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Q P - P Q
Volume V I I
M a n a g i n g 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 . Sinha, W . von Waldenfels Advisory Board
L . Gross, K . H i d a , A . Verbeure
h
World Scientific
II
Singapore
• New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Lid. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Q U A N T U M PROBABILITY & R E L A T E D TOPICS V O L . VII Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyanymeans, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.
ISBN 981-02-1011-6 ISBN 981-02-1979-2 (pbk)
Printed in Singapore by Utopia Press.
CONTENTS
F r o m M a r k o v i a n a p p r o x i m a t i o n t o a n e w t y p e of q u a n t u m stochastic calculus L.
Accardi
and
Y.
G.
Lu
Semigroups of positive-definite m a p s on *-bialgebras A.
Barchielli
and
G.
Lupieri
K e r n e l representation of *-semigroups associated w i t h infinitely divisible states V. P.
Belavkin
Minuscule weights and r a n d o m walks on lattices P.
Biane
A n e x a m p l e of a generalized B r o w n i a n m o t i o n II M.
Bozejko
and
R.
Speicher
M i n i m a l solutions i n classical a n d q u a n t u m stochastics A.
M.
Chebotarev
M a r k o v i a n i t y for states on v o n N e u m a n n algebras C.
Cecchini
Quantum
flows
w i t h infinite degrees o f f r e e d o m
and
their perturbations P.
K.
Das
Kolmogorov G.
G.
and K.
flows,
B.
Sinha
d y n a m i c a l entropies and mechanics
Emch
U n i t a r i t y of solutions to q u a n t u m stochastic differential equations and conservativity of the associated F.
semigroups
Fagnola
Free noise d i l a t i o n of semigroups of countable state M a r k o v processes F. Fagnola
and
M.
Mancino
A q u a n t u m c h a r a c t e r i z a t i o n of G a u s s i a n n e s s G.
C.
Hegerfeldt
vi Time-ordered exponentials in q u a n t u m stochastic calculus A.
S.
S t r u c t u r e relations for F e r m i o n i c R.
L. Hudson
and
P.
A.
flows
Lindsay
and P.-A.
203
Shepperson
Fermionic hypercontractivity /. M.
175
Holevo
211 Meyer
O n a c l a s s o f g e n e r a l i s e d E v a n s - H u d s o n flows r e l a t e d t o classical M a r k o v processes A.
Mohari
and
K.
R.
221 Parthasarathy
Note on 5-entropy N.
251
Muraki
Applications of
finitely
correlated states to the
ground
state p r o b l e m of q u a n t u m s p i n chains B.
261
Nachtergaele
Entropy in quantum probability I D.
275
Petz
F r o m classical geometry J.-L.
to q u a n t u m stochastic
flows:
T h e A z e m a m a r t i n g a l e s as c e n t r a l l i m i t s M.
R.
317
331
Speicher
W h i t e noise analysis theory a n d a p p l i c a t i o n s
337
Stmt
Large deviations and mean-field q u a n t u m systems R.
299
Schurmann
W h i t e noises fulfilling certain relations
L.
an example
Sauvageot
F.
Werner
349
Q u a n t u m P r o b a b i l i t y a n d R e l a t e d T o p i c s V o l . V I I ( p p . 1-14) ©
1
1992 W o r l d Scientific P u b l i s h i n g C o m p a n y
F r o m M a r k o v i a n approximation to a new type of q u a n t u m stochastic calculus
L.Accardi
Y.G.Lu *
Centro Matematico V . Volterra, Dipartimento di matematica
Universita di Roma II
ABSTRACT
The attempt to extend our results on the Markovian approximation of quantum Hamiltonian systems to quantum electro-dynamics leads to the introduction of a new type of quantum stochastic calculus, which is carried on the Fock space over a H i l b e r t module rather than a Hilbert space. In this calculus the initial space is no longer independent of the noise space, i.e. the inner product of two elements in the Hilbert module can be a linear map on the initial space.
§1 A n outline of M a r k o v i a n approximation
In this note we start from Markovian approximation of quantum Hamiltonian system. Let be given a quantum system, i.e. a Hilbert space Ho; a Hamiltonian Hs which is a self adjoint operator on Hq; an initial state i^o which is usually given in trace form: ^o(-) = tr(p-). Essentially it is equevalent for us to know a family of linear functions < £,-)? >; t,V e H . Suppose that the system is open, i.e. there exists an interaction between the given system and another system, usually a quantum field, called Reservoir, or heat bath, or field, ... depending on the interpretation. 0
^Supported in part by C N R - G N A F A , Bando N.211.01.24 * 0 n leave of absence from Beijing Normal University
2 Denote R the Reservoir; we assume that its state space is T(Hi): the Fock space over Hi which is interpreted as one particle Hilbert space. The one particle Hamiltonian is given and the unitary group generated by the Hamiltonian will be denoted by {S'J'JteR. This unitary group defines a new unitary group { r ( S ° ) } i R on the Hilbert space r(Hi). Its generator, denoted by Ha, is the Hamiltonian of the Reservoir. The initial state of Reservoir is a quasi-free state (z: fugacity) whose GNS representation is < >• Now let be given the interaction Hj 6
z
Hi := i(D ® C
+
- h.c.)
(1.1)
where, D,C are densely defined linear operators on Ho and T(H\) respectively. The composite system System-)-Reservoir (S+R) is described on the Hilbert space H ®T(H\) and the Hamiltonian is given as follows: 0
!= H ® 1+ 1® H
Htotal := fffrec +
S
where A > 0 is a coupling constant. At each time t one can define the following objects: i) wave operator: U, := e "" e **"""* itH
-
R
+ XHj
(1.2)
(1.3)
1
ii) Forward evolution on a bounded operator X £ B(Ho) X,{+) := U+{X ® 1)U,
(1.4a)
and backward evolution on a bounded operator X 6 B{Ho) X (-):=U (X®l)U+ t
(1.46)
t
If one wants to know how does the Reservoir influence the system, i.e. to look the system part of X (±) one finds that it is a solution of integro-differential equation — it remembers all information from past. In order to take away the memory effects, some limit procedures are introduced, i.e. Markovian appoximation. In the quantum case, the first step of the Markovian approximation is to require a limit semigroup and several types of limit procedures are designed. The typical ones are the weak coupling and the low density limits. The result in this first step can be formulated briefly as follows: under some conditions the limit of t
,
0
(1.5)
exsits and is equal to <e,p, (x), > ±
(i.6)
?
as A —» 0 in W . C . L . (with the time scaling t := t/\ ) and z —t 0 in L . D . L (with the time scaling t\ := t/z). Where {P^JieR^. is a completely positive, identity preserving semigroup on B(H ). A z
t2
0
2
3 The second step consists in investigating not only the limit semigroup but also the limit stochastic process. More precisely one requires the following: i) In a certain sense the wave operator Ut (with a suitable time scaling t tx, ) tends to a stochastic process {U(t)} ^. ii) In the same sense the forward evolution Xt (+) tends to x
z
t
x
U+(t)(X®
1)17(0 =:
t
X{t,+)
and the backward evolution Xt . (—) tends to X(t,—), where {t/( $, -n ® $ > new family (which we called collective coherent (or number) vectors) whose form is suggested by first order pertubation theory (cf. [14]). For the unitarity of the limit, the choice of the collective coherent (or number) vectors is very delicate. For example in the situation stated above, one should replace the vacuum vector $ by a new one which reflects all degrees of freedom. A possible choice is D
T/X
* 0
v
a
2
/ Js/\>
yy""*S?/ e^'> e-^'T, 2
= (2n) f
dw}{u>)M'< ( - "\) (w
d
;
i
e
A
+ %/2y)
JR-
1
= (2*-)'' / jR
a^eV)e l 'l 'e-'
lt=-oo
A 2
,7\/A
T /\ I "/ '
2
X
r
N
dui•• •
I
2N
t
e€S„
Js
T'JX duN I V*
2
r
X
T' /\ dtli • • • I s>t,f»
2
r
Js
K
2
dvN
J
A=l t,/fc' = -oo where denotes the N-permutation group. With the change of variables: A u 2
= r„,
A
h=l,---,iV
(4.4)
,
h = 1, • •,7Y
(4.5)
f
V
and ^(k) - r /X
2
h
= a
eW
(4.3) becomes equal to
$N,N> J2
f
< fh, S°,, f' w
eW
iVi
' I
dr
N
>iJ i ,
(
,
da
l 4
^»,
l 4
.»» >
e(fc)
(4.6)
8
J—OO
L_
Jt=-oo
or equivalently, one can write, in the weak topology of B(Ho), N_
K m < [f[* m < I A-o
[f[A
/
d=l
T
f
=
'
A
M
/
^ (5,°/< ,*)®e
E
+
i
i , p
e ^6 , ]*, i l
< i
f c
£
A+(S?/i
i t
)® " ' l
e
e
i t o
*6i, ]*> 1
k=-oo
2
E II
Sn.n'
£
*
S d / x
•' i/A S
TtlX
f ' j
< X[s»,T»].X(s; ,^ „) (t)
^ < / M , ^ W >
' "
e
(
((
P
+
>L e^+^a+b
k
J—
(4.11)
oo
Thanks to the concept of Hilbert module one can write (4.9) as N
N'
< II (Ms ,T„] h=l A+
® © A , * O b ,k)*, I] (xW,n) k£Z h=l
k
A+
h
Where the direct sum ©
i
g
z
® © / i , kez
t
Ok
>
(4.12)
K Q V of the pre-Hilbert modules K Q V is a pre-Hilbert k
k
module with the natural inner product ( © A kez
©it "k\© 9k Ok b ) := Yjifk kez kez k
Ok a \g Q k
k
k
b) k
where the right hand side makes sense since except for finitely many k £ Z, f
k
Q b = 0. k
k
Also £ ( R ) 0 ( 0 i g z K Ok V) has a natural structure of pre-Hilbert module, with inner product 2
UOx\ 'Ox')
:=< ,x'
x
X
>L'(R) (*!*')
i
X,x' € L (R);
x,x' £ ($KQ V kez
2
k
and on the Fock Hilbert module T ^ L ( R ) 0 ( 0 K Ok V)j we can define a representation 2
of the " C C R " , with ^-valued commutator (recall that V C
B(H )) 0
[A(X ® © fk Ok bk), A+(x' ® © f'k Ok b' )\ =< X, X' >L(R) ( © fk Ok bk)\ © fk Ok b' k
kez
kez
2
kez
kez
4.2 T h e weak coupling limit In order to consider W . C . L . problem we start from (2.4). First of all, by (3.9) we expand the product .H>(t.)
(4.13)
into £
t€{0,l}»
i"D (U)---D (t )®A^\Slg)---A«"HSlg) c(1)
cin)
n
(4.14)
10 where A°(f\ — } W< ~\A+(f), A
i
f
£
=
(4.15o)
0
ife = l
Uh
K
and
Since for any n £ N , f, J? € Ho, sup
| < f , D i ) ( t i ) •••£»«(„)(*„)»; > | < C < o o
(4.16)
e(
ii,--,, 5 - > l (4.19) c
P
Since (4.16), we have L E M M A 4.1. The limit of
< T T A / tJ[ JS f» L
dt T
A+(5?/ , )(8.e ''( +^6 ,J$,A rf
,
t
p
(i
1
t
/I.(n) [ J ] A / i=l
Js
T
'd/
j
X2
/
" dt f)
n
/ Jo
*, / Jo
*,-••/ Jo
A+(S?/i ) ® e-« +->6',J $ > i 4
p
k=-oo
is equal to zero. P R O O F Combine (4.16) with the proof of Lemma (4.2) in [1]. By a similar argument one has the following uniform estimate:
dt.
(4.20)
11 L E M M A 4.2. For each N, N' € N , there exists a constant n e N and t e R + , r
"
< I1
f A
it*
/
Jg.
,
E
d
(4.22)
it
it=-co
Finilly we have the follwing W . C . L . theorem T H E O R E M 4.4. The limit, as A —> 0, of (4.22) and (4.2) exist. P R O O F (sketch) In order to control the limit (4.22), the first step is to apply formulae (4.18), (4.14) and (3.10) to (4.22). Since {b ,kh
JJ
$®TT6 , e,A"(-i)" / d=i • d
1
t
/o
A (S ° +
,
> f f
)®e'' '>( " ") ,
p +
m
dt
1
/ J a
dt,--
dt„ J a
12
n
o, SS
< 5?
g > ®e- '"i-* i i
( p + w m
' ' - e ' ' * < """''*' l )
k
i t
p+
A=l
J]
A (5 +
(
0
a f f
)®e''°( +' 'p
j m
)
°6{l, -,«}\({j*}Ll 0r -l}r ) U
[ft
A
/
T
'
/
h
=l
® e''(
A
p+wt
V ri
i ) ] * ® Q„(e)
>
iy
(4.23)
where, k, k',k , k' , £ Z. By the standard technique used in [1], one can check that the limit exists. d
d
§5 T h e q u a n t u m stochastic differential equation
Having proved the W . C . L . exists we shall answer the question: which type of quantum stochastic process is the limit of the wave operator U /\2? t
T H E O R E M 5.1. The limit, as A —• 0, of (4.2) exists and is equal to N
< n
® 0 / m ©* &*,*)* ® £.
(xis ,n]
A+
h
N'
V{i) I] (x\s' ,T'] h=i A+
h
® 0 f' ,k Qk 6 ' ) * ® n > tez h
(5-1)
M
where, U(t) is the solution of the quantum stochastic differential equation dU{t) = (dA+(
0 0© te(-N)
ff
©o 1 © 0 0)Q - Q+<M ( 0 O © s © tgN *e(-N) (
o
l © 0 O ) fc€N
- Q ( f f O o llffSo l)_Qdi)t/(t)
(5.2)
U(0) = 1
(5.3)
+
on the Fock module r ( L ( R ) ® 0 2
t
g
z
K Q V) ® /Jo described in Section 4, where k
A ( F ) := A(x[o, e^ t
'a+b
p+uk
( .5) 5
13 and ^ (x ® © +
A 0*
i)Q = Q^ (x +
0 ©
kez
A
©HI
i)
(5.6)
kez
P R O O F The main idea is to notice that (5.6) is nothing but the limit form of the commutation relation ,T/X
/
7
oo
dt V
7y 2
oo
A
A (S°Jk)®e « »VQ +
i
p+
= Q /
>lt V
i t
+
( S j f t ) 8 e
i
W
W
)
)
(5.7) By combining the formula (5.7) and the argument of Section 6 in [1], one can finish the proof easily. A simple computation shows that for each k € Z, (g ®k Mg Ok i ) _ =
©* Mg o
i) + iim(s ©* Mg ©* i ) .
h
(5.8)
where for an operator B, i l m B denotes the skew-adjoint part of B. By (5.8) one can rewrite (5.2) as du(t) = (dA+(
0
*e(-N)
~Q (\(g +
oe o„i©0o)Q-Q avi ( +
o
Jt6N
©o
(
0
*6(-N)
o ©
9
©o 1) + ilm(, (c) a l i n e a r m a p TJ : B —> T> s u c h t h a t , V a , 6 € B, (1.7)
n(a) (b)=rj(ab)- (a)6(b), V
0
G i v e n if'
o
n
V
(a)V>(&) - if>(ab) + iP(a)6(b) = -(r,(a*)\r,(b)).
