Chapter VI Representation of Commutation Relations in Bose-Einstein Fields

CHAPTER

REPRESENTATION OF COMMUTATION RELATIONS IN BOSE-EINSTEIN FIELDS

As we mentioned in Chapter 111, some of the problems which led to the study of quasi-invariant measures arose in the quantum theory of fields, in particular, boson fields or Bose-Einstein fields. In the present chapter, we shall point out the connection between Bose-Einstein fields and the theory of quasi-invariant measures. In order to assist the reader in understanding the case of infinitely many degrees of freedom, we shall first (in $6.1) discuss in detail representations of the commutation relations in quantum mechanics (i.e., the case of finitely many degrees of freedom). Comparing the situation discussed in $6.1 with the corresponding situation in $6.2 and $6.3, the reader will appreciate the difficulties of the infinite-dimensional case; these are, in fact, the crucial mathematical difficulties in quantum-field theory. T h e discussion in $6.2 is rather general, whereas in $6.3 we study more specific representations and point out their connection with Gaussian measures, and so on. Finally, although the present chapter is concerned only with representations of the commutation relations, we again emphasize that the appli335

336

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COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

cations of harmonic analysis on quasi-invariant measure spaces are by no means confined to representations of the commutation relations: Applications to interacting field equations will be discussed in Volume 11. 56.1. Representations of the Commutation Relations in Quantum Mechanics

l o Basic Properties When dealing with quantum-mechanical systems having one degree of freedom, one must consider operators p , q satisfying the relation

P!7 - 4P

= 1,

(6.1.1)

where I is the identity operator. First, we point out that such operators cannot be bounded. I n fact, we have the following result.

Theorem 6.1.1. Let E be a Banach space. Then, (6.1.1) is not satisfied by any two bounded linear operators p , q on E . PROOF. Suppose that p , q are bounded linear operators on E, satisfying condition (6.1.1). For any bounded linear operator B on E, let B' = Bq - qB, B" = (B')',..., H n )= (B(n-l))',.... Then p' = I and p" = 0. We shall now show that

(P")(")= %!I.

(6.1.2)

+

Since (AB)' = A'B AB', we have the following relation, in analogy with the Newton-Leibnitz formula for differentiation: (6.1.3)

Formula (6.1.2) obviously holds for 1z = 1. Assume that (6.1.2) holds for n - 1. Then, setting A = pn-l, B = p in (6.1.3), and using the fact that p" = 0, we obtain (6.1.2). 2 (1 q 11 11 B 1 . Hence, by (6.1.2), we deduce that Obviously, 11 B' 11

<

n!

< (2 II 4 It)" II P" I1 < (2 /IP II /I 4 11)"

(6.1.4)

for all a, which contradicts the fact that 1imn+a((2 11 p 11 11 q Il)"/n!)= 0. ] In what follows, we shall confine ourselves to the case where p , q are unbounded operators on a Hilbert space; this is, in fact, the situation

6.1. Commutation Relations in Quantum Mechanics actually encountered in quantum mechanics, where operators satisfying the relation

qp - pq

337

p , q are self-adjoint (6.1.5)

= il.

This relation is not essentially different from (6.1.1), indeed, it may be obtained from (6.1.1) simply by replacing q by -iq. By Theorem 6.1.1, the operators p , q in (6.1.5) cannot be bounded, hence (6.1.5) should more properly be written as

qp - pq c il.

(6.1.6)

Here, the notation A C B means that the domain nAof the operator A is contained in the domain 3, of the operator B , that 9, is dense in a,, and that A x = Bx for all x E 3, . Now, let p be a self-adjoint operator on a Hilbert space H , and let ( E A, -a < X < CO} be the resolution of the identity corresponding to p , that is,

p =

j

a,

AdE,.

-a,

Form a one-parameter group of unitary operators (U(t);- co as follows: eitAdE,

U(t)=

< t < CO},

.

-02

This construction will be indicated by the concise notation U ( t ) = eipl.

Theorem 6-1.2, Let p and q be self-adjoint operators on a Hilbert space. Then, relation (6. I .6) is equivalent to e5Pteiqs

= eitseiqseivt

-a < s, t < cr;f.

The proof of this theorem will not be given here, since it is rather lengthy and employs methods not closeiy related with the subject of the present book. Henceforth, when dealing with the commutation relations, we shall always express them in the form given by Theorem 6.1.2, thus avoiding the complications connected with the manipulation of unbounded operators. 20 Quantum-Mechanical Systems Having Finitely Many Degrees of Freedom Let H be a Hilbert space, and let p , ,...,p , , q1 ,..., q, be self-adjoint operators on H , satisfying the following commutation relations:

VI. COMMUTATION

338

RELATIONS IN BOSE-EINSTEIN FIELDS

(i) p , ,...,p, all commute with each other; (ii) q1 ,...,qn all commute with each other; (iii) p , and q,, commute when p # v, and

We then say that {p,}, {q,} satisfy the Heisenberg commutation relations. I n view of Theorem 6.1.2, {p,), (4.) satisfy the Heisenberg commutation relations if and only if the unitary operators exp(ipyt,), exp(iqus,,)( - co < t,, s, < co) satisfy the so-called Weyl commutation relations, namely, (i) all the {exp(ip,t,)} commute with each other, (ii) all the {exp(iqyS,)} commute with each other, and (iii) exp($,t,) commutes with exp(iqySy) when p # v, and eXP(ipYtY) exp(iq,s,) = exp(it,s,) exp(iqYFy) eXP(ip,&)-

We shall presently express these commutation relations in a more concise form. Let R, denote the ordinary n-dimensional real vector space of real n-tuples t = (tl ,..., t,), with the inner product (t, s) defined by n

(2,

$1 = 1 t$,

7

”=l

where s = (sl ,..., s,). Form two families of unitary operators, U(t) and V(s),parameterized by the vectors of R, , as follows:

,..., t,J, = (sl ,...,sn),

for t = ( t l

let

U ( t ) = exp(ipltl)

exp(ip,t,);

(6.1.7)

for s

let

V(s) = exp(iq,sl)

exp(iq,s,).

(6.1.8)

It is easily proved that, for {p,,}, {qv} to satisfy the Weyl commutation relations, it is necessary and sufficient that both {U(t), t E R,} and { V(s),s E R,} are weakly continuous unitary representations of the additive topological group R, in H, and that U ( t ) V(s) = ei(t*s)V(s) U(t).

(6.1.9)

We shall refer to such a system {U(s), V(t); s, t E R,} as a weakly continuous unitary representation of the commutation relations. Actually, for any n, the Weyl commutation relations can be expressed in terms of unitary representations of a single group r, . I n fact, let r, be the totality of triples (x, y , a ) , x, y E R, , a E C, where C is the set of all complex numbers of unit modulus, and define multiplication in I‘, by

6.1. Commutation Relations in Quantum Mechanics

339

r,

Clearly, forms a group with respect to this operation. Take the Euclidean topology on R, and C,and let r, = R, x R, x C have the corresponding product topology. Then T, becomes a topological group; indeed, r, is a (2n 1)-dimensional Lie group. Consider a weakly continuous unitary representation T of the topological group F, in a Hilbert space H. If T satisfies the condition

+

T(0,0, a ) = aI,

where I is the identity operator, let

Straightforward calculation then shows that { U ( x ) ,x E R,}, { V(y ) ,y E R,} are both weakly continuous unitary representations of R , in H, and satisfy relation (6.1.9). Conversely, let { U(x),x E R,}, { V(y ) ,y E R,} be any two weakly continuous unitary representations of R, in H, satisfying relation (6.1.9). Then, T(x,y , 4

=

.U(x)

W)

is a weakly continuous unitary representation of T, in H, and T(0,0, a ) = ai.

We shall now give an important example of a pair of unitary operator groups { U(t)} ,{ V(s)}satisfying the Weyl commutation relations. Let L2(R,) be the Hilbert space formed by the totality of quadratically integrable Lebesgue measurable complex-valued functions on R , , with the inner product

For any t , s E R, follows:

, form unitary operators Uo(t)and V,(s) on L2(R,), as (U,,(t)f)(x)= e*"t-"'f(x),

(~Il(W(4 = f ( x - 4. I t is easily verified that ( U o ( t ) t, E R,} and {Vo(s),s E R,} are weakly continuous unitary representations of R, in L2(Rn),and that they satisfy (6.1.9). The system { U o ( t) ,Vo(s),t , s E R } is called the Schrodinger representation of the commutation relations, or a system of Schrodinger operators.

340

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

We may also express the Weyl commutation relations in another convenient form, as follows. Let C, be the ordinary n-dimensional complex vector space, and define an inner product in C, by (z, z') = zpl'

+ + z,Z,' .'*

where z = (zl,..., z,), z' = (xl',...,x,'). Now, let { U(t), V(s),t , s E R,} be a system of unitary operators, and let ~

(

+tis) = ~ ( t~ )( s exp ) [- i ( t ,

513.

(6.1.lo)

I t is easily proved that the system { U(t), V(s),t , s E R,} satisfies the Weyl commutation relations if and only if the operators { W(z),z E C,} depend continuously (in the weak topology) on the parameter z, and satisfy

Conversely, given a family of unitary operators (W(z),x E C,}, satisfying (6.1.11), we may define U ( t ) = W(t),V(s)= W(is).Then, { U ( t ) ,t E R,} and (V(s),s E R,} are obviously one-parameter groups of unitary operators, satisfying (6.1.9), and are related to W ( z ) by the equation (6.1.10). Moreover, if W ( z )is weakly continuous in z, then {U(t),t E R,} and { V, , s E R,} are weakly continuous. Formula (6.1.1 1 ) is known as the von Neumann form of the commutation relations.

Theorem 6.1.3. Let H be a Hilbert space, W = {W(z),z E C,} a family of unitary operators in H , satisfying (6.1.11), and such that W ( z )is weakly continuous with respect to z. Then, there exists a family {Ha , a: E A } of mutually orthogonal subspaces of H , invariant with @ H , and the restriction of W to any H , respect to W, such that H = is unitarily equivalent to the Schrodinger representation. PROOF. Define a bilinear functional I,(.$,7) on H , as follows:

xa

where

, is a bounded continuous function of x E C, ,the integral Since ( W ( x ) f 7) (6.1.12) exists, moreover

I L(6,?)I G C II E II II 7 II,

6.1. Commutation Relations in Quantum Mechanics

34 1

where 1 (2r)"

c = __

J exp (-

$11

z

112)

dz = 2".

Furthermore, by (6.1.11), we have W(z)*= W(-z), whence it follows easily that L((, .I) = L(q, 5). Therefore, there exists a bounded selfadjoint operator P such that

q 5 , r l ) = (P5,rl).

(6.1.13)

We assert that, for any z E C, , PW(z)P = exp(-

In fact, for any 8, q

E

4 1) z jJz)P.

