finds that every f ∈ G∞ (U ) admits an expansion X X z(fµ )(u)ξ µ (1.6.1) f (u|ξ) = z(fµ1 ...µn )(u)ξ1µ1 · · · ξnµn = which not accidentally looks like (1.3.2), but here with suitable fµ1 ...µn ∈ C ∞ ((U )). Unfortunately, here the truncation effects make their second appearance: the fµ1 ...µn are uniquely determined only for µ1 + · · · + µn ≤ k; otherwise, they are completely undetermined since ξ1µ1 · · · ξnµn vanishes anyway. Thus, we get an epimorphism of Z2 -graded algebras C ∞ ((U )) ⊗ B ⊗ Λ[ξ1 , . . . , ξm ] → G∞ (U ) ,

f ⊗ ξ1µ1 · · · ξnµn 7→ z(fµ1 ...µn )ξ1µ1 · · · ξnµn . (1.6.2)

Its kernel is generated by all products ξ1µ1 · · · ξnµn with µ1 +. . .+µn > k; in particular, (1.6.2) is an isomorphism iff n ≤ k. 0 0 A map B0m × B1n ⊇ U → B0m × B1n , where U is open in the natural topology, is called G∞ iff its m0 + n0 component maps are G∞ . Now, globalizing in a standard way, one gets the notion of a G∞ supermanifold which is a paracompact Hausdorff space together with an atlas of B0m × B1n -valued charts such that the transition maps are G∞ . On such a G∞ smf, there lives the sheaf G∞ (·) of B-valued functions of class ∞ G ; one could also define a G∞ smf equivalently as a paracompact Hausdorff ringed space which is locally isomorphic to the model space (B0m × B1n , G∞ (·)). Now there exists a second reasonable notion of superfunctions and supermanifolds: Let H ∞ (U ) be the subalgebra of all f ∈ G∞ (U ) for which the fµ1 ...µn can be choosen R-valued. This is still a Z2 -graded commutative algebra over R but not over B, i.e. there are no longer anticommuting constants. Now (1.6.2) restricts to an algebra epimorphism C ∞ ((U )) ⊗ Λ[ξ1 , . . . , ξn ] → H ∞ (U ) .

(1.6.3)

Using H ∞ transition maps instead of G∞ ones one gets the notion of a H ∞ smf. On a H ∞ smf, there lives the sheaf H ∞ (·) which consists of B-valued functions. However, it is no longer a sheaf of algebras over B because H ∞ (U ) ∩ B = R within G∞ (U ) for any open U . One could also define a H ∞ smf equivalently as a paracompact Hausdorff ringed space which is locally isomorphic to the model space (B0m × B1n , H ∞ (·)).

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Now every H ∞ supermanifold is also a G∞ supermanifold in a natural way but not conversely; for the sheaves one finds G∞ (·) ∼ = H ∞ (·) ⊗ B. A further ramification arises if one uses instead of the natural topology on B0m × B1n the deWitt topology which is the inverse image topology arising from : B0m × B1n → Rm ; note that it is not Hausdorff. However, although it is much coarser than the natural topology, the loss of information is smaller than one would think: It follows from the expansion (1.6.1) that once U ⊆ V are open in the natural topology with (U ) = (V ) the restriction maps G∞ (V ) → G∞ (U ) and H ∞ (V ) → H ∞ (U ) are bijective. So one defines a H ∞ -deWitt supermanifold as a paracompact space together with an atlas of B0m × B1n -valued charts, which are now local homeomorphisms w. r. to the deWitt topology, and the transition maps are H ∞ . Indeed, we will see that the smf arising from a Berezin smf is naturally of this type. 1.7. From Berezin to deWitt Rogers The formal similarity of (1.6.1) to (1.3.2) suggests the existence of a connection between both approaches. Indeed, it is possible to assign to any Berezin smf a Rogers H ∞ smf which, however, will depend also on the choice of the Grassmann algebra of constants B. While [32] gave a chart-by-chart construction, we give here, following [2], a more intrinsic mechanism. It will show that the Berezin smfs are the fundamental ones; moreover, it will be in some sense prototypical for the connection of the naive notion of configurations to our one. We first recall some well-known higher nonsense from category theory which today has become a basic tool in algebraic geometry. Given a category C, every object X generates a cofunctor (=contravariant functor) X(·) : C → Sets ,

Z 7→ X(Z)

where X(Z) is simply the set of all morphisms from Z to X. Also, every morphism X → X 0 generates a natural transformation X(·) → X 0 (·). Thus, we get a functor C → {category of cofunctors C → Sets} ,

X 7→ X(·) ,

(1.7.1)

and it is a remarkable observation that this functor is faithfully full, that is, every natural transformation X(·) → X 0 (·) is generated by a unique morphism X → X 0 . Moreover, let be given some cofunctor F : C → Sets. If there exists an object X of C such that F is isomorphic with X(·) then this object is uniquely determined up to isomorphism; thus, one can use cofunctors to characterize objects. In algebraic geometry, one often calls the elements of X(Z) the Z-valued points of X. Indeed, an algebraic manifold X usually encodes a system of equations in affine or projective space which cut it out, and if Z happens to be the spectrum of a ring, Z = Spec(R), then X(Z) is just the set of solutions of this system with values in R. One also writes simply X(R) for this. Now fix a finite-dimensional Grassmann algebra B = Λk . This is just the algebra of global superfunctions on the linear smf L(R0|k ); the underlying manifold is simply

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a point. L(R0|k ) should be viewed as C ∞ super variant of Spec(B), and therefore it is natural to denote, for any Berezin–Leites–Kostant smf X, by X(B) the set of all smf morphisms L(R0|k ) → X. The set X(B) will be the underlying set of the H ∞ smf we are going to construct. In fact, using Theorem 1.3.1, any such morphism is uniquely characterized by the pullback of global superfunctions; thus, X(B) identifies with the set of all algebra homomorphisms O∞ (X) → B. Given a morphism f : X → Y of Berezin smfs, we get a map f : X(B) → Y (B), u 7→ u ◦ f ∗ . In particular, for open U ⊆ X, we get an injective map U (B) → X(B) which we will view as inclusion. The U (B) are the open sets of a non-Hausdorff topology which is just a global variant of the deWitt topology mentioned above. φ

Now fix a superchart on X, i.e. an isomorphism U → U 0 where U is open in X φ and U 0 is open in L(Rm|n )x|ξ . We get a map U (B) → U 0 (B); the target of this map identifies naturally with a “Rogers superdomain”: U 0 (B) ∼ = −1 (U 0 ) ⊆ B0m × B1n ,

f 7→ (f (x1 ), . . . , f (xm ), f (ξ1 ), . . . , f (ξn )) .

φ

Denote the composite U (B) → U 0 (B) → −1 (U 0 ) by cφ ; it will become a superchart on X(B). Equip X(B) with the strongest topology such that all cφ arising from all possible supercharts on X are continuous (of course, an atlas is sufficient, too). It follows that, since the transition map between any two such supercharts is H ∞ , these supercharts equip X(B) with the structure of a H ∞ smf, and thus, a fortiori, a G∞ smf. Also, the epimorphism (1.6.3) globalizes: For f ∈ O∞ (U ) with open U ⊆ X, we get a map f 0 : U (B) → B, u 7→ u(f ), which is a H ∞ superfunction. One gets an algebra epimorphism O∞ (U ) → H ∞ (U (B)) which is an isomorphism iff n ≥ k. Finally, given a morphism f : X → Y of Berezin smfs, the map f : X(B) → Y (B) considered above is of class H ∞ . Altogether, we get for fixed B a functor Ber Smfs = {category of Berezin smfs} → {category of H ∞ smfs}, X 7→ X(B) . (1.7.2) In particular, any ordinary smooth manifold X can be viewed as a Berezin smf and therefore gives rise to a H ∞ smf X(B). Roughly speaking, while X is glued together from open pieces of Rm with smooth transition maps, X(B) arises by replacing the open piece U ⊆ Rm by −1 (U ) ⊆ B0m , and the transition maps by their Grassmann analytic continuations. Remark. (1) Apart from the fact that B is supercommutative instead of commutative, the functor (1.7.2) is just the Weil functor considered in [22]. (2) Looking at the Z-valued points of an smf is often a useful technique even if one stays in the Berezin framework. For instance, a Lie supergroup G, i.e. a group object in the category of supermanifolds, turns in this way to a functor with

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values in the category of ordinary groups (by the way, this property is not shared by quantum groups, which makes them much more difficult objects). While, using the diagrammatic definition of a group object, it is by no means trivial that the square of the inversion morphism ι : G → G is the identity, this fact becomes an obvious corollary of the corresponding property of ordinary groups by using Z-valued points. Cf. [36, 25]. 1.8. Molotkov’s approach and the rˆ ole of the algebra of constants For a review of Molotkov’s approach [29] to infinite-dimensional supermanifolds, and a comparison with our one, cf. [42]. Here we give an interpretation of the idea in categorial terms. Roughly said, this approach relies on the description of smfs by their points with values in 0|k-dimensional smfs, where, however, all k’s are considered simultaneously. The basic observation to understand Molotkov’s approach is the following: Let Ber Smfs be the category of all Berezin smfs, and let Gr be the category the objects of which are the real Grassman algebras Λk = Λ[ξ1 , . . . , ξk ] (one for each k ≥ 0), with all even algebra homomorphisms as morphisms. Assigning to every Λk the 0|k-dimensional smf Spec(B) := L(R0|k ) = (point, Λk ) we get a cofunctor Spec : Gr → Ber Smfs which establishes an equivalence of the opposite category GrOp with the full subcategory Spec(Gr) of Ber Smfs of those smfs which are connected and have even dimension zero. On the other hand, it is easy to see that the subcategory Spec(Gr) is sufficient to separate morphisms, that is, given two distinct smf morphisms X → →X 0 there exists some k > 0 and a morphism Spec(Λk ) → X so that the composites Spec(Λk ) → X→ →X 0 are still different. Thus, if assigning to any smf X the composite functor Spec

X(·)

Gr −−→ Ber Smfs −−→ Sets , we get a functor Ber Smfs → {category of functors Gr → Sets} ,

X 7→ X(·) ,

(1.8.1)

and, due to the separation property mentioned above, this functor is still faithful, i.e. injective on morphisms. (It is unlikely that it is full, but I do not know a counterexample.) Thus, an smf is characterized up to isomorphism by the functor Gr → Sets it generates. However, there exist functors not representable by smfs. Roughly said, a given functor F : Gr → Sets is generated by an smf iff it can be covered (in an obvious sense) by “supercharts”, i.e. subfunctors which are represented by superdomains. However, as a consequence of the lack of fullness of the functor (1.8.1), we cannot guarantee a priori that the transition between two supercharts is induced by a morphism of superdomains — we simply have to require that.

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Explicitly, given a linear smf L(E) where E ∼ = Rm|n is a finite-dimensional Z2 -graded vector space, one has for each B := Λk L(E)(B) = (B ⊗ E)0 ∼ = B0m × B1n ; we encountered this already in the previous section. Now, Molotkov’s idea is to replace here E by a Banach space and define just the linear smf generated by E as the functor FE : Gr → Sets, B 7→ (B ⊗ E)0 . He then defines a “superregion” alias superdomain as an in a suitable sense open subfunctor of this, and an smf as a functor F : Gr → Sets together with an atlas of subfunctors such that the transition between two of them is supersmooth in a suitable sense. Actually, his functors have their values not in the set category but in the category of smooth Banach manifolds, but this is a structure which cares for itself, thanks to the use of atlasses. Resuming we find that • the Berezin–Kostant approach works without any “algebra of constants” except R or C, i.e. without an auxiliary Grassmann algebra; • the deWitt–Rogers approach works with a fixed Grassmann or Grassmann-like algebra of constants; • Molotkov uses all finite-dimensional Grassmann algebras at the same time as algebras of constants, and thus achieves independence of a particular choice. My personal point of view is that any differential-geometric concept which makes explicit use of the algebra of constants B, and hence is not functorial w.r. to a change of B, is suspicious. Although the possibility of choice of a concrete B allows new differential-geometric structures and phenomena, which by some people are claimed to be physically interesting, there is up to now no convincing argument for their potential physical relevance. In particular, this is connected with the fact that there exist Rogers smfs which do not arise from a Berezin smf by the construction given above. A prototypical example is a 1|1-dimensional supertorus, in which also the odd coordinate is wrapped around. It is clearly non-functorial in B because it needs the distinction of a discrete subgroup in B1 . Also, if one defines G∞ Lie supergroups in the obvious way, the associated Lie superalgebra will be a module over B, with the bracket being B-linear. It follows that the definition is bound to a particular B. In particular, if G is a Lie supergroup in Berezin’s sense, with Lie superalgebra g, then, for a Grassmann algebra B, the Rogers Lie supergroup G(B) will have Lie superalgebra g ⊗ B; Now there is an abundance of Lie superalgebras over B which are not of this form; cf. [14, 4.1] for an example of a 0|1-dimensional simple Lie superalgebra which cannot be represented as g ⊗ B. However, none of these “unconventional” Lie superalgebras seems to be physically relevant. A general metamathematical principle, which is up to now supported by experience, is: Any mathematical structure which does not make explicit use of the algebra of constants B can be (and should be) formulated also in Berezin terms.

