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Boson-fermion relations in several dimensions

K R PARTHASARATHY and K B SINHA

Indian Statistical Institute, New Delhi 110016, India

Abstract. Zero and positive temperature fermion field operators in several dimensions are constructed as stochastic integrals of certain reflection valued processes with respect to the corresponding boson field operator processes.

Keywords. Boson and fermion Fock spaces; boson stochastic calculus; quantum Ito's formula; canonical anticommutation relation; canonical commutation relations.

PACS No. 05.50; 05-30; 03-65

1. Introduction

Zero temperature fermion field operators in one space dimension were constructed by Hudson and Parthasarathy (1986) as quantum-stochastic integrals of a certain reflection valued process with respect to the boson field operator processes using the boson stochastic calculus developed by the same authors Hudson and Parthasarathy (1984). This leads to a canonical unitary isomorphism between the boson and Fermion Fock spaces over L2 (R). Parthasarathy and Sinha (1986) showed that the stochastic integral representation of the fermion field with respect to the boson field over R is unique subject to the requirement of irreducibility, martingale property and existence of a vacuum. Here we extend this construction and some of the results to the case of arbitrary dimension and arbitrary temperature. As a consequence we obtain a new reducible, cyclic, non-Fock (nonzero temperature) fermion representation in terms of a reducible, cyclic, non-Fock boson representation in a boson Fock space. There have been other constructions of fermion operators as functionals of boson operators in the literature (Dell' Antonio et al 1972; Coleman 1975; Garbaczewski 1975; Carey and Hurst 1985).

It was observed by Dell' Antonio et al and Coleman that in some models in 1 + 1 dimension (for example, massless Thirring model) certain formal expressions of boson fields can be formed having the vacuum expectation values and statistics of fermion fields. In Carey et al, this process is made rigorous for the canonical anticommutation relation (CAR) algebra over L 2 (S 1, C). However, the constructions employed by both Carey et al and Garbaczewski are complicated and the fermion operators so obtained could not be expressed in terms of an operator martingale process. The quantum- stochastic calculus used here as well as in Hudson and Parthasarathy (1986) makes the construction of fermion operator martingales in terms of the boson operator

The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.

105

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martingales not only transparent but also keeps the relationship entirely kinematic and hence totally independent of any model.

Section 2 is devoted to the construction of an abstract fermion representation without reference to any dimension. The uniqueness of such a representation upto unitary equivalence is expected to be true but still remains an open question. In § 3 we extend the construction to the positive temperature case in several dimensions. This leads to a direct relation between the canonical commutation relations (cc~) representations of Araki and Woods (1963) and the CAR representations of Araki and Wyss (1964) and Dell' Antonio (1968).

2. Zero temperature boson-fermion relations in several dimensions

Let ~ be any complex separable Hilbert space and let P be a continuous spectral measure on R whose values are orthogonal projection operators in ¢~. In the boson Fock space F(,~) = C ( ~ ) , ~ ) . . . t ~ ) A ~ t ~ ) . . , over ~ were t~) n denotes n-fold symmetric tensor product, we consider the annihilation and creation operators a (u), a t (u), u e ~ and define for every t e R

Ap.,(t) = a ( P ( - oo, t]u), A*p,~(t) = a ( P ( - oo, t]u). (1) Writing ¢¢(u) for the coherent vector 1 ~ u ~ ) . . . O(n!)- 1/2 u ®" ~ . . . define the second-quantized reflection operators Jr (t) by the relations

Jp(t) gl(u) = g~(Rtu),

R,u = - P ( - oo, t]u + P(t, oo)u. (2)

Then, in the language of Hudson and Parthasarathy (1984) Ae.u, Ate, u and Jr can be interpreted as adapted processes. Furthermore

jr(s) Jr(t) = Jr(t) Jr(s) for all s, t,

(3) s~(t) ~ = 1, s ~ ( t ) = s~(t).