(1.8)
i t i s a l w a y s p o s s i b l e t o c o n s t r u c t a p r e - H i l b e r t s p a c e T>, a * -
r e p r e s e n t a t i o n ir a n d a l i n e a r m a p TJ s u c h t h a t (ir,T],ip)
is a generating triple o n
B w i t h r e s p e c t t o T>. O n a n o t h e r s i d e , l e t tt b e a * - r e p r e s e n t a t i o n o f B o n a p r e H i l b e r t s p a c e V, l e t TJ b e a l i n e a r m a p f r o m B i n t o V s u c h t h a t (1.7) h o l d s a n d l e t V' b e a h e r m i t i a n f u n c t i o n a l o n B s u c h t h a t (1.8) h o l d s . T h e n , o n e h a s i m m e d i a t e l y t h a t if' i s p o s i t i v e - d e f i n i t e a n d v a n i s h e s o n t h e u n i t a n d t h a t a l s o 77 v a n i s h e s o n t h e u n i t . I n t h e l a n g u a g e o f c o h o m o l o g y t h e o r y , (1.7) says t h a t 7/ i s a
1-cocycle
w i t h r e s p e c t t o t h e r e p r e s e n t a t i o n w a n d (1.8) s a y s t h a t t h e b i l i n e a r f u n c t i o n a l — (77 o I n v ( )\TJ( )) i s t h e coboundary
o f if>.
T h e r e is a class of * - b i a l g e b r a s for w h i c h a l l *-representations
are surely
b o u n d e d . L e t ir b e a ^ - r e p r e s e n t a t i o n o f B o n a p r e - H i l b e r t s p a c e V a n d l e t H b e t h e c o m p l e t i o n o f V. L e t { a ^ , i,j
= 1 , . . . , n } b e a set o f n
2
elements of B such
that n 22
n ij ij* = 22 i * i' 3=1 i-i a n d i n t e r p r e t II := {n{aij) e q u a t i o n s (*(a*)h'\ir(b)h)
a
: i,j
a
a
i
a
=
U- )
^'^
9
= 1 , . . . , n } as a n o p e r a t o r o n ')j.tW for a l l d i s j o i n t i n t e r v a l s (s,t)
a n d (s',t')
?tiuM---j*«l))-~n))
+
for a l l n € I N ,
,t
n + 1
i n IR a n d a l l b,b' G B,
G IR w i t h < ! < • • • < t
n+1
t h e i n c r e m e n t p r o c e s s i s c a l l e d a stationary Q (b) £ C(£(H )) is norm-continuous for all b £ B and satisfies t
t
equations Proof.
Q
(3.7) and (3.8) with f = 0. T h e n o r m - c o n t i n u i t y f o l l o w s f r o m t h e fact t h a t t h e g e n e r a t o r ( 3 . 8 ) i s
b o u n d e d . T h e o t h e r s t a t e m e n t s f o l l o w f r o m E x a m p l e 2.6 a n d T h e o r s . 2 . 7 , 3.2.
•
25
4 T h e Case of the *-Bialgebra of a Group L e t u s c o n s i d e r n o w t h e case o f t h e coefficient
algebra
of a group. I n t h i s case i t i s
possible to give a p r o b a b i l i s t i c i n t e r p r e t a t i o n to the semigroups of positive-definite m a p s as F o u r i e r t r a n s f o r m o f c o n v o l u t i o n s e m i g r o u p s of c e r t a i n o p e r a t o r - v a l u e d m e a s u r e s ( T h e o r . 4.5). T h e r e i s a s t a n d a r d d e f i n i t i o n o f r e p r e s e n t a t i v e b i a l g e b r a of a g r o u p , o r e v e n o f a s e m i g r o u p [21]. N o t o p o l o g i c a l n o t i o n i s n e e d e d . H o w e v e r , for t h e a s s o c i a t i o n w i t h m e a s u r e s , i t i s b e t t e r t o c o n s i d e r a r e s t r i c t e d class o f topological groups a n d a s u b - b i a l g e b r a m a d e of continuous functions. L e t u s d e n o t e b y A t h e class o f a l l l o c a l l y c o m p a c t t o p o l o g i c a l H a u s d o r f f g r o u p s G s u c h t h a t t h e set o f a l l
finite-dimensional,
unitary, continuous representations
of G s e p a r a t e s t h e p o i n t s o f G ( m a x i m a l l y a l m o s t p e r i o d i c g r o u p s : see [31], p p . 1 2 - 1 4 ) . F o r G G A , w e d e n o t e b y R e p ( G ) t h e set o f a l l n - d i m e n s i o n a l u n i t a r y n
c o n t i n u o u s r e p r e s e n t a t i o n s o f G. T h e n , 1C(G) (coefficient a l g e b r a ) i s t h e c o m m u t a t i v e * - a l g e b r a o f t h e c o m p l e x - v a l u e d f u n c t i o n s f(x) f(x)
on G such that (4.1)
= (D(x)£\r,)
for s o m e n 6 I N , D G R e p ( G ) , (,rj G C . T h e i n v o l u t i o n i s d e f i n e d b y c o m p l e x n
n
conjugation a n d the algebra multiplication by pointwise multiplication. Moreover, AC(G) c a n b e t u r n e d i n a * - b i a l g e b r a . T h e c o u n i t is d e f i n e d b y *(/) = / ( « ) ;
(4-2)
e is t h e n e u t r a l e l e m e n t o f t h e g r o u p . L e t u s w r i t e f(x)
i n t h e f o r m (4.1) a n d l e t
{fi'. i = 1 , . . . , n } b e a c o m p l e t e o r t h o n o r m a l set i n C " . T h e n , t h e c o m u l t i p l i c a t i o n is d e f i n e d b y n
^(/) = E ^ i ^ ) ® ^ i ? ) i
(->
7
4 3
t=i
If w e i d e n t i f y / C ( G ) ® / C ( G ) w i t h a s p a c e o f f u n c t i o n s o n G x G, (4.3) b e c o m e s A(f)(x,y)
= f(yx),
w h i c h s h o w s t h a t t h e d e f i n i t i o n (4.3) does n o t d e p e n d o n t h e
r e p r e s e n t a t i v e (4.1) o f / . B y d e f i n i n g t h e l i n e a r m a p S: IC(G) —• IC(G) S(f)(x)
= fix' )
,
1
(4.4)
we a l s o h a v e t h a t t h e i d e n t i t y M o ( S ® I d ) o A = M o ( I d ® S ) o A = m oo is s a t i s f i e d . T h e n , fC(G) b e c o m e s a n involutive ([21], p . 6 1 ) . T r i v i a l l y , K(G) I n K,{G)
S is called
Hop} algebra;
(4.5) antipode
is essentially u n i t a r i l y generated.
we consider t h e F o u r i e r topology T
F
(see [31], D e f . 1.3.10). A n e t i s
0, any u E there exists a compact set K, K C A, such t.hat. IIP(A \ K)ull < E. Int.egrals with respect to projection-valued measures can be defined as usual in t.he T.-t.opology and the equation
j(f) =
fa
f(x) P(dx),
(4.6)
Vf E IC(G),
defines t.he Fourier transform of the projection-valued measure P. Proposition 4.1 Let G E A. Equation (4 .6) defines a one-to-one correspon· denC/o' bdween regular projection-valued measures P on rand T,.-T.-continuous *-representations j of IC(G) into £(r).
Proof.
o
This is Proposition 1.8 of [10).
Definition 4.2 [16,10) An instrument on M with value space (G, E) is a map T from E int.o £( M)" such that (1) T(B) is complet.ely positive, VB E B; (2) T(·) is weakly* O'-additive; (3) for any normal state eon M and any positive element a E M , the positive and finit.e measure (e, T( · )[al) is regular; (4) T(G)[R) = R. By defining integrals in the weak* topology, the equation
Q(f) =
fa
f(x)T(dx),
Vf E IC(G),
(4.7)
defines the characteristic operator (or Fourier transform) of the instrument T. Theorem 4.3 Let GE A. A linear map Q from IC(G) into .c(M)" is the characteristic operator of a unique instrument T iff it satisfies conditions (b) and (d) of Def· 3.1 and the function I -+ Q(f)[X) is r,.-w*-continuous lor any X E M.
Proof. This is Theor . 1.5 of [10) . For the continuity properties see also Proposition 1.2 of [10) . 0 By t.he previous theorems one can show t.hat, under continuit.y assumptions, both t.he *-represent.ations j.t in Proposition 2.5 and the positive-definite maps Qt in Theor. 3.2 are Fourier transforms of operator-valued measures, when B = IC( G).
=
Theorem 4.4 In the hypotheses of Proposition 2.5, with B IC(G), G E A and j.t 1',. -1'. -continuous , for all s, t E IR (s :S t) a unique regular p/'ojection-valued mcaSILre p. t on G with t1alues in £(r) exists , such that
(4.8)
27 Moreover,
P, (B)
€ Af(s,t),
t
a°(P, {B))
a < t, B £ E, and the following +r t+T
/ P ,(dx)P {Bx- ) JG r
= Prt(B),
1
lt
s < i , v e Wi,
= P, , {B),
t
P„(B)
seIR,
e
hold: (4.9)
BeE,
r < s < t ,
= S (B)t,
properties
(4.10)
BeE,
(4.11)
BeE.
Proof. E q u a t i o n (4.8) i s d u e t o P r o p o s i t i o n 4 . 1 . T h e l o c a l i z a t i o n p r o p e r t i e s f o l l o w f r o m ( 2 . 1 6 ) . E q u a t i o n s ( 4 . 9 ) - ( 4 . 1 1 ) a r e e a s y c o n s e q u e n c e s o f ( 2 . 1 7 ) , (2.1) a n d t h e * - b i a l g e b r a s t r u c t u r e i n t r o d u c e d o n K.(G). • T h e o r e m 4.5 In the hypotheses of Theors. 3.2 and 4-4, equation (3.1) defines the characteristic operators of a family {It : t > 0} of instruments on M with value space (G, E) such that, V 5 £ E, J ( B ) = S {B)\d and e
0
It+,{B)= Moreover,
j X {dx)ol,(Bx- ), JG
t
representation ® XU{t,
= E (U(t, oyPotiB) 0
T h e f a m i l y {It
(4.12)
t,s>0,
1
t
we have the
11 ® l (B)[X]
M
0)) ,
(4.13)
X e M , BeE.
: t > 0} c a n b e c a l l e d a convolution
semigroup
of
instruments
[14]. I n t e g r a l s i n (4.12) a r e d e f i n e d i n t h e w * - t o p o l o g y . Proof. T r i v i a l l y , Q is t h e characteristic o p e r a t o r of some i n s t r u m e n t I. E q u a t i o n (4.13) f o l l o w s f r o m (3.1) a n d T h e o r . 4.4. E q u a t i o n (4.12) a n d t h e i n i t i a l c o n d i t i o n t
t
for To f o l l o w f r o m p r o p e r t i e s (c) a n d (a) i n D e f . 3 . 1 , r e s p e c t i v e l y . • F i n a l l y l e t u s c o n s i d e r (2.19) a n d (2.20) i n t h e case o f B = K(G). L e t D £ R e p ( C ? ) a n d {e^ : i = 1 , . . . , n } b e t h e s t a n d a r d o r t h o n o r m a l set i n o S((D \ )) ei
ej
: i,j
a r e u n i t a r y o p e r a t o r s , J(D)
l,...,n}
= 1 , . . . ,n} a n d 0(D)
1
, are selfadjoint
m a t r i c e s . M o r e o v e r , f r o m (1.7) a n d (1.8) we o b t a i n n r,((D \ )) ei
ej
= -(LT(D)T(D))..,
(©(£>)).. = £
((T{D)) \(T(D)),.) K
(4.15)
28 (cf. [10], (2.32) and (2.38)). B y identifying the operators A ,. t
• • having a matrix as
argument with the corresponding matrices of operators, (2.19) a n d (2.20) become dV (D) t
= V (D) t
-dA\(n(D)T(D))
{dA {n(D) t
- 1) +
+ [U(D)
dA (T(D)) t
- i@(D)]d 0 , what follows
p ' ( b ) . L e t us f a c t o r i z e 9
over
f r o m the
the subspace
9 ={ae9l(aip) = 0 , V p e 9 } , 1
considering i.e. a e 8
1
a
as e q u i v a l e n t to z e r o , i f
means, i n particular, a (ala)=
X a,ce!B
°£
+
a 6 m\
T h e condition
= ( a I 8 ) = 0 , and e
( a c ) a = (a°la°) = 0 , < r
c
a = 0,
38 where
a
= a
D
V p , then
for a l l b * e
0
p
+
= 1 = P , a -
l(ot° I P ° ) l classes
< (a
2
+
e
=
(a
0
0
e
+ a P +
I p°) = 0
0
=
+
a_,
d u e t o the
Schwarz
9 Ia - Pe
9
+
1
space of
i n the p s e u d o - E u c l i d e a n s p a c e
}
, w i t h the E u c l i d e a n s u b s p a c e
+
= ( P ° l , c o r r e s p o n d i n g t o the " k e t " - v e c t o r s
0
inequality
. It h e l p s t o r e p r e s e n t the c o m p l e x
, e ] , e. e C 3 e
0
a_p-
I P°) +
and ( a
(P° I P°)
Ia )
0
[e. , e
bra-vectors
£.
£
of
0
e ° = e* e 9 ° / 9 -
the p r o d u c t ( 1 . 7 ) (a
by
= 0
( P I = {a*
of triples
and
« b • H e n c e i f (a I p ) - 0 ,
e
a . = 0 , because (alp)
if
a n d ag = a - £
the
I y)
=
I Y°) +
(a°
a_Y"
+ a Y*"
=
+
+
a_Y*.
( « °
1
Y°) +
(P
[ p . , (po|, p ] «
I'-»
+
p*=sp =p , +
0
P ! = X M P )
beB
( « . • P 17) = I
M b ) ( ( S • p) ft Y)b A
=I
b«S
a
e
V Y
=>
9
o n the c o m p o n e n t s
where
a
a
B
8 •P = 0 , if p = 0 : a
( 8 • p I Y) = a
(P
It g i v e s the p o s s i b i l i t y t o d e f i n e f o r a n y a e
(P
(B -> 8
Xc
finite
for
;
is conditionally
Ka a.(a*.b.c)K
C
i . If
all
additive
=
a,ce®
J
{c;} c
family the
c > 0 ;
conditionally
in the
X
A > C ,
;
and
and A > C means
the
positive-
A - C on 0£.
immidiately
from
the construction
of
representations
of
the m i n i m a l
JC .
Fock
space
infinitely
states.
w e shall
describe
ft-semigroup
Araki-Woods bialgebra
s
representation
unital
^ ' ^
i=l
of the operator
Kernel
Now
e
follows
divisible
the
=
in (Hi) of the Theorem
defined
b =Xb =>B=X B
b
b'—»B
representation
n
X i
K
beB
0^ with the properties,
where
X
K : I K I < «o ,
i=l
construction
C(B.