(6.1 .14)

H,

( P W 4 Pt9 7) = - q W ( z )P t , 17)

1 (24"

J exp (-

- 11 z'

112)

( ~ ( z ' ~) ( z ) 7) dz'

(-

11 z'

112)

~ ( 5 ~, ( z >~*( z ' ) * y )dz'

=-

1

I

=-

(2T)n 1 -(242"

exp

1 4 1

1exp (-

(11

z'

112

~ t ,

+ 11 z" 1 2)

(6.1.15)

Again, using (6.1.1 I), we have ~ ( z '~) ( z~ )( z " = ) exp

+

(-

(W(z')W(z)W(z")t,7) dz'dz".

~ [ ( z 'z,)

+ (z', z") + (z, z")]) ~ ( +zz' + z").

+ + Z" as the variables of integration

Taking z1= z z", z2 = x x' in (6.1.19, and using the equality

342

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

+

+

+

where (ul iv, ,..., u, iv,) = z1, ( jil ,fi2 ,...,fin)= z2 z, we calculate that (6.1.15) is equal to exp(-& 11 z 1I2)(Pt,7). Hence, (6.1.14) is established. I n particular, setting z = 0 in (6.1.14), we see that P is a projection operator. Let 9JI = PH, and choose a complete orthonormal system { v a a, E ‘ill}in 93. Let Ha be the smallest closed linear subspace of H which contains g,, and is invariant under all the operators W(z),z E Cn We shall first prove that Hu 1Ha, , 01 # a‘. In fact, since Pcpa = cpa , Pg,,.= vat , it follows by (6.1.14) that

.

( w ( z )p a w(z’)?a,) 1

=

(w(z)Pva

=

(Pw(-z’) w(z)PvLy 9 v a , )

[

i

[-

51 II z - z‘ 112 - -22 3(z, 2’11 (vu, v,#)

= exp = exp

w(z’)%a,)

3(z, z’)] ( P W ( ~ z’)PV,

,pa,)

(6.1.1 6)

for any z, 2’ E Cn . Thus, W ( z )g,, 1 W ( d )rpa’ when 01 # a’. But Ha is the closure of the set of all linear combinations of the vectors W ( z )ya , z E C, , consequently Ha IHat ( a # a’). Next, we proceed to prove H , is the entire space H . Clearly, that the closed linear hull 6 of 6 is invariant under all the operators W(z),x E C, , and !Dl C G. Therefore, H @ 6 = G1 is also invariant under all the W(z), z E Cn , moreover, Pg,= 0 for all E 61. Suppose that GL # (0), and choose any g, E G1,g, # 0. Then W(x)g,also belongs to 61, and so PW(z)g,= 0. Hence, by virtue of (6. I. 12), (6.1.13), and (6.1. I I), we obtain

u,

E Cn

.

(xl

Since 3(z’,z) =

C (yv’x, - y,x,I),

where z = the above relation shows that the Fourier transform of exp(-t 11 z‘ [Iz)(W(z‘)g,, g,) (regarded as a function of the 2n real variables x,’, y,‘, v = I, 2,..., n) is identically zero. Consequently, ( W(x’)g,,g,) = 0, z‘ E C, . I n particular, taking z‘ = 0, we get g, = 0, which contradicts our assumption. Hence, @ H, . we conclude that H = for all x

+ ;rl ,..., x, + +,),

x‘

xa

= (xl‘

+ iyl’,..., x,‘ + iy,’),

6.1. Commutation Relations in Quantum Mechanics

343

We now consider the manner in which W(x)operates on H, . Writing W ( z )v, = T,,~ , we have, by (6.1.1 1) and (6.1.16),

w4

Pu.2

= exp

[

1

-

2 W', 41 Pm.z+z'

(6.1.17)

and (Pa., 9 P,,Z*)

for all x, x' E C, . Define a function cPs(U)

=

]

[

1 i exp - 4 II z - z' 112 - - 3(2,4 2

vz €L2(R,)(with z E C,

(6.1.18)

as parameter), as follows:

1 1 1 i =rr"/4exp [ - ~ l l ~ l 1 2 + ~ ~ ~ + r ~ ~ ~ - ~ I l (6.1.19) Y l 1 2 - ~ ~ ~ ~

+

where x, y E R, , x = x iy. It is easy to check that the system of operators Wo(x iy) = Uo(x)V o ( y )exp[-(i/2)(x, y)], corresponding to the Schrodinger operators U,,(x) and Vo(y), satisfies

+

[

i

4

W o ( 4 PZ = exp - 2 3(2',

%+2'

(6.1.20)

and

(vz, Tz,)= exp [- 1 11 z - z' 112 - -2i 3 ( z , 41.

(6.1.21)

Construct a linear operator U, from H , to L2(R,) as follows: let U,y,,, = y z , and extend U, by linearity to all linear combinations of the T,,~, z E C, . By virtue of (6.1.18) and (6.1.21), U, is isometric, and since the totality of linear combinations of the T,.~, z E C, , is dense in H , , the mapping U, can be uniquely extended to an isometry of H , into L2(R,). Moreover, it is clear that totality of linear combinations of the functions {vz , x E C,} is dense in L2(R,).Hence, U , is a unitary operator, furthermore, it follows easily from (6.1.17) and (6.1.20) that U,W(Z) UL1 = WO(Z),

that is, the restriction W [ H, is unitarily equivalent to the Schrodinger representation. ] Let H be a Hilbert space, and let {W(z),z E Rn}be a family of unitary operators on H , satisfying the commutation relations (6.1.1 1). If no closed linear subspace of H , other than (0) and H itself, is invariant under all the operators W ( z ) ,then we say that { W(x),z E R,} is irreducible in H . If U ( t ) = W(t), V(s)= W(is),then this irreducibility condition simply

344

v1. COMMUTATION

RELATIONS I N BOSE-EINSTEIN

FIELDS

means that the unitary representation1 of the group r, [i.e., T(x,y, a ) = aU(x) V ( y ) ]is irreducible in the usual sense.

Theorem 6,1.4.

irreducible.

The Schrodinger representation in L2(R,) is

PROOF. Let 9Jl # (0) be a closed linear subspace of L2(R,) which is invariant under all the operators Wo(z),z E C, . By virtue of Theorem 6.1.3, we may assume that the restriction of {Wo(z),z E C,} to %I is unitarily equivalent to the Schrodinger representation, for otherwise, we could select a nontrivial subspace of 'JJZ for which this is the case. Now, for the Schrodinger representation, there exists a vector cp [e.g., p(u) = exp(-4 1) u /I2)] such that the totality of linear combinations of the set { Uo(x)cpI x E R,} is dense in the entire space. Obviously, any family of operators which is unitarily equivalent to the Schrodinger representation must also have this property. Thus, there exists a vector $ in %TI such that the totality of linear combinations of the set {Uo(x)t,41 x E R,) is dense in 9Jl. Let E = {u I $(u) # 0}, and let L2(E)denote the totality of functions in L2(R,) which vanish almost everywhere outside of E. We assert that W = L2(E).In fact, it clearly follows from the foregoing remarks that W C L2(E).Suppose that W # L2(E),that is, that there is an f E L2(E)0'JJZ, f # 0. Then, ( U o ( x ) $ , f )=

ei@.")$(u)f(u>du = 0

, in other words, the Fourier transform of the function $(u) fo €L1(R,) is identically zero. Hence, +(u) f<.> vanishes almost for all x E R,

everywhere, and so f(u) vanishes almost everywhere in E. But f also vanishes almost everywhere outside of E, thereforef = 0 almost everywhere, which contradicts the choice of j.Thus, we must havem = L2(E). On the other hand, 9Jl = L2(E)is also invariant under all the operators Vo(y ) , y E R, . Consequently, for every y E R, , $(u - y ) vanishes almost everywhere outside of E. This means that E - ( E y ) is a null set for every y E R, , that is, E is quasi-invariant with respect to translations. But Lebesgue measure is ergodic (see Lemma 3.1.31), hence, either E or R, - E is a null set. Since'JJZ # (0),E cannot be a null set. Therefore, R, - E is a null set, and 'JJZ = L2(R,). ] From Theorems 6.1.3 and 6.1.4, we immediately obtain the following result.

+

Corollary 6.1.5. Let {W(z),x E C,} be a family of unitary operators on H , satisfying the commutation relations (6.1.11). Then H can be

6.1. Commutation Relations in Quantum Mechanics

345

decomposed into an orthogonal sum of closed linear subspaces H, , E 2l, which are invariant under { W ( z ) z , E C,}, and such that the restriction of {W(z),z E C,} to any H, is irreducible. a

Corollary 6.1.6, If a family { W ( z ) ,z E C,} of unitary operators on H satisfies the commutation relations (6.1.1 1) and is irreducible, then it is unitarily equivalent to the Schrodinger representation. Thus, an irreducible representation of the commutation relations is uniquely determined up to unitary equivalence. Let 'illbe the weakly closed operator algebra in B ( H ) generated by the operators {U(x), V(x') I x, x' E Rn}. We call 2I a concrete Weyl algebra (with n degrees of freedom) on H. In general, the structure of this algebra depends upon the representation { U ( x ) , V ( d ) I x, xf E Rn}.However, we have the following result. Lemma 6.1.7. If the representation { U ( x ) , V(x') I x, xf E R,} is irreducible in H, then the corresponding concrete Weyl algebra 2l is just b ( H ) (see $2.3). PROOF. Let P E (2l')P. Then the subspace PH is invariant under the operators { U(x), V(x') 1 x, xf E Rn),whence it follows by irreducibility that either P = 0 or P = I. Therefore, by Corollary 2.3.5, 21f = {AI I h E F } and '2I = 8 ( H ) (see Example 2.3.2). ] Lemma 6.1.8, Let H be a Hilbert space and 'p a symmetric automorphism of b(H). Then, there exists a unitary operator U on H such that (6.1.22)

?(A) = UAU-'

for all A E b ( H ) . PROOF. For any unit vector ( in H , let P, be the projection operator defined by P,x = (x, ()(, X E H . Note that, if P is any projection operator in H , then 'p(P)z= 'p(P2)= 'p(P), 'p(P)*= 'p(P*) = 'p(P), hence 'p(P)is also a projection operator; conversely, if q ( P )is a projection operator, then so is P . Since P , cannot be expressed as the sum of two nonzero projection operators, neither can q(P,). Consequently, 'p(P,)H is also a one-dimensional space, and so there exists a unit vector T E H such that q(P,) = P , . Now, arbitrarily choose a fixed unit vector (,, , and let yo be a fixed unit vector such that 'p(P,,) = Pn, . For any vector ( E H, form an operator P,,, as follows:

P,,l(Go) pE,(c) = 0

= At,

when

c 1t o .