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In particular, this applies to the structures appearing in geometric models in quantum field theory. 1.9. Comparison of the approaches The deWitt–Rogers approach has quite a few psychological advantages: Superfunctions are now B-valued maps, i.e. genuine functions on this space instead of elements of an abstract algebra. Also, instead of morphisms of ringed spaces one has genuine superdifferentiable maps. Also, from an aesthetic point of view, even and odd degrees of freedom seem to stand much more on equal footing: While the underlying space of a Berezin smf encodes only even degrees of freedom, the underlying space of a Rogers-like smf is a direct product of even and odd part. Finally, in the Taylor expansion (1.6.1) of a G∞ function at ξ = 0, the functions z(fµ )(u) are still B-valued, and thus they may perfectly well anticommute (they do so iff they are B1 -valued). At the first glance, this approach appears to be ideally suited for the implementation of the anticommutativity of fermionic field components. These appealing features tempted quite a few physicists to prefer this approach to the Berezin approach (cf. in particular [14]). But one has to pay for them: “This description, however, is extremely uneconomical because in the analysis do the ξ’s” (here θ’s) “explicitly enter. So, the ξ’s are really unnecessary” [18, 6.5.3]. For instance, the 4|4-dimensional super Minkowski space, that is, the space-time of e.g. N = 1 super Yang–Mills theory with superfields, is modelled as X = B04 ×B14 . Thus, one has an “inflation of points”, and even the most eager advocates of this approach have no idea how to distinguish all these points physically. In fact, the algebra B plays an auxiliary role only. Notwithstanding which of the modern, mainly heuristic, concepts of quantization of a field theory one uses, B does not appear at all at the quantized level. In other words, B is an unphysical “addendum” used to formulate the classical model. This becomes even clearer if we try to model fermion fields on the ordinary Minkowski space within this approach: Space-time would be described by B04 instead of the usual (and certainly more appropriate) R4 . On the other hand, one can obtain anticommuting field components necessary for modelling fermion fields also in the framework of Berezin’s approach: Instead of real- or complex-valued functions or superfunctions f ∈ O∞ (X) one uses “Bvalued” ones: f ∈ O∞ (X) ⊗R B where B is a Grassmann algebra or some other suitable Z2 -commutative algebra, and the tensor product is to be completed in a suitable sense if B is infinite-dimensional. This removes the inflation of points — but the unphysical auxiliary algebra B remains. At any rate, this argument shows that the deWitt-Rogers approach has not the principal superiority over Berezin’s approach which is sometimes claimed by its advocates. The problem of “B-valued configurations” will be discussed in the next section.

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The question arises, what algebra of constants B should we take, how “large” should it be? As we saw in 1.6, in particular, in considering the epimorphism (1.6.2), a finite-dimensional B leads to various unwanted truncation effects; these can be avoided by taking B infinite-dimensional. But even this decision leaves still room; instead of a Banach completion of a Grassmann algebra with a countable set of generators, one could use e.g. one of the Banach algebras P(p; R) introduced in [44] (with a purely odd field target space). On the other hand, an m|n-dimensional smf will now become as topological space m · dim(B0 ) + n · dim(B1 )-dimensional. One feels uneasy with such an inflation of dimensions. Moreover, we cannot use the manifolds of ordinary differential geometry directly — we have to associate to them the corresponding smfs (although it is an almost everywhere trivial process to “tensor through” ordinary structures to their B-analogues, for instance, this applies to metrics, connections, vector bundles, etc). Still worse, the automorphism group AutR (B), which is at least for Grassmann algebras pretty big, appears as global symmetry group; thus, we have spurious symmetries. Indeed, given a Lagrange density which is a differential polynomial L ∈ ChΦ|Ψi and an automorphism α : B → B, the field equations will stay invariant under (Φ, Ψ) 7→ (αΦ, αΨ). (This is analogous to the process in which the only continuous automorphism of C, the conjugation, induces the charge conjugation for a complex field.) Of course, this symmetry could be inhibited by using a differential polynomial with coefficients in B, i.e. L ∈ BhΦ|Ψi, but all usual models have coefficients in C. One feels that the algebra of constants B is an addendum, something which is not really a part of the structure, and that it should be thrown out. This impression is still stronger if we look at a quantized theory: Sometimes, for instance in [14], it is claimed that the result of quantization should be a a “super Hilbert space”, i.e. a B-module H together with a skew-linear B-valued scalar product h·|·i such that the body of hφ|φi for φ ∈ H is non-negative, and is positive iff the body of φ is non-vanishing. However, no one has ever measured a Grassmann number, everyone measures b real numbers. Moreover, while a bosonic field strength operator Φ(x) (or, more R b exactly, the smeared variant dx g(x)Φ(x)) encodes in principle an observable (apart from gauge fields, where only gauge-invariant expressions are observable), a R b b fermionic field strength operator Ψ(x) (or dx g(x)Ψ(x)) is only a “building block” R b for observables. That is, the eigenvalues of dx g(x)Ψ(x) do not have a physical meaning. So it is in accordance with the quantized picture to assume on the classical level that bosonic degrees of freedom are real-valued while the fermionic ones do not take constant values at all. (In fact, the picture is somewhat disturbed by the fact that, in the quantized picture, multilinear combinations of an even number of fermion fields do take values, like e.g. the electron current, or the pion field strength; that is, although classical fermion fields have only “infinitesimal” geometry, their quantizations generate non-trivial, non-infinitesimal geometry.)

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1.10. B-valued configurations Suppose we have a classical field model on R4 . Usually, the theory is described by saying what boson fields Φ(·) and fermion fields Ψ(·) appear, and a Lagrange density L[Φ, Ψ], which, at the first glance, is a functional of these fields. In the language of [44], the model is given by fixing a field target space V , which is a finite-dimensional Z2 -graded vector space on which the two-fold cover of the Lorentz group acts, and a differential polynomial L[Φ, Ψ] ∈ ChΦ|Ψi. The most obvious way to implement (1.1.1) is to assume that the fermion fields Ψ(·) are functions on R4 with values in the odd part of a Z2 -graded commutative algebra B, for instance, a Grassmann algebra Λn . Now, if one takes the field equations serious (indeed, they should give the first order approximation of the quantum dynamics) then it follows that boson fields cannot stay any more real- or complex-valued; instead, they should have values in the even part of B. Cf. e.g. [10] for a discussion of the classical field equations of supergravity within such an approach. Also, if the model under consideration works with superfields, then these should be implemented as B-valued superfunctions. For instance, the chiral superfield (1.1.2) is now implemented as an element Φ ∈ O∞ (L(R4|4 )) ⊗ B 0 which satisfies the constraint Dα˙ Φ = 0; the requirement that Φ be even leads to the required anticommutativity of the Weyl spinor ψ ∈ C ∞ (R4 , C2 ) ⊗ B1 . Now there are arguments which indicate that the “B-valued configurations” should not be the ultimate solution of the problem of modelling fermion fields: First, as in 1.9, it is again not clear which algebra of constants B one should take. Also, any bosonic field component will have dim(B0 ) real degrees of freedom instead of the expected single one. Thus, we have fake degrees of freedom, quite analogous to the “inflation of points” observed in 1.9. Our remarks in 1.9 on spurious symmetries as well as on the inobservability of Grassmann numbers apply also here. Also, the B0 -valuedness of the bosonic fields makes their geometric interpretation problematic. Of course, it is still possible to interpret a B0 -valued gauge field as B0 -linear connection in a bundle of B0 -modules over R4 (instead of an ordinary vector bundle, as it is common use in instanton theory). But what is the geometric meaning of a B0 -valued metric gµν ? For every finite-dimensional B there exists some N such that the product of any N odd elements vanishes, and this has as consequence the unphysical fake relation Ψj1 (x1 ) · · · ΨjN (xN ) = 0 . This fake relation can be eliminated by using an infinite Grassmann algebra, like e.g. the deWitt algebra Λ∞ , or some completion of it; but the price to be paid is that we now use infinitely many real degrees of freedom in order to describe just one physical d.o.f.

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Finally, in the functional integral Z

1 [DΦ][DΨ][DΨ] exp(i S[Φ, Ψ, Ψ]/~)A[Φ, Ψ, Ψ] Aˆ vac = N — however mathematically ill-defined it may be — the bosonic measure [DΦ] still runs (or is thought to do so) over R-valued configurations, not over B0 -valued ones. All these arguments indicate that B-valued configurations should not be the ultimate solution to the problem of modelling fermionic degrees of freedom. Nevertheless, we will see in a moment that they do have a satisfactory geometric interpretation, at least if B = Λk is a finite-dimensional Grassmann algebra: They should be thought of as families of configurations parametrized by the 0|k-dimensional smf L(R0|k ). 1.11. Families of configurations Historically, the “family philosophy” comes from algebraic geometry; it provides a language for the consideration of objects varying algebraically (or analytically) with parameters. It turns out to be rather useful in supergeometry, too; cf. [25, 36, 42]. We begin with the standard scalar field theory given by the Lagrangian density L[Φ] =

3 X

∂a Φ∂ a Φ + m2 Φ2 + V (Φ) ,

a=0

where V is a polynomial, e.g. V (Φ) = Φ4 . Thus, a configuration is simply a real function on R4 ; for convenience, we will use smooth functions, and the space of all configurations is the locally convex space M = C ∞ (R4 ). For any bounded open region Ω ⊆ R4 , the action on Ω is a realanalytic function Z SΩ [·] : M → R , φ 7→ SΩ [φ] := d4 xL[φ] . (1.11.1) Ω

Now suppose that φ = φ(x0 , . . . , x3 |ζ1 , . . . , ζn ) depends not only on the spacetime variables x0 , . . . , x3 but additionally on odd, anticommuting parameters ζ1 , . . . , ζn . In other words, φ ∈ O∞ (R4 × L(R0|n ))0,R is now a family of configurations parametrized by the 0|n-dimensional smf Zn := L(R0|n ) (we require φ to be even in order to have it commute). P As in (1.3.2) we may expand φ(x|ζ) = µ∈Zn , |µ|≡0(2) φµ (x)ζ µ , i.e. we may view 2 φ as a smooth map φ : R4 → C[ζ1 , . . . , ζn ]0,R . That is, φ is nothing but a B-valued configuration with the Grassmann algebra B := C[ζ1 , . . . , ζn ] as auxiliary algebra! Encouraged by this, we look at a model with fermion fields, say L[Φ|Ψ] :=

4 4 X iX (Ψγ a ∂a Ψ − ∂a Ψγ a Ψ) − mΨ ΨΨ − ∂a Φ∂ a Φ − (mΦ )2 Φ2 − i gΨΨΦ 2 a=0 a=0 (1.11.2)

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where Φ is a scalar field, Ψ a Dirac spinor, and the thick bar denotes the Dirac conjugate (in fact, (1.11.2) is a slightly dismantled version of the Yukawa model of meson-nucleon scattering). Note that (1.11.2) is well defined as a differential polynomial L[Φ|Ψ] ∈ ChΦ|Ψi0,R . While a naive configuration for the fermion field, ψ = (ψα ) ∈ C ∞ (R4 , C4 ), would violate the requirement of anticommutativity of the field components ψα , this requirement is satisfied if we look for a family (ψα ) = (ψα (x0 , . . . , x3 |ζ1 , . . . , ζn )), which now consists of odd superfunctions ψα ∈ O∞ (R4 × L(R0|n ))1 . Looking again at the expansion, we can reinterpret ψα as a smooth map ψα : R4 → C[ζ1 , . . . , ζn ]1 , that is, ψ is nothing but a B-valued configuration again. Resuming, we can state that if B = C[ζ1 , . . . , ζn ] is a Grassmann algebra then B-valued configurations can be viewed as families of configurations parametrized by the smf L(R0|n ). Now there is no need to use only 0|n-dimensional smfs as parameter space. Thus, for the model (1.11.2), a Z-family of configurations where Z is now an arbitrary Berezin supermanifold, is a tuple (φ, ψ1 , . . . , ψ4 ) ∈ O∞ (R4 × Z)0,R ×

4 Y

4 O∞ (R × Z)1 ,

(1.11.3)

and the action over any compact space-time domain Ω ⊆ R4 becomes an element Z d4 xL[φ|ψ] ∈ O∞ (Z)0,R . (1.11.4) SΩ [φ|ψ] := Ω

Also, if µ : Z 0 → Z is some smf morphism then (1.11.3) can be pullbacked along the morphism 1R4 × µ : R4 × Z 0 → R4 × Z to give a Z 0 -family (φ0 , ψ10 , . . . , ψ40 ). The simplest case is that Z = P is a point: Since O∞ (R4 × P ) = C ∞ (R4 ), every P -family has the form (φ|0, 0, 0, 0) where φ ∈ C ∞ (R4 ). Thus it encodes simply a configuration for the bosonic sector. More generally, if Z has odd dimension zero, i.e. is essentially an ordinary manifold, then for Z-family we have ψα = 0 by evenness. Hence we need genuine supermanifolds as parameter spaces in order to describe non-trivial configurations in the fermionic sector. If Z is a superdomain with coordinates z1 , . . . , zm |ζ1 , . . . , ζn then (1.11.3) is simply a collection of one even and four odd superfunctions which all depend on (x0 , . . . , x3 , z1 , . . . , zm |ζ1 , . . . , ζn ), and the pullbacked family now arises simply by substituting the coordinates (zi |ζj ) by their pullbacks in O∞ (Z 0 ); this process is what a physicist would call a change of parametrization. This suggests to look for a universal family from which every other family arises as pullback. This universal family would then encode all information on classical configurations.

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Of course, the parameter smf MC ∞ of such a family, i.e. the moduli space for configuration families, is necessarily infinite-dimensional; hence we will have to extend the Berezin–Leites–Kostant calculus (or, strictly spoken, its hermitian variant as sketched in 1.5) to a calculus allowing Z2 -graded locally convex spaces as model spaces; we will do so in Sec. 2. So, sticking to our example Lagrangian (1.11.2), we are looking for an ∞|∞dimensional smf MC ∞ and a tuple of smooth superfunctions (Φ, Ψ1 , . . . , Ψ4 ) ∈ O∞ (R4 × MC ∞ )0,R ×

4 Y

4 O∞ (R × MC ∞ )1

(1.11.5)

such that for any other family (1.11.3) there exists a unique morphism of smfs µ(φ,ψ1 ,...,ψ4 ) : Z → M such that (1.11.3) is the pullback of (1.11.5) along this morphism. We call µ(φ,ψ1 ,...,ψ4 ) the classifying morphism of the family (1.11.3). However, we will not implement (1.11.5) in the verbal sense since a definition of smooth superfunctions in the infinite-dimensional case, which is needed to give the symbol “O∞ ” in (1.11.5) sense, is technically rather difficult, and we will save much work by sticking to real-analytic infinite-dimensional supermanifolds. Fortunately, this difficulty is easily circumvented: We simply consider only those families (1.11.3) for which Z is actually a real-analytic smf, and the Z-dependence is real-analytic (note that if Z = L(R0|n ) is purely even then this requirement is ∞ 4 empty since we have actually polynomial dependence). Once the sheaf OC (R ) (·) of superfunctions on Z with values in the locally convex space C ∞ (R4 ) has been defined (cf. 2.4 below), we can then rewrite (1.11.3) to an element (φ, ψ1 , . . . , ψ4 ) ∈ O

C ∞ (R4 )

(Z)0,R ×

4 Y

O

C ∞ (R4 )

(Z)1 .

