Quantum uantum ^Probability Probability X i JL
& Related Topics & RplafpH Topics Tonics * v & Related
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Quantum uantum ^Probability Probability X i JL
& Related Topics & RplafpH Topics Tonics * v & Related
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QP-PQQ Volume VIII
1
uantum Q uantum ^* -*-p irobability - K ^
1
1
(
& Related Topics Topics JL JL & & Related Related Topics ** Managing Editor
L. Accardi Editorial Board
V. Belavkin, A. Chebotarev, A. Frigerio, R. Hudson, B. Kummerer, M. Lindsay, H. Maassen, K. R. Parthasarathy, D. Petz, K. B. Sinha, W. von Waldenfels Advisory Board
L. Gross, T. Hida, A. Verbeure
World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
QUANTUM PROBABILITY & RELATED TOPICS VOL. VIII Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orby any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, USA. ISBN 981-02-1140-6
Printed in Singapore.
V
CONTENTS
Wiener noise versus Wigner noise in quantum electrodynamics L. Accardi and Y. G. Lu
1
Quantum stochastic flows on manifolds I D. Applebaum
19
Characterizations of some operators on Fock space S. Attal
37
Characterizations of operators commuting with some conditional expectations on multiple Fock spaces S. Aiial
47
A note on processes on bialgebras, quantum flows, and convolution semigroups of instruments A. BarchieUi and G. Lupieri
71
The quantum functional Ito formula has the pseudo-Poisson structure: df(X) = [f(X [f(X + B)-f(X)]*dA; df(X) B)-f(X)]*dA; V. P. Belavkin Belavkin
81
The kernel of a Fock space operator II V. P. Belavkin and J. M. Lindsay
87
An application of fixed point theory to quantum SDE A. Boukas
95
Matrix elements stability of quantum stochastic differential equations A. Boukas
99
A non-commutative Markov equivalence theorem C. Cecchini A note on the cohomology of quantum flows P. K. Das
109 flows
Chebotarev's sufficient conditions for conservativity of quantum dynamical semigroups F. Fagnola
119
123
VI
Characterization of isometric and unitary weakly differentiable cocycles in Fock space F. Fagnola
143
On quantum resonance C. Fernandez and R. Rebolledo
165
Quantum stochastic integrals on a general Fock space W. Freudenberg
189
The statistical invariants in quantum probability and De Finetti's coherence principle A. Gilio Some remarks on trace of operator logarithm and relative entropy F. Hiai Quantum topological entropy: first steps of a "pedestrian" approach T. Hudetz
211 223
237
On Feynman-Kac cocycles R. L. Hudson
263
The kernel of a Fock space operator I /. M. Lindsay
271
Braided groups and braid statistics S. Majid
281
A quantum probability theory for the TV-level atom L. Rondoni
297
Nonlinear Boltzmann maps in classical and quantum probability L. Rondoni
313
Weak coupling and low density limits in terms of squeezed vectors S. Rudnicki, S. Sadowski and R. Alicki
329
The lattice of admissible partitions R. Speicher
347
Phonion limits and macroscopic quasi-particle spectrum for the BCS-model A. Verbeure, M. Broidioi and B. Momont
353
Quantum Probability and Related Topics Vol. VIII (pp. 1-18) ©1993 World Scientific Publishing Company
W I E N E R N O I S E V E R S U S W I G N E R N O I S E IN Q U A N T U M
L. A C C A R D I
Y.G.
ELECTRODYNAMICS
LU
Centro V. Volterra Dipartimento di Matematica — Universita di R o m a II Via Orazio Raimondo, 00173 Roma - Italy
ABSTRACT
We show t h a t the investigation of the weak coupling limit (WCL) of the quantum electro-magentic ( Q E M ) field naturally leads to the study of the free q u a n t u m stochastic calculus on a Hilbert module. A new type of Fock space, the so-called interacting Fock space also comes out from the limit.
§1.