This
a n d f o r the p r o o f
of
indefinite
[7-9] to the case
representation
i n the correspondent
construction
a natural
"B a n d i t s r e l a t i o n
pseudo-Fock
the solution
o f the
has a
of
space
the generalized
of
a cocommutative
nice
space,
decomposability whiftti w e u s e d
the q u a n t u m
ft-homomorphism
representation o f
with
property
Langevi n of
the
[11] ftpro-
[6] f o r equation
correspon-
41 ding Hudson Our sentation
even is
i n the n o n a d a p t e d
to
define
JI: !B -> . ! ? ( £ )
associated r(£ )
flow
purpose
with
an
Eo"
i n f i n i t e l y d i v i s i b l e state n = 0, 1, . . .
o f the E u c l i d e a n s p a c e
5
(a? , c®)
=
£
-V
n=o on
the g e n e r a t i n g
be
the
.=n-
v = - ,
on
n
X :II
= X
v
r(£.)
- » 3^( »
+
n
n
^ ( n ) _ { be tkdecomposable
( 2 . 1 ) , given in the
operator
pseudo-Fock space r ( J C ) over K,=C © "K® C by a linear span of the kernels (2.2). Then the kernel operator K J , defined by the pseudo-isometry oo
[ J y ] ( n% n ° , n
+
) = 8 (n") y ( n ° ) ,
[ J * V ] (n) = X ^ V < > - °> m=o m ! m
0
of the pre-Hilbert space to a continuous operator
< - )
n
T(30 into r ( J C ) , can be uniquely
2
3
extended
oo
[ \ (K)
¥
X („") X
] (m) =
n+n+=m
on the completion
n =
~h-
( " " ' ' +> n
+ ") .
n
(2-4)
!
n
0
K
ll\jrll(a) =
J of F(30 with respect to the poly norm
II a * \f II, a e 3 , where g/ven by fhe kernel
b = t ( b ® ) . Moreover,
\:K
the map
t (K),
oo
K(n-,n,n )
= X
+
n
defines a ^-representation a ^-semigroup associated
B
"
n
,
;
n", n, n
= 0 , 1, 2 , . . .
+
of the decomposable
in the pseudo-Fock S%7) of poly-Hilbert
CB
for
9
b -* b ® ,
representation
p ( b ) = e*- , into the
with an infinitely divisi ble state
operator ^-algebra
(2.5)
^-algebra (b)
space
J, such that
t (I) = I , t ( K * K ) = t ( K ) * i ( K ) ,
where
(n) = K ( n * ) * ,
n *= [
n +
°
(K*K)(n)
|i
£
b
(n) £
m
jmj { b)®( - > ® ( B k ) ® n
m
^
{
b
®°
)
\j/(n) = k ® , n
®
+
the
opera-
B
k
®
m
J =
k®
](m)
•
n
= e
m
of
o f the d e c o m p o s a b l e
^
n"=o
+
X(b) + (bk
a ft - r e p r e s e n t a t i o n [
T ( K ) . L e t us p r o v e it o n the e x p o n e n t i a l v e c t o r s
n+n =m
e
8
X ( b )
(B k + b))® .
+
m
n=o
A p p l y i n g this f o r m u l a [a~ft~c k® ](m)
due b
for
=
b = a ft c , w e o b t a i n
e < * x
a
c
>
+
< * a
(a~ft"c k + a ft c ) ) ®
k
e
X ( a ) * + X(c) + (a*c) + (a^Ck + (ck tf^
* Q ^
e
X ( a ) * + (a*(Ck + e» ( *
+ a*))®
A
to
k + c))
( C
t h e p r o p e r t i e s (1.5)
a n d (1.6)
b ) , (b. H e n c e o n the s p a n o f t (a®*c®)
that i s
= t
((a*c)®)
t ( b ® ) = J * b® J
operators
b®
on
p r e - F o c k space sequences
n
n
c = a ft b e
e
p = e*.
The
is a
e W
of
x
(b)
#)-)®m
+
a
+
b
associated
product formula operators
b®
(2.6)
T
c a n be
n
i
definition
s
II = II y
n
n
II (c)
the
sequences defines a closed infinitely
the k e r n e l obtained by
of
divisible
a linear
state
span
n i
{ a ft c ) ® ° ® ( A * C ) ® ° ® (a ft c ® " ° ) = n
n
( ( a * ) + ( a * c ) + ( c ) ) * { [ A * c) + a * ) ] ® " " ® ( A * C ) ® ° n
n
of
the l i n e a r e x t e n t i o n
of ( a ft c )
=
i n t o the c o n t i n u o u s o p e r a t o r s
with for
II \f II (a)
i s a l s o f u n d a m e n t a l due to
of a l l fundamental
f
of
a l s o o n the f u n d a m e n t a l
n
n
2*. H e n c e the m a p
of
i n the ^ - a l g e b r a
{\|f }-» {t ( b ® ) y j .
{t ( b ® ) \)/ ) n
maps
= t (a®)* t ( c ® ) , of
b = i (b®)
25 as the o p e r a t o r
the
JQ w e h a v e
ft-representation
II (a) = II J * b®J v
of
decomposable
m
c
=
m
w i t h r e s p e c t to a l l the s e m i n o r m s
r(2Q
of p o l y - H i l b e r t space ft-representation
* )
= a~ft~c = a*c
i s c o r r e c t , b e c a u s e the s e q u e n c e II i ( b ® ) v
^
{k® I k e
L e t us d e f i n e
r(9Q.
{\p }
+
the p s e u d o - F o c k s p a c e
II J * a * ® J v II , a e
for
c
® [(c + ( a * C ] ® ° } n
46 y
=
_ n ^ L _
( a*)® r+°
*)r
( a
*
c )
(
where
+
a
•
+
means
£
® (a*
C®
- ^ 7
° " ® (c®
m
}
s
,
+
a
+
( ®(
c
+
n«"-mo") } .
C
a pointwise multiplication
o f the elementary kernels
a n d c ® , d e f i n e d as
£ ® |mo- + m - , liU° + m ° ,
ri
+
{ a*)®
Remark
] . /
®
r ?
1
0
(b)
0
1
0
0
0
1
(b
0
1
0
0
1
satisfying
with
,
® (a*
C® ° M
®
(c® °} . (2.7) s
b
D
=
+
e
( c ) _ (a)+(c)
(cA? _
e
1
0
0
1
b)
0
0
1
0
properties
e o
b
= e
block-operators
-
1
0
0
0
B
0
0
0
1
.
atct =
with
with
b t
B =
A
( b ) = (a) + ( c ) ;
C;a°c° =b°
with
denote e
e
" (c)«
(2.7)
« b ) = b* = < > ,
the operators
(aA?
the elementary
b ) = a) + c ) , a ° c 8 = b °
(b = ( a + (c , and
(a)
m +
+
NS
the exponential
ao"Co- = b o
e
0
m
(a^c)
si \ _ t^) nV+m °,/
® / si, \ no + m i ,
2.
=
e
c
A * c ) ® ° ® (A * C)®
tef us introduce
e
£
S
A*®( n i + m ° ) g , ( i!r®( nii+ m ) } . { ) ® ( m £ + m - ) g,
+
b°.
)
m+
C®(ni'+m„") g
a*®
c
® A * c ) ® ° ® (A * O ®
. { i!r)®(n °-m °) g a
( a
i
i
A
+ b
having
(b^®), e
) = b : ,
B
N
= b ° ,
[ (a + (c ] A°
j^e/i
=
b
l
(
on 7 the corresponding
A ^ a ) A ^ c ) _ A£ [a) + c)] e
eC"A°
C
>
AN C
representation
N
=
2
g
)
properties
(AC)
N
,
b e 2'->b = i ( b
8
)
47 of arbitrary
2 » , associated
it-semigroup
p=e
state
can be represented
x
b = e
X ( b
It f o l l o w s
> e °> B A
from
b
N
(2.9)
the decomposition 1
(b
0
B
b)
0
0
1
0
(b)
1
0
0
0
1
0
0
1
b)
0
0
1
0
0
1
-
b
Example
(b) "
i
.
=
1
0
0
1
(b
0
0
B
0
0
1
0
0
0
1
0
0
1
-
b. bo bo b ° +
the triangular block-operator
£ ,
of the
b A
1
Let
divisible
product
e< °
b =
of
an infinitely ordered
(2.8):
operators
the m a p
with
as the normal
b a n d the multiplicative property
of
b = i (b®).
1. £
be a complex
E u c l i d e a n space w i t h a n i n v o l u t i o n
= % , ( 2 otj %{f = E a*£? , V § i e
the c o m p l e x
space o f pairs
£ , a; e C
^ - » %* e
a n d « = Cx £
b = (P,r|),peC,r|e £
be
w i t h the b i n a r y
ft-operation ( a , S ) f t ( Y , q ) = ( Y + S # c + a * , q + S ) = (P, defined
b y the scalar product
semigroup
= (5 I q) = £ * c , i "
w i t h the neutral element
b * = (P*, T| ) #
a n d the associative (b. t l ) - ( Y > S ) = (P+
The
product
•
the s i m p l e c t i c f o r m selfadjoint funciton on
p ({$, t i ) = eP
- T h e n ® i s a ft-
under the i n v o l u t i o n
operation
;
i f the i n v o l u t i o n
otherwise
# is isometric
i t represents
:
the canonical
TJ - q - q - T| = i s ( q , T i ) , c o r r e s p o n d i n g t o
s (c,,r|) = 21m ( n I q)
n = n *, | = q
the C C R algebra,
operators
£
relations ( C C R )
e = (0,0) = 0
E
T l ' S + Y . Tl + S ) , i l - q = ( T l * l q )
is commutative,
II i;* II = II i; II , V !; e commutation
n)
#
J
§
.
o n the real space
T h e positive-definite
R e £ of
and normalized
i s t h e g e n e r a t i n g f u n c t i o n a l o f a G a u s s i a n state generated
i n the G N S representation
b y the
48 b = e representing
< ^e
e
+
A
b = (p ,TI)
v a c u u m state operators,
A +
, A + ( ri) = A ( r i ) * #
i n the F o c k s p a c e f
J , where
A, A
are
+
the
w i t h respect
annihilation and
The projection
< j
function
A.(P , n ) = P
(a* + \ # ? + 7 ) S , =
X
;
The
ft-semigroup
and
X (0,0)
devisible generating
indefinite-form representation
of
space
triangular block
with
a£ = 0 ,
nil
M
* 0,
2
=
0,
hence
function. , n ) = (e. , 7t(P , n )
A.(P
E. = C © £
is g i v e n
© C
0
by
matrices
u < v , a : = 1 = at
ag = 1 ,
,
a? = E / ,
a ; = p*.
0
Example
2.
Let
S
be an operator
*-algebra
c l o s e d u n d e r the p o l y n o r m
vectors
, n)*,
= X (P
o f 2J i n the p s e u d o - E u c l i d e a n
a =^,
k e
3C,
br|(a) = n ( a b ) a*
= P*
i s an i n f i n i t e l y
canonical
7t|(p , n ) = [ a£ ]
Ki
n#)
X (P*,
p
the
creation
s a t i s f i e s the c o n d i t i o n s o f ( i i ) :
a.ye C
e.)
the
H 1-5)1.
+
if
to
s a t i s f y i n g the C C R [ A ( r,*), A (q) ] = (
X
over E
defines
and
for a l l
on
2
8 £ ^(JQ
i n a p r e - H i l b e r t space
II k II (b) = II b * k II , b e
«
o f the
a —>r|(a) be a c o n t i n u o u s l i n e a r m a p a,b e
S . The Hermitian operation
the s t r u c t u r e o f
the u n i t a l
"kef-
H -»
1
with
a ft c = c + a*c +
ft-semigroup
with
e = 0 , b * = b * , a • b = b + ab + a , V a,b e <S. D u e to not
b • c = (1 + b)(l
necessary
associative and
to
a Poissonian
in
the
the
with
X(b)
of
b
®,
1 , where one
state
p(b)
+
symmetric
e
x
p
on
{
Fock
x ( b )
the
N
by
, defined
(]n(i
space
II
2
a continuous
This
generating algebra
is
n*(b)
= T)
• is
definite positive
functional generated
operators
+ b)) + X(b)}
J
by
i s the
current algebra. the
that the p r o d u c t
is A b e l i a n . The positive
H
X ( b * b ) = II n ( b )
G N S representation = e A (ti(b))
= e
1 i s the i d e n t i t y o p e r a t o r i n 5£,
can easily find,
a n d it i s c o m m u t a t i v e i f
normalized function
linear f o r m
in
+ c) -
belonging
over
e *
creation
£
, where
f
and
N(b)
annihilation
commutation
i s the n u m b e r
operators
operator,
A (k°),
satisfying
A . ( k ° ) , k ° , k*, e
+
with
%_
the
relations [ N ( b ) , A ( k ° ) ] = A (bk°) , [ A . ( k ) , N(b) ] = A ( k b ) , +
+
0
0
[ N ( b * ) , N ( b ) ] = N ( [ b*,b ]) , [ A . ( k ) , A ( k ° ) ] = k k ° I . +
0
The
linear
respect to
*-function
X.(b)
the o p e r a t i o n
K a X.(a* + a*c + c) K =
Z
K
= 0 . Hence
p(b)
K
c
a,c e
is
an
infinitely
devisible
B
be
generating
function.
The
canonical
indefinite-form
= (e. , J t ( b ) e . ) o f "S i n the p s e u d o - E u c l i d e a n
\(b)
positive-definite
K * X(a*c) K = II £
£
c
Z
condionally
ft:
a,c e (A ) is 1 the positive type function - x t r a c e , these two Markov chains c a n be r e l a t e d by 8
some
probabilistic
generalise sections.
this
operations
relation
to
(killing, other
and
Doob's
L i e groups
and
h - t r a n s f orm). states
in
the
We next
will 3
53 2)
Some notations
and facts
from
L i e group
and L i e algebra
theory:
Before s t a t i n g the results on quantum random walks, we need some notions which can
be
found
representations
in
any
textbook
(see f o r example
L e t G be a compact,
connected,
on
compact
Lie
group
and
Lie
algebras
[61, [71, [8), [9], [13]). simply connected
semi-simple
L i e group,
with
Lie algebra 9 . Let T c G be a m a x i m a l t o r u s , and P be the weight lattice associated to T . W is the Weyl group o f G. We choose Let
{0^,
a Weyl chamber i n o, i=l,..l),
(w,
i=l,...l)
and C
be
the
weights of G w i t h respect to C . {<x ,
its closure.
positive
roots
and the
fundamental
i=l,..l} is the dual basis of
V
Let p be the h a l f sum of positive roots,
then 1
p = 1=1 Let P
= PnC, P
= P n C , then
1 and
so that:
(2.1)
P
= p + P
++
+
F o r any x e P , let e(x) be the corresponding c h a r a c t e r of T . J-r
e(x)(8)
X 6
de
is a
minimal
projector
7,
of
and corresponds
to
1
(the
(x)
indicator f u n c t i o n of x) in the isomorphism between J and 1 (T). From
Cartan-Weyl
irreducible an
element
irreducible and l*(r}.
theory,
representation of
P .