Comparing with Corollary 1.3.3 and its infinite-dimensional version Theorem 2.8.1 below it is now easy to see how to construct the universal family: Its parameter smf is simply a linear smf, MC ∞ = L(E), with the Z2 -graded locally convex space E = C ∞ (R4 ) ⊕ C ∞ (R4 ) ⊗ C0|4 | {z } | {z } even part

odd part

as model space, and the universal family (1.11.5) is now just the standard coordinate on this infinite-dimensional smf: (Φ, Ψ1 , . . . , Ψ4 ) ∈ OC

∞

(R4 )

(MC ∞ )0,R ×

4 Y

O

C ∞ (R4 )

(MC ∞ )1 .

Comparing with 1.7, we see that the set of B-valued configurations stands in the same relation to our configuration smf MC ∞ as a finite-dimensional Berezin smf X to the associated H ∞ -deWitt smf X(B). That is, all information is contained in MC ∞ , which therefore should be treated as the fundamental object. For instance, as special case of (1.11.4) and as supervariant of (1.11.1), the action on a bounded open region Ω ⊆ R4 is now a real-analytic scalar superfunction on M: Z SΩ [Φ|Ψ] := d4 xL[Φ|Ψ] ∈ O (MC ∞ )0,R , (1.11.6) Ω

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and the action (1.11.4) of any family (φ|ψ) is just the pullback of (1.11.6) along its classifying morphism. Thus, the element (1.11.6) can be called the universal action on Ω. Also, the field strengthes at a space-time point x ∈ R4 become superfunctions: Φ(x) ∈ O (MC ∞ )0,R ,

Ψα (x) ∈ O(MC ∞ )1 .

It is now the Z2 -graded commutativity of O(MC ∞ ) which ensures the validity of (1.1.1). 1.12. Classical configurations, functionals of classical fields, and supermanifolds Here we formulate our approach programmatically; an exposition of bourbakistic rigour is given in the successor papers [45, 46]. 1.12.1. Classical configurations The keystone of our approach is the following: The configuration space of a classical field model with fermion fields is an ∞|∞-dimensional supermanifold M. Its underlying manifold is the set Mbos of all configurations of the bosonic fields while the fermionic degrees of freedom are encoded as the odd dimensions into the structure sheaf OM . The functionals of classical fields, which we described in [44] as superfunctionals, are just the superfunctions on M. If such a functional describes an observable it is necessarily even and real. Suppose that our model describes V -valued fields on flat space-time Rd+1 , as in [44] (this is the case for almost all common models in Minkowski space; in Yang– Mills theory, it is the case after choosing a reference connection; however, it is no longer true for σ models or for models including gravitation in the usual way). In that case, the “naive configuration space” is a suitable (cf. Rem. (1) below) admissible space E with respect to the setup (d, V ) (we recall that this means D(Rd+1 ) ⊗ V ⊆ E ⊆ D0 (Rd+1 ) ⊗ V with dense inclusions), and the supermanifold of configurations is just the linear smf L(E) := (E0 , OE ) , i.e. it has underlying space E0 , while the structure sheaf is formed by the superfunctionals introduced in [44] (cf. 2.7 below for the precise definition of smfs). For more general models, e.g. σ models, the configuration space has to be glued together from open pieces of linear superdomains. The flaw of the usual attempts to model fermion fields lies in the implicite assumption that the configuration space M should be a set, so that one can ask for the form of its elements. However, if M is a supermanifold in Berezin’s sense then it has no elements besides that of its underlying manifold Mbos , which just correspond to configurations with all fermion fields put to zero.

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However, although “individual configurations” do not exist (besides the purely bosonic ones), families of configurations parametrized by supermanifolds Z do exist. Precise definitions will be given in the successor papers. Here we note: A Z-family of configurations encodes the same information as a morphism of supermanifolds Z → M . Thus, M can be understood as the representing object for the cofunctor Smfs → Sets ,

Z 7→ {Z-families of configurations} .

In particular, let us look for field configurations with values in a finite-dimensional complex Grassmann algebra Λn := C[ζ1 , . . . , ζn ]. Since Λn = O (L(R0|n )), such a Λn -valued configuration should be viewed as L(R0|n )-family of configurations, and thus encodes a morphism of supermanifolds L(R0|n ) → M . Thus, the set of all Λn -valued configurations is in natural bijection with the set of all morphisms from L(R0|n ) to M . This shows that the really fundamental object is M , and it explains the arbitrariness in the choice of the “auxiliary algebra” B in any Rogers-like formulation. With this point of view, we can do without Rogers’ supermanifolds at all. Even better, one has a functor from “Berezin things” to “Rogers things”: given a Berezin supermanifold X the set of all L(R0|n )-valued points of X forms in a natural way a Rogers supermanifold. Remark. (1) The precise meaning of the notion “configuration space” depends on the choice of the model space E, i.e. on the functional-analytic quality of the configurations to be allowed. If E is too large (for instance, the maximal choice is E = D0 (Rd+1 ) ⊗ V ) then we have few classical functionals (in our example, all coefficient functions have to be smooth, and there are no local functionals besides linear ones), and the field equations may be ill-defined. On the other hand, if E is too small then we have a lot of classical functionals but possibly few classical solutions of the field equations; e.g., for E = D(Rd+1 ) ⊗ V , there are no nonvanishing classical solutions. If necessary one can consider different configuration spaces for one model; the discussion at the end of 1.4 suggests that this might be appropriate. In the case of models with linear configuration space, each choice of E defines its own configuration supermanifold L(E). We note that a continuous inclusion of admissible spaces, E ⊆ E 0 , induces a morphism of supermanifolds L(E) → L(E 0 ). (2) The question arises whether the supermanifold M should be analytic, or smooth? Up to now, we stuck to the real-analytic case; besides of the fact that, in an infinite-dimensional situation, this is technically easier to handle, this choice is motivated by the fact that the standard observables (like action, four-momentum, spinor currents, etc.) are integrated differential polynomials and hence real-analytic superfunctionals, and so are also the field equations which cut out the solution supermanifold M sol (cf. below). Unfortunately, the action of the symmetry groups of the model will cause difficulties, as we will discuss in the successor paper [46]. Therefore it is not superfluous to remark that a C ∞ calculus of ∞-dimensional supermanifolds is possible.

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Fortunately, if the supermanifold of classical solutions (cf. 1.12.2 below) exists at all, it will be real-analytic. (3) Already B. DeWitt remarked in [13] that the configuration space of a theory with fermions should be regarded as a Riemannian supermanifold. Strangely enough, the same author wrote the book [14] in which he elaborated a Rogers-like approach with all details. Cf. also p. 230 of [14] in which he emphasizes again that the configuration space is a supermanifold C – of course, in a Rogers-like sense again. Now his algebra Λ∞ of “supernumbers” turns out to be the algebra of global superfunctions on the 0|∞-dimensional smf L(ΠR∞ ), and C is again the set of all L(ΠR∞ )-valued points of M . Cf. [45] for details. The “supermanifold of fields” M is (somewhat incidentally) mentioned also in [18, 18.1.1]. (4) Although our point of view could look crazy in the eyes of physicists, it is in perfect agreement with their heuristic methods to handle classical fermion fields. In particular, this applies to the fermionic variant of the Feynman integral over configurations: Consider the Gaussian integral Z (1.12.1) [DΨ][DΨ] exp(−ΨAΨ) = DetA where Ψ is a fermion field. The usual, heuristic justification of the setting (1.12.1) amounts in fact to an analogy conclusion from the model case of the integral over a volume form over a 0|n-dimensional complex supermanifold (cf. [3] or modern textbooks on quantum field theory. From this point of view, it is very natural to think that the left-hand side of (1.12.1) is in fact the integral of a volume form on a 0|∞-dimensional supermanifold M . (5) It is interesting to note that we make contact with supergeometry whenever a classical field model contains fermion fields — irrespective of whether supersymmetries are present or not. Perhaps, it is not devious to view this as a “return to the sources”. Indeed, the source of supergeometry was just the anticommutativity of classical fermion fields. (6) Of course, the programme just being presented applies equally well to string models since the latter can be viewed as two-dimensional field models. On the other hand, we neglect all models with more complicated (“plecton” or “anyon”) statistics; although they nowadays have become fashionable, they have up to now not been proven to have fundamental physical relevance for particle physics. In order to implement our approach also for more general models, one needs a theory of infinitedimensional supermanifolds. In [40, 41, 42], the present author constructed a general theory of infinite-dimensional real or complex analytic supermanifolds modelled over arbitrary locally convex topological vector spaces. In fact, [44] presented a specialization of elements of this theory to the case that the model space is an admissible function spaces on Rd . In Sec. 2, we abstract from the function space nature of E, and we globalize the theory from superdomains to supermanifolds. Thus, we will give an alternative description of the theory mentioned above.

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1.12.2. Classical action, field equations, and classical solutions Turning to the action principle, we begin with a naive formulation which is suitable for Euclidian models: The classical action S is an observable; thus, S ∈ O(M )0,R . Moreover, the condition δS = 0 should cut out in M a sub-supermanifold M sol , the supermanifold of the classical solutions. Here  Φ  Z N NΨ X X δ δ S δΦi (x) + S δΨj (x) ∈ Ω1 (M ) dx  δS = δΦ (x) δΨ (x) i j Rd+1 i=1 j=1 is the total variation (cf. 3.8), alias exterior differential, of S. The precise meaning of the phrase “cutting out” will be explained in 2.12. Note that this naive formulation of the action principle is doomed to failure for models in Minkowski space: there is not a single nonvanishing solution of the free field equations for which the action over the whole space-time is finite. Thus, there is no element of O(M )0,R which describes the action over the whole spacetime, and one has to use a “localized” variant of the action principle: Usually, the Lagrangian is given as a differential polynomial LΞ = L[Φ|Ψ] (cf. [44, 2.8] for the calculus of differential polynomials). What we have is the action over bounded open R space-time regions SΩ Ξ := Ω dxL[Ξ](x) ∈ O (M )0,R , and we may call a Z-family of configurations Ξ0 = (Φ0 |Ψ0 ) a Z-family of solutions iff (χSΩ )[Ξ0 ] = 0 for all vector fields χ on the configuration smf which have their “target support” in the interior of Ω (cf. [46] for precise definitions). Now a Z-family of configurations Ξ0 is a family of solutions iff it satisfies the arising field equations δ L[Ξ0 ] = 0 δΞi we call it a Z-family of solutions. Note that we take here not functional derivatives, but variational ones; they are defined purely algebraically. ˇ0 : Z → Now, recalling that a Z-family Ξ0 is in essence the same as a morphism Ξ M , the smf of classical solutions will be characterized by the following “universal property”: A Z-family of configurations Ξ0 is a family of solutions iff the corresponding ˇ 0 : Z → M factors through the sub-smf M sol . morphism Ξ It follows that M sol can be understood as the representing object for the cofunctor Smfs → Sets , Z 7→ {Z-families of solutions} . Roughly spoken, the superfunctions on M sol are just the classical on-shell observables (i.e. equivalence classes of observables with two observables being equivalent iff they differ only by the field equations). In the context of a model in Minkowski space, it is a natural idea that any solution should be known by its Cauchy data, and that the “general” solution,

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i.e. the universal solution family to be determined, should be parametrized by all possible Cauchy data. Thus, one introduces an smf of Cauchy data M Cau the underlying manifold of which will be the manifold of ordinary Cauchy data for the bosonic fields, and one constructs an smf morphism Ξsol : M Cau → M

(1.12.2)

with the following property: Given a morphism ζ : Z → M Cau , i.e. effectively a Z-family of Cauchy data, the composite Ξsol ◦ ζ : Z → M describes the unique Zfamily of solutions with these Cauchy data. Thus, Ξsol is now the universal family of solutions from which every other family of solutions arises as pullback. The image of the morphism (1.12.2) will be just the sub-smf M sol of solutions. The successor papers [45, 46] implement this point of view. 1.12.3. Symmetries of the classical theory First, let us look at Euclidian models, where the global action exists as a superfunction S ∈ O(M )0,R : The (super) Lie algebra of the infinitesimal symmetries of the theory manifests itself as (super) Lie algebra of vector fields on M which leave S invariant. These vector fields are tangential to M sol and hence restrict to it. The discussion of [14, 3.13] yields an example for the appearance of supersymmetry as an algebra of vector fields on a configuration supermanifold (which, however, is so small that it allows a globally defined action but no non-trivial solutions of the field equations). On the other hand, on-shell (super-) symmetry manifests itself as linear space of vector fields on M which still leave S invariant, but which only on M sol form a representation of the super Lie algebra of supersymmetry. Unfortunately, in the Minkowski situation, the non-existence of a globally defined action makes it difficult to give a general definition of symmetries of the action; we will discuss that elsewhere. Naively, one should expect that any infinitesimal (super-) symmetry algebra integrates to an action α : G × M → M of the corresponding connected, simply connected (super) Lie group G on M . For Yang-Mills gauge symmetry, this will work perfectly well; on the other hand, for space-time symmetry groups, there will arise the obstacle that we have only a real-analytic calculus while α can only be expected to be smooth. A detailed discussion will be given elsewhere. 1.12.4. Outlook The classical, non-quantized version of the canonical (anti-) commutation relations, the canonical Poisson brackets {Ξi (x), Σj (y)} = δij δ(x − y) ,

(1.12.3)

1022

T. SCHMITT

where Σj := ∂L/∂(∂0 Ξj ) [Ξ] are the canonical momenta, suggests the introduction of the two form NΦ +N Ψ Z X Cau = dxδΞi (x)δΣi (x) ω i=1

R3

on the smf of Cauchy data M Cau ; here δ is the exterior derivative for forms on the smf M Cau (cf. [42] for a detailed theory), and the product under the integral is the exterior product of one forms. This equips the smf M Cau with a symplectic structure, and (1.12.3) holds in “smeared” form: Z Z Z dxf (x)Ξi (x), dxg(x)Σj (x) = δij dxf (x)g(x) for f, g ∈ D(R3 ) (all integrals over R3 ). The necessity of the buffer functions f, g is connected with the fact that on an infinite-dimensional symplectic (super-)manifold, the Poisson bracket is defined only on a subalgebra of the (super-)function algebra. Thus, the non-smeared writing (1.12.3) used in the textbooks of physicists is highly symbolic since the r.h.s. is not a well-defined superfunction. Ξsol

With the aid of the isomorphism M Cau −→ M sol , one can carry over ω Cau to a symplectic structure ω sol on M sol ; we will show elsewhere that this is Lorentz invariant, and hence an intrinsic structure (cf. also [47] for an alternative construction. This paper constructs heuristically a pseudo-K¨ ahler structure on M sol ; however, the well-definedness of the latter is not clear). This symplectic structure makes it possible to rewrite the field equations in Hamiltonian form. In view of this, it is natural to call the smf M sol also the covariant phase space of the theory (cf. [18, 17.1.2], [11]). The symplectic smf M Cau might be the starting point for a geometric quantization. Of course, it is a rather tricky question what the infinite-dimensional substitute for the symplectic volume needed for integration should be; we guess that it is some improved variant of Berezin’s functional integral (cf. [3]). Note, however, that although the integration domain M sol is isomorphic to the linear supermanifold M Cau , this isomorphism is for a model with interaction highly Lorentz-non-invariant, and Berezin’s functional integral makes use of that linear structure. Also, in the interacting case, the usual problems of quantum field theory, in particular renormalization, will have to show up on this way, too, and the chances to construct a Wightman theory are almost vanishing. Nevertheless, it might be possible to catch some features of the physicist’s computational methods (in particular, Feynman diagrams), overcoming the present mathematician’s attitude of contempt and disgust to these methods, and giving them a mathematical description of Bourbakistic rigour. What certainly can be done is a mathematical derivation of the rules which lead to the tree approximation Stree of the scattering operator. Stree should be at least a well-defined power series; of course, the wishful result is that in a theory without bounded states, Stree is defined as an automorphism of the solution smf M free of the free theory.