INTRODUCTION
It is well known that in quantum electro-dynamics (QED), an explicit solution of the equations of motion is not known and, for their study, several types of approximations are introduced. Among these approximations, probably the best known is the so called dipole approximation (c.f. [10]) in which the response term which couples m a t t e r , represented by an electron in position q, to the k-th mode of the EM-field is assumed to be 1. In the present investigation this approximation will not be assumed. Consider a free particle, called the s y s t e m , and characterized by its Hilbert space L2(Rd) and Hamiltonian H3 := —A/2 = p 2 / 2 , where A is the Laplacian a n d p = {p\,P2,pz)
2 the momentum operator. This system interacts with a quantum field in Fock representation with Hamiltonian informally written as:
HR := J^ fcgi
(1.1)
\k\a+ak i
where to each mode k €T€ 2dd it \i is associated a representation of the C C R with creation and annihilation operators a£, a* respectively (ait = ((*l,fcio%,k>03,*))' a i , t , a2.1t, a 3,it))T h e interaction between the system and the field The field is (neglicting the the A2 A2(q) term):
Afl> = * *p p ® -^= -^= ++ h.c. h.c. == X(A(q) X(A(q) -p -p ++pp- A(q)) A(q)) \Hj =A \J2] T e ikg K kit V l1*1 l
(1.2) (1.2)
±k
where p.±''kii is the response term; A(q) A(q) is the vector potential and the tensor product p ® ak has to be meant in the sense: d
a p ®® aakk == 222J Pj P Pi ® ® aj,k i,k
(1.3) (1-3)
i=i
and where the factor A is a small scalar (coupling constant). T h e total Hamiltonian we are going to consider is then H = Hs + HR + XSf. The most important object for such interacting model is the wave operator at time t which, in the interacting picture, can be written as JJW IfW Ut Ut
._ it(He>t(H s+HR)e-it(Hs+HR+XH,) ._ + HR + \H,) e 5+HR)e-it(Hs
Q 4j (1.4)
is the solution of Schordinger equation: is the solution of Schordinger equation:
jtU[x)x) jU[
= -iXHr(t)U^ = -iXHfflU}*
where the evolved Hamiltonian
, ,
tf> :=
Hj(t) H^t)
(1.6)
Replacing the sum in (1.1), (1.2) by a. continuous integral, and replacing the factor -j= by a cut-off function g(k) (z is the complex conjugate of 2 6 C) one obtains the expression: dke > l e> -'e-^l\-ip)* (-ip)$ ■'[/.'dke'^^e'^e-'^/\-ip)
Hj(t) Hj(t) = i[J i[J
u
2 2
k
tp /2
® a+(S h.c] a+(Stg)(k) .c] tg)(k) - h
(1.7) (1.7)
3 where g is good function (e.g. a Schwartz function) and {St}t>o L2(R ) given, in m o m e n t u m representation, by (Stg)(k)
is the unitary group on
= e'Wgik)
(1.8)
Using the C C R , one can rewrite in t h e form (1.7) Hi(t)
= i[f
dke-ik-*e-itk-'t-i'^tf7(-ip)
® a+(Stg)(k)
- h.c]
(1.9)
where, here and in the following, integrals should be meant in the weak sense on the domain S(Rd) of Schwartz functions. Moreover, without loss of generality we can simply forget the factor e~lt'k' I2 by transfering it into S 0, + + + A (S,TJ)-^A (S,TJ); A+xX(S,TJ)-^A (S,Tjy,
Ax(S,TJ)—+A(S,T,f) Ax(S,TJ)—+A(S,T,f)
(2.6) (2.6) V '
5 i.e. t h e collective operators converge to some kind of creation and annihilation operators A+(S,T, f) and A(S, T, f) acting on some limit Hilbert space. T h e consideration of the two point function is naturally suggested by the fact t h a t the vacuum distribution of the creation and annihilation fields is gaussian. Consider, for each s,t > 0, < £ ® $ , Ax(t)A+(s)