The stochastic calculus and the quantum Ito's formula, developed by Hudson and Parthasarathy (1984) for the case where the spectral measure P is absolutely continuous with respect to the Lebesgue measure, can easily be extended to the present more general case. Therefore following the central idea in Hudson and Parthasarathy (1986), we define:

Fp, u(t) = Je(s)dAe, u(s), F~.u(t) = Je(s)dA~,u(s); (4)

Fp(u) = Fp, u(~), F*p(u) = ,%.u(~9. + (5)

Our aim is to show that {Fe(u), F~(u), u ~ } is a representation of CAR and establish some of its basic properties. We start with the observation that (4) and (5) define the operators in the domain ~' which is the linear manifold generated by all coherent vectors.

Proposition 1 For every t ~ R, u e,~

Je(t)Fe, u(t)+ Fp,,(t) Jr(t) = 0 on the domain 8. (6)

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Proof

For any v, w e ~, (2), (3) and (4) imply

(~b(v),

Jp(t)Fe, u(t)g/(w) ) = ~(Rtv), Jp(S) dAe.u(S)~(w)

oO

= [-t {g/(R,v),~(Rsw)) (P(ds)u,w) d-

o O

and on the other hand

( ~b(v), Fe, u(t) Je(t)~b(w) ) = (~(v), fi ® Je(s)dAe, u(S) ~b ( R,w) )

j"

= (O(v), ¢(Rs R,w)) <P(ds)u, R,w)

- - 0 0

f:

= <O(Rtv),

¢(Rsw)) (P(ds)u, w).

or.)

Adding (7) and (8) we conclude (6).

Proposition

2 The operators

Fe, u (t)

are bounded for all t ~< oo, u e A.

Proof

We have the stochastic differential equations

dFs, u = JpdAe, u, dF~, u = Jr dAfe, u .

By the quantum Ito's formula

< ~ , . (t) ~, (w,), F~,o (t) ~, (w~))

f

= (F;,u(S)g/(Wl), Je(s)

0(w2))

(P(ds)v, wx)

- - 0 0

+ <Je(s)~l(Wx),F*e,v(s)~l(w2)) (P(ds)u, w2)

oO

+ <~(w~), g,(w2) > <u, P ( - o0, (Iv).

Similarly,

IFp, v (t) g/(w 1), Fe, u ))

\

(t)

g,

(w2

/

f_

= (Fe, v(S)O(wl ), Je(S)g/(w2))~'(P(ds)u, w2 )

oO

f

+ (Je(s)~b(wl),Fe, u(S)~b(Wx)) (P(ds)v, wl ).

- - 0 0

Adding (9) and (10) we obtain

< F~,.(t)~ (w, ), F~,v(t) g, (w2) ) + ( Fe.v(t) g/ (w, ), Fp.u(t) ~b (w2) )

= (~(wl),

~(w2)) (u, P ( - 00, t]v)

for all t.

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(8)

(9)

(10)

(11)

(4)

Then for any ¢ ~ 8 , we obtain by putting u = v in (11)

IlF~,,(t)¢ll 2 + IlFp,.(t)¢ll 2 = I1¢112 ( u, e ( - o0, t]u ). (12) This completes the proof.

Corollary The operators Fe, u(t) and F~,.(t) can be extended uniquely to the whole space F(}[). If these extensions are denoted by the same symbols then Fte,.(t) is the adjoint of Fv, u (t).

Proof This is immediate from the definition of stochastic integrals (4) and the density o f g .

Hereafter we define the operators Fv u (t) and Ftv ~ (t) to be the extensions of (4) to the whole boson Fock space F(~) and pu't Fv, u ( - oo') =ftv, u(_ oo) = O.

Proposition 3 The operators {Fe,,,(t), F~,,,(t), u ~ } obey the cAa [Fp,,, (t), Fe, v (t)] + - Fe, u (t)Fp, v (t) + Fp,~ (t) Fp,. (t) = 0,

[Fe,.(t), Ffe, v(t)]+ = (u, P ( - oo, t ] v ) for all u, ve~, - oo ~< t ~< oo.

Proof The second relation is immediate from (11). In order to prove the first relation we deduce from Proposition 1 and the quantum Ito's formula the identity

d(Fv, ~ Fp, v + Fp, v Fp, u) = .Iv Fp, v dAy,. + Fp,. ,Iv dAy, v + jeFv.udAv, v + Fv, vJpdAv, u = O.

Since Fp,,, and Fp,~ vanish at - oo the proof is complete.