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We define

ut = d P t t , ) 7 0

*

I t is easily verified that U is a unitary operator, and that relation (6.1.22) is satisfied for any operator of the form P, . I t follows immediately that (6.1.22) is satisfied for any projection operator A which is of finite rank. Furthermore, since any projection operator is the supremum of a family B implies that of projection operators of finite rank, and since A v ( B ) - v(A) = v ( ( B - A)1/2)22 0, relation (6.1.22) holds for any projection operator A. Next, observe that, for any A E b(H), the spectra of A and y ( A ) coincide, hence

<

II A I1 = (I1 A*A

= (I1 dA*A)11)”2 = II rp(4II.

(6.1.23)

Applying (6.1.22) to linear combinations of projection operators, and then taking limits, it follows by (6.1.23) that (6.1.22) holds for any selfadjoint operator A , hence also for any A E d(H). ]

Lemma 6.1.9. Let {Uo(x),Vo(x‘)I x, x’ E R,} be the Schrodinger representation in L2(R,), arid let g, be a symmetric automorphism of B(L2(R,))such that ?( uo(X))

=

uo(x),

y( F‘~(x’)) = P‘o(x’),

X, X‘ E

R,

.

(6.1.24)

Then y is just the trivial automorphism A -+ A. PROOF. By Lemma 6.1.8, there exists a unitary operator U on L2(R,) such that (6.1.22) holds for all A E d(L2(R,)), and by (6.1.24), we know that U E { Uo(x),Vo(x’)I x, x’ E R,}‘. Using Theorem 6.1.4 and Lemma 6.1.7, it follows by $2.3,2O, (iv) and Example 2.3.2 that U E B(L2(R,))’= { X I I X E F } . Thus, U = X I , therefore, by (6.1.22), y ( A ) = A. ]

Theorem 6.1.10.

Let Htk), K

=

1, 2, be Hilbert spaces,

{ W ) ( x ) , V ‘ k ) ( x ’ ) I x, x‘ E R,}

unitary representations of the commutation relations in the respective spaces H ( k ) ,and ‘ill(k) the corresponding concrete Weyl algebras. Then, there exists a unique symmetric isomorphism J/,I from onto ‘ill(2) such that, for any bounded Baire functionlf, +(f(u(l)(x))) = f(U(2’(X>),

#(f(V(’)(X‘))) = f (VL2)(x’)),

x E R, X’ E

R,

(6.1.25)

I

.

(6.1.26)

The notation used here refers to the usual operational calculus (see, e.g., Riesz and Sz.-Nagy [I]).

6.1. Commutation Relations in Quantum Mechanics

347

PROOF. By Theorem 6.1.3, there exists a family of closed linear subspaces {HLk),a E A k } in Hck) such that H ( k )= C, @ Hik),each Hi k ) is invariant under { Vk), Vk)}, and the restriction of { Uk), Vk)} to Hik) is unitarily equivalent to the Schrodinger representation (K = 1,2). Let Qik)be the appropriate unitary operator from L2(R,) to Hik’, and let PLk) denote the projection operator from H C konto ) Then U‘”(X) =

c QP)U0(X)Qy-P?),

(6.1.27)

(I

(6.1.28) For any A E 23(L2(R,)), form the operator2 Ip””‘(A)=

1

Q;k)AQp-qJ;k).

a

Using (6.1.27) and (6.1.28), it is easily verified that y ( k )is a symmetric isomorphism of 23(L2(R,)) onto such that, for every bounded Baire function f, P)(f( uo(~))) = f(U ( k ) ( ~ ) ) , x E Rn (6.1.29) 9

cp‘”‘(f( V0(x‘)))= f(

X‘ E

+

Rn .

(6.1.30)

it follows at once that fulfils the requirements Taking $ = ~$~)(y(l))-l, of the theorem. Now, suppose that $’ is another isomorphism satisfying the conditions of the theorem. Consider the mapping = (+2))-1

*’Ip(1).

(6.1.31)

Obviously, y is a symmetric automorphism of 23(L2(R,)), moreover, by virtue of (6.1.25), (6.1.26), (6.1.29), and (6.1.30), y satisfies (6.1.24). Therefore, is the trivial automorphism of 23(L2(R,)), whence, by (6.1.31), $’ = $. ]

30 Another Type of Representation We shall now consider another type of representation which will be required later on. This type of representation differs from the Schrodinger representation in that it is based upon a Gaussian measure (which is only The sums appearing here and in (6.1.27), (6.1.28) are to be interpreted as strong limits.

348

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FIELDS

quasi-invariant under translations) in R, , rather than Lebesgue measure (which is invariant under translations). Let L2(R,, G) denote the totality of Lebesgue measurable complexvalued functions f on R, such that

L2(R,, G) forms a Hilbert space with respect to the ordinary linear operations and the inner product 1

(f,g) = ,n12 /f(=)g(ld) =P(- II Il’)

du-

For x, x‘ E R, , define ( U ( x ) f ) ( u )= e i ( U . 2 ) f (

4 1

T h e system { U(x), V(x’)I x, x’ E R,} clearly satisfies the Weyl commutation relations. Actually, this system is unitarily equivalent to the Schrodinger representation. I n fact, such an equivalence is realized by the unitary operator

from L2(R,, G) onto L2(R,). I t follows that the representation { U(x), V(x’)I x, x’ E R,} is also irreducible. In the present case, the corresponding self-adjoint operators p , and q, [see (6.1.7) and (6.1 A)] are defined as follows: the domain of p , is

D ( P ~= ) {V I F EL2(% G),uvdu) EL2(& G)), 9

9

where u

= (ul

,..., u,),

and, when

E

D(p,), (6.1.32)

6.1. Commutation Relations in Quantum Mechanics

349

and, when g, E D(qy),

(!?vd(4

=

- ;Uv?(4

a + . au, dU>-

(6.1.33)

I n fact, it is easily shown that (6.1.32) defines a self-adjoint operator, and, when 9) E D(p,),

Consequently, U ( x ) and ( p , ,...,pn} are related as in (6.1.7). A similar argument applies to (ql ,..., q,J. Let c, be the closure of the operator ( p , iqY)/l/Z.I t is easily verified that the domain of c, is

+

and, when

9) E

D(c,), (6.1.34)

T h e adjoint c,* of c, is the closure of the operator ( p y - iq,,)/dZ; its domain is D(cv*)=

1.

a v EL’(&, G ) ,-a% du)

€ L 2 ( R , ,G ) / ,

and, when g, E D(c,*), (6.1.35)

A simple calculation yields the relation CR*CI

- C&*

c 6,J.

Consider the complete orthonormal system {hk(U,

1);

R

=

( k , , k,

)...,kn),

R,

= 0,

1 , 2,...}

VI.

3 50

COMMUTATION RELATIONS IN BOSE-EINSTEIN FIELDS

in L2(R,, G) [see (5.4.15)]; hereafter, we shall simply write hk(u)in place of h,(u, 1). For each nonnegative integer m, let H ( m , n) be the finitedimensional linear subspace of L 2 ( R , , G) spanned by the vectors {h,(u) I I k I = m}. Then, m

L2(R,, G ) =

1 0H(m, n).

m=O

Lemma 6.1.11.

and

T h e operator c, maps H(m, n) into H(m

+ 1, n),

c , ~ , ( u )= (kv‘)1’2hk,(u>,

where k’

=

(kl’,..., k,’) and k k’=

kl,

’

Ik,

=

(k, ,..., k,) are related by

+ 1,

when 1 # V, when 1 = v.

(6.1.36)

T h e operator c,* maps H(m, n) into H(m - 1, n) [letting H ( - 1, n) = (O)], and c,*hk(u) = (k,)1’2h k * ( ~ ) , where k’ = (k,’,..., k,’) and k k 1‘

=

=

(k, ,..., k,) are related by

lkz’

max(k, - 1, 0),

when 1 # v, when I = V.

(6.1.37)

PROOF. Note that the Hermite polynomials [see (5.4.I l)] satisfy the

relations

h,’(x)

= 2mhm-,(x),

h,+,(x) - 2xh,(x)

+ 2mhm-,(x) = 0.

Using these relations, (6.1.34) and (6.1.35), the assertions of the lemma follow immediately. ] We now form system { W(z),z E C,}, in accordance with (6.1.10).

Lemma 6.1.12. Let k = (k, ,..., k,), k’ = (k,’,..., k,’) be any two n-tuples of nonnegative integers, n 2 1, and 01 any real number. Then (W(eimx) hk(.),hk,(.))= (W(x)hk(.),hk,(.))e i u ( ~ k ~ - ~ k ’ ~ ) .(6.1.38)

PROOF. First, notice that, for any A, A’ (W(x)exp(2(h, t ) ) ,exp(Z(2,t))) =

&

E

C,

,

exp ( - d t ; x, A, V )dt, (6.1.39)

6.1. Commutation Relations in Quantum Mechanics where ~ ( tz,; A,

A') = II t

112

- (ix

351

+ y + 2h + 2h', t )

Calculating (6.1.39) by means of the Gaussian integral formula (5.1.5), we obtain

+ 2(X, t)I, exp[-(k,

( W ( z )exp[-(k A) =

exp

[-

4

A')

+ 2(2,t)I)

+ 2(h, A') + i(h, + i(h', z ) ] . Z)

(6.1.40)

Now, the replacement of z by eiaz in the right-hand member of (6.1.40) yields the same result as the replacement of h by heia and A' by h'eciD. Consequently, (W(efaz)exp[-(h, A)

+ 2(h, t ) ] ,exp[-(J',

= ( ~ ( zexp[-e2"(AX, )

A)

+ 2(2,t)I)

+ 2ei=(h,t ) ] ,exp[-

eZia(A', A')

+ 2eia(2, t ) ] ) . (6.1.41)

Using (5.4.13), we expand both sides of (6.1.41) as power series in A, A'; comparing coefficients of like monomials XkXIk', we obtain (6.1.38). 3 40 Gradient Transformations

Let {W(z),z E C,} be an irreducible family of unitary operators on a Hilbert space H , weakly continuous in the parameter z, and satisfying the commutation relations (6.1 .I 1). Let U be a unitary operator on C, . Form the family of unitary operators W ( z )= W(Uz),

zE

c,

in H . Clearly, { W'(z),z E C,} is also weakly continuous in z , and satisfies the commutation relations

[

W ( z )W ( z ' ) = exp - - 3 ( z , z 7 ] W ( z + z'), 2 E

moreover, { W'(z),z E C,} is also irreducible in H. Therefore, by virtue of Corollary 6.1.6, { W ( z ) z, E C,} and {W'(z),z E C,} are unitarily

3 52

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN

FIELDS

equivalent, that is, there exists a uniqueg unitary operator I ‘( U ) on H , such that z E c, . W(Uz) = r(u)W(Z)q u y , I n particular, let us consider the case in which U is of the form eiaI : z +-eiax, x E C, , where a is a real number. I n this case, F( U ) is called a gradient transformation.