rj ® $ >
(2.7)
which is equal to r i/A
2
A / Jo
,»/A2
dtx I Jo
dt2 I dki / dk2 < 0\akla% |0 > Jt * J* *
<e,e-^^'Pe^'e«l^l2/2(-ip)(5tlS)(Jfci)e"**='»ert*^e-'= = A2/ dh / dt2 / difce-^-*')!*! 2 / 2 < ( i p ) ^ e i " 2 - ' l ) * ' " ( i p ) I 7 > 7 ? ® $ > (2.8) JO ./0 Ja d and goes, as A —> 0, to tAs
dr [ dkg(k)(STg)(k)
f J-oo
./« d
(2.9)
T h e above result can be rephrased as follows: the approximate two point {unction (2.7) tends to an object which for some aspects, is very similar to a two point functions. DEFINITION ( 2 . 1 ) A collective number vector is a vector of L2(Rd) ® T(L2(Rd)) ob tained by applying a (finite) product of collective creation operators to a vector of the form $ ® £ where $ is t h e vacuum in T(L2(Rd)) and £ is an arbitrary vector in (L2(Rd)). Such a vector will have the form A+iS^T.J,) ■... A$(S„,TnJn)t®$
(2.10)
In the following, when no confusion can arise, we shall use the simplified notation n
n2)$®n>
(3.1)
By t h e definition of collective creation and annihilation operators, (3.1) is equal t o /•7\/A
tTiJ\
2 '>'e'e' >> e-' e-' >'>e' '>e't*t*k*>'
( 5 ((ll / 11)(fc (5 ) ( f c11)(5 ) ( 5f2 ) ( f c22)a+a+|0>®f, )a+a+|0>®f, f 2//22)(fc
22
22
n/*
rr n/x
2
A /
n/\ r rn/\
dsi /
Jo Jo
ds2
,, ,,
e-fc.-v^VPe-Vv****-* e-'k1-ie»lk1-pe-,k3.ge..Ik^P
Jo Jo J* ** J* J* ** (S33J[)(k[)(S J[)(k[)(S = sJs2J)(k 2)(k 2)a+ 2)a+ iat,JO®r iat,JO®r l> l>=
rTJX2
= A44 // = A Jo Jo
yT22/A 22 rT'JX rT'jS 2 j-Tl/X* ,Tl/\2 , dt\ I1 dt22 I1 dsi // ds22 II dk^dk dt\ dt dsi ds dk^dk22 Jo Jo Jo J Jo Jo Jo J2 d ■ ,2d
xt2k2P 2k2P k2q 11itlklP itiklP isiklP isiklP ik2ikq ik2 ,S2k2P2k: 2k: 2k p lk2 k2 itiklP isiklP
[[< < ^^ ^e-" * ee' e' --'-'e'e-e- t e e e-e- -e-*-'>e" e -'>e" ri '-l'T} '-l'T} ee-"
> >
f11(k(k1)(S ffl(h)(S, J22)(k Sl-Sl tJ[)(k 1)f2(k 2)(S^2)+ )f )(S, -Hf22)+ ){k2)+ 1)(S 1)f 2(k 2t)(S^tJ3 2)(k l--tJ[)(k tJi)(k 1 2{k | k2q ' P e ie-itikiP l - i ' l eeik- liqle-ik2q i ' i2qPeeisik2P i * i ' ee)-ik ieetisis2k r2kf 1ep1ip> e-'t2k2P eisik2P e -iiqq
i*!fe-i'iieiiiti-j.
/u1 (*i)(5 (eh, * i ) (vs,„_,, 5 JJtt _, _ , 1 ^)(fci)/ ^ )f')(h) ( f c i ) / a (*»)(S« (h( * »k*)(s,. ) ( S « ll___,. / I )/■(;*WMI ,}] tttt /I)(*,}]
>
(3.2) (3.2)
By t h e C C R , one can move the 5—factors together in t h e right h a n d side of (3.2). Thus one can rewrite (3.1) as l-Ti/X2 /•Ti/A
A A44 // Jo Jo
.Tj/A ,T 2 /A2
dtj dtj //
Jo Jo
r< ^ - ' ' I ' l ' f
,7\/A2
dt dt22 //
Jo Jo
yT rT 2 /A 2
ds ds t t//
Jo Jo
,
ds dkidk ds 2 2//2 J dkidk 22 Jm Jm 2 J
.e " ! t ! - P e - ' ( i * r P e - » i * r P e i / 2 ( J i - I , ) l i ! - t i
>
h(k11)(S, h(k )(S,l-lt-JtJl)(k )+2)+ l)(k l)Mh)(S, l)Mh)(S, a-tJa2-)(k tJ22)(k _| pP 4._ < £cj-> A )= lim(* *'A> A^0
of t h e scalar p r o d u c t of two collective vectors. According t o o u r choice of the collective vectors this amounts t o study t h e limit, as A —» 0, of scalar products of the form:
,,T>])- / Aunn / d dfci dfci...dfc k,TA, *"»> 9e"" ,,i*™''* i*™''* . . . ee -- "" »» ' ** » - V * » , s . . . e~'*""> V ' " 11 " * ' "" "" ■»«) ■»«)
11
n ^ ^ * )(*»*)•
n
n
hh=1 =1
o£{l
2n}\{m!
m„}
n) = (l
2
" ) \ ( ' » I . ) J = 1 h=l
mk<mk.k =l
A2"/ J 70
-°
dtx...
