An Invitaiton To The Algebra Of Canonical Commutation Relations

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Leuven Notes in Mathematical

and ‘Theoretical Physics

Volume Series A: Mathematical

2

Physics

An Invitation to the Algebra of Canonical Commutation

Relations

Dénes

Leuven University Press

Petz

Contents

Preface

1

Vv

Representations

of canonical commutation

relations

The C*-algebra of the canonical commutation

relation

1 9

States and fields

17

Fock states

29

Fluctuations and central limit

39

The KMS

49

condition

The duality theorem

59

8

Completely positive maps

71

9

Equivalence of states

81

10 The selfdual approach

95

Bibliography

101

ill

lv

Preface This volume contains the written notes of a series of lectures delivered at the University of Leuven in the academic year 1988/89. The aim of the lectures was to give an introduction to the rigorous treatment of the canonical commutation relation and to demonstrate that the method of operator algebras in a suitable tool for this purpose. These notes cover quasifree states, the K MS-condition, central limit theorems and the equivalence of Fock states. Fundamentals of operator algebras and functional analysis are required to follow the presentation. Physical motivation and relevance are slightly commented. The author is grateful to Mark Fannes and André Verbeure for their hospitality at the Institute for Theoretical Physics. Without their expertise on the subject the notes would not have been completed. It is a pleasure to thank Anita Raets for her skilful wordprocessing. Leuven

1989.

Dénes Petz Mathematical Institute

Hungarian Academy of Sciences, Budapest

Chapter

1

Representations of canonical commutation relations In the Hilbert space formulation of quantum mechanics one considers the abstract selfadjoint operators q1, G2,---,Qn; P1,P2;---;Pn acting on a Hilbert space H and satisfying the Heisenberg commutation relations for n degrees of freedom :

laa] lai,Pj)]

= [pis pj] = 0 = id(2,9)1

(i,j =1,2,...,n)

(1.1)

where for any pair of operators A and B on H the symbol [A, B] stands for the commutator AB — BA; I is the identity operator. There is no essential difference between the case n = 1 and the general n, therefore we restrict ourself for one degree of freedom. Then gq is associated to the position and p to the momentum of a single particle. Since the early days of quantum mechanics it has been a basic problem to find concrete operators satisfying the Heisenberg commutation relations.

If one takes the complex Hilbert space L?(R), g as the multiplication operator by the variable, i.e.

(qs f)(t) =t F(é)

(1.2)

and p the differentiation, i.e.

(psf)(t) = 4 f'(t)

(1.3)

2

CHAPTER

1

then these operators form a representation of the Heisenberg commu-

tation relation on L?(R). dense set of functions, This representation is A problem to solve selfadjoint matrices q

(Formally gs and ps may be defined on some

for example the C™ ones with compact support.) called the Schrodinger representation. in the matrix mechanics of Heisenberg is to find and p satisfying the relation

(1.4)

lq,p] =i.

If the Schrodinger representation is unique in a suitable sense, then all results of the matrix theory must be equivalent to those of the Schrodinger theory. If p and q were finite matrices satisfying (1.4) then the trace of the left hand side would be zero in contradiction with the nonzero trace of the right hand side. Hence Heisenberg did not find a solution within the set of finite matrices but found a solution in the form of infinite matrices :

(1.6) This representation (qz, pz) is called the Heisenberg representation. The Schrodinger and Heisenberg solutions of the representation problem coincide. One can find an orthonormal basis of the Hilbert

space L7(R) such that the matrix of qs (ps) in the given basis is qy (py). For any real number ¢ one deduces from equation (1.4) that

CO ps Peal +.

e' P q e UP

=

qt+tl

For U(a) = exp(iap) and V(b) =

U(a) V(b) U(-a

(1.7) exp(tbg) Oy b € R) we have

@y @n—0

=

REPRESENTATIONS

OF THE C.C.R.

» wy (q+al)”

3

= exp(ib(q+al)) =eV(b).

Therefore,

U(a) V(b) = eV (b) U(a)

(a,b € R)

(1.8)

which is called the Weyl form of the canonical commutation relation.

(1.8) contains only unitary operators and hence it will be suitable for a C*-algebraic approach. It was von Neumann’s idea to compress the two families of operators, U and V, in a single one. If one takes

S(a, 6) = exp (— sia) U(a) V(b)

(a,b € R)

then U and V may be obviously recovered from S and (1.8) is equivalent to the following relation

S(a, b) S(c, d) = exp (Fila

— be)

S(a+c,b+d)

Let H be a Hilbert space and assume that S(a,b) € B(H).

(1.9) We say

that S is a representation of the canonical commutation relation if the following conditions hold.

(i) S(a, 6) is a unitary (a,b € R). (ii) S(a, b)* = S(—a, —b) (a,b € R). (iii) S(0,0) is the identity. (iv) The relation (1.9) is satisfied for all a, b,c,d € R. (v) The mapping (a,b) ++ S(a,b) is continuous in the weak operator topology. Remember that on the group of unitaries the weak and strong operator

topologies coincide. So (v) may be formulated also another way. From the Schrodinger representation one gets a representation of the

canonical commutation relation (CCR) in the above sense on L?(R).

4

CHAPTER

1

Since U and V are given by exponentiation of selfadjoint operators, they are strongly continuous. So is their product. A representation of the CCR on a Hilbert space H is called irreducible if only the trivial closed subspaces of H are invariant under all

S(a, 6). Proposition ducible.

1.1

The Schrodinger representation of the CCR

ps and gg are related by the Fourier transform.

1s trre-

If 7 f denotes the

Fourier transform of f € L?(IR) defined as Ff(t)= 7a

/ e 5 F(s)ds

then Fqs F* = ps.

We show that if f € L?(R) is a nonzero vector and (g, S(a,6)f) = 0 for every (a,b) € R? then necessarily g = 0. This gives that the smallest

invariant closed subspace containing f is L?(R) itself. Since the first factor in

(g, S(a,b) f) = e219 (ePg, ef) is nonzero, we concentrate on the second one. we have In(e *?g,

e°4 f)

_

fe

[em

By direct computation

[eeta(s)

Faas

du dt

Assuming that this is 0 we may refer to the uniqueness of the Fourier transform and conclude that

e™

F(u) / e

5 9(s)ds = 0

for almost all t,u € R. As f 40 we arrive at

[e*a(s)as = 0 for almost all ¢ € R. So g must be 0. The uniqueness theorem of von Neumann asserts that up to a unitary equivalence the Schrodinger representation is the only irreducible

(continuous) representation of the CCR.

REPRESENTATIONS

OF THE C.C.R.

5

Theorem 1.2 Jf R’ 3 (a,b) + S(a,b) € B(H)

and R’ 3 (a,b)

Bb

S'(a,b) € B(K) are irreducible (continuous) representations of the CCR on the Hilbert spaces H and K, respectively, then there exists a unitary U:H—->K such that

S(a,b) = U* S"(a,b)U

(a,b € R).

For h € L'(R’) the formula

s(f,9) = / h(a,b)(S(a,b)f,g)dadb—

(f,g EH)

sets a bounded sesquilinear form and there is an operator S(h) € B(H) such that

(Sth) f.9)=sf9)

(fg € H)

and ||S(h)|| < ||A|;. We write simply

S(h) = / h(a, b) Sa, b) dadb. It is clear that for a real valued h the operator S(h) is selfadjoint. We show that the linear correspondence ht+> S(h) is an injection.

Assume that S(h) = 0 and compute (S(—a, —b)S(h)S(a, b)g,f) = [| e MeMh(u, v)(S(u,v)g, fydu dv. By our hypothesis this vanishes for every f, g € H and every (a, b) € R’. Hence

h(u,v)(S(u, v)g, f) = 0

(1.10)

for almost all u,v € R and for every f,g € H. (If you are really pedantic you should argue by the separability of H at this point.) (1.10) readily gives h = 0. Now set a bounded selfadjoint operator A on H as

A= | [ex

1

(—F (lal? + P))

S(a, b) da db

(1.11)

and one checks that

A S(a,b)A = 2m e alla? +10?) 4 |

(1.12)

6

CHAPTER

1

In particular, P = 5- A is a projection and it can not be 0. It isa projection of rank one.

Let f and g 4 O be orthogonal vectors such

that Pf = f and Pg = g. Due to the relation (1.12) we have (S(a, b)g,f) = (P S(a,b) Pg, f) = constant (4g, f) and the irreducibility of the representation (a,b) +> S(a,b) yields that f = 0. P is a rank one projection indeed and we may assume that

IIgl| = 1. We construct similarly a vector 7 € K starting from the irreducible

representation (a, b) +> S’(a, 6). Set U S(a, b)g = S"(a, b)n

(a,b € R)

(1.13)

and extend U by linearity to the linear span of {S(a,b)g : a,b € R}. So U will be an inner product preserving densely defined operator with dense range. It follows immediately from (1.13) that U intertwines between the representations S and S”. Von Neumann’s uniqueness theorem is true in any finite dimension and the above proof with small alteration will go through. (The inter-

ested reader may consult with the nicely written original paper [vN].) Let C(R) stand for the C*-algebra of bounded continuous function on R (endowed with the sup norm) It contains the characters

Xa(x) = e* (\ € R). Set A(R) as the closure of the linear span of the characters. A(R) is called the algebra of almost periodic functions. possesses a natural inner product

It

(F,9) = Jim 5p | Fle) ale) ae and its completion is a Hilbert space. {x, : 4 € R} forms an orthonormal basis in H and in particular, H is not separable. Define the unitary operators

Vo(@)xa = Xata

(aeR)

Up(0)xa = exp(ibaA)x,

(b€ R).