By (2.1),
representation For xeP
we
know
of
we w i l l
the
highest
weight
we denote by r
and d
on G ,
X
corresponding
irreducible
equivalence
an element
x
class
weight of
of
an
which is
P ^ w i t h the
x - p , and so we identify
1° (I >
>
)
the character and the dimension
X
representation,
x
is
a
central
function
and i f
= d f F T g T X dg
n x
then
the
by its highest
identify
++
of
that
of G is determined
n
is a minimal c e n t r a l
x J
projection
X
isomorphism between
c
I
^
g
of 6",
which correspond
to 1
in the '
iS" and 1 (P
).
x
'
54 The r e s t r i c t i o n of x
to T is given by Weyl's f o r m u l a : X
L
X
(2.2)
= — )
x
det(w)
e(w(x))
det(w)
e(w(p))
f urthermore:
L We shall use the notation
det(w) e(w(p)) = n(e(a ) - e ( - a )). Y
det(w) e(w(p)) = ?,
wWV We now a r r i v e to the definition of a minuscule weight: An
element
u of P
is a minuscule weight
if
it s a t i s f i e s
one of the f o l l o w i n g
equivalent conditions: i)
the set of weights
weights w(u), weW,
of
1, weW,
ii) f o r any i=l
the representation
with
dominant
weight
u a r e the
and each weight has m u l t i p l i c i t y one. a^(w(u)) e {-1,0,1}.
(see Bourbaki [7], p 127 proposition 6 , properties i) and iii) ) The important property of minuscule weights
is the f o l l o w i n g ,
which is a
consequence of i i ) : (2.3) i f x e P
then x+w(u) e P
++
f o r any
weW.
+
The list of a l l minuscule weights of simple L i e algebras is given in [71 pl29.
3) Comparaison of two Markov chains: In this p a r a g r a p h , G is a compact
connected,
simply connected
semi-simple Lie
group, and we f i x u a minuscule weight of G. L e t K be the normalized c h a r a c t e r of the representation w i t h highest
weight u.
By
to
property
i)
of
minuscule
weights, in
\W\
Y
u,„ weW
the
restriction
of
K
T
the
restrictions
is
e(w(u))
L e t v be the weight given by v(A ) = K(g). g
We
first
of the (j )
determine
explicitly
the
transition
operators
of
• to ST and £ .
Theorem 3.1: i) T h e r e s t r i c t i o n of (j ) • to S defines k k€[N increments being
.,),.
V
| 1 V |
wW
a random walk on P, the law of its
S W
l
»
ii) If p(x,y) a r e the t r a n s i t i o n probabilities of this Markov c h a i n , p(x,y) = iii) F u r t h e r m o r e , f o r any weW,
[ e(x)(9) e(y)(6) ic(e) de. p(x,y) = p(w(x)),w(y))
then
55 proof: the i s o m o r p h i s m between In the
y
and l^IP),
A ey
corresponds to the
f l
function
x-»e(x)(8) on P . QU )
= K(6)A
e
= -p-L- £
e
e(w( ))(8)A M
Qn^pjIet.JteJKy) =
so
e
£^e(w(u)-y)(e)
Since the A ^ generate
7,
e(y)(6)
|J" e(y)O')
K(6) =
t h i s proves the f i r s t p a r t . K ( 9 ' ) e(x)(6')
orthogonality r e l a t i o n s of the functions
X
V(P) V = (
Ep (J
=
P
e ( y ) l e , )
that:
( x
f ^ < >( '! 'j J
- y ' ^x)
K(e,)
e
=
x
i
T
e
de'j
e(x)(9)
by
the
e(x)(.). 1 5 1
de
K ( e )
x
e
d
e(x)(8) A
e
de
f l
which proves i i ) . The
last
relation follows
process is i n v a r i a n t by
Theorem The
from
the
fact
that
the
law
of
the
increments
of
the
W.
3.2:
restriction
of
the
(j ) • k keIN
to
B
defines
a
Markov
chain
on
P
++
with
transition probabilities: d q(x,y) = - p x
r
x (g) X (g) J
G
x
*(g)
dg.
y
proof: QOI ) = y
Y L xeP
1< .y> n ^ x x
J
= d f x~Tg) yj~ y
K(g) A dg = g
d I xeP
'
++
J
x
G
*
{
g
,
)
^y' '' 8
K
(
g
>
)
orthogonality r e l a t i o n s between
Let of
p°(x,y) p(x,y)
be
to
the
P
.
p°
transition defines
a
d g >
]
d
'
xf
**
(
g
)
G
the
\
d
g
b
y
the c h a r a c t e r s .
probabilities
on
P
+ +
obtained
submarkovian kernel on
P
+ +
by
which
restriction corresponds
to the r a n d o m walk defined by p k i l l e d at i t s f i r s t e x i t time of P ..
Lemma
F o r any
3.1:
x,y€F
tt
p°(x.y) =
^ d e t ( w ) p(x,w(y)) = -^j
£^
V ^ d e t ( v w ) p(v(x),w(y))
proof: Let W
x,y
eP
+ +
a c t s simply t r a n s i t i v e l y on the Weyl chambers,
so that w(y)*P^
+
for
w*id.
56 By property (2.3) and theorem the
f
sum
det(w)
3.1) i) p(x.z) = 0 i f x e P ^
p(x,w(y))
a l l the terms
are 0
and z £ P ,
so t h a t i n
+
except
w=id,
a n d the
vkw f i r s t equality holds. The second equality follows f r o m i i i ) o f theorem 3.1. L e m m a 3.2: = [ x (g) X (g) K(g) dg. J x y
p°(x,y)
G
proof: Applying lemma 3.1, then theorem 1 i i ) , we have: p°(x,y)
= y i j
\W\
L
r
£^det(vw)
d e t ( v w )
J
d e t l v )
= TiJ7T f X (9) x~Te) I W I J ^ . x y =
J Q
I
< v,®...
4.3:
i) L e t ce6\ a n d m...j (c )> = u t j (c )...j t c ) 0 ® * W r W r 0 0 0 n n 0 0 n n = w(j (C )...j (C )j (0) 0 ( 0 ) " ' f(0)~ ) 0 0 n n n = u ( j (c (c 0) i/>(0f W0)~ ) 0 0 n n = w(j (A )j (c (c 0) 0(0)"' (A -l)) n
n
)
=
n
1
n
g
O
O
n
n
n
n g
f o r any g e G , since v is c e n t r a l = P is isomorphic to the
is a bialgebra f o r
take
the
finite
subspaces,
algebra
TJPs'P
coproduct 3:
x.
f where V is the space of xer x x dual is then a direct product
sum
Its
H i l b e r t subspace of J
f (g) G
The dual of
4(G)-> We
direct
representation
space
is
of
given
Pigs').
an a l g e b r a i c
the
T
inductive topology.
identification, T
With this
functions
of f i n i t e dimensional representations of the group G x G .
by: 3(P)(g,g') = T
of
representations.
y
for
x
X
dg
g
e
L (G), M ,
we
x
have
we
an
identify
£(?> ,C),
an element
i d e n t i f i c a t i o n of
4(G)
of
each f
of
with
the m u l t i p l i c a t i o n map on T is then the coproduct A :
* JE(7'®7 ,C)«.4(G) 8
4(G).
,
G
and if
Z
coefficients
a
compact,
semi-simple
connected,
simply
connected
Lie
group K. Let
k
be
its
Lie
algebra,
and
let
K
c
be
the
complex
simply
connected
group w i t h L i e a l g e b r a keik. Kj, is a complex algebraic group. By
Weyl's
"unitary
representation Kj,,
so
the
algebraic
on T,
that
of T
trick"
K can be
can
be
group K
(cf
(131)
extended
identified
we
uniquely to
with
Let xe4(G)
know
be
and A(x)(P®0) = x(P)x(Q) = x(PQ),
the
that
any
finite
an algebraic
algebra
of
representation
polynomial
a primitive element,
dimensional
x
functions
is a
linear
of on map
so that x is an a l g e b r a morphism f r o m
64 'P to
c. By elementary algebraic geometry, there exist a unique element g of
KCsuch that x(P) = P(g) for every P (x is evaluation at g), and conversely, every element of KC determines such an algebra morphism, so that the set of primitive
elements
can
be
identified
with
KC'
.
Every
element
of
k
is
a
derivation on 'P, which composed with evaluation at zero gives an element of A(G)"'!e('P,C) such that .II(a) = a®id + id®a, and a
be written exp(a+ib), and only
if
it
is
= -a. Every element of KC can
and exp(a+ib)' = exp(-a+ib),
of
the form
exp(ib) .
Hence,
so that it is positive if the
set
of
exponentials
is
identified with the homogeneous space KC/K '" IS, by polar decomposition.
Theorem 5.1: The set of primitive elements of A(K) is in one to one correspondence with K ' C the correspondance being as above; in this correspondance, exponentials correspond to elements of the form exp(ib) where bek.
Corollary 5 .1: Using Cart an coordinates on KC '" KAK, we see that ISIK '" AlW (the action of K on IS being by conjugation).
I would like to thank Y.Benoist and P.Polo for some useful discussions.
References: [II E.Abe: Hopf algebras, Cambridge tracts in mathematics, vol 74, Cambridge Univ .press, 1980.
[2]
L.Accardi,
A.Frigerio,
J .T.Lewis:
Quantum
stochastic
processes,
Pub!.
R.I.M.S. Kyoto Univ . 18 p 97-133 (1982).
[3] Ph. Biane: Marches de Bernoulli quantiques, Seminaire de Probabilites XXIV, Lecture notes in Mathematics, Springer 1990.
[4] Ph.
Biane:
Quantum random walks on the dual of SU(n) , Probability
theory and related fields, 89, p 117-129, Springer 1991.
[5]
Ph.
Biane:
Equation
de choquet
Deny sur
Ie
dual
d'un
groupe
compact, pre print, Universte Paris 6, 1991. [6],
[7],
[8]
N.
Bourbaki:
algebres de Lie, chapitres 4
Elements
de
mathematiques,
Groupes
et
a 6; chapitres 7,8; chapitre 9 , Hermann.
[9] T.Brocker, T.tom Dieck: Representations of compact Lie groups , Graduate texts in Mathematics n098, Springer.
[10]
J.Dixmier:
Les
C' -algebres
et
leurs
representations.
1964.
[11] J .Dixmier: Algebres enveloppantes, Gauthier-Villars, 1974.
Gauthier-Villars,
65 (12]
A.W.Knapp:
Mathematics [13]
Representation
series,
Naimark,
n'36,
Stern:
theory
of
semi
simple
Lie
Princeton
groups,
1986.
Theorie
des
representations
des
groupes,
Editions
MIR, Moscou. [14]
K.R.
Parthasarathy:
de Probabilites [15]
J.W.
[16]
W.
Probabilites
Lecture
Pitman,
M.
of
Durham
Proceedings Springer,
XXIV,
A
Yor:
generalized
notes Bessel
Biane's
in mathematics,
Springer,
processes
indefinitely
and
conference,
Lecture
notes
The
process
of
Seminaire
process,
in
1991. divisible
Mathematics
laws, n°851,
1981. von
Waldenfels: XXIV,
Lecture
notes
Markov
in mathematics,
total
Springer,
spin, 1991.
Seminaire
de
Q u a n t u m P r o b a b i l i t y a n d R e l a t e d Topics V o l . V I I (pp. 67-77) © 1992 W o r l d Scientific P u b l i s h i n g C o m p a n y A N
E X A M P L E
Abstract.
O F A G E N E R A L I Z E D
67
B R O W N I A N
Marek
Bozejko
Roland
Speicher
M O T I O N
II
We continue our investigations of the generalized Brownian motion con-
sidered in the first part of this paper. of the operator Pi
Namely, we show that a non-trivial kernel
can only occur for z a root of unity and we characterize the
Gaussian distribution of our random
1.
variables.
INTRODUCTION
In [ B S p ] we p r e s e n t e d a r e p r e s e n t a t i o n o f the r e l a t i o n s (for a fixed fi € ( — 1,1)) c(f)c (g) +
-
M
c ( p ) c ( / ) = < f,g +
> 1
(f,g
€ £ (2R)), 2
w h e r e c ( p ) i s t h e a d j o i n t o f c(g)
(in a full Fock space equipped w i t h a twisted
scalar procduct)
is linear.
+
a n d g >—» c (g) +
W e such received a n i n t e r p o l a t i o n
b e t w e e n t h e b o s o n i c , free a n d f e r m i o n i c r e l a t i o n s . I n d e p e n d e n t l y f r o m o u r w o r k , G r e e n b e r g [ G r e , F i v ] p r o p o s e d the s a m e r e l a t i o n s as a first ( n o n - r e l a t i v i s t i c ) field t h e o r y t h a t a l l o w s s m a l l v i o l a t i o n s of t h e e x c l u s i o n p r i n c i p l e (i.e., o f F e r m i s t a t i s t i c s ) o r o f B o s e s t a t i s t i c s a n d he a r r i v e d at s i m i l a r r e s u l t s as i n [ B S p ] . T h e r e l a t i o n for fixed f = g w i t h ||/|| = 1, n a m e l y c c p e r a r e d i n t h e c o n t e x t of t h e t w i s t e d SU (2) a //-oscillator.
+
— / i c c = 1, h a s also a p +
(e.g. [ B i e , M a c ] ) a n d i t i s t h e r e c a l l e d
q
I n t h i s r e s p e c t we a r e d e a l i n g w i t h a n ensemble of / / - o s c i a l l a t o r s
which are //-independent.
N o t e t h a t t h i s f o r m of i n d e p e n d e n c e is m o r e
adequate
for t h e / / - o s c i l l a t o r t h a n t h e t e n s o r p r o d u c t . H e r e we w a n t t o c o n t i n u e o u r i n v e s t i g a t i o n s . the ' t w i s t e d ' s y m m e t r i z a t i o n operator
I n p a r t i c u l a r , we w i l l e x a m i n e
o f [BSp] m o r e c a r e f u l l y a n d show t h a t
a n o n - t r i v i a l k e r n e l c a n o n l y o c c u r at r o o t s o f u n i t y . c h a r a c t e r i z e t h e s p e c t r a l m e a s u r e o f c(f)
2.
S O M E INVESTIGATIONS
F u r t h e r m o r e , we w a n t to
+ c (/). +
O NT H EO P E R A T O R
P
Z
T h e m a i n o b j e c t i n o u r i n v e s t i g a t i o n s i n [BSp] w a s t h e o p e r a t o r P^ = 0 P ^ d e f i n e d o n t h e f u l l F o c k s p a c e T o f L (IR). 2
n
'
It w a s - for each n - g i v e n b y a
r e p r e s e n t a t i o n o f s o m e e l e m e n t o f t h e g r o u p a l g e b r a C 5 „ o f the s y m m e t r i c g r o u p S „ . S i n c e t h i s r e p r e s e n t a t i o n i s a f a i t h f u l o n e , we c a n r e s t r i c t o u r a t t e n t i o n t o t h i s g r o u p a l g e b r a itself a n d w i l l from now on denote by P
Z
= © pj
the element o f
t h e g r o u p a l g e b r a . W e h a v e a l s o r e p l a c e d \i b y z i n o r d e r to i n d i c a t e t h a t we d o
68 n o t r e s t r i c t to \i € ( - 1 , 1 ) a n y m o r e but c o n s i d e r g e n e r a l z € C . e x a m i n e t h e c o m p o n e n t s pf™' for a r b i t r a r y , b u t f i x e d F i r s t we r e c a l l t h e d e f i n i t i o n of P^ \ S
n
now
D e n o t i n g the n u m b e r of i n v e r s i o n s
n
T6
We will
n. of
b y z ( x ) , i.e. * M
:
= # { ( » . ; ' ) e { 1 , • • •, n}
| i < j , «•(») > 7 r ( j ) } ,
2
we p u t p(»)
:
=
^
2
»"(»)»
CS„.
e
O u r m a i n p r o b l e m w i l l c o n s i s t i n f i n d i n g the z S C , for w h i c h Pi™' is n o t i n v e r t i b l e . W e w i l l s h o w t h a t t h i s c a n o n l y o c c u r at r o o t s o f u n i t y . F o r p r o v i n g t h i s we n e e d some facts about the s y m m e t r i c g r o u p S W e s h o u l d n o t e t h a t for 7r, C 6 5
and its length function I(TT).
n
we d e n o t e b y ircr t h e p e r m u t a t i o n g i v e n by
n
(7rcr)(i) = cr(7r(i)), i . e . we t h i n k of a p e r m u t a t i o n as a r e a r r a n g i n g of l a b e l s , n o t of numbers.