,.
dfc
dt2n ■>•"' •/""«
J
^°
V <S (SiS
»,H. »,H.
,.KW , . K W !=> !=>
m t < m k , k = l,. ,a .a
((, ee-il^™i' klV ^ 'eikV>i.* 1 " . .. .. ee --i, mm'i V V''mm''**ll""ee--, ,' '' '™ ™""**""VV' :' ": "»» ......
,k q " e"'"»k"pr)) e'^^e"-***'^)
e-
nTl n7mt(*fc)(5|mik-^/mJ(fc*) h=l
(4.14) (4.14)
LEMMA ( 4 . 4 ) . In the limit A —> 0 the only terms of the summation (4.14) which give a non zero contribution are those corresponding to the choice of those n-uples ( m j , . . . , rnn) which satisfy t h e equation X[m r + l , 2 n - l ] ( " l A ) - X[m r + l,2n-l]("1fc) l,2n-l]("l*) +X[mrr ++ l,2n-l](rn +X[m l,2n-l](mhh))
~ ~ X X[m [ m r ,,2n-l]("»ft) 2n-l]("»ft) = 0
(4.15) (4.15)
Proof. Remember t h a t the indices rrij in (4.14) correspond to annihilators in (4.11) and are associated to exponential factors of the form _i ee-t*m,itit '™> *e'a,.t "V*'•'
(4.16) (4.16)
in (4.14). While the indices rrij correspond to creators and are associated to exponential factors of the form (4.17) - + 1 ' 2 "- 1 ]( m ^ t™M-kT r=l
fc=l
n
n
-]TX]x[m,,+i,2,.-i](mr) r=l
(4.18c)
tmrK-kh
(4.18d)
fc=l
The sum of the four expressions (4.18a, 6, c, d) can be rewritten as Y2 [X[nv+l,2n-l](»"*)*m/i + X[m r + l , 2 n - l ] ( ' " k ) < m k lk'f(k)(Sug)(k)
(5.3)
In the following we shall identify with the eqivalence class of / with respect to the bi-linear form given by the right hand of (5.3). Thus £ 2 ( R ) 0 T becomes a P - p r e - H i l b e r t module. T h e positivity of the right hand side of (5.3) follows from Theorem (4.1). For each n £ N, we denote by ( i 2 ( R ) ©/") the algebraic tensor product of n-copies of L2(H)QT and define P - r i g h t bi-linear, P - valued form: (■10 : (L 2 (R) © ^ )
0
" x ( i 2 ( R ) © ffn
—
P
(5.4)
by ((«i 0 h) O • • • O ( « , O / » ) | ( A 0 h) © ' ' ' 0 (fin 0 §„)) : =
"
: = I I 2n 2 n 0
(5.14)
where, n G N, aj Gl 2 (R),/> 6 T (j = l,---,2n),£ 6 {0, l } 2 n and A0 := A,
A1 := A+
It is sufnccient to consider the case: £(1) = 0 ,
£(2n) c(2n) = 1
(5.15)
LEMMA (5.5) The matrix element (5.14) is not equal to zero only if 2n
E£W ="
(5.16) (5-16)
17
Proof. The Lemma can be easily verified if n = 1. Suppose by induction that J2h=i e C 0 7^ n" implies that (5.14) is equal to zero, and consider c 2 +i+ 1 < *, A'Wfo., A^\a, A<M" 0 //x )0- -. A (
(5.17)
Denote h : = min{x £ e {1, ■ • •, 2n}; e(i) = 1} 1}
(5.18) (5.18)
the the position position where where the the first first creator creator is. is. By By Theorem Theorem (5.4), (5.4), (5.17) (5.17) is is equal equal to to < A'^-2\ah-2 h-2 OOA/ f c-_22))(a _ , \ah O fh) < *, *, A'W( A«l>(aiai O 0 // ,, )) ■ ■■ ■■ • A'V>-V(a K k-_, , OOfA k-i \ak O fk) £ +1 2 +1 A "'»(a'(a/ i 0 f Hi) I ) ■- ■^ ■ 0 / )9 )* > 2 ( t l + 1 )> A«h+ ' 2 (A 9uYZ for some g e L (M,u).