Then Up(a) and Vo(b) satisfy the Weyl commutation relation Up(a)Vo(b) = exp(i a b)Vo(b)Up(a)

(a,b € R).

REPRESENTATIONS

OF THE C.C.R.

7

This representation of the Weyl commutation relation is irreducible and can not be unitarily equivalent to the Schrodinger representation since H is not separable. It is easy to see that

000) -Drall={ Pt poy hence the strong continuity of the group Vo(0) fails completely.

Let (a,b) +» S(a,b) be a representation of the CCR. The linear span of {S(a,b) a,b € R} is a *-algebra. (It is closed under the product operation due to (1.9).) Its norm closure is a C*-algebra, call it for a while A(S). It follows from the uniqueness theorem that for two irreducible representations S and S” there is a C*-algebra isomorphism

a:

A(S’) —

A(S) such that a(S’(a,b))

= S(a,b)

(a,b € R).

The

main subject of the next chapter is to show that the isomorphism exists independently of the irreducibility of the representations (and even

finite dimensional ”testfunction space” will not be required). The C*algebraic point of view prefers the Weyl form of the canonical commutation relation. ‘To show that the question of unicity for the Heisenberg

form is more subtle we describe an example of Fuglede ({Ful). As a common dense domain the span of the functions {2 exp(—ax* + br) All these functions

PD for the operators g and p he takes

: n€ Zr,

are entire analytic and

a>0O,

be C}

for f

€

D

and

z €

C

f(z) makes sense. Denote by T and M the following operators

(Tf)\(z) = f(x+iv2r) (Mf)(z) = exp(V27r 2) f(z) and define

dr

=

Qst+T

Pr

=

Dst+tM

(Here (qs, ps) is the Schrédinger representation.) Fuglede proved that (ii) | The closure of pr and gr is selfadjoint

8

CHAPTER

1

and the pair (pr, GF) is not unitarily equivalent to any direct sum of Schrodinger representations. Under additional hypothesis the uniqueness of the Schrodinger representation may be proven in the frame of selfadjoint operators. This subject is out of the scope of the present lecture and we refer to the

monograph [Pu]. In the selection of the material in this chapter, we benefited from

the unpublished notes [Ve]. x

Chapter

2

The C*-algebra of the canonical relation Let H be a

commutation

real linear space.

A bilinear form o is called symplectic

form if o(z,y) = —o(y,x) for every x,y € H. o is nondegenerate if o(x,y) = 0 for every y € H implies that x = 0. The pair (H,c) will be refered to as a symplectic space. If it is not stated explicitly otherwise, all symplectic spaces will be assumed to be nondegenerate. In Chapter 1 we met already a symplectic form. The bilinear form

((a,8), (c,d) + 5 (ad ~ be) appearing in (1.9) is a nondegenerate

symplectic form on R’.

More

generally, if # is a complex Hilbert space then o(f,g) = Im(f,g) is a nondegenerate symplectic form on the real linear space H. In finite dimension this is the typical way to define a symplectic space. (Hence the dimension of a nondegenerate symplectic space is even or infinite.)

Let (H,o) be a symplectic space. The C*-algebra of the canonical commutation relation over (H,o), written as CCR(H, 0), is by definition a C*-algebra generated by elements {W(f) : f € H} such that

Gi) W(-f)=Wf) 9 (fe #) (ii) = W(f)W(g) =expliol(f,g9))WFt+9) 9

(fg €)

10

CHAPTER

2

Condition (ii) tells us that W(f)W(0) = W(0)W(f) = W(f). Hence W (0) is the unit of the algebra and it follows that W(/) is a unitary for every f € H. Comparing with the previous definition for a (continuous) representation of the CCR you see that the continuity assumption (v) is missing here. Weak and strong continuity do not make sense here

(and norm continuity would be really too much as it turns out later.) Theorem 2.1 For any nondegenerate symplectic space (H,o) the C*algebra CCR(H, co) exists and unique up to isomorphism. To establish the existence will be easier than proof of the uniqueness. Consider H as a discrete abelian group (with the vectorspace addition).

?(H) = 1

-HOC:

S- JF (a)/? < +0] xeH

is a Hilbert space. (Any element of /?(H) is a function with countable support.) Setting

(R(z)F)(y) =expioly,2))F@+y)

(#,y € #)

(2.1)

we get a unitary R(x) on /?(H) and one may check that R(a#1) R(x) = exp(io (x1,

2) R(a1 + 22).

The norm closure of the set

{So

Ree

>

NEC,

1: i W (yi)

(2.2)

= >: i (Yi)

holds. Let H stand for the dual group of the discrete group H. H consists of characters of H and endowed by the topology of pointwise convergence forms a compact topological group. We consider the normalized

Haar measure on H.

The spaces /?(H) and L?(H)

are isomorphic by

the Fourier transformation, which establishes the unitary equivalence between the above 7 and 7 defined below.

ayAn=xyWwAon Hence

Wwe,

xeH,

AE D(H,H))

> Aim (Yi)|| = > ditt (y)||.

(2.3)

A closer look at the definition of 7 gives that 7(y) is essentially a multiplication operator (by x(y)W(y)) and its norm is the sup norm. That is,

>

i (Ys)

= sup

>>

dix (yi) W (yi)

xe

Ah,

(2.4)

12

CHAPTER

2

Since the right hand side is the sup of a continuous function over H this sup may be taken over any dense set. Let us set

G = {exp(2io(az,-) Clearly, GC

Hisa

subgroup.

: « € A}.

The following result is at our disposal

(see (23.26) of [H-R]). If K C His a proper closed subgroup then there exists 0 4h

€ H

such that k(h) = 1 for every k € K. Assume that exp(2io(x,y)) = 1 for every x € H.

Then for every

t € R there exists an integer 1! € Z such that to(z,y) =I.

This is

possible if o(a, y) = 0 (for every x € H) and y must be 0. According to the above cited result of harmonic analysis the closure of G must be the whole H. Now we are in a position to complete the proof. For x(-) = exp(2ia(z,-)) EG we have

> di x(yi) W (ya)

~

|e)

»

= |X wow

AW (y)W(-2)|

=

and this is the supremum in (2.4). Through (2.4) we arrive at (2.2). The previous theorem is due to Slawny

proof that CCR(H,c)

[Sl].

We learnt from the

has a representation on /?(H) given by (2.1).

The subalgebra

»

A(z) R(x) : A: H > C has finite spor |

xceH

is dense in CCR(H, co) and there exists a state 7 on CCR(H,c) that

7 (> A(z) R(2)) = X(0).

such

(2.5)

It is simple to verify that 7(ab) = 7(ba). Therefore, 7 is called the tracial state of CCR(H,o). We can use 7 to prove the following.

THE C*-ALGEBRA

OF THE C.C.R.

13

Proposition 2.2 If f,g © H are different then

|W (Ff) — W(g)|| > v2. For h, # hg , we have 7(W(h,)W(—he)) = 0.

Hence ||W(f) — W(g)|? = 7((W(f) — W(g)*(W(F) — W(g))) = 2. It follows from the Proposition that the unitary group t 4» W(tf)

is never normcontinuous and the C*-algebra CCR(H,c)

can not be

separable. Slawny’s theorem has also a few important consequences. Clearly for (Hi,0,) C (He,02) the inclusion CCR(Ai,0,) C CCR(Apo, 02) must hold. (If H, is a proper subspace of Hy then CCR(H;, 02) is

a proper subalgebra of CC'R(A2,02).)

If T : H — A is an invertible

linear mapping such that

o(f,g) =o(Tf,T9)

(2.6)

then it may be lifted into a *-automorphism of CCR(H,o). there exists an automorphism yr of CC R(H, oc) such that

yr(W(f)) = W(PP)

Namely,

(2.7)

A simple example is the parity automorphism

mW(f)) =W(-f)

(f € #1).

(2.8)

Let (H,o) be asymplectic space. A real linear mapping J : H — H is called a complex structure if

(i) (ii)

J? =—id off, f) 0 for every A; € C and h; € A if and only if the function F:

hw

oy(Wwth))

(h € H)

is positive-definite. Due to Bochner’s theorem ([H-R], 33.1) there is a probability measure yz on the compact dual group A such that

F(h)= f x(h)du(x)

(he H).

Hence

sup {p(a*a)'/? : y isastate} = sup {x(a"a)'? where fora = 5> \; W(h;) € A(H,o)

:xXE a}

x(a) (or a(y)) is defined as

So Aix(hi) In this way every element a of A(H,c) may be viewed to be a continuous function on H and

all =sup {la sxe A}

(ae ACH, 0).

A(H, oc) evidently separates the points of H and the Stone-Weierstrass theorem tells us that CC R(H,c) is isomorphic to the C*-algebra of all continuous functions on the compact space H. The case of a vanishing symplectic form does not occur frequently, however, it may happen that H = Hp @ A, and o(ho

Bh,

,

hy

®

hi)

=

01

(hi

@ h})

16

CHAPTER

with a nondegenerate symplectic form o,; on H,.