Theorem 6.1.13. If H = L2(R,, G ) and {W(x)} are the unitary operators introduced in 3O, then, for any real number a, we have (6.1.42)

where P,,denotes the projection of H onto H(m, n). PROOF. Since the operator defined by (6.1.42) has the property r(k.1)h k ( ’ )

= e-ialWzk(.),

it follows that (6.1.38) may be rewritten as (r(ei.1) W ( z )T(e”I)-l h k , hk,) = (W(e%) h k , h k ’ ) .

But {hk} is a complete orthonormal system in L2(R, , G), hence W(e%)

= r(ei.1)

W ( z )T(&I)-l.

This proves that (6.1.42)is, in fact, the required operator.

]

56.2. Quasi-Invariant Measures Applied t o Representationsof the Commutation Relations in Bose-Einstein Fields

lo Representations of the Commutation Relations: Various Equivalent Formulations We now turn to the consideration of the general case (including that of infinitely many degrees of freedom).

Definition 6.2.1. Let $3 and $3‘ be real linear spaces, and let B(x, x’)(x E 43, x’ E $3’) be a nondegenerate real-valued bilinear functional on ($3, 5’), that is, the following conditions are satisfied. (i) For any fixed x E $3, B(x, x’) is linear in x’; for any fixed x’ E &’, B(x, x’) is linear in x. Translator’s note: That is, up to a numerical factor of unit modulus.

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

353

(ii) For any x E $3, x # 0, there exists an x‘ E $3’ such that B(x, x’) # 0; conversely, for any x’ E $3’, x’ # 0, there exists an x E Sj such that B(x, x ’ ) # 0. Then, Z = ($3, sj‘, B ) is called a single-particle (state vector) system. We shall have occasion to imbed $3’ in $3” as follows: for every x‘ E Sj‘, define a functional f,. E $3” by f&(X)

=

B(x, x’),

x E $3.

Clearly, the correspondence x‘ --+ fx# is an isomorphism of $3‘ onto a certain linear subspace of $3”. Thus, by identifying every x’ with the corresponding f,, ,we may regard $3 as a linear subspace of $3”.

Example 6.2.1. Let sj be a real linear space, and $3’ a sufficient subspace of $3”. For any ~ € 5x ,‘ E $ ~ ’ ,let b(x, x’) = x’(x). Then ($3, sj’, B ) forms a single-particle system. Example 6.2.2. Let $5 be a real inner product space, let sj’ = 8, and let B(x, x ’ ) be the inner product on $3. Then (a, Sj, B ) forms a single-particle system. Definition 6.2.2. Let Z = ($3, sj’, 23) be a single-particle system, and C the totality of complex numbers of unit modulus. Let r(Z) = $3 x sj‘ x C, and denote the elements of r(Z)by ( x , x’, a),x E $3, x’ E $3’, a E C. Define a multiplication operation in r(Z)as follows: (x, x’, a ) ( y ,y‘, 8) = (x

+ y , x’ + y’, aPe-iBcu-s’)).

Then r(Z)forms a group, which we call the group associated with the system Z. Let S : y +S(y) be a unitary representation of the group r(Z) in a complex Hilbert space H , such that S(O,O, a) = aI,

where I denotes the identity operator. Suppose that, for any pair of finite-dimensional linear subspaces W C $3 and W’C $3‘, the mapping y --+ S(y) is weakly continuous on W x W’x C with respect to the product topology induced by the ordinary Euclidean topologies on W, W’and C. Then S is called a canonical unitary representation of r(Z)in H. The above definition is equivalent to the following one.

Definition 6.2.3. Let .Z = ($3, $3‘, B ) be a single-particle system, and H a complex Hilbert space. Let U : x + U ( x ) and V : x’ -+ V(x’)

354

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

be representations of sj and sj‘, respectively, in the group of unitary operators on H , such that the commutation relations U ( x ) V(x’) = eiB(zs*‘)V(x‘)U(x),

x E sj,

x‘

E Jj‘

(6.2.1)

are satisfied, Moreover, suppose that each of these representations is quasi-continuous, that is, if x(x’) is restricted to any finite-dimensional linear subspace of $($’), then the mapping x -+ U(x)(x’ -+ V(x’)) is weakly continuous (with respect to the Euclidean topology on the subspace). Then { U , V }is called a (Weyl) canonical system (in H ) over Z. In fact, given any canonical system { U,V } over Z,one may construct a canonical unitary representation of r(Z)in H , as follows: for any y = (x, x’, a ) , let S(x, x’,

E)

=

.U(x) V(x’).

We call S the canonical unitary representation corresponding to {U, V}. Conversely, given any canonical unitary representation S of r ( Z ) in H , let U ( x ) = S(x, 0, l),

V(x‘) = S(0, x’, 1).

Then { U(x), V(x’)I x E Jj, x’ E 9’) is a canonical system over Z, to which S corresponds in the above sense. Let { U , V } be a canonical system. Choosing any x E $, x’ E Jj‘, we form the infinitesimal generators I d t=0

(6.2.2)

of the one-parameter unitary groups {U(tx) [ -00 < t < 00) and { V ( t x ’ ) I -00 < t < a}, respectively (see 511.3, 2O). Consideration of the properties of these infinitesimal generators leads to the following definition.

Definition 6.2.3’. Let Z = ($, $‘, B ) be a single-particle system and H a complex Hilbert space. Let p : x -+ p ( x ) , q : x’ --t q(x’) be linear mappings of sj and Jj’, respectively, into the set of self-adjoint operators in H . Suppose that, for all x, y E 8,x’,y‘ E sj‘, (i) p ( x ) commutes4 with Here, as in 56.1, the cornmutability of two (unbounded) self-adjoint operators p ( x ) and p ( d ) means that their spectral functions (i.e., resolutions of the identity) commute, and the “sum” of p ( x ) and p(x’) means the closure of their linear sum (of course, we assume that this closure exists).

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

355

p ( y ) , (ii) q(x’) commutes with q(y’), and (iii) p(x), q(x’) satisfy the commutation relation

q ( x ‘ ) p ( x ) - p ( x ) q(x’)

c iB(x, x’)L

(6.2.3)

Then, we say that { p ( x ) ,q(x‘) I x E $j, x‘ E $’} is an infinitesimal canonical system (in H ) over Z. If { U , V } is a Weyl canonical system, then, using Theorem 6.1.2 and the facts stated in $11.3, 2O, it is easily shown that the { p , q} defined by (6.2.2) form an infinitesimal canonical system. Conversely, given any infinitesimal canonical system { p , q}, the unitary operators defined by

u(x) = p i P ( r ) ,

qx‘) = ei4(S’)

(6.2.4)

form a Weyl canonical system {U, V}.5 In what follows, we shall most often be concerned with the case mentioned in Example 6.2.2. Let A be a complex inner product space, and denote its inner product by ( z , z’), x, x’ E A. Let J : x -+ x* be a one-to-one mapping from A onto A, satisfying the conditions (x*)* = x,

and

(ax

+ @)* = Ex* + by*,

a, /3

complex numbers,

(X*,Y*) = ( Y , 4.

Then Jis called an involution in A. Let $j= {x I x* = x,x E A}. Then 5 forms a real inner product space with respect to the given inner product (x,y ) , x, y E 5. Moreover, every z E A can be uniquely expressed in the form z=x+iy,

x,yE$j.

A is said to be the complexiJication of 5. We note that involutions do exist in every complex inner product space A. I n fact, one need only choose an arbitrary complete orthonormal system in A, and let 5 denote the cIosure of the set of all real linear combinations of this system. Obviously, each z E A has a unique decomposition z = x iy, x,y E $5, and we set z* = x - iy.

+

Translator’s note: The verification of equivalence between the notions of Weyl canonical system and infinitesimal canonical system (as defined by the author) involves certain technical difficulties. Since the definition is, in any case, an ad hoc one, perhaps it might be better to simply say that { p ( x ) , q(x’)} is an infinitesimal canonical system provided that {eip(z’, ei9(*‘)} is a Weyl canonical system.

356

VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

Then z ---t z* is an involution. Furthermore, given any real inner product space sj, there clearly exists a complex inner product space A which is a complexification of fi, and A is unique up to an isomorphism which leaves 4, invariant. Now, if 2 = (a, $3, B), where sj is a real inner product space with the inner product B , we form the complexification A of 8,and call it the state vector space of 2.

Definition 6.2.4. Let R be a complex inner product space, with inner product (z, z‘), z, z’ E A. Let { W ( z )1 a E A) be a family of unitary operators on a complex Hilbert space H , satisfying the von Neumann commutation relations

Suppose that the mapping z -+ W ( z ) ,when restricted to any finitedimensional linear subspace %Jl, is weakly continuous (with respect to the Euclidean topology on IIJZ). Then we say that {W(z)I z E A} is a (von Neumann) canonical system over the state vector space A. If sj is a real inner product space, with inner product B(x, x’), and A is the complexification of sj, and if { U , V > is a Weyl canonical system over 2 = (@, $3, B), form the system

Then {W(z)I z E A} is a von Neumann canonical system. Conversely, if (W(z)1 z E A} satisfies the conditions of Definition 6.2.3, choose any real linear subspace sj of A such that (x, y ) is real on $3, and such that R is the complexification of sj. Let V(x) = W(x),

V(x’) = W(ix’), x, x’ E fi.

Then, it is easily verified that { U, V }constitutes a Weyl canonical system over Z = (6, fi, B), B(x, x‘) = (x, x‘).

20 The General Form of a Canonical System Definition 6.2.5. Let { U k , Vk},k = 1, 2, be canonical systems, in the respective Hilbert spaces H , (k = 1,2), over the same single-particle

6.2. Quasi-Invariant Measures and Bose-Einstein Fields system

357

Z.If there exists a unitary operator Q from H , onto H , such that QU1(x)Q-’

=

UZ(x),

QV1(x’)Q-’

=

V2(x’)

(6.2.5)

for all x E 5, x‘ E $j‘, then we say that the systems { U , , Vk},k = 1, 2, are unitarily equivalent.s Insofar as their essential properties are concerned, any two unitarily equivalent canonical systems may be regarded as identical. Accordingly, we shall proceed to describe a certain type of concrete canonical system such that every canonical system is unitarily equivalent with one of this type, and thereby establish the general form of a canonical system. Let Z = (5,b’,B ) be a single-particle system, and let { U , V } be a canonical system over Zin the complex Hilbert space H . As in $3.4, we let 2l and 6 be the weakly closed operator algebras generated by { U ( x ) I x E 5) and {V(x’)I x‘ E B}, respectively. Clearly, both % and 6 are commutative.