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

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1.13. Example: Fermions on a lattice As an interesting example for the description of configuration spaces as supermanifolds, we re-describe the standard framework of Euclidian Yang–Mills– Higgs–Dirac theory on a finite lattice (cf. [48]) in supergeometric terms. Here the finite-dimensional smf calculus is still sufficient. Let Λ ⊂ Zd be a finite lattice. Let L∗ := {(x, y) ∈ Λ2 : kx − yk = 1} the set of its links, and let L := {(x, y) ∈ L∗ : ∀i yi ≥ xi }. Let G be a compact Lie group; we recall that G has a canonical real-analytic structure. A configuration of the gauge field is a map g : L∗ → G, (x, y) 7→ gxy with gxy gyx = 1; thus, the configuration space is the real-analytic manifold GL . The Wilson–Polyakov action is the real-analytic function SYMW : GL → R ,

g 7→ −

1X χ(gvw gwx gxy gyu ) 2

where the sum runs over all plaquettes of Λ, i.e. all tuples (x, y, u, v) ∈ G4 for which (x, y), (y, u) ∈ L, (u, v), (v, w) ∈ L∗ ; tuples which differ only by cyclic permutations are identified. Also, χ : G → R is some character of G belonging to a locally exact representation. (We ignore all wave function renormalization and coupling constants.) For the Higgs field, we need a finitedimensional real Hilbert space VH and an orthogonal representation UH : G → O(VH ). A configuration of the Higgs field is a map φ : Λ → VH ; thus, the configuration space for the Higgs field is the vector space VH Λ . The Higgs action is the real-analytic function S

H R, (g, φ) 7→ − Mbos = GL × VH Λ →

X 1 X (φ(x), UH (gxy )φ(y)) + V (|φ(x)|) 2 (x,y)∈L

x∈Λ

where V is a fixed polynomial of even degree ≥ 4 with positive highest coefficient. Turning to the Dirac field, we assume to be given a Clifford module VS (“spinor space” in [48]) over Cliff(Rd+1 ); that is, VS is a finitedimensional complex Hilbert space together with selfadjoint operators γi ∈ EndC (VS ) (i = 0, . . . , d) such that γi γj + γj γi = 2δij . Also, we need a finitedimensional real Hilbert space VG (“gauge” or “colour” space) and an orthogonal representation UG : G → O(VG ). Let VF := VS ⊗R VG , which is a complex Hilbert space, and consider it purely odd. Let (eα ), (fa ) be orthonormal bases of VS and VG , respectively; thus, the elements Ψαa := eα ⊗ fa form an orthonormal basis of VF . Now take one copy VF (x) of VF for each lattice site x ∈ Λ, with basis (Ψαa (x)), and let VF (x) be the “exterior” conjugate of VF (x), i.e. the Hilbert space consisting of all elements v with v ∈ VF (x); Hilbert space structure and G-action are fixed by requiring that the map VF (x) → VF (x), v 7→ v, be antilinear, norm-preserving, and G-invariant.

1024

T. SCHMITT

Setting V :=

M

VF (x) ,

V :=

x∈Λ

M

VF (x)

x∈Λ

(of course, the “⊗” of [48] is a misprint), the space V ⊕ V has a natural hermitian structure (conjugation acts as the notation suggests). Let Vr := {v + v; v ∈ V} be its real part; thus, V identifies with the complexification of Vr . Now the Grassmann algebra ^ GΛ := (V ⊕ V) , is nothing but the algebra of superfunctions on the hermitian supermanifold Mferm := L(Vr ): GΛ = O (Mferm) (recall that Vr , being a real Hilbert space, identifies canonically with its dual), thus, Mferm is the configuration supermanifold for the fermionic field strengthes. The field strengthes now appear as odd superfunctions: Ψαa (x), Ψαa (x) ∈ O (Mferm)1 ; in the language of [39], the Ψαa (x) form a chiral coordinate system on Mferm. The configuration space for the whole system is the hermitian supermanifold M = Mbos × Mferm = GL × VH Λ × L(Vr ) . The algebra of global superfunctions is the Grassmann algebra generated by Ψαa (x), Ψαa (x) with the coefficients being real-analytic functions of the gxy , φ(x). (Actually, [48] uses instead of this the Grassmann algebra AΛ with the same generators but with the coefficients being continuous bounded functions of the gxy , φ(x); but that difference does not really matter.) Turning to the fermionic action, fix parameters r ∈ (0, 1, θ ∈ 0, π/2); for their meaning, we refer to the literature. For (x, y) ∈ L set Γxy αβ := r exp(i θγd ) +

d X

(xj − yj )(γj )αβ ;

j=0

since x, y are neighbours, only one term of the sum is non-zero. Also, observe that g 7→ UG (gxy )ab is a real-analytic function on GL . Therefore we can form the superfunction 1 X X Ψαa (x)Γxy SF := αβ UG (gxy )ab Ψβb (y) 2 (x,y)∈L a,b,α,β

+

1X X Ψαa (x) (M − rd exp(i θγd )αβ ) Ψβa (x) , 2 x∈Λ a,α,β

which is the action for the fermion field. Here M is the mass of the Dirac field. The total action is now a superfunction on M : Stot := SYMW + SH + SF ∈ O(M )0 .

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

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We look at the action of the group of gauge transformations GΛ , which is the Lie group of all maps h : Λ → G: By (h · g)xy := hx gxy hy −1 ,

(h · φ)(x) := UH (hx )φ(x) ,

we get an action GΛ × Mbos → Mbos

(1.13.1)

of the group of gauge transformations on the configuration space for the gauge and Higgs fields. On the other hand, we have a representation of UF : GΛ → O(Vr ) by restricting the natural unitary action of GΛ on V to the real part (this works due to the reality of the representation UG ). We get a representation (cf. [42, 5.18–20] for some basic notions) a : GΛ × Mferm → Mferm ,

a∗ (Ψαa (x)) = UG (hx )Ψαa (x)

(1.13.2)

of GΛ on the linear supermanifold Mferm . (1.13.1), (1.13.2) together yield an action a : GΛ × M → M of the group of gauge transformations on the configuration supermanifold, and S is now an invariant function, i.e. a∗ (S) = pr2 ∗ (S). Let Ob (Mbos ) be the subspace of all f ∈ O (Mbos ) which grow only polynomially in Higgs direction, i.e. there exist C, p > 0 such that |f (g, φ)(x)| ≤ C(1 + |φ(x)|p ) for x ∈ Λ, and let Ob (M ) := Ob (Mbos ) ⊗ O (Mferm) ⊆ O (M ) . Now we can interpret the mean value as a Berezin integral: For P ∈ Ob (M ), we may form Z Z Y Y dφk (x) dΨαa (x)dΨαa (x) P exp(−Stot /~) . (1.13.3) hP iΛ := dg k,x

α,a,x

Here the inner integral is Berezin integration along the fibres of the super vector pr bundle M = Mbos × L(Vr ) → Mbos , producing an ordinary function on Mbos . The φk (x) (k = 1, . . . , dim VH ) are orthonormal coodinates on the x-component of VH Λ ; thanks to the exponential factor and our growth condition, the integral over them is finite. Finally, dg is the normalized Haar measure on GL . The sign ambiguity arising from the missing order of the φk (x) is resolved by fixing h1 iΛ > 0. Now we suppose the existence of time reflection as a fixpoint-free involutive map r : Λ → Λ which respects the link structure, (r × r)(L∗ ) ⊆ L∗ . By permuting the factors, r yields an involution r : Mbos → Mbos . Also, we get a morphism X r : Mferm → Mferm , r∗ (Ψαa (x)) = Ψβa (rx)(γ0 )αβ . β

1026

T. SCHMITT

Altogether, we get an involutive morphism r : M → M and a new hermitian law θ(P ) := r∗ (P ) . (Actually, the notation is unlucky in view of 1.5, because it hides the skew-linearity.) We also need a decomposition of the lattice Λ = Λ+ ∪ Λ− ,

Λ+ ∩ Λ− = ∅

such that r(Λ± ) ⊆ Λ∓ (cf. [48] for the background). We get a projection morphism M = MΛ → MΛ+ , and hence an embedding Ob (MΛ+ ) ⊆ Ob (M ). Osterwalder– Schrader positivity now states that hP θ(P )i ≥ 0 for all p ∈ Ob (MΛ+ ). Also, the scalar product Ob (MΛ+ ) × Ob (MΛ+ ) → C ,

(P, Q) 7→ hP θ(Q)i ,

satisfies hQθ(P )i = hP θ(Q)i ;

(1.13.4)

hence, it equips Ob (MΛ+ )/{P : hP θ(P )i = 0} with the structure of a pre-Hilbert space the completion H of which is the Euclidian state space. (Thus, Ob (MΛ+ ) plays here the same role as the space of polarized sections in geometric quantization.) Remark. (1) Note that working with a conventional, non-hermitian calculus would bring trouble here since (1.13.4) would acquire an additional factor (−1)|Q||P | , and hence hP θ(P )i would be for odd P not positive but imaginary. (2) It would be interesting to know something about the “supermanifold of classical solutions” of the action S. That is, we may form the ideal sheaf J (·) ⊆ OM (·) generated by the “field equations” ∂ ∂ ∂ ∂ S, S, S, S ∈ O(M ) , i ∂gxy ∂φk (x) ∂Ψαa (x) ∂Ψαa (x) i are local coordinates on the (x, y)-component of GL ) and take the factor (here gxy sol := (supp O /J , O /J ). This is at least a real superanalytic space space M which, however, might have singularities. Note that, in contrast to the non-super situation, where the singular locus of an analytic space has codimension ≥ 1, it may on a superanalytic space be the whole space; it would be nice to show that this does not happen here. Also, one should prove that in the classical limit ~ → 0, the integral (1.13.3) becomes asymptotically equal to an integral over M sol (or its non-singular part). Indeed, in the bosonic sector, the exponential factor makes the measure accumulate ] sol of the minima of the action; however, in the fermionic sector, on the subspace M the picture is less clear.

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

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2. Infinite-Dimensional Supermanifolds An attentive reader of [44] will have noted that most of its material (with the exception of local functionals, differential polynomials a.s.o.) does not really depend on the fact that E is a function space. In fact, the theory can be developed in an abstract context, and one really should do so in order to get conceptual clarity which may be useful if a concrete situation does not fit into our framework. In that way, we will also establish the connection with usual, finite-dimensional supergeometry a la Berezin. Some work has been done on infinite-dimensional supergeometry and its application onto classical fields in quantum field theory. Apart from the implicite appearance of infinite-dimensional supermanifolds in [3] (cf. [43]), the first work on the mathematical side is Molotkov [29]; however, his approach is not well suited for physical purposes. Cf. below for a discussion. [1] uses an ad-hoc definition of smooth Banach smfs in order to describe mathematically the fermionic Faddeev–Popov ghost fields used by physicists for quantization of the Yang–Mills field. In [24], infinite-dimensional supergeometry makes an implicite appearance in a more general approach to the quantization of systems with first-class constraints (cf. also the comment in [16]). [34] and [16] use, on a physical level of rigour, ad-hoc generalizations of the Berezin–Kostant supergeometry framework for applying geometric quantization in field theory, in particular to fermionic fields. Finally, the present author constructed in [40, 41, 42], the predecessors of the present paper, a rather general theory of complex- and real-analytic supermanifolds modelled over locally convex spaces. Here, we will give an alternative description of this theory, using a traditional treatment via ringed spaces and charts. Also, we will treat only real-analytic supermanifolds with complete model spaces. We will give only a short account on the abstract variant since the details should be clear from the material of [44]. 2.1. Formal power series Let E, F be complete Z2 -lcs, and define the space P k|l (E; F ) of F -valued k|l-forms on E as the space of all (k + l)-multilinear continuous maps u(k|l) :

k Y

E0 ×

l Y

E1 → FC

which satisfy the symmetry requirement u(k|l) (eσ(1) , . . . , eσ(k) , e0π(1) , . . . , e0π(l) ) = sign(π)u(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) for all permutations σ, π. P k|l (E; F ) is a Z2 -graded vector space; note that we do not distinguish a topology on it.