2

Then the *-algebra

A(H, c) is the algebraic tensor product of A(Hp,0) and A(H;,0,) and CCR(H,c) will be

(Note that since CC-R(Hp,0) is commutative, the C*-norm on the tensor product is unique.) Now we review briefly the general case. For a degenerate symplectic form o we set

Hp={xEH:

o(2t,y)=0

forevery

ye H}

for the kernel of c. A(Hp,0) is the center of the *-algebra A(H,o) and there exists a natural projection & given by

E (x:

sow)

ceH

= S> XA(«)W(c)

(2.17)

x€ Ho

and mapping A(H,o) onto A(Ao,0). Having introduced the minimal regular norm we observe that CC'R(Ho,0) is the center of CCR(H, 0c) and F is a conditional expectation. The maximal *-ideals of CC R(H, co) are in one-to-one correspondence with those of CCR(Hp,0). In par-

ticular, CCR(H,o) is simple if and only if Hp = {0}, that is, o is nondegenerate. Concerning the details we refer to [M-S-T-V]. For a nondegenerate

symplectic

readily that CC R(H,c) is simple.

form

Slawny’s

theorem

provides

Chapter States

3

and

fields

Let (H,o) be a symplectic space and 7 : CCR(H,o) > B(H) a representation.

If

try (m(W(tf))§, 0)

(€,n € H)

is continuous then there is a selfadjoint B,(f) due to the Stone theorem such that

mW(tf)) =exp(itBr(f))

(te R)

(3.1)

B,(f) is called field operator. Since the unitary group 7(W(tf)) is not normcontinuous (cfr. Proposition 2.2), B,(f) is unbounded. From physical point of view the field operators are more relevant in may cases

than elements of CCR(H,oc).

The representation 7 is called regular if

the field B(f) exists for every f in the test function space.

Let us consider the representation R given by (2.1). One computes that

(R(t2)8,.8)={ 1 425

(av eH).

So it is not regular. We saw another non-regular representation in Chapter 1 on the Hilbert space of almost periodic functions. On the other hand, the Schrodinger representation is regular.

A linear functional y of a C*-algebra A is called state if y(J) = 1 and yp(a*x) > 0 for every x € A. (The latter condition may be replaced by ||y|| = 1.) Every state gives rise to a representation through the GNS construction. A state y of CCR(H,0o) is called regular if the corresponding GNS-representation is regular. 17

18

CHAPTER

3

Let X be an arbitrary (nonempty) set. A function F : X x X 4C is called a positive definite kernel if and only if n

S-

Cj Ce (Xj, 2%) = O

j,k=1

for alln EN, {21,%0,..-,%n} C X and {c1,c2,...,¢n} CC. Proposition 3.1 Let (H,c) be a symplectic space andG function. There exists a state yp on CCR(H,o) such that

y(W(f)) = Gf)

: H+

Ca

(f € H)

if and only if G(0) = 1 and the kernel

(f,9) + G(f — g) exp(—io(f,9)) is positive definite.

For z = 5) c; W(f;) we have a

S-

Cj Cy Wf;

—

frye

Site)

.

Since for a state y y(xz*x) > 0 we see that the positivity condition is necessary. On the other hand, the positivity condition allows us to define a positive functional on the linear hull of the Weyl operators and a con-

tinuous extension to CCR(H, oc) supplies a state. Lemma 3.2 Let (H,c) be a symplectic space. (It might be degenerate.) If a(-,-) is a positive symmetric bilinear form on H then the following conditions are equivalent.

(i) The kernel (f,9g) 4 a(f,g) —io(f,g) is positive definite. (ii) a(z,z) a(x,x) > o(z, x)? for every x,z € H.

STATES AND FIELDS

19

Both condition (i) and (ii) hold on A if and only if they hold on all finite dimensional subspaces. dimension.

Hence we may assume that # is of finite

If a(x, x) = 0 then both condition (i) and (ii) imply that o(z, y) = 0 for every y € H. Due to possible factorization we may assume that a is strictly positive and it will be viewed as an inner product. There is an operator @ such that

a(x, y) = a(Qz, y) and Q*

=

—@Q follows from o(z,y)

(c,yeH). =

—o(y,xz).

According to lin-

ear algebra in a certain basis the matrix of Q has a diagonal form

Diag(A1, Ao,..., Ax) where A; is a 1 x 1 0-matrix or

a=("

0

6).

Oy

(The first possibility occurs only if o is degenerate.) It is easy to see that condition (i) is equivalent to |a;| < 1 and so is condition (ii). Lemma 3.3 IfA = (a;;) and B = (b,;) are n x n positive definite matrices then the matrix C = (a;;;;);,; 1s positive definite, too. The entrywise product may be defined for any two matrices of the same type (and it is called frequently Hadamard product). One can check that

(CE,n) = S$“ (B(E * te), (0 * te)

(3.2)

k

where * stands for the Hadamard product and ¢, is the th column of

the matrix A!/?. From (3.2) the Lemma follows. Theorem 3.4 Let (H,o) be a symplectic space anda: Hx H>

Ra

symmetric positive bilinear form such that

o(f,9) (« exp (~Salti 1)

(a exP Gu

f)))

x exp(a(fj, fe) — 10 (Fj, fx)

— So bibs exp(a( Sj, fx) — 1o(F;, fx) should be shown to be nonnegative.

According to Lemma 3.2

(a(fj, fu) — 10S, Fr)55m is positive definite and entrywise exponentiation preserves this property as it follows from Lemma 3.3.

A state gy on CCR(H,c)

determined in the form (3.4) is called

quasifree. A quasifree state is regular.

Proposition 3.5 Let y be a state on CCR(H, oc). If

limy(W(tf))=1 then

(fe #)

ts regular.

We set G(f) = o(W(f)) (f € H). According to Proposition 3.1 the matrix

1 ( G(fi) G( fe)

G(—fi)

G(—f2)

1

G(fi — fe) exp(—to(fi, fa)

G(fe — fi) exp(io(fi, fe))

1

is positive definite. From this we obtain

IG (fo) — G(fi)| < 4|1 — G(fe — fi) exp(—to(fe, fi))| -

(3.5)

Combining (3.5) with the hypothesis we arrive at the continuity of the function

try» G(tf +g)

(t € R)

STATES AND FIELDS

21

for every f,g € H. Let (ay, Hy, ®) stand for the GN'S-triple. We verify by computation the continuity of the function tr

(ty(tf)To(g1)®, To(g2)®)

(¢€ R)

and the regularity of y is proven.

A state y on CCR(H, c) is said to be analytic if the numerical function

tH oWtf))

(t € R)

is analytic. Quasifree states are obviously analytic. Assume that 7 is a regular representation of CCR(H,o). operator B(g) is obtained by differentiation of the function

tr>a(W(tg))n =exp(tB(g))n

The field

(ER).

(3.6)

More precisely, if (3.6) is weakly differentiable at t = 0 and the derivative is € € H, then 7 is in the domain of B(g) and

iB(g)n = €. Proposition 3.6 Let y be an analytic state on CCR(H,c) triple (Ty, Hy, ®). Then 7,(W(g))® is in the domain of

with GNS

B(fn)B(fn—1) --- B(fi) for every 9, fi, fo,...,fn € H andn EN. We apply induction and suppose that

n= BUfn-1)-.. BU fi)to(W(g))® makes sense. For the sake of simpler notation we omit 7, in the proof. It suffices to show that

limt™((W (fn) — In,€) = F€) t0 exists if € is in a dense subset Dp of Hy, and |F(€)| < Clé||. ensures that

try W(tfa)n

(3.7) This

22

CHAPTER

is differentiable in the weak sense. (3.7) equals to O

—1)°

3

Since for € = W(h)® the limit in

ent

— —________

—

n):- Wt

fi)W

at the point t = thp_1 = tp_g =... = t; = 0, the function F' is defined on the linear hull Dy of the vectors

{W(h)®:he H}. F is linear on Dy

and by differentiation one can see that

C = lim 51 |\(W (tha) — Dl exists and it fulfils |F'(€)| < Cl|€|| for € € Dy. Although B(f) ¢ CCR(H,c)

it will be rather convenient to write

o(B(fn)BUfn-1) --- B(fi)) instead of (B( fn) B(fn-1) --- BUfi)®, ®). We shall keep also the notation Dy from the above proof. Remember that Dw as well as the superset Hilbert space 7, depend on the state » even if it is excluded from the notation.

Proposition 3.7 Let y be an analytic state on CCR(H, 0c). Then for f,g€ H andt € R

the following relations hold on Dy.

(i) (it)

B(tf) =tB(f) B(f +9) = BUf) + BY)

(iti)

[B(f), W(g)] = 20(f, 9) W (9)

(iv)

[B(f), B(g)] = —2i0(f, 9)

(i)-(iv) are deduced by derivation from (the Weyl] form of) the CCR. One gets similarly that

o(B(f)B(g)) = af, 9) — to(f, 9) if y is a quasifree state given by (3.4). Since

(>

aB(fk) >> GB(fi)) > 0

(3.8)

STATES AND FIELDS

23

the kernel (f,g) » a(f,g) — io(f,g) is positive definite. Therefore, (3.3) in Theorem 3.4 is not only sufficient but also necessary for the existence of a quasifree state.

Proposition 3.8 Let » be a quasifree state on CCR(H,o) (8.4) and fi, fo,...,fn € H. Then

given by

P(Bfn)B(fn-1)---B(fi)) = 0 if n is odd.

For an even n we have n/2

¢(B(fn)B (fn— 1)

-L Il

O(fems Fim) — 10 (Flems tim ))

where the summation is over all partitions {H), Ho,...,Hnj2} of {1,2,...,n} such that Hn = {jm km} with jm < ky (m=1,2,...,n/2). We benefit from the formula nm

9(B( fn) B(fn-1)-.- B(fi)) = ("5 ° O, O(W (tnin)-.. W(t fi))Since we have W (tn fn) W (tn—1fn—1)

x exp i (x:

tee W

(tif)

—

W (fn fn

+

tn—1fn—1

+...

tif)

tito (fi, n]

I>k

(3.4) yields Q(W (tn fn) -..W(fifi))= exp (-3>do tne nf) exp

(>:

tit, (—a(fi,

fr) + io(fi,

1)

.