Lemma 6.2.1. For every cardinal number n, there exists a projection operator P, E 2l‘ n 6’such that the restriction of 2l to P,H has uniform multiplicity n. Moreover, En P, = I . PROOF. Applying Theorem 2.4.3 to the commutative weakly closed operator ring ‘2l, we see that there exists a unique system of projections (P,}satisfying all the requirements of the Lemma, except possibly the condition P, €6‘.As in the proof of Theorem 3.4.1, we consider the mapping T(x’): A V ( x ’ )AV(x’)-1, A E Iu, --f

where x’ is an arbitrary element of 5’.Using relation (6.2.1), it is easily proved that T(x’)% = ‘3. By the uniqueness of the projections {P,}, it follows that T ( x ’ )P, = P, , whence P, E 6’. ] I n view of Lemma 6.2.1, we may, in the ensuing discussion, assume that % has uniform multiplicity n in H . When this is the case, we shall say that (U(x) I x E &} has uniform multiplicity n. Also, we shall restrict our considerations to the case where H is separable, this being sufficient for applications to quantum field theory. Theorem 6.2.2. Let Z = {$, b‘,B} be a single-particle system, and { U , V>a canonical system over Z in the separable Hilbert space H. Suppose that { U ( x ) I x E $j}has uniform multiplicity n. Then, there 23, p), where SZ is a linear subspace exists a linear measure space S = (0, of $”, L? 3 b‘,23 is the totality of weak Bore1 sets in SZ, p(Q) < OC), and S is quasi-invariant under b‘,such that { U , V }is unitarily equivalent This simply amounts to the unitary equivalence, in the usual sense, of the corresponding canonical unitary representationsof the group r(Z).

358

VI .

COMMUTATION RELATIONS IN BOSE-EINSTEIN

to the canonical system For any E E Qn2(S),

FIELDS

{o,P} in en2(S)which is defined as follows. x E $5,

Q ( x ) 5 ( f ) = eif(*’5(f), x’) ,”I)*( dcL

(6.2.6) x’ E 8‘.

(f)

(6.2.7)

Here, for each x’ E fi’, x( f;x’) is a unitary operator-valued measurable function’ on S, such that, for any given pair of elements x1 , x2 E fi’, the relations

z ( f ;xi

+ xJ

= z ( f ;xi)

z ( f - x i ; xz),

~ ( f0); == 1

(6.2.8)

hold for almost allf. PROOF, Since H is separable, it is obvious that n K O . Now, 2l has uniform multiplicity n, that is, there exist closed linear subspaces H, of H , a = 1, 2,..., n, such that H = @ H , , each H, is invariant under ‘u, each restriction ‘u, = % I H , is maximal commutative, and all the ‘u, are unitarily equivalent. Since H , is separable, it follows from Corollary 2.4.9 that ‘u, has a cyclic element in H, . Choose a system of cyclic elements {e, E H , , a = 1, 2,..., n} which correspond with one another under the unitary equivalences between the a,. Consider the function

<

x,

4(.)

=

( We,

9

em).

It is easily verified that +(x) is a positive definite quasi-continuous function on 6.By Theorem 4.3.5, there exists a linear measure space S = (9,23, p), where 9 is a linear subspace of $j”,9 3 &’ and y ( 9 ) < 00, such that #(x) =

1

a

eif‘”)

dp(f).

Proceeding as in the proof of Theorem 3.4.14, one can show that S is quasi-invariant under fi‘, and that there exists a unitary operator Q from H to gn2(S)such that = QU(x)Q-I. T h e remaining assertions can then be established by following the proof of Theorem 3.4.5. ] Remark. Suppose we are given a linear measure space* S = (52, 23, p), where fi’ C 52 C $”, 123 is the totality of weak Borel sets in 52, p ( 9 ) < co, and S is quasi-invariant under $5’. Also, Iet there be given a system of

o(x)

’

That is, taking values in the group of unitary operators on n-dimensional complex Hilbert space (see Definition 2.4.4). Translator’s note: Assuming only that 9 is a linear subspace of $?A and that 23 is the totality of weak Borel sets in Sa, one can then deduce that the conditions of Definition 4.2.3 are necessarily satisfied.

359

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

unitary operator-valued measurable functions z( f ; x’), f E SZ, x‘ E $3’, satisf ing conditions (6.2.8). Then, it is easily verified that the operators d efined by (6.2.6) and (6.2.7) constitute a canonical system9 over Z in Qfl2(S>. Thus, Theorem 6.2.2 gives the general form of a canonical system in the case of uniform multiplicity.

{a6

Corollary 6.2.3. Under the hypotheses of Theorem 6.2.2, (i) if $3 = $3’ is a nuclear space, B(x, x‘) is a continuous inner product on $3, and !lJ is the completion of $3 relative to B(x, x’), form the rigged Hilbert space $3 c(nl c $3+. Suppose also that x + U ( x )is weakly continuous relative to the topology of $3. Then, we may choose SZ = $3+. (ii) If $3 = $3’ is an inner product space, with the inner product B(x, x’), and if x --t U ( x ) is weakly continuous relative to the topology of $3, then we may choose as SZ any Hilbert space containing Sj such that the inclusion map of $3 into SZ is a Hilbert-Schmidt type operator. PROOF, I n the proof of Theorem 6.2.2, instead of applying Theorem 4.3.5 to the positive definite continuous function #, we apply Corollaries 4.3.14 and 4.3.15, thereby deriving conclusions (i) and (ii),l0 respectively. ] T h e case of greatest interest to us is n = 1 and z ( f , x’) = 1 (this is the case which suits the requirements of quantum field theory).

Theorem 6,2.4. Let Z = ($3, a’, B ) be a single-particle system. Let SZ be a linear subspace of containing $3’, let 23 be the totality of weak Bore1 sets in L2, and let (a, 8,p k ) , K = 1,2, be finite measure spaces which are quasi-invariant under $3’. Suppose that { u k , vk}, k = 1,2, are canonical systems over 2 inL2(SZ, 23, pk),defined by uk(X)

[ ( f ) = eif(r)Hf),

(6.2.9)

x E $3,

~ ~ ( [(f) ~ ’ =1 [(f - x’) ( d p k r ‘ ( f ))1’2,

X’

E

$3’.

(6.2.10)

dpk(f)

Translator’s note: I t appears that some additional condition on S is necessary to ensure the weak continuity of V ( x ’ ) (on finite-dimensional subspaces of $’). lo Translator’s note: Whether or not part (ii) of Corollary 6.2.3 is true, it is hard to see how it can be deduced from Corollary 4.3.15. If 5’= $ is a real separable Hilbert space which is a dense linear subspace of another separable real Hilbert space Q, and if the imbedding of $ into Q is of Hilbert-Schmidt type, then by Corollary 5.3.3 it follows that there exists a finite measure space (Q, b,p ) which is quasi-invariant under 8.Thus, one can obtain a canonical system (6.2.6), (6.2.7), but it is not clear why it should necessarily be equivalent to the a priori given canonical system ( U , V}.

360

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Then the measures p1 and pa are equivalent if and only if the canonical systems {Ul , Vl} and { U 2 , Vz}are unitarily equivalent. PROOF. Assume that pl and pa are equivalent. Form a unitary operator Q, from L2(Q,23, p,) to L2(Q,23, pa), as follows: (6.2.1 1)

Then, using (6.2.9), (6.2.10), and (6.2.11), it is easy to check that (6.2.5) holds. Conversely, assume that Q is a unitary operator from L2(Q,8,pl) toL2(Q, 23,pz) such that (6.2.5) holds. Write &(f) = Ql. Then, for any finite set of elements xl, x2 ,..., x, E sj and any n complex numbers A, ,..., A, , we have

Introducing the finite measure

the foregoing relation may be written in the form

Now, since 23 is the smallest a-algebra with respect to which all the functions { f ( x ) I x E sj} are measurable, the algebra

constitutes a determining set on (Q, 23). Hence, by Corollary 1.1.6, ID is pa), dense inL2(SZ, 23,pl p3). ConsequentIy, for any 5 eL2(Q,23, p , we see that

+

+

6.2. Quasi-Invariant Measures and Bose-Einstein Fields I n particular, letting E E b,we obtain

5 be the

361

characteristic function of an arbitrary set

Thus, pl Q p 2 . Similarly, p2 equivalent. ]

< pl.

Therefore, pl and pa are

Definition 6.2.6. Let ,Z be a single-particle system, and {U, V } a canonical system over ,Z in the complex Hilbert space H. If no nontrivial closed linear subspace of H is invariant under all the operators U(x), V(x’),ll then { U , V } is said to be irreducible. [This is equivalent to the irreducibility of the corresponding canonical unitary representation of the group r(Z)]. Let A be a complex Hilbert space, and let U denote the group of all unitary operators on R. In quantum field theory, one must consider not only a canonical system { W ( z )I z E R} in some Hilbert space H , but also a unitary representation r : U -+r(U ) of U in H. This representation has the property

r(U ) W(z)r(U)-1 = W(Uz).

(6.2.12)

We note that, if { W ( z )1 z E R) is irreducible, then (6.2.12) uniquely determines r ( U ) up to a scalar factor a ( U ) , where U + a ( U ) is a representation of U in the group of complex numbers of unit modulus. I n fact, suppose that U 3 r’(U ) is another unitary representation of U in H such that

r‘(U ) W(z)a(U)-1 = W(Uz) for all z E R, and let To(U ) = r’(U)-lr( U). Then

U )E { W ( z )I z E R}’. Since {W(z)I z E R} is for all z E R. Hence, To( irreducible, To(U ) must be of the form a( U ) I , where a( U ) is a complex number of unit modulus.

Theorem 6.2.5. Let ,Z = (9,9’,B) be a single-particle system. Let Q be a linear subspace of $ A such that 9‘ C Q, let b be the totality of weak Bore1 sets in Q, and let S = (Q, 8,p ) be a finite measure space l1

Or, more concisely, { U ( x ) ,V(x’)I x E $, x’

E

$’}’ = {UI X a complex number}.