T h i s s h o u l d be k e p t i n m i n d i f one w o n d e r s a b o u t s o m e
discrepancies
i n t h e o r d e r b e t w e e n o u r f o r m u l a s a n d the c o r r e s p o n d i n g f o r m u l a s i n t h e b o o k of C a r t e r [Car]. F i r s t , we f i x s o m e n o t a t i o n s . W e w i l l use a f i x e d n o t a t i o n for s o m e e l e m e n t s S „ , n a m e l y we d e n o t e t h e t r a n s p o s i t i o n (j j+1) e l e m e n t of S
n
of
b y itj (j = 1 , . . . , n — 1), t h e u n i t
b y e a n d the m i r r o r i n g p e r m u t a t i o n Tr Tr 7r . . . 7 r „ _ 7 r _ 2 . . . i r j T i b y 1
7Tn, i . e . 7To(fc) = n — (k — 1). It is easy to see t h a t T w i t h t h e greatest l e n g t h , n a m e l y i(iro)
=
«
2
1
1
n
is t h e u n i q u e e l e m e n t o f
0
S
n
•'.
n
T h e s i t u a t i o n t h a t A i s a s u b s e t of B is d e n o t e d b y A C B , w h e r e a s t h e s y m b o l * C ' i s r e s e r v e d for p r o p e r
subsets.
N o w p u t S := { 1 , 2 , . . . , n -
1} a n d d e n o t e b y Wj
5 „ w h i c h i s g e n e r a t e d b y 7T; for i £ J. subgroups.
For example,
we have Wq> =
i s o m o r p h i c to a p r o d u c t of p e r m u t a t i o n Wj
£ 5
f c l
x • • •x 5
for J C S t h e s u b g r o u p
S u c h s u b g r o u p s of S „ a r e c a l l e d {e}
and W
=
s
of
parabolic
N o t e t h a t Wj
S. n
is
groups: w i t h fc! + • • • + k
f c r
= |J|.
r
B y | J | we d e n o t e t h e n u m b e r of e l e m e n t s o f t h e set J . W e p u t
Pi (J)
•=
n)
I n p a r t i c u l a r , we h a v e P , P
2
( n )
(0) =
L E M M A
1.
( n )
(5) =
Yl
P i
z
n
i
(
)
T
)
%
, P i
G
n
C
)
W
j
£
C
( l , . . . , -
5
"-
1) =
(k
e. a) F o r a i l
J
b) W e Jiave E J C S ( - 1 )
C S there exists R (J) n) z
| J |
^ " ( J ) = z*' }
(,roL
=
€ CS
n
such
that
(J)- ). n
1
J
l
1
JCS
b) P , Let
P
( n )
2
is i n v e r t i b l e => z £ T ( '
( n )
n
be i n v e r t i b l e . Hence, by P .
( n )
= Pz
( n )
( J ) • i t i ( J ) > aU P i n )
n )
( J ) for J
C
5 : = { 1 , . . . , n — 1} a r e i n v e r t i b l e , t o o . T h u s , as i n p a r t a ) , we get (£ _i)mpM(j)-i)p(»)
=
(
z
*(^)
T o
_
(-i)i i . s
e
JCS
N o w assume z € Z * * ) , i.e.
there exists a k € { l , . . . , n -
1} w i t h z ^ *
- 1
) =
W i t h o u t r e s t r i c t i o n w e c a n a s s u m e k = n — 1, i . e . z ^ " " " ) = 1. T h i s i m p l i e s 1
(-i)i i )(£(-i)
(***>*b +
5
e
| / |
1
Pi" (^)" )^ ' )
1
(
l )
=
JCS
a n d b y t h e i n v e r t i b i l i t y o f P^™* (
^ o )
T
o
+
(
_
1 )
|5|
e )
(^ _ (
JCS
1 )
Ulp(n)
( J )
-l)
=
0.
o
1.
72 W e w i l l s h o w t h a t t h i s is n o t p o s s i b l e . W e h a v e o f c o u r s e t h a t P ^ j y and
1
e
CWj
thus £ ( - l ) l l p ^ ( J ) J
where W
:= U ; s ¥ j . c
= : w € CW
1
Note that W
:= { £
|a . 6
aw w
C},
i s o n l y a s m a l l p a r t of S , for e x a m p l e , for n
n = 3 we have W
= {e,^,^} C S
I n d e e d , we h a v e ir such that TTJ/ =
=
3
0
•W n W
=
{e,-K ,TT ,^1^2,^2^1,^1^2^l}1
0, i . e .
2
t h e r e e x i s t n o e l e m e n t T T J - i n s o m e WJI
7Tn7T/ for 7Tj i n s o m e W j w i t h J , J ' C S.
T h i s is only another
f o r m u l a t i o n of l e m m a l c ) . F r o m t h i s f o l l o w s t h a t o u r a b o v e e q u a l i t y (T ±e)w 0
= 0
c a n o n l y h o l d for w = 0, w h i c h w o u l d y i e l d z ( °^7r — ( — l ) ' ' e = 0. T h u s we a r r i v e I
7r
5
0
at the desired c o n t r a d i c t i o n a n d c a n conclude t h a t z ^
T* K n
0 R E M A R K S :
1) T h e p r o p o s i t i o n i m p l i e s t h a t Pj?^
Since P Q " ^ = i d and Pji
h a s n o k e r n e l for /x £
(—1,1).
d e p e n d s c o n t i n u o u s l y o n jx, we get a n o t h e r p r o o f for the
s t r i c t p o s i t i v i t y o f P^™^ i n t h e i n t e r v a l \i G ( — 1 , 1 ) . 2) B y l e m m a I d ) , s y m m e t r i z e d a n d a n t i - s y m m e t r i z e d v e c t o r s a r e a l w a y s eigenvect o r s o f P i " ^ a n d a r e c a n o n i c c a n d i d a t e s for t h e k e r n e l o f P i " ^ i n t h e case z € B u t t h e k e r n e l i n t h i s case is m u c h b i g g e r a n d w e h a v e n o t y e t b e e n a b l e t o d e r i v e its exact f o r m .
3.
D E T E R M I N A T I O N OF
T H E G A U S S I A N
DISTRIBUTION
N o w we c o m e b a c k t o fj, £ (—1,1) a n d w a n t t o f i n d t h e s p e c t r a l m e a s u r e of c(/)
+ c (/) +
w i t h respect
to the v a c u u m expectation
state, i.e.
the G a u s s i a n
d i s t r i b u t i o n b e l o n g i n g t o o u r B r o w n i a n m o t i o n . W e w i l l r e s t r i c t t o t h e case ||/|| = 1, w h i c h c o r r e s p o n d s t o v a r i a n c e = l . F o r t h i s , t h e w h o l e o f o u r r e l a t i o n s d o e s n o t p l a y a n y r o l e , we a r e o n l y d e a l i n g w i t h one o p e r a t o r c : = c(f)
and its adjoint c cc
+
— fic c
+
fulfilling the relation
= 1.
+
O p e r a t o r s f u l f i l l i n g t h i s r e l a t i o n ( w h e r e u s u a l l y q is u s e d i n s t e a d of u) h a v e a l s o o c c u r e d i n c o n n e c t i o n w i t h t h e t w i s t e d SU (2) q
under the n a m e of 'g-analogue
of
h a r m o n i c o s c i l l a t o r ' (see for e x a m p l e [ M a c , B i e , C K L , V a S ] ) a n d s o m e of t h e f o l l o w i n g c o n s i d e r a t i o n s h a v e a p p e a r e d t h e r e i n a m o r e o r less e x p l i c i t f o r m . In o r d e r to f i n d the s p e c t r a l m e a s u r e of c + c expectation
+
w i t h respect
to the
s t a t e we s h a l l m a k e a c o n n e c t i o n o f t h i s o p e r a t o r w i t h
p o l y n o m i a l s a n d t h e n l o o k u p t h e c o r r e s p o n d i n g m e a s u r e i n [Asl]
vacuum
orthogonal
73 L e t us denote by p
= /
:= (c ) U +
n
n
0
t h e n - p a r t i c l e s t a t e , t h e n we h a v e
n
1 — u — — P n - U 1 - fi n
(c + c )p +
+(1
=
n
Pn+1
+ n + ••• + / i
where we agree u p o n p _ i = 0 . of o u r s t a t e s , n a m e l y (n >
n _ 1
n
1
=p„+i +
I n o r d e r t o c o m e c l o s e r to [ A s l ] we m a k e a c h a n g e
0) *
1
i=0
In t e r m s o f t h e q
)p _
**
t h e a b o v e r e l a t i o n reads as
n
(c + c + ) q „ = (1 -
u
n + 1
)q
+
n + 1
N o t e t h a t t h e s p e c t r a l m e a s u r e of c + c
—^—q -i, u n
1 -
w i t h respect
+
to the s t a t e q n
for n ^ m a n d t h e of t h e f o r m
0
=
0
d e t e r m i n e d b y t h i s r e l a t i o n t o g e t h e r w i t h t h e fact t h a t q
and q
m
Q is
are o r t h o g o n a l
I n d e e d , this a l l o w s to c a l c u l a t e a l l m o m e n t s
a n d since c + c
+
is a b o u n d e d
operator,
this
d e t e r m i n e s t h e m e a s u r e u n i q u e l y . B u t the a b o v e r e c u r s i o n f o r m u l a is e x a c t l y t h e s e t t i n g o f t h e t h e o r y of o r t h o g o n a l p o l y n o m i a l s . W e o n l y h a v e t o t h i n k of the as o r t h o g o n a l p o l y n o m i a l s Q (x)
the c o n s t a n t f u n c t i o n 1 a n d of c + c
+
c o r r e s p o n d i n g H i l b e r t s p a c e L (IR,
v).
2
c + c . +
n
as t h e m u l t i p l i c a t i o n o p e r a t o r w i t h x i n t h e T h e n v w o u l d b e t h e s p e c t r a l m e a s u r e of
T h u s we h a v e t o d e a l w i t h the r e c u r s i o n f o r m u l a
xQ (x) n
These
q
w i t h respect t o s o m e m e a s u r e v o n IR, of fl as
n
= (1 - n )Q (x)
orthogonal
n+1
+ - i - Q 1— P
n+1
polynomials
((?_!
- i ( x )
n
are well k n o w n .
They
= 0, Q
1
=
1).
are c a l l e d c o n t i n u o u s
5-
H e r m i t e p o l y n o m i a l s , w h i c h c a n be c o n s i d e r e d o n one side as a s p e c i a l case of t h e A l - S a l a m - C h i h a r a p o l y n o m i a l s ( w h i c h are t h e m s e l v e s q-Wilson
ultraspherical polynomials. s p e c t r a l measure of c + c is g i v e n b y dv(x)
A*)
a s p e c i a l c a s e of
p o l y n o m i a l s ) , o n t h e o t h e r side as a s p e c i a l case of the c o n t i n u o u s +
/
V
w i t h density
^ - ) ( * )
•
±
^ r
L
s
i
n
*
n ^
1
-
where - cos0. F o r fi = 0 t h i s r e d u c e s
q-
T h e corresponding measure v - w h i c h is the wanted
- c a n be r e a d off f r o m f o r m u l a 3.30 or 3.31 i n [Asl] a n d
= u'(x)dx
= X ( - 3 / V T O
the
to
^(x)
=
X(-2.2)(^)^V l-(»/2) , /
2
-
e-
2
V I
2
} ,
74 w h i c h i s - as we a l r e a d y k n o w ( [ V o i , S p e ] ) - t h e d i s t r i b u t i o n o f / + I*, n a m e l y W i g n e r ' s s e m i c i r c l e d i s t r i b u t i o n . (It s h o u l d be r e m a r k e d t h a t t h e c o r r e s p o n d i n g f o r m u l a i n [Spe] i s n o t q u i t e c o r r e c t . ) F o r fi - » 1, t h e d e n s i t y v'(x)
tends to l / v 2 i r e x p ( - i / 2 ) , i.e. i n this l i m i t we /
2
a l s o get t h e r i g h t d i s t r i b u t i o n , n a m e l y t h e G a u s s i a n o n e . F o r n —• - 1 , t h e m e a s u r e v t e n d s t o l / 2 ( 5 _ b + b, +
+ £x), i.e. to the d i s t r i b u t i o n of
t
where 6 is the fermionic annihilation operator.
I n t h e a p p e n d i x we h a v e p l o t t e d t h e f u n c t i o n v'(x)
f o r different v a l u e s o f q = ji
i n o r d e r t o see t h e i n t e r p o l a t i o n b e t w e e n t h e f e r m i o n i c , t h e free a n d t h e b o s o n i c case.
4.
A N O T H E R
R E P R E S E N T A T I O N O F O U R RELATIONS
W o r k i n g i n the Fock space of a one-dimensional H i l b e r t space (i.e. considering o n l y o n e / / - o s c i l l a t o r c c — / / c c = 1) o n e w o u l d define t h e c r e a t i o n a n d a n n i h i l a t i o n +
+
operators by the following ladder
U _ _ J i S i n c e t h i s a l l o w s n o c a n o n i c e x t e n s i o n t o t h e case o f m o r e / / - o s c i l l a t o r s , w e c h a n g e d this p i c t u r e to the equivalent one
1 l~]
1 + // + / i
2
1 I T1+ M 1 i _ J l Instead of this we could also have used the l a d d e r
1 + / / + //
2
Q
1 + pt i_J
i 1
U _ J i H e r e we want to consider the question h o w the canonic extension of this l a d d e r ( d e s c r i b e d b y o p e r a t o r s d(f))
i s r e l a t e d to t h e o n e u s e d t i l l n o w ( d e s c r i b e d b y c ( / ) ) .
W e s h a U see t h a t t h e c o n n e c t i o n i s v e r y t r i v i a l , n a m e l y g i v e n b y a d j u n g a t i o n i n the full F o c k space. L e t u s r e c a l l t h e s e t t i n g o f o u r u s u a l p i c t u r e : W e h a v e i n t h e f u l l F o c k s p a c e !F o p e r a t o r s c ( / ) , c + ( / ) , s u c h t h a t / H-> C + ( / ) i s l i n e a r a n d / K * c(f) T h e y fulfill the relations c(f)c (g)-pc ( )c(f)=l, +
+
9
is anti-linear.