1028

T. SCHMITT

The space of F-valued formal power series on E is defined by Y P k|l (E; F ) ; Pf (E; F ) := k,l≥0

thus, its elements are formal sums u = rule

P

k,l≥0

u(k|l) where u(k|l) ∈ P k|l (E; F ). The

u(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) := u(k|l) (e1 , . . . , ek , e0l , e0l−1 , . . . , f1 ) l = (−1)(2) u(k|l) (e1 , . . . , ek , e01 , . . . , e0l )

turns Pf (E; F ) into a hermitian vector space. The product 0

0

0

0

b 0) P k|l (E; F ) × P k |l (E; F 0 ) → P k+k |l+l (E; F ⊗F is defined by (u ⊗ v)(k+k0 |l+l0 ) (e1 , . . . , ek+k0 , e01 , . . . , e0l+l0 ) −1 −1 X l + l0 k + k0 u(k|l) (ep1 , . . . , epk , e0q1 , . . . , e0ql ) = (±) k l ⊗ v(k0 |l0 ) (ep01 , . . . , ep0k , e0q0 , . . . , e0q0 ) 1

l

(the sign “⊗” on the l.h.s. is somewhat abusive). Here the sum runs over all k+k0 l+l0 partitions k l {1, . . . , k + k 0 } = {p1 , . . . , pk } t {p01 , . . . , p0k0 } , {1, . . . , l + l0 } = {q1 , . . . , ql } t {q10 , . . . , ql00 } ,

p1 ≤ . . . ≤ pk ,

p01 ≤ . . . ≤ p0k0 q10 ≤ . . . ≤ ql00

q1 ≤ . . . ≤ ql ,

and (±) is given by the sign rule: (±) := (−1)|v|(|q1 |+...+|ql |) sign(π) , where π is the permutation (q1 , . . . , ql , q10 , . . . , ql00 ) of {1, . . . , l + l0 } (cf. [41, Prop. 2.3.1]). The product turns Pf (E; R) into a Z2 -commutative hermitian algebra and each Pf (E; F ) into a hermitian module over that algebra. In both situations, we will usually write simply uv instead of u ⊗ v. Remark. An F -valued formal power series in the sense of [44, 2.3], k l X 1 XZ Y Y dXdY K I|J (X|Y ) Φim (xm ) · Ψjn (yn ) ∈ P(F ) , K[Φ|Ψ] = k!l! m=1 n=1 k,l≥0

I|J

defines an F -valued formal power series K ∈ P(D; F ) in the sense above, K = P k,l K(k|l) with K(k|l) :

k Y

D0 ×

Z ×

dXdY

l Y

D1 → FC , (φ1 , . . . , φk , ψ 1 , . . . , ψl ) 7→

X I|J

K I|J (X|Y )

k Y m=1

φm im (xm ) ·

l Y n=1

1 (−1)l(l−1)/2 k!l!

Πψjnn (yn ) ,

(2.1.1)

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

1029

where D = D(Rd ) ⊗ V , and V = V0 ⊕ V1 is the field target space. (The apparently strange parity shift Π was motivated by the wish to have Ψ as an odd symbol.) Moreover, if E is an admissible function space in the sense of [44, 3.1], i.e. E is a Z2 -graded complete locally convex space with continuous inclusions D ⊆ E ⊆ D0 (Rd ) ⊗ V , then we have K ∈ Pf (E; F ) iff for all k, l ≥ 0, (2.1.1) extends to a Nl Nk E0 ⊗ E1 → FC . In that way, we get a natural identification continuous map between Pf (E; F ) as defined in [44, 3.1] and the Pf (E; F ) defined here. (In [44, 3.1], we had also assigned to K(k|l) [Φ|Ψ] the linear map Nk Nk m=1

φm ⊗

Nl n=1

D0 ⊗

Nl

ΠD1 → FC ,

Πψ n 7→ (−1)l(l−1)/2 · (second line of (2.1.1)) .

Note, however, that the parity of this map differs from that of K[Φ|Ψ] by the parity of l. 2.2. Analytic power series Let be given continuous seminorms p ∈ CS(E), q ∈ CS(F ). We say that u ∈ Pf (E; F ) satisfies the (q, p)-estimate iff we have for all k, l with k + l > 0 and all e1 , . . . , ek ∈ E0 , e01 , . . . , e0l ∈ E1 the estimate q(u(k|l) (e1 , . . . , ek , e01 , . . . , e0l )) ≤ p(e1 ) · · · p(ek )p(e01 ) · · · p(e0l ) (we extend every q ∈ CS(F ) onto FC by q(f + if 0 ) := q(f ) + q(f 0 )). We call u analytic iff for each q ∈ CS(F ) there exists a p ∈ CS(E) such that u satisfies the (q, p)-estimate. Now every k|l-form u(k|l) ∈ P k|l (E; F ) is analytic, due to its continuity property, and analyticity of a formal power series is just a joint-continuity requirement onto its coefficients u(k|l) . The analytic power series form a hermitian subspace P(E; F ) of Pf (E; F ). Moreover, tensor product of analytic power series and composition with linear maps in the target space produce analytic power series again. For e00 ∈ E, the directional derivative ∂e00 is defined by (∂e00 u)(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) := ( (k + 1)u(k+1|l) (e00 , e1 , . . . , ek , e01 , . . . , e0l ) (−1)|u| (l + 1)u(k|l+1) (e1 , . . . , ek , e001 , e01 , . . . , e0l )

for |e00 | = 0 for |e00 | = 1.

∂e00 maps both Pf (E; F ) and P(E; F ) into themselves, and it acts as derivation on products: ∂e (u ⊗ v) = ∂e u ⊗ v + (−1)|e||u| u ⊗ ∂e v . The abstract analogon of the functional derivative of K is the linear map E → Pf (E; F ),

e 7→ ∂e K .

1030

T. SCHMITT

Suppose that F is a Z2 -graded Banach space, and fix p ∈ CS(E). Set ku(k|l) kp := inf{c > 0 : u(k|l) satisfies a (c−1 k · k, p)-estimate} for k + l > 0, and define k · k on 0|0-forms to be the norm in FC . Then X

P(E, p; F ) = {u ∈ P(E; F ) : kukp :=

ku(k|l) k < ∞}

k,l≥0

is a Banach space. Moreover, for any E we have [ P(E, p; F ) . P(E; F ) = p∈CS(E)

Now if F 0 is another Z2 -graded Banach space then ku ⊗ vkp ≤ kukp kvkp for u ∈ P(E; F ), v ∈ P(E; F 0 ). In particular, P(E, p; R) is a Banach algebra. Note that for every p ∈ CS(E), c > 1, the directional derivative ∂e maps P(E, p; F ) → P(E, cp; F ) for any e ∈ E. Remark. (1) A k|l-form u(k|l) lies in P(E, p; F ) iff it factors through that case, ku(k|l) k is just its supremum on the k + l-fold power of the unit this space. (2) Remark 3.2.2 of [44] carries over, linking the approach here with [41]: u ∈ Pf (E; F ) and k, l ≥ 0 we get a continuous map Sk E0,C · Sl E1,C → FC ,

ˆp ; in E ball of Fixing

e1 · · · ek e01 · · · e0l 7→ k!l!u(e1 , . . . , ek , e01 , . . . , e0l )

(using notations of [41]; the topology on the l.h.s. is induced from the embedding into SEC ). Using Remark 2.1.(2) of [41] we get a bijection Pf (E; F ) →

Y

L(Sk EC , FC )

k≥0

(symmetric algebra in the super sense). The r.h.s. is somewhat bigger than L(SEC , FC ) = P(E; F ), due to the absence of growth conditions. Having identified both sides, one shows for u ∈ Pf (E; F ) the estimates kukU

p/2

≤ kukp ≤ kukU2p

(cf. [41, 2.5] for the notations k · kU , P(E, U ; F )), and hence P(E, Up/2 ; F ) ⊆ P(E, p; F ) ⊆ P(E, U2p ; F ), *

P(E; F ) = P(E; F ) .

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

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2.3. Insertions On the level of formal power series, we will define uv ∈ Pf (E 0 ; F ) with the data v ∈ Pf (E 0 ; E)0 ,

u ∈ Pf (E; F ) ,

v(0|0) = 0 .

We split v = v(0) + v(1) with v(i) ∈ Pf (E 0 ; Ei )0 (i = 0, 1), and we set u[v] :=

X

* u(k|l) ,

k O

v(0) ⊗

l O

+ v(1)

.

(2.3.1)

k,l≥0

N N N N Here k v(0) ⊗ l v(1) ∈ Pf (E 0 ; k E0 ⊗ l E1 ) is the product, and u(k|l) is viewed N N as continuous linear map k E0 ⊗ l E1 → FC , hu(k|l) , e1 ⊗ · · · ⊗ ek ⊗ e01 ⊗ · · · ⊗ e0l i := u(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) . Thus, each term of (2.3.1) makes sense as element of Pf (E 0 ; F ), and by the same arguments as in [44, 3.3], (2.3.1) is a well-defined formal power series. If we want to lift the condition v(0|0) = 0 we have to introduce more hypotheses. Suppose that (1) v ∈ P(E 0 ; E)0 , u ∈ P(E, p; F ) where F is a Banach space and p ∈ CS(E), and (2) there exists some q ∈ CS(E 0 ) with ip v ∈ P(E 0 , q; Eˆp ) ,

(2.3.2)

kip vkq < 1 .

(2.3.3)

Defining u(k|l) v ∈ P(E 0 ; F ) as above, one has ku(k|l) [v]k ≤ kuk [v]kp · kip v(0) kkq · kip v(1) klq , and therefore the series (2.3.1) converges in P(E 0 , q; F ). Thus, for varying u, we get a linear map of Banach spaces P(E, p; F ) → P(E 0 , q; F ) ,

u 7→ u[v] ,

of norm ≤ 1/(1 − kip vkq ); for F = R, this is a homomorphism of Banach algebras. More generally, let again the data (1) be given, and replace (2) by the weaker condition p(v(0|0) ) < 1 . Then u[v] ∈ P(E 0 ; F ) still makes sense: We can choose q ∈ CS(E 0 ) with (2.3.2). Now, replacing q by cq with a suitable c > 1, we can achieve (2.3.3). Using these results it is easy to show that the insertion u[v] where u, v are analytic and v(0|0) = 0 is analytic again. Remark. The connection of the insertion mechanism with [41] is the following: If u, v are formal power series and v(0|0) = 0 then u[v] here is nothing but u ◦ exp(v) of [41, 2.3]. More generally, if u, v are analytic and φ := v(0|0) 6= 0 then u[v] =

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(tφ u)◦ exp(v − φ) where exp(v − φ) is the cohomomorphism SEC0 → SEC determined by v − φ. Whenever insertion is defined, it is associative, i.e. u[v[w]] makes unambiguous sense if it is defined. (A direct proof is sufficiently tedious. Cf. the approach of [41] which saves that work.) The power series x = xE ∈ P(E; E) defined by x(1|0) (e0 ) = e0 ,

x(0|1) (e1 ) = e1

(ei ∈ Ei ), x(k|l) = 0 for all other k, l, is the unit element under composition: for u ∈ P(E; F ), u[xE ] = xF [u] = u . x is the abstract analogon to the Ξ considered in [44, 3.3]. Sometimes, we call x the standard coordinate, and we will write power series in the form u = u[x]. Suppose again that F is a Banach space, and let u ∈ P(E, p; F ) ,

e ∈ E0 ,

c := 1 − p(e) > 0 .

We define the translation te u ∈ P(E, cp; F ) ⊆ P(E; F ) by te u[x] := u[x + e] . Here e is viewed as constant power series e ∈ P(E; E0 ). We have the Taylor formula te u =

X 1 ∂ku ; k! e

k≥0

the sum absolutely converges in P(E, cp; F ). Thus (te u)(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) X k + k 0 u(k+k0 |l) (e, . . . , e, e1 , . . . , ek , e01 , . . . , e0l ) . = | {z } k 0 k ≥0

k0 times

Sometimes, we will write u ◦ v instead of uv, and call this the composition of u with v. 2.4. Superfunctions Let E, F be complete Z2 -lcs and U ⊆ E0 open. The material of [44, 3.5] carries over verbally. Thus, an F -valued (real-analytic) superfunction on U is a map u : U → P(E; F ) ,

e 7→ ue ,

(2.4.1)

which satisfies the following condition: whenever ue satisfies a (q, p)-estimate we have for all e0 ∈ U with p(e0 − e) < 1 iq ue0 = te0 −e iq ue .

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We call ue the Taylor expansion of u at e. We get a sheaf O F (·) of hermitian vector spaces on E, and the product globalizes to a bilinear map F G F ⊗G (U ) , O (U ) × O (U ) → O

(u, v) 7→ u ⊗ v .

In particular, the sheaf O(·) := OR (·) of scalar superfunctions is a sheaf of Z2 -commutative hermitian algebras, and each O F (·) becomes a hermitian O -module sheaf. If the odd part of E is finitedimensional, we get for U ⊆ E0 a canonical isomorphism (2.4.2) = An(U, C) ⊗ SE1,C ∗ OE (U ) ∼ where SE1 ∗ is the symmetric algebra over the complexified dual of the odd part of E in the supersense, i.e. the exterior algebra in the ordinary sense. Also, the Propositions 3.5.1, 3.5.2 from [44] carry over: Proposition 2.4.1. (i) Suppose that u ∈ OF (U ), ue = 0 for some e ∈ U, and that U is connected. Then u = 0. (ii) Every Banach-valued analytic power series defines a “function element”: Assume that F is Banach, and let be given an element v ∈ P(E, p; F ), e ∈ E0 . Setting U := e + {e0 ∈ E0 : p(e0 ) < 1} there exists a unique F -valued superfunctional u ∈ O F (U ) with ue = v. Explicitly, it is given by (2.4.3) ue+e0 := te0 v for p(e0 ) < 1. (iii) If v ∈ Ppol (E; F ) is an analytic polynomial where E, F may be arbitrary Z2 lcs there exists a unique global F -valued superfunctional u ∈ O F (E0 ) with u0 = v. Explicitly, it is given by (2.4.3) again. We conclude with a useful criterion for a map (2.4.1) to be a superfunction. Fixing a Z2 -lcs F , we call a subset V ⊆ F0∗ ∪ F1∗ strictly separating iff there exists a defining system of seminorms C ⊆ CS(F ) such that for all p ∈ C, f ∈ F with p(f ) 6= 0 there exists some f ∗ ∈ V which satisfies |f ∗ | ≤ Kp with some K > 0 and f ∗ (f ) 6= 0 (in other words, we require that for every p ∈ C the set of that f ∗ ∈ V which factorize through Fˆp separate that Banach space). Obviously, every strictly separating V is separating (i.e. it separates the points of F ). Conversely, if F is a Z2 -Banach space then every separating V ⊆ F ∗ is strictly separating. On the other hand, if F carries the weak topology σ(F, F ∗ ) then V ⊆ F0∗ ∪ F1∗ is strictly separating iff it generates F ∗ algebraically. Proposition 2.4.2. Let U ⊆ E0 be open, let F be a Z2 -lcs, let V ⊆ F ∗ be strictly separating, and let be given a map (2.4.1) such that for all f ∗ ∈ V, the

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map e 7→ hf ∗ , ue i ∈ P(E; C) is an element of O(U ). Then (2.4.1) is an element of OF (U ). Proof. First we treat the case that F is a Banach space. Fix e ∈ E and choose p ∈ CS(E) with ue ∈ P(E, p; F ). For e0 ∈ E0 with p(e0 ) < 1, we get hf ∗ , ue+e0 − te0 ue i = 0 for all f ∗ ∈ V ; since V separates F we get ue+e0 = te0 ue , and the assertion follows. Now let F be an arbitrary Z2 -lcs. It follows from the Banach case and the definition of being strictly separating that for every p ∈ C the assignment e 7→ ip ◦ue ˆ is an element of OFp (U ). The assertion follows. Surprisingly, the conclusion becomes false if V is only supposed to be separating. For a counterexample, cf. [42, 4.1]. 2.5. Superfunctions and ordinary functions Before proceeding, we recall from [44] the definition of real-analytic maps between locally convex spaces: Definition 2.5.1. Let E, F be real locally convex vector spaces (no Z2 grading), and let U ⊆ E be open. A map v : U → F is called real-analytic iff there exists a family (uk )k≥0 of continuous maps · · × E} → F uk : U × |E × ·{z k times

which are symmetric and multilinear in the last k arguments such that for each q ∈ CS(F ), e ∈ U there exists some open V 3 0 with e + V ⊆ U such that for all e0 ∈ V X iq (uk (e, e0 , . . . , e0 )) = iq v(e + e0 ) k≥0

with absolute convergence in the Banach space Fˆq , which is uniform in e0 , i.e. X sup q(uk (e, e0 , . . . , e0 )) → 0 e0 ∈V

k≥k0

for k 0 → ∞. Of course, the uk are the Gateaux derivatives of v. We denote by An(U, F ) the set of all real-analytic maps from U to F . Remark. (1) Of course, if F is Banach then it is sufficient to check the condition only for q being the original norm, i.e., iq is the identity map. On the other hand, there exist (somewhat pathological) examples where it is impossible to find a common V for all q ∈ CS(F ). Cf. [41] for a discussion and for the connection of real-analytic maps with holomorphic maps between complex locally convex spaces. (2) The assignment U 7→ An(U, F ) is a sheaf of vector spaces on E. Also, the class of real-analytic maps is closed under pointwise addition and tensor product as well as under composition.