22 Note that the inner product in 5 is given by
< gX /2 '9A / 2 " for g
l,
g
?
I^
g
dy
...(2.5)
2
e L (M,jn) .
Let B denote t h e dense l i n e a r manifold in 5 given by o o {|A e S
; i4
f/Li1/2
0
f o r some f s C * ( M ) } .
V
continuous, one
We o b t a i n a
strongly
K
parameter group of unitary operators in 6 ,
V = (V(t), t € R) by continuous extension of the prescription V(t)0
for ill r
fu 1 / 2 e D where f v
o
j ( (f) (5 t (M v )>
...(2.6)
i s t h e p u l l b a c k of t h e flow t o the
^t
c
bundle of d-forms. Note that for f e C™(M), we then have jt(f)
V(t) f V(t)" 1
. ..(2.7)
Strong differentiation in (2.6) shows that the infinitesimal generator of V
(V(t), t c K) is -iT
where T
is the unique
skew-adjoint extension of the operator which acts on 9
as
1 ° 1 ° Y + -y div (Y) where div (Y) is the divergence (with respect to u ) Y + -j div (Y) where div (Y) is the divergence (with respect to u of the vector field Y. Hence we may write, for t e K, V(t) = exp(tTy) ...(2.8) 3
Brownian Flows ([App 1])
Let B (B(t), t E K ) denote canonical one-dimensional Brownian motion with the underlying probability space (n,g,P) so that for u € fi, t e K+, we have
23 B(t)u = w(t) Now for each s,t c (R* with s * t define $
•„t(x'w> for x c M, u e n, then $
: M x fi -> M
= W)-u(s) ( x > ($ s
...(3.1) by
-..(3.2)
, s,t s K*, s a t) is a stochastic it
flow of diffeomorphisms of M, in fact it is the simplest example of a Brownian flow in the sense of [Kun]. Our argument will lose none of its generality if we restrict our attention to maps of the form * which we will denote by $ .(Note that these do not satisfy a group law as in (2.3)) F o r e a c h t e R*, d e f i n e
J
: C™(M) -> L°°(M x t
Jf(r)
fi
H
X
P)
by
K
=
f ° *t
. . .(3.3) A.
for each f e C„(M), then it follows by (3.1) and Ito's formula that is. dJt(f)
=
Jf(Y(f)) dB(t) + \ Jt(Y2(f)) dt
...(3.4)
so $ satisfies the stochastic differential equation d*
=
Y($J o dB(t)
. . .(3.5)
where ° here denotes the Stratonovitch differential. Let f) denote the Hilbert space L2(n, 5, P;5)Q) and 3 be the dense linear manifold in h comprising square integrable maps from £2 into 3 We obtain a family of unitary operators U = (U(t), t e \R*) in I) by continuous extension of the prescription (U(t) t)(u)
=
V(u(t)) *(u)
...(3.6)
24 for * e 3, U 6 Q.
Hence we see that for each t € \R*, f c C^(M) we
have U(t) f U(t)" 1
J (f)
... (3.7)
(c.f. (2.7)). For every * e d,
we have *((j)
f(u)(!
for each u s !i, where
f is a sguare integrable map from f2 into C™(M)
Hence by (2.6) and
(3.3) we find that (U(t)*)(cj) for each * e 3, w e n
=
Jt(f(u))((j)($*(^/2))((J)
...(3.8)
(c.f. (2.6)).
A further application of Ito's formula then shows that U(t) satisfies the operator valued stochastic differential eguation dU(t)
U(t)(T
dB(t) + 1 T 2 dt)
...(3.9)
(see [Sau] for an alternative approach). Using the canonical isomorphism between L (fi,3,P;5 ) and h
a L2(Q,g,P) to identify the two spaces it is not difficult to
verify that for each t e IRT, (J(t)
exp(Ty ® B(t) )
... (3.10)
where B(t) is here acting as a self-adjoint multiplication operator in L (n,5,P).