75 a n d c(f)
a n d c (f)
c a n be made to adjoints of one a n o t h e i by choosing a new
+
s c a l a r p r o d u c t < 77, { > : = < 7>,P„£ > , w h e r e
is a strictly positive operator for
M
u £ ( - 1 , 1 ) . T h e a d j o i n t n e s s o f c(f)
a n d c + ( / ) is equivalent to c ( / ) * P
M
= P c+(/). < 1
T h e s y m b o l * denotes here a d j u n g a t i o n i n the full Fock space w i t h respect to t h e u s u a l s c a l a r p r o d u c t < , > . F u r t h e r m o r e c + ( / ) = l*(f)
w a s d e f i n e d as t h e left
creation operator. N o w w e c a n e a s i l y d e s c r i b e t h e o p e r a t o r s d(f)
belonging to the third ladder
above. W e define := c
d(f)
+
( / r = 1(f)
«,+(/) := c(/)*. M o r e e x p l i c i t e l y , d(f)
1(f) i s t h e left a n n i h i l a t i o n o p e r a t o r a n d d (f)
=
is a
+
' t w i s t e d ' left c r e a t i o n o p e r a t o r g i v e n b y d (f)h +
® • • • ® h
l
= f ® hi ® • • • ® h
n
+ phi ® f ® h ® • • • ® h + • • • +
n
+ / i "
_
2
n i ® • • • ®
1
® /
® / i
T h e n e x t l e m m a s h o w s t h a t t h e o p e r a t o r s d(f),d (f)
n
n
+ M / i i ® • • •® h n
n
® /.
b e h a v e e x a c t l y as t h e o p e -
+
rators c ( / ) , c + ( / ) . 2 . a ) The operators
L E M M A
c(f),c+(g),
namely
(f,g
d(f),d (g)
The operators a scalar
c) If we denote a bounded
< , >
measures
PROOF:
11
with
M
to adjoints
respect
to < , >
by
as c(f)
on T^.
M
and has the same norm are the
to the vacuum
by
1
1
of T with
of one another
< 77, £ >**:=< 77, P " ^ > .
the help of P'' :
with respect
+
for
can be made
+
a n d of d(f) + d (f)
+
b)
and d (f)
the completion
d) The spectral c (/)
+
d(f)
on T
operators
=< f,g > 1.
- pd (g)d(f)
product
operator
as the
2
+
b)
the same relations
L (IR))
d(f)d (g)
choosing
MM
+
e
expectation
, then d(f)
is
state of c(f)
+
same.
a) T h i s follows b y a d j u n g a t i o n ( w i t h respect t o < , > ) of the equation c(f),c (g). +
W e have
dlfyP'
1
c(/)*P„
=
P c+(/),
i.e.
M
d (/)P 4
M
=
which implies
P^d(f)*,
= P 7 d + ( / ) . T h i s i s e q u i v a l e n t t o t h e a s s e r t i o n . N o t e t h a t P' /
1
1
for p. e ( - 1 , 1 ) c) C o n s i d e r 77 € T a n d p u t £ : = P~ Tj.
T h e n we h a v e
l
< 77,77 > " = < n,P^n
> = < P^VtP^n
>=
" =< d+WP^P^d+iftP^
>
=
d(fn,p; p d(fn> i
l
=
+
=„.
exists
76 T h i s implies the assertion. d ) W e o n l y have t o n o t i c e t h a t t h e s p e c t r a l m e a s u r e i s u n i q u e l y d e t e r m i n e d b y d(f)Q
= 0 a n d the relation d(f)d (f)-pd (f)d(f) +
=
References A s k e y R . , I s m a i l M . : Recurrence
relations,
continued
fractions
and
ortho-
M e m o i r s of the A M S , v o l u m e 49, n u m b e r 300, 1984
gonal polynomials.
B i e d e n h a r n L . C . : T h e q u a n t u m g r o u p SU (2) q
a n d a q-analogue of the boson
o p e r a t o r s . J o u r n . o f P h y s i c s A 22 ( 1 9 8 9 ) , L 8 7 3 Bozejko M . , Speicher R . : A n E x a m p l e of a G e n e r a l i z e d B r o w n i a n M o t i o n . C o m m u n . M a t h . P h y s . 1 3 7 , 519-531 ( 1 9 9 1 ) C a r t e r R . W . : Simple
groups
Chaichian M . , Kulish
of Lie type. J o h n W i l e y & S o n s , 1972
P . , L u k i e r s k i J . : 5 - d e f o r m e d J a c o b i i d e n t i t y , q-
oscillators a n d g-deformed infinite-dimensional algebras.
Phys.
Letters B
237 ( 1 9 9 0 ) , 4 0 1 - 4 0 6 F i v e l D.I.: Interpolation between Fermi a n d Bose Statistics U s i n g Generalized Commutators. Greenberg
P h y s . R e v . L e t t . 65, 3361-3364 (1990)
O . W . : (J-mutators a n d violations of statistics.
U n i v e r s i t y of
M a r y l a n d P r e p r i n t 9 1 - 0 3 4 , 1990 Macfarlane A . J . :
O n g-analogues of the q u a n t u m h a r m o n i c oscillator a n d
t h e q u a n t u m g r o u p SU(2) . q
J o u r n . of P h y s i c s A 22 ( 1 9 8 9 ) , 4581
Speicher R . : A N e w E x a m p l e of 'Independence' a n d ' W h i t e N o i s e ' .
Probab.
T h . R e l . F i e l d s 84 ( 1 9 9 0 ) , 141-159 Vaksman L . L . , Soibel'man g r o u p SU(2).
S.Ya.:
A l g e b r a of functions
on the quantum
Translation from Funktsional'nyi A n a l i z i E g o Prilozheniya
V o l . 2 2 , N o . 3 ( 1 9 8 8 ) , 1-14 Voiculescn
D.:
I n Operator Theory,
Symmetries
Algebras
of some reduced
and their
Busteni, Romania,
Connection
free p r o d u c t with
Topology
C*-algebras. and
Ergodic
1983, L e c t u r e Notes i n M a t h e m a t i c s 1132,
S p r i n g e r V e r l a g , H e i d e l b e r g 1985
Instytut Matematyczny,
Uniwersytet
Wroclawski, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Po-
land Institut fur Angewandte Mathematik, Universitat Heidelberg, Im Neuenheimer Feld 294, W-6900 Heidelberg, Federal Republic of Germany
77
Q u a n t u m P r o b a b i l i t y a n d R e l a t e d Topics V o l . V I I (pp. 79-91) ©
1992
79
W o r l d Scientific P u b l i s h i n g C o m p a n y
M i n i m a l
s o l u t i o n s
i n classical a n d
q u a n t u m
stochastics
A . M . Chebotarev Moscow USSR
Institute
Applied
1.
for Electronic
109028 Moscow,
Engineering
B.Vusovski
Mathematics
,
S/12 ,
Department
T h e i d e a o f m i n i m a l s o l u t i o n a p p e a r e d o r i g i n a l l y i n t h e t h e o r y of g e n e r a l i z e d
P o i s s o n processes [1], w h e r e a t r a n s i t i o n p r o b a b i l i t y is d e s c r i b e d b y a K o l m o g o r o v F e l l e r e q u a t i o n . I n t h e s i m p l e s t case t h i s e q u a t i o n l o o k s as follows j P{x, t
t |r, 0) = j
* ) | ( P ( x + q, t\T, 0) - P(x,t\F,
p(dq)\C(q,
0)),
2
(1.1
w h e r e P\t=o = lr(«r)i -fr is t h e i n d i c a t o r f u n c t i o n of a set T 6 S , (X, E ) is t h e m e a s u r a b l e p h a s e s p a c e of a process.
P a r a m e t e r s of t h i s process are a n i n t e n s i t y
k of j u m p s a n d a d i s t r i b u t i o n m o f j u m p s f r o m a p o i n t x £ X t o a set B £ E : k(x)
= J n{dq)\C{q,x)\ ,
m(B\x)
2
= J n(dq)\C(q,
X
x)\ /k(x). 2
B
T h e M a r k o v j u m p p r o c e s s is c a l l e d regular
i f i t p r o d u c e s a finite n u m b e r of j u m p s
o n a n y f i n i t e i n t e r v a l w i t h p r o b a b i l i t y one. T h e K o l m o g o r o v - F e l l e r e q u a t i o n c a n be t r a n s f o r m e d to a n i n t e g r a l f o r m w h i c h is e q u i v a l e n t t o (1.1) o n t h e set S+ of p o s i t i v e b o u n d e d s o l u t i o n s w h i c h are m e a surable w i t h respect to x a n d t P(x,t\T,0)
= +
exp(-k(x)t)I (x)+ T
f / cir e x p ( - / f c ( z ) ( t - r ) ) f c ( x )
0
exp(-k(x)t)Ir{x),
m{dq\x)P(x,T\T,0)
(1.2)
x
o If o n e t a k e s as P (x,t\T,Q)
r
t h e s o l u t i o n of t h e d i s s i p a t i v e p a r t of e q u a t i o n (1.1)
t h e n t h e sequence
P „ ( x , t | r , 0 ) = Po(x,
g
•>0 I a n d a
closed densely defined s y m m e t r i c operator H w h i c h characterize t h e i n f i n i t e s i m a l
83 operator — G of a strongly continuous one parameter contractive semigroup W
:
t
contractions W
= exp(-Gt),
t
W> 6 D(H)
Gifi = iH4> +
n £>($),
2 ess d o m G C D(H)
(2.3)
n Z?(*)
w h e r e ess d o m G i s a n y e s s e n t i a l d o m a i n o f t h e c l o s e d o p e r a t o r G . M o r e o f t h i s , let C J J p r ) n D f » C D(G),
D(G ) 2
O u r last a s s u m p t i o n requires t h a t W
t
DjG )
is b o u n d e d
(2.4)
= H.
2
a n d continuous i n T .
This
a s s u m p t i o n i s f u l f i l l e d i f T is a B a n a c h s p a c e s u c h t h a t
l M l r =
+ W i l l i
>GZ>(G).
(2.5)
if $ = $ ( / ) = [$,(/)] a n dif the family
mappings
*«(•) = e x p { - e $ ( / ) } $ ( - ) e x p { - e $ ( / ) }
G GPn(7Y)
is n o r m a l f o r a l l e G ( 0 , 1 ] . L e t t h e a s s u m p t i o n s ( 2 . 3 ) , ( 2 . 4 ) , (2.5) h o l d a n d $ , G CPn.[t,oo)(/),
(3-2)
w h e r e W ^ [ T , t ) is a t w o - p a r a m e t e r s f a m i l y of l i n e a r b o u n d e d o p e r a t o r s i n Ti w h i c h depends o n r e s t r i c t i o n /(-)l[r,t)S u p p o s e t h a t u[r,t)
is b o u n d e d a n d i n t e r v a l - a d a p t e d f a m i l y , S[T, t) is b o u n d e d
v a c u u m f a m i l y a n d n[r,t)
= L A[r,t) 0
+ L A [r t) i
9
i
6 B"(D,E ) a
a r e o p e r a t o r s o f c r e a t i o n , a n n i h i l a t i o n a n d c o n s e r v a t i o n ([7]).
w h e r e A*,
A,
A
86 Let U[T, t)g, g eh
L e m m a 3.1.
in h with respect
and LiS[r,
i)q, q e L(D,
to r and t (0 < T < t < oo) | | ( u M ) - I)g\\ -
0,
h
and
m [ r , t) = S[T, t)+
be norm
a
t
h
stochastic
integral
U[T, O ( £ o A ( d £ ) + L A*(d€))S[Z,
f
continuous
- I)q\\ - » 0
\\Li{S[T t)
as t — T J. 0. T h e n t h e r e exists interval-adapted
E)
let
t)
x
=
J[r,t)
n lim
S[r,t)+
V«[T f )/ife,{ l
n—*oo '
w h i c h h a s t h e following (1)
the
limit
L(D,E )
(3.1)
i)5Ky+i,*)eB'(A««.)
m [ r , t ) i s interval-adapted
in B (D,E ); s
a
; u[r,t)q,
q
e
in h a n d ||(m[r,
= {g(t, r],X®
Ig(t,
(M*(dOmr},P -il^t)(X)M*(dO~g((,r}), n
+ (
)
-
3
n
)
2
)
r])+ (
3
'
1
89 where g€L(D(G),E ),
g[r,t)
a
=
S[r,t)g; g eL(D(G),E ),~g(t,r}
=
a
F r o m ( 3 . 9 ) , ( 3 . 1 1 ) a n d (3.12) i t f o l l o w s t h a t P ,
P „ are contractive a n d monotone
n
(g,P (X)g)
< \\X\\• \\g\\l (g,P (X)g)
n
< \\X\\ • \\g\\l
n
0 < P _ ! ( X ) < P ( X ) < \\X\\-I, n
0 < P „ _ ! ( X ) < P „ ( X ) < ||X|| • / ,
n
for a l l X 6 B(H)
S*(t,r]~g.
u n i f o r m l y w i t h r e s p e c t t o r a n d t . H e n c e t h e r e e x i s t t h e least
+
upper bounds P(t,r](X) = l.u.b.P (f,r](X), B
P [ r , t)(X)
= l . u . b . P [ r , t)(X),
V X e
n
T h e o r e m 3 . 3 . L e t t h e a s s u m p t i o n s (2.3)-(2.5) G
c
and G . c
which l.u.b.P .
(4)
for
operators
Langeven
equations,
and (3.12) for P = l.u.b.P ,
P
n
=
properties to r and t
r ] ( - ) , P [ r , 0 , w e d e n o t e b y [TJI/ co]
the e l e m e n t o f M o b t a i n e d b y e v a l u a t i n g i n the p o i n t i / 2 the a n a l y t i c a l e x t e n s i o n o f the C o n n e s c o c y c l e (Di|>: D c o ) (cfr. t
[10]).
c l o s u r e o f the set { [ijV co] Q : Tp E S(M),
So [ ip/ co]QE
and
V^
is the
i|i s t x t o f o r a > 0 } . T h e n o t a t i o n co
f o r co G S(M), a G M , w i l l denote the f u n c t i o n a l co (•) = c o ( a a
+
a
• a). W h e n u s i n g
94 t h i s n o t a t i o n w e s h a l l f u r t h e r a s s u m e u n l e s s e x p l i c i t e l y stated that co(|a| }) =
L
so that c o E S ( M ). a
Let now N
be a v o n N e u m a n n s u b a l g e b r a o f M , co^= u)/
N
Q a vector i n H
co £ S (M),
and
i m p l e m e n t i n g co. W e s h a l l o f t e n i d e n t i f y w i t h { a Q : a E N}
M
G . N . S . H i l b e r t space
the
for N and Q .