Proposition 4 Let U be a unitary operator on ,~ and let F (U) be its second quantization defined by

Then

r ( u ) g , ( u ) = ¢,(Uu) for all u6,~.

F(U)Fp, u(t)F(U) -1 = Fvru-,uu(t) for all - o o ~< t ~< ~ , ue,~,

where UPU-1 is the spectral measure defined by ( U P U - t ) (E) = U P ( E ) U - 1 for any

Borel set E c R.

Proof We havo from definitions

( g/ (v), F (U) f v, u(t)F (U-1)~k (w) ) = ( ~ (U- l v), Fe, u(t)~ (U -1 w) )

f,

= ( O ( U - l v ) , J v ( s ) O ( U - l w ) ) ( P ( d s ) u , U - l w )

- - 0 0

f_

= (O(v), Jvev-,(s)O(w)) ( U P ( d s ) U -1 gu, w )

= (O(v), Fupv-,,w(t)g'(w) ).

From now on we fix the spectral measure P in ,~ and drop the suffix P from Ap, u, Jr, Fp, u etc.

Proposition 5 Let f~ = g,(0) be the vacuum vector in F(~).

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Then F.(t)f~ = O,

F~, (t) . . . F~. (t)fl = Z e(a)

~ ' ¢ S .

A*~,(t)'"

A~.(t)n =

o ~ S .

f

dA~,,,(sl)'"dA~.,.,(s,)~, (13)

- a o < ~ 1 < . . . < s n < / .

f dF~,.,(sO'"dF~,.,(s.lf~ (14)

--cX3 < $ 1 < . . . <$n< ~

for every positive integer n, ua, u2 . . . u,e~, - o o <~ t <~ ~ , where S. is the permutation group acting on {1,2 . . . n} and e(a) denotes the parity of the permutation tr.

Proof This is proved exactly along the same lines as in Hudson and Parthasarathy (1985) by using induction, quantum Ito's formula and the relation (6) on the whole Fock space.

Proposition 6 Let ,~t denote the range of the projection P ( - 0% t] and let H, denote the closed linear span of the set {ff (u), u e ~, }. Then each of the sets

{ n } w { A ~ t ( t ) . . . A u t ( t ) f ~ , ux, u z , . . . , u . ~ , n = l , 2 , . . . } , { f ~ } w { F f u , ( t ) . . . F t , ( t ) f 2 , ux, u 2 . . . u , e ~ , n = 1 , 2 . . . } is total in H,.

Proof The first set contains the vacuum vector and all the n-particle vectors arising from ~,. Hence it spans the Fock space F(~t) = H,. It follows from (14) that the second set is total in H,.

Theorem 1 The operators { Fu (t), F*~(t), u ~ ~t } restricted to the subspace H, constitute an irreducible CAR representation of the Hilbert space ,~t for each - oo < t ~< oo.

Proof Let Fa(~) = C ~ @ . . . O ~ ®" ~ ) . . . be the fermion Fock space where (~)"

denotes n-fold skew symmetric tensor product. The canonical irreducible representa- tion of CAR over ~ in Fa(~) is completely characterized upto unitary equivalence by the existence of a vacuum which is cyclic for the algebra generated by the creation operators.

Thus the required result follows from Proposition 3, 5 and 6.

Theorem 2 Let f~ and f~_ be the vacuum vectors respectively in r ( ~ ) and r~(~).

Suppose P is a continuous spectral measure on R whose values are orthogonal projections on ~. Then there exists a unitary isomorphism E p: Fo (~) ---, F (,*) satisfying

=-eft- = f~

Ep X P(-oo, t-luj=(n!) -1/2

~, ~(~)

f

j = 1 e ~ S .

- ~ < S 1 . . . < S a < t

dA*~.,,(s,).., dAtu.,.,(s.)f~

for n = 1, 2 . . . ul, u2 . . . u.e,~' and - ~ ~< t ~< oo, where /~ denotes the skew

j = l

symmetric tensor product in the order 1, 2 . . . n.

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Proof By theorem 1 we have for every n

(F~, ( t ) . . . F~.(t)f~, F~, (t) . . . F~, (t)[2 )

= d e t ( ( ( u i , P ( - ~ , t ] v , ) ) ) - - n ' ( ; S =t; P ( - ~ , t ] v j ) . The required result follows immediately from Proposition 6.