(Here we have abused notation to the extent of
identifying the essentially skew-adjoint operator T
® B(t) with its
closure). We close this section by noting the following result from [Kun] which we will use in section 5 below. Let Y ,...,Y O
be complete, smooth vector fields on M which generate a
n
finite - dimensional Lie algebra, then the unique solution of the stochastic differential equation given below yields a stochastic flow of diffeomorphisms on M
25 Y (* ) o dBJ + Y (* ) dt
d*
...(3.11)
where (BJ, 1 < j < n) are independent Brownian motions. If we now define unitary operators (U(t),t e R*) as in (3.8), it follows that these are a solution of n
ddU(t) U(t) dU(t)
U(t) U ( t )) U(t)
J T y dB dBJJ + 'TT dB ++ Y
L J J
SI* jtr
T T + 1 Y T Yy +
I » 0
n
J = l
Ty22
YjJ
J
ddt dtt
J
.. .. .. (( 33 .. 11 22 )) ...(3.12)
4. Quantum Stochastic Flows Let Y
and Y
be complete smooth vector fields on M for which [ V, , Y
1
=
0
(4.1)
Let T
(j 1,2) denote the corresponding skew-adjoint operators in J f) defined as in § 2. o T
Lemma 1
v 1
Proof
and T
commute
Y
2
Let (J1, t e R) and (£ , t e R) be the one parameter groups
of diffeomorphisms generated by Y
and Y
respectively.
By (4.1) it
follows that these commute and hence so also do their pullbacks to d-forms.
Then by (2.6) the one-parameter unitary groups associated
to these flows will also commute and the required result follows D Now let D denote the duality transform from L (Q,3,P) onto r(L (R*)) which maps exponential martingales to exponential vectors.
From now
on we identify h with its image under the isomorphism I ® D. Note that for each t e R D B(t)
Q(t)
=
A(t)+ A(t)' V2
(4.2)
26 Let T(il) denote the second quantisation of the operator of multiplication by i in L2(R+) acting in r(L2(R*)) and note that
r in (4.3) and (4.4) by an arbitrary state vector g e K. Writing af(Z,g) = T a a(g) + ...(4.26) we see that we have the commutation relations [ a(Z,f) , a+(Z,g) ]
T T* ® I
...(4.27)
for f,g e H, which suggests a physical interpretation of a (Z,g) as an operator which creates a particle in the state g which is constrained in such a way that successive position measurements can only be made along the integral curves of Y and successive momentum measurements can only be made along the integral curves of Y 2
5 Some Remarks About the General Case Let Y°,Y°, . . . ,Y",Y" be 2n + 2 complete, smooth vector fields on M and
33 let !£ denote the Lie algebra which they generate.
The most general
quantum stochastic flow on a manifold driven by a finite number of annihilation and creation processes has the infinitesimal form Jt(ZJ(f))dAT f Jt(ZJ(f))dA
djt(f) where f e A,
t e K+ and each ZJ
+ j (Z°(f) + T(f))dt
YJ + Y j for 0 £ j ^ n.
...(5.1)
Formally we
may obtain j as a spatial quantum flow driven by the unitary process whose infinitesimal form is !KS(TJ dA +
d!S
T° ♦ 1 V
dA
7
Ti
TJ
(5.2)
dt
2 ^ 2 2 J=l
so that x in (5.1) is a superposition of terms of the form given by (4.21). In the case where £ is commutative we may directly construct Ef as a finite product of unitaries of the form (4.12).
In the general case,
we can only prove existence and unitarity of solutions to (5.2) if the coefficients (TJ, 0 < j < n} satisfy an analyticity condition of the type discussed in [Fag]. We would like to find some more natural condition (from a geometrical point of view) for the existence of quantum stochasic flows on manifolds.
To this end let us suppose that the Lie algebra
£ is finite dimensional.