(3.9)

I>k

What

we need is the coefficient of t,t2...t, in the power series expan-

sion. Such term comes only from the second factor of (3.9) and only in

24

CHAPTER

3

the case of an even n. In the claim it is described exactly the possibilities for getting tyt2...t, as a product of factors tit, (1 > k).

By means of (3.8) we have also

o(B( fn) B(fn1)-

=S°[[¢ (Un) Bin)

(3.10)

were summation and product are similar to those in Proposition 3.8.

The expression (3.10) makes clear that the value of a quasifree state y on any polynomial of field operators is completely determined by the

two-point-functions y(B(f)B(g)) (f,g € A). Let H be a real Hilbert space with inner product (-,-) and let o be a nondegenerate symplectic form on AH such that

lof OP ,

(fo). BY (fie) ® IP? . . BY (fx)"* OI?

ma) BY (fi BY (fo). BY (fa) * OP? = lin?

Lemma 4.2 and the explicit norm expression (4.2) have been used. Let the Fock representation act on a Hilbert space H containing the

vacuum vector ®. The linear span of the vectors B(g:)B(g2)...B(gn)® (91, 92,---;9n © H, nEN) and B*(g,)B*(go)...B*(gn)® (91, 92,---, gn © H, n EN) coincide. It will be denoted by Dg. So far it is not clear whether Dg is complete. ‘This is what we are going to show. Let A be a linear operator on a Hilbert space K. A vector € € K is

called entire analytic (for A) if € is in the domain of A” for every n € N and

te

oO

yo AlA*éll < +00 k=0

for every t > 0. If € is an entire analytic vector then exp(zA)& makes sense for every z € C and it is an entire analytic function of z.

Theorem 4.6 Dg consists of entire analytic vectors for B(f) (f € H). Let € = B(f,)B(fo)...B(fn)® € Dg. By a repeated application of Lemma 4.5 we have

|B(A)"El] < 2°4/

(n+ hy

—llE|

and it is straightforward to check that the power series

5 SIBLE oO

k=0

FOCK STATES

33

converges for every ¢. Since the entire analytic vectors form a linear subspace, the proof is complete.

Due to Theorem 4.6 every vector W(f)®

= exp(iB(f))® can be

approximated (through the power series expansion of the exponential function) by elements of Dg. This yields, immediately that Dg is dense

in H. According to Nelson’s theorem on analytic vectors (see [R-S], X.39), Dg isacore for B(f) (f € H), in other words, B(f) is the closure of its restriction to Dg. It follows also that Dg is core for B*(f) and

Bv(f)* = B'(f).

Assume now for a while that H is of one dimension (over C). Fix a unit (basis) vector 7 in H and set

fx=aBi(n"® = — (ne Z,) n

Vni

Then {fo, fi,...} is an orthonormal basis in H. B*(n) and a for B(n) then a” fn =

(4.3)

+}:

v2 + 1fr+1

afn ={

.

If we write a* for

vinfot

nee

and la,at]=1. With the choice I + q=Flata’)

p=~

—(

gta

)

the Heisenberg commutation relation (1.4) is satisfied and in the basis (fn) the matrices of q and p are given by (1.5) and (1.6), respectively. The vector f, is called n-particle vector in the physics literature. Transforming f, into f,,; the operator at increases the number of particles. ‘This is the origin of the term ”creation operator”. ‘The operator a annihilates in the similar sense.

Let {n; : i € I} be an orthonormal basis in the complex Hilbert space H. We set n

ne

n

les Ms 565M)

1

= Tan?

Ma)

n

a

Br (fi,)

n

kp.

(4.4)

34

CHAPTER

4

So for every choice of different indices 71, 22,...,%, in J and n1, n2g,...,ne E N we get to a unit vector in H. The vectors lng;

me)

and

nts.

me)

are different if ((m1,71),..., (Mx, %)) is not a permutation of ((mj, 71), ., (mj, j,)) and in this case they are orthogonal. All such vectors form a canonical orthonormal basis in H. Theorem

4.7

The Fock representation 1s irreducible.

We have to show that for any 0 4 7 € H the closed linear subspace

H, generated by {W(f)n : f € H} is H itself. Let M be the von Neumann algebra generated by the unitaries {W(f): f € H} in B(H). Clearly, Mn

C Hy.

We consider a canonical basis in H consisting of vectors (4.4). 7 € H has an expansion as (countable) linear combinations of basis vectors. Assume that a vector

frts fo?

Se)

(4.5)

has a nonzero coefficient.

The operator

BY (f)B (fi)-.-B (fk) B (fe)

(4.6)

is selfadjoint and (4.5) is its eigenvector with eigenvalue n1+n2+...+ng. Since (4.6) is affiliated with M, its spectral projections are in M. In this way we conclude that the vector (4.5) lies in H. It is easy to see that Bt (f)Hy

CH,

for every f € H. By application of the annihilation operators B(f;) (1 (aj1(1)@j1(2)) « - - P(ay1(1) 1/2) now

where the summation is over all partition of {1,2,...,k} into {j1(1)
0 such that for « > 0

small

||L(a, 6)|| < Ce? provided that |la|| < € and ||b|| < e«. Lemma 5.3 If |a;,6;| = [a:, aj] = [b;,b;] = 0 fori #7 then

(as + a5, bs + ba) < [La s) +L C0a5 0) exp (lan, ball) 1

Under the hypothesis the equality

D(a, + d2,6, +62)

=

Lay, b,) exp(taz) exp(ibe)

+

exp(ia, + ib,) exp (~sla.

1

])

L (ag, be)

holds and provides clearly the estimate of the norm.

Recall that for a € A and n € Z, we write s,(a) for

a+ q(a) +--+ 7a). Proposition 5.4 Let a,b € A®*.

Then

[z (sala), Z-sn(0)) | -+0

and noo.

We represent n in the form n = k-m+l where logn < m< 1+logn (i.e., m is the integer part of logn) and 0 (U, a) (dU, v) Therefore we have

(bv, Ea) = (V, a) (bw, v)

A6

CHAPTER

5

for every a,b € B. Being w cyclic

Ea must hold and F

= (aV, WV) = (a)

is really of rank one.

Let a € A and (b,) C B be a bounded sequence. We show that ;

1

Jim

;

@ exp (-s9(a)) } = exp y(a) Jim w(by)

(5.9)

whenever the right hand side makes sense. Indeed,

(bn exp(n™*8n(a) — y(a))V, W) _

(b,W, v)

4

(n-18n(a)V

_

y(a)¥,

S-

(n~1s,,(a*)

oe)

yey)

(k+1)!

where the second term converges to 0 obviously.

Benefiting from the

induction hypothesis and (5.9) we obtain

Jim wiexp(iF,,(a1)) ...exp(2Fn (ax)) exp(?Fn (ax+1))

= lim p(exp(iFn(a1)) . exp (iFn(ax-1))

x exp (iF, (ay + apy1)) exD (+s. )) = lim p(exp(iFa(as)) -.-exp(iFa(ax-1)) x exp(iF;, (ax + Q%41))) exp(—io (ax, Ar+1)) =

p(W (az)

wee W

(azp_1)W

=

p(W (a1)

sae W

(ax)W (a441)

(a,

+

G41)

exp(—io(ax,

x41)

.

It was used that P([ax, @k+1]) = —to (pe, Le41) -

It is worthwhile to point out that the above proof of ‘Theorem 5.4 has two critical points.

(i)

lim wiexpiF,(a)) = exp Gua with

(ii)

«))

x € A™

w(a)=0

lim La +V4+---V"™")c¥ =y(z)V

no

forevery

nN

forevery

rE A.

FLUCTUATIONS

AND

CENTRAL

LIMIT

AT

In the language of ergodic theory (ii) means that the isometry V is ergodic. The proof works with a nonproduct state ~ if conditions (i) and (ii) are satisfied. Let A be a parameter set. A process (indexed by A) means that for

every a € A an element 7(a) of an algebra BG is given and a

state w of

B is fixed. It is said that the sequence

(1: Bn; Yn) of process converges in law to a process (7, 8, w) if for every a1, a9,..., a, © A and k €N

the limit relation

Jim tn (7n(01)n(@2) -- - M(x) = Y(n(ar)n(a2) ---m(ae)) holds. Due to Theorem 5.4 the process

(a + exp(iF,(a)), B, y) converges to the Weyl process

(at+ W(a),CCR(A*,o), p) where p is a certain quasifree state and

A= {ae A®: g(a) = 0}. Quasifree states are analogues of Gaussian distributions. (The central state is not quasifree in the sense of our definition. It would correspond to a degenerate normal distribution in probability theory.)

The presentation of this chapter benefited from [G-vW], [A-B] and [G-V-V].

48

CHAPTER

5

Chapter The

6

KMS

condition

Let A be a C*-algebra and ¢t ++ aq; a homomorphism of the additive group of reals into the automorphisms of A. A state » of A satisfies

the Kubo-Martin-Schwinger, or KMS, condition if for every a,b € A there is a function Ff,» such that

(i) Fy» is defined on the strip {z € C:0 < Imz < 1}. Gi) F,» is continuous and bounded. (iii) F» is analytic on the open strip {z € C:0 < Imz < 1}. (iv) Fo4(t) = y(aa,(b)) for every te R. (v) Foo(t+t) = v(az(b)a) for every t € R. On

the boundary

call fy,

Fy,

KMS-function.

is fixed by y and In mathematical

(q;). physics

Sometimes

we shall

the KMS-condition

describes the equilibrium states for the dynamics (a;). In particular, a KMS-state is time invariant. We can see this by choosing a = J. The

function Fy, takes the same value at ¢ and ¢+7 and may be continued periodically to the whole complex plane into an entire analytic function which is bounded. Liouville theorem says that such a function must be

constant. So y(a;(b)) is independent of t € R. In this chapter we see quasifree states on the CC R-algebra satisfying the KMS-condition. Let # be a complex Hilbert space and L

a selfadjoint operator on H.