We now specialise to the case when ,~ = L2(R~). Express any point in W as (s, xx, x2 . . . x~_ ~) and consider the absolutely continuous spectral measure P on R in A defined by

[ P ( E ) f ] (s, xx . . . x~- a) = XE(s)f( s, x t , . . . , x,- 1), where Xn denotes the indicator of the Borel set E c R.

In view o f Theorems 1 and 2 we can realise the fermion field operators in v variables in the boson Fock space over L2 (W) through (4) and (5). In such a construction we have taken the first coordinate as a distinguished one but in view of Proposition 4 change of coordinates through permutations or rotations yields only an equivalent fermion field.

3. Positive temperature boson-fermion relations in several dimensions

As at the end of the last section we consider A = L2(R v) and the spectral measure P of multiplication by the indicator in the first variable. In order to construct the positive temperature boson and fermion fields we introduce the Fock spaces

/ I = r ( ~ ) ® r ( A ) = r ( , ~ , ~ ) , ~ , = r ( ~ , ) Q r ( , ~ , ) =

r(~,~A,)

(15) where ,~, is the range of the projection P ( - ~ , t]. At is to be looked upon as a subspace of R. For any ~b e A let

A~l)(t) -- A , ( t ) ® l , A~2)(t)= l ® A ~ ( t ) (16) where A~ (t) -- At, ~ (t) is defined by (1) and 1 denotes the identity operator in F (~). Let ~, fl be two bounded complex valued measurable functions on R v satisfying the conditions

Iml 2 - IPl 2 -- 1, ImPl > 0 everywhere. (17) Define the operators

/]~b(t) = A ~ (t) + A ~ t t ) , (18)

~[~(t) = A(l}t t,~ ± ~{2) ~ ~ , ~ - . ~ (t), (19)

~(~) = ~ , ( ~ ) , ~*(~) = ~ ( ~ ) . (20)

Then the following commutation relations hold:

[~,(t), ~,(t)] = 0, C~**, ~ ] = 0,

[ ] , ( t ) , , ~ (t)] : <~, P( - ~ , t]~0 ) (21)

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for all - ~ ~< t ~< ~ and q~, ff e 4. In particular, { g (~b),/] t (g,), ~b, ~, e 4 } is a representa- tion of cce in/-~. Let

fl/¢ = W~¢® W_/~, 4)e4, (22)

where ~b ~ IV, is the Weyl representation of 4 in F(,~) defined by the relations Wot~(f) = exp(-½il~bll z - (dp,f>)qt(f+d?) for all f e 4 . (23) Then we have the Weyl commutation relations

W¢ W~ = We ÷ v, exp - i Im < 4~, ~O >; (24)

f l / , f l / , = ~ + ~ , e x p - i l m < ~ b , qJ>, ~b,~,e~ (25) in F ( ~ ) a n d F ( 4 ) ® F ( 4 ) respectively. Whereas

<tl, W~tl> = exp-~lt~l[ ~, (26)

we have

<fi, fl/~,fi > = exp [-½(ll~4)[I 2 + 11/~ll2)], • = ~ ( ~ 1 . (27) Since ]ctl 2 + ]/~ ] 2 = 1 + 21/~] 2 > 1, it is clear that IT' is a quasi-free non-Fock representa- tion of positive temperature.

Following the notations at the end of the last section we write for any ~b, ¢, e 4

<4,,,/'>o(s)= f x= (x,

. . .

Xv-1),

Rv-|

114'1102 (s) = 4'>o(S).

Let Y/~ (t) = fl/p¢_ ~. t]~. Then { ff'~ (t), t e R } is an adapted unitary process satisfying t h e , quantum-stochastic differential equation

d ff'~(t) = {dTi~(t)-dTt~(t)-½[ll~qall~(t)+ 11/~4,@(t)]dt} fl/~(t).

Furthermore 4~ --* fl/¢(t), 4~ e4, satisfies the Weyl commutation relations in/~,.

Proposition 7 The set {l,~,(t)fi, ~be4,} is total in the subspace /~,. The map 4~ ~ ff'~(t), 4~ e4, is a reducible projective unitary representation in/4, of the additive group 4, for each - ~ < t ~< oc.