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which is quasi-invariant under 8‘. Then the canonical system in L2(S) defined by U(x) S ( f ) = e i f ( q f ) , x E sj, (6.2.13)

is irreducible if and only if S is ergodic with respect to the group of translations sj’. PROOF. Assume that { U , V } is irreducible. If S were not ergodic, there would exist a quasi-invariant set E E 23 such that p ( E ) > 0, p(Q - E ) > 0. Let M = {[ I (( f ) = 0 for almost everyfE E}. Then M is a closed linear subspace of L2(S)which is invariant under all the operators { U ( x ) ,V(x’)},and L2(S) # M # (0).This contradicts the irreducibility of { U , V } . Conversely, assume that { U , V }is not irreducible, so that H contains a nontrivial closed linear subspace M which is invariant under all the operators U(x), V(x’). Let P denote the projection operator from H onto M ; then P E rU‘ n E’. Now, since {eif(r)I x E sj} is a determining set on S , it follows by Lemma 2.4.4 that !!Icontains the multiplication algebra mZ(S); but p is finite, hence, by Lemma 2.4.10, mZ(S) is maximal commutative, and therefore !!I = W(S). Hence 9l is maxima1 commutative, and so we have P E rU’ = !!I = %R(S).Consequently, there exists a bounded measurable function v(.) on S such that P is the multiplication operator corresponding to -q(.). Since P is a projection operator, it is easily seen that q(-) is equal almost everywhere to the characteristic function of some set E E 23. Moreover, using the fact that P EE‘, it is easy to deduce that E is quasi-invariant. Furthermore, since I # P # 0, # p ( E ) # 0. Thus, S is not ergodic. ] Recall that, in the case of finitely many degrees of freedom (i.e., when sj is finite-dimensional), all irreducible canonical systems are (by Corollary 6.1.6) unitarily equivalent. However, in the case of infinitely many degrees of freedom, the situation becomes very complicated. For then, the number of mutually inequivalent ergodic measures is very great indeed (for example, there is at least a continuum of mutually inequivalent Gaussian measures on a countably dimensional space S!). Consequently, even if we confine ourselves to those having the relatively simple form (6.2.13)-(6.2.14), there are still a vast number of unitarily inequivalent irreducible canonical systems. I n this context, the import of the following problem is clear.

,(a)

Problem. Let Q, sj be Hilbert spaces such that !jj is a linear subspace of Q and the operator imbedding $5 into Q is of Hilbert-Schmidt

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

363

type. Determine the general form of all measures on (Q, 23) which are quasi-invariant and ergodic with respect to fj.12 We shall now express the von Neumann commutation relations in terms of quasi-invariant measures. Let 52 be a complex Hilbert space with inner product ( x , x’), x , z’ E R, and write I x I = (2,~ ) l /Now, ~ . R may also be regarded as a real linear space, and is, in fact, a real Hilbert space with respect to the inner product [z, 2’3 = %(z, z’),

z, z’ E R.

We denote this real Hilbert space by sj to distinguish it from the underlying complex Hilbert space R.

Theorem 6.2.6.

Let S = (SZ,%, p ) be a standard quasi-invariant measure space associated with fj (see 54.2). For each z E 52, let W ( z )be the unitary operator on L2(S)defined by

(6.2.15)

where (iz)(w), w E SZ denotes the quasi-linear functional13 corresponding to the element i z of 9. Then { W(x)I z E R} satisfies the von Neumann commutation relations. PROOF.For any 2, z’ E R, we have

(w4 u.’(z‘lf)(w) = exp

(-

+

[(iz)(w) (iz‘)(w

+ z)l)f(w + z + z’> (

dp(w)

(6.2.16)

Since (id)( .) is the quasi-linear functional corresponding to id, the relation (iz’)(w + z ) = (iz‘)(w) + [iz’, z] (6.2.17) la Translator’s note: Although it is not explicitly stated, the author presumably intends that 0, 5 are real and separable, that 8 is dense in 0, that b is the totality of weak Bore1 sets in 52, and that only finite measures are to be considered. l3 Translator’s note: Unless some additional hypothesis is imposed (see Lemma 4.2.4 and the remark following it), these functionals need not be unique. However, a system (6.2.15) satisfying the von Neumann commutation relations can- be constructed by choosing any subspace W of 0, which maps isomorphically onto 9’’ under the natural homomorphism (4.2.7), and, for each z E 9, letting (iz)(.)be unique antecedent of iz in W.

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holds almost everywhere. But [id, x] = %(iz’,z ) = 3(z, z‘), hence the required conclusion follows directly from (6.2.16) and (6.2.17). ]

30 Algebras of Observables Let Z = ($3, sj‘, B) be a single-particle system and { U, V} a canonical system over Z in the complex Hilbert space H. Let M be the weakly closed operator algebra generated by % u K. If (U, V } is irreducible, then M’ = {M I h a complex number), hence M = B(H). Suppose that { U , , V,} is another irreducible canonical system over Z, in another complex Hilbert space H,; in general, there will not exist a symmetric isomorphism q~ from B(H) onto B(Hl) such that v(U(x))= Ul(x), y ( V ( x ‘ ) )= Vl(xf). For, by Lemma 6.1.8, such an isomorphism must be of the form v(A) = UAU-l, where U is a unitary operator, that is, {U, V } and {Ul , V,} would have to be unitarily equivalent. Thus, it is apparent that the above definition of the algebra M is of little use. Accordingly, we proceed to reduce the size of M, in the following manner. Let %It, %Ituz’be finite-dimensional linear subspaces of $3, sj‘, respectively, and suppose that B(x, x’) is nondegenerate on (%It, %It’) (in particular, this implies that YJl and %It’ have the same dimension). We then say that For each nondegenerate pair of finitethe pair (%It, %Itf) is nondegene~ate.’~ dimensional subspaces (%It, %It’), let Wmm*denote the weakly closed ring generated by the family of unitary operators { U(x),V(x‘)I x E %It, x’ EYJ~’} on H, and let 2B be the smallest uniformly closed operator algebra which contains the union of all such rings 2Bm,mf .

Definition 6.2.7. Let ,Z be a single-particle system, { U,V} a Weyl canonical system over Z in the Hilbert space H , and {p, q} the corresponding infinitesimal canonical system. Then, the operator ring 2B described above is called the concrete WeyZ algebra over Z (on H) associated with {U, V } (or with {p, q}). The operators in 2B are called the observables of Z.

Theorem 6.2.7, Let Z = (8, sj’, B) be a single-particle system, let { p , , q,}, { p , , q,} be infinitesimal canonical systems over Z in the Hilbert spaces Hl , H , , respectively, and let a, , 2& be the associated concrete Weyl algebras. Then, there exists a unique symmetric isomorphism from a,onto 912 such that, for any bounded Baire functionf, d f ( P 1 W ) =

f(PZ(4,

d f ( q , ( x ‘ ) ) ) =f ( q 2 ( 4 ) ,

x E $3,

x’

E

5’-

l4 Note that, since B(x, x’) is nondegenerate on (5, $’), it follows that, for each finitedimensional linear subspace 1112 C $, there exists a finite-dimensional linear subspace 901’ C such that the pair (1112, !W) is nondegenerate.

a’

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

365

PROOF. Let (1131, 1131’)(1131 C fi, 1131’ C 8’) be any nondegenerate pair of finite-dimensional linear subspaces. Choose bases {el ,..., e,} and {el’,..., e i } for 1131 andm’, respectively, such that, when x = CL1 x,e, €1131 and x f = CL1 x,’e,’ E‘W, B(x, x’)

For any t

=

(tl ,..., t,), s

= xlxl’

= (sl

+ + x,x,’.

,...,s,)

u7&) = exP(iP,(e,)

tl)

Vk(4 = exp(Me1’) 51)

***

E ..* **.

R, , form the operators exP(iP,(en)

tn),

exp(iq,(en‘) sn),

= 1, 2. Then { Uk(t) I 2 E R,} and { v k ( s ) I s e R,} are weakly continuous unitary representations of R, in H k , and satisfy the commutation denote the weakly closed operator relations (6.1.9), k = 1, 2. Let %B$,! ring on Hk generated by the family of operators { Uk(t), v k ( s ) 1 t , s E R,}. By virtue of Theorem 6.1.10, there exists a unique symmetric isofrom 2BG:,, onto 2B!$!m, such that, for every bounded morphism prm,, Baire function f, ml,w,(f(P,(x))) = f(P&)), (6.2.18)

k

m,wlc(f(ql(x’)))= f(q&’)).

(6.2.19)

Let A denote the totality of nondegenerate pairs of finite-dimensional subspaces (YR,1131‘). If (%, %’), (mZ, 9X‘) E A and 9X C %, ‘33’ C W’, then we write (1131,1131’) < (%, %’); the set A is directed by this ordering relation. Now, ! l B ( k ) is the uniform closure of the ring

mp =

(J

m#,. .

(YJ31.W)€A

Clearly, if (a,3 ’ ) < (1131, W‘), then ?lBRhjC 2 B ~ ~ m . We m . assert that the restriction of F ~ , , to~ !Elk,&,is just rpw,w, . In fact, since (6.2.18) and (6.2.19) are, in particular, valid for all x E %, x’ E a’, it can be seen from to !lB!&, is also the proof of Theorem 6.1.10 that the restriction of pm,mJ a symmetric isomorphism from m#,&,to $2llg,&.such that (6.1.25) and (6.1.26) hold for all x E %, x‘ E %’. But Theorem 6.1.10 states that there exists just one such isomorphism, namely, q ~ ~ . ~ , ~ ~ , We can now define a mapping qo from ‘2Bf)to mi2),as follows. Given any A E 213h1’, there exists (ml,9X‘) E A such that A E 2BDl,m,. Define TOW) = m . m n ’ ( 4

Then rpo is a symmetric isomorphism from ‘ 9 3 ~ onto ) mi2’.Since ‘lxil),

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'r23h2' are dense in 'r23D(2),respectively, v0 can be extended to a symby (6.2.18) and (6.2.19), metric isomorphism g, from 'r23(') onto g, satisfies the requirements of the theorem. Finally, the uniqueness of g, follows at once1&from the uniqueness of the g,w,ml. ] 40 The Characteristic Functional of a Canonical System $3, B) be a single-particle system, Definition 6.2.8, Let Z = (8, where Sj is a real inner product space and B( *, -)is the inner product on Sj. Let 3 be the complexification of $3, and let { W ( z )I z E R} be a von Neumann canonical system over 2 in the Hilbert space H . Suppose there exists an rlo E H such that { W(z)rlo I z E R) generates H ; then, the canonical system { W ( z )I z E S} is said to be cyclic, rlo is called a cyclic element for the system, and

d'(.)

= ( w ( zTO ) 9 TO),

9

is called the characteristic functional of the canonical system (associated with v0). Following the proof of Lemma 2.4.6, we easily obtain the following result.

Lemma 6.2.8. Let Z be a single-particle system, and let { W ( z )I z E R} be a von Neumann canonical system over Z in a separable Hilbert space H . Then, H can be decomposed into an orthogonal sum

where each H E is invariant under the system { W ( z )I z E R}, and the restriction of ( W ( z )I z E R} to H , is cyclic. By virtue of the above lemma, we may henceforth restrict our considerations to cyclic canonical systems. If two canonical systems are unitarily equivalent, and one of them is cyclic, then obviously the other is also cyclic, moreover, the characteristic functionals associated with corresponding cyclic elements are equal. Conversely, we have the following result.

Lemma 6.2.9. Let Z be a single-particle system, and let {W,(z) I x E R), K = 1, 2, be cyclic canonical systems in the respective Hilbert spaces Hk , k = 1, 2. Suppose that there exist cyclic elements l6 Translator's note: If, in the statement of Theorem 6.2.7, one adds the condition that 'p maps each q),m, onto B3#,m,,then the uniqueness follows easily, as asserted. Otherwise, it seems difficult to prove.