We consider the C*-algebra CC'R(H, 0c) 49

50

CHAPTER

6

where o(f, 9) = Im(f,9) (f,9 € H). The automorphism a; is given by the Bogoliubov transformation exp(—2itL), that is, a.(W(f)) = W(exp(-2itL)f)

(fE€R, feH).

(6.1)

Theorem 6.1 If there is ane > 0, such that L > eI then a quasifree state ~ given by

p(W(f)) = exp (—Fe(coth Lf. fy) satisfies the KMS

condition for (a,).

(FeH)

Moreover,

(6.2)

y is the only KMS

state such that f H y(W(f)) is continuous. First we note that coth L > J is a bounded selfadjoint operator and for the positive form

a(f,g) = Re(coth Lf, g) the condition (3.3) holds. Therefore (6.2) determines a state y. Denote by D the set of all entire analytic vectors of L. For h € D the function

4

= |E"h|

(6.3)

is analytic on the whole complex plain. Lemma

6.2

Letn € H andheD.

The function

F(t) = a(n, exp(—2itL)h)

(t € R)

admits an entire analytic extension and

F(i/2) F(-i/2) We work the function

= =

a(n, cosh Lh) — ia(n,isinh Lh) a(n, cosh Lh) + ia(n,isinh Lh).

with the real linear structure

> 2!" nL")

of H.

Since a is bounded

(6.4)

THE KMS

CONDITION

51

=

nh

3]5

F(-i/2)

g LIM]8

is an entire analytic function and supplies the extension of Ff’. Furthermore,

a(n, i" Lh) 1

n

1

neven

=

rn

n odd

a(n,cosh Lh) + ia(n,isinh Lh).

The computation for F'(i/2) is similar. Let us write 7; for exp(—2itL). Since D is stable under T;, we obtain from Lemma 6.2 that

F(i/2+S) F(-1/2+ S$)

= =

a(n, coshLTsh) —ia(n,isinh LTsh) a(n,coshLTsh)+ia(n,isinhLTsh)

(6.5) (6.6)

for every SER Set fort Ee R

Gi(t) = y(W(h)W(Tih'))

and

Go(t) = p(W(Tih’)W(h))

where h, h’ € Hare fixed. By simple computation we have

Galt) = exp (SAI? — Slim'l? — a(t, Te) + éa(tgh Lh, eT) Go(t)

1

1

1

1

= exp (~5 lal? — gllF'? — a(t, Teh!) — ta(tgh Lh, inst)

In the light of Lemma 5.2 the function G,; and G» admit entire analytic

extensions. From (6.5) and (6.6) we obtain

1

G'\(i/2 + s) = exp (5a?

1

— llr’ +A+C+4+D+ B)

(6.7)

and

1 1 Go(—i/2 + s) = exp (Fla? — s\n? +A-C-D+ B)

(6.8)

52

CHAPTER

6

where

A = —a(h, cosh LT;h’)

C = ia(h,isinh LT,h')

B=o(h,isinh LT,h’) D io(h, cosh LT;h') .

Since C'+ D = 0 we arrive at the equality G,(i/2 + s) = Go(—i/2 + s) for every s € R. It follows from the uniqueness of analytic continuations

that Go(z) = Gi(z +7) for every z € C. For a = W(h) and b = W(h’) we have succeeded in the construction of the kMS-function. (In fact, our F,, is an entire analytic function.) The next task is to obtain the kK MS-function for arbitrary

a,b € CCR(H, oc) by approximation. When a and 6 are linear combinations of Weyl operators the kK MSfunction f,,» is at our disposal. We carry out the GNS construction with y and get the triple

(Hy, T,, ?). The representation 7, is injective since CCR(H, c) is simple and we simply omit 7, from the formulas. Lemma

6.3

Let &,,€ € H and assume that €, — € in the norm of H.

Then W(E,) + W(€) in the strong operator topology. On the unitaries the strong and weak operator topologies coincide. It is sufficient to check that

(W (En) S> A(g)W(g)®, S$" AF)W(F)®) converges to

(W(E) S-A(g)W(g)®, S>ACFYW(F)®) for arbitrary linear combinations $5 A(g)W(g) and 5) A(f)W(f). This is, however, clear from the explicite expressions.

For a while we denote by Ap the linear hull of the set {W(h) D}.

: hE

Kaplansky’s density theorem tells us that for a,b € CCR(H,0c)

THE KMS

CONDITION

d3

(or rather a,b € m,(CCR(H,o))")

there are sets (a;) and (6;) in Ap

such that the following conditions hold.

(i)

las] < lal], [bell < [ll

(ii) (iii)

aja b;>b

Let fj = Fo,,

and and

a;—a’* 6; 46°

in the strong operator topology. in the strong operator topology.

be the A MS-function

for a; and b;.

We

prove that

lim F;(z) = F(z) exists of 0 < Imz < 1 and F is the KMS-function of a and 0. One can see that there exists a unitary u, on H, for every t € R such that u,W (h)® = W(T;h)® (hREH). To prove that F; converges uniformly on the strip we estimate on the boundary as follows. Fy(t)— F(t)

=

vlaar(d:)) — y(ajar(d;))

(upbi®, af ®) — (ub;®, a; ®)

((b:® — bj), uza;®) + (bj, uz (an — dn)®) Hence

|Fi(t) — Fy) < |](i — 53) ®|flal] + |[(az — aj) ®|/6]| and similarly,

\Fx(i +t) — F(t + #)| < ||(ax — a5) ®I] [Il] + |], — 95) ®llflal]. Fi, — F; is a bounded analytic function which for big 2 and 7 is small on the boundary. Therefore, it must be small everywhere and F; converges uniformly on the strip to a bounded analytic function F’. Since

(aj (bi)) = (usbi®, a; ®) > (usb®, a°®) = p(aay(6)) and similarly

p(a4(bi)ai) 4 y(ar(b)a) , the function F' satisfies the necessary boundary conditions (iv) and (v).

54

CHAPTER

6

We have not checked the boundedness of the K MS-functions yet.

It is enough to do this in the case a = W(h) h' € D. Returning to (6.7) we have Si je

(s

and b = W(h’) with

1 1 — exp (~ Fal? - simi?) ~ a(h, cosh 5LT,h’)

+ s)

+ o(h,i sinh 6LT,h’) IA IA

lo(h, i sinh 6LTsh’)|

|(coth Lh, cosh 6bLT sh’) |

IA

|a(h, cosh 6LTsh')|

| coth LAl||| cosh 6Lh’|| |(h, isinh 6LT sh’) |

IN

We estimate in terms of the inner product.

|||||| sinh 6h’ ||

Since h! € D the norm |le°”h’|| is bounded if 6 is in any compact interval. It follows that | cosh 6Lh’|| and | sinh 6Lh’|| are bounded as well. It is worthwhile to note that we showed a bit more than the boundedness of the KMS-function.

Namely, we obtained that

e * < |G,(z)| < e*

(0 < Rez < 1)

(6.9)

for certain K > 0.

So far we have proven that (6.2) really gives a K MS-state and now we turn to the uniqueness.

Suppose that w is another KMS-state. the KM S-functions

F°(f,g)

and

We consider for f,g € H

F*(f,g)

corresponding to the operators a = W(f) and b = W(qg). Set

v, (2) = FY (f,9)(2)

PON

Fe(f,9)(2)

Due to the estimate (6.9) for g € D the function vy, is bounded.

It is

also clear that it is analytic on the open strip and continuous on the closed strip. On the one hand we have for t € R

w(W(f)W(Tig)) _ wo(W(f + Tig)

*60) = COW(AW(Ta)) ~ o(W(F+ Tg)

THE KMS

CONDITION

D0

and on the other hand Vpglitt)

Since vsg(t +t)

=

w(W(T)W(f)) _ wWUE + Ta) eW(Tig)W(f))

(Wf +Tig))

= vyz,(t) we may apply the same trick used in the

proof of the invariance of the

Kk MS-state.

vy, has an entire bounded

analytic continuation which should be constant.

If

u(f) = UA) AD) a— sawp) exw (Sal) then we have

u(f +g) =ulf + Tig)

(t € R)

We have to show that under the continuity condition on h

(6.10) w(W(h))

the function wu is identically 1. First of all, u is continuous on H. A repeated application of (6.10) yields that

u(g) =u (S70)

(g€H,tER).

(6.11)

=u (- [ Tigdt)

6.12)

Using (6.11) once more we have

u(g) =u (250 ( [0 The ergodic theorem tells us that 1

n

“ff Nn

Ti.gdt > Eg Jo

56

CHAPTER

6

strongly where & is the orthogonal projection onto the kernel of L. Since L is injective and u is continuous from (6.12) we conclude that

u(g) = u(0) for every g € H. Now we sumption L the Hilbert L. Then we

try to follow the lines of the above proof without the as> ¢. Assume that L is a positive selfadjoint operator on space H. Let H be the set of all entire analytic vectors for have

(i) T7.H CH

(t € R)

(ii) tanh LH CH since tanhL is a bounded operator.