Proof Let

Then ~, is a real linear manifold in 4, (~¢I, and by (17) #, + / / t is a dense linear manifold in 4,(~)4,. Hence the set {qc(u), ueA,} is total in/-],. Since

fl/~(t)O = c(t)qJ( P ( - ~ , t ] ~ q ~ @ P ( - oo, t ] ( - / ~ ) )

= c(t)g'(84)@(-.fl~)) for 4~e4,,

where c(t) is a nonvanishing scalar, this proves the first part. To prove the second part we

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have only to note that the unitary operators I ~ and Wa~ ® W_~ commute for all In order to construct the positive temperature fermion field operators in terms of the positive temperature boson field operators ,,] (~b) and ,it (4}) we introduce the stochastic integrals

where

Pc(t) = ,7(s) dSp~.(s). / ~ ( t ) = ~(s) d.~t~¢ (s), (28)

p = ([0c[ 2 +

I/~12) -'/2, 3(s)

= J(s)(~J(s), (29) (J(s) = Je(s) being defined by (2) in F(A)). It is to be noted that (28) defines the processes Y¢ and P~ on the domain 2 = d~ ® g which is the linear manifold generated by exponential vectors in ~ . Furthermore they are adjoint to each other on o ~.

Proposition 8 For any q~ ~ ~, - oo ~< t ~<

•(t) P~tt) + Y~(t),7(t) = 0 on g. (30)

Proof Since/~ = F ( ~ G , ~ ) we have for any f = J l ~ f 2 , g = ffiGg2 ( O ( f ) , J(t)P4,(t)~/(g)) = < 0 ( R , f ) , ~k(R~g)) (p~b, ugl -flf2 )ots)ds

o 0

and

(~b(f), F,(t),7(t)~,(g)) = f ' (~k(f), ~k(R~R,9)) (pc#, - a g , +flf2 )o(s)ds.

.j- at)

Adding these two relations we obtain (30).

Proposition 9 The operators/~¢ (t) are bounded for every q~ e ~, - oo 4 t 4 oo.

Proof We proceed along the same lines as in the proof of Proposition 2 and use the quantum Ito's formula

d a , ) a .~o) = dA~) t daO)* n ,~o)* a .,~) dA~} dA~ ~' = fi,i( qb, P(dt)~b ), i,j = 1,2.

We get

( Y~(t)O(f), P;(t)¢(g) ) + ( Y~,(t)¢(f), P¢(t)¢(g) )

= ( ¢ ( f ) , • (,q)) (q~, P ( - oo, t ] ¢ ) for all f, g a . ~ G ~ , - ~ ~< t ~< ~ . Hence

llP~(t)~Iq 2 + [[ff¢(t)~lJ 2 = Iq~[I 2 (~b, P ( - ~ , t]~b), ~ o ~.

This completes the proof.

Proposition 9 enables us to extend the operators Pc (t), F~ (t) uniquely to the whole space/4. Hereafter we denote these extensions by the same symbols.

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Theorem 3 The operators {jr,(t),

Jr,(t),

qSe~,} satisfy the following for every - o o ~<t~< oo:

(i) /~(t) is the adjoint of Jr,(t),

(ii) [jr, (t), jr, (t)] + = 0,

(iii) [jr,(t), jr~(t)]+ = (~b, P ( - oe, t]tp ).

Proof This is proved exactly like Proposition 3.

Proposition 10 For any q~l, • • •, q~,~, ~/'1 .. . . . ¢,.ed, - oo ~< t ~< oo

( f i , F + +,(t).. ~'t,.(t)P,,(t).. . P , . ( t ) ~ ) = O i f m ¢ n

= det ( ( ( ~ j , P ( - ~ , t] l[312p2~pi ))) otherwise. (31) Proof Let m =/= n. Suppose H, denotes the n-particle subspace ,~®" in F(~). Then

~ , , ( t ) . . . jr,.(t)~e H o @ H , . Since different particle subspaces are mutually ortho- gonal the proposition is proved in this case.