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As in [44, 3.6], every u ∈ O F (U ) determines a real-analytic underlying function u ˜ : U → FC ,

e 7→ u ˜(e) := ue [0] .

In terms of the Taylor expansion u0 ∈ P(E, p; F ) at the origin, one has the explicit formula X (u0 )(k|0) (e, . . . , e) u ˜(e) = | {z } k≥0

k factors

valid for e ∈ E0 , p(e) < 1; if u is polynomial then this formula holds for all e ∈ E0 . [44, Theorem 3.6.2] carries over: Theorem 2.5.2. (i) The map F O (U ) → An(U, FC ) ,

u 7→ u ˜,

(2.5.1)

is surjective, i.e. every real-analytic map is the underlying functional of some superfunctional. Moreover, (2.5.1) turns products into pointwise tensor products, and it turns hermitian conjugation into conjugation within FC : ^ (u ⊗ v)(e) = u˜(e) ⊗ v˜(e),

˜ (u)(e) =u ˜(e) .

Also, it turns even directional derivative into the Gateaux derivative: For e0 ∈ E0 we have ∂g u(e, e0 ) . e0 u(e) = D˜ (ii) If E1 = 0 then (2.5.1) is bijective. Explicitly, for a given real-analytic map v : U → FC the corresponding superfunctional u ∈ OF (U ) is given by (ue )(k|l) = 0 for l 6= 0, (ue )(k|0) (e1 , . . . , ek ) = Dk v(e, e1 , . . . , ek ) where Dk v is the kth Gateaux derivative of v. We will denote by MF (·) := O F (·)0,R the even, real part of OF . The material of [44, 3.7] carries over: 0

F 0 ˜ ⊆ U 0 , the Proposition 2.5.3. Given v ∈ ME E (U ), u ∈ OE 0 (U ) with Im v map U → P(E; F ) , e 7→ (u ◦ v)e := uv˜(e) ve − v˜(e)

is an element u ◦ v = u[v] ∈ O F (U ) called the composition of u with v. The standard coordinate, which was introduced in 2.3 only as power series, now globalizes by assertion (iii) of the Proposition to a superfunction x = xE ∈ ME (E0 ) which we call again the standard coordinate. As in [44, 3.5], it is often advisable to distinguish the “expansion parameter” xe at a point notationally from the standard coordinate x.

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2.6. Superdomains Given a complete Z2 -lcs, we can assign to it the hermitian ringed space (cf. 1.5) L(E) := (E0 , O ) ; we call it the linear supermanifold modelled over E. The assignment E 7→ L(E) is the superanalogon of the usual procedure of viewing a vector space as manifold. Also, we call every open subspace U = (space(U ), O |U ) of it a superdomain modelled over E, and we write (symbolically) U ⊆ L(E). A morphism of superdomains φ : (U, OU ) → (V, OV )

(2.6.1)

which are modelled over Z2 -lcs E, F , respectively, is a morphism of hermitian ringed spaces which satisfies the following condition: There exists an element φˆ ∈ MF (U ) such that ˜ (1) the underlying map of φ is given by the underlying function φˆ : U → F of ˆ and φ, ˆ For open V 0 ⊆ V , (2) the superfunction pullback is given by composition with φ: φ∗ : OV (V 0 ) → OU (φ−1 (V 0 )) maps

u 7→ u ◦ φˆ .

(2.6.2)

Using the standard coordinates xU ∈ ME (U ), xV ∈ MF (V ), this rewrites to ˆ U ]] . u[xV ] 7→ u[φ[x

(2.6.3)

ˆ = We note that φˆ is uniquely determined by φ since for all f ∗ ∈ F ∗ we have hf ∗ , φi ∗ ∗ ∗ φ (f ) ∈ O (U ) (on the r.h.s., we view f as linear superfunction on V ). Remark 2.6.1. We note that in case that both E, F are finitedimensional, the additional requirement of the existence of φˆ is automatically satisfied and hence redundant; that is, every morphism (2.6.1) of hermitian ringed spaces is a morphism of superdomains. Indeed, if fi is a basis of F and f i the left dual bases of F ∗ then it follows from the hermitian, real-analytic version of Corollary 1.3.2 that φˆ = P ∗ i φ (f )fi satisfies our requirements. However, in the general case, its lifting would allow “nonsense morphisms”, and there would be no analogon to Corollary 1.3.2. Given a fixed morphism (2.6.1), the formulas (2.6.2), (2.6.3) are applicable also to G-valued superfunctions where G is an arbitrary complete Z2 -lcs. We get a pullback map G . (2.6.4) φ∗ : OVG → φ∗ OU ˆ In particular, φ is now simply given as the pullback of the standard coordinate: φˆ = φ∗ (xV ) . Conversely, given an element φˆ ∈ MF (U ), it determines a unique superdomain morphism φ : (U, OU ) → L(F ). Thus, we get a bijection of sets F M (U ) → {morphisms (U, OU ) → L(F )} ,

φ 7→ φˆ .

(2.6.5)

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Now, given (2.6.1) and another morphism of superdomains ψ : (V, OV ) → (W, OW ) then the composite ψ ◦ φ is a morphism, too; we have ˆ . ψ[ ◦ φ = φ∗ (ψ) Hence the superdomains form a category. Remark. (2.6.5) shows that for each F , the cofunctor (U, OU ) → MF (U ) is represented by the linear superspace L(F ); the universal element is just the standard coordinate x ∈ MF (L(F )). Fix Z2 -lcs’ E, F . Every even, continuous map α : E → F yields a morphism of superdomains [ := α ◦ xE ∈ MF (L(E)) . L(α)

L(α) : L(E) → L(F ) ,

Of course, the underlying map is the restriction of α to E0 while the superfunction pullback is given by ∗

(L(α) (u)e )(k|l) :

k Y

E0 ×

l Y

Qk+l

α E1 −−−−−→

k Y

F0 ×

l Y

(ue )(k|l)

F1 −−−−−→ C

for u ∈ O(α(U )), e ∈ U , k, l ≥ 0. φ

We call any morphism φ of superdomains L(E) ⊇ U → V → L(F ) which is the restriction of some L(α) a linear morphism. 2.7. Supermanifolds In principle, a supermanifold (abbreviated smf) X is a hermitian ringed space which locally looks like a superdomain. However, such a definition would be insufficient because we cannot guarantee a priori that the arising transition map (U, OU ) → (U 0 , OU 0 ) between two local models is a morphism of superdomains. Therefore we add to the structure an atlas of local models. That is, we define a supermanifold X = (space(X), OX , (ci )i∈I ) (modelled over the complete Z2 -lcs E ) as consisting of the following data: (1) a Hausdorff space space(X), (2) a sheaf of hermitian algebras O = OX on it, (3) a family of charts, i.e. of isomorphisms of hermitian ringed spaces ci : (Ui , OX |Ui ) → (ci (Ui ), O ) ,

(2.7.1)

where (Ui )i∈I is an open covering of X, and the (ci (Ui ), O ) are superdomains modelled over E; in particular, ci (Ui ) ⊆ E0 is open. Setting Uij := Ui ∩ Uj , there arise transition morphisms between the local models, (2.7.2) gij := ci c−1 j : (cj (Uij ), O ) → (ci (Uij ), O ) and we require them to be morphisms of superdomains.

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Remark. For a formulation without charts cf. [41]. For the purposes of the “synthetic” approach presented here, which explicitly indicates the form of morphisms to be allowed, the point distributions of [41] are not needed at all. One could also use an approach via Douady’s “functored spaces” (cf. [15]) in order to save charts. The simplest example arises when there is just one chart, i.e. space(X) is effectively an open set X ⊆ E0 . We then call X a superdomain; for X = E0 we get the linear supermanifold L(E). Fix an smf X. In order to define the sheaf OF of F -valued superfunctions on X we follow the usual “coordinate philosophy” of differential geometry: Working on an open Ui ⊆ X means actually to work on the superdomain ci (Ui ) ⊆ L(E); the cocycle (gij ) tells us how to pass from Ui to Uj . F Thus, for open U ⊆ X, an element of OX (U ) is a family (ui )i∈I of elements ui ∈ OF (ci (U ∩ Ui ))

(2.7.3)

∗ which satisfies the compatibility condition gij (ui ) = uj on cj (U ∩ Uij ). To make that more explicit, let xi ∈ ME (ci (Ui )) be the standard coordinate. Then u is given on U ∩ Ui by a superfunction (2.7.3) in the sense of 2.4, ui = ui [xi ], and compatibility now means

gij [xj ]] uj [xj ] = ui [ˆ ∗ on cj (U ∩ Uij ). Here gˆij = gij (xi ) ∈ M(cj (Uij )). F OX is a sheaf of hermitian vector spaces. For fixed j we get an isomorphism of sheaves −1 F F OX |Ui → cj (OE ) ,

u 7→ c−1 j (ui ) .

Also, we get an isomorphism R O=O ,

∗ v 7→ (c−1 i ) (v) ,

and we can identify both sides. The product of 2.1 globalizes in an obvious way to a bilinear map F G F ⊗G (U ) . O (U ) × O (U ) → O

(2.7.4)

Now, given a bilinear map α : F × G → H with a third Z2 -lcs H, we can compose (2.7.4) with the map induced by α : F ⊗ G → H to get a bilinear map F G H O (U ) × O (U ) → O (U ) .

It is natural to denote the image of (u, v) simply by α(u, v).

(2.7.5)

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2.8. Morphisms of smf ’s A morphism of smfs µ : (X, OX ) → (Y, OY ) is a morphism of hermitian ringed spaces which is “compatible with the chart structure”. That is, we require that if ci : (Ui , OX |Ui ) → (ci (Ui ), O ) ,

dj : (Vj , OY |Vj ) → (dj (Vj ), O )

are charts on X, Y then the arising composite morphism of hermitian ringed spaces c−1 i

µ

(ci (µ−1 (Vj ) ∩ Ui ), O ) −→ X → Y

dj − →

(dj (Vj ), O )

(2.8.1)

is a morphism of superdomains. It is easy to check that the supermanifolds form a category; the superdomains now form a full subcategory. Fix a morphism µ : X → Y . Globalizing (2.6.4), we define the pullback of OG , G ) µ∗ : OYG → µ∗ (OX

as follows: Let v ∈ OYG (V ), i.e. v = (vj ) with vj = vj (xj ) ∈ OG (dj (V ∩Vj )). We put together the pullbacks of the vi under the superdomain morphisms (2.8.1): Fixing i, the elements ∗ G −1 (V ∩ Vj ) ∩ Ui )) (dj µc−1 i ) (vi ) ∈ O (ci (µ with running j fit together to an element ui ∈ OG (ci (µ−1 (V ) ∩ Ui )) ; and µ∗ (v) := (ui ) ∈ O G (µ−1 (V )) is the pullback wanted. The bijection (2.6.5) globalizes to smfs, yielding the infinite-dimensional version of Corollary 1.3.3: Theorem 2.8.1. A morphism of an smf X into a linear smf L(E) is characterized by the pullback of the standard coordinate xE . That is, we have a bijection Mor(X, L(E)) → ME (X) ,

µ 7→ µ ˆ := µ∗ (xE ) .