We denote tanh L(H) by Hp and

set

ao(fo,g) = Re(cothLfo,g) Theorem

(fo € Ho, 9 € A)

(6.13)

6.4 If there exists a positive symmetric nondegenerate bilin-

ear forma:

H x H +R

which is the extension of ap given by (6.13),

then the quasifree state

ew) =e (Zan)

Few

is defined on CCR(H, co) and satisfies the K MS-condition for the group

an(W (f)) = W(exp(—2itL)) Lemma 6.5 |o(f,9)/? < a(f, fa(g,g) Let D = —itanh L|H and D|Hy = Do.

(eR, fed). (f,9 € A) Simple verification shows

that

a(Df,9) =o(f,9)

€ H) a(Df.9) = —a(f,Dg) (f,9 (f.g € H). Since

a(Dofo, Dofo) = (fo, tanh Lfo) < (coth Lfo, fo) = a(fo, fo)

6.14)

THE KMS

CONDITION

57

for every fo € Ho, we get that Do is a contraction for a. Forming the completion H of H with respect to a we obtain a real Hilbert space and we denote by P the orthogonal projection onto the closure of Ho.

Let D' be DoP where Dp stands for the closure of Do (defined on Ho). For f,g € H we have

a(D'g,Df)

=

a(DoPg, Df) = —a(P9, DoDf) —a(g, PDy)Df) = —a(g, DoD f) = —a(g, D’f) = a(Dg, Df)

that is,

a(D'g — Dg, Df) =0

(f,9 € H).

This gives that D'g — Dg | Hp. But D'g — Dg € Ho, therefore D'g = Dg for every g € H. D’ is a contraction on H and the Schwarz inequality yields

lo(f, 9)?

la(Df,9)|? = |a(D'f, 9) |’
A strongly in the resolvent sense where A denotes the modular operator corresponding

to (M,H,Q). Let H’' be the domain of the closure S of the operator aQ > a*Q (a € M). So H’' becomes a Hilbert space with the scalar product

(E, n)s = (€,n) + (Sn, SE) We show that U{H,,

(En EH).

: n € N} isacore for S, or equivalently, it is dense

in H’ with respect to the graph norm | - ||>5. Fix a to to Kaplansky’s density theorem ([S-Zs], 3.10) we and (6;) from the selfadjoint part of U{M,, : € N} 1 . a; > 5 (4 +a*) and b> sla" —

€ M. According can find nets (a;) such that a)

strongly. ‘Then

(a; +1b,)24aQ This means that U{H,

and

(a; —ib,)Q > a*Q.

: n € N} is really a core for S.

Let Q, be the orthogonal projection of H’ onto H, with respect to

the inner product (-,-)s. There exists a contraction T on H’ such that

(€,7) = (TE,n)s

(En EH’).

We have

(I+ A)"€=TE

(€ EH’)

and

(EE Ha, DEK).

(I+ An)*€ = (AnTQn + QnJE Since

QnTQn + Qi 9 T strongly in the graph norm we obtain

(I+A,) "E> (T+ A)tE in the norm of H.

Being UH,

(€ €U{H,

: n € N})

total and the sequence

bounded we have completed the proof of the Lemma.

(J + A,)~

THE DUALITY

THEOREM

63

According to a simple topological argument it is sufficient to prove

(7.4) fora € M, (k EN). It follows from Lemma 7.3 that AY > A” strongly for every t € R (cf. [R-S], VIHI.7). Therefore AX aA," > AX ad”

(t € R)

strongly. However

A“ ad'P,

€ MP,

(n > k)

and (7.4) has been shown. Lemma

7.4 If J is the modular conjugation operator corresponding to

(M,H, Q) then

; InPn > J

strongly. Since the strong convergence is equivalent to weak convergence and convergence of the norms, we shall prove weak convergence only. For £,n € H, and n > k we have

(InPn€n)

= (Iné,n) = (Ar/*SE,n)

_ =f 7 Lemma

(I +

A,)-4S€,n)dt

7.3 tells us that

(An(I+ An) *SE,n) 3 (AU + A) *SE,7) and using Lebesgue’s theorem on the convergence of integrals we get

(InPr€,1) > ==f aA(I+VA)*SE, n)dt The limit is just (Jf, 7) and the proof is complete. In order to complete the proof of Theorem 7.2 we show do this it suffices to prove that [Jaz J,

ay|

=

0

(7.5).

To

(7.6)

64

CHAPTER

for a, € M;

7

and a; € Mj,. Since

Ja,Ja, = lim InagInaPn nw

and ayJagJ =

oo lim adnagdnPn nw

as a consequence of Lemma 7.4, (7.6) follows from its finite dimensional version. It is worthwhile to emphasize that we proved the continuity of the main ingredients, the modular operator and the modular conjugation, of the Tomita-Takesaki theory with respect to an increasing sequence of subalgebras.

Let K be a complex Hilbert space with inner product (-,-). Setting o(f,g)

=

—Im(f, 9)

(f,9g €K)

the C*-algebra CCR(K, c) and its Fock state

eW(f)) =e

(-FN)

(eX)

are at our disposal. If kK C K is a subspace then we write CCR(K) for the subalgebra of CCR(K) = CC'R(K,c) generated by the unitaries {W(f): f € K} and we assume that all these algebras act on the Fock space H containing the cyclic vector ®. Lemma

7.5

The mapping (f1, fe, ---) fr) A BY (fi) Be (fo)...

is normcontinuous from Kx Kx...

BY (fa)®

x K into H for every k EN.

To show the Lemma it is enough to see that (91; 92,--+,9n) > 9(B(g1)B(g2).-- BUfn)) is normcontinuous.

This is obvious from Proposition 3.8.

THE DUALITY Lemma

7.6

THEOREM

65

Let K C K be a real subspace and set Ho for the closure

of CCR(K)®.

Then

BY (fi)BY (fa)... B' (fe)® © Ho whenever fi, fo,..-,f,eE

HK andk EN.

We apply induction on k. For k = 0 the statement is trivial. order to carry out the induction step we write

In

BY (fru) Br (fe)... B(fi)® = B(far1)B" (fe) ---Bfi)o — Bo (far) B" (fe)... B(A)®. Due to the induction hypothesis

BY (f,) BY (fra)... BY(fi)® € Ho and differentiating th

W (tfrr1)B

(fr)

wee B*(f,)®

we obtain that

B(fev1)B" (fe) Br (fra)... BT (fi) ® © Ho. If we use the commutation relation

IB (f), B*(9)] = (9, f) repeatedly, we find that the vector

B (few) Br (fr) B* (fea) --. BY (fi) ® is a linear combination

of the vectors

BY (fi) B" (fri) --- BY (fu) B (fir)... BY (fi)® where 1 a*®

(ae M(K)),

which is central in Tomita’s theorem, leaves K invariant. Since

t 1 (W(4tf) — D® > +i1B(f)® as t + 0, we obtain that B(f)®

€ D(S) and SB(f)®

= B(f)® for

every f € K. It follows that

SBY(ftigh®=Bi(f—ig)®.

(f,g€ K)

(7.9)

Using the assumption that K is closed it is easy to see that the operator

5, : B(ft+ig)hb4 Bf —ig)¢ is closed.

Hence 5S; is the restriction of S to K.

(f,9€ K) One finds that

i=(7 "0 ) 0 T

with respect to the decomposition K = K, ®K,.

Let S, = TAY”?

(Note that kK = L+.)

and S = JA‘/? be the polar decompositions.

In

order to conclude that

Ji=J|K

and

A, =A\K

(7.10)

we have to establish that S* leaves K invariant, too. Denote by £ the

orthogonal complement of K in H, ie. £= {E € H : (€,n) = 0 for every 7 € K}. Let 7 € D(S)NK and € € D(S)NL. Then

(S"n,€) = (n, SE)

68

CHAPTER

7

and we need to know that D(S)M CL is dense in £ and S leaves £L invariant. This requires a bit more analysis and we sketch how to do it. First by differentiation we get

B( fi) BUf2)--- BUfn)® = BUfn)BUfna)--. BUfi)® for every fi, fo,...,fn € K and n € N. Then we deduce

SB'(A)B'(f2)...B(fn)® = B'A)B'(fe).--B'(fn)® — (7.11) (The detailed proof of (7.11) may be similar to that of Lemma 7.6.) The linear subspace spanned by the vectors B* (91) B* (g2)

ee B* (gn)®

(where n = 0 or n > 2 and g; € K +iK)

is in the domain of S,

contained £ and stable under S. It is straightforward to see that

n=(4

OL

5 |

T?

and

a= (4

0

re |

(7.12)

So J; is a bijection of K onto L. We prove that

JB(f)J = B(A;f).

(f € K)

(7.13)

For f € K the selfadjoint operator B(f) is affiliated with the von Neumann algebra M(K). Tomita’s theorem tells us that JB(f)J is affiliated with the commutant M(k)'.

For f,g € K we have

IB(f)IW(g)® W(g)JB(f)®

= =

W(g)JB(f)J® = W(g)JB*(f)®=

W(g)Br (A f)®

=

W (g)B(Af)®

Since B(Jif) is affiliated with M(L) C M(K)' we obtain

JB(f)IW(g)® = BU f)W(g)®

(7.14)

whenever f,g € K. Here we interrupt the proof of the theorem to state a lemma which is interesting also in its own right.

THE DUALITY Lemma

THEOREM

7.9 Assume

69

that K

C K is a real linear subspace such that

K+iK is dense in K. Then the linear hull D of the set {W(h)® K} is a core for every field operator B(f) (f € K). This statement improves Proposition 7.7.