Let m = n. Since J'(s)O = ~ we have from (28) and (18)

~,(t)fi = f2® A ~pp~ (t)n =

n® F~o~(t)fl,

where Ft,(t) is defined by (4). By quantum Ito's formula d/~,(t)dA,(t) = 0 and hence

. . . . . . . . . ( 1 ) ( 2 k t

d/~, P,~ /~,fi = ~ (-- 1)ll--J~l~ I ]1~(~0_ 1

Jr,,+,

P,o(dA~p,j +dAp0¢)f~®fl J

= ~ (-1)~-Jjr,, . . . . Jr%_,P,,, . . . Jr,.dA~*$~ ~.

j = l

Hence by induction and Proposition 5

Jr,,It)... Jr,°It)O = n ® y~ ~(~)

O ' E S n

f dAmps., Is1)... dA~pT~.,°, (s.)~

--'39_.<51 < . . . <Sn<t

= f~®F,p~i ( t ) . . . F,pL(t)f~.

Thus the left hand side of (31) is equal to

< F,p~, (t)... F,o:~ ,(t)n, F,,~, (t)... F,,~.(t)n

= det ( ( ( P ( - ~ , t]~p~i, P ( - oo, t]l~p~j >))

= det ((<tpj, e ( - ~ , t] lfll2p2 4~i ))).

Proposition 11 The set °)t = {•} c) {P~l ( t ) . . . Jr,* (t)P,l ( t ) . . . P , (t)~, ~bl . . . . ,4~,,,

~'t

. . . 0ne~, m, n = 0, 1, 2 . . . . } is total in/~t for every t.

Proof Let H, (t) be the n-particle subspace of F (~t). We write Hm. ~ (t) = Hm (t) ® H~ (t) and denote by S, the closed linear span of ~t. By Proposition 6 and (32) it follows that Ho,.(t) c St for all n. We now proceed by induction. Suppose H~(t)®H~(t) ~ St for j ~ m and n = 0, 1, 2 . . . . It is clear that Jr** (t) maps Hm..(t) into Hm+ 1, n(t)@H,., ._ l (t).

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Let u be a vector in Hm+ t, .(t) which is orthogonal to ~ ( t ) { H m , n(t)} for all ~b e~.. Then by induction hypothesis

u, J ( s ) d A ~ g ( s ) = 0 for all v~Hm.o(t), d?~/t.

oO

(33) The element u can be expressed as a function u (~ t . . . ~,. ÷ t, ~/t . . . . , r/,) in m + n + 1 variables from R ~, which is symmetric in the first m + 1 variables and the last n variables separately. Similarly v can be expressed as v(¢t . . . . , ~,., rh . . . . ,r/,) which is symmetric in the first m and the last n variables separately. We denote the first coordinate of the points ~J, rh 6 R~ by s j, tk respectively and put

~o(¢x . . . . , ¢ . + , , ~, . . . . , , o ) = [ ~, z( .... , . , , l I s j ) + ~

x~_~,,..,lIt,),

j = l k=l

X denoting indicator. Then by the definitions of J (s) and A ~1: the left hand side of (33) is equal to

( r e + l ) t/2 / ~(~1, " ' " , ~m+l,?~l . ' . ~n)U(~l . . . ~m, /'/X .. . . ,~n)

t t

s z < ~ t , . . . , s = + l < ~ t t t ~ t . . . . , t ~ t

x otpc~ (~m + ,) ( - 1)°'(¢ .. . . ~ .... n,... n.) d ~ . . . d~m ~ 1 drlt • • • dtl, = O.

Since a) is symmetric in (~t . . . ~,~) and (t/t . . . . , t/,), v is arbitrary in/-/,,, ,(t), ~ is arbitrary in ~, and [ctp] > 0 everywhere it follows that u = O. In other words n . ÷ ~, ~(t) ~ S , . .

Proposition 12 The CAR representation {P,(t),/?~(t), $ ~ t } restricted to R, is reducible for every t.

Proof Consider the unitary operator S defined by S ~ ( f @ g ) = ~ ( - g G - f ) , f, g e 4 , and put

8 , ( 0 = S~,(t)S- x.