This is the superanalogon of the (tautological) non-super fact that a real-analytic function on a manifold M with values in a vector space F is the same as a realanalytic map M → F (cf. also [41, Theorem 3.4.1]). 2.9. Supermanifolds and manifolds We now turn to the relations of smfs with ordinary real-analytic manifolds modelled over locally convex spaces; since we defined in 2.5 real-analytic maps, this notion makes obvious sense. Fix an smf X and charts (2.7.1). It follows from 2.5 that the underlying map g˜ij : cj (Uij ) → ci (Uij ) of each transition morphism (2.7.2) is real-analytic; therefore the maps c˜i : Ui → E0 equip space(X) with the structure of a real-analytic manifold

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˜ with model space E0 which we call the underlying manifold of X and denote by X. Assigning also to every smf morphism its underlying map we get a functor Smfs → Mfs ,

˜ X 7→ X

(2.9.1)

from the category of supermanifolds to the category of manifolds. We want to construct a right inverse to (2.9.1), i.e. we want to assign to every manifold Y a supermanifold, denoted Y again, the underlying manifold of which is Y again. First we note that given an lcs F , we may view it as Z2 -lcs by F1 := 0. Let AnF (·) be the sheaf of all real-analytic C-valued functions on Y with the obvious hermitian algebra structure. We recall from Theorem 2.5.2 (ii) that we have an isomorphism of hermitian algebra sheaves AnF (·) ∼ = O(·). Now let Y be given a real-analytic manifold with model space F . We want to assign to Y a supermanifold Y = (space(Y ), OY , (ci )i∈I ) with model space F (which we view by F = F0 as Z2 -lcs). Of course, we take the space Y as underlying space, and the sheaf OY (·) := An(·, C) of real-analytic C-valued functions on Y with the obvious hermitian algebra structure as structure sheaf. In order to get the needed atlas (ci )i∈I , we choose an atlas on the manifold Y in the usual sense, i.e. a S collection of open injective maps c0i : Vi → F where Y = i∈I Vi such that both c0i and (c0i )−1 : c0i (Vi ) → Vi are real-analytic. Using the usual function pullback we can view the ci as morphisms of hermitian ringed spaces ci : (Vi , OY |Vi ) = (Vi , AnVi (·, C)) → (c0i (Vi ), Anc0i (Vi ) (·, C)) = (c0i (Vi ), OF |c0i (Vi ) ) ; we claim that these morphisms equip Y with an smf structure. It is sufficient to show that given a real-analytic map φ : U → U 0 with U, U 0 ⊆ F open, the arising morphism of hermitian ringed spaces φ : (U, OF |U ) → (U 0 , OF |U 0 ) is a morphism of superdomains. Indeed, recalling that MF is just the sheaf of F -valued analytic maps, we can view φ as an element of MF (U ), and this is the element required in the definition of superdomain morphisms. One easily shows that this construction is functorial and provides a right inverse Mfs → Smfs

(2.9.2)

to (2.9.1). Henceforth, we will not make any notational distinction between a manifold Y and the corresponding smf. In particular, any purely even locally convex space E can be viewed as manifold, and hence as smf E. In fact, the functor (2.9.2) identifies the category of real-analytic manifolds with the full subcategory of all those supermanifolds the model space of which is purely even. Now let X again be an smf. For each Z2 -lcs F , we get by assigning to each F -valued superfunction its underlying function a sheaf morphism F O → An(·, FC ) ,

u 7→ u ˜;

(2.9.3)

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

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by Theorem 2.5.2 (ii), it is surjective on every open piece of X which is isomorphic with a superdomain. Also, (2.9.3) restricts to a sheaf morphism MF → An(·, F ). Now, applying onto X first the functor (2.9.1) and then the functor (2.9.2), we ˜ and, specializing (2.9.3) to F := R, we get a sheaf morphism get another smf X, OX → An(·, C) = OX˜ , and hence a morphism of hermitian ringed spaces ˜ O ˜ ) → (X, OX ) (X, X

(2.9.4)

the underlying point map of which is the identity. In fact, one shows by looking at the local model that (2.9.4) is an smf morphism ˜ →X X

(2.9.5)

which we call the canonical embedding. It is functorial again: For a given morphism µ : X → Y of smfs we get a commutative diagram µ

X −−−−→ x  

Y x  

µ ˜

˜ −−−−→ Y˜ . X Remark. (1) Of course, the term “canonical embedding” is slightly abusive as long as the notion of embedding for smfs has not been defined. (2) The functoriality of (2.9.5) is somewhat deceptive because it does not extend to families of morphisms: Given a Z-family of morphisms from X to Y , i.e. a morphism µ : Z × X → Y (cf. below for the definition of the product), there is no ˜ to Y˜ , i.e. a morphism natural way to assign to it Z-family of morphisms from X ˜ → Y˜ . All what one has is µ ˜ → Y˜ , which is only a Z-family. ˜ Z ×X ˜ : Z˜ × X 2.10. Cocycle description; products Let us describe supermanifolds in terms of cocycles of superfunctions: Assume ˜ with model space E0 together that we are given the model space E and a manifold X with charts (2.10.1) c˜i : Ui → c˜i (Ui ) ⊆ E0 . ˜ij ◦ g˜jk = g˜ik on c˜k (Ui ∩ Uj ∩ Uk ). Let g˜ij := c˜i c˜−1 j ; thus g ˜ the underlying manifold of a supermanifold X which Now we want to make X is on each Ui isomorphic to a superdomain in L(E) such that the maps (2.10.1) are the underlying maps of corresponding charts on X. For this purpose, we have to cj (Uij )) with underlying function choose a family (ˆ gij )i,j∈I of elements gˆij ∈ ME (˜ g˜ij and such that gˆii = xE , gˆij ◦ gˆjk = gˆik on c˜k (Ui ∩ Uj ∩ Uk ). For constructing an smf out of these data, we adapt the gluing prescription given in 2.7 to produce a hermitian algebra sheaf OX : For open U ⊆ X, an element

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of OX (U ) is a family (ui )i∈I of elements ui ∈ OL(E) (˜ ci (U ∩ Ui )) which satisfies ui ◦ gˆij = uj on c˜j (U ∩ Uij ). Moreover, we construct a chart ci (Ui ), OL(E) |c˜i (Ui ) ) ci : (Ui , OX |Ui ) → (˜ by taking, of course, c˜i as underlying map and u 7→ (u ◦ gˆij )j∈I as the superfunction pullback. In particular, we get the local coordinate xi := c∗i (xE ), so that xj = gji ◦ xi .

(2.10.2)

˜ OX , (ci )i∈I ) is the smf wanted; we call (ˆ gij ) the defining It is clear that X := (X, cocycle of X. ˜ c˜i : Ui → c˜i (Ui ) ⊆ E0 as above, one obvious Remark. Given the data E, X, lifting g˜ij 7→ gˆij of the transition maps does always exist; it is easily seen to produce ˜ × L(E1 ) (cf. below for the definition). the product smf X We now show that the category of smfs has finite products: Proposition 2.10.1. Given smfs X, Y there exists an smf X × Y together with morphisms prX : X × Y → X, prY : X × Y → Y such that the following universal property is satisfied: Given an smf Z and morphisms α : Z → X, β : Z → Y, there exists a unique morphism ζ : Z → X × Y such that prX ◦ ζ = α, prY ◦ ζ = β. We call X × Y the product of X and Y, and prX , prY the projections. Proof. We first treat the case that X, Y are superdomains U ⊆ L(E), V ⊆ L(F ). Of course, the product sought is just the superdomain U × V ⊆ L(E ⊕ F ) the underlying domain of which is space(U ) × space(V ); we now define the projection prU : U × V → U to be the linear morphism (cf. 2.6) induced by the projection E ⊕ F → E. prV is given quite analogously. Now, given α : Z → U , β : Z → U , we get elements α ˆ ∈ ME (Z) ⊆ ME⊕F (Z), F E⊕F ˆ (Z), and it is easy to see that the morphism ζ : Z → U × V β ∈ M (Z) ⊆ M given by ζˆ := α ˆ + βˆ implements the universal property wanted. Turning to the case of arbitrary smfs, we use the cocycle construction of above: Let be given smfs X, Y with model spaces E, F , and charts ci on Ui , dj on Vj , respectively. We get cocycles −1 E ck (Uik )) , gˆik := c[ i ck ∈ M (˜

−1 ˆ jl := d\ h ∈ MF (d˜l (Vjl )) . j dl

˜ × Y˜ is a manifold with charts c˜i × d˜j : space(Ui ) × space(Vj ) → E0 ⊕ F0 Now X and transition functions ˜ jl . ck × d˜l )−1 = g˜ik × h c˜i × d˜j (˜ ˆ jl to elements gˆik , h ˆ jl ∈ Using the lifting mechanism of above, we can lift gˆik , h E⊕F ˆ jl as the defining cocycle of the (˜ ck (Uik ) × d˜l (Ujl )), and we take gˆik h M

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˜ × Y˜ . This smf has charts supermanifold X × Y with underlying manifold X eij : (space(Ui ) × space(Vj ), OX×Y ) → ci (Ui ) × dj (Vj ); the compositions pr1 ◦eij

c−1 i

⊆

space(Ui ) × space(Vj ) −−−−→ ci (Ui ) −→ Ui → X agree on the overlaps and therefore glue together to an smf morphism prX : X ×Y → X; one defines prY analogously. Using the superdomain case and some chart juggling, it follows that our requirements are satisfied. Remark. The category of smfs has also a terminal object P = L(0) which is simply a point. 2.11. Comparison with finite-dimensional Berezin smfs Let X be an smf with model space E, and F be any Z2 -lcs. We get an embedding of sheaves F u ⊗ f 7→ (−1)|f ||u| f · u (2.11.1) O(·) ⊗ F → O (·) , where, of course, on every coordinate patch ((f · u)e )(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) = f · (ue )(k|l) (e1 , . . . , ek , e01 , . . . , e0l ) for e1 , . . . , ek ∈ E0 , e01 , . . . , e0l ∈ E1 . For arbitrary F , (2.11.1) is far away from being isomorphic; in fact, the image consists of all those u ∈ O F (·) for which there exists locally a finite-dimensional 0 Z2 -graded subspace F 0 ⊆ F such that u ∈ OF (·). It follows that if F is itself finite-dimensional then (2.11.1) is an isomorphism; this is the reason why the sheaves OF (·) become important only in the infinitedimensional context. From Theorem 2.8.1 we now get the usual characterization of morphisms by coordinate pullbacks (cf. Theorem 1.3.1). Corollary 2.11.1. Let X be an smf, let F be a finite-dimensional Z2 -graded vector space, and let f1 , . . . , fk , f10 , . . . , fl0 ∈ F ∗ ⊆ O(L(F )) be a basis of the dual F ∗. Given elements u1 , . . . , uk ∈ M(X), v1 , . . . , vl ∈ O (X)1,R there exists a unique smf morphism µ : X → L(F ) with the property µ∗ (fi ) = ui , µ∗ (fj0 ) = vj (i = 1, . . . , k, j = 1, . . . , l). It is given by µ ˆ=

k X i=1

f i · ui +

l X

(f 0 )i · vi ∈ MF (X)

j=1

where f 1 , . . . , f k , (f 0 )1 , . . . , (f 0 )l ∈ F is the left dual basis. We now turn to the comparison of our smf category with with the category of finite-dimensional real-analytic Berezin smfs. Adapting the definition given in 1.5 to

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our real-analytic situation, we define a (hermitian) real-analytic Berezin smf X as a hermitian ringed space (space(X), O), where space(X) is required to be Hausdorff, and which is locally isomorphic to the model space O (·) = An(·, C) ⊗C C[ξ1 , . . . , ξn ]

(2.11.2)

where ξ1 , . . . , ξn is a sequence of Grassmann variables. The hermitian structure on (2.11.2) is given by (1.5.1) again. Note that we have silently dropped the usual requirement of paracompactness since we do not know an infinite-dimensional generalization of it. Now, given an smf (space(X), OX , (ci )i∈I ) in the sense of 2.7 whose model space is finite-dimensional, it follows from the isomorphism (2.4.2) that we only need to forget the charts to get a real-analytic Berezin smf (space(X), OX ). Conversely, given such a real-analytic Berezin smf, we can take as atlas e.g. the family of all isomorphisms c : U → L(Rm|n ) where U ⊆ X is open; since, as already observed in Remark 2.6.1, the morphisms of finitedimensional superdomains “care for themselves”, this is OK. Corollary 2.11.2. We have an equivalence of categories between • the full subcategory of the category of smfs formed by the smfs with finitedimensional model space, and • the category of hermitian real-analytic Berezin smfs, which acts on objects as (space(X), OX , (ci )i∈I ) 7→ (space(X), OX ) . Finally, we consider a (fairly simple) functional calculus for scalar superfunctions: Fix an smf X. Every element f ∈ M(X) can be interpreted as a morphism f : X → L(R) = R (on the other hand, any odd superfunction f gives rise to a morphism Πf : X → L(ΠR), but we will not use that). Now if F : U → R is an analytic function on an open set U ⊆ Rn , and if f1 , . . . , fn ∈ M(X) then one can make sense of the expression F (f1 , . . . , fn ) provided that (f˜1 (x), . . . , f˜n (x)) ∈ U for all x ∈ X. Indeed, in that case, F (f1 , . . . , fn ) ∈ M(X) is the superfunction corresponding to the composite morphism (f1 ,...,fn )

F

X −−−−−−→ U → R . One easily shows: Corollary 2.11.3. (i) If F = F (z1 , . . . , zn ) = P nomial then F (f1 , . . . , fn ) = |α|≤N cα f1α1 · · · fnαn .

P |α|≤N

cα z1α1 · · · znαn is a poly-

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(ii) For i = 1, . . . , k let Ui be open in Rni , let V be open in Rk . Let Fi ∈ An(Ui , R), G ∈ An(V, R), suppose that (F1 (z1 ), . . . , Fk (zk ) ∈ V for all (z1 , . . . , zk ) ∈ U1 × . . . × Uk , and set H(z1 , . . . , zk ) := G(F1 (z1 ), . . . , Fk (zk )) Q for all (z1 , . . . , zk ) ∈ U1 × · · · × Uk . Let fi = (fi1 , . . . , fini ) ∈ ni M(X) for i = 1, . . . , k, and suppose that the image of fi : X → Rni lies in Ui . Then we have the identity H(f1 , . . . , fk ) := G(F1 (f1 ), . . . , Fk (fk )) in M(X). Corollary 2.11.4. An even scalar superfunction f ∈ O(X)0 is invertible iff ˜ f(x) 6= 0 for all x ∈ X. Proof. The condition is clearly necessary. Set U := R \ {0}, F : U → U, c 7→ c−1 . We have f f ∈ M(X) for f ∈ O(X)0 , and if f satisfies our condition then we may form f −1 := f · F (f f ); using the Corollary above we get f −1 f = 1. Thus, each stalk Ox of the structure sheaf is a (non-Noetherian) local ring. Also, exp(f ) is defined for any f ∈ M(X), and we have the identity exp(f +g) = exp(f )exp(g). 2.12. Sub-supermanifolds Essentially, we will follow here the line of [25]. Let be given smf’s X, Y with model spaces E, F , respectively, and an smf morphism φ : X → Y . We call φ linearizable at a point x ∈ X iff, roughly spoken, it looks at x like a linear morphism (cf. 2.6), i.e. iff there exist neighbourhoods U 3 x, ˜ V 3 φ(x), superdomains U 0 ⊆ L(E), V 0 ⊆ L(F ), and isomorphisms iU : U → U 0 , iV : V → V 0 such that the composite φ0 := iV φ(iU )−1 : U 0 → V 0 ⊆ L(F ) is a linear morphism L(α). We then call α : E → F the model map of φ at x. (We note that the model map is uniquely determined up to automorphisms of E, F since it can be Y ; cf. [42].) identified with the tangent map Tx X → Tφ(x) ˜ We call an even linear map F → E of Z2 -lcs a closed embedding iff it is injective, its image is closed, and the quotient topology on the image is equal to the topology induced by the embedding into F . We call F → E a split embedding iff it is a closed embedding, and there exists a closed subspace E 0 ⊆ E such that E = E 0 ⊕ F in the topological sense. We recall that this is equivalent with the existence of a linear continuous projection E → F . Thus, if F is finite-dimensional and E a Fr`echet Z2 -lcs then every injective linear map F → E is a split embedding. We call a morphism of smf’s φ : X → Y a regular closed embedding if its underlying point map is injective with closed image, and φ is linearizable at every point x ∈ X, with the model map at x being a closed embedding.