We know that Dg

: he

is a

core for B(f). So it suffices to show that the closure of D with respect to the graph norm associated with B(f) contains Dg. We may follow the proof of Proposition 7.7 and get that

B' (fi) BY (fo)... B" (fn)® is in the closure of D for fi, fo,...,fn € K +121Kk.

(7.15) Then by means of

Lemma 7.5 we conclude that the vector (7.15) is in the closure of D also for every fi, fo,.--, fn © K. (It is rather straightforward that the continuity in Lemma 7.5 holds if on H the graph norm is considered.)

Now we resume the proof of Theorem 7.8. Let Do be the (complex) linear subspace spanned by {W(g)¢ : g € K}. It follows from (7.14) that

JB(f)J > BUA)|Do-

Since the latter operator is B(J, f) according to Lemma 7.9 and J B(f)J is selfadjoint we arrive at (7.13). Finally we infer

IW (f)J = J

y —B(f)"J =B

S-

i"

BUN”

for every f € K. So the mapping A » CC R(L) and the proof is completed.

— W(-JSif)

JAJ sends CCR(K)

into

70

CHAPTER

7

Chapter

8

Completely

positive maps

Let A be a C*-algebra. We denote by M,,(.A) the set of all nx n-matrices

a = (a;;) with and involution Let A and define a linear

entries a;; in A. With the obvious matrix multiplication M,,(A) is a *-algebra and may be identified with A®M,,. B C*-algebras. For each linear map a : A — B we map a, : M,(A) > M,,(B) by

On (aij) = (a(aij))-

(8.1)

If a, is positive for all n, then a is said to be completely positive. Completely positivity is the ’right” generalization of a positive functional both from mathematical and physical point of view. Assume

that the C*-algebra acts on a Hilbert space H, that is, B C B(H). Then a : A — B is completely positive if and only 1, 02,...,0, € A, &, &o,...,€ € H and for allk Ee N

if for every

k

So w=1

(al a(aja,;)&, &) >

(8.2)

j=1

holds.

Let (H,c) and (H',o') be symplectic spaces. A linear map ( : CCR(H, oc) > CC R(H', 0’) is called quasifree if there are a linear map B:

H > H' and a function F : H > C

B(W(h)) = F(h)W'(Bh)

such that

(he BH).

(8.3)

Clearly, the composition of quasifree maps is quasifree. In this chapter we shall deal with completely positive quasifree maps. 71

72

CHAPTER

Theorem

8.1

A quasifree map 8 : CCR(H,c)

8

~ CCR(H', 0’) given

by (8.3) is completely positive if and only if the kernel

(f,9) > F(g— flexp i(o(9, f)-o'(Bg, Bf))

= (f,9 € H)

(8.4)

is positive-definite.

First we assume that (8.4) is positive-definite and show completely positivity in the form of (8.2). It suffices to choose

ds = >> AskW (fer)

(l CCR(H',o’) such that

a(W(A) =W'Afew (S14Fl? = 51/12)

(86)

for every f € H. In the light of Theorem 8.1 we need to show that the kernel

(f.g) +9 exp (SIA(9— AI? = Slla = FIP +io(4, A) ~io'(Ag, Af) is positive-definite. It is so if and only if the kernel

(f,g) > exp (Re((I — A*A)g, f) +iIm((I — A*A)g, f)) is positive-definite. Lemma 3.2 tells us that

(f,9) 4 Re((I — A*A)g, f) +ilm{(I — A*A)g, f)

74

CHAPTER

8

is positive-definite and the pointwise exponentiation preserves this property.

On the language of categories the above example hides a functor. Let us consider the category whose objects are complex Hilbert spaces and whose morphisms are contractions. The correspondence A +> a, is functorial in the sense that

QAB = AAQB whenever

B

: H;

>

Ho

and A

: He.

(8.7) Hz

are contractions.

To the

zero operator a@ associates the Fock state

ealW()=ex (-sIFI?)

Fem),

Since 0A = 0 we have

Pu, 0 AB = YH, (8.8) [‘(H) is a frequently used notation for the GNS Hilbert space corresponding to CCR(H, co) and yy. It follows from (8.9) and the Schwarz inequality that the linear application W

(hy) ®1

>

ap(W

(h1))®o

(hy

€ H,)

extends to a contraction [(B) : [(H,) > (He). (Here ®, and ®» are the cyclic vectors.) B ++ I(B) is also a functorial correspondence and it is called Fock functor.

It is convenient to know the action of ['(B)

on exponential vectors (see (4.10)). We have

['(B)e(f) = e(Bf)

(f € 71).

(8.9)

Let A; be a C*-algebra and y; a faithful state on it (¢ = 1,2).

If

the unital linear map T : A; - Ao satisfies the inequality

T(a*a) > T(a)*T(a) then it is called a Schwarz map. positive map

is of Schwarz type.

(a € A;)

It is well known

(8.10) that a completely

For a;,b; € A; and i = 1,2 we set

ar; (a;, 0;)

=

Re

(0; a;)

o;(a;,b;)

=

Im —;(bFa;)

COMPLETELY POSITIVE MAPS

19

and denote by CC R(A;, y;) the CC R-algebra associated with the symplectic space (A;,0;) (i = 1,2). On the algebra CC R(A;, y;) the Fock state

oi(W (ai) = exp (—Jevlas a)

(a; € A;, i= 1,2)

is at our disposal. Assume that 7’ is a Schwarz map and y oT’ = y,. Then T becomes a contraction with respect to the GN S-norms and there exists a quasifree

completely positive map a : CCR(Aj,¢1) ~ CCR(Ape, Y2) such that

o(W(a1)) = W(Tay) exp 5 Blas, a1)

(a € Ay)

(8.11)

where

B(ar, 61) = po(T(01)"T (a1)) — yi (bja1) We are going to central limit. Consider the Ap @...@ Ay. Yo @ Yo @...@ defined as 1

prove that the quasifree mapping @ is in some sense a n-bold tensorproducts A? = A; ®...@ A; and A} = Let T, : A? — A} be TOT ®...@T and uw, = Yo. We recall that for a € A, its nth fluctuation is n

Fr (a1) = = 01M @...@ 1° @ (a

vn is

Theorem 8.2 Let a1, d9,...,a, (1 yu > 1 and argue by contradiction. Suppose that a is a von Neumann algebra isomorphism

of m(CCR(H))” onto 7,(CCR(H))". Since the field operators are affiliated with 7,(CCR(H))”, a acts on them. Clearly,

a(By(f))= Bf)

(fF eX).

Denote by h* the characteristic function of the interval [(k—1)/n, k/n] and set

Ay = 3

1

(By(h,)? +... + By(hn)*)

exp Aj is a positive contraction in ™(CCR(H))"

and Aj is defined

similarly. We have

a(exp AX) = exp Aj. Compute

(exp ASW (g)®,, W(g)®a) = (2r)-"/? [ (eo (1320104) k=1 1

x exp (-}

n

yt] k=1

dr,dxr2...ALp

W(g)®, W(g)®))

84

CHAPTER

9

By evaluation of the Gaussian integrals we obtain —n/2

(exp ALW (g)®,, W(g)@y) = (: + * Since

n

exp (-s

Ss o(g, i)

,

S| a(g, hk) = 0 k=0

for every real function g we have

(exp ARW(g)®, W(o)®) -» exp (—3) One gets in this way that

exp A? — eV?

and

expAl—

eH?

in the weak operator topology. Since a is continuous in this topology (on bounded sets) we arrive at a contradiction. Let H be a real Hilbert space with scalar product (-,-). On the space H = HOA

o(fi ® fa,91 ® 92) = (fa, 91) — (ft, 92) defines a nondegenerate symplectic form.

The standard Fock state y

on CCR(H, c) is given by

AW @ fh) =e

(Fh)

5Unfe))

(04)

Let T be a positive bounded operator on H and suppose that 7’ has a

bounded inverse. The map T> : (f; ® fo) 6 Tf, @T and

eWD)=en(-3RAN)

| fo is symplectic

(FEM

8)

defines another Fock state. Our next aim will be to discuss the equivalence of y and yr. Below we shall assume that H and H are infinite dimensional separable Hilbert spaces.

EQUIVALENCE Theorem

and a

OF STATES

9.4 Let (Ho,00)

: CCR(Hp,00)

>

89

be a finite dimensional symplectic

B(K)

a continuous representation.

space

Then

m is the direct sum of (irreducible) Fock representations. Let Pp be the projection in CC'R(Ap, oo) given by (9.3) and (€;) an orthonormal basis in the subspace Ky = PoK.. K; = [a(CCR(Hp, 00))& and 7; = 7|K;. Since

(W (FYE, W (g)&;) = "Poo we get K; | K; ifi #7.

(W (F — 9))(&, &)

From the fact that 7;(Po) is a projection of

rank one it follows that 7; must be irreducible.

Let £ be the orthogonal complement of the sum S- ®@K;,.

the place to use the continuity hypothesis. the proof of Theorem 1.2.).

This is

It yields that £ = {0} (cf.

One can guess from the above proof that every representation of CC R(HAbo, oo) is the direct sum of a continuous and a ”singular” one. Lemma

9.5

Let Ho

be a finite dimensional subspace

representation of CCR(H)

of H

and 7 a

such that 1|CCR(Ho) is continuous.

Then

m(CCR(Ho))' N 7(CCR(H))" = 1(CCR(H5))” The point is the containment C. It follows from the previous theo-

rem that the von Neumann algebra My = 7(CCR(Ho))” is the direct sum of type J factors. Therefore, the identity is a sum of pairwise orthogonal minimal projections of Mg. It suffices to show that for every minimal projection & of My. We have

m(CCR(H)) Na(CCR(H))"E Cc n(CCR(Ht))"E.