Then { ~ (t), ( ~ (t), t# e ~ } is another CAR representation for ~,. F r o m the two relations drY, = ) ' ( d A ~ + ,~(2~_ ~

(2) AA(I~-'~

and q u a n t u m Ito's formula we have

d [ P , ( t ) , (~,(t)] ÷ = - 2 ( p ~ c k , P f l ~ ) o dt for all •, t # ~ . Hence

[P0 (t), (7,, (t)] + -- - 2 (P~$, P f l ~ ) o (s)ds

= - 2 I' ( dp, pmotfl~ )o (s) ds.

j - ob

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Since

[P,~(t), r~(t)] = ( 4 ~, ~ )o (s) ds,

af~

it follows that

[L~(t), d,(t)

+ 2 F p 2 ~ ( t ) ] + = 0 for all ~b, g, eA,.

-,

Once again by quantum ltffs formula [gg,(t ), d~(t)]+ = 0.

Combining the last two relations we conclude that

[P,(tt, { dr, (t) + 2Fp~/~ (t) / { d, It/+ 2Pp2~ (t//] = 0

for all ~b, ~ ~h?. Since the second operator in the above commutator is self-adjoint the required result follows.

We now summarize our conclusions in the following theorem.

Theorem 4 Let the operators {P~,(t), P~(t), ~b e~, - oo <~ t ~< o~} be defined in terms of the positive temperature boson field operators {,4,~ (t), , ~ (t), ~b ~,~, - oo ~< t ~< oc9 } and the reflection operators { J(t), - oo ~< t ~< oo} by the stochastic integrals (28), where

= L2(RV). Let P denote the spectral measure of multiplication by the indicator in the first coordinate in R v. Then for each fixed - o o < t ~ < oo, {P,(t), P~(t), ~be P( - oo, t],~ } restricted to the boson Fock space/~f = F (P( - oo, t] ~ G P( - oct, t]~,)is a reducible CAR representation for P ( - ~ , t]~ which has the vacuum ~ in/4t as a cyclic vector. Furthermore

( ~ , ~ l ( t ) . . . r ~ ( t ) F , , ( t ) . . , le~ (t)fi) -- 0 if m ¢ n,

= det (((Oj, P( - oo, t] lfll2p2q~))) if m = n. (33) Proof This is just a restatement of theorem 3, Proposition 10-12 put together.

Now we shall compare the CAR representation {/~(4~),/~, (4~), ~b e ~ } obtained from Theorem 4 when t = oo and the positive temperature CAR representation of Araki and Wyss (1964). Using the isomorphism E e of Theorem 2 the Araki-Wyss representation may be defined by

F(q~) = F(p~tq~)@ 1 + J (oo)@ Fr (pfl~), (34)

where ~t, fl satisfy 07), p is given by (29), F(~b), Ft(~b) denote the zero temperature irreducible CAR representation in F(/~) and J(oo) is defined by (2) by putting t = oo.

It has been shown by Araki and Wyss that the operators defined by (34) and their adjoints constitute a reducible representation of CAR with ~ as cyclic vector and expectation values (~,/¢* (~bl)... p t (~b,,)P(~b~)... F(ff~)~ ) are given by the right hand side of (33) when t = oo. In other words the CAR representation {/~(~b), r ~" (q~), 4)e ~ } described in Theorem 4 when t = oo is unitarily equivalent to the Araki-Wyss representation. Since formula (34) involves the reflection J (oo) it is not possible to replace 4) by P ( - oo, t]q~ in F(q~) and localize it to an adapted process. On the contrary formula (28) localizes P(qS) and at the same time realizes it in terms of the positive temperature boson field operators.

(12)

References

Araki H and Woods E 1963 J. Math. Phys. 4 637 Araki H and Wyss W 1964 Heir. Phys. Acta 37 136 Carey A L and Hurst C A 1985 Commun. Math. Phys. 95 435 Coleman S 1975 Phys. Rev. D I I 2086

Dell' Antonio G 1968 Commun. Math. Phys. 9 81

Dell' Antonio G, Frishman Y and Zwanziger D 1972 Phys. Rev. D6 988 Garbaczewski P 1975 Commun. Math. Phys. 43 131

Hudson R L and Parthasarathy K R 1984 Commun. Math. Phys. 93 301 Hudson R L and Parthasarathy K R 1986 Commun. Math. Phys. 104 457 Parthasarathy K R and Sinha K B 1986 J. Funct. Analys. 66

References

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