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One easily shows that if φ : X → Y is a regular closed embedding then φ is a monomorphism in the sense of category theory, i.e. given two distinct morphisms φ Z→ →X the composites Z → →X → Y are still different. Two regular closed embeddings φ : X → Y, φ0 : X 0 → Y are called equivalent iff there exists an isomorphism ι : X → X 0 with φ0 ◦ι = φ; because of the monomorphic property, ι is uniquely determined. We call any equivalence class of regular closed embeddings into Y a subsupermanifold (sub-smf ) of Y , and we call the equivalence class of a given regular closed embedding φ : X → Y the image of φ. (Sometimes, by abuse of language, one calls X itself a sub-smf, with φ being understood.) Finally, we call a sub-smf φ : Y → X a split sub-smf if its tangential map is everywhere a split embedding. ˜ a split sub-smf of X, For example, the canonical embedding (2.9.5) makes X with the model map being the embedding E0 ⊆ E. Note that the notion of a regular closed embedding is presumably in general not transitive (although we do not know a counterexample). However, it is easy to see that if φ : X → Y is linearizable at x, and Y ⊆ Z is a split sub-smf then φ the composite X → Y is linearizable at x, too. In particular, the notion of a split sub-smf is transitive. Let be given an smf X and a family of superfunctions ui ∈ O Fi (X), i ∈ I, where Fi are Z2 -lcs. We call a sub-smf represented by φ : Y → X the sub-smf cut out by the superfunctions ui iff the following holds: We have φ∗ (ui ) = 0; moreover, if φ0 : Y 0 → X is any other smf morphism with 0 ∗ (φ ) (ui ) = 0 for all i then there exists a morphism ι : Z → X with φ ◦ ι = φ0 . (Note that by the monomorphic property of φ, ι is uniquely determined.) Of course, for a given family (ui ), such a sub-smf needs not to exist; but if it does, it is uniquely determined. Also, using the universal property with Y 0 := P being a point we get that if Y exists then φ˜ : space(Y ) → space(X) maps space(Y ) homeomorphically onto {x ∈ space(X) :

uei (x) = 0 ∀i ∈ I} .

However, note that the superfunction pullback OX → OY need not be locally surjective; it is so only in a finite-dimensional context. Remark. One can show that appropriate formulations of the inverse and implicite function theorems hold provided that the model spaces of the smf’s involved are Banach spaces. 2.13. Example: the unit sphere of a super Hilbert space As a somewhat academic example, let us consider the super variant of the unit sphere of a Hilbert space: As in [39], a super Hilbert space is a direct sum of two ordinary Hilbert spaces, H = H0 ⊕ H1 ; the scalar product will be denoted by H × H → C,

(g, h) 7→ hg|hi

SUPERGEOMETRY AND QUANTUM FIELD THEORY, OR:. . .

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(the unusual notation helps to keep track of the action of the second sign rule). Let Hr denote then underlying real vector space of H. Let x ∈ MHr (L(Hr )) denote the standard coordinate. Specializing (2.7.5) to the R-bilinear pairing Hr × Hr → C ,

(g, h) 7→ hg|hi

we get a bilinear pairing O

Hr

(·) × O Hr (·) → O (·) .

(2.13.1)

Applying this to (x, x) yields a superfunction denoted by kxk2 ∈ M(L(Hr )) (the notation looks abusive, but can be justified, cf. [41, 4.4]). The Taylor series of kxk2 at zero is given by (kxk2 )0 = u(2|0) + u(0|2) where u(2|0) : (Hr )0 × (Hr )0 → C ,

(g, h) 7→ hg|hi + hh|gi = 2 · Rehg|hi ,

u(0|2) : (Hr )1 × (Hr )1 → C ,

(g, h) 7→ hg|hi − hh|gi = 2 · Imhg|hi .

Of course, the underlying map is (Hr )0 → R, g 7→ kgk2 . Proposition 2.13.1. Let H be a super Hilbert space, and suppose that H0 6= 0. There exists a sub-smf S → L(Hr ) cut out by the element kxk2 − 1 ∈ M(L(Hr )). Moreover, we have an isomorphism ∼ =

ι : S × R+ → L(Hr ) \ 0 where R+ = {c ∈ R : sphere of L(Hr ).

(2.13.2)

c > 0} is viewed by 2.9 as smf. We call S the super unit

Proof. Looking at the situation with Z = P being a point, we see that if ˜ of S exists its underlying manifold can be identified with the usual unit sphere S (Hr )0 . In order to use stereographic projection, we fix an element h ∈ H0 , khk = 1; set E := {ξ ∈ Hr : Rehξ|hi = 0} . Note (H1 )r ⊂ E; the real Z2 -graded Hilbert space E will be the model space for S. ˜ consisting of two homeomorphisms We get an atlas of the manifold S ˜ \ {±h} → E0 , c˜± : S ξ 7→ ±h +

1 (±h ±Rehξ|hi−1

− ξ) ;

(2.13.3)

2 the inverse maps are c˜−1 ± (η) = η 7→ ±h + 1+kηk2 (η ∓ h). The transition between the charts is η 7→ η/kηk2 . g˜ := c˜+ c˜−1 − : E0 \ {0} → E0 \ {0} ,

In accordance with (2.10), we lift g˜ to the superfunction g[y] := y/kyk2 ∈ ME (L(E) \ {0}) ,

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T. SCHMITT

where y ∈ ME (L(E)) is the standard coordinate. We get an smf S with underlying ˜ and model space E. The c˜± become the underlying maps of two charts manifold S c± : S \ {±h} → L(E) , ∗ E and we get coordinates y± := cc ± = c± (y) ∈ M (S \ {±h}); from (2.10.2) we get

y− = y+ /ky+ k2 . Set e± := ±h +

(2.13.4)

2 (y± ∓ h) ∈ MHr (S \ {±h}) . 1 + ky± k2

Using (2.13.4), one computes that the restrictions of e− , e+ onto S\{h, −h} coincide; hence we can define the smf morphism : S → L(Hr ) by ˆ|S\{±h} = e± . Of course, ˜ ⊂ H0 . Using again the pairing (2.13.1) one the underlying map ˜ is the inclusion S computes |ˆ i2 − 1 = 0 . (2.13.5) ∗ (kxk2 − 1) = hˆ Now we can define the morphism (2.13.2) by ˆι := tˆ (if being pedantic, one should write pr∗1 (t)pr∗2 (ˆ ) instead) where t ∈ M(R+ ) is the standard coordinate. In order to show that (2.13.2) is an isomorphism, we construct morphisms κ± : L(Hr ) \ (±R+ h) → R+ × (S \ {±h}),   x 1  ±h − κ± ∗ ((t, y± )) = kxk , ±h + Rehx|hi kxk ± kxk − 1 p where the superfunction kxk := kxk2 ∈ M(L(Hr ) \ 0) is defined by the functional calculus of 2.11. Now κ+ , κ− coincide on the overlap L(Hr ) \ Rh: One computes kκ+ ∗ (y+ )k2 =

−Rehx|hi − kxk Rehx|hi − kxk

(2.13.6)

and hence ∗

κ+ (y− ) = κ+ =

∗

1 ky+

y+ k2

=

1 kκ+

∗ (y

+ )k

2

κ+ ∗ (y+ )

Rehx|hi − kxk κ+ ∗ (y+ ) = κ− ∗ (y− ) . −Rehx|hi − kxk

Hence κ+ , κ− glue together to a morphism κ : L(Hr ) \ {0} → R+ × S and one shows by a brute force calculation that ικ = 1L(Hr )\{0} , κι = 1R+ ×S . Hence (2.13.2) is indeed an isomorphism, and it follows that is a regular closed embedding which makes S a sub-supermanifold of L(H).

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We claim that the composite pr2

κ

ω : L(Hr ) \ 0 → R+ × S → S → L(Hr ) is the “normalization morphism” ω ˆ=ω ˆ [x] =

1 x. kxk

Indeed, ω ˆ = ω (x) = κ pr2 (x) = κ ±h + ∗

∗

∗

∗

2

= ±h +

1+

kκ∗ (y

± )k

2

2 (y± ∓ h) 1 + ky± k2

(κ∗ (y± ) ∓ h) ;

investing (2.13.6), the result follows after some more steps. In view of (2.13.5), it remains to prove: Given some smf morphism φ : Z → L(Hr ) with φ∗ (kxk2 − 1) = 0 , ˆ − 1) ˆ φi ˆ = kφk ˆ 2 , i.e. (kφk φ factors through S. Indeed, we have 1 = φ∗ (hx|xi) = hφ| ˆ + 1) = 0; by Corollary 2.11.4, the second factor is invertible, and hence (kφk ˆ = 1. kφk

(2.13.7)

˜ ⊆ S ˜ ⊆ L(Hr ) \ 0. Now we claim that φ coincides with the In particular, Im(φ) composite φ

ω

Z → L(Hr ) \ 0 → L(Hr ) \ 0 . Indeed, using (2.13.7), c= ωφ

1 φ∗ (kxk)

φ∗ (x) =

1 kφ∗ (x)k

φ∗ (x) = φ∗ (x) = φˆ ,

and now φ = ωφ = pr2 κφ provides the factorization wanted.

Remark. The whole story becomes much more transparent if one looks at the Z-valued points in the sense of 1.7. Given any smf Z, we get from Theorem 2.8.1 an identification of the set L(Hr )(Z) of Z-valued points of L(Hr ) with MHr (Z), and thus the structure of a (purely even) vector space on L(Hr )(Z). Of course, thus structure varies functorially with Z. Moreover, (2.13.1) yields a bilinear map L(Hr )(Z) × L(Hr )(Z) = MHr (Z) × MHr (Z) → M(Z), i.e. a scalar product on L(Hr )(Z) with values in the algebra M(Z). Now the Z-valued points of S identify with those Z-valued points of L(Hr )(Z) which have squared length equal to one under this scalar product: S(Z) = {ξ ∈ L(Hr )(Z) : hξ|ξi − 1 = 0} .

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Now the process of stereographic projection can be applied in the vector space ±h L(Hr )(Z) for each Z, using the constant morphisms ±h : Z → P → L(Hr ) as projection centers and the subspace L(E)(Z) as screen. One gets maps cZ ± : {ξ ∈ S(Z) : (±Rehξ, hi − 1) is invertible} = (S \ {±h})(Z) → L(E)(Z) given by the same formula (2.13.3). These maps vary functorially with Z again; on the other hand, we noted already in (1.7) that the functor (1.7.1) is faithfully full, that is, “morphisms between representable functors are representable”. Thus, the maps cZ ± must be induced by suitable morphisms, and these are indeed just our charts c± . The other morphisms ι, , κ, ω can be interpreted similarly. References [1] W. A. Abramow and Iu. G. Lumiste, “Superspace with underlying Banach fibration of connections and supersymmetry of the effective action (in Russian)”, Izw. wyssh. utch. zaw. 1 (284) (1986) 3–12. [2] M. Batchelor, “Two approaches to supermanifolds”, Trans. Amer. Math. Soc. 258 (1979) 257–270. [3] F. A. Berezin, “The Method of Second Quantization”, Pure and Appl. Phys. 24, Academic Press, New York – London 1966. [4] , Introduction into the Algebra and Analysis with Anticommuting Variables (in Russian), Izd. Mosk. Un-ta 1983. [5] F. A. Berezin and M. S. Marinov, “Particle spin dynamics as the Grassmann variant of classical mechanics”, Ann. of Phys. 104, (2) (1977), 336–362. [6] I. N. Bernshtejn and D. A. Leites, “How to integrate differential forms on a supermanifold” (in Russian), Funkts. An. i pril. t. II w. 3 (1977) 70–71. [7] , “Integral forms and the Stokes formula on supermanifolds” (in Russian), Funkts. An. i pril. t. II w. 1 (1977) 55–56. [8] N. N. Bogoliubov, A. A. Logunov, A. I. Oksak and I. T. Todorov, “General principles of quantum field theory” (in Russian), “Nauka”, Moskwa 1987. [9] C. Bartocci, U. Bruzzo and D. Hern´ andez Ruip`erez, The Geometry of Supermanifolds, Kluwer Acad. Publ., Dordrecht, 1991. [10] Y. Choquet-Bruhat, “Classical supergravity with Weyl spinors”, Proc. Einstein Found. Intern. 1 (1) (1983) 43–53. [11] C. Crnkovic and E. Witten, “Covariant description of canonical formalism in geometrical theories”, in Three Hundred Years of Gravitation, eds. S. W. Hawking and W. Israel, Cambridge 1987. [12] J. Dell and L. Smolin, “Graded manifold theory as the geometry of supersymmetry”, Commun. Math. Phys. 66 (1979) 197–221. [13] B. DeWitt, “Quantum Gravity: A new synthesis”, in General Relativity., eds. S. W. Hawking and W. Israel, Cambridge Univ. Press, Cambridge, 1979. [14] , Supermanifolds, Cambridge Univ. Press, Cambridge, 1984. [15] A. Douady, “Le probl`eme des modules pour les sous-espaces analytiques compacts d’un espace analytique donn´e”, Ann. Inst. Fourier 16 (1966) 1–95. [16] H. Ewen, P. Schaller and G. Schwarz, “Schwinger terms from geometric quantization of field theories”, J. Math. Phys. 32 (1991) 1360–1367. [17] J. Glimm and A. Jaffe, “Quantum Physics. A Functional Integration Point of View”, Springer-Verlag, Berlin, 1981. [18] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, Princeton, New Jersey, 1992.

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