By Kunita's result discussed in §3 and
using the n-dimensional versions of (4.2) and (4.3), we can assert the existence of unitary processes satisfying the equations
dlt
K T j dQ. +
T o + j =
T j Y 1
dt
.(5.3)
and dB
=
55(T j dP
T o + Y ?
dt )
(5.4)
34 Now suppose there exists a unitary process B
satisfying the
equation n
i
u dB
u =
*
8 ( 1 T ) 11 dP
U Tyo U* + \ Yj
*\
I'Ol 'I'
dt)
...(5.5)
+
then we have (c.f. [Ev Hull that 33 is given by the prescription 33U(t) U(t)
!BS(t) for each t e K+.
...(5.6
A proof of the existence and unitarity of B
will
be postponed to a future article. References [AMR] R.Abraham, J.E.Marsden, T.Ratiu, Manifolds, Tensor Analysis and Applications, Addison-Wesley
(1983), Springer-Verlag
(1988)
[AFL] L.Accardi, A.Frigerio, Y.G.Lue, The Weak Coupling Limit as a Quantum Functional Central Limit, Commun.Math.Phys. 131, 537-70 (1990) [App 1] D.Applebaum, An Operator Theoretic Approach to Stochastic Flows on Manifolds, to appear in Seminaire de Probability [App 2] D.Applebaum, Towards a Quantum Theory of Classical Diffusions on Riemannian Manifolds, Quantum Probability and Applications 6, 93-111, World Scientific (1991) [App 3] D.Applebaum On a Class of Stochastic Flows Driven by Quantum Brownian Motion, to appear in Journal of Theoretical Probability [App 4] D.Applebaum, Unitary Evolutions and Horizontal Lifts in Quantum Stochastic Calculus, Commun.Math.Phys. 140, 63-80 (1991)
35 [EvHu] M.P.Evans, R.L.Hudson, Perturbations of Quantum Diffusions, J.London Math.Soc. 41_, 373-84 (1990) [Fag] F.Fagnola, On Quantum Stochastic Differential Equations With Unbounded Coefficients Probab.Th.Rel.Fields 8^, 501-16 (1990) [HuPa 1] R.L.Hudson, K.R.Parthasarathy, Construction of Quantum Diffusions, Quantum Probability and Applications 1, 173-94, Springer LNM 1055, (1984) [HuPa 2] R.L.Hudson, K.R.Parthasarathy, Quantum Ito's Formula and Stochastic Evolution, Commun.Math.Phys. £3, 301-23, (1984) [Kun] H.Kunita, Stochastic Flows and Stochastic Differential Equations, C.U.P. (1990) [Sau] J.L.Sauvageot, From Classical Geometry to Quantum Stochastic Flows : An Example, to appear in Quantum Probability and Applications 7 (World Scientific)
Quantum Probability and Related Topics Vol. VIII (pp. 37-46) ©1993 World Scientific Publishing Company
C H A R A C T E R I Z A T I O N S OF SOME O P E R A T O R S ON F O C K SPACE Stephane ATTAL Universite Louis Pasteur Departement de mathematiques 7, rue Rene Descartes 67084 Strasbourg Cedex, France
I
Introduction
On the boson Fock space o- One has the following interpretations $ t ] = L2(Q.,Ft,P) and Et = E[- /Ft], 1 e R+. We denote by 1 the vacuum element of $, and by / the identity operator on $. For / in L2(M+) we define 2
ft] = /%>,*], f[t =
f\,+oo[
e(f), the associated coherent vector. Let L2b{M+) be the space of locally bounded elements of L2(]R+). Let £ (resp. £lb) be the linear space generated by the vectors e{f) when / ranges over L2{M+) (resp. L2b(M+)). The annihilation, creation and number processes denned in Hudson & Parthasarathy [1], will respectively be denoted A, A^ and A. In Hudson & Parthasarathy [1] and Meyer [4] it is proved that if K, L and H are adapted processes, defined on £, which verify
(ii.i)
y"(iiA' s£ (/ sl )n 2 +III S £(/ S] )H 2 +i/( 5 )i 2 n# S c-(/ s] )n 2 )ds < oo
38
for all t in M+ and / in L2(R+), It=
f K.dA, Jo
then the stochastic integral
+ f L„dAl+ Jo
,teR+,
I HsdAs Jo
is well defined as an adapted process of operators on £ which verifies for all t and / , (11.2) Ite(ft])
= / f(s)I3e(fs])dWa+ Jo
f f(s)K3e(fs])ds
+ f
Jo
L3e(fs])dW3+
Jo
+ f Jo
f(s)H3£(f3])dWs.