6.2. Quasi-Invariant Measures and Bose-Einstein Fields

367

v k in Hk, k = 1 , 2, such that the associated characteristic functionah are equal. Then the two canonical systems are unitarily equivalent. PROOF. Let $(z) denote the common characteristic functional associated with ql and q2 . Since

for any z, x’

E

A, it follows that

(6.2.21)

Let Mk be the linear hull of the set of vectors { wk(z)q k , z E A}, k! = 1, 2, and let Q be the linear mapping from Ml onto M , defined by Q Wl(z)ql = W,(z) q 2 . By (6.2.21), Q is an isometry, and since Mk is dense in Hk (k = 1, 2), Q can be uniquely extended to a unitary operator from Hl onto H , . Moreover,

Consequently, for any

4 E M , , we have

This proves that QW,(x) Q-’ = W2(z),z E A. 3 Next, we establish a necessary and sufficient condition for a function to be the characteristic functional of some canonical system.

Lemma 6.2.10. Let R be a complex inner product space and $(z) a complex-valued function on A. Then, #(z) is the characteristic functional of a von Neumann canonical system in some Hilbert space if and only if: (i) # is quasi-continuous, and (ii) the function

368

m.COMMUTATION

RELATIONS I N BOSE-EINSTEIN FIELDS

is a positive definite kernel, that is, for any finite set of complex numbers A, ,..., A, , and any n vectors z1,..., z, E A,

PROOF. Suppose that I/J is the characteristic functional of a canonical system { W ( z )I z E A}. Then, using (6.2.20), we see that the left-hand side of (6.2.22)is equal to

Thus, condition (ii) is satisfied. Furthermore, since W ( z ) is weakly continuous on any finite-dimensional linear subspace of A, it follows immediately that t,h is quasi-continuous. Conversely, assume that zL, satisfies conditions (i) and (ii). Let Ho denote the totality of complex-valued functions on A which vanish everywhere except possibly on some finite set of points x1 ,...,z, E A. Obviously, H, is a linear space relative to the ordinary linear operations. For any f,g E H, , define

(f,g)

=

C

Z.Z'ER

[ f 3(z, z')] f(z) g(z').

#(z - 2') exp -

(6.2.23)

If we identify elements of Ho whose difference f is such that ( f , f ) = 0, then (6.2.23) induces an inner product ( f , g ) on H, . Let H be the completion of H, relative to this inner product. For each zo E A, define an operator W,(z,) on H, as follows: (WO(ZOlf)(4

= exp

[; w, a,)]f(z - zo).

(6.2.24)

I t is easily verified that (WO(ZO)f?

Wo(z0)g)= (f!g).

I n particular, if ( f , f ) = 0, then ( Wo(z,)f, W,(?,)f) = 0. Hence, Wo(zo) is an isometric operator from H, into H, . But since W,(z,) Wo(-z,) = I , it follows that Wo(zo) maps H, onto H , , and can therefore be uniquely extended to a unitary operator W(zo)from H to H. Moreover, it is easy to calculate from (6.2.24) that the operators Wo(z),and hence also W(z),satisfy the von Neumann commutation relations. Next, we must show that Wo(z)is weakly continuous when z is

6.3. Gaussian Measures and Convential Free-Field Systems

369

restricted to any finite-dimensional linear subspace of A. Since Ho is dense in H , it suffices to show that, iff, g E Ho , then the function ( Wo(z)f, g) =

$(u

- u’) exp [-

f 3(u, u‘ - z)]f ( u - z)g(u’>

(6.2.25)

is quasi-continuous. Let

Then, (6.2.25) becomes

Since +h is quasi-continuous, it is clear that (6.2.26) is also quasicontinuous. Finally, let 7 E H, be defined by ~ ( 0 = ) 1, ~ ( x = ) 0 for x # 0. It is easily verified that is a cyclic element for the system {W(z)1 z E A>, and that (W(x)q,3) = +h(x). ]

56.3. The Relation of Gaussian Measures and Rotationally Invariant Measures to Conventional Free-Field Systems

l o The Fock-Cook Free-Field System We now proceed to formulate the Fock-Cook free-field system commonly used in quantum field theory. Let A be an infinite dimensional (complex) Hilbert space. Let R‘O) denote the one-dimensional Hilbert space formed by the totality of complex numbers with the inner product (A, p) = Aji, and let denote the tensor product 0R of n copies of R (see Appendix I1 for the terminology of tensor product spaces and operators in such spaces); in particular, A(1)= R. Let M , be the space of symmetric tensors of order n over R (n 3 l), Mo = R(O), and let

n:-,

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This is a Hilbert space. We consider the following closed linear subspace of 2:

Let X denote the linear subspace of 2 spanned by all the R(,), n = 0, 1,..., and let M = X n H; M is called the symmetric tensor algebra over R. For each x E R, form the operator C,(x) on M, as follows: for z E M, z, , z k E 5Vk),define z = zo z1

+ + +

where x 0 z k denotes the tensor product of the vector x and the tensor z k of rank K, so that x @ x k E W+l).Clearly, C,(x) is a linear operator from M to M, and acts as a bounded linear operator from M , to M,+, . We also form an operator C,(x) on M, defined as follows: for z E M ,

x

= 20

+ + ". f z,, z1

zk

E fi(k),

(6.3.2)

where zk-l

E

Mk-l is such that, for all y (4-1

I

Y) = ( Z k

I t is easily seen that such a z;-l Z = s k ( x 1 @ * * * @ X k ) , then

9

E

Sk(X

Mk-l ,

0Y ) ) .

(6.3.3)

exists and is unique. For example, if

C,(x) is also a linear operator from M to M, and acts as a bounded linear operator from M , to M,.-l (n 3 I), with C,R(O) = 0. T h e physical significance of these spaces and operators is as follows. M , (n 1) represents the vector space of all n-particle states, and Mo represents the vacuum vector space; C,(x)(C,(x))represents the operator corresponding to the creation (annihilation) of a particle with wavefunction x .

>

Lemma 6.3.1.

C,(x) and C,(X) are related by

6.3. Gaussian Measures and Conventional Free-Field Systems

37 1

PROOF. I t suffices to prove that (6.3.4) holds for X E M , , Y E M,. . Since C,(x) M , C M,,, , C2(x) M,, C M,,-, , it suffices, in fact, to prove (6.3.4) for x E M k , y E Mk+, . But in this case, C,(x)z = (k 1)1/2Sk+lx0z , hence, by (6.3.3),

+

(cl(x)z,y) = ( ( k

+ l)l’’

c

sk+lx

0z,y> = (z, cZ(x)y)- 3

(6.3.5)

By virtue of (6.3.5) and the mapping properties of C,(x) and C,(x), it is easily proved that the closure C(x) of C,(x) exists, and that C(x)* [the adjoint of C(x)] is the closure of C2(x). We call C(x), C(x)* the creation and annihilation operators, respectively, corresponding to x. Set

Choose any real inner product subspace $3 of 53 such that si is the complexification of Sj. We shall prove below that { P ( x ) ,Q(x’) I x, xf E Sj} is an infinitesimal canonical system on ($3,$3, B ) ;we call it theFock-Cook system. We proceed to construct a canonical system of the form (6.2.13)(6.2.14) which is unitarily equivalent with the Fock-Cook system. Let S = (Q, 23,N ) be a standard Gaussian measure space, with parameter 1, associated with 6,and let { ~ ( ux) E , $3) be the corresponding standard Gaussian process. We now define a linear mapping D of H into L2(S), as follows: arbitrarily choose a complete orthonormal system {g,} in $3; then, by Lemma 11.2.4 of Appendix 11, the vectors of H may be expressed in the form x=

1

k=(k,...-.k,)

akSlklcg:log,k” 0

I uk 1’ We define16 Dx(w) =

= 11

1...

k=( k l ,

,k,)

--*

Og?)(l k I !/k!)l’z,

1 ’ < 03*

u k h k ( g l ( w ) ~ * * *g!n ( w ) ) .

By Lemma 5.4.6, we have

11 DX 1 ’

=

1I

uk

1’

11

11’9

and, since {hk} is complete, D ( H ) = L2(S),thus, D is a unitary operator l6

Here, as in $6.1. we denote hL(u1,..., u, ; 1) by hk(u1,..., u-).

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FIELDS

from H to L2(S).It can be shown that this unitary operator D is independent of the choice of the complete orthonormal system {g,}. We denote by ‘p the totality of polynomial functionals of the form P ( g l ( w ),..., g,(w)) [where P(tl ,..., tn) is a polynomial in t , ,..., t,]; then ’$ is a dense subspace of L2(S).Also, denote by H(m) the closed linear subspace spanned by all functionals of the form hk(gl(cu),...,g,(w)), I k I = m; we call H(m) the m-particle space. In fact,

H(m) = DM,

.

We now set up a representation of the commutation relations analogous to (6.2.13) and (6.2.14). Namely, iffEL2(S), let ( U ( x ) f ) ( w )= eaZcm)f ( w ) , (V(x’)f)(w)= exp [x’(w) - &(x’, x‘)] f ( w - x’).

(6.3.6) (6.3.7)

Note that this V(x‘)can also be written as (6.3.8)

( V(X’)f) ( w >

From (6.3.6) and (6.3.81, it is easily seen that (U(x), V(x’) 1 x, x’ E $1 satisfies the-Weyl ’commutation relations (6.2.1), and that U , V are weakly continuous on any finite-dimensional linear subspace of $. We note that, in analogy with (6.1.32), the operator a dt

may be expressed as follows:

W P ( 4 = {p’ I P EL2(% x(*) d.) E w P ( X ) 9J(w> = x ( w ) PP(W)I

9JE

m I

Ww).

(6.3.9)

However, the expression corresponding to (6.1.33) is more complicated. Using again the complete orthonormal system {g,}, and the notation of (6.2.2), we let p , = p(g,), qm = q(gm).Then, by (6.3.9), we have

l W”)

= {p’ I 9J E w 3 9 g u ( * ) d.1EL2(S)l, PuPbJ) = g d w ) ?+)I PE W P J ,

and, in analogy with (6.1.33), qy is the closure of the operator qy’, where B(qy’)= ‘p and %’p’(W)

=

-k&)

dw)

a +idw), agu

P E cp.

6.3. Gaussian Measures and Conventional Free-Field Systems

373

Set C”‘

1

= -( p ,

42

+ iq”),

1

c,*‘ = d z ( P , - iq,);

and let c, , c,* be the closures of c,‘, c,*‘, respectively. Then c,* is the adjoint of c, ,and, corresponding to Lemma 6.1.1 1, we have the following result.