W e shall denote b y E
Q
'
M
N
the o r t h o g o n a l p r o j e c t i o n f r o m H
to H Q
M
N
, and
b y e ^ ' ^ the g e n e r a l i z e d c o n d i t i o n a l e x p e c t a t i o n f o r co f r o m M to N (cfr. [ 4 ] , [ 7 ] , [ 8 ] , [ 9 ] ) . I f p i s the s u p p o r t o f co, t h e n , f o r a E M , e ^ ( a ) i n pN
i s the u n i q u e e l e m e n t
p s a t i s f y i n g the e q u a l i t y :
w i t h Q a v e c t o r i n H ^ s u c h that co(a ) = < Q , p a p Q . > . W e r e c a l l that w h e n e v e r there i s a co p r e s e r v i n g n o r m o n e p r o j e c t i o n
from
M
to N ( i n p a r t i c u l a r i n the c o m m u t a t i v e c a s e ) it c o i n c i d e s w i t h E ^ ' ^ . W e s h a l l denote b y
the v o n N e u m a n n s u b a l g e b r a o f N o f the p o i n t s w h i c h r e m a i n
f i x e d u n d e r e^' M
to
. T h e r e is a n o r m one p r o j e c t i o n ( c o n d i t i o n a l e x p e c t a t i o n )
r
p r e s e r v i n g co^. O n {p
apQ, a E
} the o p e r a t o r s J ^ Q
from
and T Q
A™ 7
c o i n c i d e , a n d they c o i n c i d e a l s o w i t h the i s o m e t r i c a l i n v o l u t i o n We
shall
associated p%
M
to
: S(N)
call
canonical
the c o n e
VQ
state
extension
(see [ 7 ] ) PQ
M
J fa. p
from
M
to N
( Q s e p a r a t i n g f o r N) the m a p p i n g
S ( M ) d e f i n e d as
[pQ' (^iv)](a)=
(
M
N
a E M )
w i t h Wjy E V Q t h e v e c t o r r e p r e s e n t a t i v e o f
ty
N
i n VQ.
A s this d e f i n i -
t i o n d o e s not d e p e n d o n the c h o i c e o f the c o n e , w e s h a l l u s u a l l y s h o r t e n our
n o t a t i o n to O Q '
M
if
Q is
a
vector
i m p l e m e n t i n g co i n U
M
A
N
D
95
F r o m n o w o n w e s h a l l c o n s i d e r a set o f i n d i c e s A , to e a c h o f w h i c h w e associate a v o n N e u m a n n a l g e b r a Aj B C A w e denote b y A
(j G A ) a c t i n g o n a H i l b e r t s p a c e H
the v o n N e u m a n n
B
algebra
generated
A
. For
b y [J
Aj ;
/SB i f B = {j} o u r n o t a t i o n w i l l be s i m p l i f i e d to Ay W e s h a l l a s s u m e that i f B , C C A , B f i C = 0 , then A
flA
B
W h e n dealing with
i s the t r i v i a l a l g e b r a o f the m u l t i p l e s o f the identity.
Q
the v o n N e u m a n n algebras A
B
( B C A ) , w e shall simplify
o u r notations b y u s i n g s i m p l y B as a n i n d e x instead o f A w i l l b e s h o r t e n e d to E ^ ' ) . 8
A
and A- are m u t u a l l y c o m m u t i n g (i.e. f o r a^ G A
i
2.1 L e m m a .
Let
co G
b
B,
c
co (c) = co(fe
+
j ;
(e.g. e ^ B ^ C ( B 3 C )
aj G A j a a- = a^ a ^ i
C C A ,
c
P r o o f . W e have, for c G ( A ) So
S(A^,
= s u p p (u> ) . T h e n p™c s p "0= s u p p c o
p™c
B
W e s h a l l a l s o a s s u m e n o w that f o r i j E A , i / j ,
0
fc
B f l C = 0,
b £ A
B
,
c
c b ) «: || b || co ( c ) ;
+
2
co (c) = 0 i m p l i e s co (c) = 0 fc
I n this s i t u a t i o n (cfr. [6]), f o r C , D C B C A , C D D = 0 , a
canonical
state
extension
from
A
to
c
C U D = B and A
B
,
the
p
c , B
mapping
cp - * p ' ( c p ) | A f o r c p G S ( A ) a d m i t s a n e x t e n s i o n to a l i n e a r p o s i t i v e c o n c
c
B
c
D
c
C
t i n u o u s m a p p i n g f r o m (A ) c
to (A )
t
D
t
. Its d u a l m a p p i n g i s a l i n e a r , p o s i t i v e ,
a n t i c o m p l e t e l y p o s i t i v e ( i . e . i t s c o m p o s i t i o n w i t h the a d j u n c t i o n i s a n t i l i n e a r completely from co
c
co
B
G
Any
A
positive),weak to A .
D
c
operator
It w i l l
be
continuous
denoted
Xj"'
D
unity preserving
S ( A ) a n d s o m e c a n o n i c a l state e x t e n s i o n p f
c
contraction
i f f o r s o m e co s u c h c , B
w e have p
that
(co ) =
c , B
c
linear, positive, anticompletely positive, w e a k operator continuous unity
p r e s e r v i n g c o n t r a c t i o n from A c o u p l e ( A , A ). Q
D
co(c d ) = < J
Q
D
to A
I n p a r t i c u l a r Xj*'
c
c
w i l l b e c a l l e d a s t o c h a s t i c c o u p l i n g f o r the is c h a r a c t e r i z e d b y the e q u a l i t y :
A.^ (d) Q , c Q > c c C
c
with c G A , d G A , c
D
Q
c
a vector i n H
the i s o m e t r i c a l i n v o l u t i o n l e a v i n g it i n v a r i a n t .
A
i m p l e m e n t i n g the state c o a n d c
J
Q
c
96 Conversely, if K ' D
i s a s t o c h a s t i c c o u p l i n g f o r the c o u p l e (A , A )
C
D
ping cd — < J ^ X ' f d ) Q D
£
n o r m a l f a i t h f u l states c o tensor product o f A
C
a )
,c£2
f 0 c
on A
c
and A
>(cGA ,deA ) c
C
the m a p -
c a n be extended for a l l
D
to a state co o n t h e p r o j e c t i v e C * - a l g e b r a
C
(cfr. [12] I V ) ; the v o n N e u m a n n a l g e b r a A
D
g e n e r a t e d b y its G . N . S . r e p r e s e n t a t i o n w i t h respect t o co h a s a n o r m a l state c o w h i c h i s the e x t e n s i o n o f
co a n d h?' = X / c o
i f w e identify A
and A
C
B
B
with
D
t h e i r f a i t h f u l r e p r e s e n t a t i o n s w h o s e u n i o n generates A . B
W e note that i n the a b e l i a n case w e h a v e
^
= E / |A
C
3
0
D
F o r f u r t h e r p r o p e r t i e s a n d results o n s t o c h a s t i c c o u p l i n g s see [ 6 ] . 2.2 P r o p o s i t i o n (cfr. [6]). L e t A • B D C , D , C D D = 0 , C U D = B , C D G , B D G U D
= F . Then for co E S ( A ) with co = co |A B
B
c
B
Q
E
E S ( A ) w e have f
c
>?' = e ' * G
X'
c G
CO
D C
CO
CO
P r o o f . W e have, for d E A , g E A D
< ( J
= < J if
O
ti*
G
K° - (d)Q,g£2> 1
on
{g Q : g E A } G
H
Q
G
,
this
separating for A
A
c
G
i m p l e m e n t i n g the state c o of
i s dense i n / Y implies
A . ° ' ( d ) a, g Q > =
= <E°'%
the c l o s u r e
G
:
= <x>(dg)=
c
Q is a vector i n H
involution
in
(d)Q,gQ>
G
Q
G
{g Q , and J
2^ (d)£2 G
, and both
=
: g E A A
( £
( f ^
G
G
}
G
i s the i s o m e t r i c a l
leaving Q
• }^ ) (d)
invariant. A s
Q , J ^ ' ( d ) Q are
C
G
G
G
(d)Q for all d E A
' £ f )
>^' (d) a n d ( e £ ' « G
G
and J
Q
X°' )(d)
D
are i n A
c
; since Q is G
, w e have
proved our claim.
2.3 P r o p o s i t i o n . co
B
Let A D B D
E S ( A ) , co = co | A B
(£; )(|c| )= D
2
c
B
c
E
|[(|co ) /co ]| c
D
D
C, D,
C U D = B,
S f ( A ) . T h e n for c E c
2
A
c
C n D =
w i t h co (| c |
2
)
=
0 1
97 P r o o f . N o t e first that f o r d E ( A ) D
w e have
+
\ c |
( c o ^ (d) = co (
)
s||c|| co(d); s o [ ( c o ^ / c o j i s w e l l d e f i n e d . N o w , f o r a l l d E A , i f Q i s a v e c t o r 2
D
i n R~ i m p l e m e n t i n g c o : < J p X A
(| c 1 )Q, d Q >= 2
w
=co ( | c 1 d)=(co ) ( d ) = c o ( [ ( c o ) / c o ] d [(co y c o j 2
B
c
D
D
c
[ K ^ H R d j O =
= c
I K ^ c )D^ D 1 1 ^
^ D
D
+
a
C0
n
D
c
D
'
a
s
p r o p . 2.1 this i m p l i e s o u r c l a i m .
m
3. Markovianity for triples of von Neumann algebras.
U n l e s s otherwise
stated i n this s e c t i o n A = { 1,2,3}; w e s h a l l a l s o o f t e n o m i t the i n d e x A f o r the sake o f conciseness.
3.1 P r o p o s i t i o n . L e t A b e a b e l i a n a n d co E S ( A ) . T h e n e £ ^ ' ^ ( a ) E 2
all a £ A 3
iff e^
3
2,3
*' does not depend o n a 2
Proof. L e t f £ ' ' ( a ) { 1
one
projection
^{2,3}-
W
e
h
2 }
from 3
V
e
^
A
2 3
for a l l a
2
j to A
^ { 2 , 3 } ( 2,3}) a
=co( ) = a
E(Aj )
+
a
2
98 T h e first c o n d i t i o n i n p r o p . 3 . 1
is a standard definition o f m a r k o v i a n i t y a n d
therefore i f c o G S ( A ) satisfies i t w e c a l l i t m a r k o v i a n w i t h r e s p e c t t o t h e l o c a l i z a t i o n (A., A A^)-
T h e second condition i s m o r e suitable f o r generalization to the
2
non-commutative situation.
3.2 T h e o r e m .
L e t to G S ( A ) w i t h C 0 j £ S ^ A j ) c O j
c o m m u t e w i t h A^ y 2
3
2
3
j G S (A { 3})> f
2
A ^ b e the v o n N e u m a n n s u b a l g e b r a o f t h e f i x e d
2i
under E ^ ' ^
2
and
the c o ^ j p r e s e r v i n g n o r m o n e p r o j e c t i o n f r o m A ^ 2 3
A ^ . T h e n the f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t
3
)
K
H
b)
{
2
^
3
K
c) o X a a 1
d ) X1'™
e) * 4
2
{ 2 3 }
K
( {2,3}))
^
= ^
a
3
}
'
2
1
1
0
r
a
l
l
a
i
e
A
l '
a
{ 2 , 3 }
3
2
{ 2 3 }
))
fora^A^
a
1
a l){2,3}
:
D
M
{2,3})t £ A
f
CA
2
{
2 3
}
which implies b). b ) => c ) F o r a G ( A j ) , t
a i
a
{
2 3 }
{
2
3
}
G A
{
i m p l e m e n t i n g co
a) =>b) B y a) f o r a l l a G A j
co(
{2,3}
f o r ^ G A ,
, { 2 , 3 }
m
A
2
P r o o f . I n the f o l l o w i n g let Q b e a v e c t o r i n
(° (
E
f o r a , G A,
2
)=to(a Ej ' >' (a
= X -
f
+
) = co | ( a a
{ 2 3 }
a
{
2
3
) = co
}
{ 1 2 }
and o u r c l a i m follows b y linearity
G A (
3 l
{
2
3
}
:
ej ^2
2
(a
{
2 3 }
))
2
3
}
A
i
points 2 3
j to
99 c) = > d) L e t p j
j
2 3
be the s u p p o r t o f { 3} 2
a n c
* ^
a
y ''
c
c
c
a
n
separating
u
v e c t o r f o r A i n H s u c h that co(a^ j ) = 23
= < Q,
P
{
2 3
a
}
{
23
p
}
F o r each a G
{2,3}
X
co ( a a t
=<J
i {
< 2 , 3 >
£2 >
}
+
(
3
l)
j £
2 3
a i
)Q,J
2
3
E < ' >' J 2
Q
3
A J
2
2
w e get J c ^ ' *
J
3
1
linearity for a G
A
x
(a
2
N o w , b y the d e n s i t y o f { a j rating for
{
3
a
}
(
) ) =
e) => a) F o r a E L A :
co(
a
3 l
= < A' J
2
{ 2 3 }
) = <J _
^'
{ 2
{ 2
'
*
Q
e (cfr. l e m m a 2 . 1 ) :
= C O
£ J ( 1 P{2,3> )= A
3
)) =
{ 2 3 }
Q . >=<J
E AJ
2 3
{2)3}
\J>
2
( )Q, h
J } in
a
{
2
3
}
Q >.
a n d as £2 i s s e p a -
* (aA = X ^ ' (aA first f o r a 2
G
x
A *
and then b y
a
^1,2
)
(
=
GAy
1
1,12,3}
x
Z{ 3} ^ ^ { 2 3} *
x
o
2
X ^
3 }
( i)
3 }
a v
y
1,{2,3}
( X
"
e
j Q-.a^^
2 3
2 , 3
2
{ 23 }
d ) => e) B y p r o p . 2.2 w e h a v e f o r a {2.3},2
w
A
2
(
^ { 2 3}
'P{23} ( a , ) fi, a
• « £ ( (2,3 )
Q
A
}
"
R
3
}
(
)
u
r
a
*
r e a
fi >=
= co (
f l j
^ y established notations: ^
E/ ( a
(
{ 2 3 }
a
A
Q
^
fi
>=
)).
I f A is a b e l i a n the e q u i v a l e n t c o n d i t i o n s o f t h e o r e m 3.2 i n c l u d e the s e c o n d c o n d i t i o n i n p r o p . 3.1 c h a r a c t e r i z i n g the m a r k o v i a n i t y o f co ( c o n d . b ) , w i t h r e s p e c t to t h e l o c a l i z a t i o n (A., A , A ) . I n t h i s s p i r i t w e g i v e the f o l l o w i n g . 2
3
3.3 D e f i n i t i o n . A state co G S ( A ) w i l l b e c a l l e d m a r k o v i a n w i t h respect to the l o c a l i z a t i o n (A
v
A , A j ) i f it satisfies the e q u i v a l e n t c o n d i t i o n s o f t h . 3 . 2 . 2
W e n o t e that c o n d i t i o n c ) i n t h . 3.2 i s s t r o n g e r t h a n the c o n d i t i o n u s e d i n [5] to d e f i n e M a r k o v states f o r m a t r i x a l g e b r a s ; there it i s a s s u m e d that
w
( i a
A
{2,3}) =
w
( i a
e { 2
'
3 >
'
2
( {2,3})) A
100 with E ^ ' ^ ' 2
3
a general completely positive linear contraction (not necessarily
2
the C0p j - g e n e r a l i z e d c o n d i t i o n a l e x p e c t a t i o n ) f r o m A ^ 3
In
the c l a s s i c a l case (A
abelian)
m a r k o v i a n f o r the l o c a l i z a t i o n (A localization
(A
A,
y
a s t a n d a r d fact
t
2
3
general v o n N e u m a n n algebras the equivalence g e n e r a l i z a t i o n o f t h i s fact, as a b e l i a n s i t u a t i o n A ^ ' * ' * (aA 1
2
i s that i f a state co i s
A ) , t h e n i t i s a l s o m a r k o v i a n f o r the
A,
v
j to A j .