In the following, each time that one of these stochastic integrals appears, we implicitly suppose that the processes K, L and H verify (II.1). Let F be any element of $. By the classical previsible representation theorem, F can be written „ F = JE[F] + / Gs dW3, Jo where G is a previsible process. For every t, let Ft = EJtF. If F belongs to £, equation (II.2) proves that, for every i, (11.3)
ItFt=
f I3G3dW3+ Jo
[ K.G.ds+ Jo
f L3F3dW3+ Jo
I
H3G3dW3.
Jo
If, furthermore, the operators It, Kt, Lt, Ht, t € M. , are bounded and extended to all 4>, if their extensions verify (II.3) for every F in $ , we will say that It=
[ K3 dAs + f L3 dA\ + / H3 dk3 Jo Jo Jo
on all $ . In the part dealing with Maassen-Meyer kernels we use the classical short notations of Maassen [2] and Meyer [3]. Let V be the set of finite subsets of M+ Let ^J3 be the set of elements (U,V,W) in V3 such that U, V and W are two by two disjoint. We will use the sign sum to denote a union of disjoint elements of V. Let us recall that a kernel operator is an operator on $ of the form K = J J\@ where the set function K satisfies
aK(U,V,W)dA\jd\vdAw
39 i) a compact time support condition: K(U, V, W) = 0 unless U, V and W are contained in some bounded interval [0,T] ii) a majoration of the form \K(U,V,W)\
0
be the (bounded) martingale associated to T, that is
Mt = (Et T ) | # | ] ® /,#„ = ( r ^ « ) | « t ] ® /|#[, = T|#„ ® / , # t i . For alia < t and all u in # a ] , wehave||(M« - M 0 )u|| 2 = ||Tu - T « | | 2 = 0. But, since each Et is self adjoint, if T verifies condition i) its adjoint T* verifies i) too. So if M* denotes the adjoint of Mt, the process M* is also the martingale associated to T*. Hence, for every a = * > ■> + If T satisfies ii), by (II.2), we have, for every / in Z/(2fT) and every t in R , +
Te(f*) Te(h)
+ /7(
= 2Tl£(/«])+ f/ / ( a ) Jf(s)H 1 £ ( / , ] ) + / /f{s){f ( « ) ( / 7H/udk A u a]) )dW r . £ (a/e(f . ] a] ) «)dW W .a. u du)e{f £ ( / 3a+ l)^3+ Jo Jo Jo Jo Jn ./n ^0 JO
42 So (Te(ft])) ^ is a square integrable martingale which converges to Te(f). There fore Te(ft]) = Ei Te{f), for every t. Of course, one also has Te(/ ( ] ) = T Ete(f). So i) is verified on the dense subset £, hence everywhere.
■
In section V we will need to characterize when such operators are isometries. Proposition I V . 3 - L e t T be an operator on $ such that TEt = Et T, for all t in M+. Using the same notations as in theorem III.l, the following two assertions are equivalent. i) The operator T is an isometry ii) The process Lt = Et + Mt, t € M+, is a process of (adapted)
isometries.
Proof We know from theorem III.l, that for each t 6 1R+, dMt = EtdAt. dM* = E* dh.t and, by the quantum Ito formula, d(M* Mt) = {H*t Mt + Mt Et + Et Et)
So,
dkt.
We easily deduce that the process M is a process of isometries if and only if M0 is an isometry and, for every t, Ef Mt + Mt Et + Et
Et=0
i.e. i * Lt = Mt* Mt. Hence, Lt is an isometry for every t if and only if Mt is an isometry for every t. But, as (Mte{g),
Mte(f)}
= (Ete(gt]),
Ete(ft]))
(e(g[t),
e(f[t)),
for every f,g € L2(M+), it is easy to verify that T is an isometry if and only if Mt is an isometry for every t.
■ IV
Maassen-Meyer kernels
With theorem III.l, we know that all the operators which commute with the Et admit an integral representation. We are now going to characterize those which admit a Maassen-Meyer kernel. T h e o r e m I V . l - £ e < T be an operator on $ admitting a Maassen-Meyer i) All the spaces ImEt are stable under T if and only if T(U,V,W) every (U, V, W) in