Lemma 6.3.2. T h e operator c, maps H(m) into H ( m bounded on H(m),and satisfies

where k and k’ are related as in (6.1.36). Moreover, H ( m - 1),17 is bounded on H ( m ) , and satisfies

c,*

+ l),

is

maps H(m)into

where k and K‘ are related by (6.1.37). Using (6.3.1), (6.3.2), Lemma 6.3.2, the boundedness of c,on H(m) and the boundedness of C(g,) on M , , one can calculate that c, = DC(g,) D-1.

Similarly, c,*

I n general, writing c(x)

=

=

DC(gy)*D-1.

(1/42)(p(x)

c(x) =

+ iq(x)), we have

DC(x)D-1,

c(x)* = DC(x)* D-I.

Consequently,

p ( ~= ) D P ( x )D-l,

q(x) = DQ(x)D-’.

Thus, since { p , q} is an infinitesimal canonical system, it follows that (P,Q} is an infinitesimal canonical system, moreover, the Weyl form

of the Fock-Cook system is unitarily equivalent to (6.3.6)-(6.3.7), via the unitary transformation D.Since the standard Gaussian measure space is ergodic, it follows as in the proof of Theorem 6.2.5 that the Fock-Cook l’

Here, H ( - 1) is to be understood as meaning (0).

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system is irreducible. Corresponding to Theorem 6.1.13, one can also prove the following result.

Theorem 6.3.3. Let P , denote the projection of H onto M,; then, for any real number a,

C W

r(&I) =

m

(6.3.10)

m=O

is a unitary operator on H , moreover, if { W(x)I x E R} is the von Neumann form of the Fock-Cook system, then r(eiaI)W(x)r(eial)-l = W(e%). In other words, (6.3.10) defines a gradient transformation on the FockCook system. It is easily seen that the Fock-Cook system is cyclic, with the vacuum state 1 as cyclic element. We now write out the characteristic functional of this system corresponding to the cyclic element 1. To do this, it suffices to consider the unitarily equivalent system defined by (6.3.6)(6.3.7). Then

and we may suppose that x # 0. Now, there exist mutually independent Gaussian variables U(w), V ( w ) , each having mathematical expectation zero and variance 1/2, and such that XbJ)

= I1 x I/ U ( w ) ,

Substituting (6.3.12) into (6.3.1I), we obtain (W(Z)L1)

1 2

- - ( y ,y ) =

exp[-t(ll x 112

‘i( x ,Y )

+ VP,]

- ( u2

+ IIY 11”l.

6.3. Gaussian Measures and Conventional Free-Field Systems Therefore, the characteristic functional of the Fock-Cook corresponding to the vacuum state 1, is

375 system,

We next proceed to formulate a class of canonical systems, similar to the Fock-Cook system, and also describable in terms of Gaussian measures. Let R be a complex Hilbert space, let $3 be the real Hilbert space formed by A with respect to the inner product [z, z’] = A(z, z’), and let S, = (Q, 23, N,) be a standard Gaussian measure space associated with $3. Using (6.2.15), for each z E R, we form the unitary operator

(6.3.14)

in the Hilbert space L2(S,).Following the proof of Theorem 6.2.6, we see that { W,.(z) [ z E R) satisfies the von Neumann commutation relations; moreover, one may verify that the correspondence z -+ W,(z) is weakly continuous on any finite-dimensional subspace of R. Therefore, {W,(z) I x E R} constitutes a canonical system. Furthermore, it is easily shown that it is irreduciblelO and cyclic, with 1 as cyclic element, and that the corresponding characteristic functional is (WC(z)1, 1)

=

1

R

exp

[-

i

1 (iz)(w)- z(w) C 2c

dNc(w). (6.3.15)

Since [z, z] = [iz,iz] = [ j z

the two Gaussian variables (6.3.15) becomes

112,

z(w),

[z, iz] = [iz,z] = %(z, iz) = 0,

(iz)(w)are mutually independent, and

(6.3.16)

I n particular, when c = 2, formula (6.3.16) reduces to (6.3.13). Thus, lo Translutor’s note: Details would be welcome, since the proof of Theorem 6.2.5 is not applicable in this case.

376

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COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

by virtue of Lemma 6.2.9, the Fock-Cook system is a special case of the type under consideration. We also note that, if one sets

and writes W,(x) in place of W,(x) in (6.3.16), then (6.3.16) can be written in the form (W&)l, 1)

20

A

= exp(-

II z 112 A),

a
00.

(6.3.17)

More General Type of System

Let R be a complex Hilbert space, let U be the Definition 6.3.1. group of all unitary operators on R. Let {W(z)I x E R} be a canonical system over R in the Hilbert space H , and let I' : U -P I'(U) be a unitary representation of U in H , satisfying the condition

r(U ) W(z)r(U)-1 = W(VZ).

(6.3.18)

Suppose, moreover, that ( W ( z )I z E R} is cyclic and that there is a cyclic element 7 such that 1'(U)q = 7 for all U E U. Then we call { W(x)I z E R} a conventional free-jield system, and 7 a vacuum state vector. We observe that the system {W,(x) I x E R} defined by (6.3.14) is a conventional free-field system. I n fact, letting B denote the linear space spanned by the totality of vectors of the form Wc(z)l,Z E ~ it, can be shown by methods similar to those used in 55.4 that 2)is dense inL2(Sc). For each U E U, we define a linear operator P( U) on B as follows:

T(U)WC(X)l = Wc(Uz)l.

(6.3.19)

By (6.2.20)and (6.3.16) we have, for any x, x' E A,

(r(u)Wo(413 T(U)WC(41)

= (Wc(z')L

u.',(Z)l).

Hence, P ( U ) is an isometric mapping. Since I'(U)-l = I'(U-l), r ( U ) can be uniquely extended to a unitary operator on L2(Sc);we also denote U). From (6.3.19), it is easily seen that U r(U) this extension by is a unitary representation of the group U,and that (6.3.18)holds. I n this case, 1 is a vacuum state vector.

r(

--f

6.3. Gaussian Measures and Conventional Free-Field Systems

377

We shall now investigate the general form of the characteristic functional of a conventional free-field system corresponding to an arbitrary vacuum state vector. For this purpose, we first prove the following lemma, which provides a characterization of rotationally invariant measures.

Lemma 6.3.4. Let & be an infinite dimensional real inner product space, and let +(.$),.$ E 5,be a positive definite quasi-continuous function on & such that the value of $ ( f ) depends only upon 11 g 11 = (6, E)l/2 [here (., .) denotes the inner product in 931. Then there exists a finite measure rn on [0, 00) such that

PROOF. Since +(() depends only upon 11 5 11, it may be written in the form ~ ( 1 1 f 11). Arbitrarily choose a unit vector e in &; then q(t) = +(te), = (~(11 6 11) is a continuous t 2 0, is a continuous function oft, hence +(t) functional on &. If we choose any orthonormal sequence of vectors {en, n = 1, 2, ...} in &, then $(fiel fne,) is a positive definite continuous function of the real variables t1,..., . Let I be the totality of real number sequences, and let 23 be the a-algebra generated by the family of all Bore1 cylinders in I; by the Kolmogorov theorem (see Corollary 1.3.5’), there exists a finite measure P on (I, 23) such that

+ +

+(tiel

+ + a * *

tnen) =

S,exp(i

”4

tpv)dp(x),

x = (xi

,.-, x n ,.--)(6.3.21)

Let s2, be the unit sphere in n-dimensional Euclidean space, and write

Denote the coordinates of the points of Qn by forming to spherical coordinates, we have

The element of area on f i n is dQ,(w) = sinn-2vl

w = (wl

*..

,..., w,);

trans-

sin ~ p ~ - ~ d a 7dvn-l ~

,

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VI.

COMMUTATION RELATIONS I N BOSE-EINSTEIN

FIELDS

and the total area of SZ, is 2 ~ ~ l ~ / l r ( nwhich / 2 ) , we denote by I SZ, set 5, = pwy , v = 1, 2,..., n, then the value of the integral

1. If we (6.3.22)

depends only upon p and T , hence, in the evaluation, we may choose ,...,xn) = ( T , O,...,0). Therefore, (6.3.22) becomes

(xl

--.dQ,(w)

Denote the above quantity by #,(p~). Applying the integral J.,

to both sides of (6.3.21), we obtain

where u,(T) = P({x I CL1x,2 < 7”). Make the substitution 7 = l / nu on the right-hand side of (6.3.23); since u,(l/iu), n = 1, 2,..., is a uniformly bounded sequence of monotonically increasing nonnegative functions on [0, co], it follows by Helly’s theorem that there exists a subsequence (a,,(d/nku)} which converges at all points of [0, a]to a bounded monotonically increasing nonnegative function a(u). On the other hand, sincez0

*O

Here, we use Stirling’s formula

1-;

(-

4)

cos PTV dw

-+

r(s + 1)

( 2 7 r ~ ) ~ ~ * s ~ e s- ’ ,-+

N

exp

cos p r dv ~

W.

6.3. Gaussian Measures and Conventional Free-Field Systems

379

therefore,

Similarly, one can show that21 there exist positive numbers c, cf such that I 4 J d i i p ~ ) I c exp(--Cfp2T2). Hence, taking n = nk in (6.3.23), and letting k + 00, we obtain

<

where m(t) = ~ 4 ( 2 t ) ' / ~ ]) > . If the characteristic function of a measure is of the type described by Lemma 6.3.4, then this measure is said to be rotationally invariant. I t can be seen from (6.3.20) that any rotationally invariant measure is a superposition of Gaussian measures.22

Theorem 6.3.5. Let R be a complex infinite-dimensional Hilbert space, { W ( z )1 z E A} a conventional free-field system, and 7 any vacuum state vector for this system. Then there exists a finite measure m on co) such that the characteristic functional of {W(z)I z E R) corresponding to 7 is

[t,

(6.3.24)

PROOF. Since r(U)7 = 7, the characteristic functional +(z) has the following property: for U E U, z E R,

Now, if 11 z 11 = 11 zf 1 , there must be some U E U such that z' = Uz, hence, by (6.3.25), I,!J(z) depends only upon 11 z 11. We choose any real linear subspace $j of R such that $jis a real inner product space and R is the complexification of 9. If z is restricted to 5, the function #(z) stiI1 depends only upon 11 z 11. Since 9 is also infinite-dimensional, we know that there is a finite measure m on [0, CQ) such that (6.3.20) holds for all 5 E 9. But since +((), on the entire space R, depends only upon 11 5 11, *l Translator's note: Actually, it suffices to show that the convergence of + , , ( z / ; T p ~ ) is uniform with respect to 7 . See Umemura [ 13 for further details.

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COMMUTATION RELATIONS I N BOSE-EINSTEIN FIELDS

it follows that (6.3.20) holds for all f E R. Using the condition in Lemma 6.2.10 and carrying out a rather intricate calculation, it can be proved that23

M O , t))= 0,

whence we obtain (6.3.24). ]