( " t i m e r e v e r s a l " ) . I n the c a s e o f m a r k o v i a n i t y f o r
A)
2
23
C
=
3
E ^
2
'
3
o f c o n d i t i o n s b ) a n d e) i s a
a n d , as a l r e a d y r e m a r k e d , i n the
A^
(aA. It i s p o s s i b l e , h o w e v e r , t o g e n e r a l -
*
i z e to n o n c o m m u t a t i v e v o n N e u m a n n a l g e b r a s t h e m a r k o v i a n i t y i n a d i f f e r e n t direction w h i c h is invariant under "time reversal"
3.4 T h e o r e m . L e t A
v
A, A 2
be mutually c o m m u t i n g . T h e f o l l o w i n g
3
condi-
tions are equivalent.
3
) [(»«, J * /
K
J
13
J =
tK
h 1
3
s u p p o r t o f (co ) a
/C0 2
]
Pu
3
) I K ^ / ^ l
c
) [C ,,
= [(
C O
£
, )2 i
/ C 0
2J
tK
commutes with
00
x
1
3
l
V
2 ]
W
f
0
r
3 1 1
A
T
i
a
6
3
3
G
3
A
a
2
P 3
A
[(co ) / c o ] a n d X,^' 2
3
K P
a
i
s
t
h
e
£
j
3 -
4
does not depend o n
2
„
3
a G A ).
V
3
3
3
G A
3
[(co )2/co ] a
2
c o m m u t e s w i t h [(CO ) / c o ] a n d A ' q
1
depend o n a j G
e
6
3
EA
d) f o r a
1
2
b
(a
3
0j
1
a
for
A
) [ K ^ V ^ J
2
3
2
2
does not
3
v
= [K ) / 1
0 tK^VK^] =
2
( 0
2] [ K
3
)
2
/
W
2
]
f
0
r
f o r 3 1 1
i
a
a
i
G
E
A
A
V
3
3
G
v 3 a
A
e
3"
A
3 '*P„ i s the
s u p p o r t o f (co ) . fl
2
P r o o f : W e s h a l l a s s u m e i n the f o l l o w i n g c o to b e f a i t h f u l w i t h 2
vector
corresponding
Q . T h e extension to the general situation is straightforward. W e shall
o f t e n use p r o p . 2 . 3 .
101 a) b ) ( a n d s y m m e t r i c a l l y , f ) e) ) . W e h a v e , b y the c h a i n r u l e
O n the other h a n d l(Pay
^ P * I
3
M = [M /o> ]
M
2
[(co ) / co ],
2
1
flj
fl
2
2
1
3
and our claim follows b ) => c ) ( a n d e) =>f)).
=tK ) i
/ "2] t K )
2
3
/ co ] Q, 2
2
w h i c h i m p l i e s the c o m m u t a t i v i t y o f [(to jj/coj fl
w
( I i I\1 31 ) = ^ K ^ V a
= < K
W
a
2
a ) M [K ) 3
=
>
3
2
{2,3> " 5
1
3
, a^ j
1
J l
a
E
2 3
3
r 3
,
1 3
1 ^
A
l - {2,3} 3
e A
{2,3}-
A J
2
3
J
(JI^AJ),
w e have:
3
) ) ) =
}
2
{ 2 3 }
( ^{1,2} ( l( i))) ' {2,3}
w h i c h i m p l i e s kj-™
0
° < 1 {2,3} ) =
=
( A ^ ( ^ ( a ^ Q . e f >' ( a
1
f
1
El A
1
2
X
a
b e the s t o c h a s t i c c o u p l i n g f o r the c o u p l e
2
n ^ A j ) ) a n d cp"*^ ^ . T h e n , f o r a
J
( {2,3}»
2
a
a =>b L e t A^*
* {2,3} < o
ffi'
(«l( j>2(
Q
a
(a,) = ( i t "
1
) Q >=
Q
•A^
>
2
'
{ 1 2 }
• i t ^ ) GA
2
b => a . S e t «PMCV ) = <J ^ 2
2
| 2 3 |
(a )0,
then ;
1
1 2
j b e the v o n N e u m a n n a l g e b r a g e n e r a t e d b y its
G . N . S . r e p r e s e n t a t i o n f o r cpJJ j a n d Jij ( A j ) ( i = l , 2 ) the c o r r e s p o n d i n g 2
presentations o f A
v
tensor
T h e n c l e a r l y t p =co 2
2
b y construction and
subre-
104
W
( i {2,3>) = a
=
a
6 ^ ( b
1
2,3}(*T ™
*
3 )
We have, since for each b
b
'
W e only need
3
3
1
S
f o r the c o u p l e ( A , , A ^ j ) . W
c o
- * E^' ^(* ) a
2
2
= co ( e ( | a | ) a e ( 2
1
|a | a )=co(e(|a | 2
3
2
1
| a | ) ) = a^ffi
2
3
2
2
(a^.
^
2
3
|a | a ))
2
O n the other h a n d : co(e(|a | )a
2 E
= co (E( |
E ( a ^ E (| a 1 ) ) = t o
2
1
As
[ K
E|
V
A s co
f
B
f
2 )
2
3
E
3
E
1
2
2
E ( | f l i f t
2
|a | ))) = 3
i^
2
(E ( a , ) ) .
i s a n o r m o n e p r o j e c t i o n this i m p l i e s that
a
A
|
H
(|a | ))=co( ( |a | a E(
l
2
]
=
K ^ i a j ) ! s(\aj)i 2
)
/
2
=
K^s)
^
W •
is m a r k o v i a n o n B abelian our c l a i m follows b y 3.5.
A s i m p l e e x a m p l e o f this s i t u a t i o n o c c u r s i f M i s the q x q m a t r i x a l g e b r a , N the s u b a l g e b r a o f d i a g o n a l m a t r i c e s i n M i n a g i v e n b a s i s , A
1
= M®I®I,
trace
A
2
= I®M®I,
preserving norm
A
3
= I®I®M,
A
B = N®N®N
= M®M®M, a n d E i s the
one projection f r o m A to B .
A k n o w l e d g m e n t . T h i s p a p e r w a s b e g u n d u r i n g a o n e y e a r v i s i t at the Institut f u r A n g e w a n d t e M a t h e m a t i k o f the H e i d e l b e r g U n i v e r s i t y t o w h o m , as w e l l as to t h e S o n d e r f o r s c h u n g s b e r e i c h
1 2 3 t h e a u t h o r w o u l d l i k e t o t h a n k f o r the s u p -
p o r t . It i s a p l e a s u r e t o t h a n k a l s o P r o f . W . v o n W a l d e n f e l s a n d P r o f . M . L e i n e r t for their most k i n d hospitality.
Bibliography 1.
L . A c c a r d i . N o n commutative
M a r k o v chains. International S c h o o l
of
Mathematical Physics, Camerino (1974) 268- 295 2.
L . A c c a r d i . Topics i n Quantum Probability, Physics Reports 169-192
77 (1981)
107 3.
L . A c c a r d i . C e c c h i n i ' s transition expectations
and M a r k o v chains. Q u a n -
tum Probability and Applications IV. Springer Verlag Lecture Notes
1396
( 1 9 8 9 ) 1-6. 4.
L . A c c a r d i - C . C e c c h i n i . C o n d i t i o n a l expectations
in von Neumann
alge-
bras and a theorem o f Takesaki. J.Funct. A n a l . 45 (1982) 2 4 5 - 2 7 3 5.
L A c c a r d i , A . Frigerio. M a r k o v i a n cocycles, Proc. R o y a l Irish A c a d .
83
A(1983) 251-263 6.
C . C e c c h i n i . Stochastic couplings Probability
for v o n N e u m a n n algebras.
and Applications IV. Springer
Verlag Lecture
Quantum
Notes
1396
(1989) 128-142 7.
C . C e c c h i n i . A n abstract c h a r a c t e r i z a t i o n o f c o - c o n d i t i o n a l
expectations.
M a t h . Scand. 66 (1990) 155-160 8.
C . C e c c h i n i - D . P e t z State e x t e n s i o n s a n d R a d o m - N i k o d y m t h e o r e m conditional
expectations
on
von
Neumann
algebras,
Pac.
J.
for
Math.
138(1989) 9-23 9.
C.
Cecchini-
D.
Petz.
Classes
of
conditional
expectations
over
von
N e u m a n n algebras. J . F u n c t . A n . 9 2 (1990) 8-29 10.
A . C o n n e s . S u r le t h e o r e m e de R a d o n - N i k o d y m p o u r le p o i d s
normaux
fideles semifinis. B u l l S c i M a t h S e c H I 97(1973) 2 5 3 - 2 5 8 11.
A . C o n n e s , V . Jones. Property T for v o n N e u m a n n algebras. B u l l . L o n d o n M a t h . S o c . 17 ( 1 9 8 5 ) 5 7
12.
M . Takesaki. Theory o f Operator Algebras I. Springer Verlag 1979
Quantum Probability and Related Topics Vol. VII (pp. 109-123) © 1992 World Scientific Publishing Company Q U A N T U M O F
F L O W S
F R E E D O M
A
N
W I T H
I N F I N I T E
D T H E E f t
P.K. Das
and
Kalyan B. Sinha Indian Statistical Institute
September 1990
D E G R E E S
P E R T U R B A T I O N S
Indian Statistical Institute Calcutta - 700 035
New Delhi 110016
109
110
1
Introduction
The general theory of quantum stochastic flow (or quantum diffusion) has recently been studied by Evans [1] for finite degrees of noise freedom and by Mohari and Sinha [2] for countably infinite degrees of freedom. In both cases the structure maps are bounded and in the second instance, they satisfy moreover a strong summability conditions. On the other hand, Evans and Hudson [3] considers stochastic perturbations of quantum stochastic differential equation (q.s.d.e.) of Hudson-Parthasarathy (H.P.) type [4] with time dependent coefficient operators. Here the time dependence of the coefficients are given exactly by quantumflowsof the type studied by Evans in [1]. This iB the stochastic equivalent of the following problem: let U and V satisfy in a Hilbert space the differential equations;
dV
=
iHoVdt
and
dU
=
iV* HVU
dt
with
U{0) = V(0) = / and H and Ho bounded selfadjoint. Then it is easy to see that VU satisfies: d(VU) = i(H + H)VU dt, i.e. VU gives the perturbed 0
motion due to the perturbation of H by HQ. Here we study the same question as in [3] i.e. the perturbation of a q.s.d.e. of the H.P. type but in the background of the theory of Mohari-Sinha [2] with countably infinite degrees of freedom of noise.
2
Notations and Preliminaries
All the Hilbert spaces that appear here are assumed to be complex and separable with scalar product < •, • > linear in second variable. For any Hilbert space Ti we write T{71) and B[f{) respectively for the boson Fock space over H and the C"-algebra of all bounded linear operat ors in H. Let W arid A." be twofixedHilbert spaces and let % = X (iR )\@ /C. and 2
0
W = «o©r(7*).
+
(2.1)
111
For Rn
y / € "H we denote by e(/) the exponential or coherent vector in r(7i)
associated with / and by £ the linear manifold of all vectors of the form u©e(/) with ti € He and / € W. Also we adopt the convention of writing ue(/) in place of u © e(/). Note that £ is dense in H. Choose and fix an orthonormal basis {cy/yij of K and set E'
k
=
|ej. > < ej|,j, k > 1. The basic quantum stochastic processes of the theoryare: A(M, =
|0I)
©£?) if , « , ! / > !
® *j'UM*^)
^ - '" ° ifi/>l,u = 0
£(*)} 0 corresponding to the decomposition :
U = (Tic © r ( i [ 0 , 2
t] © £)) © T(L*[t,
oo) S) £ ) , Z(
0: /' £
\\lt{,)ue{f)\\*d» {.),
(2.4)
f
where u,(t) = jftl + ||/(*)||) 0. 3
A B in [4], one may form the (quantum) stochastic integral of a square integrable adapted process L :
In the sequel we shall restrict the K -valued functions / to belong to a dense subspace
M =
{/ € L (2!? ) © K\P{») 3
+
=
= c
0 except for a finite
number N(f) of indices j and for all t > 0}. We shall write /,(*) — fi(g) for j > 1 and /(«) = /°f j) = 1 for all s > 0. We shall also denote by E(M) the linear manifold generated by ue(f),
€ M and u € Wo-
f
Then one has the following properties of the stochastic integral XU){u,v€H ,f, eM): 0
9
(i) A (t) exists as a strong integral on £{A4) and defines an adapted process, < ue(f), X(t)ve(g)
(ii)
(iii) If X'{t)
L' S
>=
t
< »e{f),
L*(s)ve{g)
> ds.
(2.6)
is another adapted square integrable process and
{L'f[s)}
=
f* f A*)g"(s) Jo
torn
<X'(i)ue(f),X(t)ve{g)>=
d*f, (s)g"(s)
f
t
•
Jo {< X'{*)ue(f
),Lt(s)ve(g)
< L*{ )ue(f).Ji{*)ve(a) a
(iv) ||A'(0«(/)|| < 2 exp (^(0) 3
> + < !*(•)«(/).X{i)vc(g)
> +
>}.
(2.7)
£ »>«
/*IWW«(/)l| A'/(«). a
•'
Cl
For the proofs of these properties, the reader is referred to [4] and [6].
( -«) 2
113
3
Unitary stochastic evolutions and quantum stochastic flows
Here we collect some of the results of [2] that will be needed in the next two sections. Lemma 3.1 : Suppose {A*} and {Bj,} be two families of bounded operators in a Hilbert space such that £ -AJ.-U and £ #JSjt converge strongly. Then £AJ5/t converges strongly. Suppose Lfc,5* be bounded operators in 7io and H be bounded selfadjoint such that : £ \\Lku\\ < C\\u\\ 2
3
for some
C >
0 and all u € «o-
S f 5* = SjSi* = 4l,
(3.1)
where the second sum is assumed to be strongly convergent. Next we set: St - 6*1
if
I„
if p>
1,^ =
- L ; S J
if ^ >
Uft
iff Then it is easy to verify the
\L\L
k
fi, v > 1
0
(3.2)
=o
if ^ = i/ = 0.
uniiaritg condition:
L% + V£ + L?Lt
= Lt + V;
+ LIU?
= 0,
(3.3)
where
EPC«lt 0 c o n s t a n t s a a n d a f a m i l y {D^}, i
l€ r
v
> 0 , a c o u n t a b l e i n d e x set J „
o f b o u n d e d o p e r a t o r s i n Ho s u c h t h a t f o r x € A, u € Tto
£
PfrHP
(ii) jt[I) = I and if(ar) is an adapted process, (iii) there exist structure maps AJ obeying (3.6) such that j (z) satisfy the f
q.s.d.e. = ix(Aj(*))dA;;(i).
(3.8)
Theorem 3.3 : Suppose that structure maps AJ satisfy (3.6) and (3.7). Then there exists a quantum stochastic flow j on A satisfying (3.8). Furthermore,
115
the map (t,*) -+ j (x) is jointly continuous in the strong topology of Ti with t
respect, to the strong topology of .4, and j satisfies the estimate for 0 < t < T t
£
\\*D»,lKX---D»*\r}Mf)\\*-
(3-9)
The proofs of Lemma 3.1 and of theorems 3.2 and theorem 3.3 can be found in PI
4
Perturbation of quantum flows
Given a quantumflowjt on .4 constructed as in theorem 3.3 with structure maps A£ satisfying (3.7) and assuming that the operators Z£ € .4 for all ft, v > 0, it follows that the time dependent coefficients jt(ZJJ) also satisfy a relation similar to (3.3). Then we may try to solve the q.s.d.e. similar to (3.5) with J