(9.6)

Evidently,

m(CCR(Ho))’ A a(CCR(H))"E C Ex(CCR(H))"E. The mapping 6 : T 4 LTE is strongly continuous.

(9.7)

The von Neumann

algebra 7(CCR(H))” is generated by 7(CCR(Ho)) and 7(CCR(H;)).

For A € 7(CCR(Hpo)) and B € 7(CCR(Hj)) we have

8(AB) = EAEB € r(CCR(Hp))"E.

86

CHAPTER

9

(because HAF is a multiple of £.) Hence En(CCR(H))"E C n(CCR(Hj))"E.This combined with (9.7) yields (9.6). We say that (H,) is an absorbing net of subspaces if (i) For every a H, is a finite dimensional complex linear subspace of H. (ii) For every a and

there is a y such that H,.,Hg C Hy.

(iii) |_}Ha is dense in H. Proposition 9.6

Let (H,) be an absorbing net of subspaces and yy, We

factor states of CCR(H).

Assume that ty, and Ty, are continuous.

Then Ww, and we are quasi-equivalent if and only if for every €« > 0 there is an a such that

Ilr — We) |CCR(H)||

is compact as well as J —T and J —T™!. Let {e1, €2,...} bea basis of the real Hilbert space H such that every

e, is an eigenvector of 7’. Set H7” for the complex linear subspace of H spanned by the vectors {€n, €n41,---,;€m} (n n. The next elementary inequality may be varified easily.

1 1 1+ 5¥yi < (4 + 5) < exp (521

(9.13)

Combination of (9.11), (9.12) and (9.13) yields that 1 ~ L+ 5D Mi < pr(Pry?m\—2 < (1 +e)

2

i=n oO

if n is big enough.

This shows that Shi i=1

is convergent.

Since

2m, < (1 — Ai)? < 2M;

(9.14)

CO

for M

>

;

>

m

we conclude

that

So(l

— ,,)? is convergent,

or in

i=1 other words, J — T is a Hilbert-Schmidt operator. Before proving the converse we state a lemma. Lemma

9.9

Jf w; and we are states of B(K)

given by vectors &

&> then

llr. — wal] < 2(1 — |(Er, &2)|?)'/?.

and

90

CHAPTER

9

Let P; be the projection onto Cé; (¢ = 1,2). It is easy to compute that P, — Py has spectral decomposition

AQ: —AQz

with = A= (1 — |(E1, &)|?)"”.

Hence

Now we go back to the proof of Theorem 9.8. Let (7,0, K) be the GNS-triplet associated with CCR(H™) and yr|CCR(H™). It follows from Lemma 9.1 that

e(W(h)) = gr( Pr") * (a (W (h)) (Pr) Q, 7 (Pr")Q) Lemma 9.9 is applicable and yields

Ile — er)|ICCR(H;,)|| < 20. — vr (Pr)? . Suppose that J — T is Hilbert-Schmidt.

Then

(9.15)

“jp; < +oo thanks

to (9.14). We are going to benefit from Proposition 9.6 and to verify condition (9.8). On the one hand

II(v — gr)|CCR(H,)||?_

< limsup ||(v — gr)|CCR(H?)||? Mm CO


0 has a bounded inverse. It follows from the decomposition

A=ViVoLV3 that y4

& yy.

Therefore,

y = yy, if and only if y = yy.

On the

other hand, A*A — J = V,(L* — I)V, and A*A — I is Hilbert-Schmidt if and only if L? — I is so. It is not difficult to see that L? — J is in the Hilbert-Schmidt class if and only if J’ — J is Hilbert-Schmidt.

This

way we have succeeded in reducing the equivalence of (i) and (ii) to Theorem 9.8.

The equivalence of (ii) and (iii) is based on the fact that y and ya are pure states. Set (7,H,,®,) for the GNS-triplet of y4. that there exists a unitary V : K — K, such that

m(W(h)) = V*m(W(A))V

Assume

(h EH).

Then we have

ya(W(h)) = (2 (W(h))Q, Q) for Q = V*®,.

(h € H)

Define a unitary by the formula

U : r(W(h))Q 6 7(W(ATtR))® and simple computation shows the U implements the automorphism

ya. The inverse implication (iii) > (ii) is trivial. The study of quasi-equivalence of non-Fock quasifree states may go

through the purification as it is done in the papers [VD] and [Hol].

EQUIVALENCE

OF STATES

93

We do not enter this subject but give another proof of the fact that on

an infinite dimensional Hilbert space H the states y, in (9.1) are not quasi-equivalent. More generally, let A and B be bounded operators on 1 such that

A,B >

(1+e)I for some ¢ > 0. Consider the quasifree states

va(W(h)) = exp (= (4h, ty

on(W(h)) = exp (

* (Bh, t)

2

on CC'R(H). Using Theorem 6.1 one can get that y4 and yz are factor states. (A nontrivial central element gives rise to another K MS-state for the appropriate group.) We are in a position to apply Proposition 9.6 and can use the argument in Lemma 9.7. Instead of (9.9) we have

exp (54h, ty

~ exp (—5 (Bh,

| yu > 1.

94

CHAPTER

9

Chapter The

10

selfdual approach

In this section we present the selfdual approach to the algebra of the canonical commutation relation. This formalism was developed mainly

by Araki ([Ar2],[/A-S]) and lays the emphasis on creation and annihilation operators rather than on Weyl unitaries. A triplet (K,y7,T) is called phase space if K is a complex linear space, y is a symmetric sesquilinear form and [ is a conjugate linear

involution (i.e., [? = id) such that

y(UFf, 9) = -r(9, f)

(f,g€K).

(10.1)

If (K,¥,I°) is a phase space then on the real linear subspace {f € K

[f = f} 7 is a symplectic form (it may be degenerate).

On the other

hand, if (H,o) is a symplectic space then H @ H may be endowed with a complex linear structure by letting a(f, 9)

The operator [(f,g)

=

(-g, f).

= (f,—-g) is an involution and o extends to a

symmetric form y on H @ H. Hence phase space and symplectic space are somewhat close notions.

The CC R-algebra A(K,~7,I) over the phase space (K,y,I°) is the quotient of the free *-algebra generated by a(f), its conjugate a*(f) and the identity over the two-sided *-ideal generated by the following relations :

(i) a(f) is complex linear in f, 95

:

96

CHAPTER

10

(ii) a(f)a*(g) - a*(g)a(f) = —(f,9), (iii) a(Pf) = a°(f). A linear mapping U of K into itself satisfying y(Uf,Ug) = y(f,g) and [TU = UT preserves the relations (i)-(iii) and there exists a *endomorphism 7(U) of A(K,y,T) such that 7(U)a(f) = a(Uf) holds. An operator P on K satisfying

(i) P?=P (ii) y(f, Pf) > 0 if Pf £0 (iii) y(Pf,9) = (Ff, Pg) (iv) T(Pr=1—-P is called a basis projection. Let P be a basis projection and consider the complex linear subspace H = rng P. It follows from property (ii) that is a separating inner product on H. So we may consider the C*-algebra

CCR(H, Imvy) over the symplectic space (H, Im)

and we let it act

in the standard Fock representation where creation and annihilation operators appear. Denote by I'(H) the Fock space. For f € K we define an application a by the formula

a(a(f)) = B'(Pf)+ B (PTS).

(10.2)

Since

[B* (Pf) + B-(PIf), B* (PT) + B-(P9)| = —(Pf, Pg) + (PTgq, PTf)

= —(Pf,g) —((1—- P)f,9) —

—(f,

9)

and

a(a"(f)) = ala f)) = a(a(f))” a extends to a *-algebra homomorphism of A(K,7,T). When A is a *-algebra then by a state y of A we mean

a linear

functional such that y(/) = 1 and y(a2*xz) > 0 for every z € A. A state y on A(K, 7,1) is called quasifree if

THE SELFDUAL

APPROACH

97

(i) yla(fi)a(fe).-.a(forri)) =O (ii) p(a(fi)a(fe)...(for)) = (KEN,

fi € K)

the

summation

where

is over

(kKEZ", fie K) TTF-1 9(@( Fim) @(Fim)) all

partitions

{H, Ho,...,H,}

of

{1,2,...,2k} such that Hy, = {jm,ln} with jm < lm. Proposition

10.1

Jf P

is a basis projection

over

the phase

space

(kK, y,T) then there exists a unique quasifree state yp on A(K,y,T) such that

y(a*(g)a(f)) = (Pf, g9)-

(10.3)

The uniqueness is obvious. ‘To establish the existence we use the *-algebra homomorphism a defined above. Setting y(x) = (a(x)®, &)

we obtain a state on A(K,y7,I).

(x € A(K,y,T))

(10.4)

(¢ is the vacuum vector here.)

It

follows from Proposition 3.8 and simple properties of the Fock states that vy is a quasifree state in the above sense and its two-point function

is really given by (10.3). It is worthwile to point out that the basis projection P determines that a(f) becomes creation or annihilation operator in the corresponding representation. Let s be a positive hermitian form on K such that

s(f,g) —s(Ug, Tf) = y(f,9).

(f,9 © K)

(10.5)

Then gs is called a polarization of y. Observe that if P is a basis projection then

se(f,g) =V(PF,9) is a polarization. Lemma

10.2

/fs is a polarization ofy then

(9, f)s = s(g, f) + sf, Tg)

98

CHAPTER

10

is a positive form such that

(10.6)

ly(9, A)? < Cf, F)s(9, 9)s holds.

IA

Ka fls + KES 9)Is

IA

(9,.9)5((f, F)3/?+ FTF)

IN

We estimate simply as follows

lngA